The effects of surface roughness in the transitionally rough regime on the overlying near-wall turbulence are modelled using quasi-linear approximations proposed recently: minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, 2020, J. Fluid Mech., vol. 894, A23), data-driven quasi-linear approximation (DQLA) (Holford et al., 2024, J. Fluid Mech., vol. 980, A12) and a newly established variant of MQLA (M2QLA, minimal two-mode quasi-linear approximation). The transpiration-resistance model (TRM) for boundary conditions is applied to account for the surface roughness (Lācis et al., 2020, J. Fluid Mech., vol. 884, A21). It is shown that many essential near-wall turbulence statistics are fairly well captured by the quasi-linear approximations in a wide range of slip and transpiration lengths for the TRM boundary conditions. In particular, the virtual origins and the resulting roughness functions are well predicted, showing good agreement with those from previous direct numerical simulations (DNS) in mild roughness cases. The DQLA and M2QLA, which incorporate streamwise-dependent Fourier modes in the approximations, are also shown to perform a little better than MQLA, especially with DQLA reproducing the two-dimensional energy spectra qualitatively consistent with the DNS. Finally, with a computational cost much lower than DNS, it is shown that the proposed quasi-linear approximation frameworks offer an efficient tool to rapidly explore the roughness effects within a large parameter space.

We report an anomalous capillary phenomenon that reverses typical capillary trapping via nanoparticle suspension and leads to a counterintuitive self-removal of non-aqueous fluid from dead-end structures under weakly hydrophilic conditions. Fluid interfacial energy drives the trapped liquid out by multiscale surfaces: the nanoscopic structure formed by nanoparticle adsorption transfers the molecular-level adsorption film to hydrodynamic film by capillary condensation, and maintains its robust connectivity, then the capillary pressure gradient in the dead-end structures drives trapped fluid motion out of the pore continuously. The developed mathematical models agree well with the measured evolution dynamics of the released fluid. This reversing capillary trapping phenomenon via nanoparticle suspension can be a general event in a random porous medium and could dramatically increase displacement efficiency. Our findings have implications for manipulating capillary pressure gradient direction via nanoparticle suspensions to trap or release the trapped fluid from complex geometries, especially for site-specific delivery, self-cleaning, or self-recover systems.

Efficient tools for predicting the drag of rough walls in turbulent flows would have a tremendous impact. However, accurate methods for drag prediction rely on experiments or numerical simulations which are costly and time consuming. Data-driven regression methods have the potential to provide a prediction that is accurate and fast. We assess the performance and limitations of linear regression, kernel methods and neural networks for drag prediction using a database of 1000 homogeneous rough surfaces. Model performance is evaluated using the roughness function obtained at a friction Reynolds number $Re_\tau$ of 500. With two trainable parameters, the kernel method can fully account for nonlinear relations between the roughness function $\Delta U^+$ and surface statistics (roughness height, effective slope, skewness, etc.). In contrast, linear regression cannot account for nonlinear correlations and displays large errors and high uncertainty. Multilayer perceptron and convolutional neural networks demonstrate performance on par with the kernel method but have orders of magnitude more trainable parameters. For the current database size, the networks’ capacity cannot be fully exploited, resulting in reduced generalizability and reliability. Our study provides insight into the appropriateness of different regression models for drag prediction. We also discuss the remaining steps before data-driven methods emerge as useful tools in applications.

In this study, we investigate the stability of a film that is attached to a corner between a cylinder and a substrate, using a combination of theoretical and numerical approaches. Notably, we place our focus on flat and thin films where the contact line is almost perpendicular to the cylinder wall whereas a small angle forms between the contact line and the substrate, and the film size is smaller than the cylinder radius. The film stability, which depends on the film size and the wall wettability, is first predicted by a standard linear stability analysis (LSA) within the long-wave theoretical framework. We find that the film size plays the most important role in controlling the film stability. Specifically, the thicker the film is, the less sensitive it becomes to the large-wavenumber perturbation. The wall wettability mainly impacts the growth rates of perturbations and slightly influences the marginal stability and postinstability patterns of wrapping films. We compare the LSA predictions with numerical results obtained from a disjoining pressure model (DPM) and volume-of-fluid (VOF) simulations, which provide more insights into the film breakup process. At the early stage there is a strong agreement between the LSA predictions and the DPM results. Notably, as the perturbation grows, thin film regions connecting two neighbouring satellite droplets form which may eventually lead to a stable or temporary secondary droplet, an aspect which the LSA is incapable of capturing. In addition, the VOF simulations suggest that beyond a critical film size, merging between two neighbouring drops becomes involved during the breakup stage. Therefore, the LSA predictions are able to provide only an upper limit on the final number of satellite droplets.

Turbulent flows over porous lattices consisting of rectangular cuboid pores are investigated using scale-resolving direct numerical simulations. Beyond a certain threshold which is primarily determined by the wall-normal Darcy permeability, ${{\mathsf{K}}_y}$, near-wall turbulence transitions from its canonical regime, marked by the presence of streak-like structures, to another marked by the presence of Kelvin–Helmholtz-like (K–H-like) spanwise-coherent structures. The threshold agrees well with that previously established in studies where permeable-wall boundary conditions had been used as surrogates for a porous substrate (Gómez-de Segura & García-Mayoral, J. Fluid Mech., vol. 875, 2019, pp. 124–172). In the smooth-wall-like regime, none of the investigated substrates demonstrate any reduction in drag relative to a smooth-wall flow. At the permeable surface, a notable component of the flow is that which adheres to the pore geometry and undergoes modulation by the turbulent scales of motions due to the interaction mechanism described by Abderrahaman-Elena et al. (J. Fluid Mech., vol. 865, 2019, pp. 1042–1071). Its resulting effect can be quantified in terms of an amplitude modulation (AM) using the approach of Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337). This pore-coherent flow component persists throughout the porous substrate, highlighting the importance of a given substrate's microstructure in the presence of an overlying turbulent flow. This geometry-related aspect of the flow is not accounted for when continuum-based models for a porous medium or effective representations of them, such as wall boundary conditions, are used. The intensity of the AM effect is enhanced in the K–H-like regime and becomes strengthened with larger permeability. As a result, structured porous materials may be designed to exploit or mitigate these flow features depending upon the intended application.

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$. The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$. High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

Using numerical simulations, we investigate the effects of Marangoni stresses induced by surfactants on the effective slip length of liquid-infused surfaces (LIS) with transverse grooves. The surfactants are assumed soluble in the external liquid shearing the surface and can adsorb onto the interfaces. Two different adsorption models are used: a classical Frumkin model and a more advanced model that better describes the decrease of surface tension for minuscule concentrations. The simulations show that LIS may face even more severe effects of surfactants than previously investigated superhydrophobic surfaces. Constructing an analytical model for the effective slip length, we can predict the critical surfactant concentration for which the slip length decreases significantly. This analytical model describes both adsorption models of LIS on a unified framework if properly adjusted. We also advance the understanding of when surfactant advection gives rise to highly skewed interfacial concentrations: the so-called partial stagnant cap regime. To a good approximation, this regime can only exist below a specific surfactant concentration given by the Marangoni number and the strength of the surfactants.

A set of boundary conditions called the transpiration-resistance model (TRM) is investigated in altering near-wall turbulence. The TRM was proposed by Lacis et al. (J. Fluid Mech., vol. 884, 2020, p. A21) as a means of representing the net effect of surface micro-textures on their overlying bulk flows. It encompasses conventional Navier-slip boundary conditions relating the streamwise and spanwise velocities to their respective shears through the slip lengths $\ell _x$ and $\ell _z$. In addition, it features a transpiration condition accounting for the changes induced in the wall-normal velocity by expressing it in terms of variations of the wall-parallel velocity shears through the transpiration lengths $m_x$ and $m_z$. Greater levels of drag increase occur when more transpiration takes place at the boundary plane, with turbulent transpiration being predominately coupled to the spanwise shear component for canonical near-wall turbulence. The TRM reproduces the effect of a homogeneous and structured roughness up to $k^{+}\approx 18$, encompassing the regime of smooth-wall-like turbulence described using virtual origins (Luchini, 1996 Reducing the turbulent skin friction. In Computational Methods in Applied Sciences’ 96 (Paris, 9–13 Sept. 1996), pp. 465–470. Wiley; Ibrahim et al., J. Fluid Mech., vol. 915, 2021, p. A56) and slightly beyond it. The transpiration factor is defined as the product of the slip and transpiration lengths, i.e. $(m\ell )_{x,z}$. This factor contains the compound effect of the wall-parallel velocity occurring at the boundary plane and increased permeability, both of which lead to the transport of momentum in the wall-normal direction. A linear relation between the transpiration factor and the roughness function is observed for regularly textured surfaces in the transitionally rough regime of turbulence. This shows that such effective flow quantities can be suitable measures for characterizing rough surfaces in this flow regime.

Liquid-infused surfaces can reduce friction drag in both laminar and turbulent flows. However, the heat transfer properties of such multi-phase surfaces have still not been investigated to a large extent. We use numerical simulations to study conjugate heat transfer of liquid-filled grooves. It is shown that heat transfer can increase for both laminar and turbulent liquid flows due to recirculation in the surface texture. Laminar flow simulations show that for the increase to be substantial, the thermal conductivity of the solid must be similar to the thermal conductivity of the fluids, and the recirculation in the grooves must be sufficiently strong (Péclet number larger than 1). The ratio of the surface cavity to the system height is an upper limit of the direct contribution from the recirculation. While this ratio can be significant for laminar flows in microchannels, it is limited for turbulent flows, where the system scale (e.g. channel height) usually is much larger than the texture height. However, heat transfer enhancement of the order of $10\,\%$ is observed (with a net drag reduction) in a turbulent channel flow at a friction Reynolds number ${{Re}}_\tau \approx 180$. It is shown that the turbulent convection in the bulk can be enhanced indirectly from the recirculation in the grooves.

The motion of the three-phase contact line between two immiscible fluids and a solid surface arises in a variety of wetting phenomena and technological applications. One challenge in continuum theory is the effective representation of molecular motion close to the contact line. Here, we characterize the molecular processes of the moving contact line to assess the accuracy of two different continuum two-phase models. Specifically, molecular dynamics simulations of a two-dimensional droplet between two moving plates are used to create reference data for different capillary numbers and contact angles. We use a simple-point-charge/extended water model. This model provides a very small slip and a more realistic representation of the molecular physics than Lennard-Jones models. The Cahn–Hilliard phase-field model and the volume-of-fluid model are calibrated against the drop displacement from molecular dynamics reference data. It is shown that the calibrated continuum models can accurately capture droplet displacement and droplet break-up for different capillary numbers and contact angles. However, we also observe differences between continuum and atomistic simulations in describing the transient and unsteady droplet behaviour, in particular, close to dynamical wetting transitions. The molecular dynamics of the sheared droplet provide insight into the line friction experienced by the advancing and receding contact lines. The presented results will serve as a stepping stone towards developing accurate continuum models for nanoscale hydrodynamics.

Recent developments in neural networks have shown the potential of estimating drag on irregular rough surfaces. Nevertheless, the difficulty of obtaining a large high-fidelity dataset to train neural networks is deterring their use in practical applications. In this study, we propose a transfer learning framework to model the drag on irregular rough surfaces even with a limited amount of direct numerical simulations. We show that transfer learning of empirical correlations, reported in the literature, can significantly improve the performance of neural networks for drag prediction. This is because empirical correlations include ‘approximate knowledge’ of the drag dependency in high-fidelity physics. The ‘approximate knowledge’ allows neural networks to learn the surface statistics known to affect drag more efficiently. The developed framework can be applied to applications where acquiring a large dataset is difficult but empirical correlations have been reported.

Liquid-infused surfaces (LISs) are a promising technique for reducing friction, fouling and icing in both laminar and turbulent flows. Previous work has demonstrated that these surfaces are susceptible to shear-driven drainage. Here, we report a different failure mode using direct numerical simulations of a turbulent channel flow with liquid-infused longitudinal grooves. When the liquid–liquid surface tension is small and/or grooves are wide, we observe travelling-wave perturbations on the interface with amplitudes larger than the viscous sublayer of the turbulent flow. These capillary waves induce a roughness effect that increases drag. The generation mechanism of these waves is explained using the theory developed by Miles for gravity waves. Energy is transferred from the turbulent flow to the LIS provided that there is a negative curvature of the mean flow at the critical layer. Given the groove width, the Weber number and an estimate of the friction Reynolds number, we provide relations to determine whether a LIS behaves as a smooth or rough surface in a turbulent flow.

The surface texture of materials plays a critical role in wettability, turbulence and transport phenomena. In order to design surfaces for these applications, it is desirable to characterise non-smooth and porous materials by their ability to exchange mass and momentum with flowing fluids. While the underlying physics of the tangential (slip) velocity at a fluid–solid interface is well understood, the importance and treatment of normal (transpiration) velocity and normal stress is unclear. We show that, when the slip velocity varies at an interface above the texture, a non-zero transpiration velocity arises from mass conservation. The ability of a given surface texture to accommodate a normal velocity of this kind is quantified by a transpiration length. We further demonstrate that normal momentum transfer gives rise to a pressure jump. For a porous material, the pressure jump can be characterised by so-called resistance coefficients. By solving five Stokes problems, the introduced measures of slip, transpiration and resistance can be determined for any anisotropic non-smooth surface consisting of regularly repeating geometric patterns. The proposed conditions are a subset of the effective boundary conditions derived from formal multi-scale expansion. We validate and demonstrate the physical significance of the effective conditions on two canonical problems – a lid-driven cavity and a turbulent channel flow, both with non-smooth bottom surfaces.

Surfaces with filamentous structures are ubiquitous in nature on many different scales, ranging from forests to micrometre-sized cilia in organs. Hairy surfaces are elastic and porous, and it is not fully understood how they modify turbulence near a wall. The interaction between hairy surfaces and turbulent flows is here investigated numerically in a turbulent channel flow configuration at friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\approx 180$. We show that a filamentous bed of a given geometry can modify a turbulent flow very differently depending on the resonance frequency of the surface, which is determined by the elasticity and mass of the filaments. Filaments having resonance frequencies lower than the main frequency content of the turbulent wall-shear stress conform to slowly travelling elongated streaky structures, since they are too slow to adapt to fluid forces of higher frequencies. On the other hand, a bed consisting of stiff and low-mass filaments has a high resonance frequency and shows local regions of increased permeability, which results in large entrainment and a vast increase in drag.

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

Interfacial boundary conditions determined from empirical or ad hoc models remain the standard approach to model fluid flows over porous media, even in situations where the topology of the porous medium is known. We propose a non-empirical and accurate method to compute the effective boundary conditions at the interface between a porous surface and an overlying flow. Using a multiscale expansion (homogenization) approach, we derive a tensorial generalized version of the empirical condition suggested by Beavers & Joseph (J. Fluid Mech., vol. 30 (01), 1967, pp. 197–207). The components of the tensors determining the effective slip velocity at the interface are obtained by solving a set of Stokes equations in a small computational domain near the interface containing both free flow and porous medium. Using the lid-driven cavity flow with a porous bed, we demonstrate that the derived boundary condition is accurate and robust by comparing an effective model to direct numerical simulations. Finally, we provide an open source code that solves the microscale problems and computes the velocity boundary condition without free parameters over any porous bed.

A reactive control technique with localised actuators and sensors is used to delay the transition to turbulence in a flat-plate boundary-layer flow. Through extensive direct numerical simulations, it is shown that an adaptive technique, which computes the control law on-line, is able to significantly reduce skin-friction drag in the presence of random three-dimensional perturbation fields with linear and weakly nonlinear behaviour. An energy budget analysis is performed in order to assess the net energy saving capabilities of the linear control approach. When considering a model of the dielectric-barrier-discharge (DBD) plasma actuator, the energy spent to create appropriate actuation force inside the boundary layer is of the same order as the energy gained from reducing skin-friction drag. With a model of an ideal actuator a net energy gain of three orders of magnitude can be achieved by efficiently damping small-amplitude disturbances upstream. The energy analysis in this study thus provides an upper limit for what we can expect in terms of drag-reduction efficiency for linear control of transition as a means for drag reduction.

In boundary-layer flows, one may reduce skin-friction drag by delaying the onset of laminar-to-turbulent transition via the attenuation of small-amplitude Tollmien–Schlichting (TS) waves. In this work, we use numerical simulations and experiments to compare the robustness of adaptive and model-based techniques for reducing the growth of two-dimensional TS disturbances. In numerical simulations, the optimal linear quadratic Gaussian (LQG) regulator shows the best performance under the conditions it was designed for. However, it is found that the performance deteriorates linearly with the drift of the Reynolds number from its nominal value. As a result, an order-of-magnitude loss of performance is observed when applying the computation-based LQG controller in wind-tunnel experiments. In contrast, it is shown that the adaptive filtered-X least-mean-squares (FXLMS) algorithm is able to maintain an essentially constant performance for significant deviations of the nominal values of the disturbance amplitude and Reynolds number.

Active linear control is applied to delay the onset of laminar–turbulent transition in the boundary layer over a flat plate. The analysis is carried out by numerical simulations of the nonlinear, transitional regime. A three-dimensional, localized initial condition triggering Tollmien–Schlichting waves of finite amplitude is used to numerically simulate the transition to turbulence. Linear quadratic Gaussian controllers based on reduced-order models of the linearized Navier–Stokes equations are designed, where the wall sensors and the actuators are localized in space. A parametric analysis is carried out in the nonlinear regime, for different disturbance amplitudes, by investigating the effects of the actuation on the flow due to different distributions of the localized actuators along the spanwise direction, different sizes of the actuators and the effort of the controllers. We identify the range of parameters where the controllers are effective and highlight the limits of the device for high amplitudes and strong control action. Despite the fully linear control approach, it is shown that the device is effective in delaying the onset of laminar–turbulent transition in the presence of packets characterized by amplitudes $a\approx 1\hspace{0.167em} \% $ of the free stream velocity at the actuator location. Up to these amplitudes, it is found that a proper choice of the actuators positively affects the performance of the controller. For a transitional case, $a\approx 0. 20\hspace{0.167em} \% $ , we show a transition delay of $\Delta {\mathit{Re}}_{x} = 3. 0\times 1{0}^{5} $ .

The Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow trajectory. In this paper, we perform a Koopman analysis of the first Hopf bifurcation of the flow past a circular cylinder. First, we decompose the flow into a sequence of Koopman modes, where each mode evolves in time with one single frequency/growth rate and amplitude/phase, corresponding to the complex eigenvalues and eigenfunctions of the Koopman operator, respectively. The analytical construction of these modes shows how the amplitudes and phases of nonlinear global modes oscillating with the vortex shedding frequency or its harmonics evolve as the flow develops and later sustains self-excited oscillations. Second, we compute the dynamic modes using the dynamic mode decomposition (DMD) algorithm, which fits a linear combination of exponential terms to a sequence of snapshots spaced equally in time. It is shown that under certain conditions the DMD algorithm approximates Koopman modes, and hence provides a viable method to decompose the flow into saturated and transient oscillatory modes. Finally, the relevance of the analysis to frequency selection, global modes and shift modes is discussed.