Lecture Objectives
-
Upload
lucius-oneal -
Category
Documents
-
view
15 -
download
2
description
Transcript of Lecture Objectives
![Page 1: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/1.jpg)
Lecture Objectives
• Review
• SIMPLE CFD AlgorithmSIMPLE Semi-Implicit Method for Pressure-Linked Equations
• Define Residual and Relaxation
![Page 2: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/2.jpg)
Review
• Conservation equations
• Turbulent flow and turbulence modeling
• RANS Equation
• Discretization
• System of equation and solution methods– Accuracy– Numerical stability of solution procedure
• Solution algorithm (new today)
![Page 3: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/3.jpg)
Navier Stokes Equations
0z
v
y
v
x
v zyx
)(Sz
vμ
y
vμ
x
vμ
y
p)
z
vv
y
vv
x
vv
τ
vρ( yM2
y2
2
y2
2
y2
yz
yy
yx
y
TTgρ
xM2x
2
2x
2
2x
2x
zx
yx
xx S
z
vμ
y
vμ
x
vμ
x
p)
z
vv
y
vv
x
vv
τ
vρ(
zM2z
2
2z
2
2z
2z
zz
yz
xz S
z
vμ
y
vμ
x
vμ
z
p)
z
vv
y
vv
x
vv
τ
vρ(
In order to use linear equation solver we need to solve two problems:
1) find velocities that constitute in advection coefficients2) link pressure field with continuity equation
This velocities that constitute advection coefficients: F=V
Pressure is in momentum equations which already has one unknown
Continuity equation
Momentum x
Momentum y
Momentum z
![Page 4: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/4.jpg)
Pressure and velocities in NS equations
How to find velocities that constitute in advection coefficients?
xM2x
2
2x
2
2x
2x
zx
yx
xx S
z
vμ
y
vμ
x
vμ
x
p)
z
vv
y
vv
x
vv
τ
vρ(
fVaVaVaVaVaVaVaLPx,LHx,HNx,NSx,SWx,WEx,EPx,P
................................
a ,V
a
VVV6a
2Wx
2E
zyx
2P
xxx
xx
For the first step use Initial guessAnd for next iterative steps usethe values from previous iteration
![Page 5: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/5.jpg)
Pressure and velocities in NS equations
How to link pressure field with continuity equation?
SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm
The momentum equations can be solved only when the pressure field is given or is somehow estimated. Use * for estimated pressure and the corresponding velocities
P EW
x
x x
xxx
)/2P– (P
)/2P (P– )/2P (P
P– P
x
p EWEPPWew
xM2x
2
2x
2
2x
2x
zx
yx
xx S
z
vμ
y
vμ
x
vμ
x
p)
z
vv
y
vv
x
vv
τ
vρ(
sideEW
LxLHxHNxNSxSWxWExEPxP
)/2P– (P fVaVaVaVaVaVaVa A
x
AeAw
Aw=Ae=Aside
We have two additional equations for y and x directions
![Page 6: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/6.jpg)
SIMPLE algorithmGuess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L
1) For this pressure field solve system of equations:
Solution is:
sideEW
LxLHxHNxNSxSWxWExEPxP
)/2P– (P fVaVaVaVaVaVaVa A
xx:
y:
z:
………………..………………..
LxHxNxSxWxExPx *V,*V,*V,*V,*V,*V,*V
P = P* + P’
2) The pressure and velocity correction
P’ – pressure correction
V = V* + f(P’)
For all nodes E,W,N,S,…
V’ – velocity correction
Substitute P=P* + P’ into momentum equations (simplify equation) and obtain
3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V
V’=f(P’)
V = V* + V’
4) Solve T , k , equations
![Page 7: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/7.jpg)
SIMPLE algorithm
Step1: solve V* from momentum equations
Step2: introduce correction P’ and express V = V* + f(P’)
Step3: substitute V into continuity equation solve P’ and then V
Step4: Solve T , k , e equations
Guess p*
start
end
Converged (residual check)
yes
no
p=p*
![Page 8: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/8.jpg)
Other methods
SIMPLERSIMPLEC variation of SIMPLEPISO
COUPLED - use Jacobeans of nonlinear velocity functions to form linear matrix ( and avoid iteration )
![Page 9: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/9.jpg)
ResidualExample:
x-exp(1/x)-2=0
Find x using iteration
Explicit form 1:
x=exp(1/x)+2
Explicit form 2:
x=1/(ln(x)-ln(2))
Solution process:
Guess x0
Iteration :
x1=exp(1/x0)+2 , R1=x1-x0
X2=exp(1/x1)+2 , R2=x2-x1
……..…….
Not all iteration process converge!
See the example for the same equation
![Page 10: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/10.jpg)
Convergence example
Explicit form 2:
x=1/(ln(x)-ln(2)
![Page 11: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/11.jpg)
Residual calculation for CFD
• Residual for the cell
Rijk=kijk-k-1
ijk
• Total residual for the simulation domain
Rtotal=Rijk|
• Scaled (normalized) residual R=Rijk|/F
iteration
cell positionVariable: p,V,T,…
For all cells
Flux of variable used for normalizationVary for different CFD software
![Page 12: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/12.jpg)
RelaxationRelaxation with iterative solvers:
When the equations are nonlinearit can happen that you get divergency in iterative procedure for solving consideredtime step
Under-Relaxation is often required when you have nonlinear equations!
iteration
convergence
variabledivergence
solution
Solution is Under-Relaxation:
Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor
For our example Y*in iteration 101=f·Y(100)+(1-f) ·Y(99)
f = [0-1] – under-relaxation -stabilize the iterationf = [1-2] – over-relaxation - speed-up the convergence
Value which is should be used for the next iteration
![Page 13: Lecture Objectives](https://reader036.fdocuments.us/reader036/viewer/2022082818/56812e5b550346895d9402cc/html5/thumbnails/13.jpg)
Example of relaxation(example from homework 3 assignment)
N1NNNN1-NN fTcTbTa
Example: Advection diffusion equation, 1-D, steady-state, 4 nodes
1NNN1-NNNNNN T/bcT/bafb/1T 1 2 3 4
1) Explicit format:
2) Guess initial values:
..T ..,T ..,T ...,T 04
03
02
01
3) Substitute and calculate:
20
111111 T/bcfb/1T
30
2211
222221 T/bcT/bafb/1T
40
3321
333331 T/bcT/bafb/1T
31
444441 T/bafb/1T
..T ..,T ..,T ...,T 14
13
12
11
Substitute and calculate:4) ..T ..,T ..,T ...,T 24
23
22
21
………………………….
.... ,f)T-(1fTT ,f)T-(1fTT 02
12
1r2
01
11
1r1
.... ,f)T-(1fTT ,f)T-(1fTT 12
22
2r2
11
21
2r1 Substitute and calculate: