A High-Order Finite-Difference Method for Linear Stability Analysis and Bi-orthogonal Decomposition of Hypersonic Boundary Layer Flow

Abstract

In the stability analysis of hypersonic flow, modal analysis theories such as the linear stability theory (LST) and bi-orthogonal decomposition play an important role in the instability characteristic analysis including receptivity and linear growth. The essence of this modal analysis lies in the mode decomposition into discrete modes S and F as well as the continuous spectrum, including acoustic, entropy and vorticity modes. Currently, numerical methods used in such analysis involving linear stability theory and multimode decomposition are shooting method or finite difference and spectral collocation methods. The linear stability theory approach used by Malik (1990) deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for LST. However, Malik's method is on the computation of discrete modes only. In order to obtain the continuous spectrum for his multimode decomposition frame work, Tumin (2007) relies on a shooting method based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the Gram-Schmidt orthonormalization is required at every step of the integration in order to minimize the accumulation of numerical errors. To overcome the drawbacks of the two approaches, this paper introduces a general very high-order finite difference method for obtaining the discrete and continuous modes eigenfunctions based on a non-uniform grid method proposed by Zhong and Tatineni (2003). Under the finite difference framework, both discrete and continuous modes can be obtained by imposing proper freestream boundary conditions, including a far field extrapolation boundary condition and the asymptotic boundary condition, which is based on the freestream fundamental solutions. The far field extrapolation boundary condition is efficient and able to solve to both discrete and continuous acoustic modes which have distinct eigenvalues. In addition, the asymptotic boundary condition is used for obtaining continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. The finite difference method can be applied in various stability analysis for hypersonic flow such as LST, $e^N$ method, and bi-orthogonal decomposition of DNS result in receptivity computation. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes as well as the bi-orthogonal decomposition with a supersonic boundary layer flow over a flat plate by comparing with the results from Malik, Tumin, and Miselis (2016). Subsequently, the new method has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone studied by He and Zhong (2022) and modal contributions are computed by the new method as coefficients to be used in the receptivity analysis.