Mathematical behaviour of pde's

Mathematical
behaviour of PDE’s
Arvind Deshpande

Governing equations of fluid flow
(Navier Stokes equations)

 div ( V )  0( Mass)
t
 ( u ) p
 div ( Vu )    div ( gradu)  S Mx ( X  momentum)
t x
 ( v) p
 div ( Vv )    div ( gradv)  S My (Y  momentum)
t y
 ( w) p
 div ( Vw)    div ( gradw)  S Mz ( Z  momentum)
t z
 ( i )
 div ( Vi )   pdivV  div (kgradT )    Si ( InternalEn ergy )
t
P  RT & i  CvT ( Equationso fstate )
3/7/2012 Arvind Deshpande (VJTI) 2

Governing equations of fluid flow
(Euler equations)

 div ( V )  0( Mass)
t
 ( u ) p
 div ( Vu )    S Mx ( X  momentum)
t x
 ( v) p
 div ( Vv )    S My (Y  momentum)
t y
 ( w) p
 div ( Vw)    S Mz ( Z  momentum)
t z
 ( i )
 div ( Vi )   pdivV  Si ( InternalEn ergy )
t
P  RT & i  CvT ( Equationso fstate )

Comments on governing equations
1. They are a coupled system of nonlinear partial differential equations and
hence are very difficult to solve analytically.
2. For the momentum and energy equations, difference in the conservation
form and non conservation form is the just the left hand side.
3. Conservation form contains terms on the left side which include the
divergence of some quantity.
4. In CFD literature, entire block of equations are called Navier-Stoke’s
equations.
5. All equations for inviscid flow are called Euler’s equations.
6. Conservation form of the governing equations provides a numerical and
computer programming convenience in that the continuity, momentum
and energy equations in conservation form can all be expressed by the
same generic equation. This helps to simplify and organize the logic in a
given computer program.
Governing equations are “bread and butter” of CFD-learn them well.


General Transport equation in
Differential form

(  )
 div ( V )  div (grad )  S
t
Rate of increase
Rate of Net rate of flow Rate of increase of φ
of φ due to
increase of φ of φ out of the due to diffusion
sources
of fluid fluid element
element


Integral form

 
  dV    n.( V )dA   n.(grad )dA   S dV
t  CV


 A A CV

Net Rate of
Rate of increase Net rate of Net rate of increase
creation of φ
of φ fluid element decrease of φ of φ due to diffusion
due to convection across the
across the boundaries
boundaries


Integral form

 n.( V )dA   n.(grad )dA   S dV (steady )
A A CV

 
t t  CVdV dt  t  n.( V )dAdt  t  n.(grad )dAdt  t CVS dVdt (unstady )




 A A



Shock capturing
 In flow fields involving shock waves, there are sharp discontinuous
changes in flow field variables, P,ρ, V, T etc. across the shocks.
 Shock capturing methods are designed to have the shock waves
appear naturally within the computational space as a direct result of
the general algorithm, without any special treatment to take care of
the shocks themselves.
 Ideal for complex flow problems involving shock waves for which we
do not know either the location or the number of shocks.
 Numerically obtained shock thickness bears no relation whatsoever
to the actual physical shock thickness and precise location of the
shock is uncertain.
 Conservation form of the governing equations is suitable for shock
capturing


Shock fitting
 In shock fitting methods, shocks are explicitly introduced into the
flow field solution. Analytical relations are used to relate the flow
immediately ahead of and behind the shock and governing
equations are used to calculate the remainder of the flow field
between the shock and some other boundary such as surface of a
aerodynamic body.
 Shock is always treated as discontinuity and its location is well
defined numerically.
 For a given problem, you have to know in advance approximately
where to put the shock waves and how many they are. For complex
flows, this can be a distinct disadvantage.
 For shock-fitting method, satisfactory results are usually obtained for
either form of the equations, conservation or non conservation.


Boundary conditions
 Real driver for any particular solution
 Dirichlet boundary condition - Specification of
dependent variables along the boundary
e.g. For Viscous flow, Wall boundary condition
V = Vw at the surface (No slip)
For a stationary wall, V = 0
Known wall temperature, T= Tw


Newmann boundary condition
 Specification of  T 
q .   K 
derivatives of dependent  n  n
variables along the  T  q.
boundary   
 n  n K
e.g. 1) if wall temperature
is changing due to heat
transfer from or to the
surface
2) Adiabatic wall
 T 
  0
 n  n

Robbins Condition

The Derivative of the dependent variable is given as a
function of the dependent variable on the boundary.


Inlet and outlet
 Inlet – Density, velocity and temperature at inlet
 Outlet – location where flow is approximately
unidirectional and where surface stresses take
known values.
 For external flows away from solid objects and
for internal flow, at a location where no change
in any of the velocity components in direction
across the boundary and Fn = -P & Ft = 0
 Specified pressure, un  0, T  0
n n


Other boundary conditions
 Open boundary
u n
condition 0
n

 Symmetry boundary 0
condition n
Cyclic boundary
1  2

condition


Initial conditions
 Everywhere in the solution region ρ, V and
T must be given at time t = 0
 The Initial and Boundary conditions must
be specified to obtain unique numerical
solutions to PDEs
 Well posed problem


Partial Differential Equations
Classifications of PDE’s according to order

 
First Order G 0
x y

 2 
Second Order  0
x 2
y

2
  3    2  
Third Order  3  
 x   xy   x  0

   


Mathematical behavior of PDE’s
 2  2  2  
a 2 b c 2 d e  f  g  0
x xy y x y
2
 dy   dy 
a   b   c  0
 dx   dx 

 If b2-4ac < 0, elliptic equation (Imaginary characteristics)
 If b2-4ac = 0, parabolic equation (1 real characteristics)
 If b2-4ac > 0, hyperbolic equation (2 real and distinct
characteristics)


Eigen value method
N
 2
N

 Ajk x x  H  0
j 1 k 1 j k
det [Ajk-λI] = 0
 If any eigen value λ = 0, the equation is parabolic.
 If all eigen value λ ≠ 0 and they are all of the same sign,
the equation is elliptic.
 If all eigen value λ ≠ 0 and all but one are of the same
sign the equation is hyperbolic.


Elliptic PDE

Typical Examples are
 2  2
 2  0 ( Laplace’s Equation – Irrotational flow of
x 2
y an incompressible fluid, steady state
conductive heat transfer)
 2  2
and  2  g ( x, y ) ( Poisson’s Equation)
x 2
y
Note that In both of the eqns, b=0, a=1, c=1 which makes
b 2  4ac  4 which is < 0
The solution domain of Elliptic Eqn has closed ended nature


Domain of
dependence

Pictorial Representation of Elliptical Problem

Elliptic PDE
 Characteristics are imaginary/complex
 Information propagates everywhere
 Equilibrium problems (div grad φ = 0)
 Boundary value problems
 Smooth solution
 Steady state temperature distribution of a
insulated solid rod
 Steady viscous flow
 Steady, subsonic inviscid flow


Parabolic PDE

A Typical Example is
  2

t x 2 ( Heat Conduction or Diffusion Eqn.)

 divgrad ( )
t

Where  is positive, real constant
In above eqn. b=0, c=0, a =  which makes b 2  4ac  0

 The solution advances outward indefinitely from Initial Condition
 This is also called as marching type problem
 The solution domain of Parabolic Eqn has open ended nature


Domain of dependence

Pictorial Representation of Parabolic Problem

Parabolic PDE
 Information travels in one particular direction
(downstream)
 Time dependent problems which involve significant
amount if dissipation
 Initial-Boundary value problems
 Smooth solution
 Unsteady heat conduction
 Unsteady viscous flows
 Thin shear layers – Boundary layers, jets, mixing layers,
wakes, fully developed duct flows


Hyperbolic PDE
A typical example is
 2  2
2
c ( Wave Equation)
t2
x 2
 2
 c 2 divgrad
t 2
Where c2 is real constant and always positive
In above eqn b = 0, a = c2, c = -1 which makes b2  4ac  4c 2
which is >0

 The solution domain of Hyperbolic Eqn has open ended nature
 Two Initial conditions are required to start the solution of
Hyperbolic eqn in contrast with Parabolic eqn where only one Initial
conditions is required.

Domain of
influence

Boundary conditions
Boundary conditions P(x’,t’)

Domain of
dependence

Initial conditions
Pictorial Representation of Hyperbolic Problem


Hyperbolic PDE
 Characteristics are real and distinct
 Information propagates along these characteristics
 Time dependent problems which involve negligible
dissipation
 Initial-Boundary value problems
 Solution may be discontinuous
 Compressible fluid flows at speeds close to or above the
speed of sound
 Steady, inviscid supersonic flow
 Unsteady inviscid flow


Mathematical behaviour of pde's

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