Energy cascade in turbulence

Question

I have studied that energy is transferred from larger eddy to smaller eddy and so on until that energy is converted into heat via viscous dissipation. 1. What i am not able to understand is why is there even an energy transfer? Why energy transfer takes place? 2. Turbulent flows have a family of eddies of different length scales. If this energy cascade does not occur, what would happen?

Answer 1

Viscous dissipation is the manifestation of collisions between molecules of the gas that have different coherent velocities. To picture what is happening, imagine you have two streams of molecules on top of each other -- the bottom stream is moving slower than the faster stream. Due to random thermal motion of the molecules, eventually a slower molecule from the bottom stream will move into the faster stream above it. Likewise, a faster molecule from the top stream will move into the slower stream below it. Those molecules will then collide with the other molecules in that stream and they will transfer momentum. So the slower particle in the fast stream will get a little faster than it was while slowing down the molecules it collided with in the faster stream, and the inverse will be true in the slower stream.

Okay, so why does that matter? What it is telling us is that viscosity operates on the molecular scale. In terms of a spectrum, this means viscosity operates at the largest wavenumbers.

But then in a turbulent flow, the turbulent energy is deposited at the large scales*. We know this by looking at a lot of different flows -- for example, imagine flow over a circular cylinder. There are vortices shed that are comparable in size to the cylinder diameter. This is a very large scale. But we also know that, at some point, that vortex dissipates due to viscosity -- which happens at the smallest scales.

So, there has to be something that connects energy at the large scale to the viscous dissipation at the small scale. And we also know that if we continuously pump energy into the large scale, say by a large fan running at constant speed, the energy in the flow eventually stabilizes. That means there is something transferring energy from the large scales to the molecular scales.

If there wasn't something doing that transfer, then continuously adding energy at the large scale would mean the energy in the flow is always increasing. That clearly can't happen. Or, it would mean viscosity has to operate on large scales, but that goes against how viscosity manifests due to molecular motion. So, the cascade is really the only way to connect energy across the wide range of scales between energy deposition and energy removal.

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* Energy is deposited at the large scale for traditional flows studied in turbulence. But, for many real flows, that isn't always true. Flow over rough walls for example, energy is created at the scale of the surface roughness, which may not be large. Or in combustion, energy is deposited at the flame scale, which is quite small. These may still lead to the formation of large scale turbulence due to a process called backscatter, or also known as the inverse cascade. This is an active area of research but it is beyond what Kolmogorov was considering at the time.