The energy cascade

The foregoing development suggests that spectra are decompositions of a measured function into waves of different periods or wavelengths. Spectra provides us a means of defining transfer of energy among different waves. One could think of eddies we have been frequently talking about as being associated with waves. Each eddy could be represented by a definite wave with an associated energy. Thus we have now at our disposal a tool that can describe the energy transfer among different eddies.

Earlier we suggested that vortex stretching is the mechanism for the process of energy exchange among eddies. Ignoring viscosity, we can write the vorticity equation as

(23.1)

This equation connects the vorticity field with the strain rate field

We restrict ourselves to a 2D strain field shown in Fig. 23.1. For this case s₁₁= −s₂₂=s , and s₁₂=0. We assume s to be constant, for all t >0 , and = = at t=0 . Eqn. (23.1) then simplifies to

(23.2)

Integration of (23.2) leads to

(23.3)

or,

(23.3)

Except at small st , the total amount of vorticity increases with st . The vorticity component in the direction of stretching ( ) increases rapidly. However decreases slowly in the direction of compression (shrinking) at large st . (see Fig. 23.1).

Figure 23.1: Vorticity stretching in a strain-rate field: (a) before stretching, (b) after stretching (after Tennekes and Lumley [1]).