A unified SHTC multiphase model of continuum mechanics

TL;DR

The paper develops a unified Symmetric Hyperbolic Thermodynamically Compatible (SHTC) multiphase continuum model capable of describing an arbitrary number of phases that may be heat-conducting inviscid/viscous fluids or elastoplastic solids, with diffuse interfaces represented by volume fractions. It reformulates the SHTC equations into a Baer-Nunziato–type structure to facilitate numerical solution via a robust second-order path-conservative MUSCL-Hancock finite-volume method, while treating stiff relaxation terms with implicit/semi-analytical integrators. The authors provide a detailed closure framework for internal, elastic, thermal, and kinetic energies, define dissipative mechanisms (strain relaxation, interfacial friction, temperature and pressure relaxation, phase transformation), and ensure consistency with the first and second laws of thermodynamics. A multi-distortion BN-type extension and a three-phase model are presented, along with an explicit FV scheme and a comprehensive suite of numerical experiments (convergence, Riemann problems, shear, Rayleigh-Taylor, wedge water entry, and multi-material impacts) to validate the approach and illustrate its capability to capture complex multiphase and multiphysics phenomena. The work demonstrates the practical viability of a monolithic, thermodynamically coherent framework for simulating fluid-structure interactions across fluids and solids with diffuse interfaces, and it lays the groundwork for future extensions to phase transitions and electrodynamics within the SHTC paradigm.

Abstract

In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) equations, can describe the arbitrary number of phases that can be heat-conducting inviscid and viscous fluids, as well as elastoplastic solids. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the SHTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the SHTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging, we restrict our numerical examples to a simplified version of the model, focusing on the isentropic limit for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications for compressible multiphase problems. Notably, results are presented for multiphase flows in all the relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids.

Figures (21)

• Figure 1: Numerical solution at time $t = 0.4$ obtained with the explicit FV scheme for compressible multiphase fluid and solid mechanics applied to a simple shear flow in fluids and in an elastic solid. Results for the solid limit (top left) and for fluids with different viscosities $\nu_1 = 10^{-2}$ (top right), $\nu_1 = 10^{-2}$ (bottom left) and $\nu_1 = 10^{-2}$ (bottom right). For fluids, the analytical solution of the first problem of Stokes is used as the reference solution.
• Figure 2: Numerical results (dashed line) for density $\rho_1$, velocity component $v_{1,1}$ and pressure $p_1$ in the inviscid limit $\tau_1 = 10^{-14}$, for the Riemann problem RP1 $(x_d = 0)$ (top left, bottom left and right), for the Riemann problem RP2 $(x_d = -0.2)$ (top right). The exact solution of the compressible Euler equations (black solid line).
• Figure 3: Numerical results for density $\rho_1$ and velocity component $v_{1,2}$ in the inviscid limit $\tau_1 = 10^{-14}$, at time $t = 0.2$, for the Riemann problem RP3 $(x_d = 0)$ (dashed line). The exact solution of the compressible Euler equations (black solid line).
• Figure 4: Filled contours of one component of the distortion field ${\boldsymbol{A}}_1$, namely of the $A_{1,12}$ component, for the double shear layer problem at times $t=1.2$ (top), $t=1.6$ (center) and $t=1.8$ (bottom); for two values of kinematic viscosity $\nu_1= 2\times10^{-3}$ ($\mathbb{R}\mathrm{e} \simeq 1000$) (left) and $\nu_1= 2\times10^{-4}$ ($\mathbb{R}\mathrm{e} \simeq 10000$) (right).
• Figure 5: Lid driven cavity at Reynolds number $\mathbb{R}\mathrm{e} = 100$. Numerical results obtained at time $t=10.0$. Colour contours of the velocity module (left), and a comparison with the reference solution of Ghia et al. Ghia1982 of the velocity components $v_{1,1}$ and $v_{1,2}$ for 1D cuts along the $x$ and $y$ axis.