Vorticity 2d - Cfd - Matlab

8/17/2019 Vorticity 2d - Cfd - Matlab

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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

v x

u

ut

u

∂

∂

∂

∂ ν ∂

∂

ρ∂

∂

∂

∂

∂

∂

Nondimensional N-S Equation

Incompressible N-S Equation in 2-D

Nondimensionalization

,/~ U uu = ,/~ L x x = 2/~ U p p ρ=,/~ LtU t =

++−=

++

2

2

2

2

2

22

~

~

~

~

~

~

~

~~

~

~~

~

~

y

u

x

u

L

U

x

p

L

U

y

uv

x

uu

t

u

L

U

∂

∂

∂

∂ ν

∂

∂

∂

∂

∂

∂

∂

∂

++−=++ 2

2

2

2

~

~

~

~

~

~

~

~~

~

~~

~

~

y

u

x

u

UL x

p

y

uv

x

uu

t

u

∂

∂

∂

∂ ν

∂

∂

∂

∂

∂

∂

∂

∂

Re

1


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Computational Fluid Dynamics IPIW

∂ u

∂ t + u∂ u

∂ x + v∂ u

∂ y = −∂ p

∂ x + 1

Re

∂ 2u

∂ x2 + ∂ 2u

∂ y2

∂ v

∂ t + u∂ v

∂ x + v∂ v

∂ y = −∂ p

∂ y + 1

Re

∂ 2v

∂ x2 +∂ 2v

∂ y2

− ∂

∂ y

∂ ∂ x

∂ω ∂ t

+ u ∂ω ∂ x

+ v ∂ω ∂ y

= 1Re

∂ 2

ω ∂ x

2 + ∂ 2

ω ∂ y

2

ω = ∂ v∂ x − ∂ u∂ y

The vorticity/stream function equations

The Vorticity Equation


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0=+

y

v

x

u

∂

∂

∂

∂

xv

yu

∂

∂ψ

∂

∂ψ −== ,

which automatically satisfies the incompressibility conditions

Define the stream function

∂

∂ x

∂ψ

∂ y −

∂

∂ y

∂ψ

∂ x = 0

by substituting


The Stream Function Equation


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∂ 2ψ

∂ x2 +

∂ 2ψ

∂ y2 = −ω

ω = ∂ v

∂ x

− ∂ u

∂ y

u = ∂ψ ∂ y

; v = − ∂ψ ∂ x

Substituting

into the definition of the vorticity

yields



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∂ω ∂ t

= − ∂ψ ∂ y

∂ω ∂ x

+ ∂ψ ∂ x

∂ω ∂ y

+ 1Re

∂ 2

ω ∂ x

2 + ∂ 2

ω ∂ y

2

∂ 2ψ

∂ x2 +

∂ 2ψ

∂ y2 = −ω

The Navier-Stokes equations in vorticity-stream

function form are:

Elliptic equation

Advection/diffusion equation



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Computational Fluid Dynamics IPIWBoundary Conditions for the Stream Function

u = 0 ⇒ ∂ψ

∂ y = 0

⇒ψ

= Constant

At the right and

the left boundary:


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v = 0 ⇒ ∂ψ

∂ x = 0

⇒ ψ = Constant

At the top and the

bottom boundary:

Boundary Conditions for the Stream Function


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Since the boundaries

meet, the constantmust be the same on

all boundaries

ψ = Constant

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 ⇒ ∂ψ ∂ x

= 0 u = 0 ⇒ ∂ψ ∂ y

= 0

At the bottom boundary:

wall wall U y

U u =∂

⇒= ∂ψ

PIW


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At the right and the left boundary:

Similarly, at the top and the bottom boundary:

⇒ ω wall = − ∂ 2

ψ ∂ x 2

⇒ ω wall = −∂ 2ψ

∂ y2

The wall vorticity must be found from the streamfunction.

The stream function is constant on the walls.

∂ 2

ψ ∂ x2

+ ∂ 2

ψ ∂ y2

= −ω

∂ 2ψ

∂ x2 +

∂ 2ψ

∂ y2 = −ω

Boundary Conditions for the Vorticity

PIW


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∂ψ

∂ y= U wall ; ω wall = −

∂ 2ψ

∂ y2

∂ψ ∂ y

= 0; ω wall = − ∂ 2

ψ ∂ y2

∂ψ

∂ x = 0

ω wall

= − ∂ 2ψ

∂ x2

Summary of Boundary Conditions

ψ = Constant

∂ψ

∂ x= 0

ω wall = −∂

2ψ

∂ 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=1 j=2

j=NY

ψ i, j and ω

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

Taylor series:

Start by expressing the value of f ( x+h) and f ( x-h)

in terms of f ( 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

x f

h

h x f h x f

x

x f

∂

∂

∂

∂

Finite difference approximations


!++−+−+=12

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

4

4

22

2 h

x

x f

h

h x f h f h x f

x

x f

∂

∂

∂

∂

∂ f (t )∂ t

= f (t + ∆t ) − f (t )∆t

+ ∂ 2 f (t )

∂ t 2∆t 2

+!

2nd order in space, 1st order in time

CPIW


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For a two-dimensional flow discretize the variables on

a two-dimensional grid

f i , j = f ( x, y) f i +1, j = f ( x + h, y)

f i , j +1 = f ( x, y + h)

( x, y)

i -1 i i+1

j+1

j

j-1

C t ti l Fl id D i IPIW


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Laplacian

∂ 2 f

∂ x2 + ∂

2 f

∂ y2 =

f i +1, jn − 2 f i, jn + f i−1, jn

h2

+ f i, j +1n − 2 f i, jn + f i , j −1n

h2

=

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

n ji

n ji

ω ω ω ω ω

ω ω ψ ψ ω ω ψ ψ

ω ω

Using these approximations, the vorticity equation becomes:


+++−= 22

2

2

Re

1

y x y x x yt ∂

ω ∂

∂

ω ∂

∂

∂ω

∂

∂ψ

∂

∂ω

∂

∂ψ

∂

∂ω



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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

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 jin

ji

n

ji

ω ω ω ω ω

ω ω ψ ψ

ω ω ψ ψ ω ω

The vorticity at the new time is given by:




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

h2

= −ω i, j

n

The stream function equation is:

∂ 2ψ

∂ x2 +

∂ 2ψ

∂ y2 = −ω




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Discretize the domain

ψ i, j = 0

i=1 i=2 i=nx j=1

j=2

j=ny

i =1i = nx

j =1 j = ny

Discretized Domain

for



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i -1 i i+1

Discrete Boundary Condition

j=3

j=2

j=1

ψ i, j = 2 = ψ i, j =1 + ∂ψ i, j =1

∂ yh + ∂ ψ i , j =1

∂ y2

h2

2+ O(h3 )

ω wall = ω i, j =1U wall



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ψ i, j = 2 = ψ i, j =1 + U

wall h

−ω wall h2

2 + O(h3)

ω wall = ψ i , j =1 − ψ i , j =2( )2

h2 +U wall

2

h +O(h)

ω wall = −

∂ ψ i, j =1

∂ y2 ; U wall =

∂ψ i, j =1

∂ y

Solving for the wall vorticity:

Using:

This becomes:


ψ i, j=2 = ψ i, j

=1 +

∂ψ i, j =1

∂ yh +

∂ ψ i , j =1

∂ y2

h2

2+ O(h3 )



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ω wall =

ψ i , j =1 −

ψ i , j =2( )

2

h2 +U wall 2

h +O(h)

At the bottom wall ( j=1):


Similarly, at the top wall ( j=ny):

At the left wall (i=1):

At the right wall (i=nx):

Fill the blank

Fill the blank

( ) )(22

21,, hOhU h wall ny jiny jiwall +−−= −== ψ ψ ω



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ψ i, jn+1 = 0.25 (ψ i +1, j

n + ψ i−1, jn + ψ i, j +1

n + ψ i, j −1n + h2ω i, j

n)

ψ i +1, jn + ψ i −1, j

n + ψ i, j +1n + ψ i, j −1

n − 4ψ i, jn

h2 = −ω

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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ψ i, jn+1 = β 0.25 (ψ i+1, j

n + ψ i−1, jn +1 + ψ i, j +1

n + ψ i , j −1n+1 + h2ω i , j

n )

+ (1− β )ψ i, jn

Successive Over Relaxation (SOR)

If the grid points are done in order, half of the points have

already been updated:

j

j-1

j+1

i i+1i-1

Solving the Stream Function Equation



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Limitations on the time step

ν ∆t h2 ≤

1

4

(|u | + | v |)∆t ν ≤ 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 + u∂ T

∂ x + v∂ T

∂ y = α ∂ 2T

∂ x2 +

∂ 2T

∂ y2

α = Dρc p

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

Vorticity 2d - Cfd - Matlab

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