computational fluid dynamics introduction

7/31/2019 Computational Fluid Dynamics Introduction

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26.03.2004

Sl ide 1 Hchstleistungsrechenzentrum Stut tgar t

C.-D. Munz1, S. Roller2, M. Dumbser1

University of Stuttgart1Institute for Aerodynamics and Gas Dynamics (IAG)

www.iag.uni-stuttgart.de2High-Performance Computing-Center Stuttgart (HLRS)

www.hlrs.de

Introduction to Computational Fluid Dynamics

1The Underlying Equations

2

Contents

1. Equations

2. Finite Volume Schemes

3. Linear Advection Equation

4. Systems of Advection Equations

5. Scalar Conservation Law

6. One-dimensional Euler Equations7. Godunov-Type Schemes

8. Flux-Vector Splitting Schemes

9. Second Order Accuracy MUSCL Schemes

10. Boundary Conditions

11. Finite-Volume Schemes in Multi-Dimensions

12. ENO-/ WENO Schemes

13. Discontinuous Galerkin Finite Element Methods


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1. Equations

Conservation equation

e.g.: Mass conservation

: normal

mass in V :

continuum assumption, density

nr

nr

dVV

4

1. Equations

Conservation equation

e.g.: Mass conservation

: normal

mass in V :

continuum assumption, density

nr

nr

No mass can appear or disappear

dSnfdVdt

d

V

m

V

=r

dVV


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5

Change of mass in V flux through boundary

continuously differentiable

(Gauss theorem)

V arbitrary

dSnfdVdt

d

Vm

V =r

mf,

( ) =+V

mt0dVf

( ) 0=+ vt

6

The Compressible Navier-Stokes Equations

Equation of state:

( ) 0v t =+

( ) ( )( ) pvvvt

=++ o

( )( ) qv)(peve t +=++

( ) ( )( )221 ve11p ==

pressure

viscous stress tensor

heat flux vector

p

q

density

speed

total energy

v

e


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Euler Equations

Equation of state:

( ) 0v t =+

( ) ( )( ) 0pvvv t =++ o

( )( ) 0pevet

=++

( ) ( )( )221 ve11p ==

pressurepdensity

speed

total energy per unit

volume

v

e

8

Equations of Fluid Dynamics

Compressible Navier-Stokes equations

Conservation equations for mass, momentum and energy

with viscosity and heat conduction

hyperbolic parabolic

Incompressible Navier-Stokes

equations

parabolic elliptic

0v

=c

M

Euler equations

gas dynamics

hyperbolic

Re


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2. Finite Volume Schemes

( ) [ ]T,0Din0ufu t =+

smoothpiecewiseboundary

fr,

j

kjj

j

C

kjCCCD

==U

iC

Discretization of space

Grid

10

( )( ) dtdSntxufuCuC nn j

t

t C

n

jj

n

jj +

+ =1

,1 r

[ ]1nnj t,tCovernIntegratio +

Evolution equations for mean values

Direct approximation of the integral conservation laws:

Finite volume scheme

Basic part: Appropriate approximation of the flux,

numerical flux


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Finite Volume Scheme in One Dimension

x

t

1ix ix 1+ix

1+nt

nt

+ +

+ +

=+

1n

n

1/2i

1/2i

1n

n

1/2i

1/2i

t

t

x

x

x

t

t

x

x

t 0dxdtt))f(u(x,t)dx(x,u

[ ] [ ]1nn1/2i1/2-i t,tx,xovernIntegratio ++

12

))dtt,f(u(xt

1esapproximatg

)dxtu(x,x

1esapproximatu

1n

n

1/2i

1/2i

t

t

1/2i1/2i

x

x

n

n

i

+

+

++

)g-(gx

tuu n1/2-i

n

1/2i

n

i

1n

i ++ =

with

Finite Volume Scheme Scheme in Conservation Form


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3. Linear Advection Equation

=+ auau xt ,0

sticscharacteri

( ) ( ) adt

tdxtxx == with:C

:Solution

satisfiesttxuufunctionA ,=

( )( ) xt uauttxudt

d+=,

CalongconstantSolution = uu

14

Solution of Initial Value Problem

( ) ( ) = xxqxu allfor0,

x

t

P

00 tax 0x

( )a

dt

tdx=

( ) ( ) txtaxqtxu ,allfor, =


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Upwind Scheme

x

t

1ix ix 1+ix

1+nt

nt

Grid

1forstable

0

:0

1

1

+

axt

x

uua

t

uu

a

n

i

n

i

n

i

n

i

(CFL-Condition)

16

CIR Method

( )

( )nin

i

n

i

n

i

n

i

n

i

n

i

n

i

n

i

n

i

n

in

i

n

i

uuux

ta

uux

tauu

auu

auu

x

tauu

11

11

1

1

11

22

2

ionReformulat

0for

0for

(1946)ReesIsaacson,Courant,

+

++

+

+

+

+

=

=

dissipation

central difference


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Reformulation of the CIR-Method I, Godunovs Idea

x

u

xi-1/2 xi+1/2 xi+3/2

n

iu

[ ]1/2i1/2-inini x,xfor xu(x)u +=

Godunovs Idea

constantpiecewiseu n

18

1. Solve the initial value problem

[ ]1/2i1/2-in

i

n

n

xt

x,xfor xu(x)u

with

Rfr x(x)uu(x,0),0auu

+=

==+

x

u

xi-1/2 xi+1/2 xi+3/2


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1ax

t:conditionCFL,t)dxu(x,

x

1:u

1/2i

1/2i

x

x

n

i = +

Result: CIR-method

x

u

xi-1/2 xi+1/2 xi+3/2

2. Average the exact solution

20

Reformulation of the CIR-Method II

x

t

1ix ix 1+ix

1+nt

nt

[ ] [ ]

+ +

+ +

=+

++1n

n

1/2i

1/2i

1n

n

1/2i

1/2i

t

t

x

x

t

t

x

x

1nn1/2i1/2-i

0t)dxdtau(x,t)dxu(x,

t,tx,xovernIntegratio


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n

1/2-i

n

1/2i

n

i

1n

i gt-gtuxux ++ =

with

)u,g(ug:fluxNumerical

)dtt,au(xt

1esapproximatg

)dxtu(x,x

1esapproximatu

1ii1/2i

t

t

1/2i1/2i

x

x

n

n

i

1n

n

1/2i

1/2i

++

++

=

+

+

22

Upwind-type Flux Calculation

)u(uax

t)u(ua

x

t-uu

formonconservatiinScheme

uaua

0aforau

0aforau)u,g(ug

n

i

n

1i

n

1i

n

i

n

i

1n

i

1i

-

i

1i

i

1ii1/2i

=

+=

==

+

++

++

+

++

Result: CIR-method


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Reformation of the CIR-Method III

Riemann Problem

>


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A matrixmm

4. Systems of Advection Equations

0=+ xt Auu

eigenvectors

= mrrrR ...21

Diagonalisation ( )maadiagARR ,...,: 11 ==

maaa


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15/18


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Riemann problem

Example 1:

( )

>

shock wave

32

Riemann problem

Example 2:

( )

>+

rarefaction wave

1=t

x1=

t

x


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Solution of the Riemann problem

waveshock:rl uu >

( )

( ) ( )

lr

lr

r

l

uu

ufufs

st

xu

st

xu

txu

=

>

computational fluid dynamics introduction

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