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Stokes Flow arround a single sphere

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Old   November 8, 2021, 10:38
Default Stokes Flow arround a single sphere
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Niko
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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):
\bf{v}\cdot \nabla\bf{v}= -\nabla p - \bf{S}
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:
\nabla p \approx \bf{S}


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:
K = \frac{F_{D}}{6\pi \mu U r}
with F_{D}= F_{pressure}+F_{viscous} which we will get from postProcess -func forcesIncompressible, U is the darcy velocity and r the radius of the sphere, \mu 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
File Type: jpg Sphere.jpg (31.3 KB, 15 views)
File Type: png K_Sim_Analytical_VolumeAveragedUx.png (20.6 KB, 17 views)
File Type: jpg Pressure_Comparison.jpg (46.6 KB, 20 views)
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Old   March 12, 2022, 02:07
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Uwe Pilz
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Cyclic b.c. arise problems. I recommend working first with an ordinary inlet and outlet first.
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