Phase appearance or disappearance in two-phase flows

This paper is devoted to the treatment of specific numerical problems which appear when phase appearance or disappearance occurs in models of two-phase flows. Such models have crucial importance in many industrial areas such as nuclear power plant safety studies. In this paper, two outstanding problems are identified: first, the loss of hyperbolicity of the system when a phase appears or disappears and second, the lack of positivity of standard shock capturing schemes such as the Roe scheme. After an asymptotic study of the model, this paper proposes accurate and robust numerical methods adapted to the simulation of phase appearance or disappearance. Polynomial solvers are developed to avoid the use of eigenvectors which are needed in usual shock capturing schemes, and a method based on an adaptive numerical diffusion is designed to treat the positivity problems. An alternate method, based on the use of the hyperbolic tangent function instead of a polynomial, is also considered. Numerical results are presented which demonstrate the efficiency of the proposed solutions

Shape Sensitivity of Free-Surface Interfaces Using a Level Set Methodology

In this paper we develop the continuous adjoint methodology to compute shape sensitivities in free-surface hydrodynamic design problems using the incompressible Euler equations and the level set methodology. The identification of the free-surface requires the convection of the level set variable and, in this work, this equation is introduced in the entire shape design methodology. On the other hand, an alternative continuous adjoint formulation based in the jump condition across the interface, and an internal adjoint boundary condition is also presented. It is important to highlight that this new methodology will allow the specific design of the free-surface interface, which has a great potential in problems where the target is to reduce the wave energy (ship design), or increase the size of the wave (surfing wave pools). The complete continuous adjoint derivation, the description of the numerical methods (including a new high order numerical centered scheme), and numerical tests are detailed in this paper. I