Potential Flow around a cylinder

[Martin Garnier]

Hello everyone,

I am currently trying to simulate a potential flow around different geometries. The final goal is to simulate the case from this publication of R.Auvray Determination of 2D Quasi Incompressible Flow around a Recorder Labium. 

First I am trying to simulate a potential flow on simple geometries. The first case I tried was a cylinder with a flow coming from left to right. I tried with Neumann and Dirichlet boundary conditions. I also tried with a half cylinder. With these two cases everything was fine (You can se the results in my Sandbox here My Sandbox in the PotentialCylinder directory). 

Now I am trying to compute a "U" shaped flow around a half cylinder. I put an input flow beneath the cylinder on the right boundary and an output flow above the cylinder on the right boundary. 

I tried two different simulations. It tried to resolve the Poisson equation on the stream function with appropriate boundary conditions, I also tried to resolve the Poisson equation on the potential function and then compute the stream function. Here are the results in my Sandbox “U” Shaped flow around a cylinder. 

The computation with the potential function works fine. But I have some problems with the stream function equation. Here are the problems : 

- I need to impose Dirichlet boundary, so I use the Dirichlet function to impose the value of the stream function. But the result is that I ended up with a solution twice as high. I looked at the wiki for the Dirichlet function and it seems that there is a coefficient 2 in the expression. Can somebody explains to me how does the Dirichlet function work ? 

- Secondly I do not understand the role of the homogeneous boundary conditions function (with "psip.boundary_homogeneous[left] = dirichlet_homogeneous_bc;).

- To finish my solution found by resolving the Poisson equation on the stream function is false, the stream lines are not normal to the right boundary. 

I Hope somebody can help to resolve all these problems. I attached the code that I used in this message. 

Thank you !

[j.a.v...@gmail.com]

Hallo Martin,

> Can somebody explains to me how does the Dirichlet function work ?

It assigns a fixed value to a centered scalar field at a boundary by setting the ghost-cell values such that the interpolated face value is equal to the prescribed value. 

> I do not understand the role of the homogeneous boundary conditions function

These are the boundary conditions for the "correction field" introduced by the Poisson solver. They are chosen such that the solution field statisfies the boundary condition after a multigrid cycle.

> The stream lines are not normal to the right boundary. 

The symmetry is broken because you impose a linear profile for psi at the inlet (and outlet) whilst setting the vorticity value (rigt-hand side of the Poisson equation) to zero. If you want the streamfunction's flow to correspond to the potential flow solution,(?) you should not impose the linear profile. (the potential flow velocity profile goes with "1/r" if I am not mistaken).

 

Antoon

Op maandag 3 juni 2024 om 16:04:18 UTC+2 schreef Martin Garnier: