Self-similar blast wave

A self-similar blast wave is an expanding shock wave whose post-shock profiles retain their shape in a rescaled radius or distance. A fixed explosion energy, ambient mass density and geometry set the front scaling through conservation of total energy. The dimensionless coefficient requires the profiles and an energy-normalization convention.

Use the fields of pressure similarity in a spherical blast wave, with front radius $R={\xi }_0\ell$. Since ${\rho }_0{\ell }^5/t^2=E$, changing the radial integration variable to $\xi$ givesBoth are time independent for fixed integrable similarity profiles and constant front coordinate ${\xi }_0$. With explosion energy $E$ and negligible external energy input or losses, their sum equals $E$. This normalization, together with shock matching and regularity, fixes the dimensionless blast coefficient rather than dimensional analysis alone.

For a spherical self-similar blast wave, put $\ell =(E{t}^2/{\rho }_0{)}^{1/5}$, $\xi =r/\ell$, $\rho ={\rho }_0{f}_{\rho }(\xi )$ and $u=(r/t)f_u(\xi )$. The radial Euler equation then requires $p={\rho }_0(\ell /t{)}^2{f}_p(\xi )$, whereThis follows from ${\partial }_t\xi =-2\xi /(5t)$ and ${\partial }_r\xi =1/\ell$. The momentum equation alone leaves a spatially uniform additive pressure undetermined. The scale-free strong-shock boundary data fix that freedom consistently with the displayed similarity scaling; an independently prescribed ambient pressure introduces another scale.

Let a strong shock wave of radius $R$ and speed $V$ sweep a uniform mass density ${\rho }_0$ into a shell of mass $4\pi {\rho }_0{R}^3/3$ and speed $2V/(\gamma +1)$. If the cavity pressure is $\alpha$ times the immediate post-shock pressure, shell momentum balance gives $R\dot{V}=3(\alpha -1)V^2$. Constant total energy gives $R^3{V}^2=\mathrm{constant}$ and hence $\alpha =1/2$. Including shell internal and kinetic energy, and treating the cavity volume as $4\pi R^3/3$ to thin-shell accuracy, yields $R=\xi (E/{\rho }_0{)}^{1/5}{t}^{2/5}$ with ${\xi }^5=75(\gamma -1)(\gamma +1{)}^2/[16\pi (5\gamma -3)]$. This coefficient is approximate; resolving the shell's finite volume changes it.

For an energy-conserving strong spherical shock wave expanding into mass density $C{r}^{-\beta }$, $0\le \beta <3$, dimensional analysis gives the displayed radius law. Neglect ambient pressure, radiation losses and gravity in this blast-dominated approximation. Write $u=\dot{R}U(\xi )$, $\rho =C{R}^{-\beta }D(\xi )$ and $p=C{R}^{-\beta }{\dot{R}}^{2}P(\xi )$, with $\xi =r/R$. The radial continuity equation, momentum equation and adiabatic pressure equation reduce toThe Strong-shock Rankine-Hugoniot conditions give $U(1)=P(1)=2/(\gamma +1)$ and $D(1)=(\gamma +1)/(\gamma -1)$. The explosion energy fixes the radius coefficient through $E=4\pi C{R}^{3-\beta }{\dot{R}}^2{\int }_0^1[D{U}^2/2+P/(\gamma -1)]{\xi }^{2}d\xi$.

The similarity profiles $U=2\xi /(\gamma +1)$, $D=(\gamma +1)\xi /(\gamma -1)$ and $P=2{\xi }^3/(\gamma +1)$ solve the spherical blast equations at the displayed ambient exponent. For $\gamma =5/3$, $\beta =2$ and $R^3=3E{t}^2/(2\pi C)$. The dimensional fields are $u=r/(2t)$, $\rho =8\pi C^{2}r/(3E{t}^2)$ and $p=2\pi C^2{r}^3/(9E{t}^4)$ for $0<r<R$. Their kinetic and internal energies are equal. Substituting these profiles into all three equations and the shock conditions verifies the solution.

For a planar adiabatic explosion with energy per unit area $E$ in a uniform medium of mass density ${\rho }_0$, the conserved energy has form $E=I{\rho }_{0}Z{\dot{Z}}^2$, with a finite positive integral $I$ of the similarity profiles. Hence $Z=C(E/{\rho }_0{)}^{1/3}{t}^{2/3}$. If $E$ is shared by two fronts, its fraction assigned to either side is absorbed into $C$.