Bulk scaling in wall-bounded and homogeneous vertical natural convection

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ( $\mathit{Nu}$ ) and the Reynolds number ( $\mathit{Re}$ ) vary with the Rayleigh number ( $\mathit{Ra}$ ) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ( $\mathit{Pr}$ ). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$ , $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$ -power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$ -power-law scaling. These results suggest that the $1/2$ -power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$ , $\mathit{Re}$ and $\mathit{Nu}$ . From a stability perspective, at low- to moderate- $\mathit{Ra}$ , we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

Related papers:

title: New perspectives in turbulent Rayleigh-Bénard convection
abstract: Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Bénard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Bénard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

title: Vertical natural convection: application of the unifying theory of thermal convection
abstract: Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$ – $1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$ -power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

title: Changes in the boundary-layer structure at the edge of the ultimate regime in vertical natural convection
abstract: In thermal convection for very large Rayleigh numbers ( $Ra$ ), the thermal and viscous boundary layers are expected to undergo a transition from a classical state to an ultimate state. In the former state, the boundary-layer thicknesses follow a laminar-like Prandtl–Blasius–Polhausen scaling, whereas in the latter, the boundary layers are turbulent with logarithmic corrections in the sense of Prandtl and von Kármán. Here, we report evidence of this transition via changes in the boundary-layer structure of vertical natural convection (VC), which is a buoyancy-driven flow between differentially heated vertical walls. The numerical dataset spans $Ra$ values from $10^{5}$ to $10^{9}$ and a constant Prandtl number value of $0.709$ . For this $Ra$ range, the VC flow has been previously found to exhibit classical state behaviour in a global sense. Yet, with increasing $Ra$ , we observe that near-wall higher-shear patches occupy increasingly larger fractions of the wall areas, which suggest that the boundary layers are undergoing a transition from the classical state to the ultimate shear-dominated state. The presence of streaky structures – reminiscent of the near-wall streaks in canonical wall-bounded turbulence – further supports the notion of this transition. Within the higher-shear patches, conditionally averaged statistics yield a logarithmic variation in the local mean temperature profiles, in agreement with the log law of the wall for mean temperature, and an $Ra^{0.37}$ effective power-law scaling of the local Nusselt number. The scaling of the latter is consistent with the logarithmically corrected $1/2$ power-law scaling predicted for ultimate thermal convection for very large $Ra$ . Collectively, the results from this study indicate that turbulent and laminar-like boundary layer coexist in VC at moderate to high $Ra$ and this transition from the classical state to the ultimate state manifests as increasingly larger shear-dominated patches, consistent with the findings reported for Rayleigh–Bénard convection and Taylor–Couette flows.

title: Transient features of natural convection in a cavity
abstract: Comparisons of numerical and experimental results for transient two-dimensional natural convection initiated by instantaneously heating and cooling the opposing vertical walls of a square cavity containing a stationary and isothermal fluid are presented. The good comparisons indicate that the simulation is capturing the important features of the flow. Several features are identified and discussed in detail; in particular, the presence of travelling wave instabilities on the vertical-wall boundary layers and horizontal intrusions, the existence of a rapid flow divergence in the region of the outflow of the intrusions, and the presence of cavity-scale oscillations, caused by the interaction of the intrusions with the opposing vertical boundary layer. The utilization of both numerical and experimental investigations has allowed a more complete exploitation of the available resources than would have been possible had each been conducted separately.

title: Scaling in laminar natural convection in laterally heated cavities: is turbulence essential in the classical scaling of heat transfer?
abstract: We analyze heat transfer and flow properties in laminar natural convection driven by a horizontal temperature gradient in a closed cavity and propose that for the classical scaling of heat transfer turbulence does not play a decisive role. Direct numerical simulations were performed with the Rayleigh number (Ra) from 1 to 10(8) and the Prandtl number Pr = 0.71. In the laminar steady flow regime with the Ra approximately from 10(3) to 10(7), power-law scalings of heat transfer and maximum velocity with Ra have exponents of 0.31 and 0.54, respectively. The scalings agree well with results obtained in turbulent Rayleigh-Bernard convection, turbulent convection in laterally heated cavities and laminar convection in inclined enclosures, etc., which, with some simple physical arguments and reviews of the literature, leads us to propose that turbulence is not essential for the classical near 1/3 power-law scaling of Nu.

title: Thermal convection in inclined cylindrical containers
abstract: By means of direct numerical simulations (DNS) we investigate the effect of a tilt angle ${\it\beta}$ , $0\leqslant {\it\beta}\leqslant {\rm\pi}/2$ , of a Rayleigh–Bénard convection (RBC) cell of aspect ratio 1, on the Nusselt number $\mathit{Nu}$ and Reynolds number $\mathit{Re}$ . The considered Rayleigh numbers $\mathit{Ra}$ range from $10^{6}$ to $10^{8}$ , the Prandtl numbers range from 0.1 to 100 and the total number of the studied cases is 108. We show that the $\mathit{Nu}\,({\it\beta})/\mathit{Nu}(0)$ dependence is not universal and is strongly influenced by a combination of $\mathit{Ra}$ and $\mathit{Pr}$ . Thus, with a small inclination ${\it\beta}$ of the RBC cell, the Nusselt number can decrease or increase, compared to that in the RBC case, for large and small $\mathit{Pr}$ , respectively. A slight cell tilt may not only stabilize the plane of the large-scale circulation (LSC) but can also enforce an LSC for cases when the preferred state in the perfect RBC case is not an LSC but a more complicated multiple-roll state. Close to ${\it\beta}={\rm\pi}/2$ , $\mathit{Nu}$ and $\mathit{Re}$ decrease with increasing ${\it\beta}$ in all considered cases. Generally, the $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ dependence is a complicated, non-monotonic function of ${\it\beta}$ .

title: Momentum and heat transport scalings in laminar vertical convection.
abstract: We derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr in laminar vertical convection (VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: Nu∼Pr^{1/4}Ra^{1/4}, Re∼Pr^{-1/2}Ra^{1/2} for Pr≪1 and Nu∼Pr^{0}Ra^{1/4}, Re∼Pr^{-1}Ra^{1/2} for Pr≫1. These theoretical results are in excellent agreement with direct numerical simulations for Ra from 10^{5} to 10^{10} and Pr from 10^{-2} to 30. The transition between the regimes takes place for Pr around 10^{-1}.

title: Turbulent Thermal Convection at Arbitrary Prandtl Number
abstract: The mixing‐length theory of turbulent thermal convection in a gravitationally unstable fluid is extended to yield the dependence of Nusselt number H/H0 on both Prandtl number σ and Rayleigh number Ra. The analysis assumes a layer of Boussinesq fluid contained between infinite, horizontal, perfectly conducting, rigid plates. Also obtained is the dependence of mean temperature deviation T(z), rms temperature fluctuation ψ(z), and rms velocity upon height z above the bottom plate. The theory gives H/H0 ∝ Ra1/3 (high σ), H/H0 ∝ (σ Ra)1/3 (low σ), and H/H0 ∼ 1 (very low σ). The boundaries of the several σ ranges are determined. At one intermediate Prandtl number only, the behavior of T(z) and ψ(z) reduces to that previously found by Priestly. At high σ, there is a range of z, outside the molecular conduction region, where T(z) ∝ z−1, ψ(z) ∝ z−1. The results at very low σ reduce to those of Ledoux, Schwarzschild, and Spiegel. The dynamics are found to be importantly modified at extremely large Ra because of the stirring action of small‐scale turbulence generated in shear boundary layers attached to the eddies of largest scale. The consequent corrected asymptotic law of heat transport at fixed σ is H/H0 ∝ [Ra/(In Ra)3]1/2.

title: Scaling in thermal convection: a unifying theory
abstract: A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 1011) the leading terms are Nu ∼ Ra1/4 Pr1/8, Re ∼ Ra1/2 Pr−3/4 for Pr [lsim ] 1 and Nu ∼ Ra1/4 Pr−1/12, Re ∼ Ra1/2 Pr−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra1/2 Pr1/2, Re ∼ Ra1/2 Pr−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra1/5 Pr1/5, Re ∼ Ra2/5 Pr−3/5 for small Pr, a regime with Nu ∼ Ra1/3, Re ∼ Ra4/9 Pr−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra3/7 Pr−1/7, Re ∼ Ra4/7 Pr−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.

title: Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection.
abstract: The Prandtl and Rayleigh number dependences of the Reynolds number in turbulent thermal convection following from the unifying theory by Grossmann and Lohse [ (2000);  (2001)] are presented and compared with various recent experimental findings. This dependence Re(Ra,Pr) is more complicated than a simple global power law. For Pr=5.5 and 10(8)<Ra<10(10) the effective or local power law exponent of Re as a function of Ra is definitely less than 0.50, namely, Re approximately Ra(0.45), in agreement with Qiu and Tong's experimental findings . We also calculated the kinetic boundary layer width. Both in magnitude and in Ra scaling it is consistent with the data.

title: Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes
abstract: Our unifying theory of turbulent thermal convection [Grossmann and Lohse,  (2000);  (2001);  (2002)] is revisited, considering the role of thermal plumes for the thermal dissipation rate and addressing the local distribution of the thermal dissipation rate, which had numerically been calculated by Verzicco and Camussi [ (2003);  (2003)]. Predictions for the local heat flux and for the temperature and velocity fluctuations as functions of the Rayleigh and Prandtl numbers are offered. We conclude with a list of suggestions for measurements that seem suitable to verify or falsify our present understanding of heat transport and fluctuations in turbulent thermal convection.

title: Multiple scaling in the ultimate regime of thermal convection
abstract: Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse,  (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu∝Ra0.14, Nu∝Ra0.22, and Nu∝Ra0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the ...

title: Ultimate state of thermal convection.
abstract: The ultimate regime of thermal convection, the so-called Kraichnan regime [R. H. Kraichnan, Phys. Fluids 5, 1374 (1962)]], hitherto has been elusive. Here numerical evidence for that regime is presented by performing simulations of the bulk of turbulence only, eliminating the thermal and kinetic boundary layers and replacing them with periodic boundary conditions.

title: Exponentially growing solutions in homogeneous Rayleigh-Bénard convection.
abstract: It is shown that homogeneous Rayleigh-Bénard flow, i.e., Rayleigh-Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full nonlinear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.

title: Axially homogeneous Rayleigh–Bénard convection in a cylindrical cell
abstract: Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like $\mathit{Nu}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $ and turbulence intensity as $\mathit{Re}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $, where the Rayleigh number $\mathit{Ra}$ indicates the strength of the driving force (for fixed values of $\mathit{Pr}$, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as $\mathit{Ra}$ is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find $\mathit{Nu}\ensuremath{\sim} \mathit{Re}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $ at $\mathit{Pr}= 1$. Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with $\mathit{Ra}$ as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-$\mathit{Ra}$ regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As $\mathit{Ra}$ is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

title: High-rayleigh-number convection in a vertical channel.
abstract: We measure the relation between convective heat flux and temperature gradient in a vertical channel filled with water, the average vertical mass flux being zero. Compared to the classical Rayleigh-Bénard case, this situation has the advantage of avoiding plates and, thus, their neighborhood, in which is usually concentrated most of the temperature gradient. Consequently, inertial processes should control the convection, with poor influence of the viscosity. This idea gives a good account of our observations, if we consider that a natural vertical length, different from the channel width, appears. Our results also suggest that heat fluxes can be deduced from velocity measurements in free convective flows. This confers to our results a wide range of applications.

title: Heat convection in a vertical channel: Plumes versus turbulent diffusion
abstract: Following a previous study [Gibert et al.,  (2006)], convective heat transfer in a vertical channel of moderate dimensions follows purely inertial laws. It would be therefore a good model for convective flows of stars and ocean. Here we report new measurements on this system. We use an intrinsic length in the definition of the characteristic Rayleigh and Reynolds numbers. We explicit the relation between this intrinsic length and the thermal correlation length. Using particle imaging velocimetry, we show that the flow undergoes irregular reversals. We measure the average velocity profiles and the Reynolds stress tensor components. The momentum flux toward the vertical walls seems negligible compared to the shear turbulent stress. A mixing length theory seems adequate to describe the horizontal turbulent heat and momentum fluxes, but fails for the vertical ones. We propose a naive model for vertical heat transport inspired by the Knudsen regime in gases.

title: Convection in a vertical channel
abstract: The flow generated by heat convection in a long, vertical channel is studied by means of particle imagery velocimetry techniques, with the help of the thermal measurements from a previous paper (Gibert et al 2009 ). We analyse the mean velocity profiles and the Reynolds stresses, and compare the present results with the previous ones obtained in a larger cell and at a larger Reynolds number. We calculate the horizontal temperature profile and the related horizontal heat flux. The pertinence of effective turbulent diffusivity and viscosity is confirmed by the low value of the associated mixing length. We study the one-point and two-point statistics of both velocity components. We show how the concept of turbulent viscosity explains the relations between the local probability density functions (pdf) of fluctuations for temperature, vertical and horizontal velocity components. Despite the low Reynolds number values explored, some conclusions can be drawn about the small scale velocity differences and the related energy cascade.

title: Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe
abstract: We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

title: Wind and boundary layers in Rayleigh-Bénard convection. I. Analysis and modeling.
abstract: The aim of this paper is to contribute to the understanding of and to model the processes controlling the amplitude of the wind of Rayleigh-Bénard convection. We analyze results from direct simulation of an L/H=4 aspect-ratio domain with periodic sidewalls at Ra=(10(5), 10(6), 10(7), 10(8)) and at Pr=1 by decomposing independent realizations into wind and fluctuations. It is shown that, deep inside the thermal boundary layer, horizontal heat fluxes exceed the average vertical heat flux by a factor of 3 due to the interaction between the wind and the mean temperature field. These large horizontal heat fluxes are responsible for spatial temperature differences that drive the wind by creating pressure gradients. The wall fluxes and turbulent mixing in the bulk provide damping. Using the direct numerical simulation results to parametrize the unclosed terms, a simple model capturing the essential processes governing the wind structure is derived. The model consists of two coupled differential equations for wind velocity and temperature amplitude. The equations indicate that the formation of a wind structure is inevitable due to the positive feedback resulting from the interaction between the wind and temperature field. Furthermore, the wind velocity is largely determined by the turbulence in the bulk rather than by the wall-shear stress. The model reproduces the Ra dependence of wind Reynolds number and temperature amplitude.

title: Turbulent convective heat transfer in a long cylinder with liquid sodium
abstract: Turbulent convective heat transfer in a closed cylinder with aspect ratio L = 5D (D is the diameter and L is the cylinder length) filled with liquid sodium, heated at one end face and cooled at the other, is studied experimentally for three different positions: vertical, inclined at 45 degrees to the vertical and horizontal. The Rayleigh number, which is determined by the superimposed temperature difference and the cylinder diameter, varies within the range . It is shown that the convective heat transfer along the cylinder is most effective in the inclined cylinder, where an intense large-scale circulation exists on a background of developed small-scale turbulence. In the horizontal cylinder, the turbulence is weak, but the large-scale circulation provides moderate heat transfer. In the vertical cylinder, the large-scale circulation is absent, the turbulent fluctuations are most active, but the heat transfer is the weakest. The dependence of the Nusselt number on the Rayleigh and the Prandtl numbers, and the dependence of the Reynolds number on the Grashof number are shown and discussed.

title: Direct numerical simulation of stationary homogeneous stratified sheared turbulence
abstract: Using direct numerical simulation, we investigate stationary and homogeneous shear-driven turbulence in various stratifications, ranging from neutral to very stable. To attain and maintain a stationary flow, we throttle the mean shear so that the net production stays constant for all times. This results in a flow that is characterized solely by its mean shear and its mean buoyancy gradient, independent of initial conditions. The method of throttling is validated by comparison with experimental spectra in the case of neutral stratification. With increasing stratification comes the emergence of vertically sheared large-scale horizontal motions that preclude a straightforward interpretation of flow statistics. However, once these motions are excluded, simply by subtracting the horizontal average, the underlying flow appears amenable to the standard methods of turbulence analysis. It is shown that a direct acknowledgement of the confining influence of the periodic simulation box can lead to a meaningful physical interpretation of the large scales. Once an appropriate confinement scale is identified, many features, including horizontal spectra, flux–gradient relationships and length scales, of stratified sheared turbulence can be readily understood, both qualitatively and quantitatively, in terms of Monin–Obukhov similarity theory. Finally, the similarity-theory framework is used to interpret the scaling of the vertical diapycnal diffusivity in stratified turbulence.

title: Mean Temperature Profiles in Turbulent Thermal Convection
abstract: To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al.,  (2015), the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated respectively as a cubic and a linear function of the distance from the plate. In the present work we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10^7 to 10^9.

title: Length scales of turbulence in stably stratified mixing layers
abstract: Turbulence resulting from Kelvin–Helmholtz instability in layers of localized stratification and shear is studied by means of direct numerical simulation. Our objective is to present a comprehensive description of the turbulence evolution in terms of simple, conceptual pictures of shear–buoyancy interaction that have been developed previously based on assumptions of spatially uniform stratification and shear. To this end, we examine the evolution of various length scales that are commonly used to characterize the physical state of a turbulent flow. Evolving layer thicknesses and overturning scales are described, as are the Ozmidov, Corrsin, and Kolmogorov scales. These considerations enable us to provide an enhanced understanding of the relationships between uniform-gradient and localized-gradient models for sheared, stratified turbulence. We show that the ratio of the Ozmidov scale to the Thorpe scale provides a useful indicator of the age of a turbulent event resulting from Kelvin–Helmholtz instability.

title: Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection
abstract: We report high-resolution measurements of the properties of the velocity boundary layer in turbulent thermal convection using the particle image velocimetry (PIV) technique and measurements of the temperature profiles and the thermal boundary layer. Both velocity and temperature measurements were made near the lower conducting plate of a rectangular convection cell using water as the convecting fluid, with the Rayleigh number Ra varying from 109 to 1010 and the Prandtl number Pr fixed at 4.3. From the measured profiles of the horizontal velocity we obtain the viscous boundary layer thickness δυ. It is found that δυ follows the classical Blasius-like laminar boundary layer in the present range of Ra, and it scales with the Reynolds number Re as δυ/H = 0.64Re−0.50±0.03 (where H is the cell height). While the measured viscous shear stress and Reynolds shear stress show that the boundary layer is laminar for Ra < 2.0 × 1010, two independent extrapolations, one based on velocity measurements and the other on velocity and temperature measurements, both indicate that the boundary layer will become turbulent at Ra ~ 1013. Just above the thermal boundary layer but within the mixing zone, the measured temperature r.m.s. profiles σT(z) are found to follow either a power law or a logarithmic behaviour. The power-law fitting may be slightly favoured and its exponent is found to depend on Ra and varies from −0.6 to −0.77, which is much larger than the classical value of −1/3. In the same region, the measured profiles of the r.m.s. vertical velocity σw(z) exhibit a much smaller scaling range and are also consistent with either a power-law or a logarithmic behaviour. The Reynolds number dependence of several wall quantities is also measured directly. These are the wall shear stress τw ~ Re1.55, the viscous sublayer δw ~ Re−0.91, the friction velocity uτ ~ Re0.80, and the skin-friction coefficient cf ~ Re−0.34. All of these scaling properties are very close to those predicted for a classical Blasius-type laminar boundary layer, except that of cf. Similar to classical shear flows, a viscous sublayer is also found to exist in the present system despite the presence of a nested thermal boundary layer. However, velocity profiles normalized by wall units exhibit no obvious logarithmic region, which is likely to be a result of the very limited distance between the edge of the viscous sublayer and the position of the maximum velocity. Compared to traditional shear flows, the peak position of the wall-unit-normalized r.m.s. profiles is found to be closer to the plate (at z+ = z/δw ≃ 5). Our overall conclusion is that a Blasius-type laminar boundary condition is a good approximation for the velocity boundary layer in turbulent thermal convection for the present range of Rayleigh number and Prandtl number.

title: Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection
abstract: The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Bénard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense.

title: Fine-scale statistics of temperature and its derivatives in convective turbulence
abstract: We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh–Bénard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 107 and 109. The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similarly to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.

title: Rayleigh-Taylor turbulence in two dimensions.
abstract: The first consistent phenomenological theory for two- and three-dimensional Rayleigh-Taylor (RT) turbulence has recently been presented by Chertkov [Phys. Rev. Lett. 91, 115001 (2003)]. By means of direct numerical simulations, we confirm the spatiotemporal prediction of the theory in two dimensions and explore the breakdown of the phenomenological description due to intermittency effects. We show that small-scale statistics of velocity and temperature follow Bolgiano-Obukhov scaling. At the level of global observables, we show that the time-dependent Nusselt and Reynolds numbers scale as the square root of the Rayleigh number. These results point to the conclusion that RT turbulence in two and three dimensions, thanks to the absence of boundaries, provides a natural physical realization of the Kraichnan scaling regime hitherto associated with the elusive "ultimate state of thermal convection."