This chapter is devoted to a detailed presentation of the linear interaction approximation (LIA) theory mentioned in Chapter 11. The main assumptions that underlie the LIA are discussed in Subsection 11.2.2 and are not duplicated here. We just recall here that the LIA holds if the following constraints are fulfilled:

1. The fluctuations must be weak in the sense that the distorted shock wave must remain well defined. Numerical experiments led Lee and co-workers (Lee, Lele, and Moin (1993) to propose the following empirical criterion for the linear regime:

   
   

   where M _t and M ₁ are the upstream turbulent and mean Mach numbers, respectively, and ? ? 0.1.

2. The time required for turbulent events to cross the shock must be small compared with the turbulence time scale / ? (with and ? the turbulent kinetic energy and the turbulent kinetic-energy dissipation rate, respectively), so that nonlinear mechanisms cannot have significant effects.

12.1 Shock Description and Emitted Fluctuating Field

We consider here the interaction of a plane shock with a normal 2D flow in the ( x, y) plane. Let the undisturbed shock normal vector and the mean flow be oriented along the x axis. The disturbed shock front is defined as

The position of the undisturbed shock is arbitrarily chosen to be x = 0. The local instantaneous normal and tangential vectors, n and t, are equal to

The shock speed in the reference frame associated with the...