Classical Rayleigh-Benard Convection Research Articles

On convective instabilities in a rotating fluid with stably stratified layer and thermally heterogeneous boundary

The onset of convection in a rotating plane layer due to a vertical temperature gradient is studied in this paper. The background stratification is modulated by lateral temperature variations and stable stratification aimed at understating the Earth's outer core convection subject to thermal core–mantle interaction. At the top boundary, sinusoidal and Gaussian temperature variations are imposed apart from the reference case of isothermal condition used in the classical Rayleigh–Benard convection. The additional modulating conditions break the top–bottom flow symmetry leading to flow localization and asymmetry that exhibit modified temporal dynamics unlike that of the classical Rayleigh–Benard cells. The threshold for convection is lowered with flows occurring in surplus heat flux regions caused by the imposed conditions. Despite flow suppression in the stable layer, rapid rotation favors the penetration of convection rolls with smaller wavelengths. The lateral variations in temperature imposed at the top boundary enhance such axial penetration with a laterally varying penetrative extent resulting in a modified clustered flow structure unlike the reference case. With both modulating conditions imposed, the onset of overstable modes is favored for low Prandtl numbers, a regime which is relevant to the Earth's core conditions. With rapid rotation, a novel mode of traveling wave instability occurs at the onset of convection, the propagation direction of which is controlled by the lateral temperature gradients at the top boundary. The onset of oscillatory modes is suppressed by the imposition of the modulating conditions indicated by the significant lowering of the transition Prandtl number.

Evolution of solid–liquid interface in bottom heated cavity for low Prandtl number using lattice Boltzmann method

The influence of natural convection cells on heat transfer and the evolution of melt interface is studied for low Prandtl number fluid (Pr = 0.025) in phase-change Rayleigh–Benard convection using the lattice Boltzmann method. The thermal lattice Boltzmann model is used to evaluate the effect of Rayleigh number (Ra = 6708, 11 708, and 21 708) and cavity aspect ratio (γ = 0.062 5, 0.125, 0.25, 0.5, and 1) on the onset of convection, number of convection cells, and Nusselt number in the classical Rayleigh–Benard convection. The modified equilibrium distribution function-based thermal lattice Boltzmann model is applied to evaluate the effect of Stefan number (Ste = 0.025, 0.05, and 0.1) in the phase change Rayleigh–Benard convection. Distinct flow configurations depend on the Rayleigh number, aspect ratio, and Stefan number. The number of convection cells follows an inverse relation with the aspect ratio. Nusselt number increases with decreasing cavity aspect ratio and increasing Rayleigh number in the classical Rayleigh–Benard convection. With the variation in the aspect ratio based on the melt layer height during melting of phase change material, the number of convection cells changes resulting in the change in the evolution of the melt interface and convective heat transfer. Melting in a cavity of aspect ratio less than 0.5, the evolution of melt interface remains symmetrical. For an aspect ratio greater than 0.5, the interface evolution becomes unsymmetrical depending on the transition to single convection cell-dominated heat transfer.

Thermal boundary layer structure in convection with and without rotation

Convection occurs in many settings from metal production to planetary interiors and atmospheres. To understand the dynamics of these systems it is vital to be able to predict the heat transport which is controlled by the thermal boundary layers (TBL). An important issue in the study of convective fluid dynamics is then to determine the temperature distribution within these thin layers in the vicinity of the bounding walls. Deviations from the classical Rayleigh-Benard convection paradigm such as the addition of rotation or fixed heat-flux (rather than fixed temperature) boundaries compromise the standard ways of defining the width of the TBL. We propose an alternative method for defining the TBL using the location at which the advective and conductive contributions to the heat transport become equal. We show that this method can be robustly applied to two-dimensional (2D) nonrotating convection between no-slip boundaries with fixed temperature or fixed heat-flux thermal boundary conditions and three-dimensional (3D) rotating convection simulations with free-slip boundaries.

Natural convection in cylindrical containers with isothermal ring-shaped obstacles

By means of three-dimensional direct numerical simulations, we investigate the influence of the regular roughness of heated and cooled plates on the mean heat transport in a cylindrical Rayleigh–Benard convection cell of aspect ratio one. The roughness is introduced by a set of isothermal obstacles, which are attached to the plates and have a form of concentric rings of the same width. The considered Prandtl number $Pr$ equals 1, the Rayleigh number $Ra$ varies from $10^{6}$ to $10^{8}$ , the number of rings on each plate is 1, 2, 4, 8 or 10, the height of the rings is varied from 1.5 % to 49 % of the cylinder height and the gap between the rings is varied from 1.5 % to 18.8 % of the cell diameter. Totally, 135 different cases are analysed. Direct numerical simulations show that with small $Ra$ and wide roughness rings, a small reduction of the mean heat transport (the Nusselt number $Nu$ ) is possible, but, in most cases, the presence of the heated and cooled obstacles generally leads to an increase of $Nu$ , compared to the case of classical Rayleigh–Benard convection with smooth plates. When the rings are very tall and the gaps between them are sufficiently wide, the effective mean heat flux can be several times larger than in the smooth case. For a fixed geometry of the obstacles, the scaling exponent in the $Nu$ versus $Ra$ scaling first increases with growing $Ra$ up to approximately 0.5, but then smoothly decreases back towards the exponent in the no-obstacle case.

Modified Rayleigh–Bénard convection driven by long-wavelength heating from above and below

Classical Rayleigh–Benard convection occurs when a horizontal fluid layer is heated sufficiently strongly from below. In more recent times, there has been increased interest in structured convection, when the heating that is applied is no longer uniform but rather spatially varying in some way. Here, we examine the effect of structured convection in a fluid layer when both the lower and upper boundaries are heated so that their temperatures fluctuate sinusoidally over a common long length scale offset by some phase $$\varOmega $$ . While this non-uniform heating can be shown to induce small fluid motions by way of a primary form of convection, some previous computations have shown that the layer is also susceptible to a stronger, secondary form of convection. When the heating is just sufficient to induce this secondary motion, the cells tend to conglomerate near the local hot spots where the underlying heating is at its most intense. The strength of the secondary convection falls off away from the hot spot on a length scale which is appreciably longer than the wavelength of the individual cells but also much shorter than the wavelength of the underlying applied heating. In this work, we derive a second-order amplitude equation that describes the secondary convection and discuss the important features of its solution. These are compared with some direct numerical simulations and are likely to apply for other situations in which long-scale heating gives rise to structured convection patterns.

Numerical simulations of thermal convection on a hemisphere

In this paper we present numerical simulations of two-dimensional turbulent convection on a hemisphere. Recent experiments on a half soap bubble located on a heated plate have shown that such a configuration is ideal for studying thermal convection on a curved surface. Thermal convection and fluid flows on curved surfaces are relevant to a variety of situations, notably for simulating atmospheric and geophysical flows. As in experiments, our simulations show that the gradient of temperature between the base and the top of the hemisphere generates thermal plumes at the base that move up from near the equator to the pole. The movement of these plumes gives rise to a two-dimensional turbulent thermal convective flow. Our simulations turn out to be in qualitative and quantitative agreement with experiments and show strong similarities with Rayleigh-Benard convection in classical cells where a fluid is heated from below and cooled from above. To compare to results obtained in classical Rayleigh-Benard convection in standard three-dimensional cells (rectangular or cylindrical), a Nusselt number adapted to our geometry and a Reynolds number are calculated as a function of the Rayleigh number. We find that the Nusselt and Reynolds numbers verify scaling laws consistent with turbulent Rayleigh-Benard convection: Nu ∝ Ra 0.31 and Re ∝ Ra 1/2. Further, a Bolgiano regime is found with the Bolgiano scale scaling as Ra −1/4. All these elements show that despite the significant differences in geometry between our simulations and classical 3D cells, the scaling laws of thermal convection are robust.

On the onset of natural convection in differentially heated shallow fluid layers with internal heat generation

This paper reports the results of an analytical and numerical investigation to determine the effect of internal heat generation on the onset of convection, in a differentially heated shallow fluid layer. The case with the bottom plate at a temperature higher than the top plate mimics the classical Rayleigh Benard convection. However, internal heat generation adds a new dimension to the problem. Linear stability analysis is first carried out for the case of an infinitely wide cavity. The effect of aspect ratio on the onset of convection is studied by solving the full Navier–Stokes equations and the equation of energy and observing the temperature contours. A bisection algorithm is used for an accurate prediction of the onset. The numerical results are used to plot the stability curves for eight different aspect ratios. A general correlation is developed to determine the onset of convection in a differentially heated cavity for various aspect ratios. For an aspect ratio of 10, it is seen that the cavity approaches the limit of an infinite cavity. Analytical results obtained by using linear stability analysis agree very well with the “full” CFD simulations, for the above aspect ratio.

Numerical Method of Modeling Thermal Creeping Flow in Heterogeneous Medium

AbstractA hybrid method for modeling the creeping flow is proposed. In our method the so‐called marker‐in‐cell (MIC) and the Finite Element Method (FEM) algorithm are combined together to simulate the thermal creeping flow concerning heterogeneous medium deformation. In particular, the unknown parameters at the Euler mesh‐nodes are calculated using the FEM. The cell‐markers in each element carry the material composition and history variables during the flowing process. The momentum and continuity equation are solved in terms of the pressure‐stabilizing Petrov‐Galerkin method (PSPG) with the equal‐order interpolation of the velocity and pressure, and the energy equation is solved using the streamline upwind Petrov‐Galerkin method (SUPG). In the MIC algorithm, the bilinear interpolation corresponds to the interpolation function in the finite elements. The FEM and MIC algorithm are independent from each other. The data in these two processes communicate through the nodal points. In addition, the triangular element algorithm makes possible to solve the problems with irregular mesh‐grid in complex structures. Our computation program has been verified with the classical Rayleigh‐Benard convection problem. As an example, the descent of an irregular geometry block is calculated. The stability of numerical solution is also investigated.

Effects of solid layer thickness and nominal composition on double-diffusive instabilities during solidification of a binary alloy cooled from the top

When a horizontal layer of fluid is cooled and solidified from the top, a cellular form of natural convection may occur inside the liquid, similar to that observed during classical Rayleigh–Benard convection. As a result of the interaction between the cellular convection and directional solidification, the phase-change interface tends to get wavy. The above situation can be mathematically represented by considering two horizontal parallel plates at z = 0 and z = h. The lower plate at z = h is fixed at temperature $T = T_1$ and the upper plate at z=0 is kept at a temperature $T = T_0$. Due to occurrence of solidification on account of cooling from the top, there is a solid–liquid interface at $z=\eta \leq \eta \leq h)$, which is assumed to be at a quasi-steady state. The physical situation is shown schematically in Fig. 1. As a result of interface perturbation on account of convection occurring below, the interface is likely to assume an irregular shape. It can be noted here that when the solidifying medium is a binary alloy that solidifies nonisothermally, the natural convection in the liquid is double-diffusive in nature (as a consequence of concentration and temperature gradients prevailing in the solidifying domain). Instabilities in thermo-solutal convection in such cases have attracted considerable attention in the literature [1,2]. However, the effects of nominal alloy composition and solid layer thickness on associated double-diffusive instabilities are yet to be addressed in the literature, to the best of our knowledge.

Pattern formation in rayleigh-Benard convection in a cylindrical container

We report on numerical investigations of pattern formation in the classical Rayleigh-Benard convection with cylindrical geometry in the regime of low Prandtl numbers and moderate aspect ratio. Beyond the onset of convection, we found straight and bent rolls as stable patterns. By increasing the Rayleigh number, we studied the generation of defects, their dynamics in the form of climbing and gliding, the existence of stable targets and spirals as well as the occurrence of core instabilities, a variety of pattern types that were also observed in experiments.

Convection induced by selective absorption of radiation: A laboratory model of conditional instability

When there is internal heating in a fluid layer, convection can occur even if the static state is one of stable stratification. We have been investigating through laboratory experiments such a stably stratified layer of water which is heated above and cooled below. The water contains in dilute solution thymol blue (a pH indicator), which normally colors the water orange. It turns yellow if the pH is low, blue if the pH is high. A small DC voltage is applied across the layer, by using the bottom boundary as the positive electrode, the top boundary as the negative electrode. The hydroxyl ions formed near the bottom boundary cause the orange fluid to turn blue. The fluid layer is uniformly and steadily illuminated from above with light from a sodium vapor lamp. This radiation travels with negligible absorption through the orange fluid but is strongly absorbed by the blue fluid. The resultant warming of the blue fluid can lead to convective instability, with the blue fluid rising into warm upper layers, which it would continue to penetrate as long as it remains blue and as long as the radiative heating is sufficient to exceed the higher ambient temperatures above. This radiative heating occurs only in the blue rising flow; the sinking fluid is orange and is not heated. We have found that with a strongly stably stratified layer, convective plumes are unable to penetrate far and they remain shallow. However, for a weakly stratified layer, plumes grow tall and furthermore collect into a large convective cluster which persists as a steady coherent structure. The present paper deals also with the formulation of the governing equations to include the fluid-state-dependent heat source. A linear stability analysis shows that the critical Rayleigh number for onset of motion is drastically reduced. Furthermore, the cell size at onset is larger by a factor of √32 than in the classical Rayleigh-Benard convection problem. However, the laboratory fluid cells were much further broadened (by a factor of 8 or 10) when they penetrated into the stably stratified fluid above. In this case, the rising region is narrow and the sinking region is broad, so that downward vertical velocities are correspondingly small. In this way, the downwards-forced warm fluid has time to cool by conduction to the cold boundary. Steady finite amplitude solutions and their stability are analyzed and it is shown that there is a parameter range in which finite amplitude hexagonal cells are stable.

Cellular convection embedded in the convective planetary boundary layer surface layer

Abstract Cellular convection was first studied in the laboratory by Benard [Ann. Chim. Phys. 23 (1901) 62–144] and Rayleigh [Phil. Mag. Ser. 6 (1916) 529–546] investigated these motions from a theoretical perspective. He defined a dimensionless number, now called the Rayleigh number, which is the ratio of convective transport to molecular transport, and found that if a certain critical value is exceeded, cellular convection occurs. Mesoscale cellular convection (MCC) is a common occurrence in the planetary boundary layer. Agee [Dyn. Atmos. Oceans 10 (1987) 317–341] discussed the similarities and differences of MCC and classical Rayleigh-Benard convection. A similar cellular pattern can be seen in the convective boundary layer (CBL) surface layer. It is known that in the CBL, air near the surface converges into thermals producing updrafts. This produces a ‘spoke’ type pattern similar to the mesoscale cellular or Rayleigh-Benard convection. This paper will focus on applying Rayleigh-Benard convection criteria, using a linearized perturbation method, to the CBL surface layer produced by Large Eddy Simulation (LES). We will investigate the length scales of turbulence in the CBL surface layer and compare them to those predicted from linear theory. Similarities and differences will be discussed between the LES produced surface layer and classical Rayleigh-Benard convection theory.

Effects of an almost resonant spatial thermal modulation in the Rayleigh-Benard problem: quasiperiodic behaviour

The authors consider the effect of prescribed spatially periodic temperatures at the bounding walls of a 2D Boussinesq fluid on 1D pattern forming transitions. They determine the normal-form equation for the slowly varying pattern amplitude. Quasiperiodic behaviour is found when one takes into account the deviation of the external wavelength modulation from the critical value for the onset of classical Rayleigh-Benard convection.