Violent flows in aqueous foams III: physical multi-phase model comparison with aqueous foam shock tube experiments

Abstract

Mitigation of blast waves in aqueous foams is a problem that has a strong dependence on multi-phase effects. Here, a simplified model is developed from the previous articles treating violent flows (D’Alesio et al. in Eur J Mech B Fluids 54:105–124, 2015; Faure and Ghidaglia in Eur J Mech B Fluids 30:341–359, 2011) to capture the essential phenomena. The key is to have two fluids with separate velocities to represent the liquid and gas phases. This allows for the interaction between the two phases, which may include terms for drag, heat transfer, mass transfer due to phase change, added mass effects, to be included explicitly in the model. A good test for the proposed model is provided by two experimental data sets that use a specially designed shock tube. The first experiment has a test section filled with spray droplets, and the second has a range of aqueous foams in the test section. A substantial attenuation of the shock wave is seen in both cases, but a large difference is observed in the sound speeds. The droplets cause no observable change from the air sound speed, while the foams have a reduced sound speed of approximately 50–75 $\hbox {m}/\hbox {s}$. In the model given here, an added mass term is introduced in the governing equations to capture the low sound speed. The match between simulation and experiment is found to be satisfactory for both droplets and the foam. This is especially good when considering the complexity of the physics and the effects that are unaccounted for, such as three-dimensionality and droplet atomisation. The resulting statistics illuminate the processes occurring in such flows.

Appendix

Here we consider the basic two-fluid model of [22], which is written as ($\ell =1$ or 2):

$$\begin{aligned}&\frac{\partial (\alpha _\ell \rho _\ell )}{\partial t} + \mathrm{div}\, (\alpha _\ell \rho _\ell u_\ell ) = 0, \end{aligned}$$

(22)

$$\begin{aligned}&\frac{\partial (\alpha _\ell \rho _\ell u_\ell )}{\partial t} + \mathrm{div}\, (\alpha _\ell (\rho _\ell u_\ell \otimes u_\ell ))+\alpha _\ell \nabla p= f_\ell , \end{aligned}$$

(23)

$$\begin{aligned}&\frac{\partial (\alpha _\ell \rho _\ell E_\ell )}{\partial t} + \mathrm{div}\, (\alpha _\ell \rho _\ell H_\ell u_\ell ) + p \frac{\partial \alpha _\ell }{\partial t} = f_\ell \cdot u_\ell , \end{aligned}$$

(24)

where

$$\begin{aligned} f_1\equiv -\kappa \frac{\alpha _1\alpha _2\rho _1\rho _2}{\alpha _1\rho _1+\alpha _2\rho _2}\frac{\partial (u_1-u_2)}{\partial t}, \text{ and } f_2=-f_1. \end{aligned}$$

(25)

These differential equations are supplemented by three algebraic relations:

$$\begin{aligned} \alpha _1+\alpha _2=1,\,\,\,\rho _1=\rho _1(p,s_1),\,\,\,\rho _2=\rho _2(p,s_2). \end{aligned}$$

(26)

Systems (22) to (24) can be written as a quasilinear system:

$$\begin{aligned} \frac{\partial v_i}{\partial t}+ \sum ^3_{k = 1} \sum ^{10}_{j = 1}A_{i,j}^k(v)\frac{\partial v_j}{\partial x_k} = 0, \end{aligned}$$

(27)

and given a vector $\xi \in {{\mathbb {R}}}^3$. We are interested in the eigenvalues of the $10\times 10$ matrix

$$\begin{aligned} A(v,\xi )=\sum ^3_{k=1} \xi _k A_{i,j}^k(v). \end{aligned}$$

(28)

Since these numbers do not depend on the chosen set of dependent variables, we rewrite this system using the physical variables:

$$\begin{aligned} w\equiv (\alpha ,p,u_1,u_2,s_1,s_2). \end{aligned}$$

(29)

We also introduce the two classical thermodynamic coefficients

$$\begin{aligned} c_i\equiv \left( \sqrt{\left( \frac{\partial \rho _i}{\partial p}\right) _{s_i}}\,\right) ^{-1} \text{ and } \quad \beta _i\equiv \left( \frac{\partial \rho _i}{\partial s_i}\right) _{p}. \end{aligned}$$

(30)

Using Gibbs relation: $T_i \mathrm {d}s_i=\mathrm {d}e_i-\frac{p}{\rho _i^2}\mathrm {d}\rho _i,$ the previous equations are combined to produce:

$$\begin{aligned}&(\alpha _1\rho _2c_2^{2} +\alpha _2\rho _1c_1^{2}) \frac{\partial p}{\partial t} + (\alpha _1\rho _2c_2^{2}u_1 +\alpha _2\rho _1c_1^{2}u_2)\nabla p \nonumber \\&\quad + \rho _1\rho _2c_1^{2}c_2^{2} \,\mathrm{div}\,(\alpha _1u_1+\alpha _2u_2)=0, \end{aligned}$$

(31)

$$\begin{aligned}&(\alpha _1\rho _2c_2^{2} +\alpha _2\rho _1c_1^{2})\frac{\partial \alpha _1}{\partial t} + \alpha _1\alpha _2(u_1-u_2)\nabla p \nonumber \\&\quad + \alpha _2\rho _1c_1^{2} \,\mathrm{div}\,(\alpha _1u_1)- \alpha _1\rho _2c_2^{2} \,\mathrm{div}\,(\alpha _2u_2)=0, \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial s_\ell }{\partial t}+u_\ell \nabla s_\ell =0, \ell = 1 \text{ or } 2, \end{aligned}$$

(33)

$$\begin{aligned}&\frac{\partial u_1}{\partial t} +\frac{\rho +\kappa \rho _1}{\rho (1+\kappa )\rho _1}\nabla p +\frac{\rho +\kappa \alpha _1\rho _1}{\rho (1+\kappa )}u_1\nabla u_1 \nonumber \\&\quad + \frac{\kappa \alpha _2\rho _2}{\rho (1+\kappa )}u_2\nabla u_2=0, \end{aligned}$$

(34)

$$\begin{aligned}&\frac{\partial u_2}{\partial t} +\frac{\rho +\kappa \rho _2}{\rho (1+\kappa )\rho _2}\nabla p +\frac{\kappa \alpha _1\rho _1}{\rho (1+\kappa )}u_1\nabla u_1 \nonumber \\&\quad +\frac{\rho +\kappa \alpha _2\rho _2}{\rho (1+\kappa )}u_2\nabla u_2=0, \end{aligned}$$

(35)

where $\rho \equiv \alpha _1\rho _1+\alpha _2\rho _2$. Let us now write (31) to (33) as a quasilinear system:

$$\begin{aligned} \frac{\partial w_i}{\partial t}+ \sum ^3_{k = 1} \sum ^{10}_{j = 1}B_{i,j}^k(w)\frac{\partial w_j}{\partial x_k} = 0. \end{aligned}$$

(36)

By taking ${{\underline{w}}}$ as a given constant state, the linearisation of (36) around this state is:

$$\begin{aligned} \frac{\partial \omega _i}{\partial t}+ \sum ^3_{k = 1} \sum ^{10}_{j = 1}B_{i,j}^k({{\underline{w}}})\frac{\partial \omega _j}{\partial x_k} = 0. \end{aligned}$$

(37)

Hence, the eigenvalues we are looking for are complex numbers $\lambda ({{\underline{w}}},\xi )$ such that there exists ${\varOmega }\in {{\mathbb {R}}}^{10}$ for which the plane wave:

$$\begin{aligned} \omega (x,t)\equiv {\varOmega }\exp {i(\xi \cdot x-\lambda ({{\underline{w}}},\xi )t)}, \end{aligned}$$

(38)

is a nonzero solution of (37).

$$\begin{aligned} N=\left[ \begin{array}{cccc} (\alpha _{{2}}\rho _{{1}}c_1^2u_{{1}} +\alpha _{{1}}\rho _{{2}}c_2^2u_{{2}})\cdot \xi &{}\alpha _{{1}}\alpha _{{2}} \left( u_{{1}}-u_{{2}}\right) \cdot \xi &{}(\alpha _{{1}}\alpha _{{2}}\rho _{{1}}c_1^2){^t}\xi &{}-\left( \alpha _{{1}}\alpha _{{2}}\rho _{{2}}c_2^2\right) {^t}\xi \\ \rho _{{1}}\rho _{{2}}c_1^2c_2^2 \left( u_{{1}}-u_{{2}}\right) \cdot \xi &{}(\alpha _{{1}}\rho _{{2}}c_2^2u_{{ 1}}+\alpha _{{2}}\rho _{{1}}c_1^2u_{{2}})\cdot \xi &{}(\alpha _{{1}}\rho _{{1}} \rho _{{2}}c_1^2c_2^2){^t}\xi &{}(\alpha _{{2}}\rho _{{1}}\rho _{{2}}c_1^2c_2^2){^t}\xi \\ 0&{}{\frac{{\rho }+\kappa \,\rho _{{1}}}{{\rho }\,\left( 1+\kappa \right) \rho _{{1}}}}\xi &{}{\frac{\left( {\rho }+\kappa \,\alpha _{{1}}\rho _{{1}} \right) u_{{1}}\cdot \xi }{{\rho }\,\left( 1+\kappa \right) }}{} \mathbf{I}&{}{\frac{\kappa \,\alpha _{{2}}\rho _{{2}}u_{{2}}\cdot \xi }{{\rho }\,\left( 1+\kappa \right) }}{} \mathbf{I} \\ 0&{}{\frac{{\rho }+\kappa \,\rho _{{2}}}{{\rho }\,\left( 1+\kappa \right) \rho _{{2}}}}\xi &{}{\frac{\kappa \,\alpha _{{1}}\rho _{{1}}u_{{1}}\cdot \xi }{{\rho }\,\left( 1+\kappa \right) }}{} \mathbf{I} &{}{\frac{\left( {\rho }+\kappa \,\alpha _{{2}}\rho _{{2}}\right) u_{{2 }}\cdot \xi }{{\rho }\,\left( 1+\kappa \right) }}{} \mathbf{I}\end{array}\right] . \end{aligned}$$

(39)

First, we observe that for this change of variables the system is block-diagonalised in the sense that equations (33), $\ell = 1$ or 2,  are, for the linearised system, uncoupled. Hence, $\lambda ({{\underline{w}}},\xi )=u_{\text {i}}\cdot \xi$, where $i=1,2$ are eigenvalues. Then, we can study the system in $(\alpha ,p,u_1,u_2)$, which corresponds to the isentropic model. By substituting (38) into (37), the following characteristic equation is obtained for $\lambda =\lambda ({{\underline{w}}},\xi )$:

$$\begin{aligned} {\mathrm{det}}\,(\lambda M-N)=0, \end{aligned}$$

(40)

where

$$\begin{aligned} M=\left[ \begin{array}{cccc} \alpha _{{1}}\rho _{{2}}c_2^2+\alpha _{{2}}\rho _{{1}}c_1^2&{}0&{}0&{}0 \\ 0&{}\alpha _{{1}} \rho _{{2}}c_2^2+\alpha _{{2}}\rho _{{1}}c_1^2&{}0&{}0 \\ 0&{}0&{}\mathbf{I}&{}0\\ 0&{}0&{}0&{}\mathbf{I}\end{array} \right] . \end{aligned}$$

(41)

To determine the speed of sound in the mixture, we set $u_1=u_2=0$, and in this case matrix N in (39) becomes:

$$\begin{aligned} N_0=\left[ \begin{array}{cccc} 0&{}0&{}(\alpha _{{1}}\alpha _{{2}}\rho _{{1}}c_1^2){^t}\xi &{}-(\alpha _{{1}}\alpha _{{2}}\rho _{{2}}c_2^2){^t}\xi \\ 0&{}0&{}\left( \alpha _{{1}}\rho _{{1}} \rho _{{2}}c_1^2c_2^2\right) {^t}\xi &{}(\alpha _{{2}}\rho _{{1}}\rho _{{2}}c_1^2c_2^2){^t}\xi \\ 0&{}{\frac{{\rho }+\kappa \,\rho _{{1}}}{{\rho }\,\left( 1+\kappa \right) \rho _{{1}}}}\xi &{}\mathbf{0}&{}\mathbf{0} \\ 0&{}{\frac{{\rho }+\kappa \,\rho _{{2}}}{{\rho }\,\left( 1+\kappa \right) \rho _{{2}}}}\xi &{}\mathbf{0} &{}\mathbf{0}\end{array} \right] . \end{aligned}$$

It can easily be shown that the equation for $\lambda$, (40), is

$$\begin{aligned} \lambda ^{4}\left( \lambda ^2-c_\kappa ^2\right) =0, \end{aligned}$$

(42)

where

$$\begin{aligned}&c_\kappa ^2\equiv \left( \frac{\alpha _1(\rho +\kappa \rho _1)}{\rho (1+\kappa )\rho _1} +\frac{\alpha _2(\rho +\kappa \rho _2)}{\rho (1+\kappa )\rho _2}\right) \nonumber \\&\quad \quad \quad \times \frac{\rho _1\rho _2 c_1^2c_2^2}{\alpha _1\rho _2 c_2^2+\alpha _2\rho _1c_1^2}. \end{aligned}$$

(43)