rayleigh–taylor instability - wikipedia, the free encyclopedia

RT fingers evident in the Crab Nebula

From Wikipedia, the free encyclopedia (Redirected from Rayleigh-Taylor instability)The RayleighTaylor instability, or RT instability (afterLord Rayleigh and G. I. Taylor), is an instability of aninterface between two fluids of different densities whichoccurs when the lighter fluid is pushing the heavier fluid.[1][2] Examples include supernova explosions in whichexpanding core gas is accelerated into denser shellgas,[3][4] instabilities in plasma fusion reactors,[5] and thecommon terrestrial example of a denser fluid such aswater suspended above a lighter fluid such as oil in theEarth's gravitational field.[2]

To model the last example, consider two completelyplane-parallel layers of immiscible fluid, the more denseon top of the less dense one and both subject to the Earth'sgravity. The equilibrium here is unstable to anyperturbations or disturbances of the interface: if a parcel ofheavier fluid is displaced downward with an equal volumeof lighter fluid displaced upwards, the potential energy ofthe configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release ofpotential energy, as the more dense material moves down under the (effective) gravitational field, and the lessdense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh.[2] The importantinsight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids areaccelerated, with the less dense fluid accelerating into the more dense fluid.[2] This occurs deep underwater onthe surface of an expanding bubble and in a nuclear explosion.[6]

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linearor "exponential" growth phase, eventually developing "plumes" flowing upwards (in the gravitationalbuoyancy sense) and "spikes" falling downwards. In general, the density disparity between the fluidsdetermines the structure of the subsequent non-linear RT instability flows (assuming other variables such assurface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum isdefined 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-likeplumes.[1]

This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but alsoin astrophysics and electrohydrodynamics. RT instability structure is also evident in the Crab Nebula, in whichthe expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from thesupernova explosion 1000 years ago.[7] The RT instability has also recently been discovered in the Sun's outeratmosphere, or solar corona, when a relatively dense solar prominence overlies a less dense plasma bubble.[8]This latter case is an exceptionally clear example of the magnetically modulated RT instability.[9][10]

Note that the RT instability is not to be confused with the Plateau-Rayleigh instability (also known asRayleigh 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 samevolume but lower surface area.Many people have witnessed the RT instability by looking at a lava lamp, although some might claim this is

RayleighTaylor instability - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability

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Base state of the RayleighTaylor instability. Gravity pointsdownwards.

more accurately described as an example of RayleighBnard convection due to the active heating of the fluidlayer at the bottom of the lamp.

1 Linear stability analysis2 Late-time behaviour3 See also4 Notes5 References

5.1 Original research papers5.2 Other

6 External links

The inviscid two-dimensional RayleighTaylor (RT) instability provides anexcellent springboard into themathematical study of stability because ofthe exceptionally simple nature of the basestate.[11] This is the equilibrium state thatexists before any perturbation is added tothe system, and is described by the meanvelocity field

where thegravitational field is Aninterface at separates the fluids ofdensities in the upper region, and in the lower region. In this section it isshown that when the heavy fluid sits ontop, the growth of a small perturbation atthe interface is exponential, and takes place at the rate[2]

where is the temporal growth rate, is the spatial wavenumber and is the Atwood number.

Details of the linear stability analysis[11] A similar derivation appears in,[9] 92, pp. 433435.The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,

Because the fluid is assumed incompressible, this velocity field has thestreamfunction representation


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where the subscripts indicate partial derivatives. Moreover, in an initially stationary incompressible fluid, there isno vorticity, and the fluid stays irrotational, hence . In the streamfunction representation,

Next, because of the translational invariance of the system in the x-direction, it is possible to make theansatz

where is a spatial wavenumber. Thus, the problem reduces to solving the equation

The domain of the problem is the following: the fluid with label 'L' lives in the region , while thefluid with the label 'G' lives in the upper half-plane . To specify the solution fully, it is necessary tofix conditions at the boundaries and interface. This determines the wave speed c, which in turn determines thestability properties of the system.The first of these conditions is provided by details at the boundary. The perturbation velocities should satisfy ano-flux condition, so that fluid does not leak out at the boundaries Thus, on , and

on . In terms of the streamfunction, this is

The other three conditions are provided by details at the interface .Continuity of vertical velocity: At , the vertical velocities match, . Using the streamfunctionrepresentation, this gives

Expanding about gives

where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.The free-surface condition: At the free surface , the kinematic condition holds:

Linearizing, this is simply

where the velocity is linearized on to the surface . Using the normal-mode and streamfunctionrepresentations, this condition is , the second interfacial condition.Pressure relation across the interface: For the case with surface tension, the pressure difference over the interfaceat is given by the YoungLaplace equation:

where is the surface tension and is the curvature of the interface, which in a linear approximation is


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Thus,

However, this condition refers to the total pressure (base+perturbed), thus

(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form

this becomes

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of thelinearised Euler equations for the perturbations,

with

to yield

Putting this last equation and the jump condition on together,

Substituting the second interfacial condition and using the normal-mode representation, this relationbecomes

where there is no need to label (only its derivatives) because at SolutionNow that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation

with the boundary conditions has the solution

The first interfacial condition states that at , which forces The thirdinterfacial condition states that

Plugging the solution into this equation gives the relation


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The time evolution of the free interface elevation initially at is given by:

which grows exponentially in time. Here B is the amplitude of the initial perturbation, and denotes thereal 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 arange of short waves for which the surface tension stabilizes the system and prevents the instability forming.

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

The A cancels from both sides and we are left with

To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then,

and clearlyIf , and c is real. This happens when the

lighter fluid sits on top;If , and c is purely imaginary. This happens

when the heavier fluid sits on top.Now, when the heavier fluid sits on top, , and

where is the Atwood number. By taking the positive solution, we see that the solution has the form

and this is associated to the interface position by: Now define


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Hydrodynamics simulation of a single "finger" of the RayleighTaylor instability[12] Note the formation of KelvinHelmholtzinstabilities, in the second and later snapshots shown (startinginitially around the level ), as well as the formation of a"mushroom cap" at a later stage in the third and fourth frame in thesequence.

RichtmyerMeshkov instabilityKelvinHelmholtz instabilityMushroom cloudPlateauRayleigh instabilitySalt fingeringHydrodynamic stabilityKrmn vortex streetFluid thread breakup

^ a b Sharp, D.H. (1984). "An Overviewof Rayleigh-Taylor Instability". PhysicaD 12: 318.Bibcode:1984PhyD...12....3S(http://adsabs.harvard.edu/abs/1984PhyD...12....3S).doi:10.1016/0167-2789(84)90510-4(http://dx.doi.org/10.1016%2F0167-2789%2884%2990510-4).

1.

^ a b c d e Drazin (2002) pp. 5051.2. ^ Wang, C.-Y. & Chevalier R. A. (2000). "Instabilities and Clumping in Type Ia Supernova Remnants".arXiv:astro-ph/0005105v1 (http://arxiv.org/abs/astro-ph/0005105v1).

3.

^ Hillebrandt, W.; Hflich, P. (1992). "Supernova 1987a in the Large Magellanic Cloud". In R. J. Tayler. StellarAstrophysics. CRC Press. pp. 249302. ISBN 0-7503-0200-3.. See page 274.

4.

^ Chen, H. B.; Hilko, B.; Panarella, E. (1994). "The RayleighTaylor instability in the spherical pinch". Journal ofFusion Energy 13 (4): 275280. Bibcode:1994JFuE...13..275C (http://adsabs.harvard.edu/abs/1994JFuE...13..275C). doi:10.1007/BF02215847 (http://dx.doi.org/10.1007%2FBF02215847).

5.

^ John Pritchett (1971). "EVALUATION OF VARIOUS THEORETICAL MODELS FOR UNDERWATEREXPLOSION" (http://www.dtic.mil/dtic/tr/fulltext/u2/737271.pdf). U.S. Government. p. 86. Retrieved October 9,2012.

6.

^ Hester, J. Jeff (2008). "The Crab Nebula: an Astrophysical Chimera". Annual Review of Astronomy andAstrophysics 46: 127155. Bibcode:2008ARA&A..46..127H (http://adsabs.harvard.edu/abs/2008ARA&A..46..127H). doi:10.1146/annurev.astro.45.051806.110608 (http://dx.doi.org/10.1146%2Fannurev.astro.45.051806.110608).

7.

^ Berger, T. E. et al.; Slater, Gregory; Hurlburt, Neal; Shine, Richard; Tarbell, Theodore; Title, Alan; Lites, BruceW.; Okamoto, Takenori J. et al. (2010). "Quiescent Prominence Dynamics Observed with the Hinode Solar OpticalTelescope. I. Turbulent Upflow Plumes". The Astrophysical Journal 716 (2): 12881307.Bibcode:2010ApJ...716.1288B (http://adsabs.harvard.edu/abs/2010ApJ...716.1288B). doi:10.1088/0004-637X/716/2/1288 (http://dx.doi.org/10.1088%2F0004-637X%2F716%2F2%2F1288).

8.

^ a b Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Instabilties. Dover. ISBN 0-486-64071-X.. SeeChap. X.

9.

^ Hillier, A. et al.; Berger, Thomas; Isobe, Hiroaki; Shibata, Kazunari. "Numerical Simulations of the Magnetic10.


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Wikimedia Commons hasmedia related to RayleighTaylor instability.

Rayleigh-Taylor Instability in the Kippenhahn-Schl{\"u}ter Prominence Model. I. Formation of Upflows". TheAstrophysical Journal 716: 120133. Bibcode:2012ApJ...746..120H (http://adsabs.harvard.edu/abs/2012ApJ...746..120H). doi:10.1088/0004-637X/746/2/120 (http://dx.doi.org/10.1088%2F0004-637X%2F746%2F2%2F120).^ a b Drazin (2002) pp. 4852.11. ^ Li, Shengtai and Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations" (http://math.lanl.gov/Research/Highlights/amrmhd.shtml). Los Alamos National Laboratory. Retrieved 2006-09-05.

12.

Original research papersRayleigh, Lord (John William Strutt) (1883). "Investigation of the character of the equilibrium of anincompressible heavy fluid of variable density". Proceedings of the London Mathematical Society 14:170177. doi:10.1112/plms/s1-14.1.170 (http://dx.doi.org/10.1112%2Fplms%2Fs1-14.1.170). (Originalpaper is available at: https://www.irphe.univ-mrs.fr/~clanet/otherpaperfile/articles/Rayleigh/rayleigh1883.pdf .)Taylor, Sir Geoffrey Ingram (1950). "The instability of liquid surfaces when accelerated in a directionperpendicular to their planes". Proceedings of the Royal Society of London. Series A, Mathematical andPhysical Sciences 201 (1065): 192196. Bibcode:1950RSPSA.201..192T (http://adsabs.harvard.edu/abs/1950RSPSA.201..192T). doi:10.1098/rspa.1950.0052 (http://dx.doi.org/10.1098%2Frspa.1950.0052).

OtherChandrasekhar, Subrahmanyan (1981). Hydrodynamic and Hydromagnetic Stability. DoverPublications. ISBN 978-0-486-64071-6.Drazin, P. G. (2002). Introduction to hydrodynamic stability. Cambridge University Press.ISBN 0-521-00965-0. xvii+238 pages.Drazin, P. G.; Reid, W. H. (2004). Hydrodynamic stability (2nd ed.). Cambridge: Cambridge UniversityPress. ISBN 0-521-52541-1. 626 pages.

Java demonstration of the RT instability in fluids(http://acg.media.mit.edu/people/fry/mixing/)Actual images and videos of RT fingers (http://www.enseeiht.fr/hmf/travaux/CD0001/travaux/optmfn/hi/01pa/hyb72/rt/rt.htm)Experiments on Rayleigh-Taylor experiments at the University of Arizona (http://fluidlab.arizona.edu/)plasma Rayleigh-Taylor instability experiment at California Institute of Technology(http://media.caltech.edu/press_releases/13496)

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Categories: Fluid dynamics Fluid dynamic instability

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rayleigh–taylor instability - wikipedia, the free encyclopedia

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