Static/Total pressure BCs for both inlet and outlet
 

I have a general question about inlet and outlet BCs.

I know that static pressure BCs for both inlet and outlet are mathematically inappropriate.
However, I don't understand why it is wrong very well.

Could you show me the reason?
Does it depend on compressible or incompressible analysis?

Similarly, total pressure BCs for both inlet and outlet are appropriate?

With regards.

First, you have to consider the mathematical character of the PDEs to set the BCs that makes the problem well posed.
Thus, incompressible or compressible flows models are very different in the setting of the BCs.

For incompressible flows there is no thermodinamic meaning in the variable called "pressure", you set the BCs in such a way to fulfill the divergence-free constraint. You can fix a "pressure difference" between inlet and outlet and let the velocity to be congruently computed.

Yes, it is a matter of incoming and outcoming waves for a given flow state (subsonic vs supersonic). Very roughly speaking, at any boundary you have three waves traveling at the local speed u normal to it, plus two waves traveling, respectively, at u + c and u - c, where c is the local speed of sound.

Now, for a subsonic inlet (0<u<c) you have 4 entering waves and 1 exiting wave. At a subsonic outlet (-c < u < 0) it is the reverse, 4 exiting waves and 1 incoming wave. This, in practice, translates in specifying 4 inlet conditions and 1 outlet condition, respectively.

In order to finally grasp the matter, you then just need to understand that total pressure is not a real variable, it is a sum of different variables. So, what typically happens when you specify total pressure is that you are actually specifying 4 variables with some assumptions (e.g., velocity normal to the boundary and bernoully like conditions).

So, you should see now how total pressure (4 variables) is ok for subsonic inlets but not ok for subsonic outlets. Static pressure (which, in contrast to total pressure, is a single variable) is ok for subsonic outlet but not ok for subsonic inlet.

For supersonic inlet/outlet things are easier, as no wave leaves from the inlet and no wave enters from the outlet, so you specify all the variables at inlet and none at outlet.

Of course, in real world scenarios of serious solvers you also have to take into account mixed scenarios, where an inlet can temporarily become an outlet and an outlet can temporarily become an inlet, so that you need some backup info specified for those boundaries (e.g., total pressure specified at subsonic outlet in case of reverse flow)

Thank you very much for your comment.

I understand only the pressure difference, not an absolute value, is meaningful for incompressible flows.
In order to compute velocity congruently, we should set total pressure BC including velocity information for inlet or outlet at least, right?

Thank you very much for your clear answer.
I probably catch the point.
I have never seen such a concise and clear explanation.

[emoji1] [emoji1] thanks. I have no idea why he is so upset.

Obviously, it is not the full story, I didn't even mention what those waves are and how they relate to the classical variables, but hopefully it was clear enough to grasp the reason behind what you observed

EDIT: I just realized that I made confusion between EricSlater and you in my previous post.

Paolo already addressed the case of the compressible flow problem.


Working with the incompressible flow model is very different and somehow more complex. You have the parabolic equation for the velocity field with the constraint of being divergence-free. The pressure is only an auxiliary variable, that is a lagrangian multiplier.

The BCs for the momentum equations are prescribed in terms of natural conditions for the velocity.
The divergence-free condition is transformed in an elliptic equation according to the Hodge decomposition. Neumann boundary conditions are prescribed.
From the decomposition you see that either pressure or velocity can be prescribed.

I would add that artificial compressibility and related methods (i.e., preconditioned density based) would still work as compressible methods. That is, even for incompressible flows they introduce an artificial speed of sound whose related Mach number is, by construction, sufficiently high to not blow up the computation.

Your professor mentioned incompressible flows for which c->\infty where the pressure is only artificial to guarantee divergence free flow-field there is nothing to be added to his answer, so who cares about the methods your refer to?

Because, with respect to the physical problem at hand, the ellipticity mentioned by Filippo is just as artificial as the compressibility in the method I mentioned. This, with respect to the original question, which is about generic boundary conditions, is indeed relevant, because boundary conditions are artificial themselves.

With respect to Filippo himself, I ensure you that there is no other person on this planet that has gone trough each one of his works more than I did. But even if this was not the case, I would still be free to think that his answer is too biased and add my contribution. There is nothing that you, Filippo or anyone not a moderator here can do about it.

But, honestly, why?

Do you know any textbooks or references about it?
I would like to study further about BCs mathematically.

Wow lot of discussions on this topic. If you are mathematician then refer books on hyperbolic conservation laws. In case of engineer refer books written by Laney, Hrisch, etc.
I'll try to put some simple layman explanation on this. In CFD we are solving primarily for three variables velocity, pressure and temperature. Ignore temperature if energy equation is not solved. So, effectively boundary condition is required for both pressure and velocity.

Having static pressure at both inlet and exit would result in unconstrained mass flow or multiple solutions on mass flow, which could result in divergence or oscillating solution.
Some kind of velocity fixation is necessary at the boundary to indicate the order of mass flow or velocity in the equation. The static pressure on boundaries would fix the change in velocity but there is no information in the solver to convert change in velocity to absolute velocity and therefore mass flow.

Have a look at:
Joel H. Ferziger, Milovan Perić, Robert L. Street - Computational Methods for Fluid Dynamics- Springer International Publishing (2020)


and Sec. 7.1.5.2 can clarify the approach for incompressible flows.


Much more details (numerical and mathematical topics) can be found in articles published on several journals, for example have a look to the references in my article:
https://www.researchgate.net/publica...ary_conditions