Spanwise Length Scale Research Articles

Effects of streamwise-elongated and spanwise-periodic surface roughness arrays on supersonic boundary-layer instability

We investigate the influence of streamwise-elongated and spanwise-periodic roughness arrays on the supersonic boundary-layer instability under the assumption of a high Reynolds number. The main focus is on the lower-branch viscous instability modes, and the spanwise spacing of the roughness arrays is taken to be comparable to the characteristic wavelength of the modes (which is on the triple-deck scale), so that most significant effects can be generated. The streamwise length scale of the elements is much greater than the spanwise length scale. The roughness height is determined by requiring the change of the wall shear to be O(1). The equations governing the nonlinear roughness-induced streaky flow are deduced from the standard triple-deck theory. These equations are parabolic in the streamwise direction and are solved using a streamwise marching method to characterize the evolution of streaky structures. The linear stability of the streaky flow is analyzed. By exploiting the asymptotic structure, the bi-global eigenvalue problem is reduced to a one-dimensional one, where the stability is found to be controlled by the spanwise-dependent wall shear. The reduced eigenvalue problem is solved numerically. The results show that roughness arrays inhibit instability modes with moderate frequencies but promote high-frequency modes. Roughness elements of greater height have stronger effects on the linear stability. The shape of roughness elements plays an important role. A significant feature, different from the subsonic case, is that fundamental and superharmonic resonance modes radiate sound waves spontaneously into the far field.

Inter-scale Causality Relations in Wall Turbulence

We investigate the inter-scale energy transfer in the buffer layer between different streamwise and spanwise lengthscales. In a periodic channel, inter-scale transfer of momentum in the periodic directions is only possible through wavenumber interactions in the non-linear advection term. The aim of this study is to map the inter-scale interactions which transfer energy through advection to different lengthscales of interest. The non-linear advection term has nine contributions, which correspond to three different advecting and advected velocity components. Because the flow is anisotropic, each of these contributions can be viewed as portraying a different physical mechanism. Maps of inter-scale transfer are therefore constructed for each of the nine contributions to the advection term in order to understand the relative importance of advecting and advected velocity components in wall turbulence. As a preliminary exploration, two maps of the inter-scale energy transfer in the buffer layer are presented for Reτ = 180, one for transfer to a ‘target’ lengthscale dynamically relevant in the near-wall cycle, λx+=188 , λz+=94 and one for transfer to a ‘target’ lengthscale in the dissipative range, λx+=42 , λz+=21 . In the first case, we observe that the transfer of energy is dominated by transfer into streamwise-velocity (u) by spanwise-elongated structures of spanwise-velocity (w) that advect streamwise-elongated structures of streamwise-velocity (u). This is nothing but the process of streak meandering induced by spanwise-elongated (w) structures. In the second case, the inter-scale transfer can be split into three dominant mechanisms. The first and second mechanisms are observed across many of the nine contributions to the advection term. Velocity structures with size similar or larger than streamwise streaks either advect (mechanism 1) or are advected by (mechanism 2) velocity structures with lengthscales close to the target one. The third mechanism is only significant for streamwise (u) velocity structures advecting streamwise (u) and spanwise (w) velocity, and involves structures with an intermediate scale which is in-between the dissipation scale and the near-wall cycle dynamic lengthscale, but still in the dissipation range. In this mechanism, intermediate-scale streamwise-velocity structures advect intermediate-scale streamwise- and spanwise-velocity structures.

Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Wall heat flux in supersonic turbulent expansion flow with shock impingement

We perform direct numerical simulations to investigate the characteristics of wall heat flux (WHF) in the interaction of an oblique shock wave at an angle of 33.2° and free-stream Mach number M ∞ = 2.25 impinging on supersonic turbulent expansion corners with deflection angles of 0o (flat plate), 6o and 12o. The effect of the expansion on the WHF characteristics is analysed by comparing it to the interaction with the flat plate under the same flow conditions and a fixed shock impingement point. In the post-expansion region, the decreased mean WHF is found to collapse onto the flat plate case when scaled with the mean wall pressure. The statistical properties of the WHF fluctuations, including probability density function, frequency spectra, and space–time correlations, are comparatively analysed. The expansion causes an increase in the occurrence probability of negative extreme events, an enhancement of high-frequency energy, and an inhibition of intermediate-frequency energy. The increased expansion angle also results in a faster recovery of characteristic spanwise length scales and an increase in convection velocity. We use the mean WHF decomposition method in conjunction with bidimensional empirical mode decomposition to quantitatively analyse the impact of expansion on scale contributions. It is demonstrated that the presence of the expansion corner has no significant impact on the decomposed results, but it significantly reduces the contribution associated with outer large-scale structures.

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.

Trailing edge noise reduction using bio-inspired finlets

Inspired by the silent flight of the owl, the current work focuses on the reduction of trailing edge noise using 2D finlets. Overset-Large Eddy Simulations are carried out of two configurations, (i) the baseline configuration consisting of a NACA0012 airfoil and (ii) an identical configuration, however, now with finlets installed near the trailing edge. The considered chord based Reynolds number Rec equals 4.2×105 for a non-zero angle of attack α of 4°. The resulting turbulence statistics in the boundary layer and the power spectral density (PSD) spectrum of the surface pressure fluctuations show good agreement with the experimental data. Detailed analysis of the unsteady flow data revealed various noise reduction mechanisms for the here considered case. Firstly, finlets provide a ‘lifting-up’ effect for the most energetic eddies in the boundary layer, which prolongs downstream of the finlets. As a result, the edge scattering is significantly weakened. Secondly, the finlets mounted on the airfoil provide an additional wetted area to the flow which in-turn has a dissipating effect on the surface pressure fluctuations, thereby again weakening the edge scattering phenomena. A reduction in the flow velocity is observed for the flow exiting the finlets channel and towards the trailing edge, which also has a favourable effect in reducing the trailing-edge noise. Finally, a marginal breakdown of the span-wise coherent length scale for frequencies between 2.5 kHz to 9 kHz is also observed.

Characteristic spanwise length scales of streamwise vortical structures in the wake of a circular cylinder at Re = 1500 measured via global and local approaches

Wake characteristics of the flow past a circular cylinder are analysed in detail at Reynolds number Re=1500 via direct numerical simulation. A periodic spanwise domain of length 1.5πD has been found to yield correct first- and second-order wake statistics in remarkable agreement with published results at the same and closeby Re. A Kelvin–Helmholtz instability with a frequency fKH≃0.666 is observed to occur intermittently in the shear layers issued from the top and bottom of the cylinder. The three-dimensional patterns in the wake have an estimated spanwise length scale lz1/D≃0.70 (D is the cylinder diameter) in the near-wake at (x,y)/D=(3,0.5), downstream from the cylinder, when quantified by autocorrelation (global approach). When using the Hilbert-transform (local approach) instead, the predicted length scale of streamwise vortical structures is distributed around λz/D≃0.33 at the same sampling location. Our results show that the two approaches measure different aspects of three-dimensionality: while the former informs of the typical spanwise spacing of streamwise vortices, the latter quantifies the local spanwise size of these same flow structures.

Effects of Geometry of Wings Submerged in Turbulent Bluff-Body Wake

The effects of the aspect ratio, the sweep angle, and the leading-edge geometry of wings placed in a turbulent wake were investigated in wind-tunnel experiments at a chord Reynolds number of . The poststall lift enhancement due to the leading-edge vortex formation was studied at optimal locations in the wake. The effects of the strength of the leading-edge vortices, the ratio of the spanwise length scale of the incident vortex to the wingspan, and the degree of two-dimensionality of the wake–wing interaction were studied. The competition between the effects of the spanwise length scale of the incident wake and the strength of the leading-edge vortices determined the optimal aspect ratio, which was found to be around four. Increasing the sweep angle decreased the mean lift due to the decreased two-dimensionality of the vortex formation. Airfoils with sharp leading edge produced the strongest leading-edge vortices but further away from the wing surface, resulting in lower maximum lift. Relative to the performance in the undisturbed freestream, the increases in the stall angle and maximum lift coefficient were not significantly affected by the leading-edge shape.

The state space and travelling-wave solutions in two-scale wall-bounded turbulence

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.

Hypersonic shock wave and turbulent boundary layer interaction in a sharp cone/flare model

A direct numerical simulation of hypersonic Shock wave and Turbulent Boundary Layer Interaction(STBLI) at Mach 6.0 on a sharp 7° half-angle circular cone/flare configuration at zero angle of attack is performed. The flare angle is 34° and the momentum thickness Reynolds number based on the incoming turbulent boundary layer on the sharp circular cone is Reθ = 2506. It is found that the mean flow is separated and the separation bubble occurring near the corner exhibits unsteadiness. The Reynolds analogy factor changes dramatically across the interaction, and varies between 1.06 and 1.27 in the downstream region, while the QP85 scaling factor has a nearly constant value of 0.5 across the interaction. The evolution of the reattached boundary layer is characterized in terms of the mean profiles, the Reynolds stress components, the anisotropy tensor and the turbulence kinetic energy. It is argued that the recovery is incomplete and the near-wall asymptotic behavior does not occur for the hypersonic interaction. In addition, mean skin friction decomposition in an axisymmetric turbulent boundary layer is carried out for the first time. Downstream of the interaction, the contributions of transverse curvature and body divergence are negligible, whereas the positive contribution associated with the turbulence kinetic energy production and the negative spatial-growth contribution are dominant. Based on scale decomposition, the positive contribution is further divided into terms with different spanwise length scales. The negative contribution is analyzed by comparing the convective term, the streamwise-heterogeneity term and the pressure gradient term.

Linearised Reynolds-averaged predictions of secondary currents in turbulent channels with topographic heterogeneity

A rapid predictive tool based on the linearised Reynolds-averaged Navier–Stokes equations is proposed in this work to investigate secondary currents generated by streamwise-independent surface topography modulations in turbulent channel flow. The tool is derived by coupling the Reynolds-averaged momentum equation to the Spalart–Allmaras transport equation for the turbulent eddy viscosity, using a nonlinear constitutive relation for the Reynolds stresses to capture correctly secondary motions. Linearised equations, describing the steady flow response to arbitrary surface modulations, are derived by assuming that surface modulations are shallow. Since the equations are linear, the superposition principle holds and the flow response induced by an arbitrary modulation can be obtained by combining appropriately the elementary responses obtained over sinusoidal modulations at multiple spanwise length scales. The tool permits a rapid exploration of large parameter spaces characterising structured surface topographies previously examined in the literature. Here, channels with sinusoidal walls and with longitudinal rectangular ridges are considered. For sinusoidal walls, a large response is observed at two spanwise wavelengths scaling in inner and outer units respectively, mirroring the amplification mechanisms in turbulent shear flows observed from transient growth analysis. For longitudinal rectangular ridges, the model suggests that the analysis of the response and the interpretation of the topology of secondary structures is facilitated when the ridge width and the gap between ridges are used instead of other combinations proposed in the literature.

Optical Diagnostics of Spanwise-Uniform Flows

Aerooptical aberrations along the spanwise direction of a canonical subsonic turbulent boundary layer were measured and studied nonintrusively using a Shack–Hartmann wave front sensor. It was demonstrated that in this case some important fluidic statistics in the wall-normal direction, like the mean velocity profiles, the local skin friction, and the spanwise integral scales, can be directly extracted from aerooptical aberrations. To avoid various spectral contamination in optical data at low frequencies, a model function for the deflection angle autospectral density at low frequencies was proposed. The spectral cross-correlation method and the dispersion method were used to extract the local convective velocities, and the dispersion analysis was demonstrated to be most accurate in computing the velocities. It was shown that it is possible to reconstruct the spectra above the Nyquist frequencies through the newly proposed stacking method. Convective velocities in the log region of the turbulent boundary layer were found to agree well with the direct measurements using a single hot wire. From the convective velocities, the local wall shear stress was nonintrusively extracted, using the Clauser method. Using corrected deflection angle spectra and the convective velocity, the local values of aerooptical aberrations were reconstructed. Finally, using the strong Reynold’s analogy, a wall-normal profile of the spanwise density correlation length was estimated and shown to be in good qualitative agreement with velocity-based spanwise length scales observed in the literature.

De-asymmetry of small-scale motions in wall-bounded turbulence

The present work focuses on the symmetry-breaking phenomenon in large-to-small amplitude modulation (AM) of wall-bounded turbulence. Using the recently proposed multi-component variational mode decomposition method, the volumetric velocity fields of a spatially developing turbulent boundary layer being obtained by direct numerical simulation are decomposed into four three-dimensional intrinsic mode functions (IMFs), whose spanwise length scales are fully separated from each other. It is found that the first IMF mainly characterizes the streamwise large-scale (LS) and very-large-scale turbulent motions. Splatting and sputtering events are observed in the second IMF (2IMF), leading to the biased conditional probability density functions of near-wall u2IMF and v2IMF under the condition of extreme large-scale motions. This is in distinct contrast to the hierarchical self-similarity of the wall-normal profiles of Reynolds shear stress (RSS) of the last two IMFs (3IMF and 4IMF). When treating 2IMF as components of small-scale (SS) turbulent motions, such splatting and sputtering events lead to asymmetric AM effect; that is, the AM coefficients corresponding to positive and negative LS motions are asymmetric to each other. The underlying reason is that these strong quadrant events are spatially asymmetric and are tightly coupled with local LS motions. Based on this observation, a de-spatial-asymmetry (DSA) method is proposed to obtain asymmetry-free “universal” SS turbulent motions. This method includes the removal of the transitional 2IMF from SS motions, the de-amplitude modulation, and the length-scale rescaling. Analysis of single-point velocity statistics, RSS, as well as velocity spectrum, shows that the “universality” of SS motions derived from the DSA method is remarkably improved.

Wall heat flux in a supersonic shock wave/turbulent boundary layer interaction

The characteristics of wall heat flux (WHF) beneath a supersonic turbulent boundary layer interacting with an impinging shock wave with a 33.2° angle at Mach 2.25 are analyzed using direct numerical simulation. It is found that the QP85 scaling, defined as the ratio of the mean WHF and wall pressure, changes across the interaction. The probability density function of the WHF fluctuations normalized by the local root-mean-squared value is similar to that of wall shear stress. Comparing the WHF and wall pressure spectra shows that the low-frequency shock unsteadiness exhibits little influence on the spectrum. The space–time correlation of the fluctuating WHF reveals that both the streamwise correlation length scale and the convection velocity experience a sharp decrease in the separation region and subsequent recovery in the downstream region. Moreover, the mean WHF in an incident shock interaction is decomposed for the first time. An analysis of the velocity and temperature fluctuations based on bidimensional empirical mode decomposition is performed to evaluate the contribution of turbulent structures with specific spanwise length scales to the mean WHF generation. The decomposed results indicate that the contribution associated with the large-scale structures in the outer region is greatly amplified by the shock interaction and has the leading role in the generation downstream of the interaction.

Large eddy simulations and modal decomposition analysis of flow past a cylinder subject to flow-induced vibration

Large eddy simulations (LES) are carried out to investigate the flow around a vibrating cylinder in the subcritical Reynolds number regime at Re = 3900. Three reduced velocities, Ur = 3, 5, and 7, are chosen to investigate the wake structures in different branches of a vortex-induced vibration (VIV) lock-in. The instantaneous vortical structures are identified to show different coherent flow structures in the wake behind the vibrating cylinder for various branches of VIV lock-in. The combined effects of the frequency and amplitude of the oscillation on the flow pattern in the wake region, the hydrodynamic quantities of the cylinder, and the spanwise length scale of the energetic wake flow structures are discussed in detail. It is found that the typical spanwise lengths of the flow structures are 0.22D at Ur = 5 and 0.3D at Ur=[3,7] in the near-wake region and level out at 0.5D further downstream. Furthermore, multiscale proper orthogonal decomposition (mPOD) is used to analyze the dominant flow features in the wake region. With the increasing Ur, the total kinetic energy contribution of superharmonic modes increases and the contribution of subharmonic modes decreases. The dominant flow characteristics associated with the vortex shedding and their super harmonics, and the low-frequency modulation of the wake flow can be captured by the mPOD modes.

Aerodynamics of a wing in turbulent bluff body wakes

The aerodynamics of a stationary wing in a turbulent wake are investigated. Force and velocity measurements are used to describe the unsteady flow. Various wakes are studied with different dominant frequencies and length scales. In contrast to the pre-stall angles of attack, the time-averaged lift increases substantially at post-stall angles of attack as the wing interacts with the von Kármán vortex street and experiences temporal variations of the effective angle of attack. At an optimal offset distance from the wake centreline, the time-averaged lift becomes maximum despite of small amplitude oscillations in the effective angle of attack. The stall angle of attack can reach 20° and the maximum lift coefficient can reach 64 % higher than that in the freestream. Whereas large velocity fluctuations at the wake centreline cause excursions into the fully attached and separated flows during the cycle, small-amplitude oscillations at the optimal location result in periodic shedding of leading edge vortices. These vortices may produce large separation bubbles with reattachment near the trailing-edge. Vorticity roll-up, strength and size of the vortices increase with increasing wavelength and period of the von Kármán vortex street, which also coincides with an increase in the spanwise length scale of the incident wake, and all contribute to the remarkable increase in lift.

Turbulent coherent structures in channel flow with a wall-mounted hemisphere

Turbulent channel flows around a wall-mounted hemisphere numerically are investigated by large eddy simulation, and the Reynolds number based on the hemisphere’s diameter is 3 × 104. The statistical characteristics and turbulent structure evolution are revealed in the Eulerian frameworks and Lagrangian frameworks. The vortex identification and Dynamic Mode Decomposition (DMD) are used to study the evolution of turbulent structure in the Eulerian frameworks, and the finite-time Lyapunov exponents are applied to identify Lagrangian coherent structures (LCS) in the Lagrangian framework. It is found that the developing angle of the hairpin vortex is ∼7° at two frameworks. What is more, there are some hairpin vortices formed behind the hemisphere and some turbulent structures formed near the wall by DMD method. The correlation analysis is applied to investigate the angle variation and scale variation of turbulent structures, and it is observed that the angle of turbulent structures is negative at Y/d ≥ 1.2 and the spanwise length scales of turbulent structures increase as it moves downstream. By studying the LCS behind a wall-mounted hemisphere, there is formation of “kink” caused by viscous interaction between some hairpin vortex legs, which is the characteristic of hairpin vortex deformation. The comparisons of statistical characteristics between Eulerian frameworks and Lagrangian frameworks are conducted by the correlation analysis, the spectrum analysis, and the structure functions.

Stable channel flow with spanwise heterogeneous surface temperature

Recent studies have demonstrated that large secondary motions are excited by surface roughness with dominant spanwise length scales of the order of the flow's outer length scale. Inspired by this, we explore the effect of spanwise heterogeneous surface temperature in weakly to strongly stratified closed channel flow (at$Ri_\tau =120$, 960;$Re_\tau = 180$, 550) with direct numerical simulations. The configuration consists of equally sized strips of high and low temperature at the lower and upper boundaries, while an overall stable stratification is induced by imposing an average temperature difference between the top and bottom. We consider the influence of the width of the strips (${\rm \pi} /8 \leq \lambda /h \leq 4{\rm \pi} $), Reynolds number, stability and upper boundary condition on the mean flow structure, skin friction and heat transfer. Results indicate that secondary flows are excited, with alternating high- and low-momentum pathways and vortices, similar to the patterns induced by spanwise heterogeneous surface roughness. We find that the impact of the surface heterogeneity on the outer layer depends strongly on the spanwise heterogeneity length scale of the surface temperature. Comparison to stable channel flow with uniform temperature reveals that the heterogeneous surface temperature increases the global friction coefficient and reduces the global Nusselt number in most cases. However, for the high-Reynolds cases with$\lambda /h \geq {\rm \pi} /2$, we find a reduction of the friction coefficient. At stronger stability, the vertical extent of the vortices is reduced and the impact of the heterogeneous temperature on momentum and heat transfer is smaller.

Wall shear stress and wall heat flux in a supersonic turbulent boundary layer

We report the characteristics of wall shear stress (WSS) and wall heat flux (WHF) from direct numerical simulation (DNS) of a spatially developing zero-pressure-gradient supersonic turbulent boundary layer at a free-stream Mach number M∞ = 2.25 and a Reynolds number Reτ = 769 with a cold-wall thermal condition (a ratio of wall temperature to recovery temperature Tw/Tr = 0.75). A comparative analysis is performed on statistical data, including fluctuation intensity, probability density function, frequency spectra, and space–time correlation. The root mean square fluctuations of the WHF exhibit a logarithmic dependence on Reτ similar to that for the WSS, the main difference being a larger constant. Unlike the WSS, the probability density function of the WHF does not follow a lognormal distribution. The results suggest that the WHF contains more energy in the higher frequencies and propagates downstream faster than the WSS. A detailed conditional analysis comparing the flow structures responsible for extreme positive and negative fluctuation events of the WSS and WHF is performed for the first time, to the best of our knowledge. The conditioned results for the WSS exhibit closer structural similarities with the incompressible DNS analysis documented by Pan and Kwon [“Extremely high wall-shear stress events in a turbulent boundary layer,” J. Phys.: Conf. Ser. 1001, 012004 (2018)] and Guerrero et al. [“Extreme wall shear stress events in turbulent pipe flows: Spatial characteristics of coherent motions,” J. Fluid Mech. 904, A18 (2020)]. Importantly, the conditionally averaged flow fields of the WHF exhibit a different mechanism, where the extreme positive and negative events are generated by a characteristic two-layer structure of temperature fluctuations under the action of a strong Q4 event or a pair of strong oblique vortices. Nevertheless, we use the bi-dimensional empirical decomposition method to split the fluctuating velocity and temperature structures into four different modes with specific spanwise length scales, and we quantify their influence on the mean WSS and WHF generation. It is shown that the mean WSS is mainly related to small-scale structures in the near-wall region, whereas the mean WHF is associated with the combined action of near-wall small-scale structures and large-scale structures in the logarithmic and outer regions.

Characteristics of reattached boundary layer in shock wave and turbulent boundary layer interaction

The reattached boundary layer in the interaction of an oblique shock wave with a flat-plate turbulent boundary layer at Mach number 2.25 is studied by means of Direct Numerical Simulation (DNS). The numerical results are carefully compared with available experimental and DNS data in terms of turbulence statistics, wall pressure and skin friction. The coherent vortex structures are significantly enhanced due to the shock interaction, and the reattached boundary layer is characterized by large-scale structures in the outer region. The space-time correlation of fluctuating wall shear stress and streamwise velocity fluctuation reveals that the structural inclination angle exhibits a gradual decrease during the recovery process. The scale interactions are analyzed by using a two-point amplitude modulation correlation. A possible mechanism is proposed to account for the strong amplitude modulation in the downstream region. Moreover, the mean skin-friction is decomposed to understand the physically informed contributions. Unlike the upstream Turbulent Boundary Layer (TBL), the contribution associated with the Turbulence Kinetic Energy (TKE) production is greatly amplified, while the spatial growth contribution induced by the pressure gradient largely inhibits skin-friction generation. Based on bidimensional empirical mode decomposition, the turbulence kinetic energy production contribution is further split into different terms with specific spanwise length scales.