[praveen]

Here is an intro to a paper by Marshall Slemrod

https://www.quantamagazine.org/20150...er_Feb_27_2015

The paper is here

http://www.mmnp-journal.org/articles...p201510p6.html

and you can get a preprint here

https://www0.maths.ox.ac.uk/system/f...xPDE_14.09.pdf

Boltzmann equation implies the Korteweg equations and not the Navier-Stokes equations. The Korteweg equation is also a system of 5 equations (one mass, 3 momentum, one energy) and has dispersive terms apart from the usual Navier-Stokes terms.

The above articles/paper talk of some physical situations where Navier-Stokes gives wrong solution but the Korteweg system gives the correct solution. These are all low density situations.

We can get the Euler eqns in the limit of vanishing knudsen number only for regular solutions. So in general the Euler eqns are not the correct model for zero viscosity limit.

Now what does this all mean for turbulence problem ?

[FMDenaro]

Originally Posted by praveen

Now what does this all mean for turbulence problem ?

What do you mean? if we assume valid the continuous framework, viscous flows, I don't think dispersive terms are in effect

[praveen]

Slemrod refers to papers by Gorban & Karlin and Saint-Raymond where they derive the Korteweg model starting from Boltzmann equation. This model is similar to Navier-Stokes but has additional Korteweg terms which are dispersive. If you accept Boltzmann as a good model of reality, then you get Korteweg and not Navier-Stokes.

[FMDenaro]

Originally Posted by praveen

Slemrod refers to papers by Gorban & Karlin and Saint-Raymond where they derive the Korteweg model starting from Boltzmann equation. This model is similar to Navier-Stokes but has additional Korteweg terms which are dispersive. If you accept Boltzmann as a good model of reality, then you get Korteweg and not Navier-Stokes.

yes, I see but what I mean is that classical NS are a model, they do not consider many effects ..
and I think that even including the dispersive terms in a computation they are not relevant in changing the solution for many flows...If I am right, dispersive term should be relevant when capillarity effect are present, this is the case of microfluidic.

[FMDenaro]

do you know some DNS performed solving a 3D Korteweg-like model and compare relevant differences in the solution? I am quite curious to know if someone explored that...

[praveen]

To quote Slemrod

If the capillarity coefficient K is sufficiently (small) viscosity would dominate and passage to the limit might be accomplished as in the paper of Schonbek [36]. But alas I have shown in [39] based on the results of Gorban & Karlin that while it is appealing mathematically to make such an assumption it does not follow from Gorban & Karlin's results. In fact just the reverse is true: capillarity dominates viscosity and our Korteweg system can be expected to generate oscillations just as in the Lax-Levermore limiting process.

I think he is saying that at very small scales, capillarity dominates viscosity.

This is all new stuff to me, so I dont know what people have done in terms of solving the Korteweg system and what it means for turbulence.

[FMDenaro]

Originally Posted by praveen

To quote Slemrod



I think he is saying that at very small scales, capillarity dominates viscosity.

This is all new stuff to me, so I dont know what people have done in terms of solving the Korteweg system and what it means for turbulence.

yes, as I can remember, microfluidic is the typical framework where such effect happens... however, owing to the very small scales, no relevance on turbulence is highlighted in such problems...

for standard turbulence, the question would turn to be if the smallest, viscous scales of turbulence, say behind the Taylor microscale, would have a new behaviour due to inclusion of capillarity. That could have great interest since an SGS model in LES should also take into account such effects

[truffaldino]

Originally Posted by praveen

Now what does this all mean for turbulence problem ?

I think it has no significant effect on turbulence theory.

At the Kolmogorov scale the Knudsen number is still huge and not close to zero, so there is no problem with Euler equation.

[FMDenaro]

Originally Posted by truffaldino

I think it has no significant effect on turbulence theory.

At the Kolmogorov scale the Knudsen number is still huge and not close to zero, so there is no problem with Euler equation.

I think the same, too...
however, the question to be explored could be if some capillary effects cohexist at small scales and are somehow relevant in the dynamic of the largest...I doubt