A local discontinuous Galerkin method for the (non)-isothermal Navier–Stokes–Korteweg equations

Abstract

In this article, we develop a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations in conservative form. These equations are used to model the dynamics of a compressible fluid exhibiting liquid-vapour phase transitions. T he NSK-equations are closed with a Van der Waals equation of state and contain third order nonlinear derivative terms. These contributions frequently cause standard numerical methods to violate the energy dissipation relation and require additional stabilization terms to prevent numerical instabilities. In order to address these problems we first develop an LDG method for the isothermal NSK equations using discontinuous finite element spaces combined with a time-implicit Runge-Kutta integration method. Next, we extend the LDG discretization to the non-isothermal NSK equations. An important feature of the LDG discretizations presented in this article is that they are relatively simple, robust and do not require special regularization terms. Finally, computational experiments are provided to demonstrate the capabilities, accuracy and stability of the LDG discretizations.

Simulations of droplet collisions with a Diffuse Interface Model near the critical point

“…Numerical simulations of liquid-vapor flows with DIM have previously been performed by Jamet et al [16], Yue et al [47], Onuki [25], Lamorgese and Mauri [19], Pecenko et al [29], Desmarais and Kuerten [9], Liu et al [21,22] and Tian et al [43]. Simulations of droplet collisions with DIM in two spatial dimensions were done by Yue et al [47] and Pecenko et al [29].…”

Computational Phase‐Field Modeling

“…Therefore, this model opens the possibility to solve vexing problems in computational mechanics such as for example, cavitation, film and nucleate boiling or the phasechange-driven implosion of thin structures (Bueno et al, 2014). The Navier-Stokes-Korteweg equations are rather new in the field, but there are interesting works of theoretical (Dunn and Serrin, 1986) and computational nature (Gomez et al, 2010b;Liu et al, 2013Liu et al, , 2015Tian et al, 2015;Giesselmann et al, 2014;Jamet et al, 2001). Our point of departure to derive the Navier-Stokes-Korteweg equations are classical balance laws for mass, linear momentum, angular momentum and energy, which may be written as followṡ…”

Liquid-vapor transformations with surfactants. Phase-field model and Isogeometric Analysis

“…Although the current form of the system has been known for several years, computational methods for the NSK equations are still in their infancy. Some noteworthy publications are, for example, [32,45,[60][61][62][63].…”

“…Additionally, our model involves third-order partial-differential spatial operators both in the NSK system and in the equation that describes the behavior of the surfactant. This fact significantly limits the use of finite element methods since we need to employ basis functions with C 1 global continuity, which is very difficult or even impossible 60 in 3D complicated geometries.…”