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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Rayleigh-Taylor instabilities ： ウィキペディア英語版 Rayleigh–Taylor instability The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.〔 Drazin (2002) pp. 50–51.〕 Examples include the behavior of water suspended above oil in the gravity of Earth,〔 mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions,〔http://gizmodo.com/why-nuclear-bombs-create-mushroom-clouds-1468107869〕 supernova explosions in which expanding core gas is accelerated into denser shell gas,〔. See page 274.〕 and instabilities in plasma fusion reactors. Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh.〔 The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the more dense fluid.〔 This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion. As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear or "exponential" growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the Atwood number, A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.〔 This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics. RT instability structure is also evident 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. The RT instability has also recently been discovered in the Sun's outer atmosphere, or solar corona, when a relatively dense solar prominence overlies a less dense plasma bubble. This latter case is an clear example of the magnetically modulated RT instability.〔. See Chap. X.〕 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. Many people have witnessed the RT instability by looking at a lava lamp, although some might claim this is more accurately described as an example of Rayleigh–Bénard convection due to the active heating of the fluid layer at the bottom of the lamp. ==Linear stability analysis== The inviscid two-dimensional Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state.〔Drazin (2002) pp. 48–52.〕 This is the equilibrium state that exists before any perturbation is added to the system, and is described by the mean velocity field $U(x,z)=W(x,z)=0,\backslash ,$ where the gravitational field is $\backslash textbf=-g\backslash hat\backslash quad\; \backslash gamma=\}\; \backslash quad\backslash text\backslash quad\; \backslash mathcal=\backslash frac\}\}\}\},\backslash ,$ where $\backslash gamma\backslash ,$ is the temporal growth rate, $\backslash alpha\backslash ,$ is the spatial wavenumber and $\backslash mathcal\backslash ,$ is the Atwood number. The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, $(u\text{'}(x,z,t),w\text{'}(x,z,t)).\backslash ,$ Because the fluid is assumed incompressible, this velocity field has the streamfunction representation :$\backslash textbf\text{'}=(u\text{'}(x,z,t),w\text{'}(x,z,t))=(\backslash psi\_z,-\backslash psi\_x),\backslash ,$ where the subscripts indicate partial derivatives. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence $\backslash nabla\backslash times\backslash textbf\text{'}=0\backslash ,$. In the streamfunction representation, $\backslash nabla^2\backslash psi=0.\backslash ,$ Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the ansatz :$\backslash psi\backslash left(x,z,t\backslash right)=e^\backslash Psi\backslash left(z\backslash right),\backslash ,$ where $\backslash alpha\backslash ,$ is a spatial wavenumber. Thus, the problem reduces to solving the equation :$\backslash left(D^2-\backslash alpha^2\backslash right)\backslash Psi\_j=0,\backslash ,\backslash ,\backslash ,\backslash \; D=\backslash frac,\backslash ,\backslash ,\backslash ,\backslash \; j=L,G.\backslash ,$ The domain of the problem is the following: the fluid with label 'L' lives in the region The first of these conditions is provided by details at the boundary. The perturbation velocities $w\text{'}\_i\backslash ,$ should satisfy a no-flux condition, so that fluid does not leak out at the boundaries $z=\backslash pm\backslash infty.\backslash ,$ Thus, $w\_L\text{'}=0\backslash ,$ on $z=-\backslash infty\backslash ,$, and $w\_G\text{'}=0\backslash ,$ on $z=\backslash infty\backslash ,$. In terms of the streamfunction, this is :$\backslash Psi\_L\backslash left(-\backslash infty\backslash right)=0,\backslash qquad\; \backslash Psi\_G\backslash left(\backslash infty\backslash right)=0.\backslash ,$ The other three conditions are provided by details at the interface $z=\backslash eta\backslash left(x,t\backslash right)\backslash ,$. ''Continuity of vertical velocity:'' At $z=\backslash eta$, the vertical velocities match, $w\text{'}\_L=w\text{'}\_G\backslash ,$. Using the streamfunction representation, this gives :$\backslash Psi\_L\backslash left(\backslash eta\backslash right)=\backslash Psi\_G\backslash left(\backslash eta\backslash right).\backslash ,$ Expanding about $z=0\backslash ,$ gives :$\backslash Psi\_L\backslash left(0\backslash right)=\backslash Psi\_G\backslash left(0\backslash right)+\backslash text,\backslash ,$ where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition. ''The free-surface condition:'' At the free surface $z=\backslash eta\backslash left(x,t\backslash right)\backslash ,$, the kinematic condition holds: :$\backslash frac+u\text{'}\backslash frac=w\text{'}\backslash left(\backslash eta\backslash right).\backslash ,$ Linearizing, this is simply :$\backslash frac=w\text{'}\backslash left(0\backslash right),\backslash ,$ where the velocity $w\text{'}\backslash left(\backslash eta\backslash right)\backslash ,$ is linearized on to the surface $z=0\backslash ,$. Using the normal-mode and streamfunction representations, this condition is $c\; \backslash eta=\backslash Psi\backslash ,$, the second interfacial condition. ''Pressure relation across the interface:'' For the case with surface tension, the pressure difference over the interface at $z=\backslash eta$ is given by the Young–Laplace equation: :$p\_G\backslash left(z=\backslash eta\backslash right)-p\_L\backslash left(z=\backslash eta\backslash right)=\backslash sigma\backslash kappa,\backslash ,$ where ''σ'' is the surface tension and ''κ'' is the curvature of the interface, which in a linear approximation is :$\backslash kappa=\backslash nabla^2\backslash eta=\backslash eta\_.\backslash ,$ Thus, :$p\_G\backslash left(z=\backslash eta\backslash right)-p\_L\backslash left(z=\backslash eta\backslash right)=\backslash sigma\backslash eta\_.\backslash ,$ However, this condition refers to the total pressure (base+perturbed), thus :$\backslash left()-\backslash left()=\backslash sigma\backslash eta\_.\backslash ,$ (As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using hydrostatic balance, in the form :$P\_L=-\backslash rho\_L\; g\; z+p\_0,\backslash qquad\; P\_G=-\backslash rho\_G\; gz\; +p\_0,\backslash ,$ this becomes :$p\text{'}\_G-p\text{'}\_L=g\backslash eta\backslash left(\backslash rho\_G-\backslash rho\_L\backslash right)+\backslash sigma\backslash eta\_,\backslash qquad\backslash textz=0.\backslash ,$ The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations, : $\backslash frac\; =\; -\; \backslash frac\backslash frac\backslash ,$ with $i=L,G,\backslash ,$ to yield :$p\_i\text{'}=\backslash rho\_i\; c\; D\backslash Psi\_i,\backslash qquad\; i=L,G.\backslash ,$ Putting this last equation and the jump condition on $p\text{'}\_G-p\text{'}\_L$ together, :$c\backslash left(\backslash rho\_G\; D\backslash Psi\_G-\backslash rho\_L\; D\backslash Psi\_L\backslash right)=g\backslash eta\backslash left(\backslash rho\_G-\backslash rho\_L\backslash right)+\backslash sigma\backslash eta\_.\backslash ,$ Substituting the second interfacial condition $c\backslash eta=\backslash Psi\backslash ,$ and using the normal-mode representation, this relation becomes :$c^2\backslash left(\backslash rho\_G\; D\backslash Psi\_G-\backslash rho\_L\; D\backslash Psi\_L\backslash right)=g\backslash Psi\backslash left(\backslash rho\_G-\backslash rho\_L\backslash right)-\backslash sigma\backslash alpha^2\backslash Psi,\backslash ,$ where there is no need to label $\backslash Psi\backslash ,$ (only its derivatives) because $\backslash Psi\_L=\backslash Psi\_G\backslash ,$ at $z=0.\backslash ,$ ; Solution Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation $\backslash left(D^2-\backslash alpha^2\backslash right)\backslash Psi\_i=0,\backslash ,$ with the boundary conditions $\backslash Psi\backslash left(\backslash pm\backslash infty\backslash right)\backslash ,$ has the solution :$\backslash Psi\_L=A\_L\; e^,\backslash qquad\; \backslash Psi\_G\; =\; A\_G\; e^.\backslash ,$ The first interfacial condition states that $\backslash Psi\_L=\backslash Psi\_G\backslash ,$ at $z=0\backslash ,$, which forces $A\_L=A\_G=A.\backslash ,$ The third interfacial condition states that :$c^2\backslash left(\backslash rho\_G\; D\backslash Psi\_G-\backslash rho\_L\; D\backslash Psi\_L\backslash right)=g\backslash Psi\backslash left(\backslash rho\_G-\backslash rho\_L\backslash right)-\backslash sigma\backslash alpha^2\backslash Psi.\backslash ,$ Plugging the solution into this equation gives the relation :$Ac^2\backslash alpha\backslash left(-\backslash rho\_G-\backslash rho\_L\backslash right)=Ag\backslash left(\backslash rho\_G-\backslash rho\_L\backslash right)-\backslash sigma\backslash alpha^2A.\backslash ,$ The ''A'' cancels from both sides and we are left with :$c^2=\backslash frac\backslash frac+\backslash frac.\backslash ,$ To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then, :$c^2=\backslash frac\backslash frac,\backslash qquad\; \backslash sigma=0,\backslash ,$ and clearly * If , $c^2>0\backslash ,$ and ''c'' is real. This happens when the lighter fluid sits on top; * If $\backslash rho\_G>\backslash rho\_L\backslash ,$, and ''c'' is purely imaginary. This happens when the heavier fluid sits on top. Now, when the heavier fluid sits on top, , and :$c=\backslash pm\; i\; \backslash sqrt\},\backslash qquad\; \backslash mathcal=\backslash frac,\backslash ,$ where $\backslash mathcal\backslash ,$ is the Atwood number. By taking the positive solution, we see that the solution has the form :$\backslash Psi\backslash left(x,z,t\backslash right)=Ae^\backslash exp\backslash left()=A\backslash exp\backslash left(\backslash alpha\backslash sqrt\}t\backslash right)\backslash exp\backslash left(i\backslash alpha$ x-\alpha|z|\right)\, and this is associated to the interface position ''η'' by: $c\backslash eta=\backslash Psi.\backslash ,$ Now define $B=A/c.\backslash ,$ The time evolution of the free interface elevation $z\; =\; \backslash eta(x,t),\backslash ,$ initially at $\backslash eta(x,0)=\backslash Re\backslash left\backslash ,\backslash ,$ is given by: :$\backslash eta=\backslash Re\backslash left\backslash \backslash ,t\backslash right)\backslash exp\backslash left(i\backslash alpha\; x\backslash right)\backslash right\backslash \}\backslash ,$ which grows exponentially in time. Here ''B'' is the amplitude of the initial perturbation, and $\backslash Re\backslash left\backslash \backslash ,$ 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 ''c''2 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rayleigh–Taylor instability」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.