A closed macroscopic model for quasi-steady, inertial, incompressible, two-phase generalised Newtonian flow in rigid and homogeneous porous media is formally derived. The model consists of macroscopic equations for mass and momentum balance as well as an expression for the macroscopic pressure difference between the two fluid phases. The model is obtained by upscaling the pore-scale equations, employing a methodology based on volume averaging, the adjoint method and Green's formulation, only assuming the existence of a representative elementary volume and the separation of scales between the microscale and the macroscale. The average mass equations coincide with those for Newtonian flow. The macroscopic momentum balance equation in each phase expresses the seepage velocity in terms of a dominant and a coupling Darcy-like term, a contribution from interfacial tension effects and another one from interfacial inertia. Finally, the expression of the macroscopic pressure difference is obtained in terms of the macroscopic pressure gradient and body force in each phase, and interfacial terms that account for capillary effects and inertia, if present when the interface is not stationary. All terms involved in the macroscale equations are predicted from the solution of adjoint closure problems in periodic representative domains. Numerical predictions from the upscaled models are compared with direct numerical simulations for two-dimensional configurations, considering flow of a Newtonian non-wetting fluid and a Carreau wetting fluid. Excellent agreement between the two approaches confirms the pertinence of the macroscopic models derived here.

The structure and impact of thermally induced secondary motions in stably stratified channel flows with two-dimensional surface temperature inhomogeneities is studied using direct numerical simulation (DNS). Starting from a configuration with only spanwise varying surface temperature, where the streamwise direction is homogeneous (Bon & Meyers, J. Fluid Mech., 2022, pp. 1–38), we study cases where the periodic temperature strip length $l_x/h$ (with $h$ the half-channel height) assumes finite values. The patch width ($l_y/h =\{{\rm \pi} /4, {\rm \pi}/8$}) and length are varied at fixed stability and two different Reynolds numbers. Results indicate that for the investigated patch widths, the streamwise development of the secondary flows depends on the patch aspect ratio ($a=l_x/l_y$), while they reach a fully developed state after approximately $25l_y$. The strength of the secondary motions, and their impact on momentum and heat transfer through the dispersive fluxes, is strongly reduced as the length of the temperature strips decreases, and becomes negligible when $a\lesssim 1$. We demonstrate that upward dispersive and turbulent heat transport in locally unstably stratified regions above the high-temperature patches lead to reduced overall downward heat transfer. Comparison to local Monin–Obukhov similarity theory (MOST) reveals that scaled velocity and temperature gradients in homogeneous stably stratified channel flow at $Re_\tau =550$ agree reasonably well with empirical correlations obtained from meteorological data. For thermally heterogeneous cases with strips of finite length, the similarity functions only collapse higher above the surface, where dispersive fluxes are negligible. Lastly, we show that mean profiles of all simulations collapse when using outer-layer scaling based on displacement thickness.

Surface differential heating on a stably stratified fluid body drives an overturning circulation confined to the upper fluid region – here coined stratified horizontal convection (SHC). In this manuscript, we investigate the dynamics of SHC via laboratory experiments, exploring local and global flow properties. By considering the available potential energy of the system, we derive a unique length scale of SHC and introduce the Péclet number $Pe$ that captures both the stabilising effect of stratification and the destabilising effect of the baroclinic adjustment. We found that $Pe$ characterises local and global flow properties, including the fluid transport of the overturning circulation, the available mechanical energy and the flow dimensionality. Our study provides insights into the fluid dynamics of stratified environments that experience horizontal convection, such as lakes, oceans and atmospheres.

Two-dimensional simulations are conducted to investigate the direct initiation of cylindrical detonation in hydrogen/air mixtures with detailed chemistry. The effects of hotspot condition and mixture composition gradient on detonation initiation are studied. Different hotspot pressures and compositions are first considered in the uniform mixture. It is found that detonation initiation fails for low hotspot pressures and the critical regime dominates with high hotspot pressures. Detonation is directly initiated from the reactive hotspot, whilst it is ignited somewhere beyond the non-reactive hotspots. Two cell diverging patterns (i.e. abrupt and gradual) are identified and the detailed mechanisms are analysed. Moreover, cell coalescence occurs if many irregular cells are generated initially, which promotes the local cell growth. We also consider non-uniform detonable mixtures. The results show that the initiated detonation experiences self-sustaining propagation, highly unstable propagation and extinction in mixtures with a linearly decreasing equivalence ratio along the radial direction, i.e. 1 → 0.9, 1 → 0.5 and 1 → 0. Moreover, the hydrodynamic structure analysis shows that, for the self-sustaining detonations, the hydrodynamic thickness increases at the overdriven stage, decreases as the cells are generated and eventually becomes almost constant at the cell diverging stage, within which the sonic plane shows a ‘sawtooth’ pattern. However, in the detonation extinction cases, the hydrodynamic thickness continuously increases, and no ‘sawtooth’ sonic plane can be observed.

We propose an experimental study on the gravitational settling velocity of dense, sub-Kolmogorov inertial particles under different background turbulent flows. We report phase Doppler particle analyser measurements in a low-speed wind tunnel uniformly seeded with micrometre scale water droplets. Turbulence is generated with three different grids (two consisting of different active-grid protocols while the third is a regular static grid), allowing us to cover a very wide range of turbulence conditions in terms of Taylor-scale-based Reynolds numbers ($Re_\lambda \in [30\unicode{x2013}520]$), Rouse numbers ($Ro \in [0\unicode{x2013}5]$) and volume fractions ($\phi _v \in [0.5\times 10^{-5}\unicode{x2013}2.0\times 10^{-5}]$). We find, in agreement with previous works, that enhancement of the settling velocity occurs at low Rouse number, while hindering of the settling occurs at higher Rouse number for decreasing turbulence energy levels. The wide range of flow parameters explored allowed us to observe that enhancement decreases significantly with the Taylor–Reynolds number and is significantly affected by the volume fraction $\phi _v$. We also studied the effect of large-scale forcing on settling velocity modification. The possibility of changing the inflow conditions by using different grids allowed us to test cases with fixed $Re_\lambda$ and turbulent intensity but with different integral length scale. Finally, we assess the existence of secondary flows in the wind tunnel and their role on particle settling. This is achieved by characterising the settling velocity at two different positions, the centreline and close to the wall, with the same streamwise coordinate.

In this article, we study the behaviour of the Abels–Garcke–Grün Navier–Stokes–Cahn– Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit $\varepsilon \to +0$, the so-called mobility parameter $m$ in the diffuse-interface model must scale appropriately with the interface-thickness parameter $\varepsilon$. In the literature various scaling relations in the range $o(1)$ to $O(\varepsilon ^3)$ have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the $m$–$\varepsilon$ scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier–Stokes–Cahn–Hilliard equations by means of an adaptive finite-element method. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime.

The response to harmonic horizontal oscillations of a stably stratified fluid-filled two-dimensional square container is examined as the forcing amplitude is increased. For the studied forcing frequency, the response flow at very small forcing amplitudes is a synchronous periodic flow with piecewise-constant vorticity in regions delineated by the characteristics emanating from the corners of the container, regularized by viscosity. The second temporal harmonic of the forced response flow resonantly excites an intrinsic mode of the stratified container, whose magnitude grows as the square of the forcing amplitude. Above a critical forcing amplitude, a sequence of pairs of other container modes are excited via triadic resonances with the second-harmonic-driven mode. The flows are computed from the Navier–Stokes–Boussinesq equations and the ensuing dynamics is analysed using Fourier techniques, providing a comprehensive picture of the transition to internal wave turbulence.

Helicity, an invariant under ideal-fluid (Euler) evolution, has a topological interpretation in terms of writhe and twist for a closed vortex tube, but accurately quantifying twist is challenging in viscous flows. With a novel helicity decomposition, we present a framework to construct the differential twist that establishes the theoretical relation between the total twisting number and the local twist rate of each vortex surface. This framework can characterize coiling vortex lines and internal structures within a vortex – important in laminar–turbulence transition, and in vortex instability, reconnection and breakdown. As a typical example, we explore the dynamics of vortex rings with differential twist via direct numerical simulation (DNS) of the Navier–Stokes equations. Two twist waves with opposite chiralities propagate towards each other along the ring and then collide whence the local twist rate rapidly surges. Local vortex surfaces are squeezed into a disk-like dipole structure containing coiled vortex lines, leading to vortex bursting. We derive a Burgers-equation-like model to quantify this process, which predicts a bursting time that agrees well with DNS.

In multispecies electrolyte solutions, even in the absence of an external electric field, differences in ion diffusivities induce an electric potential and generate additional fluxes for each species. This electro-diffusion process is well-described by the advection Nernst–Planck equation. This study aims to analyse the long-time behaviour of the governing equation under electroneutrality and zero current conditions, and to investigate how the diffusion-induced electric potential and shear flow enhance the effective diffusion coefficients of each species in channel domains. The exact solutions of the effective equation with certain special parameters, as well as the asymptotic analyses for ions with large diffusivity discrepancies, are presented. Furthermore, there are several interesting properties of the effective equation. First, it is a generalization of the Taylor dispersion, with a nonlinear diffusion tensor replacing the scalar diffusion coefficient. Second, the effective equation exhibits a scaling relation, revealing that the system with a weak flow is equivalent to the system with a strong flow under scaled physical parameters. Third, in the case of injecting an electrolyte solution into a channel containing well-mixed buffer solutions or electrolyte solutions with the same ion species, if the concentration of the injected solution is lower than that of the pre-existing solution, then the effective equation simplifies to a multi-dimensional diffusion equation. However, when introducing the electrolyte solution into a channel filled with deionized water, the ion–electric interaction results in several phenomena not present in the advection–diffusion equation, including upstream migration of some species, spontaneous separation of ions, and non-monotonic dependence of the effective diffusivity on Péclet numbers. Finally, the dependence of effective diffusivity on concentration and ion diffusivity suggests a method to infer the concentration ratio of each component and ion diffusivity by measuring the effective diffusivity.

In this paper, we construct an accurate linear model describing the propagation of both acoustic and gravity waves in water. This original model is obtained by the linearization of the compressible Euler equations, written in Lagrangian coordinates. The system is studied in the isentropic case, with a free surface, an arbitrary bathymetry, and vertical variations of the background temperature and density. We show that our model is an extension of some models from the literature to the case of a non-barotropic fluid with a variable sound speed. Other models from the literature are recovered from our model through two asymptotic analyses, one for the incompressible regime and one for the acoustic regime. We also propose a method to write the model in Eulerian coordinates. Our model includes many physical properties, such as the existence of internal gravity waves or the variation of the sound speed with depth.

This paper investigates the effect of anisotropic turbulence on generating leading-edge aerofoil–turbulence interaction noise. Thin aerofoil theory is used to model an aerofoil as a semi-infinite plate, and the scattering of incoming turbulence is solved via the Wiener–Hopf technique. This theoretical solution encapsulates the diffraction problem for gust–aerofoil interaction and is integrated over a wavenumber–frequency spectrum to account for general incoming anisotropic turbulence. We develop a novel axisymmetric wavenumber–frequency model that captures the wall-normal variation in turbulence characteristics, differing from previous approaches. Then, the method of Gaussian decomposition, in which the generalised spectra are approximated through the weighted sum of individual Gaussian eddy models, is applied to fit the turbulence model to experimental data. Comparisons with experimental data show good agreement for a range of anisotropic ratios.

Wind tunnel experiments are performed in both neutral and stable boundary layers to study the effect of thermal stability on the wake of a single turbine and on the wakes of two axially aligned turbines, thereby also showing the influence of the second turbine on the impinging wake. In the undisturbed stable boundary layers, the turbulence length scales are significantly smaller in the vertical and longitudinal directions (up to 50 % and $\approx$40 %, respectively), compared with the neutral flow, while the lateral length scale is unaffected. The reductions are larger with the imposed inversion of a second stable case, except in the near-wall region. In the neutral case, the length scales in the wake flow of the single turbine are reduced both vertically and laterally (up to 50 % and nearly 40 %, respectively). While there is significant upstream influence of a second turbine (on mean and turbulence quantities), there is virtually no upstream effect on vertical length scales. However, curiously, the presence of the second turbine aids length-scale recovery in both directions. Longitudinally, each turbine contributes to successive reduction in coherence. The effect of stability on the turbulence length scales in the wake flows is non-trivial: at the top of the boundary layer, the reduction in the wall-normal length scale is dominated by the thermal effect, while closer to the wall, the wake processes strongly modulate this reduction. Laterally, the turbines’ rotation promotes asymmetry, while stability opposes this tendency. The longitudinal coherence, significantly reduced by the wake flows, is less affected by the boundary layer's thermal stability.

In this paper the three-dimensional finite-time Lyapunov exponent (FTLE) field of a direct numerical simulation of a flat-plate turbulent boundary layer is analysed in several wall-parallel sections. The data consider a case at a low subsonic Mach number with a moderate positive pressure gradient in the streamwise direction. In contrast to other studies mainly focusing on the maxima of the FTLE field, particular emphasis is placed on the regions of minimal stretching between the vortices and shear layers of the three-dimensional turbulent flow field. These visually appear as contiguous islands or ‘valleys’ between the ‘ridges’ of the FTLE maxima, both at forward and backward integration of the flow field in time. To clearly distinguish the structures investigated from their more common counterparts (e.g. Lagrangian coherent structures, LCS), the acronym LAMS (Lagrangian areas of minimal stretching) is proposed to denote the associated cohesive fluid regions. Consistent with intuition, the largest LAMS occur near the boundary-layer edge, where large regions of homogeneous laminar external flow coexist with upwelling turbulent structures. Compensating for turbulent regions pushing upward, they sink from there down toward the wall, becoming smaller and longer. This process is associated with an increased relative velocity of the LAMS compared with the mean flow, which is observed over the whole boundary layer in the range $y^+ \gtrsim 10$. Furthermore, it is observed that the Q4 (sweep) events contained in the LAMS clearly dominate over Q2 (ejection) events above $y^+ \approx 10$. Thereby, local maxima occur at $y^+ \approx 20$ and near the boundary-layer edge. Below $y^+ \approx 10$, the relationship reverses. Sweeping LAMS from above $y^+ \approx 10$ and ejecting LAMS from below meet in the layer where the maximal vortical activity occurs. The latter is caused by mostly streamwise oriented vortices with maximal vortex stretching in the streamwise direction. Overall, LAMS are associated with cohesive fluid regions between the surrounding vortices and shear layers that both drop down from the boundary-layer edge toward the wall in the outer region of the boundary layer and lift from the wall in the near-wall region.

A large number of turbulence models (stochastic and large-eddy simulation (LES) models) developed to describe the dynamics of particle-laden turbulent flows are based on the assumption of local isotropy and use the Kolmogorov constant that correlates the spectral distribution of turbulent kinetic energy with the turbulent dissipation rate. Compilation of a large number of experimental data for different flow configurations has revealed that the Kolmogorov constant is independent of Reynolds number in the limit of high Reynolds number (Sreenivasan, Phys. Fluids, vol. 7, no. 11, 1995, pp. 2778–2784). However, several numerical studies, majorly in the area of multiphase flows at low and intermediate Reynolds numbers, consider that the Kolmogorov constant remains unchanged irrespective of whether the flow is single phase or multiphase. In this article, we assess the variation of local isotropy of the fluid fluctuations with the increase in particle loading in particle-laden turbulent channel flows. We also estimate the Kolmogorov constant using second-order velocity structure functions and compensated spectra in the case of low-Reynolds-number turbulent flows. Our study reveals that the Kolmogorov constant decreases in the channel centre with an increase in the particle volume fraction for the range of Reynolds numbers investigated here. The estimated variation of the Kolmogorov constant is used to express the Smagorinsky coefficient as a function of solid loading in particle-laden flows. Then, a new modelling technique is adopted using the large-eddy simulation (LES) to predict the fluid phase statistics without solving simultaneous particle phase equations. The new methodology also helps to qualitatively understand the phenomena of drastic collapse in turbulence intensity.

The temporally developing self-similar turbulent jet is fundamentally different from its spatially developing namesake because the former conserves volume flux and has zero cross-stream mean flow velocity whereas the latter conserves momentum flux and does not have zero cross-stream mean flow velocity. It follows that, irrespective of the turbulent dissipation's power-law scalings, the time-local Reynolds number remains constant, and the jet half-width $\delta$, the Kolmogorov length $\eta$ and the Taylor length $\lambda$ grow identically as the square root of time during the temporally developing self-similar planar jet's evolution. We predict theoretically and confirm numerically by direct numerical simulations that the mean centreline velocity, the Kolmogorov velocity and the mean propagation speed of the turbulent/non-turbulent interface (TNTI) of this planar jet decay identically as the inverse square root of time. The TNTI has an inner structure over a wide range of closely spatially packed iso-enstrophy surfaces with fractal dimensions that are well defined over a range of scales between $\lambda$ and $\delta$, and that decrease with decreasing iso-enstrophy towards values close to $2$ at the viscous superlayer. The smallest scale on these isosurfaces is approximately $\eta$, and the length scales between $\eta$ and $\lambda$ contribute significantly to the surface area of the iso-enstrophy surfaces without being characterised by a well-defined fractal dimension. A simple model is sketched for the mean propagation speeds of the iso-enstrophy surfaces within the TNTI of temporally developing self-similar turbulent planar jets. This model is based on a generalised Corrsin length, on the multiscale geometrical properties of the TNTI, and on a proportionality between the turbulent jet volume's growth rate and the growth rate of $\delta$. A prediction of this model is that the mean propagation speed at the outer edge of the viscous superlayer is proportional to the Kolmogorov velocity multiplied by the $1/4$th power of the global Reynolds number.

When a fast droplet impacts a pool of the same fluid, a thin ejecta sheet that dominates the early-time dynamics emerges within the first few microseconds. Fluid and impact properties are known to affect its evolution; we experimentally reveal that the pool depth is a critical factor too. Whilst ejecta sheets can remain separate and subsequently fold inwards on deeper pools, they instead develop into outward-propagating lamellae on sufficiently shallow pools, undergoing a transition that we delineate by comprehensively varying impact inertia and pool depth. Aided by matching direct numerical simulation results, we find that this transition stems from a confinement effect of the pool base on the impact-induced pressure, which stretches the ejecta sheet to restrict flow into it from the droplet on sufficiently shallow pools. This insight is also applied to elucidate the well-known transition due to Reynolds number.

We investigate experimentally the two-dimensional flow of a shear-thickening suspension around a rotating cylinder to which a constant torque is applied. While for low torques both the drag and the flow are steady and close to those for a Newtonian fluid, above the onset torque for discontinuous shear thickening the average velocity of the cylinder saturates and large periodic oscillations of the cylinder velocity are observed. The oscillations result from a hydrodynamic instability of the flow: slow-acceleration phases are followed by high-deceleration phases, triggered by the propagation of a thickening front, and so on. The slow-acceleration phases set the oscillation period, which is limited by the cylinder inertia and inversely proportional to the applied torque. Combined analyses of the cylinder motion and the flow reveal that the front typically nucleates when the shear rate at the cylinder surface reaches the discontinuous shear-thickening threshold. In addition, the characteristics (duration, stress) of the deceleration are set by the interplay between the thickening front propagation and the suspension and cylinder inertiae or the container size. Since for a slow acceleration the shear rate at the cylinder surface is essentially the cylinder angular velocity, this description of the unsteadiness elucidates the saturation of the average velocity. More generally, it illustrates how the hydrodynamics of a shear-thickening suspension with a strongly re-entrant rheology can lead to a marginally re-entrant, although steep, drag curve.

In this work, a near-wall model, which couples the inverse of a recently developed compressible velocity transformation (Griffin et al., Proc. Natl Acad. Sci., vol. 118, 2021, p. 34) and an algebraic temperature–velocity relation, is developed for high-speed turbulent boundary layers. As input, the model requires the mean flow state at one wall-normal height in the inner layer of the boundary layer and at the boundary-layer edge. As output, the model can predict mean temperature and velocity profiles across the entire inner layer, as well as the wall shear stress and heat flux. The model is tested in an a priori sense using a wide database of direct numerical simulation high-Mach-number turbulent channel flows, pipe flows and boundary layers (48 cases, with edge Mach numbers in the range 0.77–11, and semi-local friction Reynolds numbers in the range 170–5700). The present model is significantly more accurate than the classical ordinary differential equation (ODE) model for all cases tested. The model is deployed as a wall model for large-eddy simulations in channel flows with bulk Mach numbers in the range 0.7–4 and friction Reynolds numbers in the range 320–1800. When compared to the classical framework, in the a posteriori sense, the present method greatly improves the predicted heat flux, wall stress, and temperature and velocity profiles, especially in cases with strong heat transfer. In addition, the present model solves one ODE instead of two, and has a computational cost and implementation complexity similar to that of the commonly used ODE model.

The interactions of upper (lighter) and lower (heavier) gravity currents are closely related to fluid-phase resource recovery in porous layers and cleaning of confined spaces. The addition of a second current increases the sweep efficiency of fluid displacement. In this paper, we first derive two ordinary differential equations to describe the interaction of gravity currents in the quasi-steady regime. Two asymptotic regimes are identified, characterised by whether or not the two currents attach to each other, depending on whether the source fluxes are large enough. In the attached regime, a symmetry condition is also identified that describes whether or not the pumping and buoyancy forces balance each other. The model also leads to analytical solutions for the interface shape of the interacting currents in both the detached and attached regimes and for both symmetric and asymmetric currents. For symmetric currents, analytical solutions can also be obtained for the pressure distribution along cap rocks and the sweep efficiency of flooding processes. A particularly interesting aspect is that the displaced fluid remains quiescent at any steady state, regardless of whether the currents attach to each other. Correspondingly, the interface shape of the currents can be described by relatively simple equations and solutions, as if the currents propagate independently in unconfined porous layers. Time transition towards quasi-steady solutions is provided, employing time-dependent numerical solutions of two coupled partial differential equations for dynamic current interaction.

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.