Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

Abstract

SUMMARYThe Navier-Stokes-Korteweg (NSK) system is a classical diffuse-interface model for compressible twophase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower-order relaxation of the NSK system with hyperbolic firstorder part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi-discrete finite-volume scheme that satisfies a discrete entropy inequality.

A Relative Entropy Approach to Convergence of a Low Order Approximation to a Nonlinear Elasticity Model with Viscosity and Capillarity

“…Theorem 3.12 guarantees strong convergence of solutions, provided that the initial data are sufficiently smooth. However, in most numerical examples higher orders of convergence are observed; see [23]. We expect that in these cases some additional terms are uniformly bounded in α, while it is not clear how to uniformly bound these terms in general.…”

Pore-scale modeling of phase change in porous media

“…As the continuum thermomechanical theory of a van der Waals fluid, the Navier-Stokes-Korteweg (NSK) equations [58] represent perhaps the earliest of diffuse-interface models, describing the dynamics of a single-component two-phase system. Simulations of liquidvapor flow using the NSK model have been performed, with a focus on bulk processes (boiling and cavitation), in [59][60][61][62][63][64][65][66][67][68][69][70], and on the interaction with solids to model phase-change-driven implosion [71]. Phase separation in liquid-vapor systems with other cubic equations of state has been simulated by [72][73][74][75].…”

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

“…The AP property of the new scheme is achieved by splitting the relaxation system into a non-stiff nonlinear, compressible hyperbolic Navier-Stokes like system and a system that can be treated by a Poisson solver, and allows the use of time and spatial steps that are independent of the Korteweg parameter. As the result the proposed numerical scheme is very efficient for small values of the parameter α, which is a significant improvement compared to an explicit scheme from [34].…”

“…solutions were computed in [34] using an explicit local discontinuous Galerkin (LDG) method, see, e.g., [3,10,16].…”

“…These properties can be exploited to construct robust numerical schemes for the relaxation system. In [34], it has been shown that the overall approach is robust for problems with large density ratios and small interfacial widths. However, the numerical scheme proposed in [34] is an explicit scheme and thus the time steps decrease as the Korteweg parameter α tends to zero.…”