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.

Cross-flow transition over a delta wing is systematically studied in a Mach 6.5 hypersonic wind tunnel, employing the Rayleigh scattering flow visualisation, high-speed schlieren and fast-response pressure sensors. Direct numerical simulations and analysis based on linear stability theory under the same flow conditions are applied to analyse the transition mechanism. Three unstable modes are identified: the travelling cross-flow instabilities, the second mode and the low-frequency waves. It is shown that the travelling cross-flow vortices first appear in the cross-flow region near the leading edge of the model. These vortices can modulate the mean profile of the flow, which benefits the growth of second mode. A phase-locked interaction mechanism transfers energy from the cross-flow instabilities to the high-frequency second mode, leading to amplification at the expense of the cross-flow instability. As the second mode grows to a critical amplitude, it triggers a $z$-type secondary instability within a similar frequency range, which introduces secondary finger-like structures connecting to the cross-flow vortex. It is further found that the generation of these finger-like structures is related to the expansion and compression of the second mode. These finger vortices further evolve along the streamwise direction into low-frequency waves and corresponding hairpin-like structures that finally trigger turbulence. An interaction mechanism likely exists between the secondary instability and the low-frequency waves, since their phase speeds are approaching each other. These observations of the interaction mechanism are consistent with those of previous studies on hypersonic boundary layers (Zhang et al., Phys. Fluids, vol. 32 (7), 2020, 071702; Li et al., Phys. Fluids, vol. 32 (5), 2020, 051701).

Kinks commonly appear on the convergent shock surface when an internal conical flow deviates from the axisymmetric state. In this paper, the formation mechanisms of kinks on internal conical shocks (ICSs) generated by elliptical ring wedges with typical entry aspect ratios ($AR{\rm s}$) in a Mach 6 flow are revealed using a theoretical method, in which the spatial evolution of the three-dimensional elliptical ICS is converted into a temporal evolution of a two-dimensional elliptical moving shock (EMS) using the hypersonic equivalence principle. To simultaneously track the shock front of the EMS and the disturbances propagating along it, a front-disturbance tracking method (FDTM) based on geometrical shock dynamics is proposed. It is found that the shock–compression disturbances from the same family initially near the major axis catch up with the disturbance initially emitted from the major axis to form kinks on the EMS. The equivalent kink formation positions predicted by the FDTM always lag behind the real kink formation positions on the elliptical ICS because the applicability of the hypersonic equivalence principle decays as the shock strengthens along the incoming flow direction. The accuracy of the equivalent kink formation positions predicted by the FDTM gradually declines with the reduction in $AR$, but it can be significantly improved for all $AR{\rm s}$ after a modification of the equivalent relationship using the shock angle in the major plane of the elliptical ICS, which provides a new way to solve the kinks on the elliptical ICS.

We propose a constitutive equation for flows of gases in high-rate regimes where the Navier–Stokes theory breaks down. The model generalizes the Navier–Stokes relation and agrees well with that model in all lower rate flows examined. Our proposed constitutive relation is calibrated with the method of objective molecular dynamics (OMD) using families of compressible and incompressible flows of Lennard-Jones argon. The constitutive relation makes use of the higher-order objective strain rates due to Rivlin and Ericksen (J. Rat. Mech. Anal., vol. 4, 1955, pp. 323–425). The constitutive relation is fully frame-indifferent, and the macroscopic flows corresponding to the OMD simulations are exact solutions for the proposed model. The model is shown to agree with atomistic results much better than the Navier–Stokes equations in the transition regime. The success of our model indicates that it is not higher gradients that become important in the high-rate regime, but rather higher rates of change of the strain rate tensor. While somewhat more complicated to implement than the Navier–Stokes relation, the proposed model is expected to be compatible with existing methods of computational fluid dynamics and may extend those methods to higher rate regimes, while preserving their ability to handle large spatial scales.

The present paper finds that the coexistence of multiple primary instability waves may cause a non-trivial nonlinear interaction and breakdown process, which has not been reported before. In the considered Mach 6 flat-plate boundary layer, a global resolvent analysis reports three optimal disturbances (local maxima): a high-frequency planar wave, a low-frequency oblique wave and a stationary streak. For the dominant planar and oblique waves, a parabolised stability equation analysis identifies the initial non-modal transient growth and downstream modal growth. Initiated by these two optimal disturbances jointly, the complete linear and nonlinear instability processes until breakdown to turbulence are shown with direct numerical simulation. Owing to the transient growth, the oblique wave may be more significant than the planar wave in the breakdown. The oblique wave and scales of nonlinear interactions are pronounced in the outer layer, whose significance may not be comprehensively characterised by the wall pressure measurement. Fourier modes characterising the oblique-wave oblique breakdown, the planar-wave fundamental resonance, the planar-wave subharmonic resonance and the combination resonance related to a detuned mode are observed successively. The detuned mode seems to dominate the near-wall dynamics in the late nonlinear stage, characterised by $\varLambda$-like structures. Meanwhile, the existence of this detuned mode is independent of the initial amplitude ratio and the absolute amplitude of the oblique and planar waves. Weakly nonlinear stability analyses demonstrate that the detuned mode is mainly a consequence of the secondary instability under the combination of planar and oblique primary waves. Wave vector plots reveal the resonant state of multiple triads. Energy budget and amplitude-correction analyses provide a clear physical image of energy transfer.

Lateral jets are used to control the missiles and re-entry vehicles at high altitudes. The objective of the research paper is to investigate the effect of lateral jet interaction with the external flow on a blunted nose cone re-entry vehicle configuration and its flight speed. Structured mesh is used for the simulations, and the computational analysis is carried out by Ansys Fluent solver. The simulation results are validated, and the same methodology used for the parametric analysis. Simulations have been carried out at an external flow Mach number 6, 8.1, 12 and 16 at five degree angle-of-attack for jet-off and jet-on conditions. At 8.1 Mach number, the normal force coefficient is decreased by 45.6% due to jet interaction. The lateral jet interaction effectively reduces the nose down pitching moment. At 8.1 Mach number, the pitching moment coefficient was reduced by 48% with the jet-on condition compared to the jet-off condition.

Instability measurements of an axisymmetric, laminar separation bubble were made over a sharp cone-cylinder-flare with a $12^{\circ }$ flare angle under hypersonic quiet flow. Two distinct instabilities were identified: Mack's second mode (which peaked between 190 and 290 kHz) and the shear-layer instability in the same frequency band as Mack's first mode (observed between 50 and 150 kHz). Both instabilities were measured with surface pressure sensors and were captured with high-speed schlieren. Linear stability analysis results agreed well with these measured instabilities in terms of both peak frequencies and amplification rates. Lower-frequency fluctuations were also noted in the schlieren data. Bicoherence analysis revealed nonlinear phase-locking between the shear-layer and second-mode instabilities. For the first time in axisymmetric, low-disturbance flow, naturally generated intermittent turbulent spots were observed in the reattached boundary layer. These spots appeared to evolve from shear-layer-instability wave packets convecting downstream. This work presents novel experimental evidence of the hypersonic shear-layer instability contributing directly to transition onset for an axisymmetric model.

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.