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In the discussion of incompressible laminar boundary layers in Chapter 4, it was assumed that the typical temperature difference in gas flow was a small fraction of the absolute temperature so that fluid properties such as density and dynamic viscosity were assumed to be constant. When this is not the case, it is necessary to solve the momentum and energy equations simultaneously together with the continuity equation.

As in Chapter 6, we begin with the statement that the main difference between laminar flows and turbulent flows is that the effective diffusivities in turbulent flow are unknown. In Chapter 6 the temperature differences were small enough not to affect the mean velocity field, and it was assumed without explicit comment that the fluctuating velocity field, which controls the turbulent transport of momentum, heat, or mass, was also unaffected.

Equations (10.5.5) to (10.5.12) represent three-dimensional laminar and turbulent compressible boundary layers in a body-oriented coordinate system. Their solution requires boundary and initial conditions together with a turbulence model for the Reynolds stresses and can be obtained either in standard or inverse modes as with the two-dimensional incompressible flows discussed in Chapter 9. For the standard problem, that is for a given external velocity distributions of u _e( x, z ) and w _e( x, z ), the procedure is similar to that described for three-dimensional incompressible flows in Chapter 7; the main difference relates to the energy equation and whether it is solved together with the continuity and momentum equations or independently from them. In the latter case, the continuity and momentum equations are solved and then the energy equation. The solutions of the velocity and temperature fields are iterated until convergence.

The approach and many of the ideas presented for stability and transition in Chapters 5 and 8 for incompressible flows remain the same for compressible flows. Again the linear and parabolized stability equations can be obtained from the Navier-Stokes equations with procedures similar to those for incompressible flows, subsection 14.2.1, and the inviscid stability equations from the linear stability equations by neglecting viscous effects, subsection 14.2.2. The resulting equations reveal the differences between incompressible and compressible forms and contribute to understanding of compressible stability theory. For example, inviscid stability increases with Mach number and the mean flow relative to the disturbance phase velocity can be supersonic as discussed in subsection 14.4.2.