Continuum simulation of the discharge of the granular silo: a validation test for the mu(I)-visco-plastic flow law

Using both a continuum Navier-Stokes solver, with the mu(I)-flow-law implemented to model the viscous behavior, and the discrete Contact Dynamics algorithm, the discharge of granular silos is simulated in two dimensions from the early stages of the discharge until complete release of the material. In both cases, the Beverloo scaling is recovered. We first do not attempt quantitative comparison, but focus on the qualitative behavior of velocity and pressure at different locations in the flow. A good agreement is obtained in the regions of rapid flows, while areas of slow creep are not entirely captured by the continuum model. The pressure field shows a general good agreement. The evolution of the free surface implies differences, however, the bulk deformation is essentially identical in both approaches. The influence of the parameters of the mu(I)-flow-law is systematically investigated, showing the importance of the dependence on the inertial number I to achieve quantitative agreement between continuum and discrete discharge. The general ability of the continuum model to reproduce qualitatively the granular behavior is found to be very encouraging.Comment: 12 pages, 15 figure

Adaptive modelling of long-distance wave propagation and fine-scale flooding during the Tohoku tsunami

The 11 March 2011 Tohoku tsunami is simulated using the quadtree-adaptive Saint-Venant solver implemented within the Gerris Flow Solver. The spatial resolution is adapted dynamically from 250 m in flooded areas up to 250 km for the areas at rest. Wave fronts are tracked at a resolution of 1.8 km in deep water. The simulation domain extends over 73° of both latitude and longitude and covers a significant part of the north-west Pacific. The initial wave elevation is obtained from a source model derived using seismic data only. Accurate long-distance wave prediction is demonstrated through comparison with DART buoys timeseries and GLOSS tide gauges records. The model also accurately predicts fine-scale flooding compared to both satellite and survey data. Adaptive mesh refinement leads to orders-of-magnitude gains in computational efficiency compared to non-adaptive methods. The study confirms that consistent source models for tsunami initiation can be obtained from seismic data only. However, while the observed extreme wave elevations are reproduced by the model, they are located further south than in the surveyed data. Comparisons with inshore wave buoys data indicate that this may be due to an incomplete understanding of the local wave generation mechanisms

Transition in a numerical model of contact line dynamics and forced dewetting

We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle $\theta_\Delta$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^\circ$ to $110^\circ$ and capillary numbers from $0.00085$ to $0.2$ where the mesh size $\Delta$ is varied in the range of $0.0035$ to $0.06$ of the capillary length $l_c$. To interpret the results, we use Cox's theory which involves a microscopic distance $r_m$ and a microscopic angle $\theta_e$. In the numerical case, the equivalent of $\theta_e$ is the angle $\theta_\Delta$ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function $\phi$ so that $r_m = \Delta/\phi$ and estimate this gauge function by comparing our numerics to Cox's theory. The comparison provides a direct assessment of the agreement of the numerics with Cox's theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on $\theta_\Delta$ and the viscosity ratio $q$. In the case of small $\theta_e$, we use the prediction by Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of the critical capillary number for the Landau-Levich-Derjaguin forced dewetting transition. We generalize this prediction to large $\theta_e$ and arbitrary $q$ and express the critical capillary number as a function of $\theta_e$ and $r_m$. An analogy can be drawn between $r_m$ and the numerical slip length.Comment: This version of the paper includes the corrections indicated in Ref. [1

The granular silo as a continuum plastic flow: the hour-glass vs the clepsydra

The granular silo is one of the many interesting illustrations of the thixotropic property of granular matter: a rapid flow develops at the outlet, propagating upwards through a dense shear flow while material at the bottom corners of the container remains static. For large enough outlets, the discharge flow is continuous; however, by contrast with the clepsydra for which the flow velocity depends on the height of fluid left in the container, the discharge rate of granular silos is constant. Implementing a plastic rheology in a 2D Navier-Stokes solver (following the mu(I)-rheology or a constant friction), we simulate the continuum counterpart of the granular silo. Doing so, we obtain a constant flow rate during the discharge and recover the Beverloo scaling independently of the initial filling height of the silo. We show that lowering the value of the coefficient of friction leads to a transition toward a different behavior, similar to that of a viscous fluid, and where the filling height becomes active in the discharge process. The pressure field shows that large enough values of the coefficient of friction ($\simeq$ 0.3) allow for a low-pressure cavity to form above the outlet, and can thus explain the Beverloo scaling. In conclusion, the difference between the discharge of a hourglass and a clepsydra seems to reside in the existence or not of a plastic yield stress.Comment: 6 pages, 6 figure

An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of slightly viscoelastic droplets

We propose an adaptive numerical solver for the study of viscoelastic 2D two-phase flows using the volume-of-fluid method. The scheme uses the robust log conformation tensor technique of Fattal & Kupferman (2004,2005} combined with the time-split scheme proposed by Hao & Pan (2007}. The use of this time-split scheme has been proven to increase the stability of the numerical computation of two-phase flows. We show that the adaptive computational technique can be used to simulate viscoelastic flows efficiently. The solver is coded using the open-source libraries provided by the \basilisk \cite{Basilisk} platform. In particular, the method is implemented for Oldroyd-B type viscoelastic fluids and related models (FENE-P and FENE-CR). The numerical scheme is then used to study the splashing of weakly viscoelastic drops. The solvers and tests of this work are freely available on the Basilisk web sit

Breakup of finite-size liquid filaments: Transition from no-breakup to breakup including substrate effects

This work studies the breakup of finite-size liquid filaments, when also including substrate effects, using direct numerical simulations. The study focuses on the effects of three parameters: Ohnesorge number, the ratio of the viscous forces to inertial and surface tension forces, the liquid filament aspect ratio, and where there is a substrate, a measure of the fluid slip on the substrate, i.e. slip length. Through these parameters, it is determined whether a liquid filament breaks up during the evolution toward its final equilibrium state. Three scenarios are identified: a collapse into a single droplet, the breakup into one or multiple droplets, and recoalescence into a single droplet after the breakup (or even possibly another breakup after recoalescence). The results are compared with the ones available in the literature for free-standing liquid filaments. The findings show that the presence of the substrate promotes breakup of the filament. The effect of the degree of slip on the breakup is also discussed. The parameter domain regions are comprehensively explored when including the slip effects. An experimental case is also carried out to illustrate the collapse and breakup of a finite-size silicon oil filament supported on a substrate, showcasing a critical length of the breakup in a physical configuration. Finally, direct numerical simulations reveal striking new details into the breakup pattern for low Ohnesorge numbers, where the dynamics are fast and the experimental imaging is not available; our results therefore significantly extend the range of Ohnesorge number over which filament breakup has been considered

Adaptive Cartesian meshes for atmospheric single-column models: a study using Basilisk 18-02-16

It is well known that the representation of certain atmospheric conditions in climate and weather models can still suffer from the limited grid resolution that is facilitated by modern-day computer systems. Herein we study a simple one-dimensional analogy to those models by using a single-column model description of the atmosphere. The model employs an adaptive Cartesian mesh that applies a high-resolution mesh only when and where it is required. The so-called adaptive-grid model is described, and we report our findings obtained for tests to evaluate the representation of the atmospheric boundary layer, based on the first two GEWEX ABL Study (GABLS) inter-comparison cases. The analysis shows that the adaptive-grid algorithm is indeed able to dynamically coarsen and refine the numerical grid whilst maintaining an accurate solution. This is an interesting result as in reality, transitional dynamics (e.g. due to the diurnal cycle or due to changing synoptic conditions) are the rule rather than the exception.</p

A projection method for multiphase flows

An Eulerian projection approach for incompressible variable-density two-phase flows is presented. The Navier-Stokes equations governing these flows are reformulated to take the form of the corresponding equations for the lighter phase with a constant density, which can be efficiently solved using standard numerical methods. The effect of the additional mass in the heavier phase is accounted for by a forcing term, which is determined from the solution of an artificial velocity field. This artificial field is subjected solely to inertial and gravity forces as well as the force coupling the flow field and the artificial field. The phase interface in this purely Eulerian approach is described using the level-set method. Results for two-dimensional simulations of the Rayleigh-Taylor instability are presented to validate the new method