Linear and nonlinear mechanisms governing the growth of second-mode waves are analysed using a newly derived disturbance energy conservation equation that highlights the physical processes responsible for fluctuation energy production, flux-based transport and destruction. Axisymmetric direct numerical simulations (DNS) data from a Mach 6 hypersonic boundary layer, simulated over a $3^\circ$ half-angle sharp cone at zero angle of attack, is used as a reference. A Legendre polynomial-based forcing methodology is used to trigger transition in the DNS over a range of various amplitude levels and different forcing frequency content. Closure of the disturbance energy budgets is demonstrated numerically using the DNS data. The terms responsible for the amplification of the disturbance are identified, and nonlinear attenuation effects are discussed. We show that the interaction between the entropy/velocity fluctuations and the base temperature gradient governs the second-mode growth in the linear regime. Energy production occurs in the critical layer due to non-isentropic processes and accumulates in acoustic form below the relative sonic line through downward transport. At higher forcing amplitudes, nonlinear spectral broadening is observed, with simultaneous thermoviscous diffusion attenuating the disturbance energy growth. This effect is responsible for the non-monotonic streamwise variation of the wall-pressure spectrum. Phase speed and growth rate analyses, informed by linear stability theory (LST), reveal wave steepening effects preceding this nonlinear attenuation effect. The disturbance energy is observed to match the LST predictions at lower forcing amplitudes, deviating, as expected, at higher amplitudes.

Large-eddy simulations (LES) of a hypersonic boundary layer on a $7^\circ$-half-angle cone are performed to investigate the effects of highly cooled walls (wall-to-recovery temperature ratio of $T_w / T_r \sim 0.1$) on fully developed turbulence and to validate a newly developed rescaling method based on volumetric flow extraction. Two Reynolds numbers are considered, $Re_m = 4.1 \times 10^6\ \text {m}^{-1}$ and $6.4 \times 10^6\ \text {m}^{-1}$, at free-stream Mach numbers of $M_\infty = 7.4$. A comparison with a reference laminar-to-turbulent simulation, capturing the full history of the transitional flow dynamics, reveals that the volumetric rescaling method can generate a synthetic turbulent inflow that preserves the structure of the fluctuations. Equilibrium conditions are recovered after approximately 40 inlet boundary layer thicknesses. Numerical trials show that a longer streamwise extent of the rescaling box increases numerical stability. Analyses of turbulent statistics and flow visualizations reveal strong pressure oscillations, up to $50\,\%$ of local mean pressure near the wall, and two-dimensional longitudinal wave structures resembling second-mode waves, with wavelengths up to 50 % of the boundary layer thickness, and convective Mach numbers of $M_c \simeq 4.5$. It is shown that their quasi-periodic recurrence in the flow is not an artefact of the rescaling method. Strong and localized temperature fluctuations and spikes in the wall-heat flux are associated with such waves. Very high values of temperature variance near the wall result in oscillations of the wall-heat flux exceeding its average. Instances of near-wall temperature falling below the imposed wall temperature of $T_w=300$ K result in pockets of instantaneous heat flux oriented against the statistical mean direction.

This study investigates the heating issue associated with a V-shaped blunt leading edge (VBLE) in a hypersonic flow. The heat flux generation on the VBLE is highly correlated with the shock interaction configurations in the crotch region, determined by the relative position of the triple point T and the curved shock (CS). The primary Mach reflection (MR), accompanied by a series of secondary shock–shock interactions and shock wave–boundary layer interactions, can produce extremely high heating peaks on the crotch. To configure the shock wave structures and reduce the heat flux, a shock-controllable design approach is developed based on the simplified continuity method. The strategy involves the inverse design of the crotch sweep path, according to the location of the triple point and the contour of the CS. The comparisons between the pre-designed shock configurations and the numerical results demonstrate the reliability of the design approach across various free stream Mach numbers ranging from 6 to 10. A VBLE model designed with the shock configuration of regular reflection from the same family (sRR) at a free stream Mach number of 8 is examined. Under the design conditions, the outermost heat flux peak is reduced by 80 % compared with the baseline case. The heating reduction capabilities of the model under varying free stream Mach numbers and sideslip angles are also evaluated, confirming its robustness under undesigned operating scenarios.

The dominant mode instability in hypersonic boundary-layer transition is the so-called second-mode instability, which has a peculiar nature strongly coupled with thermoacoustic phenomena. In linear stability theory, the unstable wave is associated with one of the two eigenvalues that originate from the acoustic branches, referred to as slow and fast modes. Interestingly, the unstable mode (slow or fast) reaches its maximum amplification as the other mode (fast or slow) attains a minimum. The phase velocity of the two modes is then very close, and this phenomenon is called synchronization. The aim of the present study is to unravel the physical mechanism that explains the second-mode growth. To that aim, second-order nonlinear equations are written for the disturbances given by linear stability. In this framework, entropy, kinetic energy and temperature energy budgets are obtained up to second order. The budgets are scrutinized for various Mach numbers and for adiabatic and cold-wall thermal conditions. Perturbation entropy budgets clearly show the process is a reversible one. An energy exchange between kinetic energy and temperature energy of the weakly nonlinear modes is driven by pressure–dilatation terms. As underlined in previous studies, the unstable mode experiences an alternate heating and cooling near the wall, which is shown to be a rather nonlinear process. The change in fluctuating thermal energy in the form of a dilatational wave is sustained by pumping disturbance kinetic energy through the pressure–dilatation term, the direction of the conversion being driven by the relative phase between pressure and dilatation. This process is similar for the slow and fast modes, the unstable mode being amplified and the other being damped. No change in the process has been noted at the location of the synchronization, suggesting that the modes have the same nature but evolve independently.

This study investigates the flow structures and combustion regimes in an axisymmetric cavity-based scramjet combustor with a total temperature of 1800 K and a high Reynolds number of approximately 1 × 107. The hydroxyl planar laser-induced fluorescence technique, along with the broadband flame emission and CH* chemiluminescence, is employed to visualize the instantaneous flame structure in the optically accessible cavity. The jet-wake flame stabilization mode is observed, with intense heat release occurring in the jet wake upstream of the cavity. A hybrid Reynolds-averaged Navier–Stokes/large-eddy simulation approach is performed for the 0.18-equivalent-ratio case with a pressure-corrected flamelet/progress variable model. The combustion regime is identified mainly in the corrugated or wrinkled flamelet regime (approximately 102 < Da < 104, 103 < Ret < 105 where $Da$ is the Damköhler number and $Re_t$ is the turbulent Reynolds number). The combustion process is jointly dominated by supersonic combustion (which accounts for approximately 58 %) and subsonic combustion, although subsonic combustion has a higher heat release rate (peak value exceeding 1 × 109 J (m3s)−1). A partially premixed flame is observed, where the diffusion flame packages a considerable quantity of twisted premixed flame. The shockwave plays a critical role in generating vorticity by strengthening the volumetric expansion and baroclinic torque term, and it can facilitate the chemical reaction rates through the pressure and temperature surges, thereby enhancing the combustion. Combustion also shows a remarkable effect on the overall flow structures, and it drives alterations in the vorticity of the flow field. In turn, the turbulent flow facilitates the combustion and improves the flame stabilization by enhancing the reactant mixing and increasing the flame surface area.

In hypersonic flight the shock wave and turbulent boundary layer interaction (STBLI) sharply increases wall heat transfer that intensifies the aerodynamic heating problems. In this work the STBLI is modelled by compression ramp flow with a Mach number of 5, a Reynolds number based on momentum thickness of 4652 and a wall to recovery temperature ratio of 0.5. The aerodynamic heat generation and transport mechanisms are investigated in the interaction based on theoretical analysis and direct numerical simulation (DNS) that agrees with previous studies. A prediction correlation of wall heat flux in STBLI is deduced theoretically and validated by some representative data including the present DNS, which improves the prediction accuracy and can be applied to a wider $Ma$ range compared with the canonical Q-P theory. The correlation indicates that the sharp increase of wall heat transfer in the STBLI can be explained by the boundary layer compression and the convection transport enhancement. Based on the DNS results, the aerodynamic heat generation and transport mechanisms are revealed in the separation, recirculation and reattachment zones in the STBLI. From this perspective, the peak heat flux can be further explained by the enhancement of near-wall turbulent energy dissipation, compression aerodynamic heat generation and the near-wall turbulent transport. The generation and transport of compression aerodynamic heat reveal the underlying mechanism of the strong correlation between the peak heat flux ratios and the pressure ratios in STBLIs.

Three-dimensional numerical simulations of hypersonic boundary layer transition delay due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) were carried out on a 7$^{\circ {}}$ half-angle cone for unit Reynolds numbers $Re_m=2.43 \times 10^6$–$6.40\times 10^6\ \text {m}^{-1}$, at the free-stream Mach number $M_\infty =7.4$, for both sharp and 2.5 mm nose tip radii. A broadband time-domain impedance boundary condition was used to model the acoustic effects of the porous surface on the flow field. A quasi-spectral sub-filter-scale dynamic closure was adopted to stabilize the computations upon turbulent breakdown under extreme cooling conditions, with wall-to-adiabatic temperature ratio of $T_{w}/ T_{ad} \simeq 0.08 $, while accurately recovering the growth rates of the unstable modes present in the early transition stages. Good agreement is observed with the reference experimental data, both in terms of the predicted extent of the transition delay and the measured second-mode frequency spectrum. The latter is strongly modulated by the formation of near-wall low-temperature three-dimensional streaks. Pressure disturbances concentrate in corridors of locally thickened boundary layer, with frequencies lower than what predicted by linear theory. Here, trapped wavetrains are formed, which can persist long into the turbulent region. Finally, it is shown that the presence of a porous wall simply shifts the onset of turbulence downstream, without affecting its structure.

Solving the three-dimensional boundary layer equations carries theoretical significance and practical applications, which also poses substantial challenges due to its inherent complexity. In this paper, the laminar boundary layer equations for the symmetry plane of three-dimensional bodies are derived in an orthogonal curvilinear coordinate system associated with the principal curvatures. The derivation of the boundary layer equations is based not only on the common symmetric properties of the flow, as given by Hirschel et al. (Three-Dimensional Attached Viscous Flow, 2014, Academic Press, pp. 183–187), but also incorporates the geometric symmetry properties of the body. The derived equations are more representative and simplified. Notably, these equations can degenerate to a form consistent with or equivalent to the commonly used boundary layer equations for special bodies such as flat plates, cones and spheres. Furthermore, for hypersonic flows, the crossflow velocity gradient at the boundary layer edge on the symmetry plane is derived based on Newtonian theory. Subsequently, this parameter can provide the necessary boundary condition needed for solving the boundary layer equations using existing methods. Finally, as examples, the equations developed in this paper are solved using the difference-differential method for several typical three-dimensional blunt shapes that appeared on hypersonic vehicles. They prove to be useful in the analysis and interpretation of boundary layer flow characteristics in the symmetry plane of blunt bodies.

We present a Mach 15 air flow over a blunt two-dimensional wedge simulated using the direct molecular simulation method. As electronically excited states are not modelled, the resulting air mixture around the wedge contains the electronic ground states only, namely ${\rm N}_2(\text {X}^1 \varSigma _g^{+})$, ${\rm O}_2(\text {X}^3 \varSigma _g^{-})$, ${\rm NO}(\text {X}^2\varPi _r)$, ${\rm N}(^4{\rm S})$ and ${\rm O}(^3{\rm P})$. All the potential energy surfaces (PESs) that are used to model the various interactions between air particles are ab initio, with two notable exceptions, namely ${\rm N}_2+{\rm NO}$ and ${\rm O}_2+{\rm NO}$. At the selected free-stream conditions, strong vibrational non-equilibrium is observed in the shock layer. The flow is characterized by significant chemical activity, with near-complete oxygen dissociation, considerable formation of NO and minimal molecular nitrogen dissociation. Complex mass diffusion kinetics, driven by composition, temperature and pressure gradients, are identified in the shock layer. All these physical phenomena are directly coupled to, and responsible for, the mechanics of the gas flow and are all solely traceable to the PESs’ inputs, without the need for any thermochemical models, mixing rules or constitutive laws for transport properties. Because the flow is entirely at near-continuum conditions, it is a gas-phase thermophysics benchmark that is useful to enhance the fidelity of continuum models used in computational fluid dynamics of hypersonic flows.

Direct numerical simulations (DNS) were carried out to investigate flow control for transition delay using steady blowing/suction strips at the wall of a flared cone at Mach 6 and zero angle of attack. For the numerical investigations of the transition control strategy, the flared cone geometry and the flow conditions of the experiments in the Boeing/Air Force Office of Scientific Research (AFOSR) Mach 6 Quiet Tunnel (BAM6QT) at Purdue University were chosen. For the DNS, transition was initiated by introducing random disturbances at the inflow of the computational domain, emulating ‘natural’ transition in wind-tunnel experiments caused by free-stream noise. In both wind-tunnel experiments and numerical simulations, streamwise ‘hot’ streaks were found on the surface of the flared cone, which are caused by a nonlinear interaction of an axisymmetric second-mode wave and a pair of oblique waves of the same frequency (‘fundamental resonance’). The objective of the flow control strategy proposed here is to delay the transition onset, and thus mitigate the negative consequences associated with the nonlinear transition stages, i.e. the development of hot streaks and large wall-pressure amplitudes that were observed in experiments and DNS. Our previous so-called ‘controlled’ transition simulations have shown that flow control using steady blowing and suction strips can lead to a significant delay of the hot streak development on the surface of the flared cone. The simulation results presented in this paper show that this flow control strategy remains effective, even in a natural transition scenario characterized by broadband disturbances.

Wall temperature has a significant effect on shock wave/turbulent boundary layer interactions (STBLIs) and has become a non-negligible factor in the design process of hypersonic vehicles. In this paper, direct numerical simulations are conducted to investigate the wall temperature effects on STBLIs over a 34° compression ramp at Mach number 6. Three values of the wall-to-recovery-temperature ratio (0.50, 0.75 and 1.0) are considered in the simulations. The results show that the size of the separation bubble declines significantly as the wall temperature decreases. This is because the momentum profile of the boundary layer becomes fuller with wall cooling, which means the near-wall fluid has a greater momentum to suppress flow separation. An equation based on the free-interaction theory is proposed to predict the distributions of the wall pressure upstream of the corner at different wall temperatures. The prediction results are generally consistent with the simulation results (Reynolds number Reτ ranges from 160 to 675). In addition, the low-frequency unsteadiness is studied through the weighted power spectral density of the wall pressure and the correlation between the upstream and downstream. The results indicate that the low-frequency motion of the separation shock is mainly driven by the downstream mechanism and that wall cooling can significantly suppress the low-frequency unsteadiness, including the strength and streamwise range of the low-frequency motions.

In this paper, direct numerical simulations in hypersonic turbulent boundary layers over a $24^{\circ }$ compression ramp at Mach 6.0 are performed. The wall skin friction and its spanwise non-homogeneity in the interaction region are analysed via the spectral analysis and drag decomposition method. On the compression ramp, the premultiplied spanwise energy spectrum of wall shear stress $\tau _{w}$ reveals two energetic spanwise length scales. One occurs in the region of $x/\delta _{ref}=0\unicode{x2013}3$ ($x=0$ lies in the compression corner; $\delta _{ref}$ is the boundary layer thickness upstream of the interaction region) and is consistent with that of the large-scale streamwise vortices, indicating that the fluctuation intensity of $\tau _{w}$ is associated with the Görtler-type structures. The other one is observed downstream of $x/\delta _{ref}=3.0$ and corresponds to the regenerated elongated streaky structures. The fluctuation intensity of $\tau _{w}$ peaks at $x/\delta _{ref}=3.0$, where both the above energetic length scales are observed. The drag decomposition method proposed by Li et al. (J. Fluid Mech., vol. 875, 2019, pp. 101–123) is extended to include the effects of spanwise non-homogeneity so that it can be used in the interaction region where the mean flow field and the mean skin friction $C_f$ exhibit an obvious spanwise heterogeneity. The results reveal that, in the upstream turbulent boundary layer, the drag contribution arising from the spanwise heterogeneity can be neglected, while this value on the compression ramp is up to 20.7 % of $C_f$, resulting from the Görtler-type vortices. With the aid of the drag decomposition method, it is found that the main flow features that contribute positively to the amplification of $C_f$ and its rapid increase on the compression ramp includes: the density increase across the shock, the high mean shear stress and turbulence amplification around the detached shear layer and the Favre-averaged downward velocity towards the ramp wall. Compared with the spanwise-averaged value, $C_f$ and its components at the spanwise station where the downwash and upwash of the Görtler-type vortices occur reveal a spanwise variation exceeding 10 %.

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.

This experimental work explores the flow field around a three-dimensional expansion–compression geometry on a slender cone at Mach 8 using high-frequency pressure sensors, high-frame-rate schlieren, temperature-sensitive paint, shear-stress measurements and oil-flow visualizations. The $7^\circ$ cone geometry has a hyperbolic slice which acts as an expansion corner and suppresses the disturbances present in the upstream boundary layer. Downstream of the cone-slice corner, high-frequency boundary-layer disturbances attenuate in all cases. Under laminar conditions, second-mode instabilities from the cone diminish and lower-frequency second-mode waves develop on the slice at a frequency commensurate with the increased boundary layer thickness. For fully turbulent cases, the boundary layer over the slice shows evidence of a two-layered nature with a turbulent outer region and a near-wall region with strong attenuation of high-frequency disturbances and reappearance of lower-frequency instability waves. When a downstream compression ramp is added to the slice, the expanded boundary layer shows enhanced susceptibility to separation such that separation is observed at a $10^{\circ }$ deflection, which is smaller than expected for turbulent conditions. For a $30^{\circ }$ ramp, boundary-layer separation occurs further upstream where the heat flux contours show a decrease in heating that is characteristic of a transitional separation. These results demonstrate the effect of relaminarization caused by an upstream expansion on a subsequent shock-wave/boundary-layer interaction.

Direct numerical simulations of hypersonic turbulent Couette flows are performed for top-wall Mach numbers of 6, 7 and 8, inspired by non-reactive high-enthalpy wind tunnel free-stream conditions, with the goal of analysing the physical processes driving the sub-filter-scale (SFS) stresses to inform development of large-eddy simulation techniques for hypersonic wall-bounded flows. Semi-local scaling laws collapse mean profiles and second-order turbulent statistics well, in spite of the strong wall-normal gradients of temperature and density. On the other hand, the SFS shear stresses exhibit an unexpected profile characterized by a region of pronounced shear stress deficit, which becomes more pronounced for higher Mach numbers. Instantaneous visualizations suggest that the SFS shear stress deficit is induced by the counter-gradient resolved momentum transport driven by residual velocity motions at the interface between high-density low-speed streaks being ejected away from the wall, and low-density high-speed ones replacing the displaced fluid, qualifying this as a compressibility effect. It is shown that the SFS shear stresses are primarily driven by second-order interactions between residual velocities, in spite of their triply nonlinear nature. This, in turn, motivated a statistical quadrant analysis revealing the presence of a SFS shear stress deficit, that is, SFS processes driving momentum transport of the resolved field towards the top wall. Additionally, higher-Reynolds-number simulations reveal that there is an upper limit of spatial filter widths to resolve large-scale structures, and such deficit is observable at any Reynolds number when a reasonable spatial filter is applied.

Boundary-layer disturbances are analysed on a $5^{\circ }$ half-angle blunted cone in Mach 5, high-enthalpy flow ($h_0 = 9\ {\rm MJ}\ {\rm kg}^{-1}$) with a low wall-to-edge temperature ratio, $T_w/T_e = 0.18$. Schlieren and focused laser differential interferometry (FLDI) are utilized to assess the structures and frequency content associated with disturbances. Wave packets are identified from bursts of modal content on time-resolved spectrograms. Bandpass filtering, proper orthogonal decomposition (POD) and space–time POD are then applied to the schlieren data. Bandpass filtering suggests the presence of wave packet dispersion and elongation indicative of slow-acoustic-wave synchronization. Modal reconstruction techniques indicate the radiation of content outside the boundary layer and distinct orientation changes within disturbances, potentially the first experimental evidence of the supersonic-mode instability in such a flow field. Cross-bicoherence computations are carried out for discrete time segments of data from both schlieren and FLDI data. They demonstrate that the most dominant nonlinear interactions are the fundamental–first-harmonic and the fundamental–low-frequency interactions.

Flow physics vary in different regimes across the full Mach number range, with our knowledge being particularly poor about the hypersonic regime. An Eulerian realization of the particles on demand method, a kinetic model formulated in the comoving reference frame, is proposed to simulate hypersonic compressible flows. The present model allows for flux evaluation in different reference frames, in this case rescaled and shifted by local macroscopic quantities, i.e. fluid speed and temperature. The resulting system of coupled hyperbolic equations is discretized in physical space with a finite volume scheme ensuring exact conservation properties. Regularization via Grad expansion is introduced to implement distribution function and flux transformation between different reference frames. It is shown that the proposed method possesses Galilean invariance at a Mach number up to $100$. Different benchmarks including both inviscid and viscous flows are reproduced with the Mach number up to $198$ and pressure ratio up to $10^5$. Finally, the new model is demonstrated to be capable of simulating hypersonic reactive flows, including one-dimensional and two-dimensional detonations. The developed methodology opens up possibilities for the simulation of the full range of compressible flows, without or with chemical reactions, from the subsonic to hypersonic regimes, leading to enhanced understanding of flow behaviours across the full Mach number range.

The impact of bulk viscosity is unclear with considering the increased dilatational dissipation and compressibility effects in hypersonic turbulence flows. In this study, we employ direct numerical simulations to conduct comprehensive analysis of the effect of bulk viscosity on hypersonic turbulent boundary layer flow over a flat plate. The results demonstrate that the scaling relations remain valid even when accounting for large bulk viscosity. However, the wall-normal velocity fluctuations $v_{rms}^{\prime \prime }$ decrease significantly in the viscous sublayer due to the enhanced bulk dilatational dissipation. The intensity of travelling-wave-like alternating positive and negative structures of instantaneous pressure fluctuations $p_{rms}^{\prime }$ in the near-wall region decreases distinctly after considering the bulk viscosity, which is attributed mainly to the reduction of compressible pressure fluctuations $p_{c,rms}^{+}$. Furthermore, the velocity divergence $\partial u_{i} / \partial x_{i}$ undergoes a significant decrease by bulk viscosity. In short, our results indicate that bulk viscosity can weaken the compressibility of the hypersonic turbulent boundary layer and becomes more significant as the Mach number increases and the wall temperature decreases. Notably, when the bulk-to-shear viscosity ratio of the gas reaches a few hundred levels ($\mu _b/\mu =O(10^2)$), and mechanical behaviour of the near-wall region ($\kern 0.06em y^+\le 30$) is of greater interest, the impact of bulk viscosity on the hypersonic cold-wall turbulent boundary layer may not be negligible.

Experiments were performed to document the complex flow field around and over a $70^{\circ }$ swept fin mounted on a $7^{\circ }$ half-angle right-circular cone in a Mach 6 free-stream. Of particular interest is the turbulent transition of the boundary layer over the swept fin, which is expected to be dominated by a cross-flow instability. Stationary features in the surface temperature distribution over the fin are documented using infrared thermal imaging. These were processed further to determine average spatial Stanton number distributions over the fin. Wavelet analysis of the Stanton number distributions revealed stationary patterns with wavelengths near the fin leading edge that were consistent with linear theory predictions of stationary cross-flow modes. Further from the leading edge, the wavelength of the stationary patterns was observed to increase prior to turbulence onset. Based on these observations, specially designed arrays of discrete roughness elements (DREs) were investigated as a means of delaying turbulence transition with the objective of reducing surface heat flux on the swept fin. The DRE designs followed our previous approach used for cross-flow transition control (Corke et al., J. Fluid Mech., vol. 856, issue 10, 2018, pp. 822–849; Arndt et al., J. Fluid Mech., vol. 887, 2020, A30). These focused on either the shorter wavelengths near the leading edge, or the longer wavelengths that developed near turbulence onset. With regard to delaying transition and reducing the surface heat flux, the DREs that focused on the larger wavelengths of stationary modes were most effective. The fin included an array of pressure sensors that were used to document travelling disturbances that could include those associated with travelling cross-flow modes. Phase analysis of the pressure fluctuation time series was used to determine the wavelength, wave angle and phase speed that were consistent with the travelling cross-flow modes. Cross-bicoherence analysis between the stationary and travelling disturbances indicates a nonlinear phase locking that can account for the development of the longer-wavelength stationary features in the surface heat flux, presumed to be due to stationary cross-flow modes, prior to turbulence onset.

A new wall function method for hypersonic laminar boundary layers (HLBLs) is proposed to reduce the near-wall grid dependence of skin friction $c_f$ and wall heat flux $q_w$ in numerical simulations, aiming for fast and accurate predictions. First, an analytic laminar velocity law of the wall is derived, which achieves a universal scaling of the near-wall velocity of HLBLs. Then an accurate temperature–velocity relation is deduced by introducing the general recovery factor to address the invalidation of the Walz relation under the cold wall effect. Based on the laminar laws of the wall, a new wall function method for HLBLs is proposed. To avoid introducing the boundary layer edge quantities, the laminar laws of the wall are reformed by modifying the outer boundary conditions of the differential equation in deriving the temperature–velocity relation. Unlike the wall function method in turbulence, the new wall function obtains directly the accurate $c_f$ and $q_w$ by post-processing without being involved in the simulation iteration. The numerical experiments of a Mach 8 HLBL over the flat plate show that effectively, the new wall function can enlarge the distance of the first grid point off the wall $\Delta y_1$ from $10^{-6}$ m to $10^{-3}$ m, which brings a 50 times enhancement of the simulation efficiency. Meanwhile, the simulation errors of $c_f$ and $q_w$ of the mesh with $\Delta y_1=10^{-3}$ m are reduced significantly from 24.2 % and 18.5 % to 0.5 % and 0.1 %, respectively. Due to the new wall function removing the boundary layer edge quantities, success is also achieved under the curved walls.