Self-similar solutions to the hydrodynamic equations describing adiabatic one-dimensional flows of an ideal gas are of interest for mainly two reasons. First, the nonlinear partial differential hydrodynamic equations are reduced for self-similar flows to ordinary differential equations, which greatly simplifies the mathematical problem of solving the equations and in certain cases allows one to find analytic solutions. Second, self-similar solutions often describe the limiting behavior approached asymptotically by flows which take place over a characteristic scale, R, which diverges or tends to zero (see Sedov 1959;Zel'dovich & Raizer 1967;Barenblatt 1996, for reviews).
It is reasonable to assume that in the limit R → ∞(0) the flow becomes independent of any characteristic length scale. Using dimensional arguments, it is possible to show that in this case the flow fields must be of the self-similar form (Zel’dovich & Raizer 1967;Waxman & Shvarts 2010) u(r, t) = ṘξU (ξ), c(r, t) = ṘξC(ξ), ρ(r, t) = BR ǫ G(ξ),
(1) where u, c, and ρ are the fluid velocity, sound speed, and density, respectively (the pressure is given by p = ρc 2 /γ), and Ṙ = AR δ , ξ(r, t) = r/R(t)
(for a somewhat different approach to self-similarity, based on Lie group methods, see Coggeshall & Axford 1986;Coggeshall 1991;Coggeshall & Meyer-ter-Vehn 1992). For a self-similar solution of the form given in Equations ( 1) and (2), the hydrodynamic equations, Equations (A1), are replaced with a single ordinary differential equation, Equation (A2),
and one quadrature, Equation (A3),
∆, ∆ 1 , and ∆ 2 are given by Equations (A6). As illustrated in Section 2 (see also Guderley 1942;Meyer-ter-Vehn & Schalk 1982;Waxman & Shvarts 1993), many of the properties of self-similar flows may be inferred by analyzing the contours in the (U, C)-plane determined by Equation (A2).
In this paper, we revisit the “strong explosion problem”, which is one of the most familiar problems where asymptotic self-similarity is encountered. Consider the blast wave produced by the deposition of energy E within a region of characteristic size d at the center of an initially cold (p = 0 at r > d) gas sphere with initial density ρ 0 = Kr -ω (at r > d). As the shock radius R diverges, we expect the flow to approach a self-similar solution of the form given by Equations ( 1) and (2). The asymptotic flow is described by the Sedov-von Neumann-Taylor (ST) solutions (Sedov 1946;von Neumann 1947;Taylor 1950) for ω < 3, and by the solutions derived by Waxman & Shvrats (WS; Waxman & Shvarts 1993, 2010) for ω > 3. The ST solutions describe decelerating shocks (δ = (ω -3)/2 < 0) and are of the “first-type”, where the similarity exponents, δ and ǫ, are determined by dimensional considerations. The WS solutions describe accelerating shocks (δ > 0) and are of the “second-type”, where the similarity exponents are determined by the condition that the solutions must pass through a singular point of Equation (A2).
We revisit the strong explosion problem for several reasons. First, there exists a “gap” in the (γ, ω)-plane, 3 ≤ ω ≤ ω g (γ), where neither the ST nor the WS solutions describe the asymptotic flow (ω g is increasing with γ, ω g = 3 for γ = 1 and ω g ≃ 3.26 for adiabatic index γ = 5/3; Waxman & Shvarts 1993). Our first goal is to close this “gap”. We argue in Section 2 that second-type solutions should not be required in general to include a sonic point, and that it is sufficient to require the existence of a characteristic line r c (t), such that the energy in the region r c (t) < r < R approaches a constant as R → ∞. We show that the two requirements coincide for ω > ω g and that the latter requirement identifies δ = 0 solutions as the asymptotic solutions for 3 ≤ ω ≤ ω g . This result is in agreement with that of Gruzinov (2003), who suggested based on heuristic arguments that the R ∝ t solutions are the correct asymptotic solutions in the gap. As we explain in some detail at the end of Section 2.2, the validity of the heuristic arguments is not obvious. We use a different reasoning, based on an extension of the analysis of Waxman & Shvarts (1993).
In Section 3.1, we compare the asymptotic, R/d ≫ 1, behavior of numerical solutions of the hydrodynamic equations, Equations (A1), to that expected based on the δ = 0 self-similar solutions. We show that the convergence to self-similarity is very slow for ω ∼ 3. Hence, it is difficult to check using numerical solutions whether the flow indeed approaches a δ = 0 self-similar behavior as R → ∞. We show in Section 3.2 that in this case the flow may be described by a modified self-similar solution, d ln Ṙ/d ln R = δ with slowly varying δ(R), η ≡ dδ/d ln R ≪ 1, and spatial profiles given by a sum of the self-similar solution corresponding to the instantaneous value of δ and a self-similar correction linear in η. The modified self-similar solutions provide an excellent approximation to numerical solutions obtained for ω ∼ 3 at large R, wi
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