Asymptotic Analysis of Wave Packets in High-Speed Boundary Layers

Abstract

Asymptotic methods are utilized as computationally inexpensive tools to analyze the linear evolution of three-dimensional and two-dimensional wave packets in compressible boundary layers. Parallel and quasi-parallel spatial Linear Stability Theory (LST) are employed to formulate integrals that describe wave packet development. The Steepest Descent method and a related but simplified so-called ‘Gaussian Model’ are applied to asymptotically evaluate the integral with respect to frequency for the case of two-dimensional wave packets and the double integral with respect to frequency and spanwise wavenumber for the case of three- dimensional wave packets. Focus is placed on supersonic and hypersonic flat plate boundary layers, where wave packets are dominated by first mode and Mack’s second mode instabilities, respectively. In the case of wave packets dominated by Mack’s second mode, limitations of the ‘Gaussian Model’ become pronounced. Conversely, and as expected, the Steepest Descent method offers higher accuracy, but unexpected challenges emerge for quasi-parallel boundary layers due to the presence of synchronization between discrete modes—a phenomenon unique to Mack’s second mode instability. Discussions of these challenges along with efforts to address them are presented. Since discrete mode synchronization is absent in first mode instabilities, both Steepest Descent and the ‘Gaussian Model’ can be successfully applied for wave packets dominated by first mode instabilities. Finally, for the case of second mode dominated wave packets, asymptotic predictions for wave packets are compared with results from Linearized Navier-Stokes computations. Good agreement is found for the dominant frequency and time of wave packet arrival, as a function of downstream distance, whereas wave packet amplitudes are not predicted as accurately.