Rayleigh-Taylor instability and mixing

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Author: Dr. Andrew W. Cook, Lawrence Livermore National Laboratory, Livermore CA
Author: Dr. David Youngs, AWE plc, Aldermaston, UK

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.

Figure 1: The mixing region from a 3072x3072x3072 point simulation of Rayleigh-Taylor instability between fluids of nondimensional density 3 (red) and 1 (blue).

1 Examples
2 Historical development
3 Governing equations
4 Experiments
5 Simulations
6 References
7 Recommended reading
8 External links
9 See also

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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).

Figure 2: The Crab nebula is Rayleigh-Taylor unstable.

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 (Calvin 1998).

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).

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Historical development

Figure 3: Sir Geoffrey Ingram Taylor (7 March 1886 - 27 June 1975)

Figure 4: Lord Rayleigh (12 November 1842 - 30 June 1919)

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.

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Governing equations

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Conservation laws

RTI, arising at the interface between two miscible fluids, obeys the following equations of motion:

(1)

$\frac{\partial \rho Y_i}{\partial t} + {\nabla\cdot} (\rho Y_i \vec{u} + \vec{J}_i) = 0 \ ; \ (i=1,2)$

(2)

$\frac{\partial \rho \vec{u}}{\partial t} + {\nabla\cdot} [\rho \vec{u} \vec{u} + p \vec{\vec{\delta}} - \vec{\vec{\tau}} ] = \rho \vec{g}$

(3)

$\frac{\partial E}{\partial t} + {\nabla\cdot} [(E + p)\vec{u} - \vec{\vec{\tau}} \cdot \vec{u} + \vec{q}_c + \vec{q}_d] = \rho \vec{g} \cdot \vec{u}$

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. (1) reduces to

(4)

$\frac{\partial \rho}{\partial t} + \vec{u} \cdot \nabla \rho = \rho \nabla \cdot \left( \frac{D}{\rho} \nabla \rho \right)$

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.

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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

(5)

$\frac{d\eta}{dt} = (A g k)^{1/2} \eta$

where $g$ is a constant acceleration and

(6)

$A \equiv \frac{\rho_2 - \rho_1}{\rho_2 + \rho_1}$

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.,

(7)

$\eta(t) = \eta_o \exp[(A g k)^{1/2}t]$

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

(8)

$\eta(t) = \eta_o \exp\left\{\left[ ( A g k/\psi + \nu^2 k^4 )^{1/2} - (\nu+D)k^2 \right] t \right\}$

where $\nu$ is kinematic viscosity and $\psi$ is a function of $A$ , $k$ and the initial diffusion thickness of the interface (Duff 1962).

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Self-similarity

At late times, the flow may forget its initial conditions and enter a self-similar growth phase described by the following equation

(9)

$\frac{dh}{dt} = 2 (\alpha A g h)^{1/2}$

Here $h$ is the height of the mixing region and $\alpha$ is a dimensionless growth parameter. Eq. (9) 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. (9) is (taking only the positive root as physically realizable)

(10)

$h(t) = \alpha A g t^2 + 2(\alpha A g h_o)^{1/2} t + h_o$ .

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. (10); hence the mixing thickness becomes

(11)

$h(t) \approx \alpha A g t^2$ .

In order for Eq. (11) 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. (7) will compete with the quadratic growth of Eq. (11). The validity of Eq. (11) 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

(12)

$\lambda_m<<h<<L$ .

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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. (12). 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).

Figure 5: Photographs taken at 33, 53 and 79 ms of RTI in an accelerating tank. The density ratio is 8.5 to 1.(Youngs 1989).

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Simulations

Figure 6: A 2D slice from a 3D simulation of RTI with A=1/2.

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. (12), 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. (11).

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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

Calvin, W.H. (1998) The great climate flip-flop. The Atlantic Monthly, 281:47-64 http://williamcalvin.com/1990s/1998AtlanticClimate.htm

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

Halliday, A.N. (2004) Mixing, volatile loss and compositional change during impact-driven accretion of the earth. nature, 427:505-509

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

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External links

Wikipedia article

RTI/supernova article

Rayleigh-Taylor experiments at the University of Arizona

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Invited by:	Dr. Norman J. Zabusky, Weizmann Institute of Science, Rehovot, Israel; Rutgers University, USA
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Rayleigh-Taylor instability and mixing

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Contents

Examples

Historical development

Governing equations

Conservation laws

Linear stability theory

Self-similarity

Experiments

Simulations

References

Recommended reading

External links

See also

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