Aiaa 2005-0099 Three-dimensional Wave Packet in a Hypersonic Boundary Layer

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 receptivity 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 deform 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 c = phase speed, mode weight M = Mach number Pr = Prandtl number ___________________________________________ *Graduate Student, Program in Applied Mathematics. †Associate Professor, Department of Aerospace and Mechanical Engineering. Senior Member AIAA. Copyright © 2005 by the authors. Published by the American Inst itute of Aeronautics and Astronautics, Inc., with permission. Re = Reynolds number u = streamwise velocity disturbance v = normal velocity disturbance w = spanwise velocity disturbance x = streamwise coordinate 0 Y = location of temperature spot y = normal coordinate z = spanwise coordinate α = streamwise wave number β = spanwise wave number γ = specific heat ratio θ = temperature disturbance μ = viscosity disturbance π = pressure disturbance ρ = density disturbance ω = frequency Subscripts e = upper boundary layer edge p = Laplace transform sp = saddle point α = streamwise Fourier transform β = spanwise Fourier transform Superscripts T = transposed * = dimensional Introduction The transition process from la minar to turbulent flow in hypersonic boundary layers has been studied for many years. However, our understanding of this phenomenon is still very poor compared to the low speed case. Several reasons exist for this difference. For example, experimental conditions are severe in hypersonic wind tunnels. Because of high levels of