9.1.1 Statement of the Problem

A natural question that arises when dealing with compressible turbulent flows is this: How does one characterize the compressibility effects on turbulence? Or, in an equivalent way, what are the differences between the compressible turbulent fluctuations and the incompressible ones? To answer this question, it is first important to remark that in incompressible flows the full solution is contained in the sole velocity field because the pressure is nothing but an enslaved Lagrange multiplier. In compressible turbulence, this is no longer true because pressure is now an autonomous variable and at least one additional physical variable is required for describing the solution. ^[*] The basic governing equations for such flows are

where R, p, ?, T, u, s, , and ? denote the perfect gas constant, pressure, density, temperature, velocity, entropy, coefficients of viscosity, and heat conduction, respectively. The additional variables m, f, and Q are related to the rate of mass injection per unit volume, the body force per unit mass, and the rate of heat addition per unit volume, respectively. Both the viscosity and the heat conduction are assumed to be monotonic functions of the temperature, i.e., = ( T) and ? = ?( T). The system is supplemented by the perfect gas law

and the definition of the entropy

where s _r, p _r, and ?