Generally in experiments and/or in numerical simulations, the turbulence level of a fluid flow is reported by the Reynolds number based on the Taylor microscale, which its square, in homogeneous isotropic turbulence, is equal to 15 times the classical Reynolds number defined on the basis of the root-mean-squared-velocity of the turbulent fluctuation and by the length of the largest scales. Is there any physical interpretation of this preference? Thank you!
30 years ago Re_lambda was easy/accurate to calculate, and Re_turbulent was not.
In the good old days of experiments with a hot-wire you would get a spectra and/or a two-point correlation (one is the FFT of the other).
Finding the Taylor scale from the two-point correlation was easy and accurate
(fit a parabola to the function at small separations, see Pope text book for more info).
However, the large-eddy length scale is quite difficult and usually quite inaccurate.
It is computed from an integral of the two-point correlation from 0 to infinity. And very few experiments had data at infinity (or anywhere close to it) since that requires a very large wind tunnel (and they all use quite finite sized wind tunnels).
This was perpetuated by early DNS as well. Because their domains were all far too small. The first 'real' DNS is probably de Bruyn Kops and Riley. I love Wray, but it was all junk. When your domain is too small, you can't calculate the two-point correlations at large separations with any accuracy.
If you ever see a eps or L from 20+ years ago, it is most often fairly inaccurate.
But feel free to do things properly now.
Re = k*k/(nu*eps) makes more sense. And it is what I use.
30 years ago Re_lambda was easy/accurate to calculate, and Re_turbulent was not.
In the good old days of experiments with a hot-wire you would get a spectra and/or a two-point correlation (one is the FFT of the other).
Finding the Taylor scale from the two-point correlation was easy and accurate
(fit a parabola to the function at small separations, see Pope text book for more info).
However, the large-eddy length scale is quite difficult and usually quite inaccurate.
It is computed from an integral of the two-point correlation from 0 to infinity. And very few experiments had data at infinity (or anywhere close to it) since that requires a very large wind tunnel (and they all use quite finite sized wind tunnels).
This was perpetuated by early DNS as well. Because their domains were all far too small. The first 'real' DNS is probably de Bruyn Kops and Riley. I love Wray, but it was all junk. When your domain is too small, you can't calculate the two-point correlations at large separations with any accuracy.
If you ever see a eps or L from 20+ years ago, it is most often fairly inaccurate.
But feel free to do things properly now.
Re = k*k/(nu*eps) makes more sense. And it is what I use.
I think that Tennekes and Lumley has an explanation that complements the one above. These two dimensionless parameters show different things and in flows with more than one characteristic scale also do not relate to each other in a simple way.
This is an excellent question! I had always wondered this myself. The problem is that (unlike the integral or Kolmogorov scales) nothing special happens at the Taylor scale, which makes it difficult to explain to students. Thanks to Prof. Perot, as Paul Harvey [for those of you old enough to know who he was] would have said, "... and now you know the rest of the story."
Actually, something very physical might be happening at the Taylor scale.
Think of turbulence as a tangle of spaghetti vortices.
The Kolmogorov scale is roughly the spaghetti radius.
The large eddy scales are roughly the length from end to end of the spaghetti
(but remember the spaghetti is tangled and not straight).
Then the Taylor length scale might be the average wavelength of the
wiggles within each strand of the spaghetti.
This is obviously not a proof. But could be a reasonable model.
Most important in this model, is the ability to show how something with
two lengthscales (spaghetti) can easily get a third lengthscale
when it becomes tangled. This model also shows how that
third scale (the Taylor scale in our case) MUST lie between the other two.
Actually I have also heard that the Taylor scale is the largest scale that can be generated directly by an integral-scale eddy. If you think of the integral-scale eddy as a solid body and start rotating it, by analogy with a flat plate after one revolution the ratio of the boundary layer thickness to the eddy circumference would be on the order of 1/Re^0.5, where Re is the integral-scale Reynolds number.
Personally I never found that interpretation very compelling because the eddy isn't in solid-body rotation and (more importantly) one should think of it as a turbulent boundary layer, not a laminar one, in which case the ratio would be more like 1/Re^0.2, that is, the boundary layer thickness would be almost as big as the eddy itself.
The above answers have been informative for me, so thanks to all. A short answer is that the Taylor microscale is a well-defined quantity, so it is convenient to use for something like HIT that has many different length scales to choose from. It is also easy to get from the Taylor microscale to the characteristic timescale of the small eddies, so maybe that's another reason.
Bill George has a coupled of papers on turbulence decay that assert a stronger physical basis to the importance of the Taylor scale (thus, Re_lambda) than might normally be assumed (prioritising Kolmogorov length scales). Mazellier and Vassilicos (2010) develop some of these ideas. Hence, Re_lambda has more physical interest that it once had.
As a comment to Prof. Perot: spaghetti, yes, but also lasagne, fettuccine, bucatini, spaghettini, capellini, and so on and so forth...it is a jumble of scales, until they become so thin that the boiling water dissolves them as soon as they are formed...
I like the Italian cuisine analogy. It's better than the "zoo of animals" that comes to mind when people talk about "eddies" or "coherent structures" :)
I think that Taylor approach is more physical , but the direct Reynolds number estimation using rms of turbulent velocity fluctuations and the largest scale of these turnulent fluctuations is more easy to understand (see Tennekes and Lumley a first course on turbulence) but the Reynolds number estimation is I think a "raw estimate" as these two scales are not precise. Anyway the Reynolds number approach with this direct method shows that the turbulent scale is more efficient than the molecular diffusion scale though the physical approach by the Taylor scale has to be considerd to be probably more precise and related to a better insight into the turbulence structure.
The direct Reynolds numer estimate also corresponds to historical point of view of turbulence considerations in wind channels.
Based on " Dimotakis, P. (2005), Turbulent mixing, Annu. Rev. Fluid. Mech, 37, 329{356.", different flows transition and exhibit a similar behaviour as expected for high Reynolds number flows as the Reynolds number based on the Taylor microscale exceeds 100. Thus the Reynolds number based on the Taylor microscale can be used to identify whether turbulence is fully developed or not. See below for a more detailed discussion:
Transition in fluid behavior is observed in gas- and liquid-phase shear
layers , jets, pipe flow, boundary layers, bluff-body wakes, grid turbulence, Couette-Taylor flow, and Richtmyer-Meshkov instability flow. This transition manifests itself as
an increase in the spectrum of eddying scales and emergence of a near -5/3 power-law regime in the velocity spectrum as proposed by the Kolmogorov theory. Dimotakis suggests that this transition to a fully-developed turbulence state occurs approximately at
\begin{equation}
Re_{tr}\approx 10^4, \quad Re_{tr,\lambda}\approx 100
\end{equation}
where $Re_{tr}$ is the local (outer) transition Reynolds number and $Re_{tr,\lambda}$ is the transition Reynolds number based on the Taylor microscale. The $Re$ dependence of different phenomena is expected to decrease as the Reynolds number increases beyond $Re_{tr}$. At higher Reynolds numbers, a sufficient scale separation occurs between the inner dissipation scale and the Liepmann-Taylor scale to provide room for the quasi-inviscid motion of the eddies in the inertial range responsible for the non-dissipative cascade of energy from large scales to small scales.
Hi Abdallah, first of all thanks for this challenging question! The Taylor microscale tells us that turbulence is colored noise, not a white noise process. As this scale describes the smooth (parabolic) behavior for r => 0 in the (spatial) auto-correlation function of the turbulent fluctuations, analogously in the temporal ACF. So it has to do with the micro-canonical behavior of the turbulent field. In a way I would call it in the moment a secondary parameter. Of primary interest is the macro-canonical behavior which was not understood for a long time. We only knew semi-empirical fractions, but the full picture couldn't be drawn. But then, about 15 years ago, an idea similarly to the spaghettis of Blair Perot, surfaced in form of a "spaghetti arrabiata". The question was what is the sauce between the spaghettis, the sauce arrabiata? It is an in principal inviscid fluid whereas the spaghettis are solids - in a way. So we get a two-fluid concept similar to the one by Feynman to explain super fluidity. Please read and comment the attached paper. Then you know the rest of the story! Thanks.
What can we learn from the spaghetti story? To be successful in turbulence on a fundamental level, we have to leave the Navier-Stokes dogma. We have to deal with the elementary elements of turbulence: vortices, and we need to know how these vortices move in space and how they interact. And we have to do statistics with them, eg. in the sense of Maxwell, Boltzmann, Langevin, Fokker and Planck. Here is also the key of the quest for a mechanical (micro-canonical) interpretaion of the Taylor scale. I would be happy to launch a project about this fundamental problem. Thanks.
Well - this is quite obvious. All quantities can be measured directly and easily. The Taylor scale can be derived from the curvature of the velocity autocorrelation function for short times, which can be measured easily.
Thank you Prof. Bodenschatz for your contribution to this discussion. I am so excited! Thank you. a
By the way - when people talk about the Richardson-Kolmogorov cascade they use the poem from Richardson which implies the evolution of scales from large to small. This of course only in a statistical sense. Today we know that the cascade is only found when we observe the statistics in space and not in time. In other words in the Eulerian frame. In other words the poem applies to the statistics of one realization of a very large junk of turbulence. It does not address the time evolution of the flow, if one follows fluid elements. In find the poem misleading.
So far there is no evidence that K-scaling can be applied to the Lagrangian picture. All the similarities rely on dimensional analysis and no exact derivation form the Navier Stokes equation. We are currently working on finding other quantities than the velocities and the latest PNAS paper may give a hint. - So long Eberhard
The Betchov (1976) poem removes the implication in the Richardson "original":
Big whirls lack smaller whirls
To feed on their velocity.
They crash and form the finest curls
Permitted by viscosity.
Hi Eberhard,
I love poetry and namely Richardson's poem! He is eventually right. If one goes to the limit of vanishing viscosity ("plus zero"), then Richardson's magic clockwork doesn't need any inner friction, except at the scale "plus zero" where all the kinetic energy of the clockwork is dissipated (in a finite-viscosity case: the micro- or Kolmogorov scale, slightly greater than "plus zero"). This is the ideal, asymptotic or universal case, described in a recent paper (doi:10.1088/0031-8949/2013/T155/014001).
The downhill-uphill cascade controversy reminds of a discussion initiated early by Carl Gibson who replaced Richardson's view by its opposite: He claimes that turbulence begins at small scales, and energy is transferred upwards the wavenumber spectrum [Keeler, Gibson, Pierce: "Helicity and the Inverse Turbulence Cascade" J. Cosmol. 2013 Vol 21(55), pp 9816‐9828]. But how can excitation and dissipation of perturbations take place at the same place, i.e. at vanishing scales?
Unfortunately the asymptotic case (viscosity = "plus zero") can principally not be described by solutions of Navier-Stokes (NS) because this equation belongs to the singularly perturbed class. (See e.g. Feireisl & Novotny about "Singular limits in thermodynamics of viscous fluids", Birkhäuser, Basel 2009, 382 pp). In the NS equation a transition to vanishing viscosity is easy - we simply delete the parabolic term and get the Euler equation. In the solutions of NS it is impossible to perform this transition because they carry (in addition to the mere equation) also certain initial and boundary conditions. And the latter are different for NS and Euler as the equations belong to qualitatively different math classes (parabolic vs. hyperbolic).
Therefore a universal or asymptotic turbulence theory (for developed turbulence, not for transition) should start from Euler rather from NS. This exactly is a key issue in the 1m USD problem No. 6 of the CMI (http://www.esi2.us.es/~mbilbao/claymath.htm) wherein properties of the NS equation shall be studied to help for a better understanding of turbulence. Unfortunately the problem author, Charles Fefferman, was not aware that a universal solution of the turbulence problem cannot be derived from NS. Maybe Fefferman went in a typical mathematicians' trap because "The diversity of problems in turbulence should not obscur the fact that the heart of the subject belongs to physics." (Falkovich & Sreenivasan 2006). That's abolutely right! Possibly we are in a situation like couple years ago the geocentrics ("world is very complex, needs lots of complicated math and thus high priests...") fought the heliocentrics ("world is simple, circles vs, hypercycles, look at ALL data ...").
The a.m. asymptotic or universal theory not only gives extremely accurate answers to very old questions (e.g. the values of the universal constants of von Karman and Kolmogorov), it also opens a broad and easily passable doorway to intermittency and sub-scale Lagrangian motions. The latter may be understood as superposition of Brownian motion sensu Einstein (2005) and a quasi-localized Appolonian clockwork ("dissipative patch") sensu Hans Herrmann (1990). Unfortunately in these summer months I am too busy with garden work and cannot do a study in this direction by myself. But I am ready to support students. Good luck, Helmut
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Despite various opinions heavyweights who wrote above, I will dare to give my opinion (the opinion of a lightweight compared to the other).
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When root-mean-squared-velocity fluctuation of the turbulent and is used by the length of the largest scales, there always remains a small ambiguity, the length of the largest scales using the Reynolds number based on the Taylor microscale, has a unique and accurate result, no matter what type of flow.
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Hi Rogerio, you met the point: obesity. In my case ~ 25 kg above optimum!
Abdallah, regarding Taylor's microscale, there are two of them, a time scale and a spatial scale. Both of them follow inversely the curvature of the corresponding auto-correlations functions, R ~ , around their origin x'=0. If these functions are "sharp" at the origin, e.g. R ~ Exp[-Abs(x)], then Y(x) is a non-differentiable stochastic process. If these functions are "smooth" then Y(x) is differentiable. Solutions of Euler are not necessarily smooth and may exhibit non-differentiable parts. Solutions of NS however are generally differentiable. Regarding the application of Taylor's scales in Re_T to characterize the overall picture of a turbulent flow, I dont see big adavantages re traditional Re. A matter of taste, comfort or convention. And another point: they all do not uniquely characterize turbulence intensity nor turbulence level. Note that in superfluid He we also may have turbulence while Re = infty so that Re cannot be a reasonable measure of turbulence level or intensity. The role of the various Re forms in turbulence theory is in my view limited: they all should be "high enough". Besides, based on the theory of inviscid turbulence we got access (interestingly via a non-Newtonian detour) to a closed theory of viscous turbulence, i.e. for finite microscales (Taylor and Kolmogorov). To appear soon ...
Dear all, the amount of discussion above - even by some famous colleagues - shows to me that not all about the physical interpretation of Taylor's length scale seems to be settled. Now, I just want to draw attention to a paper of Lueck, Renner, Peinke and Friedrich (Phys Lett A 359 , 2006) who gave (experimental) evidence that Taylor's length might be seen an a "Markov-Einstein coherence length". Nice idea, isn't it ?What does the community think about it ?
Alois, the topic is clearly relevant in many aspects, e.g. with consequences for the Prandtl number etc.
Coud you send a copy of Lueck etal 2006 to baumert at iamaris dot org ?? (I am a poor east-german pensioner who neither can travel to big conferences nor afford charges for papers. It is for this personal discussion only.
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