Efficiency of heat transfer in turbulent Rayleigh-Bénard convection

We present an experimental study of turbulent Rayleigh-Bénard convection (RBC) in a cylindrical cell of height 0.3 m, diameter 0.3 m. It is designed to minimize the influence of its structure on the convective flow of cryogenic (4)He gas of Prandtl number Pr≈1, with the aim of resolving existing contradictions in Nusselt (Nu) versus Rayleigh number (Ra) scaling. For 7.2×10(6)≤Ra≤10(11) our data agree with suitably corrected data from similar cryogenic experiments and are consistent with Nu∝Ra(2/7). On approaching Ra≈10(11) our data display a crossover to Nu∝Ra(1/3) that approximately holds up to Ra=4.6×10(13); there is no sign of a transition to the ultimate Kraichnan regime. Differences in Nu(Ra) scaling observed in similar RBC experiments for Ra≥10(11) cannot be explained due to the difference in Pr, but seem to depend also on experimental details.     

Plumes and waves in two-dimensional turbulent thermal convection

We have conducted a high-resolution, two-dimensional direct numerical simulation of Rayleigh-Bénard convection with stress-free and periodic boundary conditions at a Rayleigh (Ra) number of 10(8) and Prandtl (Pr) number of unity. An aspect-ratio three box has been considered. A single cell has been used as the initial condition. First, the flow develops into time-dependent convection with a strong asymmetry and highly convoluted thermal plumes delineating a large-scale circulation. Smaller thermal plumes detach from the boundary layer and extend over the entire cell, creating a local inversion of the temperature gradient adjacent to the boundary layers. Then the conditions leading to the formation of internal waves are fulfilled, as the local Richardson number decreases sufficiently small to cross the linear threshold of Ri=0.25. Together with the strong shear, convective rolls with a Kelvin-Helmholtz wavelike character are produced. The secondary boundary layer itself becomes unstable and produces smaller plumes. At later times, the large-scale circulation is destroyed and the internal waves disappear. A Reynolds number, based on the global scale, of Re=500, is attained at this stage. Only isolated thermal plumes and vortices are present. Thus, internal waves can be generated at finite Prandtl number fluids for sufficiently high Ra in the presence of a large-scale circulation. Spectral analysis reveals that the kinetic energy decays with a logarithmic slope of -3, while the logarithmic slope of the thermal variance has a value of around -5 / 3.     

Upper bound on the heat transport in a layer of fluid of infinite prandtl number, rigid lower boundary, and stress-free upper boundary

We obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite Prandtl number with rigid lower boundary and stress-free upper boundary. Because of the asymmetric boundary conditions the solutions of the Euler-Lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid. The obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress-free boundaries.     

Extraction of plumes in turbulent thermal convection

We present a scheme to extract the velocity of buoyant structures in turbulent thermal convection from simultaneous local velocity and temperature measurements. Applying this scheme to measurements taken at positions within the convection cell where the buoyant structures are dominated by plumes, we obtain the temperature dependence of the plume velocity and understand our results using the equations of motion. We further obtain the scaling behavior of the average local heat flux in the vertical direction at the cell center with the Rayleigh number and find that the scaling exponent is different from that measured for the Nusselt number. This difference leads to the conclusion that heat cannot be mainly transported through the central region of the convection cell.     

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.     

Scaling in convective evaporation and sidewall boundary layer

Turbulent convection at aspect ratios from 0.06 to 2 is investigated in the laboratory with evaporation experiments from vertical cylinders having different diameters and liquid levels. With alcohol, only diffusive evaporation takes place. With water, for small diameters, evaporation proceeds by diffusion whereas convective evaporation develops when the diameter is increased. This onset can be effectively interpreted in terms of a viscous sidewall boundary layer, whose thickness δ varies with respect to the available height h according to δ/h = 3.4 Ra^-0.28±0.01 versus Rayleigh number Ra. The Sherwood number Sh, analog of the Nusselt number, exhibits a power law variation Sh = 0.6 Ra^0.27±0.02 for Ra varying from 10⁴ to 3 ×10⁸. The scaling observed in this case of an open boundary is thus similar to the scaling measured in confined Rayleigh-Bénard convection.     

湍流热对流Prandtl数效应的数值研究

本文采用直接数值模拟的并行直接求解方法,计算了Ra=10~(10),0.05≤Pr≤20的系列Prandtl(Pr)数二维湍流热对流.通过流动显示技术,讨论了Pr数对羽流形态和大尺度环流结构的影响.在Ra=10~(10)时,随着Pr数减小,羽流的运动和分布表现出更强的湍流性质,较高Pr数的羽流则表现出较强的规律性,当Pr4.3时,流场中存在明显的大尺度环流和角涡结构.不同Pr数的温度边界层厚度差异不大,并随Pr数存在标度率变化关系.当Pr数较低时,系统的传热Nusselt(Nu)数随着Pr数增加而增加,当Pr数较高时,Nu数随Pr数的变化不敏感.靠近底板处速度脉动随Pr数有显著的变化,Pr数越低速度波动越剧烈.通过底板中心位置水平脉动速度和平均场水平速度最大值给出的雷诺数Re_((u))和Re_(U_(max)),两种Re数随Pr数的变化满足同一标度律,为Re～Pr~(-0.81).     

竖直恒温面自然对流边界层不稳定性研究

在无扰动、随机式扰动以及正弦式扰动下,通过对竖直恒温面处状态Ra为1.328×10^9、Pr为6.24的自然对流进行模拟,探索了热边界层的不稳定性和共振强化自然对流换热。结果表明:(1)竖直自然对流边界层上游位置的随机式扰动对热边界层的影响主要体现在稳定阶段;(2)该状态下的竖直自然对流边界层的特征频率为15 067,且相比于无扰动状态,频率为15 067的正弦式扰动能在竖直恒温面处提高5.15%的换热量;(3)在竖直自然对流边界层上游位置加入特征频率的正弦式扰动,竖直恒温面处的局部努塞尔数Nu均出现明显波动,且波动随着边界层高度的增加而增大。     

Thermal convection for large Prandtl numbers

The Rayleigh-Bénard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers Pr. The Nusselt number Nu is found here to be independent of Pr. However, for fixed Rayleigh numbers Ra a maximum in the Nu(Pr) dependence is predicted. We moreover offer the full functional dependences of Nu(Ra,Pr) and Re(Ra,Pr) within this extended theory, rather than only give the limiting power laws as done in J. Fluid. Mech. 407, 27 (2000). This enables us to more realistically describe the transitions between the various scaling regimes.     

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.     

Upper bounds on the convective heat transport in a rotating fluid layer of infinite Prandtl number: case of large Taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a horizontal fluid layer heated from below and rotating about a vertical axis. We consider the interval of large Taylor numbers where the intermediate layers of the optimum fields expand in the direction of the corresponding internal layers. We consider the 1 - α-solution of the arising variational problem for the cases of rigid-stress-free, stress-free, and rigid boundary conditions. For each kind of boundary condition we discuss four cases: two cases where the boundary layers are thinner than the Ekman layers of the optimum field and two cases where the boundary layers are thicker than the Ekman layers. In most cases we use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate layers of the optimum fields. This solution leads to corrections of the thicknesses of the boundary layers of the optimum fields and to lower upper bounds on the convective heat transport in comparison to the bounds obtained by Chan [J. Fluid Mech. 64, 477 (1974)] and Hunter and Riahi [J. Fluid Mech. 72, 433 (1975)]. Compared to the existing experimental data for the case of a fluid layer with rigid boundaries the corresponding upper bounds on the convective heat transport is less than two times larger than the experimental results, the corresponding upper bound on the convective heat transport, obtained by Hunter and Riahi is about 10% higher than the bound obtained in this article. When Rayleigh number and Taylor number are high enough the upper bound on the convective heat transport ceases to depend on the boundary conditions. Received 30 January 2001 and Received in final form 28 May 2001     

Laminar free convection heat transfer between vertical isothermal plates

The paper represents results on numerical investigation of flow and heat transfer between two isothermal vertical plates under laminar natural convection. A system of complete Navier–Stokes equations is solved for a two-dimensional gas flow between the plates along with additional rectangular regions (connected to inlet and outlet sections), whose characteristic sizes are much greater than the spacing between the plates. The calculations were performed over very wide ranges of Rayleigh number Ra = 10 ÷ 10⁵ and a relative channel length AR = L/w = 1 ÷ 500. The influence of the input parameters on the gas-dynamic and thermal structure of thermogravitational convection, the local and mean heat transfer, and also the gas flow rate between the plates (convective draft. We determined sizes of the regions and regime parameters when the local heat flux on the walls tends to zero due to the gas temperature approach to the surface temperature. It is shown that the mean heat transfer decreases as the relative channel length AR grows, whereas the integral gas flow rate (convective draft) and Reynolds number in the channel Re = 2wUm/ν increase. The use of a modified Rayleigh number Ra* = Ra · (w/L) (Elenbaas number) leads to generalization of calculation data on mean heat transfer. These data are in good agreement with the correlations for heat transfer [1, 2] and gas flow rate [3]. The reasons of variation of the data in the range of low Rayleigh numbers are discussed in detail.     

On scaling laws in turbulent magnetohydrodynamic Rayleigh-Benard convection

We invoke the concepts of magnetic boundary layer and magnetic Rayleigh number and use the rates of magnetic energy dissipation in the bulk and the boundary layers to derive some scaling laws expressing how the Nusselt number depends on the magnetic Rayleigh number, Prandtl number and magnetic Prandtl number for the simple case of turbulent magnetohydrodynamic Rayleigh-Benard convection in the presence of a uniform vertical magnetic field.     

Wind reversals in turbulent Rayleigh-Bénard convection

The phenomenon of irregular cessation and subsequent reversal of the large-scale circulation in turbulent Rayleigh-Bénard convection is theoretically analyzed. The force and thermal balance on a single plume detached from the thermal boundary layer yields a set of coupled nonlinear equations, whose dynamics is related to the Lorenz equations. For Prandtl and Rayleigh numbers in the range 10(-2) < or = Pr < or = 10(3) and 10(7) < or = Ra < or = 10(12), the model has the following features: (i) chaotic reversals may be exhibited at Ra > or = 10(7); (ii) the Reynolds number based on the root mean square velocity scales as Re(rms) approximately Ra([0.41...0.47]) (depending on Pr), and as Re(rms) approximately Pr(-[0.66...0.76]) (depending on Ra); and (iii) the mean reversal frequency follows an effective scaling law omega/(nu L(-2)) approximately Pr(-(0.64 +/- 0.01))Ra(0.44 +/- 0.01). The phase diagram of the model is sketched, and the observed transitions are discussed.     

正交梯度下双扩散对流的数值模拟

本文采用有限单元法求解涡量、流函数、能量和组分控制方程,数值模拟了在温度梯度和浓度梯度正交情况下竖直环形容器内的双扩散对流结构。通过改变浮升力比N=GrS／GrT和Le数的大小,分析了溶质浮升力方向及其大小和Le数对容器内双扩散对流的影响。结果表明:当溶质浮升力改变时,壁面处的流体流动状况、边界层厚度、Nu数和Sh数都发生了变化;随着Le数的增大,竖直壁面处Nu数逐渐降低而水平壁面处的Sh数逐渐增大。     

Scaling of the local convective heat flux in turbulent Rayleigh-Bénard convection

Local convective heat flux J(r) in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in a cylindrical cell filled with water. The measured J(r) in the bulk region shows a different scaling behavior with varying Rayleigh numbers compared with that measured in the plume-dominated regions near the sidewall and near the lower conducting plate. The local transport measurements thus allow us to disentangle boundary and bulk contributions to the total heat flux and directly check their respective scaling behavior against the theoretical predictions.     

滑移边界对二维Rayleigh-Bénard热对流流态和传热特性的影响

采用并行直接数值模拟（PDM-DNS）计算无滑移和滑移边界二维Rayleigh-Bénard热对流.与无滑移边界形成的随机羽流运动的湍流热对流不同,滑移边界热对流最终形成湍流特征消失,且温度仅分布于四壁的一个大尺度环流的流动形态.平均场近底板的温度分布特性,无滑移边界逐渐变化而滑移边界出现过冲现象.宽高比Γ=1时,Nusselt数（Nu）随Rayleigh数（Ra）的变化具有相同标度指数,Nu~Ra^0.3.滑移边界热对流具有传热增强作用.滑移边界热对流Nu随Γ变化明显,并分为两个阶段,在Γ=0.5时出现Nu_max≈250,是无滑移边界热流Nu的5倍.     

Localized waves without the existence of extended waves: oscillatory convection of binary mixtures with strong soret effect

Spatially confined solutions of traveling convection rolls are determined numerically for binary mixtures such as ethanol-water. The appropriate field equations are solved in a vertical cross section of the rolls perpendicular to their axes subject to realistic horizontal boundary conditions. The localized convective states are stably and robustly sustained by strongly nonlinear mixing and complex flow-induced concentration redistribution. We elucidate how this enables their existence for strongly negative separation ratios at small subcritical heating rates below the saddle node of extended traveling convection rolls where the quiescent fluid is strongly stable.     

Critical stability of almost adiabatic convection in a rapidly rotating thick spherical shell

In this work, the convection equations in the almost adiabatic approximation is studied for which the choice of physical parameters is primarily based on possible applications to the hydrodynamics of the deep interiors of the Earth and planets and moons of the terrestrial group. The initial system of partial differential equations (PDEs) was simplified to a single second-order ordinary differential equation for the pressure or vertical velocity component to investigate the linear stability of convection. The critical frequencies, modified Rayleigh numbers, and distributions of convection are obtained at various possible Prandtl numbers and in different thick fluid shells. An analytical WKB-type solution was obtained for the case when the inner radius of the shell is much smaller than the outer radius and convective sources are concentrated along the inner boundary.     

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.