We present high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for cylindrical samples of water (Prandtl number sigma = 4.38) with diameters D = 49.7, 24.8, and 9.2 cm, all with aspect ratio Gamma = D/L = 1 (L is the sample height). In addition, we present data for D = 49.7 and Gamma = 1.5, 2, 3, and 6. For each sample the data cover a range of a little over a decade of R. For Gamma = 1 they jointly span the range 10^7 < R < 10^11. Where needed, the data were corrected for the influence of the finite conductivity of the top and bottom plates and of the side walls on the heat transport in the fluid to obtain estimates of N_infinity for plates with infinite conductivity and sidewalls of zero conductivity. For Gamma = 1 the effective exponent gamma_eff of N_infinity = N_0 R^gamma_eff ranges from 0.28 near R = 10^8 to 0.333 near R = 7 times10^10. For R < 10^10 the results are consistent with the Grossmann-Lohse model. For larger R, where the data indicate that N_infinity(R) = R^1/3, the theory has a smaller gamma_eff than 1/3 and falls below the data. The data for Gamma > 1 are only a few percent smaller than the Gamma = 1 results.
Turbulent Rayleigh-Bénard convection is often modelled with a constant surface temperature. However, the surface temperature of many geophysical systems, such as lakes, is coupled to the atmospheric forcing. In this paper, we account for this dynamic surface temperature through an additional parameter $β$. Using an appropriately defined dynamical Rayleigh number $\RaD$, we recover many of the results from the standard Rayleigh-Bénard model. We hope that this work will simplify the application of Rayleigh-Bénard theory in geophysical contexts, such as lakes.
The claim by He et al. [Physical Review Letters 108, 024502 (2012)] that their experiment reached the 'ultimate' regime of turbulent Rayleigh-Bénard convection is not justified by their data.
Direct numerical simulations are performed to study turbulent Rayleigh-Bénard convection in a vertical cylindrical cavity with uniform axial magnetic field. Flows at high Hartmann and Rayleigh numbers are considered. The calculations reveal that, similarly to the behavior observed in Rayleigh-Bénard convection with strong rotation, flows at strong magnetic field develop a central vortex, while the heat transfer is suppressed.
Temporal and spatio-temporal (turbulence) distributed chaos in Bénard-Marangoni and Rayleigh-Bénard convection have been studied using results of laboratory experiments and direct numerical simulations in the terms of effective chaotic diffusivity (viscosity) and action. It is shown that for the both cases decaying part of the power spectra has stretched exponential form - for temporal spectrum $E(f) \propto \exp-(f/f_β)^{1/2}$ and for spatial spectrum $E(k) \propto \exp-(k/k_β)^{2/3}$, where the $f_β$ and $k_β$ represent low-frequency (large-scale) coherent oscillations.
The transition from deterministic to distributed chaos/turbulence at the increase of Rayleigh number (from $10^4$ to $10^8$) in Rayleigh-Bénard convection, controlled by generalized Birkhoff-Saffman invariant, has been studied using the results of direct numerical simulations. The applications of this approach to rotating Rayleigh-Bénard convection, to stably stratified flows, and to observations in the atmosphere and in the solar photosphere have been briefly discussed.
We present a fluid dynamics video which illustrates the dynamics of the velocity field in the boundary layer in a turbulent Rayleigh-Benard convection flow. The data are obtained from direct numerical simulation.
Measurements of heat transport, as expressed by the Nusselt number $Nu$, are reported for turbulent Rayleigh-Bénard convection of water containing up to 120 ppm by weight of poly-[ethylene oxide] with a molecular weight of $4\times10^6$ g/mole. Over the Rayleigh number range $ 5\times 10^9 \alt Ra \alt 7 \times 10^{10}$ $Nu$ is smaller than it is for pure water by up to 10%.
In Rayleigh-Bénard convection, it has been found that the amount of heat passing through the fluid has a power law dependence on the imposed temperature difference. Modifying this dependence, either enhancing or reducing the heat transfer capability of fluids, is important in many scientific and practical applications. Here, we present a simple means to control the vertical heat transfer in Rayleigh-Bénard convection by injecting heat through one lateral side of the fluid domain and extracting the same amount of heat from the opposite side. This horizontal heat flux regulates the large-scale c…
Experimental and numerical data for the heat transfer as a function of the Rayleigh-, Prandtl-, and Rossby numbers in turbulent rotating Rayleigh-Benard convection are presented. For relatively small $Ra ~ 10^8$ and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small Pr = 0.7. there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.
We present results of the study of a turbulent air flow with a large scale circulation in Rayleigh-Benard rectangular convective cell with a heated bottom wall and a cooled top wall. The mean horizontal velocity of the main roll and the mean vorticity of eddy rings are almost aligned in a large part of the flow. The helicity of the mean flow is quite high, and is the source of turbulent helicity. Since helicity of the mean flow and turbulence is quite large, the flow in Rayleigh-Benard convective cell is well suited to study properties of helical turbulence.
The Boussinesq equations for Rayleigh-Benard convection are simulated for a cylindrical container with an aspect ratio near 1.5. The transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25,000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with O(2) symmetry.
Transition to hyperchaotic regimes in Rayleigh-Benard convection in a square periodicity cell is studied by three-dimensional numerical simulations. By fixing the Prandtl number at P=0.3 and varying the Rayleigh number as a control parameter, a bifurcation diagram is constructed where a route to hyperchaos involving quasiperiodic regimes with two and three incommensurate frequencies, multistability, chaotic intermittent attractors and a sequence of boundary and interior crises is shown. The three largest Lyapunov exponents exhibit a linear scaling with the Rayleigh number and are positive in t…
Upper bounds on the heat transport and other quantities of interest in Rayleigh-Bénard convection are derived in previous work from constraints resulting from the equations of time evolution for kinetic energy, the root mean square of temperature, and the temperature averaged over horizontal planes. Here, we investigate the effect of a new constraint derived from the time evolution equation for the advective heat transport. This additional constraint leads to improved bounds on the toroidal dissipation.
We apply Maxwell and Cattaneo's relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell's relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced subsonic problems, as is shown here for two-roll and four-roll Rayleigh-Benard flows.
Multiple chaotic and turbulent regimes in Rayleigh-Bénard convection have been studied and classified from the onset of deterministic chaos to the fully developed turbulence using the distributed chaos approach supported by results of laboratory experiments and numerical simulations. It is shown that the regimes can replace each other depending on value of the Rayleigh number and location in the convection cell (in the bulk of the flow or in the boundary layers). The inertial-buoyancy and helicity based invariants play crucial role in the formation of the regimes. Transition from the classical…
We invoke the concepts of magnetic boundary layer and magnetic Rayleigh number and use the magnetic energy dissipation rates in the bulk and the boundary layers to derive some scaling laws expressing how Nusselt number depends on magnetic Rayleigh number, Prandtl number and magnetic Prandtl number for the simple case of turbulent magnetohydrodynamic Rayleigh-Benard convection in the presence of uniform vertical magnetic field.
The boundary layer flow in a Rayleigh-Bénard convection cell of rectangular shape has been visualized in this fluid dynamics video. The experiment has been undertaken in air at a Rayleigh number $Ra=1.3\times 10^{10}$ and a Prandtl number $Pr=0.7$. Various sequences captured at selected positions of the heating plate show that the boundary layer is a very transient flow region characterized by coherent structures that permanently evolve. It becomes fully turbulent in the areas where the large-scale circulation impinge or leave the bottom plate.
We present the adaptation to non--free boundary conditions of a pseudospectral method based on the (complex) Fourier transform. The method is applied to the numerical integration of the Oberbeck--Boussinesq equations in a Rayleigh--Bénard cell with no-slip boundary conditions for velocity and Dirichlet boundary conditions for temperature. We show the first results of a 2D numerical simulation of dry air convection at high Rayleigh number ($R\sim10^9$). These results are the basis for the later study, by the same method, of wet convection in a solar still.
The evolution of three-dimensional, cellular convective flows in a plane horizontal layer of a Boussinesq fluid heated from below is well studied. Here we review results from the investigation of this system as well as a number of related and novel numerical findings. We present theoretical results for pattern formation in Rayleigh-Bénard convection with solving NS equation under the Boussinesq approximation . System of equations were reduced to 2-dimension form to simplify the analysis.The evolution of the flow agree with the idea of the flow achieving an optimal form.
In this contribution we have briefly introduced the problem of turbulent thermal convection with a particular look at its transition to the ultimate regime and the resolution requirements needed for the direct numerical simulation of this flow.
This work is equivalent to that in {\em Phys. Rev. Lett.} {\bf 123}, 259401 (2019), however, Physical Review Letters prohibited reference to the additional two points in the analysis published by Zhu et al., in {\em Phys. Rev. Lett.} {\bf 123}, 259402 (2019).
The spectra of turbulent heat flux $\mathrm{H}(k)$ in Rayleigh-Bénard convection with and without uniform rotation are presented. The spectrum $\mathrm{H}(k)$ scales with wave number $k$ as $\sim k^{-2}$. The scaling exponent is almost independent of the Taylor number $\mathrm{Ta}$ and Prandtl number $\mathrm{Pr}$ for higher values of the reduced Rayleigh number $r$ ($ > 10^3$). The exponent, however, depends on $\mathrm{Ta}$ and $\mathrm{Pr}$ for smaller values of $r$ ($<10^3$). The probability distribution functions of the local heat fluxes are non-Gaussian and have exponential tails.
The shape of velocity and temperature profiles near the horizontal conducting plates in turbulent Rayleigh-Bénard convection are studied numerically and experimentally over the Rayleigh number range $10^8\lesssim Ra\lesssim3\times10^{11}$ and the Prandtl number range $0.7\lesssim Pr\lesssim5.4$. The results show that both the temperature and velocity profiles well agree with the classical Prandtl-Blasius 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.
We report a new thermal boundary layer equation for turbulent Rayleigh-Benard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr>1, Pr~1 and (b) Pr>>1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol) respectively.
Rayleigh-Benard convection in a cylindrical container can take on many different spatial forms. Motivated by the results of Hof, Lucas and Mullin [Phys. Fluids 11, 2815 (1999)], who observed coexistence of several stable states at a single set of parameter values, we have carried out simulations at the same Prandtl number, that of water, and radius-to-height aspect ratio of two. We have used two kinds of thermal boundary conditions: perfectly insulating sidewalls and perfectly conducting sidewalls. In both cases we obtain a wide variety of coexisting steady and time-dependent flows.
Direct numerical simulations are carried out to study flow structure and transport properties in turbulent Rayleigh-Bénard convection in a cylindrical cell of aspect ratio one with an imposed axial magnetic field. Flows at the Prandtl number 0.025 and the Rayleigh and Hartmann numbers up to $10^9$ and 1400 are considered. The results are consistent with those of earlier experimental and numerical data. As anticipated, the heat transfer rate and kinetic energy are suppressed by strong magnetic field. At the same time, their growth with the Rayleigh number is found to be faster in flows at high …
We have developed a small, neutrally buoyant, wireless temperature sensor. Using a camera for optical tracking, we obtain simultaneous measurements of position and temperature of the sensor as it is carried along by the flow in Rayleigh-Bénard convection, at $Ra \sim 10^{10}$. We report on statistics of temperature, velocity, and heat transport in turbulent thermal convection. The motion of the sensor particle exhibits dynamics close to that of Lagrangian tracers in hydrodynamic turbulence. We also quantify heat transport in plumes, revealing self-similarity and extreme variations from plume t…
Weakly-rotating turbulent Rayleigh-Benard convection was studied experimentally and numerically. With increasing rotation and large enough Rayleigh number an abrupt transition from a turbulent state with nearly rotation-independent heat transport to another turbulent state with enhanced heat transfer is observed at a critical inverse Rossby number $1/Ro_c \simeq 0.4$. Whereas for $1/Ro < 1/Ro_c$ the strength of the large-scale convection-roll is either enhanced or essentially unmodified depending on parameters, its strength is increasingly diminished beyond $1/Ro_c$ where it competes with Ekma…
We show, using direct numerical simulations with experimentally realizable boundary conditions, that wall modes in Rayleigh-Bénard convection in a rapidly rotating cylinder persist even very far from their linear onset. These nonlinear wall states survive in the presence of turbulence in the bulk and are robust with respect to changes in the shape of the boundary of the container. In this sense these modes behave much like the topologically protected states present in two-dimensional chiral systems even though rotating convection is a three-dimensional nonlinear driven dissipative system. We s…
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 $\mathrm{Ra}$ and Prandtl number $\mathrm{Pr}$. 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 …
A quantitative theory is developed for the vertical mean temperature profile (MTP) in turbulent Rayleigh-Benard convection (RBC), which explains the recent experimental and numerical observations of a logarithmic law by Ahlers et al.(Phys. Rev. Lett., 2012). A multi-layer model is formulated and quantified, whose predictions agree with DNS and experimental data for the Rayleigh-number (Ra) over seven decades. In particular, a thermal buffer layer follows a 1/7 scaling like the previously postulated mixing zone (Procaccia et al, Phys. Rev. A,1991), and yields a Ra-dependent log law constant. A …
Turbulent Rayleigh-Bénard convection in slender cylindrical cells exhibits rich dynamics of the large-scale circulation (LSC), with several rolls stacked on top of each other. We propose that the elliptical instability is the mechanism which causes the twisting and breaking of the LSC into multiple rolls and that the volume-averaged heat and momentum transport, represented by the Nusselt number and Reynolds number, is generally weaker for larger number $n$ of the LSC rolls. This is supported by direct numerical simulations for $Ra=5\times10^5$, $Pr=0.1$, $H=5D$ and $1\leq~n\leq4$.
The corner roll (CR) in the Rayleigh-Bénard (RB) convection accounts for the behaviors of heat transport and convection flow at the corner. Streamlines of the three-dimensional direct numerical simulations for $10^8<Ra<5\times10^9$ show that CR presents well-defined similarity and multi-layer structure. A stream function for CR is developed by homotopy and the structure ensemble dynamics. The model presents the scaling of Reynolds number of corner roll $Re_{cr}\sim Ra^{1/4}$. Scaling of CR scale $r = 0.77 Ra^{-0.085}$ indicates strong near-wall shearing induced by wind and provides a probabili…
A low-dimensional model of large Prandtl-number ($P$) Rayleigh Bénard convection is constructed using some of the important modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the low-dimensional model for $P=6.8$ and aspect ratio of $2\sqrt{2}$ reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Bifurcation analysis also reveals multiple co-existing attractors, and a window with time-periodicity after chaos. The results of the low-dimensional model matches quite closely with some…
A detailed study of the Rayleigh-Bénard convection in two-dimensions with free-slip boundaries is presented. Pseudo-spectral method has been used to numerically solve the system for Rayleigh number up to $3.3 \times 10^7$. The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The `travelling rolls' instability is observed in the chaotic regime. Scaling of Nusselt number shows an exponent close to 0.33. Studies on energy spectrum and flux show an inverse cascade of kinetic energy and a forward cascade of entropy. This is consistent with the shell-to…
Convective turbulence in a Rayleigh Benard system has shown a marked reluctance to exhibit clear scaling in the energy or entropy spectrum. The recent numerical simulation of Pandey, Verma and Mishra has shown significantly better evidence of scaling in the infinite Prandtl number limit. This prompted us to look at this limit analytically. We find that the inevitable presence of sweeping helps give a very good understanding of the results of Pandey et. al. In the presence of rotation, the Rayleigh number dependence of the Nusselt number shows a strong increase in slope in recent experiments an…
A model for the large-scale circulation (LSC) dynamics of turbulent Rayleigh-Benard convection is presented. It consists of two stochastic ordinary differential equations motivated by the Navier-Stokes equation, one each for the strength and the azimuthal orientation of the LSC. Stochastic forces represent phenomenologically the action of the turbulent fluctuations on the LSC. Without adjustable parameters, the model yields a meandering LSC with occasional rotations, and with more rare cessations that occur a few times per day -- differing only by about a factor of two from experiment. Also as…
We present measurements of the orientation theta_0(t) of the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in cylindrical cells of aspect ratio 1. theta_0(t) undergoes irregular reorientations. It contains reorientation events by rotation through angles Delta theta with a monotonically decreasing probability distribution p(Delta theta), and by cessations (where the LSC stops temporarily) with a uniform p(Delta theta). Reorientations have Poissonian statistics in time. The amplitude of the LSC and the magnitude of the azimuthal rotation rate have a negative correlation.
In this numerical study on two-dimensional Rayleigh-Bénard convection we consider $10^7 \leq Ra \leq 10^{12}$ in aspect ratio $0.23 \leq Γ\leq 13$ samples. We focus on several cases. First we consider small aspect ratio cells, where at high Ra number we find a sharp transition from a low Ra number branch towards a high Ra number branch, due to changes in the flow structure. Subsequently, we show that the influence of the aspect ratio on the heat transport decreases with increasing aspect ratio, although even at very large aspect ratio of $Γ\approx10$ variations up to 2.5% in the heat transport…
Rayleigh-Benard convection is not only a classical problem in fluid dynamics but plays also an important role in many metallurgical and crystal growth applications. The measurement of the flow field and of the dynamics of the emerging large-scale circulation in liquid metals is a challenging task due to the opaqueness and the high temperature of the melts. Contactless inductive flow tomography is a technique to visualize the mean three-dimensional flow structure in liquid metals by measuring the flow induced magnetic field perturbations under the influence of one or several applied magnetic fi…
We explore the chaotic dynamics of Rayleigh-Bénard convection using large-scale, parallel numerical simulations for experimentally accessible conditions. We quantify the connections between the spatiotemporal dynamics of the leading-order Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the spatiotemporal features of the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. Using p…
We measured Reynolds numbers R_e of turbulent Rayleigh-Benard convection over the Rayleigh-number range 2 times 10^8 < R < 10^11 and Prandtl-number range 3.3 < sigma < 29 for cylindrical samples of aspect ratio Gamma = 1. For R < R_c = 3 times 10^9 we found R_e ~ R^beta_eff with beta_eff = 0.46 < 1/2. Here both the sigma- and R-dependences are quantitatively consistent with the Grossmann-Lohse (GL) prediction. For R > R_c we found R_e = 0.106 sigma^-3/4 R^1/2, which differs from the GL prediction. The relatively sharp transition at R_c to the large-R regime suggests a qualitative and sudden ch…
We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh-Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $Nu$ is bounded by $Nu \leq 0.5999 R^{1/3}$ uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $Ra$, we obtain $Nu \leq 0.4646 Ra^{1/2}$. …
Measurements of the Nusselt number $Nu$ and of temperature variations $ΔT_b$ in the bulk fluid are reported for turbulent Rayleigh-Bénard convection of a cylindrical sample. They cover the Rayleigh-number range $10^{9} \alt Ra \alt 3\times 10^{14}$ using He (Prandtl number $Pr = 0.67$), N$_2$ ($Pr = 0.72$) and SF$_6$ ($Pr = 0.79$ 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,…
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…
Liquid Metal Batteries (LMBs) are a promising grid-scale energy storage technology that offer low costs per kilowatt-hour, high energy and current densities, as well as low fade rates. The all-liquid composition of the batteries, as well as the presence of temperature gradients and electric and magnetic fields, result in the occurrence of multiple fluid phenomena. These can affect the hydrodynamic stability of the battery, thereby making their interactions critical to understand. In this work, the interaction of Rayleigh-Bénard convection and Electro-vortex flow is investigated as these types …
Rayleigh-Bénard convection in the turbulent regime is studied using statistical methods. Exact evolution equations for the probability density function of temperature and velocity are derived from first principles within the framework of the Lundgren-Monin-Novikov hierarchy known from homogeneous isotropic turbulence. The unclosed terms arising in the form of conditional averages are estimated from direct numerical simulations. Focusing on the statistics of temperature, the theoretical framework allows to interpret the statistical results in an illustrative manner, giving deeper insight into t…
We have found a multi-scale steady solution of the Boussinesq equations for Rayleigh-Bénard convection in a three-dimensional periodic domain between horizontal plates with a constant temperature difference by using a homotopy from the wall-to-wall optimal transport solution given by Motoki et al. (J. Fluid Mech., vol. 851, 2018, R4). The connected steady solution, which turns out to be a consequence of bifurcation from a thermal conduction state at the Rayleigh number $Ra\sim10^{3}$, is tracked up to $Ra\sim10^{7}$ by using a Newton-Krylov iteration. The exact coherent thermal convection exhi…
This letter confirms the existence of heat transfer enhancement (HTE) and reduction (HTR) in turbulent natural convection with polymer additives. HTE and HTR were numerically predicted by Benzi et al.(PRL, 104, 024502, 2010) in homogenous turbulent convection, but experiments by Ahlers & Nikolaenko(PRL, 104, 034503, 2010) in turbulent natural convection observed HTR only. Using direct numerical simulation of natural convection, the present study reconciles earlier numerical and experimental work on the basis of the dominant role of polymer length in the polymer dynamics in extensional flows.
Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in a cylindrical cell for a broad range of Rayleigh, Ekman, and Prandtl numbers from the onset of wall modes to the geostrophic regime, an extremely important one in geophysical and astrophysical contexts. We connect linear wall-mode states that occur prior to the onset of bulk convection with the boundary zonal flow that coexists with turbulent bulk convection in the geostrophic regime through the continuity of length and time scales and of convective heat transport. We quantitatively collapse drift frequency, bo…
Rayleigh--Bénard convection, which is the buoyancy-induced motion of a fluid enclosed between two horizontal plates, is an idealised setup to study thermal convection. We analyse the modes that transport the most heat between the plates by computing the proper orthogonal decomposition (POD) of numerical data. Instead of the usual POD ansatz of finding modes that describe the energy best, we propose a method that is optimal in describing the heat transport. Thereby, we can determine the modes with the major influence on the heat transport and the coherent structures in the convection cell. We a…
The aspect ratio (Γ) dependence of the heat transfer (Nusselt number Nu in dimensionless form) in turbulent (two-dimensional) Rayleigh-Bénard convection is numerically studied in the regime $0.4 \le Γ\le 1.25$ for Rayleigh numbers $10^7 \le Ra \le Ra^{9}$ and Prandtl numbers Pr =0.7 (gas) and 4.3 (water). Nu (Γ) shows a very rich structure with sudden jumps and sharp transitions. We connect these structures to the way the flow organizes itself in the sample and explain why the aspect ratio dependence of Nu is more pronounced for small Pr. Even for fixed Γdifferent turbulent states (with differ…
The non-hydrostatic, quasigeostrophic approximation for rapidly rotating Rayleigh-Bénard convection admits a class of exact `single mode' solutions. These solutions correspond to steady laminar convection with a separable structure consisting of a horizontal planform characterized by a single wavenumber multiplied by a vertical amplitude profile, with the latter given as the solution of a nonlinear boundary value problem. The heat transport associated with these solutions is studied in the regime of strong thermal forcing (large reduced Rayleigh number $\widetilde{Ra}$). It is shown that the N…
Rayleigh-Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised setup to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to work out similarities and differences. To this end, we derive an exact evolution equation for the temperature probability densit…
