[ff3d-users] Force in a pressure-driven Stokes flow

[Semin]

Dear Stéphane Del Pino,

I would like to compute the hydrodynamic force acting on a motionless
cylinder, which is set in the centre of a rectangular pipe where a
steady Stokes flow is established. As a first step, I performed
computation of the hydrodynamic (pressure + friction forces) forces (per
unit length) acting on an infinite cylinder; in that case the problem is
2D and I successfully used FreeFem++.


In the 3D case and in order to smooth the shape of the cylinder, I added
two half spheres at both ends of the cylinder. I modified the Stokes
file provided in one of your example file (see attached files).


Unfortunately, I get the following problems:


1. I set the pressure to range between -10 and 10: yet, some values of
the pressure are out of this range (it goes from -12 to 12).


2. The pressure gradient and the velocity are in the same direction,
what is not physically correct. Yet, the order of magnitude of the
velocity seems to be ok.


3. The pressure force (even its sign) is consistent with the 2D
computation [the length of the cylinder is approximately 6 ; this helps
to compare the two simulations]. Yet, it is not consistent with the
pressure field. [Please note that I am not sure that my computation of
the surface integral is good.]


4. The order of magnitude and the sign of the friction force are not
correct; its sign is consistent with the velocity field.


5. I try to use the force computation method you proposed in a
ff3d-users mail (see “other evaluation of the total hydrodynamic
force”). It does not work. I do not understand this method, and do not
know if it is relevant here. I would like to know when it is relevant,
and to have a reference about this method.


I use the ff3d-win32-20071108.zip
<http://www.freefem.org/ff3d/binaries/ff3d-win32-20071108.zip> release.


Thanks a lot for helping a young experimenter.

Best regards

Benoît Semin

[Stephane Del Pino]

Dear Benoît.

First, it seems that in FreeFem++, you are solving a potential flow. Did you
succeed with ff3d doing the same?

Second, since ff3d's discretization is based on a Q1-Q1 element
(velocity-pressure), it needs to be stabilized with a 'penalty' term (eps in
this case). The pressure solution depends a lot on the choice of this term.
ff3d does not support Q2-Q1 coupled discretization yet. But, you can already
use a projection algorithm to do so. I can send you an example of such a
scheme which provides much better solution than the direct resolution.
I am quite busy now but I can send you the example by the end of the week.
Remind me if I forget.

Concerning the second way of approximating the force, it requires that the
null velocity is imposed weakly using a volume penalty (in your case, you
used a strong formulation --- elimination --- for Dirichlet on the cylinder).
If I remember, this is described in this paper:
http://www.math.u-psud.fr/~maury/Files/OP60DelpinoMaury.pdf
If not, you can check B. Maury's web page, you should find an answer.

I hope this gives you enlightenment...

Best regards,
Stéphane.