Steady two-dimensional Rayleigh--Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios $π/5\leΓ\le4π$, where $Γ$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $Γ$ and is largest at $Γ\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $Γ\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini \& Cox (2009). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $Γ$ is smaller.
We consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity $Ω$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $Ω$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at …
We experimentally test the effects of tilting a turbulent Rayleigh-B{é}nard convection cell on the dynamics of the large-scale circulation (LSC) orientation $θ_0$. The probability distribution of $θ_0$ is measured, and used to obtain a tilt-induced potential acting on $θ_0$, which is used in a low-dimensional model of diffusion of $θ_0$ in a potential. The form of the potential is sinusoidal in $θ_0$, and linear in tilt angle for small tilt angles, which is explained by a simple geometric model of the vector direction of the mean buoyancy force acting on the LSC. However, the magnitude of the …
We test the ability of a low-dimensional turbulence model to predict how dynamics of large-scale coherent structures such as convection rolls change in different cell geometries. We performed Rayleigh-Bénard convection experiments in a cubic container, in which there is a single convection roll known as the large-scale circulation (LSC). The model describes the motion of the orientation $θ_0$ of the LSC as diffusion in a potential which is predicted as a function of the shape of the cell from an approximate solutions of the Navier-Stokes equations. The model predicts advected oscillation modes…
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.
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.
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…
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…
Bounds for the poloidal and toroidal kinetic energies and the heat transport are computed numerically for rotating convection at infinite Prandtl number with both no slip and stress free boundaries. The constraints invoked in this computation are linear or quadratic in the problem variables and lead to the formulation of a semidefinite program. The bounds behave as a function of Rayleigh number at fixed Taylor number qualitatively in the same way as the quantities being bounded. The bounds are zero for Rayleigh numbers smaller than the critical Rayleigh number for the onset of convection, they…
Turbulent thermal convection is characterized by the formation of large-scale structures and strong spatial inhomogeneity. This work addresses the relative heat transport contributions of the large-scale plume ejecting versus plume impacting zones in turbulent Rayleigh-Bénard convection. Based on direct numerical simulations of the two dimensional (2-D) problem, we show the existence of a crossover in the wall heat transport from initially impacting dominated to ultimately ejecting dominated at a Rayleigh number of $Ra\approx 3 \times 10^{11}$. This is consistent with the trends observed in 3-…
We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection i…
