Post on 09-Sep-2019
Remark: foils with „black background“could be skipped, they are aimed to the more advanced courses
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
Solvers, schemes SIMPLEx, upwind,…
Computer Fluid Dynamics E181107CFD7
2181106
FINITE VOLUME METHODCFD7
x
δx
y
z
δzSouthWest
Top
E
North
Bottom
δx
δy
FINITE CONTROL VOLUME δδδδx = Fluid element dx
Finite size Infinitely small size
FVM diffusion problem 1DCFD7
CV - control volume
A – surface of CV
[ ( ) ] 0
( ) 0
CV
A CV
S dV
n dA S dV
Φ
Φ
∇ Γ∇Φ + =
Γ∇Φ + =
∫
∫ ∫
i
�
i
Gauss theorem
1D case ( ) 0 ( ) ( )A B
d dS A B
dx dx
ΦΓ + = Φ =Φ Φ =Φ
A W P E B
w e
|2
P E E Pe e
PE
d
dx xδΓ +Γ Φ −ΦΦΓ = = |
2P W P W
w wWP
d
dx xδΓ +Γ Φ −ΦΦΓ = =
Diffusive flux
FVM diffusion problem 1DCFD7
( ) 0A CV
n dA S dVΦΓ∇Φ + =∫ ∫�
i
( ) ( ) 0
0
( ) ( ) ( )
e w
P WE Pe e w w U P P
PE WP
e e w w w w e eP P W E U
PE WP WP PE
d dA A S V
dx dx
A A S Sx x
A A A AS S
x x x x
δ δ
δ δ δ δ
Φ ΦΓ − Γ + ∆ =
Φ −ΦΦ −ΦΓ −Γ + + Φ =
Γ Γ Γ Γ+ − Φ = Φ + Φ +
aPaW
aE
Check properties
�aP = aW+aE for S=0 (without sources)
�All coefficients are positive
Overall flux fromthe west side
Linearization of source term
Tutorial – sphere dryingCFD7
W P E
0P WE Pe e w w
PE WP
T TT Tr k r k
x xδ δ−− − =
e eA r∼
202
202
1( ) 0 | 0 | ( ( ))
1( ) 0 | 0 | ( ( ))
wr r R o
w w wr r R e w
dd dT dT dTr k k T T R hD
r dr dr dr dr drd d dd
r D k Rr dr dr dr dr
ωα
ω ω ω β ω ω
= =
= =
= = = − +∆
= = = −
Heat transfer from air at temperature To
Enthalpy of evaporation
Diffusion coefficientMass transfer
coefficient
∆x ∆x/2
W P R
Internal control volumes
( ) 0P Wo R w w
WP
T TR T T r k
xα
δ−− − − =
TR temperature at surface of sphere must be extrapolated (using TW and TP)
FVM diffusion problem 2DCFD7
, ,....
P P W W E E N N S S U
w w e eW E
WP PE
P W E N S P
a a a a a S
A Aa a
x x
a a a a a S
δ δ
Φ = Φ + Φ + Φ + Φ +Γ Γ= =
= + + + −
( ) ( ) 0Sx x y y
∂ ∂Φ ∂ ∂ΦΓ + Γ + =∂ ∂ ∂ ∂
W P E
N
S
A
Boundary conditions
( )
|
| ( ( ) )
A
A A
A e
A
qx
k Ax
α
Φ = Φ∂Φ =∂
∂Φ = Φ − Φ∂
�First kind (Dirichlet BC)
�Second kind (Neumann BC)
�Third kind (Newton’s BC)
Example: insulation q=0
Example: Finite thermal resistance (heat transfer
coefficient α)
FVM unstructural mesh generationCFD7
i
j
Voronoi control volume associated with node i. Set of points x,yhaving distance to I less than the distance to any other node j.
Voronoi polygons
Delaunay triangulation
Circumcenterremains inside
Delaunay triangle
Finite Volume Method
Finite Element Method
FVM structural mesh generationCFD7
0
1
0
1
0
1
0
1
0X∇ ∇ =i
0Y∇ ∇ =i
Laplace equation solvers
FVM convection diffusion 1DCFD7
CV - control volume
A – surface of CV
( ) [ ( ) ]
( ) ( )
CV CV
A A CV
u dV S dV
n u dA n dA S dV
ρ
ρ
Φ
Φ
∇ Φ = ∇ Γ∇Φ +
Φ = Γ∇Φ +
∫ ∫
∫ ∫ ∫
�
i i
� � �
i i
Gauss theorem
1D case ( ) ( )d d d
u Sdx dx dx
ρ ΦΦ = Γ +
A W P E B
w e
( ) ee e e
PE
F u Dx
ρδ
Γ= =( ) ww w w
WP
F u Dx
ρδΓ= =
( ) ( )e w e E Pe w w P W U P PF F D D S S− = Φ −Φ − Φ −Φ + +Φ ΦΦ
FVM convection diffusion 1DCFD7
Relative importance of convective transport is characterised by Pecletnumber of cell (of control volume, because Pe depends upon size of cell)
w ew e
WP PE
D Dx xδ δΓ Γ= =
( ) ( ) w w e eF u F uρ ρ= =F-mass flux (through faces A)
D-diffusion conductanceContinuity equation (mass flowrateconservation) Fe=Fw
2
3
Re [ . ]
RePr [ ]/
Re [ ]
p p
m ABAB
u x F kgPe Pa s
D m s
u x k W kg K kgPe
k c c m K J m s
u x kg m kgPe Sc D
D m s m s
ρ δ µµ
ρ δ
ρ δ ρρ
= = = Γ = =⋅
⋅= = Γ = =⋅ ⋅
= = Γ = =⋅
Prandtl
SchmidtBinary diffusion coef
Thermal conductivity
FPe
D=
FVM convection diffusion 1DCFD7
Methods differ in the way how the unknown transported values at the control
volume faces (ΦΦΦΦw , ΦΦΦΦe) 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)
�Transportivity (schemes should depend upon the Peclet number of cell)
P P nb nbneighbours
a aΦ = Φ∑
FVM central scheme 1DCFD7
�Conservativeness (yes)
�Boundedness (positivity of coefficients)
�Transportivness (no)
2 2
P P W W E E
w eW w E e
a a a
F Fa D a D
Φ = Φ + Φ
= + = −
1 1( ) ( )
2 2e P E w P WΦ = Φ +Φ Φ = Φ +Φ
| |2e
ee
FPe
D= <
Very fine mesh is necessary
FVM upwind 1 st order 1DCFD7
�Conservativeness (yes)
�Boundedness (positivity of coefficients) YES for any Pe
�Transportivness (YES)
( ) ( )max ,0 max ,0P P W W E E
W w w E e e
a a a
a D F a D F
Φ = Φ + Φ= + = + −
for flow direction to right 0 else
for 0 else e P e e E
w P w w W
F
F
Φ = Φ > Φ = ΦΦ = Φ < Φ = Φ
But at a prize of decreased order of accuracy (only 1st
order!!)
FVM hybrid upwind Spalding 1972 1DCFD7
�Conservativeness (yes)
�Boundedness (positivity of coefficients) YES for any Pe
�Transportivness (YES)
max , ,0 max , ,02 2
P P W W E E
w eW w w E e e
a a a
F Fa F D a F D
Φ = Φ + Φ
= + = − −
But at a prize of decreased order of accuracy at high
Pe
�Central scheme for Pe<2
�Upwind with diffusion supressed for Pe>2
FVM power law Patankar 1980 1DCFD7
�Conservativeness (yes)
�Boundedness (positivity of coefficients) YES for any Pe
�Transportivness (YES)
5
5
max (1 ) ,0 max( ,0) 10
max (1 ) ,0 max( ,0)10
P P W W E E
wW w w
eE e e
a a a
Pea D F
Pea D F
Φ = Φ + Φ
= − +
= − + −
�Polynomial flow for Pe<10
�Upwind with zero diffusion for Pe>10
5(1 ) 110 2
Pe Pe− ≅ −
FVM QUICK Leonard 1979 1DCFD7
�Third order of accuracy very low numerical diffusion comparing with other s chemes
�Scheme is not bounded (stability problems are frequently encountered)
3 3+ ( +1) (1 )- (2- )
8 8 8 8
(1 ) 8 8
P P W W E E WW WW EE EE
e w w wW e w w E w e e
w eWW w EE e
a a a a a
F F F Fa D a D
F Fa a
α α α α
α α
Φ = Φ + Φ + Φ + Φ
= + = − − +
= − = −
Quadratic Upwind Interpolation – more nodal points necessary
WW W P E EE
w e
αe=1 for Fe>0 else αe=0. αw=1 for Fw>0 else αw=0.
FVM exponential Patankar 1980 1DCFD7
�Conservativeness (yes)
�Boundedness (positivity of coefficients) YES for any Pe
�Transportivness (YES)
/2 /2
/2 /2 /2 /2
P P W W E E
Pe Pe
W EPe Pe Pe Pe
a a a
e ea a
e e e e
−
− −
Φ = Φ + Φ
= =+ +
�Exact solution for constant velocity and diffusion coefficient Γ
FPe
D=
FVM exponential proofCFD7
1 1 2 1( )x
PexF x
c c c e cx x x F x Pe x
δδ∂Φ ∂ ∂Φ Γ ∂Φ ∂Φ= Γ → Φ = + → Φ = + → Φ = +∂ ∂ ∂ ∂ ∂
2 1
2 1
PeE
PeW
c e c
c e c−
Φ = +
Φ = +
�Exact solution for constant velocity and diffusion coefficient Γ
Assume constant mass flux F and constant Pe=F/D.
1
2
Pe PeE W
Pe Pe
E WPe Pe
e ec
e e
ce e
−
−
−
Φ − Φ=−
Φ − Φ=−
(1 ) (1 )Pe Pe Pe PeE W E W E W
P Pe Pe Pe Pe Pe Pe
e e e e
e e e e e e
− −
− − −
Φ − Φ Φ − Φ Φ − + Φ −Φ = + =− − −
/2 /2 /2 /2
/2 /2 /2 /2 /2 /2
/2
/2 /2
1 ( )
( )( )
Pe Pe Pe Pe Pe
E Pe Pe Pe Pe Pe Pe Pe Pe
Pe
W Pe Pe
e e e e ea
e e e e e e e e
ea
e e
− − − −
− − − −
−
− −= = =− − + +
=+
Analytical solution
Boundary conditions
Identified constants
Exact solution at central point
FVM central scheme with modified viscosityCFD7
Almost any scheme (with the exception of QUICK) can be converted to central scheme introducing modified transport coefficient Γ
Modified diffusion conductances are
and
.
* * 2 2w e
P P W W E E
w eW E
a a a
F Fa D a D
Φ = Φ + Φ
= + = −
*** * e
w e
w
WP PE
D Dx xδ δ
ΓΓ= =
*
*
*
1
| |1
2/2
tanh /2
Pe
Pe
Pe
Γ =ΓΓ = +ΓΓ =Γ
Central scheme
Upwind scheme
Exponential scheme
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.
FVM dispersion / diffusionCFD7
0 - boundary condition, all values are zero in the right triangle (without diffusion)
1Numerical diffusion
Numerical dispersion
FVM Navier Stokes equationsCFD7
2
2
( ) ( )
( ) ( )
0
u
v
u uv p u uS
x y x x x y y
uv v p v vS
x y y x x y y
u v
x y
ρ ρ µ µ
ρ ρ µ µ
ρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂+ =∂ ∂
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
FVM checkerboard patternCFD7
10
10
10
10
10
10
10
10 10
1000
0
0
0
0
0
00 0
0
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
W P E
2
( ) ( ) u
vS
x y x x x
u u up
y
u
y
ρ ρ µ µ∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
10 10| 0
2 2E W
P
p pp
x x xδ δ−∂ −≅ = =
∂
FVM staggered gridCFD7
10
10
10
10
10
10
10
10 10
1000
0
0
0
0
0
00 0
0
Different control volumes for different equations
Control volume for momentum balance in
y-direction
Control volume for momentum balance in
x-direction
Control volume for continuity equation,
temperature, concentrations
2
0 10
( ) ( ) u
p
u u u
x x
vS
x y x x x
p u
y y
δ
ρ ρ µ µ
∂ −≅∂
∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
FVM staggered gridCFD7
Location of velocities and pressure in staggered grid
I-2 I-1 I I+1 I+2i-1 i i+1 i+2
J+1
J
J-1
J-2
j+2
j+1
j
j-1
pu
v
Pressure I,J
U i,J
V I,j
Control volume for continuity equation,
temperature, concentrations
Control volume for momentum balance in
x-direction
Control volume for momentum balance
in y-direction
FVM continuity equationCFD7
I-2 I-1 I I+1 I+2i-1 i i+1 i+2
J+1
J
J-1
J-2
j+2
j+1
j
j-1
pu
v
Pressure I,J
U i,J
V I,j
Control volume for continuity equation,
temperature, concentrations
, 1 ,1, , 0I j I ji J i J v vu u
x yδ δ++ −−
+ =
δx
0u v
x y
∂ ∂+ =∂ ∂
FVM momentum xCFD7
I-2 I-1 I I+1 I+2i-1 i i+1 i+2
J+1
J
J-1
J-2
j+2
j+1
j
j-1
pu
Pressure I,J
U i,J
V I,jp
Control volume for momentum balance in
x-direction
, , , 1, ,( )i J i J nb nb i J I J I J iJneighbours
a u a u A p p b−= + − +∑
2
( ) ( ) u
vS
x y x x x
u u up
y
u
y
ρ ρ µ µ∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
FVM momentum yCFD7
I-2 I-1 I I+1 I+2i-1 i i+1 i+2
J+1
J
J-1
J-2
j+2
j+1
j
j-1
p
v
Pressure I,J
U i,J
V I,j
Control volume for momentum balance
in y-direction
p
( ) ( ) v
uS
x y y x
v vv v
x y y
vpρ ρ µ µ∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂
, , , , 1 , ,( )I j I j nb nb I j I J I J I jneighbours
a v a v A p p b−= + − +∑
FVM SIMPLE step 1 (velocities)CFD7
, , 1 ,
* * * *, , ,( )
I j nb I J I JI j nb I j I jneighbours
a v va A p p b−
= + − +∑
Input: Approximation of pressure p*
, 1, ,
* * * *, , ( )
i J nb I J I Ji J nb i J iJneighbours
a u Au a p p b−
= + − +∑
FVM SIMPLE step 2 (pressure correction)CFD7
* * *' ' 'p p p u u u v v v= + = + = +
' ' 'IJ IJ nb nb IJneighbours
a p a p b= +∑
, 1, ,
* * * *, , ( )
i J nb I J I Ji J nb i J iJneighbours
a u Au a p p b−
= + − +∑
, , , 1, ,( )i J i J nb nb i J I J I J iJneighbours
a u a u A p p b−= + − +∑
“true”solution
, 1, ,
' ' ' ', , ( )
i J nb I J I Ji J nb i Jneighbours
a au u A p p−
= + −∑
subtract Neglect!!!
, 1, , , 1 ,
' ' ' ' ' ', , ,( ) ( )
i J I J I J I J I Ji J I j I jd p p d p pu v− −
= − = −Substitute u*+u’ and v*+v’ into continuity equation
Solve equations for pressure corrections
FVM SIMPLE step 3 (update p,u,v)CFD7
*, , ,
*, , , 1, ,
*, , , , 1 ,
'
( ' ' )
( ' ' )
I J I J I J
i J i J i J I J I J
I j I j I j I J I J
p p p
u u d p p
v v d p p
−
−
← +
← + −
← + −
Continue with the improved pressures to the step 1
Only these new pressures are necessary for next iteration
FVM SIMPLEC (SIMPLE corrected)CFD7
, 1, ,
, , , 1, ,
, 1, ,
' ' ' ', ,
' ' ' ' ' ', ,
,' ' '
,
( )
( )
( )
i J nb I J I J
i J i J nb i J I J I J
i J I J I J
i J nb i Jneighbours
i J nb nb nb i Jneighbours neighbours neighbours
i J
i J nbneighbours
a a A p p
a a a a A p p
Ap p
a a
u u
u u u u
u
−
−
−
= + −
− = − + −
= −−
∑
∑ ∑ ∑
∑Improved diJ
Neglect!!! Good estimate as soon as neighbours are not
far from center
FVM SIMPLER (SIMPLE Revised)CFD7
Idea: calculate pressure distribution directly from the Poisson’s equation
2
2
2
( )
( )
( , )
uu p u
uu p u
p f u v
ρ µρ µ
∇ = −∇ + ∇∇ ∇ = −∇ ∇ + ∇ ∇∇ =
�� �
i
�� �
i i i i
we have at each cell discretised equation in this form,
For continuity we have
where
This interpolation of variables H and based on coefficients al forpressure velocity couplingis called Rhie-Chow interpolation . See also
FVM Rhie Chow (Fluent)CFD7
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
The way how to avoid staggering (difficult implementation in unstructured meshes) was suggested in the paper
FVM solversCFD7
�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: eachiteration calculates increment of vector Φ in the direction of gradient of minimised function – square ofresidual vector)
�Multigrid method (Fluent)
Solution of linear algebraic equation system Ax=b
Solver TDMA tridiagonal system MATLABCFD7
function x = TDMAsolver(a,b,c,d)%a, b, c, and d are the column vectors for the comp ressed tridiagonal matrixn = length(b); % n is the number of rows
% Modify the first-row coefficientsc(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:nid = 1 / (b(i) - c(i-1) * a(i)); % Division by zero risk. c(i) = c(i)* id; % Last value calcul ated 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);endend
Forward step (eliminationentries bellow diagonal)
1 1 1,2,...,i i i i i i ia x b x c x d i n− ++ + = =
Solver TDMA applied to 2D problems (ADI)CFD7
x
y
Y-implicit step (repeated application ofTDMA for 10 equations, proceeds from
left to right) X-implicit step (proceedsbottom up)
Solver PDMA pentagonal system MATLABCFD7
function x=pentsolve(A,b)
% Solve a pentadiagonal system Ax=b where A is a strongly nonsingular matrix% Reference: G. Engeln-Muellges, F. Uhlig, "Numerical Algorithms with C"% Chapter 4. Springer-Verlag Berlin (1996)%% Written by Greg von Winckel 3/15/04% Contact: gregvw@chtm.unm.edu%[M,N]=size(A);x=zeros(N,1);% Check for symmetryif A==A' % Symmetric Matrix Scheme
% Extract bandsd=diag(A);f=diag(A,1);e=diag(A,2); alpha=zeros(N,1);gamma=zeros(N-1,1);delta=zeros(N-2,1);c=zeros(N,1);z=zeros(N,1); % Factor A=LDL'alpha(1)=d(1);gamma(1)=f(1)/alpha(1);delta(1)=e(1)/alpha(1); alpha(2)=d(2)-f(1)*gamma(1);gamma(2)=(f(2)-e(1)*gamma(1))/alpha(2);delta(2)=e(2)/alpha(2); for k=3:N-2
alpha(k)=d(k)-e(k-2)*delta(k-2)-alpha(k-1)*gamma(k-1)^2;gamma(k)=(f(k)-e(k-1)*gamma(k-1))/alpha(k);delta(k)=e(k)/alpha(k);
end alpha(N-1)=d(N-1)-e(N-3)*delta(N-3)-alpha(N-2)*gamma(N-2)^2;gamma(N-1)=(f(N-1)-e(N-2)*gamma(N-2))/alpha(N-1);alpha(N)=d(N)-e(N-2)*delta(N-2)-alpha(N-1)*gamma(N-1)^2; % Update Lx=b, Dc=z z(1)=b(1);z(2)=b(2)-gamma(1)*z(1); for k=3:N
z(k)=b(k)-gamma(k-1)*z(k-1)-delta(k-2)*z(k-2);end c=z./alpha;
% Backsubstitution L'x=cx(N)=c(N);x(N-1)=c(N-1)-gamma(N-1)*x(N); for k=N-2:-1:1
x(k)=c(k)-gamma(k)*x(k+1)-delta(k)*x(k+2);end
else % Non-symmetric Matrix Schemed=diag(A);e=diag(A,1);f=diag(A,2);h=[0;diag(A,-1)];g=[0;0;diag(A,-2)]; alpha=zeros(N,1);gam=zeros(N-1,1);delta=zeros(N-2,1);bet=zeros(N,1); c=zeros(N,1);z=zeros(N,1); alpha(1)=d(1);gam(1)=e(1)/alpha(1);delta(1)=f(1)/alpha(1);bet(2)=h(2);alpha(2)=d(2)-bet(2)*gam(1);gam(2)=( e(2)-bet(2)*delta(1) )/alpha(2);delta(2)=f(2)/alpha(2); for k=3:N-2
bet(k)=h(k)-g(k)*gam(k-2);alpha(k)=d(k)-g(k)*delta(k-2)-bet(k)*gam(k-1);gam(k)=( e(k)-bet(k)*delta(k-1) )/alpha(k);delta(k)=f(k)/alpha(k);
endbet(N-1)=h(N-1)-g(N-1)*gam(N-3);alpha(N-1)=d(N-1)-g(N-1)*delta(N-3)-bet(N-1)*gam(N-2);gam(N-1)=( e(N-1)-bet(N-1)*delta(N-2) )/alpha(N-1);bet(N)=h(N)-g(N)*gam(N-2);alpha(N)=d(N)-g(N)*delta(N-2)-bet(N)*gam(N-1);% Update b=Lcc(1)=b(1)/alpha(1);c(2)=(b(2)-bet(2)*c(1))/alpha(2);
for k=3:Nc(k)=( b(k)-g(k)*c(k-2)-bet(k)*c(k-1) )/alpha(k);
end
% Back substitution Rx=cx(N)=c(N);x(N-1)=c(N-1)-gam(N-1)*x(N);for k=N-2:-1:1
x(k)=c(k)-gam(k)*x(k+1)-delta(k)*x(k+2); end
end
Solver CGM conjugated gradient GNUCFD7
function [x] = conjgrad(A,b,x)r=b-A*x;p=r;rsold=r'*r;
for i=1:size(A)(1)Ap=A*p;alpha=rsold/(p'*Ap);x=x+alpha*p;r=r-alpha*Ap;rsnew=r'*r;if sqrt(rsnew)<1e-10
break;endp=r+rsnew/rsold*p;rsold=rsnew;
endend
Solution of linear algebraic equation system Ax=b
Minimisation of quadratic function
Residual vector in k-thiteration
Vector of increment conjugated with previous increments
Solver MULTIGRIDCFD7
Rough grid (small wave numbers) Fine grid (large wave numbers details)
Interpolation fromrough to fine grid
Averaging from fineto rough grid
Solution on a rough grid takes into account very quickly long waves (distantboundaries etc), that is refined on a finer grid.
Unsteady flowsCFD7
Discretisation like in the finite difference methods discussed previously
n
n+1
∆x
∆tnPTn
WT nET
1nPT +
n-1
n
n+1
∆x
∆tnPT
1nWT + 1n
ET +1nPT +
n-1
n
n+1
∆x
∆tnPTn
WT nET
1nPT +
n-1
1nWT + 1n
ET +
Example: Temperature field
EXPLICIT scheme IMPLICIT scheme CRANK NICHOLSON scheme
2
2
xt
a
∆∆ <
2
2
T Ta
t x
∂ ∂=∂ ∂
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 bounded2x
ta
∆∆ <only if
FVM Boundary conditionsCFD7
Boundary conditions are specified at faces of cells (not at grid points)
�INLET specified u,v,w,T,k,ε (but not pressure!)
�OUTLET nothing is specified (p is calculated from continuity eq.)
�PRESSURE only pressure is specified (not velocities)
�WALL zero velocities, kP, εP calculated from the law of wall
PyP
Recommended combinations for several outlets
INLET
PRESSURE
PRESSURE
INLET
OUTLET
OUTLET
PRESSURE
PRESSURE
PRESSURE
Forbidden combinations for several outlets
PRESSURE
PRESSURE
OUTLET
there is no unique solutionbecause no flowrate is specified
FVM Boundary conditionsCFD7
�SYMMETRY
�PERIODICAL
Copied from Kozubkova: Fluent-textbook