填充型I型强爆炸中的离散自相似性
Discrete Self Similarity in Filled Type I Strong Explosions
AI总结：
本文提出了非幂律密度分布下强爆炸问题的新解，研究了密度扰动对数周期性导致的离散自相似性，并通过数值积分验证，扩展了先前对II型解的研究。
摘要：
我们提出了非幂律密度分布下强爆炸问题的新解。由Sedov、Taylor和Von Neumann发展的未扰自相似解描述了传播进入冷气体的强牛顿激波，该气体密度分布按 $r^{-ω}$ 衰减，其中 $ω\le\frac{7-γ}{γ+1}$（填充型I型解），$γ$ 为气体的绝热指数。我们考虑的扰动是球对称的，且在半径上呈对数周期性。虽然未扰解是连续自相似的，但密度扰动的对数周期性导致了扰动的离散自相似性，即解在离散时间间隔上除缩放因子外重复自身。我们讨论了这些解，并通过对含时流体力学方程的数值积分进行了验证。这是对先前关于II型解研究的扩展，有助于阐明I型自相似解扰动的边界条件。
英文摘要：
We present new solutions to the strong explosion problem in a non power law density profi{}le. The unperturbed self similar solutions developed by Sedov, Taylor and Von Neumann describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-ω}$, where $ω\le\frac{7-γ}{γ+1}$ (filled type I solutions), and $γ$ is the adiabatic index of the gas. The perturbations we consider are spherically symmetric and log periodic with respect to the radius. While the unperturbed solutions are continuously self similar, the log periodicity of the density perturbations leads to a discrete self similarity of the perturbations, i.e., the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. This is an extension of a previous investigation on type II solutions and helps clarifying boundary conditions for perturbations to type I self similar solutions.
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