Rayleigh-Taylor instability and mixing
| Andrew W. Cook and David Youngs (2009), Scholarpedia, 4(2):6092. | doi:10.4249/scholarpedia.6092 | revision #91699 [link to/cite this article] |
Rayleigh-Taylor instability (RTI) is the interpenetration of materials that occurs whenever a light fluid pushes on a heavy fluid (Rayleigh 1883, Taylor 1950). It is a dynamic process by which the two fluids seek to reduce their combined potential energy. The ensuing turbulence and mixing has far-reaching consequences in many natural and man-made flows, ranging from supernovae to inertial confinement fusion (ICF) capsules.
Contents |
Examples
RTI occurs at very large scales; e.g., interstellar gas, pushed out of the galactic plane by magnetic fields and cosmic rays, becomes RT-unstable if driven beyond its natural scale height (Zweibel 1991). It also occurs at very small scales; e.g., snapping shrimp, which produce most of the ambient noise in subtropical shallow waters throughout the world, produce their sound through collapse of a cavitation bubble, generated by a water jet formed from rapid claw closure; the bubble is destroyed through a Rayleigh-Taylor instability (Versluis et al. 2000).
RTI has a profound effect on Earth's climate. Winds off the coasts of Greenland and Iceland drive evaporation of the ocean surface making the upper layers saltier and heavier. This generates descending plumes that drive the abyssal ocean circulation, which in turn acts as a heat pump transporting warm equatorial waters up north. Without ocean overturning from RTI, northern Europe would likely face drastic drops in temperature .
RTI plays a crucial role in all known forms of fusion, whether confinement be magnetic (Bateman 1979), inertial (Petrasso 1994) or gravitational (Burrows 2000). It is even thought to limit the driving pressure in the violent collapse of acoustically forced bubbles in sonoluminescence and "sonofusion" phenomena (Lin et al. 2002, Taleyarkhan et al. 2002). Regarding gravitational fusion, much work over recent years points to RTI as the dominant acceleration mechanism for thermonuclear flames in type Ia supernovae (Zingale et al. 2005, Schmidt 2006).
Historical development
In 1880, as part of an effort to better understand the formation of cirrus clouds, Lord Rayleigh first derived the interfacial motion that occurs when a heavy fluid is supported by a lighter one. He considered the idealized case of two incompressible immiscible fluids in a constant gravitational field. In 1950, Sir. G.I. Taylor recognized that Rayleigh's interfacial instability also occurs for accelerations other than gravity. The classical case involves constant acceleration; however, RTI also occurs for time-dependent accelerations whenever the acceleration is directed from the light to the heavy fluid. Since the 1950's numerous experiments and numerical simulations have been conducted for both constant and variable accelerations in an effort to better understand RTI scaling properties and mixing rates.
Governing equations
Conservation laws
RTI, arising at the interface between two miscible fluids, obeys the following equations of motion:
- <math mass>
\frac{\partial \rho Y_i}{\partial t} + {\nabla\cdot} (\rho Y_i \vec{u} + \vec{J}_i) = 0 \ ; \ (i=1,2)
</math>
- <math momentum>
\frac{\partial \rho \vec{u}}{\partial t} + {\nabla\cdot}
[\rho \vec{u} \vec{u} + p \vec{\vec{\delta}} - \vec{\vec{\tau}} ]
= \rho \vec{g}
</math>
- <math energy>
\frac{\partial E}{\partial t}
+ {\nabla\cdot} [(E + p)\vec{u} - \vec{\vec{\tau}} \cdot \vec{u} + \vec{q}_c + \vec{q}_d]
= \rho \vec{g} \cdot \vec{u}
</math> where \(\rho\) is density, \(Y_i\) is the mass fraction of species \(i\), \(\vec{u}\) is velocity, \(\vec{J}_i\) is a diffusional mass flux, \(p\) is pressure, \(\vec{\vec{\delta}}\) is the unit tensor, \(\vec{\vec{\tau}}\) is the viscous stress tensor, \(\vec{g}\) is gravity (or frame acceleration), \(E = \rho(e+\vec{u}\cdot\vec{u}/2)\) is total energy (with \(e\) being internal energy), \(\vec{q}_c\) is the conductive heat flux and \(\vec{q}_d\) is enthalpy diffusion (Bird et al. 1960). Closure of the above equations requires an equation of state for both fluids.
For incompressible fluids, Eq. (<ref>mass</ref>) reduces to
- <math density>
\frac{\partial \rho}{\partial t} + \vec{u} \cdot \nabla \rho
= \rho \nabla \cdot \left( \frac{D}{\rho} \nabla \rho \right)
</math> where \(D\) is diffusivity (Joseph 1990), and the pressure field satisfies a Poisson equation (Cook and Dimotakis 2001). Most of the theoretical work on RTI growth rates, at both early and late times, has focused on the incompressible case.
Linear stability theory
For two incompressible immiscible fluids with an interfacial perturbation of wavenumber \(k\) and amplitude \(\eta\), such that \(\eta<<\lambda\) (where \(\lambda=2\pi/k\) is wavelength), inviscid linear stability theory (Rayleigh 1883) predicts that \(\eta\) will satisfy the following ordinary differential equation
- <math LSTODE>
\frac{d\eta}{dt} = (A g k)^{1/2} \eta
</math> where \(g\) is a constant acceleration and
- <math Atwood>
A \equiv \frac{\rho_2 - \rho_1}{\rho_2 + \rho_1}
</math> is the Atwood number, with \(\rho_1\) and \(\rho_2\) being the densities of the light and heavy fluids, respectively (Chandrasekhar 1955). Thus, the initial growth rate is exponential, i.e.,
- <math LST>
\eta(t) = \eta_o \exp[(A g k)^{1/2}t]
</math> where \(\eta_o=\eta(0)\).
Viscosity and diffusivity inhibit high-wavenumber growth, producing a "most dangerous" mode, which outpaces all other modes. Diffusion stabilizes the flow above a critical wavenumber. Including viscous/diffusive effects, the growth rate derived from a linear stability analysis is
- <math Duff>
\eta(t) = \eta_o \exp\left\{\left[ ( A g k/\psi + \nu^2 k^4 )^{1/2} - (\nu+D)k^2 \right] t \right\}
</math> where \(\nu\) is kinematic viscosity and \(\psi\) is a function of \(A\), \(k\) and the initial diffusion thickness of the interface (Duff 1962).
Self-similarity
At late times, the flow may forget its initial conditions and enter a self-similar growth phase described by the following equation
- <math hsim>
\frac{dh}{dt} = 2 (\alpha A g h)^{1/2}
</math> Here \(h\) is the height of the mixing region and \(\alpha\) is a dimensionless growth parameter. Eq. (<ref>hsim</ref>) can be derived via a self-similarity assumption (Ristorcelli and Clark 2004) or from an energy argument (Cook et al. 2004). For constant \(\alpha\), \(A\) and \(g\), the solution to Eq. (<ref>hsim</ref>) is (taking only the positive root as physically realizable)
- <math hsol>
h(t) = \alpha A g t^2 + 2(\alpha A g h_o)^{1/2} t + h_o
</math>. Taking \(t=0\) as moment in time when the flow first achieves self-similarity, \(h_o\) then corresponds to the thickness of the mixing region at that instant. At very late times, the first term dominates the right-hand side of Eq. (<ref>hsol</ref>); hence the mixing thickness becomes
- <math gt2>
h(t) \approx \alpha A g t^2
</math>. In order for Eq. (<ref>gt2</ref>) to become applicable, \(h\) must be much larger than the viscous and diffusion scales. Otherwise, the fluids viscosity and/or diffusivity will introduce additional length and time scales into the problem. Additionally, the initial perturbations must be band limited; otherwise, for long-wavelength perturbations, the exponential growth of Eq. (<ref>LST</ref>) will compete with the quadratic growth of Eq. (<ref>gt2</ref>). The validity of Eq. (<ref>gt2</ref>) thus requires \(h>>\lambda_{m}\), where \(\lambda_{m}\) is the maximum perturbation wavelength present in the initial conditions. Additionally, since the growth-controlling eddies in the flow grow horizontally as well as vertically, self-similar growth can only occur as long as the flow remains free from the influence of side boundaries, as well as top and bottom boundaries. Hence, unrestrained growth requires \(h<<L_s\) and \(h<<L_v\), where \(L_s\) is the distance between side (horizontal) boundaries and \(L_v\) is the distance between vertical (top and bottom) boundaries. Assuming \(L_v \sim L_s \sim L\), a necessary condition for self-similar growth is
- <math similarity>
\lambda_m<<h<<L </math>.
Experiments
Experimentalists have long sought to measure a "universal" value for \(\alpha\). This quest has been complicated by the fact that interfacial perturbations occur natually at all wavelengths up to the dimensions of any experimental apparatus. This makes it impossible to strictly satisfy the first inequality in Eq. (<ref>similarity</ref>). Additionally, limitations on diagnostics as well as boundary layer effects make it extremely difficult to satisfy the second inequality. Nevertheless, many experiments have demonstrated repeatability and have provided useful data for validating hydro codes.
Some strategies that experimentalists have employed are: (a) accelerating a tank of light-over-heavy fluids downward (Read 1984, Youngs 1989), (b) rotating a stably stratified tank of fluids by 180 degrees (Andrews and Spalding 1990) (c) withdrawing a plate separating a heavy fluid above a lighter one (Dalziel 1993), (d) flowing two fluids past a splitter plate (Ramaprabhu and Andrews 2004) and (e) magnetic levitation of paramagnetic or ferro fluids (Pacitto et al. 2000, Huang et al. 2007). Most experiments performed to date have returned values for \(\alpha\) in the range \(0.03<\alpha<0.07\) (Dimonte and Schneider 2000, Dimonte et al. 2004).
Simulations
Numerical simulations have a key advantage over experiments in that the amplitude of perturbations above any given wavelength can be set to exactly zero. This makes it possible to meet the first similarity requirement, i.e., \(h>>\lambda_m\). However, an enormous number of grid points is required in order to place the boundaries far enough apart to also meet the second similarity requirement, \(h<<L\). The largest simulations performed to date, which appear to satisfy Eq. (<ref>similarity</ref>), predict \(\alpha \approx 0.027\) (Cabot and Cook 2006).
The fact that RTI grows up from small scales presents a challenge to numerical computations employing the large-eddy simulation (LES) methodology. Subgrid-scale (SGS) models that assume a turbulent cascade of energy from large to small scales do not apply at early times when kinetic energy is generated at perturbation length-scales far below the grid scale. RTI also presents a challenge to computations employing the Reynolds-averaged Navier-Stokes (RANS) method since "turbulent diffusion" models, which naturally give \(t^{1/2}\) growth, must be coaxed to provide \(t^2\) growth, as required by Eq. (<ref>gt2</ref>).
References
- Andrews, M.J. and Spalding, D.B. (1990) A simple experiment to investigate two-dimensional mixing by Rayleigh-Taylor instability. Phys. Fluids A, 2:922-927
- Bateman, G. (1979) MHD Instabilities. MIT Press
- Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (1960) Transport phenomena, John Wiley & Sons, Inc. ISBN 0-471-07392-X
- Burrows, A. (2000) Supernova explosions in the universe. nature, 403:727-733
- Cabot, W.H. and Cook, A.W. (2006) Reynolds number effects on Rayleigh-Taylor instability with possible implications for type-Ia supernovae. nature physics, 2:562:568
- Chandrasekhar, S. (1955) The character of the equilibrium of an incompressible heavy viscous fluid of variable density. Proc. Camb. Phil. Soc., 51:161-178
- Cook, A.W., Cabot, W. and Miller, P.L. (2004) The mixing transition in Rayleigh-Taylor instability. J. Fluid Mech., 511:333-362
- Cook, A.W. and Dimotakis, P.E. (2001) Transition stages of Rayleigh-Taylor instability between miscible fluids. J. Fluid Mech., 443:69-99
- Dalziel, S.B. (1993) Rayleigh-Taylor instability: experiments with image analysis. Dynamics of Atmospheres and Oceans, 20:127-153
- Dimonte, G. and Schneider, M. (2000) Density ratio dependence of Rayleigh-Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids, 12:304-321
- Dimonte, G., Youngs, D.L., Dimits, A. et al. (2004) A comparative study of the turbulent Rayleigh-Taylor (RT) instability using high-resolution 3d numerical simulations: The alpha-group collaboration. Phys. Fluids, 16:1668-1693
- Duff, R.E., Harlow, F.H. and Hirt, C.W. (1962) Effects of diffusion on interface instability between gases. Phys. Fluids, 5:417-425
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- Huang, Z.B., De Luca, A., Atherton, T.J., Bird, M. and Rosenblatt, C. (2007) Rayleigh-Taylor instability experiments with precise and arbitrary control of the initial interface shape. Phys. Rev. Lett., 99:204502
- Joseph, D.D. (1990) Fluid dynamics of two miscible liquids with diffusion and gradient stresses. Eur. J. Mech. B/Fluids, 9:565-596
- Lin, H., Storey, B.D. and Szeri, A.J. (2002) Rayleigh-Taylor instability of violently collapsing bubbles. Physics of Fluids, 14:2925-2928
- Pacitto, G., Flament, C., Bacri, J.-C. and Widom, M. (2000) Rayleigh-Taylor instability with magnetic fluids: experiment and theory. Phys. Rev. E, 62:7941-7948
- Petrasso, R.D. (1994) Rayleigh's challenge endures. nature, 367:217-218
- Ramaprabhu, P. and Andrews, M.J. (2004) Experimental investigation of Rayleigh-Taylor mixing at small Atwood numbers. J. Fluid Mech., 502:233-271
- Rayleigh, Lord (1883) Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proceedings of the London Mathematical Society, 14:170-177 https://www.irphe.univ-mrs.fr/~clanet/otherpaperfile/articles/Rayleigh/rayleigh1883.pdf
- Read, K.I. (1984) Experimental investigation of turbulent mixing by Rayleigh-Taylor instability. Physica D, 12:45-58
- Ristorcelli, J.R. and Clark, T.T. (2004) Rayleigh-Taylor turbulence: Self-similar analysis and direct numerical simulations. J. Fluid Mech., 507:213-253
- Schmidt, W. (2006) From tea kettles to exploding stars. nature physics, 2:505-506
- Stevenson, D.J. (1995) Light from tungsten on core construction. nature, 378:763-764
- Taylor, G.I. (1950) The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proceedings of the Royal Society of London, A201:192-196
- Taleyarkhan, R.P. et al. (2002) Evidence for nuclear emissions during acoustic cavitation. Science, 295:1868-1873
- Versluis, M., Schmitz, B., von der Heydt, A. and Lohse, D. (2000) How snapping shrimp snap: Through cavitating bubbles. Science, 289:2114-2117
- Youngs, D.L. (1989) Modelling turbulent mixing by Rayleigh-Taylor instability. Physica D37: 264-269.
- Zingale, M., Woosley, S.E., Rendleman, C.A., Day, M.S. and Bell, J.B. (2005) Three-dimensional numerical simulations of Rayleigh-Taylor unstable flames in type Ia supernovae. Astrophysical Journal, 632:1021-1034
- Zweibel, E. (1991) Spinning a tangled web. nature, 352:755-756
Recommended reading
- Chandrasekhar, S. (1981) Hydrodynamic and hydromagnetic stability. Dover Publications, Inc., New York, ISBN 0-486-64071-X
- Drazin, P.G. and Reid, W.H. (1991) Hydrodynamic stability. Cambridge University Press, ISBN 0-521-28980-7
- Lamb, H. (1932) Hydrodynamics. Dover Publications, Inc., New York, 6th ed.
External links
Rayleigh-Taylor experiments at the University of Arizona
See also
Inertial confinement fusion, Richtmyer-Meshkov instability, Turbulence, Type Ia supernovae
