lecture 1000

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Computational Fluid Dynamics IPPPPIIIIWWWW

A Finite Difference Code

for the Navier-StokesEquations in

Vorticity/Stream Function

FormulationInstructor: Hong G. Im

University of Michigan

Fall 2001

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Develop an understanding of the steps involved in

solving the Navier-Stokes equations using a numerical

method

Write a simple code to solve the driven cavity

problem using the Navier-Stokes equations in vorticity

form

Objectives:

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The Driven Cavity Problem

The Navier-Stokes Equations in Vorticity/Stream

Function form Boundary Conditions

Finite Difference Approximations to the Derivatives

The Grid

Finite Difference Approximation of the

Vorticity/Streamfunction equations

Finite Difference Approximation of the Boundary

Conditions Iterative Solution of the Elliptic Equation

The Code

Results Convergence Under Grid Refinement

Outline

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Moving wall

Stationarywalls

The Driven Cavity Problem

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++=++ 2

2

2

21

y

u

x

u

x

p

y

u

vx

u

ut

u

Nondimensional N-S Equation

Incompressible N-S Equation in 2-D

Nondimensionalization

,/~ Uuu = ,/~ Lxx = 2/~ Upp =,/~ LtUt =

++=

++

2

2

2

2

2

22

~

~

~

~

~

~

~

~~

~

~~

~

~

y

u

x

u

L

U

x

p

L

U

y

uv

x

uu

t

u

L

U

++=++ 2

2

2

2

~

~

~

~

~

~

~

~~

~

~~

~

~

y

u

x

u

ULx

p

y

uv

x

uu

t

u

Re

1

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Computational Fluid Dynamics IPIW

u

t + uu

x + vu

y = p

x +1

Re

2u

x2 +2u

y2

v

t + uv

x + vv

y = p

y +1

Re

2v

x2 +2v

y2

y

x

t

+ u x

+ v y

= 1Re

2

x

2 + 2

y

2

= vx uy

The vorticity/stream function equations

The Vorticity Equation

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0=+

y

v

x

u

xv

yu

== ,

which automatically satisfies the incompressibility conditions

Define the stream function

x

y

y

x =0

by substituting


The Stream Function Equation

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2

x2 +

2

y2 =

=v

x

u

y

u = y

; v = x

Substituting

into the definition of the vorticity

yields


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t

= y

x

+ x

y

+ 1Re

2

x

2 + 2

y

2

2

x2 +

2

y2 =

The Navier-Stokes equations in vorticity-stream

function form are:

Elliptic equation

Advection/diffusion equation


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v = 0

x = 0

= Constant

At the top and the

bottom boundary:

Boundary Conditions for the Stream Function

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Computational Fluid Dynamics IPIWBoundary Conditions for the Vorticity

At the right and left boundary:

At the top boundary:

The normal velocity is zero since the stream function

is a constant on the wall, but the zero tangential

velocity must be enforced:

v = 0 x

= 0 u = 0 y

= 0

At the bottom boundary:

wallwall Uy

Uu =

=

PIW

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At the right and the left boundary:

Similarly, at the top and the bottom boundary:

wall = 2

x 2

wall = 2

y2

The wall vorticity must be found from the streamfunction.

The stream function is constant on the walls.

2

x2

+ 2

y2

=

2

x2+

2

y2=

Boundary Conditions for the Vorticity

PIW

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y= Uwall; wall =

2

y2

y

= 0; wall = 2

y2

x = 0

wall

= 2

x2

Summary of Boundary Conditions

= Constant

x= 0

wall =

2

x2

PIW

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Computational Fluid Dynamics IPIWFinite Difference Approximations

To compute an approximate solution numerically,the continuum equations must be discretized.

There are a few different ways to do this, but we

will use FINITE DIFFERENCE approximations

here.

PIW

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i=1 i=2 i=NX

j=1j=2

j=NY

i,j

and i ,j

Stored ateach grid

point

Discretizing the Domain

Uniform mesh (h=constant)

PIW

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When using FINITE DIFFERENCE approximations,

the values offare stored at discrete points and thederivatives of the function are approximated using a

Taylor series:

Start by expressing the value off(x+h) andf(x-h)

in terms off(x)

Finite Difference Approximations

h h

f(x-h) f(x) f(x+h)

PIW

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!+++=6

)(

2

)()()( 2

3

3 h

x

xf

h

hxfhxf

x

xf

Finite difference approximations


!++++=12

)()()(2)()( 2

4

4

22

2 h

x

xf

h

hxfhfhxf

x

xf

""

f(t)t

= f(t+ t) f(t)t

+ 2f(t)

t2t2

+!

2nd order in space, 1st order in time

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C t ti l Fl id D i IPIW

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Computational Fluid Dynamics IPIWFinite Difference Approximations

Laplacian

2f

x2+

2f

y2=

fi +1, jn 2 fi,jn + fi1,jn

h2

+ fi,j +1n 2fi,jn + fi ,j 1n

h2

=

fi +1, jn + fi 1,jn + fi, j +1n + fi ,j 1n 4fi ,jnh2


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++++

+

=

++

++++

+

2,1,1,,1,1

1,1,,1,1,1,11,1,

,1

,

4Re1

2222

h

hhhh

t

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

n

ji

nji

nji

Using these approximations, the vorticity equation becomes:


+++= 22

2

2

Re

1

yxyxxyt


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++++

=

++

++

+++

2

,1,1,,1,1

1,1,,1,1

,1,11,1,

,

1

,

4

Re

1

22

22

h

hh

hht

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

njin

ji

n

ji

The vorticity at the new time is given by:



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i +1,jn + i 1,jn + i,j +1n + i,j 1n 4i,jn

h2

= i,j

n

The stream function equation is:

2

x2

+2

y2

=



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Discretize the domain

i,j = 0

i=1 i=2 i=nxj=1

j=2

j=ny

i =1i = nx

j =1j = ny

Discretized Domain

for


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i -1 i i+1

Discrete Boundary Condition

j=3

j=2

j=1

i,j = 2 = i,j =1 +i,j =1

yh + i ,j =1

y2

h2

2+ O(h3 )

wall = i,j =1Uwall


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i,j = 2 = i,j =1 +U

wallh

wallh2

2 +O(h3)

wall = i ,j =1 i ,j =2( )2

h2 +Uwall

2

h +O(h)

wall =

i,j =1

y2 ; Uwall =

i,j =1

y

Solving for the wall vorticity:

Using:

This becomes:

Discrete Boundary Condition

i,j=2 = i,j

=1 +

i,j =1

yh +

i ,j =1

y2

h2

2+ O(h3 )

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i,jn+1 = 0.25 (i +1,j

n + i1,jn + i,j +1

n + i,j 1n + h2i,j

n)

i +1,jn + i 1,j

n + i,j +1n + i,j 1

n 4i,jn

h2=

i,j

n

Solving the elliptic equation:

Rewrite as

j

j-1

j+1

i i+1i-1

Solving the Stream Function Equation

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Limitations on the time step

th2

1

4

(|u | + | v |)t 2

Time Step Control


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for i=1:MaxIterations

for i=2:nx-1; for j=2:ny-1s(i,j)=SOR for the stream function

end; end

end

for i=2:nx-1; for j=2:ny-1

rhs(i,j)=Advection+diffusion

end; end

Solution Algorithm

Solve for the stream function

Find vorticity on boundary

Find RHS of vorticity equation

Initial vorticity given

t=t+t

Update vorticity in interior

v(i,j)=

v(i,j)=v(i,j)+dt*rhs(i,j)


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p yPIW

1 clf;nx=9; ny=9; MaxStep=60; Visc=0.1; dt=0.02; % resolution & governing parameters

2 MaxIt=100; Beta=1.5; MaxErr=0.001; % parameters for SOR iteration

3 sf=zeros(nx,ny); vt=zeros(nx,ny); w=zeros(nx,ny); h=1.0/(nx-1); t=0.0;

4foristep=1:MaxStep, % start the time integration

5 foriter=1:MaxIt, % solve for the streamfunction6 w=sf; % by SOR iteration

7 fori=2:nx-1;forj=2:ny-1

8 sf(i,j)=0.25*Beta*(sf(i+1,j)+sf(i-1,j)...

9 +sf(i,j+1)+sf(i,j-1)+h*h*vt(i,j))+(1.0-Beta)*sf(i,j);

10 end; end;

11 Err=0.0;fori=1:nx;forj=1:ny, Err=Err+abs(w(i,j)-sf(i,j));end; end;

12 ifErr

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p yPIW

nx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0

Solution Fields


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p yPIW

nx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0

Solution Fields


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p yPIW

Stream Function at t = 1.2

Accuracy by Resolution

x

y

0 0.25 0.5 0.750

0.5

1

9 9 Grid

x

y

0 0.25 0.5 0.750

0.5

1

17 17 Grid


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PIW

Vorticity at t = 1.2

Accuracy by Resolution

x

y

0 0.25 0.5 0.750

0.5

1

9 9 Grid

x

y

0 0.25 0.5 0.750

0.5

1

17 17 Grid


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PIWMini-Project

Add the temperature equation to the vorticity-

streamfunction equation and compute the

increase in heat transfer rate:

Mini-project:


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PIWConvection

T

t + uT

x + vT

y = 2T

x2 +

2T

y2

= Dcp

where


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PIWNatural convection

Natural convection in a

closed cavity. The left

vertical wall is heated.

0


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PIWNatural convection




125


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Natural convection




250


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Natural convection




375


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Natural convection




500


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Natural convection




750


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Natural convection




1000


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200 400 600 800 10000

0.05

0.1

0.15

0.2

Natural convection

Heat Transfer Rate

Conduction only

Natural convection

lecture 1000

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