Gravity currents are a ubiquitous density-driven flow occurring in both the natural environment and in industry. They include: seafloor turbidity currents, primary vectors of sediment, nutrient and pollutant transport; cold fronts; and hazardous gas spills. However, while the energetics are critical for their evolution and particle suspension, they are included in system-scale models only crudely, so we cannot yet predict and explain the dynamics and run-out of such real-world flows. Herein, a novel depth-averaged framework is developed to capture the evolution of volume, concentration, momentum and turbulent kinetic energy from direct integrals of the full governing equations. For the first time, we show the connection between the vertical profiles, the evolution of the depth-averaged flow and the energetics. The viscous dissipation of mean-flow energy near the bed makes a leading-order contribution, and an energetic approach to entrainment captures detrainment of fluid through particle settling. These observations allow a reconsideration of particle suspension, advancing over 50 years of research. We find that the new formulation can describe the full evolution of a shallow dilute current, with the accuracy depending primarily on closures for the profiles and source terms. Critically, this enables accurate and computationally efficient hazard risk analysis and earth surface modelling.

Turbidity currents, which are stratified, sediment-laden bottom flows in the ocean or lakes, can run out for hundreds or thousands of kilometres in submarine channels without losing their stratified structure. Here, we derive a layer-averaged, two-layer model for turbidity currents, specifically designed to capture long-runout. A number of previous models have captured runout of only tens of kilometres, beyond which thickening of the flows becomes excessive, and the models without a lateral overspill mechanism fail. In our framework, a lower layer containing nearly all the sediment is a faster, gravity-driven flow that propels an upper layer, where sediment concentration is nearly zero. The thickness of the lower layer is controlled by competition between interfacial water entrainment due to turbulent mixing and water detrainment due to sediment settling at the interface. The detrainment mechanism, first identified in experiments, is the key feature that prevents excessive thickening of the lower layer and allows long-runout. Under normal flow conditions, we obtain an exact solution to the two-layer formulation, revealing a constant velocity and a constant thickening rate in each of the two layers. Numerical simulations applied

to gradually varied flows on both constant and exponentially declining bed slopes, with boundary conditions mimicking field observations, show that the predicted lower layer thickness after 200 km flow propagation compares with observed submarine channel depths, whereas previous models overestimate this thickness three- to fourfold. This formulation opens new avenues for modelling the fluid mechanics and morphodynamics of long-runout turbidity currents in the submarine setting.

Experiments are presented to explore the non-axisymmetric instabilities of spreading films of aqueous suspensions of Carbopol and Xanthan gum floating on a bath of perfluoropolyether oil. The experimental observations are compared against theoretical predictions exploiting a shallow-film model in which the viscoplastic rheology is captured by the Herschel–Bulkley constitutive law. With this model, we construct axisymmetric base states that evolve from the moment that the film floats onto the bath, out towards long times at which spreading becomes self-similar, and then test their linear stability towards non-axisymmetric perturbations. In the geometry of a thinning expanding film, we find that shear thinning does not drive a loss of axisymmetry at early times (when the degree of expansion is small), but when the film has expanded in radius by a factor of two or so, shear-thinning hoop stresses drive non-axisymmetric instabilities. Unstable modes possess relatively low angular wavenumber, and the loss of symmetry is not particularly dramatic. When the oil in the bath is replaced by salty water, the experiments are completely different, with dramatic non-axisymmetric patterns emerging from interfacial effects.

We consider the flow of a volume $\mathcal {V} = q t^\alpha$ of viscous fluid injected into a gap $H$ between two horizontal plates ($q$ and $\alpha$ are positive constants, $t$ is time). When the viscosity of the displaced fluid is negligible, the injected fluid forms a slug in contact with both plates connected (at a moving grounding line) to a gravity current (GC) with a downward-inclined interface. Hutchinson et al. (J. Fluid Mech., 598, 2023, pp. A4–1–13) considered a constant source ($\alpha = 1$) of Newtonian fluid at the center of an axisymmetric gap; the flow, governed by the parameter $J$ (the height ratio of the unconfined GC to $H$), admits a similarity solution. Here, the self-similar flow theory is (a) extended to rectangular geometry and power-law fluids, and (b) simplified. Similarity appears when $\alpha = n/(n+1)$ (two-dimensional) and $\alpha = 2n/(n+1)$ (axisymmetric), with propagation $\sim t^\beta$, where $\beta /\alpha = 1$ and $1/2$, respectively, and $n-1$ is the power of the shear in the viscosity law ($n=1$ for Newtonian fluid). The flow is governed by a single parameter $J$, representing the above-mentioned ratio. For small $J$, the GC is mostly unconfined; for large $J$, almost all the injected fluid is in contact with both boundaries of the gap. For given geometry and $n$, we solve one ordinary differential equation (ODE) for the reduced thickness over the reduced length $0\lt y \leqslant 1$, with a singular-regular condition at $y=1$. The details of the confined GC, functions of $J$, follow by simple formulae.

Numerical simulations are conducted to investigate particle suspension and deposition within turbidity currents. Utilizing Lagrangian particle tracking and a discrete element model, our numerical approach enables a detailed examination of autosuspension, deposition and bulk behaviours of turbidity current. We specifically focus on flow regimes where particle settling and buoyancy-induced hydrodynamics play equally important roles. Our discussion is divided into three parts. Firstly, we examine the main body of the current formed by suspended particles, revealing a temporal evolution consisting of initial slumping, propagation and dissipation stages. Our particle calculation allows for the tracking of autosuspended particles, enabling a deeper understanding of the connection between autosuspension and current propagation through energy budget analysis. In the second part, we delve into particle deposition, highlighting transverse and longitudinal variations. Transverse variations arise from lobe-and-cleft (LC) flow features, while longitudinal variations result from vortex detachment, particularly notable with large-sized particles. We observe that as particle size increases, leading to a particle Stokes number greater than 0.1, rapid particle settling suppresses the LC flow structure, resulting in wider lobes at the deposition height. Lastly, we propose a new scaling law for the propagation speed and current length. Our simulation results demonstrate close agreement with this new scaling law, providing valuable insights into turbidity current dynamics.

Granular column collapse is a simple but important problem to the granular material community, due to its links to dynamics of natural hazards, such as landslides and pyroclastic flows, and many industrial situations, as well as its potential of analysing transient and non-local rheology of granular flows. This article proposes a new dimensionless number to describe the run-out behaviour of granular columns on inclined planes based on both previous experimental data and dimensional analysis. With the assistance of the sphero-polyhedral discrete element method (DEM), we simulate inclined granular column collapses with different initial aspect ratios, particle contact properties and initial solid fractions on inclined planes with different inclination angles ($2.5^{\circ }\unicode{x2013}20.0^{\circ }$) to verify the proposed dimensional analysis. Detailed analyses are further provided for better understanding of the influence of different initial conditions and boundary conditions, and to help unify the description of the run-out scaling of systems with different inclination angles. This work determines the similarity and unity between granular column collapses on inclined planes and those on horizontal planes, and helps investigate the transient rheological behaviour of granular flows, which has direct relevance to various natural and engineering systems.

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.