The Blast Wave Problem Revisited

The Taylor-von Neumann-Sedov solution for a blast wave generated by \textit{instantaneous }deposition of energy at a \textit{point} is a paradigm example of rapid energy addition to a compressible gas. The traditional intuitive blast wave model (Barenblatt, \textit{Scaling, self-similarity, and intermediate asymptotics}, 47-50, Cambridge University Press, 1996) can be reformulated for \textit{time resolved} dimensional energy deposition (E$\prime )$ into a \textit{finite volume} V$\prime $ (initially containing fluid with a relatively small internal energy E$_{0}\prime $ at a modest initial temperature T$_{0}\prime )$ with systematic asymptotic methods based on a small parameter $\varepsilon $=E$_{0}\prime $/E$\prime <<$1. The energy deposition occurs on a time scale t$_{H}\prime $, short compared to the initial acoustic time t$_{a}\prime $= l$\prime $/a$_{0}\prime $ (l$\prime $ is the characteristic length of the finite volume V$\prime $, a$_{0}\prime $ is the initial acoustic speed). The large local nondimensional temperature T$\prime $/T$_{0}\prime $=O(1/$\varepsilon )$ and speed u$\prime $/a$_{0}\prime $ =O(1/$\varepsilon ^{1/2})$ imply a large local acoustic speed and a significant local Mach number M$_{l}$=O(1), respectively, such that the kinetic and internal energies are commensurate. The shock Mach number, M$_{s}$=(1/$\varepsilon ^{1/2})$, is asymptotically large for the strong blast wave. It also follows that the relatively short local acoustic time t$_{al}\prime $= l$\prime $/a$\prime =\varepsilon ^{1/2}$t$_{a}\prime $ is commensurate with the energy addition time t$_{H}\prime $. The classical similarity solution for point deposition is obtained by seeking variable combinations independent of the vanishingly small artificial length scale l$\prime $.