[gautcho]

Hello, for a little project I want to implement a really easy example for the SIMPLE-algorithm with matlab (speed doesn't matter and with matlab plotting is easy). I started with an example from a book (CFD from Versteeg et al., example 6.2) and I get every value right, but the example ends before the loop of SIMPLE is really starting and I cant find my errors. So I have a few general questions about SIMPLE:

1. There are two linear equations systems to solve: one to find the u* out of the p* with the momentum equations and one to find the p' with the contnuity. Am I correct, if I say that the matrices, lets call them A and P, are the same in every inner loop? And only the solution-vector differs?
2. Is there any difference from the first time to go thru the loop to the nth (with n>=2) time besides the fact that you dont have to guess the p* because you get it from the further loop-run?
3. u* is always been calculated by the momentum equation and never guessed, right?
4. If I have got a steady and incompressible flow, is there any other variable beside the velocities and the pressure I have got to calculate?


Thank you for any answers

[gautcho]

Ok, let me simplify it: Do I have to calculate new coefficients (e.i. a_p, a_e, a_w) in every inner loop?

[CFDtoy]

Hi,
Let me try to find solutions to your problem.

Originally Posted by gautcho

Hello, for a little project I want to implement a really easy example for the SIMPLE-algorithm with matlab (speed doesn't matter and with matlab plotting is easy). I started with an example from a book (CFD from Versteeg et al., example 6.2) and I get every value right, but the example ends before the loop of SIMPLE is really starting and I cant find my errors. So I have a few general questions about SIMPLE:

1. There are two linear equations systems to solve: one to find the u* out of the p* with the momentum equations and one to find the p' with the contnuity. Am I correct, if I say that the matrices, lets call them A and P, are the same in every inner loop? And only the solution-vector differs?

Me: For u* eqn: the ap and other An are constructed from Navier momentum eqn

For p' eqn: ap, au comes from continuity eqn. so, ap's and other coefficient are different

2. Is there any difference from the first time to go thru the loop to the nth (with n>=2) time besides the fact that you dont have to guess the p* because you get it from the further loop-run?

Me: Is this steady or transient?
for transient: Each time there is an inner iteration loop..and everytime, p is different since you update p = p*+urf*p'
So, next time u visit finding u*, your right side has a new p and hence new gradient p source term

3. u* is always been calculated by the momentum equation and never guessed, right?

Me: u* is initialized thats about it (t = 0). Computed using mom eqn

4. If I have got a steady and incompressible flow, is there any other variable beside the velocities and the pressure I have got to calculate?

Me: single phase steady incompressible..u and p what more do u want?
u residuals give momentum balance and p residuals gives mass imbalance...


Thank you for any answers

[gautcho]

Thank you for your answers, they helped me a lot. Yet another question: In almost every book about CFD you can read: You need SIMPLE to compute incompressible flows, because you dont have rho in the continuity-eq, so you cant use p=rho*R*T for calculating p and the velocities are the dominant variables in the momentum eq, so you dont have got a eq for p.

But whats the problem? I have got 4 eq (3D): 3 momentum + conti and 4 variables (3 velocities + pressure), so i can solve the problem explicit or implicit... What's about this "dominant variables", why couldnt I compute one velocitie from the continuity?

[DoHander]

Because the continuity equations acts like a boundary condition, and not like a real separate equation for velocities. In incompressible flows.

If you don't like the idea of using a pressure equation, you can use the full NS equations preconditioned for low speed flows, just like for a compressible flow. Or Chorin's method where you use a pseudo-density equation instead of a continuity without the density like you have for the incompressible case.

[CFDtoy]

Hi there, (gautcho)
Let me try to get your questions straight. You have 4 eq (3 momentum + continuity) and 4 variablbes (3 vel and pressure)?

Ok lets see. for incompressible part: (btw are you looking at incomp or comp?)

continuity => rho*Area*vel = constant (without src) ok - eqn 1
Momentum eqns: u1 = dP/dx + .. - eqn 2
u2 = dP/dy + .. - eqn 3
u3 = dP/dz + ... - eqn 4

Now, you want to solve u from u1,u2,u3 right? ok..so what is your "good value" for p such that your u1, u2, u3 are good ? By saying good,

when you solve u1,u2,u3 ...it should automatically give you

Ai*ui = constnat (rho constant and no src) eqn 1 (Ai is the area vector)

Can you make sure that del (dot) (A1u1+A2u2+A3u3 ) = 0 (divergence of this vector is zero) ?

May be not. Somehow, your del (dot) (ui) gives some "extra value" on right side ..for the guessed P...so you correct p and try solving u1,u2,u3 again ...and check if del . aiui = 0 (or close to) ...this is basically SIMPLE method

Basically, since you are guessing the initial p to get u and that divergence is not maintained..you correct it. ..this is an iterative process..

/CFDtoy

Originally Posted by gautcho

Thank you for your answers, they helped me a lot. Yet another question: In almost every book about CFD you can read: You need SIMPLE to compute incompressible flows, because you dont have rho in the continuity-eq, so you cant use p=rho*R*T for calculating p and the velocities are the dominant variables in the momentum eq, so you dont have got a eq for p.

But whats the problem? I have got 4 eq (3D): 3 momentum + conti and 4 variables (3 velocities + pressure), so i can solve the problem explicit or implicit... What's about this "dominant variables", why couldnt I compute one velocitie from the continuity?

[gautcho]

@DoHander: It have to use the SIMPLE-algorithm because it's the theme of my project. The question is: Why do I have to use it, if I want to compute incompressible flows. Maybe your first sentence give the right answer, but I'm not very experienced in CFD and FD and so it's a little bit too inexact for me. Why acts the continuity only like a BC? For me it's an equation in a system of equations and after discretisation I can solve this system...

@CFDtoy: I understand the method of SIMPLE, but it's the same question here: Why not solving it directly? And I'm looking at the incompressible part...

[monkeytron]

HI gautcho,

For incompressible flow you want to calculate four quantities for 3D flow - the three velocity components (u,v,w) and the pressure (p). You therefore need four equations - you've got the 3 NS equations, one for the velocity in each direction and you need another one to calculate the pressure. A common approach is to reformulate the continuity eqn into a correction to a guessed pressure field. This is what SIMPLE is. There is no point using continuity to get a velocity, that is what the NS eqns give you.