Rapid growth of unsteady finite-amplitude perturbations in a supersonic boundary-layer flow

Abstract

Spatial numerical simulations (SNS) are performed to study transition to turbulence in a supersonic flat plate boundary layer forced by finite amplitude perturbations, where the freestream Mach numbers is 2.5 with the isothermal wall condition. Three-dimensional isotropic disturbances are superimposed on the laminar profile for Reynolds number based on the displacement thickness of 1000 at the inlet plane of the computational domain. The effects of the peak location of the disturbance spectrum and the magnitude distribution in the wall normal direction are analyzed. Numerical results indicate that energy spectra with a peak located at around the wavenumber of kmax=1.0 play an important role to induce transition to turbulence. The disturbance at lower wavenumbers with concomitant turbulence in the free stream shows an early appearance of hairpin-like structure in the transition region. The vortical structures on set of transition to turbulence rapid evolving to a hairpin packet with fine secondary structures downstream are shown in this paper, which is an inherent structure induces transition to turbulence in the supersonic boundary layer. INTRODUCTION A deep understanding of the principal route to turbulence in the wall-bounded shear flows is of great fundamental and practical interest. Various disturbances affect transition to turbulence in a laminar boundary layer, which has been investigated in detail so far. The primary stage of transition in a low-turbulence environment has been extensively studied by stability theories (Mack 1975) and flow fluctuation measurements by Kendall (1975) or Graziousi & Brown (2002). Understanding of the late stage of transition scenario or the nonlinear transition (Graziousi & Brown 2002) due to high level disturbances for boundary layers is even less pronounced especially for supersonic one due to experimental difficulties. In this study, a spatial simulation of a supersonic, isothermal flat plate boundary layer flow at Mach 2.5 is analyzed. Growth of finite amplitude disturbances on transition to turbulence in a developing boundary layer downstream require the application of fully spatial formation in a numerical simulation without extended temporal simplifying assumptions. The emphasis of this study is to assess the transitional scenarios for supersonic boundary layer and the late stage of streak breakdown and observing the transitional structures in a Lagrangian tracking manner, which leads to finding an inherent structure generated from finite-amplitude perturbations which induces transition to turbulence in the supersonic boundary layer. SMULATION DETAILS In the SNS of spatially developing boundary layer, the non-dimensional equations governing the conservation of mass, momentum, and energy for a compressible Newtonian fluid are solved. Note that displacement thickness at the inlet boundary layer (δ) is chosen to be a representative length scale. The computational domain size is Lx×Ly×Lz= 300×30×33. The form of the solenoidal disturbance is given as