The ability of streamwise-travelling waves of spanwise velocity to reduce the turbulent skin-friction drag is assessed in the compressible regime. Direct numerical simulations are carried out to compare drag reduction in subsonic, transonic and supersonic channel flows. Compressibility improves the benefits of the travelling waves, in a way that depends on the control parameters: drag reduction becomes larger than the incompressible one for small frequencies and wavenumbers. However, the improvement depends on the specific procedure employed for comparison. When the Mach number is varied and, at the same time, wall friction is changed by the control, the bulk temperature in the flow can either evolve freely in time until the aerodynamic heating balances the heat flux at the walls, or be constrained such that a fixed percentage of kinetic energy is transformed into thermal energy. Physical arguments suggest that, in the present context, the latter approach should be preferred. This provides a test condition in which the wall-normal temperature profile more realistically mimics that in an external flow, and also leads to a much better scaling of the results, over both the Mach number and the control parameters. Under this comparison, drag reduction is only marginally improved by compressibility.

High-speed vehicles experience a highly challenging environment in which the freestream Mach number and surface temperature greatly influence aerodynamic drag and heat transfer. The interplay of these two parameters strongly affects the near-wall dynamics of high-speed turbulent boundary layers (TBLs) in a non-trivial way, breaking similarity arguments on velocity and temperature fields, typically derived for adiabatic cases. We present direct numerical simulations of flat-plate zero-pressure-gradient TBLs spanning three freestream Mach numbers $[2,4,6]$ and four wall temperature conditions (from adiabatic to very cold walls), emphasising the choice of the wall-cooling parameter to recover a similar flow organisation at different Mach numbers. We link qualitative observations on flow patterns to first- and second-order statistics to explain the decoupling of temperature–velocity fluctuations that occurs at reduced wall temperatures and high Mach numbers. For these cases, we discuss the formation of a secondary peak of thermal production in the viscous sublayer, which is in contrast with the monotonic behaviour of adiabatic profiles. We propose different physical mechanisms induced by wall-cooling and compressibility that result in apparently similar flow features, such as a higher peak in the streamwise velocity turbulence intensity, and distinct features, such as the separation of turbulent scales.

A direct numerical simulation of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.28 is carried out at moderate Reynolds number, simulating flow conditions similar to those of the experiment by Dupont et al. (J. Fluid Mech., vol. 559, 2006, pp. 255–277). The low-frequency shock unsteadiness, whose characteristics have been the focus of considerable research efforts, is here investigated via the Morlet wavelet transform. Owing to its compact support in both physical and Fourier spaces, the wavelet transformation makes it possible to track the time evolution of the various scales of the wall-pressure fluctuations. This property also makes it possible to define a local intermittency measure, representing a frequency-dependent flatness factor, to pinpoint the bursts of energy that characterise the shock intermittency scale by scale. As a major result, wavelet decomposition shows that the broadband shock movement is actually the result of a collection of sparse events in time, each characterised by its own temporal scale. This feature is hidden by the classical Fourier analysis, which can only show the time-averaged behaviour. Then, we propose a procedure to process any relevant time series, such as the time history of the wall pressure or that of the separation bubble extent, in which we use a condition based on the local intermittency measure to filter out the turbulent content in the proximity of the shock foot and to isolate only the intermittent component of the signal. In addition, wavelet analysis reveals the intermittent behaviour also of the breathing motion of the recirculation bubble behind the reflected shock, and allows us to detect a direct, partial correspondence between the most significant intermittent events of the separation region and those of the wall pressure at the foot of the shock.

We study the structure of high-speed zero-pressure-gradient turbulent boundary layers up to friction Reynolds number $Re_{\tau } \approx 2000$ using direct numerical simulation of the Navier–Stokes equations. Both supersonic and hypersonic conditions with nominal free-stream Mach numbers $M_{\infty }=2$, $M_{\infty }=5.86$ and heat transfer at the wall are considered. The present simulations extend the database currently available for wall-bounded flows, enabling us to explore high-Reynolds-number effects even in the hypersonic regime. We first analyse the instantaneous fields to characterize the structure of both velocity and temperature fluctuations. In all cases elongated strips of uniform velocity and temperature (superstructures) are observed in the outer portion of the boundary layer, characterized by a clear association between low-/high-speed momentum and high/low temperature streaks. The results highlight important deviations from the typical organization observed in the inner region of adiabatic boundary layers, revealing that the near-wall temperature streaks disappear in strongly non-adiabatic flow cases. We also focus on the structural properties of regions of uniform streamwise momentum (De Silva, Hutchins & Marusic, J. Fluid Mech., vol. 786, 2016, pp. 309–331) observed in turbulent boundary layers, confirming the presence of such zones in the high-speed regime at high Reynolds number and revealing the existence of similar regions for the temperature field. The accuracy of different compressibility transformations and temperature–velocity relations is assessed extending their range of validation to moderate/high Reynolds numbers. Spanwise spectral densities of the velocity and temperature fluctuations at various wall distances have been calculated revealing the energy content and the size of the turbulent eddies across the boundary layer. Finally, we propose a revised scaling for the characteristic length scales, that is based on the local mean shear computed according to the recent theory by Griffin, Fu & Moin [Proc. Natl Acad. Sci. USA, vol. 118 (34)].

We perform direct numerical simulation of supersonic turbulent channel flow over cubical roughness elements, spanning bulk Mach numbers $M_b=0.3$–$4$, both in the transitional and fully rough regime. We propose a novel definition of roughness Reynolds number which is able to account for the viscosity variations at the roughness crest and should be used to compare rough-wall flows across different Mach numbers. As in the incompressible flow regime, the mean velocity profile shows a downward shift with respect to the baseline smooth wall cases, however, the magnitude of this velocity deficit is largely affected by the Mach number. Compressibility transformations are able to account for this effect, and data show a very good agreement with the incompressible fully rough asymptote, when the relevant roughness Reynolds number is used. Velocity statistics present outer layer similarity with the equivalent smooth wall cases, however, this does not hold for the thermal field, which is substantially affected by the roughness, even in the channel core. We show that this is a direct consequence of the quadratic temperature–velocity relation which is also valid for rough walls. Analysis of the heat transfer shows that the relative drag increase is always larger than the relative heat transfer enhancement, however, increasing the Mach number brings data closer to the Reynolds analogy line due to the rising relevance of the aerodynamic heating.

We study turbulent flows in pressure-driven planar channels with imposed unstable thermal stratification, using direct numerical simulations in a wide range of Reynolds and Rayleigh numbers and reaching flow conditions which are representative of fully developed turbulence. The combined effect of forced and free convection produces a peculiar pattern of quasi-streamwise rollers occupying the full channel thickness, with aspect ratio considerably higher than unity; it has been observed that they have an important redistributing effect on temperature and momentum, providing for a substantial fraction of the heat and momentum flux at bulk Richardson numbers larger than $0.01$. The mean values and the variances of the flow variables do not appear to follow Prandtl’s scaling in the free-convection regime, except for the temperature and vertical velocity fluctuations, which are more directly affected by wall-attached turbulent plumes. We find that the Monin–Obukhov theory nevertheless yields a useful representation of the main flow features. In particular, the widely used Businger–Dyer flux-profile relationships are found to provide a convenient way of approximately accounting for the bulk effects of friction and buoyancy, although the individual profiles may have wide scatter from the alleged trends. Significant deviations are found in direct numerical simulations with respect to the commonly used parametrization of the momentum flux in the light-wind regime, which may have important practical impact in wall models of atmospheric dynamics. Finally, for modelling purposes, we devise a set of empirical predictive formulae for the heat flux and friction coefficients, which are within approximately $10\,\%$ standard deviation from the numerical results in a wide range of flow parameters.

We study passive scalars in turbulent plane channels at computationally high Reynolds number, thus allowing us to observe previously unnoticed effects. The mean scalar profiles are found to obey a generalized logarithmic law which includes a linear correction term in the whole lower half-channel, and they follow a universal parabolic defect profile in the core region. This is consistent with recent findings regarding the mean velocity profiles in channel flow. The scalar variances also exhibit a near universal parabolic distribution in the core flow and hints of a sizeable log layer, unlike the velocity variances. The energy spectra highlight the formation of large scalar-bearing eddies with size proportional to the channel height which are caused by a local production excess over dissipation, and which are clearly visible in the flow visualizations. Close correspondence of the momentum and scalar eddies is observed, with the main difference being that the latter tend to form sharper gradients, which translates into higher scalar dissipation. Another notable Reynolds number effect is the decreased correlation of the passive scalar field with the vertical velocity field, which is traced to the reduced effectiveness of ejection events.

Direct numerical simulation of the spatially developing mixing layer issuing from two turbulent streams past a splitter plate is carried out under mild compressibility conditions. The study mainly focuses on the early evolution of the mixing region, where transition occurs from a wake-like to a canonical mixing-layer-like behaviour, corresponding to the filling-up of the initial momentum deficit. The mixing layer is found to initially grow faster than linearly, and then at a sub-linear rate further downstream. The Reynolds stress components are in close agreement with reference experiments and follow a continued slow decay till the end of the computational domain. These observations are suggestive of the occurrence of incomplete similarity in the developing turbulent mixing layer. Coherent eddies are found to form in the close proximity of the splitter plate trailing edge, that are mainly organized in bands, initially skewed and then parallel to the spanwise direction. Dynamic mode decomposition is used to educe the dynamically relevant features, and it is found to be capable of singling out the coherent eddies responsible for mixing layer development.

We present an extensive compilation of direct numerical simulation (DNS) data for Poiseuille and Couette flows, from the laminar into the fully turbulent regime, with the goal of highlighting similarities and differences. The data suggest that, for a given bulk velocity, Couette flow yields less resistance than Poiseuille flow and greater turbulence kinetic energy, which may be beneficial for more efficient diffusion, thus suggesting the effectiveness of fluid transport devices based on moving belts as opposed to classical ducts. Both flows exhibit similar trends for the wall-parallel velocity variances, which increase logarithmically with the Reynolds number. The shear stress and the wall-normal stress tend to saturate faster in Couette flow, which can thus be regarded as a limit to which Poiseuille flow tends, in the limit of high Reynolds number. Excess production over dissipation is found in the outer part of Poiseuille and Couette flow, which is responsible for non-local transfer of energy. However, the structure of the core flow seems to attain an asymptotic state which consists of a parabolic and linear mean velocity profile, respectively, and it seems unlikely that substantial changes to this scenario will occur at Reynolds numbers reachable by DNS in the foreseeable future.

We investigate the behaviour of the canonical turbulent Couette flow at computationally high Reynolds number through a series of large-scale direct numerical simulations. We achieve a Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = h/\delta _v \approx 1000$, where $h$ is the channel half-height and $\delta _v$ is the viscous length scale at which some phenomena representative of the asymptotic Reynolds-number regime manifest themselves. While a logarithmic mean velocity profile is found to provide a reasonable fit of the data, including the skin friction, closer scrutiny shows that deviations from the log law are systematic, and probably increasing at higher Reynolds numbers. The Reynolds stress distribution shows the formation of a secondary outer peak in the streamwise velocity variance, which is associated with significant excess of turbulent production as compared to the local dissipation. This excess is related to the formation of large-scale streaks and rollers, which are responsible for a substantial fraction of the turbulent shear stress in the channel core, and for significant increase of the turbulence intermittency in the near-wall region.

The effect of the Reynolds number on the behaviour of inertial particles in wall-bounded turbulent flows is investigated through large-scale direct numerical simulations (DNS) of particle-laden canonical channel flow spanning almost a decade in the friction Reynolds number, from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = 150$ to $\mathit{Re}_{\tau } = 1000$. Lagrangian particle tracking is used to study the motion of six different particle sets, described by a Stokes number in the range $\mathit{St} = 1\text {--}1000$. At all Reynolds numbers a strong segregation in the near-wall region is observed for particles characterized by intermediate Stokes number, in the range $\mathit{St} =10\text {--}100$. The wall-normal concentration profiles of such particles collapse in inner scaling, thus suggesting the independence of the turbophoretic drift from the large-scale outer motions. This observation is also supported by the spatial organization of the suspended phase in the inner layer, which is found to be universal with the Reynolds number. The deposition rate coefficient increases with $\mathit{Re}_{\tau }$ for a given $\mathit{St}$. Suitable inner and outer scalings are proposed to collapse the deposition curves across the available ranges of Reynolds and Stokes numbers for the different deposition regimes.

The high-Reynolds-number behaviour of the canonical incompressible turbulent channel flow is investigated through large-scale direct numerical simulation (DNS). A Reynolds number is achieved ($Re_{\tau } = h/\delta _v \approx 4000$, where $h$ is the channel half-height, and $\delta _v$ is the viscous length scale) at which theory predicts the onset of phenomena typical of the asymptotic Reynolds number regime, namely a sensible layer with logarithmic variation of the mean velocity profile, and Kolmogorov scaling of the velocity spectra. Although higher Reynolds numbers can be achieved in experiments, the main advantage of the present DNS study is access to the full three-dimensional flow field. Consistent with refined overlap arguments (Afzal & Yajnik, J. Fluid Mech. vol. 61, 1973, pp. 23–31; Jiménez & Moser, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007, pp. 715–732), our results suggest that the mean velocity profile never achieves a truly logarithmic profile, and the logarithmic diagnostic function instead exhibits a linear variation in the outer layer whose slope decreases with the Reynolds number. The extrapolated value of the von Kármán constant is $k \approx 0.41$. A near logarithmic layer is observed in the spanwise velocity variance, as predicted by Townsend’s attached eddy hypothesis, whereas the streamwise variance seems to exhibit a shoulder, perhaps being still affected by low-Reynolds-number effects. Comparison with previous DNS data at lower Reynolds number suggests enhancement of the imprinting effect of outer-layer eddies onto the near-wall region. This mechanisms is associated with excess turbulence kinetic energy production in the outer layer, and it reflects in flow visualizations and in the streamwise velocity spectra, which exhibit sharp peaks in the outer layer. Associated with the outer energy production site, we find evidence of a Kolmogorov-like inertial range, limited to the spanwise spectral density of $u$, whereas power laws with different exponents are found for the other spectra. Finally, arguments are given to explain the ‘odd’ scaling of the streamwise velocity variances, based on the analysis of the kinetic energy production term.

We study the fluid dynamics of rolling wheels at Reynolds number (where is the Reynolds number based on the wheel diameter), with the objective of characterizing the various regimes of steady and unsteady motion. Regardless of the Reynolds number, the flow is found to separate approximately upstream of the apex of the wheel, where a saddle point in the pseudo-streamtrace pattern is observed. Under the flow conditions here essayed, the drag coefficient steadily decreases with , and the lift coefficient remains strictly positive. The positive lift provided by the rolling wheel is associated with the presence of a strong (positive) peak of the static pressure in the upstream proximity of the contact point with the ground, which we interpret as the result of the impingement of flow particles entrained in the boundary layer that develops on the front part of the wheel. Steady laminar flow is observed up to , which is characterized by a three-dimensional wake whose length increases with the Reynolds number. Unsteadiness is first observed at , under which conditions the flow retains planar symmetry, and is characterized by the quasi-periodic shedding of hairpin vortices. Transition to three-dimensional flow happens at , in which case a sinuous mode of instability in the wheel wake is established, which modulates the shedding of the hairpins, and which causes the onset of a non-zero side force. At the highest Reynolds number considered here () the wake exhibits some characters of turbulence, with wide-band frequency spectra, and its topology entirely changes, becoming split into two parts, and being much shortened compared to the lower- cases. Despite the limitation of the study to low Reynolds numbers we find that, once significant three-dimensionality and scale separation are established in the wheel wake, the nature of the flow becomes qualitatively similar to fully developed turbulent flow. In perspective, this observation opens interesting avenues for the prediction of unsteady flow around rotating tyres at moderate computational cost.

We study the organization of turbulence in supersonic boundary layers through large-scale direct numerical simulations (DNS) at , and momentum-thickness Reynolds number up to (corresponding to ) which significantly extend the current envelope of DNS in the supersonic regime. The numerical strategy relies on high-order, non-dissipative discretization of the convective terms in the Navier–Stokes equations, and it implements a recycling/rescaling strategy to stimulate the inflow turbulence. Comparison of the velocity statistics up to fourth order shows nearly exact agreement with reference incompressible data, provided the momentum-thickness Reynolds number is matched, and provided the mean velocity and the velocity fluctuations are scaled to incorporate the effects of mean density variation, as postulated by Morkovin’s hypothesis. As also found in the incompressible regime, we observe quite a different behaviour of the second-order flow statistics at sufficiently large Reynolds number, most of which show the onset of a range with logarithmic variation, typical of ‘attached’ variables, whereas the wall-normal velocity exhibits a plateau away from the wall, which is typical of ‘detached’ variables. The modifications of the structure of the flow field that underlie this change of behaviour are highlighted through visualizations of the velocity and temperature fields, which substantiate the formation of large jet-like and wake-like motions in the outer part of the boundary layer. It is found that the typical size of the attached eddies roughly scales with the local mean velocity gradient, rather than being proportional to the wall distance, as happens for the wall-detached variables. The interactions of the large eddies in the outer layer with the near-wall region are quantified through a two-point amplitude modulation covariance, which characterizes the modulating action of energetic outer-layer eddies.

We investigate the organization of the momentum-carrying eddies in turbulent Couette–Poiseuille flows. The study relies on a direct numerical simulation (DNS) database covering a wide range of flow configurations from pure Couette to pure Poiseuille flows, at Reτ ≈ 250 (based on the flow properties at the stationary wall). The study highlights the occurrence of streaky patterns of alternating high and low momentum throughout the channel for all flow configurations, except near zeros of the mean shear, where streaks are suppressed. The mean streak spacing shows a relatively universal distribution in the core of the channel, where it ranges from 50 to 100 local viscous units. The validity of the local viscous scaling in collapsing flow features at different wall distances is confirmed by the analysis of the spanwise velocity spectra, which also highlights (in the case of Couette-like flows) the onset of a secondary low-wavenumber flow mode, superposed on the high-wavenumber flow mode that is responsible for the inner-layer dynamics. The effect of the former mode on the latter is studied by means of the two-point amplitude modulation coefficient, which brings to light a nonlinear modulation phenomenon. Physical mechanisms to explain the modulation effect are proposed, based on the interpretation of the conditional average events. Note that, although similar mechanisms have been previously observed in high-Reynolds-number turbulent boundary layers and channels, the modulation effect is here rather associated with the intrinsic large-scale dynamics of Couette-like flows, and takes place at DNS-accessible Reynolds numbers. We thus believe that the study of Couette-like flows may give an alternative avenue for probing inner/outer interaction effects than canonical channel flows.

The structure of wall pressure fluctuations beneath a turbulent boundary layer interacting with a normal shock wave at Mach number M∞ = 1.3 is studied exploiting a direct numerical simulation database. Upstream of the interaction, in the zero-pressure-gradient region, pressure statistics compare well with canonical low-speed boundary layers in terms of fluctuation intensities, space–time correlations, convection velocities and frequency spectra. Across the interaction zone, the root-mean-square wall pressure fluctuations attain very large values (in excess of 162 dB), with a maximum increase of about 7 dB from the upstream level. The two-point wall pressure correlations become more elongated in the spanwise direction, indicating an increase of the pressure-integral length scales, and the convection velocities (determined from space–time correlations) are reduced. The interaction qualitatively modifies the shape of the frequency spectra, causing enhancement of the low-frequency Fourier modes and inhibition of the higher ones. In the recovery region past the interaction, the pressure spectra collapse very accurately when scaled with either the free-stream dynamic pressure or the maximum Reynolds shear stress, and exhibit distinct power-law regions with exponent −7/3 at intermediate frequencies and −5 at high frequencies. An analysis of the pressure sources in the Lighthill's equation for the instantaneous pressure has been performed to understand their contributions to the wall pressure signature. Upstream of the interaction the sources are mainly located in the proximity of the wall, whereas past the shock, important contributions to low-frequency pressure fluctuations are associated with long-lived eddies developing far from the wall.

The interaction of a normal shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number M∞ = 1.3 and Reynolds number Reθ ≈ 1200 (based on the momentum thickness of the upstream boundary layer) is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations. The computational methodology is based on a hybrid linear/weighted essentially non-oscillatory conservative finite-difference approach, whereby the switch is controlled by the local regularity of the solution, so as to minimize numerical dissipation. As found in experiments, the mean flow pattern consists of an upstream fan of compression waves associated with the thickening of the boundary layer, and the supersonic region is terminated by a nearly normal shock, with substantial bending of the interacting shock. At the selected conditions the flow does not exhibit separation in the mean. However, the interaction region is characterized by ‘intermittent transitory detachment’ with scattered spots of instantaneous flow reversal throughout the interaction zone, and by the formation of a turbulent mixing layer, with associated unsteady release of vortical structures. As found in supersonic impinging shock interactions, we observe a different amplification of the longitudinal Reynolds stress component with respect to the others. Indeed, the effect of the adverse pressure gradient is to reduce the mean shear, with subsequent suppression of the near-wall streaks, and isotropization of turbulence. The recovery of the boundary layer past the interaction zone follows a quasi-equilibrium process, characterized by a self-similar distribution of the mean flow properties.

The dynamical relevance of vortex tubes and vortex sheets in a wall-bounded supersonic turbulent flow at Mach number M = 2 and Reynolds number Reθ ≈ 1350 is quantitatively analysed. The flow in the viscous sublayer and in the buffer region is characterized by intense, elongated vorticity tongues forming a shallow angle with respect to the wall, whose characteristic length is O(200) wall units and whose size in the cross-stream direction is O(50) wall units. The formation of vortex tubes takes place starting from y+ ≈ 10, and it is mainly associated with the roll-up and the interaction of vortex sheets. The analysis of the non-local dynamical effect of tubes and sheets suggests that the latter have a more important collective effect, being closely associated with low-speed streaks, and being responsible for a substantial contribution to the mean momentum balance and to the production of turbulence kinetic energy and enstrophy.

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier--Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20° with respect to the wall, and with an overall size of approximately 30 dissipative length scales.