Неустойчивость Рэлея - Тейлора

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Развитие нестабильности Рэлея — Тейлора.

Неустойчивость Рэлея — Тейлора — одна из характерных форм неустойчивости, рассматриваемая в гидродинамике и математической теории устойчивости. Название неустойчивости дано в честь Лорда Рэлея и Дж. И. Тейлора).

Неустойчивость Рэлея — Тейлора, наблюдаемая в Крабовидной туманности

Нестойчивость Рэлея — Тейлора природе[править]

Неустойчивость Рэлея — Тейлора возникает, например, между двумя контактирующими потоками жидких сплошных сред, когда поверхность раздела жидкостей меняется под действием движения потока. Примером такой неустойчивости может служить неустойчивость струи воды, попадающей в масло, или легко моделируемое движение капли молока в сосуд с водой, либо капли сахарного сиропа, варенья, упавшей в чай.

Математика явления[править]

Основным параметром, определяющим скорость развития этой нестабильности является число Атвуда.

Внешние ссылки[править]

}}</ref> The upward-moving, lighter material is shaped like mushroom caps.[1][2] 

This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics. RT fingers are especially obvious in the Crab Nebula, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago.[3]

Note that the RT instability is not to be confused with the Plateau-Rayleigh instability (also known as Rayleigh instability) of a liquid jet. This instability, sometimes called the hosepipe (or firehose) instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same volume but lower surface area.

Linear stability analysis[править]

Файл:Rti base.png
Base state of the Rayleigh–Taylor instability. Gravity points downwards.

The inviscid two-dimensional Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the exceptionally simple nature of the base state.[4] This is the equilibrium state that exists before any perturbation is added to the system, and is described by the mean velocity field \(U(x,z)=W(x,z)=0,\,\) where the gravitational field is \(\textbf{g}=-g\hat{\textbf{z}}.\,\) An interface at \(z=0\,\) separates the fluids of densities \(\rho_G\,\) in the upper region, and \(\rho_L\,\) in the lower region. In this section it is shown that when the heavy fluid sits on top, the growth of a small perturbation at the interface is exponential, and takes place at the rate[5] $$\exp(\gamma\,t)\;, \qquad\text{with}\quad \gamma={\sqrt{\mathcal{A}g\alpha}} \quad\text{and}\quad \mathcal{A}=\frac{\rho_{\text{heavy}}-\rho_{\text{light}}}{\rho_{\text{heavy}}+\rho_{\text{light}}},\,$$

where \(\gamma\,\) is the temporal growth rate, \(\alpha\,\) is the spatial wavenumber and \(\mathcal{A}\,\) is the Atwood number.

Hydrodynamics simulation of a single "finger" of the Rayleigh–Taylor instability[7] Note the formation of Kelvin–Helmholtz instabilities, in the second and later snapshots shown (starting initially around the level \(y=0\)), as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.

The time evolution of the free interface elevation \(z = \eta(x,t),\,\) initially at \(\eta(x,0)=\Re\left\{B\,\exp\left(i\alpha x\right)\right\},\,\) is given by: $$\eta=\Re\left\{B\,\exp\left(\sqrt{\mathcal{A}g\alpha}\,t\right)\exp\left(i\alpha x\right)\right\}\,$$

which grows exponentially in time. Here B is the amplitude of the initial perturbation, and \(\Re\left\{\cdot\right\}\,\) denotes the real part of the complex valued expression between brackets.

In general, the condition for linear instability is that the imaginary part of the "wave speed" c be positive. Finally, restoring the surface tension makes c2 less negative and is therefore stabilizing. Indeed, there is a range of short waves for which the surface tension stabilizes the system and prevents the instability forming.

Late-time behaviour[править]

The analysis of the previous section breaks down when the amplitude of the perturbation is large. The growth then becomes non-linear as the spikes and bubbles of the instability tangle and roll up into vortices. Then, as in the figure, numerical simulation of the full problem is required to describe the system.

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  1. Шаблон:Cite arxiv
  2. Stellar Astrophysics. — CRC Press, 1992. — С. 249–302. — ISBN 0750302003>. See page 274.
  3. Hester, J. Jeff (2008). "The Crab Nebula: an Astrophysical Chimera". Annual Review of Astronomy and Astrophysics 46: 127–155. DOI:10.1146/annurev.astro.45.051806.110608.
  4. а б Drazin (2002) pp. 48–52.
  5. Ошибка цитирования Неверный тег <ref>; для сносок Drazin_50_51 не указан текст
  6. A similar derivation appears in Chandrasekhar (1981), §92, pp. 433–435.
  7. Li, Shengtai and Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations". Los Alamos National Laboratory. Retrieved 2006-09-05.