Introduction

blastFoam is Synthetik's free and open source CFD airblast code available on GitHub: https://github.com/synthetik-technologies/blastfoam blastFoam is a solver for multi-component compressible flow with application to high-explosive detonation, explosive safety and airblast. blastFoam is developed by Synthetik Applied Technologies: https://www.synthetik-technologies.com

Governing Equations

The evolution of a two phase, compressible, and inviscid mixture can be defined by a set of coupled evolution equations for mass, momentum, and energy.

$${\partial }_t\mathbf{U}+\nabla \cdot \mathbf{F}=\mathbf{S}$$ $$\mathbf{U}=\left(\begin{array}{c}{\alpha }_1\\ {\alpha }_1{\rho }_1\\ {\alpha }_2\rho 2\\ \rho \mathbf{u}\\ \rho E\end{array}\right)\mathbf{F}=\left(\begin{array}{c}{\alpha }_1\mathbf{u}\\ {\alpha }_1{\rho }_1\mathbf{u}\\ {\alpha }_2{\rho }_2\mathbf{u}\\ \rho \mathbf{u}\otimes \mathbf{u}+p\mathbf{I}\\ (\rho E+p)\mathbf{u}\end{array}\right)\mathbf{S}=\left(\begin{array}{c}{\alpha }_1\nabla \cdot \mathbf{u}\\ 0\\ 0\\ 0\\ 0\end{array}\right)$$

where $\rho$ is the mixture density, $\mathbf{u}$ the mixture velocity, $E$ the total energy, $p$ the pressure, and $rh{o}_i$ and ${\alpha }_i$ are the density and volume fraction of each phase.

$${\alpha }_2=1-{\alpha }_1$$ $$\rho =\sum _i{\alpha }_i{\rho }_i$$ $$\rho E=\rho e+\frac{1}{2}\rho |\mathbf{u}{|}^2$$ $$\rho e=\sum _i{\alpha }_i{\rho }_i{e}_i$$

The pressure will be defined using a specified equation of state where the mixture internal energy, densities, and volume fraction are used to calculate the total pressure. The equations of state will be used in the Mie-Gruneisen form.

$$p_i({\rho }_i,e_i,\rho )=(\Gamma ({\rho }_i)-1){\rho }_i{e}_i-\Pi ({\rho }_i)$$

Equations of State

The mixture pressure is defined using the Mie-Gruneisen from using

$$p=\frac{\rho e}{{\sum }_i{\alpha }_i{\xi }_i}-\frac{{\sum }_i{\alpha }_i{\xi }_i{\Pi }_i}{{\sum }_i{\alpha }_i{\xi }_i}$$

where

$${\xi }_i({\rho }_i)=\frac{1}{{\Gamma }_i-1}$$

and ${\Pi }_i$ is dependent on the equation of state.

The speed of sound within a give phase is give by

$$c_i=\sqrt{\frac{{\sum }_i{y}_i{\xi }_i{c}_i^2}{{\sum }_i{\xi }_i}}$$

with

$$y_i=\frac{{\alpha }_i{\rho }_i}{\rho }$$ $$c_i^2=\frac{h_i-{\delta }_i}{{\xi }_i}$$ $$h_i=\frac{{\Gamma }_{i}p+{\Pi }_i}{({\Gamma }_i-1){\rho }_i}$$

and ${\delta }_i$ is again dependent on the equation of state.

Generalized van der Waals gas

For a gas described by the generalized van der Waals equation of state, the pressure is defined as

$$p_i=\frac{{\gamma }_i-1}{1-b_i{\rho }_i}({\rho }_i{e}_i+a_i{\rho }_i^2)-(a_i{\rho }_i^2+c_i)$$

where $a_i$, $b_i$, $c_i$, and ${\gamma }_i$ are material parameters. Writing the above equation in M.G. form, we obtain

$${\Gamma }_i=\frac{{\gamma }_i-1}{1-b_i{\rho }_i}+1$$ $${\Pi }_i=\left[1-\frac{{\gamma }_i-1}{1-b_i{\rho }_i}\right]a_i{\rho }_i^2+\left[\frac{{\gamma }_i-1}{1-b_i{\rho }_i}+1\right]c_i$$

and

$${\delta }_i=-b_i\frac{p_i+a_i{\rho }_i^2}{{\gamma }_i-1}+\left(\frac{1-b_i{\rho }_i}{\gamma -1}-1\right)2a_i{\rho }_i$$

Tait's equation of state

For a material obeying the Tait EOS, the pressure is defined as

$$p_i=({\gamma }_i-1){\rho }_i{e}_i-{\gamma }_i(b_i-a_i)$$

where $a_i$, $b_i$, and ${\gamma }_i$ are material properties. In M.G. form, we have

$${\Gamma }_i={\gamma }_i$$ $${\Pi }_i={\gamma }_i(b_i-a_i)$$

and

$${\delta }_i=0$$

Stiffened gas

For a material obeying the stiffened EOS, the pressure is defined as

$$p_i=({\gamma }_i-1){\rho }_i{e}_i-{\gamma }_i{a}_i$$

where $a_i$ and ${\gamma }_i$ are material properties, and

$${\Gamma }_i={\gamma }_i$$ $${\Pi }_i={\gamma }_i{a}_i$$ $${\delta }_i=0$$

Jones Wilkins Lee (JWL)

The more complicated JWL EOS is often used to define energetic materials, and has a reference pressure given by

$$p_{ref,i}=A_i{e}^{-\frac{R_{1,i}{\rho }_{0,i}}{{\rho }_i}}+B_i{e}^{-\frac{R_{2,i}{\rho }_{0,i}}{{\rho }_i}}$$

$A_i$, $B_i$, $R_{1,i}$, $R_{2,i}$, ${\rho }_{0,i}$, and ${\Gamma }_{0,i}$ are the material properties. The functions of the EOS are given by

$${\Gamma }_i={\Gamma }_{0,i}+1$$ $${\Pi }_i={\Gamma }_{0,i}{\rho }_i\left(\frac{A_i}{R_{1,i}{\rho }_{0,i}}{e}^{-\frac{R_{1,i}{\rho }_{0,i}}{{\rho }_i}}+\frac{B_i}{R_{2,i}{\rho }_{0,i}}{e}^{-\frac{R_{2,i}{\rho }_{0,i}}{{\rho }_i}}+e_{0,i}\right)-p_{ref,i}$$

and

$${\delta }_i=A_i{e}^{-\frac{R_{1,i}{\rho }_{0,i}}{{\rho }_i}}\left[{\Gamma }_{0,i}\left(\frac{1}{R_{1,i}{\rho }_{0,i}}+\frac{1}{{\rho }_i}\right)-\frac{R_{1,i}{\rho }_{0,i}}{{\rho }_i^2}\right]\frac{1}{{\Gamma }_{0,i}}+B_i{e}^{-\frac{R_{2,i}{\rho }_{0,i}}{{\rho }_i}}\left[{\Gamma }_{0,i}\left(\frac{1}{R_{2,i}{\rho }_{0,i}}+\frac{1}{{\rho }_i}\right)-\frac{R_{2,i}{\rho }_{0,i}}{{\rho }_i^2}\right]\frac{1}{{\Gamma }_{0,i}}+e_{0,i}$$

$e_0$ is anther material parameter which denotes a reference energy state.

Cochran Chan

The Cochran Chan EOS can be used to describe solid material, and has a reference pressure given by

$$p_{ref,i}=A_i{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{1-{\mathcal{E}}_{1,i}}-B_i{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{1-{\mathcal{E}}_{2,i}}$$

$A_i$, $B_i$, ${\mathcal{E}}_{1,i}$, ${\mathcal{E}}_{2,i}$, ${\rho }_{0,i}$, and ${\Gamma }_{0,i}$ are the material properties. The functions of the EOS are given by

$${\Gamma }_i={\Gamma }_{0,i}+1$$ $${\Pi }_i={\Gamma }_{0,i}{\rho }_i\left(-\frac{A_i}{({\mathcal{E}}_{1,i}-1){\rho }_{0,i}}{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{1-{\mathcal{E}}_{1,i}}+\frac{B_i}{({\mathcal{E}}_{2,i}-1){\rho }_{0,i}}{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{1-{\mathcal{E}}_{2,i}}+e_{0,i}\right)-p_{ref,i}$$ $${\delta }_i=\frac{A_i}{{\mathcal{E}}_{1,i}}\left[{\mathcal{E}}_{1,i}{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{-{\mathcal{E}}_{1,i}}\frac{{\mathcal{E}}_{1,i}-{\Gamma }_{0,i}-1}{{\rho }_i}+\frac{{\Gamma }_{0,i}}{{\rho }_{0,i}}\right]\frac{1}{{\Gamma }_{0,i}}+\frac{B_i}{{\mathcal{E}}_{2,i}}\left[{\mathcal{E}}_{2,i}{\left(\frac{{\rho }_{0,i}}{{\rho }_i}\right)}^{-{\mathcal{E}}_{2,i}}\frac{{\mathcal{E}}_{2,i}-{\Gamma }_{0,i}-1}{{\rho }_i}+\frac{{\Gamma }_{0,i}}{{\rho }_{0,i}}\right]\frac{1}{{\Gamma }_{0,i}}+e_{0,i}$$

where again, $e_0$ is a material parameter which denotes a reference energy state.