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The Rayleigh–Taylor instability, or RT instability (after
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
and G. I. Taylor), is an
instability In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
of an interface between two
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s of different
densities Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector quant ...
, mushroom clouds like those from
volcanic eruption Several types of volcanic eruptions—during which lava, tephra (ash, lapilli, volcanic bombs and volcanic blocks), and assorted gases are expelled from a volcanic vent or fissure—have been distinguished by volcanologists. These are oft ...
s and atmospheric
nuclear explosion A nuclear explosion is an explosion that occurs as a result of the rapid release of energy from a high-speed nuclear reaction. The driving reaction may be nuclear fission or nuclear fusion or a multi-stage cascading combination of the two, ...
s,
supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or whe ...
explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion. 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 Miscibility () is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration), forming a homogeneous mixture (a solution). The term is most often applied to liquids but also appli ...
fluid, the denser fluid 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 In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
, as the denser 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 denser 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 growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
s, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
equations are required to describe fluid motions. 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 dome A salt dome is a type of structural dome formed when salt (or other evaporite minerals) intrudes into overlying rocks in a process known as diapirism. Salt domes can have unique surface and subsurface structures, and they can be discovered usi ...
s to weather inversions, but also in
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
and
electrohydrodynamics Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. It is the study of the motions of ionized particles or molecules and their interactions with ...
. For example, RT instability structure is evident in the
Crab Nebula The Crab Nebula (catalogue designations M1, NGC 1952, Taurus A) is a supernova remnant and pulsar wind nebula in the constellation of Taurus. The common name comes from William Parsons, 3rd Earl of Rosse, who observed the object in 1842 u ...
, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the
supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or whe ...
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 resembles magnetically modulated RT instabilities.. 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 total volume but higher 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.


Stages of development and eventual evolution into turbulent mixing

The evolution of the RTI follows four main stages. In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage. Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large.


Linear stability analysis

The
inviscid The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the in ...
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,\, where the
gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
field is \textbf=-g\hat.\, An interface at z=0\, separates the fluids of
densities Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
\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 Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
, and takes place at the rate :\exp(\gamma\,t)\;, \qquad\text\quad \gamma= \quad\text\quad \mathcal=\frac,\, where \gamma\, is the temporal growth rate, \alpha\, is the spatial
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
and \mathcal\, is the Atwood number. The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, (u'(x,z,t),w'(x,z,t)).\, Because the fluid is assumed incompressible, this velocity field has the streamfunction representation :\textbf'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\, where the subscripts indicate
partial derivatives In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence \nabla\times\textbf'=0\,. In the streamfunction representation, \nabla^2\psi=0.\, Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the ansatz :\psi\left(x,z,t\right)=e^\Psi\left(z\right),\, where \alpha\, is a spatial wavenumber. Thus, the problem reduces to solving the equation :\left(D^2-\alpha^2\right)\Psi_j=0,\,\,\,\ D=\frac,\,\,\,\ 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\leq z<\infty\,. 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\, should satisfy a no-flux condition, so that fluid does not leak out at the boundaries z=\pm\infty.\, Thus, w_L'=0\, on z=-\infty\,, and w_G'=0\, on z=\infty\,. In terms of the streamfunction, this is :\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=\eta\left(x,t\right)\,. ''Continuity of vertical velocity:'' At z=\eta, the vertical velocities match, w'_L=w'_G\,. Using the stream function representation, this gives :\Psi_L\left(\eta\right)=\Psi_G\left(\eta\right).\, Expanding about z=0\, gives :\Psi_L\left(0\right)=\Psi_G\left(0\right)+\text,\, where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition. ''The free-surface condition:'' At the free surface z=\eta\left(x,t\right)\,, the kinematic condition holds: :\frac+u'\frac=w'\left(\eta\right).\, Linearizing, this is simply :\frac=w'\left(0\right),\, where the velocity w'\left(\eta\right)\, is linearized on to the surface z=0\,. Using the normal-mode and streamfunction representations, this condition is c \eta=\Psi\,, the second interfacial condition. ''Pressure relation across the interface:'' For the case with
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
, the pressure difference over the interface at z=\eta is given by the Young–Laplace equation: :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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
of the interface, which in a linear approximation is :\kappa=\nabla^2\eta=\eta_.\, Thus, :p_G\left(z=\eta\right)-p_L\left(z=\eta\right)=\sigma\eta_.\, However, this condition refers to the total pressure (base+perturbed), thus :\left _G\left(\eta\right)+p'_G\left(0\right)\right\left _L\left(\eta\right)+p'_L\left(0\right)\right\sigma\eta_.\, (As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using hydrostatic balance, in the form :P_L=-\rho_L g z+p_0,\qquad P_G=-\rho_G gz +p_0,\, this becomes :p'_G-p'_L=g\eta\left(\rho_G-\rho_L\right)+\sigma\eta_,\qquad\textz=0.\, The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations, : \frac = - \frac\frac\, with i=L,G,\, to yield :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 together, :c\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\eta\left(\rho_G-\rho_L\right)+\sigma\eta_.\, Substituting the second interfacial condition c\eta=\Psi\, and using the normal-mode representation, this relation becomes :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 \Psi\, (only its derivatives) because \Psi_L=\Psi_G\, at z=0.\, ; Solution Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation \left(D^2-\alpha^2\right)\Psi_i=0,\, with the boundary conditions \Psi\left(\pm\infty\right)\, has the solution :\Psi_L=A_L e^,\qquad \Psi_G = A_G e^.\, The first interfacial condition states that \Psi_L=\Psi_G\, at z=0\,, which forces A_L=A_G=A.\, The third interfacial condition states that :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 :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=\frac\frac+\frac.\, To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then, :c^2=\frac\frac,\qquad \sigma=0,\, and clearly * If \rho_G<\rho_L\,, c^2>0\, and ''c'' is real. This happens when the lighter fluid sits on top; * If \rho_G>\rho_L\,, c^2<0\, and ''c'' is purely imaginary. This happens when the heavier fluid sits on top. Now, when the heavier fluid sits on top, c^2<0\,, and :c=\pm i \sqrt,\qquad \mathcal=\frac,\, where \mathcal\, is the Atwood number. By taking the positive solution, we see that the solution has the form :\Psi\left(x,z,t\right)=Ae^\exp\left \alpha\left(x-ct\right)\rightA\exp\left(\alpha\sqrtt\right)\exp\left(i\alpha x-\alpha, z, \right)\, and this is associated to the interface position ''η'' by: c\eta=\Psi.\, Now define B=A/c.\, The time evolution of the free interface elevation z = \eta(x,t),\, initially at \eta(x,0)=\Re\left\,\, is given by: :\eta=\Re\left\\, which grows exponentially in time. Here ''B'' is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the initial perturbation, and \Re\left\\, 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'' is 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. When the two layers of the fluid are allowed to have a relative velocity, the instability is generalized to the Kelvin–Helmholtz–Rayleigh–Taylor instability, which includes both the Kelvin–Helmholtz instability and the Rayleigh–Taylor instability as special cases. It was recently discovered that the fluid equations governing the linear dynamics of the system admit a parity-time symmetry, and the Kelvin–Helmholtz–Rayleigh–Taylor instability occurs when and only when the parity-time symmetry breaks spontaneously.


Vorticity explanation

The RT instability can be seen as the result of baroclinic
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
created by the misalignment of the pressure and density gradients at the perturbed interface, as described by the two-dimensional
inviscid The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the in ...
vorticity equation, \frac=\frac\nabla \rho \times \nabla p , where ω is vorticity, ρ density and ''p'' is the pressure. In this case the dominant pressure gradient is hydrostatic, resulting from the acceleration. When in the unstable configuration, for a particular harmonic component of the initial perturbation, the torque on the interface creates vorticity that will tend to increase the misalignment of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
vectors. This in turn creates additional vorticity, leading to further misalignment. This concept is depicted in the figure, where it is observed that the two counter-rotating vortices have velocity fields that sum at the peak and trough of the perturbed interface. In the stable configuration, the vorticity, and thus the induced velocity field, will be in a direction that decreases the misalignment and therefore stabilizes the system.


Late-time behaviour

The analysis in 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 Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
of the full problem is required to describe the system.


See also

* Saffman–Taylor instability *
Richtmyer–Meshkov instability The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which ...
* Kelvin–Helmholtz instability * Mushroom cloud * Plateau–Rayleigh instability * Salt fingering * Hydrodynamic stability * Kármán vortex street * Fluid thread breakup


Notes


References


Original research papers

* (Original paper is available at: https://www.irphe.fr/~clanet/otherpaperfile/articles/Rayleigh/rayleigh1883.pdf .) *


Other

* * xvii+238 pages. * 626 pages.


External links


Java demonstration of the RT instability in fluids



Experiments on Rayleigh–Taylor instability at the University of Arizona

plasma Rayleigh–Taylor instability experiment at California Institute of Technology
{{DEFAULTSORT:Rayleigh-Taylor instability Fluid dynamics Fluid dynamic instabilities Plasma instabilities