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.

In many practical applications, one is interested only in the average or expected value of flow quantities, such as aerodynamic forces and heat transfer. Governing equations for these mean flow quantities may be derived by averaging the Navier-Stokes and temperature or scalar transport equations. Reynolds averaging introduces additional unknowns owing to the nonlinearity of the equations, which is known as the closure problem in the turbulence literature. Turbulence models for the unclosed terms in the averaged equations are a way to manage the closure problem, for they close the equations with phenomenological models that relate the unknown terms to the solution variables. It is important that these models do not alter the conservation and invariance properties of the original equations of motion. We take a closer look at the equations of motion to understand these fundamental qualities in more depth. We describe averaging operators for canonical turbulent flows at the core of basic turbulence research and modeling efforts, and discuss homogeneity and stationarity. We also examine the Galilean invariance of the equations of motion and the role of vorticity in turbulence dynamics.