Gallium melt flow

[Alex]

Hello everyone!
I am working on natural convection simulation via the hybrid lattice Boltzmann method. I have solved the benchmark problem of natural convection of air (Prandtl number Pr=0.71) inside the closed differentially heated cavity. In this case, the code works fine even with high Rayleigh numbers. However, when I change the Prandtl number to 0.02 which corresponds to Gallium melt, the code is numerically unstable. I have tried different collision models such as two relaxation time approximation, multiple relaxation time approximation, high order regularization procedure and even LES formulation, but still it has no effect on stability. Have someone ever encountered the same problem?
Thanks,
Alex

[mathias]

Hi Alex,

some hints:
– choose larger git sizes, smaller dx
– uMax should stay below 0.12, if choose smaler time step sizes or set smaller uMax (depending on the used unit converter)
– check the source of unstability (stop simulation if uMax>0.3 for the first time in the main loop and write results, output the velocity and locate where the great values orrur.

Best
Mathias

[Alex]

Dear Prof. Mathias,

Many thanks for your hints. Grid size variation has no effect on stability. I thought about to capture the moment when crucial error occurs. But what I should do next I don’t know since, for example, in BGK model relaxation time is constant all over the domain. An interesting note that when I apply magnetic field with high Hartmann number, code becomes stable.
kind regards,
Alex

[mathias]

ok, in that case

– check the source of unstability (stop simulation if uMax>0.3 for the first time in the main loop and write results, output the velocity and locate where the great values orrur.

[Alex]

Many thanks. I have found that the problem of low and high Pr flow simulation via LBM is partially discussed in [https://doi.org/10.3390/fluids6040148]. As Prof. Mathias noted, relaxation time needs to be non-uniformly treated for the grid points.

[mathias]

Yes, this was also done by Max Gaedtke (google for his PhD thesis). There is a link to the code in the appandix. Best Mathias

[Alex]

Unfortunately, i could not find the thesis of Dr. Gaedtke. But I found the paper of his with the TRT model for gallium and parafin melting. However, the low Pr treatment is not discussed in this work. If someone encounteres the same problem, you can refer to the work of Bawazeer et. al. [https://doi.org/10.1016/j.ijheatmasstransfer.2019.118562]. The authors added a correction term to the force term in order to make the model stable with low Prandtl numbers.

[mathias]

Gaedtke thesis: https://publikationen.bibliothek.kit.edu/1000126904 .

Interesting paper! If you find out more, you could give a short talk in our seminar. That might also be interesting for Fedor, talk with him about it.

Best
Mathias

[jjessberger]

Dear Alex,

thanks for your comment. Here, you also find the thesis: https://publikationen.bibliothek.kit.edu/1000126904

Yours, Julius

[Alex]

Dear Prof. Mathias,

Thank you for the link!

kind regards,
Alex

[Alex]

Dear Julius,

Many thanks for the link!

kind regards,
Alex

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