We use direct numerical simulation (DNS) to study drag reduction in a lubricated channel, a flow instance in which a thin layer of lubricating fluid is injected in the near-wall region so as to favour the transportation of a primary fluid. In the present configuration, the two fluids have equal density but different viscosity, so that a viscosity ratio $\lambda =\eta _1/\eta _2$ can be defined. To cover a meaningful range of possible situations, we consider five different $\lambda$ in the range $0.25 \le \lambda \le 4$. All DNS are run using the constant power input (CPI) approach, which prescribes that the flow rate is adjusted according to the actual pressure gradient so as to keep constant the power injected into the flow. The CPI approach has been purposely extended here for the first time to the case of multiphase flows. A phase-field method is used to describe the dynamics of the liquid–liquid interface. We unambiguously show that a significant drag reduction (DR) can be achieved for $\lambda \le 2.00$. Reportedly, the observed DR is a non-monotonic function of $\lambda$ and, in the present case, is maximum for $\lambda = 1.00$ (${\simeq }13\,\%$ flow-rate increase). Upon a detailed analysis of the energy budgets, we are able to show the existence of two different DR mechanisms. For $\lambda =1.00$ and $\lambda =2.00$, DR is purely due to the effect of the surface tension – a localized elasticity element that separates the two fluids – which, decoupling the wall-normal momentum transfer mechanisms between the primary and the lubricating layer, suppresses turbulence in the lubricating layer (laminarization) and reduces the overall drag. For $\lambda < 1.00$, turbulence can be sustained in the lubricating layer, because of the increased local Reynolds number. In this case, DR is simply due to the smaller viscosity of the lubricating layer that acts to decrease directly the corresponding wall friction. Finally, we show evidence that an upper bound for $\lambda$ exists, for which DR cannot be observed: for $\lambda =4.00$, we report a slight drag enhancement, thereby indicating that the turbulence suppression observed in the lubricating layer cannot completely balance the increased friction due to the larger viscosity.

The study of a turbulent premixed flame often involves analysing quantities conditioned to different iso-surfaces of a reactive scalar field. Under the influence of turbulence, such a surface is deformed and translated. To track the surface motion, the displacement speed ($S_d$) of the scalar field respective to the local flow velocity is widely used and this quantity is currently receiving growing attention. Inspired by the apparent benefits from a simple decomposition of $S_d$ into contributions due to (i) curvature, (ii) normal diffusion and (iii) chemical reaction, this work aims at deriving and exploring new evolution equations for these three contributions averaged over the reaction surface. Together with a previously obtained $S_d$-evolution equation, the three new equations are presented in a form that emphasizes the decomposition of $S_d$ into three terms. This set of equations is also supplemented with a curvature-evolution equation, hence providing a new perspective to link the flame topology and its propagation characteristics. Using two direct numerical simulation databases obtained from constant-density and variable-density reaction waves, all the derived equations and the term-wise decomposition relations are demonstrated to hold numerically. Comparison of the simulated results indicates that the thermal expansion weakly affects the key terms in the considered evolution equations. Thermal expansion can cause variations in the averaged $S_d$ and its decomposed parts through multiple routes more than introducing a dilatation term. The flow plays a major role to influence the key terms in all equations except the curvature one, due to a cancellation between negatively and positively curved surface elements.

We develop a continuous adjoint formulation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow. The focus is on surface tension-driven flows where the interface is deformed by the local fluid velocity. We apply results from shape optimization to rigorously derive the optimality conditions for a range of interfacial problems. In the cases of interest, we make use of boundary integral methods as a natural choice for the numerical discretization of the flow variables. In the static case, our methodology is tested on a tracking-type cost functional and corresponds to classic shape optimization problems. We show agreement with black-box finite difference-based gradients and accurate minimization of the cost functionals. Finally, we demonstrate the methodology on the control of unsteady droplet deformation through external forcing.

Extreme wave runup around a simplified representative floating production, storage and offloading vessel hull with a vertical bow is studied using computational fluid dynamics, complemented by experimental and diffraction analysis. This is a highly nonlinear system involving large vessel motions and extreme surface waves, and the behaviour is important for offshore design and operations. A separation method based on phase manipulation is carried out to facilitate the extraction of harmonics associated with the Stokes expansion of nonlinear waves. The separation method is applied to numerical and experimental data, and found to work well even for a highly nonlinear wave field scattered from a freely floating ship-shaped body. It is found that both low- and high-frequency second harmonic components can lead to wave runup at significantly higher levels than predicted by a linear analysis, while the vessel motions are very close to linear. The nonlinearity in the local wave field rather than vessel motion is key for the excitation of nonlinear extreme runup.

Modal decomposition techniques have become important mathematical methodologies in analysing complex flow physics. The data-driven methods, such as proper orthogonal decomposition and dynamic mode decomposition are used to extract coherent structures in the form of spatial modes. These methods can be applied to both numerical and experimental data to identify the characteristic dynamics of the system. However, the classical data-driven modal decomposition methods are only applicable to problems with a fixed shape and are not suitable for systems with fluid–structure interaction or systems with shape-changing geometries. In this paper we propose a novel method utilizing a conformal mapping technique to solve this issue. Through different examples with deforming geometry, the capability of the method to accurately capture the flow features is demonstrated. The proposed method is found to be suitable for a wide range of applications and a possible candidate for reduced-order flow modelling of complex shape-changing systems.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$, with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

Transient behaviours during spin-up in rotating Rayleigh–Bénard convection (RBC) with imposed rotation were quantitatively investigated in laboratory experiments. Horizontal and vertical velocity fields were measured by particle image velocimetry with water as the test fluid. Varying the aspect ratio, Rayleigh number and Taylor number, a total of twenty parameters were systematically explored. Toroidal and spiral rolls were formed when the flow reached the rigid-body rotation state, and creation of these structures propagated from the rim towards the internal regions together with the development of the spin-up. Alternate alignments of rolls with opposite meridional circulation transported azimuthal momentum in the rigid-body rotation, and a meandering velocity profile in the radial direction, induced Kelvin–Helmholtz (KH) instability generating azimuthally aligned vortices. The vortices progressively decreased in horizontal dimensions with the wall-to-centre propagation of the vortex formation, but the vortical structures remain larger than the columnar vortices formed in the equilibrium state of a rotating RBC. At the intermediate radius of the fluid layer, the wall-to-centre propagation of the roll formation was overtaken by that of the KH vortex formation. Farther into the interior region, thermal plumes forming columnar vortices were generated as separations of the thermal boundary layers, and the system reached an equilibrium state of rotating RBC dominated by columnar vortices. Use of a fluid vessel with a moderate aspect ratio clarified these transitions to occur in a stepwise fashion, and a spin-up time scale unique in the rotating RBC was found to be from a few to 10 times the Ekman time scale.

We report an idealized numerical study of a melting and freezing solid adjacent to a turbulent, buoyancy-affected shear flow, in order to improve our understanding of topography generation by phase changes in the environment. We use the phase-field method to dynamically couple the heat equation for the solid with the Navier–Stokes equations for the fluid. We investigate the evolution of an initially flat and horizontal solid boundary overlying a pressure-driven turbulent flow. We assume a linear equation of state for the fluid and change the sign of the thermal expansion coefficient, such that the background density stratification is either stable, neutral or unstable. We find that channels aligned with the direction of the mean flow are generated spontaneously by phase changes at the fluid–solid interface. Streamwise vortices in the fluid, the interface topography and the temperature field in the solid influence each other and adjust until a statistical steady state is obtained. The crest-to-trough amplitude of the channels is larger than approximately 10$\delta _{\nu }$ in all cases, with $\delta _{\nu }$ the viscous length scale, but is much larger and more persistent for an unstable stratification than for a neutral or stable stratification. This happens because a stable stratification makes the cool melt fluid buoyant such that it shields the channel from further melting, whereas an unstable stratification makes the cool melt fluid sink, inducing further melting by rising hot plumes. The statistics of flow velocities and melt rates are investigated, and we find that channels and keels emerging in our simulations do not significantly change the mean drag coefficient.

We investigate flow fields and trajectories of sonic jets in a supersonic cross-flow with different injectant properties. The cross-flow is held at a fixed condition with Mach number 1.71, static temperature $\sim$375 K and static pressure $\sim$76 kPa. Jet conditions cover momentum flux ratios $J$ from 1 to 6, molecular weights from $\sim$4 to 44 g mol$^{-1}$ and specific heat ratios from $\sim$1.24 to 1.66. Mie-scattering images are used to study turbulent mixing and trajectory development. Qualitative trends suggest that, at $J=4\text {--}6$, the convective Mach number concept applies as discussed in previous literature. At lower $J$, however, trends for changing molecular weights seem to reverse and the boundary layer might influence turbulent mixing. Analytically estimated jet velocities suggest the suppression of hydrodynamic instabilities changes at different rates for different injectants, as $J$ increases. A newly developed, semi-empirical jet trajectory scaling explicitly considers the momentum flux ratio, boundary layer effects and the existence of the jet bow shock. For validation, this scaling is applied to our trajectory data and those of existing literature, extending the covered parameter space. Quantifying the degree of trajectory correlation shows the scaling is specifically relevant at $J \leq 6$ in this study, where the boundary layer and bow shock influence are important. On the other hand, at higher $J$ and for thin boundary layers, the momentum flux ratio plays a more dominant role. The results in this study can guide the design of injection systems for supersonic applications and improve prediction of jet trajectories.

An understanding of pressure propagation mechanisms is crucial in finding suitable conditions for the temporary storage, transmission and attenuation of a large pressure signal in a multi-plug gelled pipeline. The current work investigates the transient flow start in a multi-plug gelled pipeline, filled with a weakly compressible, thixotropic waxy gelled crude oil. A multi-plug gel can form either naturally or artificially. We solve time-dependent mass and momentum conservation equations for the gas phase and conservation equations together with a shear-history-dependent constitutive model for the gel phase to capture the transient phenomena. An advection equation traces the motion of the gel–gas interface using the volume of fluid method. The results of this work show that pressure propagation in the first gel plug compresses it. The compressed first gel plug further compresses the gas pocket, which delays pressure propagation in the downstream gel. The delay in the pressure propagation through the second gel plug creates a high-pressure gradient in the first gel plug. A high-pressure gradient in the first gel results in faster degradation of the first gel. Degraded gel offers a substantially lower resistance to flow, allowing a high-pressure gradient in the second gel. A steeper pressure gradient assists the breakage of the next gel plug and results in a sequential gel breakage. This sequential gel breakage allows flow restart in a long pipeline. The results also indicate that the compressibility of the gas pocket dominates the overall compressibility of the system, controlling both pressure propagation and flow-restart time.

The jet wiping process is a cost-effective coating technique that uses impinging gas jets to control the thickness of a liquid layer dragged along a moving strip. This process is fundamental in various coating industries (mainly in hot-dip galvanizing) and is characterized by an unstable interaction between the gas jet and the liquid film that results in wavy final coating films. To understand the dynamics of the wave formation, we extend classic laminar boundary layer models for falling films to the jet wiping problem, including the self-similar integral boundary layer and the weighted integral boundary layer models. Moreover, we propose a transition and turbulence model to explore modelling extensions to larger Reynolds numbers and to analyse the impact of the modelling strategy on the liquid film dynamics. The validity of the long-wave formulation was first analysed on a simpler problem, consisting of a liquid film falling over an upward-moving wall, using volume of fluid simulations. This validation proved the robustness of the integral formulation in conditions that are well outside their theoretical limits of validity. Finally, the three models were used to study the response of the liquid coat to harmonic and non-harmonic oscillations and pulsations in the impinging jet. The impact of these disturbances on the average coating thickness and wave amplitude is analysed, and the range of dimensionless frequencies yielding maximum disturbance amplification is presented.

This work addresses the question of a pertinent macroscale model describing creeping, incompressible and single-phase flow of a Newtonian fluid in an exuding, rigid and homogeneous porous medium. The macroscopic model is derived by upscaling the pore-scale Stokes equations considering a normal mass flux at the solid–fluid interface. The upscaled mass equation shows that the average velocity is non-solenoidal. In addition, the macroscopic momentum equation involves a Darcy term with the classical permeability tensor accounting for macroscopic drag and a correction velocity vector which is a signature of the local fluid displacements induced by the exuding phenomenon. This correction is the sum of a term accounting for the local exuding effect and a compensation term associated with the assumption of spatial periodicity. Both the first term and the permeability tensor are obtained from the solution of the same unique and intrinsic closure problem, which corresponds to that involved in the classical Darcy's law. The upscaled model is validated by comparisons with pore-scale numerical simulations in several illustrative examples. The different configurations evidence the richness of the problem, despite the apparent simplicity of its formulation. The results of this work motivate further investigation about the influence of internal flow sources in transport phenomena in porous media.

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

We present results from open-channel, Euler–Lagrange (EL) simulations of turbulent flow over an erodible particle bed at a shear Reynolds number of $Re_{\tau }=180$. Upon space and time averaging, our simulations correctly reproduce the Wong & Parker (ASCE J. Hydraul. Engng, vol. 132, issue 11, 2006, pp. 1159–1168) bedload transport relation (WP). However, local and instantaneous sediment flux shows orders of magnitude scatter around the WP prediction. Visualization of the vortical structures using swirling strength shows the existence of bed-penetrating Kelvin–Helmholtz (KH) vortex packets, which coupled with particle inertia are primarily responsible for the large scatter. The results also show that Euler–Euler (EE) simulations, where the individual sediment grains are not distinguished, are still able to reliably capture the turbulent shear stress variation, however, they do not capture the wide distribution of sediment flux indicative of saltating transport. The KH vortices induce non-zero streamwise and bed-normal velocities at the upper surface of the bed, which must be considered in EE simulations.

A novel mechanism for the amplification of distortions to the material interface of liquid jets is identified. The mechanism is independent of the exponential instability of the flow and can intensify small perturbations to the material interface by several orders of magnitude. Depending on the parameters, it can amplify interfacial distortions at a faster pace than modal mechanisms such as the Kelvin–Helmholtz instability. The study is based on spatial linear stability theory in a two-fluid formulation that accounts for the effects of both viscosity and surface tension. The analysis of the mechanism is cast into an optimization problem in the surface tension energy of the interface distortion and discounts the trivial redistribution of perturbation kinetic energy. The identified mechanism is related to the Orr mechanism, and amplifies distortions to the material interface via a reorientation of perturbations by the mean shear. Analyses of the linearized energy budgets show that energy is extracted from the mean shear by the production term of the streamwise perturbation velocity component and subsequently transferred to the radial perturbation velocity component, where it is absorbed by the surface tension potential of the interface. The gain in surface tension energy attributable to the mechanism is shown to scale linearly with the Reynolds number. A critical Weber number is identified as a lower bound beyond which the mechanism becomes active, and a power-law relation to the Reynolds number is established. Nonlinear simulations based on the full two-fluid Navier–Stokes equations substantiate the observability and realizability of the mechanism.

We carry out direct numerical simulations of turbulent Rayleigh–Bénard convection in a square box with rough conducting plates over the Rayleigh number range $10^7\leqslant Ra\leqslant 10^9$ and the Prandtl number range $0.01\leqslant Pr\leqslant 100$. In Zhang et al. (J. Fluid Mech., vol. 836, 2018, R2), it was reported that while the measured Nusselt number $Nu$ is enhanced at large roughness height $h$, the global heat transport is reduced at small $h$. The division between the two regimes yields a critical roughness height $h_c$, and we now focus on the effects of the Prandtl number ($Pr$) on $h_c$. Based on the variations of $h_c$, we identify three regimes for $h_c(Pr)$. For low $Pr$, thermal boundary layers become thinner with increasing $Pr$. This makes the boundary layers easier to be disrupted by rough elements, leading to the decrease of $h_c$ with increasing $Pr$. For moderate $Pr$, the corner-flow rolls become much more pronounced and suppress the global heat transport via the competition between the corner-flow rolls and the large-scale circulation (LSC). As a consequence, $h_c$ increases with increasing $Pr$ due to the intensification of the corner–LSC competition. For high $Pr$, the convective flow transitions to the plume-controlled regime. As the rough elements trigger much stronger and more frequent plume emissions, $h_c$ again decreases with increasing $Pr$.

This paper solves an approximate form of conservation of mass and momentum for a turbine in a wind farm array. The solution is a fairly simple explicit relationship that predicts the streamwise velocity distribution within a wind farm with an arbitrary layout. As this model is obtained by solving flow-governing equations directly for a turbine that is subject to upwind turbine wakes, no ad hoc superposition technique is needed to predict wind farm flows. A suite of large-eddy simulations (LES) of wind farm arrays is used to examine self-similarity as well as validity of the so-called conservation of momentum deficit for turbine wakes in wind farms. The simulations are performed with and without the presence of some specific turbines in the wind farm. This allows us to systematically study some of the assumptions made to develop the analytical model. A modified version of the conservation of momentum deficit is also proposed to provide slightly better results at short downwind distances, as well as in the far wake of turbines deep inside a wind farm. Model predictions are validated against the LES data for turbines in both full-wake and partial-wake conditions. While our results highlight the limitation in capturing the flow speed-up between adjacent turbine columns, the model is overall able to acceptably predict flow distributions for a moderately sized wind farm. Finally, the paper employs the new model to provide insights on the accuracy of common wake superposition methods.

This work provides evidence that anisotropic drainage in sheared foam is at the origin of convective instability in very long foam channels. Convective instability occurs in foam under forced drainage when a critical liquid fraction is exceeded. Liquid spontaneously accumulates at one side of the channel. The weight imbalance induces convection rolls in the foam. Experiments in a very long vertical foam channel demonstrate that the critical liquid fraction is smaller than in previous findings by a factor of five. The critical liquid fraction depends on both the channel length and the inhomogeneity of the liquid feed. Well below the critical liquid fraction, a static, elastic shear deformation of the foam structure occurs. At the critical liquid fraction, initial steady convection rolls are located at the lower region of the channel and expand as the liquid fraction further increases. Combining the drainage equation with both the elastic response of the foam and a model for anisotropic drainage, a critical liquid fraction for the growth of an initial liquid imbalance is derived analytically, which corresponds very well to experimental findings. Numerical simulations of the drainage equation and the elastic response of the foam reproduce these experimental and analytical findings.

Interscale energy transfer in wall turbulence has been intensively studied in recent years, and both forward (i.e. from larger to smaller scales) and reversed transfers of turbulent energy have been found, although their corresponding physical phenomena have not been revealed. In the present study, we perform direct numerical simulations of turbulent plane Couette flow with reduced-size computational domains, where either the streamwise or the spanwise domain size is reduced to its minimal length, aiming to elucidate the role of scale interactions in each direction. Our computational results with the streamwise-minimal domain suggest that the interplays between streamwise-elongated streaks and vortices smaller than the streamwise minimal length are the essential scale interactions for both the inner and the outer structures. We further show that these streamwise-independent and smaller-scale structures exchange energy through forward and reversed interscale energy transfers, and the reversed energy transfer maintains the energy production at larger scales. Based on the resemblance between the observed Reynolds-stress transport and the scenario of the self-sustaining cycle, we conjecture that the forward and reversed energy transfers mainly represent the streak instabilities and regeneration of streamwise vortices, respectively. Furthermore, the computation with the spanwise-minimal domain indicates that the interscale energy transfers observed by one-dimensional spanwise spectral analysis are likely related to the individual dynamics of each inner and outer structure, rather than representing their interactions.

We elucidate the effect of the secondary baroclinic vorticity (SBV) on the Richtmyer–Meshkov instability (RMI) accelerated by a weak incident shock and develop a vortex-based model for spike and bubble growth rates. Two major mechanisms of the single-mode RMI, the primary baroclinic vorticity (PBV) and the pressure perturbation, are distinguished by simplified models with the vortex-surface field. We find that the effect of the pressure perturbation can be neglected in the present RMI, and the growth of the interface or vortex surface is first driven by the PBV. Subsequently, the SBV, generated by the misalignment between the density gradient across the interface and the pressure gradient produced by the PBV-induced velocity, leads to the nonlinear growth of the interface with the generation of spikes and bubbles. Inspired by this mechanism, we develop a predictive model of spike and bubble growth rates using the motion of viscous vortex rings. The circulation of the vortex ring is modelled with the SBV effect. This model is validated by five data sets of direct numerical simulations and experiments of the single-mode RMI with various initial conditions.