[FMDenaro]

Originally Posted by samycfd

I know it was just because there formula were written clearly.

In a lot of great books I found the Pressure poisson equation in 2D for a divergence-free flow as the following : ( approx ~ 2nd order )

laplacien(p) = - rho * ( (u_x)^2 + u_x * v_y + (v_y)^2 )
+ Boundary conditions

I want to be certain, you told me that it is not correct ?

the mathematical equivalence among several formulations is correct, but what is not is the numerical treatment of them....
for example Lap p and Div Grad p are numerically not always equivalent...
the same happens if you discretize v*Grad v or Div (vv).
This is especially relevant when you cannot ensure that Div v =0.


the momentum equation

d(rho*v)/dt + Div (rho*vv)+ Grad p = Div (2*mu*Grad v)

becomes

rho*dv/dt + rho*v Grad v+ Grad p = mu Lap v

only if you use the continuity constraint. But if it is not verified by your velocity field you can not

[samycfd]

Okay yes i agree. And if the continuity constraint is ensured regarded as a specific tolerance( the divergence is nearly equal to zero but not zero exactly) then the results won't be exact, but not totally true.

Is there a better numerical treatment of this formulation ?

I suppose it's really difficult to obtain good results when the continuity constraint is not satisfied. Maybe using a SIMPLE algorithm to correct the velocity in order it satisfy the continuity constraint ..

[FMDenaro]

Originally Posted by samycfd

Okay yes i agree. And if the continuity constraint is ensured regarded as a specific tolerance( the divergence is nearly equal to zero but not zero exactly) then the results won't be exact, but not totally true.

Is there a better numerical treatment of this formulation ?

I suppose it's really difficult to obtain good results when the continuity constraint is not satisfied. Maybe using a SIMPLE algorithm to correct the velocity in order it satisfy the continuity constraint ..

in any numerical solution, the velocity is computed in such a way to ensure it is divergence-free (in approximate o exact way). But if I got correctly your problem, the velocity field was prescribed to you, you don't have a real 3D numerical solution coming from computation, right?

[samycfd]

No most of the time I only have a 2D vector field .. and maybe I could access to the gradient of the 3rd component relative to this planar field.

[FMDenaro]

In my opinion this is the real problem, you have access a planar velocity field that, however, is a 3D field, not a 2D. In this plane do you know the value of the third component (the normal to plane)?

[samycfd]

Yes i know it but not the gradient of this 3rd component..

But on all my test flows I computed the divergence and it's close to zero on each nodes. But i know it will be like that every time.

[FMDenaro]

Originally Posted by samycfd

Yes i know it but not the gradient of this 3rd component..

But on all my test flows I computed the divergence and it's close to zero on each nodes. But i know it will be like that every time.

well, actually you know the gradient....assume the plane be (x,y), you have from the divergence-free constraint

dw/dz = - (du/dx+dv/dy)

[samycfd]

If it's divergence free yes .. but it doesn't change anything I still have an experimental flow which may not respect this condition..even if the fluid is incompressible.

So what the advantage of knowing the gradient ?

[FMDenaro]

Originally Posted by samycfd

If it's divergence free yes .. but it doesn't change anything I still have an experimental flow which may not respect this condition..even if the fluid is incompressible.

So what the advantage of knowing the gradient ?

you can force a local reconstruction of the velocity field respecting the divergence-free constraint. If you know dw/dz and w in the plane (x,y). This way you can have a local 3D velocity and computing a local 3D pressure field.
However, the approximations can be meaningful... I don't know details of your problem and cannot say more...

[samycfd]

It's a kind of Simple algorithm where the pressure correction enforced the continuity equation

There are no other details :/ These are my hypothesis.

So if I judge that the divergence free is verified then I solved the Poisson equation and if not I can attempt to force the field to be divergence-free..and then solve the pressure poisson equation ?

[FMDenaro]

Originally Posted by samycfd

It's a kind of Simple algorithm where the pressure correction enforced the continuity equation

There are no other details :/ These are my hypothesis.

So if I judge that the divergence free is verified then I solved the Poisson equation and if not I can attempt to force the field to be divergence-free..and then solve the pressure poisson equation ?

yes ... in the framework of lot of hypothesis...

[samycfd]

I am more focused on the divergence-free case, so what are these hypothesis you are talking about it or the issues I can get during the resolution ?

[FMDenaro]

I still have not a clear idea of how your velocity field is obtained, is numerical or experimental measurement? And what about the geometry of the flow problem?

[samycfd]

Sorry..It's an experimental flow obtained with Particle Image Velocimetry method(PIV).
So whatever the flow I get a rectangular ( or square ) velocity field ( because the PIV method deals with images ) but not always fully populated. For example in the Von Karman experiment the cylinder could be part of the image and there is no velocity inside of course.

So The data I am working on are these kind of velocity fields added with a boolean matrix which help me to know which vector is inside outside or not relevant. So I can define a mesh with boundaries and use finite element method. I already implemented that but I am not sure of the Pressure poisson equation I have to solve.

You know the whole story know, I guess

[FMDenaro]

Originally Posted by samycfd

Sorry..It's an experimental flow obtained with Particle Image Velocimetry method(PIV).
So whatever the flow I get a rectangular ( or square ) velocity field ( because the PIV method deals with images ) but not always fully populated. For example in the Von Karman experiment the cylinder could be part of the image and there is no velocity inside of course.

So The data I am working on are these kind of velocity fields added with a boolean matrix which help me to know which vector is inside outside or not relevant. So I can define a mesh with boundaries and use finite element method. I already implemented that but I am not sure of the Pressure poisson equation I have to solve.

You know the whole story know, I guess

If the flow problem is quasi-2D you can implement periodical BC.s in the normal-to-plane direction

[samycfd]

I understand the idea but if I just have the out-of-plane velocity in the (x,y) plane I still can treat the problem in 3D so why BC in that way and not just treat the problem in 2D since the case is quasi-2D ?

[samycfd]

these are my results for the pressure for the following velocity field
in the lid driven cavity case :

Originally Posted by samycfd

I have found an error ! So my pressure field is

, for this velocity field :

[FMDenaro]

yes, the pressure field seems quite reasonable... what kind of bc.s for pressure did you use?

[samycfd]

I used Neumann boundary conditions everywhere because the velocity is set on each edges and I have set one point to zero.(dirichlet value )

[FMDenaro]

Originally Posted by samycfd

I used Neumann boundary conditions everywhere because the velocity is set on each edges and I have set one point to zero.(dirichlet value )

does the solver converge without fixing the value in one point?