Direct Numerical Simulation and the Theory of Receptivity in a Hypersonic Boundary Layer

Abstract

Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of half-angle 5:3 degrees was carried out with two-dimensional perturbations introduced into the ∞ow by periodic-in-time blowing-suction through a slot. The free stream Mach number was equal to 8. The perturbation ∞ow fleld downstream from the slot was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples ilustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results. I. Introduction The progress being made in computational ∞uid dynamics provides an opportunity for reliable simulation of such complex phenomena as laminar-turbulent transition. The dynamics of ∞ow transition depends on the instability of small perturbations excited by external sources. Computational results provide complete information about the ∞ow fleld, which would be impossible to measure in real experiments. However, validation of the results might be a challenging problem. Sometimes, numerical simulations of small perturbations in boundary layers are accompanied by comparisons with results obtained within the scope of the linear stability theory. In principle, this is possible in the case of a ∞ow possessing an unstable mode. Far downstream from the actuator, the perturbations might be dominated by the unstable mode, and one may compare the computational results for the velocity and temperature perturbation proflles and their growth rates with the linear stability theory. This analysis does not work when the amplitude of the unstable mode is comparable to that of other modes, or when one needs to evaluate the amplitude of a decaying mode. Recently, a method of normal mode decomposition was developed for two- and three- dimensional perturbations in compressible and incompressible boundary layers. 1{3 The method is based on the expansion of solutions of linearized Navier{Stokes equations for perturbations of prescribed frequency into the normal modes of discrete and continuous spectra. The instability modes belong to the discrete spectrum, whereas the continuous spectrum is associated with vorticity, entropy, and acoustic modes. Because the problem of perturbations within the scope of the linearized Navier{Stokes equations is not self-adjoint, the eigenfunctions representing the normal modes are not orthogonal. Therefore, the eigenfunctions of the adjoint problem are involved in the computation of the normal modes’ weights. Originally, the method based on the expansion into the normal modes was used for analysis of discrete modes (Tollmien{Schlichting{like modes) only. After clariflcation of uncertainties associated with the continuous spectra in Ref. 1, the method was also applied to the analysis of roughness-induced perturbations. 4{6 In order to flnd the amplitude of a normal mode, one needs proflles of the velocity, temperature, and pressure perturbations, together with some of their streamwise derivatives given at only one station downstream from the disturbance source. Because computational results can provide all the necessary information about the perturbation fleld, the application of the multimode decomposition is straightforward. However, the flrst