We investigate the effect of external oscillatory forcing on evolving two-dimensional (2-D) gravity currents, resulting from the well-known lock-exchange set-up, by superimposing a horizontally uniform oscillating pressure gradient. This pressure gradient generates a 2-D horizontally uniform laminar oscillating flow over the flat no-slip bottom that interacts with the evolving gravity current. We explore the effect of the velocity amplitude of the applied oscillating flow and its period of oscillations on the behaviour of the evolving gravity currents. A key element introduced by the external forcing is the Stokes boundary layer near the no-slip bottom wall generating differential advection near the bottom wall when the propagation direction of the gravity current and the oscillating externally imposed flow are in the same direction. This results in a phenomenon that we refer to as lifting of the gravity current, which clearly distinguishes the oscillatory forced gravity current from the freely evolving case. This phenomenon induces fine-scale density structures when the externally imposed flow is opposite to the propagation direction of the gravity current a semi-period later. We have explored the effect of lifting on the current propagation and the density structure of the gravity current front. Three separate regimes are distinguished for the evolution of the density structure in the front of the gravity current depending on the period of forcing, including a regime reminiscent of tidally forced estuarine flows. The present study shows the existence of significant effects of an oscillatory forcing on the dynamics, advection and density distribution of gravity currents.

We develop a simple model which describes the repeated injection and extraction of hydrogen in a permeable water-saturated rock which has the form of an anticline. We demonstrate that the flow is controlled by the dimensionless ratio of the square of the buoyancy speed to the product of the two-dimensional volume injection rate and the injection–extraction frequency, and we explore the cases in which this ratio is large and small. Over the first few cycles, the volume of hydrogen in the system gradually builds up since during the extraction phase, some of the water eventually reaches the extraction well, and in our model the system ceases to extract fluid for the remainder of this extraction phase. After many cycles, there is sufficient hydrogen in the system that a quasi-equilibrium state develops in which the mass of fluid injected matches the mass extracted over the course of a cycle. We show that in this equilibrium, the ratio between the mass of gas remaining in the aquifer at the end of the extraction phase, known as the cushion gas, to the mass of gas injected, known as the working gas, decreases if either the flow rate or frequency of the cycles decrease or the buoyancy speed increases, leading to more efficient storage.

We examine the gravity-driven flow of a thin film of viscous fluid spreading over a rigid plate that is lubricated by another viscous fluid. We model the flow over such a ‘soft’ substrate by applying the principles of lubrication theory, assuming that vertical shear provides the dominant resistance to the flow. We do so in axisymmetric and two-dimensional geometries in settings in which the flow is self-similar. Different flow regimes arise, depending on the values of four key dimensionless parameters. As the viscosity ratio varies, the behaviour of the intruding layer ranges from that of a thin coating film, which exerts negligible traction on the underlying layer, to a very viscous gravity current spreading over a low-viscosity, near-rigid layer. As the density difference between the two layers approaches zero, the nose of the intruding layer steepens, approaching a shock front in the equal-density limit. We characterise a frontal stress singularity, which forms near the nose of the intruding layer, by performing an asymptotic analysis in a small neighbourhood of the front. We find from our asymptotic analysis that unlike single-layer viscous gravity currents, which exhibit a cube-root frontal singularity, the nose of a viscous gravity current propagating over another viscous fluid instead exhibits a square-root singularity, to leading order. We also find that large differences in the densities between the two fluids give rise to flows similar to that of thin films of a single viscous fluid spreading over a rigid, yet mobile, substrate.

Viscous gravity currents play a fundamental role in many natural and industrial applications, where practical scenarios often involve the current propagating over rigid curvilinear surfaces. In this study, we employ lubrication theory to develop low-dimensional models for such two-dimensional and axisymmetric propagation, resulting from the release of a finite volume of viscous fluid. A key dimensionless parameter is identified, representing the volume ratio between the released fluid and the curvilinear surface, which governs the current evolution. By simplifying the curvilinear surface with linear–exponential and sinusoidal shapes, we observe distinct flow behaviours. Over linear–exponential surfaces, the current may become trapped, bypass the peak or flow downward, while over sinusoidal surfaces, the propagation is hindered compared with the behaviour over horizontal straight surfaces. The low-dimensional models are validated using the volume of fluid method in computational fluid dynamics, showing consistent predictions of the current evolution over rigid curvilinear surfaces.

We consider the initial ‘slumping phase’ of a lock-release gravity current (GC) on a down slope with focus on particle-driven (turbidity) flows, in the inertia–buoyancy (large Reynolds number) and Boussinesq regime. We use a two-layer shallow-water (SW) model for the depth-averaged variables, and compare the predictions with previously published experimental data. In particular, we analyse the empirical conclusion of Gadal et al. (J. Fluid Mech., vol. 974, 2023, A4) that the slumping displays a constant speed for a significant range of slopes and particle-sedimentation speeds. We emphasize the physical definition of the slumping phase (stage): the adjustment process during which (a) the fluid in the lock is set into motion by the dam break, then (b) forms a tail from the backwall to the nose. We focus on the question of if and when the propagation speed $u_N$ of the nose (front) of the GC is constant during this process (there is consensus that a significant deceleration of $u_N$ appears in the post-slumping stage.) The SW theory predicts correctly the adjustment of the flow field during the slumping stage, but indicates that a constant $u_N$ appears only for the classical case ($\gamma =E=c_D=\beta =0$) where $\gamma, E, c_D, \beta$ are the slope, entrainment and drag coefficients, and the scaled particle settling speed for a particle-driven GC. However, since $\gamma, E, c_D, \beta$ are typically small, the change of $u_N$ during the slumping phase is also small in many cases of interest. The interaction between the various driving and hindering mechanisms is elucidated. We show that, in a system with a horizontal (open) top (typical laboratory experiments), the height of the ambient increases along the slope, and this compensates for buoyancy loss due to particle sedimentation. We point out the need for further experimental and simulation studies for a better understanding of the slumping phase and transition to the next phases, and further assessment/improvement of the SW predictions.

We theoretically and experimentally study gravity currents of a Newtonian fluid advancing in a two-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid and the lighter, immobile ambient fluid. The current is taken to be in the Darcy–Forchheimer regime, where a term quadratic in the seepage velocity accounts for inertial contributions to the resistance. The volume of fluid of the current varies as a function of time as $\sim T^{\gamma }$, where the exponent parameterizes the case of constant volume subject to dam break ($\gamma =0$), of constant ($\gamma =1$), waning ($\gamma <1$) and waxing inflow rate ($\gamma >1$). The nonlinear governing equations, developed within the lubrication theory, admit self-similar solutions for some combinations of the parameters involved and for two limiting conditions of low and high local Forchheimer number, a dimensionless quantity involving the local slope of the current profile. Another parameter $N$ expresses the relative importance of the nonlinear term in Darcy–Forchheimer's law; values of $N$ in practical applications may vary in a large interval around unity, e.g. $N\in [10^{-5},10^{2}]$; in our experiments, $N\in [2.8,64]$. Sixteen experiments with three different grain sizes of the porous medium and different inflow rates corroborate the theory: the experimental nose speed and current profiles are in good agreement with the theory. Moreover, the asymptotic behaviour of the self-similar solutions is in excellent agreement with the numerical results of the direct integration of the full problem, confirming the validity of a relatively simple one-dimensional model.

The effect of microbial activity on buoyancy-driven flow within a porous layer is analysed. The input fluid provides an energy source for the growth of biofilms on the porous rock. At each location within the porous layer, the porosity and permeability begin to decrease once the input fluid has invaded. This leads to an evolving rock heterogeneity that depends on the passing time of the input fluid. Hence, the evolution of the flow is partly controlled by its own history. We present an axisymmetric gravity current model, accounting for this effect. In general, a reduction in permeability leads to the flow having a lesser extent in the radial direction and greater thickness (extent in the cross-flow direction), whilst a reduction in porosity has negligible effect on the thickness but leads to a much greater radial extent. The flow is fastest near the free surface where the permeability is greatest. In the case where the porosity and permeability reduce as power-law functions of fluid residence time, the evolution of the flow and the rock properties are self-similar. Consumption of the input fluid by the microbes is also incorporated in the model and it generally leads to flows with lesser radial extent but little change in the thickness. The three impacts of microbial growth (volume loss owing to consumption and the reduction in permeability and porosity) each influence the flow in substantially different ways and the interplay is analysed. A motivation of the study, the underground storage of hydrogen, is briefly discussed.

The entrainment hypothesis states that the mean inflow velocity across the boundary of a turbulent flow is proportional to a characteristic velocity of the flow. Proposed by G. I. Taylor approximately 80 years ago, it is still a common model of turbulence closure widely used in environmental engineering and geophysical fluid mechanics. Although it is a very simple concept and mathematical model, it has proven to be able to predict the entrainment in a variety of geophysical flows, e.g. convective clouds and plumes from erupting volcanoes in the atmosphere; dense water overflows and turbidity currents in the ocean; magma injection in a magma chamber in the interior of the Earth, to name just a few. In a seminal paper, Turner (J. Fluid Mech., vol. 173, 1986, pp. 431–471) presents a variety of laboratory and geophysical flows to illustrate the success of the entrainment hypothesis and discusses why such a simple hypothesis works so well even when the original assumptions are no longer valid.

We present a thin-film viscoplastic fluid model of a mountain range, where a uniform fluid layer is deformed by a vertical backstop moving at constant speed. This represents a simplification of the geometry in subduction zones, where an overlying tectonic plate scrapes sediments off the underthrusting plate, thereby forming an accretionary wedge. By using a viscoplastic rheology, we aim to generalise Newtonian models that capture the effective viscous behaviour of rock at large length scales. The model system is characterised by the dimensionless Bingham number, which is the ratio of the yield stress to characteristic shear stress. At low and high Bingham numbers and at early and late times the system is found to be asymptotically self-similar, which we confirm by solving the governing equations numerically. In addition, we test the high-Bingham-number results experimentally using ultrasound gel as the working fluid. The experiments reproduce many features of the theoretically predicted behaviour. The size of the observed wedge grows in the manner predicted, and the fluid surface profile is found to collapse to a universal shape. The viscoplastic fluid wedge exhibits features of both viscous continuum and Coulomb models of accretionary wedges, suggesting that a viscoplastic rheology may provide quantitative insights into the dynamics of real accretionary wedges found in convergent tectonic settings.

High-resolution simulations of gravity currents in the lock-exchange configuration are conducted to study the flow within the head. The simulations exhibit the geometric features of the head as reported in the laboratory experiments and numerical simulations, and provide more detailed information on the flow within the head of a gravity current. The flow in the lower part of the head, where the lobes and clefts are forming at the leading edge, is qualitatively different from but interconnected to the flow in the upper part of the head, where steepening bulges are protruding from the upright surface above the clefts. Interestingly, regions of positive and negative streamwise vorticity are observed not only in the lower part of the head but also in the upper part of the head at staggered spanwise locations. We have shown that both the streamwise vorticity at the leading edge of the lobes in the lower part of the head and the streamwise vorticity at the steepening bulges in the upper part of the head are contributed from the twisting of spanwise vorticity into the streamwise direction, due to the geometric features of the lobes and the steepening bulges, and contributed from the baroclinic production of vorticity. Our results from visualization using tracers indicate that the ambient fluid ingested in and rising from the clefts is being swept towards the leading edge of a gravity current before being carried upwards from the leading edge to the upright surface above the left and right neighbouring lobes. Furthermore, the heavy fluid inside a lobe may descend towards the bottom boundary, move forward towards the leading edge and outwards towards the neighbouring clefts, and ultimately be carried upwards to the upright surface above the left and right neighbouring lobes. With the knowledge that the erosive power of a gravity current is concentrated in the head region, it is plausible that the bed material, once resuspended by a gravity current, may be lifted up away from the bottom boundary and be dispersed in both the streamwise and spanwise directions. The present study complements existing findings in the literature and provides new insights into the three-dimensional flow field within the head of a gravity current.

We consider the long-time propagation of a Boussinesq inertia–buoyancy (large-Reynolds- number) gravity current released from a lock over a downslope of angle $\gamma$, affected by entrainment and drag. We show that the shallow-water (depth-averaged) equations with a Benjamin-type front-jump condition admit a similarity solution $x_N(t) = K t^{2/3}$ while $h, \phi, u$ change like $t$ to the power of $2/3, -4/3, -1/3$, respectively; here $x_N, h, \phi, u$ and $t$ are the position of the nose (distance from backwall), thickness, concentration of dense fluid, velocity and time, respectively, and K is a constant. Assuming that $\gamma$ and the coefficients of entrainment and drag are constant, we derive an analytical exact solution for the similarity profiles and show that $K \propto (\tan \gamma )^{1/3}$; the driving of the slope is balanced by entrainment and/or drag. The predicted $t^{2/3}$ propagation is in agreement with previously published experimental data but a conclusive quantitative assessment of the present theory cannot be performed due to various uncertainties (discussed in the paper) that must be resolved by future work.

The two-dimensional gravity-driven motion of a relatively dense viscous liquid at the base of a granular mush is investigated using a model that exploits the relative shallowness of the flow. The granular mush obeys a $\mu (I)$-rheology, and we assume that the two phases are segregated throughout the motion. The viscous liquid spreads under gravity, carrying the granular mush above and transporting it outwards as levees at either end of the flow. The accumulation of granular material away from the centre of the deposit produces hydrostatic pressure gradients that retard the viscous gravity current. At later times, the granular mush is quasi-static relative to the moving liquid owing to the balance of outward granular transfer by the liquid and inward hydrostatic pressure gradients associated with the granular free surface. The viscous liquid exhibits a Poiseuille-like flow structure with negligible velocity at both the base and the granular interface. The flow of a fixed volume of viscous liquid becomes self-similar with the effective viscosity quadrupled relative to a classical viscous gravity current owing to the retarding effects of the granular mush. The case of constant input flux of viscous liquid is also analysed. The qualitative features are akin to the fixed volume case with the granular mush forming levees and slowing the viscous spreading. The case in which the upper medium is a Bingham material rather than a granular mush is also discussed, and the same features are observed, demonstrating the importance of the yield criterion in the upper medium.

Motivated by buoyancy-driven flows within geological formations, we study the evolution of a (dense) gravity current in a porous medium bisected by a thin interbed layer. The gravity current experiences distributed drainage along this low-permeability boundary. Our theoretical description of this flow takes into account dispersive mass exchange with the surrounding ambient fluid by considering the evolution of the bulk and dispersed phases of the gravity current. In turn, we model basal draining by considering two bookend limits, i.e. no mixing versus perfect mixing in the lower layer. Our formulations are assessed by comparing model predictions against the output of complementary numerical simulations run using COMSOL. Numerical output is essential both for determining the value of the entrainment coefficient used within our theory and for assessing the reasonableness of key modelling assumptions. Our results suggest that the degree of dispersion depends on the dip angle and the depth and permeability of the interbed layer. We further find that the nose position predictions made by our theoretical models are reasonably accurate up to the point where the no mixing model predicts a retraction of the gravity current front. Thereafter, the no mixing model significantly under-predicts, and the perfect mixing model moderately over-predicts, numerical data. Reasons for the failure of the no mixing model are provided, highlighting the importance of convective instabilities in the lower layer. A regime diagram is presented that defines the parametric region where our theoretical models do versus do not yield predictions in good agreement with numerical simulations.

Direct numerical simulations (DNSs) of three-dimensional cylindrical release gravity currents in a linearly stratified ambient are presented. The simulations cover a range of stratification strengths $0< S\leq 0.8$ (where $S=(\rho _b^*-\rho _0^*)/(\rho _c^*-\rho _0^*), \rho _b^*, \rho _0^*$ and $\rho _c^*$ are the dimensional density at the bottom of the domain, top of the domain and the dense fluid, respectively) at two different Reynolds numbers. A comparison between the stratified and unstratified cases illustrates the influence of stratification strength on the dynamics of cylindrical gravity currents. Specifically, the front velocity in the slumping phase decreases with increasing stratification strength whereas the duration of the slumping phase increases with increments of $S$. The Froude number calculated in this phase shows a good agreement with models proposed by Ungarish & Huppert (J. Fluid Mech., vol. 458, 2002, pp. 283–301) and Ungarish (J. Fluid Mech., vol. 548, 2006, pp. 49–68), originally developed for planar gravity currents in a stratified ambient. In the inertial phase, the front velocity across cases with different stratification strengths adheres to a power-law scaling with an exponent of $-$1/2. Higher Reynolds numbers led to more frequent lobe splitting and merging, with lobe size diminishing as stratification strength increased. Strong interactions among inner vortex rings occurred during the slumping phase, leading to the early formation of hairpin vortices in weakly stratified cases, while strongly stratified cases exhibited delayed vortex formation and less turbulence.

The column collapse experiment is a simplified version of natural and industrial granular flows. In this set-up, a column built with grains collapses and spreads over a horizontal plane. Granular flows are often studied with a monodisperse distribution; however, this is not the case in natural granular flows where a variety of grain sizes, known as polydispersity, is a common feature. In this work, we study the effect of polydispersity, and of the inherent changes that polydispersity causes in the initial packing fraction, in dry and immersed columns. We show that dry columns are not significantly affected by polydispersity, reaching similar distances at similar times. In contrast, immersed columns are strongly affected by the polydispersity and packing fraction, and the collapse sequence is linked to changes of the basal pore fluid pressure $P$. At the collapse initiation, negative changes of $P$ beneath the column produce a temporary increase of the column strength. The negative change of $P$ lasts longer in polydisperse columns than in monodisperse columns, delaying the collapse sequence. Conversely, during the column spreading, positive changes of $P$ lead to a decrease of the shear strength. For polydisperse collapses, the excess of $P$ lasts longer, allowing the material to reach farther distances, compared with the collapses of monodisperse materials. Finally, we show that a mobility model that scales the final runout with the collapse kinetic energy remains true for different polydispersity levels in a three-dimensional configuration, capturing the scaling between the micro to macro controlling features.

We develop a theoretical and experimental framework for generating slip underneath thin-film flows of viscous fluids in the laboratory, with the ability to control slip as desired. Such a framework is useful for large-scale fluid-mechanical experiments in which basal sliding is important. In particular, we consider the flow of a thin film of viscous fluid spreading over a structured, slippery substrate, involving a sequence of two-dimensional cavities that are prewetted with a fluid of smaller viscosity. By averaging over small-scale inhomogeneities, we demonstrate that such a substrate gives rise to a macroscopic linear sliding law, or Navier slip condition, that is effectively homogeneous on the large scale. The slip length, determining the slipperiness of the substrate, is proportional to the viscosity ratio and width of each cavity. As such, the slipperiness of the substrate can be controlled by altering the viscosity ratio, as desired. Two asymptotic regimes arise, describing flow over very slippery substrates and flow over no-slip substrates. The former regime is valid for early times, when the depth of the overlying fluid is much less than the slip length, and the latter is valid for late times, when the depth is much greater than the slip length. Solutions to the full model approach similarity solutions describing the two regimes for early and late times. We confirm our theoretical predictions by conducting a series of analogue laboratory experiments.

We present the results from a series of experiments investigating the dynamics of gravity currents which form when a dense saline or particle-laden plume issuing from a moving source interacts with a horizontal surface. We define the dimensionless parameter $P$ as the ratio of the source speed, $u_a$, to the buoyancy speed, $(B_0/z_0)^{1/3}$, where $B_0$ and $z_0$ are the source buoyancy flux and height above the horizontal surface, respectively. Using our experimental data, we determine that the limiting case in which $P=P_c$ the gravity current only spreads downstream of the initial impact point occurs when $P_c=0.83\pm 0.02$. For $P< P_c$, from our experiments we observe that the plume forms a gravity current that spreads out in all directions from the point of impact and the propagation of the gravity current is analogous to a classical constant-flux gravity current. For $P>P_c$, we observe that the descending plume is bent over and develops a pair of counter-rotating line vortices along the axis of the plume. The ensuing gravity current spreads out downstream of the source, normal to the motion of the source. Analogous processes occur with particle-laden plumes, but there is a second dimensionless parameter $S$, the ratio of the particle fall speed, $v_s$, to the vertical speed of a plume in a crossflow, $(B_0/u_a z_0)^{1/2}$. For $S\ll 1$, particles remain well mixed in the plume and a particle-driven gravity current develops. For $S\gg 1$, particles separate from the plume prior to impacting the boundary which leads to a fall deposit and no gravity current. We discuss these results in the context of deep-sea mining.

We study the dynamic interaction of two gravity currents in a confined porous layer, one heavier and one lighter, partly inspired by the practice of geological $\mathrm {CO}_2$ sequestration in oil fields. Two coupled nonlinear advective-diffusive equations are derived to describe the time evolution of the profile shape of both the upper (lighter) and lower (heavier) currents. At early times, the upper and lower currents remain separated and propagate independently. As time progresses, the currents approach each other and start to interact. We have identified eight different regimes of gravity current interaction at late times, impacted by four dimensionless parameters, representing the flow rate partition, ratio of buoyancy over the injection force, and the viscosity contrasts between the two injecting and displaced fluids. By defining appropriate similarity variables at either the early or late times, the governing partial differential equations (PDEs) reduce to different ordinary differential equations (ODEs), corresponding to the classic nonlinear diffusion solutions at early times and eight different self-similar solutions at late times when the currents attach to each other. It is of interest to note that in four of the eight regimes of late-time interaction (regimes 2, 6–8), self-similar solutions can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction and travelling wave solutions) identified during single fluid injection in confined porous layers. In the four other regimes (regimes 1, 3–5), implicit solutions in the form of logarithm or error functions are obtained due to the influence of flow confinement and interaction of gravity currents. Potential implications of the model and solutions are also briefly discussed in the context of ${\rm CO}_2$-water co-flooding for simultaneous ${\rm CO}_2$ sequestration and oil recovery.

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has a higher viscosity than the ambient fluid, thus avoiding Saffman–Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al. (J. Fluid Mech., vol. 747, 2014, pp. 218–246) studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow 1$ and $n\rightarrow 3$, when fluid flows into and out of the channel, respectively. No divergence is found in the shallow case presented here up to the breakdown of the geometric assumption. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming an inverse dependence on the ratio disagreement between the current front using the method of lines. An inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase-plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta }$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ for $n\gg 1$.

We develop a theoretical model to study (dense) two-dimensional gravity current flow in a laterally extensive porous medium experiencing leakage through a discrete fissure situated along this boundary at some finite distance from the injection point. Our model, which derives from the depth-averaged mass and buoyancy equations in conjunction with Darcy's law, considers dispersive mixing between the gravity current and the surrounding ambient by allowing two different gravity current phases. Thus do we define a bulk phase consisting of fluid whose density is close to that of the source fluid and a dispersed phase consisting of fluid whose density is close to that of the ambient. We characterize the degree of dispersion by estimating, as a function of time, the buoyancy of the dispersed phase and the separation distance between the bulk nose and the dispersed nose. On this basis, it can be shown that the amount of dispersion depends on the flow conditions upstream of the fissure, the fissure permeability and the vertical and horizontal extents of the fissure. We also show that dispersion is larger when the gravity current propagates along an inclined barrier rather than along a horizontal barrier. Model predictions are fitted against numerical simulations. The simulations in question are performed using COMSOL and consider different inclination angles and fissure and upstream flow conditions. Our study is motivated by processes related to underground $\mathrm {H}_2$ storage e.g. an irrecoverable loss of $\mathrm {H}_2$ when it is injected into the cushion gas saturating an otherwise depleted natural gas reservoir.