Understanding the hysteretic behaviour in fluid–fluid displacement processes in porous media is critical in many engineering applications. In this work, we study the quasi-static immiscible displacement process in two-dimensional porous media during cyclic injections in the context of carbon geosequestration. The role of wettability on the residual trapping of CO$_2$ is investigated numerically using an extended interface tracking algorithm. Despite that higher CO$_2$ saturation can be achieved in CO$_2$-wet porous media after the first CO$_2$ injection, the majority of CO$_2$ is found to be unstable and can be mobilised during subsequent water injection processes. An improvement in the residual trapping of CO$_2$ is observed as the number of injection cycles increases, which is associated with the dispersion of continuous CO$_2$ ganglia into numerous smaller blobs. Compared with either water-wet or CO$_2$-wet porous media, it is found that less CO$_2$ is trapped within the neutral-wet ones at equilibrium state after a sufficient number of injection cycles. The hysteretic behaviour of saturation between water/CO$_2$ injection cycles is found to follow an exponential decay, which eventually reaches a finite value. This process corresponds to the shift of the mobile region during displacement from typical capillary fingering to a less ramified regime, which ultimately converges towards main flow channels. This work highlights the hysteretic behaviour during cyclic injections, providing insights on the wettability impacts on multiphase flow in porous media, which is of great importance in applications such as carbon geosequestration and geological hydrogen storage.

Understanding how turbulence enhances irreversible scalar mixing in density-stratified fluids is a central problem in geophysical fluid dynamics. While isotropic overturning regions are commonly the focus of mixing analyses, we here investigate whether significant mixing may arise in anisotropic statically stable regions of the flow. Focusing on a single forced direct numerical simulation of stratified turbulence, we analyse spatial correlations between the vertical density gradient $\partial \rho /\partial z$ and the dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$, the latter quantifying scalar mixing. The domain is characterized by relatively well-mixed density layers separated by sharp stable interfaces that are correlated with high vertical shear. While static instability is most prevalent within the mixed layers, much of the scalar mixing is localized to the intervening interfaces, a phenomenon not apparent if considering local static instability or $\epsilon$ alone. While the majority of the domain is characterized by the canonical flux coefficient $\varGamma \equiv \chi /\epsilon =0.2$, often assumed in ocean mixing parametrizations, extreme values of $\chi$ within the statically stable interfaces, associated with elevated $\varGamma$, strongly skew the bulk statistics. Our findings suggest that current parametrizations of turbulent mixing may be biased by undersampling, such that the most common, but not necessarily the most significant, mixing events are overweighted. Having focused here on a single simulation of stratified turbulence, it is hoped that our results motivate a broader investigation into the role played by stable density interfaces in mixing, across a wider range of parameters and forcing schemes representative of ocean turbulence.

Direct numerical simulation data obtained from four pairs of turbulent, lean hydrogen–air, complex-chemistry flames are analysed to explore the influence of molecular diffusion on flame surface density, displacement speed $S_d$ and the flame surface density transport equation terms. Each pair involves (i) a flame where mixture-averaged molecular diffusivities are adopted and Lewis number $Le$ is significantly less than unity and (ii) an equidiffusive flame where all molecular diffusivities are set equal to molecular heat diffusivity of the mixture and $Le=1$, with other things being equal. Reported results show that significantly higher turbulent burning rates simulated in the former flames result mainly from an increase in the local fuel consumption rate, whereas an increase in flame surface area plays a secondary role, especially in more intense turbulence. The rate increase stems from (i) an increase in the peak local fuel consumption rate and (ii) an increase in a width of a zone where the rate is significant. The latter phenomenon is of more importance in richer flames and both phenomena are most pronounced in the vicinity of the flame leading edges, thus indicating a crucial role played by the leading edge of a premixed turbulent flame in its propagation. Moreover, mean displacement speed differs significantly from the laminar flame speed even in the equidiffusive flames, varies substantially across flame brush and may be negative at the leading edges of highly turbulent flames.

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$, compares to the basic-state amplitude parameter, $\epsilon \ll 1$. For $\mu ={O}(1)$, the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$, this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

The evolution of a turbulent boundary layer over truncated cones in staggered and random configurations is investigated using direct numerical simulations. Three random planform densities and one staggered, which closely match the configuration in the recent experiment by Womack et al. (J. Fluid Mech., vol. 933, 2022, A38), are considered. The experiment is also used to validate the present computations which are found to reproduce the flow measurements very accurately. Spanwise heterogeneity in the mean streamwise velocity in the form of high and low momentum pathways was detected in both the staggered and random arrangements, indicating the presence of secondary flows in the cross-stream plane. The momentum pathways found in the staggered arrangements are much weaker and confined near the roughness crest, while in the random arrangements they approach the edge of the boundary layer. We found a clear correlation of the momentum pathways with the leading edge of the roughness, while the correlation to the local topography was very weak for all the cases studied. Vorticity transport analysis is also employed to explore the nature of the secondary flows formed, highlighting the strong presence of vortex stretching near the leading edge of the roughness. The impact of the momentum pathways on the flow statistics, especially in the random arrangements, is demonstrated by the breakdown of outer-layer similarity.

An analytical model is developed which explores the impact of irregular sea-floor roughness on large-scale oceanic flows. The previously reported asymptotic ‘sandpaper’ theory of flow-topography interaction represents relatively swift currents and exhibits singular behaviour in the weak flow limit. The present investigation systematically spans a wider parameter space and identifies the principal dissimilarities in the topographic regulation of slow and fast currents. The fast flows are controlled by the Reynolds stresses produced by topographically generated eddies. In contrast, relatively weak flows are more affected by the eddy-induced bottom form drag. The asymptotic models for fast and slow currents are then combined to arrive at a concise description of flow forcing by small-scale topography in homogeneous and multilayer models. The proposed closure is validated by comparing corresponding topography-resolving and parametric simulations.

The electromagnetically driven anticyclonic flow in the wide gap of a rotating concentric spheres system is studied experimentally and numerically in the laminar regime. The working fluid is an electrolyte contained in the spherical gap. The outer sphere rotates at constant angular speed, and the inner sphere is at rest. The electromagnetic stirring is generated due to the interaction of a direct current, which is injected radially through ring-shaped electrodes located at the equatorial zone of each sphere, and a dipolar magnetic field produced by a permanent magnet located inside the inner sphere. Experimental velocity fields in the equatorial plane were obtained with particle image velocimetry. Additionally, full three-dimensional numerical simulations were performed. The high shearing at the equatorial region promotes an instability that can be perceived as a triggering mechanism of four tornado-like vortical structures tilted and entangled in the polar direction. The instability structure rotates in either the cyclonic or the anticyclonic direction, depending on the flow parameters, namely, the angular velocity of the outer sphere and the applied electric current. To the best knowledge of the authors, this is the first time the phenomenon under study is reported with electromagnetic forcing. The numerical results agree quantitatively with the experimental observations.

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$, with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

In this paper, we report on a direct numerical simulation (DNS) study of turbulent thermal convection in mixed porous–pure fluid domains. The computational domain consists of a cavity that contains a porous medium placed right above the bottom wall. The solid matrix is internally heated which, in turn, induces the convective motions of the fluid. The Rayleigh number of the flow in the pure fluid region above the porous medium is of the order of $10^7$. In our study, we consider cases of different sizes of the porous medium, as well as cases with both uniform and non-uniform heat loading of the solid matrix. For each case, we analyse the convective structures in both the porous and the pure fluid domains and investigate the effect of the porous medium on the emerging flow patterns above it. Results for the flow statistics, as well as for the Nusselt number and each of its components, are also presented herein. Further, we make comparisons of the flow properties in this pure fluid region with those in Rayleigh–Bénard convection. Our simulations predict that, depending on the area coverage, the large-scale circulation above the porous medium can be in a single-roll, dual-roll or intermediate state. Also, when the area coverage increases, the temperatures inside it increase due to reduced fluid circulation. Accordingly, when the area coverage increases, then the Nusselt number becomes smaller whereas the Rayleigh number is increased.

We investigate the equilibrium of an axisymmetric system consisting of sessile and pendent drops on pre-stretched nonlinear elastic membranes. The membrane experiences large deformations due to the drop's weight and interfacial interactions. We first show that force balance alone leads to non-unique equilibrium solutions. Identifying the system's equilibrium with the minimum of its free energy, we then demonstrate that the equilibrium solution is made unique by requiring the continuity of meridional stretches across the three-phase contact circle. For a special class of nonlinear elastic materials – $I_2$ materials – we then compute the equilibrium configurations of the drop–membrane system for a range of drop volumes and membrane pre-tensions. Finally, the present work facilitates two important applications: (a) the membrane's pre-tension and current tension are related exactly to help in utilizing the system as an elastocapillary probe for membrane pre-tension and (b) we suggest an experimental protocol for measuring the membrane's surface properties.

The actuator line method (ALM) is used extensively in wind turbine and rotor simulations. However, its original uncorrected formulation overestimates the forces near the tip of the blades and does not reproduce well forces on translating wings. The recently proposed vortex-based smearing correction for the ALM is a correction based on physical and mathematical properties of the simulation that allows for a more accurate and general ALM. So far, to correct the forces on the blades, the smearing correction depended on an iterative process at every time step, which is usually slower, less stable and less deterministic than direct methods. In this work, a non-iterative process is proposed and validated. First, we propose a formulation of the nonlinear lifting line that is equivalent to the ALM with smearing correction, showing that the results are practically identical for a translating wing. Then, by linearizing the lifting line method, the iterative process of the correction is substituted by the direct solution of a small linear system. No significant difference is observed in the results of the iterative and non-iterative corrections, in both wing and rotor simulations. Additional contributions of the present work include the use of a more accurate approximation for the velocity induced by a smeared vortex segment and the implementation of a free-vortex wake model to define the vortex sheet, which contribute to the accuracy and generality of the method. The results presented here may motivate the adoption of the ALM by other communities, for example, in fixed-wing applications.

The formation of step-like ‘density staircase’ distributions induced by stratification and turbulence has been widely studied, and can be explained by the ‘instability’ of a sufficiently strongly stably stratified turbulent flow due to the decrease of the turbulent density flux with increasing stratification via the ‘Phillips mechanism’ (Phillips, Deep Sea Research and Oceanographic Abstracts, vol. 19, 1972, pp. 79–81. Elsevier). However, such density staircases are not often observed in ocean interiors, except in regions where double-diffusion processes are important, leading to thermohaline staircases. Using reduced-order models for the evolution of velocity and density gradients, we analyse staircase formation in stratified and sheared turbulent flows. Under the assumption of inertial scaling $\epsilon \sim U^3/L$ for the kinetic energy dissipation rate $\epsilon$, where $U$ and $L$ are characteristic velocity and length scales, we determine ranges of bulk Richardson numbers ${Ri}_{b}$ and turbulent Prandtl numbers ${Pr}_{T}$ for which staircases can potentially form and show that the Phillips mechanism only survives in the limit of sufficiently small turbulent Prandtl numbers. For relevant oceanic parameters, a range of turbulent Prandtl numbers above which the system is not prone to staircases is found to be ${Pr}_T \simeq 0.5\unicode{x2013}0.8$. Since several studies indicate that the turbulent Prandtl number in stably stratified turbulence and in ocean interiors is usually above this threshold, this result supports the empirical observation that staircases are not favoured in ocean interiors in the presence of relatively homogeneous and sustained turbulence. We also show that our analysis is robust to other scalings for $\epsilon$ (such as the more strongly stratified scaling $\epsilon \sim U^{2}N_{c}$, where $N_{c}$ is a characteristic value of the buoyancy frequency), supporting our results in both shear-dominated and buoyancy-dominated turbulent regimes as well as in weakly and strongly stratified regimes.

We study the mixing processes inside a forced fountain using data from direct numerical simulation. The outer boundary of the fountain with the ambient is a turbulent/non-turbulent interface. Inside the fountain, two internal boundaries, both turbulent/turbulent interfaces, are identified: (i) the classical boundary between upflow and downflow which is composed of the loci of points of zero mean vertical velocity; and (ii) the streamline that separates the mean flow emitted by the source from the entrained fluid from the ambient (the separatrix). We show that entrainment due to turbulent fluxes across the internal boundary is at least as important as that by the mean flow. However, entrainment by the turbulence behaves substantively differently from that by the mean flow and cannot be modelled using the same assumptions. This presents a challenge for existing models of turbulent fountains and other environmental flows that evolve inside turbulent environments.

In the present study an optimisation problem is formulated to determine the forcing of an eddy-viscosity-based linearised Navier–Stokes model in channel flow at $Re_\tau \approx 5200$ ($Re_\tau$ is the friction Reynolds number), where the forcing is white-in-time and spatially decorrelated. The objective functional is prescribed such that the forcing drives a response to best match a set of velocity spectra from direct numerical simulation (DNS), as well as remaining sufficiently smooth. Strong quantitative agreement is obtained between the velocity spectra from the linear model with optimal forcing and from DNS, but only qualitative agreement between the Reynolds shear stress co-spectra from the model and DNS. The forcing spectra exhibit a level of self-similarity, associated with the primary peak in the velocity spectra, but they also reveal a non-negligible amount of energy spent in phenomenologically mimicking the non-self-similar part of the velocity spectra associated with energy cascade. By exploiting linearity, the effect of the individual forcing components is assessed and the contributions from the Orr mechanism and the lift-up effect are also identified. Finally, the effect of the strength of the eddy viscosity on the optimisation performance is investigated. The inclusion of the eddy viscosity diffusion operator is shown to be essential in modelling of the near-wall features, while still allowing the forcing of the self-similar primary peak. In particular, reducing the strength of the eddy viscosity results in a considerable increase in the near-wall forcing of wall-parallel components.

An analytical Green's function is developed to study the acoustic scattering by a flat plate with a serrated edge. The scattered pressure is solved using the Wiener–Hopf technique in conjunction with the adjoint technique. It is shown that the kernel decomposition proposed in recent literature appears only valid at high frequencies. We focus on this high-frequency regime and obtain the scattered pressure in the form of a contour integral. We show that such an integral, although complicated, can be evaluated exactly for any arbitrary piecewise linear serrations, for which closed-form analytical Green's functions are obtained. The derivation is validated by performing numerical integration of the contour integral showing excellent agreement. The Green's function is shown to agree well with the numerical results obtained using the finite element method at high frequencies. The noise directivity patterns are studied as a function of the frequency, serration amplitude, source position and Mach number. It is found that noise is often enhanced at low observer angles and may be slightly reduced at high observer angles, which may be understood from the perspective of an extended or removed rigid reflection surface. It is found that increasing the mean-flow Mach number leads to increasingly evident noise amplification at side angles, a seemingly strange Doppler behaviour exhibited in source-fixed coordinate frames. The analytical Green's function is applicable to both leading- and trailing-edge scatterings, and is particularly suitable for developing a three-dimensional trailing-edge noise model that is not only highly efficient but also capable of including non-frozen turbulence effects.

Experimental measurements in a wind tunnel of the unsteady force and moment that a fluid exerts on flexible flapping aerofoils are not trivial because the forces and moments caused by the aerofoil's inertia and others structural tensions at the pivot axis have to be obtained separately and subtracted from the direct measurements with a force/torque sensor. Here we derive from the nonlinear beam equation general relations for the force and torque reactions at the leading edge of a pitching aerofoil in terms of the fluid force and moment on the aerofoil and its kinematics, involving geometric and structural parameters of the flexible aerofoil. These relations are validated by comparing high-resolution numerical simulations of the flow–structure interaction of a two-dimensional flexible aerofoil pitching about its leading edge with direct force and torque measurements in a wind tunnel.

Driving oscillatory flow around an obstacle generates, due to inertial rectification, a steady ‘streaming’ flow that is useful in a host of microfluidic applications. While theory has focused largely on two-dimensional flows, streaming in many practical microfluidic devices is three-dimensional due to confinement. We develop a three-dimensional streaming theory around an obstacle in a microchannel with a Hele-Shaw-like geometry, where one dimension (depth) is much shorter than the other two dimensions. Utilizing inertial lubrication theory, we demonstrate that the time-averaged streaming flow has a three-dimensional structure. Notably, the flow reverses direction across the depth of the channel, which is a feature not observed in less confined streaming set-ups. This feature is confirmed by our experiments of streaming around a cylinder sandwiched in a microchannel. Our theory also predicts that the streaming velocity decays as the inverse cube of the distance from the cylinder, faster than that expected from previous two-dimensional approaches. We verify this velocity decay quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.