## 6 Discussion
### Grid Refinement  
We also explored the effect of grid refinement on scaling. We doubled the resolution and re-ran the HE fireball case shown in Figures 3-15. Table 3 summarizes the profile comparisons of 1\(\times\) versus 2\(\times\) zoning. We found that a factor of 2 increase in grid resolution had no effect on the _mean_ and _RMS_ profiles of thermodynamic variables, \(T,\ \rho,\ m_{F}(t)\), and _mean_ velocity-related profiles of \(\omega,\mathbf{v}_{r},\mathbf{v}_{\theta},\mathbf{v}_{\phi}\). The _RMS_ profiles also scaled with \(r_{0}\), but had about 30% variation due to grid resolution. gaseous explosions to show the same scaling relation. The two solutions were identical when displayed in scaled variables (i.e., the flow field profiles only differed by machine roundoff). Numerical simulations of pyrotechnic fireballs performed at two different scales confirmed this observation; their azimuthally averaged scaled profiles overlayed, proving that they scaled with \(r_{0}\). However, the DLP flow fields and the wake flow fields of pyrotechnic explosions did not scale with \(r_{0}\) because particle sizes introduced another length scale that broke similarity. For example, scaled mist temperature, density, and velocity profiles can differ by a factor of 10 in certain regions. Thus, calculations and numerical simulations of pyrotechnic explosions must be performed at each scale of interest. This has not been recognized in the past. [PERSON] was responsible for the gas-dynamic formulation of the three-phase models used in this research; [PERSON] performed the numerical simulations reported here; [PERSON] led the development of the high-order Godunov algorithms and Adaptive Mesh Refinement methods used in the simulations reported here. All authors have read and agreed to the published version of the manuscript. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07 NA27344. This work was funded by the Office of Defense Nuclear Nonproliferation Research and Development. The authors declare no conflict of interest. Definition  
\(\rho\)  
gas density  
\(u\)  
gas velocity vector  
\(p\)  
gas pressure  
\(E\)  
gas total energy: \(E=e+0.5u\): \(e\)  
gas internal energy  
\(\gamma_{D}\)  
mass fraction detonation products DP  
\(Y_{A}\)  
mass fraction of air  
\(Y_{F}\)  
mass fraction of fuel (detonation products)  
\(Y_{O}\)  
mass fraction of oxidizer (air)  
\(Y_{P}\)  
mass fraction of combustion products  
\(\phi\)  
combustion rate  
\(\alpha\)  
stoichiometric air/fuel ratio  
\(\omega\)  
vorticity vector  
\(\dot{\omega}\)  
baroclinic vorticity production: \(\dot{\omega}=\
abla\rho\times\
abla p/\rho^{2}\)  
\(x\)  
nondimensional radius: \(x=r/r_{0}\)  
\(\tau\)  
nondimensional time: \(\tau=t/t_{0}\)  
\(r_{0}\)  
explosion length scale: \(r_{0}=(m\cdot\Delta H_{d}/p_{a})^{1/(j+1)}\)  
\(t_{0}\)  
explosion time scale: \(t_{0}=r_{0}/a_{a}\)  
\(m\)  
mass of the high explosives charge  
\(\Delta H_{d}\)  
heat of detonation  
\(j\)  
geometry index \(j=0\), 1 or 2 for planar, cylindrical, or spherically symmetric flow  
\(R\)  
nondimensional density function: \(R=\varrho/\varrho_{a}\)  
\(U\)  
nondimensional velocity function: \(U=u/a_{a}\)  
\(c\)  
nondimensional internal energy function: \(\varepsilon=e/\varrho_{a}\)  
\(P\)  
nondimensional pressure function: \(P=p/p_{a}\)  
\(\Theta\)  
nondimensional temperature function: \(\Theta=T/T_{a}\)  
\(U_{\Delta}\)  
nondimensional dilatational velocity: \(U_{\Delta}=u_{\Delta}/a_{a}\)  
\(U_{\omega}\)  
nondimensional rotational velocity: \(U_{\omega}=u_{\omega}/a_{a}\)  
\(x\), \(y\), \(z\)  
nondimensional Cartesian coordinates: \(x=x/r_{0}\), \(y=y/r_{0}\), \(z=z/r_{0}\)  
\(R_{FB}\)  
fireball radius  
\(R_{c}\)  
high explosives charge radius  
\(R_{B}\)  
liquid ball radius