Thermodynamically Compatible Discretization of a Compressible Two-Fluid Model with Two Entropy Inequalities

Abstract

We present two methods for the numerical solution of an overdetermined symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flows which has the peculiar feature that it is endowed with two entropy inequalities as primary evolution equations. The total energy conservation law is an extra conservation law and is obtained via suitable linear combination of all other equations based on the Godunov variables (main field). In the stiff relaxation limit the SHTC model tends to an asymptotically reduced Baer–Nunziato-type (BN) limit system with a unique choice for the interface velocity and the interface pressure, including parabolic heat conduction terms and additional lift forces that are not present in standard BN models. Both numerical schemes directly discretize the two entropy inequalities, including the entropy production terms, and obtain total energy conservation as a consequence. The first method is of the finite volume type and makes use of a thermodynamically compatible flux recently introduced by Abgrall et al. that allows to fulfill an additional extra conservation law exactly at the discrete level. The scheme satisfies both entropy inequalities by construction and can be proven to be nonlinearly stable in the energy norm. The second scheme is a general purpose discontinuous Galerkin method that achieves thermodynamic compatibility merely via the direct solution of the underlying viscous regularization of the governing equations. We show computational results for several benchmark problems in one and two space dimensions, comparing the two methods with each other and with numerical results obtained for the asymptotically reduced BN limit system. We also investigate the influence of the lift forces.

Appendix: Polar Coordinate Representation of the Two-fluid Model

We consider a continuous solution of the homogeneous (black) part of system (3). Let the Cartesian coordinates in 2D be denoted by $x = (x_1,x_2)$. Then we can define the polar coordinates in terms of radius r and angle $\theta$ as

$$\begin{aligned} x_1 = r \cos (\theta ), \quad x_2 = r \sin (\theta ). \end{aligned}$$

(62)

The velocity and thermal impuls vectors are defined by

$$\begin{aligned} v_1&= v_r \cos (\theta ) - v_\theta \sin (\theta ), \quad&v_2&= v_r \cos (\theta ) + v_\theta \sin (\theta ), \end{aligned}$$

(63a)

$$\begin{aligned} w_1&= w_r \cos (\theta ) - w_\theta \sin (\theta ), \quad&w_2&= w_r \cos (\theta ) + w_\theta \sin (\theta ), \end{aligned}$$

(63b)

$$\begin{aligned} j_1^1&= j_r^1 \cos (\theta ) - j_\theta ^1 \sin (\theta ), \quad&j_2^1&= j_r^1\cos (\theta ) + j_\theta ^1 \sin (\theta ), \end{aligned}$$

(63c)

$$\begin{aligned} j_1^2&= j_r^2 \cos (\theta ) - j_\theta ^2 \sin (\theta ), \quad&j_2^2&= j_r^2\cos (\theta ) + j_\theta ^2 \sin (\theta ). \end{aligned}$$

(63d)

Using

$$\begin{aligned} \frac{\partial x_1}{\partial r} = \cos (\theta ), \quad \frac{\partial x_1}{\partial \theta } = -\frac{\sin (\theta )}{r}, \quad \frac{\partial x_2}{\partial r} = \sin (\theta ), \quad \frac{\partial x_1}{\partial \theta } = \frac{\cos (\theta )}{r}, \end{aligned}$$

(64)

we obtain the following system in polar coordinates for

$$\begin{aligned} \textbf{q}= (\alpha ^1, \alpha ^1 \rho ^1, \alpha ^2 \rho ^2, \rho v_r, \rho v_\theta ,\rho w_r, \rho w_\theta , \rho j^1_r, \rho j^1_\theta , \rho j^2_r, \rho j^2_\theta , \alpha ^1\rho ^1s^1, \alpha ^2\rho ^2s^2)^T \end{aligned}$$

(65)

as follows

$$\begin{aligned}{} & {} \frac{\partial \alpha ^1}{\partial t}+\frac{v_r}{r}\frac{\partial }{\partial r} \left( r \alpha ^1 \right) + \frac{v_\theta }{r}\frac{\partial }{\partial \theta } \alpha ^1 = 0, \end{aligned}$$

(66a)

$$\begin{aligned}{} & {} \frac{\partial (\alpha ^1\rho ^1)}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r} \left( r \alpha ^1 \rho ^1 v_r^1 \right) + \frac{1}{r}\frac{\partial }{\partial \theta } \left( \alpha ^1\rho ^1 v_\theta ^1 \right) =0, \end{aligned}$$

(66b)

$$\begin{aligned}{} & {} \frac{\partial (\alpha ^2\rho ^2)}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r} \left( r \alpha ^2 \rho ^2 v_r^2 \right) + \frac{1}{r}\frac{\partial }{\partial \theta } \left( \alpha ^2 \rho ^2 v_\theta ^2 \right) =0, \end{aligned}$$

(66c)

$$\begin{aligned}{} & {} \begin{aligned} \frac{\partial (\rho v_r)}{\partial t}&+\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho v_r^2 + \rho c^1(1-c^1) w_r w_r + p\right) \right) \\&+ \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho v_rv_\theta + \rho c^1c^2 w_r w_\theta \right) = \frac{\rho v_\theta ^2 + \rho c^1(1-c^1) w_\theta ^2 + p}{r}, \end{aligned} \end{aligned}$$

(66d)

$$\begin{aligned}{} & {} \begin{aligned} \frac{\partial (\rho v_\theta )}{\partial t}&+\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho v_rv_\theta + \rho c^1(1-c^1) w_r w_\theta \right) \right) \\&+ \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho v_\theta ^2 + p + \rho c^1(1-c^1) w_\theta ^2\right) = -\frac{\rho v_rv_\theta + \rho c^1c^2 w_r w_\theta }{r}, \end{aligned} \end{aligned}$$

(66e)

$$\begin{aligned}{} & {} \begin{aligned} \frac{\partial w_r}{\partial t}&+\frac{\partial }{\partial r} \left( v_rw_r + v_\theta w_\theta + (1-2c^1) \frac{w_r w_r + w_\theta w_\theta }{2} + \mu _1 - \mu _2\right) \\&+ v_\theta \left( \frac{1}{r}\frac{\partial }{\partial \theta } w_r - \frac{1}{r} \frac{\partial }{\partial r}\left( r w_\theta \right) \right) = 0, \end{aligned} \end{aligned}$$

(66f)

$$\begin{aligned}{} & {} \begin{aligned} \frac{\partial w_\theta }{\partial t}&+\frac{1}{r}\frac{\partial }{\partial \theta } \left( v_rw_r+ v_\theta w_\theta + (1-2c^1) \frac{w_rw_r+ w_\theta w_\theta }{2} + \mu _1 - \mu _2 \right) \\&+v_r\left( \frac{1}{r} \frac{\partial }{\partial r}\left( r w_\theta \right) - \frac{1}{r}\frac{\partial }{\partial \theta } w_r\right) = 0, \end{aligned} \end{aligned}$$

(66g)

$$\begin{aligned}{} & {} \frac{\partial (\rho j_r^1)}{\partial t} +\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho j_r^1v_r+ T^1\right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho j_r^1 v_\theta \right) = \frac{\rho j_\theta ^1 v_\theta + T^1}{r}, \end{aligned}$$

(66h)

$$\begin{aligned}{} & {} \frac{\partial (\rho j_\theta ^1)}{\partial t} +\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho j_\theta ^1v_r\right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho j_\theta ^1 v_\theta + T^1\right) = - \frac{\rho j_\theta ^1 v_r}{r}, \end{aligned}$$

(66i)

$$\begin{aligned}{} & {} \frac{\partial (\rho j_r^2)}{\partial t} +\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho j_r^2v_r+ T^2\right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho j_r^2 v_\theta \right) = \frac{\rho j_\theta ^2 v_\theta + T^2}{r}, \end{aligned}$$

(66j)

$$\begin{aligned}{} & {} \frac{\partial (\rho j_\theta ^2)}{\partial t} +\frac{1}{r}\frac{\partial }{\partial r}\left( r\left( \rho j_\theta ^2v_r\right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta }\left( \rho j_\theta ^2 v_\theta + T^2\right) = - \frac{\rho j_\theta ^2 v_r}{r}, \end{aligned}$$

(66k)

$$\begin{aligned}{} & {} \frac{\partial (\alpha ^1\rho ^1 s^1)}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r} \left( r \left( \alpha ^1 \rho ^1 s^1 v_r+ A^1 j_r^1 \right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta } \left( \alpha ^1\rho ^1 s^1 v_\theta + A^1 j_\theta ^1\right) =0, \end{aligned}$$

(66l)

$$\begin{aligned}{} & {} \frac{\partial (\alpha ^2\rho ^2 s^2)}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r} \left( r \left( \alpha ^2 \rho ^2 s^2 v_r+ A^2 j_r^2 \right) \right) + \frac{1}{r}\frac{\partial }{\partial \theta } \left( \alpha ^1\rho ^2 s^2 v_\theta + A^2 j_\theta ^2\right) =0. \nonumber \\ \end{aligned}$$

(66m)