Hypersonic Turbulent Boundary Layers Research Articles

Hypersonic turbulent boundary layer over the windward side of a lifting body

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2 $^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

On the generation of near-wall dilatational motions in hypersonic turbulent boundary layers

Dilatational motions in the shape of travelling wave packets have been identified recently to be dynamically significant in hypersonic turbulent boundary layers. The present study investigates the mechanisms of their generation and their association with the solenoidal motions, especially the well-recognized near-wall self-sustaining process of the regeneration cycle between the velocity streaks and quasi-streamwise vortices. By exploiting the direct numerical simulation databases and orchestrating numerical experiments, we explore systematically the near-wall flow dynamics in the processes of the formation and transient growth of low-speed streaks. We conclude via theoretical ansatz that the nonlinearity related to the parallel density and pressure gradients close to the wall due to the restriction of the isothermal boundary condition is the primary cause of the generation of the dilatational structures at small scales. In fully developed turbulence, the formation and the existence of healthy dilatational travelling wave packets require the participation of the turbulence at scales larger than those of the near-wall regeneration cycles, especially the occurrence of the bursting events that generate vortex clusters. This is proven by the less intensified dilatational motions in the numerical experiments in which the Orr mechanism is alleviated and the vortical structures and turbulent bursts are weakened.

Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers

An improved resolvent analysis is proposed in the regime of compressible turbulent boundary layers. To better model nonlinear processes in the input, the resolvent framework is augmented by adding eddy viscosity. To this end, we propose two eddy-viscosity models: a modified Cess eddy-viscosity model coupling the compressibility transformation and outer-layer correction, and a new eddy-viscosity model based on an empirical relationship and mixing-length theory. Both are incorporated into the resolvent operator to examine the performance of the eddy-viscosity-improved resolvent-based reduced-order modelling. Results of the augmented resolvent analysis are compared qualitatively and quantitatively with the first leading mode of spectral proper orthogonal decomposition, by checking the profiles and cross-spectral densities of velocities, density and temperature in two hypersonic turbulent boundary layers under different wall conditions. Higher accuracy of the turbulence prediction is achieved by adding the proposed eddy-viscosity models, particularly for the energetic cycle in the outer-layer region where strong nonlinear energy transfer exists.

Effect of bulk viscosity on the hypersonic compressible turbulent boundary layer

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.

Hybrid RANS/LES modeling of hypersonic turbulent boundary layers with cold walls

Various strategies for hybrid RANS/LES modeling of hypersonic turbulent boundary layers subjected to strong wall-cooling are developed and analyzed. By defining a turbulent thermal length scale, the eddy viscosity and diffusivity of the hybrid model are calculated by the same formula, ℓ2ω. This allows a similar blending of the length scales and the development of analogous dynamic models for turbulent stress and heat flux. The dynamic procedure is applied to heat transfer by equating norms, rather than invoking least-squares with a lower bound, which improves robustness. The shielding function, as used in detached eddy simulations models, is revised to prevent a spurious early switch from the RANS to the LES mode. This revision significantly improves the accuracy of hybrid models on coarse grids. Finally, the proposed models are tested and validated for Mach numbers between 2.5 and 14, and wall-to-recovery temperature ratios between 0.18 and 1.

Characterization of very-large-scale motions in supersonic and hypersonic turbulent boundary layers

Very-large-scale motions are commonly observed in moderate- and high-Reynolds-number wall turbulence, constituting a considerable portion of the Reynolds stress and skin friction. This study aims to investigate the behaviour of these motions in high-speed and high-Reynolds-number turbulent boundary layers at varying Mach numbers. With the aid of high-precision numerical simulations, numerical experiments and theoretical analysis, it is demonstrated that the very-large-scale motions are weakened in high-Mach-number turbulence at the same friction Reynolds numbers, leading to the reduction in turbulent kinetic energy in the outer region. Conversely, the lower wall temperature enhances the very-large-scale motions but shortens the scale separation between the structures in the near-wall and outer regions.

Measurement of the Convection Velocities in a Hypersonic Turbulent Boundary Layer Using Two-Point Cylindrical-Focused Laser Differential Interferometer

A two-point cylindrical-focused laser differential interferometer (2P-CFLDI) system and a conventional Z-type Schlieren were used to measure the hypersonic turbulent boundary layer on a flat plate at Mach number Ma = 6 and Reynolds number Re = 1.08 × 106 m−1. The boundary layer thickness at the measurement location and the noise radiation angle were obtained by post-processing the Schlieren image. The 2P-CFLDI data underwent cross-correlation analysis to calculate the mean convective velocities at different heights and compared with previous experimental and numerical results. The experimentally measured mean convective velocities agree with the trend of available DNS and experimental results. The mean convective velocity near the wall is significantly larger than the local mean velocity and is the main noise source region. Further filtering treatment shows that the convective velocity of the disturbed structure decreases gradually with the increase in the disturbance scale. The differences between convective velocities at different scales are significantly larger outside the boundary layer than inside the boundary layer, which is in agreement with the findings of the previous hot wire experiments. Near the wall, large-scale disturbances mainly determine the localized mean convective velocity, which are the main source of noise radiation for the hypersonic turbulent boundary layer.

Turbulent heat flux and wall heat transfer in hypersonic turbulent boundary layers with wall disturbances

Hypersonic turbulence over disturbed walls can be encountered when the fuselage of the vehicles is ablated or installed with transpiration-cooling porous media, resulting in the variation of the wall heat transfer that complicates its accurate predictions. In this paper, we study the turbulent heat flux and wall heat transfer in hypersonic turbulent boundary layers at the free-stream Mach number of 6.0 subject to wall disturbances with different wall temperatures, with our focus on the compressibility effects. We found that the temperature fluctuation intensity and the turbulent heat flux are enhanced in the outer region, which is attributed to the higher mean temperature gradient induced by the wall disturbances. The solenoidal and dilatational components of the turbulent heat flux, obtained by the Helmholtz decomposition that splits the velocity fluctuations into the solenoidal and dilatational components, are of opposite signs, with the latter smaller in magnitude but showing the tendency of mitigating the former. The integration formula that decomposes the wall heat transfer into various physical processes reveals that the viscous dissipation generates the wall heat transfer, while the turbulent heat flux, work of pressure on dilatation and the mean flow convection transport it towards the free stream. The contribution of the dilatational motions to the viscous dissipation and turbulent heat flux is negligible compared with that of the solenoidal components.

Transducer Resolution Effect on Pressure Fluctuations Beneath Hypersonic Turbulent Boundary Layers

The size of a pressure transducer is known to affect the accuracy of measurements of wall-pressure fluctuations beneath a turbulent boundary layer because of spatial averaging over the sensing area of the transducer. In this paper, the effect of finite transducer size is investigated by applying spatial averaging or wavenumber filters to a database of hypersonic wall pressure generated from a direct numerical simulation (DNS) that simulates the turbulent portion of the boundary layer over a sharp 7° half-angle cone at nominally Mach 8. A good comparison between the DNS and the experiment in the Sandia Hypersonic Wind Tunnel at Mach 8 is achieved after spatial averaging is applied to the DNS data over an area similar to the sensing area of the transducer. The study shows that a finite sensor size similar to that of the PCB132 transducer can cause significant attenuation in the root-mean-square and power spectral density (PSD) of wall-pressure fluctuations, and the attenuation effect is identical between cone and flat plate configurations at the same friction Reynolds number. The Corcos theory is found to successfully compensate for the attenuated high-frequency components of the wall-pressure PSD.

Wall cooling effect on spectra and structures of thermodynamic variables in hypersonic turbulent boundary layers

The wall cooling effect on the spectra and structures of thermodynamic variables are investigated in hypersonic turbulent boundary layers. The density and temperature can be divided into the acoustic and entropic modes based on the Kovasznay decomposition. The intensities of the pressure and the acoustic modes of density and temperature attain the maximum values near the wall, while those of the entropy and the entropic modes of density and temperature achieve their primary peaks near the edge of the boundary layer. In the near-wall region, the pressure and the acoustic modes of density and temperature are significantly enhanced when the wall is strongly cooled, which can be attributed to the appearance of the travelling-wave-like alternating positive and negative structures. Moreover, the intensities of the entropy and the entropic modes of density and temperature become stronger near the wall as the wall temperature decreases, due to the appearance of the streaky entropic structures (SES). The SES are mainly caused by the advection effect of the strong positive wall-normal gradient of the mean temperature associated with ejection and sweep events. It is also found that the profiles of the intensities of the entropy, density and temperature are similar to each other far from the wall, which is mainly due to the reason that the entropic modes are dominant in the fluctuating density and temperature in the far-wall region. The acoustic modes of density and temperature only have significant contributions in the near-wall region.

Conditional analysis on extreme wall shear stress and heat flux events in compressible turbulent boundary layers

This study presents a comprehensive analysis on the extreme positive and negative events of wall shear stress and heat flux fluctuations in compressible turbulent boundary layers (TBLs) solved by direct numerical simulations. To examine the compressibility effects, we focus on the extreme events in two representative cases, i.e. a supersonic TBL of Mach number $M=2$ and a hypersonic TBL of $M=8$ , by scrutinizing the coherent structures and their correlated dynamics based on conditional analysis. As characterized by the spatial distribution of wall shear stress and heat flux, the extreme events are indicated to be closely related to the structural organization of wall streaks, in addition to the occurrence of the alternating positive and negative structures (APNSs) in the hypersonic TBL. These two types of coherent structures are strikingly different, namely the nature of wall streaks and APNSs are shown to be related to the solenoidal and dilatational fluid motions, respectively. Quantitative analysis using a volumetric conditional average is performed to identify and extract the coherent structures that directly account for the extreme events. It is found that in the supersonic TBL, the essential ingredients of the conditional field are hairpin-like vortices, whose combinations can induce wall streaks, whereas in the hypersonic TBL, the essential ingredients become hairpin-like vortices as well as near-wall APNSs. To quantify the momentum and energy transport mechanisms underlying the extreme events, we proposed a novel decomposition method for extreme skin friction and heat flux, based on the integral identities of conditionally averaged governing equations. Taking advantage of this decomposition method, the dominant transport mechanisms of the hairpin-like vortices and APNSs are revealed. Specifically, the momentum and energy transports undertaken by the hairpin-like vortices are attributed to multiple comparable mechanisms, whereas those by the APNSs are convection dominated. In that, the dominant transport mechanisms in extreme events between the supersonic and hypersonic TBLs are indicated to be totally different.

Compressibility effects in supersonic and hypersonic turbulent boundary layers subject to wall disturbances

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.

Coherent structures and turbulent model refinement in oblique shock/hypersonic turbulent boundary layer interactions

In the present study, we investigate the evolution of turbulent statistics and coherent structures in hypersonic turbulent boundary layers at the Mach number of 5 impinged by oblique shock waves generated by the wedge with the angles of 14°, 10°, and 6°, inducing strong, mild, and incipient flow separation, by exploiting direct numerical simulation databases, for the purpose of revealing the underlying flow physics that are of significance to turbulent modeling. We found that the large-scale structures are amplified within the interaction zone, manifested in the form of large-scale low- and high-speed streaks with the spanwise length scale of boundary layer thickness, and gradually decay downstream, the process of which is extremely long. The abrupt variation in the characteristic length, time, and velocity scales as well as the incompatible viscous dissipation of the mean and turbulent kinetic energy results in the incorrect predictions by the Reynolds-Averaged Navier–Stokes (RANS) equation simulations, provided the models are established based on solving the transport equations of the turbulent kinetic equation and its viscous dissipation (k−ε or k−ω models, for instance). To amend this issue, we propose to refine the parameters in the model as the functions of wall pressure, the flow quantities related to multiple flow features. The RANS simulations with the k−ω SST model utilizing the proposed refinement improve greatly the accuracy of the skin friction, wall heat flux, and Reynolds shear stress downstream of the interaction zone, and the wall pressure distributions in hypersonic turbulence over compression ramp, suggesting its promising prospect in engineering applications.

Combined Experimental and Numerical Investigation of a Hypersonic Turbulent Boundary Layer by Means of FLDI and Large-Eddy Simulations

This work investigates a hypersonic turbulent boundary layer over a 7° half angle cone at a wall-to-total temperature ratio of 0.1, M∞=7.4 and Re∞m=4.2×106 m−1, in terms of density fluctuations and the convection velocity of density disturbances. Experimental shock tunnel data are collected using a multi-foci Focused Laser Differential Interferometer (FLDI) to probe the boundary layer at several heights. In addition, a high-fidelity, time-resolved Large-Eddy Simulation (LES) of the conical flowfield under the experimentally observed free stream conditions is conducted. The experimentally measured convection velocity of density disturbances is found to follow literature data of pressure disturbances. The spectral distributions evidence the presence of regions with well-defined power laws that are present in pressure spectra. A framework to combine numerical and experimental observations without requiring complex FLDI post-processing strategies is explored using a computational FLDI (cFLDI) on the numerical solution for direct comparisons. Frequency bounds of 160 kHz <f<1 MHz are evaluated in consideration of the constraining conditions of both experimental and numerical data. Within these limits, the direct comparisons yield good agreement. Furthermore, it is verified that in the present case, the cFLDI algorithm may be replaced with a simple line integral on the numerical solution.

Reynolds number and wall cooling effects on correlations between the thermodynamic variables in hypersonic turbulent boundary layers

The Reynolds number and wall cooling effects on correlations between the thermodynamic variables are systematically investigated in hypersonic turbulent boundary layers by direct numerical simulations. The correlations between the thermodynamic variables and the streamwise velocity are also analysed. The Kovasznay decomposition is introduced to decompose the fluctuating density and temperature into the acoustic and entropic modes. It is found that in the strongly cooled wall cases, the travelling-wave-like alternating positive and negative structures (TAPNSs) are found in the fluctuating pressure and the acoustic modes of density and temperature, and the streaky entropic structures (SESs) are identified in the fluctuating entropy and the entropic modes of density and temperature near the wall. Furthermore, both the acoustic and the entropic modes of density and temperature give significant contributions to the correlations involving density and temperature in the near-wall region, while these correlations are almost totally contributed by the entropic modes in the far-wall region. The entropic modes of the density and temperature are almost linearly correlated with the fluctuating entropy. Therefore, the fact that the fluctuating entropy is strongly correlated with the fluctuating density and temperature far from the wall is mainly due to the dominance of the entropic modes in the fluctuating density and temperature. Moreover, the fluctuating temperature is strongly positively correlated with the fluctuating streamwise velocity near the wall in strongly cooled wall cases, which can be ascribed to the appearance of the TAPNSs and SESs.

Profiles of streamwise velocity and fluctuations in a hypersonic turbulent boundary layer using acetone tagging velocimetry

Profiles of streamwise velocity and fluctuations in a hypersonic turbulent boundary layer using acetone tagging velocimetry

Spectral analysis of turbulence kinetic and internal energy budgets in hypersonic turbulent boundary layers

The spectral transport equations of turbulence kinetic and internal energy (TKE and TIE) are rigorously derived in the compressible turbulent flows, exhibiting a full structural similarity in terms of two-point correlations of velocity and a sound-speed-like variable. The scale dependence of TKE and TIE flow across space and scales, and among components is revealed comprehensively in a similar manner, in hypersonic turbulent boundary layers. Special attention is paid to the effect of wall cooling on the energy evolution, which is observed to exist throughout the boundary layer primarily by means of modifying the small-scale motions.

Amplification of turbulent kinetic energy and temperature fluctuation in a hypersonic turbulent boundary layer over a compression ramp

In this paper, direct numerical simulations in a Mach 6.0 hypersonic turbulent boundary layer over a 30 ° compression ramp are performed. The influence of shock wave/boundary layer interactions on the amplification of turbulent kinetic energy (TKE) and temperature fluctuation (TF) is explored, to provide an insight into the physical mechanism. In the initial part of the interaction region before the detachment of the shear layer, the amplification of the TKE and TF is found, via a frequency spectrum analysis, to be closely related to the low-frequency unsteadiness of the shock wave. Once the free shear layer is established, the shear component of the TKE production defined in the shear layer coordinate appears to act as the main contributor for the TKE amplification, owing to the mixing layer turbulence and the resultant Kelvin–Helmholtz instability. This is consistent with the result from the spectrum analysis that the TKE and TF amplification and their streamwise evolution are dominated by the spectral energy in the median-frequency range, arising from the mixing layer turbulence. As the flow moves downstream along the shock wave, the high-frequency spectral energy content of TF shows a decreasing trend, while the low-frequency spectral energy tends to increase gradually, implying that the shock wave low-frequency unsteadiness exists not only in the initial stage of the interaction region but also around the main shock wave. Under the combined influence of the shock wave intensity and interaction intensity, the median-frequency content appears to weaken first and then tends to increase before decreasing again. The variation amplitude appears to be small and generally dominates the distribution of the TF intensity.

Resolvent-based analysis of hypersonic turbulent boundary layers with/without wall cooling

The ability of the low-rank approximation of hypersonic turbulent boundary layers with/without wall cooling is examined with the linear resolvent operator in a compressible form. The freestream Mach number of the base flow is 5.86, and the friction Reynolds number is 420. The wall-to-recovery temperature ratio is set as 1.0 and 0.25, respectively, corresponding to an adiabatic wall condition and a cold-wall condition. Different from the resolvent analysis of incompressible turbulent boundary layers, the optimal response mode in the wave-parameter space exhibits a relatively subsonic and a relatively supersonic region [Bae et al., “Resolvent-based study of compressibility effects on supersonic turbulent boundary layers,” J. Fluid Mech. 883, A29 (2020)], divided by the freestream relative Mach number of unity. The features of energy distribution of the optimal response mode in space and scales are examined, and the energy spectra of streamwise velocity and temperature fluctuations, carried by the optimal response mode, are discussed with typical subsets of streamwise and spanwise wavelengths. This reveals the dynamics of the near-wall small-scale and outer larger-scale motions and the distinction in the relatively subsonic/supersonic region. Moreover, the coherent structures, including the velocity and temperature streaks, quasi-streamwise vortices, and large-scale/very-large-scale motions, are identified in the optimal response mode. Special attention is paid to the effects of wall cooling.

Turbulence Effect on Transpiration Cooling Effectiveness Over a Flat Plate in Hypersonic Flow and Sensitivity to Injection Parameters

This work presents a numerical study of coolant porous injection in hypersonic turbulent boundary layer, with an analysis of blowing ratio and pore diameter effects on the cooling performance. Direct numerical simulations (DNS) are carried out for a Mach 5 flow over a flat plate with induced transition, and with a porous injection model to mimic injection from a bed of equally-spaced circular pores. The cooling performance in turbulent flow is compared to laminar 2D flow cases. Results show downstream development of a turbulent wedge-shaped structure, where a dramatic decay of the near-wall coolant concentration is observed. Blowing ratio and pore size are seen to affect the calmed and transitional regions, however they have a marginal or negligible effect within the turbulent region. A cooling effectiveness deficit/reduction of 15% to 30% for the turbulent cases, with respect to the laminar 2D cases, is observed above the injection region due to the 3D flow effects associated with the porous injection, whereas it reaches values as high as 80% in the developed turbulent region due to the turbulent convective effects. The present results shed light on the effects of turbulence on porous wall cooling and clearly indicate that alternative (ad-hoc) injection strategies and parameter calibration are needed to guarantee appropriate wall cooling in a turbulent flow.