Improved nonlinear parabolized stability equations approach for hypersonic boundary layers* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12072232 and 11672351) and the National Key Project of China (Grant No. GJXM92579).

Abstract

The nonlinear parabolized stability equations (NPSEs) approach is widely used to study the evolution of disturbances in hypersonic boundary layers owing to its high computational efficiency. However, divergence of the NPSEs will occur when disturbances imposed at the inlet no longer play a leading role or when the nonlinear effect becomes very strong. Two major improvements are proposed here to deal with the divergence of the NPSEs. First, all disturbances are divided into two types: dominant waves and non-dominant waves. Disturbances imposed at the inlet or playing a leading role are defined as dominant waves, with all others being defined as non-dominant waves. Second, the streamwise wavenumbers of the non-dominant waves are obtained using the phase-locked method, while those of the dominant waves are obtained using an iterative method. Two reference wavenumbers are introduced in the phase-locked method, and methods for calculating them for different numbers of dominant waves are discussed. Direct numerical simulation (DNS) is performed to verify and validate the predictions of the improved NPSEs in a hypersonic boundary layer on an isothermal swept blunt plate. The results from the improved NPSEs approach are in good agreement with those of DNS, whereas the traditional NPSEs approach is subject to divergence, indicating that the improved NPSEs approach exhibits greater robustness.