The impact of intrinsic compressibility effects – changes in fluid volume due to pressure variations – on high-speed wall-bounded turbulence has often been overlooked or incorrectly attributed to mean property variations. To quantify these intrinsic compressibility effects unambiguously, we perform direct numerical simulations of compressible turbulent channel flows with nearly uniform mean properties. Our simulations reveal that intrinsic compressibility effects yield a significant upward shift in the logarithmic mean velocity profile that can be attributed to the reduction in the turbulent shear stress. This reduction stems from the weakening of the near-wall quasi-streamwise vortices. In turn, we attribute this weakening to the spontaneous opposition of sweeps and ejections from the near-wall expansions and contractions of the fluid, and provide a theoretical explanation for this mechanism. Our results also demonstrate that intrinsic compressibility effects play a crucial role in the increase in inner-scaled streamwise turbulence intensity in compressible flows, as compared with incompressible flows, which was previously regarded to be an effect of mean property variations alone.

This study investigates how the spatial configuration of submerged three-dimensional patches of vegetation impacts turbulence. Laboratory experiments were conducted in a channel with submerged patches of model vegetation configured with different patch area densities ($\phi _{p}$), representing the bed area fraction occupied by patches, ranging from 0.13 to 0.78, and different spatial patterns transitioning from two dimensional (channel-spanning patches) to three dimensional (laterally unconfined patches). These configurations produced a range of flow regimes within the canopy, from wake interference flow to skimming flow. At low area density ($\phi _{p}\lt0.5$), turbulence within the canopy increased with increasing $\phi _{p}$ regardless of spatial configuration, while at high area density ($\phi _{p}\gt0.5$), the relationship between turbulence and $\phi _{p}$ depended on the spatial configuration of the patches. For the same patch area density, the configuration with smaller lateral gaps generated stronger turbulence within the canopy. The relative contributions of wake and shear production also varied with the spatial configuration of the patches. At low area densities, wake production dominated over shear production, while at high area densities, shear production was more dominant due to an enhanced shear layer at the top of the canopy and reduced mean velocity within the canopy. A new predictive model for the channel-averaged turbulent kinetic energy (TKE) was developed as a function of channel-averaged velocity, canopy geometry, and patch area density, which showed good agreement with the measured TKE.

The turbulent boundary layer is a region where both preferential dissipation of energy and the production of significant vorticity arises as a consequence of the strong velocity gradients. Previous work has shown that, following a Reynolds decomposition, the purely fluctuating component of the enstrophy production is the dominant term. Near the wall this varies in a complex manner with height. In this study, we additionally decompose the strain rate and vorticity terms into normal and non-normal components using a Schur decomposition and are able to explain all these features in terms of contributions at different heights from constituents involving different combinations of normal and non-normal quantities. What is surprising about our results is that, while the mean shear and the action of larger-scale structures should mean that non-normal effects are of over-riding importance at the wall, the most important individual term involves the fluctuating normal strain rate in the transverse direction. In part, this is because of a strong correlation between this term and the non-normal vorticity with a transverse axis, but it is also the case that individual components of the purely non-normal enstrophy production are negative in the mean. Hence, a local strain rate that is orthogonal to the direction of the dominant mean and fluctuating shear plays a crucial role in amplifying vorticity that is yet to have developed a local component. These conclusions support the emphasis in the control literature on the transverse velocity components at the wall.

The instantaneous structure of a turbulent boundary layer (TBL) subjected to freestream turbulence (FST) is investigated at several streamwise locations downstream of an active turbulence-generating grid. Using planar particle image velocimetry, three grid sequences are tested at four streamwise locations with FST intensities up to 10.9 %. A low-turbulence reference case is included for comparison. A novel method is proposed to separate the instantaneous TBL and FST flows by identifying a distinct interface for each realisation using probability density functions of the vorticity field. Two alternative approaches are used to define the interfaces, based on either constant velocity contour lines or constant vorticity magnitude contour lines. The former is found to highlight the momentum events in the velocity fields, whereas the latter outlines the vortical features of the flow. Regardless of the interface choice, when faced with FST, the interface moves closer to the wall on average, and its location fluctuates more. When FST is present, the shear and mean spanwise vorticity magnitudes increase on the TBL side of the interface. Uniform momentum zones (UMZs) beneath the velocity interfaces are identified. In the presence of FST, UMZs located closer to the wall appear to be compressed, resulting in fewer identified UMZs. Moving downstream, the FST intensity decays while the TBL develops. As a result, many characteristics of the TBL recover to an undisturbed state, with the interface moving away from the wall, vorticity and turbulent fluctuations returning to their natural state undisturbed by FST and the number of detected UMZs increasing.

Hydrodynamic consequences of using simpler geometric shapes to represent coral canopies are examined through a laboratory study. A canopy composed of cylinders is compared with a canopy composed of 3-D-printed, scaled down coral heads in a recirculating flume. Vertical velocity profiles are measured at four horizontal locations for each canopy type, and mean velocity and turbulence statistics are compared both within and above the canopy. A narrow, defined wake on the scale of the canopy element is present behind the cylinder canopy elements that is absent in the coral canopy. There is also a peak in shear stress at the top of the cylinder canopy, likely due to the sharp edge at the top of the cylinder. Above the canopy, however, turbulence statistics and friction velocities behave similarly for both canopy types. Therefore, the results indicate we may reasonably get coral reef drag estimates from canopies with simpler geometric surrogates, especially when the mean free-stream and within-canopy flow speeds are matched to environmental conditions.

Large-eddy simulations (LES) of a hypersonic boundary layer on a $7^\circ$-half-angle cone are performed to investigate the effects of highly cooled walls (wall-to-recovery temperature ratio of $T_w / T_r \sim 0.1$) on fully developed turbulence and to validate a newly developed rescaling method based on volumetric flow extraction. Two Reynolds numbers are considered, $Re_m = 4.1 \times 10^6\ \text {m}^{-1}$ and $6.4 \times 10^6\ \text {m}^{-1}$, at free-stream Mach numbers of $M_\infty = 7.4$. A comparison with a reference laminar-to-turbulent simulation, capturing the full history of the transitional flow dynamics, reveals that the volumetric rescaling method can generate a synthetic turbulent inflow that preserves the structure of the fluctuations. Equilibrium conditions are recovered after approximately 40 inlet boundary layer thicknesses. Numerical trials show that a longer streamwise extent of the rescaling box increases numerical stability. Analyses of turbulent statistics and flow visualizations reveal strong pressure oscillations, up to $50\,\%$ of local mean pressure near the wall, and two-dimensional longitudinal wave structures resembling second-mode waves, with wavelengths up to 50 % of the boundary layer thickness, and convective Mach numbers of $M_c \simeq 4.5$. It is shown that their quasi-periodic recurrence in the flow is not an artefact of the rescaling method. Strong and localized temperature fluctuations and spikes in the wall-heat flux are associated with such waves. Very high values of temperature variance near the wall result in oscillations of the wall-heat flux exceeding its average. Instances of near-wall temperature falling below the imposed wall temperature of $T_w=300$ K result in pockets of instantaneous heat flux oriented against the statistical mean direction.

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress (${\tau }_w$) and wall-pressure ($\,{p}_w$) spectra of canonical turbulent boundary layers, both of which are well known to increase with Reynolds number, albeit in a distinct manner. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97–120) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired ${\tau }_w$, ${p}_w$ and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number ($Re_{\tau }$) range of $ {O}(10^3) \lesssim Re_{\tau } \lesssim {O}(10^6)$. A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021, A5) is implemented to reveal the active and inactive contributions to the ${\tau }_w$- and $p_w$-spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, issue 2, 1967, pp. 241–258) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the ${\tau }_w$-spectrum, which grow with $Re_{\tau }$ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the $p_w$-spectrum. Both these contributions are found to increase with increasing $Re_{\tau }$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau }$-growth of the $p_w$-spectra relative to ${\tau }_w$, given the dependence of the latter only on the inactive contributions.

In this study, we investigate the properties of energy thickness $\delta _3$ in turbulent boundary layer (TBL) flows, a parameter derived solely from the mean streamwise velocity ($U$) profile. Through an analysis of the energy integral equation for zero pressure gradient TBLs, we establish a close relationship between turbulent kinetic energy (TKE) production and $\delta _3$, offering a practical method to estimate TKE production, which is particularly useful in physical experiments where direct measurements are challenging. The significance of $\delta _3$ becomes even more pronounced in TBLs under pressure gradient. Through extensive analysis of numerical and experimental data, we show that the ratio between $\delta _3$ and the momentum thickness $\delta _2$ is a promising criterion for predicting flow separation. Moreover, we derive a new energy integral equation for TBLs under arbitrary pressure gradients, and provide approximations for TKE productions terms by $R_{uv}\,\partial U/\partial y$ and $R_{uu}\,\partial U/\partial x$, and dissipation term by the mean shear. Here, $x, y$ represent the streamwise and wall-normal directions, respectively, and $R_{uu}$ and $R_{uv}$ are the Reynolds normal and shear stresses. The accuracy and robustness of the new energy integral equation and the approximation equations are validated using direct numerical simulations data. Our results show that the TKE production by $R_{uv}\,\partial U/\partial y$ and the overall productions consistently remain positive, reflecting a continuous conversion of mean kinetic energy into TKE across all TBLs. However, under strong favourable pressure gradients, TKE production by $R_{uu}\,\partial U/\partial x$ becomes negative, indicating a reverse energy transfer from TKE to mean kinetic energy.

Adverse pressure gradient (APG) turbulent boundary layers (TBL) require an understanding of the details of the pressure gradient, or history effect, to characterize the associated variation of spatiotemporal turbulent statistics. The streamwise-varying mean pressure gradient is reflected in the streamwise developing mean flow field and thus resolvent analysis, which captures the amplification of the Navier–Stokes equations linearized about the turbulent mean, can be used to understand linear amplification in APG TBLs. In particular, by using a biglobal approach in which the amplification is characterized by a temporal frequency and spanwise wavenumber, the streamwise and wall-normal inhomogeneities of the APG TBL can be resolved and related to the APG history. The linear response is able to identify multiscale phenomena, identifying a near-wall peak with $\lambda _{z}^+\approx 100$ for zero pressure gradient TBLs and mild to moderate APG TBLs as well as large-scale modes whose amplification increases with APG strength and Reynolds number. It is shown that the monotonic growth in the turbulent statistics with increasing APG is reflected in the linear growth in the associated resolvent amplification. Collapse in the Reynolds stresses is obtained through an augmented hybrid velocity scale, which replaces the local APG strength measure in the hybrid velocity scale presented in Romero et al. (Intl J. Heat Fluid Flow, vol. 93, 2022, 108885) with a velocity that encapsulates the pressure gradient history. While this resolvent approach is applicable to any APG TBL, it is shown from a scaling analysis of the linearized Navier–Stokes equations that the linear growth observed in the resolvent amplification with the history effect is limited to near-equilibrium APG TBLs.

Turbulent boundary layers on immersed objects can be significantly altered by the pressure gradients imposed by the flow outside the boundary layer. The interaction of turbulence and pressure gradients can lead to complex phenomena such as relaminarization, history effects and flow separation. The angular momentum integral (AMI) equation (Elnahhas & Johnson, J. Fluid Mech., vol. 940, 2022, A36) is extended and applied to high-fidelity simulation datasets of non-zero pressure gradient turbulent boundary layers. The AMI equation provides an exact mathematical equation for quantifying how turbulence, free-stream pressure gradients and other effects alter the skin friction coefficient relative to a baseline laminar boundary layer solution. The datasets explored include flat-plate boundary layers with nearly constant adverse pressure gradients, a boundary layer over the suction surface of a two-dimensional NACA 4412 airfoil and flow over a two-dimensional Gaussian bump. Application of the AMI equation to these datasets maps out the similarities and differences in how boundary layers interact with favourable and adverse pressure gradients in various scenarios. Further, the fractional contribution of the pressure gradient to skin friction attenuation in adverse-pressure-gradient boundary layers appears in the AMI equation as a new Clauser-like parameter with some advantages for understanding similarities and differences related to upstream history effects. The results highlight the applicability of the integral-based analysis to provide quantitative, interpretable assessments of complex boundary layer physics.

The amplitude modulation coefficient, $R$, that is widely used to characterize nonlinear interactions between large- and small-scale motions in wall-bounded turbulence is not compatible with detecting the convective nonlinearity of the Navier–Stokes equations. Through a spectral decomposition of $R$ and a simplified model of triadic convective interactions, we show that $R$ suppresses the signature of convective scale interactions, but is strongly influenced by linear interactions between large-scale motions and the background mean flow. We propose an additional coefficient that is specifically designed for the detection of convective nonlinearities, and we show how this new coefficient, $R_T$, quantifies the turbulent kinetic energy transport involved in turbulent scale interactions and reveals a classical energy cascade across widely separated scales.

Linear non-modal analyses are performed to study the mechanism of how deformable free surfaces influence very-large-scale motions (VLSMs) in turbulent open channel flows. The mean velocity and eddy viscosity profiles obtained from direct numerical simulations are used in the generalised Orr–Sommerfeld and Squire equations to represent background turbulence effects. Solutions of surface-wave eigenmodes and shear eigenmodes are obtained. The results indicate that at high Froude numbers, free surfaces enhance the maximum transient growth rate of VLSMs through surface-wave eigenmodes. We then analyse the energy budget equation to reveal the underlying mechanism. For streamwise-uniform motions, the energy growth rate is enhanced by an energy production term associated with the correlation between the streamwise velocity, which is generated by the lifting-up effect of streamwise vortices composed of shear eigenmodes, and the vertical velocity, which is induced by a spanwise standing wave composed of surface-wave eigenmodes. For streamwise-varying motions, the energy growth rate is enhanced by a standing wave moving with a pair of vortices that travel at a speed approximately equal to the projection of the mean surface velocity along the wavenumber vector direction. Finally, an analytical expression of the energy production term is derived to provide the initial conditions for the maximum transient growth and explain the weak free-surface effect observed at large spanwise wavenumbers and low Froude numbers. The results demonstrate a linear non-modal mechanism in interactions between free surfaces and VLSMs in open channel flows.

The eddy-viscosity model, as initially used to model the mean Reynolds stress, has been widely used in the linear analysis of turbulence recently by direct extension. In this study, the mechanism of the eddy viscosity in improving the prediction of fluctuation structures with linear analysis is clarified by investigating the statistical properties of forcing, eddy-viscosity term and their correlations. From the direct numerical simulation (DNS) results of turbulent channel flows with $Re_{\tau }=186$–$2003$, the spatial correlation of forcing is partially cancelled due to its interaction with eddy-viscosity terms. The stochastic forcing after excluding the eddy-viscosity term is nearly uncorrelated spatially, which better matches the condition where the resolvent modes are consistent with the spectral proper orthogonal decomposition (SPOD) modes from DNS. With the above findings, an optimisation framework is developed by minimising the spatial correlations of the stochastic forcing. The optimised eddy-viscosity profiles nearly overlap with the mean-quantity-based model in the near-wall region, but have different maximum values. Compared with the mean-quantity-based model, the optimised results enhance the consistency between the resolvent and DNS results significantly. Based on the optimised results, a new modelling framework is further abstracted, leaving only one to-be-modelled parameter of the maximum value of the eddy-viscosity profile. This parameter follows distinctive rules with spanwise flow scales, based on which a simplified predictive model is constructed. The resolvent modes predicted by the new model exhibit high consistency with SPOD modes, which are essentially comparable to the optimised results for wide ranges of streamwise and spanwise scales.

Empirical evidence is provided that within the inertial sublayer (i.e. logarithmic region) of adiabatic turbulent flows over smooth walls, the skewness of the vertical-velocity component $S_w$ displays universal behaviour, being a positive constant and constrained within the range $S_w \approx 0.1\unicode{x2013}0.16$, regardless of flow configuration and Reynolds number. A theoretical model is then proposed to explain this behaviour, including the observed range of variations of $S_w$. The proposed model clarifies why $S_w$ cannot be predicted from down-gradient closure approximations routinely employed in large-scale meteorological and climate models. The proposed model also offers an alternative and implementable approach for such large-scale models.

Prandtl's secondary flows of the second kind generated by laterally varying roughness are studied using the linearised Reynolds-averaged Navier–Stokes approach proposed by Zampino et al. (J. Fluid Mech., vol. 944, 2022, p. A4). The momentum equations are coupled to the Spalart–Allmaras model while the roughness is captured by adapting established strategies for homogeneous roughness to heterogeneous surfaces. Linearisation of the governing equations yields a framework that enables a rapid exploration of the parameter space associated with heterogeneous surfaces, in the limiting case of small spanwise variations of the roughness properties. Channel flow is considered, with longitudinal high- and low-roughness strips arranged symmetrically. By varying the strip width, it is found that linear mechanisms play a dominant role in determining the size and intensity of secondary flows. In this setting, secondary flows may be interpreted as the time-averaged output response of the turbulent mean flow subjected to a steady forcing produced by the wall heterogeneity. In fact, the linear model predicts that secondary flows are most intense when the strip width is about 0.7 times the half-channel height, in excellent agreement with available data. Furthermore, a unified framework to analyse combinations of heterogeneous roughness properties and laterally varying topographies, common in applications, is discussed. Noting that the framework assumes small spanwise variations of the surface properties, two separate secondary-flow-inducing source mechanisms are identified, i.e. the lateral variation of the virtual origin from which the turbulent structure develops and the lateral variation of the streamwise velocity slip, capturing the acceleration/deceleration perceived by the bulk flow over troughs and crests of non-planar topographies.

Pore-resolved direct numerical simulations have been performed to investigate the turbulent open-channel flow over a rough and permeable sediment bed, represented by a mono-disperse random sphere pack. After a careful validation, eleven cases were simulated to systemically sample a parameter space spanned by a friction Reynolds number $Re_\tau \in [150, 500]$ and a permeability Reynolds number $Re_K \in [0, 2.8]$. By varying the ratio of flow depth to sphere diameter within a range of $h/D \in \{ 3,5,10,\infty \}$, the influence of both Reynolds numbers on the flow field and the turbulence structure could be investigated independently. The simulation results are analysed within a time–space double-averaging framework, whereas flow visualizations provide insight into instantaneous fields. Based on the drag distribution, we propose a consistent interface description, which can be used to define both near-interface and outer-flow coordinates. In these near-interface coordinates, the profiles of the mean velocity and the total shear stress collapse. Furthermore, the proposed interface definition yields outer-layer coordinates, in which the flow and turbulence statistics over a rough and permeable bed reveal similarity to a smooth-wall flow at a similar $Re_\tau$. Within the parameter space, $Re_\tau$ has a strong influence on the wake region of the velocity profile. In contrast, $Re_K$ changes the wall-blocking effect and the shear intensity, which is reflected by the turbulence structure and vortex orientation in the near-interface region. As streamwise velocity streaks disappear and the vortex inclination increases with higher $Re_K$, differences between near-interface and outer-layer turbulence structure are reduced.

To aid in prediction of turbulent boundary layer flows over rough surfaces, a new model is proposed to estimate hydrodynamic roughness based solely on geometric surface information. The model is based on a fluid-mechanics motivated geometric parameter called the wind-shade factor. Sheltering is included using a rapid algorithm adapted from the landscape shadow literature, while local pressure drag is estimated using a piecewise potential flow approximation. Similarly to evaluating traditional surface parameters such as skewness or average slope magnitude, the wind-shade factor is purely geometric and can be evaluated efficiently from knowing the surface elevation map and the mean flow direction. The wind-shade roughness model is applied to over 100 different surfaces available in a public roughness database and some others, and the predicted sandgrain-roughness heights are compared with measured values. Effects of various model ingredients are analysed, and transitionally rough surfaces are treated by adding a term representing the viscous stress component.

We establish a theoretical framework for predicting friction and heat transfer coefficients in variable-property forced air convection. Drawing from concepts in high-speed wall turbulence, which also involves significant temperature, viscosity and density variations, we utilize the mean momentum balance and mean thermal balance equations to develop integral transformations that account for the impact of variable fluid properties. These transformations are then applied inversely to predict the friction and heat transfer coefficients, leveraging the universality of passive scalars transport theory. Our proposed approach is validated using a comprehensive dataset from direct numerical simulations (DNS), covering both heating and cooling conditions up to a friction Reynolds number $\textit {Re}_\tau \approx 3200$. The predicted friction and heat transfer coefficients closely match the DNS data with accuracy margin 1–2 %, representing a significant improvement over the current state of the art.

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

The spherical Couette system consists of two differentially rotating concentric spheres with the space in between filled with fluid. We study a regime where the outer sphere is rotating rapidly enough so that the Coriolis force is important and the inner sphere is rotating either slower or in the opposite direction with respect to the outer sphere. We numerically study the sudden transition to turbulence at a critical differential rotation seen in experiments at BTU Cottbus-Senftenberg, Germany, and investigate its cause. We find that the source of turbulence is the boundary layer on the inner sphere, which becomes centrifugally unstable. We show that this instability leads to generation of small-scale structures which lead to turbulence in the bulk, dominated by inertial waves, a change in the force balance near the inner boundary, the formation of a mean flow through Reynolds stresses and, consequently, to an efficient angular momentum transport. We compare our findings with axisymmetric simulations and show that there are significant similarities in the nature of the flow in the turbulent regimes of full three-dimensional and axisymmetric simulations but differences in the evolution of the instability that leads to this transition. We find that a heuristic argument based on a Reynolds number defined using the thickness of the boundary layer as a length scale helps explain the scaling law of the variation of critical differential rotation for transition to turbulence with rotation rate observed in the experiments.