Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection

Optical phase and amplitude measurements of underwater turbulence via self-heterodyne detection

The creation of underwater optical turbulence is driven by density variations that lead to small changes in the water's refractive index, which induce optical path length differences that affect light propagation. Measuring a laser beam's optical phase after traversing these turbulent variations can provide insight into how the water's turbulence behaves. The sensing technique to measure turbulent fluctuations is a self-heterodyne beatnote enhanced by light's orbital angular momentum (OAM) to obtain simultaneous optical phase and amplitude information. Experimental results of this method are obtained in a water tank that creates a thermally driven flow called Rayleigh-Bénard (RB) convection. The results show time-varying statistics of the beatnote that depend on the incident OAM mode order and the strength of the temperature gradient. Beatnote amplitude and phase power spectral densities are compared to analytic theory to obtain estimates of the turbulent length scales using the Taylor hypothesis that include mean flow speed, turbulent strength, and length scales, and flow dynamics due to intermittency in the RB process.

Synchronous optical intensity and phase measurements to characterize Rayleigh-Bénard convection

Propagation of a laser beam through the Rayleigh-Bénard (RB) convection is experimentally investigated using synchronous optical wavefront and intensity measurements. Experimental results characterize the turbulence strength and length scales, which are used to inform numerical wave optic simulations employing phase screens. Experimentally found parameters are the refractive index structure constant, mean flow rate, kinetic and thermal dissipation rates, Kolmogorov microscale, outer scale, and shape of the refractive index power spectrum using known models. Synchronization of the wavefront and intensity measurements provide statistics of each metric at the same instance in time, allowing for two methods of comparison with numerical simulations. Numerical simulations prove to be within agreement of experimental and published results. Synchronized measurements provided more insight to develop reliable propagation models. It is determined that the RB test bed is applicable for simulating realistic undersea environments.

Experimental study on the flow structures and dynamics of turbulent Rayleigh-Bénard convection in an annular cell

We conduct an experimental study on the flow structures and dynamics of turbulent Rayleigh-Bénard convection in an annular cell with radius ratio η≃0.5 and aspect ratio Γ≃4. The working fluid is water with a Prandtl number of Pr≃5.4, and the Rayleigh number (Ra) ranges from 5.05×10^{7} to 5.05×10^{8}. The multithermal-probe method and the particle image velocimetry technique are employed to measure the temperature profiles and the velocity fields, respectively. Two distinct states with multiroll standing waves are observed, which are the quadrupole state (QS) characterized by a four-roll structure and the sextupole state (SS) by a six-roll structure. The scaling exponents of Reynolds number Re with Ra are different for the two states, which are 0.56 for QS and 0.41 for SS. In addition, the standing waves become unstable upon tilting the cell by 1^{∘} in relation to the horizontal plane, and they evolve into traveling waves. At relatively high Ra, for instance, Ra⩾2.55×10^{8}, it is observed that the traveling wave state SS undergoes a transition to the traveling wave state QS. However, the opposite transition from QS to SS is not observed in our experiments. Our findings provide insights into the flow structures and dynamics in the convection flow with rotation symmetry.

Reynolds number scaling and energy spectra in geostrophic convection

We report flow measurements in rotating Rayleigh-Bénard convection in the rotationally-constrained geostrophic regime. We apply stereoscopic particle image velocimetry to measure the three components of velocity in a horizontal cross-section of a water-filled cylindrical convection vessel. At a constant, small Ekman number Ek = 5 × 10^-8 we vary the Rayleigh number Ra between 10¹¹ and 4 × 10¹² to cover various subregimes observed in geostrophic convection. We also include one nonrotating experiment. The scaling of the velocity fluctuations (expressed as the Reynolds number Re) is compared to theoretical relations expressing balances of viscous-Archimedean-Coriolis (VAC) and Coriolis-inertial-Archimedean (CIA) forces. Based on our results we cannot decide which balance is most applicable here; both scaling relations match equally well. A comparison of the current data with several other literature datasets indicates a convergence towards diffusion-free scaling of velocity as Ek decreases. However, the use of confined domains leads at lower Ra to prominent convection in the wall mode near the sidewall. Kinetic energy spectra point at an overall flow organisation into a quadrupolar vortex filling the cross-section. This quadrupolar vortex is a quasi-two-dimensional feature; it only manifests in energy spectra based on the horizontal velocity components. At larger Ra the spectra reveal the development of a scaling range with exponent close to -5/3, the classical exponent for inertial-range scaling in three-dimensional turbulence. The steeper Re(Ra) scaling at low Ek and development of a scaling range in the energy spectra are distinct indicators that a fully developed, diffusion-free turbulent bulk flow state is approached, sketching clear perspectives for further investigation.

Effect of Gap Length and Partition Thickness on Thermal Boundary Layer in Thermal Convection

Two-dimensional direct numerical simulations of partitioned thermal convection are performed using the thermal lattice Boltzmann method for the Rayleigh number (Ra) of 10⁹ and the Prandtl number (Pr) of 7.02 (water). The influence of the partition walls on the thermal boundary layer is mainly focused on. Moreover, to better describe the spatially nonuniform thermal boundary layer, the definition of the thermal boundary layer is extended. The numerical simulation results show that the gap length significantly affects the thermal boundary layer and Nusselt number (Nu). The gap length and partition wall thickness have a coupled effect on the thermal boundary layer and the heat flux. Based on the shape of the thermal boundary layer distribution, two different heat transfer models are identified at different gap lengths. This study provides a basis for improving the understanding of the effect of partitions on the thermal boundary layer in thermal convection.

Steady thermal convection representing the ultimate scaling

Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier-Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance [Formula: see text] and a constant temperature difference [Formula: see text]. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. [Formula: see text] (Jiménez et al. 2001 J. Fluid Mech., 442, 89-117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton-Krylov iteration up to the Rayleigh number [Formula: see text] for the Prandtl number [Formula: see text], the horizontal period [Formula: see text] and the permeability parameter [Formula: see text]-[Formula: see text], [Formula: see text] being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady 'wall-bounded' thermal convection. The ultimate scaling [Formula: see text] has been observed for [Formula: see text] at high [Formula: see text], where [Formula: see text] is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity [Formula: see text] as well as intense temperature variation of [Formula: see text] even in the bulk region. As a consequence, the wall-to-wall heat flux scales with [Formula: see text] independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Computation by Convective Logic Gates and Thermal Communication

We demonstrate a novel computational architecture based on fluid convection logic gates and heat flux-mediated information flows. Our previous work demonstrated that Boolean logic operations can be performed by thermally driven convection flows. In this work, we use numerical simulations to demonstrate a different , but universal Boolean logic operation (NOR), performed by simpler convective gates. The gates in the present work do not rely on obstacle flows or periodic boundary conditions, a significant improvement in terms of experimental realizability. Conductive heat transfer links can be used to connect the convective gates, and we demonstrate this with the example of binary half addition. These simulated circuits could be constructed in an experimental setting with modern, 2-dimensional fluidics equipment, such as a thin layer of fluid between acrylic plates. The presented approach thus introduces a new realm of unconventional, thermal fluid-based computation.

Dynamics and length scales in vertical convection of liquid metals

Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$ ) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number $\textit {Re}$ collapse if the side height $L$ is used as the length scale rather than the distance $H$ between heated and cooled vertical plates. These results are obtained for various Rayleigh numbers $5\times 10^3\leq {\textit {Ra}}_H\leq 10^8$ (based on $H$ ) and the aspect ratios $L/H=1, 2, 3$ and $5$ . Furthermore, we present a novel method to extract the wind-based Reynolds number, which works particularly well with the experimental Doppler-velocimetry measurements along vertical lines, regardless of their horizontal positions. The extraction method is based on the two-dimensional autocorrelation of the time–space data of the vertical velocity.

Anomalous Convective Flows Carve Pinnacles and Scallops in Melting Ice

We report on the shape dynamics of ice suspended in cold fresh water and subject to the natural convective flows generated during melting. Experiments reveal shape motifs for increasing far-field temperature: Sharp pinnacles directed downward at low temperatures, scalloped waves for intermediate temperatures between 5 °C and 7 °C, and upward pointing pinnacles at higher temperatures. Phase-field simulations reproduce these morphologies, which are closely linked to the anomalous density-temperature profile of liquid water. Boundary layer flows yield pinnacles that sharpen with accelerating growth of tip curvature while scallops emerge from a Kelvin-Helmholtz-like instability caused by counterflowing currents that roll up to form vortex arrays. By linking the molecular-scale effects underlying water's density anomaly to the macroscale flows that imprint the surface, these results show that the morphology of melted ice is a sensitive indicator of ambient temperature.

Differential-pressure fiber-optic airflow sensor for wind tunnel testing

A differential-pressure fiber-optic airflow (DPFA) sensor based on Fabry-Perot (FP) interferometry for wind tunnel testing is proposed and demonstrated. The DPFA sensor can be well coupled with a Pitot tube, similar to the operation of the differential diaphragm capsule in the airspeed indicator on the aircraft. For differential pressure sensing between total pressure and static pressure in the airflow, an FP cavity is formed between the sensing diaphragm and a fiber end-face, and a tubule is inserted into the FP cavity. According to the principle of differential pressure derived from Bernoulli's equation, the airflow velocity can be determined by monitoring the change of the FP cavity length. The experimental results demonstrate that a DPFA sensor with 0∼11 kPa measurable range, 826.975 nm/kPa sensitivity, and 0.008% (0.89 Pa) resolution can be realized. Combined with a 100 Hz-sweep frequency self-developed white light interferometric (WLI) interrogator and a Pitot tube, the DPFA sensor can be used for measuring the airflow velocity of 2.0∼119.24 m/s with an accuracy of 0.61%. The system is applied to the analysis of the flat-plate boundary layer, a wind tunnel experimental model, where the results are consistent with those of the theoretical analysis and from the standard electronic pressure transducer. With the large measurable range, high sweep frequency, and high precision, the system has potential application value for wind tunnel experimental investigation and in-flight measurement of airspeed.

Numerical study of heat transfer in Rayleigh-Bénard convection under rarefied gas conditions

The focus of this research is to delineate the thermal behavior of a rarefied monatomic gas confined between horizontal hot and cold walls, physically known as rarefied Rayleigh-Bénard (RB) convection. Convection in a rarefied gas appears only for high temperature differences between the horizontal boundaries, where nonlinear distributions of temperature and density make it different from the classical RB problem. Numerical simulations adopting the direct simulation Monte Carlo approach are performed to study the rarefied RB problem for a cold to hot wall temperature ratio equal to r=0.1 and different rarefaction conditions. Rarefaction is quantified by the Knudsen number, Kn. To investigate the long-time thermal behavior of the system two ways are followed to measure the heat transfer: (i) measurements of macroscopic hydrodynamic variables in the bulk of the flow and (ii) measurements at the microscopic scale based on the molecular evaluation of the energy exchange between the isothermal wall and the fluid. The measurements based on the bulk and molecular scales agreed well. Hence, both approaches are considered in evaluations of the heat transfer in terms of the Nusselt number, Nu. To characterize the flow properly, a modified Rayleigh number (Ra_{m}) is defined to take into account the nonlinear temperature and density distributions at the pure conduction state. Then the limits of instability, indicating the transition of the conduction state into a convection state, at the low and large Froude asymptotes are determined based on Ra_{m}. At the large Froude asymptote, simulations following the onset of convection showed a relatively small range for the critical Rayleigh (Ra_{m}=1770±15) that flow instability occurs at each investigated rarefaction degree. Moreover, we measured the maximum Nusselt values Nu_{max} at each investigated Kn. It was observed that for Kn≥0.02, Nu_{max} decreases linearly until the transition to conduction at Kn≈0.03, known as the rarefaction limit for r=0.1, occurs. At the low Froude (parametric) asymptote, the emergence of a highly stratified flow is the prime suspect of the transition to conduction. The critical Ra_{m} in which this transition occurs is then determined at each Kn. The comparison of this critical Rayleigh versus Kn also shows a linear decrease from Ra_{m}≈7400 at Kn=0.02 to Ra_{m}≈1770 at Kn≈0.03.

Boolean logic by convective obstacle flows

We present a potential new mode of natural computing in which simple, heat-driven fluid flows perform Boolean logic operations. The system comprises a two-dimensional single-phase fluid that is heated from below and cooled from above, with two obstacles placed on the horizontal mid-plane. The obstacles remove all vertical momentum that flows into them. The horizontal momentum extraction of the obstacles is controlled in a binary fashion, and constitutes the 2-bit input. The output of the system is a thresholded measure of the energy extracted by the obstacles. Due to the existence of multiple attractors in the phase space of this system, the input–output relationships are equivalent to those of the OR, XOR or NAND gates, depending on the threshold and obstacle separation. The ability to reproduce these logical operations suggests that convective flows might have the potential to perform more general computations, despite the fact that they do not involve electronics, chemistry or multiple fluid phases.

Elusive transition to the ultimate regime of turbulent Rayleigh-Bénard convection

By using cryogenic ^{4}He gas as the working fluid in a cylindrical cell 0.3 m in both height and diameter, we study the influence of non-Oberbeck-Boussinesq (NOB) effects on the heat transfer in turbulent Rayleigh-Bénard convection (RBC). We show that the NOB effects increase the heat transfer efficiency when the top plate temperature closely approaches the saturation vapor curve even far away from the critical point. Viewed in this light, our analysis points to the likelihood that the claim of having observed the transition to Kraichnan's ultimate regime, under nominally similar conditions in the experiments with SF_{6} [Phys. Rev. Lett. 108, 024502 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.024502], is probably an NOB effect and the important issue of the transition to the ultimate state of RBC remains open.

Jump rope vortex in liquid metal convection

Understanding large-scale circulations (LSCs) in turbulent convective systems is important for the study of stars, planets, and in many industrial applications. The canonical model of the LSC is quasi-planar with additional horizontal sloshing and torsional modes [Brown E, Ahlers G (2009) J Fluid Mech 638:383-400; Funfschilling D, Ahlers G (2004) Phys Rev Lett 92:194502; Xi HD et al. (2009) Phys Rev Lett 102:044503; Zhou Q et al. (2009) J Fluid Mech 630:367-390]. Using liquid gallium as the working fluid, we show, via coupled laboratory-numerical experiments in tanks with aspect ratios greater than unity ([Formula: see text]), that the LSC takes instead the form of a "jump rope vortex," a strongly 3D mode that periodically orbits around the tank following a motion much like a jump rope on a playground. Further experiments show that this jump rope flow also exists in more viscous fluids such as water, albeit with a far smaller signal. Thus, this jump rope mode is an essential component of the turbulent convection that underlies our observations of natural systems.

Torque scaling in small-gap Taylor-Couette flow with smooth or grooved wall

The torque in the Taylor-Couette flow for radius ratios η≥0.97, with smooth or grooved wall static outer cylinders, is studied experimentally, with the Reynolds number of the inner cylinder reaching up to Re_{i}=2×10^{5}, corresponding to the Taylor number up to Ta=5×10^{10}. The grooves are perpendicular to the mean flow, and similar to the structure of a submersible motor stator. It is found that the dimensionless torque G, at a given Re_{i} and η, is significantly greater for grooved cases than smooth cases. We compare our experimental torques for the smooth cases to the fit proposed by Wendt [F. Wendt, Ing.-Arch. 4, 577 (1993)10.1007/BF02084936] and the fit proposed by Bilgen and Boulos [E. Bilgen and R. Boulos, J Fluids Eng. 95, 122 (1973)10.1115/1.3446944], which shows both fits are outside their range for small gaps. Furthermore, an additional dimensionless torque (angular velocity flux) Nu_{ω} in the smooth cases exhibits an effective scaling of Nu_{ω}∼Ta^{0.39} in the ultimate regime, which occurs at a lower Taylor number, Ta≈3.5×10^{7}, than the well-explored η=0.714 case (at Ta≈3×10^{8}). The same effective scaling exponent, 0.39, is also evident in the grooved cases, but for η=0.97 and 0.985, there is a peak before this exponent appears.

Effect of polymer additives on heat transport and large-scale circulation in turbulent Rayleigh-Bénard convection

The present paper presents direct numerical simulations of Rayleigh-Bénard convection (RBC) in an enclosed cell filled with the polymer solution in order to investigate the viscoelastic effect on the characteristics of heat transport and large-scale circulation (LSC) of RBC. To overcome the difficulties in numerically solving a high Weissenberg number (Wi) problem of viscoelastic fluid flow with strong elastic effect, the log-conformation reformulation method was implemented. Numerical results showed that the addition of polymers reduced the heat flux and the amount of heat transfer reduction (HTR) behaves nonmonotonically, which firstly increases but then decreases with Wi. The maximum HTR reaches around 8.7% at the critical Wi. The nonmonotonic behavior of HTR as a function of Wi was then corroborated with the modifications of the period of LSC and turbulent energy as well as viscous boundary layer thickness. Finally, a standard turbulent kinetic energy (TKE) budget analysis was done for the whole domain, the boundary layer region, and the bulk region. It showed that the role change of elastic stress contributions to TKE is mainly responsible for this nonmonotonic behavior of HTR.

Experimental velocity study of non-Boussinesq Rayleigh-Bénard convection

The effects of strongly varying fluid properties, beyond the validity range of the so-called Boussinesq approximation, were experimentally studied in Rayleigh-Bénard (RB) convection. Two experiments were conducted in the same cubical RB convection cell at similar Rayleigh and Prandtl numbers. In one experiment water was used as working fluid and the imposed temperature difference between the top and bottom plates of the cell was such to ensure non-Boussinesq conditions. In the other experiment, taken as a reference for Boussinesq conditions, methanol was used as working fluid, allowing a smaller temperature difference between the plates. In both experiments the instantaneous and time-averaged flow fields were determined experimentally in a vertical cross section of the cell by using particle image velocimetry. Results show a non-Boussinesq effect that manifests itself as an increase of the time-averaged horizontal velocity component close to the bottom wall of the cell and as a global top-bottom asymmetry of the velocity field. This is an experimental study of the whole velocity field of RB convection at non-Boussinesq conditions.

Dynamics of large-scale quantities in Rayleigh-Bénard convection

In this paper we estimate the relative strengths of various terms of the Rayleigh-Bénard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity U or the Péclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of Ra^{1/2} to the experimentally observed scaling of nearly Ra^{0.3}.

Vertically Bounded Double Diffusive Convection in the Finger Regime: Comparing No-Slip versus Free-Slip Boundary Conditions

Vertically bounded fingering double diffusive convection is numerically investigated, focusing on the influences of different velocity boundary conditions, i.e., the no-slip condition, which is inevitable in the lab-scale experimental researches, and the free-slip condition, which is an approximation for the interfaces in many natural environments, such as the oceans. For both boundary conditions the flow is dominated by fingers and the global responses follow the same scaling laws, with enhanced prefactors for the free-slip cases. Therefore, the laboratory experiments with the no-slip boundaries serve as a good model for the finger layers in the ocean. Moreover, in the free-slip case, although the tangential shear stress is eliminated at the boundaries, the local dissipation rate in the near-wall region may exceed the value found in the no-slip cases, which is caused by the stronger vertical motions of horizontally focused fingers and sheet structures near the free-slip boundaries. This counterintuitive result might be relevant for properly estimating and modeling the mixing and entrainment phenomena at free surfaces and interfaces widespread in oceans and geophysical flows.

Transitions in turbulent rotating convection: A Lagrangian perspective

Using measurements of Lagrangian acceleration in turbulent rotating convection and accompanying direct numerical simulations, we show that the transition between turbulent states reported earlier [e.g., S. Weiss et al., Phys. Rev. Lett. 105, 224501 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.224501] is a boundary-layer transition between the Prandtl-Blasius type (typical of nonrotating convection) and Ekman type.

From convection rolls to finger convection in double-diffusive turbulence

Double-diffusive convection (DDC), which is the buoyancy-driven flow with fluid density depending on two scalar components, is ubiquitous in many natural and engineering environments. Of great interests are scalars' transfer rate and flow structures. Here we systematically investigate DDC flow between two horizontal plates, driven by an unstable salinity gradient and stabilized by a temperature gradient. Counterintuitively, when increasing the stabilizing temperature gradient, the salinity flux first increases, even though the velocity monotonically decreases, before it finally breaks down to the purely diffusive value. The enhanced salinity transport is traced back to a transition in the overall flow pattern, namely from large-scale convection rolls to well-organized vertically oriented salt fingers. We also show and explain that the unifying theory of thermal convection originally developed by Grossmann and Lohse for Rayleigh-Bénard convection can be directly applied to DDC flow for a wide range of control parameters (Lewis number and density ratio), including those which cover the common values relevant for ocean flows.

Turbulent boundary layer in high Rayleigh number convection in air

Flow visualizations and particle image velocimetry measurements in the boundary layer of a Rayleigh-Bénard experiment are presented for the Rayleigh number Ra=1.4×1010. Our visualizations indicate that the appearance of the flow structures is similar to ordinary (isothermal) turbulent boundary layers. Our particle image velocimetry measurements show that vorticity with both positive and negative sign is generated and that the smallest flow structures are 1 order of magnitude smaller than the boundary layer thickness. Additional local measurements using laser Doppler velocimetry yield turbulence intensities up to I=0.4 as in turbulent atmospheric boundary layers. From our observations, we conclude that the convective boundary layer becomes turbulent locally and temporarily although its Reynolds number Re≈200 is considerably smaller than the value 420 underlying existing phenomenological theories. We think that, in turbulent Rayleigh-Bénard convection, the transition of the boundary layer towards turbulence depends on subtle details of the flow field and is therefore not universal.

Scaling of heat flux and energy spectrum for very large Prandtl number convection

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ(rms). We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra(γ) with γ ≈ (0.30-0.32), Pe ∼ Ra(η) with η ≈ (0.57-0.61), and θ(rms) ∼ const. The Nusselt number is observed to be an intricate function of Pe, θ(rms), and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E(u)(k) ∼ k(-13/3), which is in a very good agreement with our numerical results. The entropy spectrum E(θ)(k), however, exhibits dual branches consisting of k(-2) and k(0) spectra; the k(-2) branch corresponds to the Fourier modes θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.

Completing the mechanical energy pathways in turbulent Rayleigh-Bénard convection

A new, more complete view of the mechanical energy budget for Rayleigh-Bénard convection is developed and examined using three-dimensional numerical simulations at large Rayleigh numbers and Prandtl number of 1. The driving role of available potential energy is highlighted. The relative magnitudes of different energy conversions or pathways change significantly over the range of Rayleigh numbers Ra ~ 10(7)-10(13). At Ra < 10(7) small-scale turbulent motions are energized directly from available potential energy via turbulent buoyancy flux and kinetic energy is dissipated at comparable rates by both the large- and small-scale motions. In contrast, at Ra ≥ 10(10) most of the available potential energy goes into kinetic energy of the large-scale flow, which undergoes shear instabilities that sustain small-scale turbulence. The irreversible mixing is largely confined to the unstable boundary layer, its rate exactly equal to the generation of available potential energy by the boundary fluxes, and mixing efficiency is 50%.

Dissipation layers in Rayleigh-Bénard convection: a unifying view

Boundary layers play an important role in controlling convective heat transfer. Their nature varies considerably between different application areas characterized by different boundary conditions, which hampers a uniform treatment. Here, we argue that, independent of boundary conditions, systematic dissipation measurements in Rayleigh-Bénard convection capture the relevant near-wall structures. By means of direct numerical simulations with varying Prandtl numbers, we demonstrate that such dissipation layers share central characteristics with classical boundary layers, but, in contrast to the latter, can be extended naturally to arbitrary boundary conditions. We validate our approach by explaining differences in scaling behavior observed for no-slip and stress-free boundaries, thus paving the way to an extension of scaling theories developed for laboratory convection to a broad class of natural systems.

Spatial distribution of heat flux and fluctuations in turbulent Rayleigh-Bénard convection

We numerically investigate the radial dependence of the velocity and temperature fluctuations and of the time-averaged heat flux j ¯(r) in a cylindrical Rayleigh-Bénard cell with aspect ratio Γ=1 for Rayleigh numbers Ra between 2×10^{6} and 2×10^{9} at a fixed Prandtl number Pr=5.2. The numerical results reveal that the heat flux close to the sidewall is larger than in the center and that, just as the global heat transport, it has an effective power law dependence on the Rayleigh number, j ¯(r)∝Ra{γ{j}(r)}. The scaling exponent γ{j}(r) decreases monotonically from 0.43 near the axis (r≈0) to 0.29 close to the sidewalls (r≈D/2). The effective exponents near the axis and the sidewall agree well with the measurements of Shang et al. [Phys. Rev. Lett. 100, 244503 (2008)] and the predictions of Grossmann and Lohse [Phys. Fluids 16, 1070 (2004)]. Extrapolating our results to large Rayleigh number would imply a crossover at Ra≈10^{15}, where the heat flux near the axis would begin to dominate. In addition, we find that the local heat flux is more than twice as high at the location where warm or cold plumes go up or down than in plume depleted regions.

Logarithmic temperature profiles in turbulent Rayleigh-Bénard convection

We report results for the temperature profiles of turbulent Rayleigh-Bénard convection (RBC) in the interior of a cylindrical sample of aspect ratio Γ≡D/L=0.50 (D and L are the diameter and height, respectively). Both in the classical and in the ultimate state of RBC we find that the temperature varies as A×ln(z/L)+B, where z is the distance from the bottom or top plate. In the classical state, the coefficient A decreases in the radial direction as the distance from the side wall increases. For the ultimate state, the radial dependence of A has not yet been determined. These findings are based on experimental measurements over the Rayleigh-number range 4×10(12)≲Ra≲10(15) for a Prandtl number Pr≃0.8 and on direct numerical simulation at Ra=2×10(12), 2×10(11), and 2×10(10), all for Pr=0.7.

New perspectives in turbulent Rayleigh-Bénard convection

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.

Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample

We numerically investigate the structures of the near-plate temperature profiles close to the bottom and top plates of turbulent Rayleigh-Bénard flow in a cylindrical sample at Rayleigh numbers Ra = 10(8) to Ra = 2 × 10(12) and Prandtl numbers Pr = 6.4 and Pr = 0.7 with the dynamical frame method [Zhou and Xia, Phys. Rev. Lett. 104, 104301 (2010)], thus extending previous results for quasi-two-dimensional systems to three-dimensional systems. The dynamical frame method shows that the measured temperature profiles in the spatially and temporally local frame are much closer to the temperature profile of a laminar, zero-pressure gradient boundary layer (BL) according to Pohlhausen than in the fixed reference frame. The deviation between the measured profiles in the dynamical reference frame and the laminar profiles increases with decreasing Pr, where the thermal BL is more exposed to the bulk fluctuations due to the thinner kinetic BL, and increasing Ra, where more plumes are passing the measurement location.

Transition to the ultimate state of turbulent Rayleigh-Bénard convection

Measurements of the Nusselt number Nu and of a Reynolds number Re(eff) for Rayleigh-Bénard convection (RBC) over the Rayleigh-number range 10(12)≲Ra≲10(15) and for Prandtl numbers Pr near 0.8 are presented. The aspect ratio Γ≡D/L of a cylindrical sample was 0.50. For Ra≲10(13) the data yielded Nu∝Ra(γ(eff)) with γ(eff)≃0.31 and Re(eff)∝Ra(ζ(eff)) with ζ(eff)≃0.43, consistent with classical turbulent RBC. After a transition region for 10(13)≲Ra≲5×10(14), where multistability occurred, we found γ(eff)≃0.38 and ζ(eff)=ζ≃0.50, in agreement with the results of Grossmann and Lohse for the large-Ra asymptotic state with turbulent boundary layers which was first predicted by Kraichnan.

Effect of aspect ratio on vortex distribution and heat transfer in rotating Rayleigh-Bénard convection

Numerical and experimental data for the heat transfer as a function of the Rossby number Ro in turbulent rotating Rayleigh-Bénard convection are presented for the Prandtl number Pr=4.38 and the Rayleigh number Ra=2.91×10(8) up to Ra=4.52×10(9). The aspect ratio Γ≡D/L, where L is the height and D the diameter of the cylindrical sample, is varied between Γ=0.5 and 2.0. Without rotation, where the aspect ratio influences the global large-scale circulation, we see a small-aspect-ratio dependence in the Nusselt number for Ra=2.91×10(8). However, for stronger rotation, i.e., 1/Ro>>1/Ro(c), the heat transport becomes independent of the aspect ratio. We interpret this finding as follows: In the rotating regime the heat is mainly transported by vertically aligned vortices. Since the vertically aligned vortices are local, the aspect ratio has a negligible effect on the heat transport in the rotating regime. Indeed, a detailed analysis of vortex statistics shows that the fraction of the horizontal area that is covered by vortices is independent of the aspect ratio when 1/Ro>>1/Ro(c). In agreement with the results of Weiss et al. [Phys. Rev. Lett. 105, 224501 (2010)], we find a vortex-depleted area close to the sidewall. Here we show that there is also an area with enhanced vortex concentration next to the vortex-depleted edge region and that the absolute widths of both regions are independent of the aspect ratio.

Local energy dissipation rate balances local heat flux in the center of turbulent thermal convection

The local kinetic energy dissipation rate ε(u,c) in Rayleigh-Bénard convection cell was measured experimentally using the particle tracking velocimetry method, with varying Rayleigh number Ra, Prandtl number Pr, and cell height H. It is found that ε(u,c)/(κ(3)H(-4))=1.05×10(-4)Ra(1.55±0.02)Pr(1.15±0.38). The Ra and H dependencies of the measured results are found to be consistent with the assumption made for the bulk energy dissipation rate ε(u,bulk) in the Grossmann-Lohse model. A remarkable finding of the study is that ε(u,c) balances the directly measured local Nusselt number Nu(c) in the cell center, not only scalingwise but also in magnitude.

The Twente turbulent Taylor-Couette (T3C) facility: strongly turbulent (multiphase) flow between two independently rotating cylinders

A new turbulent Taylor-Couette system consisting of two independently rotating cylinders has been constructed. The gap between the cylinders has a height of 0.927 m, an inner radius of 0.200 m, and a variable outer radius (from 0.279 to 0.220 m). The maximum angular rotation rates of the inner and outer cylinder are 20 and 10 Hz, respectively, resulting in Reynolds numbers up to 3.4 × 10(6) with water as working fluid. With this Taylor-Couette system, the parameter space (Re(i), Re(o), η) extends to (2.0 × 10(6), ±1.4 × 10(6), 0.716-0.909). The system is equipped with bubble injectors, temperature control, skin-friction drag sensors, and several local sensors for studying turbulent single-phase and two-phase flows. Inner cylinder load cells detect skin-friction drag via torque measurements. The clear acrylic outer cylinder allows the dynamics of the liquid flow and the dispersed phase (bubbles, particles, fibers, etc.) inside the gap to be investigated with specialized local sensors and nonintrusive optical imaging techniques. The system allows study of both Taylor-Couette flow in a high-Reynolds-number regime, and the mechanisms behind skin-friction drag alterations due to bubble injection, polymer injection, and surface hydrophobicity and roughness.

Flow reversals in thermally driven turbulence

We analyze the reversals of the large-scale flow in Rayleigh-Bénard convection both through particle image velocimetry flow visualization and direct numerical simulations of the underlying Boussinesq equations in a (quasi-) two-dimensional, rectangular geometry of aspect ratio 1. For medium Prandtl number there is a diagonal large-scale convection roll and two smaller secondary rolls in the two remaining corners diagonally opposing each other. These corner-flow rolls play a crucial role for the large-scale wind reversal: They grow in kinetic energy and thus also in size thanks to plume detachments from the boundary layers up to the time that they take over the main, large-scale diagonal flow, thus leading to reversal. The Rayleigh vs Prandtl number space is mapped out. The occurrence of reversals sensitively depends on these parameters.

Measured instantaneous viscous boundary layer in turbulent Rayleigh-Bénard convection

We report measurements of the instantaneous viscous boundary layer (BL) thickness delta(v)(t) in turbulent Rayleigh-Bénard convection. It is found that delta(v)(t) obtained from the measured instantaneous two-dimensional velocity field exhibits intermittent fluctuations. For small values, delta(v)(t) obeys a lognormal distribution, whereas for large values, the distribution of delta(v)(t) exhibits an exponential tail. The variation of delta(v)(t) with time is found to be driven by the fluctuations of the large-scale mean-flow velocity, and the local horizontal velocities close to the plate can be used as an instant measure of this variation. It is further found that in the present parameter range of the experiment, the mean velocity profile measured in the laboratory frame can be brought into coincidence with the theoretical Prandtl-Blasius laminar BL profile, if it is resampled relative to the time-dependent frame of delta(v)(t).

Structure of viscous boundary layers in turbulent Rayleigh-Bénard convection

Highly resolved profiles of the mean velocity in turbulent Rayleigh-Bénard convection in air are presented and discussed. The present work extends our recently performed experiments at constant aspect ratio [Phys. Rev. Lett. 99, 234504 (2007)] to variable aspect ratios. The experiments cover a range of Rayleigh numbers 10(9)<Ra<10(12) and aspect ratios 1.13<Gamma<11.3 whereas the Prandtl number is fixed at Pr=0.7. The major finding of the present work is that the profiles of the mean horizontal velocity and its fluctuations are virtually invariant against the variation in Ra or Gamma if the wall distance is scaled by the displacement thickness of the boundary layer. Furthermore we have studied typical length scales of the boundary layer and their scaling with Ra and Gamma. Regarding a potential transition of the heat transport toward the ultimate regime we found that the boundary layer Reynolds number remains below Redelta=250 which is significantly lower than the predicted limit.

Heat transport measurements in turbulent rotating Rayleigh-Bénard convection

We present experimental heat transport measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with a Prandtl number (sigma) of about 6, was confined in a cell with a square cross section of 7.3 x 7.3 cm2 and a height of 9.4 cm. Heat transport was measured for Rayleigh numbers 2 x 10(5)<Ra<5 x 10(8) and Taylor numbers 0<Ta<5 x 10(9). We show the variation in normalized heat transport, the Nusselt number, at fixed dimensional rotation rate OmegaD, at fixed Ra varying Ta, at fixed Ta varying Ra, and at fixed Rossby number Ro. The scaling of heat transport in the range of 10(7) to about 10(9) is roughly 0.29 with a Ro-dependent coefficient or equivalently is also well fit by a combination of power laws of the form a Ra1/5+b Ra1/3. The range of Ra is not sufficient to differentiate single power law or combined power-law scaling. The data are roughly consistent with an assumption that the enhancement of heat transport owing to rotation is proportional to the number of vortical structures penetrating the boundary layer. We also compare indirect measures of thermal and Ekman boundary layer thicknesses to assess their potential role in controlling heat transport in different regimes of Ra and Ta.

Search for the "ultimate state" in turbulent Rayleigh-Bénard convection

Measurements of the Nusselt number Nu and of temperature variations DeltaTb in the bulk fluid are reported for turbulent Rayleigh-Bénard convection of a cylindrical sample. They cover the Rayleigh-number range 10(9) less than or similar to Ra less than or similar to 3x10(14) using He (Prandtl number Pr=0.67), N2 (Pr=0.72) and SF6 (Pr=0.79 to 0.84) at pressures up to 15 bars and near-ambient temperatures. The sample had a height L=2.24 m and diameter D=1.12 m and was located in a new High-Pressure Convection Facility (HPCF) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen, Germany. The data do not show the transition to an "ultimate regime" reported by Chavanne et al. and are consistent with the measurements of Niemela et al.

Measurements of the thermal dissipation field in turbulent Rayleigh-Bénard convection

A systematic study of the thermal dissipation field and its statistical properties is carried out in turbulent Rayleigh-Bénard convection. A local temperature gradient probe consisting of four identical thermistors is made to measure the normalized thermal dissipation rate epsilonN(r) in two convection cells filled with water. The measurements are conducted over varying Rayleigh numbers Ra (8.9x10(8)<approximately Ra<approximately 9.3x10(9)) and spatial positions r across the entire cell. It is found that epsilonN(r) contains two contributions; one is generated by thermal plumes, present mainly in the plume-dominated bulk region, and decreases with increasing Ra. The other contribution comes from the mean temperature gradient, being concentrated in the thermal boundary layers, and increases with Ra. The experiment provides a complete physical picture about the thermal dissipation field and its statistical properties in turbulent convection.

Ultimate-state scaling in a shell model for homogeneous turbulent convection

An interesting question in turbulent convection is how the heat transport depends on the strength of thermal forcing in the limit of very large thermal forcing. Kraichnan predicted [Phys. Fluids 5, 1374 (1962)] that for fluids with low Prandtl number (Pr), the heat transport measured by the Nusselt number (Nu) would depend on the strength of thermal forcing measured by the Rayleigh number (Ra) as Nu approximately Ra(1/2) with logarithmic corrections at very high Ra. According to Kraichnan, the shear boundary layers play a crucial role in giving rise to this so-called ultimate-state scaling. A similar scaling result is predicted by the Grossmann-Lohse theory [J. Fluid Mech. 407, 27 (2000)], but with the assumption that the ultimate state is a bulk-dominated state in which both the average kinetic and thermal dissipation rates are dominated by contributions from the bulk of the flow with the boundary layers either broken down or playing no role in the heat transport. In this paper, we study the dependence of Nu and the Reynolds number (Re) measuring the root-mean-squared velocity fluctuations on Ra and Pr, for low Pr, using a shell model for homogeneous turbulent convection where buoyancy is acting directly on most of the scales. We find that Nu approximately Ra(1/2)Pr(1/2) and Re approximately Ra(1/2)Pr(-1/2) , which resemble the ultimate-state scaling behavior for fluids with low Pr, and show that the presence of a drag acting on the large scales is crucial in giving rise to such scaling. As a large-scale drag cannot exist by itself in the bulk of turbulent thermal convection, our results indicate that if buoyancy acts on most of the scales in the bulk of turbulent convection at very high Ra, then the ultimate state cannot be bulk dominated.

Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point

As shown in earlier work [Ahlers, J. Fluid Mech. 569, 409 (2006)], non-Oberbeck-Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Bénard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point, the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient beta(T). More precisely, it is the nonlinear T dependence of the density rho(T) in the buoyancy force that causes another type of NOB effect. We demonstrate this through a combination of experimental, numerical, and theoretical work, the last in the framework of the extended Prandtl-Blasius boundary-layer theory developed by Ahlers as cited above. The theory comes to its limits if the temperature dependence of the thermal expension coefficient beta(T) is significant. The measurements reported here cover the ranges 2.1<or similar to Pr<or similar to 3.9 and 5x10(9)<or similar to Ra<or similar to 2x10(12) and are for cylindrical samples of aspect ratios 1.0 and 0.5.

Measured thermal dissipation field in turbulent Rayleigh-Bénard convection

The time-averaged local thermal dissipation rate epsilonN(r) in turbulent convection is obtained from direct measurements of the temperature gradient vector in a cylindrical cell filled with water. It is found that epsilonN(r) contains two contributions. One is generated by thermal plumes, present mainly in the plume-dominated bulk region, and decreases with increasing Rayleigh number Ra. The other contribution comes from the mean temperature gradient, being concentrated in the thermal boundary layers, and increases with Ra. The experiment thus provides a new physical picture about the thermal dissipation field in turbulent convection.

Large-scale circulation model for turbulent Rayleigh-Bénard convection

A model for the large-scale-circulation (LSC) of turbulent Rayleigh-Bénard convection in cylindrical samples is presented. It consists of two physically motivated stochastic ordinary differential equations, one each for the strength and the azimuthal orientation of the LSC. Stochastic forces represent phenomenologically the influence of turbulent fluctuations. Consistent with measurements, the model yields an azimuthally meandering LSC with occasional rotations, and with more rare cessations. As in experiment, cessations have a uniform distribution of LSC orientation changes.

Bubbly turbulent drag reduction is a boundary layer effect

In turbulent Taylor-Couette flow, the injection of bubbles reduces the overall drag. On the other hand, rough walls enhance the overall drag. In this work, we inject bubbles into turbulent Taylor-Couette flow with rough walls (with a Reynolds number up to 4 x 10(5), finding an enhancement of the dimensionless drag as compared to the case without bubbles. The dimensional drag is unchanged. As in the rough-wall case no smooth boundary layers can develop, the results demonstrate that bubbly drag reduction is a pure boundary layer effect.

Non-oberbeck-boussinesq effects in gaseous Rayleigh-Bénard convection

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Bénard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease of the central temperature as compared to the arithmetic mean of the top- and bottom-plate temperature and an increase of the Nusselt number. Both effects are of opposite sign and greater magnitude than those for NOB convection in liquids like water.