Three-dimensional Wave Packet in a Hypersonic Boundary Layer

Abstract

A three-dimensional wave packet generated by a local disturbance in a hypersonic boundary layer flow is studied with the aid of the previously solved initialvalue problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of discrete and continuous modes. A specific disturbance consisting of an initial temperature spot is considered, and the receptiv ity to this initial temperature spot is computed for both the twodimensional and three-dimensional cases. Using previous analysis of the discrete and continuous spectrum, we numerically compute the inverse Fourier transform. The two-dimensional inverse Fourier transform is found for Mode S, and the result is compared with the asymptotic approximation of the Fourier integral. Due to the synchronism between Mode F and entropy/vorticity modes, it is necessary to d eform the path of integration around the associated branch cut. Additionally, the inverse Fourier transform for a prescribed spanwise wave number is computed for three-dimensional Mode S. Nomenclature A = vector function (direct problem) j A = j th component of vector A B = vector function (adjoint problem) j B = j th component of vector B