Asymptotic self-similar solution for a finite-source spherical blast wave in power-law density media

doi:10.1007/s10483-025-3288-7

Abstract

Abstract:

This study generalizes the classical Taylor-Sedov framework to analyze finite-source spherical blast waves propagating through both uniform and power-law density media. Previous analyses have predominantly focused on the effects of varying initial conditions on blast dynamics. In contrast, this study investigates the primary shock wave evolution within different ambient gases, demonstrating the critical dependence on the initial density ratio between the blast sphere and the ambient medium, as well as the ambient density profile. We derive new scaling laws based on the density ratio, which accurately predict the dimensionless main shock distance. Furthermore, we systematically examine, for the first time, the conditions for uniform volume expansion, uniform surface area growth, and uniform shock wave propagation in power-law density media, revealing a key scaling relation associated with the power-law exponent. Numerical simulations validate these novel theoretical predictions, demonstrating excellent agreement with the normalized solutions. These findings provide new insights into blast wave dynamics in inhomogeneous media and have implications for astrophysical and laboratory plasma environments.

Key words: finite source sphere, main shock, power-law density

2010 MSC Number:

O381

Qihang MA, Bofu WANG, Quan ZHOU. Asymptotic self-similar solution for a finite-source spherical blast wave in power-law density media. Applied Mathematics and Mechanics (English Edition), 2025, 46(9): 1771-1786.

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Figures/Tables 12

Fig. 1

Table 1

Summary of characteristic scales and dimensionless variables in a finite-source spherical blast wave"

Type	Length scale	Velocity scale	Time scale	$x^$	$u^$	$t^$
I	$R s$	$c 0$	$R s / c 0$	$x / R s$	$u / c 0$	$t c 0 / R s$
II	$L$	$c 0$	$L / c 0$	$x / L$	$u / c 0$	$t c 0 / L$
III	$R s$	$c 4$	$R s / c 4$	$x / R s$	$u / c 4$	$t c 4 / R s$
IV	$L$	$c 4$	$L / c 4$	$x / L$	$u / c 4$	$t c 4 / L$

Table 1

Fig. 2

Table 2

Physical parameters of the gas phase for the cases simulated"

Case	$ρ 4 / (kg ⋅ m − 3)$	$p 4$ /Pa	$ρ 0 / (kg ⋅ m − 3)$	$p 0$ /Pa	$ρ 4 / ρ 0$	$p 4 / p 0$	$c 4 / c 0$
1	1.222	$1.222 × 107$	1.222	$1.010 × 105$	1	121	11
2	1.222	$1.222 × 107$	1.222	$1.509 × 105$	1	81	9
3	1.222	$1.222 × 107$	1.222	$2.494 × 105$	1	49	7
4	1.222	$1.222 × 107$	0.122	$1.010 × 104$	10	1 210	11
5	1.222	$1.222 × 107$	0.122	$1.697 × 104$	10	720	8.5
6	1.222	$1.222 × 107$	0.122	$3.406 × 104$	10	358.8	6
7	1.222	$1.222 × 107$	0.012	$1.010 × 103$	100	12 100	11
8	1.222	$1.222 × 107$	0.012	$2.062 × 103$	100	5 927.3	7.7
9	1.222	$1.222 × 107$	0.012	$6.314 × 103$	100	1 935.4	4.4

Table 2

Fig. 3

Table 3

Numerical results of scaling exponents in Cases 1–9 (x^∼t^K)"

Case	1	2	3	4	5	6	7	8	9
$K$	0.406 7	0.402 5	0.408 3	0.407 1	0.406 9	0.408 2	0.404 1	0.408 4	0.407 2

Table 3

Fig. 4

Table 4

Physical parameters of the gas phase for the cases simulated"

Case	$ρ 4 / (kg ⋅ m − 3)$	$p 4$ /Pa	$ρ 0 / (kg ⋅ m − 3)$	$p 0$ /Pa	$ρ 4 / ρ 0$	$p 4 / p 0$	$c 4 / c 0$
a	1.222	$1.222 × 107$	1.222	$2.494 × 105$	1	49	7
b	1.222	$1.222 × 107$	0.244	$6.791 × 104$	5	180	6
c	1.222	$1.222 × 107$	0.122	$1.697 × 104$	10	720	8.5
d	1.222	$1.222 × 107$	0.049	$7.640 × 103$	25	1 600	8
e	1.222	$1.222 × 107$	0.024	$9.779 × 103$	50	1 250	5
f	1.222	$1.222 × 107$	0.016	$3.858 × 103$	75	3 167.8	6.5
g	1.222	$1.222 × 107$	0.012	$2.062 × 103$	100	5 927.3	7.7

Table 4

Fig. 5

Table 5

Numerical results of scaling exponent with different initial density ratios (x^∼(ρ4ρ0)K)"

$t / s$	$t^$	$K$
$2 × 10 − 5$	2.734	0.192 1
$3 × 10 − 5$	4.101	0.192 5
$4 × 10 − 5$	5.468	0.192 2

Table 5

Fig. 6

Table 6

Numerical results of scaling exponent of power-law density (x^∼t^K)"

$α$	1	$− 1$	$− 3$
$K$	0.341 1	0.513 6	1.026 8

Table 6

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