Stéphane Glockner

Improvements on open and traction boundary conditions for Navier-Stokes time-splitting methods

We present in this paper a numerical scheme for incompressible Navier-Stokes equations with open and traction boundary conditions, in the framework of pressure-correction methods. A new way to enforce this type of boundary condition is proposed and provides higher pressure and velocity convergence rates in space and time than found in the present state of the art. We illustrate this result by computing some numerical and physical tests. In particular, we establish reference solutions of a laminar flow in a geometry where a bifurcation takes place and of the unsteady flow around a square cylinder.

Benchmark solution for a three-dimensional mixed convection flow - Part 1: reference solutions

A solution to a benchmark problem for a three-dimensional mixed-convection flow in a horizontal rectangular channel heated from below and cooled from above (Poiseuille-Rayleigh-Bénard flow) is proposed. This flow is a steady thermoconvective longitudinal roll flow in a large-aspect-ratio channel at moderate Reynolds and Rayleigh numbers (Re = 50, Ra = 5,000) and Prandtl number Pr = 0.7. The model is based on the Navier-Stokes equations with Boussinesq approximation. We propose reference solutions resulting from computations on large grids, Richardson extrapolation (RE), and cubic spline interpolations. The solutions obtained with one finite-difference, one finite-volume, and two finite-element codes are in good agreement, and reference values for the flow and thermal fields and for the heat and momentum fluxes are given with four to five significant digits.

A COUPLED NUMERICAL MODEL FOR TSUNAMIS GENERATED BY SUBAERIAL AND SUBMARINE MASS FAILURES

This paper presents a new numerical model simulating tsunamis generation by landslides. Water, air, and the slide (considered either as a viscous fluid or as a rigid bloc using the penalization method) are described by Navier-Stokes equations, expressed in a unified single fluid approach. The PLIC-VOF method is used to describe the motion of fluid interfaces. We present results for two test cases featuring solid blocks. The first one is that of a semi-elliptical body sliding over a plane slope, without surrounding water. We find, the slide motion is accurately reproduced by the model when the slide viscosity is around 10 Pa.s. For larger viscosity values, the conditioning of the matrix associated with the Navier-Stokes solver is degraded. In the second case, a falling rectangular block generates waves in a flume. We find, wave height and box velocity are in agreement with experiments by Monaghan and Kos (2000). The simulated flow close to the falling box is similar to that observed in experiment, except that the observed main plunging wave and associated air entrainment.do not occur. In future work, the model thus validated will be applied to simulate more realistic cases of tsunamis generated by subaerial landslides.

Benchmark Solution for a Three-Dimensional Mixed-Convection Flow, Part 2: Analysis of Richardson Extrapolation in the Presence of a Singularity

A reference solution to a benchmark problem for a three-dimensional mixed-convection flow in a horizontal rectangular channel differentially heated (Poiseuille-Rayleigh-Bénard flow) has been proposed in Part 1 of the present article (Numer. Heat Transfer B, vol. 60, pp. 325–345, 2011). Since mixed Dirichlet and Neumann thermal boundary conditions are used on the horizontal walls of the channel, a temperature gradient discontinuity is generated. The aim of this article is to analyze the consequences of this singularity on Richardson extrapolation (RE) of the numerical solutions. The convergence orders of the numerical methods used (finite difference, finite volume, finite element), observed from RE of local and integral quantities are discussed with an emphasis on singularity influence. With the grids used, it is shown that RE can increase the accuracy of the discrete solutions preferentially with the discretization methods of low space accuracy order, but only in some part of the channel and for a restricted range of the extrapolation coefficient. A correction to the Taylor expansion involved in the RE formalism is proposed to take into account the singularity and to explain the majority of the RE behaviors observed.

A fourth-order accurate curvature computation in a level set framework for two-phase flows subjected to surface tension forces

We propose an accurate and robust fourth-order curvature extension algorithm in a level set framework for the transport of the interface. The method is based on the Continuum Surface Force approach, and is shown to efficiently calculate surface tension forces for two-phase flows. In this framework, the accuracy of the algorithms mostly relies on the precise computation of the surface curvature which we propose to accomplish using a two-step algorithm: first by computing a reliable fourth-order curvature estimation from the level set function, and second by extending this curvature rigorously in the vicinity of the surface, following the Closest Point principle. The algorithm is easy to implement and to integrate into existing solvers, and can easily be extended to 3D. We propose a detailed analysis of the geometrical and numerical criteria responsible for the appearance of spurious currents, a well known phenomenon observed in various numerical frameworks. We study the effectiveness of this novel numerical method on state-of-the-art test cases showing that the resulting curvature estimate significantly reduces parasitic currents. In addition, the proposed approach converges to fourth-order regarding spatial discretization, which is two orders of magnitude better than algorithms currently available. We also show the necessity for high-order transport methods for the surface by studying the case of the 2D advection of a column at equilibrium thereby proving the robustness of the proposed approach. The algorithm is further validated on more complex test cases such as a rising bubble.

3D NUMERICAL SIMULATIONS OF WAVES GENERATED BY SUBAERIAL MASS FAILURES: APPLICATION TO LA PALMA CASE

Three-dimensional (3D) waves generated by landslides are simulated using a three-fluid Navier-Stokes VOF model. With this approach, the interaction between slide and water is implicitly solved. The model capabilities are first tested for benchmark cases featuring rigid body motion. Results are good in two dimensions (2D) and encouraging in 3D. Wave generation by a potential collapse of the Cumbre Vieja Volcano, on La Palma island, is then studied. Stability analyses show that the Cumbre Vieja flank is currently highly stable and that potential slide volumes are likely to be closer to 100 km, rather than the 500 km predicted in earlier studies. Results of the Navier-Stokes model show that waves generated are highly dependent upon the details of slide mechanism and kinematics. In our worst 3D scenario (assuming an inviscid fluid), the initial wavelength is 20 km and the wave height decrease due to lateral spreading is high.

Numerical Simulation of Bubble Formation and Transport in Cross-Flowing Streams

Numerical simulations on confined bubble trains formed by cross-flowing streams are carried out with the numerical code THETIS which is based on the Volume of Fluid (VOF) method and has been developed for two phase flow studies and especially for a gas-liquid system. The surface tension force, which needs particular attention in order to determine the shape of the interface accurately, is computed using the Continuum Surface Force model (CSF). Through the coupling of a VOF-PLIC technique (Piecewise-Linear Interface Calculation) and a smoothing function of adjustable thickness, the Smooth Volume of Fluid technique (SVOF) is intended to capture accurately strong interface distortion, rupture or reconnection with large density and viscosity contrasts between phases. This approach is extended by using the regular VOF-PLIC technique, while applying a smoothing procedure affecting both physical characteristics averaging and surface tension modeling. The front-capturing strategy is extended to gas injection. We ...

Simulation of a Fluidized Bed Using a Hybrid Eulerian-Lagrangian Method for Particle Tracking

The characterisation of fluidized beds still requires specific investigation for understanding and modelling the local coupling between the dispersed phase and the carrier fluid. The aim of this work is to simulate this type of unsteady particle laden flows via Direct Numerical Simulations in order to provide a local and instantaneous description of particle flow interactions and to extract statistical parameters useful for large scale models. A fluidized bed has been studied experimentally by Aguilar Corona ([1]). In this laboratory experiment, 3D tracking of a single bed particle provided Lagrangian properties of the discrete phase motion, while 2D PIV was used to characterize the flow of the continuum phase. This fluidized bed has been simulated during nine seconds in order to compare experimental and numerical results and to obtain some data that experimental studies can’t give.

Reduction of the discretization stencil of direct forcing immersed boundary methods on rectangular cells: The ghost node shifting method

Abstract We present an analytical study of discretization stencils for the Poisson problem and the incompressible Navier–Stokes problem when used with some direct forcing immersed boundary methods. This study uses, but is not limited to, second-order discretization and Ghost-Cell Finite-Difference methods. We show that the stencil size increases with the aspect ratio of rectangular cells, which is undesirable as it breaks assumptions of some linear system solvers. To circumvent this drawback, a modification of the Ghost-Cell Finite-Difference methods is proposed to reduce the size of the discretization stencil to the one observed for square cells, i.e. with an aspect ratio equal to one. Numerical results validate this proposed method in terms of accuracy and convergence, for the Poisson problem and both Dirichlet and Neumann boundary conditions. An improvement on error levels is also observed. In addition, we show that the application of the chosen Ghost-Cell Finite-Difference methods to the Navier–Stokes problem, discretized by a pressure-correction method, requires an additional interpolation step. This extra step is implemented and validated through well known test cases of the Navier–Stokes equations.

Moment-of-fluid analytic reconstruction on 2D Cartesian grids

Moment-of-Fluid (MoF) is a piecewise linear interface reconstruction method that tracks fluid through its volume fraction and centroid, which are deduced from the zeroth and first moments. We present a method that replaces the original minimization stage by an analytic reconstruction algorithm on bi-dimensional Cartesian grids. This algorithm provides accurate results for a lower computational cost than the original minimization algorithm. When more than two fluids are involved, this algorithm can be used coupled with the minimization algorithm. Although this paper deals with Cartesian grids, everything remains valid for any meshes that are made of rectangular cells.