# Rayleigh-Taylor instability and mixing

**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 whereby 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.

## 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). Similarly, RTI limits the driving pressure in the collapse of acoustically forced bubbles in sonoluminescence phenomena (Lin et al. 2002).

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

\[\tag{2} \frac{\partial \rho \vec{u}}{\partial t} + {\nabla\cdot} [\rho \vec{u} \vec{u} + p \vec{\vec{\delta}} - \vec{\vec{\tau}} ] = \rho \vec{g} \]

\[\tag{3} \frac{\partial E}{\partial t} + {\nabla\cdot} [(E + p)\vec{u} - \vec{\vec{\tau}} \cdot \vec{u} + \vec{q}_c + \vec{q}_d] = \rho \vec{g} \cdot \vec{u} \]

where \(\rho\) is density, \(Y_i\) is the mass fraction of species \(i\ ,\) \(\vec{u}\) is velocity, \(\vec{J}_i = -\rho D \nabla Y_i\) is a diffusional mass flux (where \(D\) is binary diffusivity), \(p\) 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, the Amagat-Leduc law of additive volumes states that \[\tag{4} \frac{1}{\rho} = \frac{Y_1}{\rho_1} + \frac{Y_2}{\rho_2} \]

where \(\rho_1\) and \(\rho_2\) are the constant densities of each fluid. Since \(Y_1 + Y_2 = 1\ ,\) the mixture density, \(\rho\ ,\) is uniquely described in terms of either \(Y_1\) or \(Y_2\ ;\) hence Eq. (1) reduces to \[\tag{5} \frac{\partial \rho}{\partial t} + \vec{u} \cdot \nabla \rho = \rho \nabla \cdot \left( \frac{D}{\rho} \nabla \rho \right) \]

(Joseph 1990). A sometimes-overlooked aspect of flows comprised of incompressible miscible fluids is that the velocity field is divergent; i.e., \[\tag{6} \nabla \cdot \vec{u} = - \nabla \cdot \frac{D}{\rho} \nabla \rho \]

Most of the theoretical work on RTI growth rates, at both early and late times, has focused on the incompressible case.

### Linear stability theory

The initial growth rate of small amplitude perturbations can be affected by many things including viscosity, diffusivity, surface tension, ablation, finite density gradients, magnetic fields etc. (For effects of compressibility, surface tension and viscosity, see Livescu 2004.) Considering only the very simplest case of 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 \[\tag{7} \frac{d\eta}{dt} = (A g k)^{1/2} \eta \]

where \(g\) is a constant acceleration and \[\tag{8} A \equiv \frac{\rho_2 - \rho_1}{\rho_2 + \rho_1} \]

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., \[\tag{9} \eta(t) = \eta_o \exp[(A g k)^{1/2}t] \]

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 \[\tag{10} \eta(t) = \eta_o \exp\left\{\left[ ( A g k/\psi + \nu^2 k^4 )^{1/2} - (\nu+D)k^2 \right] t \right\} \]

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

### Self-similarity

At late times, under certain idealized conditions, the flow may forget its initial conditions and enter a self-similar growth phase (Fermi and von Neumann 1953) described by the following equation \[\tag{11} \frac{dh}{dt} = 2 (\alpha A g h)^{1/2} \ .\]

Here \(h\) is the height of the mixing region and \(\alpha\) is a dimensionless growth parameter. Eq. (11) can be derived via a self-similarity assumption (Ristorcelli and Clark 2004) or from an energy argument (Cook et al. 2004). The similarity derivation assumes only that solutions to the moment equations can be expressed as the product of a temporal scaling function and a spatial similarity function. The energy argument notes that \(dh/dt\) is proportional to the net mass flux through the midplane and models the vertical velocity fluctuations at the midplane through a generalization of the terminal velocity equation for a falling sphere with diameter \(\propto h\ .\) For constant \(\alpha\ ,\) \(A\) and \(g\ ,\) the solution to Eq. (11) is (taking only the positive root as physically realizable) \[\tag{12} h(t) = \alpha A g t^2 + 2(\alpha A g h_o)^{1/2} t + h_o \ .\]

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. (12); hence the mixing thickness becomes \[\tag{13} h(t) \approx \alpha A g t^2 \ .\]

In order for Eq. (13) 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. (9) will compete with the quadratic growth of Eq. (13). The validity of Eq. (13) 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*
\[\tag{14}
\lambda_m<<h<<L
\ .\]

The idealized self-similar state described above rarely occurs in practice, due to the broadbanded nature of uncontrolled perturbations. Furthermore, between the exponential and self-similar growth stages there exists a transitional stage not easily described by simple ordinary differential equations. According to Dimonte, when the amplitude of individual modes become comparable to their wavelength, they enter into a constant-velocity regime which favors longer wavelength modes. This drives the instability toward self-similar growth either by mode coupling or mode competition. The former occurs when the initial perturbations are dominated solely by short wavelength perturbations and the latter when they are broadbanded. The former is found in numerical simulations that achieve a universal state independent of initial conditions. The latter is probably what occurs in most experiments and leads to an \(\alpha\) that depends weakly on initial conditions (Dimonte 2004).

## Experiments

Experimentalists have tried to measure \(\alpha\) under a variety of conditions. A major challenge in any experiment is to precisely characterise the tiny initial perturbations. Interfacial perturbations occur natually at all wavelengths up to the dimensions of the experimental apparatus; hence, most experiments don't satisfy the first inequality in Eq. (14). Additionally, limitations on diagnostics as well as boundary layer effects make it unlikely that the second inequality will be strictly satisfied. Caution must therefore be exercised in applying self-similar analysis to experimental results. Despite these difficulties however, many experiments have demonstrated good repeatability and have provided useful data for validating growth models and 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 enjoy 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. (14), predict \(\alpha \approx 0.027\) (Cabot and Cook 2006). Most large simulations have produced values for \(\alpha\) lower than experiments. This appears to be due to band-limiting of the initial perturbations (Dimonte et al. 2004).

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. (13).

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**Internal references**

- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.

- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

## Further 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

## See also

Inertial confinement fusion, Richtmyer-Meshkov instability, Turbulence, Type Ia supernovae