In this chapter, we will focus on the statistical spectral dynamics which are paramount to understanding the development of the integrated mixing quantities described in Chapter 5. Reynolds flow averaging and the turbulent kinetic energy are introduced. In addition, I will discuss how the energy of the flows is transferred from large scale to small scale modes, as well as the impact of the shockwave and gravity on the isotropy of the flows. The flow spectra allow several important length scales to be defined. Numeric simulations and experimental data will be offered to provide insights on the mixing processes.

Building on the governing equations and spectral tools introduced in earlier chapters, we analyze the energy cascade, which describes the transfer of turbulent kinetic energy from large to small eddies. This includes an estimate of the energy dissipation rate, as well as the characteristic length and time scales of the smallest-scale motions. Nonlinearity in the Navier-Stokes equations is responsible for triadic interactions between wavenumber triangles that drive energy transfer between scales. Empirical observations suggest that the net transfer of energy occurs from large to small scales. In systems where the large scales are sufficiently separated from the small scales, an inertial subrange emerges in an intermediate range of scales where the dynamics are scale invariant. Kolmogorov’s similarity hypotheses and the ensuing expressions for the inertial-subrange energy spectrum and viscous scales are introduced. The Kolmogorov spectrum for the inertial subrange, which corresponds to a -5/3 power law, is a celebrated result in turbulence theory. We further discuss key characteristic turbulence scales including the Taylor microscale and Batchelor scale.