## 2 Gas Dynamics of Explosions  
The turbulent combustion fields are governed by the inviscid conservation laws of gas dynamics. These may be expressed in strong conservation form as  
_Mass:_  
\[\partial_{t}\rho+\
abla\cdot p\mathbf{u}=0 \tag{1}\]  
_Momentum:_  
\[\partial_{t}\rho\mathbf{u}+\
abla\cdot(\rho\mathbf{uu}+p)=0 \tag{2}\]  
_Energy:_  
\[\partial_{t}\rho E+\
abla\cdot(\rho Eu+pu)=0 \tag{3}\]  
_Component DP:_  
\[\partial_{t}\rho Y_{D}+\
abla\cdot(\rho Y_{D}u)=0 \tag{4}\]  
Figure 1: Fireball image from a 15-kg COMP B spherical charge at 4 m height of burst. In the equations above, \(\rho\) is density, \(\mathbf{u}\) represents velocity, \(E\) denotes total energy (\(E=e+0.5\;u\cdot u\)), and \(Y_{D}\) is the mass fraction of detonation product species. Two components are recognized: detonation products (DP), which serve as the fuel, and air (A). Their mass fractions obey the conservation relation:  
\[Y_{D}+Y_{A}=1 \tag{5}\]  
In pure computational cells, we have either \(Y_{D}=1\) or \(Y_{A}=1\); in mixed cells, \(Y_{D}\) and \(Y_{A}<1\). We assume that the components are in a state of thermodynamic equilibrium. This allows us to compute the pressure and temperature equation of states (EOS) according to  
_Pressure:_  
\[\mathrm{P}=f_{1}(\rho,e,\;Y_{D}) \tag{6}\]  
_Temperature:_  
\[\mathrm{T}=f_{2}(\rho,e,\;Y_{D}) \tag{7}\]  
These functions are evaluated by using the thermodynamic equilibrium code Cheetah [11; 12]. Results have been tabulated for computational efficiency. We follow the evolution of component mass fractions of _fuel (detonation products)_: \(Y_{F}\), _oxidizer; (air)_: \(Y_{O}\); and _products (combustion products)_: \(Y_{P}\). They were computed according to the following conservation laws:  
_Fuel_ (\(Y_{F}\)):  
\[\partial_{t}\partial Y_{F}+\
abla\cdot U\partial Y_{F}=-\dot{\varphi} \tag{8}\]  
_Oxidizer_ (\(Y_{O}\)):  
\[\partial_{t}\partial Y_{O}+\
abla\cdot U\partial Y_{O}=-\alpha\dot{\varphi} \tag{9}\]  
_Products_ (\(Y_{P}\)):  
\[\partial_{t}\rho\mathbf{u}+\
abla\cdot(\rho\mathbf{uu}+p)=0 \tag{2}\]0  
_Conservation:_  
\[\partial_{t}\rho\mathbf{u}+\
abla\cdot(\rho\mathbf{uu}+p)=0 \tag{2}\]1  
where \(\alpha\) denotes the stoichiometric oxidizer/fuel ratio. We use the fast chemistry, infinite Damkohler number limit (i.e., gas dynamic limit), where turbulent mixing within the cell occurs according to the MILES concept of [PERSON] [13]:  
_Combustion Rate:_  
\[\dot{\mathbf{\varphi}}=\left\{ \begin{array}{l}\text{burn all Air}:\;\Delta\varrho _{A}/\partial\alpha t\quad(\text{Fuel}-\text{rich cell})\\ \text{burn all Fuel}:\;\Delta\varrho_{F}/\varrho\Delta t\quad(\text{Air}- \text{rich cell})\end{array} \right. \tag{12}\]  
The problem is initialized by a similarity solution of [PERSON] [14] for a constant velocity detonation wave propagating at the Chapman-Jouguet (CJ) speed when the detonation wave reaches the surface of the charge. See Figure 2 for the similarity profiles. (We used a perturbation on the density profile to break nonphysical numerical symmetries. We discuss HE density perturbation effects on the _mean_ and root mean squared (_RMS_) profiles in the Section 6.) The detonation products expand, doing \(p\cdot dV\) work on the surroundings, thereby creating a blast wave. The \(DP\)-air interface is unstable; perturbations grow due to vorticity creation by the inviscid baroclinic mechanism: \(\dot{\omega}=\frac{1}{\rho^{2}}\
abla\rho\times\
abla p\), thereby creating a turbulent combustion layer near the edge of the fireball. We solved the above conservation laws with an unsplit second-order Godunov method. We used the unsplit piecewise parabolic method (PPM) of [PERSON] and [PERSON] [16; 17] to advance the solution in time. PPM flattens slopes in cells near discontinuities to enforce monotonicity constraints (i.e., suppression of oscillations near discontinuities). This slope flattening induces a non-linear dissipation mechanism that acts on the cell level.