Jean-Claude Latché

Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results

We propose a numerical method to calculate unsteady flows of Bingham fluids without any regularization of the constitutive law. The strategy is based on the combination of the characteristic/Galerkin method to cope with convection and of the Fortin-Glowinsky decomposition/coordination method to deal with the non-differentiable and non-linear terms that derive from the constitutive law. For the spatial discretization, we use low order finite elements, with, in particular, linear discretization for the velocity and the pressure, stabilized by a Brezzi-Pitkaranta perturbation term. We illustrate this numerical strategy through two well-known problems, namely the hydrodynamic benchmark of the lid-driven cavity and the natural convection benchmark of the differentially heated cavity. For both, we assess our numerical scheme against previous publications, for Newtonian flow or in the creeping flow regime, and propose novel results in the case of Bingham fluid non-creeping flows.

A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state

Convergence Analysis of a Colocated Finite Volume Scheme for the Incompressible Navier-Stokes Equations on General 2D or 3D Meshes

\rho=p

A finite element penalty-projection method for incompressible flows

, based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove {\em a priori} estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.

Convergence of the MAC Scheme for the Compressible Stokes Equations

We study a colocated cell-centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressure, all of them colocated at the center of the cells of a unique mesh; such a configuration is known to lead to stability problems, hence the need for a stabilization technique, which we choose of the Brezzi-Pitka¨ranta type. The scheme features two essential properties: the discrete gradient is the transpose of the divergence terms, and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem and for the transient and the steady Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the

Analysis of the Brezzi-Pitkäranta Stabilized Galerkin Scheme for Creeping Flows of Bingham Fluids

L^2

An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

-convergence of the components of the velocity, and, in the steady case, the weak

Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

L^2

Une méthode de pénalité-projection pour les écoulements dilatables

-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov’s theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and nonlinear cases.

A multilevel local mesh refinement projection method for low Mach number flows

The penalty-projection method for the solution of Navier-Stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage, namely the solution of the momentum balance equation, to constrain the divergence of the predicted velocity field. After a presentation of the scheme in the time semi-discrete formulation, then in fully discrete form for a finite element discretization, we assess its behaviour against a set of benchmark tests, including in particular prescribed velocity and open boundary conditions. The results demonstrate that the augmentation always produces beneficial effects. As soon as the augmentation parameter takes a significant value, the projection method splitting error is reduced, pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore. For high values of the augmentation parameter, the results of coupled solvers are recovered. Consequently, in contrast with standard penalty methods, there is no need for a dependence of the augmentation parameter with the time step, and this latter can be kept to reasonable values, to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step.