Asymptotic self-similar solutions with a characteristic timescale

For a wide variety of initial and boundary conditions, adiabatic one-dimensional flows of an ideal gas approach self-similar behavior when the characteristic length scale over which the flow takes place, R, diverges or tends to zero. It is commonly assumed that self-similarity is approached since in the R → ∞(0) limit the flow becomes independent of any characteristic length or timescales. In this case, the flow fields f (r, t ) must be of the form f (r, t ) = t^αf F(r/R) with R α (±t )^α. We show that requiring the asymptotic flow to be independent only of characteristic length scales implies a more general form of self-similar solutions, f (r, t ) = R^δf F(r/R) with Ṙ αRδ , which includes the exponential (δ = 1) solutions, R α e^t/τ .We demonstrate that the latter, less restrictive, requirement is the physically relevant one by showing that the asymptotic behavior of accelerating blast waves, driven by the release of energy at the center of a cold gas sphere of initial density ρ α r ^-ω, changes its character at large ω: the flow is described by 0 ≤ d < 1, R α t ^1/(1-7δ), solutions for ω < ω_c , by δ > 1 solutions with R α (-t )^1/(δ-1) diverging at finite time (t = 0) for ω > ω_c , and by exponential solutions for ω = ω_c (ωc depends on the adiabatic index of the gas, ωc ∼ 8 for 4/3 < γ < 5/3). The properties of the new solutions obtained here for ω ≥ ω_c are analyzed, and self-similar solutions describing the t > 0 behavior for ω > ω_c are also derived.