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

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

Fluctuations of elastic interfaces in fluids: Theory and simulation

We study the dynamics of elastic interfaces-membranes-immersed in thermally excited fluids. The work contains three components: the development of a numerical method, a purely theoretical approach, and numerical simulation. In developing a numerical method, we first discuss the dynamical coupling between the interface and the surrounding fluids. An argument is then presented that generalizes the single-relaxation time lattice-Boltzmann method for the simulation of hydrodynamic interfaces to include the elastic properties of the boundary. The implementation of the new method is outlined and it is tested by simulating the static behavior of spherical bubbles and the dynamics of bending waves. By means of the fluctuation-dissipation theorem we recover analytically the equilibrium frequency power spectrum of thermally fluctuating membranes and the correlation function of the excitations. Also, the non-equilibrium scaling properties of the membrane roughening are deduced, leading us to formulate a scaling law describing the interface growth, W^2(L,T)=L^3 g[t/L^(5/2)], where W, L and T are the width of the interface, the linear size of the system and the temperature respectively, and g is a scaling function. Finally, the phenomenology of thermally fluctuating membranes is simulated and the frequency power spectrum is recovered, confirming the decay of the correlation function of the fluctuations. As a further numerical study of fluctuating elastic interfaces, the non-equilibrium regime is reproduced by initializing the system as an interface immersed in thermally pre-excited fluids.Comment: 15 pages, 11 figure

Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique

The theoretical results regarding the ``transition frequencies'' of two acoustically interacting bubbles have been verified numerically. The theory provided by Ida [Phys. Lett. A 297 (2002) 210] predicted the existence of three transition frequencies per bubble, each of which has the phase difference of $\pi /2$ between a bubble's pulsation and the external sound field, while previous theories predicted only two natural frequencies which cause such phase shifts. Namely, two of the three transition frequencies correspond to the natural frequencies, while the remaining does not. In a subsequent paper [M. Ida, Phys. Rev. E 67 (2003) 056617], it was shown theoretically that transition frequencies other than the natural frequencies may cause the sign reversal of the secondary Bjerknes force acting between pulsating bubbles. In the present study, we employ a direct numerical simulation technique that uses the compressible Navier-Stokes equations with a surface-tension term as the governing equations to investigate the transition frequencies of two coupled bubbles by observing their pulsation amplitudes and directions of translational motion, both of which change as the driving frequency changes. The numerical results reproduce the recent theoretical predictions, validating the existence of the transition frequencies not corresponding to the natural frequency.Comment: 18 pages, 8 figures, in pres

An edge-based interface tracking (EBIT) method for multiphase-flow simulation with surface tension

We present a novel Front -Tracking method, the Edge -Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front -Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity. In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for piecewise linear reconstruction and advection of the interface. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier-Stokes solver. A circle fit is now implemented to improve the accuracy of mass conservation in the reconstruction phase. Furthermore, to identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier-Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height -Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. The two-dimensional version of the EBIT method has been implemented in the free Basilisk platform, and validated with seven standard test cases: stagnation flow, translation with uniform velocity, single vortex, Zalesak's disk, capillary wave, Rayleigh -Taylor instability and rising bubble. The results are compared with those obtained with the Volume -of -Fluid (VOF) method already implemented in Basilisk

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