A model for three-dimensional Rayleigh-Bénard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $Φ$ and the convective heat flux ($Nu-1$) show scaling behaviour with $ε= r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in $r$ and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in $r$ depending on the values of Prandtl number $Pr$. They become irregular in time, with increase in $r$. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as $r$ is raised further. The variations of the critical Rayleigh number $Ra_{os}$ and the frequency $ω_{os}$ at the onset of the secondary instability with $Pr$ are also studied for different values of Chandrasekhar's number $Q$.
Floquet analysis of modulated magnetoconvection in Rayleigh-Bénard\ geometry is performed. The temperature of the lower plate is varied sinusoidally in time about a finite mean. As the Rayleigh number $\mathrm{Ra}$ is made to cross a critical value $\mathrm{Ra}_o$, the oscillatory magnetoconvection begins. The flow at the onset of magnetoconvection may oscillate either subharmonically or harmonically with the external modulation. The critical Rayleigh number $\mathrm{Ra}_o$ varies non-monotonically with $ω$ for appreciable value of $a$. The temperature modulation may either postpone or prepone…
We present the results of direct numerical simulations of power spectral densities for kinetic energy, convective entropy and heat flux for unsteady Rayleigh-Bénard magnetoconvection in the frequency space. For larger values of frequency, the power spectral densities for all the global quantities vary with frequency $f$ as $f^{-2}$. The scaling exponent is independent of Rayleigh number, Chandrasekhar's number and thermal Prandtl number.
We report experimental measurements of heat transport in rotating Rayleigh-B{é}nard convection in a cylindrical convection cell with aspect ratio $Γ= 1/2$. The fluid was helium gas with Prandtl number Pr = 0.7. The range of control parameters was Rayleigh number $4 \times 10^9 < {\rm Ra} < 4 \times 10^{11}$ and Ekman number $2 \times 10^{-7} < {\rm Ek} < 3 \times 10^{-5}$(corresponding to Taylor number $4 \times 10^9 < {\rm Ta} < 1 \times 10^{14}$ and convective Rossby number $0.07 < {\rm Ro} < 5$). We determine the crossover from weakly rotating turbulent convection to rotation dominated geos…
We present results of numerical investigation on thermal flux in Rayleigh-Bénard magnetoconvection in the presence of a uniform vertical magnetic field. We have studied thermal flux in different viscous fluids with a range of Prandtl number ($0.1 \le \mathrm{Pr} < 6.5$) and a range of Chandrasekhar number ($50 \le \mathrm{Q} \le 2.5 \times 10^4$). The power spectral density of the Nusselt number varies with frequency $f$ approximately as $f^{-2}$. The probability distribution function of the fluctuating part of the Nusselt number is nearly normal distribution with slight asymmetric tails. For …
In agreement with a recent experimental discovery by Xia et. al. (2009), we also find a sloshing mode in experiments on the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in a cylindrical sample of aspect ratio one. The sloshing mode has the same frequency as the torsional oscillation discovered by Funfschilling and Ahlers (2004). We show that both modes can be described by an extension of a model developed previously [Brown and Ahlers (2008)] which consists of permitting a lateral displacement of the LSC circulation plane away from the vertical center line of the sample…
Three-dimensional turbulent magnetoconvection at a Rayleigh number of $Ra=10^7$ in liquid gallium at a Prandtl number $Pr=0.025$ is studied in a closed square cell for very strong external vertical magnetic fields $B_0$ in direct numerical simulations which apply the quasistatic approximation. As $B_0$ or equivalently the Hartmann number $Ha$ are increased, the convection flow that is highly turbulent in the absence of magnetic fields crosses the Chandrasekhar linear stability limit for which thermal convection is ceased in an infinitely extended layer and which can be assigned with a critical…
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 …
In this article, the theoretical model on heat and momentum transfer for Rayleigh-Bénard convection in a vertical magnetic field by Zürner et al. (Phys. Rev. E 94, 043108 (2016)) is revisited. Using new data from recent experimental and numerical studies the model is simplified and extended to the full range of Hartmann numbers, reproducing the results of the Grossmann-Lohse theory in the limit of vanishing magnetic fields. The revised model is compared to experimental results in liquid metal magnetoconvection and shows that the heat transport is described satisfactorily. The momentum transpor…
We report Particle Image Velocimetry of the Large Scale Circulation and the viscous boundary layer in turbulent thermal convection. We use two parallelepipedic Rayleigh-B{é}nard cells with a top smooth plate. The first one has a rough bottom plate and the second one has a smooth one so we compare the rough-smooth and the smooth-smooth configurations. The dimensions of the cell allow to consider a bi-dimensional mean flow. Lots of previous heat flux measurements have shown a Nusselt--Rayleigh regime transition corresponding to an increase of the heat flux in presence of roughness which is highe…
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…
Coupled laboratory-numerical experiments of Rayleigh-Bénard convection (RBC) in liquid gallium subject to a vertical magnetic field are presented. The experiments are carried out in two cylindrical containers with diameter-to-height aspect ratio $Γ= 1.0$ and $2.0$ at varying thermal forcing (Rayleigh numbers $10^5 \lesssim Ra \lesssim 10^8$) and magnetic field strength (Chandrasekhar numbers $0\lesssim Ch \lesssim 3\times 10^5$). Laboratory measurements and numerical simulations confirm that magnetoconvection in our finite cylindrical tanks onsets via non-drifting wall-attached modes, in good …
Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^5$-$1.0\times 10^9$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann-Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh-B{é}nard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl-Blasius-Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$…
We present a numerical study of quasistatic magnetoconvection in a cubic Rayleigh-Bénard (RB) convection cell subjected to a vertical external magnetic field. For moderate values of the Hartmann number Ha, we find an enhancement of heat transport. Furthermore, a maximum heat transport enhancement is observed at certain optimal $Ha_{opt}$. The enhanced heat transport may be understood as a result of the increased coherency of the thermal plumes, which are elementary heat carriers of the system. To our knowledge this is the first time that a heat transfer enhancement by the stabilising Lorentz f…
Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of $Q=10^8$ over a broad range of Rayleigh numbers $Ra$. Three magnetoconvection regimes are identified: two of the regimes are magnetically-constrained in the sense that a leading-order balance exists between the Lorentz and buoyancy forces, whereas the third regime is characterized by unbalanced dynamics that is similar to non-magnetic convection. Each regime is distinguished by flow morphology, momentum and heat equation balances, and heat transport behavior. O…
The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied through a perturbation analysis. Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. Results for the case of thermally infinitely conducting boundaries and for the case of nearly thermally insulating boundaries are obtained.
A series of numerical simulations of Rayleigh-B{é}nard convection in a cubic cavity are conducted in order to examine the structure of the thermal boundary layer in case of mixed boundary conditions. The main goal of the study is the physical mechanism which provides increasing of heat flux with spatial frequency of conducting-adiabatic pattern. Different spatial configuration of conducting plates, including the fractal one, are considered for Rayleigh numbers from $\Ray=10^{7}$ to $\Ray=2.0\times 10^{9}$. We have shown that the temperature boundary layer in case of mixed boundary conditions a…
We study the onset of magnetoconvection between two infinite horizontal planes subject to a vertical magnetic field aligned with background rotation. In order to gain insight into the convection taking place in the Earth's tangent cylinder (TC), we target regimes of asymptotically strong rotation. The critical Rayleigh number $Ra_{c}$ and critical wavenumber are computed numerically by solving the linear stability problem in a systematic way. A parametric study is conducted, varying the Ekman number, $E$ (ratio of viscous to Coriolis forces) and the Elsasser number, $Λ$ (ratio of the Lorentz f…
A numerical study of convection with stress-free boundary conditions in the presence of an imposed magnetic field that is tilted with respect to the direction of gravity is carried out in the limit of small magnetic Reynolds number. The dynamics are investigated over a range of Rayleigh number $Ra$ and Chandrasekhar numbers up to $Q = 2\times10^6$, with the tilt angle between the gravity vector and imposed magnetic field vector fixed at $45^{\circ}$. For a fixed value of $Q$ and increasing $Ra$, the convection dynamics can be broadly characterized by three primary flow regimes: (1) quasi-two-d…
We have developed novel instrumentation for making Lagrangian measurements of temperature in diverse fluid flows. A small neutrally buoyant capsule is equipped with on-board electronics which measure temperature and transmit the data via a wireless radio frequency link to a desktop computer. The device has 80 dB dynamic range, resolving milli-Kelvin changes in temperature with up to 100 ms sampling time. The capabilities of these "smart particles" are demonstrated in turbulent thermal convection in water. We measure temperature variations as the particle is advected by the convective motion, a…
We investigate the onset of convective instability and subsequent transitions near it in Rayleigh-Bénard convection (RBC) of electrically conducting low Prandtl-number (Pr) fluids in the simultaneous presence of rotation about the vertical axis and external uniform horizontal magnetic field with free-slip boundary conditions. Both three dimensional direct numerical simulations (DNS) and low dimensional modeling of the system have been performed for the investigation. DNS results show a plethora of patterns near the onset of convection for weak magnetic field. Most interestingly, the Nusselt nu…
The classical Rayleigh-Bénard convection (RBC) system is known to exhibit either subcritical or supercritical transition to convection in the presence or absence of rotation and/or magnetic field. However, the simultaneous exhibition of subcritical and supercritical branches of convection in plane layer RBC depending on the initial conditions, has not been reported so far. Here, we report the phenomenon of simultaneous occurrence of subcritical and supercritical branches of convection in overstable RBC of electrically conducting low Prandtl number fluids (liquid metals) in the presence of an e…
The effects of a vertical static magnetic field on the flow structure and global transport properties of momentum and heat in liquid metal Rayleigh-Bénard convection are investigated. Experiments are conducted in a cylindrical convection cell of unity aspect ratio, filled with the alloy GaInSn at a low Prandtl number of $\mathit{Pr}=0.029$. Changes of the large-scale velocity structure with increasing magnetic field strength are probed systematically using multiple ultrasound Doppler velocimetry sensors and thermocouples for a parameter range that is spanned by Rayleigh numbers of $10^6 \le \m…
We investigate instabilities and chaos near the onset of Rayleigh-Bénard convection (RBC) of electrically conducting fluids with free-slip, perfectly electrically and thermally conducting boundary conditions in presence of uniform rotation about vertical axis and horizontal external magnetic field by considering zero Prandtl-number limit ($\mathrm{Pr} \rightarrow 0$). Direct numerical simulations (DNS) and low dimensional modeling of the system are done for the investigation. Values of the Chandrasekhar number ($\mathrm{Q}$) and the Taylor number ($\mathrm{Ta}$) are varied in the range $0 < \m…
Numerical simulation studies in 2D with the addition of noise are reported for the convection of a supercritical fluid,3He, in a Rayleigh-Be'nard cell where the fluid parameters and cell height L are the same as in published laboratory experiments. The noise addition is to accelerate the instability onset after starting the heat flow across the fluid, so as to bring simulations into better agreement with experimental observations. Homogeneous temperature noise and spatial lateral periodic temperature variations in the top plate were programmed into the simulations. A speed-up in the instabilit…
A challenge in fundamental physics and especially in thermodynamics is to understand emergent order in far-from-equilibrium systems. While at equilibrium, temperature plays the role of a key thermodynamic variable whose uniformity in space and time defines the equilibrium state the system is in, this is not the case in a far-from-equilibrium driven system. When energy flows through a finite system at steady-state, temperature takes on a time-independent but spatially varying character. In this study, the convection patterns of a Rayleigh-B{é}nard fluid cell at steady-state is used as a prototy…
