Abstract: The Bernoulli problem is rephrased into a shape optimization problem. In particular, the cost function, which turns out to be a constitutive law gap functional, is borrowed from inverse problem formulations. The shape derivative of the cost functional is explicitly determined. The gradient information is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by numerical results for both interior and exterior Bernoulli problems.

Abstract: We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor-Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained.

Abstract: In this paper, we design numerical methods for a PDE system arising in corrosion modelling. This system describes the evolution of a dense oxide layer. It is based on a drift-diffusion system and includes moving boundary equations. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance. Finally, numerical experiments with real-life data shows the efficiency of the developed methods.

Abstract: A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley-Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples.

Abstract: We consider the numerical solution of the free boundary Bernoulli problem by employing level set formulations. Using a perturbation technique, we derive a second order method that leads to a fast iteration solver. The iteration procedure is adapted in order to work in the case of topology changes. Various numerical experiments confirm the efficiency of the derived numerical method.

Abstract: This paper presents a new result concerning the perturbation theory of M-matrices. We give the proof of a theorem showing that some perturbations of irreducibly diagonally dominant M-matricies are monotone, together with an explicit bound of the norm of the perturbation. One of the assumptions concerning the perturbation matrix is that the sum of the entries of each of its row is nonnegative. The resulting matrix is shown to be monotone, although it may not be diagonally dominant and its off diagonal part may have some positive entries. We give as an application the proof of the second order convergence of an non-centered finite difference scheme applied to an elliptic boundary value problem.

Abstract: A second-order finite difference scheme for mixed boundary value problems is presented. This scheme does not require the tangential derivative of the Neumann datum. It is designed for applications in which the Neumann condition is available only in discretized form. The second-order convergence of the scheme is proven and the theory is validated by numerical examples.

Abstract: We present a numerical method based on a level set formulation to solve the Bernoulli problem. The formulation uses time as a parameter of boundary evolution. The level set formulation enables to consider non connected domains. Numerical experiments show the efficiency of the method if boundary conditions are handled accurately. In particular, the case of multiple solutions is treated.

Abstract: In this paper, we apply a spectral multilevel method in the non homogeneous direction of a channel. The spectral tau method being not well suited to separate the scales, we use a Galerkin basis in the wall normal direction. Then we can separate the scales, as in the periodic case, from the spectral decomposition of the velocity field. In this way, the quantities associated with the small and large scales verify the no slip boundary conditions. Then, we resolve the large and the small scale equations, simplifying the computation of the small scales. Indeed, we use a quasi-static approximation to compute the small scales. As for the interaction terms, we apply a quasi-static approximation to estimate the modulus, the phase being updated, at each time step, in function of the large scales. To validate the method proposed, we have done two simulations of the channel with the multilevel method. They correspond to two different choices of the total number of modes and of the coarse cut-off level for the multilevel method in the wall normal direction. The results obtained are compared with the results stemming from direct numerical simulations (DNS): one fine DNS (fine resolution) and one low DNS (coarse resolution).

Abstract: The Diffusion Poisson Coupled Model (DPCM) is presented to modelling the oxidation of a metal covered by an oxide layer. This model is similar to the Point Defect Model and the Mixed Conduction Model except for the potential profile which is not assumed but calculated in solving the Poisson equation. This modelling considers the motions of two moving interfaces linked through the ratio of Pilling-Bedworth. Their locations are unknowns of the model. Application to the case of iron in neutral or slightly basic solution is discussed. Then, DPCM has been first tested in a simplified situation where the locations of interfaces were fixed. In such a situation, DPCM is in agreement with Mott-Schottky model when iron concentration profile is homogeneous. When it is not homogeneous, deviation from Mott-Schottky model has been observed and is discussed. The influence of the outer and inner interfacial structures on the kinetics of electrochemical reactions is illustrated and discussed. Finally, simulations for the oxide layer growth are presented. The expected trends have been obtained. The steady-state thickness is a linear function of the applied potential and the steady-state current density is potential independent.

Abstract: We present a new cut-cell method, based on the MAC scheme on Cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. The discretization of the nonlinear terms, written in conservative form, is formulated in the context of finite volume methods. While first order approximations are used in cut-cells the scheme is globally second-order accurate. The linear systems are solved by a direct method based on the capacitance matrix method. Accuracy and efficiency of the method are supported by numerical simulations of 2D flows past a cylinder at Reynolds numbers up to 9 500.

Abstract: A finite difference scheme for the heat equation with mixed boundary conditions on a moving domain is presented. We use an immersed interface technique to discretize the Neumann condition and the Shortley-Weller approximation for the Dirichlet condition. Monotonicity of the discretized parabolic operator is established. Numerical results illustrate the feasibility of the approach.

Abstract: A model based on incremental scales is applied to LES of incompressible turbulent channel flow. With this approach, the resolved scales are decomposed into large and incremental scales; the incremental scales have a larger (two times) spectral support than the large ones. Both velocity components are advanced in time by integrating their respective equations. At every time step and point in the wall normal direction, the one-dimensional energy spectra of the incremental scales are corrected in order to fit the slopes of the corresponding large scale spectra. LES of turbulent channel flow at two different Reynolds numbers are conducted. Results for both simulations are in good agreement with filtered DNS data. A significant improvement is shown compared to simulations with no model at the same low resolutions as the LES. The computational cost of the incremental method is similar to that of a Galerkin approximation on the same grid.

Abstract: Subgrid-scale models based on incremental unknowns (IU) are proposed and investigated for LES of incompressible homogeneous turbulence. The aim of this approach is to derive an estimation procedure of scales (IU) smaller than the resolved ones. The IU components are solutions of an evolution equation. The SGS stress tensor is then explicitly computed. The SGS force is finally modified by phase correction procedures in order to enhance SGS dissipation. A good level of correlation between modeled and exact SGS force, as well as SGS energy transfer, is obtained. The IU models predict the right amount of SGS dissipation. A good agreement between LES results and filtered DNS is noted. In the case of decaying turbulence, IU models perform better than the dynamic model.

Abstract: Multilevel methods have been used in the numerical simul ation of turbulent flows. The separation of scales can lead to different strategies, such as large eddy simulation or adaptative schemes for example. The large eddy simulation propose to resolve the large scale equation, by modeling the subgrid stress tensor. Multilevel methods propose a different approach: by analyzing the time and space behavior of the different scales, we propose to compute them differently.
In this paper, we describe the strategy in the case of non-homogeneous turbulence (the channel flow problem), after giving some results for the one-dimensional Burgers equation.