[xchen_1123]

Hi,

I am solving a plasma flow including a time-dependent temperature field. The density of plasma is only determined by the temperature. Could I suppose the plasma flow to be an incompressible flow? If I use the Projection method to solve the fluid field, could you please introduce me some related papers? Have any more efficient method to solve this problem?

Thank you!

[FMDenaro]

Originally Posted by xchen_1123

Hi,

I am solving a plasma flow including a time-dependent temperature field. The density of plasma is only determined by the temperature. Could I suppose the plasma flow to be an incompressible flow? If I use the Projection method to solve the fluid field, could you please introduce me some related papers? Have any more efficient method to solve this problem?

Thank you!

I am not sure if you are asking for a low-Mach model but I don't think that working with the divergence-free constraint makes physical sense for plasma...you should solve the MHD equations

[xchen_1123]

Originally Posted by FMDenaro

I am not sure if you are asking for a low-Mach model but I don't think that working with the divergence-free constraint makes physical sense for plasma...you should solve the MHD equations

Thank you for your reply. The flow I have mentioned is a low-Mach (<0.3) flow. In this mean, the flow seems to be regarded as an incompressible flow. Meanwhile, the fluid of plasma is not the point. It could be replaced with metallic vapor.

[FMDenaro]

Ok, however I suggest to check in the literature to see similar existing lo-Mach models for MHD flows...

[xchen_1123]

Originally Posted by FMDenaro

Ok, however I suggest to check in the literature to see similar existing lo-Mach models for MHD flows...

Thank you for your suggestion.

[LuckyTran]

In constant density flows, you have that density derivative in time is zero and density gradients in space are zero so that the material derivative of density is zero. The side-effect is that the divergence of velocity is zero. A temperature dependent density is a pseudo-incompressible flow. Here you don't have the nice luxury that divergence of velocity is automatically zero. However, like "true" incompressible flows, acoustic waves are much lower order of magnitude because the sound speed tends towards infinity. So for temperature dependent density, you must assume the divergence of velocity is zero, or you must keep it compressible. Assuming the flow is incompressible is reasonable as long as pressure change are "small," which you can read as "Mach number is low."

Well, it depends on your problem but incompressible flow is often employed in astrophysics where they say 90% of the unknown universe is characterized by incompressible MHD. Unlike ideal gases, which do not have the long range forces, in MHD you do get long range forces (which propagate not at the sound speed, but light speed). Another way to put it is, if an ideal gas can be assumped to be incompressible while lacking means of communicating with far-off neighbors in advance, then a plasma, having this mechanism, is much more likely to be incompressible. The limitation is that incompressible MHD rules out displacement currents. The momentum equation for MHD is not terribly different, you just need an extra body force term for the Lorentz force.

[FMDenaro]

You can find example of MHD equation based on the incompressible (div v =0) assumption, one example is https://perswww.kuleuven.be/~u001654.../incompMHD.pdf
In this case, the divergene-free constraint allows to use projection-based method as for the classical NS equations.

However, in principle you could consider also a low-Mach model in which some hypotheses are relaxed. This model is often used in combustion.

[xchen_1123]

Originally Posted by LuckyTran

In constant density flows, you have that density derivative in time is zero and density gradients in space are zero so that the material derivative of density is zero. The side-effect is that the divergence of velocity is zero. A temperature dependent density is a pseudo-incompressible flow. Here you don't have the nice luxury that divergence of velocity is automatically zero. However, like "true" incompressible flows, acoustic waves are much lower order of magnitude because the sound speed tends towards infinity. So for temperature dependent density, you must assume the divergence of velocity is zero, or you must keep it compressible. Assuming the flow is incompressible is reasonable as long as pressure change are "small," which you can read as "Mach number is low."

Well, it depends on your problem but incompressible flow is often employed in astrophysics where they say 90% of the unknown universe is characterized by incompressible MHD. Unlike ideal gases, which do not have the long range forces, in MHD you do get long range forces (which propagate not at the sound speed, but light speed). Another way to put it is, if an ideal gas can be assumped to be incompressible while lacking means of communicating with far-off neighbors in advance, then a plasma, having this mechanism, is much more likely to be incompressible. The limitation is that incompressible MHD rules out displacement currents. The momentum equation for MHD is not terribly different, you just need an extra body force term for the Lorentz force.

Thank you ! Could you recommend me any numerical methods or related materials for solving the temperature-dependent density incompressible flow?

[xchen_1123]

Originally Posted by FMDenaro

You can find example of MHD equation based on the incompressible (div v =0) assumption, one example is https://perswww.kuleuven.be/~u001654.../incompMHD.pdf
In this case, the divergene-free constraint allows to use projection-based method as for the classical NS equations.

However, in principle you could consider also a low-Mach model in which some hypotheses are relaxed. This model is often used in combustion.

In my research, the variable density only caused by the changing temperature have an effect on the fluid flow (Mach< 0.3). Thus, the divergence of velocity is not zero. Anyway, thank you for your discussion and materials.

[FMDenaro]

Originally Posted by xchen_1123

In my research, the variable density only caused by the changing temperature have an effect on the fluid flow (Mach< 0.3). Thus, the divergence of velocity is not zero. Anyway, thank you for your discussion and materials.

In a real physical case, the divergence of the velocity field is never zero, you have to assess if the mathematical model of a pure incompressible flow is well suited for your problem.

[xchen_1123]

Originally Posted by FMDenaro

In a real physical case, the divergence of the velocity field is never zero, you have to assess if the mathematical model of a pure incompressible flow is well suited for your problem.

I agree with you. In my work, the influence of the density variable could not be neglected, while the Mach number is much less than 0.3. Therefore, I am searching for a numerical method to solve this temperature-dependent density incompressible flow.