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Gerry Kan April 24, 2020 09:13
Generalizing discussion of turbulence
 
Howdy Folks:

I have something of a philosophical question and would like to ask for your opinions.

I see that most texts and articles immediately use velocity component (u_i) to facilitate derivation of the important equations such as Reynolds/subgrid stress, eddy viscosity and the like.

In my case, I would to generalize the discussion treatment of turbulence (i.e., RANS averaging / subgrid scale filtering) for a given scalar (say, \phi). Would it come across to be a bit strange if I present something like:

\rho\overline{\phi'u_j'}=\Gamma_T\frac{d\overline{\phi}}{dx_j}

instead of what is traditionally presented for velocity component:

\rho\overline{u_i'u_j'}=\nu_T\frac{d\overline{u_i}}{dx_j},

or is it conceptually wrong, since turbulent diffusion for the corresponding scalar will be scaled according to the Eddy viscosity \nu_T using Reynolds' analogy anyways?

Thanks in advance, Gerry.

FMDenaro April 24, 2020 12:00
Quote:
Originally Posted by Gerry Kan (Post 767196)
Howdy Folks:

I have something of a philosophical question and would like to ask for your opinions.

I see that most texts and articles immediately use velocity component (u_i) to facilitate derivation of the important equations such as Reynolds/subgrid stress, eddy viscosity and the like.

In my case, I would to generalize the discussion treatment of turbulence (i.e., RANS averaging / subgrid scale filtering) for a given scalar (say, \phi). Would it come across to be a bit strange if I present something like:

\rho\overline{\phi'u_j'}=\Gamma_T\frac{d\overline{\phi}}{dx_j}

instead of what is traditionally presented for velocity component:

\rho\overline{u_i'u_j'}=\nu_T\frac{d\overline{u_i}}{dx_j},

or is it conceptually wrong, since turbulent diffusion for the corresponding scalar will be scaled according to the Eddy viscosity \nu_T using Reynolds' analogy anyways?

Thanks in advance, Gerry.



If you consider the internal energy equation, that is the temperature equation, you see the same concept you wrote above in terms of closure expressed by means of a Fourier-like turbulent heat flux.
However, while in case of a passive temperature that can be somehow justified, in case of a coupling to the other variables, there are further considerations.
That issue is analysed in a couple of papers by Lilly.

LuckyTran April 24, 2020 12:37
There is a tendency to do everything for velocity because we are explicitly invoking the eddy viscosity hypothesis, which is a pretty big theoretical leap.

You still need to write down the turbulent flux of other scalars at some point, which you don't have to obtain these by Reynolds analogy. But I would write down the velocity one first (put it at the top of the page so to say) just so it is clear the eddy viscosity hypothesis is the one being used.

I would not for example, write down the turbulent flux for an arbitrary scalar and pretend like I have a magic wand that can always prescribe the right turbulent diffusivity coefficient.

Gerry Kan April 24, 2020 14:31
Quote:
Originally Posted by LuckyTran (Post 767245)
I would not for example, write down the turbulent flux for an arbitrary scalar and pretend like I have a magic wand that can always prescribe the right turbulent diffusivity coefficient.
Hi Lucky:

The "magic wand" fallacy you spoke of is something I worry about very much. In the end, I realized that working out the turbulent fluxes in general isn't as useful as I first thought, and caused my discussion to lose focus and formalism very quickly. In that sense I could just stick with a relevant scalar variable (i.e., velocity component) and maintain a more concrete formulation.

It is also one of those things where if you can do this for velocity, you can transfer the principle to other scalar variables.

Thanks, Gerry.

LuckyTran April 24, 2020 14:56
Here's the thing. \overline{a'b'} is pretty much the cross-correlation between a and b (for real a and b, it is the cross-correlation). To suggest that the cross-correlation between two arbitrary random variables can be re-written as a scalar times the gradient of \overline{b} is pretty weird. If you told that someone not in the know, they'd either tell you that you're just wrong or give you really funny looks.

Context and justification are paramount in this case.


It works because one of those things is u and because we apply the gradient-diffusion model. We are imposing these behavior rather than derive it. We shouldn't even be writing it with an equal sign, because it's not an equality.

Gerry Kan April 25, 2020 11:07
Quote:
Originally Posted by LuckyTran (Post 767280)
erive it. We shouldn't even be writing it with an equal sign, because it's not an equality.
Yes, you're right! It's an \aprox on my books, not an equal! Gerry.

FMDenaro April 25, 2020 12:23
There is a strong impact from the history for the present use of the eddy viscosity concept.

It could be also questionable the fact that a term coming from the hyperbolic part is modelled as a non-linear parabolic term.

Gerry Kan May 19, 2020 13:59
Quote:
Originally Posted by FMDenaro (Post 767343)
There is a strong impact from the history for the present use of the eddy viscosity concept.

It could be also questionable the fact that a term coming from the hyperbolic part is modelled as a non-linear parabolic term.
Sorry Professore for the tardiness in my response. My professor in turbulence often said that people stopped thinking after Prandtl came up with the mixing length concept. To this day, I still can't resist using an 'eye-ball' mixing length estimate!

On your second comment ... my first CFD code (from a long time ago) modelled the hyperbolic term with a parabolic treatment, although eddy viscosity was still an elliptic term. Ironic, isn't it?

Gerry.

FMDenaro May 19, 2020 14:05
Quote:
Originally Posted by Gerry Kan (Post 771148)
Sorry Professore for the tardiness in my response. My professor in turbulence often said that people stopped thinking after Prandtl came up with the mixing length concept. To this day, I still can't resist using an 'eye-ball' mixing length estimate!

On your second comment ... my first CFD code (from a long time ago) modelled the hyperbolic term with a parabolic treatment, although eddy viscosity was still an elliptic term. Ironic, isn't it?

Gerry.



That is one of the questionable aspect of the eddy viscosity model, that is it changes the original mathematical character. That is due to the history and tradition of RANS approach.

In LES there are modern SGS models where the original character is retained (dynamic similarity model, deconvolution model and so on).


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