Self-consistent parabolized stability equation (PSE) method for compressible boundary layer

Abstract

The parabolized stability equation (PSE) method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation (DNS) as applied to a problem of compressible boundary layer with Mach number 6.

Improved nonlinear parabolized stability equations approach for hypersonic boundary layers*

“…The phase-locked phenomenon provides a satisfactory representation of the relationship between the real parts of streamwise wavenumbers, but it does not hold for their imaginary parts, and this is manifested as a lack of self-consistency. [22] Therefore, we adopt the suggestion of Zhang and Su [22] and set the imaginary part of the streamwise wavenumber (indicated by subscript i) equal to zero…”

“…Its streamwise wavenumbers for high harmonics are not self-consistent. [22] It is difficult to extend the NPSEs to three-dimensional (3D) boundary layers where the spanwise wavenumber varies along the marching line. Fortunately, there are ways to overcome these shortcomings.…”

“…Huang and Wu [23] proposed a nonperturbative approach that provided an accurate initial condition for NPSEs by taking nonparallelism into account. Zhang and Su [22] suggested that the streamwise wavenumber be kept real to ensure selfconsistency of the NPSEs. Song et al [24] applied ray tracing theory to predict the real part of the spanwise wavenumber and determined its imaginary part using the conservation relation for the generalized growth rate [25] for 3D boundary layers.…”

Reused LU factorization as a preconditioner for efficient solution of the Parabolized Stability Equations

“…Despite these drawbacks, the PSE method has been extended to compressible and hypersonic flows [20][21][22], curvilinear coordinates [23] and three-dimensional boundary layers [4,24], nonequilibrium thermo-chemical interactions and reacting flows [25][26][27][28], and surface movement [27]. Moreover, several improvements were proposed to improve the accuracy and robustness of the PSE method [4,[29][30][31][32][33][34], and it is frequently applied in boundary layer research (recently with more emphasis on hypersonic flows [35]) along with considerable success in jet noise modeling [36].…”