A Physical-Constraint-Preserving Discontinuous Galerkin Method for Weakly Compressible Two-Phase Flows

Abstract

This work focuses on the robust and high-order numerical simulations of weakly compressible two-phase flows by using the discontinuous Galerkin (DG) method combined with an explicit strong-stability-preserving Runge–Kutta scheme. In order to improve the computational robustness under large density ratios, a nonlinear weighted essentially non-oscillatory (WENO) limiter and a positivity-preserving limiter are specially designed and applied with the aim of dampening the nonphysical oscillations around the phase interface and preventing the occurrence of negative density, respectively. More importantly, we theoretically prove that the present method is able to satisfy the uniform-pressure–velocity criterion which states that uniform pressure and velocity profiles around an isolated phase interface should be preserved during the simulation. The performance of the present method is validated by a range of benchmark test cases with density ratios up to 1000:1. The results demonstrate that the present method possesses a good capability of simulating weakly compressible two-phase flows with large density ratios.

Appendix A

Here, we are going to prove that the WENO reconstructed polynomial solution preserves uniform velocity and pressure. The velocity obtained from the WENO reconstructed solution (36) is

$$\begin{aligned} \begin{aligned} u_h^{(r)}(x_i^\alpha , y) = \frac{2 u_0 {\mathcal {W}}_r([q_1]_{i,j}) + u_0 {\mathcal {W}}_r([q_2]_{i,l})}{2{\mathcal {W}}_r([q_1]_{i,j}) + {\mathcal {W}}_r([q_2]_{i,l})}\equiv u_0, \\ v_h^{(r)}(x_i^\alpha , y) = \frac{2 v_0 {\mathcal {W}}_r([q_1]_{i,j}) + v_0 {\mathcal {W}}_r([q_2]_{i,l})}{2{\mathcal {W}}_r([q_1]_{i,j}) + {\mathcal {W}}_r([q_2]_{i,l})}\equiv v_0. \end{aligned} \end{aligned}$$

Thus, uniform velocity is preserved. Next, based on the pressure equilibrium

$$\begin{aligned} p_h^{(r)}(x_i^\alpha , y) = p_0 + c_1^2\left( \frac{(\tilde{\rho }_1)_{h}^{(r)}}{(z_1)_{h}^{(r)}} - \rho _{1,0}\right) = p_0 + c_2^2\left( \frac{(\tilde{\rho }_2)_{h}^{(r)}}{(z_2)_{h}^{(r)}} - \rho _{2,0}\right) , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&(\tilde{\rho }_1)_{h}^{(r)}(x_i^\alpha , y) = \rho _{1,0}\overline{(z_1)}_{i,j} \!+\! \frac{\rho _{1,0}}{\rho _{1,0} \!-\! \rho _{2,0}}{\mathcal {W}}_r([q_2]_{i,l}),\\&(\tilde{\rho }_2)_{h}^{(r)}(x_i^\alpha , y) = \rho _{2,0}\overline{(z_2)}_{i,j} \!-\! \frac{\rho _{2,0}}{\rho _{1,0} \!-\! \rho _{2,0}}{\mathcal {W}}_r([q_2]_{i,l}), \end{aligned} \end{aligned}$$

we have the reconstructed equilibrium volume fraction

$$\begin{aligned} \begin{aligned}&(z_1)_{h}^{(r)}(x_i^\alpha , y) = \overline{(z_1)}_{i,j} \!+\! \frac{1}{\rho _{1,0} \!-\! \rho _{2,0}}{\mathcal {W}}_r([q_2]_{i,l}), \\&(z_2)_{h}^{(r)}(x_i^\alpha , y) \!=\! \overline{(z_2)}_{i,j} \!-\! \frac{1}{\rho _{1,0} \!-\! \rho _{2,0}}{\mathcal {W}}_r([q_2]_{i,l}), \end{aligned} \end{aligned}$$

which implies that $p_{h}^{(r)}(x_i^\alpha , y) \equiv p_0$. This completes the proof.