[TurbJet]

I am going through the derivation of compressible NS equations. But there is a bit of confusion: according to some sources, they claim that Stokes hypothesis holds for any circumstance; some others say that it doesn't hold when the flow is compressible.

So I am wondering which one is correct? If Stokes hypothesis does not hold for compressible flow, then I guess the bulk viscosity will come into play, and make the equations harder to solve.

[FMDenaro]

Have a look here https://link.springer.com/article/10...707-015-1380-9

[TurbJet]

Originally Posted by FMDenaro

Have a look here https://link.springer.com/article/10...707-015-1380-9

Thx very much, it is really helpful.

I am no expert in compressible flow, but from what I gathered, in practical applications, generally speaking, people still use Stokes Hypothesis, right?

[FMDenaro]

Yes correct

[mohd_umair]

I have developed a compressible flow solver to investigate compressible turbulent flows. And as far as my experience goes Stokes hyothesis works fine.

[TurbJet]

Originally Posted by mohd_umair

I have developed a compressible flow solver to investigate compressible turbulent flows. And as far as my experience goes Stokes hyothesis works fine.

Have you ever thought of trying the solver without Stokes hypothesis (sincerely asking)?

[LuckyTran]

For inviscid flows & divergence free flows (incompressible flows) the influence of the second viscosity term is none. For boundary layer flows (where shear stresses dominate compared to the normal stresses) the influence of the second viscosity term is small (due to normal stresses being small).

The most practical example that I know of where the second viscosity is important is absorption of intense (very high amplitude) sound waves (note that this isn't a velocity boundary layer problem). That begs the question: if you are interested in the fluid flow & possibly turbulent phenomenon, why you are hunting down sound attenuation?

However, (Emanuel, 1992) did investigate the influence of non-zero bulk viscosity on a hypersonic boundary layer. Furthermore it is known that certain fluids (e.g. CO2) have extraordinarily high bulk viscosity, see the recent JFM article. It is noted that the effect of the bulk viscosity is to make turbulence proceed towards the incompressible state. This result I find very interesting because that term is often grossly over-simplified in the "exact transport equation for k" or the "exact Reynolds stress transport equation." So even if you could write down and solve the correct RANS equations, you probably will need a new set of turbulence closure models.

Originally Posted by TurbJet

Have you ever thought of trying the solver without Stokes hypothesis (sincerely asking)?

It sounds like you want a super general equation that covers everything. In my opinion, completely dropping the Stokes hypothesis is no easy feat. For sure your mechanical pressure is no longer the thermodynamic pressure. To even get your state variables (density and so on) you basically need to enter the realm of statistical thermodynamics, or at least non-equilibrium thermodynamics. And then you are no longer dealing with N-S but the full Cauchy momentum equation + a bunch of constitutive relations that I'm not really sure what these even look like. It's really a lot to think about and is well beyond the scope of compressible N-S.