Hi Aditya,

In order to characterize turbulence, you may need a length scale to describe how large the eddy is and a velocity scale to describe how fast the eddy overturns. Typical two equation models (e.g k-epsilon) can give you the length scale (by using epsilon and tke) and velocity scale (using tke), which you may use to qualitatively show how eddy evolves with time and varies in space. However, in a realistic situation, you will get a spectrum of length scales and velocity scales for those eddies. Therefore, you can hardly rely on typical two-equation models to show such cascade process quantitatively. 

Other options may be LES model or DNS model that resolves turbulent motions (for LES model, resolves large scale turbulent motions). After you obtain the velocity field, you can get the instantaneous vorticity. Then you remove the mean vorticity and get the fluctuation vorticity that may reflect such eddy cascading process. 

Hi Aditya,

the Kolmogorov cascade is conceptually identical with the "devil's gear" discussed by Hans Hermann, see arxiv.org/pdf/1107.5305 Actually you may treat turbulence also as a "spaghetti clockwork" (dipoles of counter-rotating "spaghettis" in an inviscid fluid). @Dong Wang: k-eps is a fundamentally wrong concept because it takes eps as a conserved quantity. However, eps is NOT a conserved quantity! For energy and vortiticity (~omega) we do have conservation laws such that the k-omega model is a physically sound concept, goes back to Kolmogorov and Landau (1942).

Dear sir,  than you very much for you time. It is certainly penny to my thoughts. I would like to agree to concept of contra-rotating vortex or spaghetti in an inviscid flow. Besides, it also appears that i have much learn about turbulence and turbulence models. 

In particular, i was surprised to read that eddies in real flow appear in elliptical shapes. This immediately links the orbital motion of earth around the sun (Kepler's laws) to the motion of electron around the nucleus (current understanding to atomic physics).

However, from the paper cited above i worked some intuitive analysis with following results:

if two contra-rotating vortices, of same strength and same size, placed at a distance forces the surrounding medium to rotate, the strength of liquid will be zero, exactly at the central point (midway between two vortices) and will vary from center of each vortex to that point, similar to the shape of infinity. While if they are superimposed the strength will become zero. Hence, two such vortices could only fuel the strength to each other if they are in contact.

if two vortices rotating in the same direction forces the fluid to rotate, addition of such vortices will increase  the strength of rotation, if superimposed the effect will doubled. However, in contact they will oppose the motion of each other hence, they cannot be kept in contact. 

However, if the motion of two vortices placed eccentrically with motion of  of bigger vortex  fueling  motion of the smaller. If many such smaller vortices are assumed to be rotating with the strength supplied by the bigger vortex (ball bearing) and if the rotation of smaller one are ceased some how the bigger will not be able to move. With this i was trying to reach at molecular level but i was not able to correctly relate the process i suppose.

Hence, my dilemma whether to represent the vortices organized one over the another or distributed over the dissipation scale separately remains unsolved.

Aditya:

1) Better you forget the analogy between Kepler orbits and eddies, at least for the moment.

2) Ellipticity of vortices is a matter of neighboring vortex structures and pressure, kinda inner boundary condition. In my paper I mentioned the picture of a ball bearing initially originating from Hans Hermann. It's a nice picture but for turbulence maybe less fruitful because vortices, if they are finite, are typically much longer than their diameter, if their centerline does not even form a closed circle in space so that they are actually infinite.

The trick of the system to solve the case of two likewise-rotating vortices (the dissipative patch) is to spontaneously "invent" between them a number of smaller vortices with the "right" polarity - finally forming the notorious Kolmogorov cascade wherein only those vortices carry energy who exhibit the most friendly behaviour to their neighbors. Of course, energy conservation has to be observed.

Besides, if we talk about length scales and vortices or eddies, we mean always a diameter- or radius-related variable, not the "length" of a vortex. The length is typically assumed to be sufficiently greater than the diameter.

But maybe I did not understand your question and you will formulate it more precisely. I recommend to use math with WORD or LaTex and you attach the question to the next post.

The example of the sea you previously did is not well suited to explain energy cascade according to Kolmogorov theory. Why? Immagine a stable condition in sea, similar to rest in a sunny day. Now you have a wind coming a day. What kind of energy transfer you see? typically, the wind transfer energy to the sea surface by the action of the stress. It however starts generating very small waves over the surface that, slowly tend to become greater and greater. This resemble somehow an inverse cascade process, small structures take energy and they transfer to greater structure.

Filippo:

Thanks for your interest in the debate. The point here is IDEALIZED turbulence, and you talk about physical oceanography which contains much more processes, and turbulence is only one small element, however extremely important if it comes to deep-sea mixing when the time scales involved are of climatological relevance. Further, the case you skech contains thermal and maybe haline stratification etc. But the initial question was about idealized turbulence in the sense of Kolmogorov and Landau, if I am not in error. Maybe the attached papers give you a hint where to go, or you have a look into the collection of papers in the book MARINE TURBULENCE, Cambridge U. Press 2005, edited by Baumert, Simpson, Sündermann.

Helmut,

yes you're right, I was answering to the original Aditya example about seawater that indeed does not correspond to an idealize case of energy cascade.

Coming back to the original question, I suggest thinking about the case of a turbulent flow generated by a sharp change in the geometry, for example the backward facing step. Starting from the energy possessed by the structures generated by the step (integral lenght of the order of the step hight, low wavenumber), in the streamwise flow these structures correlate, are stretched and generate smaller structures (energy distributed at higher wavenumbers in the Fourier space) until they are of dimension of the order of the Taylor micro-scale where the viscosity starts to play role in the energy dissipation.

"It however starts generating very small waves over the surface that, slowly tend to become greater and greater."

Is there a proof of this?

I agree wind seems to excite low wavelengths. I thought other sources excite higher wavelengths (moon) and then a coupling starts between different equations (same as what happens on the surface of a transducer) allowing a transfer of turbulence.

If we see inverse cascade effect, is it a coupling? Does the eddies (low wavelengths) act like Kuramoto oscillators and then synchronize to a higher frequncy (high wavelength)?