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h.jordi November 20, 2001 05:35

Reynolds decompostions splits the total velocity (Utot), into mean U and fluctuating components u'.

Where Utot = U + u'.

This is simple to understand when a turbulent field is considered as a fluctuating 2D trace on a sheet of paper, but in real turbulence (where there is a full scale of 3D eddies) what then is the physical meaning of u'.

It must be the fluctuating velocity of an eddy!, but what is a fluctuation of an eddy and how can it be measured?

Most books seem to explain the 2D fluctuations around the mean very well, but then they go on to analyse the eddies - their scales and vortex stretching. Can anyone help fill the gap, by explaining the fluctuating velocity idea in terms of the 3D eddies?

Axel Rohde November 20, 2001 08:49

I don't quite understand the question. In 3-D you have,

Utot= U + u'

Vtot = V + v'

Wtot = W + w'

or do you look at Utot, U, and u' as vector quantities?

Why is it only a 2D trace?

h.jordi November 20, 2001 12:35

yes we have U V and W components in 3D,

but in a book diagram (obviously 2D) it can show as a mean line with a 'noisy' looking signal (the fluctuation) superimposed.

i.e. a fluctuation and a mean

but in 3D yes we have U V and W but we have eddies so the question is how does the fluctuting velocty relate to an eddy?

i.e. does the eddy fluctuate in size at that speed, or does it 'move' with fluctuating speed, or is u' only characteristic of the eddy.

or rephrasing the question - how can u' v' w' be interpreted in 3D with respect to the eddies?


Axel Rohde November 20, 2001 14:13

Makes sense, but the 2-D graphs in books with the 'noisy' signal is Utot versus time, i.e. Utot(t), right? Eddies, on the other hand are spatial (x,y,z) structures, and you want the relationship between the two, correct?

kalyan November 20, 2001 16:40

The term "eddy" is not often precisely defined mathematically. At the moment, I can describe an eddy simply as a flow (vortical) structure with certain degree of spatio-temporal coherence (other definitions are most welcome). Examples : lambda vortices in wall layer of turbulet flows, von-Karman vortices, cat-eye patterns resulting from K-H instability etc.

The eddies can be spatially characterized using modal decomposition (e.g. Fouries modes, spectral elements, wavelet transforms). Some, like, Fourier modes are non-local (i.e., do not have compact supports) and hence it is difficult to visualize modes using them. Transforms with localized bases (like wavelets) are easier to define mathematical eddies. Each eddy travels with the local convective speed, exchanges energy with other eddies (modal energy transfer) and the local convective speed has contributions from all eddies in the vicinity.

The u',v',w' at a given point can have contributions from many eddies of different sizes and strengths. The eddy size distribution and the corresponding amplitudes (i.e., the energy spectrum) are given by the turbulence scaling laws.

u',v',w' (or the total velocity field) can be decomposed into modes in the most optimal way using what is called the Karhunen-Loeve decomposition or proper orthogonal decomposition (POD) as it is often referred to as in turbulence literature. What it does is express the velocity field in terms of a finite basis (with say "n" basis functions). This methods guarantees that no other set of "n" basis functions can better express your field (with respect to an appropriately defined vector norm). Is simple english, it allows to decompose your velocity field into "canonical" modes. The problem with POD is that, sometimes the basis function that it predicts can be non-local (like Fourier modes for homogeneous turbulence).

Hope this hasn't added to the confusion.

h.jordi November 21, 2001 06:45

Axel Rohde, yes i think that might be what i want to know!!!

i think of u' of a fluctuating velocity in time (as you say), but in the physical sence, lets say in homogeneous turbulence we have a full range of eddies (i.e. a full range of scales).

so in the physical sence (where all we have is a full set of eddies) what is u' (v' w') then representing. a fluctuation in time of velocity, but 'what*' velocity?

*what with respect to the eddy nature.

A. confused. student. November 21, 2001 09:44

I think I udnerstand the question....but I am not sur eit has been answered yet! I expect what is being asked is the following. What does a fluctuating velocity component mean? If you are looking at a fluid what will it look a fluctuating velocity physically the velocity fluctuating at a point or is it a characteristic of the flow. WIth regard to eddies....the question seems to be, what is it about an eddy that can be described in terms of a fluctuating velocity. Any answers, oh wise ones.please!

Axel Rohde November 21, 2001 10:37

I would say that u', v', and w' represent the velocity fluctuations on the smallest scale. However, I believe that these small scale fluctuations also trigger eddies on the medium and large scale of motion. I cannot really comment on the relationship between those, because this is something that evolves over time, and it is a complicated process.

I once did a 2-D viscous computation over a cylinder (Re = 100,000). By introducing small velocity fluctuations into the far field, I was able to break up the two stable vortices and ended up with (periodic) Von Karman type vortices, after marching in time an equivalent length of about 100 cylinder diameters. I thought this was an interesting case, which clearly demonstrated the 'butterfly effect' in fluid mechanics. (The story goes that a butterfly flapping its wings in China may eventually cause a hurricane in the Caribbean).

Adrin Gharakhani November 21, 2001 15:44

The fluctuations are local. Think of it purely in mathematical terms. You have a 3-D vector in the x,y and z coordinates and each component of this vector has a mean value and a corresponding perturbation (in time) in each coordinate direction. These are pointwise values and not "integral". That is, they are used in conjunction with, say, the Navier Stokes equations to decompose the fields into some average and unsteady fluctuating terms. This has nothing to do with the flow dynamics (at this level) or its topology (e.g., whether the flow is eddy-like or not).

To put this in a form that is perhaps easier to visualize, let's say that you have an instrument (such as an anemometer) which has three little signal measring wires at the tip of the measuring device. Let's say that these wires are designed such that they can see signals coming in one direction only but not in directions that are orthogonal to it. Now if you direct each of these little thingies in the x, y and z direction, they will each measure signals from only the x, y or z direction. Note: these little prongs are assumed to be infinitely close to each other (this is for the sake of argument to demonstrate that we are assuming that the x, y, and z signals correspond to a "singular" point and not a "volume") So now you can view/record each of the measured signals in exactly the same way as you described as published in the book in "2-D" form. BTW, the data you are talking about is 2-D with respect to the velocity and TIME not space. This is perhaps where you might be confusing 2-D/3-D differences

Adrin Gharakhani

Jongdae Kim November 23, 2001 15:59

What about triple decomposition of u ?

u(t)= <u> + u' = u_mean + u_fluctuation(t) + u'(t)

<u> : ensemble or phase average of u(t) u_ mean : time average of u(t), constant u_fluctuation : periodic or fluctuating part of u(t) u' : residual or turbulent part of u(t)

This equation seems different from Reynolds decomposition.

Using this triple decomposition, we can define some large/coherent eddy structures and we can get turbulent part, u'(t) also.

There are many ways of ensemble averaging of velocity data. In my case, I use phase averaging. (ref. : Lyn & Rodi's paper, JFM, 1994, vol.267, pp353-376)

Based on triple decomposition technique, u' from Reynolds decompostion corresponds to the signal of (coherent+random) part of flow.

chidu December 1, 2001 16:25

I think Kalyan has answered the relationship between velocity fluctuations at a point and "eddies" in the flow quite well. Maybe it appears a little too technical.

The velocity fluctuations at just one point cannot give information about "eddy" structures in a flow, which is necessarily of a spatial extent. One needs atleast two-point correlation informations and so on.

Like Kalyan says, the definition of an eddy varies according to what mathematical bases you use to characterize the flow. The size and shape of an eddy depends on your mathematical looking-glass!!

regards, chidu...

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