Resolved simulations of submarine avalanches with a simple soft-sphere / immersed boundary method

Physical mechanisms at the origin of the transport of solid particles in a fluid are still a matter of debate in the physics community. Yet, it is well known that these processes play a fundamental role in many natural configurations, such submarines landslides and avalanches, which may have a significant environmental and economic impact. The goal here is to reproduce the local dynamics of such systems from the grain scale to that of thousands of grains approximately. To this end a simple soft-sphere collision / immersed-boundary method has been developed in order to accurately reproduce the dynamics of a dense granular media collapsing in a viscous fluid. The fluid solver is a finite-volume method solving the three-dimensional, time-dependent Navier-Stokes equations for a incompressible flow on a staggered. Here we use a simple immersed-boundary method consisting of a direct forcing without using any Lagrangian marking of the boundary, the immersed boundary being defined by the variation of a solid volume fraction from zero to one. The granular media is modeled with a discrete element method (DEM) based on a multi-contact soft-sphere approach. In this method, an overlap is allowed between spheres which mimics the elasto-plastic deformation of real grain, and is used to calculate the contact forces based on a linear spring model and a Coulomb criterion. Binary wall-particle collisions in a fluid are simulated for a wide range of Stokes number ranging from 10-¹ to 10⁴. It is shown that good agreement is observed with available experimental results for the whole range of investigated parameters, provided that a local lubrication model is used when the distance of the gap between the particles is below a fraction of the particle radius. A new model predicting the coefficient of restitution as a function of the Stokes number and the relative surface roughness of the particles is proposed. This model, which makes use of no adjustable constant, is shown to be in good agreement with available experimental data. Finally, simulations of dense granular flows in a viscous fluid are performed. The present results are encouraging and open the way for a parametric study in the parameter space initial aspect ratio - initial packing

A numerical investigation of high-Reynolds-number constant-volume non-Boussinesq density currents in deep ambient

The time-dependent behaviour of non-Boussinesq high-Reynolds-number density currents, released from a lock of height h0 and length x0 into a deep ambient and spreading over horizontal flat boundaries, is considered. We use two-dimensional Navier–Stokes simulations to cover: (i) a wide range of current-to-ambient density ratios, (ii) a range of length-to-height aspect ratios of the initial release within the lock (termed the lock aspect ratio λ=x0/h0) and (iii) the different phases of spreading, from the initial acceleration phase to the self-similar regimes. The Navier–Stokes results are compared with predictions of a one-layer shallow-water model. In particular, we derive novel insights on the influence of the lock aspect ratio (λ) on the shape and motion of the current. It is shown that for lock aspect ratios below a critical value (λcrit ), the dynamics of the current is significantly influenced by λ. We conjecture that λcrit depends on two characteristic time scales, namely the time it takes for the receding perturbation created at the lock upon release to reflect back to the front, and the time of formation of the current head. A comparison of the two with space–time diagrams obtained from the Navier–Stokes simulations supports this conjecture. The non-Boussinesq effect is observed to be significant. While the critical lock aspect ratio (λcrit ) is of order 1 for Boussinesq currents, its value decreases for heavy currents and increases significantly (up to about 20) for light currents. We present a simple analytical model which captures this trend, as well as the observation that for a light current the speed of propagation is proportional to λ1/4 when λ<λcrit

A numerical investigation of constant-volume non-Boussinesq density currents

The time-dependent behaviour of non-Boussinesq high-Reynolds-number density currents of density ρc, released from a lock of height h₀ and length x₀ into a ambient of height H and density ρₐ, is considered. We use two dimensional Navier-Stokes simulations to cover a wide range of density ratio ρc/ρₐ (for both "heavy"-bottom and "light"-top currents) and geometric ratios (H*=H/h₀, λ=x₀/h₀). To our knowledge, the ranges of parameters and times of propagation considered here were not covered in previous experimental or numerical studies. In the first part, we set the lock aspect ratio to λ=18.75, and vary the density ratio 10-⁴<ρc/ρₐ<10⁴ and initial depth ratio 1≤H*≤50. The Navier-Stokes results are compared with predictions of a shallow-water model, in the regime of constant-speed (slumping) phase. Good agreement is observed in a large region of the parameter space (ρc/ρₐ; H*). The larger discrepancy is observed in the range of high-H* and low-ρc/ρₐ for which the shallow-water model overpredicts the velocity of the current. Two possible reasons are suspected, namely the fluid motion in the ambient fluid which is not accounted for in the model, and the choice of the model for the front condition. In the second part, we set the initial depth ratio to H*=10, and vary the density ratio 10-²<ρc/ρₐ<10² and lock aspect ratio 0.5≤λ≤18.75. In particular, we derive novel insights on the influence of the lock aspect ratio λ=x₀/h₀ on the shape and motion of the current in the slumping stage. It is shown that a critical value exists, λcrit; the dynamics of the current is significantly influenced by λ if below λcrit. We present a simple analytical model which support the observation that for a light current the speed of propagation is proportional to λ¼ when λ<λcrit

Visualization of toner ink adsorption at bubble surfaces

Flotation deinking involves interactions between inks particles and bubbles surfaces. These interactions are very difficult to observe directly or to quantify in bench-scale experiments or mill operations, making it difficult to evaluate effects of process conditions such as bubble size and solution chemistry on deinking efficiency. This paper presents images and measurements of toner ink interactions with bubble surfaces in laboratory-scale flotation processes. Stable adsorption of toner ink was observed at surfaces of stationary and suspended bubbles for several system chemistries. Interactions of toner particles and bubbles were quantified by high magnification and high temporal resolution digital videos obtained in bubble flow facilities creating both stationary and flowing bubbles. Large (>200 micron), flat toner particles adsorbed to bubble surfaces by single contact points. Smaller toner particles formed very stable complexes in fatty acid chemistries. Desorption of toner ink from bubble surfaces was not observed, even for vigorous flows. Bubbles were observed to be fully covered with toner after 4 minutes of residence time in the suspending bubble flow facility. Initial estimates indicate that bubbles with diameters of approximately 1 mm carry more than 1 mg of ink per bubble

Hydrodynamic structures of droplets in square micro-channels

This paper reports on numerical simulations of the hydrodynamics inside droplets in rectangular micro-channels. We use a finite-volume/front-capturing method that allows us to perform two- and three-dimensional simulations with a reasonable cost. The numerical method is an interface-capturing technique without any interface reconstruction. Therefore no complex or expensive interface tracking is needed. Droplet interface deformation and velocity fields inside both droplets and continuous phase can then be followed. This study leads to important results about droplet deformation and inner streamlines for mass and heat transfer studies. More particularly, the capillary number seems to have a great influence on the liquid/liquid flow hydrodynamics whatever is the channel width

Modelling the normal bouncing dynamics of spheres in a viscous fluid

Bouncing motions of spheres in a viscous fluid are numerically investigated by an immersed boundary method to resolve the fluid flow around solids which is combined to a discrete element method for the particles motion and contact resolution. Two well-known configurations of bouncing are considered: the normal bouncing of a sphere on a wall in a viscous fluid and a normal particle-particle bouncing in a fluid. Previous experiments have shown the effective restitution coefficient to be a function of a single parameter, namely the Stokes number which compares the inertia of the solid particle with the fluid viscous dissipation. The present simulations show a good agreement with experimental observations for the whole range of investigated parameters. However, a new definition of the coefficient of restitution presented here shows a dependence on the Stokes number as in previous works but, in addition, on the fluid to particle density ratio. It allows to identify the viscous, inertial and dry regimes as found in experiments of immersed granular avalanches of Courrech du Pont et al. Phys. Rev. Lett. 90, 044301 (2003), e.g. in a multi-particle configuration

Dynamics of non-circular finite release gravity currents

The present work reports some new aspects of non-axisymmetric gravity currents obtained from laboratory experiments, fully resolved simulations and box models. Following the earlier work [Zgheib et al. 2014 Theor. Comput. Fluid Dyn. 28, 521-529] which demonstrated that gravity currents initiating from non-axisymmetric cross-sectional geometries do not become axisymmetric, nor do they retain their initial shape during the slumping and inertial phases of spreading, we show that such non-axisymmetric currents eventually reach a self-similar regime during which (i) the local front propagation scales as t^(1/2) as in circular releases and (ii) the non-axisymmetric front has a self-similar shape that primarily depends on the aspect ratio of the initial release. Complementary experiments of non-Boussinesq currents and top-spreading currents suggest that this self-similar dynamics is independent of the density ratio, vertical aspect ratio, wall friction, and Reynolds number, provided Re is large, i.e., Re≥Ο(10^4). The local instantaneous front Froude number obtained from the fully-resolved simulations is compared to existing models of Froude functions. The recently reported extended box model (EBM) is capable of capturing the dynamics of such non-axisymmetric flows. Here we use the EBM to propose a relation for the self-similar horizontal aspect ratio χ_∞ of the propagating front as a function of the initial horizontal aspect ratioχ_0, namely χ_∞=1+(1/3)ln χ_0. The experimental and numerical results are in good agreement with the proposed relation

Spreading of non-planar non-axisymmetric gravity and turbidity currents

The dynamics of non-axisymmetric turbidity currents is considered here. The study comprises a series of experiments for which a finite volume of particle-laden solution is released into fresh water. A mixture of water and polystyrene particles of diameter 280<Dp<315μm and density ρc=1007Kg/m3 is initially confined in a hollow cylinder at the center of a large tank filled with fresh water. Cylinders with four different cross-sections are examined: a circle, a plus-shape, a rectangle and a rounded rectangle in which the sharp corners are smoothened. The time evolution of the front is recorded as well the spatial distribution of the thickness of the final deposit via the use of a laser triangulation technique. The dynamics of the front and final deposit are significantly influenced by the initial geometry, displaying substantial azimuthal variation especially for the rectangular case where the current extends farther and deposits more particles along the initial minor axis of the rectangular cross section. Interestingly, this departure from axisymmetry cannot be predicted by current theoretical methods such as the Box Model. Several parameters are varied to assess the dependence on the settling velocity, initial height aspect ratio, local curvature and mixture density

Propagation and deposition of non-circular finite release particle-laden currents

The dynamics of non-axisymmetric turbidity currents is considered here for a range of Reynolds numbers of O(10^4) when based on the initial height of the release. The study comprises a series of experiments and highly resolved simulations for which a finite volume of particle-laden solution is released into fresh water. A mixture of water and polystyrene particles of mean diameter dp=300 μm and mixture density ρc=1012 kg/m^3 is initially confined in a hollow cylinder at the centre of a large tank filled with fresh water. Cylinders with two different cross-sectional shapes, but equal cross-sectional areas, are examined: a circle and a rounded rectangle in which the sharp corners are smoothened. The time evolution of the front is recorded as well as the spatial distribution of the thickness of the final deposit via the use of a laser triangulation technique. The dynamics of the front and final deposit are significantly influenced by the initial geometry, displaying substantial azimuthal variation especially for the rectangular case where the current extends farther and deposits more particles along the initial minor axis of the rectangular cross-section. Several parameters are varied to assess the dependence on the settling velocity, initial height aspect ratio and volume fraction. Even though resuspension is not taken into account in our simulations, good agreement with experiments indicates that it does not play an important role in the front dynamics, in terms of velocity and extent of the current. However, wall shear stress measurements show that incipient motion of particles and particle transport along the bed are likely to occur in the body of the current and should be accounted to properly capture the final deposition profile of particles

Simulation of an avalanche in a fluid with a soft-sphere / immersed boundary method including a lubrication force.

The present work aims at reproducing the local dynamics of a dense granular media evolving in a viscous fluid from the grain scale to that of thousands of grains, encountered in environmental multiphase flows. To this end a soft-sphere collision / immersed-boundary method is developed. The methods are validated alone through various standard configurations including static and dynamical situations. Then, simulations of binary wall-particle collisions in a fluid are performed for a wide range of Stokes number ranging in [10-1, 104]. Good agreement with available experimental data is found provided that a local lubrication model is used. Finally, three-dimensional simulations of gravity/shear-driven dense granular flows in a viscous fluid are presented. The results open the way for a parametric study in the parameter space initial aspect ratio - initial packing