BGK source terms for out-of-equilibrium two-phase flow models

Abstract

A modified form of BGK source terms is proposed for modeling two-phase flows with thermodynamical disequilibrium. The novelty is that three independent time-scales allow to manage the return to the thermodynamical equilibrium while remaining in agreement with the second law of thermodynamics. This is achieved thanks to the definition of a “local-in-time” equilibrium state which tends toward the asymptotic equilibrium state when time increases. The thermodynamical paths of the system are then modified with respect to the classical BGK source terms used for two-phase flow modeling, and the relaxation process of the system toward the asymptotic equilibrium state can be defined with additional degrees of freedom. In a numerical point of view, both the classical and the modified BGK source terms have advantages. The choice between these two forms strongly depends on the numerical strategy used to perform simulations.

Appendix: computation of the formulas of section 4.4

We present here the detailed computations that allow to retrieve equations (35) and (36). It is first recalled that perfect gas EOS are considered in Sect. 4.4; hence, we have: $P_k/T_k=\delta _k/\tau _k$, with $\delta _k=(\gamma _k-1) C_{V,k}$ and $1/T_k=C_{V,k}/e_k$. Moreover, mass fractions are assumed to be such that: $y_1=\overline{y}_1$, so that definitions (34) lead to:

$$\begin{aligned} \overline{\alpha }_1=\frac{{y}_1 \delta _1}{{y}_1 \delta _1+{y}_2 \delta _2}= \quad \text {and} \quad \overline{z}_1=\frac{{y}_1 C_{v,1}}{{y}_1 C_{v,1}+{y}_2 C_{v,2}}. \end{aligned}$$

(67)

Thanks to the definitions of the mixture pressure and temperature (6) and to the definitions (4), we have:

$$\begin{aligned} \frac{P}{T}(\overline{\alpha }_1-{\alpha }_1)= & {} \left( \alpha _1 \frac{P_1}{T_1}+\alpha _2 \frac{P_2}{T_2}\right) \left( \frac{{y}_1 \delta _1}{{y}_1 \delta _1+{y}_2 \delta _2}-{\alpha }_1\right) ,\\= & {} \left( \alpha _1 \frac{\delta _1}{\tau _1}+\alpha _2 \frac{\delta _2}{\tau _2}\right) \left( \frac{{y}_1 \delta _1}{{y}_1 \delta _1+{y}_2 \delta _2}-{\alpha }_1 \right) , \\= & {} \frac{y_1 \delta _1+y_2 \delta _2}{\tau } \left( \frac{{y}_1 \delta _1}{{y}_1 \delta _1+{y}_2 \delta _2}-{\alpha }_1\right) \\= & {} \left( \alpha _2 \frac{{y}_1 \delta _1}{\tau }-\alpha _1 \frac{{y}_2 \delta _2}{\tau }\right) =\alpha _1 \alpha _2 \left( \frac{\delta _1}{\tau _1}-\frac{\delta _2}{\tau _2}\right) , \\= & {} \alpha _1 \alpha _2 \left( \frac{P_1}{T_1}-\frac{P_2}{T_2}\right) . \end{aligned}$$

Hence, we obtain the formula of Eq. (35). Equation (36) can be found using exactly the same computations with $(\overline{z}_1-z_1)$.