Input-output analysis of the stability and free-stream receptivity of hypersonic flows over sharp and blunt cones

Abstract

Understanding the receptivity of hypersonic flows to free-stream disturbances is crucial for predicting laminar to turbulent boundary layer transition. While significant attention has been given to understanding boundary layer instabilities which amplify disturbances in the downstream flow close to the wall, significantly less attention has been given to linking these instabilities to upstream receptivity mechanisms such as shock-perturbation interaction and effects of nose-tip bluntness. Sharp flow features, such as shock waves, or other features created by complex geometry, are notoriously difficult to treat with traditional stability methods. For this reason, state-of-the-art transition prediction methods are semi-empirical, requiring experiments to infer the receptivity for each given geometry of interest. As a global method, input-output (I/O) analysis does not rely upon the assumptions underpinning traditional stability analyses and therefore is able to treat complex flow features in a natural way. I/O analysis is a first-principles-based, and thus predictive, approach linking freestream perturbations to instabilities downstream. In this work, we explore I/O analysis as a tool to understand the receptivity and instability mechanisms on sharp and blunt cones in Mach 5.8 flow. Several theoretical and computational challenges are overcome to enable this study. To more accurately capture the interaction of linear disturbances with strong shock waves, a shock-kinematic boundary condition (SKBC) is developed and implemented in the finite-volume framework, overcoming the inherent conflict between the non-linearity of shock motion in the linearization of the governing equations. Next, a new variant of I/O analysis---hierarchical I/O (H-IO) analysis---is developed. H-IO analysis is employed to compute the three-dimensional receptivity of flows over axisymmetric cones to free-stream planar waves by restricting the allowable input disturbances to be physically realizable. This approach leverages a Fourier decomposition of the azimuthal dynamics and the low-rank truncation of each I/O model in the Fourier space, before re-coupling and re-optimizing the full dynamics. The H-IO approach makes three-dimensional I/O analysis more computationally affordable and permits computations with two orders of magnitude more degrees of freedom than previous analyses. Additionally presented is the Incomplete Block Tri-Diagonal (IBT) preconditioner for use in finite-volume based geometric domain decomposition of the global resolvent operator arising from hypersonic flows. The performance of the two-level version of the preconditioner shows faster solve times and better convergence than similar domain decomposition methods. Application of I/O analysis to Mach 5.8 flows over sharp and blunt cones yielded several new physical insights. Traditional I/O analysis of flow over a blunt cone with a 3.6 mm tip shows the amplification of low-frequency disturbances via the acoustic destabilization of a slow acoustic boundary layer mode. This boundary layer mode---predicted to be stable by linear stability theory---is destabilized by shock confinement, and it is most receptive to disturbances impinging in the oblique region of the shock above the frustum. The analysis also captured the non-modal amplification of high frequency entropy and vorticity waves via a rotation and deceleration mechanism. This mechanism is most receptive to disturbances impinging on the curved portion of the bow shock, just above the nose-cone junction. Receptivity analysis using the new H-IO approach revealed highly three-dimensional receptivity processes. Analysis of Mach 5.8 flow over a sharp cone yielded results consistent with the literature, verifying that Mack's second mode is most receptive to slow acoustic waves at an incidence angle near ten degrees. Mack's first mode is also observed and is most receptive to vorticity waves at very high incidence angles. The addition of nose-tip bluntness stabilizes the Mack modes, but destabilizes the entropy layer instability, such that it becomes the dominant instability mechanism as the nose-tip bluntness increases. The observed entropy layer instability is more broadband than the modal instabilities and is most receptive to vorticity and entropy waves at shallow incidence angles. At high frequency, the entropy layer instability is also found to interact with a fast boundary layer mode, destabilizing it upstream of its synchronization with the continuous spectra. The maximum amplitude achieved by the entropy layer instability increases with increasing nose-tip bluntness, providing a possible mechanism by which transition reversal may occur. These results both confirm and extend I/O analysis as a useful and capable tool for investigating the global linear stability and receptivity of the hypersonic boundary layer, both by addressing key computational challenges faced by high-speed boundary layer analysis, and by providing valuable insight into the varied and complex receptivity mechanisms that influence stability and transition at hypersonic speeds.