Convective Instabilities in Hypersonic Compression Ramp Flows

Several transition scenarios are present in a hypersonic compression-ramp flow. In our previous work (Cao et al., J. Fluid Mech., vol. 941, 2022, p. A8), a complete transition process induced by the global instability of a compression-ramp flow was revealed. In a globally stable flow, however, the transition to turbulence can be promoted by convective instabilities, which is the focus of this work. The same flow conditions as in our previous work (Mach number 7.7, Reynolds number $8.6\times 10^5$ based on the flat-plate length) are considered here. Owing to a smaller ramp angle, a weakly separated flow forms on the compression ramp, which supports no global instability. Resolvent analysis identifies low-frequency streamwise streaks as the optimal response of base flow to upstream forcing. Local stability analysis reveals Mack's second mode in the boundary layer downstream of reattachment. By introducing random disturbances upstream of separation in direct numerical simulations, we observe breakdown to turbulence downstream of reattachment. Two transition scenarios are revealed, and they are highly dependent on the amplitude of upstream disturbances. For a large amplitude, strong streamwise streaks develop near the reattachment region, which break down to turbulence quickly. However, when the disturbance amplitude is reduced, the second-mode instability dominates the transition to turbulence.

Experiments of shock wave/boundary layer interactions over a nominally two-dimensional compression ramp are conducted in a Mach 4 Ludwieg tube tunnel. Measurements of Schlieren, Rayleigh scattering, and surface pressure are performed to present the relevant flow features. The effects of two parameters, namely the Reynolds number based on the length of the flat plate and the ramp angle, on the flow stabilities are focused on. Four ramp angles of 6°, 8°, 10°, and 12° are tested under a Reynolds number of 7.22 × 105, while two other Reynolds numbers (3.66 × 105 and 9.19 × 105) are investigated with a ramp angle of 10°. Streamwise streaks are observed downstream of the reattachment point. The spanwise wavelength of the streaks remains unchanged with different ramp angles, whereas it slightly decreases as the Reynolds number increases. Power spectral density results show that the flow is transitional in the streak region and becomes turbulent where streaks break down. When increasing the ramp angle or the Reynolds number, the streamwise length of streaks shrinks. Two different patterns are distinguished at the breakdown, resembling the two unstable modes observed in the breakdown of Görtler vortices. To clarify the underlying physics of the formation of streaks, global stability analysis and resolvent analysis are carried out. Two regions of maximum optimal gain are identified, which are associated with Mack's first mode and streaks. The former can serve as an initial seed of Görtler instability via nonlinear interaction, while the latter can be associated with transient growth due to the lift-up mechanism and Görtler instability.

Hypersonic flow over a two-dimensional compression corner with a Mach number of 7.7 and unit Reynolds number of 4.2 × 106m−1is numerically investigated. Special emphasis is given to the onset of global instability with respect to three-dimensional perturbations. Global stability analysis is performed for various ramp angles and wall temperatures. It is found that the shock-induced separated flow system becomes unstable when the ramp angle is beyond a certain value. The critical ramp angle increases slightly with the wall temperature, although the length of the separation region is significantly enlarged. The global instability is shown to be closely linked with the occurrence of secondary separation beneath the primary separation bubble. A criterion is established based on a scaled ramp angle defined in the triple-deck theory to predict the global stability boundary, which depends on the free-stream conditions and geometries only.

Abstract

Hypersonic compression corner flow with crossflow at the leading edge of the flat plate is investigated through global stability analysis (GSA) and direct numerical simulation (DNS). Base flow conditions with a Mach number of 7.7 and a unit Reynolds number of 4.2×106 m−1 are selected. The effects of varying sweep angles (0–50 deg) and ramp angles (12–15 deg) are examined. The results reveal multiple local maxima in temporal amplification rate with different spanwise wavenumbers signifying oscillatory modes in the presence of crossflow, one of which corresponds to the leading stationary mode observed in unswept flows. The introduction of crossflow induces streamwise periodicity in the reattached boundary layer. The critical ramp angle of 14 deg is further discussed, where the flow becomes globally stable at moderate sweep angles and globally unstable at higher sweep angles, with changes in the leading global mode under varying crossflow intensities. DNS of a globally unstable case demonstrates good agreement with the GSA’s most unstable mode during the linear growth stage. In the saturated stage, harmonics of the most unstable global mode are observed, followed by the emergence of broadband unsteadiness.

Baseflow computation and stability analysis were conducted for hypersonic flow over a cone-cylinder-flare (CCF) geometry for conditions that correspond to the experimental runs carried out in three wind tunnels. Owing to the presence of an attached boundary layer, a separation bubble induced by a shock–boundary-layer interaction, and a reattachment region, the chosen flow configuration is physically rich. The complexity of this flowfield encompasses a combination of convective instabilities developing on the cone, global instabilities in the separation bubble, and shear-layer modes and streaks in the reattachment region. Thus, the selected CCF configuration provides the opportunity for a comprehensive comparison of the currently available methodologies for analyzing boundary-layer instabilities. Various tools are used for the analysis, including global stability codes as well as convective instability analyses based on a local theory, a weakly nonparallel analysis, and tools that are applicable to strongly nonparallel flows. The paper presents a comparison of the convective instability characteristics based on different methodologies, such as linear stability theory, the harmonic form of linearized Navier–Stokes equations, and resolvent analysis. The CCF configuration provided an effective framework for conducting a detailed cross-validation of this type, which had not yet been addressed in existing literature. This document is accompanied by a companion paper that is focused on the experimental aspects of the CCF configuration. Both papers highlight the research activities of the NATO STO Research Task Group AVT-346.

In this work, a transition process in a hypersonic flow over a cold-wall compression ramp is studied using direct numerical simulation (DNS) and global stability analysis (GSA). The free-stream Mach number and the Reynolds number based on the flat-plate length are 7.7 and $8.6 \times 10^5$ , respectively. The shock-induced pressure rise causes the boundary layer to separate on the flat plate, forming a separation bubble around the corner. Without introducing any external disturbances, the DNS captures the transition to turbulence downstream of flow reattachment. The DNS results agree well with the experimental data as well as theoretical predictions. To uncover the intrinsic instability in the flow system, GSA is employed to investigate the three-dimensionality of the two-dimensional base flow. Several stationary and oscillatory unstable modes are revealed, which result in spanwise periodicity inside and downstream of the separation bubble. The GSA and DNS results indicate that the intrinsic instability of the flow system triggers the formation of streamwise counter-rotating vortices and boundary-layer streaks near reattachment. The downstream transition to turbulence starts from the breakdown of the streamwise vortices and streaks. Moreover, the second harmonic of the most unstable global mode and a broadband low-frequency unsteadiness occur in the saturated flow, which has a significant influence on the transition process. In summary, the present study demonstrates a transition process in a hypersonic compression-ramp flow as a result of the intrinsic instability of the flow system.

Background-oriented schlieren and infrared thermography measurements were made simultaneously on a ROTEX-T cone/flare model in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel at freestream unit Reynolds numbers from 5.8 to 12.1 M/m and nominally zero angle of attack. The surface heat-flux and Stanton number distributions were computed. Separation and reattachment locations were determined from Stanton number profiles and density fields. For a laminar boundary layer, the surface and off-wall determinations of separation and reattachment correspond well. For a transitional boundary layer, the off-wall measurements clarify otherwise ambiguous separation assessments from surface measurements. The convective and global boundary-layer instabilities of the axisymmetric laminar flow at the experimental conditions were investigated computationally. Amplification of Mack’s first and second modes with a logarithmic amplification factor of 5 to 7.5 are observed at the separation location, depending on the conditions. The flow is found to be globally unstable to stationary three-dimensional disturbances concentrated in the reattachment region. Wavelengths of the experimentally observed streamwise streaks on the flare were quantified and agreed well with predictions by global stability analysis.

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Experimental studies which focus on flow visualization and the velocity field of a supersonic laminar/turbulent flow over a compression ramp were carried out in a Mach 3.0 wind tunnel. Fine flow structures and velocity field structures were obtained via NPLS (nanoparticle-tracer planar laser scattering) and PIV (particle image velocimetry) techniques, time-averaged flow structures were researched, and spatiotemporal evolutions of transient flow structures were analyzed. The flow visualization results indicated that when the ramp angles were 25°, a typical separation occurred in the laminar flow, some typical flow structures such as shock induced by the boundary layer, separation shock, reversed flow and reattachment shock were visible clearly. While a certain extent separation occurred in turbulent flow, the separation region was much smaller. When the ramp angles were 28°, laminar flow separated further, and the separation region expanded evidently, flow structures in the separation region were complex. While a typical separation occurred in turbulent flow, reversed flow structures were significant, flow structures in the separation region were relatively simple. The experimental results of velocity field were corresponding to flow visualization, and the velocity field structures of both compression ramp flows agreed with the flow structures well. There were three layered structures in the U component velocity, and the V component velocity appeared like an oblique “v”. Some differences between these two compression ramp flows can be observed in the velocity profiles of the shear layer and the shearing intensity.

Based on the Helmholtz–Rayleigh minimal dissipation theorem, a theoretical model is proposed to predict both the plateau and peak of pressure in a compression ramp flow with large separation (CRFLS). Since the total dissipation of CRFLS is mainly contributed by the shock waves, the steady flow pattern can be determined by minimizing the shock dissipation among all the possible configurations. The predictions agree well with both experimental data and numerical simulations, covering a wide range of free-stream Mach number and ramp angle. This method could be applied to other flow systems where the dissipation is dominated by shock waves.

Deep neural networks incorporating an AutoEncoder architecture are applied to compression ramp flow with shock-wave/boundary-layer interaction. This study aims to demonstrate how the fusion of aerodynamic data distributed over the compression ramp surface enables global stability predictions of compression ramp flow using high-fidelity data from small datasets, thereby significantly reducing data acquisition costs. The deep learning model is trained on direct numerical simulations of supersonic to hypersonic compression ramp flows, with global stability assessed using global stability analysis. The predictions agree well with experimental data and numerical simulations across a wide range of freestream Mach numbers, Reynolds number, far-field flow temperature, ramp angle of the geometry, and wall temperature ratio. Furthermore, by leveraging feature extraction techniques to train the model on a limited set of critical data points, the results remain highly accurate. This highlights an effective approach for optimizing sensor quantity and placement to evaluate the global stability of flows.

In this work, the stability of hypersonic flow over a curved compression ramp is studied using several stability analysis tools and direct numerical simulations (DNS). The free-stream Mach number and the unit Reynolds number are 7.7 and $4.2 \times 10^6$ m $^{-1}$ , respectively. Corner rounding is considered to alter the separation bubble flow so as to suppress the intrinsic instability of the compression-ramp flow. The variation of intrinsic instability is confirmed by global stability analysis. Subsequently, resolvent analysis is employed to examine the response of intrinsically stable flows to external disturbances. It is shown that the considered flows strongly amplify low-frequency streamwise streaks with a preferential spanwise wavelength. This result is verified using DNS by introducing a random forcing upstream of the separation point. Furthermore, both resolvent analysis and DNS demonstrate that the separation bubble contributes little to the selection of the spanwise wavelength of streamwise streaks. The combined effects of convective and intrinsic instabilities are also explored using DNS. A better agreement with experimental data is achieved after introducing upstream disturbances in an inherently unstable flow.

A numerical analysis of boundary-layer stability over the cone–cylinder–flare geometry is conducted at Mach 6 and a Reynolds number of Re=4.62×106 m−1 in order to investigate the role of wall-to-recovery temperature ratio similarity in hypersonic boundary-layer transition. The study compares typical flight conditions with wind-tunnel experiments to assess differences caused by the missing wall-temperature similarity parameter. A second wind-tunnel case with a cooled wall is studied as a potential solution to attain Mach, Reynolds, and wall-to recovery temperature similarity with flight. Global stability and resolvent analysis are employed to characterize the dominant global and convective instabilities. The results indicate that the global bubble modes are strongly destabilized by the wall cooling. Regarding the convective modes, Mach and Reynolds number similarities alone are insufficient to replicate the linear transition mechanisms observed during hypersonic flight. In wind-tunnel conditions, both the first and second modes contribute to transition, whereas mainly the second mode is amplified in flight conditions. The third case incorporating wall-to-recovery temperature ratio similarity demonstrates that the inclusion of this parameter allows for the successful reproduction of the linear mechanisms at play in the transition process in flight conditions, even at low stagnation temperatures. These findings emphasize the critical role of wall-to-recovery temperature ratio effects in hypersonic boundary-layer transition studies and the challenges associated with replicating flight conditions in ground-test campaigns.

The linear global instability and resulting transition to turbulence induced by an isolated cylindrical roughness element of height $h$ and diameter $d$ immersed within an incompressible boundary layer flow along a flat plate is investigated using the joint application of direct numerical simulations and fully three-dimensional global stability analyses. For the range of parameters investigated, base flow computations show that the roughness element induces a wake composed of a central low-speed region surrounded by a three-dimensional shear layer and a pair of low- and high-speed streaks on each of its sides. Results from the global stability analyses highlight the unstable nature of the central low-speed region and its crucial importance in the laminar–turbulent transition process. It is able to sustain two different global instabilities: a sinuous and a varicose one. Each of these globally unstable modes is related to a different physical mechanism. While the varicose mode has its root in the instability of the whole three-dimensional shear layer surrounding the central low-speed region, the sinuous instability turns out to be similar to the von Kármán instability in the two-dimensional cylinder wake and has its root in the lateral shear layers of the separated zone. The aspect ratio of the roughness element plays a key role on the selection of the dominant instability: whereas the flow over thin cylindrical roughness elements transitions due to a sinuous instability of the near-wake region, for larger roughness elements the varicose instability of the central low-speed region turns out to be the dominant one. Direct numerical simulations of the flow past an aspect ratio ${\it\eta}=1$ (with ${\it\eta}=d/h$) roughness element sustaining only the sinuous instability have revealed that the bifurcation occurring in this particular case is supercritical. Finally, comparison of the transition thresholds predicted by global linear stability analyses with the von Doenhoff–Braslow transition diagram provides qualitatively good agreement.