This study explores partial synchronisation in turbulent channel flows using sequential variational data assimilation with sparse observations, emphasising the roles of model and observation uncertainties. Unlike previous work that focused on synchronisation using direct numerical simulation, this study considers synchronisation under imperfect models and noisy data. In the first part, a synchronisation map is constructed, revealing invariance with respect to variations in the predictive model, Reynolds number and mesh resolution. Full synchronisation emerges above a critical level of equivalent observation density. At lower observation densities, modal synchronisation is observed, where the energies of dominant modes evolve independently of initial conditions. As data become sparser, the system transitions to a non-synchronisation regime, with assimilated flows exhibiting minimal correlation with the observations. The second part of this study uses the master flow interpolated from down-sampled sparse observations. The delay-coordinate strategy is introduced to enhance the modal synchronisation. Results indicate that the optimal $\sigma$ lies near the threshold between modal synchronisation and non-synchronisation. This demonstrates that the modal synchronisation serves as a critical prerequisite for leveraging historical information in data assimilation to improve the accuracy of turbulence reconstruction. These findings extend the scope of synchronisation theory and provide valuable guidance for advancing data assimilation methodologies.

In this work, the classical Prandtl relation for the skin-friction law of incompressible turbulent channel and pipe flows is generalised to compressible cases. Specifically, based on the law of the wall and asymptotic analysis, a skin-friction transformation is proposed to map the skin-friction law of compressible turbulent channel and pipe flows to the classical Prandtl relation. It has been theoretically proven that the skin-friction coefficient $C_{\!f,i}$ and the bulk Reynolds number $\textit{Re}_{b,i}$ for compressible turbulent channel and pipe flows, where the subscript $i$ denotes the transformed quantity obtained from the proposed skin-friction transformation, adhere to the Prandtl relation, expressed as $\sqrt {2/C_{\!f,i}}\propto \ln (\textit{Re}_{b,i}\sqrt {C_{\!f,i}/2})$. Moreover, it is quantitatively verified that the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$, obtained from direct numerical simulations (DNS) of compressible turbulent channel flows with bulk Mach numbers ranging from $0.2$ to $4$, and friction Reynolds numbers from $200$ to $2000$, elegantly collapse into the Prandtl relation for the incompressible skin-friction law. Additionally, the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$ from DNS of compressible turbulent pipe flows, with bulk Mach numbers ranging from $1.5$ to $3$, and friction Reynolds numbers from $200$ to $1000$, are also unified with the Prandtl relation for the incompressible skin-friction law.

The Monin–Obukhov similarity theory (MOST) is a cornerstone of atmospheric science for describing turbulence in stable boundary layers. Extending MOST to stably stratified turbulent channel flows, however, is non-trivial due to confinement by solid walls. In this study, we investigate the applicability of MOST in closed channels and identify where and to what extent the theory remains valid. A key finding is that the ratio of the half-channel height to the Obukhov length serves as a governing parameter for identifying distinct flow regions and determining their corresponding mean velocity scaling. Hence, we propose a relation to estimate this ratio directly from the governing input parameters: the friction Reynolds and friction Richardson numbers ($\textit{Re}_{\tau }$ and $Ri_{\tau }$). The framework is tested against a series of direct numerical simulations across a range of $\textit{Re}_{\tau }$ and $Ri_{\tau }$. The reconstructed velocity profiles enable accurate prediction of the skin-friction coefficient crucial for quantifying pressure losses in stratified flows in engineering applications.

In this work we propose a neural operator-based coloured-in-time forcing model to predict space–time characteristics of large-scale turbulent structures in channel flows. The resolvent-based method has emerged as a powerful tool to capture dominant dynamics and associated spatial structures of turbulent flows. However, the method faces the difficulty in modelling the coloured-in-time nonlinear forcing, which often leads to large predictive discrepancies in the frequency spectra of velocity fluctuations. Although the eddy viscosity has been introduced to enhance the resolvent-based method by partially accounting for the forcing colour, it is still not able to accurately capture the decay rate of the time-correlation function. Also, the uncertainty in the modelled eddy viscosity can significantly limit the predictive reliability of the method. In view of these difficulties, we propose using the neural operator based on the DeepONet architecture to model the stochastic forcing as a function of mean velocity and eddy viscosity. Specifically, the DeepONet-based model is constructed to map an arbitrary eddy-viscosity profile and corresponding mean velocity to stochastic forcing spectra based on the direct numerical simulation data at $Re_\tau =180$. Furthermore, the learned forcing model is integrated with the resolvent operator, which enables predicting the space–time flow statistics based on the eddy viscosity and mean velocity from the Reynolds-averaged Navier–Stokes (RANS) method. Our results show that the proposed forcing model can accurately predict the frequency spectra of velocity in channel flows at different characteristic scales. Moreover, the model remains robust across different RANS-provided eddy viscosities and generalises well to $Re_\tau =550$.

Based on the generalised Saint-Venant equations for granular flow on an inclined chute, we show how to generate solitary waves from localised perturbations at the inlet. Such perturbations usually give rise to a group of roll waves, but by choosing the system parameters appropriately, the formation of all but the first wave can be suppressed, thus turning this first one into a solitary wave. This calls for a highly diffusive flow, which is realised for inclination angles close to the minimal angle required to keep the granular material flowing.

The interaction between the flow in a channel with multiple obstructions on the bottom and an elastic ice sheet covering the liquid is studied for the case of steady flow. The mathematical model employs velocity potential theory and fully accounts for the nonlinear boundary conditions at the ice/liquid interface and on the channel bottom. The integral hodograph method is used to derive the complex velocity potential of the flow, explicitly containing the velocity magnitude at the interface. This allows the boundary-value problem to be reduced to a system of nonlinear equations for the unknown velocity magnitude at the ice/liquid interface, which is solved using the collocation method. Case studies are carried out for a widened rectangular obstruction, whose width exceeds the wavelength of the interface, and for arrays of triangular ripples forming the undulating bottom shape. The influence of the bottom shape on the interface is investigated for three flow regimes: the subcritical regime, $F \lt F_{{cr}}$, for which the depth-based Froude number is less than the critical Froude number, and the interface perturbation decays upstream and downstream of the obstruction; the ice-supercritical and channel-subcritical regime, $F_{cr} \lt F \lt 1$, for which two waves of different wavelengths extend upstream and downstream to infinity; and the channel-supercritical regime, $F \gt 1$, for which the hydroelastic wave extends downstream to infinity. The results revealed a trapped interface wave above the rectangular obstruction and the ripple patch. The resonance behaviour of the interface over the undulating bottom occurs when the period of ripples approaches the wavelength of the ice/liquid interface.

While flow confinement effects on a shear layer of an one-sided or submerged vegetation array’s interface have been widely studied, turbulent interactions between shear layers in channels with vegetation on both sides remain unclear. This study presents laboratory experiments investigating flow adjustments and turbulent interaction within a symmetrical vegetation–channel–vegetation system, considering varying array widths and densities. In the outer shear layer, the shear stress is primarily balanced by the pressure gradient. As the array extends laterally, the outer penetration of the shear layer reduces from a fully developed thickness to the half-width of the open region, resulting in flow confinement. Flow confinement enhances the pressure gradient, which increases the interior velocity and shear stress at the interface. Despite the time-averaged shear stress being zero at the centreline when the shear layer is confined, the shear instabilities from both sides interact, producing significant turbulent events at the centreline with equal contributions from each side. Furthermore, the two parallel vortex streets self-organised and created a wave response with a $\pi$-radian phase shift , where alternating vortex cores amplify the pressure gradient, intensifying coherent structures and facilitating momentum exchange across the channel centreline. Although the turbulent intensity is enhanced, the decreased residence time for turbulent flow events may limit transport distance. Overall, the shear layer that develops on one interface acts as an additional resistance to shear turbulence on the other interface, leading to a more rapid decline of shear stress in the open region, despite a higher peak at the interface.

We investigate the energy transfer from the mean profile to velocity fluctuations in channel flow by calculating nonlinear optimal disturbances, i.e. the initial condition of a given finite energy that achieves the highest possible energy growth during a given fixed time horizon. It is found that for a large range of time horizons and initial disturbance energies, the nonlinear optimal exhibits streak spacing and amplitude consistent with direct numerical simulation (DNS) at least at ${Re}_\tau = 180$, which suggests that they isolate the relevant physical mechanisms that sustain turbulence. Moreover, the time horizon necessary for a nonlinear disturbance to outperform a linear optimal is consistent with previous DNS-based estimates using eddy turnover time, which offers a new perspective on how some turbulent time scales are determined.

A knowledge gap exists for flows and transport phenomena at the angstrom scale when the Poisson–Nernst–Planck equation based on the concept of the electrical double layer fails. We discovered that streaming conductance becomes pressure-dependent in angstrom channels using latent-track membranes. The streaming current emerges only when the applied pressure exceeds a threshold value, which is inconsistent with the existing knowledge as a constant. With increasing channel size, we found that the pressure-dependent streaming conductance phenomenon weakens and vanishes into a constant streaming conductance regime when the mean channel radius exceeds $\sim$2 nm. The effective surface potential derived from the streaming conductance that divides conduction anomalously increases as the channel narrows. We suspect that the pressure-dependent streaming current is due to the reinforced Coulomb interaction between counterions and deprotonated carboxyl groups at the surface, which is close to the ion channel but different from that of electrified two-dimensional materials. The streaming current emerged due to hydrodynamic friction when the counterions were released from the surface. We approximated the stochastic process of counterion dissociation by a one-dimensional Kramer escape theory framework and defined the Damk$\ddot {\mathrm{o}}$hler number to describe the transition from the nonlinear streaming conductance regime to the linear regime as functions of applied pressure and channel radius and well explained the enhanced effective surface potential in confinement.

We consider two-dimensional (2-D) free surface gravity waves in prismatic channels, including bathymetric variations uniquely in the transverse direction. Starting from the Saint-Venant equations (shallow-water equations) we derive a one-dimensional transverse averaged model describing dispersive effects related solely to variations of the channel topography. These effects have been demonstrated in Chassagne et al. 2019 J. Fluid Mech. 870, 595–616 to be predominant in the propagation of bores with Froude numbers below a critical value of approximately 1.15. The model proposed is fully nonlinear, Galilean invariant, and admits a variational formulation under natural assumptions about the channel geometry. It is endowed with an exact energy conservation law, and admits exact travelling-wave solutions. Our model generalises and improves the linear equations proposed by Chassagne et al. 2019 J. Fluid Mech. 870, 595–616, as well as in Quezada de Luna and Ketcheson, 2021 J. Fluid Mech. 917, A45. The system is recast in two useful forms appropriate for its numerical approximations, whose properties are discussed. Numerical results allow the verification of the implementation of these formulations against analytical solutions, and validation of our model against fully 2-D nonlinear shallow-water simulations, as well as the famous experiments by Treske 1994 J. Hyd. Res. 32, 355–370.

Long-duration and time-resolved particle image velocimetry measurements were conducted in rough-wall open channel flows (OCFs), with the friction Reynolds number ranging from 642 to 2034. The primary objective is to investigate the impacts of various turbulent motions at different scales on the mean wall-shear stress ($\langle \tau _w \rangle$). To achieve this aim, a physical decomposition of $\langle \tau _w \rangle$ was initially performed utilizing the double-averaged methodology proposed by Nikora et al. (2019 J. Fluid Mech. 872, 626–664). This method enabled the breakdown of $\langle \tau _w \rangle$ into three distinct constituents: viscous, turbulent and dispersive stress segments. The findings underscore the substantial roles that turbulent and dispersive stresses play, accounting for over 75 % and 9 % of $\langle \tau _w \rangle$, respectively. Subsequently, a scale decomposition was further applied to analyse the contributions of coherent motions at different scales to $\langle \tau _w \rangle$. Adopting typical cutoff streamwise wavelengths ($\lambda _x = 3h$ and $10h$), the contribution of large-scale motions (LSMs) and very large-scale motions (VLSMs) to the overall wall-shear stress was quantified. It was revealed that turbulent motions with $\lambda _x \gt 3h$ and $\lambda _x \gt 10h$ contribute more than 40 % and 18 % of $\langle \tau _w \rangle$, respectively. The scale decomposition of the wall-shear stress and the contribution from LSMs and VLSMs exhibit evident dependencies on the Reynolds number. The contribution of LSMs and VLSMs to $\langle \tau _w \rangle$ is lower in rough OCFs compared with those of smooth counterparts. Secondary currents induced by the rough wall are hypothesised to be responsible for the reduced strength of LSMs and VLSMs and decreases in their contribution to $\langle \tau _w \rangle$.

Power minimisation in branched fluidic networks has gained significant attention in biology and engineering. The optimal network is defined by channel radii that minimise the sum of viscous dissipation and the volumetric energetic cost of the fluid. For limit cases including laminar flows, high-Reynolds-number turbulence or smooth-channel approximations, optimal solutions are known. However, current methods do not allow optimisation for a large intermediate part of the parameter space which is typically encountered in realistic fluidic networks that exhibit turbulent flow. Here, we present a unifying optimisation approach based on the Darcy friction factor, which has been determined for a wide range of flow regimes and fluid models and is applicable to the entire parameter space: (i) laminar and turbulent flows, including networks that exhibit both flow types, (ii) non-Newtonian fluids (powerlaw, Bingham and Herschel–Bulkley) and (iii) networks with arbitrary wall roughness, including non-uniform relative roughness. The optimal channel radii are presented analytically and graphically. All existing limit cases are recovered, and a concise framework is presented for systematic optimisation of fluidic networks. Finally, the parameter $x$ in the optimisation relationship $Q\propto R^{x}$, with $Q$ the flow rate and $R$ the channel radius, was approximated as a function of the Reynolds number, revealing in which case the entire network can be optimised based on one optimal channel radius, and in which case all radii must be optimised individually. Our approach can be extended to a wide range of fluidic networks for which the friction factor is known, such as different channel curvatures, bubbly flows or specific wall slip conditions.

We investigate the drag reduction effects by two representative blowing/suction-based control methods having different drag reduction mechanisms, i.e. the opposition control and uniform blowing (UB), in a bump-installed turbulent channel flow through direct numerical simulations. We consider two different bulk Reynolds numbers ${\textit {Re}}_b = 5600$ and $12\,600$, and bump heights $h^+ \approx 20$ and $40$. In the opposition-controlled case, the friction drag reduction effect in the case with a bump is similar to that in the case without a bump, while the control effect on the pressure drag is hardly observed. The total drag reduction rate decreases for the higher bump height because the ratio of the pressure drag to the total drag increases as the bump height. In the UB case, UB at $0.1\,\%$ or $0.5\,\%$ of the bulk-mean velocity is imposed on the lower wall with a bump, while the same amount of uniform suction (US) is applied on the upper flat wall to keep the mass flow rate. Although the total friction drag increases due to a detrimental effect of US on the upper wall, the wall-normal motions due to the existence of a bump on the lower wall are suppressed by the UB, so that the pressure drag is decreased, unlike the opposition-controlled case. Due to the difference in the inherent drag reduction mechanisms, the flow separation in the region behind the bump is enhanced by the opposition control, while suppressed by UB.

Evolution of solitary waves and an undular bore intruding through an abrupt transition from a wide basin into a narrow channel with opposing current is investigated. The laboratory experiments are performed in a wave tank that is crafted to achieve a steady and symmetrical shallow-water jet in the basin. The channel has a breadth comparable to the wave lengths, and the flow has Froude number approximately 0.1. The opposing current amplifies and slows the incoming waves on the jet in the basin, but the propagation speed is faster than the local Doppler effect of the current due to the influence of the wave propagating in the flank of the jet. At the channel mouth, the wave amplitude is enhanced due to the waveform altered by the current in the basin, although the amplification in the upstream channel is similar with and without the current. The longer incident waves have greater amplification into the channel. The leading wave of the undular bore is impacted by the opposing flow and transition similarly to the solitary waves. In contrast, the subsequent waves of the undular bore have a complex phase interference on the jet that causes disconnection in the lateral wave formation across the breadth of the jet. At the transition, the subsequent waves exhibit greater amplification than the leading one due to accumulated wave energy at the channel mouth. The intrusion of the undular bore against the current further enhances a rise in mean water level in the channel.

We investigate the drag reduction effect of the streamwise travelling wave-like wall deformation in a high-Reynolds-number turbulent channel flow by large-eddy simulation (LES). First, we assess the validity of subgrid-scale models in uncontrolled and controlled flows. For friction Reynolds numbers $Re_\tau = 360$ and $720$, the Smagorinsky and wall-adapting local eddy-viscosity (WALE) models with a damping function can reproduce well the mean velocity profile obtained by direct numerical simulation (DNS) in both the uncontrolled and controlled flows, leading to a small difference in drag reduction rate between LES and DNS. The LES with finer grid resolution can reproduce well the key structures observed in the DNS of the controlled flow. These results show that the high-fidelity LES is valid for appropriately predicting the drag reduction effect. In addition, a small computational domain is sufficient for reproducing the turbulence statistics, key structures and drag reduction rate obtained by DNS. Subsequently, to investigate the trend of drag reduction rate at higher Reynolds numbers, we utilize the WALE model with the damping function to investigate the control effect at higher Reynolds numbers up to $Re_\tau = 3240$. According to the analyses of turbulence statistics and instantaneous flow fields, the drag reduction at higher Reynolds numbers occurs basically through the same mechanism as that at lower Reynolds numbers. In addition, the drag reduction rate obtained by the present LES approaches that predicted using the semi-empirical formula (Nabae et al., Intl J. Heat Fluid Flow, vol. 82, 2020, 108550) as the friction Reynolds number increases, which supports the high predictability of the semi-empirical formula at significantly high Reynolds numbers.

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.

Modern machine-learning techniques are generally considered data-hungry. However, this may not be the case for turbulence as each of its snapshots can hold more information than a single data file in general machine-learning settings. This study asks the question of whether nonlinear machine-learning techniques can effectively extract physical insights even from as little as a single snapshot of turbulent flow. As an example, we consider machine-learning-based super-resolution analysis that reconstructs a high-resolution field from low-resolution data for two examples of two-dimensional isotropic turbulence and three-dimensional turbulent channel flow. First, we reveal that a carefully designed machine-learning model trained with flow tiles sampled from only a single snapshot can reconstruct vortical structures across a range of Reynolds numbers for two-dimensional decaying turbulence. Successful flow reconstruction indicates that nonlinear machine-learning techniques can leverage scale-invariance properties to learn turbulent flows. We also show that training data of turbulent flows can be cleverly collected from a single snapshot by considering characteristics of rotation and shear tensors. Second, we perform the single-snapshot super-resolution analysis for turbulent channel flow, showing that it is possible to extract physical insights from a single flow snapshot even with inhomogeneity. The present findings suggest that embedding prior knowledge in designing a model and collecting data is important for a range of data-driven analyses for turbulent flows. More broadly, this work hopes to stop machine-learning practitioners from being wasteful with turbulent flow data.

We conduct a numerical study on the drag-reduction mechanism of an opposition- controlled turbulent channel flow from the viewpoint of a symbolic dynamics approach. The effect of the virtual wall formed by the opposition control is maximised at the location of the detection plane $y_d^+ \approx 10$. At this wall-normal location, the local link strength of the self-loop of network nodes representing the negative correlation pattern between the streamwise and wall-normal velocity fluctuations is maximised in the uncontrolled flow. In the controlled case, the multiscale complexity–entropy causality plane and the spatial permutation entropy at $y_d^+ \approx 10$ indicate that the drag-reduction effect is attributed to the reduction of the region where streaks actively coalesce and separate and the suppression of the regeneration cycle in the region near the wall.

We investigate a turbulent stratified plane Poiseuille flow using linear models and nonlinear simulations. We propose the first complete explanation for the prolific and coherent backward (BWs)- and forward-propagating waves (FWs), which have been observed in these flows. We demonstrate a significant presence of oblique waves in the channel core, particularly for the FWs. Critically, we show that neglect of spanwise structure leads to a distorted dispersion relation due to its strong dependence on the angle of obliquity. Interestingly, solutions to the Taylor–Goldstein equations show that wave dynamics is strongly dependent on shear, with only a weak dependence on buoyancy for the BWs at low and order-one wavenumbers, when the wavenumber is scaled by the channel half-height. As the wavenumber increases, waves transition from a shear-dominated regime to a buoyancy-dominated regime, with their dispersion relation tending towards that of idealised internal waves subject to a shear-free and constant-buoyancy-gradient flow, with a characteristic velocity and buoyancy frequency corresponding to respective centreline values in the channel. Finally, we show that the dominance of the BWs arises due to the external forcing of the system, whereby turbulent fluid ejected into the core has a lower momentum when compared with the local flow, therefore preferentially generating BWs in the channel. Qualitatively, channel-core dynamics can be reproduced with low-momentum forcing to a velocity profile with a velocity maximum and a corresponding negative second derivative intersecting a region of strong buoyancy gradient. This structure is inherent to a wide variety of jet-like environmental, atmospheric and industrial flows, suggesting that BWs are a critical control on dynamics of such flows.