WPI Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U Michigan... ·...
Transcript of WPI Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U Michigan... ·...
![Page 1: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/2.jpg)
Computational Fluid Dynamics IPPPPIIIIWWWW
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:
![Page 3: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/3.jpg)
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
![Page 4: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/4.jpg)
Computational Fluid Dynamics IPPPPIIIIWWWW
Moving wall
Stationary walls
The Driven Cavity Problem
![Page 5: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/5.jpg)
Computational Fluid Dynamics IPPPPIIIIWWWW
++−=++
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
![Page 6: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/7.jpg)
Computational Fluid Dynamics IPIW
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 vorticity/stream function equations
The Stream Function Equation
![Page 8: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/8.jpg)
Computational Fluid Dynamics IPIW
∂2ψ∂x2 +
∂2ψ∂y2 = −ω
ω =∂v∂x
−∂u∂y
u =∂ψ∂y
; v = −∂ψ∂x
Substituting
into the definition of the vorticity
yields
The vorticity/stream function equations
![Page 9: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/9.jpg)
Computational Fluid Dynamics IPIW
∂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
The vorticity/stream function equations
![Page 10: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/10.jpg)
Computational Fluid Dynamics IPIWBoundary Conditions for the Stream Function
u = 0 ⇒∂ψ∂y
= 0
⇒ ψ = Constant
At the right and the left boundary:
![Page 11: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/11.jpg)
Computational Fluid Dynamics IPIW
v = 0 ⇒∂ψ∂x
= 0
⇒ ψ = Constant
At the top and thebottom boundary:
Boundary Conditions for the Stream Function
![Page 12: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/12.jpg)
Computational Fluid Dynamics IPIW
Since the boundaries meet, the constant must be the same on all boundaries
ψ = Constant
Boundary Conditions for the Stream Function
![Page 13: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/14.jpg)
Computational Fluid Dynamics IPIW
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
![Page 15: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/16.jpg)
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.
![Page 17: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/17.jpg)
Computational Fluid Dynamics IPIW
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)
![Page 18: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/18.jpg)
Computational Fluid Dynamics IPIW
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)
![Page 19: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/19.jpg)
Computational Fluid Dynamics IPIW
!++−−+=6
)(2
)()()( 2
3
3 hx
xfh
hxfhxfxxf
∂∂
∂∂
Finite difference approximations
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
![Page 20: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/20.jpg)
Computational Fluid Dynamics IPIWFinite Difference Approximations
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
![Page 21: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/21.jpg)
Computational Fluid Dynamics IPIWFinite Difference Approximations
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
![Page 22: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/22.jpg)
Computational Fluid Dynamics IPIW
−++++
−
−+
−
−−
=∆−
−+−+
−+−+−+−+
+
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:
Finite Difference Approximations
+++−= 2
2
2
2
Re1
yxyxxyt ∂ω∂
∂ω∂
∂∂ω
∂∂ψ
∂∂ω
∂∂ψ
∂∂ω
![Page 23: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/23.jpg)
Computational Fluid Dynamics IPIW
−++++
−
−−
−
−∆−=
−+−+
−+−+
−+−++
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:
Finite Difference Approximations
![Page 24: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/24.jpg)
Computational Fluid Dynamics IPIW
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
Finite Difference Approximations
![Page 25: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/25.jpg)
Computational Fluid Dynamics IPIW
Discretize the domainψ i, j = 0
i=1 i=2 i=nxj=1j=2
j=ny
i =1i = nxj =1j = ny
Discretized Domain
for
![Page 26: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/26.jpg)
Computational Fluid Dynamics IPIW
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
![Page 27: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/27.jpg)
Computational Fluid Dynamics IPIW
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:
Discrete Boundary Condition
ψ i, j = 2 =ψ i, j =1 +∂ψ i, j =1
∂yh +
∂ 2ψ i , j =1
∂y2h2
2+ O(h3 )
![Page 28: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/28.jpg)
Computational Fluid Dynamics IPIW
ωwall = ψ i , j =1 −ψ i , j =2( ) 2h2 +Uwall
2h
+O(h)
At the bottom wall (j=1):
Discrete Boundary Condition
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
![Page 29: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/29.jpg)
Computational Fluid Dynamics IPIW
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
![Page 30: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/30.jpg)
Computational Fluid Dynamics IPIW
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
![Page 31: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/31.jpg)
Computational Fluid Dynamics IPIW
Limitations on the time step
ν∆th2 ≤ 1
4(| u | + | v |)∆t
ν≤ 2
Time Step Control
![Page 32: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/32.jpg)
Computational Fluid Dynamics IPIW
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)
![Page 33: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/33.jpg)
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
![Page 34: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/34.jpg)
Computational Fluid Dynamics IPIWnx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0
Solution Fields
![Page 35: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/35.jpg)
Computational Fluid Dynamics IPIWnx=17, ny=17, dt = 0.005, Re = 10, Uwall = 1.0
Solution Fields
![Page 36: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/36.jpg)
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
![Page 37: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/37.jpg)
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
![Page 38: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/38.jpg)
Computational Fluid Dynamics IPIWMini-Project
Add the temperature equation to the vorticity-streamfunction equation and compute the increase in heat transfer rate:
Mini-project:
![Page 39: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/39.jpg)
Computational Fluid Dynamics IPIWConvection
∂T∂t
+ u∂T∂x
+ v∂T∂y
= α∂ 2T∂x2 +
∂ 2T∂y2
α =D
ρcp
where
![Page 40: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/40.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
0
![Page 41: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/41.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
125
![Page 42: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/42.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
250
![Page 43: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/43.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
375
![Page 44: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/44.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
500
![Page 45: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/45.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
750
![Page 46: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/46.jpg)
Computational Fluid Dynamics IPIWNatural convection
Natural convection in a closed cavity. The left vertical wall is heated.
1000
![Page 47: 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](https://reader030.fdocuments.us/reader030/viewer/2022040718/5e26bf56062d096e666b0132/html5/thumbnails/47.jpg)
Computational Fluid Dynamics IPIW
200 400 600 800 10000
0.05
0.1
0.15
0.2
Natural convection
Heat Transfer Rate
Conduction only
Natural convection