Transient growth analysis of hypersonic flow over an elliptic cone

Abstract

Non-modal linear stability analysis results are presented for hypersonic flow over an elliptic cone with an aspect ratio of two at zero angle of attack, completing earlier modal instability analysis of flow around the same geometry. The theoretical framework to perform transient growth analysis of compressible flows on a generalized two-dimensional frame of reference is developed for the first time and is then applied to solve the initial-value problem governing non-modal linear instability on planes perpendicular to the cone axis, taken at successive streamwise locations along the elliptic cone. Parameter ranges examined here are chosen so as to model flight of the Hypersonic International Flight Research Experimentation 5 (HIFiRE-5) test geometry at altitudes of 21 km and 33 km, corresponding to Mach numbers 7.45 and 8.05 and unit Reynolds numbers $Re' = 1.07\times 10^7$ and $1.89\times 10^6$ , respectively. Results obtained indicate that the significance of the non-modal growth for laminar–turbulent transition increases with increasing flight altitude (decreasing Reynolds number). At a given set of flow parameters, transient growth is stronger in the vicinity of the tip of the cone and in azimuthal locations away from both of the minor (centreline) and major (attachment line) axes of the cone. Linear optimal disturbances calculated at conditions of maximal transient growth are found to peak in the crossflow region of the elliptic cone. These structures are elongated along the streamwise spatial direction, while being periodic along the spanwise direction with periodicity lengths of the same order of magnitude as the well-known structures identified as crossflow vortices in both experiments and simulations.

Transient Dynamics in Counter-Rotating Stratified Taylor–Couette Flow

“…The wavelengths in the homogeneous directions θ and z are L θ = 2π/n and L z = 2π/k, respectively. Thus, we proceed by substituting the above functional forms into the linearized Equations ( 22)- (24), which after further simplification become:…”

“…The non-normal route to turbulence has raised many concerns in the literature from different schools of thought [7,19,20], conversely, Reddy and Henningson [21] objectively addressed the issues with insightful results for two-and three-dimensional plane Couette and Poiseuille flows. The utilization of the transient growth method for investigating subcritical bifurcations has witnessed widespread adoption and significant advancements over the last two decades, with both theoretical and experimental justifications [8,[22][23][24].…”

Response of hypersonic compression corner flow to upstream disturbances

“…The latter can be further divided into convective-and component-type non-normalities (Chomaz 2005). If the base flow is weakly non-parallel, the convective-type non-normality is related to the 'normal-mode' instabilities obtained from local analysis, such as Mack's first and second modes (Mack 1984), while the component-type non-normality may share some similarities with the algebraic instabilities (Ellingsen & Palm 1975) obtained from transient growth analysis (Hanifi, Schmid & Henningson 1996;Tumin & Reshotko 2003;Tempelmann, Hanifi & Henningson 2012;Quintanilha et al 2022). The dynamics of the shock-induced separated flow as a noise amplifier is further complicated by its non-parallelism and has attracted much attention.…”

The Experiments and Stability Analysis of Hypersonic Boundary Layer Transition on a Flat Plate

“…Consequently, the precise determination of the transition location in the hypersonic boundary layer is of paramount importance for the meticulous design of the entire aerospace vehicle, especially concerning the thermal protection system (TPS). Nevertheless, due to the inherently highly nonlinear characteristics, the transition of the hypersonic boundary layer remains a formidable challenge in the realm of fluid mechanics [2,3].…”