Stability Of Flow Research Articles

Experimental study on overtopping failure of concrete face rockfill dam

With the frequent occurrence of natural disasters, the problem of dam failure with low probability and high risk has gradually attracted people's attention. This paper uses flume model tests to systematically analyze the overtopping failure mechanisms of concrete face rockfill dam (CFRD) and identify its failure modes. The tests reveal that the longitudinal erosion of the CFRD breach progress through stages of soil erosion, panel failures, and water flow stabilization. Meanwhile, the cross-section breach process involves the evolution of breach size in rockfill materials, including traceable erosion, lateral broadening, and breach morphology stabilization. The fracture characteristics of the water-blocking panel are primarily evident in the flow-time curve. By analyzing the breach morphology evolution processes in longitudinal and cross sections, the flow-time curve can be subdivided into stages of burst flow formation, breach expansion with flow increase, rapid increase of breach flow discharge due to panel failures, and stabilization of breach flow and size. The primary damage process of the CFRD occurs in a cyclical stage of breach expansion, flow increase, panel failure, and rapid discharge. The rigid face plate and granular body structure contribute to partial dam failure, showing a tendency for gradual expansion of the breach. The longitudinal section illustrates dam failure resulting from panel fracture and rockfill erosion interaction, while downstream slopes exhibit failure due to lateral intrusion of rockfill and cyclic instability. These research results can serve as a reference for constructing a concrete CFRD failure prediction model and conducting disaster risk assessments.

Flow boiling heat transfer enhancement of R134a in additively manufactured minichannels with microengineered surfaces

Additive manufacturing (AM) of metal components has opened new frontiers in heat transfer applications owing to its wide design freedom, which enables previously unexplored geometries with enhanced thermal performance to be developed. Open minichannels, on the other hand, consist of an extra manifold situated above the common flow channels, have been demonstrated to effectively reduce pressure drop and two-phase flow instability by mitigating backflow and maldistribution. In this paper, we synergized open minichannel design methodology with an ultrascalable surface microstructuring method for AM AlSi10Mg to improve flow boiling heat transfer performance of R134a. Experiments were conducted at the refrigerant mass flow rates (ṁ) of 0.005 kg/s to 0.01 kg/s (corresponding to mass fluxes of 73 to 146 kg/m2⋅s), and effective heat fluxes (qeff) of 1.8 kW/m2 to 141 kW/m2, by supplying 2 ℃ subcooled liquid to the minichannel inlet at saturation pressure (Psat) of 7.27 bar. The effects of heat flux, nucleation site density, mass flow rate and location of nucleation sites on the harmonic-average heat transfer coefficient (have) and pressure drop (ΔP) along the flow direction are investigated and compared against a conventionally manufactured (Al6061) counterpart. Our results show that the proposed etching process significantly improves the cooling performance of AM microstrucured minichannels, resulting in 210% higher have than Al6061 with negligible pressure drop penalty. The flow visualization results reveal a sequential order of nucleation site activation with increasing heat flux for microstructured open minichannels, which starts from the top of fins, followed by the corners and other three surfaces within the channels. This also results in the non-negligible effects of mass flow rate on cooling performance for microstructured minichannels, with approximately 10%–66% higher have at the lowest mass flow rate. Besides, the pressure drop penalty resulting from plain AM rough surface is also reduced by up to 13% through the proposed microstructuring process. In all, this work not only successfully identifies the dominant mechanisms in AM microstructured open minichannels, but it also provides useful minichannel design guidelines for high-performance two-phase cooling devices by AM.

The nonlinear analysis of Portevin-Le Chatelier (PLC) effect: An application to medium Mn steel

The instability of plastic flow in medium manganese steel was investigated through uniaxial tensile tests conducted at room temperature across various strain rates. The Portvin-Le Chatelier (PLC) effect was analyzed using statistical methods and a multifractal approach. The findings indicate that the zigzag stress–strain curves associated with instability plastic flow exhibit typical multifractal characteristics. Furthermore, the multifractal analysis quantifies the influence of strain rate and initial phase composition on the instability characteristics of the stress–strain curves. Specifically, it was observed that lower strain rates and austenite content correspond to a chaotic state, whereas higher strain rates and austenite content result in self-organizing critical dynamics. Finally, the underlying physical mechanisms driving these differing modes of behavior are discussed, with a focus on their relationship to strain rate and the initial austenitic phase fraction.

Unified framework for laser-induced transient bubble dynamics within microchannels

Oscillatory flow in confined spaces is central to understanding physiological flows and rational design of synthetic periodic-actuation based micromachines. Using theory and experiments on oscillating flows generated through a laser-induced cavitation bubble, we associate the dynamic bubble size (fluid velocity) and bubble lifetime to the laser energy supplied—a control parameter in experiments. Employing different channel cross-section shapes, sizes and lengths, we demonstrate the characteristic scales for velocity, time and energy to depend solely on the channel geometry. Contrary to the generally assumed absence of instability in low Reynolds number flows (<1000\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<1000$$\\end{document}), we report a momentary flow distortion that originates due to the boundary layer separation near channel walls during flow deceleration. The emergence of distorted laminar states is characterized using two stages. First the conditions for the onset of instabilities is analyzed using the Reynolds number and Womersley number for oscillating flows. Second the growth and the ability of an instability to prevail is analyzed using the convective time scale of the flow. Our findings inform rational design of microsystems leveraging pulsatile flows via cavitation-powered microactuation.

Linstab2D: stability and resolvent analysis of compressible viscous flows in MATLAB

Linstab2D: stability and resolvent analysis of compressible viscous flows in MATLAB

The Casimir-like effect induced by active nematics

We consider an active nematic phase and use hydrodynamical equations of it to model the activity as an internal field. The interaction of this field with the nematic director in a confined geometry is included in the Hamiltonian of the system. Based on this model Hamiltonian and the standard field theoretical approach, the Casimir-like force induced between the boundaries of such a confined film is discussed. The force depends on the geometrical shape and the dynamical character of the constituents of our active phase, as well as the anchoring conditions. For homeotropically aligned rod-like particles which in principle tend to align along a planar flow field, extensile activity enhances the attraction present in a thin nematic film. As the film thickness increases the force reduces. Beyond a critical thickness, a planar flow field instantaneous to a bend distortion sets in. Near but below the threshold of this activity-induced instability, the force crosses zero and repulsively diverges right at the critical threshold of this so-called flow instability. For contractile rods, in the same geometry as above, the structure is stable and the Casimir-like force diminishes by an exponential factor as a function of the film thickness. On the other side for a planar director alignment, rod-like contractile particles can induce opposite shear flows at the boundaries creating a splay distortion for the director between the plates. In this configuration, we obtain the same universal pretransitional behavior for the force as above. Vice versa, for extensile rod-like particles in this geometry, the director fluctuations become massive and the Casimir-like force diminishes again by an exponential factor as the film thickness increases. The effect of the active field on thermal fluctuations of the director and the fluctuation-induced Casimir force per area is derived through a "semi"-dynamical approach as well. However, the results of the calculation due to a mathematical sum over the fluctuating modes do not lead to an approved closed form.

Ecological Security Pattern Construction and Multi-Scenario Risk Early Warning (2020–2035) in the Guangdong–Hong Kong–Macao Greater Bay Area, China

The effectiveness of ecological security patterns (ESPs) in maintaining regional ecological stability and promoting sustainable development is widely recognized. However, limited research has focused on the early warning of risks inherent in ESPs. In this study, the Guangdong–Hong Kong–Macao Greater Bay Area (GHKMGBA) is taken as the study area, and ecological security risk zones are delineated by combining the landscape ecological risk index and habitat quality, and a multi-level ESP is constructed based on the circuit theory. The PLUS model was employed to simulate future built-up land expansion under different scenarios, which were then extracted and overlaid with the multi-level ESP to enable the multi-scenario early warning of ESP risks. The results showed the following: The ESP in the central plains and coastal areas of the GHKMGBA exhibits a high level of ecological security risk, whereas the peripheral forested areas face less threat, which is crucial for regional ecological stability. The ESP, comprising ecological sources, corridors, and pinch points, is crucial for maintaining regional ecological flow stability, with tertiary corridors under significant stress and risk in all scenarios, requiring focused restoration and enhancement efforts. There are significant differences in risk early warning severity within the ESP across various development scenarios. Under the ecological protection scenario, the ESP will have the best early warning situation, effectively protecting ecological land and reducing ecological damage, providing a valuable reference for regional development policies. However, it must not overlook economic development and still needs to further seek a balance between economic growth and ecological protection.

Experimental and Numerical Study of Air Flow Reversal Induced by Fire in an Inclined Mine Working

Effective fire prevention in mine workings and tunnels requires a thorough theoretical analysis of the heat and mass transfer processes within these structures. This involves using established models to calculate non-isothermal air flow dynamics in long tunnels and mine workings. While the ventilation of tunnels has been extensively studied, significant challenges persist regarding mine ventilation systems, particularly due to their complex and branched topology. This study aimed to address these challenges and gaps in mine ventilation. We designed a laboratory bench to simulate an inclined mine working with a heat source (fire) and validated a mathematical model of heat and mass transfer in such settings. Using experimental measurements, we verified the model’s accuracy. It is important to note that our experimental and theoretical analyses focused solely on the thermal effects of a fire, without considering the release of harmful impurities. Using the validated model, we conducted multiparameter simulations to identify the conditions leading to the formation of a thermal slug in an inclined mine working and the subsequent reversal of air flow. The simulation data enabled us to determine the dependency of the critical heat release rate on the aerodynamic parameters of the mine working. Additionally, we evaluated the changes in average air density within a mine working at the critical heat release rate. These findings are crucial for the further development of a network-based method to analyze air flow stability in mine ventilation networks during fires.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Mixed Rossby Gravity Waves at Middepths of the Equatorial Indian Ocean

Abstract This study examines mixed Rossby gravity (MRG) waves at middepths (500–2000 m) of the Indian Ocean, using multiyear velocity time series obtained from current-meter moorings at 77°, 83°, and 93°E along the equator over the period 2000–19. These data are analyzed in combination with a high-resolution wind-forced ocean general circulation model. The spectrum of observed meridional velocity showed elevated energy over a wide range of periods from about 10 to 100 days with the spectral peak at a period of about 30 days. The model was able to simulate the characteristics of the observed spectrum. Further diagnostics determined that the detected variability is generally consistent with theoretical MRG waves in a resting ocean. Statistical analysis and a model sensitivity experiment identified distinct variations at three periods, where meridional velocity has sizable energy. The 14-day variability is wind driven and has a long zonal (∼3300 km) and vertical (∼4200 m) wavelength. The 28-day variability is excited by the dynamical instability of the background flow in the equatorial western Indian Ocean near the surface and propagates to the study area. It is characterized by a shorter zonal (∼1100 km) and vertical (∼2800 m) wavelength compared to 14-day variability. The 43-day variability has a zonal wavelength (∼820 km) comparable to the 28-day variability but does not show the tendency of propagation and is likely generated in situ through nonlinear interactions. These results show that various processes contribute to the excitation of MRG waves at middepths of the Indian Ocean. Significance Statement Mixed Rossby gravity (MRG) waves are a prominent mode of intraseasonal variability in tropical oceans. They are of climatic importance owing to their effects on turbulent mixing in the deep sea and how they affect the large-scale ocean circulation. This study examines MRG waves at the middepths of the equatorial Indian Ocean using a unique set of in situ current-meter measurements and a high-resolution ocean general circulation model. We describe characteristics such as horizontal and vertical wavelengths, frequency spectra, dispersion properties, and energy sources. Results show that wind forcing at the sea surface, dynamical instability of the background flow near the surface in the western Indian Ocean, and nonlinear interactions generate MRG waves at distinct periods and wavelengths.

Modal and non-modal stability for Hagen–Poiseuille flow with non-ideal fluid

Modal and non-modal stability analyses are applied to Hagen–Poiseuille flow with a non-ideal fluid. The non-ideal fluid is defined as a fluid close to its vapor–liquid critical point. In this region, properties of the fluid deviate significantly from the assumptions of the ideal gas model. In this paper, the specific example of CO2 near the critical point is taken as a non-ideal fluid. We studied fluids at supercritical pressure and different wall temperatures so that the centerline temperatures can be lower, equal, and higher than the pseudo-critical temperature. Flow instability is characterized by the Reynolds number, and the product of the Prandtl and Eckert numbers. In modal stability analysis, we observe that there is no unstable mode in Hagen–Poiseuille flow with a non-ideal fluid. Regarding the growth rate, as the axial wavenumber increases, another mode becomes the least stable. The non-modal theory is employed to investigate the optimal response to harmonic external force and transient energy growth. The influence of axial and azimuthal wave numbers, Prandtl and Eckert numbers, and thermodynamic states are also taken into account. In this study, we identify an generalized inflection point in the transcritical base profile, causing the transcritical state to be the most unstable. In non-modal instability, we observe that the optimal response mainly occurs at time invariant axisymmetric disturbance. This suggests that the axisymmetric disturbance could potentially initiate the transition to turbulence.

Numerical Investigation on Hydraulic Performance of Reversible Pump Turbines with Various Wrap Angles in Turbine Mode of Operation

Abstract Reversible Pump Turbines (RPTs) are radial turbines, used in Pumped Storage Hydropower (PHS) plants. They have two modes of operation: generating and pumping. Despite of their uniqueness and importance, RPTs have typical range of operation at both modes and are prone to undesirable instabilities. The shape of the geometry plays a crucial role on reducing these characteristics and even the minor changes to the geometry can have a significant impact on its performance. In this paper, the effect of blade wrap angles on the hydraulic performance of runner was studied in an attempt to mitigate instabilities at turbine mode of operation. For that, two dimensionally similar RPT runners were designed using CFTurbo software by only varying wrap angle. The wrap angle considered for the runners are 150° and 180°. Steady state simulation was done in ANSYS-CFX for both the designs under same boundary conditions. It was found that the hydrodynamic performance of the runner gradually decreases in both cases when the operating condition deviates from normal operating range, causing flow instabilities within the runner. The study also found that varying blade wrap angles had no significant impact on the reduction of instabilities on RPTs, however it is an important parameter that comes into play while designing turbines at different head and flow conditions.

Aerodynamic shape optimization using a physics-informed hot-start method combined with modified metric-based proper orthogonal decomposition

Aerodynamic shape optimization based on computational fluid dynamics still has a huge demand for improvement in the optimization effect and efficiency when optimizing the unstable flow of airfoils. This article presents a physics-informed hot-start method combined with modified metric-based proper orthogonal decomposition (MPOD-ML-Phys). The data-based filtering strategy is a core step in the original metric-based proper orthogonal decomposition method (MPOD), but existing filtering strategies generate a significant amount of additional computational consumption. Therefore, this article applies machine learning methods to data-based filtering strategy in MPOD and establishes a modified MPOD method (MPOD-ML). In addition, during the MPOD-ML process, a lot of hidden physical knowledge that is beneficial for optimization will also be generated. This article combines Bayesian optimization to construct an MPOD-ML-Phys method, which fully utilizes the flow physical knowledge in MPOD-ML. The efficiency and effect of MPOD-ML and MPOD-ML-Phys are validated by two typical cases: inverse and direct design for airfoils. The results indicate that both MPOD-ML and MPOD-ML-Phys methods can effectively improve the overall optimization efficiency. However, the intervention of machine learning models has significantly reduced the robustness of the MPOD-ML method, while the embedding of physical knowledge makes MPOD-ML-Phys more robust. Meanwhile, the optimized airfoil obtained by MPOD-ML-Phys has better drag divergence characteristics, a later flow separation point, and better flow stability.

Thermocapillary instability of a surfactant-laden shear-imposed film flow

Thermocapillary instability of a surfactant-laden shear-imposed film flow

Effects of doors on an open cavity flow at Mach number 6

Open cavities with different door-opening angles are investigated using high-speed schlieren visualization and dynamic pressure measurements in hypersonic flow with a freestream Mach number of 6. With the help of numerical simulations, the shear layer deformation and pressure increase in the cavities due to the impingement of the door-leading-edge shocks are identified via comparison with those in the cavity without doors. As the door-opening angle decreases from 90°, the shear layer above the forepart of the cavity is gradually raised by the high pressure in the cavity. When the door-opening angle decreases to 30°and 15°, the boundary layer on the upstream flat plate of the cavity separates, and separation shock is observed. The doors enhance the instability of the cavity flow and increase the pressure fluctuations in the cavities. A new oscillation pattern, referred to as coupled oscillation, is observed in the cases with separation on the upstream flat plate, in which the separation shock oscillates at the same dominant frequency as the flow inside the cavity. Compared with the cavity without doors, this coupled oscillation causes a lower oscillation frequency and a larger overall sound pressure level. Cross-correlation analyses between pressure signals indicate that the disturbances generated at the trailing edge of the cavity can propagate to the separation on the upstream flat plate and cause coupled oscillation of the separation shock. The fundamental frequencies of the coupled oscillations can be normalized to approximately the same Strouhal number as that of the cavity without doors. These findings support that the oscillation mechanisms of hypersonic cavities without and with doors are primarily dominated by acoustic feedback.

Shear-imposed falling film on a vertical moving plate with disrupted time-reversal

We propose a mathematical model to study the stability and dynamics of a shear-imposed thin film flow on a vertical moving plate, incorporating the influence of odd viscosity. This odd viscosity effect is vital in conventional fluids when there is a disruption in time-reversal symmetry. Our motivation to study the dynamics with odd viscosity arises from recent studies (Kirkinis & Andreev, vol. 878, 2019, pp. 169–189; Chattopadhyay & Ji, vol. 455, 2023, pp. 133883) where the odd viscosity effectively reduces flow instabilities under different scenarios. Utilizing a long wave perturbation method, we derive a nonlinear evolution equation at the liquid–air interface, which is influenced by the motion of the vertical plate, imposed shear, odd viscosity, and inertia. We first perform a linear stability analysis of the model to get firsthand information on various flow parameters. Three distinct conditions for the vertical plate, quiescent, upward-moving, and downward-moving, are considered, accounting the imposed shear and odd viscosity. Additionally, employing the method of multiple scales, we conduct a weakly nonlinear stability analysis for the traveling wave solution of the evolution equation and explore its bifurcation analysis. The bifurcation analysis reveals the existence of subcritical unstable and supercritical stable zones for crucial flow parameters: odd viscosity, imposed shear, and motion of the vertical plate. Both linear and weakly nonlinear stability analyses demonstrate that the destabilizing effect induced by the upward motion of the vertical plate can be alleviated by applying uphill shear, while the destabilizing effect of downhill shear can be mitigated when the vertical plate is in a downward motion. Moreover, we define an eigenvalue problem that mirrors the Orr–Sommerfeld (OS) model for analyzing normal modes and identifying the critical Reynolds number. We investigate the dynamics of surface waves through numerical solutions of the OS eigenvalue problem using the Chebyshev spectral collocation method. We observe the consistent enhancement of stabilization in the presence of odd viscosity. In the low to moderate Reynolds number range, vertical plate motion and odd viscosity show similar behavior in OS analysis, while imposed shear exhibits distinct changes. The Benney-type model does not agree with the OS problem when the Reynolds number is moderate with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity. However, when the Reynolds number is low with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity, the Benney-type model agrees with the OS. Further, numerical simulations of the evolution equation corroborate the results obtained from linear stability, weakly nonlinear stability, and OS analyses. Finally, the Hopf bifurcation analysis of the fixed point reveals that the wave speed is influenced by both the motion of the plate and the imposed shear while it remains independent of odd viscosity.

Thermal stability and slip effects in micropolar nanofluid flow over a shrinking surface: A numerical study via Keller box scheme with block-elimination method

Thermal stability and slip effects in micropolar nanofluid flow over a shrinking surface: A numerical study via Keller box scheme with block-elimination method

Unstable buoyant viscoelastic fluid flow in a vertical porous layer with temperature-dependent viscosity

The stability of thermally driven buoyant flow of a viscoelastic fluid saturating a vertical porous layer with viscosity depending linearly on temperature is investigated numerically. The rheological behavior of the fluid is described through the Oldroyd-B model, leading to a modified Darcy's law of momentum transfer in the porous medium. The study explores the linear stability of the base flow by analyzing the behavior of normal modes of perturbation. Neutral stability curves and the critical Darcy–Rayleigh number are determined for a wide range of viscoelastic and viscosity parameters. Transition curves from stability to instability in the viscoelastic parameters space are also provided for both constant and variable viscosity cases. Additionally, the results for Newtonian, Boger, and Maxwell fluids are delineated as particular cases from this study.

Time-Delay Following Model for Connected and Automated Vehicles Considering Multiple Vehicle Safety Potential Fields

Connected and automated vehicles (CAVs) represent a significant development in the transport industry owing to their intelligent and interconnected features. Potential field theory has been extensively used to model CAV driving behaviour owing to its objectivity, universality, and measurability. However, existing car-following models do not consider the impact of time delays and the influence of information from multiple vehicles ahead and behind. This paper focuses on the driving-safety risks associated with CAVs, aiming to enhance vehicle safety and reliability during travelling. We developed a multi-vehicle car-following model based on safety potential fields (MIDM-SPF), taking into account the characteristics of multi-vehicle connected information and time delays. To enhance the model’s precision, real-world data from urban roads were employed, alongside an improved optimisation algorithm to fine-tune the car-following model. The simulation experiment revealed that MIDM-SPF significantly reduces stop-and-go traffic, thereby improving traffic flow stability in urban areas. Additionally, we validated the stability of our model under varying market penetration rates in large-scale mixed traffic. Our findings indicate that increasing the CAV proportion improves the stability of mixed traffic flows, which has important implications for alleviating traffic congestion and guiding the large-scale implementation of autonomous driving in the future.

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M &amp;gt; 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M &amp;gt; 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M &amp;lt; 2) cases.