We study the near-wall behaviour of pressure spectra and associated variances in canonical wall-bounded flows, with a special focus on pipe flow. Analysis of the pressure spectra reveals the universality of small and large scales, supporting the establishment of $ k^{-1}$ spectral layers as predicted by fundamental physical theories. However, this universality does not extend to the velocity spectra (Pirozzoli, J. Fluid Mech., vol. 989, 2024, A5), which show a lack of universality at the large-scale end and systematic deviations from the $ k^{-1}$ behaviour. We attribute this fundamental difference to the limited influence of direct viscous effects on pressure, with implied large differences in the near-wall behaviour. Consequently, the inner-scaled pressure variances continue to increase logarithmically with the friction Reynolds number as we also infer from a refined version of the attached-eddy model, while the growth of the velocity variance tends to saturate. Extrapolated distributions of the pressure variance at extremely high Reynolds numbers are inferred.

Neural network models have been employed to predict the instantaneous flow close to the wall in a viscoelastic turbulent channel flow. Numerical simulation data at the wall are used to predict the instantaneous velocity fluctuations and polymeric-stress fluctuations at three different wall-normal positions in the buffer region. Such an ability of non-intrusive predictions has not been previously investigated in non-Newtonian turbulence. Our comparative analysis with reference simulation data shows that velocity fluctuations are predicted reasonably well from wall measurements in viscoelastic turbulence. The network models exhibit relatively improved accuracy in predicting quantities of interest during the hibernation intervals, facilitating a deeper understanding of the underlying physics during low-drag events. This method could be used in flow control or when only wall information is available from experiments (for example, in opaque fluids). More importantly, only velocity and pressure information can be measured experimentally, while polymeric elongation and orientation cannot be directly measured despite their importance for turbulent dynamics. We therefore study the possibility to reconstruct the polymeric-stress fields from velocity or pressure measurements in viscoelastic turbulent flows. The neural network models demonstrate a reasonably good accuracy in predicting polymeric shear stress and the trace of the polymeric stress at a given wall-normal location. The results are promising, but also underline that a lack of small scales in the input velocity fields can alter the rate of energy transfer from flow to polymers, affecting the prediction of the polymeric-stress fluctuations.

We conduct direct numerical simulations to investigate the synchronisation of Kolmogorov flows in a periodic box, with a focus on the mechanisms underlying the asymptotic evolution of infinitesimal velocity perturbations, also known as conditional leading Lyapunov vectors. This study advances previous work with a spectral analysis of the perturbation, which clarifies the behaviours of the production and dissipation spectra at different coupling wavenumbers. We show that, in simulations with moderate Reynolds numbers, the conditional leading Lyapunov exponent can be smaller than a lower bound proposed previously based on a viscous estimate. A quantitative analysis of the self-similar evolution of the perturbation energy spectrum is presented, extending the existing qualitative discussion. The prerequisites for obtaining self-similar solutions are established, which include an interesting relationship between the integral length scale of the perturbation velocity and the local Lyapunov exponent. By examining the governing equation for the dissipation rate of the velocity perturbation, we reveal the previously neglected roles of the strain rate and vorticity perturbations, and uncover their unique geometrical characteristics.

We investigate turbulent flow between two concentric cylinders, oriented either axially or azimuthally. The axial configuration corresponds to a concentric annulus, where curvature is transverse to the flow, while the azimuthal configuration represents a curved channel with longitudinal curvature. Using direct numerical simulations, we examine the effects of both types of curvature on turbulence, varying the inner radius from $r_i=0.025\delta$ to $r_i=95.5\delta$, where $\delta$ is the gap width. The bulk Reynolds number, based on bulk velocity and $\delta$, is set at $R_b\approx 5000$, ensuring fully turbulent conditions. Our results show that transverse curvature, although breaking the symmetry of axial flows, induces limited changes in the flow structure, leading to an increase in friction at the inner wall. In contrast, longitudinal curvature has a significant impact on the structure and statistics of azimuthal flows. For mild to moderate longitudinal curvatures ($r_i\gt 1.5\delta$), the convex wall stabilises the flow, reducing turbulence intensity, wall friction and turbulent kinetic energy (TKE) production. For extreme longitudinal curvatures ($r_i\leqslant 0.25\delta$), spanwise-coherent flow structures develop near the inner wall, leading to a complete redistribution of the TKE budget: production becomes negligible near the inner wall, while pressure–velocity correlations increase substantially. As a result, the mean TKE peaks near the inner wall, thereby weakening the stabilising effect of convex curvature.

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ‘thought experiment’ via several clean numerical experiments based on the Navier–Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ‘noise-expansion cascade’ whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier–Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. In addition, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. We hope that the ‘noise-expansion cascade’, as a fundamental property of the Navier–Stokes equations, could greatly deepen our understandings about turbulence, and also be helpful for attacking the fourth millennium problem posed by the Clay Mathematics Institute in 2000.

We investigate the effects of thermal boundary conditions and Mach number on turbulence close to walls. In particular, we study the near-wall asymptotic behaviour for adiabatic and pseudo-adiabatic walls, and compare to the asymptotic behaviour recently found near isothermal cold walls (Baranwal et al. 2022. J. Fluid Mech. 933, A28). This is done by analysing a new large database of highly-resolved direct numerical simulations of turbulent channels with different wall thermal conditions and centreline Mach numbers. We observe that the asymptotic power-law behaviour of Reynolds stresses as well as heat fluxes does change with both centreline Mach number and thermal condition at the wall. Power-law exponents transition from their analytical expansion for solenoidal fields to those for non-solenoidal field as the Mach number is increased, though this transition is found to be dependent on the thermal boundary conditions. The correlation coefficients between velocity and temperature are also found to be affected by these factors. Consistent with recent proposals on universal behaviour of compressible turbulence, we find that dilatation at the wall is the key scaling parameter for these power-law exponents, providing a universal functional law that can provide a basis for general models of near-wall behaviour.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.

An important parameter characterising the synchronisation of turbulent flows is the threshold coupling wavenumber. This study investigates the relationship between the threshold coupling wavenumber and the leading Lyapunov vector using large eddy simulations and the SABRA model. Various subgrid-scale stress models, Reynolds numbers and different coupling methods are examined. A new scaling relation is identified for the leading Lyapunov exponents in large eddy simulations, showing that they approximate those of filtered direct numerical simulations. This interpretation provides a physical basis for results related to the Lyapunov exponents of large eddy simulations, including those related to synchronisation. Synchronisation experiments show that the peak wavenumber of the energy spectrum of the leading Lyapunov vector coincides with the threshold coupling wavenumber, in large eddy simulations of box turbulence with standard Smagorinsky or dynamic mixed models as well as in the SABRA model, replicating results from direct numerical simulations of box turbulence. Although the dynamic Smagorinsky model exhibits different behaviour, the totality of the results suggests that the relationship is an intrinsic property of a certain class of chaotic systems. We also confirm that conditional Lyapunov exponents characterise the synchronisation process in indirectly coupled systems as they do in directly coupled ones, with their values insensitive to the nature of the master flow. These findings advance the understanding of the role of the Lyapunov vector in the synchronisation of turbulence.

Estimation of near-wall turbulence in channel flow from outer observations is investigated using adjoint-variational data assimilation. We first consider fully resolved velocity data, starting at a distance from the wall. By enforcing the estimated flow to exactly satisfy the Navier–Stokes equations, we seek a statistically stationary turbulent state that reproduces the instantaneous outer measurements. Such an estimated state provides full access to the unknown near-wall turbulence, including the wall shear stresses and pressure. When the first observation is within 50 viscous units from the wall, the correlation coefficient between the true and estimated state exceeds 95 %. As the observations are further separated from the wall, at 90 viscous units, the accuracy of the assimilated wall stresses decreases to 40 % at the wall. This trend is nearly independent of the Reynolds number. The Fourier spectrum of the estimation error is qualitatively consistent with the coherence spectrum between the outer and the inner state variables: observed long wavelength structures in the outer flow have deeper coherence into the unobserved near-wall region and, therefore, the error is lowest at large scales. Nevertheless, the adjoint-variational approach provides a more rigorous quantification of the capacity to accurately predict the instantaneous near-wall turbulence from outer measurements. Lastly, we demonstrate the robustness of the estimation accuracy using filtered and sub-sampled outer observations.

When turbulent convection interacts with a turbulent shear flow, the cores of convective cells become aligned with the mean current, and these cells (which span the height of the domain) may interact with motions closer to the solid boundary. In this work, we use coupled Eulerian–Lagrangian direct numerical simulations of a turbulent channel flow to demonstrate that, under conditions of turbulent mixed convection, interactions between motions associated with ejections and low-speed streaks near the solid boundary and coherent superstructures in the interior of the flow interact and lead to significant vertical transport of strongly settling Lagrangian particles. We show that the primary suspension mechanism is associated with strong ejection events (canonical low-speed streaks and hairpin vortices characterised by $u'\lt 0$ and $w'\gt 0$, where $u'$ and $w'$ are the streamwise and vertical turbulent velocity fluctuations), whereas secondary suspension is strongly associated with large-scale plume structures aligned with the mean shear (characterised by $w'\gt 0$ and $\theta '\gt 0$, where $\theta$ represents temperature fluctuations). This coupling, which is absent in the limiting cases (pure channel flow or free convection) is shown to lead to a sudden increase in the interior concentration profiles as ${Ri}_\tau$, the friction Richardson number, increases, resulting in concentrations that are larger by roughly an order of magnitude at the channel midplane.

Using clean numerical simulation (CNS) in which artificial numerical noise is negligible over a finite, sufficiently long interval of time, we provide evidence, for the first time, that artificial numerical noise in direct numerical simulation (DNS) of turbulence is approximately equivalent to thermal fluctuation and/or stochastic environmental noise. This confers physical significance on the artificial numerical noise of DNS of the Navier–Stokes equations. As a result, DNS on a fine mesh should correspond to turbulence under small internal/external physical disturbance, whereas DNS on a sparse mesh corresponds to turbulent flow under large physical disturbance. The key point is that all of them have physical meanings and so are correct in terms of their deterministic physics, even if their statistics are quite different. This is illustrated herein. Our paper provides a positive viewpoint regarding the presence of artificial numerical noise in DNS.

The real-fluid effect induced by large density variation at supercritical pressure (SCP) modulates the turbulent dynamics and heat transfer, and poses challenges to existing turbulence models that are based on ideal-gas conditions. This study conducts direct numerical simulations of fully developed channel flows at SCP, with the upper and lower channel walls being isothermally heated and cooled, respectively. Emphasis is placed on examining the effects of various levels of density variations on near-wall turbulence as well as turbulent heat transfer by changing wall temperatures. The results show that the density fluctuation significantly impacts both first-order and second-order turbulence statistics near the heated wall owing to the close vicinity of pseudo-boiling point. Such real-fluid impact increases substantially with increasing density ratio, and tends to weaken the turbulent kinetic energy by damping turbulence production, while simultaneously inducing an additional turbulent mass flux that partially offsets this reduction. Detailed quadrant analysis reveals that the ‘ejection’ events dominate diverse effects of density fluctuation on Reynolds shear stresses, with density fluctuation contributing positively on the cooled wall side, and negatively on the heated wall side. Regarding the turbulent heat transfer, density fluctuation enhances the enthalpy–pressure–gradient correlation, tending to weaken the turbulent heat flux, which is slightly compensated by additional terms induced by density fluctuations. The overall negative contribution of density fluctuation to turbulent heat flux stems primarily from ‘hot ejection’ motions. Instantaneous flow characteristics provide additional support for these findings. Additionally, the mechanisms by which density fluctuations affect Reynolds shear stress and turbulent heat flux could also be extended to the skin friction coefficient and Nusselt number, respectively.

The acoustic field radiated by a system of contra-rotating propellers in wetted conditions (with no cavitation) is reconstructed by exploiting the Ffowcs Williams–Hawkings acoustic analogy and a database of instantaneous realizations of the flow. They were generated by high-fidelity computations using a large eddy simulation approach on a cylindrical grid of 4.6 billion points. Results are also compared against the cases of the front and rear propellers working alone. The analysis shows that the importance of the quadrupole component of sound, originating from wake turbulence and instability of the tip vortices, is reinforced, relative to the linear component radiated from the surface of the propeller blades. The sound from the contra-rotating propellers decays at a slower rate for increasing radial distances, compared with the cases of the isolated front and rear propellers, again due to the quadrupole component. The quadrupole sound is often neglected in the analysis of the acoustic signature of marine propellers, by considering the only linear component. In contrast, the results of this study point out that the quadrupole component becomes the leading one in the case of contra-rotating propulsion systems, due to the increased complexity of their wake. This is especially the result of the mutual inductance phenomena between the tip vortices shed by the front and rear propellers of the contra-rotating system.

The interaction between a turbulent flow and a porous boundary is analysed with focus on the sensitivity of the roughness function, $\Delta U^+$, to the upscaled coefficients characterizing the wall. The study is aimed at (i) demonstrating that imposing effective velocity boundary conditions at a virtual plane boundary, next to the physical one, can efficiently simplify the direct numerical simulations (DNS); and (ii) pursuing correlations to estimate $\Delta U^+$ a priori, once the upscaled coefficients are calculated. The homogenization approach employed incorporates near-interface advection via an Oseen-like linearization, and the macroscopic coefficients thus depend on both the microstructural details of the wall and a slip-velocity-based Reynolds number, $Re_{slip}$. A set of homogenization-simplified DNS is run to study the channel flow over transversely isotropic porous beds, testing values of the grains’ pitch within $0\lt \ell ^+\lt 40$. Reduction of the skin-friction drag is attainable exclusively over streamwise-aligned inclusions for $\ell ^+$ values up to $20{-}30$. The drag increase over spanwise-aligned inclusions (or streamwise-aligned ones at large $\ell ^+$) is accompanied by enhanced turbulence levels, including intensified sweep and ejection events. The root-mean-square of the transpiration velocity fluctuations at the virtual plane, $\tilde V_{rms}$, is the key control parameter of $\Delta U^+$; our analysis shows that, provided $\tilde V_{rms} \lesssim 0.25$, then $\tilde V_{rms}$ is strongly correlated to a single macroscopic quantity, $\Psi$, which comprises the Navier-slip and interface/intrinsic permeability coefficients. Fitting relationships for $\Delta U^+$ are proposed, and their applicability is confirmed against reference results for the turbulent flow over impermeable walls roughened with three-dimensional protrusions or different geometries of riblets.

In this study, we conducted interface-capturing high-resolution simulations of a bubbly upflow in a vertical channel to investigate the bubble distribution and its interaction with surrounding turbulence, focusing on the effects of the density ratio. A bulk Reynolds number $Re_b=2300$ was used for all simulations. The influence of density ratio on vortex structures and turbulence statistics differed between the near-wall and core regions of the channel. Adding 5.43 $\%$ gas caused an increase in wall friction. By applying a generalised FIK identity to analyse wall friction, it was determined that the drag rise in the bubbly channel was mostly due to the near-wall region. Visualisation of the bubble and vortex structures showed that small bubbles near the wall induced larger magnitude of Reynolds shear stress and increased wall friction. Bubble behaviour near the wall region was similar for density ratios above 30, leading to wall friction saturation. In the core region, large deformable bubbles created wake vortices due to slip velocity between liquid and gas phases. Wake vortices help large bubbles absorb smaller bubbles and maintain their sizes. As the density ratio increased, the slip velocity increased owing to greater difference in the gravitational acceleration between liquid and gas phases, resulting in corresponding increase in wake intensity and velocity fluctuations. However, quadrant analysis showed that Q1 and Q3 events increased together with Q2 and Q4 events in the core region, cancelling out any net effect of wake vortices on Reynolds shear stress or wall friction.

We investigate fully developed turbulent flow in curved channels to explore the interaction between turbulence and curvature-driven coherent structures. By focusing on two cases of mild and strong curvature, we examine systematically the effects of the Reynolds number through a campaign of direct numerical simulations, spanning flow regimes from laminar up to the moderately high Reynolds number – based on bulk velocity and channel height – of $87\,000$. Our analysis highlights the influence of curvature on the friction coefficient, showing that flow transition is anticipated by concave curvature and delayed by convex curvature. In the case of mild curvature, a frictional drag reduction compared with plane channel flow is found in the transitional regime. Spectral analysis reveals that the near-wall turbulence regeneration cycle is maintained in mildly curved channels, while it is absent or severely inhibited on the convex wall of strongly curved channels. Streamwise large-scale structures resembling Dean vortices are found to be weakly dependent on the Reynolds number and strongly affected by curvature: increasing curvature shifts these vortices towards the outer wall and reduces their size and coherence, limiting their contribution to streamwise velocity fluctuations and momentum transport. In the case of strong curvature, spanwise large-scale structures are also detected. These structures are associated with large pressure fluctuations and the suppression of turbulent stresses near the convex wall, where a region with negative turbulence production is observed and characterised via quadrant analysis.

We introduce a wall model for large-eddy simulation (WMLES) applicable to rough surfaces with Gaussian and non-Gaussian distributions for both the transitionally and fully rough regimes. The model is applicable to arbitrary complex geometries where roughness elements are assumed to be underresolved, i.e. subgrid-scale roughness. The wall model is implemented using a multi-hidden-layer feedforward neural network, with the mean geometric properties of the roughness topology and near-wall flow quantities serving as input. The optimal set of non-dimensional input features is identified using information theory, selecting variables that maximize information about the output while minimizing redundancy among inputs. The model also incorporates a confidence score based on Gaussian process modelling, enabling the detection of potentially low model performance for untrained rough surfaces. The model is trained using a direct numerical simulation (DNS) roughness database comprising approximately 200 cases. The roughness geometries for the database are selected from a large repository through active learning. This approach ensures that the rough surfaces incorporated into the database are the most informative, achieving higher model performance with fewer DNS cases compared with passive learning techniques. The performance of the model is evaluated both apriori and aposteriori in WMLES of turbulent channel flows with rough walls. Over 550 channel flow cases are considered, including untrained roughness geometries, roughness Reynolds numbers and grid resolutions for both transitionally and fully rough regimes. Our rough-wall model offers higher accuracy than existing models, generally predicting wall shear stress within an accuracy range of 1%–15 %. The performance of the model is also assessed on a high-pressure turbine blade with two different rough surfaces. We show that the new wall model predicts the skin friction and the mean velocity deficit induced by the rough surface on the blade within 1%–10 % accuracy except the region with transition or shock waves. This work extends the building-block flow wall model (BFWM) introduced by Lozano-Durán & Bae (2023. J. Fluid Mech. 963, A35) for smooth walls, expanding the BFWM framework to account for rough-wall scenarios.

The influence of irregular three-dimensional rough surfaces on the displacement of the logarithmic velocity profile relative to that of a smooth wall in turbulent flow, known as the roughness function, is studied using direct numerical simulations. Five different surface power spectral density (PSD) shapes were considered, and for each, several rough Gaussian surfaces were generated by varying the root mean square ($k_{rms}$) of the surface heights. It is shown that the roughness function ($\Delta U^{+}$) depends on both the PSD and $k_{rms}$. For a given $k_{rms}$, $\Delta U^{+}$ increases as the wavenumbers of the PSD expand to large values, but at a rate that decreases with the magnitude of the wavenumbers. Although $\Delta U^{+}$ generally does not scale with either $k_{rms}$ or the effective slope $ES$ when these variables are considered singularly, for PSDs with low wavenumbers, $\Delta U^{+}$ tends to scale with $ES$, whereas as wavenumbers increase, $\Delta U^{+}$ tends to scale with $k_{rms}$. An equivalent Nikuradse sand roughness of about eight times $k_{rms}$ is found, which is similar to that observed in previous studies for a regular three-dimensional roughness. Finally, it is shown that $k_{rms}$ and the effective slope are sufficient to describe the roughness function in the transitional rough regime.

Drag reduction induced by a polydisperse solution of polyethylene oxide is investigated by direct numerical simulations of the Navier–Stokes equations coupled with the Lagrangian evolution of the polymers, modelled as dumbbells. Simulation parameters are chosen to match the experimental conditions of Berman (1977), who measured the polymer molecular weight distribution. Drag reduction is induced only by the few high molecular weight polymers fully stretched by the turbulent flow, whilst the hundreds of parts per million of low molecular weight chains are ineffective.

Deep reinforcement learning (DRL) is employed to develop control strategies for drag reduction in direct numerical simulations of turbulent channel flows at high Reynolds numbers. The DRL agent uses near-wall streamwise velocity fluctuations as input to modulate wall blowing and suction velocities. These DRL-based strategies achieve significant drag reduction, with maximum rates $35.6\,\%$ at $Re_{\tau }\thickapprox 180$, $30.4\,\%$ at $Re_{\tau }\thickapprox 550$, and $27.7\,\%$ at $Re_{\tau }\thickapprox 1000$, outperforming traditional opposition control methods. An expanded range of wall actions further enhances drag reduction, although effectiveness decreases at higher Reynolds numbers. The DRL models elevate the virtual wall through blowing and suction, aiding in drag reduction. However, at higher Reynolds numbers, the amplitude modulation of large-scale structures significantly increases the residual Reynolds stress on the virtual wall, diminishing the drag reduction. Analysis of budget equations provides a systematic understanding of the underlying drag reduction dynamics. The DRL models reduce skin friction by inhibiting the redistribution of wall-normal turbulent kinetic energy. This further suppresses the wall-normal velocity fluctuations, reducing the production of Reynolds stress, thereby decreasing skin friction. This study showcases the successful application of DRL in turbulence control at high Reynolds numbers, and elucidates the nonlinear control mechanisms underlying the observed drag reduction.