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Неустойчивость Рэлея - Тейлора

Icons-mini-icon 2main.png Основная статья: Теория катастроф
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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 природеПравить

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 явленияПравить

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

Внешние ссылкиПравить

}}</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 analysisПравить

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 , U(x,z)=W(x,z)=0,\, where the gravitational field is g = − g z ^ . \textbf{g}=-g\hat{\textbf{z}}.\, An interface at z = 0 z=0\, separates the fluids of densities ρ G \rho_G\, in the upper region, and ρ L \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 ⁡ ( γ t ) , with γ = A g α and A = ρ heavy − ρ light ρ heavy + ρ light , \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 γ \gamma\, is the temporal growth rate, α \alpha\, is the spatial wavenumber and A \mathcal{A}\, is the Atwood number.

Details of the linear stability analysis[6][4] 

The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) . (u'(x,z,t),w'(x,z,t)).\, Because the fluid is assumed incompressible, this velocity field has the streamfunction representation u ′ = ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) = ( ψ z , − ψ x ) , \textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,

where the subscripts indicate partial derivatives. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence ∇ × u ′ = 0 \nabla\times\textbf{u}'=0\, . In the streamfunction representation, ∇ 2 ψ = 0. \nabla^2\psi=0.\, Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz ψ ( x , z , t ) = e i α ( x − c t ) Ψ ( z ) , \psi\left(x,z,t\right)=e^{i\alpha\left(x-ct\right)}\Psi\left(z\right),\,

where α \alpha\, is a spatial wavenumber. Thus, the problem reduces to solving the equation ( D 2 − α 2 ) Ψ j = 0 ,   D = d d z ,   j = L , G . \left(D^2-\alpha^2\right)\Psi_j=0,\,\,\,\ D=\frac{d}{dz},\,\,\,\ j=L,G.\,

The domain of the problem is the following: the fluid with label 'L' lives in the region − ∞ -\infty , while the fluid with the label 'G' lives in the upper half-plane 0 ⩽ z 0\leq z . To specify the solution fully, it is necessary to fix conditions at the boundaries and interface. This determines the wave speed c, which in turn determines the stability properties of the system.

The first of these conditions is provided by details at the boundary. The perturbation velocities w i ′ w'_i\, should satisfy a no-flux condition, so that fluid does not leak out at the boundaries z = ± ∞ . z=\pm\infty.\, Thus, w L ′ = 0 w_L'=0\, on z = − ∞ z=-\infty\, , and w G ′ = 0 w_G'=0\, on z = ∞ z=\infty\, . In terms of the streamfunction, this is Ψ L ( − ∞ ) = 0 , Ψ G ( ∞ ) = 0. \Psi_L\left(-\infty\right)=0,\qquad \Psi_G\left(\infty\right)=0.\,

The other three conditions are provided by details at the interface z = η ( x , t ) z=\eta\left(x,t\right)\, .

Continuity of vertical velocity: At z = η z=\eta , the vertical velocities match, w L ′ = w G ′ w'_L=w'_G\, . Using the streamfunction representation, this gives Ψ L ( η ) = Ψ G ( η ) . \Psi_L\left(\eta\right)=\Psi_G\left(\eta\right).\,

Expanding about z = 0 z=0\, gives Ψ L ( 0 ) = Ψ G ( 0 ) + H.O.T. , \Psi_L\left(0\right)=\Psi_G\left(0\right)+\text{H.O.T.},\,

where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.

The free-surface condition: At the free surface z = η ( x , t ) z=\eta\left(x,t\right)\, , the kinematic condition holds: ∂ η ∂ t + u ′ ∂ η ∂ x = w ′ ( η ) . \frac{\partial\eta}{\partial t}+u'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,

Linearizing, this is simply ∂ η ∂ t = w ′ ( 0 ) , \frac{\partial\eta}{\partial t}=w'\left(0\right),\,

where the velocity w ′ ( η ) w'\left(\eta\right)\, is linearized on to the surface z = 0 z=0\, . Using the normal-mode and streamfunction representations, this condition is c η = Ψ c \eta=\Psi\, , the second interfacial condition.

Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at z = η z=\eta is given by the Young–Laplace equation: p G ( z = η ) − p L ( z = η ) = σ κ , p_G\left(z=\eta\right)-p_L\left(z=\eta\right)=\sigma\kappa,\,

where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is κ = ∇ 2 η = η x x . \kappa=\nabla^2\eta=\eta_{xx}.\,

Thus, p G ( z = η ) − p L ( z = η ) = σ η x x . p_G\left(z=\eta\right)-p_L\left(z=\eta\right)=\sigma\eta_{xx}.\,

However, this condition refers to the total pressure (base+perturbed), thus [ P G ( η ) + p G ′ ( 0 ) ] − [ P L ( η ) + p L ′ ( 0 ) ] = σ η x x . \left[P_G\left(\eta\right)+p'_G\left(0\right)\right]-\left[P_L\left(\eta\right)+p'_L\left(0\right)\right]=\sigma\eta_{xx}.\,

(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form P L = − ρ L g z + p 0 , P G = − ρ G g z + p 0 , P_L=-\rho_L g z+p_0,\qquad P_G=-\rho_G gz +p_0,\,

this becomes p G ′ − p L ′ = g η ( ρ G − ρ L ) + σ η x x , on  z = 0. p'_G-p'_L=g\eta\left(\rho_G-\rho_L\right)+\sigma\eta_{xx},\qquad\text{on }z=0.\,

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations, ∂ u i ′ ∂ t = − 1 ρ i ∂ p i ′ ∂ x \frac{\partial u_i'}{\partial t} = - \frac{1}{\rho_i}\frac{\partial p_i'}{\partial x}\, Шаблон:Pad with i = L , G , i=L,G,\,

to yield p i ′ = ρ i c D Ψ i , i = L , G . p_i'=\rho_i c D\Psi_i,\qquad i=L,G.\,

Putting this last equation and the jump condition on p G ′ − p L ′ p'_G-p'_L together, c ( ρ G D Ψ G − ρ L D Ψ L ) = g η ( ρ G − ρ L ) + σ η x x . c\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\eta\left(\rho_G-\rho_L\right)+\sigma\eta_{xx}.\,

Substituting the second interfacial condition c η = Ψ c\eta=\Psi\, and using the normal-mode representation, this relation becomes c 2 ( ρ G D Ψ G − ρ L D Ψ L ) = g Ψ ( ρ G − ρ L ) − σ α 2 Ψ , c^2\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\Psi\left(\rho_G-\rho_L\right)-\sigma\alpha^2\Psi,\,

where there is no need to label Ψ \Psi\, (only its derivatives) because Ψ L = Ψ G \Psi_L=\Psi_G\, at z = 0. z=0.\,


Solution

Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation ( D 2 − α 2 ) Ψ i = 0 , \left(D^2-\alpha^2\right)\Psi_i=0,\, with the boundary conditions Ψ ( ± ∞ ) \Psi\left(\pm\infty\right)\, has the solution Ψ L = A L e α z , Ψ G = A G e − α z . \Psi_L=A_L e^{\alpha z},\qquad \Psi_G = A_G e^{-\alpha z}.\,

The first interfacial condition states that Ψ L = Ψ G \Psi_L=\Psi_G\, at z = 0 z=0\, , which forces A L = A G = A . A_L=A_G=A.\, The third interfacial condition states that c 2 ( ρ G D Ψ G − ρ L D Ψ L ) = g Ψ ( ρ G − ρ L ) − σ α 2 Ψ . c^2\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\Psi\left(\rho_G-\rho_L\right)-\sigma\alpha^2\Psi.\,

Plugging the solution into this equation gives the relation A c 2 α ( − ρ G − ρ L ) = A g ( ρ G − ρ L ) − σ α 2 A . Ac^2\alpha\left(-\rho_G-\rho_L\right)=Ag\left(\rho_G-\rho_L\right)-\sigma\alpha^2A.\,

The A cancels from both sides and we are left with c 2 = g α ρ L − ρ G ρ L + ρ G + σ α ρ L + ρ G . c^2=\frac{g}{\alpha}\frac{\rho_L-\rho_G}{\rho_L+\rho_G}+\frac{\sigma\alpha}{\rho_L+\rho_G}.\,

To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then, c 2 = g α ρ L − ρ G ρ L + ρ G , σ = 0 , c^2=\frac{g}{\alpha}\frac{\rho_L-\rho_G}{\rho_L+\rho_G},\qquad \sigma=0,\,

and clearly

  • If ρ G \rho_G , c 2 > 0 c^2>0\, and c is real. This happens when the

lighter fluid sits on top;

  • If ρ G > ρ L \rho_G>\rho_L\, , c 2 c^2 and c is purely imaginary. This happens

when the heavier fluid sits on top.

Now, when the heavier fluid sits on top, c 2 c^2 , and c = ± i g A α , A = ρ G − ρ L ρ G + ρ L , c=\pm i \sqrt{\frac{g\mathcal{A}}{\alpha}},\qquad \mathcal{A}=\frac{\rho_G-\rho_L}{\rho_G+\rho_L},\,

where A \mathcal{A}\, is the Atwood number. By taking the positive solution, we see that the solution has the form Ψ ( x , z , t ) = A e − α | z | exp ⁡ [ i α ( x − c t ) ] = A exp ⁡ ( α g A ~ α t ) exp ⁡ ( i α x − α | z | ) \Psi\left(x,z,t\right)=Ae^{-\alpha|z|}\exp\left[i\alpha\left(x-ct\right)\right]=A\exp\left(\alpha\sqrt{\frac{g\tilde{\mathcal{A}}}{\alpha}}t\right)\exp\left(i\alpha x-\alpha|z|\right)\,

and this is associated to the interface position η by: c η = Ψ . c\eta=\Psi.\, Now define B = A / c . B=A/c.\,

 
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 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 = η ( x , t ) , z = \eta(x,t),\, initially at η ( x , 0 ) = ℜ { B exp ⁡ ( i α x ) } , \eta(x,0)=\Re\left\{B\,\exp\left(i\alpha x\right)\right\},\, is given by: η = ℜ { B exp ⁡ ( A g α t ) exp ⁡ ( i α x ) } \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 ℜ { ⋅ } \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 behaviourПравить

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. -->

  1. ↑ Шаблон:Cite arxiv
  2. ↑ Stellar Astrophysics. — CRC Press, 1992. — С. 249–302. — ISBN 0750302003о книге. 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. 
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