Thank you AliE,

So if I understand correctly in the x and y momentum equations for provisional velocities u*, v*, the terms p(i+1,j)-p(i,j) and p(i,j+1)-p(i,j) remain even though I have dp/dx=0. In the case I have non zero dp/dx simply I add the term dp/dx*vol as a source term in the x-momentum. Then I solve pressure correction equation, then I calculate vcorrection and finally correct velocity and pressure field. So the pressure correction must be zero or constant everywhere resulting in all iterations in constant pressure field.

Before proceed is it right the above?

Yes correct. Leave the discretized dpdx coming from navier stokes equation and add a source term to force the flow.

I don't think that for couette flow the driving pressure gradient must be added explicitly. The slip velocity of the wall would produce the necessary force and the pressure equation would adapt.
However, I never tried personally so that this is only an idea.

Thank you AliE,

I understand. So in the simple case we have dp/dx=0 the contant pressure field will result from the calculations. Is it right?

The term related to the pressure gradient shuold be always present in your code and it is the solution process that decides its value. So do not be afraid and write your code in a general way. When you need to simulate the couette + imposed pressure gradient, add a source term to mometum equation. Think your pressure gradient as divide in two parts: one coming from the solution of the pression equation and one from external forcing. If you do a standard couette, the second is zero and the first will be decided by the code but it will be zero as well.

Ok AliE I undertstand,

Let me ask you something else. We have said that in the simple couette flow the pressure field must be calculated constant at each iteration level since there must be not pressure change. If this happens then according to the algorithm the pressure correction has to be zero or constant number everywhere in the field e.g. 5, so in the formula for the correction of velocity uc,vc woule be zero since pressure correction is everywhere constant, resulting in non changing velocity field at n+1 iteration level.

Yes, in practice you can choose the constant by fixing the pressure at some reference cell. I suggest you to do the same simulation in openFoam so you can get a very well idea of what should happen :)

Thank you AliE,

I do not understand one thing.
In my case where I have simple Couette flow with dp/dx=0 the calculated pressure field must be the same everywhere. This means that the pressure correction equation will give either zero or constant number pressure correction. If this happens then the velocity correction is zero resulting in no correction. I mean:

p(n+1)=0 everywhere at n+1 itelation level, means p'(n+1)=0 or contant number everywhere, where in turn means uc(n+1)=0, vc(n+1)=0 so u(n+1)=u*, v(n+1)=v* at each iteration level.

Is it correct?

Yes, this is what is expected :)

Is there any other way to solve this problem using SIMPLE algorithm without using periodic boundary conditions?

Unfortunately I do not think so. If you put a standard inlet/outlet you will have boubdary effect.

I understand.

Hello,

I send as an attachment the results as a graph showing the horizontal velocity u versus vertical distance for different longitudinal positions in x direction for the simple Couette flow with dp/dx=0.

How is it?

Hello AliE,

I run the program and these are the results in the above attached image.

Hello,

Sorry forgot to reply :) a couette flow shoudn't br a linear profile?

the solution is wrong...consider only the steady state and check that the linear solution does not change along x

Thank you AliE, FMDenaro,

Yes I know that simple Couette flow with dp/dx=0 must be linear in velocity distribution and not changing along x direction.

Let me ask: When I check the zero divergence of velocity field will I check it at each volume cell (i,j) separately so for this cell I will find the velocity field and pressure field or will I check the criterion for all cells together. In pseudo-algorithm what I mean:

1) cell (i,j), iterations until for that cell satisfied the criterion of zero divergence, and then continue to next cell,
or
2) for all cells and specific iteration I check the max value of divergence for all cells and if not satisfied continue to new iteration.

What is the correct way?
Thank you

Hello to everyone,

Any suggestion about this quote?

Thank you.

div(i,j) is a local variable but you have to satisfy simultaneously the divergence-free constrain in all the cells

Thank you FMDenaro.

So for all cells I check the max value of divergence and if not satisfied according to the maximum permitted value then I continue to new iteration.
Is it right?