Finite volume method for one-dimensional Euler equations and application to multi-fluid problem

This thesis is about studying the finite volume method for hyperbolic conservation laws system. Starting from the one dimensional Euler equations, we rewrite them from the form in Eulerian coordinates into the form in the Lagrangian coordinates. This technique transforms a moving grid in Eulerian co- ordinates into a fixed grid in Lagrangian coordinates, thus allowing easier imple- mentation of boundary conditions. The thesis consists of three parts: • piston problem, • multi-fluid models, • asymptotic behavior of Euler equations. In the first part, we consider the piston problem in the paper where the authors Yoshinori Inoue and Takeru Yano study the nonlinear propagation of plane waves radiated into a semi-infinite space filled with a perfect gas, by the sinusoidal motion of an infinite plate. They use the Euler equations in Eulerian coordinates describing the conservation of mass, momentum, and energy then approximate the solution. From this idea, we consider the waves propagating into a semi- infinite tube, which is filled with a perfect gas, closed by a piston on one end and extending along the x-axis at infinity. We use the mass Lagrangian coordinates to obtain the Euler equations rewritten in Lagrangian coordinates and reproduce the results following the piston problem in Yano's paper. The goal is to perform the computation in a finite computational domain, and to develop non-reflecting boundary conditions to impose on the right boundary. In order to reduce the impact of the reflected wave, we propose to combine the Burgers equation in few additional cells of the computational domain. The numerical error caused by the reflected wave is reduced by an order of magnitude by using this approach. In the second part, we consider the tube filled periodically by a large number of pairs of two immiscible fluids. We use Roe’s solver, which is described in Munz's paper, in Lagrangian coordinates, to study the motion of multi-fluid problem and then compare this detailed numerical solution with two isentropic homogeneous models. The first one is a 2 × 2 isentropic system and the second model is a 3 × 3 system which takes into account some turbulent effects. The goal is to check which homogenized model gives better prediction. We study two cases according to the ratio of densities of the two fluids: moderate ratio and large ratio. For each case, we perform the test with smooth and discontinuous initial condition in pressure and velocity. For the problem with smooth initial conditions before the shock formation, the detailed numerical solutions and the numerical results of the two isentropic homogeneous models are in very good agreement. After the shock formation, the detailed numerical solution is strongly oscillatory and we have to use the average values, namely smoothed numerical solution, for the comparison with the two models. We observe the difference between the predictions of the two models. For moderate density ratio the 2×2 model gives a better prediction of the shock position, while for large density ratio, the turbulent 3×3 model is in better agreement with a smoothed out version of the detailed numerical solution compared with the simple 2 × 2 model. In the third part, we study long time behavior of the solutions to the Euler equations by using two different numerical methods: the second order finite volume method in Lagrangian coordinates adopted in the previous chapters, and a high order finite volume method in Eulerian coordinate. In particular, the latter is based on WENO (Weighted Essentially Non-Oscillatory) reconstruction. Then we perform the comparison of the numerical solutions obtained at a final time, when pressure and velocity profiles are almost flat.