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Acceleration and turbulence
in Rayleigh–Taylor mixing
Katepalli R. Sreenivasan1 and Snezhana I. Abarzhi2
1 Department of Physics, Courant Institute of Mathematical Sciences,
rsta.royalsocietypublishing.org
Research
Cite this article: Sreenivasan KR, Abarzhi SI.
2013 Acceleration and turbulence in
Rayleigh–Taylor mixing. Phil Trans R Soc A 371:
20130267.
http://dx.doi.org/10.1098/rsta.2013.0267
One contribution of 13 to a Theme Issue
‘Turbulent mixing and beyond:
non-equilibrium processes from atomistic
to astrophysical scales II’.
Subject Areas:
applied mathematics, fluid mechanics,
complexity
Keywords:
turbulent mixing, acceleration, turbulence,
Rayleigh–Taylor mixing
Author for correspondence:
Katepalli R. Sreenivasan
e-mail: [email protected]
New York University, New York, NY, USA
2 University of Chicago and University of Illinois at Chicago,
Chicago, IL, USA
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.
2013 The Author(s) Published by the Royal Society. All rights reserved.
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In RT turbulent mixing, the velocity is v ∼ gt, and the characteristic length scale L is L ∼ h ∼
gt2 . Thus, the flow Reynolds number grows with time as Re ∼ vL/ν ∼ g2 t3 /ν, and the rate of
dissipation of specific kinetic energy increases as ε ∼ v 2 /(L/v) ∼ v 3 /L ∼ g2 t. The viscous (or the
Kolmogorov) length-scale decays with time as lε ∼ (ν 3 /ε)1/4 ∼ (ν 3 /g2 )1/4 t−1/4 and the range of
scales grows as L/lε ∼ (g2 t3 /ν)3/4 ∼ Re3/4 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 vrN , where vr is the velocity difference across an inertial-range scale r
and . . . marks the average, are of the form CN (εr)N/3 (ignoring intermittency corrections). Here,
the constants CN 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) = Cε ε1/3 k−5/3 . The so-called Kolmogorov constant Cε 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 gt3/2 /ν 1/2 ∼ Re1/2 , 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
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2. Classical dimensional arguments
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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 λm 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 fastestgrowing length scale is λm ∼ (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 ∼ gt2 . (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.
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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 μ̃ = ε̃/v is an effective buoyant force, where
ε̃ ∼ g is the rate of energy gain induced by buoyancy. The rate of momentum loss is an effective
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3. Brief consideration of the effects of acceleration
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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.
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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
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5. Discussion and conclusion
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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 C∗N (μr)N/2 instead of CN (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 Cμ μk−2 , instead of having the classical −5/3
form, again measurably different. The viscous scale, lμ , the equivalent of the Kolmogorov scale,
is given by lμ ∼ (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 λm ∼ (ν 2 /g)1/3 .
The scale range L/lμ 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 C∗N and Cμ 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 ∼ μ2 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 ∼ gt2 develops owing to the imbalance of gain and loss of specific momentum, μ = μ̃. This
imbalance may occur when the horizontal scale grows as λ ∼ gt2 . 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 ∼ gt2 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 106 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 ∼ 103 ) [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.
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Acknowledgements. S.I.A. expresses her deep gratitude to the National Science Foundation (award 1004330).
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is adequate to interpret some observed results [3,5,6,10,19]. We have provided a theoretical
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