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Assuming that a direct numerical simulation is performed, what is a good method for estimate the impact of small scale on large scale in fluid dynamics ? For example is it pertinent to compare two run with different grid size or two run with different viscosity ? Is there some relevant statistical tools for this ?

The large scale field can be defined as a coarse-grained field \begin{equation} \overline{q}_l(t,\mathbf{x})=\int G_l(\mathbf{y}) q(t,\mathbf{y}+\mathbf{x})d\mathbf{y} \end{equation} where $G_l$ is a normalized convolution kernel of scale $l$. For example the forme of $G_l$ can be $G_l(y)=l^{-3}/\sqrt{2 \pi} \exp(-((y/l)^2/2)$.

The small scales field is defined as \begin{equation} q'_l=q-\overline{q}_l \end{equation}

If at some scale $l$ we can remove the small scale of the dynamic, the impact of the small scale on the large scale, will be the difference between the field of the full dynamical system with the field of the truncated dynamical system.

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    $\begingroup$ Unless you define the terms "small scale", "large scale", and "impact" in mathematical or physical terms, I think this question is not sufficiently defined to be answerable. $\endgroup$ – David Ketcheson Sep 4 '12 at 18:06
  • $\begingroup$ I modified the question considering your comment. $\endgroup$ – ucsky Sep 4 '12 at 19:56
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    $\begingroup$ I think the answer should deal with backscatter in turbulence and the way to estimate it computationally. I'm not an expert in that question so I will just leave a comment and not give an answer. Just want to note that there are a lot of interesting results after searching for "backscatter turbulence". $\endgroup$ – Johntra Volta Sep 4 '12 at 22:45
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There are many reasons that a coarser simulation would give different results that a finer grained simulation. A few examples:

  1. are boundary layers being resolved differently?
  2. am I resolving new features (vortices / blockages to flow)

Thinking about a grid result as purely a convolution of a fine-grid result with a Gaussian will work very well in viscous dominated flows (where there is already a large-scale smoothness imposed), but may be remarkably wrong where that assumption breaks down (higher Reynolds number)

If you can pose a situation with a known symbolic solution, simulate that at several different scales. If your algorithm/implementation is good, there should be a (roughly linear) convergence in log(error) vs. log(grid size), the slope of which is the "order" of your accuracy. There are some examples of this in my thesis, and I suggest a bunch more reading if you are so inclined.

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It is imperative to compare many runs with refined meshes until you detect convergence. A single solution with no mesh refinement study shouldn't give you much confidence in your results.

Comparing runs with different fluid properties tells you something different. If you think that a set of runs with different viscosities is relevant to your ultimate science/engineering question, then you should also perform such a study. You should, of course, make sure that each point in this study is mesh-refined as well.

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  • $\begingroup$ I agree that it's imperative to compare many runs with different resolution but is the under-resolved run can give some information about the impact of the small scall ? For the viscosity I expect that increasing the viscosity (or more precisely hyper-viscosity) will act as a cut off on the small scale but how wrong is it ? $\endgroup$ – ucsky Sep 4 '12 at 16:28
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You are essentially describing a large eddy simulation (LES). The key word in LES is "large". You must resolve the "large" eddies for LES to deliver meaningful results. "Large" in this context refers to the energy-containing scales so you must resolve into the inertial subrange for LES to be valid. For many industrial flows, the Reynolds number is much too high to resolve into the inertial subrange, in which case we must resort to turbulence modeling (RANS; examples include $k-\epsilon$, $k-\omega$, Reynolds stress models, etc). These RANS models explicitly model the effect of the unresolved scales on the larger scales.

Under-resolved models that do not explicitly model these subgrid scales deliver vastly inferior results. Although some people do this anyway, mostly to avoid the complexity of an actual turbulence model, the results have almost no predictive value. You will rarely see this degree of laziness in fields with rigorous validation standards.

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