## Abstract

Past decades have significantly advanced our ability to probe turbulent mixing in Rayleigh–Taylor flows, in both experiments and simulations. Yet, our basic understanding remains elusive and requires better basis. For instance, observations do not substantiate the rudimentary dimensional arguments to the same degree of certainty as in classical three-dimensional turbulence. We provide a plausible scenario based on a momentum-driven model. The results presented are specific enough that they can be used to interpret experimental and numerical simulation data.

## 1. Introduction

The Rayleigh–Taylor (RT) instability develops in the flow of a material when the gradients of pressure and density point in opposite directions. Of particular interest here is the final state of this instability, which is the RT turbulent mixing. This topic has been considered in a large number of studies (e.g. [1]). RT mixing is inhomogeneous, anisotropic and statistically unsteady and one may expect its properties to depart from those of the standard isotropic, homogeneous and statistically steady turbulence. It is this facet that we examine here briefly.

Different physical phenomena are manifest in different examples of RT mixing. For instance, shock waves are an important part of the RT problem in inertial confinement fusion; changes in chemical composition are necessary for inclusion in the description of mixing in reactive flows; the diffusion of scalars plays an important role in interfacial mixing of miscible fluids, and so forth. Each new feature introduces a new element of physics, so it is not feasible to give a generic discussion to cover all circumstances. Nevertheless, RT flows exhibit a number of similar features during their evolution. RT instability starts to develop when the fluid interface is slightly perturbed near its equilibrium state. The flow transitions from an initial stage in which the perturbation amplitude grows exponentially, to a nonlinear stage in which the growth rate slows and the interface is composed of a large-scale coherent structure on which are superimposed the small-scale irregular structures driven by shear. The interaction among these scales leads to the state of turbulent mixing, whose dynamics is believed to be self-similar (e.g. [2]). In general, the RT mixing may be regarded as a combination of the ordered flow component at the interface and the disordered RT component in the fluid bulk [3]. Whether the latter is the primary mechanism of turbulent mixing may depend on the specifics of the flow. If the turbulent mixing in this flow should be regarded as having the degree of universality characteristic of canonical turbulence, a necessary condition is that the stretching of the interface by turbulent fluctuations should be large compared with the stretching owing to large-scale structures arising from initial instabilities. To see whether this is so, we estimate in §2 some turbulence parameters by dimensional arguments. First, we present a few key features of the initial development.

Though RT turbulent mixing involves many interacting scales, a ‘far field’ observer can identify two macroscopic length scales associated with the flow—the vertical scale in the direction of gravity and the horizontal scale in the normal plane, or equivalently, the amplitude *h* and the spatial period *λ* of the interface. The horizontal spatial scale *λ* is induced by the initial perturbation and there is a characteristic wavelength *λ*_{m} associated with the mode of fastest growth. For incompressible immiscible fluids with characteristic kinematic viscosity *ν* and gravity value *g*=|**g**|, where gravity **g** is directed initially from the heavy to the light fluid, the fastest-growing length scale is *λ*_{m}∼(*ν*^{2}/*g*)^{1/3} [4]. The scale *λ* may grow and become dominant if the initial perturbation is broadband and incoherent. The amplitude *h* is defined by the bubbles (associated with the penetration of the light fluid into the heavy fluid) and spikes (associated with the penetration of the heavy fluid into the light fluid). It is believed that in mixing regime the amplitude grows self-similarly, with *h*∼*gt*^{2}. (Sometimes in reporting results, it is convenient to introduce a virtual origin in time to account for the initial development that falls outside the self-similar behaviour; e.g. [5,6].) Measurements and simulations [7] suggest that the pre-factor in this dependence is substantially smaller than the free-fall value.

## 2. Classical dimensional arguments

In RT turbulent mixing, the velocity is *v* ∼ *gt*, and the characteristic length scale *L* is *L* ∼ *h* ∼*gt*^{2}. Thus, the flow Reynolds number grows with time as *Re*∼*vL*/*ν*∼*g*^{2}*t*^{3}/*ν*, and the rate of dissipation of specific kinetic energy increases as *ε*∼*v*^{2}/(*L*/*v*)∼*v*^{3}/*L*∼*g*^{2}*t*. The viscous (or the Kolmogorov) length-scale decays with time as *l*_{ε}∼(*ν*^{3}/*ε*)^{1/4}∼(*ν*^{3}/*g*^{2})^{1/4}*t*^{−1/4} and the range of scales grows as *L*/*l*_{ε}∼(*g*^{2}*t*^{3}/*ν*)^{3/4}∼*Re*^{3/4} with the upper limit of *Re* being set when the flow speed reaches the sound speed given by *c* ∼ *gt*. Therefore, the RT mixing flow is characterized by the increasing velocity and integral length scales, a quickly growing Reynolds number and a slowly decaying viscous scale, and by the span of scales which enlarges faster than quadratically in time. If this turbulence obeys the canonical Kolmogorov scenario, the velocity structure functions of order *N*, given by , where *Δv*_{r} is the velocity difference across an inertial-range scale *r* and 〈…〉 marks the average, are of the form *C*_{N}(*εr*)^{N/3} (ignoring intermittency corrections). Here, the constants *C*_{N} depend on the moment order *N*. The energy spectral density *ϕ*(*k*), where *k* is the wavevector magnitude, corresponds to the Fourier space representation of the second-order structure function and is of the form *ϕ*(*k*)=*C*_{ε}*ε*^{1/3}*k*^{−5/3}. The so-called Kolmogorov constant *C*_{ε} is known empirically to be about 1.5 for three-dimensional spectrum [8].

If these standard features hold true in the RT flow as well, the rate of stretching by turbulence to the mean stretching is given in the bulk of the flow by *gt*^{3/2}/*ν*^{1/2}∼*Re*^{1/2}, which, for a given fluid, increases quickly with time. Thus, one may expect the same (nearly) universal features of small-scale mixing as in other turbulent flows. However, this situation cannot be expected to hold close to the interface especially because there are possibilities of jumps there [3]. We do not yet know how to characterize the interface conditions completely and accurately, though one can presume that it may attain an approximately fractal shape as in a variety of other flows [9].

Even if we restrict attention to the bulk flow away from the interface (assuming that it is possible to do so), there is the question of whether the classical dimensional estimates discussed above are applicable. Unfortunately, experiments and simulations to date do not provide unequivocal support for these expectations. This conclusion is evident from careful assessments that can be found in the literature [7,10]. One reason for this behaviour may be that the concepts of classical turbulence may not apply to an accelerating RT flow. We briefly examine this notion below.

## 3. Brief consideration of the effects of acceleration

We may assume that turbulence with a given set of characteristics is established in the flow naturally at some point in time. This state could well be the remnant of initial conditions. For simplicity, we call this state turbulent because it may in general consist of a range of interacting scales, even if it may not have all the characteristics of hydrodynamic turbulence. We now enquire about the time evolution of this state. An important feature of RT flows is the continual presence of the acceleration owing to gravity. In a rapidly accelerated flow, turbulence does not have the time to adjust itself to the changing flow dynamics and so will ‘freeze’ [11]. The concept of ‘frozen’ turbulence applies better to the large scales of motion than to small scales. The evolution of the scales to which it applies will then be merely kinematic, much as in the rapid distortion theory, rather than an adjustment to the traditional interscale and intercomponent energy transfer. At a first glance, it appears possible to construct elementary spectral theory for RT turbulence, given the initial conditions, using arguments similar to those used by Prandtl [12]. However, reasonable estimates of the effects of acceleration appear to be orders of magnitude smaller than those that are characteristic of the frozen turbulence state. We base this remark on the magnitude of (the slightly revised definitions of) the parameter *K*=*νv*′/*v*^{3}, where *v* is the free-stream velocity and *v*′ is its rate of change with time. This parameter was used to measure the effect of free-stream acceleration in highly accelerated boundary layers (e.g. [13]). In the RT flow, the parameter *K* actually decreases rapidly with time, further making it unlikely that acceleration effects are large enough to freeze the turbulent motion. Another convenient parameter is *Λ*=*v*′*h*/T, where *h* is the thickness (width) of the mixing region, and T is the characteristic turbulent stress within the mixing region [11]. This parameter is also an order of magnitude smaller than that required for the frozen state of turbulence to occur. Thus, it appears that the acceleration effects are not strong enough to freeze the turbulent motion even on the large scale, let alone on the small scales.

In summary, it does not appear that the notion that rapid acceleration can overwhelmingly upstage the classical arguments of §2. This does not, however, imply that there are no significant effects of the acceleration on the mixing development, as we shall discuss below. One may need to take a fundamentally different outlook in order to understand these effects (see [2,14,15]).

## 4. The momentum model

The approach we discuss here accounts for the fact that, in RT mixing flow, the specific momentum is gained owing to buoyant force and is lost owing to friction force. Indeed, with RT flow evolves owing to gravity, with the heavy fluid falling down and the light fluid rising up; and the position of the centre of mass of the fluid system changes, leading to the change of potential energy. This energy change manifests as the fluid motion and dissipation. If the rates of gain and dissipation of specific energy as well as the rates of gain and loss of specific momentum are balanced, the flow is steady. The imbalance of these rates leads to flow acceleration [14]. The dynamics of each parcel of fluid in the direction of gravity is governed by a balance of the rate of momentum gain and the rate of momentum loss. The rate of momentum gain is an effective buoyant force, where is the rate of energy gain induced by buoyancy. The rate of momentum loss is an effective friction force *μ*=*ε*/*v*, where *ε* is the rate of dissipation of kinetic energy, as before. The relations between the rates of change of energy and momentum are the standard relations between power and force per unit mass [16]. The invariant of the motion is not the rate of energy change as in classical turbulence (this being essentially the power of an external ‘motor’ that supplies energy to the flow) but the rate of momentum change (caused by gravity that drives the flow dynamics), which is what controls the dimensional analysis.

As before, the Reynolds number will increase with time as its third power because the large scales of velocity and length follow the same dimensional arguments. It is easy to write down some implications of this model for turbulent quantities. For instance, the *N*-th order structure functions will be of the form instead of *C*_{N}(*εr*)^{N/3} and show a non-trivial difference in the *r*-dependence. (Incidentally, it seems straightforward to work out the intermittency corrections from the knowledge of the acceleration statistics of turbulence but this has not been done.) The spectral density would be in the form *C*_{μ}*μk*^{−2}, instead of having the classical −5/3 form, again measurably different. The viscous scale, *l*_{μ}, the equivalent of the Kolmogorov scale, is given by *l*_{μ}∼(*ν*^{2}/*μ*)^{1/3}, which has a markedly different dependence on viscosity than in the classical case and is comparable to the wavelength of the mode of fastest growth *λ*_{m}∼(*ν*^{2}/*g*)^{1/3}. The scale range *L*/*l*_{μ} increases with Reynolds number as the two-third power instead of the three-fourth power characteristic of turbulence [17,18].

It would, perhaps, be useful to make two further comments. First, as the RT turbulence is anisotropic, the constants and *C*_{μ} in the above expressions will be different in ‘spanwise’ and ‘streamwise’ directions. To be correct, we should indicate the directional dependence explicitly, but we have not done so for simplicity. Second, in the streamwise direction, it might be more useful to express the structure function results in terms of time rather than space or the spectral result in terms of frequency *f* rather than the wavenumber. When so expressed, the spectral density of kinetic energy will assume the form ∼*μ*^{2}*f*^{−3}.

The momentum model allows one to reproduce, in certain limiting cases, the quantitative and qualitative results found in previous studies [14,15]. At the same time, it identifies some new properties of RT turbulent mixing. In particular, the theory suggests that self-similar mixing with *h*∼*gt*^{2} develops owing to the imbalance of gain and loss of specific momentum, . This imbalance may occur when the horizontal scale grows as *λ*∼*gt*^{2}. It may also occur when the vertical scale *h* is the characteristic scale for energy dissipation *ε*∼*v*^{3}/*L* with *L*∼*h* and represents cumulative contributions of small-scale structures to flow dynamics. Furthermore, in accelerated RT mixing, the vortical structures are likely to be helices rather than vortices asymptotically. That is, as the length and velocity scales increase as ∼*gt*^{2} and ∼*gt*, respectively, the curl of the velocity decreases with time as |∇×**v**|∼1/*t*→0, whereas helicity approaches a constant, |**v**⋅(∇×**v**)|∼*g*. This constant is comparable to the rate of change of specific momentum as ∼*v*^{2}/*h*∼*g*.

An important qualitative result of the momentum model is the explanation of the paradox that was observed in RT mixing flows [2]. It was found that RT mixing flow maintains a significant degree of order when it is strongly accelerated, even when its Reynolds number is high on the order of 10^{6} or higher [3,19], in contrast to a statistically steady RT flow which can possess a turbulence-like disordered state even at far lower Reynolds numbers (*Re* ∼ 10^{3}) [5,6,10]. For a more detailed discussion on properties of RT mixing and on similarities and differences between RT mixing and canonical Kolmogorov turbulence, see [2,14,15] and references therein.

Finally, we should note that it is entirely conceivable that the ordered and disordered dynamics coexist during the development of RT mixing. A full-scale testing of the predictions of the momentum model is at present well open for experiments and simulations with high degree of control of flow parameters and precise diagnostics, sufficient data acquisition rate and dynamic range.

## 5. Discussion and conclusion

While a complete understanding of RT mixing requires extensive theoretical, experimental and numerical studies, it is not clear that the framework of classical Kolmogorov turbulence is adequate to interpret some observed results [3,5,6,10,19]. We have provided a theoretical framework, which explains the observed results qualitatively and produces specific results that can be applied for interpretation of accurate experimental and numerical data. The notion that the mixing region is momentum driven shows that the turbulence has stronger correlations and steeper spectral roll-off when compared with Kolmogorov turbulence.

Both RT mixing and Kolmogorov turbulence have a number of symmetries in statistical sense. In particular, they are invariant under the scaling transformations *L*→*LK*, *T*→*TK*^{1−n}, *v*→*vk*^{n} where the constant *K*>0 and *L*, *T*, *v* are characteristic length, time and velocity scales. In the limit of vanishing viscosity, the Kolmogorov turbulence corresponds to *n*=1/3 with the energy dissipation rate, given by *ε*∼*v*^{3}/*L*, as the appropriate invariant. In the RT mixing examined here, *n*=1/2 and the invariant corresponding to this symmetry is the rate of change of specific momentum (e.g. rate of momentum loss) given by *μ*∼*v*^{2}/*L* with *μ*=*ε*/*v* and *μ*∼*g*. It plays the same role as *ε* in classical phenomenology.

As a further point of difference, we consider the case of diffusion of energy for the case with constant diffusion coefficient *D*. The constancy of this parameter is reflected in the diffusion power-law *L*∼*T*^{1/2} and normal distribution of energy fluctuations [16]. In three-dimensional Kolmogorov turbulence, the diffusion coefficient is scale dependent in the inertial range of scales, whereas the energy dissipation rate *ε*∼*v*^{3}/*L*, with dimensions *ε*∼*L*^{2}/*T*^{3}, is invariant. This leads to *L*∼*T*^{3/2} with the velocity obeying the diffusion law *v*∼*T*^{1/2} [20]. These fluctuations are the internal property of the system and do not depend on initial conditions, boundary conditions or external forcing [17,18]. In the RT mixing, the diffusion coefficient and the rate of energy dissipation rate are both scale dependent, whereas the rate of momentum loss *μ*∼*v*^{2}/*L*∼*g* is invariant with respect to both time and space. As *μ*∼*L*/*T*^{2}, we are led to the result *L*∼*T*^{2} and *v*∼*T*, and to essentially kinematic (ballistic) motion of the fluid particles, whose velocity distribution is set by the initial conditions [16]. In RT mixing, the flow quantities apparently do not obey the diffusion scaling law with normal distribution for their fluctuations. Thus, for a given time scale or length scale, the velocity in standard turbulence fluctuates more strongly, making it less sensitive to initial conditions in comparison with the present scenario for RT mixing.

## Acknowledgements

S.I.A. expresses her deep gratitude to the National Science Foundation (award 1004330).

## Footnotes

One contribution of 13 to a Theme Issue ‘Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales II’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.