The axisymmetric flow in an aspect-ratio-one cylinder whose upper and lower bounding disks are maintained at different temperatures and rotate at equal and opposite velocities is investigated. In this combined Rayleigh-Benard/von Karman problem, the imposed temperature gradient is measured by the Rayleigh number Ra and the angular velocity by the Reynolds number Re. Although fluid motion is present as soon as Re is non-zero, a symmetry-breaking transition analogous to the onset of convection takes place at a finite Rayleigh number higher than that for Re=0. For Re<95, the transition is a pitchfork bifurcation to a pair of steady states, while for Re>95, it is a Hopf bifurcation to a limit cycle. The steady states and limit cycle are connected via a pair of SNIPER bifurcations except very near the Takens-Bogdanov codimension-two point, where the scenario includes global bifurcations. Detailed phase portraits and bifurcation diagrams are presented, as well as the evolution of the leading part of the spectrum, over the parameter ranges 0<Re<120 and 0<Ra<30,000.
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.
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…
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.
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 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.
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.
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.
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 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 …
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…
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 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 $Nu \leq 0.4646 Ra^{1/2}$. …
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 …
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…
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…
We study the connection between extreme events of thermal and kinetic energy dissipation rates in the bulk of three-dimensional Rayleigh-Bénard convection and the wall shear stress patterns at the top and the bottom planes that enclose the layer. Zero points of this two-dimensional vector field stand for detachments of strong thermal plumes. If their position at the opposite planes and a given time is close then they can be considered as precursors for high-amplitude bulk dissipation events triggered by plume collisions or close passings. This scenario requires a breaking of the synchronicity …
