Modelling of an Homogeneous Equilibrium Mixture Model (HEM)

Abstract

We present here a model for two phase flows which is simpler than the 6-equations models (with two densities, two velocities, two temperatures) but more accurate than the standard mixture models with 4 equations (with two densities, one velocity and one temperature). We are interested in the case when the two-phases have been interacting long enough for the drag force to be small but still not negligible. The so-called Homogeneous Equilibrium Mixture Model (HEM) that we present is dealing with both mixture and relative quantities, allowing in particular to follow both a mixture velocity and a relative velocity. This relative velocity is not tracked by a conservation law but by a closure law (drift relation), whose expression is related to the drag force terms of the two-phase flow. After the derivation of the model, a stability analysis and numerical experiments are presented.

Appendix: Proof of the Resolution of System (25)–(26) of Sect. 2.1 for the Asymptotic Resolution on u r

Proof of the obtention of the value of V (27) thanks to the resolution of the system (24)–(25)

Putting to the square Eq. (25) and defining the variables $X=(v_{1}^{-}-v_{1}^{+})^{2}$ and $Y=(v_{2}^{-}- v_{2}^{+})^{2}$, we have to solve the following equation:

(65)

in terms of the variable X with Y fixed. Since X>0 by definition, we get $X=\frac{-Y+\sqrt{Y^{2}+4\varPi_{1}^{2}}}{2}$ and as X also satisfies $X=\frac{\varPi_{1}^{2}}{X+Y}$ (cf. (65)), then $X=\frac{2\varPi_{1}^{2}}{Y+\sqrt{Y^{2}+4\varPi_{1}^{2}}}$. Replacing now X and Y by there values and since $v_{1}^{-}-v_{1}^{+}$ has the same sign as Π ₁ (cf. (25)), we have:

(66)

and thanks to (25),

(67)

As Π (22) is a constant vector, using the initial conditions (24), we deduce from (67) that Π ₁=1>0. Thus, thanks to the definition of Π (22), we have $\frac{\partial p}{\partial x_{1}} <0$ since all the quantities are supposed to be positive and ρ ⁻<ρ ⁺, which gives us the value of V (V>0 by hypothesis):

$$ V=\sqrt{-\frac{r^*\frac{\partial p }{\partial x_1}}{C^*\theta_{\rho }} \biggl(\frac{1}{\rho^{-}}-\frac{1}{\rho^{+}} \biggr)}. $$

(68)

 □

Proof of Remark 1

As Π ₁=1, and replacing in (66)–(67) $v_{2}^{-}-v_{2}^{+}$ by its value (29), we immediately get that $v_{1}^{-} -v_{1}^{+}=\exp (-\frac{\theta}{2} )$ and $|\boldsymbol{v}^{-}-\boldsymbol{v}^{+}|=\exp\frac{\theta}{2}$. □

Proof of Proposition 1

The first point is checked immediately using the definition of E (29), (32) and with the assumption (36), we have:

(69)

where L ₃ is a constant. Besides, we also get that the derivative of E tends toward 0:

(70)

These two limits will allow us to prove the second part of the Proposition using the derivative of the function w (33):

(71)

• Reasoning by contradiction, we get the value of ϵ (ϵ=±1 thanks to its definition (29)). We suppose that $\epsilon=-\frac{\varPi_{2}}{|\varPi_{2}|}$, thus in system (30)–(31), the quantity L(E) (34) writes:

  

  (72)

  Then, since we have a finite limit for E (69), the quantity L(E) also tends to a finite limit L ₄≠0 when y ₂→+∞ (since Π ₂≠0). Besides, as we know the limit of the derivative of E (70) (which is 0), using (30), we have:

  

  (73)

  But relation (71) shows that $\frac{dw}{dy_{2}}\underset {y_{2}\rightarrow+\infty}{\rightarrow} \rightarrow0$ using the condition on the velocities in the output of the cells (36), and (69)–(70). This leads to a contradiction and thus $\epsilon=\frac{\varPi_{2}}{|\varPi_{2}|}$.

It remains to get the limit of E. As $\epsilon=\frac{\varPi_{2}}{|\varPi _{2}|}$, we have thanks to (34) that $L(E) \underset {y_{2}\rightarrow+\infty}{\rightarrow} 2 (\varPi_{2} -\frac{\varPi _{2}}{|\varPi_{2}|} \sqrt{L_{3}^{2}-1} )$. Since we have $\lim_{y_{2}\rightarrow+\infty} \frac{d w}{d y_{2}}+\frac {d E}{d y_{2}} =0$ (using (70) and (71)), necessarily: $L(E)\underset{y_{2}\rightarrow+\infty}{\rightarrow} 0$, and by the definition of L(E) (34), we get that:

(74)

which ends the Proof of Proposition 1. □