Taylor–von Neumann–Sedov blast wave

Example

An energy $E > 0$ is released at a point at time $t = 0$ in a polytropic gas at rest, of uniform density $\rho_0$, with adiabatic exponent $\gamma > 1$. Assume the pressure of the undisturbed gas is negligible, so the outgoing spherical shock wave is strong. Then the flow is a self-similar solution of the Euler equations. The shock radius is $$R(t) = \xi_0 \left( \frac{E t^2}{\rho_0} \right)^{1/5},$$ where $\xi_0$ is a dimensionless constant that depends only on $\gamma$. Behind the shock, the density, velocity and pressure are fixed profiles in the variable $r / R(t)$, scaled by powers of $t$.

Check by dimensions: the only parameters are $E$, of dimension mass times length squared over time squared, and $\rho_0$, of dimension mass over length cubed. So $E t^2 / \rho_0$ has dimension length to the fifth power, and $(E t^2/\rho_0)^{1/5}$ is the only length that can be formed from $E$, $\rho_0$ and $t$. The shock speed is $R'(t) = \tfrac{2}{5} R(t)/t$, which decreases in time as the energy spreads over a growing volume.