mathematical behaviour of pde's

Mathematical

behaviour of PDE’s

Arvind Deshpande

Arvind Deshpande (VJTI) 23/7/2012

Governing equations of fluid flow

(Navier Stokes equations)

)(&

)()()()(

)()()()(

)()()()(

)()()()(

)(0)(

fstateEquationsoCvTiRTP

ergyInternalEnSkgradTdivpdivVVidivt

i

momentumZSgradwdivz

pVwdiv

t

w

momentumYSgradvdivy

pVvdiv

t

v

momentumXSgradudivx

pVudiv

t

u

MassVdivt

i

Mz

My

Mx


Governing equations of fluid flow

(Euler equations)

)(&

)()()(

)()()(

)()()(

)()()(

)(0)(

fstateEquationsoCvTiRTP

ergyInternalEnSpdivVVidivt

i

momentumZSz

pVwdiv

t

w

momentumYSy

pVvdiv

t

v

momentumXSx

pVudiv

t

u

MassVdivt

i

Mz

My

Mx


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

SgraddivVdivt

)()(

)(

Rate of

increase of φ

of fluid

element

Net rate of flow

of φ out of the

fluid element

Rate of increase of φ

due to diffusion

Rate of increase

of φ due to

sources



Integral form

dVSdAgradndAVndVt

CVAACV

)(.)(.

Rate of increase

of φ fluid element

Net rate of

decrease of φ

due to convection

across the

boundaries

Net rate of increase

of φ due to diffusion

across the

boundaries

Net Rate of

creation of φ



Integral form

)()(.)(.

)()(.)(.

unstadydVdtSdAdtgradndAdtVndtdVt

steadydVSdAgradndAVn

t CVt At At CV

CVAA


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

derivatives of dependent

variables along the

boundary

e.g. 1) if wall temperature

is changing due to heat

transfer from or to the

surface

2) Adiabatic wall

0

nn

T

K

q

n

T

n

TKq

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, 0,0

n

T

n

un


Other boundary conditions

Open boundary

condition

Symmetry boundary

condition

Cyclic boundary

condition 21

0

0

n

n

un


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


First Order 0

yG

x

Second Order 02

2

yx

Third Order 02

2

3

3

xyxx

Partial Differential Equations

Classifications of PDE’s according to order


Mathematical behavior of PDE’s

0

0

2

2

22

2

2

cdx

dyb

dx

dya

gfy

ex

dy

cyx

bx

a

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

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.

N

j

N

k kj

jk Hxx

A1 1

2

0


Elliptic PDE

Typical Examples are

( Laplace’s Equation – Irrotational flow of

an incompressible fluid, steady state

conductive heat transfer)

02

2

2

2

yx

and ),(2

2

2

2

yxgyx

( Poisson’s Equation)

Note that In both of the eqns, b=0, a=1, c=1 which makes

442 acb which is < 0

The solution domain of Elliptic Eqn has closed ended nature


Pictorial Representation of Elliptical Problem

Domain of

dependence


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

( Heat Conduction or Diffusion Eqn.)

Where is positive, real constant

In above eqn. b=0, c=0, a = which makes 042 acb

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

)(

2

2

divgradt

xt


Pictorial Representation of Parabolic Problem

Domain of dependence


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

( Wave Equation)

Where c2 is real constant and always positive

In above eqn b = 0, a = c2, c = -1 which makes

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.

divgradct

xc

t

2

2

2

2

22

2

2

22 44 cacb


Pictorial Representation of Hyperbolic Problem

Bo

un

da

ry c

on

ditio

ns

Bo

un

da

ry c

on

ditio

ns

Initial conditions

P(x’,t’)

Domain of

dependence

Domain of

influence


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

Technology

v dadt t n

p div vv div gradv s

shock waves

p div vv s

conservation form

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