Unsteady compressible laminar boundary-layer flow within an expansion wave

Numerical solutions are presented for the unsteady compressible laminar boundary-layer flow developing within a centered expansion wave for two wall thermal conditions, namely, isothermal and adiabatic. The solutions are obtained using Howarth's transformation, a similarity transformation via one-parameter groups, and a series expansion. The expansion in terms of a dimensionless similarity variable is carried up to second-order terms and it improves Hall's zero-order solution for the temperature distribution in the boundary layer. It is shown that both velocity and temperature boundary layers grow rapidly behind the expansion wavefront but attenuate further down stream. If the wall temperature is held at the undisturbed temperature, the heat transfer occurs from the wall to the expanding gas. The rate of heat transfer increases from the wavefront to approximately x =0.35 behind the wave where maximum heat transfer is observed, x being a dimensionless distance from the expansion wavefront. The isothermal wall temperature is found to be higher than the adiabatic wall temperature by a factor of (1 + 0.335x).