Heat transport and flow structure in rotating Rayleigh-B\'enard convection

Here we summarize the results from our direct numerical simulations (DNS) and experimental measurements on rotating Rayleigh-B\'enard (RB) convection. Our experiments and simulations are performed in cylindrical samples with an aspect ratio \Gamma varying from 1/2 to 2. Here \Gamma=D/L, where D and L are the diameter and height of the sample, respectively. When the rotation rate is increased, while a fixed temperature difference between the hot bottom and cold top plate is maintained, a sharp increase in the heat transfer is observed before the heat transfer drops drastically at stronger rotation rates. Here we focus on the question of how the heat transfer enhancement with respect to the non-rotating case depends on the Rayleigh number Ra, the Prandtl number Pr, and the rotation rate, indicated by the Rossby number Ro. Special attention will be given to the influence of the aspect ratio on the rotation rate that is required to get heat transport enhancement. In addition, we will discuss the relation between the heat transfer and the large scale flow structures that are formed in the different regimes of rotating RB convection and how the different regimes can be identified in experiments and simulations.Comment: 12 pages, 10 figure

Numerical simulations of rotating Rayleigh-Bénard convection

The Rayleigh-Bénard (RB) system is relevant to astro- and geophysical phenomena, including convection in the ocean, the Earth’s outer core, and the outer layer of the Sun. The dimensionless heat transfer (the Nusselt number Nu) in the system depends on the Rayleigh number Ra=ßg¿L 3/(¿¿) and the Prandtl number Pr=¿/¿. Here, ß is the thermal expansion coefficient, g the gravitational acceleration, ¿ the temperature difference between the bottom and top, and ¿ and ¿ the kinematic viscosity and the thermal diffusivity, respectively. The rotation rate H is used in the form of the Rossby number Ro=(ßg¿/L)/(2H). The key question is: How does the heat transfer depend on rotation and the other two control parameters: Nu(Ra, Pr, Ro)? Here we will answer this question by giving a summary of our result

Optimal Prandtl number for heat transfer in rotating Rayleigh-Benard convection

Numerical data for the heat transfer as a function of the Prandtl (Pr) and Rossby (Ro) numbers in turbulent rotating Rayleigh-Benard convection are presented for Rayleigh number Ra = 10^8. When Ro is fixed the heat transfer enhancement with respect to the non-rotating value shows a maximum as function of Pr. This maximum is due to the reduced efficiency of Ekman pumping when Pr becomes too small or too large. When Pr becomes small, i.e. for large thermal diffusivity, the heat that is carried by the vertical vortices spreads out in the middle of the cell, and Ekman pumping thus becomes less efficient. For higher Pr the thermal boundary layers (BLs) are thinner than the kinetic BLs and therefore the Ekman vortices do not reach the thermal BL. This means that the fluid that is sucked into the vertical vortices is colder than for lower Pr which limits the efficiency of the upwards heat transfer.Comment: 5 pages, 6 figure

Clustering of vertically constrained passive particles in homogeneous, isotropic turbulence

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force, to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulent velocity field. The model that we introduce and study is possibly the simplest description for the dynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, or any other mechanism, maintain themselves in a shallow horizontal layer below the free surface of oceans or lakes. By varying the strength of the restoring force, we are able to control the thickness of the fluid slab in which the particles can move. This allows us to analyze the statistical features of the system over a wide range of conditions going from a fully 3D incompressible flow (corresponding to the case of no confinement) to the extremely confined case corresponding to a two-dimensional slice. The background 3D turbulent velocity field is evolved by means of fully resolved direct numerical simulations. Whenever some level of vertical confinement is present, the particle trajectories deviate from that of fluid tracers and the particles experience an effectively compressible velocity field. Here, we have quantified the compressibility, the preferential concentration of the particles, and the correlation dimension by changing the strength of the restoring force. The main result is that there exists a particular value of the force constant, corresponding to a mean slab depth approximately equal to a few times the Kolmogorov length scale, that maximizes the clustering of the particles

Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh-B\'enard convection

In turbulent thermal convection in cylindrical samples of aspect ratio \Gamma = D/L (D is the diameter and L the height) the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis, because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number 1/\Ro_c, with 1/\Ro_c \propto 1/\Gamma. We present a Ginzburg-Landau like model that explains the existence of a bifurcation at finite 1/\Ro_c as a finite-size effect. The model yields the proportionality between 1/\Ro_c and $1/\Gamma$ and is consistent with several other measured or computed system properties.Comment: 4 pages, 4 figure

Regime transitions in stratified shear flows: the link between horizontal and inclined ducts

We present an analytical model that provides the transition curves between different regimes of stratified shear flows in inclined ducts for high Schmidt number values. These curves are described by constant values of a generalized Reynolds number multiplied by the aspect ratio of the duct, showing good agreement with previous experimental results. The generalized Reynolds number is obtained by extending to inclined ducts the solution of a one-dimensional model of a stratified shear flow in a horizontal duct within a regime where advection is neglected in the momentum equation but included in the density transport equation

Effect of Plumes on Measuring the Large Scale Circulation in Turbulent Rayleigh-B\'enard Convection

We studied the properties of the large-scale circulation (LSC) in turbulent Rayleigh-B\'enard (RB) convection by using results from direct numerical simulations in which we placed a large number of numerical probes close to the sidewall. The LSC orientation is determined by either a cosine or a polynomial fit to the azimuthal temperature or azimuthal vertical velocity profile measured with the probes. We study the LSC in \Gamma=D/L=1/2 and \Gamma=1 samples, where D is the diameter and L the height. For Pr=6.4 in an aspect ratio \Gamma=1 sample at $Ra=1\times10^8$ and $5\times10^8$ the obtained LSC orientation is the same, irrespective of whether the data of only 8 or all 64 probes per horizontal plane are considered. In a \Gamma=1/2 sample with $Pr=0.7$ at $Ra=1\times10^8$ the influence of plumes on the azimuthal temperature and azimuthal vertical velocity profiles is stronger. Due to passing plumes and/or the corner flow the apparent LSC orientation obtained using a cosine fit can result in a misinterpretation of the character of the large-scale flow. We introduce the relative LSC strength, which we define as the ratio between the energy in the first Fourier mode and the energy in all modes that can be determined from the azimuthal temperature and azimuthal vertical velocity profiles, to further quantify the large-scale flow. For $Ra=1\times10^8$ we find that this relative LSC strength is significantly lower in a \Gamma=1/2 sample than in a \Gamma=1 sample, reflecting that the LSC is much more pronounced in a \Gamma=1 sample than in a \Gamma=1/2 sample. The determination of the relative LSC strength can be applied directly to available experimental data to study high Rayleigh number thermal convection and rotating RB convection.Comment: 12 pages, 15 figure