Statistics of Mixing in Three-Dimensional Rayleigh–Taylor Turbulence at Low Atwood Number and Prandtl Number One G

PHYSICS OF FLUIDS 22, 035109 ͑2010͒

Statistics of mixing in three-dimensional Rayleigh–Taylor turbulence at low Atwood number and Prandtl number one G. Boffetta,1 A. Mazzino,2 S. Musacchio,3 and L. Vozella2 1Dipartimento di Fisica Generale and INFN, Università di Torino, via P. Giuria 1, 10125 Torino, Italy and CNR-ISAC, Sezione di Torino, corso Fiume 4, 10133 Torino, Italy 2Dipartimento di Fisica, INFN and CNISM, Università di Genova, via Dodecaneso 33, 16146 Genova, Italy 3CNRS UMR 6621, Lab. J. A. Dieudonnné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France ͑Received 14 October 2009; accepted 6 February 2010; published online 31 March 2010͒ Three-dimensional miscible Rayleigh–Taylor ͑RT͒ turbulence at small Atwood number and at Prandtl number one is investigated by means of high resolution direct numerical simulations of the Boussinesq equations. RT turbulence is a paradigmatic time-dependent turbulent system in which the integral scale grows in time following the evolution of the mixing region. In order to fully characterize the statistical properties of the flow, both temporal and spatial behaviors of relevant statistical indicators have been analyzed. Scaling of both global quantities ͑e.g., Rayleigh, Nusselt, and Reynolds numbers͒ and scale dependent observables built in terms of velocity and temperature fluctuations are considered. We extend the mean-field analysis for velocity and temperature fluctuations to take into account intermittency, both in time and space domains. We show that the resulting scaling exponents are compatible with that of the classical Navier–Stokes turbulence advecting a passive scalar at comparable Reynolds number. Our results support the scenario of universality of turbulence with respect to both the injection mechanism and the geometry of the flow. © 2010 American Institute of Physics. ͓doi:10.1063/1.3371712͔

I. INTRODUCTION partially confirmed by recent numerical studies.3,9–11 Other alternative phenomenological approaches ͑see, e.g., Ref. 12͒ The Rayleigh–Taylor ͑RT͒ instability is a well-known does not necessarily lead to the Kolmogorov scaling for the fluid-mixing mechanism originating at the interface between energy spectra. light fluids accelerated into a heavy fluid. It was first de- In this paper we carry out an analysis of the scaling scribed by Rayleigh1 for incompressible fluid under gravity 2 behavior of relevant observables with the aim of deepening and later generalized to all accelerated fluid by Taylor. 11 our previous investigation. Indeed, our aim is to make a RT instability plays a crucial role in many fields of sci- careful investigation of the time evolution of global observ- ence and technology. In particular, in gravitational fusion it ables and of spatial/temporal scaling and intermittency. has been recognized as the dominant acceleration mechanism Moreover we push the analogy of RT turbulence with the for thermonuclear reactions in type-Ia supernovae.3,4 The ef- usual Navier–Stokes ͑NS͒ turbulence much further. We show ficiency of inertial confinement fusion depends dramatically that small-scale velocity and temperature fluctuations de- on the ability to suppress RT instability on the interface be- velop intermittent distributions with structure function scal- tween the fuel and the pusher shell.5,6 ing exponents consistent with NS turbulence advecting a In a late stage, RT instability develops into the so-called passive scalar. RT turbulence in which a layer of mixed fluid grows in time, This paper is organized as follows. In Sec. II we formu- increasing the kinetic energy of the flow at the expenses of late the problem and outline the phenomenology. After pro- the potential energy. This process finds applications in many viding a description of the numerical setup in Sec. III, we fields, e.g., atmospheric and oceanic buoyancy driven mix- describe our results in Secs. IV and V. Section IV is devoted ing. Despite the great importance and long history of RT to the investigation of the temporal evolution of global quan- turbulence, a consistent phenomenological theory has been tities. In Sec. V we focus on the statistics at small scales. proposed only recently.7 In three dimensions, this theory pre- Finally, the conclusions are provided by summarizing the dicts a Kolmogorov-like scenario, with a quasistationary en- main results. ergy cascade in the mixing layer. The prediction is based on the Kolmogorov–Obukhov picture of turbulence in which density fluctuations are transported passively in the cascade and kinetic-energy flux is scale independent.8 Quasistation- II. EQUATION OF MOTION AND PHENOMENOLOGY arity is a consequence of the Kolmogorov scaling of charac- We consider the three-dimensional ͑3D͒ Boussinesq teristic times associated to turbulent eddies: Large scales ,equations for an incompressible velocity field ٌ͑·v=0͒ grow driven from potential energy, while small-scale struc- tures, fed by the turbulent cascade, follow adiabatically ͒ ͑ ␤ ␯ ᭝ ٌ ٌ ץ large-scale growth. These theoretical predictions have been tv + v · v =− p + v − gT, 1

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TABLE I. Parameters of the simulations: Nx, Ny,andNz spatial resolutions, been observed in numerical simulations of thermal convec- ␯ ␬ 2 ͱ /͑␯⑀͒ 21 viscosity, thermal diffusivity, and R␭ =vrms 15 Reynolds number tion at moderate Ra when boundaries are removed, while evaluated at the end of the simulation. All dimensional quantities are no indication of ultimate state regime has been observed in made dimensionless using the vertical box size Lz, the characteristic time 14 ␶ ͑ / ͒1/2 ␪ = Lz Ag , and the temperature jump 0 as reference units. Rayleigh–Bénard experiments. The ultimate state exponent is formally derived from ki- ␯=␬ netic energy and temperature balance equations.18 In the ͑ϫ −6͒ Label Nx =Ny Nz 10 R␭ present context of RT turbulence they can more easily be A 256 1024 9.5 103 obtained from the temporal scaling of h and vrms. Assuming ͗ ͘ϳ ␪ B 512 2048 4.8 196 that wT vrms 0, using the above definitions one estimates, C 1024 1024 3.2 122 Ra Ӎ͑Ag͒4t6/͑␯␬͒,ReӍ͑Ag͒2t3/␯, and ͑3͒ Nu Ӎ͑Ag͒2t3/␬, ␬⌬ ͑ ͒ from which ٌ ץ tT + v · T = T, 2 ϳ 1/2 1/2 ϳ −1/2 1/2 ͑ ͒ with T͑x,t͒ being the temperature field, proportional to the Nu Pr Ra and Re Pr Ra . 4 ␤ ␳ ␳ density via the thermal expansion coefficient as = 0 For what concerns the small-scale statistics inside the ϫ͓ ␤͑ ͔͒ ͑␳ ͒ ␯ 7 1− T−T0 0 and T0 are reference values , is the mixing zone, the phenomenological theory predicts for the kinematic viscosity, ␬ is the molecular diffusivity, and 3D case an adiabatic Kolmogorov–Obukhov scenario with a g=͑0,0,−g͒ is the gravitational acceleration. ⑀Ӎ 3 / Ӎ͑␤ ␪ ͒2 time-dependent kinetic-energy flux vrms h g 0 t. At time t=0 the system is at rest with cooler ͑heavier, Spatial-temporal scaling of velocity and temperature fluctua- ␳ ͒ ͑ ␳ ͒ density 2 fluid placed above the hotter lighter, density 1 tions is therefore expected to follow one. This corresponds to v͑x,0͒=͑0,0,0͒ and to a step ␦ ͑t͒Ӎ⑀1/3r1/3 Ӎ͑␤g␪ ͒2/3t1/3r1/3, ͑5͒ function for the initial temperature profile: T͑x,0͒ rv 0 =−͑␪ /2͒sgn͑z͒, where ␪ is the temperature jump which 0 0 ␦ ͑ ͒Ӎ⑀−1/6⑀1/2 1/3 Ӎ ␪2/3͑␤ ͒−1/3 −2/3 1/3 ͑ ͒ ͑␳ ␳ ͒/͑␳ ␳ ͒ ͑ / ͒␤␪ rT t T r 0 g t r , 6 fixes the Atwood number A= 2 − 1 2 + 2 = 1 2 0. ␦ ͑ ͒ ͑ ͒ ͑ ͒ The development of the instability leads to a mixing zone of where rv t =v x+r,t −v x,t is the velocity increment on width h that starts from the plane z=0 and is dimensionally ͑ ͒ ⑀ Ӎ␪2 −1 a separation r similarly for temperature and T 0t is the expected to grow in time according to h͑t͒=␣Agt2 ͑where ␣ temperature-variance flux. We remark that the above scaling is a dimensionless constant to be determined͒, which implies is consistent with the assumption of the theory that tempera- Ӎ ͑ the relation vrms Agt for typical velocity fluctuations root ture fluctuations are passively transported at small scales mean square velocity͒ inside the mixing zone. ͓indeed using Eqs. ͑5͒ and ͑6͒, the buoyancy term ␤gT be- The convective state is characterized by the turbulent comes subleading in Eq. ͑1͒ at small scales͔. This is the main heat flux and energy transfer as a function of mean tempera- difference with respect to the two-dimensional ͑2D͒ case in ture gradient. In terms of dimensionless variables, these which temperature fluctuations force the turbulent flow at all quantities are represented respectively by the Nusselt number scales.4,7,22 ͗ ͘ /͑␬␪ ͒͑ ͒ Nu=1+ wT h 0 w being the vertical velocity and the /␯ Reynolds number Re=vrmsh as a function of the Rayleigh ␤ ␪ 3 /͑␯␬͒ ␯/␬ III. NUMERICAL SETTING number Ra= g 0h and the Prandtl number Pr= . ͗ ͘ Here and in the following ¯ denotes spatial average inside The Boussinesq equations ͑1͒ and ͑2͒ are integrated by a the turbulent mixing zone, while the overbar indicates the standard 2/3-dealiased pseudospectral method on a 3D average over horizontal planes at fixed z. periodic domain of square basis Lx =Ly and aspect ratio / One of the most important problems in thermal convec- Lx Lz =R with uniform grid spacing at different resolutions, tion is to find the functional relation between the convective as shown in Table I. In the following, all physical quantities state characterized by Nu and Re and the parameter space are made dimensionless using the vertical scale Lz, the tem- 13 ␪ ␶ ͑ / ͒1/2 defined by Ra and Pr. The existence of an asymptotic re- perature jump 0, and the characteristic time = Lz Ag as gime at high Ra, with a simple power law dependence fundamental units. NuϳRa␰ and ReϳRa␥, is still controversial in the case of Time evolution is obtained by a second-order Runge– the Rayleigh–Bénard convection, despite the number of ex- Kutta scheme with explicit linear part. In all the runs, periments at very large Ra. Most of the experiments have ␤g=2.0 and Pr=␯/␬=1. Viscosity is sufficiently large to ␰Ӎ ͑ ͒ ͑ ␩Ӎ reported an exponent 0.3 Refs. 14 and 15 of a more resolve small scales kmax 1.2 at final time, being 16,17 ␩ ␯3/4⑀−1/4 / ͒ complex behavior partially described by a phenomeno- = the Kolmogorov scale and kmax=Nx 3 . logical theory.18 However, many years ago, Kraichnan19 pre- RT instability is seeded by perturbing the initial condi- dicted an asymptotic exponent ␰=1/2 ͑with logarithmic cor- tion with respect to the unstable step profile. Two different rections͒ associated to the now called “ultimate state of perturbations were implemented in order to check the inde- thermal convection,” while exponents ␰Ͼ1/2 are excluded pendence of the turbulent state from initial conditions. In the by a rigorous upper bound NuՅ͑1/6͒Ra1/2 −1.20 The ulti- first case the interface T=0 at z=0 is perturbed by a super- mate state regime is expected to hold when thermal and ki- position of 2D waves of small amplitude h0 =0.004Lz in an Յ Յ ͑ 2 2 netic boundary layers become irrelevant, and indeed has isotropic range of wavenumbers 32 k 64 with k =kx

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2 rms /u 1.8 rms

,w 1.6 0.4 rms rms ,w u) 1.4 0.2 x rms ∂ u /( 1.2 0.1 rms 1 2 t/τ 4 w)

z 1 ∂ ( 20 40 60 80 100 120 140 160 180 200 Rλ

͑ ͒ FIG. 3. Color online Ratio of the vertical rms velocity wrms to the hori- ͑ ͒ zontal rms velocity urms open circles and ratio of the vertical velocity ͒ ͑ ͒ ץ͑ ͒ ץ͑ gradient zw rms to the horizontal velocity gradient xu rms ﬁlled circles vs Reynolds number R␭ indicating the recovery of isotropy at small scales. ͑ ͒ Inset: temporal evolution of horizontal rms velocity urms open circles and ͑ ͒ vertical rms velocity wrms ﬁlled circles . The line represents linear scaling. Data from run B.

͑ ͒ϳ 2 ϳ 4 ⑀ ϳ⑀ ϳ 3 FIG. 1. ͑Color online͒ Snapshot of temperature field for the RT simulation E t vrmsh t and L h t , as indeed observed at late at t=2␶. White ͑black͒ regions correspond to hot ͑cold͒ fluid. Parameters in times. We also plot in Fig. 2 the total potential-energy loss, Table I, run B. defined as P͑0͒− P͑t͒ with P͑t͒=−␤g͐zT͑x͒dx, which has the same temporal scaling of E͑t͒ as it is obvious from energy balance: d͑E+ P͒/dt=−⑀ . Notice that for this nonsta- +k2͒ and random phases.23 For the second set of simulations, L y tionary turbulence the energy balance does not fix the we perturbed the initial condition by adding 10% of white ratio between the energy growth rate dE/dt and the energy noise to the value of T͑x,0͒ in a layer of width h around 0 dissipation ͑and flux͒ ⑀ . In the turbulent regime, our z=0. Figure 1 shows a snapshot of the temperature field in a L simulations show an “equipartition” between large-scale en- cubic slice around z=0 in the turbulent regime at time ergy growth and small-scale energy dissipation: dE/dtӍ⑀ t=2␶ for simulation B ͑see Table I͒. L Ӎ−͑1/2͒dP/dt. This amounts to saying that half of the power injected into the flow contributes to the growth of the IV. EVOLUTION OF GLOBAL QUANTITIES large-scale flow, and half feeds the turbulent cascade ͑see ͒ Figure 2 displays the evolution of the total kinetic energy inset of Fig. 2 . This result was found to be independent on ͑ E=͐͑1/2͒v͑x͒2dx and total kinetic-energy dissipation ⑀ as a the value of viscosity the only adjustable parameter in the L ͒ 24 function of time. After the linear instability regime, at tӍ␶ system and is consistent with previous findings. the turbulent regime sets in with algebraic time dependence. An interesting remark is that RT turbulence represents an Temporal evolution of the two quantities are easily obtained instance of the general case of a turbulent flow adiabatically I͑ ͒ recalling that, being global quantities, an additional geo- evolving under a time-dependent energy input density t , ͑ ͒͑ metrical factor h͑t͒ϳt2 due to the integration over the verti- which forces the flow at the integral scale L t concerning cal direction has to be included. Therefore the predictions are the problem of turbulent flow characterized by a time- dependent forcing, see, for example, Refs. 25 and 26 and references therein͒. Energy balance requires dE/dt=I͑t͒ -1 ⑀͑ ͒ E 10 − t , where is the kinetic-energy density. Assuming a Kolmogorov spectrum for velocity fluctuations at scales E͑ ͒Ӎ⑀2/3 2/3 -3 smaller than the integral scale, one estimates t L . 10 Therefore, in situations characterized by an algebraic growth P ␥ ∆ I͑ ͒ϳ , 6 of the energy input density t t a self-similar evolution L ␥ ε -5 L ⑀͑ ͒ϳ , 10 ε of the energy spectrum can be obtained only if t t and 4 E L͑t͒ϳt͑3+␥͒/2. This is indeed realized in RT turbulence, where 2

(dE/dt)/ 2 -7 ␥=1 and ⑀ϳt, L͑t͒ϳt . 10 0 0 2 t/τ 4 In the inset of Fig. 3 the growth of vertical and horizon- ͑ ͒ tal rms velocity wrms and urms, respectively , computed 0.2 0.5 1 2 5 within the mixing layer, is shown. Both urms and wrms grow t linearly in time, as expected, with the vertical velocity about FIG. 2. ͑Color online͒ Temporal growth of kinetic energy E ͑circles͒, twice the horizontal one, reﬂecting the anisotropy of the ⑀ ͑ ͒ ⌬ kinetic-energy dissipation L triangles , and potential-energy loss P forcing due to gravity. It is interesting to observe that aniso- ͑ ͒ ⑀ squares for run B. For clarity of the plot L has been shifted by a factor 10. tropy decays at small scales, where almost complete isotropy The two short lines represent the dimensional scaling E͑t͒ϳ⌬P͑t͒ϳt4 and ⑀ ͑t͒ϳt3. Inset: ratio of the energy growth rate dE/dt and the ﬂux ⑀ . Data is recovered, as shown in Fig. 3. The ratio of vertical to L L / Ӎ from run B. horizontal rms velocity reaches a value of wrms urms 1.8 at

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0.5 0.6 0.03 — 0.1 w2 0.02 0.5 0.25 0.01 0.05 0.4 0.03 —– 0 wT -0.2 0.0 0.2 0 0 h 0.3 T(z) 0.02 1 2 3 4 0.01 0.2 t -0.25 0 -0.2 0.0 0.2 0.1 -0.5 0 -0.4 -0.2 0 0.2 0.4 0 1 2 3 4 z t

FIG. 4. ͑Color online͒ Mean temperature profiles ¯T͑z,t͒ for a single realiza- FIG. 5. ͑Color online͒ Evolution of the mixing layer width h as a function of tion of simulation B with diffused initial perturbation at times t=1.4␶, time t for simulation B computed from the profiles of Fig. 4 with a threshold t=2.0␶, t=2.6␶, and t=3.2␶. Lower and upper insets: profiles of the heat flux s=0.8. The inset shows the compensation with dimensional prediction wT͑z,t͒ and square vertical velocity w2͑z,t͒ at times t=1.4␶, t=2.0␶, and h/͑Agt2͒ converging to a valueӍ0.036. t=2.6␶.

h˙ 2 =4␣Agh, ͑7͒ ͑ ͒ ␣ 2 ͑␣ ͒1/2 which has solution h t = Agt +2 Ah0 t+h0, where later times ͑corresponding to R␭ Ӎ200͒ while for the gradi- h is the initial width introduced by the perturbation. u͒ Ӎ1.0. 0 ץ͑/ w͒ ץ͑ ents we have z rms x rms ␣ ˙ 2 /͑ ͒ ¯͑ ͒ =h 4Agh . The idea is to get rid of the subleading terms The evolution of the mean temperature profile T z,t 2 ϵ /͑ ͒͐ ͑ ͒ and extract the t contribution at early time by using directly 1 LxLy T x,t dxdy is shown in Fig. 4. As observed in 10,11,22,27 ͑ ͒ ␣ ˙ 2 /͑ ͒ previous simulations the mean profile is approxi- Eq. 7 and evaluating =h 4Agh . ͑ ͒ mately linear within the mixing layer ͑where therefore the The growth of the mixing layer width h t , a geometrical system recovers statistical homogeneity͒. Nevertheless, sta- quantity, is accompanied by the growth of the integral scale ͑ ͒ tistical fluctuations of temperature in the mixing layer are L t , a dynamical quantity representing the typical size of relatively strong: At later time we find a flat profile of fluc- the large-scale turbulent eddies. Following Ref. 9 we define tuations. Moreover their distribution is close to a Gaussian L as the half width of the velocity correlation function ␴ ͑ ͒Ӎ ␪ ͑ ͒ f͑L͒=͗v ͑r͒v ͑r+L͒͘/͗v2͘=1/2. In the turbulent regime the with a standard deviation T z 0.25 0 not shown here . i i In Fig. 4 we also plot the profile of the heat flux wT͑z,t͒ integral scale and the mixing length are linearly related ͑see and the square vertical velocity w2͑z,t͒. Both vanish outside Fig. 6͒. A linear fit gives L/hӍ1/17 and L/hӍ1/42 for the the mixing layer and inside show a similar shape not far from integral scale based on the vertical and horizontal velocity a parabola. Of course, the time behaviors of the heat flux and components, respectively, in agreement with the results of the square vertical velocity amplitude are different. In- shown in Ref. 9 ͑of course, the precise values of the coeffi- deed, the former is expected to grow as ϰt and the latter cients depend on the definition of h͒. The anisotropy of the as ϰt2. large-scale flow is reflected in the velocity correlation length: The mean temperature profile defines the width of the The integral scale based on horizontal velocity is smaller mixing layer. Different definitions of the mixing width, h, than the one based on vertical velocity. have been proposed on the basis of integral quantities or We end this section by discussing the behavior of the threshold values ͑see Ref. 28 for a discussion of the different turbulent heat flux, the energy transfer, and the mean tem- methods͒. In the following we will use the simple definition perature gradient in terms of dimensionless variables ͑as dis- ¯͑Ϯ / ͒ ␪ / Ͻ based on a threshold value: T h 2 =s 0 2, where s 1 represents the threshold. The evolution of the mixing width for s=0.8 is shown in Fig. 5. After an initial stage ͑tϽ0.3␶͒ in which the perturba- 0.03 tion relaxes toward the most unstable direction, we observe a short exponential growth corresponding to the linear RT in- 0.02 stability. At later times ͑tϾ0.6␶͒ the similarity regime sets in L and the dimensional t2 law is observed. The naive compensation with Agt2 gives an asymptotic constant value of 0.01 h/͑Agt2͒Ӎ0.036 for tՆ3␶ and ReӍ104 ͑at which the ͒ mixing width is still below half box . For the calculation of 0 ␣, more sophisticated analysis have been proposed 0 0.1 0.2 0.3 0.4 0.5 recently3,29,30 using slightly different approaches ͑briefly, in h Ref. 29 a similarity assumption and in Ref. 30 a mass flux ͑ ͒ ͒ FIG. 6. Color online Growth of the integral scale L based on the vertical and energy balance argument . In both cases, the authors velocity ͑circles͒ and horizontal velocities ͑squares͒ as a function of the derive for the evolution of h͑t͒ the equation mixing layer h. Data from simulation B.

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1010 15 0.04 (k) T 0.02 8 Π

10 ) a 0 -3 R 10 1 10 100 e, 6 R 10 k u, (k) (x 10 N

Π 5 104

0 102 1 2 3 4 1 10 100 t k

FIG. 7. ͑Color online͒ Temporal scaling of Nusselt number Nu=1 FIG. 9. ͑Color online͒ Spectral global kinetic-energy flux ⌸͑k͒ at times ͗ ͘ /͑␬␪ ͒͑ ͒ /␯ ͑ ͒ + wT h 0 triangles , Reynolds number Re=vrmsh squares , and t=2.4␶, t=2.6␶, t=2.8␶, and t=3.0␶ ͑from bottom to top͒ and temperature- ␤ ␪ 3 /͑␯␬͒͑ ͒ Rayleigh number Ra= g 0h circles for simulation B at Pr=1. The variance flux at t=3.0␶ ͑inset͒ for simulation B. Kinetic-energy flux is de- ˆ lines are the temporal scaling predictions t3 for Nu and Re and t6 for Ra. ⌸͑ ͒ ͐ϱ ͓ ͑ Јٌ͒͑ ͒͑ Ј͔͒ Ј fined as k =− k Re vˆ i −k v· vi k dk , where is the Fourier trans- form ͑Ref. 8͒. A similar definition holds for the temperature-variance flux.

cussed in Sec II͒: Nusselt, Reynolds, and Rayleigh numbers, respectively. The temporal evolution of these numbers, shown in Fig. 7, follows the dimensional prediction ͑3͒ for Figure 9 shows the global energy flux in spectral space the temporal evolution of ␣ ͑see the inset in Fig. 5͒. The at different times in the turbulent stage of the simulation. As presence of the ultimate state of thermal convection, in the discussed above, the flux grows in time following the in- restricted case Pr=1, is also confirmed by our numerical re- crease in the input I͑t͒ at large scales and at smaller ones, sults. Data obtained from simulations at various resolutions faster scales have time to adjust their intensities to generate a ͑see Fig. 8͒ are in close agreement with the “ultimate state” scale-independent flux. scaling ͑4͒. If the analogy with NS turbulence is taken seriously, one can extend the dimensional predictions ͑5͒ and ͑6͒ to include V. SMALL-SCALE STATISTICS intermittency effects. Structure functions for velocity and temperature fluctuations are therefore expected to follow As already discussed in Sec. I, the phenomenological ␨ r p theory predicts that at small scales, RT turbulence realizes an ͑ ͒ϵ͗͑␦ ͑ ͒͒p͘Ӎ ͑ ͒pͩ ͪ ͑ ͒ Sp r,t rvʈ t vrms t , 8 adiabatically evolving Kolmogorov–Obukhov scenario of h͑t͒ NS turbulence. Here adiabatic means that because of the ␨T scaling laws, small scales have sufficient time to adapt to the r p ST͑r,t͒ϵ͗͑␦ ␪͑t͒͒p͘Ӎ␪pͩ ͪ . ͑9͒ variations of large scales, leading to a scale-independent en- p r 0 h͑t͒ ergy flux. We remark that this is not the only possibility, as in two dimensions the phenomenology is substantially differ- In Eq. ͑8͒ we introduce the longitudinal velocity differences ␦ ͑ ͒ϵ͓ ͑ ͒ ͑ ͔͒ / ent. Unlike the 3D configuration, the 2D scenario is an ex- rvʈ t v x+r,t −v x,t ·r r and the increment r is made ample of active scalar problem. Indeed, the buoyancy effect dimensionless with a characteristic large scale which, in the is leading at both large and smaller scales. An adiabatic gen- present setup, is proportional to the width of the mixing layer eralization of Bolgiano–Obukhov scaling has been predicted h͑t͒, the only scale present in the system. The two sets of 7 ␨ ␨T by means of mean-field theory and has been confirmed scaling exponents p and p are known from both 31,32 33 numerically.22 experiments and numerical simulations with good ac- ␨ ␨T / curacy for moderate p. Mean-field prediction is p = p =p 3 while intermittency leads to a deviation with respect to this linear behavior. Kolmogorov’s “4/5” law for third-order ve- ␨ locity implies the exact result 3 =1, while temperature expo- 104 nents are not fixed, apart for standard inequality 8

e requirements. Both experiments and simulations give stron- R ger intermittency in temperature than in velocity fluctuations, T Nu, 3 ␨ Ͻ␨ 10 i.e., p p for large p. We have computed velocity and temperature structure functions and spectra in our simulations of RT turbulence. To overcome the inhomogeneity of the setup, velocity and tem- 102 8 9 10 ͑ ͒ 10 10 10 perature differences at fixed time are taken between points Ra both belonging to the mixing layer as defined above. Isot- ropy is recovered by averaging the separation r over all di- ͑ ͒ ͑ ͒ ͑ FIG. 8. Color online Nusselt open symbols and Reynolds filled sym- rections. Spectra are computed by the Fourier-transforming bols͒ numbers as a function of Rayleigh number at Pr=1. The ultimate state prediction ͑lines͒ expressed by Eq. ͑4͒ is compared to numerical data ob- velocity and temperature fields on 2D horizontal planes and tained from simulations A ͑squares͒,B͑circles͒,andC͑triangles͒. then averaging vertically over the mixing layer.

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-1 -1 -1 10 2.5 10 u) 10 δ ) -3 p( -2 10 u δ 3

-2 -1/3 10 / ε σ 2 10 2 T ε 30 ε 0.15 10-5 ) -3

-3 -3 10 E(k)/ (k)/( 10 p T T 0.1 -6 -4 -2 0 2 4 6 T 20 ε 1.5 E (x 10

ζ δ σ -4 ε -4 u/ δu 10 0.05 , 10 10 1 2 3 4 1 2 3 4 p ζ 1 -5 t -5 t 10 10 -1 10 100 10 100 10 T)

(a) (b) 0.5 δ k k 10-2 p( δ σ T T/ δT δ -3 FIG. 10. ͑Color online͒͑a͒ Kinetic-energy spectra compensated with ⑀2/3 at 0 σ 10 -4 -2 0 2 4 times t=1␶ ͑crosses͒, t=1.4␶ ͑times͒, t=1.8␶ ͑stars͒, and t=3.8␶ ͑squares͒. Inset: kinetic-energy dissipation vs time. The line represents the linear grow- 1 2 3 4 5 6 7 8 ing of energy dissipation ͑see Sec. II͒. ͑b͒ Temperature-variance spectra p ⑀−1⑀1/3 compensated with T at same times. Inset: temperature-variance dissipation vs time. The line is the dimensional prediction ϳt−1 ͑see Sec. II͒. FIG. 12. ͑Color online͒ Structure-function scaling exponents for velocity ␨ ͑ ͒ ␨T ͑ ͒ Data from simulation B. increments p circles and temperature increments p triangles with abso- lute values. Red open symbols are obtained using ESS procedure ͑Ref. 35͒ ␶ ␨ ␨T on the present simulation at time t=3 , fixing the values of 3 =1 and 2 =2/3. Errors represent fluctuations observed in different realizations of A. Lower-order statistics simulation B. Blue filled symbols are taken from a stationary NS simulation at R␭ =427 ͑Ref. 33͒. Black line is Kolmogorov nonintermittent scaling p/3. ␦ ͑ ͒͑ ͒ In Fig. 10͑a͒, we plot kinetic-energy spectra at different Insets: probability density function for velocity differences rv t upper ␦ ͑ ͒ ␶ and temperature differences rT t at time t=3 and scales r=0.008Lz times in the turbulent stage, compensated with the time- ͑ ͒ ͑ ͒ circles and r=0.06Lz squares . Black lines represent a standard Gaussian. dependent energy dissipation ⑀2/3͑t͒. In the intermediate range of wavenumbers, corresponding to inertial scales, the collapse is almost perfect. The evolution of the compensated to the fact that at large-scale temperature fluctuations are spectra shows that the growth of the integral scale at small dominated by unmixed plumes which have strong correla- wavenumbers is in agreement with Fig. 6. Likewise tions with vertical velocity. temperature-variance spectra are considered in Fig. 10͑b͒. Here, the spectra are compensated with both the time- B. Spatial/temporal intermittency ⑀−1͑ ͒ dependent temperature-variance dissipation T t and the energy dissipation ⑀1/3͑t͒. The evolution of the intermediate Despite the clear scaling observable in Fig. 11, it is very range of wavenumbers follows the dimensional prediction difficult to compute scaling exponents directly from higher- ͑6͒. order structure functions because of limited Reynolds num- Figure 11 displays the third-order velocity structure ber and statistics. Therefore, assuming a scaling region as in ͑ ͒ Fig. 11, we can compute relative scaling exponents using the function S3 r , related to the energy flux by Kolmogorov’s 35 ͑ ͒ ͑ / ͒⑀ 8 so-called extended self-similarity ͑ESS͒ procedure. This 4/5 law S3 r =− 4 5 r. We also plot the mixed velocity- 2 corresponds to consider the scaling of one structure function temperature structure function S ͑r͒ϵ͗␦ vʈ͑␦ T͒ ͘, which is 1,2 r r ͓ ͑ ͒ proportional to the ͑constant͒ flux of temperature fluctuations with respect to a reference one e.g., S3 r for velocity sta- ͔ ͑ ␨ /␨ ͒ ⑀ ͑ ͒ ͑ / ͒⑀ 34 tistics , and thus to measure a relative exponent i.e., p 3 . T according to Yaglom’s law S1,2 r =− 4 3 Tr. Both the computed structure functions display a range of linear Scaling exponents obtained in this way are shown in Fig. ␨ scaling, i.e., a constant flux, in the inertial range of scales 12. Reference exponents for the ESS procedure are 3 =1 and ␨T / ͑ ͒ ϫ −3Յ / Յ ϫ −2 2 =2 3 which is not an exact result . We see that both ve- 5 10 r Lz 5 10 . It is interesting to observe that the mixed structure function S ͑r͒ seems to have a range of locity and temperature scaling exponents deviate from the 1,2 ͑ ͒ ͑ ͒ ͑ ␨ ␨T p/ ͒ scaling which extends to larger scales. This is probably due dimensional prediction of Eqs. 5 and 6 i.e., p = p = 3 indicating intermittency in the inertial range. We also observe a stronger deviation for temperature exponents, which is consistent with what is known for the statistics of a passive scalar advected by a turbulent flow.8,36 The question regarding the universality of the set of scal- 10-3 ing exponents with respect to the geometry and the large- (r)

1,2 scale forcing naturally arises. Several experimental and nu- S merical investigations in 3D turbulence support the

(r),- -4 universality scenario in which the set of velocity and passive 3 10 S - scalar scaling exponents are independent of the details of large-scale energy injection and geometry of the ﬂow. There- fore, because we have seen that in 3D RT turbulence at small 10-5 10-3 10-2 scales temperature becomes passively transported and isot- r ropy is recovered, one is tempted to compare scaling exponents with those obtained in NS turbulence. As shown in Fig. FIG. 11. ͑Color online͒ Third-order isotropic longitudinal velocity structure ͑ ͒ ͑ ͒ 12, the two sets of exponents coincide, within the error bars, function S3 r computed at a late stage in the simulation circles and mixed longitudinal velocity-temperature structure function S ͑r͒͑triangles͒.The with the exponents obtained from a standard NS simulation 1,2 33 black line represents the linear scaling. Data from simulation B. with passive scalar at comparable R␭.

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1 -1 10-2 10 10-2 rms ,t) 0 a -3 4 ) (r 10 p a

-3 ( S 10 p -4 10 rms 2 a -5 10 0 1 2 3 4 -6 t 10-4 10 410-2 610-2 -10 -5 0 5 10 a/a S2(r0,t) rms ͑ ͒ FIG. 13. ͑Color online͒ Time dependence of p-order velocity structure func- FIG. 14. Color online Probability density function of the vertical compo- ␶ ͑ ͒ ␶ ͑ tion S ͑r ,t͒ vs S ͑r ,t͒ for p=4 ͑red open circles͒ and p=8 ͑blue ﬁlled nent of the acceleration at time t=1 red, bottom tails , t=2 green, inter- p 0 2 0 ͒ ␶ ͑ ͒ circles͒ with r /L =0.012, in the middle of the inertial range for simulation mediate tails , and t=3.8 blue, upper tails normalized with rms values. 0 z ͑ ͒ B. Red, continuous lines represent the intermittent prediction ␤ =p−2␨ Inset: evolution of arms with time for the three contributions of Eq. 1 : ͒ ͑ ⌬␯ ͒ ͑ ץ p p ␨ pressure gradient zp red circles , dissipation w green open triangles , with p given by spatial structure functions; blue dashed lines are the non- ␤ / and buoyancy term ␤gT ͑blue ﬁlled triangles͒. Data from simulation B. intermittent prediction p =p 3.

the dominant one, as shown in the inset of Fig. 14. After an We remark that scaling exponents for passive scalar in initial transient, we have that for tՆ2␶ both terms grow with NS turbulence are very sensitive to the fitting procedure. .p͒ /͑␯᭝w͒ Ӎ8 ץ͑ a constant ratio Strong temporal fluctuations have been observed in single z rms rms 37 The inset of Fig. 14 suggests that the temporal growth of realization and dependence on the fitting region has been / a is faster than t3 4. Again, this can be understood as an reported.33 Indeed, different realizations of RT turbulence rms effect of intermittency which is particularly important at ͑starting with slightly different initial perturbations͒ lead to small scales. Indeed, using the multifractal model of fluctuations of scaling exponents which account for the error intermittency8 one obtains the prediction a ϳt0.86.39 bars shown in Fig. 12. rms The effect of intermittency on acceleration statistics is Figure 12 also shows probability density functions for evident by looking at the probability density function ͑pdf͒. velocity and temperature fluctuations for two different scales. Figure 14 shows that the distribution develops larger tails as Both distributions are close to a Gaussian at large scale and turbulence intensity, and Reynolds number, increases. This develop wide tails at small scales, indicating the absence of effect is indeed expected, as the shape of the acceleration pdf self-similarity thus confirming the intermittency scenario. depends on the Reynolds number and therefore no universal As a further numerical support of Eqs. ͑8͒ and ͑9͒ we form is reached. Nevertheless, given the value of R␭ as a now consider temporal behavior of structure functions. From parameter, the pdf can be predicted again using the multi- Eq. ͑8͒, taking into account the temporal evolution of large- 39 ␤ fractal model. ͑ ͒ϳ p scale quantities, we expect the temporal scaling Sp t t ␤ ␨ with p =p−2 p. With the Kolmogorov scaling one simply ␤ / VI. CONCLUSION has p =p 3 but intermittent corrections are expected to be ␤ Ӎ / important, for example, 6 2.4 instead of p 3=2. Figure 13 We have studied spatial and temporal statistics of RT ͑ ͒ ͑ ͒͑ shows the scaling of Sp r,t versus S2 r,t i.e., in the ESS turbulence in three dimensions at small Atwood number and ͒ framework for a particular value of r=r0 =0.0012Lz. The at Prandtl number one on the basis of a set of high resolution ␤ /␤ relative temporal exponents p 2 obtained from the spatial numerical simulations. RT turbulence is a paradigmatic ex- ␨ exponents p of Fig. 12 fit well the data, while nonintermit- ample of nonstationary turbulence with a time-dependent in- ␤ /␤ / tent relative scaling exponents p 2 =p 2 are ruled out. jection scale. The phenomenological theory proposed by The effects of intermittency are particularly important at Chertkov7 is based on the notion of adiabaticity where small very small scales. One important example is the statistics of scales are slaved to large ones: The latter are forced by con- acceleration which has recently been the object of experi- version of potential energy into kinetic energy and the former mental and numerical investigations.38,39 For completeness, undergo a turbulence cascade flowing to smaller scales until we briefly recall the main results obtained in those studies. molecular viscosity becomes important. In this picture, tem- The acceleration a of a Lagrangian particle transported perature actively forces hydrodynamic degrees of freedom at by the turbulent flow is by definition given by the right hand large scales while it behaves like a passive scalar field at side of Eq. ͑1͒. In the present case of the Boussinesq ap- small scales where a constant kinetic-energy flux develops. proximation, the acceleration has three contributions: pres- The above scenario suggests comparison of RT turbu- sure gradient, viscous dissipation, and buoyancy terms. Ne- lence with classical homogeneous, isotropic, and stationary glecting intermittency for the moment, dimensional scalings NS turbulence, in the general framework of the existence of Ӎ␯᭝ Ӎ␯−1/4͑␤ ␪ ͒3/2 3/4 ٌ ͒ ͑ ͒ ͑ 5 and 6 , implies that − p u g 0 t universality classes in turbulence. ␤ Ӎ␤ ␪ ͑ ͒ while gT g 0. Therefore the buoyancy term in Eq. 1 By means of accurate direct numerical simulations, we becomes subleading not only going to small scales but also provide numerical evidence in favor of the mean-field theory. at later times. Among the other two terms, we find that as in Moreover, we extend the analysis to higher-order statistics standard NS turbulence, the pressure gradient term is by far thus addressing the issue related to intermittency corrections.

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