Extend Example_StokesIteratedPenalty.jl to cover total pressure inlet condition

[ziolai]

Hello,

Could we please discuss how to extend/modify/redo the Example_StokesIteratedPenalty.jl example to implement total pressure inlet conditions? This boundary condition introduces a non-linear coupling between the pressure and velocity on the boundary. Possibly (likely) the penalty method can be used to resolve the non-linearity.

I am happy to start with stationary Stokes. I ideally would like to bring it laminar Navier-Stokes.

Some details of what, why and how are at notes on total pressure inlet conditions.

I am happy to elaborate further.

Grusse, D.

[chmerdon]

Hello,

thanks for the question!

I am not yet familiar with this boundary condition. Looking at your notes I think both possibilities (via penalty term or Lagrange multiplier) to include the condition should be possible. How exactly would the weak formulation look like in case of the incompressible stationary Stokes equation? My rough expectation on the necessary modifications are these:

• adding a NonlinearOperator on the inlet with the arguments entities = ON_BFACES and regions = [inlet_region_number] and the corresponding kernel function for the condition
• for the Lagrange Multiplier approach one needs another FESpace on the boundary grid, hopefully FESpace{FEType, ON_BFACES}(xgrid; regions = [inlet_region_number]) works

I would like to give it a try. Do you have a nice test problem with which we can use to test if it works correctly?

[ziolai]

Thank you for your reply. Very much appreciated. I understand the need for a reference solution on a model problem. Allow me to think it over and get back to you.

[ziolai]

Greetings. A reference solution is now at report_talice_part1.pdf. Happy to discuss further.

[chmerdon]

I am sorry, but I could not find anything on the total pressure inlet condition in that pdf. What I need is a weak formulation, i.e., exactly the equations (for a simple model problem) that you want to solve including the total pressure inlet boundary condition. For the stationary incompressible Stokes equation with the total pressure inlet condition (implemented via penalty method) I expect the weak formulation to look like that:

$ (\nabla u, \nabla v) - (p, \mathrm{div}(v)) = (f,v) \quad \text{for all } v \in V$

and

$ (q, \mathrm{div} u) + \frac{1}{\epsilon} (p + \frac{1}{2} \lvert u \rvert^2, q)_{\Gamma_\text{inlet}} = \frac{1}{\epsilon} (p_\text{total-pressure-inlet}, q)_{\Gamma_\text{inlet}} \text{for all } q \in Q$

for a given value $p_\text{total-pressure-inlet}$ and some small penalty value $\epsilon$. Is this correct?

[ziolai]

I am unfortunately familiar with the approach. What motivates adding (,){\Gamma{Inlet}} to the second (as opposed to the first) equation? Thank you, Domenico.