High Rayleigh number convection and passive scalar mixing

A brief review is given of recent laboratory investigations of high Ra convection and their relation to other turbulent flows. For a passive scalar we summarize an emerging body of theory for the one point distribution function and inertial range correlation functions which display non-Kolmogorov exponents. Low R a number convection has long been the system of choice for investigating the onset of turbulence in both the small and large aspect ratio limits. For large systems, where there are a multiplicity of possible patterns, experiments might never have gotten started if it were not for the very quantitative road map furnished by Busse and coworkers and its subsequent elaboration and application to large cells via amplitude expansions [1-3]. Theory, as practiced both by physicists and applied mathematicians, truly advanced in step with high quality experiments to sort out what was a very complex problem. Can we look to convection at high R a as a route out of the quagmire of fully developed turbulence? Someone remarked that turbulence is too important to ignore but too hard to solve; the latter statement being self-fulfilling prophecy in our view. Engineers concern themselves with the first part of the dichotomy and "turbulators" with the second. The sociological aspects of "pure" turbulence research are probably of more interest to a general audience than the scientific issues. The early success of Kolmogorov scal* Corresponding author. E-mail: [email protected]. ing (K41) seems to have guaranteed the acceptance of its less well-founded successor (K62) [4]. Theory defined what was to be measured; isotropic turbulence was purer than boundary layers, and point measurements plus lots of signal processing were given primacy over visualizations. Large numerical simulations were run whose only output was a spectra. It hardly mattered that fourth-order velocity derivative statistics did not all scale the same way within simulations [5] (and therefore could not be parameterized by fluctuations in the energy transfer rate E) and even isotropic flows were plagued with structures [6]. Engineers, who were not concerned with constructing a general theory of everything, have amassed considerable information about shear flows which brings us back to the subject of convection. Much effort has gone into the elucidation of the N u ( R a ) relation and potent arguments have been given for scaling according to N u ~ R a 1/3 [7]. Ultimately this relation must fail as Kraichnan long ago recognized when bulk Kolmogorov-like turbulence reached a sufficient Reynolds number so that it generated turbulent boundary layers which invaded the thermal one [8]. No one anticipated however the 0167-2789/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0 !67-27 89(96)00075-9 B.I. Shraiman, E.D; Siggia/Physica D 97 (1996) 286-290 scaling N u ",~ Ra 2/7 which Libchaber and coworkers first put on a solid footing [9,10]. In retrospect, exponents indistinguishable from 2 had long been seen in water but it was always assumed that this was a transient on the way to becoming ~ [10]. An essential element for this new scaling relation in our view is the large scale coherent shear flow first seen by Howard and Krishnamurthy [10]. It might seem paradoxical that a wind would lower the N u ( R a ) exponent but in fact the coefficient accompanying the 2 scaling is sufficiently greater than that for marginal stability theory that it overwhelms small differences in exponent. Our own theory for { scaling utilized standard engineering turbulence ideas; energy balance, nesting of the thermal boundary layer within the viscous one, and the kinetic energy dissipation rate for turbulent boundary layers [11]. As a bonus we got a very good fit to the Re(Ra) relation for the mean flow. However 2 scaling extends down to nearly Ra ~ 104 in aspect 7 ratio "-~ 6 cells, and occurs also in 2D convection [10]. In neither circumstance would a conventional turbulent boundary layer occur. Many visualizations and now more quantitative measurements of the boundary layers in pressured gas cells up to Ra ~ l0 n show that plumes are an important component of the near wall turbulence [ 12,13]. 1 There is at least a qualitative analogy here with wall bounded shear flow where Willrnarth and Lu long ago showed that most of the Reynolds ' stress is carried by the bursts [14]. Further evidence for the importance of shear in turbulent convection comes from experiments in mercury where the low Pr should enhance the velocities [15]. Here, around Ra ~ 2 x 107 a transition to higher N u was seen which may signal the crossing of the viscous and thermal boundary layers. The surprise is always that scaling arguments that do not admit structures work so well at the two point level. Their success should not blind one to the true nature of the flow. 1 We do not agree that the maximum cutoff frequency in the temporal spectrum of the scalar is a surrogate for the large scale velocity, its dependence on Ra may merely track the turbulent homogenization of the scalar. 287 If turbulent convection reduces to a complicated turbulent boundary layer with local plume forcing, are there any statistical fluid mechanics problems where analytic progress is possible? One possibility is to drop the buoyancy in the Boussinesq equations and look at passive scalar advection. One small recent success in this area was the prediction and observation of tails in the distribution function (PDF) for the scalar fluctuations in the presence of a mean gradient g (the total temperature field is 0 g . r) [16,17], OtO + v V0 = ~cV20 + g . v. (1) If the velocity field is stationary and homogeneous, g will transmit the same properties to 0. The tails of the 0 PDF measure the probability, a parcel of fluid is transported without mixing from a point Ar Sufficiently, up/down the gradient such that g Ar is the desired excursion in 0. If L is the integral scale of v or 0, (0 2} ~" L2g2; so we are looking at transport by the large scales of motion over distances of several or many integral scales. Computation of the tails of PDF is thus equivalent to asking for the fraction of velocity fields in our ensemble which will transport a parcel the desired distance with no mixing. Logically this can occur either for typical mixing rate but an atypical path or for a typical (random walk) path for which the mixing time is anomalously long. The latter effect wins. Within an integral scale, the probability that the scalar will mix with its environment depends on the shear of the large scales either through its direct action or in its role in maintaining the turbulent energy cascade and through it the eddy diffusivity. The strain is ~< 1 /T in magnitude (T >> 1 in units where (v2)/L 2 1) with probability ~ ( 1 / T ) a. Impose this condition T time in succession with a probability e a T In :r. I f there were no diffusion, the PDF from fluid parcels random walking up and down the gradient for a time T would be non-stationary and of the form exp(--O2/2T) in suitable units. Therefore: the tails are given by P(O) ~ f dTe-°2/2Te -aT In T ,.. e-101 In 10l (2) 288 B.L Shraiman, E.D. Siggia/Physica D 97 (1996) 286-290 A proper calculation is very naturally formulated as a path integral over the prior history of the parcel and eliminates the ln[0[ in (2) [17]. A related question with relevance to the treatment of small scale 0 statistics is to examine the PDF of V0. Within simulations [18], the distribution is cusped around its maximum and has stretched exponential tails. However the cusp is centered around g . This is manifest in snapshots of the total temperature field, as plateaus where the temperature is uniform, separated by "cliffs". The mean gradient is thus expelled into the cliffs. Interestingly enough, the path integral calculation of Pumir et al. [17], which had only a single scale velocity, did reproduce this feature of the simulations. A more ambitious problem is to look within an integral scale and study the scaling of 0 fluctuations advected by a velocity which itself scales. This is an appealing problem since all the Kolmogorov phenomenology carries over to the scalar. The spectrum of 0 often follows a ~law though with less precision than for the velocity [19]. The surprising fact is that Kolmogorov theory fails badly already for the third-order moment which is frequently packaged as Sd -~ ( (g . 00)3)/((00)2) 3/2. The small scales should be isotropic irrespective of large scale gradients so we expect Sd ~ g/((O0)2) 1/2 Re -1/2. Instead the experiment very unambiguously says Sd is Re independent and ~ 1 [19]. Additional encouragement to tackle this problem analytically came from simulations and wind tunnel experiments which showed that the anomalous skewness was not unique to shear flows but true even for synthetic Gaussian velocity fields [19]! The skewness is symptomatic of coherent structures in the 0 field consisting of gentle ramps and abrupt cliffs (noted already above) across which 0 may fall by a good fraction of its variance over a microscale in distance [18,19]. This is qualitatively the same property that was noticed for the coherent vortex tubes seen in isotropic simulations [5]. A velocity difference of order the RMS occurs across the vortex core which itself scales with the Kolmogorov length. Kraichnan has shown by a plausible closure and simulations that even powers of A0r ---= (O(r) -0(0)) exhibit non-Kolmogorov scaling when v is Gaussian white noise with a 5/3 spectrum [20]. The method of choice for calculating higher-order 0 correlation functions is the Hopf equation which expresses their stationary. This equation is simple only for g-correlated velocity fields where it reduces to the sum of the Richardson operator acting on all pairs of points (a, b are spacial labels). L(R ~) F2ij -~ (ga b (d + i i f ) ( 2 ~)Fijri b)O a 0 b.