![Parallel Computational Fluid Dynamics - New Frontiers and Multi-Disciplinary Applications
K. Matsuno, A. Ecer, J. Periaux, N. Satofuka and P. Fox (Editors)
9 2003 Elsevier Science B.V. All rights reserved.
Lattice Boltzmann Methods: High Performance
Computing and Engineering Applications
G. Brenner ~, Th. Zeiser~, K. Beronov ~, P. Lammers~, J. Bernsdorfb
~Institute of Fluid Mechanics, University of Erlangen-Nuremberg, Cauerstrafie 4,
91058 Erlangen, Germany
bC&C Research Laboratories, Rathausallee t0, 53757 Sankt Augustin, Germany
The development of novel numerical methods for applications in computational fluid
dynamics has made rapid progress in recent years. These new techniques include the
lattice gas and lattice Boltzmann methods. Compared to the traditional CFD methods,
the lattice Boltzmann methods are based on a more rigorous physical modelling, the
Boltzmann equation. This allows to circumvent many deficiencies inherent in existing
Navier-Stokes based approaches. Thus, the lattice Boltzmann methods have attracted
a lot of attention in the fluid dynamics community and emerged as an attractive alter-
native in many application areas. In the present paper, we discuss some perspectives
of the lattice Boltzmann methods, in particular for industrial applications and present
some successful examples from projects related to aerodynamics, chemical and process
engineering.
1. Introduction
In the past years, the methods of lattice gascellular automata (LGCA) and the lattice
Boltzmann methods (LBM) have attained a certain maturity and subsequently chal-
lenged the traditional methods of computational fluids dynamics (CFD) in many areas.
In that context, traditional methods of CFD are understood to include all numerical
schemes, that aim to solve the Navier-Stokes equations by some direct discretisation.
In contrast to that, the LBM is based on a more rigorous description of the transport
phenomena, the Boltzmann equation. Compared to other attempts, that have been
made to solve this equation in the past, the LBM makes use of several significant,
physically motivated simplifications that allow to construct efficient and competitive
or even superior computational codes as compared to the classical approaches.
Lattice gas cellular automata and even more lattice Boltzmann methods are relatively
new. Just about 15 years ago, the field of LGCA started almost out of the blue
with the now famous paper of Frisch, Hasslacher and Pomeau [1], who showed that
some simplified kind of "billiard game" representing the propagation and collision of
fluid particles leads to the Navier-Stokes equations in a suitable macroscopic limit. In
particular, the authors showed how the propagation and collisions of particles have
to be abstracted in order to conserve mass and momentum and how the underlaying
lattice has to be designed in order to provide sufficient symmetries to obtain Navier-
Stokes like behaviour. Each month, several papers appear to present new models or to](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-7-320.jpg)
![investigate existing models, to demonstrate and assess the use of LBM in application
fields or to evaluate high performance computing (HPC) aspects. Summerschools,
special conferences and LBM sessions in existing conferences have been organised to
satisfy also the growing interest of developers and potential users in this technique.
Besides that, commercial products are available with remarkable success (see e. g.
[2]).
The goal of the present paper is to show the potential of the lattice Boltzmann
method in CFD and in related areas. Besides the classical application fields, such
as aerodynamics, these are in particular problems related to chemical and process
engineering. Due to the complexity of the relevant transport and chemical conversion
mechanisms, that have to be modelled, these areas open new challenges also for the
LBM.
In the present paper, after a short summary of the basic principles of the LBM,
examples related to turbulent flows, reacting flows and the respective application fields
are discussed.
2. Lattice Gas and Boltzmann Method
From a gaskinetical, i.e. microscopic, point of view, the movement of a fluid may be
considered as the propagation and collision of molecular particles governed by funda-
mental laws of physics. The modelling of this motion may be carried out on several
levels, starting with the Hamilton equation of a set of discrete particles. Since this
approach prohibits itself because of the large number of freedoms to be considered,
several attempts have been made to simplify this picture by extracting only the essen-
tial criteria required to model e. g. the motion of a Newtonian fluid. In that context,
the lattice gas automata may be seen as an abstraction of the fluid making use of the
fact, that the statistics of the gas may be correctly described by a significantly reduced
number of molecules and by applying simplyfied dynamics of the particles. This can be
explained by the fact, that the conservation principles as well as associated symmetries
are the basic building blocks for the continuum equations of fluids. Thus, in oder to
simulate a continuum flow, the approximation of the computer gas has to recover only
these principles to a certain extend. The FHP automata, named after [1], was a first
successful attempt to construct a discrete model to compute the motion of a Newtonain
fluid. Although this discrete particle approach seems promising, there are problems
due to spurious invariants and random noise in the solutions. These deficiencies can be
overcome by applying the idea of McNamara and Zanetti [3], who considered the dis-
crete Boltzmann equation as a base for the numerical algorithm. This approach may be
briefly explained as follows: The Boltzmann equation is an integro-differential equation
for the single particle distribution function f(t, 2, g), which describes the propability
to find a particle in a volume (2, 2 + d2) and with a velocity in the range (~, ~ + dg).
Neglecting body forces one has:
cOtf + v-'Vf = Q(f) . (1)
A suitable simplification of the complicated collision integral Q(f) is the BGK approx-
imation,
1
Q(f) ~ -(f~q- f) (2)
T](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-8-320.jpg)
![which preserves the lower moments an satisfies an H-Theorem like the original equations
1. Here feq is the Maxwell equilibrium distribution. The discretisation of this equation
requires a finite representation of the distribution function in the velocity space. One
way to realise this is to introduce a finite set of velocities gi and associated distribution
functions fi(t,J, ~), which are governed by the discrete velocity Boltzmann (BGK)
equation:
Otfi _qt_~iV fi -- 1---(f'ezq -- fi) . (3)
T
Next, the discretisation in space and time is accomplished by an explicit finite difference
approximation. With a scaling of the lattice spacing, the time step and the discrete
velocities according to ~i - A2i/At, the discretised equation takes the following form:
+ zxt, t + At)- t) - _ t)).
T
(4)
The discrete values of the equilibrium functions are chosen of Maxwell type, in the sense
that their velocity moments up to fourth order are identical with the velocity moments
over the Maxwell distribution. The following definition satisfies this requirement:
-- tpp 1+ c~2 + 2c~ c~2 ~ . (5)
The discrete equilibrium functions may be computed efficiently at each time step for
each node, from the components of the local macroscopic flow velocity us, the fluid
density p, the "speed of sound cs" and a direction-dependent lattice geometry weighting
factor tp. The viscosity u of the simulated fluid can be controlled by the relaxation
time ~-, according to
. 1
3 - (6)
Further technical details of this method, in particular concerning the formulation of
boundary conditions, may be found in [4-6].
From the computational point of view the above approach is interesting as it re-
sembles a simple finite difference scheme applied to a first oder (in time and space)
hyperbolic system of equations in diagonal form. This extremely simplifies the design
of a numerical scheme. However, finally the solution of the Navier-Stokes equations
with second order accuracy in the limit of low Mach numbers (c~ > > 1~712)is recovered,
as can be shown rigorously in [7] by applying the Chapman-Enskog procedure to eq.
(4).
The approach presented above leads to the basic version of LBM. Many improve-
ments have been designed in order to broaden the methods range of applicability, see
e.g. the review article of Chen and Doolen [8]. Current issues are the multi-time re-
laxation methods to enhance the stability of the method [9,10], implicit methods [11]
or the application of nonuniform and locally refined meshes [12,13] and of improved
boundary conditions for Cartesian meshes [14]. For various applications, particularly
in chemical and process engineering, the convective and diffusive transport of energy
and species are of key importance. Examples of a few among many interesting contri-
butions made in this field by various authors can be found in [15,16]. Thermal models](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-9-320.jpg)
![have been proposed in [17,18], reaction diffusion problems have been investigated e.g.
in [19-21]. The simulation of multiphase flows and immiscible fluids were the subject of
several investigations where the LBM provided interesting alternatives to model parti-
cle interaction, surface tension, etc. In modelling suspensions of particles for example,
the interactions of the fluid and particles may be treated in different ways, either by dis-
cretising and mapping moving particles in a Lagrangian sense [22] or by a combination
of an integer lattice gas model for the disperse phase and the BCK for the fluid phase
[23]. Models for turbulent flows have been adapted from classical LES approaches,
as applied to the Navier-Stokes equations, by changing the relaxation parameter in
eq. 6 in order to scale the viscosity according to a subgrid model, such as published
in [24]. Alternatively, two-equation models have been propsed by e.g. [25-27]. The
following sections presents some practical results related to chemical engineering and
aerodynamics, that show the applicability of LBM in various technical fields.
3. Applications of LBM in chemical engineering
In chemical industries, packed beds and porous media are frequently used as reac-
tion, separation or purification units. The design of these devices is usually based on
pseudo homogeneous equations with averaged semi empirical models such as disper-
sion and mass transfer correlations. The design concepts based on these models fail
if local flow phenomena such as channeling effects become dominant. Therefore, sev-
eral attempts have been made in oder to improve these models. However, new design
methodologies are required if no or insufficent empirical data are available. Lattice
Boltzmann methods can been used to directly simulate the flow field in these configu-
rations together with chemical reactions and diffusion effects. This allows to analyse in
detail the hydrodynamic effects, e. g. the channeling due to inhomogeneous void space
distributions and other flow anomalies and to quantify their influence on the prediction
of the bulk conversion and selectivity of the reactor. The lattice Boltzmann method
has been chosen mainly because of its ability to model highly complex geometries and
fluids.
The "direct" numerical simulation of flows through packed beds uses a digitized im-
age of the structure under considerations. This may be obtained from computer tomo-
graphic data [28] of a real probe of the material or as synthetically generated geometry.
A Monte Carlo (MC) approach is used in [29] to generate randomly distributed packed
beds of spherical pellets. Following the marker and cell approach, this geometry can
easily be transferred to the uniform, Cartesian mesh, which is typically used in lattice
Boltzmann methods. Due to the low memory requirements of these methods, meshes
with several million elements may easily be used to capture the geometric details.
The fluid is modelled assuming that a species A is transported by a carrier gas through
the structure shown in Fig. 1. This species may be adsorbed by the solid particles and
convert to species B. This reaction and adsorption is assumed to be fast compared
to the convection resulting in a mass transport limitation. The DahmkShler number,
i. e. the ratio of reaction rate to flow rate is Da ~ 100; the Reynolds number based
on the particle diameter is Rep ~ 10. The structure consists of particles, randomly
distributed in a confining cylinder with a diameter ratio (cylinder/particle) of 5. The
computational domain is discretised by a 150 • 150 x 750 regular grid, i. e. more than
16.106 elements. In order to obtain a converged steady state solution, about 40,000](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-10-320.jpg)
![propagation (=iteration) steps were necessary for the present laminar flow conditions,
which took about 2h CPU time on six NBC SX 5e shared memory processors. The
simulation of the velocity field has been discussed previously [29,30] and therefore,
only the main characteristics are summarized here. In the geometries under consid-
eration, strong variations of the local porosity become dominant, in particular close
to the confining cylinder surface leading to severe flow inhomogeneities. This can be
D~/Dp=5.0
ui rm, r" i
0.8
0.6
0.4
0.2
00 1 2 3 4 5
(b) Radial position r (in multiples of the particle radius)
Figure 1. Image of the packed bed structure generated by an MC simulation (a) and the
radial voidage profile extracted from the structure (b). The tube to particle diameter
ratio is 5, leading to a severe wall effect (channeling), namely very high porosities in
the region close to the wall of the tube
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (b)
o - o x=-O. 1
- ............. 9 x=1.8
~...........~. x=3.75
-<~[~ -. ;x=5.7
'~ /1~ ..........x=+2.o / l
: - ----~--
~ 1 2 3 4 5
Distance from the wall (in multiples of the particle radius)
Figure 2. Snapshot of the flow field (a) through the MC generated geometrical structure
of the packed bed simulated by the LBA approach and radial velocity distribution (b) in
the different cross sections along the axial direction. Rep= 6.5 (laminar flow regime).
The axial positions x = -0.1 and +0.1 and +2.0 specify the distance in multiples of
the particle diameter in front/behind the packed structure. All other positions give
the distance from the beginning of the bed](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-11-320.jpg)
![(a)
+2.0
+0.I
5.7
325
1.8
-0.I
(b)
1
0.8
-9
0.8 ~'
0.4
~0.2
Z
0
0
,,................ x=1.8
....... ~, x=3.75
,. A x=5.7
1 2 3 4 5
Distance from the wall (in multiples of the particle radius)
Figure 3. Snapshot of the concentration field of reactant A (a) simulated by the LBA
approach and the radial averaged mass flux (b) in the different cross sections along the
axial direction. The corresponding flow field is shown in Fig. 2. The Schmidt number
of species A is Sc = 18.5 and the reaction occurs only at the surface of the spheres.
The reactant is continuously injected over the complete cross section at x ~ -0.5. All
axial positions are given as multiples of the particle diameter
seen from the velocity contour plots along cross sections inside the packing at various
x-positions (Fig. 2a). Predominantly close to the walls, velocity spikes are observed
with a magnitude up to eight times higher than the averaged flux. The corresponding
circumferentially averaged radial profiles of the velocity are depicted in Fig. 2b and
show the same trend known as the wall effect or wall channeling. This effect is due to
the fact that the structure of the packing close to the wall is "ordered", which leads
to high local porosities. The simulated local concentrations of species A (the reactant)
along cross sections are shown in Fig. 3. As can be seen, reactants are still present in
high concentrations close to the exit of the reactor which is due to the fact of the low
local flow residence times close to the wall and consequently, the low local conversion
rates of A. This breakthrough of species A would lead to a decrease in conversion and
degradation of selectivity.
4. Applications of LBM for turbulent flows
The application of the LBM for design tasks in aerodynamics requires to qualify the
method as a reliable tool with high predictive accuracy and efficiency. In that context,
it can be shown that the dynamics of turbulent flows can be correctly simulated, e.g.
within a DNS. Figure 4 shows a snapshot of the pressure field computed with LBM.
The geometry and flow parameters have been chosen according to the well known
benchmark test case of Kim et al. [31]. In the present calculations, the computational](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-12-320.jpg)
![mesh consists of 4096 x 256 x 256 lattice sites.
. . . . . . .
i,i)
Figure 4. Isosurfaces of the pressure computed by the lattice Boltzmann model. The
shown snapshots are at approximate times 7.57 (top) and 127 (bottom) in physical
time units 7 . Rendered pressure values on corresponding plots on top and bottom
are proportionally equal, values on the left +3 standard deviations while those on the
right are ~ 0.
The computation of a correct mean velocity is the easiest and first necessary test for a
turbulence simulation. It is passed successfully by many different numerical methods.
In that respect, the Lattice Boltzmann Method is no exception. As seen in 5, the
comparison with the reference data base [32] and also with experimental data [33] is
very good. The figure compares profiles not only of (streamwise) mean velocity but
also of mean pressure.
Also for the Reynolds stress components, the LBM and pseudo-spectral results,
shown in 6, are in excellent agreement. Some available experimental data have been
added in that figure to illustrate the expected smallness of the difference between nu-
merical results compared with the deviation of experimental data. More results related
to this test case are published in [35].
5. Conclusion
The lattice gas cellular automata and lattice Boltzmann methods emerged just 15
years ago as new techniques to describe the motion af a fluid based on the simplified
transport equations. During that time, remarkable improvements to the methods have
been introduced that allow the prediction of complex flows, i.e. fluids with chemical
reactions, diffusion processes, turbulent flows, among others. A conceptual advantage
of the LBM is the algorithmic structure, which can be efficiently implemented on digital
computers. Thus, the resulting codes require in general less memory and CPU time
compared to classical CFD methods.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-13-320.jpg)
![10
u + 10
I Illllll
9 ~ lattice Boltzmann ,
.... pseudospectral 9
Moser et al. (1999)
o Fischer et al. (2001)
J
jP
-0
Pm~. -0
-0
-0.
9 ~ I ~ lattice Boltzmann
",,~ I .... pseudospectral - -
.
I/
I/
'~. r
-II
10 100
y§
Figure 5. Comparison of LBM and a Chebyshev pseudo-spectral [34] results for mean
profiles left: streamwise velocity, right: pressure. Line styles solid: LBM ,
dashed: Chebyshev pseudo-spectral data from [32], dashed-dotted: Chebyshev pseudo-
spectral data from in-house DNS [34]. Symbols: experimental data from in-house LDA
measurements [33].
6. Acknowledgements
Financial support from the Deutsche Forschungsgemeinschft (DFG), from the Bavar-
ian Science Foundation (BFS) within FORTWIHR and from the Bavarian State Min-
istry for Science, Research and Arts within KONWIHR is gratefully acknowleged. The
calculations were mainly carried out on maschines of the Leibniz Computing Center in
Munich and the High-Performance Comnputing Centre in Stuttgart, Germany.
REFERENCES
I. U. Frisch, B. Hasslacher, and Y. Pomeau. Lattice-gas automata for the Navier-Stokes
Equation. Phys. Rev. Lett.,56(14):1505-1508, April 1986.
2. EXA Corp. Powerflow news. EXAnews, 4.0, 1998.
3. G.R. McNamara and G. Zanetti. Use of the Boltzmann equation to simulate lattice gas
automata. Phys. Rev. Lett., 61:2332-2335, 1988.
4. Y. H. Qian, D. d'Humi~res, and P. Lallemand. Lattice BGK models for Navier-Stokes
equation. Europhys. Lett., 17(6):479-484, January 1992.
5. Diether A. Wolf-Gladrow. Lattice-Gas Cellular Automata and Lattice Boltzmann Models,
volume 1725 of Lecture Notes in Mathematics. Springer, Berlin, 2000.
6. Gunther Brenner. Numerische Simulation komplexer fluider Transportvorgiinge in der
Verfahrenstechnik. PhD thesis, Technische Fakults Universits Erlangen-Niirnberg,
2002.
7. Xiaoyi He and Li-Shi Luo. Theory of the lattice Boltzmann method: From the Boltzmann
equation to the lattice Boltzmann equation. Phys. Rev. E, 56(6):6811-6817, December
1997.
8. Shiyi Chen and Gary D. Doolen. Lattice Boltzmann method for fluid flows. Annu. Rev.
Fluid Mech., 30:329-364, 1998.
9. Dominique d'Humi~res. Generalized lattice-Boltzmann equations. In Bernie D. Shizgal
and David P. Weaver, editors, Rarefied Gas Dynamics: Theory and Simulation, Progrss](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-14-320.jpg)
![14
1 INTRODUCTION
The overall motivation for current large-scale, multi-institutional Grid projects is to enable
the resource and human interactions that facilitate large-scale science and engineering such as
aerospace systems design, high energy physics data analysis [1], climatology, large-scale
remote instrument operation [2], collaborative astrophysics based on virtual observatories [3],
etc. In this context, Grids are providing significant new capabilities to scientists and engineers
by facilitating routine construction of information and collaboration based problem solving
environments that are built on-demand from large pools of resources.
Functionally, Grids are tools,
middleware, and services for:
o building the application
frameworks that allow
discipline scientists to express
and manage the simulation,
analysis, and data
management aspects of
overall problem solving
o providing a uniform look and
feel to a wide variety of
distributed computing and
data resources
o supporting construction,
management, and use of
widely distributed application
systems
o facilitating human
collaboration through
common security services,
and resource and data sharing
o providing remote access to,
and operation of, scientific
and engineering
instrumentation systems
o managing and securing this
computing and data
Application Portals / Frameworks
(problemexpression;userstatemanagement;collaboration
services;workflowengines;faultmanagement)
Web Grid Services
Applications and Utilities
(domainspecificandGridrelated)
Language Specific APIs
(Python,perl,C, C++,Java)
Grid Collective Services
(resourcebrokering;resourceco-allocation;datacataloguing,
publishing,subscribing,andlocationmanagement;collectiveI/O,
job management)
Grid Common Services
(resourcediscovery;resourceaccess;authenticationand security;
eventpublishandsubscribe;monitoring)
Communication Services
SecurityServices
Resource Managers
(exportresourcecapabilitiesto the Grid)
Physical Resources
(computers,datastoragesystems,scientificinstruments,etc.)
Figure 1. Grid Architecture
infrastructure as a persistent service
This is accomplished through two aspects: A set of uniform software services that manage
and provide access to heterogeneous, distributed resources and a widely deployed
infrastructure. The software architecture is depicted in Figure 1, and the deployment is
discussed later.
3 MULTI-DISCIPLINARY APPLICATION EXAMPLE
As the problems tackled by the science and research engineering communities become more
and more complex, the computing requirements are not just for more computing power, but for
dealing with more complex application and data systems as well. Coupling the sub-
components of a jet engine simulation in order to simulate its overall operation, and then](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-18-320.jpg)
![15
coupling that ensemble to the aerodynamic lift models of an aircraft wing, is an example of a
complex system that also requires high capability computing and data handling - multiple
computers, databases, archives, etc.- all of which must be coordinated to solve the problem.
The NPSS program [4] at NASA Glenn is working on coupling the many component
models required to simulate an operational jet engine, and integrating the resulting engine
model with operational data. The sub-system simulations have been developed over a long
time and they are written in a variety of languages (e.g. FORTRAN) and in a variety of styles.
The NPSS program has built an application framework for coupling these components
together [5], and working with NASA Ames, a wing lift model is being added to the multi-
disciplinary simulation ("MDS").
4 APPLICATION CHARACTERISTICS
From this MDS example we can enumerate a set of high-level characteristics of the
application, which turn out to be fairly general.
o system simulations are built up by coupling legacy code components
o computing capacity and simulation
expertise will come from many
different organizations
o simulation components must be
coordinated on many different
computing systems
o computational simulations must be
coupled to independent
environmental and operations data
sources that originate from hundreds
of different locations
o confidentiality of data and data
access policy enforcement is required
o security and access control for the
underlying computing and data
archive systems must prevent service
disruption
Figure 2. Multi-component, multi-disciplinary
simulation
Multiplesub-systems,e.g. a wingliftmodeloperatingat
NASAAmesand a turbo-machinemodeloperatingat
NASAGlenn,are combinedusingGlenn's NPSS
(NumericalPropulsionSystemSimulation)application
frameworkthatmanagesthe interactionsofmultiple
modelsandusesGridservicesto coordinatecomputing
and datastoragesystemsacrossNASACenters.
5 THE ROLE OF GRIDS
The application characteristics from the example above imply a collection of capabilities
that must be addressed by the distributed systems that combine computational tools with each
other, and with data and instruments.
In addition to the environment and services needed to support these applications, our
experience in working with the design engineer / analyst who must use the system to
accomplish a specific task suggests many other characteristics and requirements as well. (E.g.,
see [6].)
The desired Grid fianctionality may be represented as a hierarchically structured set of
services and capabilities that are described below, and whose interrelationship is illustrated in
Figure 1. Some of the issues that Grids currently address include:
o techniques for locating, incorporating, accessing, and managing resources in the overall
environment that are scalable to thousands of resources](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-19-320.jpg)
![16
o coordinating resource availability so that the components of an MDS that are located on
different systems can operate simultaneously
o comprehensive network monitoring to locate, analyze, and correct bandwidth
bottlenecks
o global data catalogues and data replica management
o security and management of access rights for the collaboration data and information
Additionally, scientific and engineering applications involving distributed teams and
distributed resources requires higher level services, many of which are currently being
developed in the Grid community:
o techniques for coupling heterogeneous computer codes, resources, and data sources in
ways so that they can work on integrated/coupled
o describing and managing multi-step, asynchronous component workflows, including
managing fault detection and recovery
o access to data and metadata publication and subscription mechanisms
o global event mechanisms - e.g. notification of when data or simulation results come into
existence anywhere in the space of resources of interest
o service oriented interfaces to all of the above
The NPSS MDS described above uses an existing framework (CORBA based) to
coordinate the components, and uses Grid services to access computing systems, provide
security, etc. (Figure 3). ~-
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o a common databne::: : : ......
! :1
: ! : 9 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
-::::...::~
: ~ :....... ,.,, .... ,.,,,,;~. ~
....... ...... ...,. ...... ,..:........... ,:
security approach : (Obj! ) :.: ~,
........................... .~............................
= o
and infrastructure , ........... , ........ , ............... i
o persistent Grid
services that are
used to run the ~
CORBA
applications on an
as-needed basis
The ability to use the . . . . . .
Grid to instantiate i
existing application Figure 3. NPSS / CORBA use of Grids
frameworks allows ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
users flexibility in building their applications in the framework of their choice. They do not
have to rely on that framework being provided as a persistent service on all of the computing
systems where they need to run - they can instantiate the framework environment, as needed,
using persistent Grid services.
6 COMPUTING AND DATA GRIDS
In this section be give a brief overview of the roles of the architectural components of a
Grid as illustrated in Figure 1. We will start at the lowest layer and work up. In many extant](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-20-320.jpg)
![17
Grids, the basic services are provided by Globus [7], and several of the Globus services will be
mentioned below. More detail may be found in [6].
6.1 Resource Managers
The basic function of resource managers is to export the functionality provided by the
resource - computing, storage, bandwidth, etc. - so that it is accessible to the Grid. There are
two fundamental Grid resource managers, with a third being developed.
The most basic service is the ability to authenticate and authorize a user, and then provide
to that user the mechanism to download code and run it. This is provided by the server-end of
the Grid job initiator. (See [8].) The second basic service is access to data. This is provided by
the GridFTP server, which is described below. A third basic service that is currently being
developed is a generalized event manager. This service should not only manage systems events,
but will provide user jobs with the ability to signal, e.g., a workflow manager about its state.
To support co-scheduling of multiple resources, CPU advance reservation scheduling and
network bandwidth advance reservation are essential. In addition, tape marshaling in tertiary
storage systems to support temporal reservations of tertiary storage system off line data and/or
capacity is also likely to be essential. The basic functionality for co-scheduling and/or resource
reservation must almost always be provided by the individual resource managers, however
Grid services such as Globus' GRAM/GARA serve to coordinate and provide uniform access
to these resource specific services. CPU advance scheduling services are currently provided,
e.g., by the PBSPro batch scheduling system (http://www.pbspro.com/), and is a topic in the
Grid Forum's Scheduling Working Group (see [9]).
6.2 Resource Managers
Grid security is based on cryptographic identity tokens that are issued via some
organizationally acceptable process. Many Grids use Public Key Infrastructure to provide
these tokens (called X.509 identity certificates). For more information, see [10]. The Globus,
Grid Security Infrastructure (GSI) uses the identity certificate to support a "single sign-on"
capability- that is, the user authenticates once for access to all Grid resources for which the
user is authorized. Identity certificates are also issued to computing systems and services (such
as secure FTP). The identity certificate, in conjunction with various security protocols, is used
to provide message integrity (the message that got to the receiver is guaranteed to be identical
to the one that the sender sent) and, optionally, message confidentiality. (See [11].)
6.3 Communication Services
Grid communication services provide a uniform model for communicating among
distributed job components. The Globus I/O function provides a standard interface to stream
and datagram (message) based I/O, and in each case provides transparent use of the security
services.
6.4 Grid Resource Access and Management
"Grid Resource Access" refers to the basic services that provide uniform and location
independent access and management of distributed resources. Much of the operational effort to
run Grids is involved in maintaining these services and their associated resource manager.
Many Grids, including NASA's IPG [12],DOE's Science Grid [13] and ASCI Grid [14], and
the Grids of the NSF Supercomputer centers [15], currently use Globus to provide the basic](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-21-320.jpg)
![18
services that characterize and locate resources, initiate and monitor jobs, provide secure
authentication of users, provide uniform access to data, etc.
The Grid Information Service is the Grid resource discovery mechanism. It has the task of
representing, and/or providing access to, virtually all aspects of the configuration and state of
the Grid: it maintains, or provides pointers to services that provide, all Grid resource
characteristics (e.g. CPU count, memory size, OS version, etc.), dynamic performance
information, information about current process state, user identifies, allocations and
accounting information, etc. It also responds to structured queries about the availability of
resources with certain characteristics and performance state, e.g. the computing systems
architecture needed for a particular code. The GIS is implemented as a distributed collection of
directory servers [16].
A Grid resource access service / job initiator provides a standard way of specifying a
resource and then submitting a request for service. Most commonly this is a script to be
executed on a remote system, and this service provides a uniform interface to the underlying
batch scheduling systems. The Globus GRAM service provides this function.
GridFTP is a file oriented, remote data access service that provides third party transfers,
partial file access, reliable transfer, parallel transfers and server side data striping if supported
by the data storage system, Grid security based access control, etc. The resource side of this is
a special FTP server that implements an extended FTP service. (See [17].)
Monitoring services are important for both users and the managers of the Grid in order to
do fault detection and management, system and application debugging, and performance
optimization. The Globus Heart Beat Monitor provides basic, job-level monitoring. The
NetLogger toolkit [18] provide detailed monitoring of distributed applications, and the
Network Weather Service [19] provides for network monitoring and characterization. All of
these are being integrated as Grid services, and the Grid Forum, Grid Performance Working
Group [20]is addressing this issue in a general way.
General Grid event services are being addressed in the Grid Forum, Grid Computing
Environments working group (see the GCE working group pages at www.gridforum.org).
6.5 Grid Collective Services
There are many "collective services" being developed for the Grid. Such services provide
aggregated functionality that makes it easier to use the Grid. Of the many such services being
developed, and only a few that are of interest to computational scientists are mentioned here.
A resource broker takes a requirements specification for a job (min. CPU count, min.
memory, etc.) together with a policy statement (e.g., "run this job as soon as possible" or "run
this job at the least cost") and locates a set of resources that will meet the requirements.
Resource brokers are currently under development.
A job manager automates the task of ensuring that a large collection of related job (e.g.
those in a parameter study) all execute exactly once. Condor-G [21]provides such a service for
the Grid.
High-performance applications require high-speed access to data files, and the Grid
services must be able to locate, stage, cache, and automatically manage the location of local,
remote and cached copies of files. Publishing metadata about the data is critical in large
collaborations where many different people have to use each other's data. These services are
being developed in the GriPhyN (Grid Physics Network) project [1], and several are already in
use.
MPICH-G [22] is a grid-enabled implementation of the Message Passing Interface (MPI)](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-22-320.jpg)
![19
that allows a user to run MPI programs across multiple computers at different sites using the
same commands that would be used on a parallel computer. MPICH-G extends the Argonne
MPICH implementation of MPI to use services provided by the Globus Grid toolkit.
6.6 Applications and Utilities
The components managed by application portals are constructed from are applications that
perform simulations, data transformations, data analysis, etc. The applications are typically
developed independently of the Grid milieu, but must be incorporated into it in order to
participate in Grid managed distributed applications. What has to be done in order to
accomplish this varies widely with the nature of the application and the framework that is used
to coordinate the interoperation of the components. Some of the typical issues are discussed in
[5].
There are many Grid based utilities being developed to assist in the use and management of
Grid based applications. For example, SSH is a well-known program for doing secure logon to
remote hosts over an open network. GSI enabled SSH (GSSH) is a modification of SSH that
allows SSH to use Grid certificates and designated proxy certificates for authentication. [23]
6.7 Web Services
Building on concepts and technologies from the Grid and Web services communities, the
concept of a Web Grid Service is rapidly evolving, Such a service defines standard
mechanisms for creating and discovering transient Grid Service instances, provides location
transparency and multiple binding protocols for service instances, and supports mapping
services for integration with underlying native platform facilities.
These services are defined in terms of Web Services Description Language (WSDL)
interfaces, and provide the mechanisms required for creating and composing sophisticated
distributed systems. These mechanisms include code encapsulation and interface
characterization, lifetime management, reliable remote invocation, change management,
credential management, and notification.
Discovery services, like the Universal Description, Discovery, and Integration (UDDI)
service, will provide for locating services based on their functional characteristics, and then
provide the detailed descriptions of the data types and interfaces needed to use those services
with other services.
Web Grid Services implement a service oriented architecture, allowing Grid functionality to
be incorporated into a Web services framework that provides for constructing complex
application problem solving portals based on frameworks like IBM's WebSphere [24],
Microsoft's .NET [25],Apache Tomcat [26], etc. This will allow the integration of science and
experimental engineering services with the commercial engineering, manufacturing, planning,
cost management, etc., services that are becoming available through major industry
commitments to this approach.
This is a recent development that is receiving a lot of attention in industry and in the Grid
Forum. See, e.g., the Grid Forum's OSGI working group pages, [27], [28], and [25]. For a
general introduction to Web Services, see [29].
6.8 Problem Solving Environments: Knowledge Based Queries, User Interfaces and
Workflow Management
The Knowledge Grid :It is clear that for the Grid to realize the maximum impact on science](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-23-320.jpg)
![20
and engineering that there must be mechanisms for discipline problem solvers to be able to
express a problem in terms of the knowledge framework of their discipline, and then have that
problem translated to the computational and data analysis operations of the underlying
problem solving system. There have been various discipline specific efforts to do this sort of
thing, but not much general infrastructure has been developed. The approach of Cannataro, et
al [30] suggests one way to approach at least the representation and manipulation of the
knowledge base that could translate moderately abstract queries in to sets of computations and
data analysis that resolve the query.
The User Interface- Integration with the Desktop: A number of services directly support
using the Grid by engineers or scientists. These include the toolkits for construction of
application frameworks / problem solving environments (PSEs) that integrate Grid services
and applications into the "desktop" environment. Services available in the user interface
should include, for example, the graphical components ("widgets" / applets) for building
application user interfaces; methods for control of the computer mediated; distributed human
collaboration that support interface sharing and management; the tools that access the resource
discovery and brokering services; the tools that provide generalized workflow management
services such as resource scheduling and managing high throughput jobs, etc. This will likely
come about from the Web Services frameworks mentioned above.
Workflow Management: Reliable operation of large and complex data analysis and
simulation tasks requires methods for their description and control. A workflow management
system must provide for a rich and flexible description of the analysis processes and their
inter-relationships, and also provide mechanisms for fault detection and recovery strategies in
widely distributed systems. Workflow management systems will carry out the human defined
protocols for, e.g., multi-disciplinary simulations and data analysis; global data cataloguing
and replica management systems to manage the data for these scenarios, and global event
services to manage the dynamic aspects of work protocols, will be essential adjuncts to the
workflow engines. The emerging Web Services Flow Language should assist with this. See [31].
7 SERVICES FOR OPERABILITY: OPERATIONS AND SYSTEM
ADMINISTRATION
Implementing a persistent, managed Grid requires tools for deploying and managing the
system software. In addition, tools for diagnostic analysis and distributed performance
monitoring are required, as are accounting and auditing tools. Operational documentation and
procedures are essential to managing the Grid as a robust production service.
To operate the Grid as a reliable, production environment is a challenging problem. Some
--~-~.z~:,.
..................
! i~iiiiiiiii
,~ ::~ 9, i of the identified issues include management tools for the Grid
, .......... ~ , ~
............9 ~ Information Service; diagnostic tools so operations/systems staff can
i'~ ..................=~7~,::1~;~i~i investigate remote problems, and; tools and common interfaces for
.............
~:~...........
~ system and user administration, accounting, auditing and job tracking.
I:~ Verification suites, benchmarks, regression analysis tools for
7~~ ........... ..............
.~i~!::~=~i:ii;i.~performance, reliability, and system sensitivity testing are essential
parts of standard maintenance.
Tools and documentation for operating production Grids are being developed at NCSA [32],
and in the IPG [12]and DOE Science Grid []3] projects. See [33]for the general issues involved
in implementing production Grids.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-24-320.jpg)
![21
8 CURRENT STATE OF GRIDS
There are several Grids that are at, or close, to production status. NASA's Information
Power Grid (www.ipg.nasa.gov), the DOE Science Grid (www.doesciencegrid.org) and the
UK eScience Grid (www.nesc.ac.uk/intro) are all working toward production Grid
environments.
IPG has, over the past three years, deployed a prototype production Grid. By production,
we mean that the services and resources are persistent, there are operational groups responsible
for those services and resources, and there is documentation and user support.
In the process of building this Grid environment a great deal is being learned about
integrating Grids into production supercomputing environments, and some of the issues and
lessons learned are documented in [33]and [34].
9 ACKNOWLEDGEMENTS
IPG is funded primarily by NASA's Aero-Space Enterprise, Computing, Information, and
Communication Technologies (CICT) Program (formerly the Information Technology),
Computing, Networking, and Information Systems Project. DOE's Science Grid is funded by
the U.S. Dept. of Energy, Office of Science, Office of Advanced Scientific Computing
Research, Mathematical, Information, and Computational Sciences Division
(http://www.sc.doe.gov/production/octr/mics) under contract DE-AC03-76SF00098 with the
University of California.
10 REFERENCES
[1]
[2]
[3]
[4]
[5]
GriPhyN - Grid Physics Network, GriPhyN. http://www.m'iphvn.org
Network for Earthquake Engineering Simulation Grid (NEESgrid). http:#www.neesgrid.org/
Virtual Observatories of the Future, Caltech. http://www.astro.caltech.edu/nvoeonf/
Numerical Propulsion System Simulation, NPSS. http://hpee.lerc.nasa.gov/npssintro.shtrn!
A CORBA-based Development Environment for Wrapping and Coupling Legacy Codes, G. Follen, C.
Kim, I. Lopez, J. Sang and S. Townsend. In Tenth IEEE International Symposium on High Performance
Distributed Computing. 2001. San Francisco.http://cnis.grc.nasa.gov/papers/hpdc-10_corbawrag_ping.pdf
[6] Role of Computational and Data Grids in Large-Scale Science and Engineering, W. Johnston. Int. Journal
Of High Performance Computing Applications, 2001.15(3).
http://www.itg.lbl.gov/~johnston/Grids/homepage.html#JHPCA2001
[7] The Globus Project, Globus. http://www.globus.org
[8] A Resource Management Architecture for Metacomputing Systems, K. Czajkowski, I. Foster, N. Karonis,
C. Kesselman, S. Martin, W. Smith and S. Tuecke, in The 4th Workshop on Job Scheduling Strategies for
Parallel Processing. 1998.p. 62--82. http://www.globus.org/research/papers.html#GRAM97
[9] Scheduling and Resource Management Area, Global Grid Forum.
http://www.gridforum.org/3_SRM/SRM.htm
[10] ESnet's SciDAC PKI & Directory Project - Homepage, T. Genovese and M. Helm. DOE Energy Sciences
Network. http://envisage.es.net/
[11] Design and Deployment of a National-Scale Authentication Infrastructure, R. Butler, D. Engert, I. Foster,
C. Kesselman, S. Tuecke, J. Volmer and V. Welch. IEEE Computer, 2000. 33(12): p. 60-66.
http://www.globus.org/research/papers.html#GSll
[12] NASA's Information Power Grid, IPG. http://www.ipg.nasa.gov](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-25-320.jpg)
![22
[13] DOE Science Grid. http://www.doescienc~grid.org
[14] Distance Computing and Distributed Computing (DisCom2) Program, DISCOM.
http://www.cs.sandia.gov/discom
[15] NSF PACI Program, PACI. http://www.ncsa.uiuc.edu/About/PACl/
[16] Grid Information Services for Distributed Resource Sharing, K. Czajkowski, S. Fitzgerald, I. Foster and
C. Kesselman. In HighPerformanceDistributed Computing- 10. 2001. San Francisco,
CA.http://www.globus.org/research/papers.htm!#MDS-HPDC
[17] GridFTP Update January 2002, W. Allcock, J. Bresnahan, I. Foster, L. Liming, J. Link and P. Plaszczac.
http://www.globus.org/research/papers.html#GridFTP-Update-Jan2001
[18] The NetLogger Methodology for High Performance Distributed Systems Performance Analysis, B.
Tiemey, W. Johnston, B. Crowley, G. Hoo, C. Brooks and D. Gunter. In Proc. 7th IEEESymp. on High
PerformanceDistributed Computing. 1998.http://www-didc.lbl.gov/NetLol~er/
[19] Forecasting Network Performance to Support Dynamic Scheduling Using the Network Weather Service,
R. Wolski, in Proc. 6th IEEESymp. on HighPerformanceDistributed Computing. 1997:Portland, Oregon.
[20] Grid Monitoring Architecture Paper, Grid Performance Working Group. Global Grid Forum. http://www-
didc.lbl.gov/GridPerf
[21] Condor-G, J. Frey, T. Tannenbaum, M. Livny, I. Foster and S. Tuecke. In Proceedingsof the Tenth
InternationalSymposiumon HighPerformanceDistributed Computing(HPDC-IO).2001: IEEE
Press.http://www.globus.or~research/papers.html#Condor-G-HPDC
[22] A Grid-Enabled MPI: Message Passing in Heterogeneous Distributed Computing Systems, I. Foster and
N. Karonis. In SC98. 1998.http://www.globus.org/research/Dat~ers.html#mpich98
[23] Using GSI Enabled SSH, NCSA.
http://www.ncsa.uiue.edu~serlnfo/Alliance/GridSecurity/GSl/Tools/GSSH.html
[24] Web Services, IBM. http://www-106.ibm.com/developerworks/webservices/
[25] .NET, Microsoft. http://msdn.microsot~.com/net
[26] Jakarta Tomcat, Apache. http://jakarta.apaehe.org/tomeat/index.html
[27] Towards Globus Toolkit 3.0: Open Grid Services Architecture, Globus Project.
http://www.globus.org/ogsa/
[28] IBM, Globus announce open grid services, IBM. 2002. http://www.ibm.com/news/us/2002/02/202.html
[29] BuildingWebServiceswithJava: MakingSense ofXML, SOAP, WSDL,and UDDI, S. Graham, S.
Simeonov, T. Boubez, D. Davis, G. Daniels, Y. Nakamura and R. Neyama. 2002: Sams Publishing Co.
[30] The Knowledge Grid: Towards an Architecture for Knowledge Discovery on the Grid, M. Cannataro, D.
Talia and P. Trunfio. In FIRST EUROGLOBUS WORKSHOP.2001. Robinson Club Apulia Village, Marina di
Ugento, Lecce.http://www.euroglobus.unile.it/
[31] Web Services Flow Language (WSFL), R. Cover. http://xml.coveroa~es.or~wsfl.html
[32] Alliance Computational Environments & Security, NCSA.
2001.http://www.ncsa.uiuc.edu/TechFocus/Proj cots/
[33] Implementing Production Grids, W. Johnston.
http://www.itg.lbl.gov/~johnston/Grids/homet~a~e.html#1mNement
[34] NASA's Information Power Grid: Production Grid Experience with Distributed Computing and Data
Management, W. Johnston. In Second Global GridForum Workshop(GGF2).2001. Washington,
D.C.http://www.itg.lbl,gov/~j ohnston/Grids](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-26-320.jpg)
![Parallel Computational Fluid Dynamics - New Frontiers and Multi-Disciplinary Applications
K. Matsuno, A. Ecer, J. Periaux, N. Satofuka and P. Fox (Editors)
9 2003 Elsevier Science B.V. All rights reserved. 23
A Different Approach to Large-Eddy Simulation with Advantages for Com-
puting Turbulence-Chemical Kinetics Interactions
J. M. McDonough
Departments of Mechanical Engineering and Mathematics
University of Kentucky
Lexington, KY 40506-0503, USA
E-mail: jmmcd@uky.edu
We present an alternative to usual large-eddy simulation based on filtering the solution
and modeling physical variables on the sub-grid scales. We focus on behavior of these
subgrid-scale models and show, at least qualitatively, that they are capable of model-
ing interactions between hydrodynamic turbulence and chemical kinetics. Moreover, the
overall approach exhibits potential for parallelization at several levels.
1. INTRODUCTION
It is widely accepted that direct numerical simulation (DNS) would be the preferred
approach for essentially every analysis of turbulent reacting systems, but even with par-
allelization this will be viable only as a research tool in the immediate future. Large-eddy
simulation (LES) in some form presents a feasible direction for near-term calculations;
but although much progress in LES has been made in recent years, it is still far from
being a completely reliable tool in the context of turbulent combustion---especially if any
but the simplest kinetic mechanisms are used. Much recent research in LES has focused
on the subgrid-scale (SGS) models, and these are especially problematic in the context of
finite-rate chemistry. Various approaches have been implemented ranging from laminar
flamelets, Cook et el. [1] through PDF models, e.g., Pope [2] and including extension
of scale similarity ideas to reactive scalars as in Germano et al. [3]. Giacomazzi et al.
[4] present an approach that resembles, in some respects, the treatment to be presented
here. In addition, the linear eddy models (LEMs) of Kerstein and co-workers (see e.g.,
Kerstein [5] and Echekki et el. [6]), and more recently the one-dimensional turbulence
(ODT) models of the same group of investigators, incorporate some of the ideas to be
used below, but differ considerably in detail. While much additional important work is
available in the literature, for the sake of brevity we will avoid an exhaustive review.
Our approach, while retaining the basic LES decomposition of dependent variables,
differs from usual LES in three distinct ways: i) dependent variables are filtered rather
than the governing equations; ii) physical variables are modeled on the sub-grid scales
instead of their statistics; iii) modeled quantities are directly used to enhance resolved-](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-27-320.jpg)
![24
scale solutions, rather than being discarded.
Thus, if we let Q - (Q1,Q2,..., QN,) T denote the solution vector consisting of Nv
components, we decompose this as
t) = t) + t) x e R d, d=2,3, (1)
and substitute into the transport equation(s) for Q to obtain
(q + q*)t + V . F(q + q*) = V . G(q + q*) + S(q + q*) . (2)
Here, the subscript t denotes partial differentiation with respect to time, and V- is the
divergence operator. F and G are, respectively, advective and diffusive fluxes, and S is
a general nonlinear source term.
In contrast to usual LES, we do not filter this equation. Instead, we model q* to
produce "synthetic velocity" and "synthetic scalars" of the general form
q* = A~Mi , i = 1, 2,..., ?iv. (3)
These are substituted into Eqs. (2) which are then solved for q. The results are filtered,
if necessary, to remove aliasing due to under resolution, and then the complete solution
at the current time step is obtained from Eq. (1).
In Eqs. (3) the Ais are amplitudes derived from scaling laws of Kolmogorov (see,
e.g., Frisch [7]) using scale similarity to estimate certain sub-grid variables of the Kol-
mogorov formalism. Details of these constructions will be outlined below. The M~s are
discrete dynamical systems (DDSs) representing temporal fluctuations of the SGS quan-
tities. We have recently introduced a procedure (described below) for obtaining the Mis
directly from the governing equations, and we have employed this in several recent studies,
the two most closely related to the present work being McDonough and Huang [8] and
McDonough and Zhang [9]. The purpose of the present paper is to outline the overall
procedure for constructing synthetic-variable SGS models but with particular emphasis
placed on the behavior of the DDSs. We will consider a realistic kinetic mechanism for
H2-air combustion, and provide comparisons of the temporal fluctuations produced by
the corresponding DDS with the experimental results Meier et al. [10].
2. ANALYSIS
In this section we introduce the governing equations and provide a preliminary demon-
stration of the effectiveness of filtering solutions instead of equations. We then derive a
general discrete dynamical system that can be used to model any desired kinetics and
simplify this to a form corresponding to a particular reduced mechanism for H2-air com-
bustion. Finally, we briefly describe calculation of the amplitudes in Eqs. (3).
2.1 Governing Equations
The general equations describing fluid flow, heat transfer and chemical reactions are
well known and can be found in any standard reference, e.g., Libby and Williams [11].](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-28-320.jpg)
![25
We present them here in the following form.
pt + v. (pU) = o,
DU
= -Vp + v. (#vu) + og,
P Dt
DT N, (py~ N,
i=l i=1
D(pYi)
= V. (pDiVYi) +@i, i = 1,...,]Vs.
Dt
(4a)
(4b)
(4~)
(4d)
Here,
N~
j=l
(5)
with
t,j
~j = kl,jH w~]
1=1
- k~'JH w~]
~=1
V II
l,j
(6)
In these equations notation is standard, and we refer the reader to Ref. [11] for details.
2.2 Solution Filtering
It is worthwhile at this point to briefly return to the first of the three differences
between the approach being described here and usual LES, viz., filtering solutions instead
of the governing equations. The consequence of not filtering the equations is easily seen
and impacts two specific areas associated with the solution process. First, filtering of
nonlinear terms leads to the SGS stresses and scalar fluxes that require statistical models.
This presents especially serious difficulties with regard to the reaction rate formulas.
Second, in generalized coordinates the filtering process does not exactly commute with
the differential operators, so extra errors arise even in approximating linear terms. In the
approach being considered here, neither of these problems occurs.
It should be fairly obvious that filtering solutions is little more than a signal-processing
problem, so the main difficulty is selecting an appropriate filter. We have indicated above
that the need for filtering arises from under resolution (mainly spatial) imposed by the
LES method itself; so a low-pass filter capable of removing aliasing effects is crucial. We
have employed a fairly simple one:
k+2
(7)
for a one-dimensional problem of filtering a typical grid function {fi}iN1 for a](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-29-320.jpg)
![26
data set containing N points;
the extensions to 2D and 3D
are fairly obvious. Figure 1
provides a comparison between
using this filter and employing
a usual LES procedure with a
Smagorinsky model for solving
Burgers' equation forced with a
complicated quasiperiodic func-
tion so as to yield a known an-
alytical solution (see Yang and
McDonough [12]). It is easily
seen from this figure that filter-
ing the solution produces results
that are generally as good as
those produced by the full LES
procedure, but we still have the
ability to improve this solution
further by adding the SGS syn-
thetic velocity.
2
-.j
o
2
0
-.2
-4
0.0 0.2 Q4 O~ O.B
Sc~iKILm~hI~ ilr~'Unil~
1.0
Figure 1: Solution filtering compared with equa-
tion filtering: (a) usual LES, (b) solution filtering.
2.3 The Discrete Dynamical System
The approach we employ for modeling the M~s in Eq. (3) was first introduced in [8]. It
provides a systematic technique for deriving DDSs that are directly related to the partial
differential equations (PDEs) they are to model. The premise employed to start the
procedure is that all solution variables possess Fourier series representations constructed
from a complete orthonormal basis that "behaves like" complex exponentials with regard
to differentiation. We substitute these into the governing equations (4), Construct the
Galerkin ordinary differential equations and decimate the result to a single mode per
original PDE. Then we use very simple numerical integrators (forward or backward Euler
methods) to perform the temporal discretization and obtain
a('~+1) --/~ua (n) (1 - a(n)) - %a(n)b (n), (8a)
b (n+l) : ~v b(n) (1 - b(')) - %a(~)b(~) + aTC (n), (8b)
c(n+l) = a .~(,~+1)
T~i -- %Ta (~+~) -- %Tb(~+~) c(~) -- ~ H~Lbi
i=l 9 i=1
/(1 +/~T) + Co, (8C)
d~ n+l) -- - (~]Q"~-"fu~a(~+1) + "/v~b(n+m))d~n) + &i + di,o,
i = 1, 2,..., Ns, (8d)](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-30-320.jpg)
![28
reaction (9a) is HO2 so the corresponding DDS is
d~n+l) -" - (~Y7 + "YuY7a(n+l) + ~vY~b(n+l)) d~n) -~ w7 + d7,0,
with
(11)
,, W7
&7 = CI,7,1dld2 Cf,7,1 = v7,1WIW2kf,1.
We note that introduction of the species molecular weights arises from the form of Eqs.
(5,6). The second product of reaction (9a) and the remaining reactions can be treated
in an analogous way, leading to a total of 18 iterated maps in the complete DDS for the
chemical reactions.
One of the features of the present approach is its ability to directly account for differ-
ent reaction rates of the individual elementary reactions comprising a kinetic mechanism
and relate this to the time scale of the hydrodynamic turbulence. This is done by calcu-
lating the Kolmogorov-scale DamkShler number Dag for each elementary reaction using
its specific reaction rate to obtain a chemical time scale, and employing an input hydro-
dynamic turbulence time scale. Then for each velocity time step a number of iterations
proportional to Dag is performed for each reaction.
2.5 Amplitude Factors.
The amplitude factors of Eqs. (3) are constructed for each ith solution component using
second-order structure functions (see Ref. [7]) and a generalization (to nonhomogeneous,
anisotropic cases) of Kolmogorov's inertial range energy spectrum. That is, we determine
the subgrid-scale energy for the k th Fourier mode of the ith variable from
Ei(k) - C2,i<~)~'k -(~'+1) , (12)
where <c> is a spatially-averaged turbulence kinetic energy dissipation rate obtained from
resolved-scale results via scale similarity, and the C2,i and ~i are obtained from local (in
time and space) structure function correlations.
More details can be found in Mc-
Donough [13]. The A~ are calculated
as the sum of the Ei(k) over a few
wavenumbers k in the inertial sub-
range, and possibly into the dissipation
scales.
Figure 2 provides an indication of the
potential of this approach; it displays
an instantaneous snapshot of a turbu-
lent pool fire with oscillations modeled
with Eq. (3) and applied to experimen-
tal mean data of Weckman and Strong
[14].
Figure 2: Effect of amplitude fac-
tors: instantaneous temperature fluctua-
tions about measured average.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-32-320.jpg)
![29
3. RESULTS AND DISCUSSION
In this section we present results of running the DDSs discussed above. We begin by
describing the problem setup; we then present computed results and compare these with
a portion of the experimental co-flow data, denoted H3, from reference [10]. We will, for
the sake of brevity, consider only one location within the non-premixed 50% N2-diluted
H2-air flame of this experiment. This corresponds to an axial location of x/D - 20 with
D -- 8 mm being the H2/N2 jet exit diameter; the measurement location is at a radial
distance of 10.5 mm from the jet centerline.
3.1 DDS Model Problem Setup
The full system of Eqs. (8) has been coded in Fortran 90 and for the present study
was specialized to equations of the form (11) corresponding to the nine-step mechanism
(9). The velocity time scale used in the code to set the Damkbhler number was equated
to the data sampling frequency of the H3 experiment (10 kHz), and 104 velocity time
steps were computed using Eqs. (8a,b). During each such time step a number (>_ 1) of
chemical time steps was computed for each reaction with the number being calculated
during each velocity time step using local in time temperature results to set the reaction
rate, and thus Dag, for each reaction. Following calculation of all fluctuating species, the
fluctuating temperature is updated via Eq. (8c).
2400 . . . . . .
$ 2000
1600
~ 1200
800
0.03 (b) . . . . I
O0
o18 I-(c). . . . I I.
0.1
0.06
0.02 1
0.16
'~' 0.12
s o.o8
0.04
0 0.01 0.02 0.03
Time
Figure 3: Experimental data, temperature
and mass fractions; (a) temperature, K,
(b) H2, (c)02, (d)H20.
2200
$ 1800
.,,.,
1400
~. 1000
0.006
0.0O2
0.O8
,-.-, 0.06
O~ 0.04
0.02
0.14
s o.13
0.12
0 0.01 0.02 0.03
Time
Figure 4: DDS model results; (a) through
(d), same as in Fig. 3.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-33-320.jpg)
![30
3.2 Computed Results~ Comparisons with Data
Figures 3 (experimental, Ref. [10]) and 4 (computational, present model) provide a
qualitative comparison of our results with those from a well-established data base that
was selected as a "standard flame" at the International Workshop on Measurements and
Computation of Turbulent Nonpremixed Flames, Naples, 1996. We first observe that the
qualitative appearance is very similar between corresponding parts of these figures. This
demonstrates the general ability of the DDS model to provide physically realistic turbulent
fluctuations in a nontrivial situation.
It is important to recognize that the results displayed here come from a single execution
of the DDS--they are not the result of individual runs for each species and temperature.
Moreover, additional species mass fractions for which there are no data were calculated
at the same time but are not presented here. It should also be noted that the differences
in amplitudes seen in these two sets of figures are inconsequential. In complete LES im-
plementations the DDS results would be rescaled to unity amplitude and then multiplied
by the local (in space and time) amplitude factors described above. But no spatial data
needed to construct these amplification factors were available, so the DDS results could
not be properly scaled. Thus, it is only their temporal behavior that should be compared
with the experimental results, and we see that this comparison is quite satisfactory.
3.3 Parallelizatlon
Be have earlier alluded to the fact that the form of LES we have presented has potential
for significant parallelization. We have not yet completed the parallel implementation at
this time, but the work is in its late stages. Here we will briefly describe the opportunities
for parallel coding with such an algorithm. First, the large and small scales of the LES
procedure can be done in parallel because although each depends on the other, the specific
details are not extremely important; in particular, it is possible to employ time step n
large-scale information to compute time-level n § 1 small-scale results at exactly the same
time as the large- (resolved-) scale results are being calculated because actual turbulent
solutions are so nonsmooth that high-order accuracy in time cannot be achieved under
any conditions. Computing the two scales in parallel results in only first-order temporal
accuracy but, in fact, this is all that can actually be obtained in any case.
The second parallelization opportunity is the usual domain decomposition one, and
it can be equally-well implemented at both scales of the computations. Finally, within
each domain it is possible to parallelize (or, maybe just vectorize) the specific numerical
analytic algorithms. Thus, in all, there are three possible levels of parallelization to be
exploited with an algorithm of the type we have presented.
5. CONCLUSIONS
In this paper we have presented an alternative approach to large-eddy simulation based
on unfiltered equations of motion and discrete dynamical systems SGS models. We have
noted the advantages of this for modeling interactions of hydrodynamic turbulence with
other physical fluctuations, and we have demonstrated the effectiveness of the individual
pieces of the overall procedure by comparing them with standard LES in the case of](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-34-320.jpg)
![34
production gas turbines (both power and propulsion versions), combustion always occurs
far away from the lean limit by design. On the other hand, if stable combustion can be
achieved in the lean limit then not only emission can be reduced drastically, but it may
also result in significant reduction in fuel consumption. This goal is now of particular
relevance due to the increasing environmental concerns of fossil fuel combustion.
Numerical prediction of highly swirling turbulent reacting flows is very difficult since
conventional time-averaged methods cannot predict these flows accurately. Fortunately,
a technique based on large-eddy simulations (LES) has the ability to capture the physics
of these flows. In LES, all scales of motion larger than the grid resolution are resolved by
the numerical scheme (that is of high temporal and spatial accuracy) and the scales of
motion smaller than the grid size are represented using subgrid models. The accuracy of
the LES depends not only on the resolution of the resolved scales but also on the models
used to represent the unresolved small-scales. The ability of advanced subgrid models
to simulate these complex flows using relatively coarse grids is discussed using examples
of LES of full-scale gas turbine engine combustor flows. Application of the LES to both
single and two-phase (spray) mixing and combustion is reported.
2. SIMULATION MODEL
In LES, only scales larger than the cut-off size are computed and the rest are modeled.
Therefore, computations are still tractable on massively parallel systems. Unresolved
terms at the sub-grid level arise in the filtered momentum and scalar equations. A local-
ized dynamic model for the subgrid kinetic energy is used to define an eddy viscosity that
is then used for the subgrid stress closure[l]. For premixed combustion, a thin-flame LES
approach [1,2] and new subgrid based combustion approach [3,4] have been developed.
Details are avoided here for brevity.
The numerical solver solves the Favre-filtered, unsteady, compressible, Navier-Stokes
equations along with the Lagrangian equations for the liquid phase [5]. The scheme is
fourth-order accurate in space and second order in time. Full two-way coupling is employed
for the two-phase spray studies. Although currently only dilute spray can be simulated
with this code, spray breakup and collision models are currently being included.
3. PARALLEL IMPLEMENTATION
The governing LES equations are discretized using a finite-volume formulation are in-
tegrated over a discrete control-volumes. A distributed memory parallel model is chosen
whereby the computational grid is partitioned and distributed evenly among the avail-
able processors. To make the LES algorithm portable, the standardized Message-Passing
Interface (MPI) protocol is used for parallel communication.
The amount of data transferred is dependent on the size of the finite-volume stencil
used to solve the governing equations. For the overall central, fourth-order scheme used
here, the stencil size is +/- three cells about the center cell. Therefore, each processor
must exchange three layers of computational cells to their respective neighbors. Since
communication is generally much more expensive than computation, only the flux vari-
ables are exchanged and the primitive variables are calculated. As a result, the parallel
overhead is a combination of communication and computation.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-38-320.jpg)
![35
Table 1
Parallel computing hardware comparison. Owl, Jedi and Raptor are Intel-based PC
clusters, Seymour and Habu are specialized computing platforms.
Name Total # CPU's / CPU Speed Communication
(Platform) of CPU's SMP Node (MHz) Fabric
Owl (ia32) 16 4 500 Fast-Ethernet
Raptor (ia32) 32 2 733 Fast-Ethernet &
Giganet-cLAN
Jedi (ia32) 138 8 550 Gigabit-Ethernet
Habu (IBM SP3) 1336 4 375 SP Switch Fabric
Seymour (Cray T3E) 1024 1 450 3D folded torus
4. RESULTS AND DISCUSSION
4.1. Parallel Performance Analysis
Three clusters in 2-, 4-, and 8-way SMP configurations using the Intel Pentium III
Xeon CPU are evaluated along with supercomputers. The system configurations and
identification are listed in Table 1.
All PC clusters are running the Linux operating system (2.4.x kernels) and use the
Portland Group Fortran90 compiler. MPICH, an open-source MPI implementation de-
veloped is used for all TCP/IP parallel communication. MPI/Pro, a commercial MPI
implementation from MPI Software Technologies, Inc. is used for all Virtual Interface
(VI) based communications (in Raptor).
The LESLIE3D code, a turbulent, temporal mixing layer simulation model with two
non-reacting chemical species, is used for a "realistic" benchmark. This simulation models
the time-evolution of two distinct fluids moving parallel to each other at different velocities
[6]). This situation is often encountered in combustion systems and is often used as an
idealized model for fuel-oxidizer mixing studies. Physically, the configuration is a cubic
box (with length of 27~) with periodic boundary conditions in two of the three directions
(X, Z). For a large number of processors (neglecting those along the Y-boundaries), the
inter-process communication is nearly isotropic, i.e., all processors have 6 neighbors. Grid
densities of 643 and 1283 were used to give a wider range of applicability. For example,
1283 (over two million cells) represents represents a resonable grid for a full scale gas
turbine combustor simulation.
The timing results are shown in Fig. l(a). All timings are in total-time per time-step
(seconds). For reference, Seymour achieves 76 Mflops per node (1216 Mflops aggregate)
on 16 processors with the 1283 grid. We see that Habu surpasses the competition by a
substantial margin with Raptor (running VI) coming in second. As we progress toward
8 and 12 processes, the effects of contention begin to show themselves. At saturation (16
processes), Owl is actually slower than at 8 processes for both grids.
The parallel overhead is shown in Fig. l(b). For the Cray T3E, the overhead remains
below 20% for both grids which allows for efficient scaling. We can see for the TCP/IP
communication, the overhead reaches upward of 50% and 40% for the small and large](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-39-320.jpg)
![36
50.o
45.01
4O.0
35.0 I
~30.0
25.0
~ 20.0
15.0
I0.0
5.0
0.0 2
..... , ,~
i ' i ' i ' '
[] Owl
O Raptor(Fast-Ethernet)
O Raptor(Vl-Giganet)
g:~ Jedi
9 IBM SMP Power3
A
- , . ~ , . J . , i , . I , ,
8 14 20 26 32
Number of Processors
40
[] Owl I I " " I " " | " A I
r RaP~~ I ~ 1
0 Rapt~ (Vl'GiganeO I ~ "]
I~ J~i
2 8 14 20 26 32
Number of Processors
Figure 1. Performance profiles for LESLIE3D on 1283 grid. (a) Wall-clock time per step
and (b) communication overhead (% of total time).
grid, respectively (on Raptor). When communicating over the dedicated cLAN switch,
Raptor incurs an overhead on the same order as Seymour and even superior to Habu as
the number of processors is increased.
The parallel efficiencies (scalability) for these simulations are reported in Fig. 2. Owl,
as indicted earlier, exhibits zero or even negative scalability for both grids with half the
system is used. Jedi, even with a high-speed communication fabric, also scales poorly
though (at least) remains positive throughout the spectrum. In fact, Jedi scales worse
than Raptor when using TCP/IP over Fast-Ethernet. As could be expected, Seymour
is able to scale at over 90% efficiency (for the large grid). For the 1283 grid, Raptor's
efficiency (with VI) is comparable or even surpasses that of the two comparison systems.
For LES of realistic gas turbine combustion process, the computational time can be
quite large, e.g., approximately 15,000 CPU hours on Habu using one million grid points
and an efficient combustion model to obtain sufficient data for statistical analysis [2]. The
scalability of LESLIE3D on very large-scale computational grids were also evaluated on
an IBM eower3-SMP (222 MHz) and an SGI Origin 3800 (400 MHz). The results for a
2163 grid (10+ million grid points) are shown in Fig. 2b. Even at over 300 CPU's, the
algorithm is able to scale at 83% efficiency (based on the 64 CPU time). Based on the
results for the Raptor PC-cluster which scaled superior to the IBM system, it is possible
to estimate that the same calculation could be completed in a feasible amount of time on
the high-performance (yet commodity) PC-cluster system.
4.2. Premixed Combustion in Gas Turbine Engines
Premixed combustion in a full-scale gas turbine combustor (GE LM6000) have been
studied extensively [1,2]. Some characteristic features that were captured in LES (but not
in steady-state calculations) are reported here (more details are in the cited literature).
For premixed combustion, when swirl is below a critical value, the flow entering the
combustor behaves more like a free jet and forms circular vortex rings (similar to smoke](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-40-320.jpg)
![39
~illi~: ~,ii}:.:.
..... :::!li,i~:
(a) Low-Swirl (b) (High-Swirl
Figure 4. Instantaneous Azimuthal Vorticity and Droplet Distribution in the GE DACRS
combustor.
non-reacting case, increase in swirl increases the spray dispersion and vortex breakdown
process. Large droplets tend to accumulate in regions of low vorticity (i.e., they tend to
"surround" the vortices and are not present inside them). Thus, as the vortices break
down into smaller features due to increased swirl, the droplets also gets dispersed [5].
This phenomenon impacts combustion (here only infinite-rate kinetics is studied) as
well since as the droplet vaporize and become very small, they and the vaporized fuel
get entrained into the vortices where they under mixing. Thus, in high swirl flows, flame
surfaces are highly wrinkled and localized (the primary flame zone occurs in the core
of the swirling flow) whereas in low swirl case, the mixing process is delayed, and hence,
combustion is delayed and occurs in the outer regions of the large scale structures. Increase
in the swirl also creates the vortex breakdown bubble which is not present in the low swirl
case (Fig. 5). This bubble stabilizes the flame zone in a manner somewhat similar to
the premixed case but there are many differences since spray combustion here is mixing
controlled.
4.4. LES using Lattice Boltzmann Method
A new parallel Lattice Boltzmann Equation (LBE) method for LES applications has
also been developed [7]. The LBE approach allows full resolution of very small scales of
motion and therefore, is particularly suited for resolving the flow features inside micro-
scale MEMS-scale synthetic jets [7] that have been shown to enhance fuel-air mixing
in some recent experiments. In the present effort, LES-LBE approach is employed in
full 3D (using a 19-velocity model) to simulate the flow field generated using synthetic
micro-jets, square and elliptical jets, and square jets in crossflow. A localized dynamic
algebraic subgrid eddy viscosity model is used to obtain the effective viscosity used in](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-43-320.jpg)
![40
Low Swirl - X/D = 1.0
0 .... ,/, , ,
-0.2 ~ ~ ~ r J~'
41.6 1'~I'~
-0.8 le..., Non_ReactingI
' [,,--,,Reacting I
-! ~ I I I I I I I I
-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
U/Uo
(a) Low Swirl Case
-0.2
High Swirl - X/D = 0.75
b
P
!
/
,,'/
0 0.2
U/Uo
]e..-eNon-ReactingI
,,-- Reacting [
0.4 0.6
(b) High Swirl Case
Figure 5. Streamwise Mean Velocity Profiles in the combustor. Location is chosen to
highlight the presence of the vortex breakdown bubble in the high swirl case. Here, R0 is
the radius of the combustor, and r/Ro - 0 is located at the centerline.
Outflow
d Outflow
Wall
lOd
PerJ
~ v
odic
x
Per~ odic
Figure 6. Geometry and computational domain employed for jet in crossflow.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-44-320.jpg)
![41
. . . . . , . . . . . i ' '
s-B
/~
... ........ O.J"
1 2
(a) x/D = O, U/Wj
i ,iv . . . . I . . . . . 1 , ,
t...p 6 ~,~
o 0.5 1
(c) x/D = O, U/Wj
. . . . . i . . . . . , . . . . . r . . . . . . . . . .
/r
, , , , , I . . . . .
-i o ! 2
(b) x/D = 1, W/W~
i 0 . . . . I ,
)o
o 0.5
(d) x/D = 1, W/Wj
Figure 7. Mean velocity profiles along the jet center plane (y/D=0) at different x/D. The
other curves are RANS calculations reported elsewhere[9].
the LBE-LES equation. Some results for jet in crossflow is discussed here to show the
ability of the LBELES approach. The simulation is carried out at Reynolds number of
4700 based on the jet velocity and the nozzle width D and at jet-cross-flow velocity ratio
of 0.5. The computational domain is resolved using 200 x 150 x 100 for the cross-flow
domain and 50 x 50 x 100 for jet section. Approximately, 1500 single processor hours in
the SGI Origin 2000 is needed for this simulation. Figure 6 shows the schematic of the
test case experimentally and numerically studied in the past [8,9]. Figure 7 shows some
characteristics mean velocity profiles predicted by LBE-LES method and by other RANS
based calculations. It can be clearly seen that the LBE-LES prediction is substantially
better. More details of this study will be reported elsewhere.
5. CONCLUSIONS
From the parallel scaling studies on both supercomputersand PC clusters presented
here, it is concluded that while both communicationand memorybandwidth contention
are prevalent in SMP clusters, memory contention plays a more significant role with](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-45-320.jpg)
![42
respect to LES applications. Compared to the HPC systems, Raptor,when operating with
the VI-cLAN configuration, has comparable performance and superior scalability (versus
the IBM Power3-SMP, Habu). Thus, the steady increase in processor performance and
advances in memory/communication bandwidth, LES on PC-clusters may soon become
integrated into the design process at a cost that will be affordable for even small-scale
companies.
The effects of swirl intensity and heat release are investigated. Results show that
increase in swirl has dramatic effect on the flame structure and reduces the pressure oscil-
lation amplitude in premixed systems. In spray combustion, increase in swirl drastically
increases the droplet dispersion. In all situations, vortex breakdown occurs only under
high swirl conditions. Also, combustion and heat release tends to reduce the size of this
recirculation zone, due to an increase in flow acceleration associated with heat release.
Finally, a new LES method based on the Lattice Boltzmann model is used to simulate
complex flows such as jet in cross flow. The eventual application of the LBE-LES will be
in flows where fuel-air mixing is actively controlled, for example, with synthetic micro-jets
[7]. In these flows, conventional LES of the micro-scale domain of the micro-jets would
imply an enormous grid resolution and the associated computational cost to do reacting
flow simulations would be prohibitive. By using a LBE model for the microjet region and
then coupling it to the more conventional finite-volume LES solver for the reacting flow
in the combustor, it may be possible to handle this very complex problem. An effort is
currently underway in this very effort and results will be reported soon.
Acknowledgements
This work is supported in part by US Army Research Office and General Electric Power
Systems. Computational time is provided by US Department of Defense High Performance
Computing Centers at NAVO, MS, ERDC, MS, and ARL, MD.
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Mech. 184 (1987), 207.
7. H. Wang and S. Menon, AIAA J. 39 (2001) 2308.
8. P. Ajresch, J. Zho, S. Ketler, M. Salcudean, and I. Gartshore, J. Turbomachinery, 119
(1997), 330.
9. A. Hoda,S.Acharyaand M.Tyagi,ProceedingsofASMETURBOEXPO2000(2000),
I.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-46-320.jpg)
![ParallelComputationalFluidDynamics-NewFrontiersandMulti-DisciplinaryApplications
K.Matsuno,A. Ecer,J. Periaux,N. SatofukaandP. Fox(Editors)
9 2003ElsevierScienceB.V.Allrightsreserved. 43
A Substepping Navier-Stokes Splitting Scheme for Spectral/hp Element
Discretisations
Spencer Sherwina *
Department of Aeronautics, Imperial College, London, SW7 2BY, UK
In this paper we investigate the application of a substepping advection algorithm in
conjunction with a Navier-Stokes high order splitting scheme. We initially compare the
substepping approach to the semi-Lagrangian treatment of the advection terms in the con-
text of a one-dimensional advection-diffusion equation. The choice of a modal spectral/hp
element discretisation then motivates the use of a discontinuous Galerkin formulation to
solve the hyperbolic advection component of the Navier-Stokes time stepping scheme.
This is necessary since a classical Galerkin discretisation would require a global matrix
inversion thereby destroying the potential gain of the substepping technique. Finally the
algorithm is applied to a two-dimensional model Kovasznay problem as well as the more
applied problem of flow through a stenosis.
1. Introduction
The application of spectral/hp element discretisations to a variety of industrially mo-
tivated fluid problems such as bio-fluids, bridge aerodynamics and marine technology
requires the use of efficient time-stepping techniques. An important aspect of achieving
this efficiency is overcoming the Courant Fredrichs Levy (CFL) time step restrictions aris-
ing from the explicit treatment of the convective terms in the Navier-Stokes equations.
In explicit algorithms it is possible to have a stability restriction, arising from the CFL
condition, which is overly restrictive from the point of view of accuracy. For example
problems with a large range of physical scales where only the largest scale is required or
when automated mesh design generates "slither" elements normal to the flow direction.
Two approaches have typically been adopted within the spectral/hp element community
to alleviate this problem. The first is the operator integration factor or substepping
technique (see Maday et al. [6]) and the second is the semi-Lagrangian formulation
more recently adopted in the spectral/hp element technique by Giraldo [2] and Xiu
Karniadakis [i0]. In a general context these techniques are related by considering a
characteristic treatment of the advection terms and solving the Navier-Stokes equations
in the Lagrangian frame of reference. Nevertheless a notable distinction arises in the
discretisation of the advection term. In the substepping technique the advection term is
advanced in an Eulerian frame of reference using a smaller time step than the rest of the
Navier-Stokes solver. Alternatively the semi-Lagrangian technique performs a backward
*http://www.ae.ic.ac.uk/staff/sherwin](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-47-320.jpg)
![44
x~_1 x~ x~§
Figure 1. Schematic of the standard finite difference approximation to the advection-
diffusion equations using an implicit diffusion and explicit advection discretisation.
particle tracking over the complete time step.
In the present work we investigate the substepping strategy. Previous implementations
of this technique have been adopted within the context of a full Stokes solver [6] or a
fractional step method [9], however in this paper we discuss substepping in conjunction
with a high order splitting scheme [4,7]. This scheme can be interpreted as a pure advec-
tion step followed by a Poisson pressure correction to enforce divergence and a Helmholtz
equation introducing the role of diffusion. This interpretation therefore provides a natural
setting for the substepping method. However unlike the traditional nodal spectral ele-
ment method, where a diagonal mass matrix arises from discrete orthogonality, the modal
spectral/hp approach produces a full mass matrix system. A classical Galerkin discretisa-
tion therefore results in a full matrix solution which would not provide any advantage to
using the substepping technique over the traditional splitting scheme. However applica-
tion of a discontinuous Galerkin formulation provides an appropriate setting to discretise
the hyperbolic part of the system whilst maintaining numerical efficiency.
The paper is organised in the following manner. In section 2 we will outline the for-
mulation of the substepping Navier Stokes splitting scheme. Section 2.1 discusses the
substepping approach and compares it with the semi-Lagrangian technique using one-
dimensional concepts to illustrate the different approaches. Then in section 2.2 we detail
the use of a discontinuous Galerkin formulation to efficiently solve the hyperbolic scalar
advection equation when using spectral/hp element discretisation. In section 2.3 we briefly
outline the spectral/hp element discretisation and finally in section 2.4 we apply all the
previous components to arrive at a high order substepping splitting scheme for the Navier-
Stokes equations. In section 3 we detail the application of the substepping Navier-Stokes
algorithm to a model Kovasznay problem and the more applied problem of stenotic flow.
2. Formulation
2.1. Advection Substepping
To detail the substepping and semi-Lagrangian approach we will consider, as a model
problem, the scalar advection-diffusion equation
r + a. Vr = vVur (1)
where a(x,t) is a divergence free advection field. Under any explicit discretisation the
time step restriction associated with the diffusion operator, vV2r is greater than the](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-48-320.jpg)
![45
t'+l
i/,1'
['
~_~ ,~X
d x~
1"
i"
i
lX~+I
Figure 2. Schematic of the semi-Lagrangian approximation to the advection-diffusion
equations using an implicit diffusion discretisation. The advection operator is approxi-
mated by backward particle tracking along the characteristic from x~ at t~+1 to determine
Xd at t n.
advection operator, a. Vr For this reason it is often desirable to treat the diffusion
operator implicitly and since, in general, we would like to consider non-linear convection
the advection terms are often handled explicitly. Such a strategy for a one-dimensional
finite difference scheme can be symbolically represented on a discrete x-t diagram as
shown in figure 1. In this figure the black squares represent the implicit contribution from
a centred approximation to the diffusion operator (i.e. Vr +1 -- ~z21~('~n+lw~+l
- 2r n+l +
(~n+l
~_~ ) whilst the black circles represent the contributions from an upwinded first order
approximation to the advection operator assuming a > 0 (i.e. a-Vr _ ~_~
(r n - r
Following classical analysis the size of the time step At = tn+~ - tn is restricted by the
CFL conditions At < Ax/a where Ax - xi+l - xi.
Equation (1) can equivalently be written in Lagrangian form as
De = vV 2r where D 0
Dt Dt = 0-~ + a. V (2)
where D/Dt is the Lagrangian or material derivative moving with the advection velocity
i.e.Dx/Dt = a. Following a semi-Lagrangian approach [2,10] and temporally discretising
equation (2) at point x~ using a first order implicit time discretisation we obtain
cn+l _ r
At
. v-72 ~.n-F-1
z t/v ~i "
where r - Cn(Xd, tn) and Xd is the so-called departure point.
Since the material derivative is evaluated along the characteristics we can evaluate
the departure point X d by solving the characteristic equation Dx/Dt - a backward in
time from t n+l < t <_ t ~ using the initial conditions x(t n+l) - x~. This approach is
schematically represented in figure 2 where we show the one-dimensional discretisation
on a discrete x-t diagram. Once again the implicit discretisation of the diffusion term is
represented by the black squares. However in the semi-Lagrangian approach we backward
particle track along the characteristic, Dx/Dt - a, of the hyperbolic advection operator](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-49-320.jpg)
![46
a)
" ............ ........... ,"
I: ..::. ....::2-;:.:::: ::::i:::::/::'
. :: ,:'.'."..... -::::::~ ~::::':---./::::,q
x~_~ /:
// ~+s
b)
~n+l
'l'l'l!
X~+,
Figure 3. Schematic representation of the substepping approach: (a)A hyperbolic advec-
tion equation is advanced using a smaller time step to determine r (b) Using
~(Xi,tn+l) -- ~)(~d, tn) along the characteristic the diffusion step can then be solved.
to determine point Xd, subsequently r is evaluated and the approximation to the
Lagrangian time derivative can then be determined. The complexity of backward particle
tracking depends on the form of a(x, t). If a(x, t) is time independent the calculation
is relatively straight forward on a well behaved discretisation. However if the convection
velocity is time dependent then different time integration strategies can be adopted. For
further details of this methods in the context of spectral/hp element methods see [2,10].
Finally we also note that this scheme requires an interpolation operation to evaluate
Although we will not be adopting the semi-Lagrangian approach in the current work it
does provide a convenient setting to discuss the substepping/integration operator splitting
[6] approach. The implementation of this strategy is similar to the semi-Lagrangian tech-
nique however instead of backward particle tracking to obtain ~rd and then interpolating to
obtain r we solve for the departure solution r directly. This can be achieved by solving
the advection part of the problem independently in its Eulerian form and with a smaller
time step. Therefore we introduce a pseudo solution r ~-) and solve the problems:
-- + a. Vr = O t n < T < t n+l (3)
DT O~ - -
with initial conditions r tn) = r Since this is a strictly hyperbolic equation
then along the characteristic the solution is constant and so r tn) = r t~+l). The
explicit nature of equation (3) means that from stability considerations the time step is
restricted by a CFL condition. Nevertheless providing the solution of (3) can be solved
more efficiently than the implicit diffusion operator then equation (3) can be discretised
with a time step near to the CFL limit whilst the implicit diffusion operator is solved
less frequently in time thereby reducing the cost of the overall algorithm for a fixed
integration time. Indeed it is the ratio of the cost of the explicit advection term to the
implicit diffusion operator which ultimately limits the possible speedup. We also note that
the time accuracy is governed by the larger time step applied on the implicit diffusion
operator.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-50-320.jpg)
![47
The substepping solution is schematically shown in figure 3(b) where we again denote
the discretisation of the implicit diffusion step by the black squares. To determine the
solution at the departure point we have to discretise equation (3) using a smaller, CFL
satisfying, time step as shown in figure 3(a). For a mathematical description of the
scheme see Maday et al. [6] which describes the solution of equation (3) as determining the
integration operator factor which makes the advection operator into a total or Lagrangian
derivative.
In the substepping approach we do not need to interpolate the solution field to de-
termine r and can also use standard Eulerian formulation to solve for this variable.
However substepping is still limited by an explicit time step restriction which potentially
could be quite severe and may make the solution of equation (3) overly costly. Therefore,
an efficient solution technique for the advection equation is required, such as the discon-
tinuous Galerkin method as discussed in section 2.2. We note that the semi-Lagrangian
approach is potentially quite expensive when considering the cost of evaluating the back-
ward particle tracking and then interpolation of the solution. However this cost is rel-
atively independent of At which may make this approach more attractive at very large
At. As a final point we observe that parallelisation of the substepping approach follows a
relatively traditional implementation only requiring communication of nearest neighbour
information. Parallelisation of the semi-Lagrangian approach is more involved due to the
particle tracking nature of the algorithm.
2.2. Discontinuous Galerkin Formulation for Scalar Advection
Consider the multi-dimensional scalar advection equation
0u 0u
O---t + a. Vu - --~ + V . (au) - 0 in ~ (4)
where we have assumed V. a = 0. We formulate our problem on Net non-overlapping
tIN~ F4 ~. Following the standard finite element formulation on
elements ~ such than ue=l =
a single element we take the inner product of (4) with respect to a test function r which
only has support in ~ to obtain
r Ou
+ (r V. f(u))n~ =0
where f(u) = au and (,) is the L2 inner product. Applying the divergence theorem to
the second term and summing over all element we arrive at
r - (re. f(~))~o + Cf(~). nd~ : 0 (5)
e=l Fte ~te
where cg~e represents the boundary of F4 and n is the outward normal to ~.
To obtain the discrete form of our problem we choose u to be in the finite space of
[L2(~)] D~'~ functions which are polynomial of degree P on each element. We indicate an
element of such space using the superscript ~. We also note that u ~ may be discontinuous
across inter-element boundaries. However to attain a global solution in the domain
we need to allow information to propagate between the elemental regions. Information
is propagated between elements by upwinding the boundary flux in the third term of](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-51-320.jpg)
![48
equation (5). Denoting the upwinded flux as f~, for the linear advection equation the
upwind normal flux can simply be determined by
fU(u) . n- { uinta . n
Uexta n
ifa.n>0
m
ifa.n < 0
where u~t and ue~t are respectively the values of u interior and exterior to the element
edge. The discrete weak formulation can now be written as
Following the traditional Galerkin approach, we choose the test function within each
element to be in the same discrete space as the numerical solution u ~. At this point if we
defined our polynomial basis and choose an appropriate quadrature rule we would have a
semi-discrete scheme. However, from an implementation point of view, the calculation of
the second term can be inconvenient and consequently we choose to integrate this term
by parts once more to obtain
r + (r f(u))n~ + r []~(u) - f(u)], rids = 0 (7)
e=l Fte gte
We note that the information between elements is transmitted by the third boundary
term as the difference between the upwinded and the local fluxes. This method can be
considered as a penalty method with an automatic procedure for determining the penalty
parameter. The discretisation is completed by defining a time stepping method such
as the Adams-Bashforth or Runge Kutta schemes and a spatial discretisation which is
addressed in section 2.3.
2.3. Spectral/hp Element Discretisation
In the current work we have adopted a spectral/hp element discretisation. This ap-
proach combines the exponential accuracy of Fourier methods with the geometric flexi-
bility of the finite element/volume methods. Within each local element of triangular or
quadrilateral shape in two-dimensions a polynomial expansion of arbitrary order is ap-
plied. In classical Galerkin methods this expansion is required to be C o continuous over
the whole solution domain and can be either modal or nodal in form. Traditionally nodal
elements have been referred to as spectral element methods. Within the current work
we have applied a continuous modal expansion for the elliptic operators and an orthogo-
nal modal expansion for the hyperbolic operators in conjunction with the discontinuous
Galerkin formulation.
2.4. Substepping Navier-Stokes Splitting Scheme
The Lagrangian Navier-Stokes equations can be written as:
Du
= -Vp+vV2u (8)
Dt
v. =0 (9)](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-52-320.jpg)
![49
where u = [u, v, w]T, D_UU_ O
_
_
U
_
U
+ u. Vu and p = Ps/P where ps is the static pressure. The
Dt ~ Ot
divergence conditions (9) can be equivalently expressed by a pressure Poisson equation
by taking the divergence of the momentum equation (8) and noting that V. (V2u) = 0
if V 9u = 0 to arrive at:
Pu -V. (u. Vu). (10)
V2p=-V. Dt =
Discretising equation (8) at time level n + 1 and applying an implicit scheme for the
linear diffusion term we obtain:
O~OUn+!+ EqJ=-iOLqUrd+l-q = --Vp n+l + .V2U n+l (11)
At
where C~q represents the coefficients for backward integration in time, i.e. J = 1 =a •q --
[1,-1], J- 2 =~ aq = [3/2,-2, 1/2]. In equation (11), the velocity u~ +l-q represents the
velocity at the foot of the characteristic at time level n + 1- q. In the substepping scheme
u~+l-q = ~n+l where ~+1 is determined from the integration between tn+l-q < ~- < tn+l
d
of the Eulerian advection equation
0~
-- + u. V~ = 0 (12)
0f
~(x, t"+1-~) - ~(x, t~+~-~) (13)
where u in the above equation is the divergence free velocity which is extrapolated in
time between tn < ~- < tn+l. We note that equation (12) can be put in conservative form
by using the fact that V. u = 0 which implies that u. Vfi = V 9(fiiu).
To solve equation (11) we apply a high order splitting scheme as discussed for the
Eulerian discretisation in Orszag et al. [7], Karniadakis et al. [4] and more recently by
Guermond & Shen [3] as well as for the semi-Lagrangian scheme by Xiu and Karniadakis
[10]. Following this scheme we split equation (11) into two steps
J OLqU~+l--q
+ Eq=l = -Vp ~+1 (14)
At
OL0un+l -- ?~
= ,V2u n+l (15)
At
To decouples steps (14) and (15) we take the divergence of equation (14) and require that
V. ~ = 0 to obtain
V2pn+i -- - EqJ1 ~ (16)
At
Equation (16) is a semi-discrete approximation to pressure Poisson equation (10) through
which divergence is enforced. We note from the discrete divergence analysis in [7] that
implicit inclusion of u ~ in the right hand side of equation (16) is important since the gov-
erning divergence equation is of an elliptic rather than parabolic form. This is significant
since it since it implies the divergence errors in the initial and boundary conditions are
restricted to a boundary layer.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-53-320.jpg)
![50
o
-1
~_~
X 2 4 6 8 10 -3<~ -3 -1,5
(a) (b) ,o,~.om~.,o.=or (c) ,=(A~)
: "A::;: .]
zU ]
Figure 4. Kovasznay flow: (a) Solution and computational mesh, (b) Convergence as a
function of polynomial order, (b) Convergence as a function of time step.
n+l-q
Finally equations (15),(16) together with equation (12) for the solution of u d rep-
resent the substepping Navier-Stokes splitting scheme. However to close the system we
require appropriate boundary conditions for equation (16) which are obtained by taking
the inner product of equation (11) with the outward normal to the boundary to obtain
Opn+l [
On = n" /2V2U n+l --
a~ + EqJ=lAt~
This form of the boundary condition is not typically found to be stable [4,8] and to
correct this the Laplacian term is decomposed into a divergence and solenoidal component
(V2u = V(V. u) - V x V x u) where the divergence term can be set identically to zero.
A complete discussion of boundary conditions for splitting formulations can be found
in Petersson [8]. To make the scheme explicit we still have to evaluate the solenoidal
component V x Vu at t~+1 which is evaluated by extrapolation from previous time levels.
The discrete pressure boundary conditions can therefore be written as
0--~ = -n. v ~ 7qV • Vu ~-q + ~/
q=0
where the discrete Lagrangian time derivative has been replaced by ~2 by imposing V.
u ~+1 = 0 and 7q represents the extrapolation coefficients (i.e. Je = 1 --+ 7a = [1],
Je = 2 ~ 7q = [2,-1]).
As a final comment we note that equation (12) needs to be evaluated efficiently for a
significant benefit of the substepping to be observed. This can be achieved by using the
discontinuous Galerkin formulation as discussed in section 2.2.
3. Results
As a first example we consider an analytic problem proposed by Kovasznay [5] which
can be considered as a steady laminar flow behind a two-dimensional grid. This exazt](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-54-320.jpg)
![But however this may be, we find Bolingbroke in correspondence with the Stuart
agents in the later part of 1712. Macpherson, 366. And his own correspondence
with Lord Strafford shows his dread and dislike of Hanover (Bol. Corr. ii. 487 et
alibi). The Duke of Buckingham wrote to St. Germains in July that year, with
strong expressions of his attachment to the cause, and pressing the necessity of
the prince's conversion to the protestant religion. Macpherson, 327. Ormond is
mentioned in the Duke of Berwick's letters as in correspondence with him; and
Lockhart says there was no reason to make the least question of his affection to
the king, whose friends were consequently well pleased at his appointment to
succeed Marlborough in the command of the army, and thought it portended
some good designs in favour of him. Id. 376.
Of Ormond's sincerity in this cause there can indeed be little doubt; but there is
almost as much reason to suspect that of Bolingbroke as of Oxford; except that,
having more rashness and less principle, he was better fitted for so dangerous a
counter-revolution. But in reality he had a perfect contempt for the Stuart and tory
notions of government, and would doubtless have served the house of Hanover
with more pleasure, if his prospects in that quarter had been more favourable. It
appears that in the session of 1714, when he had become lord of the ascendant,
he disappointed the zealous royalists by his delays as much as his more cautious
rival had done before. Lockhart, 470. This writer repeatedly asserts that a majority
of the House of Commons, both in the parliament of 1710 and that of 1713,
wanted only the least encouragement from the court to have brought about the
repeal of the act of settlement. But I think this very doubtful; and I am quite
convinced that the nation would not have acquiesced in it. Lockhart is sanguine,
and ignorant of England.
It must be admitted that part of the cabinet were steady to the protestant
succession. Lord Dartmouth, Lord Powlett, Lord Trevor, and the Bishop of London
were certainly so; nor can there be any reasonable doubt, as I conceive, of the
Duke of Shrewsbury. On the other side, besides Ormond, Harcourt, and
Bolingbroke, were the Duke of Buckingham, Sir William Wyndham, and probably
Mr. Bromley.
[327] It is said that the Duke of Leeds, who was now in the Stuart interest, had
sounded her in 1711, but with no success in discovering her intention.
Macpherson, 212. The Duke of Buckingham pretended, in the above-mentioned](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-57-320.jpg)
![letter to St. Germains, June 1712, that he had often pressed the queen on the
subject of her brother's restoration, but could get no other answer than, "you see
he does not make the least step to oblige me;" or, "he may thank himself for it: he
knows I always loved him better than the other." Id. 328. This alludes to the
Pretender's pertinacity, as the writer thought it, in adhering to his religion; and it
may be very questionable, whether he had ever such conversation with the queen
at all. But, if he had, it does not lead to the supposition, that under all
circumstances she meditated his restoration. If the book under the name of
Mesnager is genuine, which I much doubt, Mrs. Masham had never been able to
elicit anything decisive of her majesty's inclinations; nor do any of the Stuart
correspondents in Macpherson pretend to know her intentions with certainty. The
following passage in Lockhart seems rather more to the purpose: On his coming
to parliament in 1710, with a "high monarchical address," which he had procured
from the county of Edinburgh, "the queen told me, though I had almost always
opposed her measures, she did not doubt of my affection to her person, and
hoped I would not concur in the design against Mrs. Masham, or for bringing over
the Prince of Hanover. At first I was somewhat surprised, but recovering myself, I
assured her I should never be accessary to the imposing any hardship or affront
upon her; and as for the Prince of Hanover, her majesty might judge from the
address I had read, that I should not be acceptable to my constituents if I gave
my consent for bringing over any of that family, either now or at any time
hereafter. At that she smiled, and I withdrew; and then she said to the duke
(Hamilton), she believed I was an honest man and a fair dealer, and the duke
replied, he could assure her I liked her majesty and all her father's bairns."—P.
317. It appears in subsequent parts of this book, that Lockhart and his friends
were confident of the queen's inclinations in the last year of her life, though not of
her resolution.
The truth seems to be, that Anne was very dissembling, as Swift repeatedly says
in his private letters, and as feeble and timid persons in high station generally are;
that she hated the house of Hanover, and in some measure feared them; but that
she had no regard for the Pretender (for it is really absurd to talk like Somerville
of natural affection under all the circumstances), and feared him a great deal
more than the other; that she had, however, some scruples about his right, which
were counterbalanced by her attachment to the church of England; consequently,
that she was wavering among opposite impulses, but with a predominating
timidity which would have probably kept her from any change.
[328] The Duchess of Gordon, in June 1711, sent a silver medal to the faculty of
advocates at Edinburgh, with a head on one side, and the inscription, "Cujus est";
on the other, the British isles, with the word "Reddite." The dean of faculty,
Dundas of Arniston, presented this medal; and there seems reason to believe that
a majority of the advocates voted for its reception. Somerville, p. 452.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-58-320.jpg)
![Bolingbroke, in writing on the subject to a friend, it must be owned, speaks of the
proceeding with due disapprobation. Bolingbroke Correspondence, i. 343. No
measures, however, were taken to mark the court's displeasure.
"Nothing is more certain," says Bolingbroke in his letter to Sir William Wyndham,
perhaps the finest of his writings, "than this truth, that there was at that time no
formed design in the party, whatever views some particular men might have,
against his majesty's accession to the throne."—P. 22. This is in effect to confess a
great deal; and in other parts of the same letter, he makes admissions of the
same kind: though he says that he and other tories had determined, before the
queen's death, to have no connection with the Pretender, on account of his
religious bigotry. P. 111.
[329] Lockhart gives us a speech of Sir William Whitelock in 1714, bitterly
inveighing against the elector of Hanover, who, he hoped, would never come to
the crown. Some of the whigs cried out on this that he should be brought to the
bar; when Whitelock said he would not recede an inch; he hoped the queen would
outlive that prince, and in comparison to her he did not value all the princes of
Germany one farthing. P. 469. Swift, in "Some Free Thoughts upon the present
State of Affairs," 1714, speaks with much contempt of the house of Hanover and
its sovereign; and suggests, in derision, that the infant son of the electoral prince
might be invited to take up his residence in England. He pretends in this tract, as
in all his writings, to deny entirely that there was the least tendency towards
jacobitism, either in any one of the ministry, or even any eminent individual out of
it; but with so impudent a disregard of truth that I am not perfectly convinced of
his own innocence as to that intrigue. Thus, in his "Inquiry into the Behaviour of
the Queen's last Ministry," he says, "I remember, during the late treaty of peace,
discoursing at several times with some very eminent persons of the opposite side
with whom I had long acquaintance. I asked them seriously, whether they or any
of their friends did in earnest believe, or suspect the queen or the ministry to have
any favourable regards towards the Pretender? They all confessed for themselves
that they believed nothing of the matter," etc. He then tells us that he had the
curiosity to ask almost every person in great employment, whether they knew or
had heard of any one particular man, except professed nonjurors, that discovered
the least inclination towards the Pretender; and the whole number they could
muster up did not amount to above five or six; among whom one was a certain
old lord lately dead, and one a private gentleman, of little consequence and of a
broken fortune, etc. (vol. 15, p. 94, edit. 12mo, 1765). This acute observer of
mankind well knew that lying is frequently successful in the ratio of its effrontery
and extravagance. There are, however, some passages in this tract, as in others
written by Swift, in relation to that time, which serve to illustrate the obscure
machinations of those famous last years of the queen.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-59-320.jpg)
![[330] On a motion in the House of Lords that the protestant succession was in
danger, April 5, 1714, the ministry had only a majority of 76 to 69, several bishops
and other tories voting against them. Parl. Hist. vi. 1334. Even in the Commons
the division was but 256 to 208. Id. 1347.
[331] Somerville has a separate dissertation on the danger of the protestant
succession, intended to prove that it was in no danger at all, except through the
violence of the whigs in exasperating the queen. It is true that Lockhart's
Commentaries were not published at this time; but he had Macpherson before
him, and the Memoirs of Berwick, and even gave credit to the authenticity of
Mesnager, which I do not. But this sensible, and on the whole impartial writer, had
contracted an excessive prejudice against the whigs of that period as a party,
though he seems to adopt their principles. His dissertation is a laboured attempt
to explain away the most evident facts, and to deny what no one of either party at
that time would probably have in private denied.
[332] The queen was very ill about the close of 1713; in fact it became evident, as
it had long been apprehended, that she could not live much longer. The
Hanoverians, both whigs and tories, urged that the electoral prince should be sent
for; it was thought that whichever of the competitors should have the start upon
her death would succeed in securing the crown. Macpherson, 385, 546, 557 et
alibi. Can there be a more complete justification of this measure, which Somerville
and the tory writers treat as disrespectful to the queen? The Hanoverian envoy,
Schutz, demanded the writ for the electoral prince without his master's orders; but
it was done with the advice of all the whig leaders (Id. 592), and with the sanction
of the Electress Sophia, who died immediately after. "All who are for Hanover
believe the coming of the electoral prince to be advantageous; all those against it
are frightened at it." Id. 596. It was doubtless a critical moment; and the court of
Hanover might be excused for pausing in the choice of dangers, as the step must
make the queen decidedly their enemy. She was greatly offended, and forbade the
Hanoverian minister to appear at court. Indeed she wrote to the elector, on May
19, expressing her disapprobation of the prince's coming over to England, and
"her determination to oppose a project so contrary to her royal authority, however
fatal the consequences may be." Id. 621. Oxford and Bolingbroke intimate the
same. Id. 593; and see Bolingbroke Correspondence, iv. 512, a very strong
passage. The measure was given up, whether from unwillingness on the part of
George to make the queen irreconcilable, or, as is at least equally probable, out of
jealousy of his son. The former certainly disappointed his adherents by more
apparent apathy than their ardour required; which will not be surprising, when we
reflect that, even upon the throne, he seemed to care very little about it.
Macpherson, sub ann. 1714, passim.
[333] He was strongly pressed by his English adherents to declare himself a
protestant. He wrote a very good answer. Macpherson, 436. Madame de](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-60-320.jpg)
![Maintenon says, some catholics urged him to the same course, "par une politique
poussée un peu trop loin." Lettres à la Princesse des Ursins, ii. 428.
[334] The rage of the tory party against the queen and Lord Oxford for retaining
whigs in office is notorious from Swift's private letters, and many other authorities.
And Bolingbroke, in his letter to Sir W. Wyndham, very fairly owns their intention
"to fill the employments of the kingdom, down to the meanest, with tories."—"We
imagined," he proceeds, "that such measures, joined to the advantages of our
numbers and our property, would secure us against all attempts during her reign;
and that we should soon become too considerable not to make our terms in all
events which might happen afterwards; concerning which, to speak truly, I believe
few or none of us had any very settled resolution." P. 11. It is rather amusing to
observe that those who called themselves the tory or church party, seem to have
fancied they had a natural right to power and profit, so that an injury was done
them when these rewards went another way; and I am not sure that something of
the same prejudice has not been perceptible in times a good deal later.
[335] Though no republican party, as I have elsewhere observed, could with any
propriety be said to exist, it is easy to perceive that a certain degree of
provocation from the Crown might have brought one together in no slight force.
These two propositions are perfectly compatible.
[336] This is well put by Bishop Willis in his speech on the bill against Atterbury.
Parl. Hist. viii. 305. In a pamphlet, entitled "English Advice to the Freeholders"
(Somers Tracts, xiii. 521), ascribed to Atterbury himself, a most virulent attack is
made on the government, merely because what he calls the church party had
been thrown out of office. "Among all who call themselves whigs," he says, "and
are of any consideration as such, name me the man I cannot prove to be an
inveterate enemy to the church of England; and I will be a convert that instant to
their cause." It must be owned perhaps that the whig ministry might better have
avoided some reflections on the late times in the addresses of both houses; and
still more, some not very constitutional recommendations to the electors, in the
proclamation calling the new parliament in 1714 Parl. Hist. vi. 44, 50. "Never was
prince more universally well received by subjects than his present majesty on his
arrival; and never was less done by a prince to create a change in people's
affections. But so it is, a very observable change hath happened. Evil infusions
were spread on the one hand; and, it may be, there was too great a stoicism or
contempt of popularity on the other." "Argument to prove the Affections of the
People of England to be the best Security for the Government," p. 11 (1716). This
is the pamphlet written to recommend lenity towards the rebels, which Addison
has answered in the Freeholder. It is invidious, and perhaps secretly jacobite.
Bolingbroke observes, in the letter already quoted, that the Pretender's journey
from Bar, in 1714, was a mere farce, no party being ready to receive him; but "the
menaces of the whigs, backed by some very rash declarations [those of the king],](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-61-320.jpg)
![and little circumstances of humour, which frequently offend more than real
injuries, and by the entire change of all persons in employment, blew up the
coals."—P. 34. Then, he owns, the tories looked to Bar. "The violence of the whigs
forced them into the arms of the Pretender." It is to be remarked on all this, that,
by Bolingbroke's own account, the tories, if they had no "formed design" or
"settled resolution" that way, were not very determined in their repugnance before
the queen's death; and that the chief violence of which they complained was, that
George chose to employ his friends rather than his enemies.
[337] The trials after this rebellion were not conducted with quite that appearance
of impartiality which we now exact from judges. Chief Baron Montagu
reprimanded a jury for acquitting some persons indicted for treason; and Tindal,
an historian very strongly on the court side, admits that the dying speeches of
some of the sufferers made an impression on the people, so as to increase rather
than lessen the number of jacobites. Continuation of Rapin, p. 501 (folio edit.).
There seems, however, upon the whole, to have been greater and less necessary
severity after the rebellion in 1745; and upon this latter occasion it is impossible
not to reprobate the execution of Mr. Ratcliffe (brother of that Earl of
Derwentwater who had lost his head in 1716), after an absence of thirty years
from this country, to the sovereign of which he had never professed allegiance nor
could owe any, except by the fiction of our law.
[338] Parl. Hist. 73. It was carried against Oxford by 247 to 127, Sir Joseph Jekyll
strongly opposing it, though he had said before (Id. 67) that they had more than
sufficient evidence against Bolingbroke on the statute of Edward III. A motion was
made in the Lords, to consult the judges whether the articles amounted to
treason, but lost by 84 to 52. Id. 154. Lord Cowper on this occasion challenged all
the lawyers in England to disprove that proposition. The proposal of reference to
the judges was perhaps premature; but the house must surely have done this
before their final sentence, or shown themselves more passionate than in the case
of Lord Strafford.
[339] Parl. Hist. vii. 486. The division was 88 to 56. There was a schism in the
whig party at this time; yet I should suppose the ministers might have prevented
this defeat, if they had been anxious to do so. It seems, however, by a letter in
Coxe's Memoirs of Walpole, vol. ii. p. 123, that the government were for dropping
the charge of treason against Oxford, "it being very certain that there is not
sufficient evidence to convict him of that crime," but for pressing those of
misdemeanour.
[340] Parl. Hist. vii. 105.
[341] Parl. Hist. vi. 972. Burnet, 560, makes some observations on the vote
passed on this occasion, censuring the late ministers for advising an offensive war
in Spain. "A resolution in council is only the sovereign's act, who upon hearing his](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-62-320.jpg)
![counsellors deliver their opinions, forms his own resolution; a counsellor may
indeed be liable to censure for what he may say at that board; but the resolution
taken there has been hitherto treated with a silent respect; but by that precedent
it will be hereafter subject to a parliamentary inquiry." Speaker Onslow justly
remarks that these general and indefinite sentiments are liable to much exception,
and that the bishop did not try them by his whig principles. The first instance
where I find the responsibility of some one for every act of the Crown strongly laid
down is in a speech of the Duke of Argyle, in 1739. Parl. Hist. ix. 1138. "It is true,"
he says, "the nature of our constitution requires that public acts should be issued
out in his majesty's name; but for all that, my lords, he is not the author of them."
[342] "Lord Bolingbroke used to say that the restraining orders to the Duke of
Ormond were proposed in the cabinet council, in the queen's presence, by the
Earl of Oxford, who had not communicated his intention to the rest of the
ministers; and that Lord Bolingbroke was on the point of giving his opinion against
it, when the queen, without suffering the matter to be debated, directed these
orders to be sent, and broke up the council. This story was told by the late Lord
Bolingbroke to my father." Note by Lord Hardwicke on Burnet (Oxf. edit. vi. 119).
The noble annotator has given us the same anecdote in the Hardwicke State
Papers, ii. 482; but with this variance, that Lord Bolingbroke there ascribes the
orders to the queen herself, though he conjectured them to have proceeded from
Lord Oxford.
[343] Parl. Hist. vii. 292. The apprehension that parliament, having taken this
step, might go on still farther to protract its own duration, was not quite idle. We
find from Coxe's Memoirs of Walpole, ii. 217, that in 1720, when the first
septennial House of Commons had nearly run its term, there was a project of
once more prolonging its life.
[344] Parl. Hist. vii. 589.
[345] The arguments on this side are urged by Addison, in the Old Whig; and by
the author of a tract, entitled "Six Questions Stated and Answered."
[346] The speeches of Walpole and others, in the Parliamentary Debates, contain
the whole force of the arguments against the peerage bill. Steele in the Plebeian
opposed his old friend and coadjutor, Addison, who forgot a little in party and
controversy their ancient friendship.
Lord Sunderland held out, by way of inducements to the bill, that the Lords would
part with scandalum magnatum, and permit the Commons to administer an oath;
and that the king would give up the prerogative of pardoning after an
impeachment. Coxe's Walpole, ii. 172. Mere trifles, in comparison with the
innovations projected.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-63-320.jpg)
![[347] The letters in Coxe's Memoirs of Walpole, vol. ii., abundantly show the
German nationality, the impolicy and neglect of his duties, the rapacity and petty
selfishness of George I. The whigs were much dissatisfied; but fear of losing their
places made them his slaves. Nothing can be more demonstrable than that the
king's character was the main cause of preserving jacobitism, as that of his
competitor was of weakening it.
The habeas corpus was several times suspended in this reign, as it had been in
that of William. Though the perpetual conspiracies of the jacobites afforded a
sufficient apology for this measure, it was invidiously held up as inconsistent with
a government which professed to stand on the principles of liberty. Parl. Hist. v.
153, 267, 604; vii. 276; viii. 38. But some of these suspensions were too long,
especially the last, from October 1722 to October 1723. Sir Joseph Jekyll, with his
usual zeal for liberty, moved to reduce the time to six months.
[348] "It was first settled by a verbal agreement between Archbishop Sheldon and
the Lord Chancellor Clarendon, and tacitly given into by the clergy in general as a
great ease to them in taxations. The first public act of any kind relating to it was
an act of parliament in 1665, by which the clergy were, in common with the laity,
charged with the tax given in that act, and were discharged from the payment of
the subsidies they had granted before in convocation; but in this act of parliament
of 1665 there is an express saving of the right of the clergy to tax themselves in
convocation, if they think fit; but that has been never done since, nor attempted,
as I know of, and the clergy have been constantly from that time charged with
laity in all public aids to the Crown by the House of Commons. In consequence of
this (but from what period I cannot say), without the intervention of any particular
law for it, except what I shall mention presently, the clergy (who are not lords of
parliament) have assumed, and without any objection enjoyed, the privilege of
voting in the election of members of the House of Commons, in virtue of their
ecclesiastical freeholds. This has constantly been practised from the time it first
began; there are two acts of parliament which suppose it to be now a right. The
acts are 10 Anne, c. 23; 18 Geo. II. c. 18. Gibson, Bishop of London, said to me,
that this (the taxation of the clergy out of convocation) was the greatest alteration
in the constitution ever made without an express law." Speaker Onslow's note on
Burnet (Oxf. edit. iv. 508).
[349] The first authority I have observed for this pretension is an address of the
House of Lords (19 Nov. 1675) to the throne, for the frequent meeting of the
convocation, and that they do make to the king such representations as may be
for the safety of the religion established. Lords' Journals. This address was
renewed February 22, 1677. But what took place in consequence I am not
apprised. It shows, however, some degree of dissatisfaction on the part of the
bishops, who must be presumed to have set forward these addresses, at the](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-64-320.jpg)
![virtual annihilation of their synod which naturally followed from its relinquishment
of self-taxation.
[350] Kennet, 799, 842; Burnet, 280. This assembly had been suffered to sit,
probably, in consequence of the tory maxims which the ministry of that year
professed.
[351] Wilkins's Concilia, iv.; Burnet, passim; Boyer's Life of Queen Anne, 225;
Somerville, 82, 124.
[352] The lower house of convocation, in the late reign, among their other
vagaries, had requested "that some synodical notice might be taken of the
dishonour done to the church by a sermon preached by Mr. Benjamin Hoadley at
St. Lawrence Jewry, Sept. 29, 1705, containing positions contrary to the doctrine
of the church, expressed in the first and second parts of the homily against
disobedience and wilful rebellion." Wilkins, iv. 634.
[353] These qualities are so apparent, that after turning over some forty or fifty
tracts, and consuming a good many hours on the Bangorian controversy, I should
find some difficulty in stating with precision the propositions in dispute. It is,
however, evident that a dislike, not perhaps exactly to the house of Brunswick, but
to the tenor of George I.'s administration, and to Hoadley himself as an eminent
advocate for it, who had been rewarded accordingly, was at the bottom a leading
motive with most of the church party; some of whom, such as Hare, though
originally of a whig connection, might have had disappointments to exasperate
them.
There was nothing whatever in Hoadley's sermon injurious to the established
endowments and privileges, nor to the discipline and government, of the English
church, even in theory. If this had been the case, he might be reproached with
some inconsistency in becoming so large a partaker of her honours and
emoluments. He even admitted the usefulness of censures for open immoralities,
though denying all church authority to oblige any one to external communion, or
to pass any sentence which should determine the condition of men with respect to
the favour or displeasure of God. Hoadley's Works, ii. 465, 493. Another great
question in this controversy was that of religious liberty, as a civil right, which the
convocation explicitly denied. And another related to the much debated exercise
of private judgment in religion, which, as one party meant virtually to take away,
so the other perhaps unreasonably exaggerated. Some other disputes arose in the
course of the combat, particularly the delicate problem of the value of sincerity as
a plea for material errors.
[354] Tindal, 539.
[355] Parl. Hist. vi. 362.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-65-320.jpg)
![[356] 10 Anne, c. 2.
[357] 12 Anne, c. 7; Parl. Hist. vi. 1349. The schism act, according to Lockhart,
was promoted by Bolingbroke, in order to gratify the high tories, and to put Lord
Oxford under the necessity of declaring himself one way or other. "Though the
Earl of Oxford voted for it himself, he concurred with those who endeavoured to
restrain some parts which they reckoned too severe; and his friends in both
houses, particularly his brother auditor Harley, spoke and voted against it very
earnestly."—P. 462.
[358] 5 Geo. I. c. 4. The whigs out of power, among whom was Walpole,
factiously and inconsistently opposed the repeal of the schism act, so that it
passed with much difficulty. Parl. Hist. vii. 569.
[359] The first act of this kind appears to have been in 1727. 1 Geo. II. c. 23. It
was repeated next year, intermitted the next, and afterwards renewed in every
year of that reign except the fifth, the seventeenth, the twenty-second, the
twenty-third, the twenty-sixth, and the thirtieth. Whether these occasional
interruptions were intended to prevent the nonconformists from relying upon it, or
were caused by some accidental circumstance, must be left to conjecture. I
believe that the renewal has been regular every year since the accession of
George III. It is to be remembered, that the present work was first published
before the repeal of the test act in 1828.
[360] We find in Gutch's Collectanea Curiosa, vol. i. p. 53, a plan, ascribed to Lord
Chancellor Macclesfield, for taking away the election of heads of colleges from the
fellows, and vesting the nomination in the great officers of state, in order to cure
the disaffection and want of discipline which was justly complained of. This
remedy would have been perhaps the substitution of a permanent for a temporary
evil. It appears also that Archbishop Wake wanted to have had a bill, in 1716, for
asserting the royal supremacy, and better regulating the clergy of the two
universities (Coxe's Walpole, ii. 122); but I do not know that the precise nature of
this is anywhere mentioned. I can scarcely quote Amherst's Terræ Filius as
authority; it is a very clever, though rather libellous, invective against the
university of Oxford at that time; but from internal evidence, as well as the
confirmation which better authorities afford it, I have no doubt that it contains
much truth.
Those who have looked much at the ephemeral literature of these two reigns
must be aware of many publications fixing the charge of prevalent disaffection on
this university, down to the death of George II.; and Dr. King, the famous jacobite
master of St. Mary Hall, admits that some were left to reproach him for apostasy
in going to court on the accession of the late king in 1760. The general reader will
remember the Isis by Mason, and the Triumph of Isis by Warton; the one a severe
invective, the other an indignant vindication; but in this instance, notwithstanding](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-66-320.jpg)
![the advantages which satire is supposed to have over panegyric, we must award
the laurel to the worse cause, and, what is more extraordinary, to the worse poet.
[361] Layer, who suffered on account of this plot, had accused several peers,
among others Lord Cowper, who complained to the house of the publication of his
name; and indeed, though he was at that time strongly in opposition to the court,
the charge seems wholly incredible. Lord Strafford, however, was probably guilty;
Lords North and Orrery certainly so. Parl. Hist. viii. 203. There is even ground to
suspect that Sunderland, to use Tindal's words, "in the latter part of his life had
entered into correspondencies and designs, which would have been fatal to
himself or to the public."—P. 657. This is mentioned by Coxe, i. 165; and certainly
confirmed by Lockhart, ii. 68, 70. But the reader will hardly give credit to such a
story as Horace Walpole has told, that he coolly consulted Sir Robert, his political
rival, as to the part they should take on the king's death. Lord Orford's Works, iv.
287.
[362] State Trials, xvi. 324; Parl. Hist. viii. 195 et post. Most of the bishops voted
against their restless brother; and Willis, Bishop of Salisbury, made a very good
but rather too acrimonious a speech on the bill. Id. 298. Hoadley, who was no
orator, published two letters in the newspaper, signed "Britannicus," in answer to
Atterbury's defence; which, after all that had passed, he might better have
spared. Atterbury's own speech is certainly below his fame, especially the
peroration. Id. 267.
No one, I presume, will affect to doubt the reality of Atterbury's connections with
the Stuart family, either before his attainder or during his exile. The proofs of the
latter were published by Lord Hailes in 1768, and may be found also in Nicholls's
edition of Atterbury's Correspondence, i. 148. Additional evidence is furnished by
the Lockhart Papers, vol. ii. passim.
[363] The Stuart papers obtained lately from Rome, and now in his majesty's
possession, are said to furnish copious evidence of the jacobite intrigues, and to
affect some persons not hitherto suspected. We have reason to hope that they will
not be long withheld from the public, every motive for concealment being wholly
at an end.
It is said that there were not less than fifty jacobites in the parliament of 1728.
Coxe, ii. 294.
[364] The tories, it is observed in the MS. journal of Mr. Yorke (second Earl of
Hardwicke), showed no sign of affection to the government at the time when the
invasion was expected in 1743, but treated it all with indifference. Parl. Hist. xiii.
668. In fact a disgraceful apathy pervaded the nation; and according to a letter
from Mr. Fox to Mr. Winnington in 1745, which I only quote from recollection, it
seemed perfectly uncertain, from this general passiveness, whether the revolution](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-67-320.jpg)
![might not be suddenly brought about. Yet very few comparatively, I am
persuaded, had the slightest attachment or prejudice in favour of the house of
Stuart; but the continual absence from England, and the Hanoverian predilections
of the two Georges, the feebleness and factiousness of their administration, and
of public men in general, and an indefinite opinion of misgovernment, raised
through the press, though certainly without oppression or arbitrary acts, had
gradually alienated the mass of the nation. But this would not lead men to expose
their lives and fortunes; and hence the people of England, a thing almost
incredible, lay quiet and nearly unconcerned, while the little army of Highlanders
came every day nearer to the capital. It is absurd, however, to suppose that they
could have been really successful by marching onward; though their defeat might
have been more glorious at Finchley than at Culloden.
[365] See Parl. Hist. xiii. 1244; and other proofs might be brought from the same
work, as well as from miscellaneous authorities of the age of George II.
[366] See in the Lockhart Papers, ii. 565, a curious relation of Charles Edward's
behaviour in refusing to quit France after the peace of Aix-la-Chapelle. It was so
insolent and absurd that the government was provoked to arrest him at the opera,
and literally to order him to be bound hand and foot; an outrage which even his
preposterous conduct could hardly excuse.
Dr. King was in correspondence with this prince for some years after the latter's
foolish, though courageous, visit to London in September 1750; which he left
again in five days, on finding himself deceived by some sanguine friends. King
says he was wholly ignorant of our history and constitution. "I never heard him
express any noble or benevolent sentiment, the certain indications of a great soul
and good heart; or discover any sorrow or compassion for the misfortune of so
many worthy men who had suffered in his cause." Anecdotes of his own Times, p.
201. He goes on to charge him with love of money and other faults. But his great
folly in keeping a mistress, Mrs. Walkinshaw, whose sister was housekeeper at
Leicester House, alarmed the jacobites. "These were all men of fortune and
distinction, and many of them persons of the first quality, who attached
themselves to the P. as to a person who they imagined might be made the
instrument of saving their country. They were sensible that by Walpole's
administration the English government was become a system of corruption; and
that Walpole's successors, who pursued his plan without any of his abilities, had
reduced us to such a deplorable situation that our commercial interest was
sinking, our colonies in danger of being lost, and Great Britain, which, if her
powers were properly exerted, as they were afterwards in Mr. Pitt's administration,
was able to give laws to other nations, was become the contempt of all Europe."—
P. 208. This is in truth the secret of the continuance of jacobitism. But possibly
that party were not sorry to find a pretext for breaking off so hopeless a
connection, which they seem to have done about 1755. Mr. Pitt's great successes](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-68-320.jpg)
![reconciled them to the administration; and his liberal conduct brought back those
who had been disgusted by an exclusive policy. On the accession of a new king
they flocked to St. James's; and probably scarcely one person of the rank of a
gentleman, south of the Tweed, was found to dispute the right of the house of
Brunswick after 1760. Dr. King himself, it may be observed, laughs at the old
passive obedience doctrine (page 193); so far was he from being a jacobite of
that school.
A few nonjuring congregations lingered on far into the reign of George III.,
presided over by the successors of some bishops whom Lloyd of Norwich, the last
of those deprived at the revolution, had consecrated in order to keep up the
schism. A list of these is given in D'Oyly's Life of Sancroft, vol. ii. p. 34, whence it
would appear that the last of them died in 1779. I can trace the line a little
farther: a bishop of that separation, named Cartwright, resided at Shrewsbury in
1793, carrying on the business of a surgeon. State Trials, xxiii. 1073. I have heard
of similar congregations in the west of England still later. He had, however,
become a very loyal subject to King George: a singular proof of that tenacity of
life by which religious sects, after dwindling down through neglect, excel frogs and
tortoises; and that, even when they have become almost equally cold-blooded!
[367] Parl. Hist. viii. 904.
[368] Id. vii. 536.
[369] 8 Geo. 2, c. 30; Parl. Hist. viii. 883.
[370] The military having been called in to quell an alleged riot at Westminster
election in 1741, it was resolved (Dec. 22nd) "that the presence of a regular body
of armed soldiers at an election of members to serve in parliament is a high
infringement of the liberties of the subject, a manifest violation of the freedom of
elections, and an open defiance of the laws and constitution of this kingdom." The
persons concerned in this, having been ordered to attend the house, received on
their knees a very severe reprimand from the speaker. Parl. Hist. ix. 326. Upon
some occasion, the circumstances of which I do not recollect, Chief Justice Willis
uttered some laudable sentiments as to the subordination of military power.
[371] Lord Hardwicke threw out the militia bill in 1756, thinking some of its
clauses rather too republican, and, in fact, being adverse to the scheme. Parl.
Hist. xv. 704; H. Walpole's Memoirs, ii. 45; Coxe's Memoirs of Lord Walpole, 450.
[372] By the act of 6 Anne, c. 7, all persons holding pensions from the Crown
during pleasure were made incapable of sitting in the House of Commons; which
was extended by 1 Geo. I. c. 56, to those who held them for any term of years.
But the difficulty was to ascertain the fact; the government refusing information.
Mr. Sandys, accordingly proposed a bill in 1730, by which every member of the](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-69-320.jpg)
![Commons was to take an oath that he did not hold any such pension, and that, in
case of accepting one, he would disclose it to the house within fourteen days. This
was carried by a small majority through the Commons, but rejected in the other
house; which happened again in 1734 and in 1740. Parl. Hist. viii. 789; ix. 369; xi.
510. The king, in an angry note to Lord Townshend, on the first occasion, calls it
"this villainous bill." Coxe's Walpole, ii. 537, 673. A bill of the same gentleman to
limit the number of placemen in the house had so far worse success, that it did
not reach the Serbonian bog. Parl. Hist. xi. 328, Bishop Sherlock made a speech
against the prevention of corrupt practices by the pension bill, which, whether
justly or not, excited much indignation, and even gave rise to the proposal of a bill
for putting an end to the translation of bishops. Id. viii. 847.
[373] 25 Geo. 2, c. 22. The king came very reluctantly into this measure: in the
preceding session of 1742, Sandys, now become chancellor of the exchequer, had
opposed it, though originally his own; alleging, in no very parliamentary manner,
that the new ministry had not yet been able to remove his majesty's prejudices.
Parl. Hist. xii. 896.
[374] Mr. Fox declared to the Duke of Newcastle, when the office of secretary of
state, and what was called the management of the House of Commons, was
offered to him, "that he never desired to touch a penny of the secret service
money, or to know the disposition of it farther than was necessary to enable him
to speak to the members without being ridiculous." Dodington's Diary, 15th March
1754. H. Walpole confirms this in nearly the same words. Mem. of Last Ten Years,
i. 332.
[375] In Coxe's Memoirs of Sir R. Walpole, iii. 609, we have the draught, by that
minister, of an intended vindication of himself after his retirement from office, in
order to show the impossibility of misapplying public money, which, however, he
does not show; and his elaborate account of the method by which payments are
made out of the exchequer, though valuable in some respects, seems rather
intended to lead aside the unpractised reader.
[376] This secret committee were checked at every step for want of sufficient
powers. It is absurd to assert, like Mr. Coxe, that they advanced accusations which
they could not prove, when the means of proof were withheld. Scrope and Paxton,
the one secretary, the other solicitor, to the treasury, being examined about very
large sums traced to their hands, and other matters, refused to answer questions
that might criminate themselves; and a bill to indemnify evidence was lost in the
upper house. Parl. Hist. xii. 625 et post.
[377] See vol. i. 254, 255.
[378] Parl. Hist. vi. 1265. Walpole says, in speaking for Steele, "the liberty of the
press is unrestrained; how then shall a part of the legislature dare to punish that](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-70-320.jpg)
![as a crime, which is not declared to be so by any law framed by the whole?"
[379] Vol. i. p. 250.
[380] The instances are so numerous, that to select a few would perhaps give an
inadequate notion of the vast extension which privilege received. In fact, hardly
anything could be done disagreeable to a member, of which he might inform the
house, and cause it to be punished.
[381] 12 Will. 3, ch. 3.
[382] Journals, 11th Feb. It had been originally proposed, that the member
making the complaint should pay the party's costs and expenses, which was
amended, I presume, in consequence of some doubt as to the power of the house
to enforce it.
[383] 10 G. 3, c. 50.
[384] Resolved, That whatever ill consequences may arise from the so long
deferring the supplies for the year's service, are to be attributed to the fatal
counsel of putting off the meeting of a parliament so long, and to unnecessary
delays of the House of Commons. Lords' Journals, 23rd June 1701. The Commons
had previously come to a vote, that all the ill consequences which may at this time
attend the delay of the supplies granted by the Commons for the preserving the
public peace, and maintaining the balance of Europe, are to be imputed to those
who, to procure an indemnity for their own enormous crimes, have used their
utmost endeavours to make a breach between the two houses. Commons'
Journals, June 20th.
[385] Journals, 8th May; Parl. Hist. v. 1250; Ralph, 947. This historian, who
generally affects to take the popular side, inveighs against this petition, because
the tories had a majority in the Commons. His partiality, arising out of a dislike to
the king, is very manifest throughout the second volume. He is forced to admit
afterwards, that the house disgusted the people by their votes on this occasion. P.
976.
[386] History of the Kentish Petition; Somers Tracts, xi. 242; Legion's Paper; Id.
264; Vindication of the Rights of the Commons (either by Harley or Sir Humphrey
Mackworth); Id. 276. This contains in many respects constitutional principles; but
the author holds very strong language about the right of petitioning. After quoting
the statute of Charles II. against tumults on pretence of presenting petitions, he
says: "By this statute it may be observed, that not only the number of persons is
restrained, but the occasion also for which they may petition; which is for the
alteration of matters established in church or state, for want whereof some
inconvenience may arise to that county from which the petition shall be brought.
For it is plain by the express words and meaning of that statute that the grievance](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-71-320.jpg)
![think may affect their religion and liberties, will never, I hope, be taken from the
subject." Id. xv. 149; see also 376. And it is very remarkable that notwithstanding
the violent clamour excited by that unfortunate statute, no petitions for its repeal
are to be found in the journals. They are equally silent with regard to the
marriage act, another topic of popular obloquy. Some petitions appear to have
been presented against the bill for naturalisation of foreign protestants; but
probably on the ground of its injurious effect on the parties themselves. The great
multiplication of petitions on matters wholly unconnected with particular interests
cannot, I believe, be traced higher than those for the abolition of the slave trade
in 1787; though a few were presented for reform about the end of the American
war, which would undoubtedly have been rejected with indignation in any earlier
stage of our constitution. It may be remarked also that petitions against bills
imposing duties are not received, probably on the principle that they are intended
for the general interests, though affecting the parties who thus complain of them.
Hatsell, iii. 200.
The convocation of public meetings for the debate of political questions, as
preparatory to such addresses or petitions, is still less according to the practice
and precedents of our ancestors; nor does it appear that the sheriffs or other
magistrates are more invested with a right of convening or presiding in assemblies
of this nature than any other persons; though, within the bounds of the public
peace, it would not perhaps be contended that they have ever been unlawful. But
that their origin can be distinctly traced higher than the year 1769, I am not
prepared to assert. It will of course be understood, that this note is merely
historical, and without reference to the expediency of that change in our
constitutional theory which it illustrates.
[387] State Trials, xiv. 849.
[388] Parl. Hist. vi. 225 et post; State Trials, xiv. 695 et post.
[389] Parl. Hist. xiv. 888 et post, 1063; Walpole's Memoirs of the last Ten Years of
George II., i. 15 et post.
[390] Journals, vii. 9th July 1725.
[391] Commons' Journals, 25th Oct. 1689.
[392] Id. Dec. 5.
[393] Parl. Hist. vii. 803.
[394] Lords' Journals, 10th Jan. 1702; Parl. Hist. vi. 21.
[395] Hargrave's Juridical Arguments, vol. i. p. 1, etc.
[396] State Trials, vi. 1369; 1 Modern Reports, 159.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-73-320.jpg)
![[397] Id., xii. 822; T. Jones, Reports, 208.
[398] Journals, 10th, 12th, 19th July 1689.
[399] State Trials, xiv. 849.
[400] Id., viii. 30.
[401] This is very elaborately and dispassionately argued by Mr. Hargrave in his
Juridical Arguments, above cited; also vol. ii. p. 183. "I understand it," he says,
"to be clearly part of the law and custom of parliament that each house of
parliament may inquire into and imprison for breaches of privilege." But this he
thinks to be limited by law; and after allowing it clearly in cases of obstruction,
arrest, assault, etc., on members, admits also that "the judicative power as to
writing, speaking, or publishing, of gross reflections upon the whole parliament or
upon either house, though perhaps originally questionable, seems now of too long
a standing and of too much frequency in practice to be well counteracted." But
after mentioning the opinions of the judges in Crosby's case, Mr. H. observes: "I
am myself far from being convinced that commitment for contempts by a house of
parliament, or by the highest court of judicature in Westminster Hall, either ought
to be, or are thus wholly privileged from all examination and appeal."
[402] Mr. Justice Gould, in Crosby's case, as reported by Wilson, observes: "It is
true this court did, in the instance alluded to by the counsel at the bar (Wilkes's
case, 2 Wilson, 151), determine upon the privilege of parliament in the case of a
libel; but then that privilege was promulged and known; it existed in records and
law-books, and was allowed by parliament itself. But even in that case we now
know that we were mistaken; for the House of Commons have since determined,
that privilege does not extend to matters of libel." It appears, therefore, that Mr.
Justice Gould thought a declaration of the House of Commons was better
authority than a decision of the court of common pleas, as to a privilege which, as
he says, existed in records and law-books.
[403] "I am far from subscribing to all the latitude of the doctrine of attachments
for contempts of the king's courts of Westminster, especially the King's Bench, as
it is sometimes stated, and it has been sometimes practised." Hargrave, ii. 213.
"The principle upon which attachments issue for libels on courts is of a more
enlarged and important nature: it is to keep a blaze of glory around them, and to
deter people from attempting to render them contemptible in the eyes of the
people." Wilmot's Opinions and Judgments, p. 270. Yet the king, who seems as
much entitled to this blaze of glory as his judges, is driven to the verdict of a jury
before the most libellous insult on him can be punished.
[404] Hargrave, ubi supra.](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-74-320.jpg)
![[405] This effect of continual new statutes is well pointed out in a speech ascribed
to Sir William Wyndham in 1734: "The learned gentleman spoke (he says) of the
prerogative of the Crown, and asked us if it had lately been extended beyond the
bounds prescribed to it by law. Sir, I will not say that there have been lately any
attempts to extend it beyond the bounds prescribed by law; but I will say that
these bounds have been of late so vastly enlarged that there seems to be no great
occasion for any such attempt. What are the many penal laws made within these
forty years, but so many extensions of the prerogative of the Crown, and as many
diminutions of the liberty of the subject? And whatever the necessity was that
brought us into the enacting of such laws, it was a fatal necessity; it has greatly
added to the power of the Crown, and particular care ought to be taken not to
throw any more weight into that scale." Parl. Hist. ix. 463.
Among the modern statutes which have strengthened the hands of the executive
power, we should mention the riot act (1 Geo. I. stat. 2, c. 5), whereby all persons
tumultuously assembled to the disturbance of the public peace, and not dispersing
within one hour after proclamation made by a single magistrate, are made guilty
of a capital felony. I am by no means controverting the expediency of this law;
but, especially when combined with the aid of a military force, it is surely a
compensation for much that may seem to have been thrown into the popular
scale.
[406] 9 Geo. 2, c. 35, sect. 10, 13; Parl. Hist. ix. 1229. I quote this as I find it: but
probably the expressions are not quite correct; for the reasoning is not so.
[407] Coxe's Walpole, i. 296; H. Walpole's Works, iv. 476. The former, however,
seems to rest on H. Walpole's verbal communication, whose want of accuracy, or
veracity, or both, is so palpable that no great stress can be laid on his testimony. I
believe, however, that the fact of George I. and his minister conversing in Latin
may be proved on other authority.
[408] H. Walpole's Memoirs of the last Ten Years; Lord Waldegrave's Memoirs. In
this well written little book, the character of George II. in reference to his
constitutional position, is thus delicately drawn: "He has more knowledge of
foreign affairs than most of his ministers, and has good general notions of the
constitution, strength, and interest of this country; but, being past thirty when the
Hanover succession took place, and having since experienced the violence of
party, the injustice of popular clamour, the corruption of parliaments, and the
selfish motives of pretended patriots, it is not surprising that he should have
contracted some prejudices in favour of those governments where the royal
authority is under less restraint. Yet prudence has so far prevailed over these
prejudices, that they have never influenced his conduct. On the contrary, many
laws have been enacted in favour of public liberty; and in the course of a long
reign there has not been a single attempt to extend the prerogative of the Crown](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-75-320.jpg)
![beyond its proper limits. He has as much personal bravery as any man, though his
political courage seems somewhat problematical; however, it is a fault on the right
side; for had he always been as firm and undaunted in the closet as he showed
himself at Oudenarde and Dettingen, he might not have proved quite so good a
king in this limited monarchy,"—P. 5. This was written in 1757.
The real tories, those I mean who adhered to the principles expressed by that
name, thought the constitutional prerogative of the Crown impaired by a
conspiracy of its servants. Their notions are expressed in some "Letters on the
English Nation," published about 1756, under the name of Battista Angeloni, by
Dr. Shebbeare, once a jacobite, and still so bitter an enemy of William III. and
George I. that he stood in the pillory, not long afterwards, for a libel on those
princes (among other things); on which Horace Walpole justly animadverts, as a
stretch of the law by Lord Mansfield destructive of all historical truth. Memoirs of
the last Ten Years, ii. 328. Shebbeare, however, was afterwards pensioned, along
with Johnson, by Lord Bute, and at the time when these letters were written, may
possibly have been in the Leicester House interest. Certain it is, that the self-
interested cabal who belonged to that little court endeavoured too successfully to
persuade its chief and her son that the Crown was reduced to a state of
vassalage, from which it ought to be emancipated; and the government of the
Duke of Newcastle, as strong in party connection as it was contemptible in ability
and reputation, afforded them no bad argument. The consequences are well
known, but do not enter into the plan of this work.
[409] Many proofs of this occur in the correspondence published by Mr. Coxe.
Thus Horace Walpole writing to his brother Sir Robert, in 1739, says: "King William
had no other object but the liberties and balance of Europe; but, good God! what
is the case now? I will tell you in confidence; little, low, partial, electoral notions
are able to stop or confound the best conducted project for the public." Memoirs
of Sir R. Walpole, iii. 535. The Walpoles had, some years before, disapproved the
policy of Lord Townshend on account of his favouring the king's Hanoverian
prejudices. Id. i. 334. And, in the preceding reign, both these whig leaders were
extremely disgusted with the Germanism and continual absence of George I. (Id.
ii. 116, 297), though first Townshend, and afterwards Walpole, according to the
necessity, or supposed necessity, which controls statesmen (that is, the fear of
losing their places), became in appearance the passive instruments of royal
pleasure.
It is now, however, known that George II. had been induced by Walpole to come
into a scheme, by which Hanover, after his decease, was to be separated from
England. It stands on the indisputable authority of Speaker Onslow. "A little while
before Sir Robert Walpole's fall (and as a popular act to save himself, for he went
very unwillingly out of his offices and power), he took me one day aside, and said,
'What will you say, speaker, if this hand of mine shall bring a message from the](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-76-320.jpg)
![king to the House of Commons, declaring his consent to having any of his family,
after his death, to be made, by act of parliament, incapable of inheriting and
enjoying the crown, and possessing the electoral dominions at the same time?' My
answer was, 'Sir, it will be as a message from heaven.' He replied, 'It will be done.'
But it was not done; and I have good reason to believe, it would have been
opposed, and rejected at that time, because it came from him, and by the means
of those who had always been most clamorous for it; and thus perhaps the
opportunity was lost: when will it come again? It was said that the prince at that
juncture would have consented to it, if he could have had the credit and
popularity of the measure, and that some of his friends were to have moved it in
parliament, but that the design at St. James's prevented it. Notwithstanding all
this, I have had some thoughts that neither court ever really intended the thing
itself; but that it came on and went off, by a jealousy of each other in it, and that
both were equally pleased that it did so, from an equal fondness (very natural) for
their own native country." Notes on Burnet (iv. 490, Oxf. edit.). This story has
been told before, but not in such a manner as to preclude doubt of its
authenticity.
[410] A bill was brought in for this purpose in 1712, which Swift, in his History of
the Last Four Years, who never printed anything with his name, naturally blames.
It miscarried, probably on account of this provision. Parl. Hist. vi. 1141. But the
queen, on opening the session, in April 1713, recommended some new law to
check the licentiousness of the press. Id. 1173. Nothing, however, was done in
consequence.
[411] Bolingbroke's letter to the Examiner, in 1710, excited so much attention that
it was answered by Lord Cowper, then chancellor, in a letter to the Tatler (Somers
Tracts, xiii. 75), where Sir Walter Scott justly observes, that the fact of two such
statesmen becoming the correspondents of periodical publications shows the
influence they must have acquired over the public mind.
[412] It was resolved, nem. con., Feb. 26th, 1729, That it is an indignity to, and a
breach of the privilege of, this house, for any person to presume to give, in written
or printed newspapers, any account or minutes of the debates, or other
proceedings of this house or of any committee thereof; and that upon discovery of
the authors, etc., this house will proceed against the offenders with the utmost
severity. Parl. Hist. viii. 683. There are former resolutions to the same effect. The
speaker having himself brought the subject under consideration some years
afterwards, in 1738, the resolution was repeated in nearly the same words, but
after a debate wherein, though no one undertook to defend the practice, the
danger of impairing the liberty of the press was more insisted upon than would
formerly have been usual; and Sir Robert Walpole took credit to himself, justly
enough, for respecting it more than his predecessors. Id. x. 800; Coxe's Walpole,
i. 572. Edward Cave, the well-known editor of the Gentleman's Magazine, and the](https://image.slidesharecdn.com/2205431-250604143342-d215e317/85/Parallel-Computational-Fluid-Dynamics-2002-New-Frontiers-And-Multidisciplinary-Applications-K-Matsuno-77-320.jpg)
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- 3. Parallel Computational Fluid Dynamics 2007 Implementations And Experiences On Large Scale And Grid Computing 1st Edition Ismail H Tuncer https://ebookbell.com/product/parallel-computational-fluid- dynamics-2007-implementations-and-experiences-on-large-scale-and-grid- computing-1st-edition-ismail-h-tuncer-1082594 Parallel Computational Fluid Dynamics 2003 Advanced Numerical Methods Software And Applications 1st Edition Boris Chetverushkin https://ebookbell.com/product/parallel-computational-fluid- dynamics-2003-advanced-numerical-methods-software-and- applications-1st-edition-boris-chetverushkin-1206306 Parallel Computational Fluid Dynamics 2005 Theory And Applications A Deane https://ebookbell.com/product/parallel-computational-fluid- dynamics-2005-theory-and-applications-a-deane-1531654 Parallel Computational Fluid Dynamics 2004 Multidiscipl Applns G Winter https://ebookbell.com/product/parallel-computational-fluid- dynamics-2004-multidiscipl-applns-g-winter-4093406 Parallel Computational Fluid Dynamics Multidisciplinary Applications Proceedings Of The Parallel Cfd 2004 Conference Las Palmas De Gran Canaria Spain May 2427 2004 G Winter Et Al https://ebookbell.com/product/parallel-computational-fluid-dynamics- multidisciplinary-applications-proceedings-of-the-parallel- cfd-2004-conference-las-palmas-de-gran-canaria-spain- may-2427-2004-g-winter-et-al-4105264
- 5. PREFACE Parallel CFD 2002 conference was held in Kansai-Science City, Japan, from May 20 to 22, 2002. It was the fourteenth conference in an international series of meetings featuring computational fluid dynamics research on parallel computers. There were over 140 participants from 14 countries. During the conference, 10 invited papers and 75 contributed papers were delivered in three parallel sessions. On March 11, 2002, the Japanese ultra high-speed vector parallel computing system known as "the Earth simulator" began operations scoring peak speed over 35 TeraFlop/s. At the conference, sessions related to the Earth Simulator were organized. Grid computing is of current interest to the Parallel CFD researcher. ITBL (IT-Based Laboratory) is an ongoing Japanese project, with a similar concept to Grid Computing. Two invited papers related to Grid computing and ITBL were also presented. Multi-disciplinary applications of the parallel CFD have been constantly presented since the first conference. A tutorial session on numerical optimization was organized which attracted a large audience. These Proceedings include about 80 percent of the oral lectures presented at the conference. All published papers have been refereed. This volume provides the full papers. A paper of the tutorial session has also been included in this volume. The Editors
- 6. vii ACKNOWLEDGEMENTS Parallel CFD2002 is hosted by Kyoto Institute of Technology in collaboration with JAERI-Center for Promotion of Computational Science and Engineering, in association with RIKEN(The Institute of Physical and Chemical Research) NASA/Goddard Space Flight Center We thank the generous financial assistance of BEST SYSTEMS INC. CD-adapco JAPAN Co., Ltd. Compaq Computer K.K. Engineous Japan, Inc. FUJI RESEARCH INSTITUTE CORPORATION FUJITSU LIMITED FUJITSU INFO SOFTWARE TECHNOLOGIES LIMITED FUJITSU PRIME SOFTWARE TECHNOLOGIES LIMITED Hewlett-Packard Japan, Ltd. Hitachi, Ltd. IBM Japan, Ltd. KGT Inc. NEC Corporation Nihon ESI K.K. - a subsidiary of ESI Group Platform Computing K.K. Research Center of Computational Mechanics, Inc SGI Japan, Ltd. TOKYO ELECTRON LTD. VINAS Co., Ltd. The Parallel CFD 2002 conference is partially supported by the grant of Ministry of Education, Culture, Sports, Science and Technology The conference could not have been organized without the contribution of many people who helped in its organization and execution. We are very grateful for the help and guidance received from Pat Fox and all the members of the international scientific committee. We would like to especially thank Jacques Periaux and Isaac Lopez for proposing and organizing tutorial and special sessions. We would also like to thank Naoki Hirose and Masahiro Fukuda, the core members of the local organizing committee, for their devotion to making the conference a success. Kenichi Matsuno Chairman, Parallel CFD 2002
- 7. Parallel Computational Fluid Dynamics - New Frontiers and Multi-Disciplinary Applications K. Matsuno, A. Ecer, J. Periaux, N. Satofuka and P. Fox (Editors) 9 2003 Elsevier Science B.V. All rights reserved. Lattice Boltzmann Methods: High Performance Computing and Engineering Applications G. Brenner ~, Th. Zeiser~, K. Beronov ~, P. Lammers~, J. Bernsdorfb ~Institute of Fluid Mechanics, University of Erlangen-Nuremberg, Cauerstrafie 4, 91058 Erlangen, Germany bC&C Research Laboratories, Rathausallee t0, 53757 Sankt Augustin, Germany The development of novel numerical methods for applications in computational fluid dynamics has made rapid progress in recent years. These new techniques include the lattice gas and lattice Boltzmann methods. Compared to the traditional CFD methods, the lattice Boltzmann methods are based on a more rigorous physical modelling, the Boltzmann equation. This allows to circumvent many deficiencies inherent in existing Navier-Stokes based approaches. Thus, the lattice Boltzmann methods have attracted a lot of attention in the fluid dynamics community and emerged as an attractive alter- native in many application areas. In the present paper, we discuss some perspectives of the lattice Boltzmann methods, in particular for industrial applications and present some successful examples from projects related to aerodynamics, chemical and process engineering. 1. Introduction In the past years, the methods of lattice gascellular automata (LGCA) and the lattice Boltzmann methods (LBM) have attained a certain maturity and subsequently chal- lenged the traditional methods of computational fluids dynamics (CFD) in many areas. In that context, traditional methods of CFD are understood to include all numerical schemes, that aim to solve the Navier-Stokes equations by some direct discretisation. In contrast to that, the LBM is based on a more rigorous description of the transport phenomena, the Boltzmann equation. Compared to other attempts, that have been made to solve this equation in the past, the LBM makes use of several significant, physically motivated simplifications that allow to construct efficient and competitive or even superior computational codes as compared to the classical approaches. Lattice gas cellular automata and even more lattice Boltzmann methods are relatively new. Just about 15 years ago, the field of LGCA started almost out of the blue with the now famous paper of Frisch, Hasslacher and Pomeau [1], who showed that some simplified kind of "billiard game" representing the propagation and collision of fluid particles leads to the Navier-Stokes equations in a suitable macroscopic limit. In particular, the authors showed how the propagation and collisions of particles have to be abstracted in order to conserve mass and momentum and how the underlaying lattice has to be designed in order to provide sufficient symmetries to obtain Navier- Stokes like behaviour. Each month, several papers appear to present new models or to
- 8. investigate existing models, to demonstrate and assess the use of LBM in application fields or to evaluate high performance computing (HPC) aspects. Summerschools, special conferences and LBM sessions in existing conferences have been organised to satisfy also the growing interest of developers and potential users in this technique. Besides that, commercial products are available with remarkable success (see e. g. [2]). The goal of the present paper is to show the potential of the lattice Boltzmann method in CFD and in related areas. Besides the classical application fields, such as aerodynamics, these are in particular problems related to chemical and process engineering. Due to the complexity of the relevant transport and chemical conversion mechanisms, that have to be modelled, these areas open new challenges also for the LBM. In the present paper, after a short summary of the basic principles of the LBM, examples related to turbulent flows, reacting flows and the respective application fields are discussed. 2. Lattice Gas and Boltzmann Method From a gaskinetical, i.e. microscopic, point of view, the movement of a fluid may be considered as the propagation and collision of molecular particles governed by funda- mental laws of physics. The modelling of this motion may be carried out on several levels, starting with the Hamilton equation of a set of discrete particles. Since this approach prohibits itself because of the large number of freedoms to be considered, several attempts have been made to simplify this picture by extracting only the essen- tial criteria required to model e. g. the motion of a Newtonian fluid. In that context, the lattice gas automata may be seen as an abstraction of the fluid making use of the fact, that the statistics of the gas may be correctly described by a significantly reduced number of molecules and by applying simplyfied dynamics of the particles. This can be explained by the fact, that the conservation principles as well as associated symmetries are the basic building blocks for the continuum equations of fluids. Thus, in oder to simulate a continuum flow, the approximation of the computer gas has to recover only these principles to a certain extend. The FHP automata, named after [1], was a first successful attempt to construct a discrete model to compute the motion of a Newtonain fluid. Although this discrete particle approach seems promising, there are problems due to spurious invariants and random noise in the solutions. These deficiencies can be overcome by applying the idea of McNamara and Zanetti [3], who considered the dis- crete Boltzmann equation as a base for the numerical algorithm. This approach may be briefly explained as follows: The Boltzmann equation is an integro-differential equation for the single particle distribution function f(t, 2, g), which describes the propability to find a particle in a volume (2, 2 + d2) and with a velocity in the range (~, ~ + dg). Neglecting body forces one has: cOtf + v-'Vf = Q(f) . (1) A suitable simplification of the complicated collision integral Q(f) is the BGK approx- imation, 1 Q(f) ~ -(f~q- f) (2) T
- 9. which preserves the lower moments an satisfies an H-Theorem like the original equations 1. Here feq is the Maxwell equilibrium distribution. The discretisation of this equation requires a finite representation of the distribution function in the velocity space. One way to realise this is to introduce a finite set of velocities gi and associated distribution functions fi(t,J, ~), which are governed by the discrete velocity Boltzmann (BGK) equation: Otfi _qt_~iV fi -- 1---(f'ezq -- fi) . (3) T Next, the discretisation in space and time is accomplished by an explicit finite difference approximation. With a scaling of the lattice spacing, the time step and the discrete velocities according to ~i - A2i/At, the discretised equation takes the following form: + zxt, t + At)- t) - _ t)). T (4) The discrete values of the equilibrium functions are chosen of Maxwell type, in the sense that their velocity moments up to fourth order are identical with the velocity moments over the Maxwell distribution. The following definition satisfies this requirement: -- tpp 1+ c~2 + 2c~ c~2 ~ . (5) The discrete equilibrium functions may be computed efficiently at each time step for each node, from the components of the local macroscopic flow velocity us, the fluid density p, the "speed of sound cs" and a direction-dependent lattice geometry weighting factor tp. The viscosity u of the simulated fluid can be controlled by the relaxation time ~-, according to . 1 3 - (6) Further technical details of this method, in particular concerning the formulation of boundary conditions, may be found in [4-6]. From the computational point of view the above approach is interesting as it re- sembles a simple finite difference scheme applied to a first oder (in time and space) hyperbolic system of equations in diagonal form. This extremely simplifies the design of a numerical scheme. However, finally the solution of the Navier-Stokes equations with second order accuracy in the limit of low Mach numbers (c~ > > 1~712)is recovered, as can be shown rigorously in [7] by applying the Chapman-Enskog procedure to eq. (4). The approach presented above leads to the basic version of LBM. Many improve- ments have been designed in order to broaden the methods range of applicability, see e.g. the review article of Chen and Doolen [8]. Current issues are the multi-time re- laxation methods to enhance the stability of the method [9,10], implicit methods [11] or the application of nonuniform and locally refined meshes [12,13] and of improved boundary conditions for Cartesian meshes [14]. For various applications, particularly in chemical and process engineering, the convective and diffusive transport of energy and species are of key importance. Examples of a few among many interesting contri- butions made in this field by various authors can be found in [15,16]. Thermal models
- 10. have been proposed in [17,18], reaction diffusion problems have been investigated e.g. in [19-21]. The simulation of multiphase flows and immiscible fluids were the subject of several investigations where the LBM provided interesting alternatives to model parti- cle interaction, surface tension, etc. In modelling suspensions of particles for example, the interactions of the fluid and particles may be treated in different ways, either by dis- cretising and mapping moving particles in a Lagrangian sense [22] or by a combination of an integer lattice gas model for the disperse phase and the BCK for the fluid phase [23]. Models for turbulent flows have been adapted from classical LES approaches, as applied to the Navier-Stokes equations, by changing the relaxation parameter in eq. 6 in order to scale the viscosity according to a subgrid model, such as published in [24]. Alternatively, two-equation models have been propsed by e.g. [25-27]. The following sections presents some practical results related to chemical engineering and aerodynamics, that show the applicability of LBM in various technical fields. 3. Applications of LBM in chemical engineering In chemical industries, packed beds and porous media are frequently used as reac- tion, separation or purification units. The design of these devices is usually based on pseudo homogeneous equations with averaged semi empirical models such as disper- sion and mass transfer correlations. The design concepts based on these models fail if local flow phenomena such as channeling effects become dominant. Therefore, sev- eral attempts have been made in oder to improve these models. However, new design methodologies are required if no or insufficent empirical data are available. Lattice Boltzmann methods can been used to directly simulate the flow field in these configu- rations together with chemical reactions and diffusion effects. This allows to analyse in detail the hydrodynamic effects, e. g. the channeling due to inhomogeneous void space distributions and other flow anomalies and to quantify their influence on the prediction of the bulk conversion and selectivity of the reactor. The lattice Boltzmann method has been chosen mainly because of its ability to model highly complex geometries and fluids. The "direct" numerical simulation of flows through packed beds uses a digitized im- age of the structure under considerations. This may be obtained from computer tomo- graphic data [28] of a real probe of the material or as synthetically generated geometry. A Monte Carlo (MC) approach is used in [29] to generate randomly distributed packed beds of spherical pellets. Following the marker and cell approach, this geometry can easily be transferred to the uniform, Cartesian mesh, which is typically used in lattice Boltzmann methods. Due to the low memory requirements of these methods, meshes with several million elements may easily be used to capture the geometric details. The fluid is modelled assuming that a species A is transported by a carrier gas through the structure shown in Fig. 1. This species may be adsorbed by the solid particles and convert to species B. This reaction and adsorption is assumed to be fast compared to the convection resulting in a mass transport limitation. The DahmkShler number, i. e. the ratio of reaction rate to flow rate is Da ~ 100; the Reynolds number based on the particle diameter is Rep ~ 10. The structure consists of particles, randomly distributed in a confining cylinder with a diameter ratio (cylinder/particle) of 5. The computational domain is discretised by a 150 • 150 x 750 regular grid, i. e. more than 16.106 elements. In order to obtain a converged steady state solution, about 40,000
- 11. propagation (=iteration) steps were necessary for the present laminar flow conditions, which took about 2h CPU time on six NBC SX 5e shared memory processors. The simulation of the velocity field has been discussed previously [29,30] and therefore, only the main characteristics are summarized here. In the geometries under consid- eration, strong variations of the local porosity become dominant, in particular close to the confining cylinder surface leading to severe flow inhomogeneities. This can be D~/Dp=5.0 ui rm, r" i 0.8 0.6 0.4 0.2 00 1 2 3 4 5 (b) Radial position r (in multiples of the particle radius) Figure 1. Image of the packed bed structure generated by an MC simulation (a) and the radial voidage profile extracted from the structure (b). The tube to particle diameter ratio is 5, leading to a severe wall effect (channeling), namely very high porosities in the region close to the wall of the tube 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (b) o - o x=-O. 1 - ............. 9 x=1.8 ~...........~. x=3.75 -<~[~ -. ;x=5.7 '~ /1~ ..........x=+2.o / l : - ----~-- ~ 1 2 3 4 5 Distance from the wall (in multiples of the particle radius) Figure 2. Snapshot of the flow field (a) through the MC generated geometrical structure of the packed bed simulated by the LBA approach and radial velocity distribution (b) in the different cross sections along the axial direction. Rep= 6.5 (laminar flow regime). The axial positions x = -0.1 and +0.1 and +2.0 specify the distance in multiples of the particle diameter in front/behind the packed structure. All other positions give the distance from the beginning of the bed
- 12. (a) +2.0 +0.I 5.7 325 1.8 -0.I (b) 1 0.8 -9 0.8 ~' 0.4 ~0.2 Z 0 0 ,,................ x=1.8 ....... ~, x=3.75 ,. A x=5.7 1 2 3 4 5 Distance from the wall (in multiples of the particle radius) Figure 3. Snapshot of the concentration field of reactant A (a) simulated by the LBA approach and the radial averaged mass flux (b) in the different cross sections along the axial direction. The corresponding flow field is shown in Fig. 2. The Schmidt number of species A is Sc = 18.5 and the reaction occurs only at the surface of the spheres. The reactant is continuously injected over the complete cross section at x ~ -0.5. All axial positions are given as multiples of the particle diameter seen from the velocity contour plots along cross sections inside the packing at various x-positions (Fig. 2a). Predominantly close to the walls, velocity spikes are observed with a magnitude up to eight times higher than the averaged flux. The corresponding circumferentially averaged radial profiles of the velocity are depicted in Fig. 2b and show the same trend known as the wall effect or wall channeling. This effect is due to the fact that the structure of the packing close to the wall is "ordered", which leads to high local porosities. The simulated local concentrations of species A (the reactant) along cross sections are shown in Fig. 3. As can be seen, reactants are still present in high concentrations close to the exit of the reactor which is due to the fact of the low local flow residence times close to the wall and consequently, the low local conversion rates of A. This breakthrough of species A would lead to a decrease in conversion and degradation of selectivity. 4. Applications of LBM for turbulent flows The application of the LBM for design tasks in aerodynamics requires to qualify the method as a reliable tool with high predictive accuracy and efficiency. In that context, it can be shown that the dynamics of turbulent flows can be correctly simulated, e.g. within a DNS. Figure 4 shows a snapshot of the pressure field computed with LBM. The geometry and flow parameters have been chosen according to the well known benchmark test case of Kim et al. [31]. In the present calculations, the computational
- 13. mesh consists of 4096 x 256 x 256 lattice sites. . . . . . . . i,i) Figure 4. Isosurfaces of the pressure computed by the lattice Boltzmann model. The shown snapshots are at approximate times 7.57 (top) and 127 (bottom) in physical time units 7 . Rendered pressure values on corresponding plots on top and bottom are proportionally equal, values on the left +3 standard deviations while those on the right are ~ 0. The computation of a correct mean velocity is the easiest and first necessary test for a turbulence simulation. It is passed successfully by many different numerical methods. In that respect, the Lattice Boltzmann Method is no exception. As seen in 5, the comparison with the reference data base [32] and also with experimental data [33] is very good. The figure compares profiles not only of (streamwise) mean velocity but also of mean pressure. Also for the Reynolds stress components, the LBM and pseudo-spectral results, shown in 6, are in excellent agreement. Some available experimental data have been added in that figure to illustrate the expected smallness of the difference between nu- merical results compared with the deviation of experimental data. More results related to this test case are published in [35]. 5. Conclusion The lattice gas cellular automata and lattice Boltzmann methods emerged just 15 years ago as new techniques to describe the motion af a fluid based on the simplified transport equations. During that time, remarkable improvements to the methods have been introduced that allow the prediction of complex flows, i.e. fluids with chemical reactions, diffusion processes, turbulent flows, among others. A conceptual advantage of the LBM is the algorithmic structure, which can be efficiently implemented on digital computers. Thus, the resulting codes require in general less memory and CPU time compared to classical CFD methods.
- 14. 10 u + 10 I Illllll 9 ~ lattice Boltzmann , .... pseudospectral 9 Moser et al. (1999) o Fischer et al. (2001) J jP -0 Pm~. -0 -0 -0. 9 ~ I ~ lattice Boltzmann ",,~ I .... pseudospectral - - . I/ I/ '~. r -II 10 100 y§ Figure 5. Comparison of LBM and a Chebyshev pseudo-spectral [34] results for mean profiles left: streamwise velocity, right: pressure. Line styles solid: LBM , dashed: Chebyshev pseudo-spectral data from [32], dashed-dotted: Chebyshev pseudo- spectral data from in-house DNS [34]. Symbols: experimental data from in-house LDA measurements [33]. 6. Acknowledgements Financial support from the Deutsche Forschungsgemeinschft (DFG), from the Bavar- ian Science Foundation (BFS) within FORTWIHR and from the Bavarian State Min- istry for Science, Research and Arts within KONWIHR is gratefully acknowleged. The calculations were mainly carried out on maschines of the Leibniz Computing Center in Munich and the High-Performance Comnputing Centre in Stuttgart, Germany. REFERENCES I. U. Frisch, B. Hasslacher, and Y. Pomeau. Lattice-gas automata for the Navier-Stokes Equation. Phys. Rev. Lett.,56(14):1505-1508, April 1986. 2. EXA Corp. Powerflow news. EXAnews, 4.0, 1998. 3. G.R. McNamara and G. Zanetti. Use of the Boltzmann equation to simulate lattice gas automata. Phys. Rev. Lett., 61:2332-2335, 1988. 4. Y. H. Qian, D. d'Humi~res, and P. Lallemand. Lattice BGK models for Navier-Stokes equation. Europhys. Lett., 17(6):479-484, January 1992. 5. Diether A. Wolf-Gladrow. Lattice-Gas Cellular Automata and Lattice Boltzmann Models, volume 1725 of Lecture Notes in Mathematics. Springer, Berlin, 2000. 6. Gunther Brenner. Numerische Simulation komplexer fluider Transportvorgiinge in der Verfahrenstechnik. PhD thesis, Technische Fakults Universits Erlangen-Niirnberg, 2002. 7. Xiaoyi He and Li-Shi Luo. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 56(6):6811-6817, December 1997. 8. Shiyi Chen and Gary D. Doolen. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech., 30:329-364, 1998. 9. Dominique d'Humi~res. Generalized lattice-Boltzmann equations. In Bernie D. Shizgal and David P. Weaver, editors, Rarefied Gas Dynamics: Theory and Simulation, Progrss
- 15. u 1.5 0.5 1 V "~ 0.4 / 1_.J r~ X lattice Boltzmann pseudospectral . . . . Maser et al.(1999), .... o Fisheretal.(2001) I I I11111 10 100 y+ I III111 I J"'~ " IpaltiuCdeosBpOtttrt~nn -.---ff/ "~N [ I I[~~162 rt ai"(19991/ e / / / / / / / j" -u--W' 1 Wrms 0.5 t / /, 1 2 /f o 50 100 y* [ I I lattice Boltzmann . . . . Maser et al. (1999) o Eckelmann(1974) o Eckelmann(1974) 150 Ill[Ill pseudospect ral Maser et al.(1999) f ~ k'~r~, ..... iceBali.... n / d /' Ol I0 I00 I I0 y* y+ If 10o Figure 6. Comparison of LBM and pseudo-spectral results for the second moments of fluctuating velocity: upper left: streamwise velocity intensity, upper right: Reynolds stress, lower left: wall-normal velocity intensity, lower right: spanwise velocity intensity. Line styles as in 5. 10. 11. 12. 13. 14. 15. 16. 17. in Astronautics and Aeronautics, pages 450-458, Washington, 1992. AIAA. Pierre Lallemand and Li-Shi Luo. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E, 61(6):6546-6562, 2000. Manfred Krafczyk. Gitter-Boltzrnann-Methoden: Van der Theorie zur Anwendung. PhD thesis, Fakults fiir Bauingenieur- und Vermessungswescn der Technischen Universit/it Miinchen, 2001. J. TSlke, M. Krafczyk, M. Schulz, E. Rank, and R. Berrios. Implicit discretization and nonuniform mesh refinement approaches for FD discretizations of LBGK models. Int. J. Mad. Phys. C, 9(8):1143-1158, 1998. alga Filippova and Dieter Hs Grid refinement for lattice-BGK models. J. Comput. Phys., 147:219-228, 1998. Renwei Mei, Li-Shi Luo, and Wei Shyy. An accurate curved boundary treatement in the lattice Boltzmann method. J. Comput. Phys., 155:307-330, 1999. E. G. Flekk0y, T. Rage, U. Oxaal, and J. Feder. Hydrodynamic irreversiblility in creeping flow. Phys. Rev. Lett., 77(20):4170-4173, 1996. Harlan W. Stockman, J. T. Fredrich, R. M. O'Connor, D. Noble, R. J. Glass, and S. R. Brown. 3D and 2D lattice Boltzmann calculations of dispersion and reaction, www. Y. Chen, H. Ohashi, and M. Akiyama. Heat transfer in lattice BGK model fluid. J. Stat.
- 16. 12 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. Phys., 81(1/2):71-85, 1995. Xiaoyi He, Shiyi Chen, and Gary D. Doolen. A novel thermal model for the lattice Boltzmann method in incompressible limit. J. Comput. Phys., 146:282-300, 1998. S. Chen, S. P. Dawson, G. D. Doolen, D. R. Janecky, and A. Lawiczak. Lattice methods and their applications to reacting systems. Comput. Chem. Eng., 19(6/7):617-646, 1995. Xiaoyi He, Ning Li, and Byron Goldstein. Lattice Boltzmann simulation of diffusion- convection systems with surface chemical reaction. Molecular Simulation, Internet Con- ference, April 1999. JSrg Richard Weimar. Cellular Automata for Reactive Systems. PhD thesis, Universit~ Libre de Bruxelles, Facult(~ des Sciences, Service de Chimie Physique, 1995. A.J.C. Ladd and R. Verberg. Lattice-Boltzmann simulations of particle-fluid suspensions. J. Star. Phys., 104(5/6), 2001. A. Masselot and B. Chopard. A multiparticle lattice-gas model for hydrodynamics. Int. J. Mod. Phys. C, 9(8):1221-1230, 1998. Shuling Hou, James D. Sterling, Shiyi Chen, and Gary Doolen. A lattice Boltzmann subgrid model for high Reynolds number flow. Fields Institute Communications, 6:151- 166, 1996. Mehtab M. Pervaiz and Christopher M. Teixeira. Two equation turbulence modeling with the lattice Boltzmann method. 2nd International Symosium on Computational Technologies for Fluid/Thermal/Chemical Systems with Industrial Applications, ASME PVP Division Conference, August 1-5 1999, Bosten, MA, 1999. preprint. Christopher M. Teixeira. Incorporating turbulence models into the lattice-Boltzmann method. Int. J. Mod. Phys. C, 9(8):1159-1175, 1998. Sauro Succi, Giorgio Amati, and Robert Benzi. Challenges in lattice Boltzmann comput- ing. J. Star. Phys., 81:5-16, 1995. J. Bernsdorf, O. Giinnewig, W. Hamm, and M. Miinker. StrSmungsberechnung in por6sen Medien. GIT Labor-Fachzeitschrift, 4:389, 1999. Y.-W. Li, Thomas Zeiser, Hannsj6rg Freund, Martin Steven, Gunther Brenner, Elias Klemm, G. Emig, and Franz Durst. Direct simulation of the structure and consequen- tial flow field in a packed bed. AIChE Journal, (submitted; revised version currently prepared). Th. Zeiser, P. Lammers, E. Klemm, Y.W. Li, J. Bernsdorf, and G. Brenner. CFD- calculation of flow, dispersion and reaction in a catalyst filled tube by the lattice Boltz- mann method. Chem. Eng. Sci., 56(4):1697-1704, 2001. John Kim, Parviz Moin, and Robert Moser. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech., 177:133-166, 1987. Robert Moser, John Kim, and Nagi Mansour. Direct numerical simulation of turbulent channel flow up to Re~ = 560. Phys. Fluids, 11, 1999. M. Fischer and F. Durst J. Jovanovid. Reynolds number effects in the near-wall region of turbulent channel flows. Phys. Fluids, 13:1755-1767, 2001. Rene Volkert, Michael Breuer, and Franz Durst. Enhanced direct numerical simulations of the plane channel flow based on a spectral method. Technical report, Lehrstuhl fiir Str6mungsmechanik, Universits Erlangen-Niirnberg, 2002. Peter Lammers, Kamen Beronov, Rene Volkert, Gunther Brenner and Franz Durst. Lat- tice boltzmann direct numerical simulation of fully developed 2d-channel turbulence. Computers ~J Fluids, submitted for publication.
- 17. ParallelComputationalFluidDynamics- NewFrontiersandMulti-DisciplinaryApplications K. Matsuno,A. Ecer,J. Periaux,N. SatofukaandP. Fox(Editors) 9 2003ElsevierScienceB.V.Allrightsreserved. 13 MULTI-DISCIPLINARY SIMULATIONS AND COMPUTATIONAL AND DATA GRIDS William E. Johnston * Lawrence Berkeley National Laboratory Berkeley, California, USA email: wejohnston@lbl.gov, web page: http://www-itg.lbl.gov/---wej/ and NASA Ames Research Center Moffett Field,, California, USA Key words: Special Architectures Abstract. Computational and Data "Grids" are middleware for building dynamically constructed problem solving environments using geographically and organizationally dispersed high performance computing and data handling resources. Grids also provide infrastructurefor collaboration.
- 18. 14 1 INTRODUCTION The overall motivation for current large-scale, multi-institutional Grid projects is to enable the resource and human interactions that facilitate large-scale science and engineering such as aerospace systems design, high energy physics data analysis [1], climatology, large-scale remote instrument operation [2], collaborative astrophysics based on virtual observatories [3], etc. In this context, Grids are providing significant new capabilities to scientists and engineers by facilitating routine construction of information and collaboration based problem solving environments that are built on-demand from large pools of resources. Functionally, Grids are tools, middleware, and services for: o building the application frameworks that allow discipline scientists to express and manage the simulation, analysis, and data management aspects of overall problem solving o providing a uniform look and feel to a wide variety of distributed computing and data resources o supporting construction, management, and use of widely distributed application systems o facilitating human collaboration through common security services, and resource and data sharing o providing remote access to, and operation of, scientific and engineering instrumentation systems o managing and securing this computing and data Application Portals / Frameworks (problemexpression;userstatemanagement;collaboration services;workflowengines;faultmanagement) Web Grid Services Applications and Utilities (domainspecificandGridrelated) Language Specific APIs (Python,perl,C, C++,Java) Grid Collective Services (resourcebrokering;resourceco-allocation;datacataloguing, publishing,subscribing,andlocationmanagement;collectiveI/O, job management) Grid Common Services (resourcediscovery;resourceaccess;authenticationand security; eventpublishandsubscribe;monitoring) Communication Services SecurityServices Resource Managers (exportresourcecapabilitiesto the Grid) Physical Resources (computers,datastoragesystems,scientificinstruments,etc.) Figure 1. Grid Architecture infrastructure as a persistent service This is accomplished through two aspects: A set of uniform software services that manage and provide access to heterogeneous, distributed resources and a widely deployed infrastructure. The software architecture is depicted in Figure 1, and the deployment is discussed later. 3 MULTI-DISCIPLINARY APPLICATION EXAMPLE As the problems tackled by the science and research engineering communities become more and more complex, the computing requirements are not just for more computing power, but for dealing with more complex application and data systems as well. Coupling the sub- components of a jet engine simulation in order to simulate its overall operation, and then
- 19. 15 coupling that ensemble to the aerodynamic lift models of an aircraft wing, is an example of a complex system that also requires high capability computing and data handling - multiple computers, databases, archives, etc.- all of which must be coordinated to solve the problem. The NPSS program [4] at NASA Glenn is working on coupling the many component models required to simulate an operational jet engine, and integrating the resulting engine model with operational data. The sub-system simulations have been developed over a long time and they are written in a variety of languages (e.g. FORTRAN) and in a variety of styles. The NPSS program has built an application framework for coupling these components together [5], and working with NASA Ames, a wing lift model is being added to the multi- disciplinary simulation ("MDS"). 4 APPLICATION CHARACTERISTICS From this MDS example we can enumerate a set of high-level characteristics of the application, which turn out to be fairly general. o system simulations are built up by coupling legacy code components o computing capacity and simulation expertise will come from many different organizations o simulation components must be coordinated on many different computing systems o computational simulations must be coupled to independent environmental and operations data sources that originate from hundreds of different locations o confidentiality of data and data access policy enforcement is required o security and access control for the underlying computing and data archive systems must prevent service disruption Figure 2. Multi-component, multi-disciplinary simulation Multiplesub-systems,e.g. a wingliftmodeloperatingat NASAAmesand a turbo-machinemodeloperatingat NASAGlenn,are combinedusingGlenn's NPSS (NumericalPropulsionSystemSimulation)application frameworkthatmanagesthe interactionsofmultiple modelsandusesGridservicesto coordinatecomputing and datastoragesystemsacrossNASACenters. 5 THE ROLE OF GRIDS The application characteristics from the example above imply a collection of capabilities that must be addressed by the distributed systems that combine computational tools with each other, and with data and instruments. In addition to the environment and services needed to support these applications, our experience in working with the design engineer / analyst who must use the system to accomplish a specific task suggests many other characteristics and requirements as well. (E.g., see [6].) The desired Grid fianctionality may be represented as a hierarchically structured set of services and capabilities that are described below, and whose interrelationship is illustrated in Figure 1. Some of the issues that Grids currently address include: o techniques for locating, incorporating, accessing, and managing resources in the overall environment that are scalable to thousands of resources
- 20. 16 o coordinating resource availability so that the components of an MDS that are located on different systems can operate simultaneously o comprehensive network monitoring to locate, analyze, and correct bandwidth bottlenecks o global data catalogues and data replica management o security and management of access rights for the collaboration data and information Additionally, scientific and engineering applications involving distributed teams and distributed resources requires higher level services, many of which are currently being developed in the Grid community: o techniques for coupling heterogeneous computer codes, resources, and data sources in ways so that they can work on integrated/coupled o describing and managing multi-step, asynchronous component workflows, including managing fault detection and recovery o access to data and metadata publication and subscription mechanisms o global event mechanisms - e.g. notification of when data or simulation results come into existence anywhere in the space of resources of interest o service oriented interfaces to all of the above The NPSS MDS described above uses an existing framework (CORBA based) to coordinate the components, and uses Grid services to access computing systems, provide security, etc. (Figure 3). ~- ................................................................................................................. *IIII~I~IIIIIII I I fill I~I III I IiI I*I III I I~ ]II III ~II ll~ll} llllll I IIIIII III III iI*I III llllllll!ll IIi III Ill I ~II i~ That is, in this context NASA Ames : : NASA.Glenn :. Grids provided: : environ~t : 9 i 9i~:i,/:, ~il,i~,l:..... o standardized Webd~t : .qt--'i--- i: : ~ : : ~ : : i : : : ~ : ............... User I : i : , i i : : ~ : : : : ~ : ~ i i ! : ~ i i : : : : ~ i : i i : : : : !':~: datable . access to multi- :. ! ~ : : : ~ : ~ : : : : : : " : : : : ' ~ institutional ! ~ : ...... :N~i~ ~~!: ::!i:!:~:i::i:: ~ o~ = :: ~ ~( ~i i~ ii~,i, . resources ~t~r~ -.~ - ............... ~g~ ~i;$fi: ................. ;i!i o a common databne::: : : ...... ! :1 : ! : 9 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: -::::...::~ : ~ :....... ,.,, .... ,.,,,,;~. ~ ....... ...... ...,. ...... ,..:........... ,: security approach : (Obj! ) :.: ~, ........................... .~............................ = o and infrastructure , ........... , ........ , ............... i o persistent Grid services that are used to run the ~ CORBA applications on an as-needed basis The ability to use the . . . . . . Grid to instantiate i existing application Figure 3. NPSS / CORBA use of Grids frameworks allows ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . users flexibility in building their applications in the framework of their choice. They do not have to rely on that framework being provided as a persistent service on all of the computing systems where they need to run - they can instantiate the framework environment, as needed, using persistent Grid services. 6 COMPUTING AND DATA GRIDS In this section be give a brief overview of the roles of the architectural components of a Grid as illustrated in Figure 1. We will start at the lowest layer and work up. In many extant
- 21. 17 Grids, the basic services are provided by Globus [7], and several of the Globus services will be mentioned below. More detail may be found in [6]. 6.1 Resource Managers The basic function of resource managers is to export the functionality provided by the resource - computing, storage, bandwidth, etc. - so that it is accessible to the Grid. There are two fundamental Grid resource managers, with a third being developed. The most basic service is the ability to authenticate and authorize a user, and then provide to that user the mechanism to download code and run it. This is provided by the server-end of the Grid job initiator. (See [8].) The second basic service is access to data. This is provided by the GridFTP server, which is described below. A third basic service that is currently being developed is a generalized event manager. This service should not only manage systems events, but will provide user jobs with the ability to signal, e.g., a workflow manager about its state. To support co-scheduling of multiple resources, CPU advance reservation scheduling and network bandwidth advance reservation are essential. In addition, tape marshaling in tertiary storage systems to support temporal reservations of tertiary storage system off line data and/or capacity is also likely to be essential. The basic functionality for co-scheduling and/or resource reservation must almost always be provided by the individual resource managers, however Grid services such as Globus' GRAM/GARA serve to coordinate and provide uniform access to these resource specific services. CPU advance scheduling services are currently provided, e.g., by the PBSPro batch scheduling system (http://www.pbspro.com/), and is a topic in the Grid Forum's Scheduling Working Group (see [9]). 6.2 Resource Managers Grid security is based on cryptographic identity tokens that are issued via some organizationally acceptable process. Many Grids use Public Key Infrastructure to provide these tokens (called X.509 identity certificates). For more information, see [10]. The Globus, Grid Security Infrastructure (GSI) uses the identity certificate to support a "single sign-on" capability- that is, the user authenticates once for access to all Grid resources for which the user is authorized. Identity certificates are also issued to computing systems and services (such as secure FTP). The identity certificate, in conjunction with various security protocols, is used to provide message integrity (the message that got to the receiver is guaranteed to be identical to the one that the sender sent) and, optionally, message confidentiality. (See [11].) 6.3 Communication Services Grid communication services provide a uniform model for communicating among distributed job components. The Globus I/O function provides a standard interface to stream and datagram (message) based I/O, and in each case provides transparent use of the security services. 6.4 Grid Resource Access and Management "Grid Resource Access" refers to the basic services that provide uniform and location independent access and management of distributed resources. Much of the operational effort to run Grids is involved in maintaining these services and their associated resource manager. Many Grids, including NASA's IPG [12],DOE's Science Grid [13] and ASCI Grid [14], and the Grids of the NSF Supercomputer centers [15], currently use Globus to provide the basic
- 22. 18 services that characterize and locate resources, initiate and monitor jobs, provide secure authentication of users, provide uniform access to data, etc. The Grid Information Service is the Grid resource discovery mechanism. It has the task of representing, and/or providing access to, virtually all aspects of the configuration and state of the Grid: it maintains, or provides pointers to services that provide, all Grid resource characteristics (e.g. CPU count, memory size, OS version, etc.), dynamic performance information, information about current process state, user identifies, allocations and accounting information, etc. It also responds to structured queries about the availability of resources with certain characteristics and performance state, e.g. the computing systems architecture needed for a particular code. The GIS is implemented as a distributed collection of directory servers [16]. A Grid resource access service / job initiator provides a standard way of specifying a resource and then submitting a request for service. Most commonly this is a script to be executed on a remote system, and this service provides a uniform interface to the underlying batch scheduling systems. The Globus GRAM service provides this function. GridFTP is a file oriented, remote data access service that provides third party transfers, partial file access, reliable transfer, parallel transfers and server side data striping if supported by the data storage system, Grid security based access control, etc. The resource side of this is a special FTP server that implements an extended FTP service. (See [17].) Monitoring services are important for both users and the managers of the Grid in order to do fault detection and management, system and application debugging, and performance optimization. The Globus Heart Beat Monitor provides basic, job-level monitoring. The NetLogger toolkit [18] provide detailed monitoring of distributed applications, and the Network Weather Service [19] provides for network monitoring and characterization. All of these are being integrated as Grid services, and the Grid Forum, Grid Performance Working Group [20]is addressing this issue in a general way. General Grid event services are being addressed in the Grid Forum, Grid Computing Environments working group (see the GCE working group pages at www.gridforum.org). 6.5 Grid Collective Services There are many "collective services" being developed for the Grid. Such services provide aggregated functionality that makes it easier to use the Grid. Of the many such services being developed, and only a few that are of interest to computational scientists are mentioned here. A resource broker takes a requirements specification for a job (min. CPU count, min. memory, etc.) together with a policy statement (e.g., "run this job as soon as possible" or "run this job at the least cost") and locates a set of resources that will meet the requirements. Resource brokers are currently under development. A job manager automates the task of ensuring that a large collection of related job (e.g. those in a parameter study) all execute exactly once. Condor-G [21]provides such a service for the Grid. High-performance applications require high-speed access to data files, and the Grid services must be able to locate, stage, cache, and automatically manage the location of local, remote and cached copies of files. Publishing metadata about the data is critical in large collaborations where many different people have to use each other's data. These services are being developed in the GriPhyN (Grid Physics Network) project [1], and several are already in use. MPICH-G [22] is a grid-enabled implementation of the Message Passing Interface (MPI)
- 23. 19 that allows a user to run MPI programs across multiple computers at different sites using the same commands that would be used on a parallel computer. MPICH-G extends the Argonne MPICH implementation of MPI to use services provided by the Globus Grid toolkit. 6.6 Applications and Utilities The components managed by application portals are constructed from are applications that perform simulations, data transformations, data analysis, etc. The applications are typically developed independently of the Grid milieu, but must be incorporated into it in order to participate in Grid managed distributed applications. What has to be done in order to accomplish this varies widely with the nature of the application and the framework that is used to coordinate the interoperation of the components. Some of the typical issues are discussed in [5]. There are many Grid based utilities being developed to assist in the use and management of Grid based applications. For example, SSH is a well-known program for doing secure logon to remote hosts over an open network. GSI enabled SSH (GSSH) is a modification of SSH that allows SSH to use Grid certificates and designated proxy certificates for authentication. [23] 6.7 Web Services Building on concepts and technologies from the Grid and Web services communities, the concept of a Web Grid Service is rapidly evolving, Such a service defines standard mechanisms for creating and discovering transient Grid Service instances, provides location transparency and multiple binding protocols for service instances, and supports mapping services for integration with underlying native platform facilities. These services are defined in terms of Web Services Description Language (WSDL) interfaces, and provide the mechanisms required for creating and composing sophisticated distributed systems. These mechanisms include code encapsulation and interface characterization, lifetime management, reliable remote invocation, change management, credential management, and notification. Discovery services, like the Universal Description, Discovery, and Integration (UDDI) service, will provide for locating services based on their functional characteristics, and then provide the detailed descriptions of the data types and interfaces needed to use those services with other services. Web Grid Services implement a service oriented architecture, allowing Grid functionality to be incorporated into a Web services framework that provides for constructing complex application problem solving portals based on frameworks like IBM's WebSphere [24], Microsoft's .NET [25],Apache Tomcat [26], etc. This will allow the integration of science and experimental engineering services with the commercial engineering, manufacturing, planning, cost management, etc., services that are becoming available through major industry commitments to this approach. This is a recent development that is receiving a lot of attention in industry and in the Grid Forum. See, e.g., the Grid Forum's OSGI working group pages, [27], [28], and [25]. For a general introduction to Web Services, see [29]. 6.8 Problem Solving Environments: Knowledge Based Queries, User Interfaces and Workflow Management The Knowledge Grid :It is clear that for the Grid to realize the maximum impact on science
- 24. 20 and engineering that there must be mechanisms for discipline problem solvers to be able to express a problem in terms of the knowledge framework of their discipline, and then have that problem translated to the computational and data analysis operations of the underlying problem solving system. There have been various discipline specific efforts to do this sort of thing, but not much general infrastructure has been developed. The approach of Cannataro, et al [30] suggests one way to approach at least the representation and manipulation of the knowledge base that could translate moderately abstract queries in to sets of computations and data analysis that resolve the query. The User Interface- Integration with the Desktop: A number of services directly support using the Grid by engineers or scientists. These include the toolkits for construction of application frameworks / problem solving environments (PSEs) that integrate Grid services and applications into the "desktop" environment. Services available in the user interface should include, for example, the graphical components ("widgets" / applets) for building application user interfaces; methods for control of the computer mediated; distributed human collaboration that support interface sharing and management; the tools that access the resource discovery and brokering services; the tools that provide generalized workflow management services such as resource scheduling and managing high throughput jobs, etc. This will likely come about from the Web Services frameworks mentioned above. Workflow Management: Reliable operation of large and complex data analysis and simulation tasks requires methods for their description and control. A workflow management system must provide for a rich and flexible description of the analysis processes and their inter-relationships, and also provide mechanisms for fault detection and recovery strategies in widely distributed systems. Workflow management systems will carry out the human defined protocols for, e.g., multi-disciplinary simulations and data analysis; global data cataloguing and replica management systems to manage the data for these scenarios, and global event services to manage the dynamic aspects of work protocols, will be essential adjuncts to the workflow engines. The emerging Web Services Flow Language should assist with this. See [31]. 7 SERVICES FOR OPERABILITY: OPERATIONS AND SYSTEM ADMINISTRATION Implementing a persistent, managed Grid requires tools for deploying and managing the system software. In addition, tools for diagnostic analysis and distributed performance monitoring are required, as are accounting and auditing tools. Operational documentation and procedures are essential to managing the Grid as a robust production service. To operate the Grid as a reliable, production environment is a challenging problem. Some --~-~.z~:,. .................. ! i~iiiiiiiii ,~ ::~ 9, i of the identified issues include management tools for the Grid , .......... ~ , ~ ............9 ~ Information Service; diagnostic tools so operations/systems staff can i'~ ..................=~7~,::1~;~i~i investigate remote problems, and; tools and common interfaces for ............. ~:~........... ~ system and user administration, accounting, auditing and job tracking. I:~ Verification suites, benchmarks, regression analysis tools for 7~~ ........... .............. .~i~!::~=~i:ii;i.~performance, reliability, and system sensitivity testing are essential parts of standard maintenance. Tools and documentation for operating production Grids are being developed at NCSA [32], and in the IPG [12]and DOE Science Grid []3] projects. See [33]for the general issues involved in implementing production Grids.
- 25. 21 8 CURRENT STATE OF GRIDS There are several Grids that are at, or close, to production status. NASA's Information Power Grid (www.ipg.nasa.gov), the DOE Science Grid (www.doesciencegrid.org) and the UK eScience Grid (www.nesc.ac.uk/intro) are all working toward production Grid environments. IPG has, over the past three years, deployed a prototype production Grid. By production, we mean that the services and resources are persistent, there are operational groups responsible for those services and resources, and there is documentation and user support. In the process of building this Grid environment a great deal is being learned about integrating Grids into production supercomputing environments, and some of the issues and lessons learned are documented in [33]and [34]. 9 ACKNOWLEDGEMENTS IPG is funded primarily by NASA's Aero-Space Enterprise, Computing, Information, and Communication Technologies (CICT) Program (formerly the Information Technology), Computing, Networking, and Information Systems Project. DOE's Science Grid is funded by the U.S. Dept. of Energy, Office of Science, Office of Advanced Scientific Computing Research, Mathematical, Information, and Computational Sciences Division (http://www.sc.doe.gov/production/octr/mics) under contract DE-AC03-76SF00098 with the University of California. 10 REFERENCES [1] [2] [3] [4] [5] GriPhyN - Grid Physics Network, GriPhyN. http://www.m'iphvn.org Network for Earthquake Engineering Simulation Grid (NEESgrid). http:#www.neesgrid.org/ Virtual Observatories of the Future, Caltech. http://www.astro.caltech.edu/nvoeonf/ Numerical Propulsion System Simulation, NPSS. http://hpee.lerc.nasa.gov/npssintro.shtrn! A CORBA-based Development Environment for Wrapping and Coupling Legacy Codes, G. Follen, C. Kim, I. Lopez, J. Sang and S. Townsend. In Tenth IEEE International Symposium on High Performance Distributed Computing. 2001. San Francisco.http://cnis.grc.nasa.gov/papers/hpdc-10_corbawrag_ping.pdf [6] Role of Computational and Data Grids in Large-Scale Science and Engineering, W. Johnston. Int. Journal Of High Performance Computing Applications, 2001.15(3). http://www.itg.lbl.gov/~johnston/Grids/homepage.html#JHPCA2001 [7] The Globus Project, Globus. http://www.globus.org [8] A Resource Management Architecture for Metacomputing Systems, K. Czajkowski, I. Foster, N. Karonis, C. Kesselman, S. Martin, W. Smith and S. Tuecke, in The 4th Workshop on Job Scheduling Strategies for Parallel Processing. 1998.p. 62--82. http://www.globus.org/research/papers.html#GRAM97 [9] Scheduling and Resource Management Area, Global Grid Forum. http://www.gridforum.org/3_SRM/SRM.htm [10] ESnet's SciDAC PKI & Directory Project - Homepage, T. Genovese and M. Helm. DOE Energy Sciences Network. http://envisage.es.net/ [11] Design and Deployment of a National-Scale Authentication Infrastructure, R. Butler, D. Engert, I. Foster, C. Kesselman, S. Tuecke, J. Volmer and V. Welch. IEEE Computer, 2000. 33(12): p. 60-66. http://www.globus.org/research/papers.html#GSll [12] NASA's Information Power Grid, IPG. http://www.ipg.nasa.gov
- 26. 22 [13] DOE Science Grid. http://www.doescienc~grid.org [14] Distance Computing and Distributed Computing (DisCom2) Program, DISCOM. http://www.cs.sandia.gov/discom [15] NSF PACI Program, PACI. http://www.ncsa.uiuc.edu/About/PACl/ [16] Grid Information Services for Distributed Resource Sharing, K. Czajkowski, S. Fitzgerald, I. Foster and C. Kesselman. In HighPerformanceDistributed Computing- 10. 2001. San Francisco, CA.http://www.globus.org/research/papers.htm!#MDS-HPDC [17] GridFTP Update January 2002, W. Allcock, J. Bresnahan, I. Foster, L. Liming, J. Link and P. Plaszczac. http://www.globus.org/research/papers.html#GridFTP-Update-Jan2001 [18] The NetLogger Methodology for High Performance Distributed Systems Performance Analysis, B. Tiemey, W. Johnston, B. Crowley, G. Hoo, C. Brooks and D. Gunter. In Proc. 7th IEEESymp. on High PerformanceDistributed Computing. 1998.http://www-didc.lbl.gov/NetLol~er/ [19] Forecasting Network Performance to Support Dynamic Scheduling Using the Network Weather Service, R. Wolski, in Proc. 6th IEEESymp. on HighPerformanceDistributed Computing. 1997:Portland, Oregon. [20] Grid Monitoring Architecture Paper, Grid Performance Working Group. Global Grid Forum. http://www- didc.lbl.gov/GridPerf [21] Condor-G, J. Frey, T. Tannenbaum, M. Livny, I. Foster and S. Tuecke. In Proceedingsof the Tenth InternationalSymposiumon HighPerformanceDistributed Computing(HPDC-IO).2001: IEEE Press.http://www.globus.or~research/papers.html#Condor-G-HPDC [22] A Grid-Enabled MPI: Message Passing in Heterogeneous Distributed Computing Systems, I. Foster and N. Karonis. In SC98. 1998.http://www.globus.org/research/Dat~ers.html#mpich98 [23] Using GSI Enabled SSH, NCSA. http://www.ncsa.uiue.edu~serlnfo/Alliance/GridSecurity/GSl/Tools/GSSH.html [24] Web Services, IBM. http://www-106.ibm.com/developerworks/webservices/ [25] .NET, Microsoft. http://msdn.microsot~.com/net [26] Jakarta Tomcat, Apache. http://jakarta.apaehe.org/tomeat/index.html [27] Towards Globus Toolkit 3.0: Open Grid Services Architecture, Globus Project. http://www.globus.org/ogsa/ [28] IBM, Globus announce open grid services, IBM. 2002. http://www.ibm.com/news/us/2002/02/202.html [29] BuildingWebServiceswithJava: MakingSense ofXML, SOAP, WSDL,and UDDI, S. Graham, S. Simeonov, T. Boubez, D. Davis, G. Daniels, Y. Nakamura and R. Neyama. 2002: Sams Publishing Co. [30] The Knowledge Grid: Towards an Architecture for Knowledge Discovery on the Grid, M. Cannataro, D. Talia and P. Trunfio. In FIRST EUROGLOBUS WORKSHOP.2001. Robinson Club Apulia Village, Marina di Ugento, Lecce.http://www.euroglobus.unile.it/ [31] Web Services Flow Language (WSFL), R. Cover. http://xml.coveroa~es.or~wsfl.html [32] Alliance Computational Environments & Security, NCSA. 2001.http://www.ncsa.uiuc.edu/TechFocus/Proj cots/ [33] Implementing Production Grids, W. Johnston. http://www.itg.lbl.gov/~johnston/Grids/homet~a~e.html#1mNement [34] NASA's Information Power Grid: Production Grid Experience with Distributed Computing and Data Management, W. Johnston. In Second Global GridForum Workshop(GGF2).2001. Washington, D.C.http://www.itg.lbl,gov/~j ohnston/Grids
- 27. Parallel Computational Fluid Dynamics - New Frontiers and Multi-Disciplinary Applications K. Matsuno, A. Ecer, J. Periaux, N. Satofuka and P. Fox (Editors) 9 2003 Elsevier Science B.V. All rights reserved. 23 A Different Approach to Large-Eddy Simulation with Advantages for Com- puting Turbulence-Chemical Kinetics Interactions J. M. McDonough Departments of Mechanical Engineering and Mathematics University of Kentucky Lexington, KY 40506-0503, USA E-mail: jmmcd@uky.edu We present an alternative to usual large-eddy simulation based on filtering the solution and modeling physical variables on the sub-grid scales. We focus on behavior of these subgrid-scale models and show, at least qualitatively, that they are capable of model- ing interactions between hydrodynamic turbulence and chemical kinetics. Moreover, the overall approach exhibits potential for parallelization at several levels. 1. INTRODUCTION It is widely accepted that direct numerical simulation (DNS) would be the preferred approach for essentially every analysis of turbulent reacting systems, but even with par- allelization this will be viable only as a research tool in the immediate future. Large-eddy simulation (LES) in some form presents a feasible direction for near-term calculations; but although much progress in LES has been made in recent years, it is still far from being a completely reliable tool in the context of turbulent combustion---especially if any but the simplest kinetic mechanisms are used. Much recent research in LES has focused on the subgrid-scale (SGS) models, and these are especially problematic in the context of finite-rate chemistry. Various approaches have been implemented ranging from laminar flamelets, Cook et el. [1] through PDF models, e.g., Pope [2] and including extension of scale similarity ideas to reactive scalars as in Germano et al. [3]. Giacomazzi et al. [4] present an approach that resembles, in some respects, the treatment to be presented here. In addition, the linear eddy models (LEMs) of Kerstein and co-workers (see e.g., Kerstein [5] and Echekki et el. [6]), and more recently the one-dimensional turbulence (ODT) models of the same group of investigators, incorporate some of the ideas to be used below, but differ considerably in detail. While much additional important work is available in the literature, for the sake of brevity we will avoid an exhaustive review. Our approach, while retaining the basic LES decomposition of dependent variables, differs from usual LES in three distinct ways: i) dependent variables are filtered rather than the governing equations; ii) physical variables are modeled on the sub-grid scales instead of their statistics; iii) modeled quantities are directly used to enhance resolved-
- 28. 24 scale solutions, rather than being discarded. Thus, if we let Q - (Q1,Q2,..., QN,) T denote the solution vector consisting of Nv components, we decompose this as t) = t) + t) x e R d, d=2,3, (1) and substitute into the transport equation(s) for Q to obtain (q + q*)t + V . F(q + q*) = V . G(q + q*) + S(q + q*) . (2) Here, the subscript t denotes partial differentiation with respect to time, and V- is the divergence operator. F and G are, respectively, advective and diffusive fluxes, and S is a general nonlinear source term. In contrast to usual LES, we do not filter this equation. Instead, we model q* to produce "synthetic velocity" and "synthetic scalars" of the general form q* = A~Mi , i = 1, 2,..., ?iv. (3) These are substituted into Eqs. (2) which are then solved for q. The results are filtered, if necessary, to remove aliasing due to under resolution, and then the complete solution at the current time step is obtained from Eq. (1). In Eqs. (3) the Ais are amplitudes derived from scaling laws of Kolmogorov (see, e.g., Frisch [7]) using scale similarity to estimate certain sub-grid variables of the Kol- mogorov formalism. Details of these constructions will be outlined below. The M~s are discrete dynamical systems (DDSs) representing temporal fluctuations of the SGS quan- tities. We have recently introduced a procedure (described below) for obtaining the Mis directly from the governing equations, and we have employed this in several recent studies, the two most closely related to the present work being McDonough and Huang [8] and McDonough and Zhang [9]. The purpose of the present paper is to outline the overall procedure for constructing synthetic-variable SGS models but with particular emphasis placed on the behavior of the DDSs. We will consider a realistic kinetic mechanism for H2-air combustion, and provide comparisons of the temporal fluctuations produced by the corresponding DDS with the experimental results Meier et al. [10]. 2. ANALYSIS In this section we introduce the governing equations and provide a preliminary demon- stration of the effectiveness of filtering solutions instead of equations. We then derive a general discrete dynamical system that can be used to model any desired kinetics and simplify this to a form corresponding to a particular reduced mechanism for H2-air com- bustion. Finally, we briefly describe calculation of the amplitudes in Eqs. (3). 2.1 Governing Equations The general equations describing fluid flow, heat transfer and chemical reactions are well known and can be found in any standard reference, e.g., Libby and Williams [11].
- 29. 25 We present them here in the following form. pt + v. (pU) = o, DU = -Vp + v. (#vu) + og, P Dt DT N, (py~ N, i=l i=1 D(pYi) = V. (pDiVYi) +@i, i = 1,...,]Vs. Dt (4a) (4b) (4~) (4d) Here, N~ j=l (5) with t,j ~j = kl,jH w~] 1=1 - k~'JH w~] ~=1 V II l,j (6) In these equations notation is standard, and we refer the reader to Ref. [11] for details. 2.2 Solution Filtering It is worthwhile at this point to briefly return to the first of the three differences between the approach being described here and usual LES, viz., filtering solutions instead of the governing equations. The consequence of not filtering the equations is easily seen and impacts two specific areas associated with the solution process. First, filtering of nonlinear terms leads to the SGS stresses and scalar fluxes that require statistical models. This presents especially serious difficulties with regard to the reaction rate formulas. Second, in generalized coordinates the filtering process does not exactly commute with the differential operators, so extra errors arise even in approximating linear terms. In the approach being considered here, neither of these problems occurs. It should be fairly obvious that filtering solutions is little more than a signal-processing problem, so the main difficulty is selecting an appropriate filter. We have indicated above that the need for filtering arises from under resolution (mainly spatial) imposed by the LES method itself; so a low-pass filter capable of removing aliasing effects is crucial. We have employed a fairly simple one: k+2 (7) for a one-dimensional problem of filtering a typical grid function {fi}iN1 for a
- 30. 26 data set containing N points; the extensions to 2D and 3D are fairly obvious. Figure 1 provides a comparison between using this filter and employing a usual LES procedure with a Smagorinsky model for solving Burgers' equation forced with a complicated quasiperiodic func- tion so as to yield a known an- alytical solution (see Yang and McDonough [12]). It is easily seen from this figure that filter- ing the solution produces results that are generally as good as those produced by the full LES procedure, but we still have the ability to improve this solution further by adding the SGS syn- thetic velocity. 2 -.j o 2 0 -.2 -4 0.0 0.2 Q4 O~ O.B Sc~iKILm~hI~ ilr~'Unil~ 1.0 Figure 1: Solution filtering compared with equa- tion filtering: (a) usual LES, (b) solution filtering. 2.3 The Discrete Dynamical System The approach we employ for modeling the M~s in Eq. (3) was first introduced in [8]. It provides a systematic technique for deriving DDSs that are directly related to the partial differential equations (PDEs) they are to model. The premise employed to start the procedure is that all solution variables possess Fourier series representations constructed from a complete orthonormal basis that "behaves like" complex exponentials with regard to differentiation. We substitute these into the governing equations (4), Construct the Galerkin ordinary differential equations and decimate the result to a single mode per original PDE. Then we use very simple numerical integrators (forward or backward Euler methods) to perform the temporal discretization and obtain a('~+1) --/~ua (n) (1 - a(n)) - %a(n)b (n), (8a) b (n+l) : ~v b(n) (1 - b(')) - %a(~)b(~) + aTC (n), (8b) c(n+l) = a .~(,~+1) T~i -- %Ta (~+~) -- %Tb(~+~) c(~) -- ~ H~Lbi i=l 9 i=1 /(1 +/~T) + Co, (8C) d~ n+l) -- - (~]Q"~-"fu~a(~+1) + "/v~b(n+m))d~n) + &i + di,o, i = 1, 2,..., Ns, (8d)
- 31. 27 with ~oi = ~.= Cl,ij t=l d 't-Cb,ijt=ld * " Here, superscripts (n) denote time step index; a, b, c and the dis can be viewed as Fourier coefficients of the two velocity components, temperature and the species concentrations, respectively; the subscripted as, /3s, etc., are bifurcation parameters of the DDS, all of which are related to the various physical bifurcation parameters. For example, flu and /3v are (the same) functions of the flow Reynolds number; aT is related to the Rayleigh or Grashof number, and the various 7s correspond to velocity, temperature and species concentration gradients. For example, % ~ uv, "/uT ,.o Tx, etc. 2.4 Reduced Mechanism for H2-Air Combustion In this subsection we present the reduced kinetic mechanism and specialize Eqs. (8) to this case. We employ a nine-step mechanism for the H2-air reaction consisting of the following: H2 + 02 --+ H02 + H 02+H~OH+O H2+O-+OH+H H02 + H --+ OH + OH OH+M --+ H+O+M OH+O--+ 02 +H OH + H + M -+ H20 + M H2 + OH --+ H20 + H H20 + H --+ H2 + OH (9a) (9b) (9c) (9d) (9e) (9f) (9g) (9h) (9i) These reactions are ordered so as to allow at least partial modeling of a reaction pathway by requiring any intermediate species to already be present before it can be used in subsequent elementary reactions. To construct the DDS corresponding to this reduced mechanism we derive an iterated map for each product appearing in each elementary reaction. Each iterated map is of the general form of Eq. (Sd) and the formula for cbi following it. But now in this latter expression Nr = 1, and backward reactions are treated separately. Thus, the individual maps are quite simple. To fix notation we make the following identifications: dl ~ H2, d2 ~-' 02, da ~-' H20, d4 ~ OH, d5 N H, d6 N O, d7 ~ HO2, ds ~ N2. (10) We now carry out the details for the initiation reaction Eq. (9a). We first observe that the reaction (9a) yields two products and thus, two iterated maps. The first product in
- 32. 28 reaction (9a) is HO2 so the corresponding DDS is d~n+l) -" - (~Y7 + "YuY7a(n+l) + ~vY~b(n+l)) d~n) -~ w7 + d7,0, with (11) ,, W7 &7 = CI,7,1dld2 Cf,7,1 = v7,1WIW2kf,1. We note that introduction of the species molecular weights arises from the form of Eqs. (5,6). The second product of reaction (9a) and the remaining reactions can be treated in an analogous way, leading to a total of 18 iterated maps in the complete DDS for the chemical reactions. One of the features of the present approach is its ability to directly account for differ- ent reaction rates of the individual elementary reactions comprising a kinetic mechanism and relate this to the time scale of the hydrodynamic turbulence. This is done by calcu- lating the Kolmogorov-scale DamkShler number Dag for each elementary reaction using its specific reaction rate to obtain a chemical time scale, and employing an input hydro- dynamic turbulence time scale. Then for each velocity time step a number of iterations proportional to Dag is performed for each reaction. 2.5 Amplitude Factors. The amplitude factors of Eqs. (3) are constructed for each ith solution component using second-order structure functions (see Ref. [7]) and a generalization (to nonhomogeneous, anisotropic cases) of Kolmogorov's inertial range energy spectrum. That is, we determine the subgrid-scale energy for the k th Fourier mode of the ith variable from Ei(k) - C2,i<~)~'k -(~'+1) , (12) where <c> is a spatially-averaged turbulence kinetic energy dissipation rate obtained from resolved-scale results via scale similarity, and the C2,i and ~i are obtained from local (in time and space) structure function correlations. More details can be found in Mc- Donough [13]. The A~ are calculated as the sum of the Ei(k) over a few wavenumbers k in the inertial sub- range, and possibly into the dissipation scales. Figure 2 provides an indication of the potential of this approach; it displays an instantaneous snapshot of a turbu- lent pool fire with oscillations modeled with Eq. (3) and applied to experimen- tal mean data of Weckman and Strong [14]. Figure 2: Effect of amplitude fac- tors: instantaneous temperature fluctua- tions about measured average.
- 33. 29 3. RESULTS AND DISCUSSION In this section we present results of running the DDSs discussed above. We begin by describing the problem setup; we then present computed results and compare these with a portion of the experimental co-flow data, denoted H3, from reference [10]. We will, for the sake of brevity, consider only one location within the non-premixed 50% N2-diluted H2-air flame of this experiment. This corresponds to an axial location of x/D - 20 with D -- 8 mm being the H2/N2 jet exit diameter; the measurement location is at a radial distance of 10.5 mm from the jet centerline. 3.1 DDS Model Problem Setup The full system of Eqs. (8) has been coded in Fortran 90 and for the present study was specialized to equations of the form (11) corresponding to the nine-step mechanism (9). The velocity time scale used in the code to set the Damkbhler number was equated to the data sampling frequency of the H3 experiment (10 kHz), and 104 velocity time steps were computed using Eqs. (8a,b). During each such time step a number (>_ 1) of chemical time steps was computed for each reaction with the number being calculated during each velocity time step using local in time temperature results to set the reaction rate, and thus Dag, for each reaction. Following calculation of all fluctuating species, the fluctuating temperature is updated via Eq. (8c). 2400 . . . . . . $ 2000 1600 ~ 1200 800 0.03 (b) . . . . I O0 o18 I-(c). . . . I I. 0.1 0.06 0.02 1 0.16 '~' 0.12 s o.o8 0.04 0 0.01 0.02 0.03 Time Figure 3: Experimental data, temperature and mass fractions; (a) temperature, K, (b) H2, (c)02, (d)H20. 2200 $ 1800 .,,., 1400 ~. 1000 0.006 0.0O2 0.O8 ,-.-, 0.06 O~ 0.04 0.02 0.14 s o.13 0.12 0 0.01 0.02 0.03 Time Figure 4: DDS model results; (a) through (d), same as in Fig. 3.
- 34. 30 3.2 Computed Results~ Comparisons with Data Figures 3 (experimental, Ref. [10]) and 4 (computational, present model) provide a qualitative comparison of our results with those from a well-established data base that was selected as a "standard flame" at the International Workshop on Measurements and Computation of Turbulent Nonpremixed Flames, Naples, 1996. We first observe that the qualitative appearance is very similar between corresponding parts of these figures. This demonstrates the general ability of the DDS model to provide physically realistic turbulent fluctuations in a nontrivial situation. It is important to recognize that the results displayed here come from a single execution of the DDS--they are not the result of individual runs for each species and temperature. Moreover, additional species mass fractions for which there are no data were calculated at the same time but are not presented here. It should also be noted that the differences in amplitudes seen in these two sets of figures are inconsequential. In complete LES im- plementations the DDS results would be rescaled to unity amplitude and then multiplied by the local (in space and time) amplitude factors described above. But no spatial data needed to construct these amplification factors were available, so the DDS results could not be properly scaled. Thus, it is only their temporal behavior that should be compared with the experimental results, and we see that this comparison is quite satisfactory. 3.3 Parallelizatlon Be have earlier alluded to the fact that the form of LES we have presented has potential for significant parallelization. We have not yet completed the parallel implementation at this time, but the work is in its late stages. Here we will briefly describe the opportunities for parallel coding with such an algorithm. First, the large and small scales of the LES procedure can be done in parallel because although each depends on the other, the specific details are not extremely important; in particular, it is possible to employ time step n large-scale information to compute time-level n § 1 small-scale results at exactly the same time as the large- (resolved-) scale results are being calculated because actual turbulent solutions are so nonsmooth that high-order accuracy in time cannot be achieved under any conditions. Computing the two scales in parallel results in only first-order temporal accuracy but, in fact, this is all that can actually be obtained in any case. The second parallelization opportunity is the usual domain decomposition one, and it can be equally-well implemented at both scales of the computations. Finally, within each domain it is possible to parallelize (or, maybe just vectorize) the specific numerical analytic algorithms. Thus, in all, there are three possible levels of parallelization to be exploited with an algorithm of the type we have presented. 5. CONCLUSIONS In this paper we have presented an alternative approach to large-eddy simulation based on unfiltered equations of motion and discrete dynamical systems SGS models. We have noted the advantages of this for modeling interactions of hydrodynamic turbulence with other physical fluctuations, and we have demonstrated the effectiveness of the individual pieces of the overall procedure by comparing them with standard LES in the case of
- 35. 31 the filtering approach used, and with experimental data in the case of the SGS models. Both comparisons showed very satisfactory agreement between the new approach and the data employed. We have also noted that the form of algorithm being proposed is highly parallelizable with a hierarchy of at least three levels of parallelization possible. ACKNOWLEDGEMENTS The author gratefully acknowledges the support for these studies provided in part by AFOSR Grant #F49620-00-1-0258 and NASA/EPSCoR Grant #WKU 522635-00-10. In addition, graduate students Ms. Sha Zhang (now a student at North Carolina State) and Ms. Ying Xu performed all of the calculations and constructed much of the graphics reported herein. REFERENCES 10. 11. 1. A. W. Cook, J. J. Riley and G. Koss "A laminar flamelet approach to subgrid- scale chemistry in turbulent flows," Combust. Flame 109, 332-341 (1997). 2. S. B. Pope, "Pdf methods for turbulent reactive flows," Prog. Energy Combust. Sci. 11, 119-192 (1985). 3. M. Germano, A. Maffio, S. Sello and G. Mariotti, "On the extension of the dy- namic modeling procedure to turbulent reacting flows," in Direct and Large Eddy Simulation II (Collet et al., eds), Kluwer Academic Publishers, Amsterdam, 1997. 4. E. Giacomazzi, C. Bruno and B. Favini, "Fractal model of turbulent combustion," Combust. Theory Modelling 4, 391-412 (2000). 5. A. R. Kerstein, "Linear-Eddy Modeling of Turbulent Transport. Part 4. Structure of Diffusion Flames," Comb. Sci. and Tech. 81, 57-96 (1992). 6. T. Echekki, A. R. Kerstein and T. D. Dreeben, "'One-Dimensional Turbulence' Simulation of Turbulent Jet Diffusion Flames: Model Formulation and Illustrative Applications," Combust. and Flame 125, 1083-1105 (2001). 7. U. Frisch, TURBULENCE The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995. 8. J. M. McDonough and M. T. Huang, "A low-dimensional model of turbulence- chemical kinetics interactions," paper ISSM3-E8 in Proceedings of Third Int. Syrup. on Scale Modeling, Nagoya, Japan, Sept. 10-13, 2000. 9. J. M. McDonough and Sha Zhang, "LES subgrid-scale models of turbulence-chemical kinetics interactions based on discrete dynamical systems," AIAA Paper 2002-3172, presented at 32na AIAA Fluid Dynamics Conference, St. Louis, June 24-27, 2002. W. Meier, S. Prucker, M.-H. Cao and W. Stricker, Combust. Sci. Technol. 118, 293, 1996. P. A. Libby and F. A. Williams, Turbulent Reacting Flows, Academic Press, London, 1994.
- 36. 32 12. T. Yang and J. M. McDonough, "Solution filtering technique for solving Burgers' equation," submitted for special issue of J. Discrete Cont. Dyn. Sys., (2002). 13. J. M. McDonough, "A 'Synthetic Scalar' Subgrid-Scale Model for Large-Eddy Sim- ulation of Turbulent Combustion," paper c3.3, Proc. of 2002 Tech. Mtg. Central States Sea, the Combust. Inst., 2002. 14. E. J. Weckman and A. B. Strong, "Experimental investigation of the turbulent structure of medium-scale pool fires," Combust. Flame 105, 245-266, (1996).
- 37. ParallelComputationalFluidDynamics-NewFrontiersandMulti-DisciplinaryApplications K.Matsuno,A.Ecer,J. Periaux,N. SatofukaandP. Fox(Editors) 9 2003ElsevierScienceB.V.Allrightsreserved. 33 Simulation of Combustion Dynamics in Gas Turbine Engines S. Menon~* aSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA The ability to predict turbulence-chemistry interactions in realistic full-scale gas turbine combustors has not been feasible till now due to the lack of simulation models and com- puting power. Here, a parallel large-eddy simulation (LES) model for both premixed and spray combustion has been developed and utilized to investigate combustion dynamics in realistic gas turbine combustors. The parallel performance of this code in various systems is reported. The effects of swirl intensity and heat release are investigated. Results show that increase in swirl has dramatic effect on the flame structure and reduces the pressure oscillation amplitude in premixed systems. In spray combustion, increase in swirl dras- tically increases the droplet dispersion. In all situations, the central recirculation zone (which is a manifestation of the vortex breakdown process), occurs only under high swirl conditions. Combustion and heat release tends to reduce the size of this recirculation zone, due to an increase in flow acceleration associated with heat release. Finally, a new LES method based on the Lattice Boltzmann model is used to simulate complex flows such as jet in cross flow. Key Words Reactive Flows, Turbulence, Spray Combustion, Unsteady Flows, Lattice Boltzmann Methods, Domain Decomposition 1. INTRODUCTION For next generation combustion systems stringent emission constraints on gas (e.g., CO, NO, and unburned hydrocarbons) and solid particulates (e.g., soot) pollutants are likely to become major design criteria. For land based power generation gas turbines it appears that lean premixed combustion may be able to meet most, if not all of these emission constraints. However, design and deployment of such systems will have to deal with the consequences of lean combustion, including flame instability. Flame instability manifests itself when the low-frequency pressure (acoustic) oscillation in the combustor suddenly grows into large-amplitude non-linear waves that can cause flame extinction and under certain conditions, even structural damage. Combustion instability can also occur in spray combustion gas turbines that are used in nearly all propulsion systems (aircraft, helicopters, tanks, submarines, etc.). Therefore, in order to avoid this phenomenon in *This work is supported in part by US Army Research Office,General Electric PowerSystems
- 38. 34 production gas turbines (both power and propulsion versions), combustion always occurs far away from the lean limit by design. On the other hand, if stable combustion can be achieved in the lean limit then not only emission can be reduced drastically, but it may also result in significant reduction in fuel consumption. This goal is now of particular relevance due to the increasing environmental concerns of fossil fuel combustion. Numerical prediction of highly swirling turbulent reacting flows is very difficult since conventional time-averaged methods cannot predict these flows accurately. Fortunately, a technique based on large-eddy simulations (LES) has the ability to capture the physics of these flows. In LES, all scales of motion larger than the grid resolution are resolved by the numerical scheme (that is of high temporal and spatial accuracy) and the scales of motion smaller than the grid size are represented using subgrid models. The accuracy of the LES depends not only on the resolution of the resolved scales but also on the models used to represent the unresolved small-scales. The ability of advanced subgrid models to simulate these complex flows using relatively coarse grids is discussed using examples of LES of full-scale gas turbine engine combustor flows. Application of the LES to both single and two-phase (spray) mixing and combustion is reported. 2. SIMULATION MODEL In LES, only scales larger than the cut-off size are computed and the rest are modeled. Therefore, computations are still tractable on massively parallel systems. Unresolved terms at the sub-grid level arise in the filtered momentum and scalar equations. A local- ized dynamic model for the subgrid kinetic energy is used to define an eddy viscosity that is then used for the subgrid stress closure[l]. For premixed combustion, a thin-flame LES approach [1,2] and new subgrid based combustion approach [3,4] have been developed. Details are avoided here for brevity. The numerical solver solves the Favre-filtered, unsteady, compressible, Navier-Stokes equations along with the Lagrangian equations for the liquid phase [5]. The scheme is fourth-order accurate in space and second order in time. Full two-way coupling is employed for the two-phase spray studies. Although currently only dilute spray can be simulated with this code, spray breakup and collision models are currently being included. 3. PARALLEL IMPLEMENTATION The governing LES equations are discretized using a finite-volume formulation are in- tegrated over a discrete control-volumes. A distributed memory parallel model is chosen whereby the computational grid is partitioned and distributed evenly among the avail- able processors. To make the LES algorithm portable, the standardized Message-Passing Interface (MPI) protocol is used for parallel communication. The amount of data transferred is dependent on the size of the finite-volume stencil used to solve the governing equations. For the overall central, fourth-order scheme used here, the stencil size is +/- three cells about the center cell. Therefore, each processor must exchange three layers of computational cells to their respective neighbors. Since communication is generally much more expensive than computation, only the flux vari- ables are exchanged and the primitive variables are calculated. As a result, the parallel overhead is a combination of communication and computation.
- 39. 35 Table 1 Parallel computing hardware comparison. Owl, Jedi and Raptor are Intel-based PC clusters, Seymour and Habu are specialized computing platforms. Name Total # CPU's / CPU Speed Communication (Platform) of CPU's SMP Node (MHz) Fabric Owl (ia32) 16 4 500 Fast-Ethernet Raptor (ia32) 32 2 733 Fast-Ethernet & Giganet-cLAN Jedi (ia32) 138 8 550 Gigabit-Ethernet Habu (IBM SP3) 1336 4 375 SP Switch Fabric Seymour (Cray T3E) 1024 1 450 3D folded torus 4. RESULTS AND DISCUSSION 4.1. Parallel Performance Analysis Three clusters in 2-, 4-, and 8-way SMP configurations using the Intel Pentium III Xeon CPU are evaluated along with supercomputers. The system configurations and identification are listed in Table 1. All PC clusters are running the Linux operating system (2.4.x kernels) and use the Portland Group Fortran90 compiler. MPICH, an open-source MPI implementation de- veloped is used for all TCP/IP parallel communication. MPI/Pro, a commercial MPI implementation from MPI Software Technologies, Inc. is used for all Virtual Interface (VI) based communications (in Raptor). The LESLIE3D code, a turbulent, temporal mixing layer simulation model with two non-reacting chemical species, is used for a "realistic" benchmark. This simulation models the time-evolution of two distinct fluids moving parallel to each other at different velocities [6]). This situation is often encountered in combustion systems and is often used as an idealized model for fuel-oxidizer mixing studies. Physically, the configuration is a cubic box (with length of 27~) with periodic boundary conditions in two of the three directions (X, Z). For a large number of processors (neglecting those along the Y-boundaries), the inter-process communication is nearly isotropic, i.e., all processors have 6 neighbors. Grid densities of 643 and 1283 were used to give a wider range of applicability. For example, 1283 (over two million cells) represents represents a resonable grid for a full scale gas turbine combustor simulation. The timing results are shown in Fig. l(a). All timings are in total-time per time-step (seconds). For reference, Seymour achieves 76 Mflops per node (1216 Mflops aggregate) on 16 processors with the 1283 grid. We see that Habu surpasses the competition by a substantial margin with Raptor (running VI) coming in second. As we progress toward 8 and 12 processes, the effects of contention begin to show themselves. At saturation (16 processes), Owl is actually slower than at 8 processes for both grids. The parallel overhead is shown in Fig. l(b). For the Cray T3E, the overhead remains below 20% for both grids which allows for efficient scaling. We can see for the TCP/IP communication, the overhead reaches upward of 50% and 40% for the small and large
- 40. 36 50.o 45.01 4O.0 35.0 I ~30.0 25.0 ~ 20.0 15.0 I0.0 5.0 0.0 2 ..... , ,~ i ' i ' i ' ' [] Owl O Raptor(Fast-Ethernet) O Raptor(Vl-Giganet) g:~ Jedi 9 IBM SMP Power3 A - , . ~ , . J . , i , . I , , 8 14 20 26 32 Number of Processors 40 [] Owl I I " " I " " | " A I r RaP~~ I ~ 1 0 Rapt~ (Vl'GiganeO I ~ "] I~ J~i 2 8 14 20 26 32 Number of Processors Figure 1. Performance profiles for LESLIE3D on 1283 grid. (a) Wall-clock time per step and (b) communication overhead (% of total time). grid, respectively (on Raptor). When communicating over the dedicated cLAN switch, Raptor incurs an overhead on the same order as Seymour and even superior to Habu as the number of processors is increased. The parallel efficiencies (scalability) for these simulations are reported in Fig. 2. Owl, as indicted earlier, exhibits zero or even negative scalability for both grids with half the system is used. Jedi, even with a high-speed communication fabric, also scales poorly though (at least) remains positive throughout the spectrum. In fact, Jedi scales worse than Raptor when using TCP/IP over Fast-Ethernet. As could be expected, Seymour is able to scale at over 90% efficiency (for the large grid). For the 1283 grid, Raptor's efficiency (with VI) is comparable or even surpasses that of the two comparison systems. For LES of realistic gas turbine combustion process, the computational time can be quite large, e.g., approximately 15,000 CPU hours on Habu using one million grid points and an efficient combustion model to obtain sufficient data for statistical analysis [2]. The scalability of LESLIE3D on very large-scale computational grids were also evaluated on an IBM eower3-SMP (222 MHz) and an SGI Origin 3800 (400 MHz). The results for a 2163 grid (10+ million grid points) are shown in Fig. 2b. Even at over 300 CPU's, the algorithm is able to scale at 83% efficiency (based on the 64 CPU time). Based on the results for the Raptor PC-cluster which scaled superior to the IBM system, it is possible to estimate that the same calculation could be completed in a feasible amount of time on the high-performance (yet commodity) PC-cluster system. 4.2. Premixed Combustion in Gas Turbine Engines Premixed combustion in a full-scale gas turbine combustor (GE LM6000) have been studied extensively [1,2]. Some characteristic features that were captured in LES (but not in steady-state calculations) are reported here (more details are in the cited literature). For premixed combustion, when swirl is below a critical value, the flow entering the combustor behaves more like a free jet and forms circular vortex rings (similar to smoke
- 41. 37 ' l ' ! ' I ' i ' i ' i ' 1 ' t " Owl t t - O"O Raptor(Vl.-GiganeO /t/ ./ - ~ Io ~,o,o.~o,~0~, I --~t' [ 9 ~ 12 2 8 , I i Number of Processors Number of Processes Figure 2. (a) Scalability of LESLIE3D for 1283 grid. Dashed line is linear (ideal) speed-up. (b) Large-scale computational test on 216a grid on up to 384 processors. Solid and dashed lines represent total wall-clock time (secs) per time-step and communication overhead (% of total time). rings) that undergo a twisting shear motion leading to their eventual breakdown. The vortex shedding also forces the flame to pulsate in-phase. When swirl is increased beyond a critical value, the high swirling motion of the fuel-air mixture and the rapid expansion at the entrance of the combustor causes an adverse axial pressure gradient along the centerline. This pressure gradient causes rapid breakdown of the vortex rings, slows down the axial motion and forms a recirculating bubble near the centerline. The coupling between the vortex motion and flame is broken, and the flame is pushed radially outwards and also upstream finally stabilizing very close to the inlet. For a fixed incoming fuel-air mixture (fixed equivalence ratio), increase in swirl therefore, stabilizes the combustion process. Decreasing the equivalence ratio (i.e., in lean mixture) for a fixed Swirl Number, increases the pressure oscillation amplitude. This phenomenon is precursor to combustion instability. The current effort is now on adaptive control of the instability whereby, the pressure signal is recorded and analyzed on-line and then used to change the incoming swirl of the fuel content to stabilize combustion when the fuel-air mixture is made leaner. These results will be reported elsewhere. The formation of the Vortex-Breakdown bubble along the centerline has a dramatic effect on the flame length, as shown in Fig. 3 where the time averaged flame surface for low and high swirl, respectively, are shown. The connected flow in low swirl ease results in a long, pointed flame similar to that observed for a jet flame. The more compact flame structure observed in high swirl is a direct result of Vortex-Breakdown effects which anchors the flame upstream of the leading stagnation point. The pressure response to variation in swirl number is shown in Table 2. Strong at- tenuation in p' occurs during the transition from jet-like to re-circulating (i.e., Vortex- Breakdown) flow. When V B occurs, the flame is held stationary and does not pulse
- 42. i 38 Figure 3. Mean flame surface for low and high swirl (a) Low swirl (b) High swirl. Flow direction is from top right to bottom left. Flame base is at dump plane. Table 2 Simulation inflow and equivalence ratio conditions and resulting RMS pressure and Sound Pressure level (relative to Case I). The high laminar flame speeds, S~, are a result of the elevated operating pressure and preheat fuel-air mixture. Case Inlet Swirl p' SPL ID (Si) (% P) (dB) 1 0.56 1.95 2 0.84 1.06 -5.5 3 1.12 0.96 -6.6 longitudinally. The reduction in pulsation is accompanied by a reduction in pressure os- cillation amplitude. The drop in p~ by approximately 50% from S = 0.56 to S = 0.84 corresponds to a decrease of 5.5 dB. As S is further increased to 1.12, an additional -1.1 dB drop is achieved. The peak frequency is approximately 3000 Hz with only a small dependence on the initial swirl. A modal analysis reveal a 3/4 wave shape with a pressure node occurring slightly downstream of the dump plane. 4.3. Spray Combustion in Gas Turbine Engines For spray combustion, the gas phase LES model is coupled to a Lagrangian droplet- tracking model in which droplet groups are tracked explicitly in the flow field. The spray LES code is being used to study swirl control in a GE DACRS combustor. The liquid (n-heptane) is injected in the inlet and the droplets are entrained into the swirling air as it enters the combustor. Results show that increasing the swirl changes the spray dispersion (Fig. 4). The combustion process is also significantly effected by increase in swirl. For a
- 43. 39 ~illi~: ~,ii}:.:. ..... :::!li,i~: (a) Low-Swirl (b) (High-Swirl Figure 4. Instantaneous Azimuthal Vorticity and Droplet Distribution in the GE DACRS combustor. non-reacting case, increase in swirl increases the spray dispersion and vortex breakdown process. Large droplets tend to accumulate in regions of low vorticity (i.e., they tend to "surround" the vortices and are not present inside them). Thus, as the vortices break down into smaller features due to increased swirl, the droplets also gets dispersed [5]. This phenomenon impacts combustion (here only infinite-rate kinetics is studied) as well since as the droplet vaporize and become very small, they and the vaporized fuel get entrained into the vortices where they under mixing. Thus, in high swirl flows, flame surfaces are highly wrinkled and localized (the primary flame zone occurs in the core of the swirling flow) whereas in low swirl case, the mixing process is delayed, and hence, combustion is delayed and occurs in the outer regions of the large scale structures. Increase in the swirl also creates the vortex breakdown bubble which is not present in the low swirl case (Fig. 5). This bubble stabilizes the flame zone in a manner somewhat similar to the premixed case but there are many differences since spray combustion here is mixing controlled. 4.4. LES using Lattice Boltzmann Method A new parallel Lattice Boltzmann Equation (LBE) method for LES applications has also been developed [7]. The LBE approach allows full resolution of very small scales of motion and therefore, is particularly suited for resolving the flow features inside micro- scale MEMS-scale synthetic jets [7] that have been shown to enhance fuel-air mixing in some recent experiments. In the present effort, LES-LBE approach is employed in full 3D (using a 19-velocity model) to simulate the flow field generated using synthetic micro-jets, square and elliptical jets, and square jets in crossflow. A localized dynamic algebraic subgrid eddy viscosity model is used to obtain the effective viscosity used in
- 44. 40 Low Swirl - X/D = 1.0 0 .... ,/, , , -0.2 ~ ~ ~ r J~' 41.6 1'~I'~ -0.8 le..., Non_ReactingI ' [,,--,,Reacting I -! ~ I I I I I I I I -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 U/Uo (a) Low Swirl Case -0.2 High Swirl - X/D = 0.75 b P ! / ,,'/ 0 0.2 U/Uo ]e..-eNon-ReactingI ,,-- Reacting [ 0.4 0.6 (b) High Swirl Case Figure 5. Streamwise Mean Velocity Profiles in the combustor. Location is chosen to highlight the presence of the vortex breakdown bubble in the high swirl case. Here, R0 is the radius of the combustor, and r/Ro - 0 is located at the centerline. Outflow d Outflow Wall lOd PerJ ~ v odic x Per~ odic Figure 6. Geometry and computational domain employed for jet in crossflow.
- 45. 41 . . . . . , . . . . . i ' ' s-B /~ ... ........ O.J" 1 2 (a) x/D = O, U/Wj i ,iv . . . . I . . . . . 1 , , t...p 6 ~,~ o 0.5 1 (c) x/D = O, U/Wj . . . . . i . . . . . , . . . . . r . . . . . . . . . . /r , , , , , I . . . . . -i o ! 2 (b) x/D = 1, W/W~ i 0 . . . . I , )o o 0.5 (d) x/D = 1, W/Wj Figure 7. Mean velocity profiles along the jet center plane (y/D=0) at different x/D. The other curves are RANS calculations reported elsewhere[9]. the LBE-LES equation. Some results for jet in crossflow is discussed here to show the ability of the LBELES approach. The simulation is carried out at Reynolds number of 4700 based on the jet velocity and the nozzle width D and at jet-cross-flow velocity ratio of 0.5. The computational domain is resolved using 200 x 150 x 100 for the cross-flow domain and 50 x 50 x 100 for jet section. Approximately, 1500 single processor hours in the SGI Origin 2000 is needed for this simulation. Figure 6 shows the schematic of the test case experimentally and numerically studied in the past [8,9]. Figure 7 shows some characteristics mean velocity profiles predicted by LBE-LES method and by other RANS based calculations. It can be clearly seen that the LBE-LES prediction is substantially better. More details of this study will be reported elsewhere. 5. CONCLUSIONS From the parallel scaling studies on both supercomputersand PC clusters presented here, it is concluded that while both communicationand memorybandwidth contention are prevalent in SMP clusters, memory contention plays a more significant role with
- 46. 42 respect to LES applications. Compared to the HPC systems, Raptor,when operating with the VI-cLAN configuration, has comparable performance and superior scalability (versus the IBM Power3-SMP, Habu). Thus, the steady increase in processor performance and advances in memory/communication bandwidth, LES on PC-clusters may soon become integrated into the design process at a cost that will be affordable for even small-scale companies. The effects of swirl intensity and heat release are investigated. Results show that increase in swirl has dramatic effect on the flame structure and reduces the pressure oscil- lation amplitude in premixed systems. In spray combustion, increase in swirl drastically increases the droplet dispersion. In all situations, vortex breakdown occurs only under high swirl conditions. Also, combustion and heat release tends to reduce the size of this recirculation zone, due to an increase in flow acceleration associated with heat release. Finally, a new LES method based on the Lattice Boltzmann model is used to simulate complex flows such as jet in cross flow. The eventual application of the LBE-LES will be in flows where fuel-air mixing is actively controlled, for example, with synthetic micro-jets [7]. In these flows, conventional LES of the micro-scale domain of the micro-jets would imply an enormous grid resolution and the associated computational cost to do reacting flow simulations would be prohibitive. By using a LBE model for the microjet region and then coupling it to the more conventional finite-volume LES solver for the reacting flow in the combustor, it may be possible to handle this very complex problem. An effort is currently underway in this very effort and results will be reported soon. Acknowledgements This work is supported in part by US Army Research Office and General Electric Power Systems. Computational time is provided by US Department of Defense High Performance Computing Centers at NAVO, MS, ERDC, MS, and ARL, MD. REFERENCES 1. W.-W. Kim, S., Menon, and H. Mongia, Comb. Sci. and Tech. 143 (1999) 25. 2. C. Stone and S. Menon, J. Supercomputing 22 (2002) 7. 3. V.K. Chakravarthy and S. Menon, Flow, Turb. and Comb. 65(2000) 133. 4. V.K. Chakravarthy and S. Menon, Comb. Sci. and Tech. 162 (2001) 175-222 5. V. Sankaran and S. Menon, J. Turbulence 3 (2002) 3. 6. R. W. Metcalfe, S. A. Orszag, M. E. Brachet, S. Menon. and J. J. Riley, J. Fluid Mech. 184 (1987), 207. 7. H. Wang and S. Menon, AIAA J. 39 (2001) 2308. 8. P. Ajresch, J. Zho, S. Ketler, M. Salcudean, and I. Gartshore, J. Turbomachinery, 119 (1997), 330. 9. A. Hoda,S.Acharyaand M.Tyagi,ProceedingsofASMETURBOEXPO2000(2000), I.
- 47. ParallelComputationalFluidDynamics-NewFrontiersandMulti-DisciplinaryApplications K.Matsuno,A. Ecer,J. Periaux,N. SatofukaandP. Fox(Editors) 9 2003ElsevierScienceB.V.Allrightsreserved. 43 A Substepping Navier-Stokes Splitting Scheme for Spectral/hp Element Discretisations Spencer Sherwina * Department of Aeronautics, Imperial College, London, SW7 2BY, UK In this paper we investigate the application of a substepping advection algorithm in conjunction with a Navier-Stokes high order splitting scheme. We initially compare the substepping approach to the semi-Lagrangian treatment of the advection terms in the con- text of a one-dimensional advection-diffusion equation. The choice of a modal spectral/hp element discretisation then motivates the use of a discontinuous Galerkin formulation to solve the hyperbolic advection component of the Navier-Stokes time stepping scheme. This is necessary since a classical Galerkin discretisation would require a global matrix inversion thereby destroying the potential gain of the substepping technique. Finally the algorithm is applied to a two-dimensional model Kovasznay problem as well as the more applied problem of flow through a stenosis. 1. Introduction The application of spectral/hp element discretisations to a variety of industrially mo- tivated fluid problems such as bio-fluids, bridge aerodynamics and marine technology requires the use of efficient time-stepping techniques. An important aspect of achieving this efficiency is overcoming the Courant Fredrichs Levy (CFL) time step restrictions aris- ing from the explicit treatment of the convective terms in the Navier-Stokes equations. In explicit algorithms it is possible to have a stability restriction, arising from the CFL condition, which is overly restrictive from the point of view of accuracy. For example problems with a large range of physical scales where only the largest scale is required or when automated mesh design generates "slither" elements normal to the flow direction. Two approaches have typically been adopted within the spectral/hp element community to alleviate this problem. The first is the operator integration factor or substepping technique (see Maday et al. [6]) and the second is the semi-Lagrangian formulation more recently adopted in the spectral/hp element technique by Giraldo [2] and Xiu Karniadakis [i0]. In a general context these techniques are related by considering a characteristic treatment of the advection terms and solving the Navier-Stokes equations in the Lagrangian frame of reference. Nevertheless a notable distinction arises in the discretisation of the advection term. In the substepping technique the advection term is advanced in an Eulerian frame of reference using a smaller time step than the rest of the Navier-Stokes solver. Alternatively the semi-Lagrangian technique performs a backward *http://www.ae.ic.ac.uk/staff/sherwin
- 48. 44 x~_1 x~ x~§ Figure 1. Schematic of the standard finite difference approximation to the advection- diffusion equations using an implicit diffusion and explicit advection discretisation. particle tracking over the complete time step. In the present work we investigate the substepping strategy. Previous implementations of this technique have been adopted within the context of a full Stokes solver [6] or a fractional step method [9], however in this paper we discuss substepping in conjunction with a high order splitting scheme [4,7]. This scheme can be interpreted as a pure advec- tion step followed by a Poisson pressure correction to enforce divergence and a Helmholtz equation introducing the role of diffusion. This interpretation therefore provides a natural setting for the substepping method. However unlike the traditional nodal spectral ele- ment method, where a diagonal mass matrix arises from discrete orthogonality, the modal spectral/hp approach produces a full mass matrix system. A classical Galerkin discretisa- tion therefore results in a full matrix solution which would not provide any advantage to using the substepping technique over the traditional splitting scheme. However applica- tion of a discontinuous Galerkin formulation provides an appropriate setting to discretise the hyperbolic part of the system whilst maintaining numerical efficiency. The paper is organised in the following manner. In section 2 we will outline the for- mulation of the substepping Navier Stokes splitting scheme. Section 2.1 discusses the substepping approach and compares it with the semi-Lagrangian technique using one- dimensional concepts to illustrate the different approaches. Then in section 2.2 we detail the use of a discontinuous Galerkin formulation to efficiently solve the hyperbolic scalar advection equation when using spectral/hp element discretisation. In section 2.3 we briefly outline the spectral/hp element discretisation and finally in section 2.4 we apply all the previous components to arrive at a high order substepping splitting scheme for the Navier- Stokes equations. In section 3 we detail the application of the substepping Navier-Stokes algorithm to a model Kovasznay problem and the more applied problem of stenotic flow. 2. Formulation 2.1. Advection Substepping To detail the substepping and semi-Lagrangian approach we will consider, as a model problem, the scalar advection-diffusion equation r + a. Vr = vVur (1) where a(x,t) is a divergence free advection field. Under any explicit discretisation the time step restriction associated with the diffusion operator, vV2r is greater than the
- 49. 45 t'+l i/,1' [' ~_~ ,~X d x~ 1" i" i lX~+I Figure 2. Schematic of the semi-Lagrangian approximation to the advection-diffusion equations using an implicit diffusion discretisation. The advection operator is approxi- mated by backward particle tracking along the characteristic from x~ at t~+1 to determine Xd at t n. advection operator, a. Vr For this reason it is often desirable to treat the diffusion operator implicitly and since, in general, we would like to consider non-linear convection the advection terms are often handled explicitly. Such a strategy for a one-dimensional finite difference scheme can be symbolically represented on a discrete x-t diagram as shown in figure 1. In this figure the black squares represent the implicit contribution from a centred approximation to the diffusion operator (i.e. Vr +1 -- ~z21~('~n+lw~+l - 2r n+l + (~n+l ~_~ ) whilst the black circles represent the contributions from an upwinded first order approximation to the advection operator assuming a > 0 (i.e. a-Vr _ ~_~ (r n - r Following classical analysis the size of the time step At = tn+~ - tn is restricted by the CFL conditions At < Ax/a where Ax - xi+l - xi. Equation (1) can equivalently be written in Lagrangian form as De = vV 2r where D 0 Dt Dt = 0-~ + a. V (2) where D/Dt is the Lagrangian or material derivative moving with the advection velocity i.e.Dx/Dt = a. Following a semi-Lagrangian approach [2,10] and temporally discretising equation (2) at point x~ using a first order implicit time discretisation we obtain cn+l _ r At . v-72 ~.n-F-1 z t/v ~i " where r - Cn(Xd, tn) and Xd is the so-called departure point. Since the material derivative is evaluated along the characteristics we can evaluate the departure point X d by solving the characteristic equation Dx/Dt - a backward in time from t n+l < t <_ t ~ using the initial conditions x(t n+l) - x~. This approach is schematically represented in figure 2 where we show the one-dimensional discretisation on a discrete x-t diagram. Once again the implicit discretisation of the diffusion term is represented by the black squares. However in the semi-Lagrangian approach we backward particle track along the characteristic, Dx/Dt - a, of the hyperbolic advection operator
- 50. 46 a) " ............ ........... ," I: ..::. ....::2-;:.:::: ::::i:::::/::' . :: ,:'.'."..... -::::::~ ~::::':---./::::,q x~_~ /: // ~+s b) ~n+l 'l'l'l! X~+, Figure 3. Schematic representation of the substepping approach: (a)A hyperbolic advec- tion equation is advanced using a smaller time step to determine r (b) Using ~(Xi,tn+l) -- ~)(~d, tn) along the characteristic the diffusion step can then be solved. to determine point Xd, subsequently r is evaluated and the approximation to the Lagrangian time derivative can then be determined. The complexity of backward particle tracking depends on the form of a(x, t). If a(x, t) is time independent the calculation is relatively straight forward on a well behaved discretisation. However if the convection velocity is time dependent then different time integration strategies can be adopted. For further details of this methods in the context of spectral/hp element methods see [2,10]. Finally we also note that this scheme requires an interpolation operation to evaluate Although we will not be adopting the semi-Lagrangian approach in the current work it does provide a convenient setting to discuss the substepping/integration operator splitting [6] approach. The implementation of this strategy is similar to the semi-Lagrangian tech- nique however instead of backward particle tracking to obtain ~rd and then interpolating to obtain r we solve for the departure solution r directly. This can be achieved by solving the advection part of the problem independently in its Eulerian form and with a smaller time step. Therefore we introduce a pseudo solution r ~-) and solve the problems: -- + a. Vr = O t n < T < t n+l (3) DT O~ - - with initial conditions r tn) = r Since this is a strictly hyperbolic equation then along the characteristic the solution is constant and so r tn) = r t~+l). The explicit nature of equation (3) means that from stability considerations the time step is restricted by a CFL condition. Nevertheless providing the solution of (3) can be solved more efficiently than the implicit diffusion operator then equation (3) can be discretised with a time step near to the CFL limit whilst the implicit diffusion operator is solved less frequently in time thereby reducing the cost of the overall algorithm for a fixed integration time. Indeed it is the ratio of the cost of the explicit advection term to the implicit diffusion operator which ultimately limits the possible speedup. We also note that the time accuracy is governed by the larger time step applied on the implicit diffusion operator.
- 51. 47 The substepping solution is schematically shown in figure 3(b) where we again denote the discretisation of the implicit diffusion step by the black squares. To determine the solution at the departure point we have to discretise equation (3) using a smaller, CFL satisfying, time step as shown in figure 3(a). For a mathematical description of the scheme see Maday et al. [6] which describes the solution of equation (3) as determining the integration operator factor which makes the advection operator into a total or Lagrangian derivative. In the substepping approach we do not need to interpolate the solution field to de- termine r and can also use standard Eulerian formulation to solve for this variable. However substepping is still limited by an explicit time step restriction which potentially could be quite severe and may make the solution of equation (3) overly costly. Therefore, an efficient solution technique for the advection equation is required, such as the discon- tinuous Galerkin method as discussed in section 2.2. We note that the semi-Lagrangian approach is potentially quite expensive when considering the cost of evaluating the back- ward particle tracking and then interpolation of the solution. However this cost is rel- atively independent of At which may make this approach more attractive at very large At. As a final point we observe that parallelisation of the substepping approach follows a relatively traditional implementation only requiring communication of nearest neighbour information. Parallelisation of the semi-Lagrangian approach is more involved due to the particle tracking nature of the algorithm. 2.2. Discontinuous Galerkin Formulation for Scalar Advection Consider the multi-dimensional scalar advection equation 0u 0u O---t + a. Vu - --~ + V . (au) - 0 in ~ (4) where we have assumed V. a = 0. We formulate our problem on Net non-overlapping tIN~ F4 ~. Following the standard finite element formulation on elements ~ such than ue=l = a single element we take the inner product of (4) with respect to a test function r which only has support in ~ to obtain r Ou + (r V. f(u))n~ =0 where f(u) = au and (,) is the L2 inner product. Applying the divergence theorem to the second term and summing over all element we arrive at r - (re. f(~))~o + Cf(~). nd~ : 0 (5) e=l Fte ~te where cg~e represents the boundary of F4 and n is the outward normal to ~. To obtain the discrete form of our problem we choose u to be in the finite space of [L2(~)] D~'~ functions which are polynomial of degree P on each element. We indicate an element of such space using the superscript ~. We also note that u ~ may be discontinuous across inter-element boundaries. However to attain a global solution in the domain we need to allow information to propagate between the elemental regions. Information is propagated between elements by upwinding the boundary flux in the third term of
- 52. 48 equation (5). Denoting the upwinded flux as f~, for the linear advection equation the upwind normal flux can simply be determined by fU(u) . n- { uinta . n Uexta n ifa.n>0 m ifa.n < 0 where u~t and ue~t are respectively the values of u interior and exterior to the element edge. The discrete weak formulation can now be written as Following the traditional Galerkin approach, we choose the test function within each element to be in the same discrete space as the numerical solution u ~. At this point if we defined our polynomial basis and choose an appropriate quadrature rule we would have a semi-discrete scheme. However, from an implementation point of view, the calculation of the second term can be inconvenient and consequently we choose to integrate this term by parts once more to obtain r + (r f(u))n~ + r []~(u) - f(u)], rids = 0 (7) e=l Fte gte We note that the information between elements is transmitted by the third boundary term as the difference between the upwinded and the local fluxes. This method can be considered as a penalty method with an automatic procedure for determining the penalty parameter. The discretisation is completed by defining a time stepping method such as the Adams-Bashforth or Runge Kutta schemes and a spatial discretisation which is addressed in section 2.3. 2.3. Spectral/hp Element Discretisation In the current work we have adopted a spectral/hp element discretisation. This ap- proach combines the exponential accuracy of Fourier methods with the geometric flexi- bility of the finite element/volume methods. Within each local element of triangular or quadrilateral shape in two-dimensions a polynomial expansion of arbitrary order is ap- plied. In classical Galerkin methods this expansion is required to be C o continuous over the whole solution domain and can be either modal or nodal in form. Traditionally nodal elements have been referred to as spectral element methods. Within the current work we have applied a continuous modal expansion for the elliptic operators and an orthogo- nal modal expansion for the hyperbolic operators in conjunction with the discontinuous Galerkin formulation. 2.4. Substepping Navier-Stokes Splitting Scheme The Lagrangian Navier-Stokes equations can be written as: Du = -Vp+vV2u (8) Dt v. =0 (9)
- 53. 49 where u = [u, v, w]T, D_UU_ O _ _ U _ U + u. Vu and p = Ps/P where ps is the static pressure. The Dt ~ Ot divergence conditions (9) can be equivalently expressed by a pressure Poisson equation by taking the divergence of the momentum equation (8) and noting that V. (V2u) = 0 if V 9u = 0 to arrive at: Pu -V. (u. Vu). (10) V2p=-V. Dt = Discretising equation (8) at time level n + 1 and applying an implicit scheme for the linear diffusion term we obtain: O~OUn+!+ EqJ=-iOLqUrd+l-q = --Vp n+l + .V2U n+l (11) At where C~q represents the coefficients for backward integration in time, i.e. J = 1 =a •q -- [1,-1], J- 2 =~ aq = [3/2,-2, 1/2]. In equation (11), the velocity u~ +l-q represents the velocity at the foot of the characteristic at time level n + 1- q. In the substepping scheme u~+l-q = ~n+l where ~+1 is determined from the integration between tn+l-q < ~- < tn+l d of the Eulerian advection equation 0~ -- + u. V~ = 0 (12) 0f ~(x, t"+1-~) - ~(x, t~+~-~) (13) where u in the above equation is the divergence free velocity which is extrapolated in time between tn < ~- < tn+l. We note that equation (12) can be put in conservative form by using the fact that V. u = 0 which implies that u. Vfi = V 9(fiiu). To solve equation (11) we apply a high order splitting scheme as discussed for the Eulerian discretisation in Orszag et al. [7], Karniadakis et al. [4] and more recently by Guermond & Shen [3] as well as for the semi-Lagrangian scheme by Xiu and Karniadakis [10]. Following this scheme we split equation (11) into two steps J OLqU~+l--q + Eq=l = -Vp ~+1 (14) At OL0un+l -- ?~ = ,V2u n+l (15) At To decouples steps (14) and (15) we take the divergence of equation (14) and require that V. ~ = 0 to obtain V2pn+i -- - EqJ1 ~ (16) At Equation (16) is a semi-discrete approximation to pressure Poisson equation (10) through which divergence is enforced. We note from the discrete divergence analysis in [7] that implicit inclusion of u ~ in the right hand side of equation (16) is important since the gov- erning divergence equation is of an elliptic rather than parabolic form. This is significant since it since it implies the divergence errors in the initial and boundary conditions are restricted to a boundary layer.
- 54. 50 o -1 ~_~ X 2 4 6 8 10 -3<~ -3 -1,5 (a) (b) ,o,~.om~.,o.=or (c) ,=(A~) : "A::;: .] zU ] Figure 4. Kovasznay flow: (a) Solution and computational mesh, (b) Convergence as a function of polynomial order, (b) Convergence as a function of time step. n+l-q Finally equations (15),(16) together with equation (12) for the solution of u d rep- resent the substepping Navier-Stokes splitting scheme. However to close the system we require appropriate boundary conditions for equation (16) which are obtained by taking the inner product of equation (11) with the outward normal to the boundary to obtain Opn+l [ On = n" /2V2U n+l -- a~ + EqJ=lAt~ This form of the boundary condition is not typically found to be stable [4,8] and to correct this the Laplacian term is decomposed into a divergence and solenoidal component (V2u = V(V. u) - V x V x u) where the divergence term can be set identically to zero. A complete discussion of boundary conditions for splitting formulations can be found in Petersson [8]. To make the scheme explicit we still have to evaluate the solenoidal component V x Vu at t~+1 which is evaluated by extrapolation from previous time levels. The discrete pressure boundary conditions can therefore be written as 0--~ = -n. v ~ 7qV • Vu ~-q + ~/ q=0 where the discrete Lagrangian time derivative has been replaced by ~2 by imposing V. u ~+1 = 0 and 7q represents the extrapolation coefficients (i.e. Je = 1 --+ 7a = [1], Je = 2 ~ 7q = [2,-1]). As a final comment we note that equation (12) needs to be evaluated efficiently for a significant benefit of the substepping to be observed. This can be achieved by using the discontinuous Galerkin formulation as discussed in section 2.2. 3. Results As a first example we consider an analytic problem proposed by Kovasznay [5] which can be considered as a steady laminar flow behind a two-dimensional grid. This exazt
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. to a height." But I must confess that, although Macpherson has extracted the above passage, and a more judicious writer, Somerville, quotes the book freely as genuine (Hist. of Anne, p. 581, etc.), I found in reading it what seemed to me the strongest grounds of suspicion. It is printed in England, without a word of preface to explain how such important secrets came to be divulged, or by what means the book came before the world; the correct information as to English customs and persons frequently betrays a native pen; the truth it contains, as to jacobite intrigues, might have transpired from other sources, and in the main was pretty well suspected, as the Report of the Secret Committee on the Impeachments in 1715 shows; so that, upon the whole, I cannot but reckon it a forgery in order to injure the tory leaders.
- 57. But however this may be, we find Bolingbroke in correspondence with the Stuart agents in the later part of 1712. Macpherson, 366. And his own correspondence with Lord Strafford shows his dread and dislike of Hanover (Bol. Corr. ii. 487 et alibi). The Duke of Buckingham wrote to St. Germains in July that year, with strong expressions of his attachment to the cause, and pressing the necessity of the prince's conversion to the protestant religion. Macpherson, 327. Ormond is mentioned in the Duke of Berwick's letters as in correspondence with him; and Lockhart says there was no reason to make the least question of his affection to the king, whose friends were consequently well pleased at his appointment to succeed Marlborough in the command of the army, and thought it portended some good designs in favour of him. Id. 376. Of Ormond's sincerity in this cause there can indeed be little doubt; but there is almost as much reason to suspect that of Bolingbroke as of Oxford; except that, having more rashness and less principle, he was better fitted for so dangerous a counter-revolution. But in reality he had a perfect contempt for the Stuart and tory notions of government, and would doubtless have served the house of Hanover with more pleasure, if his prospects in that quarter had been more favourable. It appears that in the session of 1714, when he had become lord of the ascendant, he disappointed the zealous royalists by his delays as much as his more cautious rival had done before. Lockhart, 470. This writer repeatedly asserts that a majority of the House of Commons, both in the parliament of 1710 and that of 1713, wanted only the least encouragement from the court to have brought about the repeal of the act of settlement. But I think this very doubtful; and I am quite convinced that the nation would not have acquiesced in it. Lockhart is sanguine, and ignorant of England. It must be admitted that part of the cabinet were steady to the protestant succession. Lord Dartmouth, Lord Powlett, Lord Trevor, and the Bishop of London were certainly so; nor can there be any reasonable doubt, as I conceive, of the Duke of Shrewsbury. On the other side, besides Ormond, Harcourt, and Bolingbroke, were the Duke of Buckingham, Sir William Wyndham, and probably Mr. Bromley. [327] It is said that the Duke of Leeds, who was now in the Stuart interest, had sounded her in 1711, but with no success in discovering her intention. Macpherson, 212. The Duke of Buckingham pretended, in the above-mentioned
- 58. letter to St. Germains, June 1712, that he had often pressed the queen on the subject of her brother's restoration, but could get no other answer than, "you see he does not make the least step to oblige me;" or, "he may thank himself for it: he knows I always loved him better than the other." Id. 328. This alludes to the Pretender's pertinacity, as the writer thought it, in adhering to his religion; and it may be very questionable, whether he had ever such conversation with the queen at all. But, if he had, it does not lead to the supposition, that under all circumstances she meditated his restoration. If the book under the name of Mesnager is genuine, which I much doubt, Mrs. Masham had never been able to elicit anything decisive of her majesty's inclinations; nor do any of the Stuart correspondents in Macpherson pretend to know her intentions with certainty. The following passage in Lockhart seems rather more to the purpose: On his coming to parliament in 1710, with a "high monarchical address," which he had procured from the county of Edinburgh, "the queen told me, though I had almost always opposed her measures, she did not doubt of my affection to her person, and hoped I would not concur in the design against Mrs. Masham, or for bringing over the Prince of Hanover. At first I was somewhat surprised, but recovering myself, I assured her I should never be accessary to the imposing any hardship or affront upon her; and as for the Prince of Hanover, her majesty might judge from the address I had read, that I should not be acceptable to my constituents if I gave my consent for bringing over any of that family, either now or at any time hereafter. At that she smiled, and I withdrew; and then she said to the duke (Hamilton), she believed I was an honest man and a fair dealer, and the duke replied, he could assure her I liked her majesty and all her father's bairns."—P. 317. It appears in subsequent parts of this book, that Lockhart and his friends were confident of the queen's inclinations in the last year of her life, though not of her resolution. The truth seems to be, that Anne was very dissembling, as Swift repeatedly says in his private letters, and as feeble and timid persons in high station generally are; that she hated the house of Hanover, and in some measure feared them; but that she had no regard for the Pretender (for it is really absurd to talk like Somerville of natural affection under all the circumstances), and feared him a great deal more than the other; that she had, however, some scruples about his right, which were counterbalanced by her attachment to the church of England; consequently, that she was wavering among opposite impulses, but with a predominating timidity which would have probably kept her from any change. [328] The Duchess of Gordon, in June 1711, sent a silver medal to the faculty of advocates at Edinburgh, with a head on one side, and the inscription, "Cujus est"; on the other, the British isles, with the word "Reddite." The dean of faculty, Dundas of Arniston, presented this medal; and there seems reason to believe that a majority of the advocates voted for its reception. Somerville, p. 452.
- 59. Bolingbroke, in writing on the subject to a friend, it must be owned, speaks of the proceeding with due disapprobation. Bolingbroke Correspondence, i. 343. No measures, however, were taken to mark the court's displeasure. "Nothing is more certain," says Bolingbroke in his letter to Sir William Wyndham, perhaps the finest of his writings, "than this truth, that there was at that time no formed design in the party, whatever views some particular men might have, against his majesty's accession to the throne."—P. 22. This is in effect to confess a great deal; and in other parts of the same letter, he makes admissions of the same kind: though he says that he and other tories had determined, before the queen's death, to have no connection with the Pretender, on account of his religious bigotry. P. 111. [329] Lockhart gives us a speech of Sir William Whitelock in 1714, bitterly inveighing against the elector of Hanover, who, he hoped, would never come to the crown. Some of the whigs cried out on this that he should be brought to the bar; when Whitelock said he would not recede an inch; he hoped the queen would outlive that prince, and in comparison to her he did not value all the princes of Germany one farthing. P. 469. Swift, in "Some Free Thoughts upon the present State of Affairs," 1714, speaks with much contempt of the house of Hanover and its sovereign; and suggests, in derision, that the infant son of the electoral prince might be invited to take up his residence in England. He pretends in this tract, as in all his writings, to deny entirely that there was the least tendency towards jacobitism, either in any one of the ministry, or even any eminent individual out of it; but with so impudent a disregard of truth that I am not perfectly convinced of his own innocence as to that intrigue. Thus, in his "Inquiry into the Behaviour of the Queen's last Ministry," he says, "I remember, during the late treaty of peace, discoursing at several times with some very eminent persons of the opposite side with whom I had long acquaintance. I asked them seriously, whether they or any of their friends did in earnest believe, or suspect the queen or the ministry to have any favourable regards towards the Pretender? They all confessed for themselves that they believed nothing of the matter," etc. He then tells us that he had the curiosity to ask almost every person in great employment, whether they knew or had heard of any one particular man, except professed nonjurors, that discovered the least inclination towards the Pretender; and the whole number they could muster up did not amount to above five or six; among whom one was a certain old lord lately dead, and one a private gentleman, of little consequence and of a broken fortune, etc. (vol. 15, p. 94, edit. 12mo, 1765). This acute observer of mankind well knew that lying is frequently successful in the ratio of its effrontery and extravagance. There are, however, some passages in this tract, as in others written by Swift, in relation to that time, which serve to illustrate the obscure machinations of those famous last years of the queen.
- 60. [330] On a motion in the House of Lords that the protestant succession was in danger, April 5, 1714, the ministry had only a majority of 76 to 69, several bishops and other tories voting against them. Parl. Hist. vi. 1334. Even in the Commons the division was but 256 to 208. Id. 1347. [331] Somerville has a separate dissertation on the danger of the protestant succession, intended to prove that it was in no danger at all, except through the violence of the whigs in exasperating the queen. It is true that Lockhart's Commentaries were not published at this time; but he had Macpherson before him, and the Memoirs of Berwick, and even gave credit to the authenticity of Mesnager, which I do not. But this sensible, and on the whole impartial writer, had contracted an excessive prejudice against the whigs of that period as a party, though he seems to adopt their principles. His dissertation is a laboured attempt to explain away the most evident facts, and to deny what no one of either party at that time would probably have in private denied. [332] The queen was very ill about the close of 1713; in fact it became evident, as it had long been apprehended, that she could not live much longer. The Hanoverians, both whigs and tories, urged that the electoral prince should be sent for; it was thought that whichever of the competitors should have the start upon her death would succeed in securing the crown. Macpherson, 385, 546, 557 et alibi. Can there be a more complete justification of this measure, which Somerville and the tory writers treat as disrespectful to the queen? The Hanoverian envoy, Schutz, demanded the writ for the electoral prince without his master's orders; but it was done with the advice of all the whig leaders (Id. 592), and with the sanction of the Electress Sophia, who died immediately after. "All who are for Hanover believe the coming of the electoral prince to be advantageous; all those against it are frightened at it." Id. 596. It was doubtless a critical moment; and the court of Hanover might be excused for pausing in the choice of dangers, as the step must make the queen decidedly their enemy. She was greatly offended, and forbade the Hanoverian minister to appear at court. Indeed she wrote to the elector, on May 19, expressing her disapprobation of the prince's coming over to England, and "her determination to oppose a project so contrary to her royal authority, however fatal the consequences may be." Id. 621. Oxford and Bolingbroke intimate the same. Id. 593; and see Bolingbroke Correspondence, iv. 512, a very strong passage. The measure was given up, whether from unwillingness on the part of George to make the queen irreconcilable, or, as is at least equally probable, out of jealousy of his son. The former certainly disappointed his adherents by more apparent apathy than their ardour required; which will not be surprising, when we reflect that, even upon the throne, he seemed to care very little about it. Macpherson, sub ann. 1714, passim. [333] He was strongly pressed by his English adherents to declare himself a protestant. He wrote a very good answer. Macpherson, 436. Madame de
- 61. Maintenon says, some catholics urged him to the same course, "par une politique poussée un peu trop loin." Lettres à la Princesse des Ursins, ii. 428. [334] The rage of the tory party against the queen and Lord Oxford for retaining whigs in office is notorious from Swift's private letters, and many other authorities. And Bolingbroke, in his letter to Sir W. Wyndham, very fairly owns their intention "to fill the employments of the kingdom, down to the meanest, with tories."—"We imagined," he proceeds, "that such measures, joined to the advantages of our numbers and our property, would secure us against all attempts during her reign; and that we should soon become too considerable not to make our terms in all events which might happen afterwards; concerning which, to speak truly, I believe few or none of us had any very settled resolution." P. 11. It is rather amusing to observe that those who called themselves the tory or church party, seem to have fancied they had a natural right to power and profit, so that an injury was done them when these rewards went another way; and I am not sure that something of the same prejudice has not been perceptible in times a good deal later. [335] Though no republican party, as I have elsewhere observed, could with any propriety be said to exist, it is easy to perceive that a certain degree of provocation from the Crown might have brought one together in no slight force. These two propositions are perfectly compatible. [336] This is well put by Bishop Willis in his speech on the bill against Atterbury. Parl. Hist. viii. 305. In a pamphlet, entitled "English Advice to the Freeholders" (Somers Tracts, xiii. 521), ascribed to Atterbury himself, a most virulent attack is made on the government, merely because what he calls the church party had been thrown out of office. "Among all who call themselves whigs," he says, "and are of any consideration as such, name me the man I cannot prove to be an inveterate enemy to the church of England; and I will be a convert that instant to their cause." It must be owned perhaps that the whig ministry might better have avoided some reflections on the late times in the addresses of both houses; and still more, some not very constitutional recommendations to the electors, in the proclamation calling the new parliament in 1714 Parl. Hist. vi. 44, 50. "Never was prince more universally well received by subjects than his present majesty on his arrival; and never was less done by a prince to create a change in people's affections. But so it is, a very observable change hath happened. Evil infusions were spread on the one hand; and, it may be, there was too great a stoicism or contempt of popularity on the other." "Argument to prove the Affections of the People of England to be the best Security for the Government," p. 11 (1716). This is the pamphlet written to recommend lenity towards the rebels, which Addison has answered in the Freeholder. It is invidious, and perhaps secretly jacobite. Bolingbroke observes, in the letter already quoted, that the Pretender's journey from Bar, in 1714, was a mere farce, no party being ready to receive him; but "the menaces of the whigs, backed by some very rash declarations [those of the king],
- 62. and little circumstances of humour, which frequently offend more than real injuries, and by the entire change of all persons in employment, blew up the coals."—P. 34. Then, he owns, the tories looked to Bar. "The violence of the whigs forced them into the arms of the Pretender." It is to be remarked on all this, that, by Bolingbroke's own account, the tories, if they had no "formed design" or "settled resolution" that way, were not very determined in their repugnance before the queen's death; and that the chief violence of which they complained was, that George chose to employ his friends rather than his enemies. [337] The trials after this rebellion were not conducted with quite that appearance of impartiality which we now exact from judges. Chief Baron Montagu reprimanded a jury for acquitting some persons indicted for treason; and Tindal, an historian very strongly on the court side, admits that the dying speeches of some of the sufferers made an impression on the people, so as to increase rather than lessen the number of jacobites. Continuation of Rapin, p. 501 (folio edit.). There seems, however, upon the whole, to have been greater and less necessary severity after the rebellion in 1745; and upon this latter occasion it is impossible not to reprobate the execution of Mr. Ratcliffe (brother of that Earl of Derwentwater who had lost his head in 1716), after an absence of thirty years from this country, to the sovereign of which he had never professed allegiance nor could owe any, except by the fiction of our law. [338] Parl. Hist. 73. It was carried against Oxford by 247 to 127, Sir Joseph Jekyll strongly opposing it, though he had said before (Id. 67) that they had more than sufficient evidence against Bolingbroke on the statute of Edward III. A motion was made in the Lords, to consult the judges whether the articles amounted to treason, but lost by 84 to 52. Id. 154. Lord Cowper on this occasion challenged all the lawyers in England to disprove that proposition. The proposal of reference to the judges was perhaps premature; but the house must surely have done this before their final sentence, or shown themselves more passionate than in the case of Lord Strafford. [339] Parl. Hist. vii. 486. The division was 88 to 56. There was a schism in the whig party at this time; yet I should suppose the ministers might have prevented this defeat, if they had been anxious to do so. It seems, however, by a letter in Coxe's Memoirs of Walpole, vol. ii. p. 123, that the government were for dropping the charge of treason against Oxford, "it being very certain that there is not sufficient evidence to convict him of that crime," but for pressing those of misdemeanour. [340] Parl. Hist. vii. 105. [341] Parl. Hist. vi. 972. Burnet, 560, makes some observations on the vote passed on this occasion, censuring the late ministers for advising an offensive war in Spain. "A resolution in council is only the sovereign's act, who upon hearing his
- 63. counsellors deliver their opinions, forms his own resolution; a counsellor may indeed be liable to censure for what he may say at that board; but the resolution taken there has been hitherto treated with a silent respect; but by that precedent it will be hereafter subject to a parliamentary inquiry." Speaker Onslow justly remarks that these general and indefinite sentiments are liable to much exception, and that the bishop did not try them by his whig principles. The first instance where I find the responsibility of some one for every act of the Crown strongly laid down is in a speech of the Duke of Argyle, in 1739. Parl. Hist. ix. 1138. "It is true," he says, "the nature of our constitution requires that public acts should be issued out in his majesty's name; but for all that, my lords, he is not the author of them." [342] "Lord Bolingbroke used to say that the restraining orders to the Duke of Ormond were proposed in the cabinet council, in the queen's presence, by the Earl of Oxford, who had not communicated his intention to the rest of the ministers; and that Lord Bolingbroke was on the point of giving his opinion against it, when the queen, without suffering the matter to be debated, directed these orders to be sent, and broke up the council. This story was told by the late Lord Bolingbroke to my father." Note by Lord Hardwicke on Burnet (Oxf. edit. vi. 119). The noble annotator has given us the same anecdote in the Hardwicke State Papers, ii. 482; but with this variance, that Lord Bolingbroke there ascribes the orders to the queen herself, though he conjectured them to have proceeded from Lord Oxford. [343] Parl. Hist. vii. 292. The apprehension that parliament, having taken this step, might go on still farther to protract its own duration, was not quite idle. We find from Coxe's Memoirs of Walpole, ii. 217, that in 1720, when the first septennial House of Commons had nearly run its term, there was a project of once more prolonging its life. [344] Parl. Hist. vii. 589. [345] The arguments on this side are urged by Addison, in the Old Whig; and by the author of a tract, entitled "Six Questions Stated and Answered." [346] The speeches of Walpole and others, in the Parliamentary Debates, contain the whole force of the arguments against the peerage bill. Steele in the Plebeian opposed his old friend and coadjutor, Addison, who forgot a little in party and controversy their ancient friendship. Lord Sunderland held out, by way of inducements to the bill, that the Lords would part with scandalum magnatum, and permit the Commons to administer an oath; and that the king would give up the prerogative of pardoning after an impeachment. Coxe's Walpole, ii. 172. Mere trifles, in comparison with the innovations projected.
- 64. [347] The letters in Coxe's Memoirs of Walpole, vol. ii., abundantly show the German nationality, the impolicy and neglect of his duties, the rapacity and petty selfishness of George I. The whigs were much dissatisfied; but fear of losing their places made them his slaves. Nothing can be more demonstrable than that the king's character was the main cause of preserving jacobitism, as that of his competitor was of weakening it. The habeas corpus was several times suspended in this reign, as it had been in that of William. Though the perpetual conspiracies of the jacobites afforded a sufficient apology for this measure, it was invidiously held up as inconsistent with a government which professed to stand on the principles of liberty. Parl. Hist. v. 153, 267, 604; vii. 276; viii. 38. But some of these suspensions were too long, especially the last, from October 1722 to October 1723. Sir Joseph Jekyll, with his usual zeal for liberty, moved to reduce the time to six months. [348] "It was first settled by a verbal agreement between Archbishop Sheldon and the Lord Chancellor Clarendon, and tacitly given into by the clergy in general as a great ease to them in taxations. The first public act of any kind relating to it was an act of parliament in 1665, by which the clergy were, in common with the laity, charged with the tax given in that act, and were discharged from the payment of the subsidies they had granted before in convocation; but in this act of parliament of 1665 there is an express saving of the right of the clergy to tax themselves in convocation, if they think fit; but that has been never done since, nor attempted, as I know of, and the clergy have been constantly from that time charged with laity in all public aids to the Crown by the House of Commons. In consequence of this (but from what period I cannot say), without the intervention of any particular law for it, except what I shall mention presently, the clergy (who are not lords of parliament) have assumed, and without any objection enjoyed, the privilege of voting in the election of members of the House of Commons, in virtue of their ecclesiastical freeholds. This has constantly been practised from the time it first began; there are two acts of parliament which suppose it to be now a right. The acts are 10 Anne, c. 23; 18 Geo. II. c. 18. Gibson, Bishop of London, said to me, that this (the taxation of the clergy out of convocation) was the greatest alteration in the constitution ever made without an express law." Speaker Onslow's note on Burnet (Oxf. edit. iv. 508). [349] The first authority I have observed for this pretension is an address of the House of Lords (19 Nov. 1675) to the throne, for the frequent meeting of the convocation, and that they do make to the king such representations as may be for the safety of the religion established. Lords' Journals. This address was renewed February 22, 1677. But what took place in consequence I am not apprised. It shows, however, some degree of dissatisfaction on the part of the bishops, who must be presumed to have set forward these addresses, at the
- 65. virtual annihilation of their synod which naturally followed from its relinquishment of self-taxation. [350] Kennet, 799, 842; Burnet, 280. This assembly had been suffered to sit, probably, in consequence of the tory maxims which the ministry of that year professed. [351] Wilkins's Concilia, iv.; Burnet, passim; Boyer's Life of Queen Anne, 225; Somerville, 82, 124. [352] The lower house of convocation, in the late reign, among their other vagaries, had requested "that some synodical notice might be taken of the dishonour done to the church by a sermon preached by Mr. Benjamin Hoadley at St. Lawrence Jewry, Sept. 29, 1705, containing positions contrary to the doctrine of the church, expressed in the first and second parts of the homily against disobedience and wilful rebellion." Wilkins, iv. 634. [353] These qualities are so apparent, that after turning over some forty or fifty tracts, and consuming a good many hours on the Bangorian controversy, I should find some difficulty in stating with precision the propositions in dispute. It is, however, evident that a dislike, not perhaps exactly to the house of Brunswick, but to the tenor of George I.'s administration, and to Hoadley himself as an eminent advocate for it, who had been rewarded accordingly, was at the bottom a leading motive with most of the church party; some of whom, such as Hare, though originally of a whig connection, might have had disappointments to exasperate them. There was nothing whatever in Hoadley's sermon injurious to the established endowments and privileges, nor to the discipline and government, of the English church, even in theory. If this had been the case, he might be reproached with some inconsistency in becoming so large a partaker of her honours and emoluments. He even admitted the usefulness of censures for open immoralities, though denying all church authority to oblige any one to external communion, or to pass any sentence which should determine the condition of men with respect to the favour or displeasure of God. Hoadley's Works, ii. 465, 493. Another great question in this controversy was that of religious liberty, as a civil right, which the convocation explicitly denied. And another related to the much debated exercise of private judgment in religion, which, as one party meant virtually to take away, so the other perhaps unreasonably exaggerated. Some other disputes arose in the course of the combat, particularly the delicate problem of the value of sincerity as a plea for material errors. [354] Tindal, 539. [355] Parl. Hist. vi. 362.
- 66. [356] 10 Anne, c. 2. [357] 12 Anne, c. 7; Parl. Hist. vi. 1349. The schism act, according to Lockhart, was promoted by Bolingbroke, in order to gratify the high tories, and to put Lord Oxford under the necessity of declaring himself one way or other. "Though the Earl of Oxford voted for it himself, he concurred with those who endeavoured to restrain some parts which they reckoned too severe; and his friends in both houses, particularly his brother auditor Harley, spoke and voted against it very earnestly."—P. 462. [358] 5 Geo. I. c. 4. The whigs out of power, among whom was Walpole, factiously and inconsistently opposed the repeal of the schism act, so that it passed with much difficulty. Parl. Hist. vii. 569. [359] The first act of this kind appears to have been in 1727. 1 Geo. II. c. 23. It was repeated next year, intermitted the next, and afterwards renewed in every year of that reign except the fifth, the seventeenth, the twenty-second, the twenty-third, the twenty-sixth, and the thirtieth. Whether these occasional interruptions were intended to prevent the nonconformists from relying upon it, or were caused by some accidental circumstance, must be left to conjecture. I believe that the renewal has been regular every year since the accession of George III. It is to be remembered, that the present work was first published before the repeal of the test act in 1828. [360] We find in Gutch's Collectanea Curiosa, vol. i. p. 53, a plan, ascribed to Lord Chancellor Macclesfield, for taking away the election of heads of colleges from the fellows, and vesting the nomination in the great officers of state, in order to cure the disaffection and want of discipline which was justly complained of. This remedy would have been perhaps the substitution of a permanent for a temporary evil. It appears also that Archbishop Wake wanted to have had a bill, in 1716, for asserting the royal supremacy, and better regulating the clergy of the two universities (Coxe's Walpole, ii. 122); but I do not know that the precise nature of this is anywhere mentioned. I can scarcely quote Amherst's Terræ Filius as authority; it is a very clever, though rather libellous, invective against the university of Oxford at that time; but from internal evidence, as well as the confirmation which better authorities afford it, I have no doubt that it contains much truth. Those who have looked much at the ephemeral literature of these two reigns must be aware of many publications fixing the charge of prevalent disaffection on this university, down to the death of George II.; and Dr. King, the famous jacobite master of St. Mary Hall, admits that some were left to reproach him for apostasy in going to court on the accession of the late king in 1760. The general reader will remember the Isis by Mason, and the Triumph of Isis by Warton; the one a severe invective, the other an indignant vindication; but in this instance, notwithstanding
- 67. the advantages which satire is supposed to have over panegyric, we must award the laurel to the worse cause, and, what is more extraordinary, to the worse poet. [361] Layer, who suffered on account of this plot, had accused several peers, among others Lord Cowper, who complained to the house of the publication of his name; and indeed, though he was at that time strongly in opposition to the court, the charge seems wholly incredible. Lord Strafford, however, was probably guilty; Lords North and Orrery certainly so. Parl. Hist. viii. 203. There is even ground to suspect that Sunderland, to use Tindal's words, "in the latter part of his life had entered into correspondencies and designs, which would have been fatal to himself or to the public."—P. 657. This is mentioned by Coxe, i. 165; and certainly confirmed by Lockhart, ii. 68, 70. But the reader will hardly give credit to such a story as Horace Walpole has told, that he coolly consulted Sir Robert, his political rival, as to the part they should take on the king's death. Lord Orford's Works, iv. 287. [362] State Trials, xvi. 324; Parl. Hist. viii. 195 et post. Most of the bishops voted against their restless brother; and Willis, Bishop of Salisbury, made a very good but rather too acrimonious a speech on the bill. Id. 298. Hoadley, who was no orator, published two letters in the newspaper, signed "Britannicus," in answer to Atterbury's defence; which, after all that had passed, he might better have spared. Atterbury's own speech is certainly below his fame, especially the peroration. Id. 267. No one, I presume, will affect to doubt the reality of Atterbury's connections with the Stuart family, either before his attainder or during his exile. The proofs of the latter were published by Lord Hailes in 1768, and may be found also in Nicholls's edition of Atterbury's Correspondence, i. 148. Additional evidence is furnished by the Lockhart Papers, vol. ii. passim. [363] The Stuart papers obtained lately from Rome, and now in his majesty's possession, are said to furnish copious evidence of the jacobite intrigues, and to affect some persons not hitherto suspected. We have reason to hope that they will not be long withheld from the public, every motive for concealment being wholly at an end. It is said that there were not less than fifty jacobites in the parliament of 1728. Coxe, ii. 294. [364] The tories, it is observed in the MS. journal of Mr. Yorke (second Earl of Hardwicke), showed no sign of affection to the government at the time when the invasion was expected in 1743, but treated it all with indifference. Parl. Hist. xiii. 668. In fact a disgraceful apathy pervaded the nation; and according to a letter from Mr. Fox to Mr. Winnington in 1745, which I only quote from recollection, it seemed perfectly uncertain, from this general passiveness, whether the revolution
- 68. might not be suddenly brought about. Yet very few comparatively, I am persuaded, had the slightest attachment or prejudice in favour of the house of Stuart; but the continual absence from England, and the Hanoverian predilections of the two Georges, the feebleness and factiousness of their administration, and of public men in general, and an indefinite opinion of misgovernment, raised through the press, though certainly without oppression or arbitrary acts, had gradually alienated the mass of the nation. But this would not lead men to expose their lives and fortunes; and hence the people of England, a thing almost incredible, lay quiet and nearly unconcerned, while the little army of Highlanders came every day nearer to the capital. It is absurd, however, to suppose that they could have been really successful by marching onward; though their defeat might have been more glorious at Finchley than at Culloden. [365] See Parl. Hist. xiii. 1244; and other proofs might be brought from the same work, as well as from miscellaneous authorities of the age of George II. [366] See in the Lockhart Papers, ii. 565, a curious relation of Charles Edward's behaviour in refusing to quit France after the peace of Aix-la-Chapelle. It was so insolent and absurd that the government was provoked to arrest him at the opera, and literally to order him to be bound hand and foot; an outrage which even his preposterous conduct could hardly excuse. Dr. King was in correspondence with this prince for some years after the latter's foolish, though courageous, visit to London in September 1750; which he left again in five days, on finding himself deceived by some sanguine friends. King says he was wholly ignorant of our history and constitution. "I never heard him express any noble or benevolent sentiment, the certain indications of a great soul and good heart; or discover any sorrow or compassion for the misfortune of so many worthy men who had suffered in his cause." Anecdotes of his own Times, p. 201. He goes on to charge him with love of money and other faults. But his great folly in keeping a mistress, Mrs. Walkinshaw, whose sister was housekeeper at Leicester House, alarmed the jacobites. "These were all men of fortune and distinction, and many of them persons of the first quality, who attached themselves to the P. as to a person who they imagined might be made the instrument of saving their country. They were sensible that by Walpole's administration the English government was become a system of corruption; and that Walpole's successors, who pursued his plan without any of his abilities, had reduced us to such a deplorable situation that our commercial interest was sinking, our colonies in danger of being lost, and Great Britain, which, if her powers were properly exerted, as they were afterwards in Mr. Pitt's administration, was able to give laws to other nations, was become the contempt of all Europe."— P. 208. This is in truth the secret of the continuance of jacobitism. But possibly that party were not sorry to find a pretext for breaking off so hopeless a connection, which they seem to have done about 1755. Mr. Pitt's great successes
- 69. reconciled them to the administration; and his liberal conduct brought back those who had been disgusted by an exclusive policy. On the accession of a new king they flocked to St. James's; and probably scarcely one person of the rank of a gentleman, south of the Tweed, was found to dispute the right of the house of Brunswick after 1760. Dr. King himself, it may be observed, laughs at the old passive obedience doctrine (page 193); so far was he from being a jacobite of that school. A few nonjuring congregations lingered on far into the reign of George III., presided over by the successors of some bishops whom Lloyd of Norwich, the last of those deprived at the revolution, had consecrated in order to keep up the schism. A list of these is given in D'Oyly's Life of Sancroft, vol. ii. p. 34, whence it would appear that the last of them died in 1779. I can trace the line a little farther: a bishop of that separation, named Cartwright, resided at Shrewsbury in 1793, carrying on the business of a surgeon. State Trials, xxiii. 1073. I have heard of similar congregations in the west of England still later. He had, however, become a very loyal subject to King George: a singular proof of that tenacity of life by which religious sects, after dwindling down through neglect, excel frogs and tortoises; and that, even when they have become almost equally cold-blooded! [367] Parl. Hist. viii. 904. [368] Id. vii. 536. [369] 8 Geo. 2, c. 30; Parl. Hist. viii. 883. [370] The military having been called in to quell an alleged riot at Westminster election in 1741, it was resolved (Dec. 22nd) "that the presence of a regular body of armed soldiers at an election of members to serve in parliament is a high infringement of the liberties of the subject, a manifest violation of the freedom of elections, and an open defiance of the laws and constitution of this kingdom." The persons concerned in this, having been ordered to attend the house, received on their knees a very severe reprimand from the speaker. Parl. Hist. ix. 326. Upon some occasion, the circumstances of which I do not recollect, Chief Justice Willis uttered some laudable sentiments as to the subordination of military power. [371] Lord Hardwicke threw out the militia bill in 1756, thinking some of its clauses rather too republican, and, in fact, being adverse to the scheme. Parl. Hist. xv. 704; H. Walpole's Memoirs, ii. 45; Coxe's Memoirs of Lord Walpole, 450. [372] By the act of 6 Anne, c. 7, all persons holding pensions from the Crown during pleasure were made incapable of sitting in the House of Commons; which was extended by 1 Geo. I. c. 56, to those who held them for any term of years. But the difficulty was to ascertain the fact; the government refusing information. Mr. Sandys, accordingly proposed a bill in 1730, by which every member of the
- 70. Commons was to take an oath that he did not hold any such pension, and that, in case of accepting one, he would disclose it to the house within fourteen days. This was carried by a small majority through the Commons, but rejected in the other house; which happened again in 1734 and in 1740. Parl. Hist. viii. 789; ix. 369; xi. 510. The king, in an angry note to Lord Townshend, on the first occasion, calls it "this villainous bill." Coxe's Walpole, ii. 537, 673. A bill of the same gentleman to limit the number of placemen in the house had so far worse success, that it did not reach the Serbonian bog. Parl. Hist. xi. 328, Bishop Sherlock made a speech against the prevention of corrupt practices by the pension bill, which, whether justly or not, excited much indignation, and even gave rise to the proposal of a bill for putting an end to the translation of bishops. Id. viii. 847. [373] 25 Geo. 2, c. 22. The king came very reluctantly into this measure: in the preceding session of 1742, Sandys, now become chancellor of the exchequer, had opposed it, though originally his own; alleging, in no very parliamentary manner, that the new ministry had not yet been able to remove his majesty's prejudices. Parl. Hist. xii. 896. [374] Mr. Fox declared to the Duke of Newcastle, when the office of secretary of state, and what was called the management of the House of Commons, was offered to him, "that he never desired to touch a penny of the secret service money, or to know the disposition of it farther than was necessary to enable him to speak to the members without being ridiculous." Dodington's Diary, 15th March 1754. H. Walpole confirms this in nearly the same words. Mem. of Last Ten Years, i. 332. [375] In Coxe's Memoirs of Sir R. Walpole, iii. 609, we have the draught, by that minister, of an intended vindication of himself after his retirement from office, in order to show the impossibility of misapplying public money, which, however, he does not show; and his elaborate account of the method by which payments are made out of the exchequer, though valuable in some respects, seems rather intended to lead aside the unpractised reader. [376] This secret committee were checked at every step for want of sufficient powers. It is absurd to assert, like Mr. Coxe, that they advanced accusations which they could not prove, when the means of proof were withheld. Scrope and Paxton, the one secretary, the other solicitor, to the treasury, being examined about very large sums traced to their hands, and other matters, refused to answer questions that might criminate themselves; and a bill to indemnify evidence was lost in the upper house. Parl. Hist. xii. 625 et post. [377] See vol. i. 254, 255. [378] Parl. Hist. vi. 1265. Walpole says, in speaking for Steele, "the liberty of the press is unrestrained; how then shall a part of the legislature dare to punish that
- 71. as a crime, which is not declared to be so by any law framed by the whole?" [379] Vol. i. p. 250. [380] The instances are so numerous, that to select a few would perhaps give an inadequate notion of the vast extension which privilege received. In fact, hardly anything could be done disagreeable to a member, of which he might inform the house, and cause it to be punished. [381] 12 Will. 3, ch. 3. [382] Journals, 11th Feb. It had been originally proposed, that the member making the complaint should pay the party's costs and expenses, which was amended, I presume, in consequence of some doubt as to the power of the house to enforce it. [383] 10 G. 3, c. 50. [384] Resolved, That whatever ill consequences may arise from the so long deferring the supplies for the year's service, are to be attributed to the fatal counsel of putting off the meeting of a parliament so long, and to unnecessary delays of the House of Commons. Lords' Journals, 23rd June 1701. The Commons had previously come to a vote, that all the ill consequences which may at this time attend the delay of the supplies granted by the Commons for the preserving the public peace, and maintaining the balance of Europe, are to be imputed to those who, to procure an indemnity for their own enormous crimes, have used their utmost endeavours to make a breach between the two houses. Commons' Journals, June 20th. [385] Journals, 8th May; Parl. Hist. v. 1250; Ralph, 947. This historian, who generally affects to take the popular side, inveighs against this petition, because the tories had a majority in the Commons. His partiality, arising out of a dislike to the king, is very manifest throughout the second volume. He is forced to admit afterwards, that the house disgusted the people by their votes on this occasion. P. 976. [386] History of the Kentish Petition; Somers Tracts, xi. 242; Legion's Paper; Id. 264; Vindication of the Rights of the Commons (either by Harley or Sir Humphrey Mackworth); Id. 276. This contains in many respects constitutional principles; but the author holds very strong language about the right of petitioning. After quoting the statute of Charles II. against tumults on pretence of presenting petitions, he says: "By this statute it may be observed, that not only the number of persons is restrained, but the occasion also for which they may petition; which is for the alteration of matters established in church or state, for want whereof some inconvenience may arise to that county from which the petition shall be brought. For it is plain by the express words and meaning of that statute that the grievance
- 72. or matter of the petition must arise in the same county as the petition itself. They may indeed petition the king for a parliament to redress their grievances; and they may petition that parliament to make one law that is advantageous, and repeal another that is prejudicial to the trade or interest of that county; but they have no power by this statute, nor by the constitution of the English government, to direct the parliament in the general proceedings concerning the whole kingdom; for the law declares that a general consultation of all the wise representatives of parliament is more for the safety of England than the hasty advice of a number of petitioners of a private county, of a grand jury, or of a few justices of the peace, who seldom have a true state of the case represented to them."—P. 313. These are certainly what must appear in the present day very strange limitations of the subject's right to petition either house of parliament. But it is really true that such a right was not generally recognised, nor frequently exercised, in so large an extent as is now held unquestionable. We may search whole volumes of the journals, while the most animating topics were in discussion, without finding a single instance of such an interposition of the constituent with the representative body. In this particular case of the Kentish petition, the words in the resolution, that it tended to destroy the constitution of parliament and subvert the established government, could be founded on no pretence but its unusual interference with the counsels of the legislature. With this exception, I am not aware (stating this, however, with some diffidence) of any merely political petition before the Septennial bill in 1717, against which several were presented from corporate towns; one of which was rejected on account of language that the house thought indecent; and as to these it may be observed, that towns returning members to parliament had a particular concern in the measure before the house. They relate, however, no doubt, to general policy, and seem to establish a popular principle which stood on little authority. I do not of course include the petitions to the long parliament in 1640, nor one addressed to the Convention, in 1689, from the inhabitants of London and Westminster, pressing their declaration of William and Mary; both in times too critical to furnish regular precedents. But as the popular principles of government grew more established, the right of petitioning on general grounds seems to have been better recognised; and instances may be found, during the administration of Sir Robert Walpole, though still by no means frequent. Parl. Hist. xii. 119. The city of London presented a petition against the bill for naturalisation of the Jews, in 1753, as being derogatory to the Christian religion as well as detrimental to trade. Id. xiv. 1417. It caused, however, some animadversion; for Mr. Northey, in the debate next session on the proposal to repeal this bill, alluding to this very petition, and to the comments Mr. Pelham made on it, as "so like the famous Kentish petition that if they had been treated in the same manner it would have been what they deserved," observes in reply, that the "right of petitioning either the king or the parliament in a decent and submissive manner, and without any riotous appearance against anything they
- 73. think may affect their religion and liberties, will never, I hope, be taken from the subject." Id. xv. 149; see also 376. And it is very remarkable that notwithstanding the violent clamour excited by that unfortunate statute, no petitions for its repeal are to be found in the journals. They are equally silent with regard to the marriage act, another topic of popular obloquy. Some petitions appear to have been presented against the bill for naturalisation of foreign protestants; but probably on the ground of its injurious effect on the parties themselves. The great multiplication of petitions on matters wholly unconnected with particular interests cannot, I believe, be traced higher than those for the abolition of the slave trade in 1787; though a few were presented for reform about the end of the American war, which would undoubtedly have been rejected with indignation in any earlier stage of our constitution. It may be remarked also that petitions against bills imposing duties are not received, probably on the principle that they are intended for the general interests, though affecting the parties who thus complain of them. Hatsell, iii. 200. The convocation of public meetings for the debate of political questions, as preparatory to such addresses or petitions, is still less according to the practice and precedents of our ancestors; nor does it appear that the sheriffs or other magistrates are more invested with a right of convening or presiding in assemblies of this nature than any other persons; though, within the bounds of the public peace, it would not perhaps be contended that they have ever been unlawful. But that their origin can be distinctly traced higher than the year 1769, I am not prepared to assert. It will of course be understood, that this note is merely historical, and without reference to the expediency of that change in our constitutional theory which it illustrates. [387] State Trials, xiv. 849. [388] Parl. Hist. vi. 225 et post; State Trials, xiv. 695 et post. [389] Parl. Hist. xiv. 888 et post, 1063; Walpole's Memoirs of the last Ten Years of George II., i. 15 et post. [390] Journals, vii. 9th July 1725. [391] Commons' Journals, 25th Oct. 1689. [392] Id. Dec. 5. [393] Parl. Hist. vii. 803. [394] Lords' Journals, 10th Jan. 1702; Parl. Hist. vi. 21. [395] Hargrave's Juridical Arguments, vol. i. p. 1, etc. [396] State Trials, vi. 1369; 1 Modern Reports, 159.
- 74. [397] Id., xii. 822; T. Jones, Reports, 208. [398] Journals, 10th, 12th, 19th July 1689. [399] State Trials, xiv. 849. [400] Id., viii. 30. [401] This is very elaborately and dispassionately argued by Mr. Hargrave in his Juridical Arguments, above cited; also vol. ii. p. 183. "I understand it," he says, "to be clearly part of the law and custom of parliament that each house of parliament may inquire into and imprison for breaches of privilege." But this he thinks to be limited by law; and after allowing it clearly in cases of obstruction, arrest, assault, etc., on members, admits also that "the judicative power as to writing, speaking, or publishing, of gross reflections upon the whole parliament or upon either house, though perhaps originally questionable, seems now of too long a standing and of too much frequency in practice to be well counteracted." But after mentioning the opinions of the judges in Crosby's case, Mr. H. observes: "I am myself far from being convinced that commitment for contempts by a house of parliament, or by the highest court of judicature in Westminster Hall, either ought to be, or are thus wholly privileged from all examination and appeal." [402] Mr. Justice Gould, in Crosby's case, as reported by Wilson, observes: "It is true this court did, in the instance alluded to by the counsel at the bar (Wilkes's case, 2 Wilson, 151), determine upon the privilege of parliament in the case of a libel; but then that privilege was promulged and known; it existed in records and law-books, and was allowed by parliament itself. But even in that case we now know that we were mistaken; for the House of Commons have since determined, that privilege does not extend to matters of libel." It appears, therefore, that Mr. Justice Gould thought a declaration of the House of Commons was better authority than a decision of the court of common pleas, as to a privilege which, as he says, existed in records and law-books. [403] "I am far from subscribing to all the latitude of the doctrine of attachments for contempts of the king's courts of Westminster, especially the King's Bench, as it is sometimes stated, and it has been sometimes practised." Hargrave, ii. 213. "The principle upon which attachments issue for libels on courts is of a more enlarged and important nature: it is to keep a blaze of glory around them, and to deter people from attempting to render them contemptible in the eyes of the people." Wilmot's Opinions and Judgments, p. 270. Yet the king, who seems as much entitled to this blaze of glory as his judges, is driven to the verdict of a jury before the most libellous insult on him can be punished. [404] Hargrave, ubi supra.
- 75. [405] This effect of continual new statutes is well pointed out in a speech ascribed to Sir William Wyndham in 1734: "The learned gentleman spoke (he says) of the prerogative of the Crown, and asked us if it had lately been extended beyond the bounds prescribed to it by law. Sir, I will not say that there have been lately any attempts to extend it beyond the bounds prescribed by law; but I will say that these bounds have been of late so vastly enlarged that there seems to be no great occasion for any such attempt. What are the many penal laws made within these forty years, but so many extensions of the prerogative of the Crown, and as many diminutions of the liberty of the subject? And whatever the necessity was that brought us into the enacting of such laws, it was a fatal necessity; it has greatly added to the power of the Crown, and particular care ought to be taken not to throw any more weight into that scale." Parl. Hist. ix. 463. Among the modern statutes which have strengthened the hands of the executive power, we should mention the riot act (1 Geo. I. stat. 2, c. 5), whereby all persons tumultuously assembled to the disturbance of the public peace, and not dispersing within one hour after proclamation made by a single magistrate, are made guilty of a capital felony. I am by no means controverting the expediency of this law; but, especially when combined with the aid of a military force, it is surely a compensation for much that may seem to have been thrown into the popular scale. [406] 9 Geo. 2, c. 35, sect. 10, 13; Parl. Hist. ix. 1229. I quote this as I find it: but probably the expressions are not quite correct; for the reasoning is not so. [407] Coxe's Walpole, i. 296; H. Walpole's Works, iv. 476. The former, however, seems to rest on H. Walpole's verbal communication, whose want of accuracy, or veracity, or both, is so palpable that no great stress can be laid on his testimony. I believe, however, that the fact of George I. and his minister conversing in Latin may be proved on other authority. [408] H. Walpole's Memoirs of the last Ten Years; Lord Waldegrave's Memoirs. In this well written little book, the character of George II. in reference to his constitutional position, is thus delicately drawn: "He has more knowledge of foreign affairs than most of his ministers, and has good general notions of the constitution, strength, and interest of this country; but, being past thirty when the Hanover succession took place, and having since experienced the violence of party, the injustice of popular clamour, the corruption of parliaments, and the selfish motives of pretended patriots, it is not surprising that he should have contracted some prejudices in favour of those governments where the royal authority is under less restraint. Yet prudence has so far prevailed over these prejudices, that they have never influenced his conduct. On the contrary, many laws have been enacted in favour of public liberty; and in the course of a long reign there has not been a single attempt to extend the prerogative of the Crown
- 76. beyond its proper limits. He has as much personal bravery as any man, though his political courage seems somewhat problematical; however, it is a fault on the right side; for had he always been as firm and undaunted in the closet as he showed himself at Oudenarde and Dettingen, he might not have proved quite so good a king in this limited monarchy,"—P. 5. This was written in 1757. The real tories, those I mean who adhered to the principles expressed by that name, thought the constitutional prerogative of the Crown impaired by a conspiracy of its servants. Their notions are expressed in some "Letters on the English Nation," published about 1756, under the name of Battista Angeloni, by Dr. Shebbeare, once a jacobite, and still so bitter an enemy of William III. and George I. that he stood in the pillory, not long afterwards, for a libel on those princes (among other things); on which Horace Walpole justly animadverts, as a stretch of the law by Lord Mansfield destructive of all historical truth. Memoirs of the last Ten Years, ii. 328. Shebbeare, however, was afterwards pensioned, along with Johnson, by Lord Bute, and at the time when these letters were written, may possibly have been in the Leicester House interest. Certain it is, that the self- interested cabal who belonged to that little court endeavoured too successfully to persuade its chief and her son that the Crown was reduced to a state of vassalage, from which it ought to be emancipated; and the government of the Duke of Newcastle, as strong in party connection as it was contemptible in ability and reputation, afforded them no bad argument. The consequences are well known, but do not enter into the plan of this work. [409] Many proofs of this occur in the correspondence published by Mr. Coxe. Thus Horace Walpole writing to his brother Sir Robert, in 1739, says: "King William had no other object but the liberties and balance of Europe; but, good God! what is the case now? I will tell you in confidence; little, low, partial, electoral notions are able to stop or confound the best conducted project for the public." Memoirs of Sir R. Walpole, iii. 535. The Walpoles had, some years before, disapproved the policy of Lord Townshend on account of his favouring the king's Hanoverian prejudices. Id. i. 334. And, in the preceding reign, both these whig leaders were extremely disgusted with the Germanism and continual absence of George I. (Id. ii. 116, 297), though first Townshend, and afterwards Walpole, according to the necessity, or supposed necessity, which controls statesmen (that is, the fear of losing their places), became in appearance the passive instruments of royal pleasure. It is now, however, known that George II. had been induced by Walpole to come into a scheme, by which Hanover, after his decease, was to be separated from England. It stands on the indisputable authority of Speaker Onslow. "A little while before Sir Robert Walpole's fall (and as a popular act to save himself, for he went very unwillingly out of his offices and power), he took me one day aside, and said, 'What will you say, speaker, if this hand of mine shall bring a message from the
- 77. king to the House of Commons, declaring his consent to having any of his family, after his death, to be made, by act of parliament, incapable of inheriting and enjoying the crown, and possessing the electoral dominions at the same time?' My answer was, 'Sir, it will be as a message from heaven.' He replied, 'It will be done.' But it was not done; and I have good reason to believe, it would have been opposed, and rejected at that time, because it came from him, and by the means of those who had always been most clamorous for it; and thus perhaps the opportunity was lost: when will it come again? It was said that the prince at that juncture would have consented to it, if he could have had the credit and popularity of the measure, and that some of his friends were to have moved it in parliament, but that the design at St. James's prevented it. Notwithstanding all this, I have had some thoughts that neither court ever really intended the thing itself; but that it came on and went off, by a jealousy of each other in it, and that both were equally pleased that it did so, from an equal fondness (very natural) for their own native country." Notes on Burnet (iv. 490, Oxf. edit.). This story has been told before, but not in such a manner as to preclude doubt of its authenticity. [410] A bill was brought in for this purpose in 1712, which Swift, in his History of the Last Four Years, who never printed anything with his name, naturally blames. It miscarried, probably on account of this provision. Parl. Hist. vi. 1141. But the queen, on opening the session, in April 1713, recommended some new law to check the licentiousness of the press. Id. 1173. Nothing, however, was done in consequence. [411] Bolingbroke's letter to the Examiner, in 1710, excited so much attention that it was answered by Lord Cowper, then chancellor, in a letter to the Tatler (Somers Tracts, xiii. 75), where Sir Walter Scott justly observes, that the fact of two such statesmen becoming the correspondents of periodical publications shows the influence they must have acquired over the public mind. [412] It was resolved, nem. con., Feb. 26th, 1729, That it is an indignity to, and a breach of the privilege of, this house, for any person to presume to give, in written or printed newspapers, any account or minutes of the debates, or other proceedings of this house or of any committee thereof; and that upon discovery of the authors, etc., this house will proceed against the offenders with the utmost severity. Parl. Hist. viii. 683. There are former resolutions to the same effect. The speaker having himself brought the subject under consideration some years afterwards, in 1738, the resolution was repeated in nearly the same words, but after a debate wherein, though no one undertook to defend the practice, the danger of impairing the liberty of the press was more insisted upon than would formerly have been usual; and Sir Robert Walpole took credit to himself, justly enough, for respecting it more than his predecessors. Id. x. 800; Coxe's Walpole, i. 572. Edward Cave, the well-known editor of the Gentleman's Magazine, and the
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