28E29P
自己相似球収縮
自己相似球収縮衝撃波における球収縮衝撃波における流体不安定
衝撃波における流体不安定性
流体不安定性の時間発展
時間発展

Instability analysis of spherical converging shock wave
小橋 龍知, 村上匡且
Tatsushi Kobashi, Masakatsu Murakami
大阪大学 レーザーエネルギー研究センター 〒565-0871 吹田市山田丘2-6
Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan[1] G. Guderley, Luftfahrforschung 19, 302 (1942)[2] Zeldvich, Ya. B., & Raizer, Yu. P. 1960, Physics ofShock Waves and High Temperature Hydrodynamic
Phenomena(New York: Academic)[3] M. Murakami, Nagatomo H.,Nucl. Instrum. Methods
Phys. Plasmas, 7 (2000) 2978

1300.5

γ=7/520

γ=5/3
γ=7/50I m a g i n a r y p a r t o f g r o w th r a t e I m (σ )where V0, G0, and Z0, denote the dimensionlessvelocity, density, and squared sound speed,
respectively, with being the initial density.
The perturbation quantities are postulated to beexpanded in terms of spherical harmonics Ylm(θ,ψ).
We then introduce such a 1-st order variable as thetotal perturbation amplitude of the shock surface isgiven by R(t)[1+Σ(-t/τ)ση1lYlm] with normalized
lth–mode amplitude η1l, where is unkown growthrate. Moreover we formulate the other variables
such as the radial velocity v=(r/t)[V0+Σ(-t/τ)σV1lYlm],the density=[G0+Σ(-t/τ)σG1lYlm], and thesquared sound speed cs2=(r/t)2[Z0+Σ(-t/τ)σZ1lYlm].
The σ is to be found as an eigenvalue by solving thefirst-order system, derived by linearizing the one
dimensional hydrodynamic system. Meanwhile thetransverse velocity v ⊥ is not used below in its
explicit form. Instead, its divergence is used as atractable form, r ・ v⊥=(r/t)Σ(-t/τ)σD1lYlm, where
denote the transverse divergence operator withrespect to the polar/azimuthal angles (θ,ψ).
Then the one dimensional hydrodynamic system,
the conservation of mass, momentum, and entropy,
is reduced to the perturbed system, which isnumerically integrated. The integration isterminated at the singular point, such all theintegrated corves for the perturbed quantities
smoothly pass the singular point, = , asobserved for the background quantities. The fluid
becomevelocity relative to the line with =sonic,whereis a specific constant of the system.
Only an appropriate complex value σ of satisfies
this condition as the eigenvalue.
A novel compression scheme is proposed, inwhich hollow targets with specifically curved
structures initially filled with uniform matter, aredriven by converging shock wave. A linear stability
analysis for a spherical geometry reveals a newdispersion relation with cut-off mode numbers as afunction of the specific heat ratio, above which
eigenmode perturbations are smeared.
We have presented the rigorous linear
perturbation theory, for spherical converging shock.
Concerning the stability, it has been revealed that a
cut-off mode number depending on γexists, overwhich eigenmode perturbations diminish.

R e a l p a r t o f g r o w t h r a t e R e ( -σ )The shock wave is a most basic and importanthydrodynamic phenomenon in many different
branches of high energy density physics. In
converging geometries such as cylinders andspheres, a shock wave is cumulatively strengthened
towards the center and exhibits asymptotically
self-similar behavior. The self-similar solution of aspherical converging shock wave was first reported
by Guderley [1]Following Gurderley[1] and using the notationsof Ref.[2], we introduce the self-similar coordinate,
ξ=r/R=r/A|t|α, and the self-similar fluid variables,,=,and,==Unstable-0.5 010

Stableγ=5/3

10

Cut-off

101
Spherical harmonic mode l102

0100

101
Spherical harmonic mode l102

Fig. 1: Growth rate of perturbations in a sphericalconverging shock: (a) Re(-σ) (b) Im(σ)