Energy Cascade confusion
 

Dear Colleagues and CFD enthusiasts,

I am in serious confusion regarding the concept of energy cascade. It is simply the energy behaviour where larger eddiies have low frequency (on x axis) and higher energy eddies have larger frequencies. In high school physics, it is taught that the more is the frequency the higher the energy it contains. So this does not conflict with turbulence energy cascade? I mean, where is the missing point? Or in turbulence, its simply the frequency of occurrence and not the physical frequency what we encounter in wave physics. Please guide me in this regard.
I would be grateful to you.
Thanks

Shamoon Jamshed, Ph.D.

"In high school physics, it is taught that the more is the frequency the higher the energy it contains."



This concept does not apply at all in turbulence. The frequency (in time or in space) is associate to the dimension of the energy-containing eddy, in the inertial cascade kinetic energy being just transferred in a almost inviscid mechanism. Only when the vortical structures are at the dimension of the Taylor microscale the effect of the viscosity start to be relevant the the kinetic energy is converted in internal energy in an irreversible way. Thus, the higher frequencies lie in the range where dissipation of energy is the main mechanism.

Energy cascade represents a lot of different concepts. If we want to resort to the Richardson view in a more intuitive and modern but approximate setting, imagine the following.

Your car, that has a characteristic dimension L, is traveling at speed U. Once it is passed, it will leave in the wake vortices of characteristic size L and speed U. At some point (around L/U later) such size L flow structures will likely break and pass their energy to some smaller structures. Such energy passage is basically lossless. So the amount of energy in smaller structures is basically the same, but as there are more smaller structures they carry a fraction of the initial energy. Such a breakup and energy passage goes on and on until the flow structures (let's even call them vortices) are so small that viscosity becomes important and kills all of them below a certain size (for given viscosity and initial U and L).

In reality, for a given time and position, there will be an ensemble of such structures influencing the velocity at that point and time, each one coming from a different history (but also from the same larger structure that previously broke into several smaller ones).

The wavelength in the energy cascade (that's more common than the frequency) typically refers to the spatial dimension of such structures (or, less typically, to their characteristic time). So, from the above arguments, you should see that energy here is not to be understood from the signal point of view (for a given amplitude, a higher frequency signal carries more energy) but as the energy carried by flow structures of a given size, given initial U and L.

That is my point that the frequency is not meant as we understood in h/s physics so its basically the occurrence of the eddies of the flow structure.

Energy cascade confusion
 

there was also a non-traditional curve in Davidson book of turbulence whose replica is in the attached file. Here the lower energy has lower frequency then suddenly rises and then decreases as per the usual energy cascade. Can someone guide me about this curve too?

Higher frequency implying higher energy comes primarily from:
Planck's law:E=hf which is the energy of photons (which are quantized). Higher frequency photons do contain more energy.

The other is: mw^2 for the simple harmonic oscillator.

Both of these are for systems with distinct frequencies. They don't describe how systems with many many frequencies transfer energy to and from one another. Neither of them relate to the spatial scale of the oscillations.

Is this figure in log-log plot, linear-linear plot, or linear-log plot?

You can find "affordable" discussions on turbulent energy spectra in Turbulent Flows by S. Pope and An Introduction to Turbulent Flow by Mathieu and Scott. Honestly, I don't remember the details, but a very rough reasoning on the behavior of the energy at large scales is as follows:

1) First of all, energy is not magically introduced, it comes in at one or more specific scales L (we can assume one without lack of generality) because of a specific forcing at those very specific scales.

2) We already discussed how energy goes from L to the smaller scales

3) Imagine a cube (side L) of Homogeneous Isotropic Turbulence (HIT) suddenly put into a much larger ambient (whose characteristic size is much larger than L). Don't you expect that, in time, that cube will start affecting larger and larger portions of the ambient? That is, energy first present only at scales L or smaller, will also flow to larger ones?

4) I'm not making a disciplined derivation but, in HIT it turns out that: a) analiticity of the spectrum tensor for k->0 requires E(k) proportional to k^4 for k->0 b) nonetheless, there are cases where E(k) proportional to k^2 for k->0 is observed as well

So, roughly speaking, the idea is that the low wavenumber part of the energy spectrum depends from the scale at which you are injecting energy in the flow (L) and by the presence of additional larger scales (with respect to L) in the actual space window in which you are observing the spectrum. If such scales exist, the inverse energy cascade will inevitably take place (actually it continuously happens all along the whole spectrum, not only at the larger scales, but for them you see the net flow of energy as opposite with respect to the rest of the spectrum)

This is only a sketch. The peak in the energy spectrum is associated to the vortical structure at the integral lenght scale (k=pi/l), after some lenght of correlation. Have a look to real spectra in homogeneous isotropic turbulence. But be aware that for confined turbulence the spectra change their shape at different y+ values

As also mentioned by Filippo, a typical scenario is wall turbulence. In this flow the turbulent energy at walls is introduced by bursting/ejection events at very small scales (by definition, also confined by their distance to the wall). Such structures will inevitably affect larger scales (as smaller ones are already close to non-existent and strongly affected by viscosity)

What is your background? Are you reading only Davidson?

It was log-log plot. I am attaching original image from the book of Davidson. Chapter 3

Davidson and Pope

Pope is good.

What you should clearly understand is the nature of the mathematical model governing the turbulence: the quadratic non-linear term in the NSE.
Clearly, when you consider different physical problems, governed by somehow different mathematical equations, the way in which the energy is transfered from frequency to frequency, will change.

The original image from Davidson book chapter 3

Again, that is only a sketch... consider that, differently from what is shown in figure, in a log scale the transfer of energy in the inertial range has a theoretical constant slope (k^-5/3) for 3D homogeneous isotropic turbulence. The exponential decay due to the contribution of the viscosity starts at the Taylor microscale and ends at the Kolmogorov lenght scale, that is l=O(kinematic viscosity/characteristic velocity). The corresponding Kolmogorov frequency is kl=pi/l.

Hi,



The statement.
""In high school physics, it is taught that the more is the frequency the higher the energy it contains."



In addition the concept is that out of two waves having same amplitude, one which has a higher frequency will have larger energy. However, a wave with larger amplitude will have larger energy compared to one with lower amplitude. Thus, a large eddy can have more energy compared to smaller eddies.

Example. a loud sound even though having a low frequency will have more energy compared to a soft sound.


So, energy content of a wave is not just a function of frequency but also amplitude.