1. Lecture Objectives: Advance discretization methods
Review Residual, Stability, Relaxation
Simple algorithm

2. Advection diffusion equation 1-D, steady-stateW Dx P Dx E Dx w e y Q
Example:
Equation for temperature
of water flowing through hot pipe
Assume that diffusion in y direction ins negligible: This is incorrect assumption introduced jut to simplify example to 1-D problem! x
Temperature is changing along x Q
model Vx T1 T2 T3 T4 … Tn

3. Advection diffusion equation 1-D, steady-stateDx N Dx
N+1
Different notation: Dx
General equation

4. Advection equation 1-D, steady-stateW Dx P Dx E
Vx>0 Dx
1) Upwind scheme:
Vx<0
2) Central differencing scheme:
3) Hybrid of upwind and central differencing scheme
Higher order differencing scheme:
Quadratic upwind differencing
Scheme (QUICK)
N-2
N-1 N
N+1
N+2 WW P W E EE
We need to find coefficients
aP, aW, aE, aWW, aEE,

5. Central Difference and Upwind Differencing Schemes
Notation of coefficients in book
Stability of the Central Difference Schemes
Pe = Diffusion / Advection = (Vx)/(/∆x)<2
It defined by the requirement that coefficient aE>0
Requirement for “boundedness”
D=/∆x
F=Vx

6. Hybrid Differencing Schemes
Notation of coefficients in the book T N W P E S B

7. Higher order differencing scheme: Quadratic upwind differencing
Scheme (QUICK)
N-2
N-1 N
N+1
N+2 WW P W E EE
We need to find coefficients
aP, aW, aE, aWW, aEE,

8. Quadratic upwind differencing
Scheme (QUICK)
Coefficients:
Advection coefficient:
Source:
Diffusion coefficients :
For advection only:

9. Residual calculation for CFD
Residual for the cell
RFijk=Fkijk-Fk-1ijk
Total residual for the simulation domain
RFtotal=S|RFijk|
Scaled (normalized) residual
RF=S|RFijk|/FF
iteration
cell position
Variable: p,V,T,…
For all cells
Flux of variable F used for normalization
Vary for different CFD software

10. Relaxation Relaxation with iterative solvers:
When the equations are nonlinear
it can happen that you get divergency
in iterative procedure for solving considered
time step
divergence
variable
solution
convergence
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 iteration
f = [1-2] – over-relaxation - speed-up the convergence
iteration
Value which is should be used for the next iteration
Under-Relaxation is often required when you have nonlinear equations!

11. Example of relaxation
Example: Advection diffusion equation, 1-D, steady-state, 4 nodes
1) Explicit format: 1 2 3 4
2) Guess initial values: 3)
Substitute and calculate: 4)
Substitute and calculate:
Substitute and calculate:
………………………….

12. Navier Stokes Equations CFD Specific Solver
Continuity equation
This velocities that constitute advection coefficients: F=rV
Momentum x
Momentum y
Momentum z
Pressure is in momentum equations
which already has one unknown
In order to use linear equation solver we need to solve two problems:
find velocities that constitute in advection coefficients
2) link pressure field with continuity equation

13. Pressure and velocities in NS equations
How to find velocities that constitute advection coefficients?
For the first step use Initial guess
And for next iterative steps use
the values from previous iteration

14. Pressure and velocities in NS equations
How to link pressure field with continuity equation?
SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm W Dx P Dx E Dx Aw Ae
Aw=Ae=Aside
We have two additional equations for y and x directions
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

15. SIMPLE algorithm
Guess 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: x: y:
………………..
……………….. z:
Solution is:
2) The pressure and velocity correction
P = P* + P’
P’ – pressure correction
For all nodes E,W,N,S,…
V = V* + V’
V’ – velocity correction
Substitute P=P* + P’ into momentum equations (simplify equation) and obtain
V’=f(P’)
V = V* + f(P’)
3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V
4) Solve T , k , e equations

16. SIMPLE algorithm start Guess p* p=p*
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 no
Converged
(residual check)
yes
end

17. Other methods SIMPLER SIMPLEC variation of SIMPLE PISO
COUPLED - use Jacobeans of nonlinear velocity functions to form
linear matrix ( and avoid iteration )

18. Newton-Raphson method (example of Jacobean solver)
Faster convergence
Used in many professional tools (MathCAD, EES, MatLab, Mathematica, etc)
More complex for programming
Requires linear solver
Based on Taylor-Series Expansion
You need first derivative for each function to create the Jacobean matrix
Equations in the form where all side are on one side of equality sign
Our simple example: X-Y/2= → X-Y/2+1=0
X2-Y= → X2-Y+3=0

19. Newton-Raphson method
Section 6.11 of handouts
Our simple example:
f1 = X-Y/2+1=0
f2 = X2-Y+3=0
Steps:
0) Find derivatives d(f1)/dX = , d(f1)/dY =-1/2
d(f2)/dX =2X , d(f2)/dY =-1
1) Initial guess: Y(0)=2, X(0)=2
2) Find f1(Y(0),X(0))=2-2/2+1=2
f2(Y(0),X(0))=22-2+3=5
3) Using derivatives and guess values find the Jacobean matrix
4) Solve the matrix using linear solver and find DX and DY
5) Find Y(1)=Y(0)+ DY, X(1)=X(0)+ DX,
Repeat step (2) with Y(1) and X(1) … Follow the procedure till convergence
Unknowns
(correction Dxi)
Jacobean matrix
Function values
for guessed
variables

20. Course Review (so far)

21. Conservation Equations Navier Stokes Equations

22. Reynolds Averaged Navier Stokes equations
Continuity: 1)
Momentum: 2) 3) 4)
Similar is for STy and STx
4 equations 5 unknowns → We need to model

23. Modeling of Turbulent Viscosity
Fluid property – often called laminar viscosity
Flow property – turbulent viscosity
MVM: Mean velocity models
TKEM: Turbulent kinetic energy equation models
Additional models:
LES: Large Eddy simulation models
RSM: Reynolds stress models

24. Discretization and equation solver
SIMPLE algorithm
Discretization of RANS
Guess p*
p=p*
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 no
Converged
(residual check)
yes
end

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