Eric Chénier

Numerical Results Using a Colocated Finite-Volume Scheme on Unstructured Grids for Incompressible Fluid Flows

ABSTRACT This article presents numerical results using a new finite-volume scheme on unstructured grids for the incompressible Navier-Stokes equations. The discrete unknowns are the components of the velocity, the pressure, and the temperature, colocated at the centers of the control volumes. The scheme is stabilized using an original method leading to local redistributions of the fluid mass, which simultaneously yields the control of the kinetic energy and the convergence of the scheme. Different comparisons with the literature (2-D and 3-D lid-driven cavity, backward-facing step, differentially heated cavity) allow us to assess the numerical properties of the scheme.

STABILITY OF THE AXISYMMETRIC BUOYANT-CAPILLARY FLOWS IN A LATERALLY HEATED LIQUID BRIDGE

The axisymmetric steady-states solutions of buoyant-capillary flows in a cylindrical liquid bridge are calculated by means of a pseudo-spectral method. The free surface is undeformable and laterally heated. The working fluid is a liquid metal, with a Prandtl number value Pr=0.01. Particular care was taken to preserve the physical regularity in our model, by writing appropriate flux boundary conditions. The location and nature of the bifurcations undergone by the flows are investigated in the space of the dimensionless numbers (Marangoni, Ma∈[0,600]; Rayleigh, Ra∈[0,5×104]). Saddle-node and Hopf bifurcations are found. By analyzing the steady state structures and the energy budgets, the saddle-node bifurcations are observed to play a determinant role. Only two sets of stable steady-states, connected by saddle-nodes, are allowed by the coupling of buoyancy and capillarity. Most of the solutions of the explored part of the (Ma, Ra) plane belong to these states.

Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage

Abstract: The study of two phase flow in porous media under high capillary pressures, in the case where one phase is incompressible and the other phase is gaseous, shows complex phenomena. We present in this paper a numerical approximation method, based on a two pressures formulation in the case where both phases are miscible, which is shown to also handle the limit case of immiscible phases. The space discretization is performed using a finite volume method, which can handle general grids. The efficiency of the formulation is shown on three numerical examples related to underground waste disposal situations.

Stability analysis of natural convective flows in horizontal annuli: Effects of the axial and radial aspect ratios

Linear stability analyses for two-dimensional natural convection in horizontal air-filled annuli are performed for three-dimensional perturbations and radius ratios in the range 1.2⩽R⩽3. Flow transitions from moderate to large-gap annuli, which have not been reported before, are thoroughly investigated. As a result, stability diagrams are obtained for finite and for infinite length annuli. The leading disturbances and threshold values are found to agree well with experimental data and three-dimensional numerical solutions. Three-dimensional simulations were also carried out to examine the influence on the flow stability of no-slip boundary conditions at the end-walls.

Transient Rayleigh-Bénard-Marangoni solutal convection

Solutal driven flow is studied for a binary solution submitted to solvent evaporation at the upper free surface. Evaporation induces an increase in the solute concentration close to the free surface and solutal gradients may induce a convective flow driven by buoyancy and/or surface tension. This problem is studied numerically, using several assumptions deduced from previous experiments on polymer solutions. The stability of the system is investigated as a function of the solutal Rayleigh and Marangoni numbers, the evaporative flux and the Schmidt number. The sensitivity of the thresholds to initial perturbations is analyzed. The effect of viscosity variation during drying is also investigated. At last numerical simulations are presented to study the competition between buoyancy and Marangoni effects in the nonlinear regime.

A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

In order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids, we propose a method which locally reduces the curvature of the discrete solution where the loss of monotony is observed. The discrete solution is shown to fulfill a variational formulation, thanks to the use of Lagrange multipliers. We can then show its convergence to the solution of the continuous problem, and an error estimate is derived. A numerical method, based on Uzawas algorithm, is shown to provide accurate and stable approximate solutions to various problems. Numerical results show the increase of precision due to the application of the method.

Sensitivity of the liquid bridge hydrodynamics to local capillary contributions

In the usual models of thermocapillary flows, a vorticity singularity occurs at the contact free surface–solid boundaries. The steady axisymmetric hydrodynamics of the side-heated liquid bridge of molten metal is addressed here for its sensitivity to the size δ of a length scale explicitly introduced to regularize the problem. By linear stability analysis of the flows, various stable steady states are predicted: The already known steady states which are reflection-symmetric about the mid-plane, but also others which do not possess this property. The thresholds in Ma of the associated bifurcations are strongly dependent on δ, and converge with δ→0 towards low values. Published data give these results some physical relevance.

Letter to the Editor: A collocated finite volume scheme to solve free convection for general non-conforming grids

We present a new collocated numerical scheme for the approximation of the Navier-Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity-pressure unknowns. This scheme is based on a recent scheme for the diffusion terms. Stability properties are drawn from particular choices for the pressure gradient and the non-linear terms. Convergence of the approximate solutions may be proven mathematically. Numerical results show the accuracy of the scheme on irregular grids.

A Finite Volume Scheme for the Transport of Radionucleides in Porous Media

The COUPLEX1 Test case (Bourgeat et al., 2003) is devoted to the comparison of numerical schemes on a convection–diffusion–reaction problem. We first show that the results of the simulation can be mainly predicted by a simple analysis of the data. A finite volume scheme, with three different treatments of the convective term, is then shown to deliver accurate and stable results under a low computational cost.

Sensitivity of diffusive-convective transition to the initial conditions in a transient Bénard-Marangoni problem

Sensitivity of a transient Benard-Marangoni problem is studied using stochastic models to simulate the uncertainties of thermal initial conditions. Using different assumptions, three probabilistic models are developed and compared. Statistics are performed on flow velocities and temperatures. Transitions are examined with respect to the stochastic models.Copyright