= Pressure similarity in a spherical blast wave
{title2=$p=E^{2/5}\rho_0^{3/5}t^{-6/5}f_p(\xi)$}
For a spherical <self-similar blast wave>, put $\ell=(Et^2/\rho_0)^{1/5}$, $\xi=r/\ell$, $\rho=\rho_0f_\rho(\xi)$ and $u=(r/t)f_u(\xi)$. The radial <Euler equation> then requires $p=\rho_0(\ell/t)^2f_p(\xi)$, where
$$
f_p'=-\xi f_\rho\left[f_u^2-f_u+\xi\left(f_u-\frac25\right)f_u'\right].
$$
This 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.
Back to article page
