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I have seen derivations of the the length scale, time scale and characteristic velocity of eddies in the inertial or large-scale range and in the diffusive or Kolmogorov scale range. I'm interested in the region in between, which Alistair Revell in his video titled “Advanced CFD Course: Turbulence Scaling” calls the Transfer Scale.

Whereas the length and timescale in the Integral stage are functions of kinetic energy and energy dissipation rate and in the Kolmogorov scale (which Revell refers to as the "destructive stage") are functions of viscosity and energy dissipation rate, Revell states that the length and timescale in the Transfer stage or scale is a function of the length scale and the energy dissipation rate.

I have not been able to find any work on estimating representative or characteristic length, time and velocity scales in the Transfer range. Perhaps it is not possible because the eddies in this Transfer stage span a wide range of sizes. Or perhaps, since I am new to turbulence theory, I am searching for articles using the wrong keywords. If anyone could provide some assistance, I would greatly appreciate it.

Also, it is my assumption that eddies in the "Transfer" stage would equate to transition flow for the fluid. Is that correct?

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The "transfer" range as you call it is (in my experience) usually referred to as the inertial range, and it is difficult to characterize by a particular length or time scale. This is because it is the entire range of scales between the integral scales and the Kolmogorov scale, so it's composed of basically everything between the two -- which changes as a function of Reynolds number.

However, there is one more length scale that is commonly used in turbulence: the Taylor microscale. This scale basically marks the end of the inertial range. Above it, viscosity isn't very important; below, viscosity is important. It can be related to other length scales of turbulence pretty easily.

You can of course find your own scales of interest. For example, a possible Rdemyan Scale could be defined by keeping in mind that the dissipation rate is constant at all scales from integral to Kolmogorov and is defined as:

$$ \epsilon = u'^3 (l_R) / l_R $$

where $u'(l_R)$ is the velocity fluctuations at the length scale $l_R$, our Rdemyan Scale. This can be related to other length scales using their definitions (for example, the integral length scale is $l = k^{3/2}/\epsilon$) and by knowing (or identifying) a way to specify the velocity fluctuations at the Rdemyan Scale.

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  • $\begingroup$ Thanks for your response. Over thirty years ago I did some mixing studies with impinging thin sheets of liquids. Diffusive mixing at low viscosities was complete with a few milliseconds at flowrates of liters per minute because the thickness of the free sheets at impingement was of order 50 microns. The geometry of this impingement makes it quite easy to calculate the energy dissipation rate. The results at 1 cP fit a model assuming that micromixing occurred as diffusion in slabs of size the half Kolmogorov microscale. $\endgroup$
    – rdemyan
    Commented Jul 9, 2020 at 16:33
  • $\begingroup$ I also did some work at higher viscosities and the results indicated that the micromixing slab size was significantly larger than the Kolmogorov microscale. I came up with my own length scale that fit data at 4.3 cP, 9 cP and 12 cP quite well. It's a simple length scale with no experimentally determined constants, i.e. the length scale was calculated using only the thickness of the sheets at impingement, the viscosity and the y component of sheet velocity that is destroyed upon impingement and is what provides the energy dissipation. $\endgroup$
    – rdemyan
    Commented Jul 9, 2020 at 16:34
  • $\begingroup$ It's been thirty years since I did this research, but I want to start it up again. I'm thinking of writing a paper on turbulence cascade in impinging sheets with derivation of the representative scales in the integral, inertial and Kolmogorov ranges. I've been out of research for so long, however, that I don't know what to expect and if this is even publishable. $\endgroup$
    – rdemyan
    Commented Jul 9, 2020 at 16:37

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