[matdehaast]

Hello

Just got a quick question that I seem to be conflicted on. With the classic projection method will the always satisfy the incompressibility constraint? ie the divergence of the velocity field after each step be zero?

Further is this true even when starting with an arbitrary initial field?

Thanks
Matt

[mprinkey]

The projection method typically assumes that the previous timestep velocity field satisfies the divergence-free condition. This includes the initial conditions. You see this when you take the divergence of the time-discretized momentum equation...div(U_n) (and div(U_n-1), etc if you use a multi-level time discretization) will be assumed to be zero. Then div(U_n+1) is forced to zero by solution of the pressure(-like) field.

If you specify an initial velocity field that is NOT divergence free, I don't think that you have a well-posed problem. Having said that, the momentum equation will not care that the initial velocity field has a non-zero divergence and it will update the provisional velocity to the new time level. The projection method will remove the divergent portion of the provisional velocity field, so you should end up with a U_n+1 that is div() = 0. But, this will depend on the exact implementation of the fractional step method that you use.

TL;DR version. Don't specify non-physical initial conditions.

[FMDenaro]

Originally Posted by matdehaast

Hello

Just got a quick question that I seem to be conflicted on. With the classic projection method will the always satisfy the incompressibility constraint? ie the divergence of the velocity field after each step be zero?

Further is this true even when starting with an arbitrary initial field?

Thanks
Matt

No, not always... depending on the type of grid arrangement and colocation you can have either the Exact Projection Method (divergence-free constraint satisfied at machine accuracy) or the Approximate Projection Method (divergence-free constraint satisfied only up to the magnitude of the local truncation error).
You can find many papers about these methods

[matdehaast]

Thank you both for your reply. I believe I understand much better now. The real issue I am having, which I suspect it was/is to do with not satisfying the incompressibility constraint, is that for Lid Cavity I am getting really weird issues in the top corners.

Please see the attachment to see what I am talking about.

Has anyone see/had this issue before?

[FMDenaro]

you did not add any scale for the values....

1) what about the spatial discretization you use?
2) what about the grid arrangement?
3) how do you compute Div vn+1 for the plot? is the same type of discretization used for the Div Grad p term?
4) has the pressure solver converged? what about the BC.s you set?

[kmooney]

Originally Posted by matdehaast

Thank you both for your reply. I believe I understand much better now. The real issue I am having, which I suspect it was/is to do with not satisfying the incompressibility constraint, is that for Lid Cavity I am getting really weird issues in the top corners.

Please see the attachment to see what I am talking about.

Has anyone see/had this issue before?

What 'issues' are you referring to? The top right and left face is no slip right? Aren't we just seeing the post processor interpolate from the no slip wall inward?

[kaya]

ps. plotting without interpolations would provide more insight