## 4 Results
### Mean and RMS Profiles  
To quantitatively compare the flow field differences in Figures 4 and 5, we azimuthally averaged the three dimensions fields in \(\theta\)_and_\(\phi\) to produce _mean_ and _RMS_ (root mean squared) radial profiles of the turbulent variables. The averaging method is presented in Appendix A.  
Figure 4: Comparison of the cross-section of the density fields. Figure 5: Comparison of the cross-section of the vorticity fields. Figures 7 and 8 depict azimuthally averaged profiles of the thermodynamic profiles of temperature, pressure, and density from a 1 g charge at \(t=0.5015\) ms (blue curves) compared with flow fields from the 1 kg charge at the same scaled time, \(t=5.015\) ms (orange curves). The _mean_ and _RMS_ profiles from the two scales overlayed. Figures 9 and 10 present the azimuthally averaged combustion profiles of fuel (detonation products), oxidizer (air), and combustion products from the 1 g charge at \(t=0.5015\)\(ms\) (blue curves)  
Figure 6: Maximum density difference between the 1 g case and the 1 kg case as a function of time. Differences stayed at roundoff errors until 5 \(\upmu\)s, when they grew exponentially at late times. Figure 7: Comparison of the _mean_ thermodynamic profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). compared with flow fields from the 1 kg charge at the same scaled time, \(t=5.015\) ms (orange curves). The _mean_ and _RMS_ profiles from the two scales overlayed. Figure 8: Comparison of the _RMS_ thermodynamic profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). Figure 9: Comparison of the _mean_ combustion component profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). Figures 11 and 12 depict azimuthally averaged velocity profiles (\(u_{r},u_{\theta},\text{ and }u_{\phi}\)) from a 1 g charge at \(t=0.5015\) ms (blue curves) compared with flow fields from the 1 kg charge at the same scaled time, \(t=5.015\) ms (orange curves). The _mean_ and _RMS_ profiles from the two scales overlayed. Profiles of mean values for \(u_{\theta}\) and \(u_{\phi}\) were essentially zero. Figure 11: Comparison of the _mean_ velocity profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). Figure 10: Comparison of the _RMS_ combustion component profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). Figure 13 shows azimuthally averaged off-diagonal turbulent Reynolds stresses (\(\tau_{r\theta},\tau_{r\phi},\text{ and }\tau_{\theta\phi}\)) from a 1 g charge at \(t=0.5015\) ms, compared with flow fields from the 1 kg charge at the same scaled time, \(t=5.015\) ms. The profiles were essentially zero, with random fluctuations due to the small ensemble size at small radii. Figure 12: Comparison of the \(RMS\) velocity profiles at \(t=0.5015\) ms (1 g charge) and \(t=5.0\) ms (1 kg charge). Figure 14 shows the maximum density histories for a 1 g charge (blue curve) and a 1 kg charge (orange curve); the curves overlayed. Maximum densities decayed because of the adiabatic expansion of the detonation products, with a spike at \(t=0.1\) ms/g\({}^{1/3}\) due to the imploding shock [20]. Figure 15 presents the combustion histories for 1 g and (blue curve) and scaled 1 kg charge (orange curve); the curves overlayed. About 50% of the fuel mass was consumed by combustion at this time. Let us azimuthally average the fireball temperature field in \(\theta,\phi\) to find the evolution of the mean temperature field with time:  
\[\overline{T}_{\rm FB}(x,\tau)=\iint T(x,\theta,\phi,\tau)\mathrm{d}\theta \mathrm{d}\phi \tag{29}\]  
Figure 5:0 Maximum density histories for 1 g and 1 kg charges. Figure 5:1 Combustion histories of fuel and products for 1 g and 1 kg charges. Let us define the fireball radius as the radius where \(\overline{T}_{FB}(x_{FB},\tau)=T_{max}/2\) (starting from outside the fireball and working inward).