An Energy-Splitting High-Order Numerical Method for Multi-material Flows

Abstract

This chapter deals with multi-material flow problems by a kind of effective numerical methods, based on a series of reduced forms of the Baer-Nunziato (BN)-type model. Numerical simulations often face a host of difficult challenges, typically including the volume fraction positivity and stability of multi-material shocks. To cope with these challenges, we propose a new non-oscillatory energy-splitting Godunov-type scheme for computing multi-material flows in the Eulerian framework. A novel reduced version of the BN-type model is introduced as the basis for the energy-splitting scheme. In comparison with existing two-material compressible flow models obtained by reducing the BN-type model in the literature, it is shown that our new reduced model can simulate the energy exchange around material interfaces very effectively. Then a second-order accurate extension of the energy-splitting Godunov-type scheme is made using the generalized Riemann problem (GRP) solver. Numerical experiments are carried out for the shock-interface interaction, shock-bubble interaction and the Richtmyer-Meshkov instability problems, which demonstrate the excellent performance of this type of schemes.

Appendix: The 2-D GRP Solver

Since the two-dimensional case is considered, we need to solve a so-called quasi 1-D GRP of (25) by setting the adjacent interface along $x = 0$,

$$\begin{aligned} \begin{array}{l} \boldsymbol{W}_t+{\text {div}} \boldsymbol{F}(\boldsymbol{W})=\boldsymbol{0},\qquad \boldsymbol{F}=[\boldsymbol{f},\boldsymbol{g}], \\ (z_{a})_t+\boldsymbol{u} \cdot \nabla z_{a}=\Theta {\text {div}} \boldsymbol{u},\\ \boldsymbol{V}(x,y,0)=\left\{ \begin{array}{ll} \boldsymbol{V}_-(x,y), \ \ \ \ &{} x<0, \\ \boldsymbol{V}_+(x,y) &{} x>0, \end{array} \right. \end{array} \end{aligned}$$

(96)

where $\boldsymbol{V}=[\boldsymbol{W};z_a]$, $\boldsymbol{V}_-(x,y)$ and $\boldsymbol{V}_+(x,y)$ are two polynomials defined on the two neighboring computational cells at time $t = 0$, respectively. Since we just want to construct fluxes normal to cell interfaces, the tangential effect can be regarded as a source term. Therefore, we rewrite the quasi 1-D GRP (96) as

$$\begin{aligned} \begin{array}{l} \boldsymbol{W}_t+\boldsymbol{f}(\boldsymbol{W})_x=-\boldsymbol{g}(\boldsymbol{W})_y,\\ (z_{a})_t + u (z_{a})_x - \Theta u_x = \Theta v_y -v (z_{a})_y,\\ \boldsymbol{V}(x,\tilde{y},0)=\left\{ \begin{array}{ll} \boldsymbol{V}_-(x,\tilde{y}), \ \ \ \ &{} x<0, \\ \boldsymbol{V}_+(x,\tilde{y}), &{} x>0, \end{array} \right. \end{array} \end{aligned}$$

(97)

by fixing a y-coordinate. That is, we solve the 1-D GRP at a point $(0,\tilde{y})$ on the interface, by considering the transversal effect to the interface $x=0$. The value $\boldsymbol{g}(\boldsymbol{W})_y$ and $\Theta v_y -v (z_{a})_y$ at $(0,\tilde{y})$ takes account of the local wave propagation. The solution of this GRP is denoted as $\mathbf{GRP} \left( \boldsymbol{V}_-(\boldsymbol{x}),\boldsymbol{V}_+(\boldsymbol{x})\right)$ and solved by a 2-D GRP solver. This appendix introduces the 2-D GRP solver used in the coding process just for completeness and readers’ convenience. The details can be found in [49]. We notice that the equation of $z_a$ is very close to the equation of mass fraction for the burnt gas in the basic “combustion model” [7], and the GRP solver for the combustion model in [7, 9] is a heuristic form of our 2-D GRP solver.

The GRP solver for solving (96) has the following two versions, which are the acoustic version and the genuinely nonlinear version.

1.1 2-D Acoustic Case

At any point $(0,\tilde{y})$, if $\boldsymbol{V}_-(0-0,\tilde{y})\approx \boldsymbol{V}_+(0+0,\tilde{y})$ and $\big \Vert \frac{\partial \boldsymbol{V}_-}{\partial x}(0-0,\tilde{y})\big \Vert \ne \big \Vert \frac{\partial \boldsymbol{V}_+}{\partial x}(0-0,\tilde{y})\big \Vert$, we view it as an acoustic case. Denote $\boldsymbol{V}_*: = \boldsymbol{V}_-(0-0,\tilde{y})\approx \boldsymbol{V}_+(0+0,\tilde{y})$, and then linearize the governing equations (25) to get

$$\begin{aligned} \frac{\partial \boldsymbol{V}}{\partial t}+\boldsymbol{A}(\boldsymbol{V})\frac{\partial \boldsymbol{V}}{\partial x}+\boldsymbol{B}(\boldsymbol{V})\frac{\partial \boldsymbol{V}}{\partial y}=\boldsymbol{0}. \end{aligned}$$

(98)

We make the decomposition $\boldsymbol{A}(\boldsymbol{V}_*) =\boldsymbol{R} \boldsymbol{\Lambda } \boldsymbol{R}^{-1}$, where $\boldsymbol{\Lambda }=\text{ diag }\{\lambda _i\}$, $\boldsymbol{R}$ is the (right) eigenmatrix of $\boldsymbol{A}(\boldsymbol{V}_*)$. Then the acoustic GRP solver takes

$$\begin{aligned} \begin{array}{rl} \displaystyle \left( \frac{\partial \boldsymbol{V}}{\partial t}\right) _{(0,\tilde{y}, 0)} =&{} \displaystyle -\boldsymbol{R}\boldsymbol{\Lambda }^+ \boldsymbol{R}^{-1} \left( \frac{\partial \boldsymbol{V}_-}{\partial x}\right) _{(0-0,\tilde{y})}-\boldsymbol{R} \boldsymbol{I}^+ \boldsymbol{R}^{-1} \left( \boldsymbol{B}(\boldsymbol{V}_-)\frac{\partial \boldsymbol{V}_-}{\partial y}\right) _{(0-0,\tilde{y})}\\ &{}\displaystyle -\boldsymbol{R}\boldsymbol{\Lambda }^- \boldsymbol{R}^{-1} \left( \frac{\partial \boldsymbol{V}_+}{\partial x}\right) _{(0+0,\tilde{y})}-\boldsymbol{R} \boldsymbol{I}^- \boldsymbol{R}^{-1} \left( \boldsymbol{B}(\boldsymbol{V}_+)\frac{\partial \boldsymbol{V}_+}{\partial y}\right) _{(0+0,\tilde{y})}, \end{array} \end{aligned}$$

(99)

where $\boldsymbol{\Lambda }^+ =\text{ diag }\{\max (\lambda _i,0)\}$, $\boldsymbol{\Lambda }^- =\text{ diag }\{\min (\lambda _i,0)\}$, $\boldsymbol{I}^+ =\frac{1}{2} \text{ diag }\{1+\text{ sign }(\lambda _i)\}$, $\boldsymbol{I}^- =\frac{1}{2} \text{ diag }\{1-\text{ sign }(\lambda _i)\}$.

1.2 2-D Nonlinear Case

At any point $(0,\tilde{y})$, if the difference $\Vert \boldsymbol{V}_-(0-0,\tilde{y})-\boldsymbol{V}_+(0+0,\tilde{y})\Vert$ is large, we regard it as the genuinely nonlinear case and have to solve the 2-D GRP analytically. A key ingredient is how to understand $\boldsymbol{g}(\boldsymbol{W})_y$ and $\Theta v_y -v (z_{a})_y$ at $(0,\tilde{y})$. Here, we construct the 2-D GRP solver by two steps.

• (i) We solve the local planar 1-D Riemann problem

  $$\begin{aligned} \begin{array}{l} \boldsymbol{W}_t+\boldsymbol{f}(\boldsymbol{W})_x=\boldsymbol{0},\ \ \ \ t>0, \\ (z_{a})_t + u (z_{a})_x - \Theta u_x = 0,\\ \mathbf {w}(x,\tilde{y},0) =\left\{ \begin{array}{ll} \boldsymbol{V}_-(0-0,\tilde{y}), \ \ \ &{}x<0,\\ \boldsymbol{V}_+(0+0,\tilde{y}), &{} x>0, \end{array} \right. \end{array} \end{aligned}$$

  (100)

  where $\mathbf {w}=[\boldsymbol{W};z_a]$, to obtain the local Riemann solution $\boldsymbol{V}_* =\mathbf {w}(0,\tilde{y},0+0)$. Just as in the acoustic case, we decompose $\boldsymbol{A}(\boldsymbol{V}_*) =\boldsymbol{R}\boldsymbol{\Lambda }\boldsymbol{R}^{-1}$. Then we set

  $$\begin{aligned} \boldsymbol{h}(x,y)= \begin{bmatrix} -\overline{ \boldsymbol{g}(\boldsymbol{W})_y}\\ \overline{ \Theta v_y -v (z_{a})_y} \end{bmatrix} = \left\{ \begin{array}{ll} -\boldsymbol{R} \boldsymbol{I}^+ \boldsymbol{R}^{-1} \left( \boldsymbol{B}(\boldsymbol{V}_-)\frac{\partial \boldsymbol{V}_-}{\partial y}\right) _{(0-0,\tilde{y})}, \ \ \ &{}x<0,\\ -\boldsymbol{R} \boldsymbol{I}^- \boldsymbol{R}^{-1} \left( \boldsymbol{B}(\boldsymbol{V}_+)\frac{\partial \boldsymbol{V}_+}{\partial y}\right) _{(0+0,\tilde{y})}, \ \ \ &{}x>0, \end{array} \right. \end{aligned}$$

  (101)

  where $\boldsymbol{I}^\pm$ are defined the same as in (99).

• (ii) We solve the quasi 1-D GRP

  $$\begin{aligned} \begin{array}{l} \boldsymbol{W}_t+\boldsymbol{f}(\boldsymbol{W})_x=-\overline{ \boldsymbol{g}(\boldsymbol{W})_y},\ \ \ \ t>0, \\ (z_{a})_t + u (z_{a})_x - \Theta u_x = \overline{ \Theta v_y -v (z_{a})_y},\\ \mathbf {w}(x,y, 0) =\left\{ \begin{array}{ll} \boldsymbol{V}_-(x,y), \ \ \ &{}x<0,\\ \boldsymbol{V}_+(x,y), &{} x>0, \end{array} \right. \end{array} \end{aligned}$$

  (102)

  to obtain $\left( \frac{\partial \boldsymbol{V}}{\partial t}\right) _*=\frac{\partial \boldsymbol{V}}{\partial t}(0,\tilde{y}, 0+0)$. This is done by solving the 1-D GRP for homogeneous equations

  $$\begin{aligned} \begin{array}{l} \boldsymbol{W}_t+\boldsymbol{f}(\boldsymbol{W})_x=\boldsymbol{0},\ \ \ \ t>0, \\ (z_{a})_t + u (z_{a})_x - \Theta u_x = 0,\\ \mathbf {w}(x,\tilde{y}, 0) =\left\{ \begin{array}{ll} \boldsymbol{V}_-(x,\tilde{y}), \ \ \ &{}x<0,\\ \boldsymbol{V}_+(x,\tilde{y}), &{} x>0. \end{array} \right. \end{array} \end{aligned}$$

  (103)

  For the augmented Euler equations, $\left( \frac{\partial \mathbf {w}}{\partial t}\right) _*=\frac{\partial \mathbf {w}}{\partial t}(0,\tilde{y}, 0+0)$ is obtained by solving a pair of algebraic equations essentially,

  $$\begin{aligned} \begin{array}{ll} \displaystyle a_L\left( \frac{\partial u}{\partial t}\right) _* + b_L \left( \frac{\partial p}{\partial t}\right) _*=d_L,\\ \displaystyle a_R\left( \frac{\partial u}{\partial t}\right) _* + b_R\left( \frac{\partial p}{\partial t}\right) _*=d_R. \end{array} \end{aligned}$$

  (104)

  At last, we have

  $$\begin{aligned} \left( \frac{\partial \boldsymbol{V}}{\partial t}\right) _*=\left( \frac{\partial \mathbf {w}}{\partial t}\right) _*+\boldsymbol{h}(x,\tilde{y}). \end{aligned}$$

  (105)

  Here, if we ignore $\boldsymbol{h}(x,\tilde{y})$, the solver is called 1-D GRP solver. Solvers for the GRP (102) with a general source term are presented in [12]. When specified to (97), construction of the solver can be found in [49].