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Implementation of Simple Algorithm

HI to All,

I am trying to solve Lid Driven Cavity problem using Finite Volume Method, for that i want to use Simple Algorithm. I have all ready solve LDC using Artificial Compressibility now i want to use SIMPLE Algorithm to solve velocity-pressure coupling, for that i am following Versteeg's book.

I am using steady navier stokes equations for LDC.

The procedure that i have adopted is given below:

1. Initialize u*,v*,p*
2. Solve Discretised momentum equation and Calculated u*,v*

NOW comes the problem. in step 3 when i try to calculate p'.
3.
the equation is
a[i,j]p'[i,j]=a[i-1,j]p'[i-1,j]+a[i+1,j]p'[i+1,j]+a[i,j-1]p'[i,j-1]+a[i,j+1]p'[i,j+1]+b'[i,j]

a[i-1,j]=(d A)[i-1,j]

a[i+1,j]=(d A)[i+1,j]

a[i,j-1]=(d A)[i,j-1]

a[i,j+1]=(d A)[i,j+1]

b'[i,j] = (u*A)[i-1,j]-(u*A)[i+1,j]+(v*A)[i,j+1]-(v*A)[i,j-1]
In this equation i need d to be calculated which is d=A/a where a is the centeral coefficient of velocity equation.
where i try to calculate this it some how a become zero which makes d=infinity. and p' become undefined or infinity. that cause problems in step 4 i.e.

4.
p[i,j]=p*[i,j]+p'[i,j]
u[i,j]=u*[i,j]+d[i,j]*(p'[i-1,j]-p'[i,j])
u[i,j]=u[i,j]+u[i,j]*(p'[i,j-1]-p'[i,j])
5. Set Boundary condition for u and v
6. Set Wall presure gradient to zero
7. Copy Values
p*=p
u*=u
v*=v
8. Check Convergence. and goto Step 1

My main problem is step 3 any sugesstions for that? (I am using c++)

Regards
Junaid

 leflix August 12, 2012 06:37

Quote:
 Originally Posted by JunaidAhmad (Post 376579) the equation is a[i,j]p'[i,j]=a[i-1,j]p'[i-1,j]+a[i+1,j]p'[i+1,j]+a[i,j-1]p'[i,j-1]+a[i,j+1]p'[i,j+1]+b'[i,j] a[i-1,j]=(d A)[i-1,j] a[i+1,j]=(d A)[i+1,j] a[i,j-1]=(d A)[i,j-1] a[i,j+1]=(d A)[i,j+1] b'[i,j] = (u*A)[i-1,j]-(u*A)[i+1,j]+(v*A)[i,j+1]-(v*A)[i,j-1] Junaid
Hi Junaid

I do not know how you defined your notations but your implementation is a bit weird.

You should rather have something like:

ap[i,j]p'[i,j]=aw[i,j]p'[i-1,j]+ae[i,j]p'[i+1,j]+as[i,j]p'[i,j-1]+an[i,j]p'[i,j+1]+b'[i,j]

Yes your are correct yours make more sense. Mine your are same.
So any thoughts how to calculate d

 Michail August 12, 2012 18:36

Take a look here http://www.cfd-online.com/Wiki/Sampl...9_-_Fortran_90

Hope this will help

This is Pressure Correction File. First let me tell you that i have no working knowledge of FORTRAN So its hard for me to understand this. If I am not wrong then its your code. it will be very help full if you explain it and tell me which book you follow to develop your understanding. Or at least explain the Highlighted terms in this code.

Regards

Junaid

!************************************************* *********************
Subroutine Solve_Pressure_Correction
include 'icomm_1.f90'

DpU(2:NXmax,2:NYmax) = 1./Ap(2:NXmax,2:NYmax,1)
DpV(2:NXmax,2:NYmax) = 1./Ap(2:NXmax,2:NYmax,2)

!************************************************* ***************************************
! vertical faces East - e
Do 101 I=1,NXmax
Do 101 J=1,NYmax-1

If((i.ne.1).and.(i.ne.NXmax))then

DXc_e = Xc(i+1,j+1) - Xc(i,j+1)
S_e = Y(i ,j+1) - Y(i,j )

VOL_e = DXc_e * S_e

!-------------------------------------------------------------
APU_e = 0.5 * ( DpU(i,j+1) + DpU(i+1,j+1) )
Ul_e = 0.5 * ( F(i,j+1,1) + F(i+1,j+1,1) )
DPl_e = 0.5 * ( DPx_c(i,j+1) + DPx_c(i+1,j+1) )

DPx_e = ( F(i+1,j+1,4) - F(i,j+1,4) ) / DXc_e

U_e = Ul_e + APU_e * VOL_e * ( DPl_e - DPx_e)

!--------------------------------------------------------------
Con_e(i,j) = U_e * S_e

End If

Con_e(1,j) = 0.
Con_e(NXmax,j) = 0.

101 continue

!************************************************* ***************************************
! horisontal faces
Do 102 I=1,NXmax-1
Do 102 J=1,NYmax

If((j.ne.1).and.(j.ne.NYmax))then

DYc_n = Yc(i+1,j+1) - Yc(i+1,j)
S_n = X(i+1,j ) - X(i ,j)

VOL_n = DYc_n * S_n

!-----------------------------------------------------------
APV_n = 0.5 * ( DpV(i+1,j) + DpV(i+1,j+1) )
Vl_n = 0.5 * ( F(i+1,j,2) + F(i+1,j+1,2) )
DPl_n = 0.5 * ( DPy_c(i+1,j) + DPy_c(i+1,j+1) )

DPy_n = ( F(i+1,j+1,4) - F(i+1,j,4) ) / DYc_n

V_n = Vl_n + APV_n * VOL_n * ( DPl_n - DPy_n )
!-----------------------------------------------------------

Con_n(i,j) = V_n * S_n

End If

Con_n(i,1) = 0.
Con_n(i,NYmax) = 0.

102 continue

!************************************************* ***************************************
Summ = 0.

Do 200 I=2,NXmax
Do 200 J=2,NYmax

S_e = Y(i ,j) - Y(i,j-1)
S_n = X(i ,j) - X(i-1,j)

An(i,j) = 0.5 * ( DpV(i,j) + DpV(i,j+1) ) * S_n * S_n
As(i,j) = 0.5 * ( DpV(i,j) + DpV(i,j-1) ) * S_n * S_n
Ae(i,j) = 0.5 * ( DpU(i,j) + DpU(i+1,j) ) * S_e * S_e
Aw(i,j) = 0.5 * ( DpU(i,j) + DpU(i-1,j) ) * S_e * S_e

Ap(i,j,3) = (An(i,j) + As(i,j) + Ae(i,j) + Aw(i,j))

Sp(i,j,3) = -1. * ( Con_n(i-1,j) - Con_n(i-1,j-1) + Con_e(i,j-1) - Con_e(i-1,j-1) )

Summ = Summ + Sp(i,j,3) !
200 continue

write(*,*)'Sum',summ
!************************************************* **********************************
!************************************************* **********************************
write(*,*) Summ

write(*,*)'solve PP'
niter = 0

20 call Convergence_Criteria(3,Res_sum_before)
niter= niter + 1

Call TDMA_1(3)
call Convergence_Criteria(3,Res_sum_After)
If((abs(Res_sum_before-Res_sum_After).Ge.0.0000001).and.(niter.le.500).an d.(abs(Res_sum_After).ge.0.0001))go to 20
write(*,*)'Res_sum_before-Res_sum_After',Res_sum_before-Res_sum_After,niter

!************************************************* **********************************
!************************************************* **********************************
! velocities and pressure correction

Do 300 I=2,NXmax
Do 300 J=2,NYmax

DY = Y(i,j)-Y(i,j-1)
DX = X(i,j)-X(i-1,j)

PPe = 0.5 * ( F(i,j,3) + F(i+1,j,3) )
PPw = 0.5 * ( F(i,j,3) + F(i-1,j,3) )
PPn = 0.5 * ( F(i,j,3) + F(i,j+1,3) )
PPs = 0.5 * ( F(i,j,3) + F(i,j-1,3) )

F(i,j,1) = F(i,j,1) + (PPw - PPe) * DpU(i,j) * Dy
F(i,j,2) = F(i,j,2) + (PPs - PPn) * DpV(i,j) * Dx

F(i,j,4) = F(i,j,4) + 0.1 * F(i,j,3)

300 continue

!************************************************* **********************************
! correct convective fluxes through vertical East faces
Do 701 I=1,NXmax
Do 701 J=1,NYmax-1

If((i.ne.1).and.(i.ne.NXmax))then

Con_e(i,j) = Con_e(i,j) + Ae(i,j+1) * (F(i+1,j+1,3)-F(i,j+1,3) )

end if

if((i.eq.1).or.(i.eq.NXmax))Con_e(i,j)=0.

701 continue

! correct convective fluxes through horisontal North faces
Do 702 I=1,NXmax-1
Do 702 J=1,NYmax

If((j.ne.1).and.(j.ne.NYmax))then

Con_n(i,j) = Con_n(i,j) + An(i+1,j) * (F(i+1,j+1,3)-F(i+1,j,3) )

end if

if((j.eq.1).or.(j.eq.NYmax))Con_n(i,j)=0.

702 continue

!************************************************* **********************************
Do 501 I=1,NXmaxC

F(i,1,4) = F(i,2,4) !
F(i,NYmaxC,4) = F(i,NYmaxC-1,4)!

501 continue

Do 502 J=1,NYmaxC

F(1,j,4) = F(2,j,4) !
F(NXmaxC,j,4) = F(NXmaxC-1,j,4) !

502 continue

F(:,:,4) = F(:,:,4) - F(3,4,4)
!************************************************* **********************************

Return
End

 leflix August 13, 2012 04:55

Hi Junaid,

The code of Michail is really well writen and the notations are very understanable because intuitive. Greatings for him !!

You should first read a book on finite volume method and SIMPLE like the ones of Patankar, Peric or Versteek.
The one of Peric should be better because Michail used colocated variables arrangement based on Rhie and Chow interpolation. So for this purpose I think Peric's book is better oriented.

However just reading the code of Michail you understand what he is talking about...

Quote:
 Originally Posted by JunaidAhmad (Post 376718) !************************************************* ********************* Subroutine Solve_Pressure_Correction include 'icomm_1.f90' DpU(2:NXmax,2:NYmax) = 1./Ap(2:NXmax,2:NYmax,1) DpV(2:NXmax,2:NYmax) = 1./Ap(2:NXmax,2:NYmax,2) This is 1./AP coefficient involved in the derivation of the pressure equation and correction velocities. Check for simple algorithm in dedicated books. AP is the central coefficent there is one for U momentum equation --> APU and one for V-momentum APV. in fact for Michail APU(i,j) = AP(i,j,1) APV(i,j) = AP(i,j,2) because he works with vector componnents style !************************************************* *************************************** ! vertical faces East - e Do 101 I=1,NXmax Do 101 J=1,NYmax-1 If((i.ne.1).and.(i.ne.NXmax))then DXc_e = Xc(i+1,j+1) - Xc(i,j+1) this is the distance between node P and node E S_e = Y(i ,j+1) - Y(i,j ) This is the east surface S_e VOL_e = DXc_e * S_e !------------------------------------------------------------- APU_e = 0.5 * ( DpU(i,j+1) + DpU(i+1,j+1) ) this is the iterpolation of 1/APU at east location Ul_e = 0.5 * ( F(i,j+1,1) + F(i+1,j+1,1) ) this is the interpolation of F on east location.I don't knowyet what is F but it doesn't matter DPl_e = 0.5 * ( DPx_c(i,j+1) + DPx_c(i+1,j+1) ) the same for interpolation of DPx DPx_e = ( F(i+1,j+1,4) - F(i,j+1,4) ) / DXc_e this is the gradient of variable F at east location usint central difference. probaby the gradient of pressure I guess U_e = Ul_e + APU_e * VOL_e * ( DPl_e - DPx_e) This is the interpolated velocity using at east location using the interpolation of Rhie and Chow !-------------------------------------------------------------- Con_e(i,j) = U_e * S_e this is the convective flux. End If End

Really get a book on FV , SIMPLE algorithm with Rhie and Chow interpolation and everything will be clear for you because the code is really clear!!

Thanks,

I am following verseek's book, and in it d(i-1,j)=A(i-1,j)/ap(i-1,j)

A(i-1,j) is area of west face
ap(i-1,j) is central coefficient of velocity
the problem is when i calculate ap by use the value of u* and v* and use it to find p' it some how blow up the velocity field, and when i use that to calculate ap which is ap = aw+ae+an+as+dF
For Hybrid scheme in that e.g. ae=Max(De-Fe/2,-Fe,0)
and Fe=0.5*(uP+uE)
De=Gama if dx=dy
so when velocity blowup everything went side ways.

So what should i use when i calculate ap do i use u* or u the corrected velocity.
Anyways i try to find this book of Peric and see how it works.

 Michail August 13, 2012 10:18

Dear Leflix

May I'll include Your explanations into the code? I guess it'l be better for future?

Regards
Michail

F(:,:,1) - U-velocity
F(:,:,2) - V-velocity
F(:,:,3) - pressure correction
F(:,:,4) - pressure

 leflix August 13, 2012 10:49

Quote:
 Originally Posted by Michail (Post 376827) Dear Leflix May I'll include Your explanations into the code? I guess it'l be better for future?
of course Michail, you are welcome!

 michaelmoor.aero August 13, 2012 18:40

SIMPLE FVM solution

I have a working Poiseulle flow problem based on Versteeg which matches nicely to the Hagen-Poiseulle velocity profile. It is extremey well annotated and culd be of great help.

It has under-relaxation and uses the JAcobi method, and was possible due to members of this forum such as Ztdep and houkensjtu.

The only problems that i have, are that the momentum resuduals do not get much better that 1e-3, and that the largest mesh that i am able to solve for (without divergence or stagnation) is about 34*34.... I would greatly appreciate any help, but for right now, you can look at my code...

Regards,
Michael

If you implemented versteeks method then would realy like to take a look at it. you may give me the like of send me the code at tojunaidahmad@hotmail.com

Regards
Junaid

 Michail August 15, 2012 13:11

Dear Junaid

I offer to take a look through M.Peric codes

ftp://ftp.springer.de/pub/technik/peric/

There are both staggered and collocated grids codes for SIMPLE-algorithm.

As concerned M.Peric book I offer to check torrents and previous posts on CFD-online.

Fight for CFD and CFD will fight for You :)

Best wishes
Michail

Thanks Michail, i need to know what is the best time to add under relaxation factor in the calculation of p, u, and v. now when i revisit the literature i find out that it is the under relaxation factor that cause divergence to me solution.

Regards
Junaid

 michaelmoor.aero August 16, 2012 05:43

Quote:
 Originally Posted by JunaidAhmad (Post 377323) Thanks Michail, i need to know what is the best time to add under relaxation factor in the calculation of p, u, and v. now when i revisit the literature i find out that it is the under relaxation factor that cause divergence to me solution. Regards Junaid
were you using implicit under-relaxation for u and v? what values were you using for under-relaxation to make it diverge? I am having a similar issue...

what is the good method for under relaxation do i use

or use URF embedded in the discretised momentum equation

Regards
Junaid

 michaelmoor.aero August 16, 2012 15:10

Quote:
 Originally Posted by JunaidAhmad (Post 377399) what is the good method for under relaxation do i use or use URF embedded in the discretised momentum equation Regards Junaid
If you look at Versteeg page 189, you will find that by applying the under-relaxation to the momentum equations yields the under-relaxed form of equations...6.36 and 6.37, they are not separate mehtods, and i believe that it is called implicit under-relaxation, and it the same form as that of peric.

AS for pressure under-relaxation, I understand it such that only the correction is under-relaxed... i.e. pnew = p* +alphap*p'

I don't understand i did exactly what is written in book. it didn't work. I did the same think using artificial compressibility and it worked but using SIMPLE method it did not. the only thing left for me is to paste the code a let you guys decide what is wrong with it.

But first i have to go through one more time.

 michaelmoor.aero August 19, 2012 08:50

Quote:
 Originally Posted by JunaidAhmad (Post 377689) I don't understand i did exactly what is written in book. it didn't work.
It sounds that easy! ;)

Did you get the code that i sent? I have now upgraded it to SIMPLEC also, and it is working well.

Yes i got the code . and its approach and mine is the same i.e. hi-bride scheme.

why we need SIMPLE Algorithm when We have Artificial compressibility?:confused:

 michaelmoor.aero August 19, 2012 12:47

Quote:
 Originally Posted by JunaidAhmad (Post 377697) Yes i got the code . and its approach and mine is the same i.e. hi-bride scheme.
thank you would be nice.

as for artificial compressibility, that is outside of my knowledge.

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