WPI Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U Michigan... · WPI Computational Fluid Dynamics I A Finite Difference Code for the Navier-Stokes

Computational Fluid Dynamics IPPPPIIIIWWWW

A Finite Difference Code for the Navier-Stokes

Equations in Vorticity/Stream Function

FormulationInstructor: Hong G. ImUniversity of Michigan

Fall 2001


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 vorticityform

Objectives:

Computational Fluid Dynamics IPPPPIIIIWWWW• 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


Moving wall

Stationary walls

The Driven Cavity Problem


++−=++

2

2

2

21yu

xu

xp

yuv

xuu

tu

∂∂

∂∂ν

∂∂

ρ∂∂

∂∂

∂∂

Nondimensional N-S Equation

Incompressible N-S Equation in 2-D

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

++−=

++

2

2

2

2

2

22

~~

~~

~~

~~~

~~~

~~

yu

xu

LU

xp

LU

yuv

xuu

tu

LU

∂∂

∂∂ν

∂∂

∂∂

∂∂

∂∂

++−=++

2

2

2

2

~~

~~

~~

~~~

~~~

~~

yu

xu

ULxp

yuv

xuu

tu

∂∂

∂∂ν

∂∂

∂∂

∂∂

∂∂

Re1

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

∂ 2ω∂x2 +

∂ 2ω∂y2

ω =∂v∂x

−∂u∂y

The vorticity/stream function equations

The Vorticity Equation


0=+yv

xu

∂∂

∂∂

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


∂2ψ∂x2 +

∂2ψ∂y2 = −ω

ω =∂v∂x

−∂u∂y

u =∂ψ∂y

; v = −∂ψ∂x

Substituting

into the definition of the vorticity

yields



∂ω∂t

= −∂ψ∂y

∂ω∂x

+∂ψ∂x

∂ω∂y

+ 1Re

∂ 2ω∂x2 +

∂ 2ω∂y2

∂ 2ψ∂x2 +

∂ 2ψ∂y2 = −ω

The Navier-Stokes equations in vorticity-stream function form are:

Elliptic equation

Advection/diffusion equation


Computational Fluid Dynamics IPIWBoundary Conditions for the Stream Function

u = 0 ⇒∂ψ∂y

= 0

⇒ ψ = Constant

At the right and the left boundary:


v = 0 ⇒∂ψ∂x

= 0

⇒ ψ = Constant

At the top and thebottom boundary:

Boundary Conditions for the Stream Function


Since the boundaries meet, the constant must be the same on all boundaries

ψ = Constant

Boundary Conditions for the Stream Function

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:

wallwall Uy

Uu =∂

⇒= ∂ψ


At the right and the left boundary:

Similarly, at the top and the bottom boundary:

⇒ ω wall = −∂ 2ψ∂x 2

⇒ ω wall = −∂ 2ψ∂y2

The wall vorticity must be found from the streamfunction. The stream function is constant on the walls.

∂2ψ∂x2 +

∂ 2ψ∂y2 = −ω

∂ 2ψ∂x2 +

∂ 2ψ∂y2 = −ω

Boundary Conditions for the Vorticity

Computational Fluid Dynamics IPIW∂ψ∂y

= Uwall; ωwall = −∂ 2ψ∂y2

∂ψ∂y

= 0; ωwall = −∂ 2ψ∂y2

∂ψ∂x

= 0

ωwall = − ∂ 2ψ∂x 2

Summary of Boundary Conditions

ψ = Constant

∂ψ∂x

= 0

ωwall = − ∂ 2ψ∂x 2

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.


i=1 i=2 i=NXj=1j=2

j=NY

ψ i, j and ω i , j

Stored at each grid point

Discretizing the Domain

Uniform mesh (h=constant)


When using FINITE DIFFERENCE approximations, the values of f are stored at discrete points and the derivatives 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)


!++−−+=6

)(2

)()()( 2

3

3 hx

xfh

hxfhxfxxf

∂∂

∂∂

Finite difference approximations


!++−+−+=12

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

4

4

22

2 hx

xfh

hxfhfhxfx

xf∂

∂∂

∂

""

∂f (t)∂t

=f (t + ∆t) − f (t)

∆t+

∂ 2 f (t)∂t 2

∆t2

+!

2nd order in space, 1st order in time


For a two-dimensional flow discretize the variables on a two-dimensional grid

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

(x, y)

i -1 i i+1

j+1

j

j-1


Laplacian

∂ 2 f∂x2 +

∂ 2 f∂y2 =

fi +1, jn − 2 fi, j

n + fi−1, jn

h2 +fi, j +1

n − 2 fi, jn + fi , j −1

n

h2 =

fi +1, jn + fi −1, j

n + fi, j +1n + fi , j −1

n − 4 fi , jn

h2


−++++

−

−+

−

−−

=∆−

−+−+

−+−+−+−+

+

2,1,1,,1,1

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

,1

,

4Re1

2222

h

hhhh

t

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

ωωωωω

ωωψψωωψψ

ωωUsing these approximations, the vorticity equation becomes:


+++−= 2

2

2

2

Re1

yxyxxyt ∂ω∂

∂ω∂

∂∂ω

∂∂ψ

∂∂ω

∂∂ψ

∂∂ω


−++++

−

−−

−

−∆−=

−+−+

−+−+

−+−++

2,1,1,,1,1

1,1,,1,1

,1,11,1,,

1,

4Re1

22

22

h

hh

hht

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

njin

jin

ji

ωωωωω

ωωψψ

ωωψψωω

The vorticity at the new time is given by:



ψ i +1, jn +ψ i −1, j

n +ψ i, j +1n +ψ i, j −1

n − 4ψ i, jn

h2 = −ω i, jn

The stream function equation is:

∂ 2ψ∂x2 +

∂ 2ψ∂y2 = −ω



Discretize the domainψ i, j = 0

i=1 i=2 i=nxj=1j=2

j=ny

i =1i = nxj =1j = ny

Discretized Domain

for


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 +

∂ 2ψ i , j =1

∂y2h2

2+ O(h3 )

ωwall = ω i, j =1Uwall


ψ i, j = 2 =ψ i, j =1 + Uwallh −ωwallh2

2+ O(h3)

ωwall = ψ i , j =1 −ψ i , j =2( ) 2h2 +Uwall

2h

+O(h)

ωwall = −∂ 2ψ i, j =1

∂y2 ; Uwall =∂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 +

∂ 2ψ i , j =1

∂y2h2

2+ O(h3 )


ωwall = ψ i , j =1 −ψ i , j =2( ) 2h2 +Uwall

2h

+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

( ) )(2221,, hO

hU

h wallnyjinyjiwall +−−= −== ψψω


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, jn

Solving the elliptic equation:

Rewrite as

j

j-1

j+1

i i+1i-1

Solving the Stream Function Equation


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


Limitations on the time step

ν∆th2 ≤ 1

4(| u | + | v |)∆t

ν≤ 2

Time Step Control


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

s(i,j)=SOR for the stream functionend; end

end

for i=2:nx-1; for j=2:ny-1rhs(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)

Computational Fluid Dynamics IPIW1 clf;nx=9; ny=9; MaxStep=60; Visc=0.1; dt=0.02; % resolution & governing parameters2 MaxIt=100; Beta=1.5; MaxErr=0.001; % parameters for SOR iteration3 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 integration5 foriter=1:MaxIt, % solve for the streamfunction6 w=sf; % by SOR iteration7 fori=2:nx-1; forj=2:ny-18 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 <= MaxErr, break, end % stop if iteration has converged13 end;14 vt(2:nx-1,1)=-2.0*sf(2:nx-1,2)/(h*h); % vorticityon bottom wall15 vt(2:nx-1,ny)=-2.0*sf(2:nx-1,ny-1)/(h*h)-2.0/h; % vorticityon top wall16 vt(1,2:ny-1)=-2.0*sf(2,2:ny-1)/(h*h); % vorticityon right wall17 vt(nx,2:ny-1)=-2.0*sf(nx-1,2:ny-1)/(h*h); % vorticityon left wall18 fori=2:nx-1; forj=2:ny-1 % compute19 w(i,j)=-0.25*((sf(i,j+1)-sf(i,j-1))*(vt(i+1,j)-vt(i-1,j))... % the RHS20 -(sf(i+1,j)-sf(i-1,j))*(vt(i,j+1)-vt(i,j-1)))/(h*h)... % of the21 +Visc*(vt(i+1,j)+vt(i-1,j)+vt(i,j+1)+vt(i,j-1)-4.0*vt(i,j))/(h*h); % vorticity22 end; end; % equation23 vt(2:nx-1,2:ny-1)=vt(2:nx-1,2:ny-1)+dt*w(2:nx-1,2:ny-1); % update the vorticity24 t=t+dt % print out t25 subplot(121), contour(rot90(fliplr(vt))), axis('square'); % plot vorticity26 subplot(122), contour(rot90(fliplr(sf))), axis('square');pause(0.01) % streamfunction27 end;

MATLAB Code

Computational Fluid Dynamics IPIWnx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0

Solution Fields

Computational Fluid Dynamics IPIWnx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0

Solution Fields

Computational Fluid Dynamics IPIWStream Function at t = 1.2

Accuracy by Resolution

x

y

0 0.25 0.5 0.750

0.5

19 × 9 Grid

x

y

0 0.25 0.5 0.750

0.5

117 × 17 Grid

Computational Fluid Dynamics IPIWVorticity at t = 1.2

Accuracy by Resolution

x

y

0 0.25 0.5 0.750

0.5

19 × 9 Grid

x

y

0 0.25 0.5 0.750

0.5

117 × 17 Grid

Computational Fluid Dynamics IPIWMini-Project

Add the temperature equation to the vorticity-streamfunction equation and compute the increase in heat transfer rate:

Mini-project:

Computational Fluid Dynamics IPIWConvection

∂T∂t

+ u∂T∂x

+ v∂T∂y

= α∂ 2T∂x2 +

∂ 2T∂y2

α =D

ρcp

where

Computational Fluid Dynamics IPIWNatural convection

Natural convection in a closed cavity. The left vertical wall is heated.

0



125



250



375



500



750



1000


200 400 600 800 10000

0.05

0.1

0.15

0.2

Natural convection

Heat Transfer Rate

Conduction only

Natural convection

WPI Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U Michigan... · WPI Computational Fluid Dynamics I A Finite Difference Code for the Navier-Stokes

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