Jean-Yves Trépanier

A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws

In this paper, an a posteriori error estimation technique for hyperbolic conservation laws is proposed. The error distributions are obtained by solving a system of equations for the errors which are derived from the linearized hyperbolic conservation laws. The error source term is estimated using the modified equation analysis. Numerical tests for one-dimensional linear and non-linear scalar equations and systems of equations are presented. The results demonstrate that the error estimation technique can correctly predict the location and magnitude of the errors. In addition, it is shown in an example that the estimated error source terms can be used for grid adaptation to control the magnitude of error.

A finite-volume method for the Euler equations on arbitrary Lagrangian-Eulerian grids

Abstract This study presents a finite-volume method for the solution of 2-D/axisymmetric Euler equations using triangular moving grids. The flow simulation is carried out using Roes approximate Riemann solver. The importance of the implicit treatment of the space conversation laws, based on geometric analysis, is emphasized. The procedure for reconstructing Roes method for moving meshes is described and validated.

Progress and investigation on lattice Boltzmann modeling of multiple immiscible fluids or components with variable density and viscosity ratios

Lattice Boltzmann models for simulating multiphase flows are relatively new, and much work remains to be done to demonstrate their ability to solve fundamental test cases before they are considered for engineering problems. From this perspective, a hydrodynamic lattice Boltzmann model for simulating immiscible multiphase flows with high density and high viscosity ratios, up to O(1000)O(1000) and O(100)O(100) respectively, is presented and validated against analytical solutions. The method is based on a two phase flow model with operators extended to handle N immiscible fluids. The current approach is O(N)O(N) in computational complexity for the number of different gradient approximations. This is a major improvement, considering the O(N2)O(N2) complexity found in most works. A sequence of systematic and essential tests have been conducted to establish milestones that need to be met by the proposed approach (as well as by other methods). First, the method is validated qualitatively by demonstrating its ability to address the spinodal decomposition of immiscible fluids. Second, the model is quantitatively verified for the case of multilayered planar interfaces. Third, the multiphase Laplace law is studied for the case of three fluids. Fourth, a quality index is developed for the three-phase Laplace–Young’s law, which concerns the position of the interfaces between the fluids resulting from the different surface tensions. The current model is compatible with the analytical solution, and is shown to be first order accurate in terms of this quality index. Finally, the multilayered Couette’s flow is studied. In this study, numerical results can recover the analytical solutions for all the selected test cases, as long as unit density ratios are considered. For high density and high viscosity ratios, the analytical solution is recovered for all tests, except that of the multilayered Couette’s flow. Numerical results and a discussion are presented for this unsuccessful test case. It is believed that other LB models may have the same problem in addressing the simulation of multiphase flows with variable density ratios.

A comparison of three error estimation techniques for finite-volume solutions of compressible flows

Three techniques to obtain error estimates for finite-volume solutions on unstructured grids are compared in this study. The first estimation technique uses Richardson extrapolation involving three flow solutions on different grids. Error estimates on these grids are computed simultaneously with the order of convergence. The second technique is based on the difference between the computed solution and a higher-order reconstruction obtained using the least-squares method. Finally, a third technique solves an error equation driven by source terms computed from the flux jump at cell interfaces. The flows solved as test cases are governed by the two-dimensional Euler equations, and the discretization employs Roes flux difference splitting scheme. Comparisons with exact errors allow the efficiency of each error estimation technique to be assessed for various types of flows.

Modelling radiative transfer in circuit-breaker arcs with the P-1 approximation

Radiative transfer in arc plasmas is one of the key issues in modelling circuit-breaker arcs properly. For a transient arc flow interaction model, the P-1 approximation appears to be a viable alternative, both from an efficiency and from an accuracy point of view. Evaluation of this approximation against other methods, such as the net emission coefficients model and the partial characteristics method, shows that the model handles the strong self-absorption at the arc boundary well and gives results with good accuracy. The method has been successfully implemented in a Euler arc-flow solver and the complete methodology has been applied to a high-current circuit-breaker arc. The computational results are compared with published experimental data.

Airfoil Shape Optimization Using a Nonuniform Rational B-Splines Parametrization Under Thickness Constraint

Results for 2-D airfoil shape optimization in transonic regime are presented. Airfoil shapes are represented by nonuniform rational B-splines with appropriate regularity properties. A Navier-Stokes flow solver is used to compute the flow field and to obtain aerodynamic coefficients. A design of experiment is conducted to select the most sensitive design variables among the nonuniform rational B-splines parameters to reduce their number in the final optimization process. Single-point and multipoint formulations of the optimization problem are proposed and compared. The nonuniform rational B-splines parameterization guarantees smooth optimized airfoils. The multipoint optimization formulation combined with the nonuniform rational B-splines parameterization leads to airfoils with good performance over a specified Mach range.

Modelling and simulation of nozzle ablation in high-voltage circuit-breakers

In the present paper, a new approach to modelling the ablation phenomenon using computational fluid dynamics (CFD) tools coupled with a physical radiation model is proposed, implemented and validated for application in circuit-breaker arc studies. The present work shows that the principal characteristics of ablation-dominated arcs can be simulated effectively using CFD tools if the radiation incident at the wall boundaries can be predicted accurately.

Decomposition of multidisciplinary optimization problems: formulations and application to a simplified wing design

Several formulations for solving multidisciplinary design optimization (MDO) problems are presented and applied to a test case. Two bi-level hierarchical decomposition approaches are compared with two classical single-level approaches without decomposition of the optimization problem. A methodology to decompose MDO problems and a new formulation based on this decomposition are proposed. The problem considered here for validation of the different formulations involves the shape and structural optimization of a conceptual wing model. The efficiency of the design strategies are compared on the basis of optimization results.

Numerical Assessment of Error Estimators for Euler Equations

Grid convergence studies are conducted to assess four error estimators for their asymptotic behavior: explicit residual, solution reconstruction, Richardson extrapolation, and solution of the error equations. Their accuracy, reliability, and efficitivity to predict the true error are verified on the quasi-one-dimensional Euler equations solved by a second-order accurate finite volume method

Time-accurate local time stepping method based on flux updating

measured profiles of velocity components U and V, the turbulent kinetic energy k, and the dissipation rate of the turbulent energy £, which is deduced from the equilibrium relation 8 = (0.3&)//. The values of the mixing length / are calculated from the data using its definition in terms of the mean-velocity gradient and the shear stress. The boundary conditions are that U at the wake edges is equal to the measured edge velocity Ue measured in the experiment, k and e satisfy the zero-gradient conditions, Ue dk/dx = e and Ue d£/dx = Ce2e/&, where Ce2 is one of the model constants. The configuration of the flow for which the calculation is made is depicted in Fig. 1. The wake-generating model is a flexible plate whose shape is varied to produce different pressure gradients on the upper and lower sides of the plate controlling the properties of the initial wake. The characteristics of the test flow at the trailing edge including the boundary layer thickness, the friction coefficient Cy, and the momentum thickness Reynolds number Re§ are shown in Table 1. The details of the experiments and the results are given in Nakayama and Kreplin. The step size in the calculation is initially taken about 0.5 x 10~ times the momentum thickness 0, at the trailing edge and is doubled at every 50th step until the step size of 0.26, is reached. About 1000 integration steps are needed to cover a distance of about 400, from the trailing edge, which is considered a near wake region.