Freestream Unit Reynolds Number Research Articles

Boundary-layer instability on a highly swept fin on a cone at Mach 6

The growth and characteristics of linear, oblique instabilities on a highly swept fin on a straight cone in Mach 6 flow are examined. Large streamwise pressure gradients cause doubly inflected cross-flow profiles and reversed flow near the wall, which necessitates using the harmonic linearized Navier–Stokes equations. The cross-flow instability is responsible for the most-amplified disturbances, however, not all disturbances show typical cross-flow characteristics. Distinct differences in perturbation structure are shown between small ( $\sim$ 3–5 mm) and large ( $\sim$ 10 mm) wavelength disturbances at the unit Reynolds number $Re' = 11 \times 10^6$ m $^{-1}$ . As a result, amplification measurements based solely on wall quantities bias a most-amplified disturbance assessment towards larger wavelengths and lower frequencies than would otherwise be determined by an off-wall total-energy approach. A spatial-amplification energy-budget analysis demonstrates (i) that wall-normal Reynolds-flux terms dictate the local growth rate, despite other terms having a locally larger magnitude and (ii) that the Reynolds-stress terms are responsible for large-wavelength disturbances propagating closer to the wall compared with small-wavelength disturbances. Additionally, the effect of free-stream unit Reynolds number and small yaw angles on the perturbation amplification and energy budget is considered. At a higher Reynolds number ( $Re' = 22 \times 10^6$ m $^{-1}$ ), the most-amplified wavelength shrinks. Perturbations do not behave self-similarly in the thinner boundary layer, and the shift in most-amplified wavelength is due to decreased dissipation relative to the lower-Reynolds-number case. Small yaw angles produce a streamwise shift in the boundary layer and disturbance amplification. The yaw results quantify a potential uncertainty source in experiments and flight.

OCTRA as ultrasonically absorptive thermal protection material for hypersonic transition suppression

Previous investigations in the High Enthalpy Shock Tunnel Göttingen (HEG) of the German Aerospace Center (DLR) show that carbon fiber reinforced carbon ceramic (C/C) surfaces can be utilized to damp hypersonic boundary layer instabilities resulting in a delay of boundary layer transition onset. Numerical stability analyses confirmed these experimental results. However, C/C has some disadvantages, especially the limited oxidation resistance and its low mechanical strength, which could be critical during hypersonic flights. Thus, an ultrasonically absorptive fiber reinforced ceramic material based on a silicon carbide (C/C-SiC) was developed in the past years to fulfill this need. The present paper addresses the numerical rebuilding of the C/C-SiC absorber properties using impedance boundary conditions together with linear stability analysis. The focus of this paper is on the numerical comparison of the original C/C material and the improved C/C-SiC material, referred to as OCTRA in the literature. The influence on the second modes and the transition itself is investigated. The numerical results are compared with HEG wind tunnel tests. The wind tunnel model tested in HEG is a 7^circ half-angle blunted cone with an overall model length of about 1.1 ,textrm{m} and a nose tip radius of 2.5 mm. These experiments were performed at Mach 7.5 and at different freestream unit Reynolds numbers.

Modal Instabilities over Blunted Cones at Angle of Attack in Hypersonic Flow

The effects of nose radius and angle of attack on the linear modal amplification over blunt circular cones at hypersonic speeds are computationally investigated. The three-dimensional laminar flow solutions over a 1.5-m-long, 7°-half-angle cone with 5.080, 9.525, and 25.40 mm nosetip radii are computed for selected angle-of-attack values and freestream conditions that match the Mach 10 experiments at a freestream unit Reynolds number of 17.1 million per meter conducted within the Hypervelocity Wind Tunnel 9 at the Air Force Arnold Engineering Development Complex (AEDC). Results indicate that the linear amplification of stationary and traveling crossflow waves along inflection lines increases with the angle of attack and decreases with the nosetip radius. The trend in transition front with respect to increasing angle of attack and bluntness is found to be consistent with the predicted increase in the amplification factors for Mack’s second mode (MM) disturbances along the streamline trajectories for the small and medium bluntness cases. The increase in the MM amplification along the windward ray for higher angles of attack is shown to be the result of a progressively earlier entropy-layer swallowing. Computations also indicate that the transition amplification factor along the windward ray increases with the angle of attack and decreases with the nosetip radius. Furthermore, the transition -factors along the leeward ray are below 2 and, therefore, rather small to lead to transition.

Roughness effect on hypersonic second mode instability and transition on a cone

Recent studies by Fong et al. [“Finite roughness effect on modal growth of a hypersonic boundary layer,” AIAA Paper No. 2012-1086, 2012; “Stabilization of hypersonic boundary layer waves using 2-D surface roughness,” AIAA Paper No. 2013-2985, 2013; “Numerical simulation of roughness effect on the stability of a hypersonic boundary layer,” Comput. Fluids 96, 350–367 (2014); and “Second mode suppression in hypersonic boundary layer by roughness: Design and experiments,” AIAA J. 53, 1–6 (2015)] have shown that finite roughness can attenuate Mack's second mode instability when placed at the discrete mode synchronization location for two-dimensional planar flow over a flat plate. However, more practical hypersonic flows are non-planer conical flows, and the roughness effect phenomenon in conical flows has not been extensively investigated. For that reason, this investigation research the ability of finite roughness strips to attenuate the second mode instability on a Mach 8 straight blunt cone with a freestream unit Reynolds number of 9 585 000. Two roughness configurations are studied: a single roughness strip and an array of six sequential strips. N-factor calculations determine the second mode frequency most likely to lead to turbulent transition, and the linear stability theory is used to determine the mode's synchronization location. In the unsteady simulations of the roughness configurations, a blowing-suction actuator introduces an upstream broadband Gaussian pulse. Fourier decomposition of the pulse's history shows that the single roughness strip attenuates frequencies higher than 208 kHz while lower frequencies are amplified. Likewise, the roughness array exhibits similar results, attenuating frequencies higher than 164 kHz and amplifying lower frequencies downstream. The results show that both configurations can delay second mode instability growth on a hypersonic blunt cone and possibly delay turbulent transition. However, investigations of the roughness effect's behavior downstream of the roughness configurations show that disturbance growth resumes and becomes more destabilizing to the boundary layer.

Hypersonic Boundary-Layer Disturbances on a Cooled, Slender Cone at Mach 6

This paper presents an experimental study of the effect of wall cooling on the boundary-layer structures of a sharp-nosed, 5° half-angle cone in Mach 6 flow. An experimental methodology was developed that allowed quantitative measurements of second-mode disturbance growth over a range of wall-temperature ratios from without modifying the freestream disturbance environment. The cooling system comprised probe cooling, liquid nitrogen injection, and thermal insulation. The model temperature was monitored before each run, with thermocouples mounted at the base and nosetip piece agreeing within 3.3 K. Calibrated schlieren provided global and time-resolved measurements of the boundary-layer features at a freestream unit Reynolds number of . An increase in wall-cooling level was associated with a shrinking of the boundary layer and a lowering of wavepacket structure angle at the wall. Despite some outlying results, an overall trend of increasing disturbance frequency with decreasing surface temperature was observed; additionally, cooling increased the spectral amplitude and growth rate of second-mode disturbances, pushing the point of saturation upstream.

Hypersonic shock impingement studies on a flat plate: flow separation of laminar boundary layers

Interactions between shock waves and boundary layers produce flow separations and augmented pressure/thermal loads in hypersonic flight. This study provides details of Mach 7 impinging-shock-flat-plate experiments conducted in the T4 Stalker Tube. Measurements were taken at flow conditions of Mach 7.0 (2.44 MJ kg $^{-1}$ ) and Mach 7.7 (2.88 MJ kg $^{-1}$ ) flight enthalpies with a range of freestream unit Reynolds numbers from $1.43 \times 10^{6}$ m $^{-1}$ to $5.01 \times 10^{6}$ m $^{-1}$ . A shock generator at $12^{\circ }$ or $16^{\circ }$ to the freestream created an oblique shock which impinged on a boundary layer over a flat plate to induce flow separation. The flow field was examined using simultaneous measurements of wall static pressure, heat transfer and schlieren visualisation. Measured heat transfer along the flat plate without the shock impingement indicated that the boundary layer remained laminar for all flow conditions. The shock impingement flow field was successfully established within the facility test duration. The onset of separation was observed by a rise in wall pressure and a decrease in heat transfer at the location corresponding to the stem of the separation shock. Downstream of this initial rise, an increased pressure and higher heating loads were observed. The heat-transfer levels also indicated an immediate boundary layer transition due to the shock impingement. The separation data of the present work showed good agreement with our previous work on shock impingement on heated walls (Chang et al., J. Fluid Mech., vol. 908, 2021, pp. 1–13). A comparison with the previous scaling indicated that the separation also relates to the pressure ratio and the wall temperature parameter.

Experimental study on the effects of the cone nose-tip bluntness

In this Letter, hypersonic boundary-layer transition was investigated on a large-scale cone with a height of 3 m and a half-cone angle of 7° at a zero angle of attack in the JF-12 hypersonic flight duplicate shock tunnel. For the same freestream unit Reynolds number, with the increase in the bluntness Reynolds number, the transition Reynolds number has a trend of first increasing and then decreasing, showing a “transition reversal” phenomenon. As the bluntness increased, the high/low-frequency instability waves in the boundary-layer were modulated, which caused the boundary-layer transition to be delayed and then advanced.

Transition induced by an egg-crate roughness on a flat plate in supersonic flow

Hot-wire measurements in a Mach 3.5 quiet tunnel were made in the wake of a roughness patch on a flat plate. These measurements were used to determine mode shapes and frequencies of the dominant instabilities leading to boundary-layer transition. The egg-crate roughness pattern is an analytic function described by a sinusoidal equation, similar to an array of discrete elements that are positioned in a spanwise and streamwise grid, but containing both protuberances and dimples. This is an intermediate configuration towards understanding the underlying physics of a pseudorandom distributed roughness, and ultimately, the underlying physics of roughness-induced boundary-layer transition. The roughness pattern had a wavelength of 6.25 mm, with a nominal amplitude of 272 ${\rm \mu}{\rm m}$ (0.49 times the boundary layer thickness at the first row of protuberances). The roughness was positioned near the leading edge of the flat plate and contained 3.5 wavelengths in the streamwise direction and 7.5 wavelengths in the spanwise direction. The dominant instability was centred near 74 kHz at a free stream unit Reynolds number of $12.9\times 10^{6}\,{\rm m}^{-1}$ and resembled an antisymmetric mode downstream of each of the protuberances in the roughness patch. Computations using linear stability analysis based on the plane-marching parabolized stability equations (PSE) showed limited agreement with measurements when comparing the growth of the wake instability. Better agreement with the measurements was observed when considering the modification of first mode waves by the egg-crate roughness patch and the solution of the three-dimensional harmonic linearized Navier–Stokes equations was used as the in-flow to the PSE. The agreement confirms the significance of disturbance growth both upstream of and above a finite length roughness patch and the effect on the growth of instabilities in the wake.

Improved Transition Model with Mechanical Considerations for Hypersonic Flow

An improved transition model with mechanical considerations is proposed in this paper for hypersonic boundary-layer transition. The original V-SST model is not suitable for hypersonic flow with low freestream turbulence intensity. Thus, three key modifications based on the physical mechanisms and multiple unstable modes, which involve the addition of source term in the transport equation, reconstruction of pretransitional viscosity, and reformulation of the control function of effective turbulence viscosity, are constructed to make it possible to predict the hypersonic transition process. Numerous validation test cases are chosen to adequately access the prediction accuracy of the current transition model, including flat plates and blunt cones with different Reynolds numbers, a double ramp, an inlet model with composite shock interference flow, and three-dimensional X-51A forebody model. The flow structures and heat transfer distributions predicted by the numerical simulation are well consistent with experimental data, which demonstrate the capability of improved V-SST transition model in accurately predicting hypersonic transition behavior and reasonably reflecting the influence of freestream unit Reynolds number. It needs to further investigate the performance of the current model in capturing unusual transition phenomenon such as heat transfer overshoot, and consider more physical influence factors on boundary-layer transition.

Heat Flux on a Hypersonic Cone with a Highly Swept Fin

A detailed comparison between computations and experiments is given for Mach 6 flow over a straight cone with a highly swept fin. Five cases are simulated that vary the cone nose radius, fin thickness, fin sweep angle, and freestream unit Reynolds number. Experimental data are available for four cases. Surface heat flux comparisons between steady, laminar simulations and experimental measurements are made on the surface of the cone and fin. Uncertainty for both computations and experiments is estimated. Three cases are identified as transitional or turbulent, while one case is identified as laminar. Estimates of transition location are derived from experimental heat transfer and measurements of surface pressure fluctuations over a range of unit Reynolds numbers. The effect of geometric parameters is deduced. Further investigation into geometric changes demonstrates the sensitivity of the finned cone flowfield (e.g., vortex location and strength) to the shape of the fin–cone junction. Additionally, simulations demonstrate that the flow on the fin and cone is largely unaffected by truncating the fin. Changes in the flowfield remain localized to the fin truncation.

On the synchronisation of three-dimensional shock layer and laminar separation bubble instabilities in hypersonic flow over a double wedge

Linear three-dimensional instability is studied in the shock layer and the laminar separation bubble (LSB) induced by shock-wave/boundary-layer interactions in a Mach 7 flow of nitrogen over a double wedge with a$30^{\circ }\text {--}55^{\circ }$cross-sectional profile. At a free-stream unit Reynolds number$Re=5.2\times 10^{4}\,{\rm m}^{-1}$this flow exhibits rarefaction effects and has shock thicknesses comparable to the thickness of the boundary layer at separation. Flow features have been fully resolved using a high-fidelity massively parallel implementation of the direct simulation Monte Carlo method that captures the flow evolution from the inception of three-dimensionality, through linear growth of instabilities, to the early stages of nonlinear saturation. It is shown that the LSB sustains self-excited, small-amplitude perturbations that originate past the primary separation line and lead to spanwise-periodic wall striations inside the bubble and downstream of the primary reattachment line, as known from earlier experiments, simulations and instability analyses. A spanwise-periodic instability, synchronised with that in the separation zone, is identified herein for the first time, which exists in the internal structure of the separation and detached shock layers, and manifests itself as spanwise-periodic cats-eyes patterns in the global mode amplitude functions. The growth rate and the spanwise-periodicity length of linear disturbances in the shock layers and the LSB are found to be identical. Linear amplification of the most unstable three-dimensional flow perturbations leads to synchronised low-frequency unsteadiness of the triple point, with a Strouhal number of$St\approx 0.028$.

Linear Instabilities over Ogive-Cylinder Models at Mach 6

Computational investigations of a tangent ogive-cylinder geometry with varying forebodies at zero degrees angle of attack are presented. The model geometry and conditions are selected to match experiments conducted in the Air Force Research Laboratory (AFRL) Mach 6 Ludwieg Tube. Specifically, five forebodies of interest consisting of sharp and blunt ogives of 4 and 2 caliber and a hemispherical shape are studied at a freestream unit Reynolds number of . Boundary-layer-edge properties and wall-normal profiles of velocity and temperature are compared across streamwise locations past the ogive-cylinder junction. Modal stability analysis is used to characterize the most amplified frequencies corresponding to Mack’s first and second modes and those are found to agree with experimental results for the sharp forebodies. The entropy layer induced by blunt forebodies envelopes the boundary layer and stabilizes modal disturbances. Nonmodal analysis revealed a broadband set of disturbances present for the blunter forebodies, in agreement with experimental observations. Flow perturbation contours of most amplified planar and oblique disturbances are shown to qualitatively match wind tunnel schlieren images, with a switch from rope-like to elongated structures, i.e., from high-frequency Mack’s second modes to low-frequency Mack’s first modes, as the forebody angle for the sharp tip is increased, and from boundary-layer to entropy-layer disturbances as the bluntness is increased.

Fluid–structural coupling of an impinging shock–turbulent boundary layer interaction at Mach 3 over a flexible panel

We present high-fidelity numerical simulations of the interaction of an oblique shock impinging on the turbulent boundary layer developed over a rectangular flexible panel, replicating wind tunnel experiments by Daub et al. (AIAA Journal, vol. 54, 2016, pp. 670–678). The incoming free-stream Mach and unit Reynolds numbers are $M_{\infty } = 3$ and $Re_{\infty }=49.4\times 10^6 {\rm m}^{-1}$ , respectively. The reference boundary layer thickness upstream of the interaction with the shock is $\delta _0 = 4$ mm. The oblique shock is generated with a rotating wedge initially parallel to the flow that increases the deflection angle up to $\theta _{{max}} = 17.5^{\circ }$ within approximately $15$ ms. A loosely coupled partitioned flow–structure interaction simulation methodology is used, combining a finite-volume flow solver of the compressible wall-modelled large-eddy simulation equations, an isoparametric finite-element solid mechanics solver and a spring-system-based mesh deformation solver. Simulations are conducted with rigid and flexible panels, and the results compared to elucidate the effects of panel flexibility on the interaction. Three-dimensional effects are evaluated by conducting simulations with both full ( $50 \delta _0$ ) and reduced ( $5\delta _0$ ) spanwise panel width, the latter enforcing spanwise periodicity. Panel flexibility is found to increase the separation bubble size and modify its spectral dynamics. Time- and spanwise-averaged streamwise profiles of the wall pressure exhibit a drop over the flexible panel prior to the interaction and a reduced peak pressure in comparison with the rigid case. Spectral analyses of wall pressure data indicate that the low-frequency motions have a similar spectral distribution for the rigid and flexible cases, but the flexible case shows a wider region dominated by low-frequency motions and traces of the panel vibration on the wall pressure signal. The sensitivity of the interaction to small variations in the wedge extent and incoming boundary layer thickness is evaluated. Predictions obtained from lower-fidelity modelling simplifications are also assessed.

HIFiRE-5b Boundary-Layer Transition Length and Turbulent Overshoot

Hypersonic International Flight Research Experimentation 5 (HIFiRE-5) is a hypersonic flight-test experiment designed to investigate the aerothermodynamics of a 3-D geometry. The vehicle is an elliptic cone with a 2:1 aspect ratio and 2.5 mm nose radius. The HIFiRE-5b achieved a Mach number of 7.7–7.9 and freestream unit Reynolds number of to during descent. Heat flux was calculated from thermocouple pairs and boundary-layer transition locations were identified from these heating rates. A multilobed transition front was observed; it is suspected that a different mechanism is responsible for each of these lobes. The vehicle’s attitude oscillated, which has been exploited to assess the sensitivity of heat flux to yaw angle along the leading edges and ascertain the appropriateness of the filtering applied to the thermocouple signals. Spatial heat-flux profiles have been constructed for constant freestream unit Reynolds numbers. The heat-flux rate at the end of transition is examined; turbulent overshoot was present along some azimuths, but not others. This flight-test result indicates the phenomenon is not specific to ground test, freestream noise level, or wall-to-stagnation-temperature ratio. Finally, transition length (its end location normalized by its onset location) is reported and connected to turbulent spot generation rates.

Toward a Practical Method for Hypersonic Transition Prediction Based on Stability Correlations

An accurate boundary-layer transition prediction method integrated with computational fluid dynamics (CFD) solvers is pursued for hypersonic boundary-layer flows over slender hypersonic vehicles at flight conditions. The geometry and flow conditions are selected to match relevant trajectory locations from the ascent phase of the HIFiRE-1 flight experiment, namely, a 7 deg half-angle cone with 2.5 mm nose radius, freestream Mach numbers in the range of 3.8–5.5, and freestream unit Reynolds numbers in the range of . Earlier research had shown that the onset of transition during the Hypersonic International Flight Research Experimentation 1 (HIFiRE-1) flight experiment correlated with an amplification factor of for the planar Mack modes. However, to incorporate the -factor correlations into a CFD code, we investigate surrogate models for disturbance amplification that avoids the direct computation of stability characteristics. The results demonstrate that the application of the commonly used approach, which is based on a database of stability characteristics for locally similar profiles, leads to large, unacceptable errors in the predictions of amplification factors. We propose and demonstrate an alternate approach that employs the stability computations for a canonical set of blunt cone configurations to train a convolutional neural network model that is shown to provide substantially improved transition predictions. The excellent performance of the neural network model is also confirmed for cone configurations with nose radius and half-angle values outside of those used to build the database. Finally, the convolutional neural network model is found to outperform the linear stability calculations for underresolved basic states.

Unsteady characteristics of compressible reattaching shear layers

Compressible reattaching flows occur in many aerospace applications and are characterized by high aerothermal loads at reattachment and a broad range of characteristic time scales. The flowfield in this study involves a separated shear layer reattaching onto a 20° ramp at a freestream unit Reynolds number of 6.7 × 107 m−1 and Mach number of 2.9. Delayed detached eddy simulations were carried out using OVERFLOW 2.2K by Leger et al. [“Detached-eddy simulation of a supersonic reattaching shear layer,” AIAA J. 55, 3722–3733 (2017)], and the corresponding results were analyzed to determine the unsteady features of this flowfield using statistical techniques. The simulations were run for long integration times, which ensured sufficient temporal resolution of low-frequency unsteadiness in the range of St=O(0.01). The mean flow data highlighted essential flow components such as an expansion fan at the separation point, a large recirculation vortex, and a reattachment shock. Fourier analysis of wall pressure data revealed several high energy frequency bands, which appeared to correspond to separation bubble breathing, shear-layer flapping, and shedding of vortices from the recirculation zone. The spectra also highlighted the possible presence of Rossiter modes, suggesting a feedback mechanism through the recirculation zone. Correlations in the shear-layer and recirculation zone confirmed the presence of large-scale turbulence structures, with an increase in length and time scales downstream. The spectra of reattachment shock location suggested a broadband nature of the oscillations. The role of upstream events on the same was investigated by examining coherence, conditional averages, and correlations. A similar exercise was carried out to investigate the nature of unsteadiness at the mean reattachment location.

Hypersonic shock-wave/boundary-layer interactions on a cone/flare

Time-resolved surface temperature measurements have been carried out on a 7° half-angle circular cone/flare model in the Air Force Research Laboratory’s Mach-6 Ludwieg Tube using infrared thermography. Measurements were performed at zero angle of attack on a total of 16 different nose bluntness/flare angle geometries at three different freestream unit Reynolds numbers. The boundary layer entering the interaction region was turbulent for the two sharper nosetips and laminar for the two blunter nosetips. Stanton-number contours and profiles were used to investigate the effect of nose bluntness, flare angle, and freestream unit Reynolds number on the length of boundary-layer separation and the peak heating achieved upon reattachment. The classical scaling analyses of Souverein et al.and Hung & Barnett for separation length and peak heating, respectively, have been extended for the use of axisymmetric cone/flare shock-wave/boundary-layer interactions. Scaling the peak heating with the density ratio across the separation shock wave, instead of the pressure ratio, significantly improves/simplifies the analysis.

High-Speed Schlieren Analysis of Retropropulsion Jet in Mach 10 Flow

High-speed schlieren imaging of a 5-in.-diam, 70° sphere-cone model with a single centerline 15° half-angle nozzle was performed in a Mach 10 flow. Image sequences captured at 100 kHz were obtained with the nozzle plugged (fully blunted model), no nozzle flow, and with nozzle flow over a range of supply pressures. The freestream unit Reynolds number was kept constant for all runs at . Time-average information pertaining to the location, width, and stand-off distance of the bow shock, interface, terminal shock, and triple point as a function of jet pressure ratio are presented based on analysis of covariance images from each run. Measurement of jet plume boundary shape as a function of jet pressure ratio is also presented. Analyses of frequency content associated with the jet structure are shown, as are results of the proper orthogonal decomposition of these data. For all runs with nozzle flow, a dominant fundamental frequency of 2 kHz was observed. Finally, a discussion of bow shock unsteadiness resulting from interaction with freestream disturbances for the blunted and no nozzle flow is provided, as is a discussion on the formation of weak waves emanating from the bow shock resulting from this interaction.

Influence of Nose-Tip Bluntness on Conical Boundary-Layer Instabilities at Mach 10

Hypersonic boundary-layer transition experiments are performed at the von Karman Institute for Fluid Dynamics in the low-enthalpy Longshot wind tunnel. Freestream Mach numbers are within the range , and freestream unit Reynolds numbers vary within the range . The model is a 7 deg half-angle cone, which is 800 mm long, with exchangeable nose tips for which the radii vary between 0.05 and 10 mm. The natural transition process is investigated using flush-mounted thermocouples, fast-response pressure sensors, and a laser-based schlieren flow visualization technique. The most amplified frequencies retrieved experimentally from wavelet analysis techniques compare excellently with linear stability theory results and confirm the inviscid nature of the instabilities observed along geometries with a small amount of nose-tip bluntness. The corresponding factors at the transition onset are inferred from the experimental location where transition is initiated. Schlieren images exhibit the typical rope shape of second-mode instabilities for the sharpest nose tips, but not along blunter geometries where instabilities peak outward of the boundary-layer edge. The different measurement techniques used show that the maximum amplitude of the second-mode waves is reached shortly after the location with the minimum wall heat flux. Schlieren images show that wave packets break down to turbulence near the middle of the transition region.

Nonlinear behaviour of the Mack mode in a hypersonic boundary layer

Mack-mode waves are measured in a hypersonic boundary layer using high-frequency focusing schlieren deflectometry. Experiments are performed using a $5^{\circ }$ flared cone at $0^{\circ }$ angle of attack in the low-disturbance Mach 6 Quiet Tunnel at Texas A&M University across a free-stream unit Reynolds number range of $7.8\times 10^{6}~\text{m}^{-1}\leqslant \mathit{Re}^{\prime }\leqslant 11.0\times 10^{6}~\text{m}^{-1}$. The high-frequency response of the measurement system allows harmonics and other nonlinear behaviour to be measured. Mack-mode waves and several harmonics are clearly observed at a frequency of $f_{0}\approx 250~\text{kHz}$. Bispectral analysis is used to show that these waves undergo several quadratic phase-coupled sum and difference interactions with themselves to produce harmonics, as well interact with a relatively low-frequency wave that results in amplitude modulation. Bispectral analysis is used to highlight these interactions.