[FMDenaro]

Originally Posted by H0T_S0UP

Basically, what I am saying is the continuum assumption takes a time and spacial average over a very small domain, and this leads to an approximation of the momentum of the fluid in this region. You cant have a domain where dxdydz = V =0 or dt = 0, because you would not have a flux through the control volume or not be able to measure velocity. Im sure the errors are minor, but my proposition is the sum of these errors over practical length and time scales leads to small deviations from what continuum mechanics (ie, N-S) predicts. Im working on a stochastic derivative in my free time to see if this approach is fruitful. It is more of a fun project since nobody will pay you to do that kind of work, but the results are kind of neat so far.

The general continuum assumption is based on the fact that the smallest characteristic lenght of turbulence (i.e., Kolmogorov scale) is several order of magnitude greater than the mean free path. I dont know real case where such lenghts become comparable, even shock waves lenghts (Mach <3-4) have one order of magnitude greater.
So, what do you mean for deviation from continuum mechanics?

[truffaldino]

Originally Posted by H0T_S0UP

The honest answer:

We don't know.

We do not know enough about the mathematical characteristics of the N-S equations, we only superficially understand turbulence, and we know even less about
laminar-turbulent transition.

That is true. In my opinion, the more or less satisfactory advance in turbulence is the Kolmogorov's theory. The rest is handwaiwing and numerical simulations.

[H0T_S0UP]

So, what do you mean for deviation from continuum mechanics?

I am still working on quantifying this error. The deviation lies in the continuum assumption itself: recall that a differential volume in space and time is an average. Continuity, for example, can only be enforced statistically d/dt (rho) = 0 for extremely small volumes and timescales. The continuum assumption does not have an "average" and therefore must deviate from a discreet description at the smallest scales.

Quite frankly, I don't see how we can develop a full theory of turbulence while simultaneously neglecting the physical fact that fluids are made up of molecules and their ensemble behavior governs what we observe. What is the smallest scale we have generated turbulence at anyway? Have we generated turbulence in near super-fluid systems? These are the experiments that need to be considered to determine the physical limits of the governing rules of turbulent flow.

[Martin Hegedus]

I'm not sure I understand. Your statement seems to imply that we don't understand laminar flow. For example, what happens to shear flow as it gets very very thin.

[Martin Hegedus]

Another example, the viscous sub layer is between one and two orders thick in regards to Re where Re=u(+)*v(+). However inside this sublayer are very very small U shaped vortices like little hairs sticking up. Even though it is very unsteady in this region, the region is not considered turbulent. At least from what I understand. As mentioned a few times now, I'm definitely not an expert. So I guess the uncertainty is whether the Navier-Stokes equations can be correctly applied to that region?

[H0T_S0UP]

Im not expert either. I left a PhD program due to a lack of funding, but I read a lot and still teach myself things. Anyway, a really good example of where N-S fails is in filtration. Take a look at submicron scale filtration, youll find most numerical methods need to employ an adjusted Brownian motion solver. Other cool things, such as thermophoresis, I do not believe are captured by N-S, though I am not as well read on microscale flows as on larger scale topics.

[Martin Hegedus]

Does anyone know of a reference to a numerical (or real life) experiment of what happens to the boundary layer of a spinning cylinder (started impulsively or some other way) in still air as the boundary layer thickens up? I gather as the boundary layer increases in size, a wave will develop and it will eventually go unstable (depending on omega I guess).

[FMDenaro]

To the best of my knowledge, there exists studies that confirmed the correctness of NS equations from the statistically averaged molecular point of view.

As I wrote above, shock layer for air is about 10^-6 - 10^-7m and is still well governed by the NS equations as confirmed in some observations.

Kolomogorov scale is much greater, so I don't see how any correction can be relevant.

Said that, if you see difference in filtration or similar problem these are nothing to do with validity of NS for turbulence, simply such problems can require to focus on nanofluidics wherein many effects can be relevant and are erroneously disregarded in general flow problems

[robo]

I aggree with FMDenaro. The mean free path is orders of mangitude lower then the Kolmogorov scale. That is the reason you can neglect statistical mechanics in the vast majority of cases. The examples you cited may in fact be cases where some statistical mechanics are required, but those are niche cases where you are purposely introducing or changing some facet of the intermolecular forces. That does not imply that turbulence requires a statistical description, it just implies that it is required when you mess with the intermolecular forces. The turbulent flow of air over a flat plate can be described by the NS equations using continuum mechanics, including the turbulent motions, with no need for a statistical description, precisely because the mean free path is orders of magnitude lower then the smallest turbulent motion.

[FMDenaro]

Originally Posted by Martin Hegedus

Does anyone know of a reference to a numerical (or real life) experiment of what happens to the boundary layer of a spinning cylinder (started impulsively or some other way) in still air as the boundary layer thickens up? I gather as the boundary layer increases in size, a wave will develop and it will eventually go unstable (depending on omega I guess).

I remember a series of papers on JFM, I googled that


http://journals.cambridge.org/action...%20Coutanceau&