Generalized Rayleigh's problem for a compressible flow over the surface of a dihedral corner

Summary

The generalized version of Rayleigh's problem is considered, which is formulated taking into account both compressibility and heat conductivity, while the boundary surface is that of a dihedral corner of arbitrary size β. Like it was done for the case of an infinite plane surface [1], a small parameter is introduced in the form of ɛ=|T _w −T ₀|/T ₀. Under the assumption of M=O(ε ^3/4) for the Mach number, and following the same lines of analysis as in [1], a specific set of governing equations is got as a linearized transformed version of the full Navier-Stokes equations presented in the cylindrical coordinate system.

The Fourier separation method and introduction of a “similarity” argument λ=r ²/(4t) play the most important parts in the solution of the governing equations. An expression for the main (longitudinal) velocity component is obtained directly, in terms of confluent hypergeometric functions. At subsequent stages of solution one succeeds in determination of (a) velocity components for secondary flow, (b) density, (c) temperature, (d) pressure, all the results being once again expressed in terms of confluent hypergeometric functions. To satisfy the boundary conditions at the impenetrable surfaces and at the bisectorial plane, as well as that of a smooth transition to one-dimensional flow in a degenerate case of β=π, one has at his disposal certain constant coefficients in expansions. Thus analytical expressions for each of the unknown variables are obtained within any desirable range of approximation.