Understanding the Splitting of Velocity Fields in Reynolds Averaging Equations

[K41]

Okay, so the RANS equations use the idea that we can generate a statistical equation of motion from the Navier Stokes equations by using the idea that the velocity field is split into a mean field and a fluctuating field. I've had a great deal of trouble understanding this.

From a purely physical perspective, there is one velocity field and, if turbulent, we are taught it contains these "eddy" like parts to it, which come in various shapes and sizes.

So from the statistics definition, how are these eddies sort of "split up"? Is it the case that the mean velocity field is treated as purely laminar and the fluctuating part contains all these "eddies" or "turbulence" or is it that maybe large eddies are contained in the mean velocity field and the smaller eddies in the fluctuating field? I've heard a few people talk about the "fluctuating component" being the turbulence, but it confuses me because there is only one velocity field...

As another question which has stemmed from this, what therefore, is the significance of the idea that the turbulence "extracts" its energy from the mean flow? We see the equations for the transport of kinetic energy for instance. How should I interpret this when there is, quite literally (physically), only one velocity field, i.e. one instantaneous value of velocity in time and space (assuming N-S is smooth and continuous for all time)?

Sorry if its a weird question, its just been bugging me quite a lot, because a lot of papers I'm reading go off on a merry dance with the statistics, and its very difficult to interpret certain things.

 

[boneh3ad]

So from the statistics definition, how are these eddies sort of "split up"? Is it the case that the mean velocity field is treated as purely laminar and the fluctuating part contains all these "eddies" or "turbulence" or is it that maybe large eddies are contained in the mean velocity field and the smaller eddies in the fluctuating field? I've heard a few people talk about the "fluctuating component" being the turbulence, but it confuses me because there is only one velocity field...

Imagine you have an anemometer in a fluid flow measuring the velocity at a point as the fluid passes. The signal would look something like this:

Now, you can take the mean of this signal to get something like this:

After subtracting the mean (the second image) from the total velocity (the first image), you get something like this:

That first image is the velocity. The second image is the mean component of the velocity. The third image is the fluctuating component of the velocity. So yes, it is all one velocity field, but you can split the time history of a point of that field into different components.

In terms of where the eddies are in a signal like this, it comes down to the frequency components of the signal. The higher-frequency components correspond to small eddies passing by (they are small so they pass by faster and more frequently, thus the higher frequency) and lower-frequency components are large eddies.

The mean profile itself is not laminar if the flow is turbulent. It will be much "fuller" as a result of the effects of the Reynolds stresses in the RANS equations. In other words, you can't just solve a Blasius boundary layer and then superpose fluctuations on it to make it turbulent.

(The above images were acquired in an actual fluid flow with a hot-wire anemometer, though I changed the actual velocity values to be arbitrary. The flow was laminar, however, so it wouldn't show a classical -5/3 roll-off in the spectrum.)

 

[K41]

Are those frequency components calculated from a Fourier transform, because if so, each eddy would contribute to a range of frequencies :s

 

[boneh3ad]

Yes, the spectrum would typically be calculated using FFT. There will certainly be some energy in each eddy at a higher frequency than the one associated with the size of the eddy, but it will be much less than the dominant one.