# Stokes Flow arround a single sphere

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November 8, 2021, 09:38
Stokes Flow arround a single sphere
#1
New Member

Niko
Join Date: Jan 2021
Posts: 3
Rep Power: 4
Dear Foamers,

I am currently simulating Stokes Flow arround a single sphere and try to compare the results to the analytical solution.

Analytical Solution/Theory: We have incompressible flow at very low Reynolds numbers (< 1) in a cube geometry with a single sphere in the middle with periodic BC at all cube sides and noSlip on the sphere's boundary (Image attached). A force is added to the N-S-Eqn to enforce fluid flow solved with a steady-state solver (simpleFoam):

Since the flow is at very slow flow, one can neglect the left-hand-side and the pressure drop should be equal to the added source term:

We can compare the results with the analytical solution of Sangani (1982, "Slow Flow Through A Periodic Array of Spheres") by calculating the incompressible pressure and viscous force, which are used to calculate the non-dim drag K for spheres with different diameter via:

with which we will get from postProcess -func forcesIncompressible, U is the darcy velocity and r the radius of the sphere, the viscosity.
With this we can calculate the non-dim K which is in very good agreement with the analytical solution (Image attached).

So here is the problem: Even though the K values are correct, the calculated pressure drop is not even close to the added source term as mentioned above in the N-S-Eqn. Further, the pressure field does not look correct as it should be a linear gradient through the domain, but this is not what it looks like (Image attached).

Here is what I tried (next to refine the mesh): mappedOutlet BC for the inlet velocity, with zeroGradient velocity at the outlet and set a fixed pressure gradient between inlet (uniform 1E-3) and outlet (uniform 0). With this I get the correct pressure drop gradient and pressure field but the non-dim K is completely off (calculated again with postProcess -forcesIncompressible).
This leads to my dilemma: Either a correct non-dim K or a correct pressure gradient....

Any explanation or help is appreciated. Am I missing something regarding the BC or the theory?
Attached Images
 Sphere.jpg (31.3 KB, 25 views) K_Sim_Analytical_VolumeAveragedUx.png (20.6 KB, 26 views) Pressure_Comparison.jpg (46.6 KB, 29 views)

 March 12, 2022, 01:07 #2 Senior Member     Uwe Pilz Join Date: Feb 2017 Location: Leipzig, Germany Posts: 743 Rep Power: 14 Cyclic b.c. arise problems. I recommend working first with an ordinary inlet and outlet first. __________________ Uwe Pilz -- Die der Hauptbewegung überlagerte Schwankungsbewegung ist in ihren Einzelheiten so hoffnungslos kompliziert, daß ihre theoretische Berechnung aussichtslos erscheint. (Hermann Schlichting, 1950)