Numerical and experimental investigation of structure-function scaling in turbulent Rayleigh-Bénard convection

A randomly stirred model for Bolgiano-Obukhov scaling in turbulence in a stably stratified fluid

A randomly stirred model, akin to the one used by DeDominicis and Martin for homogeneous isotropic turbulence, is introduced to study Bolgiano-Obukhov scaling in fully developed turbulence in a stably stratified fluid. The energy spectrum E(k), where k is a wavevector in the inertial range, is expected to show the Bolgiano-Obukhov scaling at a large Richardson number Ri (a measure of the stratification). We find that the energy spectrum is anisotropic. Averaging over the directions of the wavevector, we find [Formula: see text], where ε_θ is the constant energy transfer rate across wavenumbers with very little contribution coming from the kinetic energy flux. The constant K₀ is estimated to be of O(0.1) as opposed to the Kolmogorov constant, which is O(1). Further for a pure Bolgiano-Obukhov scaling, the model requires that the large distance 'stirring' effects dominate in the heat diffusion and be small in the velocity dynamics. These could be reasons why the Bolgiano-Obukhov scaling is difficult to observe both numerically and experimentally. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like Ra^{1/4}Ek^{2/3} where the Ekman number Ek decreases with increasing rotation rate.

Applicability of Taylor's hypothesis in thermally driven turbulence

In this paper, we show that, in the presence of large-scale circulation (LSC), Taylor's hypothesis can be invoked to deduce the energy spectrum in thermal convection using real-space probes, a popular experimental tool. We perform numerical simulation of turbulent convection in a cube and observe that the velocity field follows Kolmogorov's spectrum (k^-5/3). We also record the velocity time series using real-space probes near the lateral walls. The corresponding frequency spectrum exhibits Kolmogorov's spectrum (f^-5/3), thus validating Taylor's hypothesis with the steady LSC playing the role of a mean velocity field. The aforementioned findings based on real-space probes provide valuable inputs for experimental measurements used for studying the spectrum of convective turbulence.

Statistics of velocity and temperature fluctuations in two-dimensional Rayleigh-Bénard convection

We investigate fluctuations of the velocity and temperature fields in two-dimensional (2D) Rayleigh-Bénard (RB) convection by means of direct numerical simulations (DNS) over the Rayleigh number range 10^{6}≤Ra≤10^{10} and for a fixed Prandtl number Pr=5.3 and aspect ratio Γ=1. Our results show that there exists a counter-gradient turbulent transport of energy from fluctuations to the mean flow both locally and globally, implying that the Reynolds stress is one of the driving mechanisms of the large-scale circulation in 2D turbulent RB convection besides the buoyancy of thermal plumes. We also find that the viscous boundary layer (BL) thicknesses near the horizontal conducting plates and near the vertical sidewalls, δ_{u} and δ_{v}, are almost the same for a given Ra, and they scale with the Rayleigh and Reynolds numbers as ∼Ra^{-0.26±0.03} and ∼Re^{-0.43±0.04}. Furthermore, the thermal BL thickness δ_{θ} defined based on the root-mean-square (rms) temperature profiles is found to agree with Prandtl-Blasius predictions from the scaling point of view. In addition, the probability density functions of turbulent energy ɛ_{u^{'}} and thermal ɛ_{θ^{'}} dissipation rates, calculated, respectively, within the viscous and thermal BLs, are found to be always non-log-normal and obey approximately a Bramwell-Holdsworth-Pinton distribution first introduced to characterize rare fluctuations in a confined turbulent flow and critical phenomena.

Probing the energy cascade of convective turbulence

The existence of a buoyancy-dominated scaling range in convective turbulence is a longstanding open question. We investigate this issue by considering the scale-by-scale energy budget in direct numerical simulations of Rayleigh-Bénard convection. We try to minimize the so-called Bolgiano length scale, the length scale at which buoyancy becomes dominant for scaling. Therefore, we deliberately choose modest Rayleigh numbers Ra=2.5×10(6) and 2.5×10(7). The budget reveals that buoyant forcing, turbulent energy transfer, and dissipation are contributing significantly over a wide range of scales. Thereby neither Kolmogorov-like (balance of turbulent transfer and dissipation) nor Bolgiano-Obukhov-like scaling (balance of turbulent transfer and buoyancy) is expected in the structure functions, which indeed reveal inconclusive scaling behavior. Furthermore, we consider the calculation of the Bolgiano length scale. To account for correlations between the dissipation rates of kinetic energy and thermal variance we propose to average the Bolgiano length scale directly. This gives an estimate, which is one order of magnitude larger than the previous estimate, and actually larger than the domain itself. Rather than studying the scaling of structure functions, we propose that the use of scale-by-scale energy budgets resolving anisotropic contributions is appropriate to consider the energy cascade mechanisms in turbulent convection.

Polymer-induced change in scaling behavior in two-dimensional homogeneous turbulent thermal convection

We study the effects of polymers in two-dimensional turbulent thermal convection using a shell model. In the absence of polymers, the inverse energy cascade in two dimensions leads to the observed Bolgiano-Obukhov scaling. When polymers are added, energy is extracted from the flow by the polymers, and as a result, the thermal balance between buoyancy and inertia in Bolgiano-Obukhov scaling is destroyed around the scales at which polymers interact strongly with the flow. This results in an increase in the Bolgiano scale and leads to a change in the scaling behavior of the velocity and temperature fluctuations for scales below the modified Bolgiano scale. We make theoretical estimates of the dependence of the mean rate of energy extracted by the polymers and the mean energy dissipation rate on the polymer relaxation time. Our theoretical analysis further leads to the prediction that the heat transport is not altered much by the polymers in two dimensions. We show that our theoretical estimates and prediction are in good agreement with the numerical results.

Flow patterns in inclined-layer turbulent convection

We study the flow patterns of turbulent convection in an inclined layer with a large aspect ratio Γ and moderately high Rayleigh numbers ranging from 9 × 10(4) to 2 × 10(7) based on three-dimensional numerical simulations. The Prandtl number is fixed at σ = 0.7 and the angles of inclination are varied between 5° ≤ θ ≤ 60°. The initial quiescent fluid layer is observed to firstly evolve into a quasi-periodical flow pattern before the turbulent convection is fully developed. The transient flow at earlier times, though elongated along the slope and anisotropic in the directions parallel to the top and bottom plates, becomes isotropic in the final statistically steady state, provided that the Rayleigh number is high (R ≃ 2 × 10(7)) and the angle of inclination is small (θ ≤ 17°). The effect of inclination on the large-scale flow is different from that on individual plumes, which exhibits isotropy and is independent of the angles of inclination for Rayleigh numbers above 5 × 10(6). The regions near the upper and lower sidewalls of the enclosure, considered as extensions of the thermal boundary layer, shrink with increasing Rayleigh numbers and the scaling exponent is about 2/7.

Entropy and energy spectra in low-Prandtl-number convection with rotation

We present results for entropy and kinetic energy spectra computed from direct numerical simulations for low-Prandtl-number (Pr < 1) turbulent flow in Rayleigh-Bénard convection with uniform rotation about a vertical axis. The simulations are performed in a three-dimensional periodic box for a range of the Taylor number (0 ≤ Ta ≤ 10(8)) and reduced Rayleigh number r = Ra/Ra(∘)(Ta,Pr) (1.0 × 10(2) ≤ r ≤ 5.0 × 10(3)). The Rossby number Ro varies in the range 1.34 ≤ Ro ≤ 73. The entropy spectrum E(θ)(k) shows bisplitting into two branches for lower values of wave number k. The entropy in the lower branch scales with k as k(-1.4 ± 0.1) for r>10(3) for the rotation rates considered here. The entropy in the upper branch also shows scaling behavior with k, but the scaling exponent decreases with increasing Ta for all r. The energy spectrum E(v)(k) is also found to scale with the wave number k as k(-1.4 ± 0.1) for r>10(3). The scaling exponent for the energy spectrum and the lower branch of the entropy spectrum vary between -1.7 and -2.4 for lower values of r (<10(3)). We also provide some simple arguments based on the variation of the Kolmogorov picture to support the results of simulations.

Scaling behavior in turbulent Rayleigh-Bénard convection revealed by conditional structure functions

We show that the nature of the scaling behavior can be revealed by studying the conditional structure functions evaluated at given values of the locally averaged thermal dissipation rate. These conditional structure functions have power-law dependence on the value of the locally averaged thermal dissipation rate, and such dependence for the Bolgiano-Obukhov scaling is different from the other scaling behaviors. Our analysis of experimental measurements verifies the power-law dependence and reveals the Bolgiano-Obukhov scaling behavior at the center of the bottom plate of the convection cell.

Vortex statistics in turbulent rotating convection

The vortices emerging in rotating turbulent Rayleigh-Bénard convection in water at Rayleigh number Ra=6.0×10{8} are investigated using stereoscopic particle image velocimetry and by direct numerical simulation. The so-called Q criterion is used to detect the vortices from velocity fields. This criterion allows distinguishing vorticity- and strain-dominated regions in the flow by decomposing the velocity gradient tensor into symmetric and antisymmetric parts. Vortex densities, mean vortex radii and mean vortex circulations are calculated at two horizontal cross-sections of the cylindrical flow domain and at several rotation rates, described by the Taylor number which takes values between 3.0×10{8} and 7.7×10{10} . Separate statistics are calculated for cyclonic and anticyclonic vortices. Vortex densities and mean vortex radii are mostly independent of the Taylor number except very close to the bottom and top plates where more vortices are detected when the Taylor number is raised (rotation increases). The vortex population close to the plate consists mostly of cyclones while further into the bulk of the domain a similar amount of cyclones and anticyclones is found. The cyclonic vortices contain more circulation than the anticyclones. The same vortex analysis of the simulation results at additional vertical positions revealed that the vortices are formed in a boundary layer on the plate with a thickness of approximately two Ekman lengths.

Energy spectra and fluxes for Rayleigh-Bénard convection

We compute the spectra and fluxes of the velocity and temperature fields in Rayleigh-Bénard convection in turbulent regime for a wide range of Prandtl numbers using pseudospectral simulations on 512(3) grids. Our spectral and flux results support the Kolmogorov-Obukhov (KO) scaling for zero Prandtl number and low Prandtl number (P=0.02) convection. The KO scaling for the velocity field in zero-Prandtl number and low-Prandtl number convection is because of the weak buoyancy in the inertial range (buoyancy is active only at the very low wave numbers). We also observe that for intermediate Prandtl numbers (P=0.2) the KO scaling fits better with the numerical results than the Bolgiano-Obukhov (BO) scaling. For large Prandtl number (P=6.8) , the spectra and flux results are somewhat inconclusive on the validity of the KO or BO scaling, yet the BO scaling is preferred over the KO scaling for these cases. The numerical results for P=1 is rather inconclusive.

Enhanced vertical inhomogeneity in turbulent rotating convection

In this Letter we report experimental evidence that rotation enhances vertical inhomogeneity in turbulent convection, in spite of the increased columnar flow ordering under rotation. Measurements using stereoscopic particle image velocimetry have been carried out on turbulent rotating convection in water. At constant Rayleigh number Ra=1.11 x 10(9) several rotation rates have been used, so that the Rossby number takes values from Ro=infinity (no rotation) to 0.09 (strong rotation). The three-component velocity data, obtained at two vertical positions, are used to investigate the anisotropy of the flow through the invariants of the Reynolds-stress anisotropy tensor and the Lumley triangle, as well as to correlate the vertical velocity and vorticity. In the center plane rotation causes the turbulence to be "rodlike," while closer to the top plate a trend toward isotropy is observed.

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.