DOCSLIB.ORG

(1976) 33--49 Institute of Geophysics and Planetary Physics, UCLA, Los

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(1976) 33--49 Institute of Geophysics and Planetary Physics, UCLA, Los Dynamics of Atmospheres and Oceans, 1 (1976) 33--49 33 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands S T A B I L I T Y O F R A Y L E I G H - - B E N A R D C O N V E C T I O N R O L L S A N D B I M O D A L F L O W A T M O D E R A T E P R A N D T L N U M B E R JOHN A. WHITEHEAD Jr. * and GERALD L. CHAN ** Institute of Geophysics and Planetary Physics, U.C.L.A., Los Angeles, Calif. (U.S.A.) School of Engineering and Applied Science, U.C.L.A., Los Angeles, Calif. (U.S.A.) (Received January 5, 1976) ABSTRACT Whitehead Jr., J.A. and Chan, G.L., 1976. Stability of Rayleigh--B~nard convection rolls and bimodal flow at moderate Prandtl number. Dyn. Atmos. Oceans, 1: 33--49. Experimental observations of cellular convection between two rigid, horizontal, con- ducting boundaries are reported for two different cases. First, the stability of two-dimen- sional roll convection of various wavenumbers and Rayleigh numbers is investigated in fluids of Prandtl numbers 16 and 2.7. The results qualitatively agree with earlier observa- tions by Busse and Whitehead of fluid with Prandtl number 126 but they differ somewhat quantitatively. Second, the stability of the bimodal flow, consisting of two rolls of differing and perpendicular wavenumber, is observed to be stable for given bandwidths and ranges of Rayleigh numbers when the configuration of the bimodal flow, consisting of two sets of rolls at right angles, is flawless -- without any disruptions in the periodic matrix. The stability range differs from earlier experiments and from our own experiments in which the bimodal planform is uncontrolled. INTRODUCTION F r o m the t i m e t h a t the e x p e r i m e n t s of B~nard (1901) were first r e p o r t e d , it has been k n o w n t h a t a h o r i z o n t a l layer of fluid heated f r o m b e l o w exhibits strikingly simple g e o m e t r i c a l c o n v e c t i o n p a t t e r n s when the t e m p e r a t u r e dif- ference is raised above a certain critical value. On the basis of o b s e r v a t i o n s of the change of slope of the heat flux vs. t e m p e r a t u r e , Malkus (1954) p r o p o s e d that the c o n v e c t i o n liberated progressively m o r e and m o r e c o n v e c t i o n m o d e s with increased t e m p e r a t u r e difference across the layer, and suggested t h a t the c o n v e c t i o n s y s t e m is an ideal subject for t u r b u l e n c e investigations because the progressive transitions can be investigated in detail. In the interior of the earth, in the ocean, and in the a t m o s p h e r e , such c o n v e c t i o n occurs, o f t e n in a relatively s t r u c t u r e d manner, y e t little is k n o w n of the details of h o w these structures c o m e a b o u t . * Present address, Department of Physical Oceanography, Woods Hole Oceanographic In- stitution, Woods Hole, Mass. (U.S.A.). ** Present address, Harvard School of Public Health, Boston, Mass. (U.S.A.). 34 Recent studies into the geometric form of convection confirm the consis- tency of Malkus' original hypothesis. Although the convection above the critical Rayleigh number consisted of two-dimensional roll-like cells, Busse (1967) showed theoretically that the roll-form of convection cannot exist above a Rayleigh number of approximately 22,000 because rolls at right angles grow. The flow was observed to become three-dimensional by Krishnamurti (1970a) and this transition was accompanied by a change in the slope of the heat-flux curve. The three-dimensional pattern was observed to consist of two sets of rolls lying at right angles and of different wavelength by Busse and Whitehead (1971), and this pattern has henceforth been called bimodal flow. Further studies by Krishnamurti (1970b) and by Willis and Deardorff (1970) reported a gradual transition to time-dependent flow in large Prandtl number fluid in the Rayleigh number range between 60,000 to somewhat over 100,000. Krishnamurti further observed accompanying changes in the heat-flux curve. Recent studies by Busse and Whitehead (1974) show a much clearer transition to a time-dependent flow when the geometry of bimodal flow is carefully controlled. The purpose of the present paper is to report the results of further studies of the stability of convection rolls and bimodal flows for various Prandtl numbers, wavenumbers, and Rayleigh numbers. To this end we first describe observations of the stability of roll convection for silicon oil with a Prandtl number of 16 and hot water with a Prandtl number of 2.7 and such transitions have been observed by us to behave qualitatively like the transitions predicted by Busse for fluid with infinite Prandtl number, and observed by Busse and Whitehead (1971) in fluid with a Prandtl number of 126. Secondly, we inves- tigate the stability of the bimodal flow for fluid with a Prandtl number of 126 and we find that the bimodal flow is stable in a given volume in Rayleigh number, and two wavenumber space. To accomplish these observations the experimental technique of Chen and Whitehead (1968) was used to generate roll and bimodal convection consisting of parallel planar rolls of a specified wavelength as an initial condition. It was found that the instability of convection in a perfect bimodal planform, that is, one consisting of two sets of planar rolls at exactly right angles without any flaws in the bimodal matrix, has stability properties unlike those reported by Krishnamurti (1970b) or Willis and Deardorff (1970). It was found, how- ever, that when the bimodal pattern was not "perfectly flawless," as is always the case when initial conditions of the convection are not controlled, a new convection pattern grew at the flaw regions, which appears to possess the os- cillatory properties reported in the above papers. We believe that this is evi- dence that such an instability does not grow from background noise in the bi- modal flow, and hence that a transition to oscillations will not be predicted by a linearized stability analysis in the ranges observed here. THE EXPERIMENTAL APPARATUS Two convection chambers of different sizes but similar designs were used in the present experiments. The larger chamber was used for Rayleigh numbers 35 above 10,000 in the Pr = 16 and the Pr = 126 case whereas the rest of the experiments were done with the smaller apparatus. A schematic diagram of the larger apparatus is shown in Fig. 1. It consists of two water channels, one placed horizontally above the other at a precise spacing, thus leaving a middle channel in between the two, which is filled with the working fluid. The bottom channel has a dimension of 100 × 91 c m 2 and is bounded by a one-half-inch-thick piece of plate glass below, and a one- quarter-inch piece of plate glass above. The top channel has a dimension of 80 × 80 c m 2 and is bounded by one-quarter-inch-thick pieces of plate glass. It is held in place by four brass brackets at its four corners. Although glass has about 5.6 better heat conductivity than the silicon oil, and 1.5 better conductivity than water, there is no other material which has higher thermal conductivity and meets the requirements of transparency and economy. In the water channels the circulating water is thermally regulated. The water in the two channels is drawn from two separate units of constant-tem- perature water baths which are capable of delivering up to five gallons of water per minute at any desired temperature between room temperature and 100 ° C. The thermostats in these baths maintain the water at a constant tem- perature to within a few hundredths of a degree Kelvin. Since the water in the two channels serves as the thermal boundary for the convection layer, it is essential that its flow be spatially homogeneous. A distributor is used both at the inlet and the outlet such that the flow may be as close as possible to being parallel. For the same reason, a rather high flow rate is employed, thus eliminating the possibility of a significant temperature gradient between the inlet and the outlet. The one-quarter-inch glass plates bounding the channels are not sufficient in strength to remain flat while supporting the weights of the fluids and their own weight. A counter pressure system is necessary to maintain the hydrostatic head in the lower, middle, and upper channels con- stant to 1 mm.
Load more

Recommended publications
  • Convection Heat Transfer
    Convection Heat Transfer Heat transfer from a solid to the surrounding fluid Consider fluid motion Recall flow of water in a pipe Thermal Boundary Layer • A temperature profile similar to velocity profile. Temperature of pipe surface is kept constant. At the end of the thermal entry region, the boundary layer extends to the center of the pipe. Therefore, two boundary layers: hydrodynamic boundary layer and a thermal boundary layer. Analytical treatment is beyond the scope of this course. Instead we will use an empirical approach. Drawback of empirical approach: need to collect large amount of data. Reynolds Number: Nusselt Number: it is the dimensionless form of convective heat transfer coefficient. Consider a layer of fluid as shown If the fluid is stationary, then And Dividing Replacing l with a more general term for dimension, called the characteristic dimension, dc, we get hd N ≡ c Nu k Nusselt number is the enhancement in the rate of heat transfer caused by convection over the conduction mode. If NNu =1, then there is no improvement of heat transfer by convection over conduction. On the other hand, if NNu =10, then rate of convective heat transfer is 10 times the rate of heat transfer if the fluid was stagnant. Prandtl Number: It describes the thickness of the hydrodynamic boundary layer compared with the thermal boundary layer. It is the ratio between the molecular diffusivity of momentum to the molecular diffusivity of heat. kinematic viscosity υ N == Pr thermal diffusivity α μcp N = Pr k If NPr =1 then the thickness of the hydrodynamic and thermal boundary layers will be the same.
    [Show full text]
  • The Influence of Magnetic Fields in Planetary Dynamo Models
    Earth and Planetary Science Letters 333–334 (2012) 9–20 Contents lists available at SciVerse ScienceDirect Earth and Planetary Science Letters journal homepage: www.elsevier.com/locate/epsl The influence of magnetic fields in planetary dynamo models Krista M. Soderlund a,n, Eric M. King b, Jonathan M. Aurnou a a Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095, USA b Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA article info abstract Article history: The magnetic fields of planets and stars are thought to play an important role in the fluid motions Received 7 November 2011 responsible for their field generation, as magnetic energy is ultimately derived from kinetic energy. We Received in revised form investigate the influence of magnetic fields on convective dynamo models by contrasting them with 27 March 2012 non-magnetic, but otherwise identical, simulations. This survey considers models with Prandtl number Accepted 29 March 2012 Pr¼1; magnetic Prandtl numbers up to Pm¼5; Ekman numbers in the range 10À3 ZEZ10À5; and Editor: T. Spohn Rayleigh numbers from near onset to more than 1000 times critical. Two major points are addressed in this letter. First, we find that the characteristics of convection, Keywords: including convective flow structures and speeds as well as heat transfer efficiency, are not strongly core convection affected by the presence of magnetic fields in most of our models. While Lorentz forces must alter the geodynamo flow to limit the amplitude of magnetic field growth, we find that dynamo action does not necessitate a planetary dynamos dynamo models significant change to the overall flow field.
    [Show full text]
  • Turning up the Heat in Turbulent Thermal Convection COMMENTARY Charles R
    COMMENTARY Turning up the heat in turbulent thermal convection COMMENTARY Charles R. Doeringa,b,c,1 Convection is buoyancy-driven flow resulting from the fluid remains at rest and heat flows from the warm unstable density stratification in the presence of a to the cold boundary according Fourier’s law, is stable. gravitational field. Beyond convection’s central role in Beyond a critical threshold, however, steady flows in myriad engineering heat transfer applications, it un- the form of coherent convection rolls set in to enhance derlies many of nature’s dynamical designs on vertical heat flux. Convective turbulence, character- larger-than-human scales. For example, solar heating ized by thin thermal boundary layers and chaotic of Earth’s surface generates buoyancy forces that plume dynamics mixing in the core, appears and per- cause the winds to blow, which in turn drive the sists for larger ΔT (Fig. 1). oceans’ flow. Convection in Earth’s mantle on geolog- More precisely, Rayleigh employed the so-called ical timescales makes the continents drift, and Boussinesq approximation in the Navier–Stokes equa- thermal and compositional density differences induce tions which fixes the fluid density ρ in all but the buoyancy forces that drive a dynamo in Earth’s liquid temperature-dependent buoyancy force term and metal core—the dynamo that generates the magnetic presumes that the fluid’s material properties—its vis- field protecting us from solar wind that would other- cosity ν, specific heat c, and thermal diffusion and wise extinguish life as we know it on the surface. The expansion coefficients κ and α—are constant.
    [Show full text]
  • On the Inverse Cascade and Flow Speed Scaling Behavior in Rapidly Rotating Rayleigh-Bénard Convection
    This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 On the inverse cascade and flow speed scaling behavior in rapidly rotating Rayleigh-B´enardconvection S. Maffei1;3y, M. J. Krouss1, K. Julien2 and M. A. Calkins1 1Department of Physics, University of Colorado, Boulder, USA 2Department of Applied Mathematics, University of Colorado, Boulder, USA 3School of Earth and Environment, University of Leeds, Leeds, UK (Received xx; revised xx; accepted xx) Rotating Rayleigh-B´enardconvection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, Ek ! 0. The Prandtl number (P r) and the asymptotically scaled Rayleigh number (Raf = RaEk4=3, where Ra is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a depth-invariant large-scale vortex (LSV). With respect to the kinetic energy, we find a transition from convection dominated states to LSV dominated states at an asymptotically reduced (small-scale) Reynolds number of Ref ≈ 6 for all investigated values of P r. The ratio of the depth-averaged kinetic energy to the kinetic energy of the convection reaches a maximum at Ref ≈ 24, then decreases as Raf is increased. This decrease in the relative kinetic energy of the LSV is associated with a decrease in the convective correlations with increasing Rayleigh number. The scaling behavior of the convective flow speeds is studied; although a linear scaling of the form Ref ∼ Ra=Pf r is observed over a limited range in Rayleigh number and Prandtl number, a clear departure from this scaling is observed at the highest accessible values of Raf .
    [Show full text]
  • Rayleigh-Bernard Convection Without Rotation and Magnetic Field
    Nonlinear Convection of Electrically Conducting Fluid in a Rotating Magnetic System H.P. Rani Department of Mathematics, National Institute of Technology, Warangal. India. Y. Rameshwar Department of Mathematics, Osmania University, Hyderabad. India. J. Brestensky and Enrico Filippi Faculty of Mathematics, Physics, Informatics, Comenius University, Slovakia. Session: GD3.1 Core Dynamics Earth's core structure, dynamics and evolution: observations, models, experiments. Online EGU General Assembly May 2-8 2020 1 Abstract • Nonlinear analysis in a rotating Rayleigh-Bernard system of electrical conducting fluid is studied numerically in the presence of externally applied horizontal magnetic field with rigid-rigid boundary conditions. • This research model is also studied for stress free boundary conditions in the absence of Lorentz and Coriolis forces. • This DNS approach is carried near the onset of convection to study the flow behaviour in the limiting case of Prandtl number. • The fluid flow is visualized in terms of streamlines, limiting streamlines and isotherms. The dependence of Nusselt number on the Rayleigh number, Ekman number, Elasser number is examined. 2 Outline • Introduction • Physical model • Governing equations • Methodology • Validation – RBC – 2D – RBC – 3D • Results – RBC – RBC with magnetic field (MC) – Plane layer dynamo (RMC) 3 Introduction • Nonlinear interaction between convection and magnetic fields (Magnetoconvection) may explain certain prominent features on the solar surface. • Yet we are far from a real understanding of the dynamical coupling between convection and magnetic fields in stars and magnetically confined high-temperature plasmas etc. Therefore it is of great importance to understand how energy transport and convection are affected by an imposed magnetic field: i.e., how the Lorentz force affects convection patterns in sunspots and magnetically confined, high-temperature plasmas.
    [Show full text]
  • Arxiv:1903.08882V2 [Physics.Flu-Dyn] 6 Jun 2019
    Lattice Boltzmann simulations of three-dimensional thermal convective flows at high Rayleigh number Ao Xua,∗, Le Shib, Heng-Dong Xia aSchool of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China bState Key Laboratory of Electrical Insulation and Power Equipment, Center of Nanomaterials for Renewable Energy, School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China Abstract We present numerical simulations of three-dimensional thermal convective flows in a cubic cell at high Rayleigh number using thermal lattice Boltzmann (LB) method. The thermal LB model is based on double distribution function ap- proach, which consists of a D3Q19 model for the Navier-Stokes equations to simulate fluid flows and a D3Q7 model for the convection-diffusion equation to simulate heat transfer. Relaxation parameters are adjusted to achieve the isotropy of the fourth-order error term in the thermal LB model. Two types of thermal convective flows are considered: one is laminar thermal convection in side-heated convection cell, which is heated from one vertical side and cooled from the other vertical side; while the other is turbulent thermal convection in Rayleigh-B´enardconvection cell, which is heated from the bottom and cooled from the top. In side-heated convection cell, steady results of hydrodynamic quantities and Nusselt numbers are presented at Rayleigh numbers of 106 and 107, and Prandtl number of 0.71, where the mesh sizes are up to 2573; in Rayleigh-B´enardconvection cell, statistical averaged results of Reynolds and Nusselt numbers, as well as kinetic and thermal energy dissipation rates are presented at Rayleigh numbers of 106, 3×106, and 107, and Prandtl numbers of arXiv:1903.08882v2 [physics.flu-dyn] 6 Jun 2019 ∗Corresponding author Email address: [email protected] (Ao Xu) DOI: 10.1016/j.ijheatmasstransfer.2019.06.002 c 2019.
    [Show full text]
  • Time-Periodic Cooling of Rayleigh–Bénard Convection
    fluids Article Time-Periodic Cooling of Rayleigh–Bénard Convection Lyes Nasseri 1 , Nabil Himrane 2, Djamel Eddine Ameziani 1 , Abderrahmane Bourada 3 and Rachid Bennacer 4,* 1 LTPMP (Laboratoire de Transports Polyphasiques et Milieux Poreux), Faculty of Mechanical and Proceeding Engineering, USTHB (Université des Sciences et de la Technologie Houari Boumedienne), Algiers 16111, Algeria; [email protected] (L.N.); [email protected] (D.E.A.) 2 Labo of Energy and Mechanical Engineering (LEMI), Faculty of Technology, UMBB (Université M’hamed Bougara-Boumerdes, Boumerdes 35000, Algeria; [email protected] 3 Laboratory of Transfer Phenomena, RSNE (Rhéologie et Simulation Numérique des Ecoulements) Team, FGMGP (Faculté de génie Mécaniques et de Génie des Procédés Engineering), USTHB (Université des Sciences et de la Technologie Houari Boumedienne), Bab Ezzouar, Algiers 16111, Algeria; [email protected] 4 CNRS (Centre National de la Recherche Scientifique), LMT (Laboratoire de Mécanique et Technologie—Labo. Méca. Tech.), Université Paris-Saclay, ENS (Ecole National Supérieure) Paris-Saclay, 91190 Gif-sur-Yvette, France * Correspondence: [email protected] Abstract: The problem of Rayleigh–Bénard’s natural convection subjected to a temporally periodic cooling condition is solved numerically by the Lattice Boltzmann method with multiple relaxation time (LBM-MRT). The study finds its interest in the field of thermal comfort where current knowledge has gaps in the fundamental phenomena requiring their exploration. The Boussinesq approximation is considered in the resolution of the physical problem studied for a Rayleigh number taken in the range 103 ≤ Ra ≤ 106 with a Prandtl number equal to 0.71 (air as working fluid).
    [Show full text]
  • Presentation
    Scaling laws for dynamos in rotating spherical shells and applications to planetary magnetic fields Ulrich Christensen Max-Planck-Institute for Solar System Research, Katlenburg-Lindau, Germany In collaboration with Julien Aubert, Carsten Kutzner, Peter Olson, Andreas Tilgner, Johannes Wicht Outline of geodynamo models Solve equations of thermal / compositional con- vection and magnetic induction in a rotating and electrically conducting spherical shell • Fundamental MHD equations • Some parameters not Earth-like Fluid outer core • Differences between models: - parameter values Solid inner core - boundary conditions - mode of driving convection Tangent cylinder Successes of geodynamo modeling Earth’s radial magnetic field at CMB Dipole tilt | 1 million years | Dynamo model field at full resolution and filtered to ℓ < 13 Tilt of dipole axis vs. time in dynamo model Control parameters Name Force Earth Model balance value values Rayleigh Buoyancy 5000 x < 50 x Ra* number Retarding forces critical critical Ekman Viscosity E 10-14 ≥ 3x10-6 number Coriolis force Prandtl Viscosity Pr 0.1 - 1 0.1 - 10 number Thermal diffusion Magnetic Viscosity Pm 10-6 0.06 - 10 Prandtl # Magnetic diffusion Models all wrong ? • Geodynamo models successfully reproduced the properties of the geomagnetic field • This is surprising, because several control parameters are far from Earth values (viscosity and thermal diffusivity too large) • Pessimistic view: Models give right answer for wrong reasons • Optimistic assumption: Models are already in regime where diffusion
    [Show full text]
  • Turbulent Superstructures in Rayleigh-Bأ©Nard Convection
    ARTICLE DOI: 10.1038/s41467-018-04478-0 OPEN Turbulent superstructures in Rayleigh-Bénard convection Ambrish Pandey 1, Janet D. Scheel 2 & Jörg Schumacher 1 Turbulent Rayleigh-Bénard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the fl 1234567890():,; onset of convection. Here we report numerical simulations of turbulent convection in uids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to 107.We identify characteristic scales and times that separate the fast, small-scale turbulent fluc- tuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale patterns, which change with Prandtl and Rayleigh number, are also correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo- and astrophysical settings. 1 Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany. 2 Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, CA 90041, USA. Correspondence and requests for materials should be addressed to J.S. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:2118 | DOI: 10.1038/s41467-018-04478-0 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04478-0 arge temperature differences across a horizontally extended be able to separate the fast, small-scale turbulent fluctuations of fluid layer induce a turbulent convective fluid motion which velocity and temperature from the gradual variation of the large- L 1 is relevant in numerous geo- and astrophysical systems .
    [Show full text]
  • Free Convective Heat Transfer from an Object at Low Rayleigh Number 23
    Free Convective Heat Transfer From an Object at Low Rayleigh Number 23 FREE CONVECTIVE HEAT TRANSFER FROM AN OBJECT AT LOW RAYLEIGH NUMBER Md. Golam Kader and Khandkar Aftab Hossain* Department of Mechanical Engineering, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh. *Corresponding e-mail: [email protected],ac.bd Abstract: Free convective heat transfer from a heated object in very large enclosure is investigated in the present work. Numerical investigation is conducted to explore the fluid flow and heat transfer behavior in the very large enclosure with heated object at the bottom. Heat is released from the heated object by natural convection. Entrainment is coming from the surrounding. The two dimensional Continuity, Navier-Stokes equation and Energy equation have been solved by the finite difference method. Uniform grids are used in the axial direction and non-uniform grids are specified in the vertical direction. The differential equations are discretized using Central difference method and Forward difference method. The discritized equations with proper boundary conditions are sought by SUR method. It has been done on the basis of stream function and vorticity formulation. The flow field is investigated for fluid flowing with Rayleigh numbers in the range of 1.0 ≤ Ra ≤ 1.0×103 and Pr=0.71. It is observed that the distortion of flow started at Rayleigh number Ra=10. It is observed that the average heat transfer remains constant for higher values of Reyleigh number and heating efficiency varies with Ra upto the value of Ra=35 and beyond this value heating efficiency remains constant. Keywords: Free convection, enclosure, distortion, Rayliegh no., vorticity.
    [Show full text]
  • Convection, Stability and Turbulence
    Convection, Stability and Turbulence Lecturer: Charlie Doering Notes written by: Chris Cawthorn & Toby Wood 18 June 2008 1 Introduction The three basic forms of heat transport are: Radiation • Conduction • Convection • In many physical systems, all of these processes may play a role (see Figures 1{2 below). Convection is buoyancy-driven flow due to density variations in a gravitational field. It is particularly important as it enhances the vertical heat transport. For example, without convection, it would take many hours to heat a saucepan of water to boiling point, simply because heat diffusion in water is so weak (κ 1:5 10−3cm2 s−1). ≈ × Figure 1: Heat transfer in a saucepan. 2 Rayleigh-B´enard convection In 1900, Henri B´enard [2] performed an experiment concerning convection cells in a thin liquid layer. He observed spontaneous pattern formation - the convection was seen to organise itself into hexagonal cells throughout the entire domain, as shown in Figure 3. 19 Figure 2: Heat transfer in the Earth and Sun. Figure 3: Convection cells in B´enard's experiment and on the surface of the Sun. Rayleigh [28] was the first to undertake a mathematical theory of fluid convection. His analysis described the formation of convection rolls for fluid confined between parallel plates held at different temperatures. The width of the rolls at convective onset is proportional to the depth of the fluid layer, but also depends on details of the boundary conditions. However, these convection rolls are a different physical phenomenon to the convection cells observed by B´enard. Such convection cells are now known to be associated with Marangoni flows, where variations of surface tension with temperature play a crucial role.
    [Show full text]
  • Experiments on Rayleigh–Bénard Convection, Magnetoconvection
    J. Fluid Mech. (2001), vol. 430, pp. 283–307. Printed in the United Kingdom 283 c 2001 Cambridge University Press Experiments on Rayleigh–Benard´ convection, magnetoconvection and rotating magnetoconvection in liquid gallium By J. M. AURNOU AND P. L. OLSON Department of Earth and Planetary Sciences,† Johns Hopkins University, Baltimore, MD 21218, USA (Received 4 March 1999 and in revised form 12 September 2000) Thermal convection experiments in a liquid gallium layer subject to a uniform rotation and a uniform vertical magnetic field are carried out as a function of rotation rate and magnetic field strength. Our purpose is to measure heat transfer in a low-Prandtl-number (Pr =0.023), electrically conducting fluid as a function of the applied temperature difference, rotation rate, applied magnetic field strength and fluid-layer aspect ratio. For Rayleigh–Benard´ (non-rotating, non-magnetic) convection 0.272 0.006 we obtain a Nusselt number–Rayleigh number law Nu =0.129Ra ± over the range 3.0 103 <Ra<1.6 104. For non-rotating magnetoconvection, we find that the critical× Rayleigh number× RaC increases linearly with magnetic energy density, and a heat transfer law of the form Nu Ra1/2. Coherent thermal oscillations are detected in magnetoconvection at 1.4Ra∼C . For rotating magnetoconvection, we find that the convective heat transfer is∼ inhibited by rotation, in general agreement with theoretical predictions. At low rotation rates, the critical Rayleigh number increases linearly with magnetic field intensity. At moderate rotation rates, coherent thermal oscillations are detected near the onset of convection. The oscillation frequencies are close to the frequency of rotation, indicating inertially driven, oscillatory convection.
    [Show full text]