Direct numerical simulation (DNS) of rotating pipe flows up to $Re_\tau \approx 3000$ is carried out to investigate drag reduction effects associated with axial rotation, extending previous studies carried out at a modest Reynolds number (Orlandi & Fatica, J. Fluid Mech., vol. 343, 1997, pp. 43–72; Orlandi & Ebstein, Intl J. Heat Fluid Flow, vol. 21, 2000, pp. 499–505). The results show that the drag reduction, which we theoretically show to be equivalent to net power saving assuming no mechanical losses, monotonically increases as either the Reynolds number or the rotation number increases, proportionally to the inner-scaled rotational speed. Net drag reduction up to approximately $70\,\%$ is observed, while being far from flow relaminarisation. Scaling laws for the mean axial and azimuthal velocity are proposed, from which a predictive formula for the friction factor is derived. The formula can correctly represent the dependency of the friction factor on the Reynolds and rotation numbers, maintaining good accuracy for low-to-moderate rotation numbers. Examination of the turbulent structures highlights the role of rotation in widening and elongating the small-scale streaks, with subsequent suppression of sweeps and ejections. In the core part of the flow, clear weakening of large-scale turbulent motions is observed at high Reynolds numbers, with subsequent suppression of the outer-layer peak in the pre-multiplied spectra. The Fukagata–Iwamoto–Kasagi decomposition indicates that, consistent with a theoretically derived formula, the outer layer yields the largest contribution to drag reduction at increasingly high Reynolds numbers. In contrast, both the inner and the outer layers contribute to drag reduction as the rotation number increases.

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number ${{Re}}_{\tau } \approx 12000$. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a $k_{\theta }^{-1+\alpha }$ decay (with $k_{\theta }$ the wavenumber in the azimuthal direction, and $\alpha \approx 0.18$), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance $(\langle{u}^2\rangle)$ from the largest scales of motion (superstructures) slowly declines as ${{Re}}_{\tau }^{-\alpha }$, and the integrated inner-scaled variance follows a defect power law of the type $\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$, with constants $A$ and $B$ depending on $y^+$. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of $\langle u^2 \rangle ^+ \approx 12.1$, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately $0.28$, hence slightly exceeding the value $0.25$ which was assumed in previous analyses (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.

We study turbulent flow in open channels with a free surface and rectangular cross-section, for various Reynolds numbers and duct aspect ratios. Direct numerical simulations are used to obtain accurate characterization of the secondary motions, which are found to be more intense than in closed ducts, and to scale with the bulk, rather than with the friction velocity. A notable feature of open-duct flows is the presence of a velocity dip, namely the peak velocity is achieved at some depth underneath the free surface. We find that the depth of the velocity peak increases with the Reynolds number, and correspondingly the flow becomes more symmetric with respect to the horizontal midplane. This is also confirmed from the change of the topology of the secondary motions, which exhibit a strong corner circulation at the free-surface/wall corners at low Reynolds number, which, however, weakens at higher $Re$. The structure of the mean velocity field is such that the log law applies with good approximation in the direction normal to the nearest wall, which allows us to explain why predictive friction formulae based on the hydraulic diameter concept are successful. Additional analysis shows that the secondary motions account for a large fraction of the frictional drag (up to $15$ %).

We carry out direct numerical simulation (DNS) of turbulent flow in rough pipes. Two types of irregular roughness are investigated, namely a grit-blasted and a graphite surface. A wide range of Reynolds numbers is tested, from the laminar up to the fully rough regime, attempting to replicate Nikuradse's pioneering study. Despite the large relative roughness, outer-layer similarity is achieved at high Reynolds number as hypothesised by Townsend, with deviations from the smooth wall case of 4 % for the grit-blasted surface and 13 % for the graphite surface. This makes it possible to define a roughness function and the equivalent sand-grain roughness. The results are compared with those obtained in plane channels, with small differences pointing to the residual influence of the duct cross-sectional shape in the presence of relatively large roughness. The computed friction factors behave similar to those Nikuradse's chart, with differences in terms of the friction factor in the laminar region and of the critical Reynolds number, which are partly absorbed by using the hydraulic radius as reference length scale. The distributions of the velocity fluctuations intensities point to a isotropisation of turbulence in the near-wall region resulting from the roughness, with influence of the roughness geometry. Comparison of the computed equivalent sand-grain roughness height suggest that existing correlations suffer from poor predictive power, at least for surfaces with large relative roughness.

We carry out direct numerical simulations of flow in a plane open channel at friction Reynolds number up to ${{Re}}_{\tau } \approx 6000$. We find solid evidence for the presence of universal large-scale organization in the outer layer, with eddies that are larger and stronger than in the closed channel flow. As a result, velocity fluctuations are found to be stronger than in closed channels, throughout the depth. The inner-layer peak of the streamwise velocity variance is observed to grow logarithmically, as in Townsend's attached-eddy model (Townsend, The Structure of Turbulent Shear Flow, 2nd edn, Cambridge University Press, 1976), but saturation of the growth cannot be discarded based on the present data. Although we do not observe a clear outer peak of the streamwise velocity variance, we present substantial evidence that such a peak should emerge at a Reynolds number barely higher than achieved herein. The most striking feature of the flow is the presence of an extended logarithmic layer, with associated Kármán constant asymptoting to $k \approx 0.375$, in line with observations made in shear-free Couette–Poiseuille flow (Coleman et al., Flow Turbul. Combust., vol. 99, issue 3, 2017, pp. 553–564). The virtual absence of a wake region and of corrective terms to the log law in the present flow leads us to conclude that deviations from the log law observed in internal flows are likely due to the effects of the opposing walls, rather than the presence of a driving pressure gradient.

We derive explicit formulae for the mean profiles of passive scalars (either temperature or concentration of a diffusing substance), and their respective wall fluxes (either heat or mass fluxes), in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. Direct numerical simulation data for turbulent flow within a smooth straight pipe of circular cross-section, at friction Reynolds number ${{Re}}_{\tau }=1140$, in the range of Prandtl numbers from ${{Pr}}=0.00625$ to ${{Pr}}=16$, are used to infer the proper analytical form of the eddy diffusivity. This is leveraged to derive accurate predictive formulae for the mean passive scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling laws result for the thickness of the conductive (diffusive) layer, and for the Nusselt number, which significantly extend the predictive envelope of classical formulae.

The structure and the frequency spectra of wall pressure fluctuations beneath a planar turbulent boundary layer interacting with a conical shock wave at Mach number $M_\infty =2.05$ and Reynolds number $\textit {Re}_\theta \approx 630$ (based on the upstream boundary layer momentum thickness) are examined to elucidate the effects of pressure gradient and flow separation on the characteristics of the wall pressure fluctuations, by exploiting a direct numerical simulation database. Upstream of the interaction, in the zero pressure gradient region, wall pressure statistics compare well with canonical compressible boundary layers in terms of fluctuation intensities and frequency spectra. Across the main interaction zone (APG1), the root-mean-square of wall pressure fluctuations becomes very large (corresponding to approximately 173.3 dB), with maximum increase approximately 12.7 dB from the incoming level. In the second adverse pressure gradient zone (APG2), the root-mean-square of wall pressure fluctuations attains a second peak (corresponding to $164.7$ dB), with an increase of 8.4 dB from the upstream level. Both the APG1 and APG2 regions feature a substantial fraction of flow reversal events, which are, however, scattered and interspersed with regions of attached flow. The wall pressure power spectral density exhibits a broadband and energetic low-frequency component associated with the global unsteadiness of the separation bubble/conical shock system. Analysis of the two-point correlations and wavenumber/frequency spectra of wall pressure fluctuations further suggests that the typical eddies become more elongated along the spanwise direction, as the flow in the separated region tends to escape the centreline, and the convection velocity is significantly reduced.

We study the statistics of passive scalars (either temperature or concentration of a diffusing substance) at friction Reynolds number ${Re}_{\tau }=1140$, for turbulent flow within a smooth straight pipe of circular cross-section, in the range of Prandtl numbers from ${Pr}=0.00625$, to ${Pr}=16$, using direct numerical simulations (DNS) of the Navier–Stokes equations. Whereas the organization of passive scalars is similar to the axial velocity field at ${Pr} = O(1)$, similarity is impaired at low Prandtl number, at which the buffer-layer dynamics is filtered out, and at high Prandtl number, at which the passive scalar fluctuations become confined to the near-wall layer. The mean scalar profiles at ${Pr} \gtrsim 0.0125$ are found to exhibit logarithmic overlap layers, and universal parabolic distributions in the core part of the flow. Near-universality of the eddy diffusivity is exploited to derive accurate predictive formulas for the mean scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling formulas are derived for the thickness of the conductive (diffusive) layer, for the peak scalar variance, and its production rate. The DNS data are leveraged to synthesize a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15, 1972, pp. 2329–2351), which is capable of accounting accurately for the dependence on both Reynolds and Prandtl numbers, for ${Pr} \gtrsim 0.25$.

We carry out direct numerical simulations (DNS) of turbulent flow and heat transfer in pressure-driven plane channels, by considering cases with heating on both walls, as well as asymmetric heating limited to one of the channel walls. Friction Reynolds numbers up to ${Re}_{\tau } \approx 2000$ are considered, and Prandtl numbers from ${Pr}=0.025$ to ${Pr} = 4$, the temperature field being regarded as a passive scalar. Whereas cases with symmetric heating show close similarity between the temperature and the streamwise velocity fields, with turbulent structures confined to either half of the channel, in the presence of one-sided heating the temperature field exhibits larger regions with coherent fluctuations extending beyond the channel centreline. Validity of the logarithmic law for the mean temperature is confirmed, as well as universality of the associated von Kármán constant, which we estimate to be $k_{\theta } = 0.459$. Deviations from the logarithmic behaviour are much clearer in cases with one-sided heating, which feature a wide outer region with parabolic mean temperature profile. The DNS data are exploited to construct a predictive formula for the heat transfer coefficient as a function of both Reynolds and Prandtl number. We find that the reduction of the thermal efficiency in the one-sided case is approximately $20\,\%$ at unit Prandtl number; however, it can become much more significant at low Prandtl number.

We derive a scaling law for the characteristic frequencies of wall pressure fluctuations in swept shock wave/turbulent boundary layer interactions in the presence of cylindrical symmetry, based on analysis of a direct numerical simulations database. Direct numerical simulations in large domains show evidence of spanwise rippling of the separation line, with typical wavelength proportional to separation bubble size. Pressure disturbances around the separation line are shown to be convected at a phase speed proportional to the cross-flow velocity. This information is leveraged to derive a simple model for low-frequency unsteadiness, which extends previous two-dimensional models (Piponniau et al., J. Fluid Mech., vol. 629, 2009, pp. 87–108), and which correctly predicts growth of the typical frequency with the sweep angle. Inferences regarding the typical frequencies in more general swept shock wave/turbulent boundary layer interactions are also discussed.

The flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically with a gas–liquid Navier–Stokes solver. The solver relies on the volume-of-fluid method to account for different phases, and the interface tracking is carried out by using novel schemes based on a tailored total-variation-diminishing limiter. The numerical solver is proved to be characterized by a low numerical dissipation, thanks to the use of a scheme that guarantees energy conservation in the discrete form. Both two- and three-dimensional simulations have been performed, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. To this purpose, coherent vortical structures, such as vortex tubes and vortex sheets, are identified, and the different behaviours of the vortex sheets and tubes at various Reynolds numbers are highlighted. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is identified primarily by the vortex sheets, whereas the vortex tubes govern mainly the intermittency.

We perform large-scale numerical simulations to study Rayleigh–Darcy convection in three-dimensional fluid-saturated porous media up to Rayleigh–Darcy number $Ra=8\times 10^4$. At these large values of $Ra$, the flow is dominated by large columnar structures – called megaplumes – which span the entire height of the domain. Near the boundaries, the flow is hierarchically organized, with fine-scale structures interacting and nesting to form larger-scale structures called supercells. We observe that the correlation between the flow structure in the core of the domain and at the boundaries decreases only slightly for increasing $Ra$, and remains rather high even at the largest $Ra$ considered here. This confirms that supercells are the boundary footprint of megaplumes dominating the core of the domain. In agreement with available literature predictions, we show that the thickness of the thermal boundary layer scales very well with the Nusselt number as $\delta \sim 1/ Nu$. Measurements of the mean wavenumber – inverse of the mean length scale – in the core of the flow support the scaling $\bar {k} \sim Ra^{0.49}$, in very good agreement with theoretical and numerical predictions. Interestingly, the behaviour of the mean wavenumber near the boundaries scales as $\bar {k} \sim Ra^{0.81}$, which is distinguishably different from the presumed linear behaviour. We hypothesize that a linear behaviour can only be observed in the ultimate regime, which we argue to set in only at $Ra$ in excess of $5\times 10^5$, whereas a sublinear behaviour is recovered at more modest $Ra$. The present results are expected to help the development of long desired reliable models to predict the large- and fine-scale structure of Rayleigh–Darcy convection in the high-$Ra$ regime typically encountered in geophysical processes, such as for instance in geological carbon dioxide sequestration.

Flow control for turbulent skin-friction drag reduction is applied to a transonic airfoil to improve its aerodynamic performance. The study is based on direct numerical simulations (with up to 1.8 billion cells) of the compressible turbulent flow around a supercritical airfoil, at Reynolds and Mach numbers of $Re_\infty = 3 \times 10^{5}$ and $M_\infty =0.7$. Control via spanwise forcing is applied over a fraction of the suction side of the airfoil. Besides locally reducing friction, the control modifies the shock wave and significantly improves the aerodynamic efficiency of the airfoil by increasing lift and decreasing drag. Hence, the airfoil can achieve the required lift at a lower angle of attack and with a lower drag. Estimates at the aircraft level indicate that substantial savings are possible; when control is active, its energy cost becomes negligible thanks to the small application area. We suggest that skin-friction drag reduction should be considered not only as a goal, but also as a tool to improve the global aerodynamics of complex flows.

We carry out direct numerical simulations (DNS) of flow in a turbulent square duct by focusing on heat transfer effects, considering the case of unit Prandtl number. Reynolds numbers up to ${{Re}}_{\tau } \approx 2000$ are considered that are much higher than in previous studies, and that yield clear scale separation between inner- and outer-layer dynamics. Close similarity between the behaviour of the temperature and the streamwise velocity fields is confirmed as in previous studies related to plane channels and pipes. Just like the mean velocity, the mean temperature is found to exhibit logarithmic layers as a function of the nearest wall, but with a different slope. The most important practical implication is the validity of the traditional hydraulic diameter as the correct reference length for reporting heat transfer data, as we show rigorously here. Temperature and velocity fluctuations also have similar behaviour, but apparently logarithmic growth of their inner-scaled peak variances is not observed here, unlike in canonical wall-bounded flows. Analysis of the split contributions to the heat transfer coefficient shows that mean cross-stream convection associated with secondary motions is responsible for about $5\,\%$ of the total. Finally, we use the DNS database to highlight shortcomings of traditional linear closures for the turbulent heat flux, and show that substantial modelling improvement in principle may be obtained by retaining at least the three terms in the vector polynomial integrity basis expansion.

We study the statistics of passive scalars at $Pr=1$, for turbulent flow within a smooth straight pipe of circular cross section up to $Re_{\tau } \approx 6000$ using direct numerical simulation (DNS) of the Navier–Stokes equations. While featuring a general organisation similar to the axial velocity field, passive scalar fields show additional energy at small wavenumbers, resulting in a higher degree of mixing and in a $k^{-4/3}$ spectral inertial range. The DNS results highlight logarithmic growth of the inner-scaled bulk and mean centreline scalar values with the friction Reynolds number, implying an estimated scalar von Kármán constant $k_{\theta } \approx 0.459$, which also nicely fits the mean scalar profiles. The DNS data are used to synthesise a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15 (12), 1972, pp. 2329–2351), which points to some shortcomings of the original formulation. Universality of the mean core scalar profile in defect form is recovered, with very nearly parabolic shape. Logarithmic growth of the buffer-layer peak of the scalar variance is found in the Reynolds number range under scrutiny, which well conforms with Townsend's attached-eddy hypothesis, whose validity is also supported by the spectral maps. The behaviour of the turbulent Prandtl number shows good universality in the outer wall layer, with values $Pr_t \approx 0.84$, as also found in previous studies, but closer to unity near the wall, where existing correlations do not reproduce the observed trends.

A new scaling is derived that yields a Reynolds-number-independent profile for all components of the Reynolds stress in the near-wall region of wall-bounded flows, including channel, pipe and boundary layer flows. The scaling demonstrates the important role played by the wall shear stress fluctuations and how the large eddies determine the Reynolds number dependence of the near-wall turbulence behaviour.

We study turbulent flows in a smooth straight pipe of circular cross-section up to friction Reynolds number $({\textit {Re}}_{\tau }) \approx 6000$ using direct numerical simulation (DNS) of the Navier–Stokes equations. The DNS results highlight systematic deviations from Prandtl friction law, amounting to approximately $2\,\%$, which would extrapolate to approximately $4\,\%$ at extreme Reynolds numbers. Data fitting of the DNS friction coefficient yields an estimated von Kármán constant $k \approx 0.387$, which nicely fits the mean velocity profile, and which supports universality of canonical wall-bounded flows. The same constant also applies to the pipe centreline velocity, thus providing support for the claim that the asymptotic state of pipe flow at extreme Reynolds numbers should be plug flow. At the Reynolds numbers under scrutiny, no evidence for saturation of the logarithmic growth of the inner peak of the axial velocity variance is found. Although no outer peak of the velocity variance directly emerges in our DNS, we provide strong evidence that it should appear at ${\textit {Re}}_{\tau } \gtrsim 10^4$, as a result of turbulence production exceeding dissipation over a large part of the outer wall layer, thus invalidating the classical equilibrium hypothesis.

Numerical simulations are used to probe Rayleigh–Darcy convection in fluid-saturated porous media towards the ultimate regime. The present three-dimensional dataset, up to Rayleigh–Darcy number $\textit {Ra}=8\times 10^4$, suggests that the appropriate scaling of the Nusselt number is $\textit {Nu}=0.0081\textit {Ra}+0.067\textit {Ra}^{0.61}$, fitting the computed data for $\textit {Ra}\gtrsim 10^3$. Extrapolation of current predictions to the ultimate linear regime yields the asymptotic law $\textit {Nu}=0.0081 \textit {Ra}$, about $16\,\%$ less than indicated in previous studies. Upon examination of the flow structures near the boundaries, we confirm previous indications of small flow cells hierarchically nesting into supercells, and we show evidence that the supercells at the boundary are the footprints of the megaplumes that dominate the interior part of the flow. The present findings pave the way for more accurate modelling of geophysical systems, with special reference to geological $\textrm {CO}_2$ sequestration.

Much of our theoretical understanding of statistically stable and unstable flows is from the classical Monin–Obukhov similarity theory: the theory predicts the scaling of the mean flow well, but its prediction of the turbulent fluctuation is far from satisfactory. This study builds on Monin–Obukhov similarity theory and Townsend’s attached-eddy hypothesis. We present a model that connects the mean flow and the streamwise velocity fluctuations in both neutral and unstable boundary-layer flows at both moderate and high Reynolds numbers. The model predictions are compared to direct numerical simulations of weakly unstable boundary layers at moderate Reynolds numbers, and large-eddy simulations of unstable boundary-layer flows at high Reynolds numbers. The flow is shear dominated. The range of stability parameter considered in this work is $L/\unicode[STIX]{x1D6FF}<-0.1$, where $L$ is the Monin–Obukhov length, and $\unicode[STIX]{x1D6FF}$ is the boundary-layer height. Reasonably good prediction of velocity fluctuations based on knowledge of the mean velocity profile is obtained.

Direct numerical simulation of the Navier–Stokes equations is carried out to investigate the interaction of a conical shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number $M_{\infty }=2.05$ and Reynolds number $Re_{\unicode[STIX]{x1D703}}\approx 630$, based on the upstream boundary layer momentum thickness. The shock is generated by a circular cone with half opening angle $\unicode[STIX]{x1D703}_{c}=25^{\circ }$. As found in experiments, the wall pressure exhibits a distinctive N-wave signature, with a sharp peak right past the precursor shock generated at the cone apex, followed by an extended zone with favourable pressure gradient, and terminated by the trailing shock associated with recompression in the wake of the cone. The boundary layer behaviour is strongly affected by the imposed pressure gradient. Streaks are suppressed in adverse pressure gradient (APG) zones, but re-form rapidly in downstream favourable pressure gradient (FPG) zones. Three-dimensional mean flow separation is only observed in the first APG region associated with the formation of a horseshoe vortex, whereas the second APG region features an incipient detachment state, with scattered spots of instantaneous reversed flow. As found in canonical geometrically two-dimensional wedge-generated shock–boundary layer interactions, different amplification of the turbulent stress components is observed through the interacting shock system, with approach to an isotropic state in APG regions, and to a two-component anisotropic state in FPG. The general adequacy of the Boussinesq hypothesis is found to predict the spatial organization of the turbulent shear stresses, although different eddy viscosities should be used for each component, as in tensor eddy-viscosity models, or in full Reynolds stress closures.