![The governing equations of fluid dynamics 5
The rates of strain are the primary cause of the general stresses, σij , and these are
defined in a manner analogous to that of infinitesimal strain in solid mechanics as
˙
ij =
1
2
∂ui
∂xj
+
∂uj
∂xi
(1.2)
This is a well-known tensorial definition of strain rates but for use later in variational
forms is written as a vector which is more convenient in finite element analysis. Details
of such matrix forms are given fully in reference 1 but for completeness we summarize
them here. Thus, the strain rate is written as a vector ( ˙
) and is given by the following
form
˙
= [˙
11, ˙
22, 2˙
12]T
= [˙
11, ˙
22, γ̇12]T
(1.3a)
in two dimensions with a similar form in three dimensions
˙
= [˙
11, ˙
22, ˙
33, 2˙
12, 2˙
23, 2˙
31]T
(1.3b)
When such vector forms are used we can write the strain rate vector in the form
˙
= Su (1.4)
where S is known as the strain rate operator and u is the velocity given in Eq. (1.1).
The stress–strain rate relations for a linear (Newtonian) isotropic fluid require the
definition of two constants. The first of these links the deviatoric stresses τij to the
deviatoric strain rates:
τij ≡ σij − 1
3
δij σkk = 2µ
˙
ij − 1
3
δij ˙
kk
(1.5)
In the above equation the quantity in brackets is known as the deviatoric strain rate,
δij is the Kronecker delta, and a repeated index implies summation over the range of
the index; thus
σkk ≡ σ11 + σ22 + σ33 and ˙
kk ≡ ˙
11 + ˙
22 + ˙
33 (1.6)
The coefficient µ is known as the dynamic (shear) viscosity or simply viscosity and
is analogous to the shear modulus G in linear elasticity.
The second relation is that between the mean stress changes and the volumetric strain
rate. This defines the pressure as
p = − 1
3
σkk = − κ ˙
kk + p0 (1.7)
where κ is a volumetric viscosity coefficient analogous to the bulk modulus K in linear
elasticity and p0 is the initial hydrostatic pressure independent of the strain rate (note
that p and p0 are invariably defined as positive when compressive).
We can immediately write the ‘constitutive’ relation for fluids from Eqs (1.5) and
(1.7)
σij = τij − δij p
= 2µ
˙
ij − 1
3
δij ˙
kk
+ κ δij ˙
kk − δij p0
(1.8a)](https://image.slidesharecdn.com/40011-250302153324-c440393d/85/The-Finite-Element-Method-for-Fluid-Dynamics-Sixth-Edition-O-C-Zienkiewicz-25-320.jpg)
![The governing equations of fluid dynamics 7
It is possible to recast the above equations in relation to a moving frame of reference
and, if the motion follows the particle, the equations will be named ‘Lagrangian’.
Such Lagrangian frame of reference is occasionally used in fluid dynamics and briefly
discussed in Chapter 6.
1.2.3 Momentum conservation: dynamic equilibrium
In the jth direction the balance of linear momentum leaving and entering the control
volume (Fig. 1.1) is to be in dynamic equilibrium with the stresses σij and body forces
ρgj . This gives a typical component equation
∂(ρuj )
∂t
+
∂
∂xi
[(ρuj )ui] −
∂
∂xi
(σij ) − ρgj = 0 (1.13)
or using Eq. (1.11a),
∂(ρuj )
∂t
+
∂
∂xi
[(ρuj )ui] −
∂τij
∂xi
+
∂p
∂xj
− ρgj = 0 (1.14)
with Eq. (1.11b) implied.
The conservation of angular momentum merely requires the stress to be symmetric,
i.e.
σij = σji or τij = τji
In the sequel we will use the term momentum conservation to imply both linear and
angular forms.
1.2.4 Energy conservation and equation of state
We note that in the equations of Secs 1.2.2 and 1.2.3 the dependent variables are ui (the
velocity components), p (the pressure) and ρ (the density). The deviatoric stresses,
dx3
x2; (y)
dx2
dx1
x1; (x)
x3; (z)
Fig. 1.1 Coordinate direction and the infinitesimal control volume.](https://image.slidesharecdn.com/40011-250302153324-c440393d/85/The-Finite-Element-Method-for-Fluid-Dynamics-Sixth-Edition-O-C-Zienkiewicz-27-320.jpg)
![Incompressible (or nearly incompressible) flows 13
where H is the enthalpy, given as H = uiui/2 + p/ρ.
If isothermal conditions pertain, the specific energy is constant and Eq. (1.39) implies
that
−
∂φ
∂t
+
1
2
uiui +
p
ρ
+ P = constant (1.40)
over the whole domain. This can be taken as a corollary of the existence of the potential
and indeed is a condition for its existence. In steady-state flows it provides the well-
known Bernoulli equation that allows the pressures to be determined throughout the
whole potential field once the value of the constant is established.
We note that the governing potential equation (1.31) is self-adjoint (see Appendix
B) and that the introduction of the potential has side-stepped the difficulties of dealing
with convective terms. It is also of interest to note that the Laplacian equation, which
is obeyed by the velocity potential, occurs in other contexts. For instance, in two-
dimensional flow it is convenient to introduce a stream function the contours of which
lie along the streamlines. The stream function, ψ, defines the velocities as
u1 =
∂ψ
∂x2
and u2 = −
∂ψ
∂x1
(1.41)
which satisfy the incompressibility condition (1.28)
∂ui
∂xi
=
∂
∂x1
∂ψ
∂x2
+
∂
∂x1
−
∂ψ
∂x1
= 0 (1.42)
For an existence of a unique potential for irrotational flow we note that ω12 = 0
[Eq. (1.34)] gives a Laplacian equation
∂2
ψ
∂xi∂xi
= ∇2
ψ = 0 (1.43)
The stream function is very useful in getting a pictorial representation of flow. In
Appendix C we show how the stream function can be readily computed from a known
distribution of velocities.
1.4 Incompressible (or nearly incompressible) flows
We observed earlier that the Navier–Stokes equations are completed by the existence
of a state relationship giving [Eq. (1.15a)]
ρ = ρ(p, T )
In (nearly) incompressible relations we shall frequently assume that:
(a) The problem is isothermal.
(b) The variation of ρ with p is very small, i.e. such that in product terms of velocity
and density the latter can be assumed constant.
The first assumption will be relaxed, as we shall see later, allowing some thermal
coupling via the dependence of the fluid properties on temperature. In such cases we](https://image.slidesharecdn.com/40011-250302153324-c440393d/85/The-Finite-Element-Method-for-Fluid-Dynamics-Sixth-Edition-O-C-Zienkiewicz-33-320.jpg)
![Numerical solutions: weak forms, weighted residual and finite element approximation 17
Na
a
x
y
~
Fig. 1.2 Basis function in linear polynomials for a patch of trianglular elements.
equal to two) and therefore reduce the constraints on choosing the basis functions to
permit integration over individual elements using Eq. (1.54). In the present case, for
instance, the weighted residual after integration by parts and introducing the natural
boundary condition becomes
∂Wb
∂xi
k
a
∂Na
∂xi
φ̃
d +
WbQ d +
q
Wb q̄n d = 0 (1.57)
The Galerkin, finite element, method
In the Galerkin method we simply take Wb = Nb which gives the assembled system of
equations
n
a=1
Kbaφ̃
a
+ fb = 0; b = 1, 2, · · · , n − r (1.58)
wherer isthenumberofnodesappearingintheapproximationtotheDirichletboundary
condition [i.e. Eq. (1.47a)] and Kba is assembled from element contributions Ke
ba with
Ke
ba =
e
∂Nb
∂xi
k
∂Na
∂xi
d (1.59)
Similarly, fb is computed from the element as
f e
b =
e
Nb Q d +
eq
Nb q̄n d (1.60)
To impose the Dirichlet boundary condition we replace φ̃a
by φ̄a for the r boundary
nodes.
It is evident in this example that the Galerkin method results in a symmetric set of
algebraic equations (e.g. Kba = Kab). However, this only happens if the differential
equations are self-adjoint (see Appendix B). Indeed the existence of symmetry pro-
vides a test for self-adjointness and also for existence of a variational principle whose
stationarity is sought.1](https://image.slidesharecdn.com/40011-250302153324-c440393d/85/The-Finite-Element-Method-for-Fluid-Dynamics-Sixth-Edition-O-C-Zienkiewicz-37-320.jpg)
![Numerical solutions: weak forms, weighted residual and finite element approximation 19
where xa
i is the i direction coordinate of node a and other coefficients are obtained by
cyclic permutation of the subscripts in the order 1, 2, 3, and
2 = det
1 x1
1 x1
2
1 x2
1 x2
2
1 x3
1 x3
2
= 2 · (area of triangle 123) (1.66)
From Eq. (1.64) we see that the shape functions are given by
Na = (aa + ba x1 + ca x2)/(2); a = 1, 2, 3 (1.67)
Since the unknown nodal quantities defined by these shape functions vary linearly
along any side of a triangle the interpolation equation (1.64) guarantees continuity
between adjacent elements and, with identical nodal values imposed, the same scalar
variable value will clearly exist along an interface between elements. We note, however,
that in general the derivatives will not be continuous between element assemblies.1
Example 1.2 Poisson equation in two dimensions: Galerkin
formulation with triangular elements
The relations for a Galerkin finite elemlent solution have been given in Eqs (1.58)
to (1.60). The components of Kba and fb can be evaluated for a typical element or
subdomain and the system of equations built by standard methods.
For instance, considering the set of nodes and elements shown shaded in Fig. 1.4(a),
to compute the equation for node 1 in the assembled patch, it is only necessary to
compute the Ke
ba for two element shapes as indicated in Fig. 1.4(b). For the Type
1 element [left element in Fig. 1.4(b)] the shape functions evaluated from Eq. (1.67)
using Eqs (1.65) and (1.66) gives
N1 = 1 −
x2
h
; N2 =
x1
h
; N3 =
x2 − x1
h
thus, the derivatives are given by:
∂N
∂x1
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
∂N1
∂x1
∂N2
∂x1
∂N3
∂x1
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
0
1
h
−
1
h
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
and
∂N
∂x2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
∂N1
∂x2
∂N2
∂x2
∂N3
∂x2
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
−
1
h
0
1
h
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
Similarly, for the Type 2 element [right element in Fig. 1.4(b)] the shape functions
are expressed by
N1 = 1 −
x1
h
; N2 =
x1 − x2
h
; N3 =
x2
h](https://image.slidesharecdn.com/40011-250302153324-c440393d/85/The-Finite-Element-Method-for-Fluid-Dynamics-Sixth-Edition-O-C-Zienkiewicz-39-320.jpg)
The Finite Element Method for Fluid Dynamics Sixth Edition O. C. Zienkiewicz
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- 5. The Finite Element Method for Fluid Dynamics Sixth Edition O. C. Zienkiewicz Digital Instant Download Author(s): O. C. Zienkiewicz, R. L. Taylor, P. Nithiarasu, ISBN(s): 9780750663229, 0750663227 Edition: 6 File Details: PDF, 23.98 MB Year: 2005 Language: english
- 7. The Finite Element Method for Fluid Dynamics Sixth edition
- 8. Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies – an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years’experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contri- butions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics – USACM (1996) and a Fellow of the Interna- tionalAssociation of Computational Mechanics – IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss–Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, avail- able from the publisher’s website, is incorporated into the book. Dr P. Nithiarasu, Senior Lecturer at the School of Engineering, University of Wales Swansea, has over ten years’experience in finite element based computational fluid dynam- ics research. He moved to Swansea in 1996 after completing his PhD research at IIT Madras. He was awarded the Zienkiewicz silver medal and prize of the Institution of Civil Engineers, UK in 2002. In 2004 he was selected to receive the European Community on Computational Methods in Applied Sciences (ECCOMAS) award for young scientists in computational engineering sciences. Dr Nithiarasu is the author of several articles in the area of fluid dynamics, porous medium flows and the finite element method.
- 9. The Finite Element Method for Fluid Dynamics Sixth edition O.C. Zienkiewicz, CBE, FRS Professor Emeritus, Civil and Computational Engineering Centre University of Wales Swansea UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona R.L. Taylor Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California P. Nithiarasu Civil and Computational Engineering Centre School of Engineering University of Wales Swansea AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
- 10. Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Copyright c 2000, 2005. O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. All rights reserved The right of O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu to be identified as the authors of this work have been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90, Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 6322 7 Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es) For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com
- 11. Dedication This book is dedicated to our wives Helen, Mary Lou and Sujatha and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Oñate and his group at CIMNE for their help, encouragement and support during the preparation process.
- 13. Contents Preface xi Acknowledgements xiii 1 Introduction to the equations of fluid dynamics and the finite element approximation 1 1.1 General remarks and classification of fluid dynamics problems discussed in this book 1 1.2 The governing equations of fluid dynamics 4 1.3 Inviscid, incompressible flow 11 1.4 Incompressible (or nearly incompressible) flows 13 1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 14 1.6 Concluding remarks 26 References 27 2 Convection dominated problems – finite element approximations to the convection–diffusion-reaction equation 28 2.1 Introduction 28 2.2 The steady-state problem in one dimension 31 2.3 The steady-state problem in two (or three) dimensions 45 2.4 Steady state -- concluding remarks 49 2.5 Transients -- introductory remarks 50 2.6 Characteristic-based methods 53 2.7 Taylor--Galerkin procedures for scalar variables 65 2.8 Steady-state condition 66 2.9 Non-linear waves and shocks 66 2.10 Treatment of pure convection 70 2.11 Boundary conditions for convection--diffusion 72 2.12 Summary and concluding remarks 73 References 74 3 The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow 79 3.1 Introduction 79 3.2 Non-dimensional form of the governing equations 81 3.3 Characteristic-based split (CBS) algorithm 82
- 14. viii Contents 3.4 Explicit, semi-implicit and nearly implicit forms 92 3.5 Artificial compressibility and dual time stepping 95 3.6 ‘Circumvention’ of the Babuška--Brezzi (BB) restrictions 97 3.7 A single-step version 98 3.8 Boundary conditions 100 3.9 The performance of two-step and one-step algorithms on an inviscid problem 103 3.10 Concluding remarks 104 References 105 4 Incompressible Newtonian laminar flows 110 4.1 Introduction and the basic equations 110 4.2 Use of the CBS algorithm for incompressible flows 112 4.3 Adaptive mesh refinement 123 4.4 Adaptive mesh generation for transient problems 131 4.5 Slow flows -- mixed and penalty formulations 131 4.6 Concluding remarks 136 References 136 5 Incompressible non-Newtonian flows 141 5.1 Introduction 141 5.2 Non-Newtonian flows -- metal and polymer forming 141 5.3 Viscoelastic flows 154 5.4 Direct displacement approach to transient metal forming 163 5.5 Concluding remarks 165 References 166 6 Free surface and buoyancy driven flows 170 6.1 Introduction 170 6.2 Free surface flows 170 6.3 Buoyancy driven flows 189 6.4 Concluding remarks 191 References 193 7 Compressible high-speed gas flow 197 7.1 Introduction 197 7.2 The governing equations 198 7.3 Boundary conditions -- subsonic and supersonic flow 199 7.4 Numerical approximations and the CBS algorithm 202 7.5 Shock capture 203 7.6 Variable smoothing 205 7.7 Some preliminary examples for the Euler equation 206 7.8 Adaptive refinement and shock capture in Euler problems 212 7.9 Three-dimensional inviscid examples in steady state 217 7.10 Transient two- and three-dimensional problems 226 7.11 Viscous problems in two dimensions 227 7.12 Three-dimensional viscous problems 240
- 15. Contents ix 7.13 Boundary layer--inviscid Euler solution coupling 241 7.14 Concluding remarks 242 References 242 8 Turbulent flows 248 8.1 Introduction 248 8.2 Treatment of incompressible turbulent flows 251 8.3 Treatment of compressible flows 264 8.4 Large eddy simulation 267 8.5 Detached Eddy Simulation (DES) 270 8.6 Direct Numerical Simulation (DNS) 270 8.7 Concluding remarks 271 References 271 9 Generalized flow through porous media 274 9.1 Introduction 274 9.2 A generalized porous medium flow approach 275 9.3 Discretization procedure 279 9.4 Non-isothermal flows 282 9.5 Forced convection 282 9.6 Natural convection 284 9.7 Concluding remarks 288 References 289 10 Shallow water problems 292 10.1 Introduction 292 10.2 The basis of the shallow water equations 293 10.3 Numerical approximation 297 10.4 Examples of application 298 10.5 Drying areas 310 10.6 Shallow water transport 311 10.7 Concluding remarks 313 References 314 11 Long and medium waves 317 11.1 Introduction and equations 317 11.2 Waves in closed domains -- finite element models 318 11.3 Difficulties in modelling surface waves 320 11.4 Bed friction and other effects 320 11.5 The short-wave problem 320 11.6 Waves in unbounded domains (exterior surface wave problems) 321 11.7 Unbounded problems 324 11.8 Local Non-Reflecting Boundary Conditions (NRBCs) 324 11.9 Infinite elements 327 11.10 Mapped periodic (unconjugated) infinite elements 327 11.11 Ellipsoidal type infinite elements of Burnett and Holford 328 11.12 Wave envelope (or conjugated) infinite elements 330 11.13 Accuracy of infinite elements 332
- 16. x Contents 11.14 Trefftz type infinite elements 332 11.15 Convection and wave refraction 333 11.16 Transient problems 335 11.17 Linking to exterior solutions (or DtN mapping) 336 11.18 Three-dimensional effects in surface waves 338 11.19 Concluding remarks 344 References 344 12 Short waves 349 12.1 Introduction 349 12.2 Background 349 12.3 Errors in wave modelling 351 12.4 Recent developments in short wave modelling 351 12.5 Transient solution of electromagnetic scattering problems 352 12.6 Finite elements incorporating wave shapes 352 12.7 Refraction 364 12.8 Spectral finite elements for waves 372 12.9 Discontinuous Galerkin finite elements (DGFE) 374 12.10 Concluding remarks 378 References 378 13 Computer implementation of the CBS algorithm 382 13.1 Introduction 382 13.2 The data input module 383 13.3 Solution module 384 13.4 Output module 387 References 387 Appendix A Non-conservative form of Navier–Stokes equations 389 Appendix B Self-adjoint differential equations 391 Appendix C Postprocessing 392 Appendix D Integration formulae 395 Appendix E Convection–diffusion equations: vector-valued variables 397 Appendix F Edge-based finite element formulation 405 Appendix G Multigrid method 407 Appendix H Boundary layer–inviscid flow coupling 409 Appendix I Mass-weighted averaged turbulence transport equations 413 Author index 417 Subject index 427
- 17. Preface The major part of this book has been derived by updating the third volume of the fifth edition. However, it now contains three new chapters and also major improvements in the existing ones. Its objective is to separate the fluid dynamics formulations and applications from those of solid mechanics and thus to reach perhaps a different interest group. It is our intention that the present text could be used by investigators familiar with the finite element method in general terms and introduce them to the subject of fluid dynamics. It can thus in many ways stand alone. Although the finite element dis- cretization is briefly covered here, many of the general finite element procedures may not be familiar to a reader introduced to the finite element method through different texts and therefore we advise that this volume be used in conjunction with the text on ‘The Finite Element Method: Its Basis and Fundamentals’by Zienkiewicz, Taylor and Zhu to which we make frequent reference. In fluid dynamics, several difficulties arise. The first is that of dealing with incom- pressibleoralmostincompressiblesituations. Theseaswealreadyknowpresentspecial difficulties in formulation even in solids. The second difficulty is introduced by the convection which requires rather specialized treatment and stabilization. Here, partic- ularly in the field of compressible, high speed, gas flow many alternative finite element approaches are possible and often different algorithms for different ranges of flow have been suggested. Although slow creeping flows may well be dealt with by procedures almost identical to those of solid mechanics, the high speed range of supersonic and hypersonic kind will require a very particular treatment. In this text we shall use the so-called Characteristic-Based Split (CBS) introduced a few years ago by the authors. It turns out that this algorithm is applicable to all ranges of flow and indeed gives results which are at least equal to those of specialized methods. We organized the text into 13 individual chapters. The first chapter introduces the topic of fluid dynamics and summarizes all relevant partial differential equations together with appropriate constitutive relations. Chapter 1 also provides a brief sum- mary of the finite element formulation. In Chapter 2 we discuss convection stabilization procedures for convection–diffusion–reaction equations. Here, we make reference to methods available for steady and transient state equations and also one and multi- dimensional equations. We also discuss the similarity between various stabilization procedures. From Chapter 3 onwards the discussion is centred around the numerical
- 18. xii Preface solution of fluid dynamic equations. In Chapter 3, the CBS scheme is introduced and discussed in detail in its various forms. Its simplicity and universality makes it highly desirable for the study of incompressible and compressible flows and in the later chap- ters we shall indicate its widely applicable use. Though not all problems are necessarily solved using this method in this book, as work of several decades are reported here, the reader shall find the CBS method in general at least as accurate as other meth- ods and that its performance is very good. For this reason we do not describe any other alternatives to make the reader’s life simple. The topic of incompressible fluid dynamics is covered in Chapters 4, 5 and 6. Chapter 4 discusses the general Newtonian incompressible flows without reference to any special problems. This chapter could be used as a validating part of any fluid dynamics code development for incompressible flows. Chapter 5 discusses the non-Newtonian flows in general and metal forming and visco-elastic flows in particular. In Chapter 6 we discuss the special topics of gravity assisted incompressible flows which include treatment of free surfaces and buoyancy driven flows. Chapter 7 is devoted to compressible gas flows. Here, we discuss several special requirements for solving Navier–Stokes equations including phenomena such as shock capturing and adaptivity. Chapters 8 and 9 are new additions to the book. In Chapter 8 we discuss various basic turbulence modelling options available for both compressible and incompressible flows and in Chapter 9 we provide a brief description of flow through porous media. Chapter 10 discusses the shallow water flow and here application of the CBS scheme to a different incompressible flow approximation is con- sidered. Although the flow is incompressible the approximations and variables involved produce a set of differential equations similar to those of compressible flows. Thus, the use of methods already derived for the solution of compressible flow is obvious for dealing with shallow water problems. Chapters 11 and 12 provide a detailed overview on the numerical treatment of long and short waves. Chapter 12 is a new chapter and both these chapters on waves are contributed by Professor Peter Bettess, University of Durham. The last chapter of this book is a brief outline on computer implementation. Further details, including source codes, are available from the author’s personal home pages www.nithiarasu.co.uk and www.elsevier.com. We hope that the book will be useful in introducing the reader to the complex subject of computational fluid dynamics (CFD) and its many facets. Further, we hope it will also be of use to the experienced practitioner of CFD who may find the new presentation of interest to practical application.
- 19. Acknowledgements The authors would like to thank Professor Peter Bettess for largely contributing the chapters on waves (Chapters 11 and 12), in which he has made so many achievements, and Dr Pablo Ortiz who with the main author was first to apply the CBS algorithm to shallow water equations and Chapter 10 of this text is partly contributed by him. Several other colleagues contributed to this text either directly or indirectly. Professors K. Morgan, N.P. Weatherill and O. Hassan, all from the University of Wales Swansea, Professors E. Onãte and R. Codina, both from CIMNE, Barcelona, Professor J. Peraire from MIT and Professor R. Löhner from George Mason University, USA, are a few to name. The third author thanks Professor P.G. Tucker, University of Wales, Swansea, and Dr S. Vengadesan, IIT, Madras, for their constructive comments on the chapter on turbulence. The third author also thanks his graduate students Ray Hickey and Chun-Bin Liu for their assistance.
- 21. 1 Introduction to the equations of fluid dynamics and the finite element approximation 1.1 General remarks and classification of fluid dynamics problems discussed in this book The problems of solid and fluid behaviour are in many respects similar. In both media stresses occur and in both the material is displaced. There is, however, one major difference. Fluids cannot support any deviatoric stresses when at rest. Thus only a pressure or a mean compressive stress can be carried. As we know, in solids deviatoric stresses can exist and a solid material can support general forms of structural forces. In addition to pressure, deviatoric stresses can develop when the fluid is in motion and such motion of the fluid will always be of primary interest in fluid dynamics. We shall therefore concentrate on problems in which displacement is continuously changing and in which velocity is the main characteristic of the flow. The deviatoric stresses which can now occur will be characterized by a quantity that has great resemblance to the shear modulus of solid mechanics and which is known as dynamic viscosity (molecular viscosity). Up to this point the equations governing fluid flow and solid mechanics appear to be similar with the velocity vector u replacing the displacement which often uses the same symbol. However, there is one further difference, even when the flow has a constant velocity (steady state), convective acceleration effects add terms which make the fluid dynamics equations non-self-adjoint. Therefore, in most cases, unless the velocities are very small so that the convective acceleration is negligible, the treatment has to be somewhat different from that of solid mechanics. The reader should note that for self-adjoint forms, approximating the equations by the Galerkin method gives the minimum error in the energy norm and thus such approximations are in a sense optimal. In general, this is no longer true in fluid mechanics, though for slow flows (creeping flows) where the convective acceleration terms are negligible the situation is somewhat similar. With a fluid which is in motion, conservation of mass is always necessary and, unless the fluid is highly compressible, we require that the divergence of the velocity vector be zero. Similar problems are encountered in the context of incompressible elasticity
- 22. 2 Introduction to the equations of fluid dynamics and the finite element approximation and the incompressibility constraint can introduce difficulties in the formulation (viz. reference 1). In fluid mechanics the same difficulty again arises and all fluid mechanics approximations have to be such that, even if compressibility is possible, the limit of incompressibility can be modelled. This precludes the use of many elements which are otherwise acceptable. In this book we shall introduce the reader to finite element treatment of the equa- tions of motion for various problems of fluid mechanics. Much of the activity in fluid mechanics has, however, pursued a finite difference formulation and more recently a derivative of this known as the finite volume technique. Competition between finite element methods and techniques of finite differences have appeared and led to a much slower adoption of the finite element process in fluid dynamics than in structures. The reasons for this are perhaps simple. In solid mechanics or structural problems, the treat- ment of continua often arises in combination with other structural forms, e.g. trusses, beams, plates and shells. The engineer often dealing with structures composed of structural elements does not need to solve continuum problems. In addition when con- tinuum problems are encountered, the system can lead to use of many different material models which are easily treated using a finite element formulation. In fluid mechanics, practically all situations of flow require a two- or three-dimensional treatment and here approximation is required. This accounts for the early use of finite differences in the 1950s before the finite element process was made available. However, as pointed out in reference 1, there are many advantages of using the finite element process. This not only allows a fully unstructured and arbitrary domain subdivision to be used but also provides an approximation which in self-adjoint problems is always superior to or at least equal to that provided by finite differences. A methodology which appears to have gained an intermediate position is that of finite volumes, which were initially derived as a subclass of finite difference methods. As shown later in this chapter, these are simply another kind of finite element form in which subdomain collocation is used. We do not see much advantage in using this form of approximation; however, there is one point which seems to appeal to some investigators. That is the fact the finite volume approximation satisfies conservation conditions for each finite volume. This does not carry over to the full finite element analysis where generally satisfaction of conservation conditions is achieved only in an assembly region of elements surrounding each node. Satisfaction of the conservation conditions on an individual element is not an advantage if the general finite element approximation gives results which are superior. In this book we will discuss various classes of problems, each of which has a certain behaviour in the numerical solution. Here we start with incompressible flows or flows where the only change of volume is elastic and associated with transient changes of pressure (Chapters 4 and 5). For such flows full incompressible constraints must be available. Further, with very slow speeds, convective acceleration effects are often negligible and the solution can on occasion be reached using identical programs to those derived for linear incompressible elasticity. This indeed was the first venture of finite element developers into the field of fluid mechanics thus transferring the direct knowledge from solid mechanics to fluids. In particular the so-called linear Stokes flow is the case where fully incompressible but elastic behaviour occurs. A particular variant of Stokes flow is that used in metal forming where the material can no longer be described by a constant
- 23. General remarks for fluid mechanics problems 3 viscosity but possesses a viscosity which is non-Newtonian and depends on the strain rates. Chapter 5 is partly devoted to such problems. Here the fluid formulation (flow formulation) can be applied directly to problems such as the forming of metals or plastics and we shall discuss this extreme situation in Chapter 5. However, even in incompressible flows, when the speed increases convective acceleration terms become important. Here often steady-state solutions do not exist or at least are extremely unstable. This leads us to such problems as vortex shedding. Vortex shedding indicates thestartofinstabilitywhichbecomesveryirregularandindeedrandomwhenhighspeed flow occurs in viscous fluids. This introduces the subject of turbulence, which occurs frequently in fluid dynamics. In turbulent flows random fluctuation of velocity occurs at all points and the problem is highly time dependent. With such turbulent motion, it is possible to obtain an averaged solution using time averaged equations. Details of some available time averaged models are summarized in Chapter 8. Chapter 6 deals with incompressible flow in which free surface and other gravity con- trolled effects occur. In particular we show three different approaches for dealing with free surface flows and explain the necessary modifications to the general formulation. The next area of fluid dynamics to which much practical interest is devoted is of course that of flow of gases for which the compressibility effects are much larger. Here compressibility is problem dependent and generally obeys gas laws which relate the pressure to temperature and density. It is now necessary to add the energy conservation equation to the system governing the motion so that the temperature can be evaluated. Such an energy equation can of course be written for incompressible flows but this shows only a weak or no coupling with the dynamics of the flow. This is not the case in compressible flows where coupling between all equations is very strong. In such compressible flows the flow speed may exceed the speed of sound and this may lead to shock development. This subject is of major importance in the field of aerodynamics and we shall devote a part of Chapter 7 to this particular problem. In a real fluid, viscosity is always present but at high speeds such viscous effects are confined to a narrow zone in the vicinity of solid boundaries (the so-called boundary layer). In such cases, the remainder of the fluid can be considered to be inviscid. There we can return to the fiction of an ideal fluid in which viscosity is not present and here various simplifications are again possible. Such simplifications have been used since the early days of aerodynamics and date back to the work of Prandtl and Schlichting.2 One simplification is the introduction of potential flow and we shall mention this later in this chapter. Potential flows are indeed the topic of many finite element investigators, but unfortunately such solutions are not easily extendible to realistic problems. A particular form of viscous flow problem occurs in the modelling of flow in porous media. This important field is discussed in Chapter 9. In the topic of flow through porous media, two extreme situations are often encountered. In the first, the porous mediumisstationaryandthefluidflowoccursonlyinthenarrowpassagesbetweensolid grains. Such an extreme is the basis of porous medium flow modelling in applications such as geo-fluid dynamics where the flow of water or oil through porous rocks occurs. The other extreme of porous media flow is the one in which the solid occupies only a small part of the total volume (for example, representing thermal insulation systems, heat exchangers etc.). In such problems flow is almost the same as that occurring in fluids without the solid phase which only applies an added, distributed, resistance to flow. Both extremes are discussed in Chapter 9.
- 24. 4 Introduction to the equations of fluid dynamics and the finite element approximation Another major field of fluid mechanics of interest to us is that of shallow water flows that occur in coastal estuaries or elsewhere. In this class of problems the depth dimension of flow is very much less than the horizontal ones. Chapter 10 will deal with such problems in which essentially the distribution of pressure in the vertical direction is almost hydrostatic. For such shallow water problems a free surface also occurs and this dominates the flow characteristics and here we note that shallow water flow problems result in a formulation which is closely related to gas flow. Whenever a free surface occurs it is possible for transient phenomena to happen, generating waves such as those occurring in oceans or other bodies of water. We have introduced in this book two chapters (Chapters 11 and 12) dealing with this particular aspect of fluid dynamics. Such wave phenomena are also typical of some other physical problems. For instance, acoustic and electromagnetic waves can be solved using similar approaches. Indeed, one can show that the treatment for this class of problems is very similar to that of surface wave problems. In what remains of this chapter we shall introduce the general equations of fluid dynamics valid for most compressible or incompressible flows showing how the par- ticular simplification occurs in some categories of problems mentioned above. How- ever, before proceeding with the recommended discretization procedures, which we present in Chapter 3, we must introduce the treatment of problems in which convection and diffusion occur simultaneously. This we shall do in Chapter 2 using the scalar convection–diffusion–reaction equation. Based on concepts given in Chapter 2, Chap- ter 3 will introduce a general algorithm capable of solving most of the fluid mechanics problems encountered in this book. There are many possible algorithms and very often specialized ones are used in different areas of applications. However, the general algo- rithm of Chapter 3 produces results which are at least as good as others achieved by more specialized means. We feel that this will give a certain unification to the whole subject of fluid dynamics and, without apology, we will omit reference to many other methods or discuss them only in passing. For completeness we shall show in the present chapter some detail of the finite element process to avoid the repetition of basic finite element presentations which we assume are known to the reader either from reference 1 or from any of the numerous texts available. 1.2 The governing equations of fluid dynamics3−9 1.2.1 Stresses in fluids As noted above, the essential characteristic of a fluid is its inability to sustain deviatoric stresses when at rest. Here only hydrostatic ‘stress’or pressure is possible. Any analysis must therefore concentrate on the motion, and the essential independent variable is the velocity u or, if we adopt indicial notation (with the coordinate axes referred to as xi, i = 1, 2, 3), ui, i = 1, 2, 3 or u = u1, u2, u3 T (1.1) This replaces the displacement variable which is of primary importance in solid mechanics.
- 25. The governing equations of fluid dynamics 5 The rates of strain are the primary cause of the general stresses, σij , and these are defined in a manner analogous to that of infinitesimal strain in solid mechanics as ˙ ij = 1 2 ∂ui ∂xj + ∂uj ∂xi (1.2) This is a well-known tensorial definition of strain rates but for use later in variational forms is written as a vector which is more convenient in finite element analysis. Details of such matrix forms are given fully in reference 1 but for completeness we summarize them here. Thus, the strain rate is written as a vector ( ˙ ) and is given by the following form ˙ = [˙ 11, ˙ 22, 2˙ 12]T = [˙ 11, ˙ 22, γ̇12]T (1.3a) in two dimensions with a similar form in three dimensions ˙ = [˙ 11, ˙ 22, ˙ 33, 2˙ 12, 2˙ 23, 2˙ 31]T (1.3b) When such vector forms are used we can write the strain rate vector in the form ˙ = Su (1.4) where S is known as the strain rate operator and u is the velocity given in Eq. (1.1). The stress–strain rate relations for a linear (Newtonian) isotropic fluid require the definition of two constants. The first of these links the deviatoric stresses τij to the deviatoric strain rates: τij ≡ σij − 1 3 δij σkk = 2µ ˙ ij − 1 3 δij ˙ kk (1.5) In the above equation the quantity in brackets is known as the deviatoric strain rate, δij is the Kronecker delta, and a repeated index implies summation over the range of the index; thus σkk ≡ σ11 + σ22 + σ33 and ˙ kk ≡ ˙ 11 + ˙ 22 + ˙ 33 (1.6) The coefficient µ is known as the dynamic (shear) viscosity or simply viscosity and is analogous to the shear modulus G in linear elasticity. The second relation is that between the mean stress changes and the volumetric strain rate. This defines the pressure as p = − 1 3 σkk = − κ ˙ kk + p0 (1.7) where κ is a volumetric viscosity coefficient analogous to the bulk modulus K in linear elasticity and p0 is the initial hydrostatic pressure independent of the strain rate (note that p and p0 are invariably defined as positive when compressive). We can immediately write the ‘constitutive’ relation for fluids from Eqs (1.5) and (1.7) σij = τij − δij p = 2µ ˙ ij − 1 3 δij ˙ kk + κ δij ˙ kk − δij p0 (1.8a)
- 26. 6 Introduction to the equations of fluid dynamics and the finite element approximation or σij = 2µ˙ ij + δij (κ − 2 3 µ) ˙ kk − δij p0 (1.8b) The Lamé notation is occasionally used, putting κ − 2 3 µ ≡ λ (1.9) but this has little to recommend it and the relation (1.8a) is basic. There is little evidence about the existence of volumetric viscosity and, in what follows, we shall take κ ˙ kk ≡ 0 (1.10) giving the essential constitutive relation as (now dropping the suffix on p0) σij = 2µ ˙ ij − 1 3 δij ˙ kk − δij p ≡ τij − δij p (1.11a) without necessarily implying incompressibility ˙ kk = 0. In the above, τij = 2 µ ˙ ij − 1 3 δij ˙ kk = µ ∂ui ∂xj + ∂uj ∂xi − 2 3 δij ∂uk ∂xk (1.11b) The above relationships are identical to those of isotropic linear elasticity as we will note again later for incompressible flow. However, in solid mechanics we often consideranisotropicmaterialswherealargernumberofparameters(i.e.morethan2)are required to define the stress–strain relations. In fluid mechanics use of such anisotropy is rare and in this book we will limit ourselves to purely isotropic behaviour. Non-linearity of some fluid flows is observed with a coefficient µ depending on strain rates. We shall term such flows ‘non-Newtonian’. We now consider the basic conservation principles used to write the equations of fluid dynamics. These are: mass conservation, momentum conservation and energy conservation. 1.2.2 Mass conservation If ρ is the fluid density then the balance of mass flow ρui entering and leaving an infinitesimal control volume (Fig. 1.1) is equal to the rate of change in density as expressed by the relation ∂ρ ∂t + ∂ ∂xi (ρui) ≡ ∂ρ ∂t + ∇T (ρu) = 0 (1.12) where ∇T = ∂/∂x1, ∂/∂x2, ∂/∂x3 is known as the gradient operator. It should be noted that in this section, and indeed in all subsequent ones, the control volume remains fixed in space. This is known as the ‘Eulerian form’and displacements of a particle are ignored. This is in contrast to the usual treatment in solid mechanics where displacement is a primary dependent variable.
- 27. The governing equations of fluid dynamics 7 It is possible to recast the above equations in relation to a moving frame of reference and, if the motion follows the particle, the equations will be named ‘Lagrangian’. Such Lagrangian frame of reference is occasionally used in fluid dynamics and briefly discussed in Chapter 6. 1.2.3 Momentum conservation: dynamic equilibrium In the jth direction the balance of linear momentum leaving and entering the control volume (Fig. 1.1) is to be in dynamic equilibrium with the stresses σij and body forces ρgj . This gives a typical component equation ∂(ρuj ) ∂t + ∂ ∂xi [(ρuj )ui] − ∂ ∂xi (σij ) − ρgj = 0 (1.13) or using Eq. (1.11a), ∂(ρuj ) ∂t + ∂ ∂xi [(ρuj )ui] − ∂τij ∂xi + ∂p ∂xj − ρgj = 0 (1.14) with Eq. (1.11b) implied. The conservation of angular momentum merely requires the stress to be symmetric, i.e. σij = σji or τij = τji In the sequel we will use the term momentum conservation to imply both linear and angular forms. 1.2.4 Energy conservation and equation of state We note that in the equations of Secs 1.2.2 and 1.2.3 the dependent variables are ui (the velocity components), p (the pressure) and ρ (the density). The deviatoric stresses, dx3 x2; (y) dx2 dx1 x1; (x) x3; (z) Fig. 1.1 Coordinate direction and the infinitesimal control volume.
- 28. 8 Introduction to the equations of fluid dynamics and the finite element approximation of course, are defined by Eq. (1.11b) in terms of velocities and hence are dependent variables. Obviously, there is one variable too many for this equation system to be capable of solution. However, if the density is assumed constant (as in incompressible fluids) or if a single relationship linking pressure and density can be established (as in isothermal flow with small compressibility) the system becomes complete and solvable. Moregenerally, thepressure(p), density(ρ)andabsolutetemperature(T )arerelated by an equation of state of the form ρ = ρ(p, T ) (1.15a) For an ideal gas this takes the form ρ = p RT (1.15b) where R is the universal gas constant. In such a general case, it is necessary to supplement the governing equation system by the equation of energy conservation. This equation is of interest even if it is not coupled with the mass and momentum conservation, as it provides additional information about the behaviour of the system. Before proceeding with the derivation of the energy conservation equation we must define some further quantities. Thus we introduce e, the intrinsic energy per unit mass. This is dependent on the state of the fluid, i.e. its pressure and temperature or e = e(T, p) (1.16) The total energy per unit mass, E, includes of course the kinetic energy per unit mass and thus E = e + 1 2 uiui (1.17) Finally, we can define the enthalpy as h = e + p ρ or H = h + 1 2 uiui = E + p ρ (1.18) and these variables are found to be convenient to express the conservation of energy relation. Energy transfer can take place by convection and by conduction (radiation generally being confined to boundaries). The conductive heat flux qi for an isotropic material is defined as qi = −k ∂T ∂xi (1.19) where k is thermal conductivity. To complete the relationship it is necessary to determine heat source terms. These can be specified per unit volume as qH due to chemical reaction (if any) and must include the energy dissipation due to internal stresses, i.e. using Eqs (1.11a) and (1.11b), ∂ ∂xi (σij uj ) = ∂ ∂xi (τij uj ) − ∂ ∂xj (puj ) (1.20)
- 29. The governing equations of fluid dynamics 9 The balance of energy in a infinitesimal control volume can now be written as ∂(ρE) ∂t + ∂ ∂xi (ρuiE) − ∂ ∂xi k ∂T ∂xi + ∂ ∂xi (pui) − ∂ ∂xi (τij uj ) − ρgiui − qH = 0 (1.21a) or more simply ∂(ρE) ∂t + ∂ ∂xi (ρuiH) − ∂ ∂xi k ∂T ∂xi − ∂ ∂xi (τij uj ) − ρgiui − qH = 0 (1.21b) Here, the penultimate term represents the rate of work done by body forces. 1.2.5 Boundary conditions On the boundary of a typical fluid dynamics problem boundary conditions need to be specified to make the solution unique. These are given simply as: (a) The velocities can be described as ui = ūi on u (1.22a) or traction as ti = nj σij = t̄i on t (1.22b) where u t = . Generally traction is resolved into normal and tangential components to the boundary. (b) In problems for which consideration of energy is important the temperature on the boundary is expressed as T = T̄ on T (1.23a) or thermal flux qn = −nik ∂T ∂xi = −k ∂T ∂n = q̄n on q (1.23b) where T q = . (c) For problems of compressible flow the density is specified as ρ = ρ̄ on ρ (1.24) 1.2.6 Navier--Stokes and Euler equations The governing equations derived in the preceding sections can be written in a general conservative form as ∂Φ ∂t + ∂Fi ∂xi + ∂Gi ∂xi + Q = 0 (1.25) in which Eq. (1.12), (1.14) or (1.21b) provides the particular entries to the vectors.
- 30. 10 Introduction to the equations of fluid dynamics and the finite element approximation Thus, using indicial notation the vector of independent unknowns is Φ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ ρu1 ρu2 ρu3 ρE ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (1.26a) the convective flux is expressed as Fi = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ρui ρu1ui + pδ1i ρu2ui + pδ2i ρu3ui + pδ3i ρHui ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ , (1.26b) similarly, the diffusive flux is expressed as Gi = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 −τ1i −τ2i −τ3i −(τij uj ) − k ∂T ∂xi ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (1.26c) and the source terms as Q = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 ρg1 ρg2 ρg3 ρgiui − qH ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (1.26d) with τij = µ ∂ui ∂xj + ∂uj ∂xi − 2 3 δij ∂uk ∂xk The complete set of Eq. (1.25) is known as the Navier–Stokes equation. A particular case when viscosity is assumed to be zero and no heat conduction exists is known as the ‘Euler equation’ (where τij = 0 and qi = 0). The above equations are the basis from which all fluid mechanics studies start and it is not surprising that many alternative forms are given in the literature obtained by combinations of the various equations.5 The above set is, however, convenient and physically meaningful, defining the conservation of important quantities. It should be noted that only equations written in conservation form will yield the correct, physically meaningful, results in problems where shock discontinuities are present. In Appendix A, we show a particular set of non-conservative equations which are frequently used. The reader is cautioned not to extend the use of non-conservative equations to problems of high speed flow.
- 31. Inviscid, incompressible flow 11 In many actual situations one or another feature of the flow is predominant. For instance, frequently the viscosity is only of importance close to the boundaries at which velocities are specified. In such cases the problem can be considered separately in two parts: one as a boundary layer near such boundaries and another as inviscid flow outside the boundary layer. Further, in many cases a steady-state solution is not available with the fluid exhibit- ing turbulence, i.e. a random fluctuation of velocity. Here it is still possible to use the general Navier–Stokes equations now written in terms of the mean flow with an additional Reynolds stress term. Turbulent instability is inherent in the Navier–Stokes equations. It is in principle always possible to obtain the transient, turbulent, solution modelling of the flow, providing the mesh size is capable of reproducing the small eddies which develop in the problem. Such computations are extremely costly and often not possible at high Reynolds numbers. Hence the Reynolds averaging approach is of practical importance. Two further points have to be made concerning inviscid flow (ideal fluid flow as it is sometimes known). First, the Euler equations are of a purely convective form: ∂Φ ∂t + ∂Fi ∂xi + Q = 0 Fi = Fi(Φ) (1.27) and hence very special methods for their solutions will be necessary. These methods are applicable and useful mainly in compressible flow, as we shall discuss in Chap- ter 7. Second, for incompressible (or nearly incompressible) flows it is of interest to introduce a potential that converts the Euler equations to a simple self-adjoint form. We shall discuss this potential approximation in Sec. 1.3. Although potential forms are also applicable to compressible flows we shall not use them as they fail in complex situations. 1.3 Inviscid, incompressible flow In the absence of viscosity and compressibility, ρ is constant and Eq. (1.12) can be written as ∂ui ∂xi = 0 (1.28) and Eq. (1.14) as ∂ui ∂t + ∂ ∂xj (uj ui) + 1 ρ ∂p ∂xi − gi = 0 (1.29) 1.3.1 Velocity potential solution The Euler equations given above are not convenient for numerical solution, and it is of interest to introduce a potential, φ, defining velocities as u1 = − ∂φ ∂x1 u2 = − ∂φ ∂x2 u3 = − ∂φ ∂x3 or ui = − ∂φ ∂xi (1.30)
- 32. 12 Introduction to the equations of fluid dynamics and the finite element approximation If such a potential exists then insertion of Eq. (1.30) into Eq. (1.28) gives a single governing equation ∂2 φ ∂xi∂xi ≡ ∇2 φ = 0 (1.31) which, with appropriate boundary conditions, can be readily solved. Equation (1.31) is a classical Laplacian equation. For contained flow we can of course impose the normal velocity un on the boundaries: un = − ∂φ ∂n = ūn (1.32) and, as we shall see later, this provides a natural boundary condition for a weighted residual or finite element solution. Of course we must be assured that the potential function φ exists, and indeed deter- mine what conditions are necessary for its existence. Here we observe that so far in the definition of the problem we have not used the momentum conservation equations (1.29), to which we shall now return. However, we first note that a single-valued potential function implies that ∂2 φ ∂xj ∂xi = ∂2 φ ∂xi ∂xj (1.33) Defining vorticity as rotation rate per unit area ωij = 1 2 ∂ui ∂xj − ∂uj ∂xi (1.34) we note that the use of the velocity potential in Eq. (1.34) gives ωij = 0 (1.35) and the flow is therefore named irrotational. Inserting the definition of potential into the first term of Eq. (1.29) and using Eqs (1.28) and (1.34) we can rewrite Eq. (1.29) as − ∂ ∂xi ∂φ ∂t + ∂ ∂xi 1 2 uj uj + p ρ + P = 0 (1.36) in which P is a potential of the body forces given by gi = − ∂P ∂xi (1.37) In problems involving constant gravity forces in the x2 direction the body force potential is simply P = g x2 (1.38) Equation (1.36) is alternatively written as ∇ − ∂φ ∂t + H + P = 0 (1.39)
- 33. Incompressible (or nearly incompressible) flows 13 where H is the enthalpy, given as H = uiui/2 + p/ρ. If isothermal conditions pertain, the specific energy is constant and Eq. (1.39) implies that − ∂φ ∂t + 1 2 uiui + p ρ + P = constant (1.40) over the whole domain. This can be taken as a corollary of the existence of the potential and indeed is a condition for its existence. In steady-state flows it provides the well- known Bernoulli equation that allows the pressures to be determined throughout the whole potential field once the value of the constant is established. We note that the governing potential equation (1.31) is self-adjoint (see Appendix B) and that the introduction of the potential has side-stepped the difficulties of dealing with convective terms. It is also of interest to note that the Laplacian equation, which is obeyed by the velocity potential, occurs in other contexts. For instance, in two- dimensional flow it is convenient to introduce a stream function the contours of which lie along the streamlines. The stream function, ψ, defines the velocities as u1 = ∂ψ ∂x2 and u2 = − ∂ψ ∂x1 (1.41) which satisfy the incompressibility condition (1.28) ∂ui ∂xi = ∂ ∂x1 ∂ψ ∂x2 + ∂ ∂x1 − ∂ψ ∂x1 = 0 (1.42) For an existence of a unique potential for irrotational flow we note that ω12 = 0 [Eq. (1.34)] gives a Laplacian equation ∂2 ψ ∂xi∂xi = ∇2 ψ = 0 (1.43) The stream function is very useful in getting a pictorial representation of flow. In Appendix C we show how the stream function can be readily computed from a known distribution of velocities. 1.4 Incompressible (or nearly incompressible) flows We observed earlier that the Navier–Stokes equations are completed by the existence of a state relationship giving [Eq. (1.15a)] ρ = ρ(p, T ) In (nearly) incompressible relations we shall frequently assume that: (a) The problem is isothermal. (b) The variation of ρ with p is very small, i.e. such that in product terms of velocity and density the latter can be assumed constant. The first assumption will be relaxed, as we shall see later, allowing some thermal coupling via the dependence of the fluid properties on temperature. In such cases we
- 34. 14 Introduction to the equations of fluid dynamics and the finite element approximation shall introduce the coupling iteratively. For such cases the problem of density-induced currents or temperature-dependent viscosity will be typical (see Chapters 5 and 6). If the assumptions introduced above are used we can still allow for small com- pressibility, noting that density changes are, as a consequence of elastic deformability, related to pressure changes. Thus we can write dρ = ρ K dp (1.44a) where K is the elastic bulk modulus. This also can be written as dρ = 1 c2 dp (1.44b) or ∂ρ ∂t = 1 c2 ∂p ∂t (1.44c) with c = √ K/ρ being the acoustic wave velocity. Equations (1.25) and (1.26a)–(1.26d) can now be rewritten omitting the energy trans- port equation (and condensing the general form) as 1 c2 ∂p ∂t + ρ ∂ui ∂xi = 0 (1.45a) ∂uj ∂t + ∂ ∂xi (uj ui) + 1 ρ ∂p ∂xj − 1 ρ ∂τji ∂xi − gj = 0 (1.45b) In three dimensions j = 1, 2, 3 and the above represents a system of four equations in which the variables are uj and p. Here 1 ρ τij = ν ∂ui ∂xj + ∂uj ∂xi − 2 3 δij ∂uk ∂xk where ν = µ/ρ is the kinematic viscosity. The reader will note that the above equations, with the exception of the convective acceleration terms, are identical to those governing the problem of incompressible (or slightly compressible) elasticity (e.g. see Chapter 11 of reference 1). 1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 1.5.1 Strong and weak forms We assume the reader is already familiar with basic ideas of finite element and finite difference methods. However, to avoid a constant cross-reference to other texts (e.g. reference 1), we provide here a brief introduction to weighted residual and finite element methods.
- 35. Numerical solutions: weak forms, weighted residual and finite element approximation 15 The Laplace equation, which we introduced in Sec. 1.3, is a very convenient example for the start of numerical approximations. We shall generalize slightly and discuss in some detail the quasi-harmonic (Poisson) equation − ∂ ∂xi k ∂φ ∂xi + Q = 0 (1.46) where k and Q are specified functions. These equations together with appropriate boundary conditions define the problem uniquely. The boundary conditions can be of Dirichlet type φ = φ̄ on φ (1.47a) or that of Neumann type qn = −k ∂φ ∂n = q̄n on q (1.47b) where a bar denotes a specified quantity. Equations (1.46) to (1.47b) are known as the strong form of the problem. Weak form of equations We note that direct use of Eq. (1.46) requires computation of second derivatives to solve a problem using approximate techniques. This requirement may be weakened by considering an integral expression for Eq. (1.46) written as v − ∂ ∂xi k ∂φ ∂xi + Q d = 0 (1.48) in which v is an arbitrary function. A proof that Eq. (1.48) is equivalent to Eq. (1.46) is simple. If we assume Eq. (1.46) is not zero at some point xi in then we can also let v be a positive parameter times the same value resulting in a positive result for the integral Eq. (1.48). Since this violates the equality we conclude that Eq. (1.46) must be zero for every xi in hence proving its equality with Eq. (1.48). We may integrate by parts the second derivative terms in Eq. (1.48) to obtain ∂v ∂xi k ∂φ ∂xi d + v Q d − v ni k ∂φ ∂xi d = 0 (1.49) We now split the boundary into two parts, φ and q, with = φ q, and use Eq. (1.47b) in Eq. (1.49) to give ∂v ∂xi k ∂φ ∂xi d + v Q d + q v q̄n d = 0 (1.50) which is valid only if v vanishes on φ. Hence we must impose Eq. (1.47a) for equivalence. Equation (1.50) is known as the weak form of the problem since only first derivatives are necessary in constructing a solution. Such forms are the basis for the finite element solutions we use throughout this book.
- 36. 16 Introduction to the equations of fluid dynamics and the finite element approximation Weighted residual approximation In a weighted residual scheme an approximation to the independent variable φ is written as a sum of known trial functions (basis functions) Na(xi) and unknown parameters φ̃a . Thus we can always write φ ≈ φ̂ = N1(xi)φ̃1 + N2(xi)φ̃2 + · · · = n a=1 Na(xi)φ̃ a = N(xi)φ̃ (1.51) where N = N1, N2, · · · Nn (1.52a) and φ̃ = φ̃1 , φ̃2 , · · · φ̃n T (1.52b) In a similar way we can express the arbitrary variable v as v ≈ v̂ = W1(xi)ṽ1 + W2(xi)ṽ2 + · · · = n a=1 Wa(xi)ṽa = W(xi)ṽ (1.53) in which Wa are test functions and ṽa arbitrary parameters. Using this form of approx- imation will convert Eq. (1.50) to a set of algebraic equations. In the finite element method and indeed in all other numerical procedures for which a computer-based solution can be used, the test and trial functions will generally be defined in a local manner. It is convenient to consider each of the test and basis functions to be defined in partitions e of the total domain . This division is denoted by ≈ h = e (1.54) and in a finite element method e are known as elements. The very simplest uses lines in one dimension, triangles in two dimensions and tetrahedra in three dimensions in which the basis functions are usually linear polynomials in each element and the unknown parameters are nodal values of φ. In Fig. 1.2 we show a typical set of such linear functions defined in two dimensions. In a weighted residual procedure we first insert the approximate function φ̂ into the governing differential equation creating a residual, R(xi), which of course should be zero at the exact solution. In the present case for the quasi-harmonic equation we obtain R = − ∂ ∂xi k a ∂Na ∂xi φ̃ a + Q (1.55) and we now seek the best values of the parameter set φ̃ a which ensures that WbR d = 0; b = 1, 2, · · · , n (1.56) Note that this is the term multiplying the arbitrary parameter ṽb . As noted previously, integration by parts is used to avoid higher-order derivatives (i.e. those greater than or
- 37. Numerical solutions: weak forms, weighted residual and finite element approximation 17 Na a x y ~ Fig. 1.2 Basis function in linear polynomials for a patch of trianglular elements. equal to two) and therefore reduce the constraints on choosing the basis functions to permit integration over individual elements using Eq. (1.54). In the present case, for instance, the weighted residual after integration by parts and introducing the natural boundary condition becomes ∂Wb ∂xi k a ∂Na ∂xi φ̃ d + WbQ d + q Wb q̄n d = 0 (1.57) The Galerkin, finite element, method In the Galerkin method we simply take Wb = Nb which gives the assembled system of equations n a=1 Kbaφ̃ a + fb = 0; b = 1, 2, · · · , n − r (1.58) wherer isthenumberofnodesappearingintheapproximationtotheDirichletboundary condition [i.e. Eq. (1.47a)] and Kba is assembled from element contributions Ke ba with Ke ba = e ∂Nb ∂xi k ∂Na ∂xi d (1.59) Similarly, fb is computed from the element as f e b = e Nb Q d + eq Nb q̄n d (1.60) To impose the Dirichlet boundary condition we replace φ̃a by φ̄a for the r boundary nodes. It is evident in this example that the Galerkin method results in a symmetric set of algebraic equations (e.g. Kba = Kab). However, this only happens if the differential equations are self-adjoint (see Appendix B). Indeed the existence of symmetry pro- vides a test for self-adjointness and also for existence of a variational principle whose stationarity is sought.1
- 38. 18 Introduction to the equations of fluid dynamics and the finite element approximation (a) Three-node triangle. (b) Shape function for node 1. y 1 3 x 2 N1 x2 x1 2 3 1 1 Fig. 1.3 Triangular element and shape function for node 1. It is necessary to remark here that if we were considering a pure convection equation ui ∂φ ∂xi + Q = 0 (1.61) symmetry would not exist and such equations can often become unstable if the Galerkin method is used. We will discuss this matter further in the next chapter. Example 1.1 Shape functions for triangle with three nodes A typical finite element with a triangular shape is defined by the local nodes 1, 2, 3 and straight line boundaries between nodes as shown in Fig. 1.3(a) and will yield the shape of Na of the form shown in Fig. 1.3(b). Writing a scalar variable as φ = α1 + α2 x1 + α3 x2 (1.62) we may evaluate the three constants by solving a set of three simultaneous equations which arise if the nodal coordinates are inserted and the scalar variable equated to the appropriate nodal values. For example, nodal values may be written as φ̃1 = α1 + α2 x1 1 + α3 x1 2 φ̃2 = α1 + α2 x2 1 + α3 x2 2 φ̃3 = α1 + α2 x3 1 + α3 x3 2 (1.63) We can easily solve for α1, α2 and α3 in terms of the nodal values φ̃1 , φ̃2 and φ̃3 and obtain finally φ = 1 2 (a1 + b1x1 + c1x2)φ̃1 + (a2 + b2x1 + c2x2)φ̃2 + (a3 + b3x1 + c3x2)φ̃3 (1.64) in which a1 = x2 1 x3 2 − x3 1 x2 2 b1 = x2 2 − x3 2 c1 = x3 1 − x2 1 (1.65)
- 39. Numerical solutions: weak forms, weighted residual and finite element approximation 19 where xa i is the i direction coordinate of node a and other coefficients are obtained by cyclic permutation of the subscripts in the order 1, 2, 3, and 2 = det 1 x1 1 x1 2 1 x2 1 x2 2 1 x3 1 x3 2 = 2 · (area of triangle 123) (1.66) From Eq. (1.64) we see that the shape functions are given by Na = (aa + ba x1 + ca x2)/(2); a = 1, 2, 3 (1.67) Since the unknown nodal quantities defined by these shape functions vary linearly along any side of a triangle the interpolation equation (1.64) guarantees continuity between adjacent elements and, with identical nodal values imposed, the same scalar variable value will clearly exist along an interface between elements. We note, however, that in general the derivatives will not be continuous between element assemblies.1 Example 1.2 Poisson equation in two dimensions: Galerkin formulation with triangular elements The relations for a Galerkin finite elemlent solution have been given in Eqs (1.58) to (1.60). The components of Kba and fb can be evaluated for a typical element or subdomain and the system of equations built by standard methods. For instance, considering the set of nodes and elements shown shaded in Fig. 1.4(a), to compute the equation for node 1 in the assembled patch, it is only necessary to compute the Ke ba for two element shapes as indicated in Fig. 1.4(b). For the Type 1 element [left element in Fig. 1.4(b)] the shape functions evaluated from Eq. (1.67) using Eqs (1.65) and (1.66) gives N1 = 1 − x2 h ; N2 = x1 h ; N3 = x2 − x1 h thus, the derivatives are given by: ∂N ∂x1 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂N1 ∂x1 ∂N2 ∂x1 ∂N3 ∂x1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0 1 h − 1 h ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ and ∂N ∂x2 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂N1 ∂x2 ∂N2 ∂x2 ∂N3 ∂x2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ − 1 h 0 1 h ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ Similarly, for the Type 2 element [right element in Fig. 1.4(b)] the shape functions are expressed by N1 = 1 − x1 h ; N2 = x1 − x2 h ; N3 = x2 h
- 40. 20 Introduction to the equations of fluid dynamics and the finite element approximation (a) ‘Connected’ equations for node 1. (b) Type 1 and Type 2 element shapes in mesh. h 5 4 3 6 1 2 7 3 2 1 1 2 3 8 9 h h h Fig. 1.4 Linear triangular elements for Poisson equation example. and their derivatives by ∂N ∂x1 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂N1 ∂x1 ∂N2 ∂x1 ∂N3 ∂x1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ − 1 h 1 h 0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ and ∂N ∂x2 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂N1 ∂x2 ∂N2 ∂x2 ∂N3 ∂x2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0 − 1 h 1 h ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ and Evaluation of the matrix Ke ba and f e b for Type 1 and Type 2 elements gives (refer to Appendix D for integration formulae) Ke φ̃ e = 1 2 k 1 0 −1 0 1 −1 −1 −1 2 ⎧ ⎪ ⎨ ⎪ ⎩ φ̃ 1e φ̃ 2e φ̃ 3e ⎫ ⎪ ⎬ ⎪ ⎭ and Ke φ̃ e = 1 2 k 1 −1 0 −1 2 −1 0 −1 1 ⎧ ⎪ ⎨ ⎪ ⎩ φ̃ 1e φ̃ 2e φ̃ 3e ⎫ ⎪ ⎬ ⎪ ⎭ ,
- 41. Numerical solutions: weak forms, weighted residual and finite element approximation 21 respectively. The force vector for a constant Q over each element is given by fe = 1 6 Q h2 1 1 1 for both types of elements. Assembling the patch of elements shown in Fig. 1.4(b) gives the equation with non-zero coefficients for node 1 as (refer to references 1 and 10 for assembly procedure) k 4 −1 −1 −1 −1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ φ̃ 1 φ̃ 2 φ̃ 4 φ̃ 6 φ̃ 8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + Q h2 = 0 Using a central difference finite difference approximation directly in the differential equation (1.46) gives the approximation k h2 4 −1 −1 −1 −1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ φ̃ 1 φ̃ 2 φ̃ 4 φ̃ 6 φ̃ 8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + Q = 0 and we note that the assembled node using the finite element method is identical to the finite difference approximation though presented slightly differently. If all the boundary conditions are forced (i.e. φ = φ̄) no differences arise between a finite element and a finite difference solution for the regular mesh assumed. However, if any boundary conditions are of natural type or the mesh is irregular differences will arise, with the finite element solution generally giving superior answers. Indeed, no restrictions on shape of elements or assembly type are imposed by the finite element approach. Example 1.3 In Fig. 1.5 an example of a typical potential solution as described in Sec. 1.3 is given. Here we show the finite element mesh and streamlines for a domain of flow around a symmetric aerofoil. Example 1.4 Some problems of specific interest are those of flow with a free surface.11,12,13 Here the governing Laplace equation for the potential remains identical, but the free surface position has to be found iteratively. In Fig. 1.6 an example of such a free surface flow solution is given.12 For gravity forces given by Eq. (1.38) the free surface condition in two dimensions (x1, x2) requires 1 2 (u1u1 + u2u2) + gx2 = 0
- 42. 22 Introduction to the equations of fluid dynamics and the finite element approximation Fig. 1.5 Potential flow solution around an aerofoil. Mesh and streamline plots.
- 43. Numerical solutions: weak forms, weighted residual and finite element approximation 23 Fig. 1.6 Free surface potential flow, illustrating an axisymmetric jet impinging on a hemispherical thrust reverser (from Sarpkaya and Hiriart12 ). Solution of such conditions involves an iterative, non-linear algorithm, as illustrated by examples of overflows in reference 11. 1.5.2 A finite volume approximation Many choices of basis and weight functions are available. A large number of procedures are discussed in reference 1. An approximation which is frequently used in fluid mechanics is the finite volume process which many consider to be a generalized finite difference form. Here the weighting function is often taken as unity over a specified subdomain b and two variants are used: (a) an element (cell) centred approach; and (b) a node (vertex) centred approach. Here we will consider only a node centred approach with basis functions as given in Eq. (1.51) for each triangular subdomain and the specified integration cell (dual cell) for each node as shown in Fig. 1.7. For a solution of the Poisson equation discussed above, integration by parts of Eq. (1.56) for a unit Ŵb gives b Q d − b ni ∂φ ∂xi d = 0 (1.68)
- 44. 24 Introduction to the equations of fluid dynamics and the finite element approximation Fig. 1.7 Finite volume weighting. Vertex centred method (b). for each subdomain b with boundary b. In this form the integral of the first term gives b Q d = Q b (1.69) when Q is constant in the domain. Introduction of the basis functions into the second term gives b ni ∂φ ∂xi d ≈ b ni ∂φ̂ ∂xi d = b ni ∂Na ∂xi d φ̃ a (1.70) requiring now only boundary integrals of the shape functions. In order to make the process clearer we again consider the case for the patch of elements shown in Fig. 1.4(a). Example 1.5 Poisson equation in two dimensions: finite vol- ume formulation with triangular elements The subdomain for the determination of equation for node 1 using the finite volume method is shown in Fig. 1.8(a). The shape functions and their derivatives for the Type 1 and Type 2 elements shown in Fig. 1.8(b) are given in Example 1.2. We note especially that the derivatives of the shape functions in each element type are constant. Thus, the boundary integral terms in Eq. (1.70) become b ni ∂Na ∂xi d = e e ne i d ∂Ne a ∂xi where e denotes the elements surrounding node a. Each of the integrals on e will be an I1, I2, I3 for the Type 1 and Type 2 elements shown. It is simple to show that the integral I1 1 = e ne 1 d = c 4 ne 1 d + 6 c ne 1 d = x6 2 − x4 2
- 45. Numerical solutions: weak forms, weighted residual and finite element approximation 25 5 4 3 7 8 9 6 1 2 (a) ‘Connected’ area for node 1. 3 6 c 4 1 5 2 I3 I2 I2 I1 I3 I1 6 1 2 4 3 5 c (b) Type 1 and Type 2 element boundary integrals. Fig. 1.8 Finite volume domain and integrations for vertex centred method. where x4 2 , x6 2 are mid-edge coordinates of the triangle as shown in Fig. 1.8(b). Similarly, for the x2 derivative we obtain I2 1 = e ne 2 d = c 4 ne 2 d + 6 c ne 2 d = x4 1 − x6 1 Thus, the integral I1 = e ni ∂Na ∂xi d = ∂Na ∂x1 (x6 2 − x4 2 ) + ∂Na ∂x2 (x4 1 − x6 1 )
- 46. 26 Introduction to the equations of fluid dynamics and the finite element approximation The results I2 and I3 are likewise obtained as I2 = ∂Na ∂x1 (x4 2 − x5 2 ) + ∂Na ∂x2 (x5 1 − x4 1 ) I3 = ∂Na ∂x1 (x5 2 − x6 2 ) + ∂Na ∂x2 (x6 1 − x5 1 ) Using the above we may write the finite volume result for the subdomain shown in Fig. 1.8(a) as k 4 −1 −1 −1 −1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ φ̃ 1 φ̃ 2 φ̃ 4 φ̃ 6 φ̃ 8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + Q h2 = 0 We note that for the regular mesh the result is identical to that obtained using the standard Galerkin approximation. This identity does not generally hold when irregular meshes are considered and we find that the result from the finite volume approach applied to the Poisson equation will not yield a symmetric coefficient matrix. As we know, the Galerkin method is optimal in terms of energy error and, thus, has more desirable properties than either the finite difference or the finite volume approaches. Using the integrals defined on ‘elements’, as shown in Fig. 1.8(b), it is possible to implement the finite volume method directly in a standard finite element program. The assembled matrix is computed element-wise by assembly for each node on an element. The unit weight will be ‘discontinuous’ in each element, but otherwise all steps are standard. 1.6 Concluding remarks We have observed in this chapter that a full set of Navier–Stokes equations can be written incorporating both compressible and incompressible behaviour. At this stage it is worth remarking that 1. More specialized sets of equations such as those which govern shallow water flow or surface wave behaviour (Chapters 10, 11 and 12) will be of similar forms and will be discussed in the appropriate chapters later. 2. The essential difference from solid mechanics equations involves the non-self- adjoint convective terms. Before proceeding with discretization and indeed the finite element solution of the full fluid equations, it is important to discuss in more detail the finite element procedures which are necessary to deal with such non-self-adjoint convective transport terms. We shall do this in the next chapter where a standard scalar convective–diffusive– reactive equation is discussed.
- 47. References 27 References 1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Funda- mentals. Elsevier, 6th edition, 2005. 2. H. Schlichting. Boundary Layer Theory. Pergamon Press, London, 1955. 3. C.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967. 4. H. Lamb. Hydrodynamics. Cambridge University Press, Cambridge, 6th edition, 1932. 5. C. Hirsch. Numerical Computation of Internal and External Flows, volumes 1 2. John Wiley Sons, New York, 1988. 6. P.J. Roach. Computational Fluid Mechanics. Hermosa Press, Albuquerque, New Mexico, 1972. 7. L.D. Landau and E.M. Lifshitz.Fluid Mechanics. Pergamon Press, London, 1959. 8. R. Temam. The Navier–Stokes Equation.North-Holland, Dordrecht, 1977. 9. I.G. Currie. Fundamental Mechanics of Fluids. McGraw-Hill, New York, 1993. 10. R.W. Lewis, P. Nithiarasu and K.N. Seetharamu. Fundamentals of the Finite Element Method for Heat and Fluid Flow. Wiley, Chichester, 2004. 11. P. Bettess and J.A. Bettess. Analysis of free surface flows using isoparametric finite elements. International Journal for Numerical Methods in Engineering, 19:1675–1689, 1983. 12. T. Sarpkaya and G. Hiriart. Finite element analysis of jet impingement on axisymmetric curved deflectors. In J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher and C.Taylor, editors, Finite Elements in Fluids, volume 2, pages 265–279. John Wiley Sons, New York, 1976. 13. M.J. O’Carroll. A variational principle for ideal flow over a spillway. International Journal for Numerical Methods in Engineering, 15:767–789, 1980.
- 48. 2 Convection dominated problems -- finite element approximations to the convection--diffusion-reaction equation 2.1 Introduction In this chapter we are concerned with steady-state and transient solutions for equations of the type ∂Φ ∂t + ∂Fi ∂xi + ∂Gi ∂xi + Q = 0 (2.1) where in general Φ is the basic dependent, vector-valued variable, Q is a source or reaction term vector and the flux matrices F and G are such that Fi = Fi(Φ) Gi = Gi ∂Φ ∂xj (2.2a) and in general Q = Q(xi, Φ) (2.2b) In the above, xi and i refer, in the indicial manner, to Cartesian coordinates and quantities associated with these. A linear relationship between the source and the scalar variable in Eq. (2.2b) is frequently referred to as a reaction term. The general equation (2.1) can be termed the transport equation with F standing for the convective and G for the diffusive flux quantities. Equation (2.1) is a set of conservation laws arising from a balance of the quantity Φ with its fluxes F and G entering and leaving a control volume. Such equations are typical of fluid mechanics which we have discussed in Chapter 1. As such equations may also arise in other physical situations this chapter is devoted to a general discussion of their approximate solution. The simplest form of Eqs (2.1), (2.2a) and (2.2b) is one in which the unknown is a scalar. Most of this chapter is devoted to the solution of such equations. Throughout this book we shall show that there is no need dealing with convection of vector type
- 49. Introduction 29 quantities. Thus, for simplicity we may consider the form above Φ → φ Q → Q(xi, φ) Fi → Fi = Uiφ Gi → Gi = −k ∂φ ∂xi (2.3) We now have in Cartesian coordinates a scalar equation of the form ∂φ ∂t + ∂(Uiφ) ∂xi − ∂ ∂xi k ∂φ ∂xi + Q = 0 (2.4) In the above equation, Ui is a known velocity field and φ is a scalar quantity being transported by this velocity. However, diffusion can also exist and here k is the diffusion coefficient. The term Q represents any external sources of the quantity φ being admitted to the system and also the reaction loss or gain which itself is dependent on the function φ. A simple relation for the reaction may be written as Q = c φ (2.5) where c is a scalar parameter. The equation can be rewritten in a slightly modified form in which the convective term has been differentiated as ∂φ ∂t + Ui ∂φ ∂xi + φ ∂Ui ∂xi − ∂ ∂xi k ∂φ ∂xi + Q = 0 (2.6) We note that the above form of the problem is self-adjoint with the exception of a convective term which is underlined. The reader is referred to Appendix B for the definition of self-adjoint problems. The third term in Eq. (2.6) disappears if the flow itself is such that its divergence is zero, i.e. if ∂Ui ∂xi = 0 (2.7) In what follows we shall discuss the scalar equation in much more detail as many of the finite element remedies are only applicable to such scalar problems and are not directly transferable to the vector form. In the CBS scheme, which we shall introduce in Chapter 3, the equations of fluid dynamics will be split so that only scalar transport occurs, where the treatment considered here is sufficient. From Eqs (2.6) and (2.7) we have ∂φ ∂t + Ui ∂φ ∂xi − ∂ ∂xi k ∂φ ∂xi + Q = 0 (2.8) With the variable φ approximated in the usual way: φ ≈ φ̂ = NΦ̃ = Naφ̃a (2.9)
- 50. 30 Convection dominated problems theproblemmaybepresentedfollowingtheusual(weightedresidual)semi-discretization process as M ˙ Φ̃ + HΦ̃ + f = 0 (2.10) where Mab = WaNb d Hab = WaUi ∂Nb ∂xi + ∂Wa ∂xi k ∂Nb ∂xi d fa = Wa Q d + q Waq̄n d Now even with standard Galerkin weighting the matrix H will not be symmetric. However, this is a relatively minor computational problem compared with inaccuracies and instabilities in the solution which follow the arbitrary use of the weighting function. This chapter will discuss the manner in which these difficulties can be overcome and the approximation improved. We shall in the main address the problem of solving Eq. (2.8), i.e. the scalar form, and to simplify matters further we shall start with the idealized one-dimensional equation: ∂φ ∂t + U ∂φ ∂x − ∂ ∂x k ∂φ ∂x + Q = 0 (2.11) The term φ ∂U/∂x has been removed for simplicity, which of course is true if U is constant. The above reduces in steady state to an ordinary differential equation: U dφ dx − d dx k dφ dx + Q = 0 or L(φ) + Q = 0 (2.12) in which we shall often assume U, k and Q to be constant. The basic concepts will be evident from the above and will later be extended to multidimensional problems, still treating φ as a scalar variable. Indeed the methodology of dealing with the first space derivatives occurring in differential equations governing a problem, which lead to non-self-adjointness, opens the way for many new physical situations.1 The present chapter will be divided into three parts. Part I deals with steady-state situations starting from Eq. (2.12), Part II with transient solutions starting from Eq. (2.11) and Part III with treatment of boundary conditions for convective–diffusive problems where use of ‘weak forms’ is shown to be desirable. Although the scalar problem will mainly be dealt with here in detail, the discussion of the procedures can indicate the choice of optimal ones which will have much bearing on the solution of the general case of Eq. (2.1). The extension of some procedures to the vector case is presented in Appendix E.
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