1. Computer Fluid Dynamics E181107
CFD7r
Solvers, schemes SIMPLEx, upwind REDUCED lecture…
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. FINITE VOLUME METHOD Top West South
CFD7r x x y z z
South
West
Top E
North
Bottom y
FINITE CONTROL VOLUME x = Fluid element dx
Finite size
Infinitely small size

3. FVM diffusion problem 2D
CFD7r N
W P e E A S
Boundary conditions
Example: insulation q=0
First kind (Dirichlet BC)
Second kind (Neumann BC)
Third kind (Newton’s BC)
Example: Finite thermal resistance (heat transfer coefficient )

4. FVM convection diffusion 1D
CFD7r
CV - control volume
A – surface of CV
Gauss theorem
1D case
F-mass flux D-diffusion conductance
A W P E B
w e

5. FVM convection diffusion 1D
CFD7r
Relative importance of convective transport is characterised by Peclet number of cell (of control volume, because Pe depends upon size of cell)
Thermal conductivity
Prandtl
Binary diffusion coef
Schmidt

6. FVM convection diffusion 1D
CFD7r
Methods differ in the way how the unknown transported values at the control volume faces (w , e) are calculated (different interpolation techniques)
Result can be always expressed in the form
Properties of resulting schemes should be evaluated
Conservativeness
Boundedness (positivity of coefficients anb)
Transportivity (schemes should depend upon the Peclet number of cell)

7. Very fine mesh is necessary
FVM central scheme 1D
CFD7r
Conservativeness (yes)
Boundedness (positivity of coefficients)
Transportivness (no)
Very fine mesh is necessary

8. But at a prize of decreased order of accuracy (only 1st order!!)
FVM upwind 1st order 1D
CFD7r
Conservativeness (yes)
Boundedness (positivity of coefficients) YES for any Pe
Transportivness (YES)
But at a prize of decreased order of accuracy (only 1st order!!)

9. FVM Upwind FLUENT CFD7r Upwind of the first order
Upwind of the second order 1 f 0

10. Projection of velocity to the direction r
FVM exponential FLUENT
CFD7r
Called power law model in Fluent
Exact solution of
Projection of velocity to the direction r 1 r 0
For Pe<<1 the exponential profile is reduced to linear profile

11. FVM dispersion / diffusion
CFD7r
There are two basic errors caused by approximation of convective terms
False diffusion (or numerical diffusion) – artificial smearing of jumps, discontinuities. Neglected terms in the Taylor series expansion of first derivatives (convective terms) represent in fact the terms with second derivatives (diffusion terms). So when using low order formula for convection terms (for example upwind of the first order), their inaccuracy is manifested by the artificial increase of diffusive terms (e.g. by increase of viscosity). The false diffusion effect is important first of all in the case when the flow direction is not aligned with mesh, see example
Dispersion (or aliasing) – causing overshoots, artificial oscillations. Dispersion means that different components of Fourier expansion (of numerical solution) move with different velocities, for example shorter wavelengths move slower than the velocity of flow.
Numerical dispersion
Numerical diffusion 1
u=v
0 - boundary condition, all values are zero in the right triangle (without diffusion)

12. FVM Navier Stokes equations
CFD7r
Different methods are used for numerical solution of Navier Stokes equations at
Compressible flows (finite speed of sound, existence of velocity discontinuities at shocks)
Incompressible flows (infinite speed of sound)
Explicit methods (e.g. MacCormax) but also implicit methods (Beam-Warming scheme) are used. These methods will not be discussed here.
Vorticity and stream function methods (pressure is eliminated from NS equations). These methods are suitable especially for 2D flows.
Primitive variable approach (formulation in terms of velocities and pressure). Pressure represents a continuity equation constraint. Only these methods (pressure corrections methods SIMPLE, SIMPLER, SIMPLEC and PISO) using staggered and collocated grid will be discussed in this lecture.
Beckman

13. FVM Navier Stokes equations
CFD7r
Example: Steady state and 2D (velocities u,v and pressure p are unknown functions)
This is equation for u
This is equation for v
This is equation for p?
But pressure p is not in the continuity equation! Solution of this problem is in SIMPLE methods, described later

14. FVM checkerboard pattern
CFD7r
Other problem is called checkerboard pattern
This nonuniform distribution of pressure has no effect upon NS equations
The problem appears as soon as the u,v,p values are concentrated in the same control volume 10
W P E

15. FVM staggered grid CFD7r
Different control volumes for different equations
Control volume for momentum balance in y-direction 10
Control volume for continuity equation, temperature, concentrations
Control volume for momentum balance in x-direction

16. FVM staggered grid j+1 J+1 j-1 J-2 I-2 I-1 I I+1 I+2 j CFD7r
Location of velocities and pressure in staggered grid
Control volume for momentum balance in y-direction
Control volume for continuity equation, temperature, concentrations
j+2
j+1 j
j-1
J+1 J
J-1
J-2
Pressure I,J
U i,J
V I,j u p v
Control volume for momentum balance in x-direction
I I I I+1 I+2
i i i i+2

17. Control volume for momentum balance in x-direction
FVM momentum x
CFD7r
j+2
j+1 j
j-1
J+1 J
J-1
J-2
Pressure I,J
U i,J
V I,j u p p
Control volume for momentum balance in x-direction
I I I I+1 I+2
i i i i+2

18. FVM SIMPLE step 1 (velocities)
CFD7r
Input: Approximation of pressure p*

19. Solve equations for pressure corrections
FVM SIMPLE step 2 (pressure correction)
CFD7r
“true” solution
Neglect!!!
subtract
Substitute u*+u’ and v*+v’ into continuity equation
Solve equations for pressure corrections

20. Only these new pressures are necessary for next iteration
FVM SIMPLE step 3 (update p,u,v)
CFD7r
Only these new pressures are necessary for next iteration
Continue with the improved pressures to the step 1

21. FVM SIMPLER (SIMPLE Revised)
CFD7r
Idea: calculate pressure distribution directly from the Poisson’s equation

22. FVM Rhie Chow (Fluent) CFD7r
The way how to avoid staggering (difficult implementation in unstructured meshes) was suggested in the paper
Rhie C.M., Chow W.L.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, Vol.21, No.11, 1983
This technique is called Rhie-Chow interpolation.

23. FVM Rhie Chow (Fluent) CFD7r
Rhie C.M., Chow W.L.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, Vol.21, No.11, 1983 P E e
How to interpolate ue velocity at a control volume face

24. Fluent - solvers CFD7r Segregated solver Coupled solver , , cp,…
w…
p… (SIMPLE…)
CFL Courant Friedrichs Lewy criterion
k,,……
T,k,,……
Nonstationary problem always solved by implicit Euler
Nonstationary problem solved by implicit Euler (CFL~5), explicit Euler (CFL~1) or Runge Kutta
Algebraic eq. by multigrid AMG
Algebraic eq. by multigrid AMG, or FAS (Full Aproximation Storage)

25. FVM solvers
CFD7r
Dalí

26. FVM solvers
CFD7r
Solution of linear algebraic equation system Ax=b
TDMA – Gauss elimination tridiagonal matrix (obvious choice for 1D problems, but suitable for 2D and 3D problems too, iteratively along the x,y,z grid lines – method of alternating directions ADI)
PDMA – pentadiagonal matrix (suitable for QUICK), Fortran version
CGM – Conjugated Gradient Method (iterative method: each iteration calculates increment of vector  in the direction of gradient of minimised function – square of residual vector)
Multigrid method (Fluent)

27. Forward step (elimination entries bellow diagonal)
Solver TDMA tridiagonal system MATLAB
CFD7r
function x = TDMAsolver(a,b,c,d)
%a, b, c, and d are the column vectors for the compressed tridiagonal matrix
n = length(b); % n is the number of rows
% Modify the first-row coefficients
c(1) = c(1) / b(1); % Division by zero risk.
d(1) = d(1) / b(1); % Division by zero would imply a singular matrix.
for i = 2:n
id = 1 / (b(i) - c(i-1) * a(i)); % Division by zero risk.
c(i) = c(i)* id; % Last value calculated is redundant.
d(i) = (d(i) - d(i-1) * a(i)) * id;
end
% Now back substitute.
x(n) = d(n);
for i = n-1:-1:1
x(i) = d(i) - c(i) * x(i + 1);
Forward step (elimination entries bellow diagonal)

28. X-implicit step (proceeds bottom up)
Solver TDMA applied to 2D problems (ADI)
CFD7r
Y-implicit step (repeated application of TDMA for 10 equations, proceeds from left to right)
X-implicit step (proceeds bottom up) y x

29. Solver MULTIGRID
CFD7r
Solution on a rough grid takes into account very quickly long waves (distant boundaries etc), that is refined on a finer grid.
Rough grid (small wave numbers)
Fine grid (large wave numbers details)
Interpolation from rough to fine grid
Averaging from fine to rough grid

30. Unsteady flows
CFD7r
Discretisation like in the finite difference methods discussed previously
Example: Temperature field n
n+1 ∆x ∆t
n-1 n
n+1 ∆x ∆t
n-1 n
n+1 ∆x ∆t
n-1
EXPLICIT scheme
IMPLICIT scheme
CRANK NICHOLSON scheme
First order accuracy in time First order accuracy in time Second order accuracy in time
Stable only if Unconditionally stable and bounded Unconditionally stable, but bounded
only if

31. there is no unique solution because no flowrate is specified
FVM Boundary conditions
CFD7r
Boundary conditions are specified at faces of cells (not at grid points)
INLET specified u,v,w,T,k, (but not pressure!)
OUTLET nothing is specified (p is calculated from continuity eq.)
PRESSURE only pressure is specified (not velocities)
WALL zero velocities, kP, P calculated from the law of wall yP P
Recommended combinations for several outlets
INLET
PRESSURE
PRESSURE
INLET
OUTLET
Forbidden combinations for several outlets
PRESSURE
OUTLET
there is no unique solution because no flowrate is specified

32. Stream function-vorticity
CFD7r
Introducing stream function and vorticity instead of primitive variables (velocities and pressure) eliminates the problem with the checkerboard pattern. Continuity equation is exactly satisfied.
However the technique is used mostly for 2D problems (it is sufficient to solve only 2 equations), because in 3D problems 3 components of vorticity and 3 components of vector potential must be solved (comparing with only 4 equations when using primitive variable formulation of NS equations).
Ghirlandaio

33. Stream function-vorticity
CFD7r
The best method for solution of 2D problems of incompressible flows
Pressure elimination
Introduction vorticity (z-component of vorticity vector)
Velocities expressed in terms of scalar stream function
Vorticity expressed in terms of stream function

34. Stream function-vorticity
CFD7r
Three equations for u,v,p are replaced by two equations for , . Advantage: continuity equation is exactly satisfied. Vorticity transport is parabolic equation, Poisson equaion for stream function is eliptic.
Lines of constant  are streamlines (at wall =const). The no-slip condition at wall (prescribed velocities) must be reflected by boundary condition for vorticity
How to do it, is shown in the next slide on example of flow in a channel

35.  Increases from 0 to w for example =uy
Stream function-vorticity
CFD7r
Stream function 
 Increases from 0 to w for example =uy
N-1=N M y H 2
N N x 1
=0
Vorticity
Zero vorticity at axis (follows from definition)
Prescribed velocity profile at inlet u1(y) and v1(y)=0
Vorticity at wall (prescribed velocity U, in this case zero)

36. Example: cavity flow 1/4 CFD7r
Cavity flow. Lid of cavity is moving with a constant velocity U. There is no in-out flow, therefore stream function (proportional to flowrate) is zero at boundary, at all walls of cavity x y H
N N 1 2 M UM
Vorticity at moving lid
Vorticity at fixed walls

37. Example: cavity flow 2/4 CFD7r UM
W P E UM N S
FVM applied to vorticity transport equation (upwind of the first order). Equidistant grid is assumed =x=y
Eliptic equation for stream function
This systém of algebraic equation (including boundary conditions) is to be solved iteratively (for example by alternating direction method ADI described previously)

38. Example: cavity flow 3/4 CFD7r Matlab
% tok v dutině. metoda vířivosti a pokutové funkice. Upwind. Metoda
% střídavých směrů.
h=0.01;
um=0.0001;
visc=1e-6;
% parametry site, a casovy krok
relax=0.5;
n=21;
d=h/(n-1);
d2=d^2;
niter=50;
psi=zeros(n,n);
omega=zeros(n,n);
u=zeros(n,n);
v=zeros(n,n);
a=zeros(n,1);
b=ones(n,1);
c=zeros(n,1);
r=zeros(n,1);
for iter=1:niter
for i=1:n
u(i,n)=um;
end
for i=2:n-1
for j=2:n-1
u(i,j)=(psi(i,j+1)-psi(i,j-1))/(2*d);
v(i,j)=(psi(i-1,j)-psi(i+1,j))/(2*d);
%stream function x-implicit
for j=2:n-1
for i=2:n-1
a(i)=-1/d2;b(i)=4/d2;c(i)=-1/d2; r(i)=(psi(i,j-1)+psi(i,j+1))/d2+ omega(i,j);
end
r(1)=0;r(n)=0;
ps=tridag(a,b,c,r,n);
psi(i,j)=(1-relax)*psi(i,j)+relax*ps(i);
%stream function y-implicit
a(j)=-1/d2;b(j)=4/d2;c(j)=-1/d2;r(j)=(psi(i-1,j)+psi(i+1,j))/d2+ omega(i,j);
psi(i,j)=(1-relax)*psi(i,j)+relax*ps(j);
% vorticity boundary conditions
for i=1:n
omega(i,n)=relax*omega(i,n)+(1-relax)*2/d2* (psi(i,n)-psi(i,n-1)-um*d);
omega(i,1)=relax*omega(i,1)+(1-relax)*2* (psi(i,1)-psi(i,2))/d2;
omega(1,i)=relax*omega(1,i)+(1-relax)*2*(psi(1,i)-psi(2,i))/d2;
omega(n,i)=relax*omega(n,i)+(1-relax)*2*(psi(n,i)-psi(n-1,i))/d2;
%vorticity x-implicit
for j=2:n-1
for i=2:n-1
up=u(i,j);vp=v(i,j);
a(i)=-visc/d2-max(up,0)/d;
b(i)=4*visc/d2+(abs(up)+abs(vp))/d;
c(i)=-visc/d2-max(-up,0)/d;
r(i)=omega(i,j-1)*(visc/d2+max(vp,0)/d)+ omega(i,j+1)*(visc/d2+max(-vp,0)/d);
end
r(1)=omega(1,j);r(n)=omega(n,j);
ps=tridag(a,b,c,r,n);
omega(i,j)=(1-relax)*omega(i,j)+relax*ps(i);
%vorticity y-implicit
a(j)=-visc/d2-max(vp,0)/d;
c(j)=-visc/d2-max(-vp,0)/d;
r(j)=omega(i-,j)*(visc/d2+max(up,0)/d) omega(i+1,j)* (visc/d2+max(-up,0)/d);
r(1)=omega(i,1);r(n)=omega(i,n);
omega(i,j)=(1-relax)*omega(i,j)+relax*ps(j);

39. Example: cavity flow 4/4
CFD7r
Re=1000
Re=1  u  v 

40. Example: cavity flow FLUENT
CFD7r
Re=1 u v