Phase Appearance or Disappearance in Two-Phase Flows

Abstract

This paper is devoted to the treatment of specific numerical problems which appear when phase appearance or disappearance occurs in models of two-phase flows. Such models have crucial importance in many industrial areas such as nuclear power plant safety studies. In this paper, two outstanding problems are identified: first, the loss of hyperbolicity of the system when a phase appears or disappears and second, the lack of positivity of standard shock capturing schemes such as the Roe scheme. After an asymptotic study of the model, this paper proposes a first step towards the design of accurate and robust numerical methods adapted to the simulation of phase appearance or disappearance. Polynomial solvers are developed to avoid the use of eigenvectors which are needed in usual shock capturing schemes, and a method based on an adaptive numerical diffusion is designed to treat the positivity problems. An alternate method, based on the use of the hyperbolic tangent function instead of a polynomial, is also considered. Numerical results are presented which demonstrate the efficiency of the proposed solutions.

Appendix

1.1 Hyperbolicity of the Two-Fluid Model

In the finite volume framework, the system can be written in the quasilinear form:

$$\begin{aligned} \frac{\partial \mathbf V }{\partial t} + \mathbb{A }_{\mathbf{n}}(\mathbf V ) \frac{\partial \mathbf V }{\partial \mathbf{n}} = 0. \end{aligned}$$

where $\mathbf{n}$ is the normal vector on the considered face. The characteristic polynomial $P(\lambda )$ of the $\mathbb{A }_{\mathbf{{n}}}$ matrix, of degree $2(2+d)$, $d$ being the space dimension writes

$$\begin{aligned} P(\lambda ) = (\lambda - u_{v n})^d (\lambda - u_{\ell n})^d P_4(\lambda ), \end{aligned}$$

(7.1)

where $u_{kn}$ is the the projection on the normal vector of the velocity of phase $k$, and $P_4$ is a polynomial of degree $4$, and $\lambda$ denotes an eigenvalue of $\mathbb{A }$. We recall the results obtained in [30, 31]. The authors rewrites

$$\begin{aligned} P_4(X)=(X-\delta )^2 (X+\delta )^2 -K_1 (X-\delta )^2 -K_2 (X+\delta )^2 +K_3, \end{aligned}$$

(7.2)

with

$$\begin{aligned}&X =\frac{\lambda - \frac{u_{vn}+u_{\ell n}}{2}}{\gamma }, \quad \gamma ^2=\frac{c_g^2 c_\ell ^2}{\alpha _v\rho _\ell c_\ell ^2+\alpha _\ell \rho _vc_v^2}, \quad K_1= \alpha _\ell \rho _v+ \frac{\alpha _v}{c_v^2} D_{pi}, \\&\quad K_2 = \alpha _v\rho _\ell + \frac{\alpha _\ell }{c_\ell ^2} D_{pi}, \quad K_3 = \frac{D_{pi}}{\gamma ^2}, \quad \delta = \displaystyle {\frac{u_{vn} - u_{\ell n}}{2 \gamma }}, \nonumber \end{aligned}$$

(7.3)

where $c_k$ is the k-phase sound velocity. Our system is said to be strictly hyperbolic if and only if $P_4$ has four distinct real roots. In this case, the matrix $\mathbb{A }$ is necessarily diagonalizable with real eigenvalues. By a careful analysis of the polynomial $P_4$, it has been shown in [30, 31] that for any value of the parameters $\alpha _v,\rho _v, \rho _\ell , c_v, c_\ell$, the hyperbolic region is an unbounded and connected subset of the $(D_{pi},(u_v-u_\ell )^2)$ plane. The location and topology of the non hyperbolic regions has been identified. It is possible from the diagram of the non hyperbolic regions to predict the effect that a given choice of $D_{pi}$ has on the hyperbolicity of the two-fluid model. Figure 12 represents the hyperbolicity diagram for relatively small relative velocities.

Fig. 12

Hyperbolicity diagram for small relative velocities; The interfacial pressure term $D_{pi}$ is denoted by $\Delta P$ and the liquid and vapor phase quantities are indexed by “f” and “g” respectively

First, the authors show that the tangent line of the double root curve at the origin is

$$\begin{aligned} D_{pi}= \displaystyle {\frac{\alpha _v\alpha _\ell \rho _v\rho _\ell }{\alpha _v\rho _\ell + \alpha _\ell \rho _v}}(u_v - u_\ell )^2, \end{aligned}$$

(7.4)

and that this curve is most of the time convex near the origin. Hence the tangent line (7.4) is in the non hyperbolic region for small relative velocities. One needs to rise the slope of the tangent line with a constant parameter $\delta > 1$ to obtain a non zero critical velocity $u_{rc}\ne 0$ (a critical value for the relative velocity for the system to be hyperbolic). This gives the minimal model for the interfacial pressure default term

$$\begin{aligned} D_{pi1} = \delta \frac{\alpha _v\alpha _\ell \rho _v\rho _\ell }{\alpha _v\rho _\ell + \alpha _\ell \rho _v} (u_v -u_\ell )^2. \end{aligned}$$

(7.5)

The larger the coefficient $\delta$ is, the larger the critical relative velocity is. Thus one should adjust the parameter $\delta$ according to the desired range of relative velocities, which is not an easy task. Such a closure law with $\delta =1$ is used in the code CATHARE as explained by Bestion in [5]. This model ensures unconditional hyperbolicity in the presence of the virtual mass force.

Second, the authors construct the line

$$\begin{aligned} D_{pi2} = \rho _v(u_v - u_\ell )^2, \end{aligned}$$

(7.6)

located on the hyperbolic region for every $\alpha _v, \rho _v, \rho _\ell , c_v, c_\ell$ as long as $(u_v - u_\ell )^2 \le c_v^2$. The closure law (7.6) provided by the equation of this hyperbolicity line is sufficient in practice to ensure hyperbolicity as relative velocities are generally less than $c_v$ in all our applications. It moreover has the advantage of allowing a large and fixed range of relative velocities. However, as it does not depend on $\alpha _k$ it does not cancel explicitly when one of the phases disappears (but only due to the fact that the relative velocity tends to zero).

Finally, in [30] a third intermediate model is proposed that is close to $D_{pi1}$ for small relative velocities, and close to $D_{pi2}$ for large ones:

$$\begin{aligned} D_{pi3} = 1.1 \frac{\alpha _v\alpha _\ell \rho _v\rho _\ell }{\alpha _v\rho _\ell + \alpha _\ell \rho _v} (u_v -u_\ell )^2 + \frac{1}{c_g^2}\left( \rho _v-1.1 \frac{\alpha _v\alpha _\ell \rho _v\rho _\ell }{\alpha _v\rho _\ell + \alpha _\ell \rho _v}\right) (u_v -u_\ell )^4. \end{aligned}$$

(7.7)

$D_{pi3}$ is a parabola in $(u_v -u_\ell )$ that is tangent to the double root curve for small relative velocities, and passes through the critical point $M=(c_v^2,\rho _vc_v^2)$ on the hyperbolicity diagram (12). This is the model used to run our test cases.

1.2 Eigenvalues of the Two-Fluid Model

We investigate the structure of the eigenvalues of the two-fluid system (including the energy equations). We recall the method employed in [27, 39]. From the characteristic polynomial 7.1, it follows immediately that $u_{v n}$ and $u_{\ell n}$ are some of the eigenvalues of the system of multiplicity $d$.

For the other eigenvalues, we look for an approximation of the roots of $P_4$ and use a perturbation method by introducing the small ratio

$$\begin{aligned} \xi =\frac{u_{r n}}{a_m}, \end{aligned}$$

(7.8)

where $u_{r n}$ is the projection of the relative velocity on the normal vector and $a_m$ is the “characteristic” speed of sound, in the two-phase mixture, given by

$$\begin{aligned} a_m&= \left( \frac{\rho _m (\alpha _v \rho _\ell + \alpha _\ell \rho _v)}{\rho _v \rho _\ell } \right) ^{1/2} c_m, \end{aligned}$$

with $c_m$ the mixture sound velocity : $c_m^2 = \frac{\rho _v\rho _\ell }{\rho _m } \gamma ^2$ and $\gamma ^2$ given by (7.3). The first order approximation of the two-fluid system eigenvalues is

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle \frac{\alpha _v \rho _\ell u_{v n} + \alpha _\ell \rho _v u_{\ell n}}{\alpha _\ell \rho _v + \alpha _v \rho _\ell } - a_m + O(\xi ^2) , } \\ {\displaystyle \frac{\alpha _v \rho _\ell u_{vn} + \alpha _\ell \rho _v u_{\ell n}}{\alpha _\ell \rho _v + \alpha _v \rho _\ell } + a_m + O(\xi ^2) , } \\ {\displaystyle \frac{\alpha _\ell \rho _v u_{vn} + \alpha _v \rho _\ell u_{\ell n}}{\alpha _\ell \rho _v + \alpha _v \rho _\ell } - \sqrt{ \frac{1}{\alpha _\ell \rho _v + \alpha _v \rho _\ell }( D_{pi} - \frac{u_{rn}^2 \alpha _v \rho _v \alpha _\ell \rho _\ell }{\alpha _\ell \rho _v + \alpha _v \rho _\ell })} + O(\xi ^2) , } \\ {\displaystyle \frac{\alpha _\ell \rho _v u_{vn} + \alpha _v \rho _\ell u_{\ell n}}{\alpha _\ell \rho _v + \alpha _v \rho _\ell } + \sqrt{ \frac{1}{\alpha _\ell \rho _v + \alpha _v \rho _\ell }(D_{pi} - \frac{u_{rn}^2\alpha _v \rho _v \alpha _\ell \rho _\ell }{\alpha _\ell \rho _v + \alpha _v \rho _\ell })} + O(\xi ^2) , } \\ \end{array} \right. \end{aligned}$$

(7.9)

with $D_{pi}$ the interfacial pressure default. The approximate formula of the eigenvalues associated with the void waves leads to the hyperbolicity condition

$$\begin{aligned} D_{pi} \ge \frac{(\mathbf{{u}}_r \cdot {\mathbf{{n}}})^2\alpha _v \rho _v \alpha _\ell \rho _\ell }{\alpha _\ell \rho _v + \alpha _v \rho _\ell } , \end{aligned}$$

which corresponds to Bestion’s model for the interfacial pressure term [5]. Expressions (7.9) can be rewritten in the following form

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle (\mathbf{{u}}_v - \frac{\kappa \alpha _\ell }{\alpha _v +\alpha _\ell \kappa } \mathbf{{u}}_r) \cdot {\mathbf{{n}}} - a_m + O(\xi ^2) , } \\ {\displaystyle (\mathbf{{u}}_v - \frac{\kappa \alpha _\ell }{\alpha _v +\alpha _\ell \kappa } \mathbf{{u}}_r) \cdot {\mathbf{{n}}} + a_m + O(\xi ^2) , } \\ {\displaystyle (\mathbf{{u}}_\ell + \frac{\kappa \alpha _\ell }{\alpha _v +\alpha _\ell \kappa } \mathbf{{u}}_r - \frac{\sqrt{ (\delta - 1) \alpha _v \alpha _\ell \kappa }}{\alpha _v+\alpha _\ell \kappa } \mathbf{{u}}_r) \cdot {\mathbf{{n}}} + O(\xi ^2) , } \\ {\displaystyle (\mathbf{{u}}_\ell + \frac{\kappa \alpha _\ell }{\alpha _v +\alpha _\ell \kappa } \mathbf{{u}}_r + \frac{\sqrt{ (\delta - 1) \alpha _v \alpha _\ell \kappa }}{\alpha _v+\alpha _\ell \kappa } \mathbf{{u}}_r) \cdot {\mathbf{{n}}} + O(\xi ^2) , } \\ \end{array} \right. \end{aligned}$$

with $\kappa =\frac{\rho _v}{\rho _\ell }$ denoting in general a small number, which allows to better realize the order of magnitude of these eigenvalues.

1.3 Void Fraction and Pressure Wave Eigenvectors of the Two-Fluid Model, and Asymptotic Behavior

A first-order approximation in $\xi$ (given by 7.8) of the eigenvectors of the two-fluid model has been given in [38] for a perfect gas of constant $\gamma$. Let us recall the expression of the right eigenvectors $R_3$ and $R_4$ associated to the eigenvalues $\lambda _3$ and $\lambda _4$ in (7.2), and which are suspected to collapse when the void fraction $\alpha _v$ tends to zero:

$$\begin{aligned} R_{3,4} = \left[ \begin{array}{c} \displaystyle 1 \\ \displaystyle -\frac{\rho _\ell }{\rho _v} \\ \displaystyle \lambda _{3,4} \\ \displaystyle -\frac{\rho _\ell }{\rho _v} \lambda _{3,4} \\ \displaystyle \frac{1}{\gamma }(H_v - \frac{1}{2}u_v^2) - u_v(\frac{1}{2}u_v-\lambda _{3,4}) \\ \displaystyle -\frac{\rho _\ell }{\rho _v} (H_\ell - \frac{p}{\rho _\ell }) \end{array} \right] . \end{aligned}$$

(7.10)

Let us now suppose that the vapor phase disappears and the vapor volume fraction $\alpha _v$ tends to zero. In this case, we assume that the relative velocity $\mathbf u _{rn}$ also tends to zero. The fast eigenvalues $\lambda _1$ and $\lambda _2$ are now equal to $u_{n} {\pm } a_m$. They remain distinct and the eigenvectors associated to these eigenvalues do not collapse. As for the intermediate eigenvalues, the void eigenvalues $\lambda _3$ and $\lambda _4$, the form of which are recalled below:

$$\begin{aligned} \lambda _{3,4}= (\mathbf{{u}}_\ell + \frac{\kappa \alpha _\ell }{\alpha _v +\alpha _\ell \kappa } \mathbf{{u}}_r {\pm } \frac{\sqrt{ (\delta - 1) \alpha _v \alpha _\ell \kappa }}{\alpha _v+\alpha _\ell \kappa } \mathbf{{u}}_r) \cdot {\mathbf{{n}}} + O(\xi ^2) , \end{aligned}$$

tend to $u_{n}$.

One can also check that the eigenvectors $R_3$ and $R_4$ have the following form $R= R^0 + \delta R+ O(\xi ^2)$, namely:

$$\begin{aligned} R= \left[ \begin{array}{c} \displaystyle 1 \\ \displaystyle -\frac{\rho _\ell }{\rho _v} \\ \displaystyle u_{n} \\ \displaystyle -\frac{\rho _\ell }{\rho _v} u_{n} \\ \displaystyle \frac{1}{\gamma }(H_v - \frac{1}{2}u^2) +\frac{1}{2}u^2 \\ \displaystyle -\frac{\rho _\ell }{\rho _v} (H_\ell - \frac{p}{\rho _\ell }) \end{array} \right] {\pm } \left[ \begin{array}{c} \displaystyle 0 \\ \displaystyle 0 \\ \displaystyle u_{rn}\sqrt{\alpha _v} \beta \\ \displaystyle -\frac{\rho _\ell }{\rho _v} u_{rn}\sqrt{\alpha _v} \beta \\ \displaystyle \mathbf u \cdot \mathbf u _r \sqrt{\alpha _v} \beta \\ \displaystyle 0 \end{array} \right] + O(\xi ^2) , \end{aligned}$$

(7.11)

with $\beta = \frac{\sqrt{ (\delta - 1) \alpha _\ell \kappa }}{\alpha _v+\alpha _\ell \kappa }$ and $\beta \rightarrow \sqrt{\frac{ \delta - 1 }{\kappa }}$ when $\alpha _v\rightarrow 0$. When $\alpha _v$ tends to zero, $\mathbf u _r$ tends to zero and so does $\xi$. Therefore, $\delta R \sim \alpha ^\frac{1}{2}u_r$ also tends to zero and $R_3$ and $R_4$ collapse.

1.4 Test-Cases

1.4.1 Boiling Channel

The model used in the boiling channel test-case is the two fluid two phase flow model presented in Sect. 2.1. Here are the modeling terms included in the case. We assume that while $h_\ell <h_\ell ^{sat}$, the liquid saturation enthalpy, the heat flux is only implied in the heating of the liquid (heat transfer). When $h_\ell >h_\ell ^{sat}$, the heat flux becomes implied in the evaporation only and therefore results in mass transfer. The mass transfer also implies a transfer of momentum and energy. All numerical values are indicated below.

• 1. The interfacial pressure term is the Bestion’s modeling term (2.17) with $\delta = 1.1$ and $\kappa =10^{-4}$.

• 2. Interfacial velocities and enthalpies:

  $$\begin{aligned}&\displaystyle \mathbf{{u}}^i= \mathbf{{u}}_\ell ,\\&\displaystyle h_v^i= h_v^{sat},\qquad h_\ell ^i = h_\ell ^{sat}. \end{aligned}$$

• 3. Wall heat transfer concentrations:

  $$\begin{aligned} Q^w_\ell&= q \qquad \text{ if } \qquad h_\ell <h_\ell ^{sat}, \\&= 0 \qquad \text{ otherwise }. \\ Q^w_v&= 0. \end{aligned}$$

• 4. Mass transfer:

  $$\begin{aligned} \Gamma&= 0 \qquad \text{ if } \qquad h_\ell <h_\ell ^{sat}, \\&= \frac{q}{L} \qquad \text{ otherwise }. \end{aligned}$$

• 5. Drag force:

  $$\begin{aligned} \mathbf{{F}}^{iD}_v = -F^{iD}_\ell = -\frac{1}{8} C_D a_i\rho _m | \mathbf{{u}}_r| \mathbf{{u}}_r. \end{aligned}$$

• 6. Wall friction:

  $$\begin{aligned} \mathbf{{F}}_k^w = \frac{f}{D_h}\frac{\alpha _k \rho _k| \mathbf{{u}}_k| \mathbf{{u}}_k}{2}. \end{aligned}$$

• 7. Gravity:

  $$\begin{aligned} \mathbf{{f}}_{ext} = \mathbf{{g}}. \end{aligned}$$

Numerical data and auxiliary relations are given in Tables 4 and 5.

Table 4 Numerical data for the boiling channel test-case

Table 5 Auxiliary relations for the boiling channel test-case

1.4.2 Tee Junction

The model used in the tee-junction test-case is the two-fluid two-phase flow model presented in Sect. 2.1. Here are the source terms included in the case:

• 1. The interfacial pressure term is the Bestion’s modeling term (2.17) with $\delta = 1.1$ and $\kappa =10^{-4}$.

• 2. Interfacial velocity:

  $$\begin{aligned} \mathbf{{u}}^i = \mathbf{{u}}_\ell , \end{aligned}$$

• 3. Drag force:

  $$\begin{aligned} \mathbf{{F}}^{iD}_v = -F^{iD}_\ell = -\frac{1}{8} C_D a_i\rho _\ell | \mathbf{{u}}_r| \mathbf{{u}}_r, \end{aligned}$$

  with $a_i= \frac{ 3 \alpha _v}{ r_i}$, $r_i = 0.3165\ 10^{-3}$, and $C_D = 0.44$.

• 4. Wall friction:

  $$\begin{aligned} \mathbf{{F}}_k^w = \frac{f}{D_h}\frac{\alpha _k \rho _k| \mathbf{{u}}_k| \mathbf{{u}}_k}{2}, \end{aligned}$$

  with $D_h=1$ m and $f=0.05$.

1.5 Coefficients of the Polynomial $P_{HDF}$

The $P_{HDF}$ polynomial is written:

$$\begin{aligned} P(x) = \sum _{k=0}^{17} a_k x^{2k}. \end{aligned}$$

(7.12)

with the $a_k$ given by:

$$\begin{aligned} a_0&= 6.209633161688544e-02\\ a_1&= 4.516480010541272e+00\\ a_1&= -3.049057345414379e+01\\ a_2&= 1.657256844603353e+02\\ a_4&= -6.133533687894306e+02\\ a_5&= 1.580698142537855e+03\\ a_6&= -2.879210705862515e+03\\ a_7&= 3.673105197391366e+03\\ a_8&= -3.121407591514732e+03\\ a_9&= 1.512887040780976e+03\\ a_{10}&= -2.111058506112595e+02\\ a_{11}&= 9.753698909265717e+01\\ a_{12}&= -6.475861637079317e+02\\ a_{13}&= 8.947647548149256e+02\\ a_{14}&= -6.303841204016171e+02\\ a_{15}&= 2.586951712420909e+02\\ a_{16}&= -5.941358894806618e+01\\ a_{17}&= 5.960406627331660e+00\\ \end{aligned}$$