DOCSLIB.ORG

Effects of Variations in Magnetic Reynolds Number on Magnetic Field Distribution in Electrically Conducting Fluid Under Magnetohydrodynamic Natural Convection

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Effects of Variations in Magnetic Reynolds Number on Magnetic Field Distribution in Electrically Conducting Fluid Under Magnetohydrodynamic Natural Convection Journal of Heat and Mass Transfer Research 4 (2017) 149-155 Journal of Heat and Mass Transfer Research Journal homepage: http://jhmtr.journals.semnan.ac.ir Effects of variations in magnetic Reynolds number on magnetic field distribution in electrically conducting fluid under magnetohydrodynamic natural convection Mohsen Pirmohammadi Department of Mechanical Engineering, Pardis Branch , Islamic Azad University, Pardis, Iran PAPER INFO A B S T R A C T This study numerically investigated the effects of variations in magnetic Reynolds History: number on magnetic field distribution in an enclosure under natural convection heat Submitted 2016-12-15 transfer. The investigated geometry was a two-dimensional enclosure with a hot left Revised 2017-10-07 wall, a cold right wall, and adiabatic top and bottom walls. The fluid was molten Accepted 2017-10-12 sodium with Pr=0.01. Natural convection heat transfer at a Rayleigh number of 105 and magnetic Reynolds numbers (Rem) of 10–1, 10–3, and 10–5 was considered in the Keywords: analysis. The governing equations adopted were continuum, momentum, energy, and Magnetic Reynolds; magnetic induction equations, which were solved concurrently using the finite volume number; method. For the coupling of velocity and pressure, the revised semi-implicit method Natural convection; for pressure linked equations algorithm was employed. Results showed that under a Magnetic field high magnetic Reynolds number, the non-dimensional magnetic fields in the X and Y directions were approximately constant because the diffusion of the magnetic potential was greater than the advection of such potential. As the magnetic Reynolds number increased, however, the magnetic field in the enclosure reached a magnitude unequal to that of the applied magnetic field, exhibited inconstancy, and increased to a value that deviated from 1. Thus, the Rem=10–1 under a non-dimensional magnetic field increased from 0.09 to 6.6 in the X direction and from –1.164 to 4.05 in Y the direction. © 2017 Published by Semnan University Press. All rights reserved. DOI: 10.22075/jhmtr.2017.1503.1100 strength is a critical factor for crystal formation. Hadid 1. Introduction et al. [2] probed into the effects of a strong vertical magnetic field on convection and segregation in The free convection heat transfer of electrically vertical Bridgeman crystal growth. Bessaih et al. [3] conducting fluids in enclosures has been the subject of numerically examined the effects of the electric a considerable number of theoretical, experimental, conductivity of walls and the direction of a magnetic and numerical investigations because of its importance field on gallium flow. The authors found a in many technological applications, such as use in considerable reduction in convection intensity as the liquid metal blankets for reactors. Another important magnetic field increases. application of the aforementioned process is crystal Ciofalo and Cricchio [4] considered the MHD growth in the industrial production of semiconductors. natural convection of liquid Pb–17Li in a cubical This particular application has been explored in a cavity with a uniform volumetric heat source and an number of works. Oreper and Szekely [1], for induced current that significantly stretches to the walls example, showed that a magnetic field suppresses normal to the applied magnetic field flux. Piazza and natural convection currents and that magnetic field Ciofalo [5, 6] developed a simple wall boundary Corresponding Author : Department of Mechanical Engineering, Pardis Branch , Islamic Azad University, Pardis, Iran Email: [email protected] 150 M. Pirmohammadi/ JHMTR 4 (2017) 149-155 condition to solve electromagnetism in a cavity magnetic Reynolds number on the magnetic field without allocating real grid points to electrically distribution and temperature and velocity profiles. conducting walls. Pirmohammadi et al. [7] numerically studied natural convection flow in the 2. Basic Equations presence of a magnetic field in a tilted enclosure heated from below and filled with liquid gallium. The The schematic of the examined enclosure is authors found that at a given inclination angle, depicted in Fig. 1. The cavity is differentially heated; convection heat transfer decreases when magnetic the left and right walls are isothermal at TH and TC, field intensity increases. Pirmohammadi et al. [8] respectively (TH>TC); and the horizontal walls are studied the effects of a magnetic field on buoyancy- adiabatic. A magnetic field was applied in the X driven convection in a differentially heated square direction. enclosure. The researchers showed that the heat transfer mechanisms and flow characteristics in the The governing equations used in this research are enclosure depend strongly on both the strength of the based on the conservation laws of mass, momentum, magnetic field and the Rayleigh number. They energy, and induction equations. The governing concluded that a magnetic field considerably equations are provided below: decreases the average Nusselt number. An extensive study of mixed convection was u v 0 (1) conducted by Selimefendigil et al. [11] for various x y geometries. Heidary et al. [12] numerically investigated the laminar flow and convective heat 2 2 transfer of Cu-water nanofluid in a channel subjected u u P u u 1 By Bx (2) (u v ) ( ) ( )B x y x x 2 y2 x y y to a uniform magnetic field. The authors revealed that 0 the thickness of the thermal boundary layer increases with the addition of nanoparticles to pure fluid. v v P 2v 2v Applying a magnetic field, however, thins the (u v ) ( ) g (T T ) 2 2 c (3) boundary layer because of the increase in velocity x y y x y B gradient near walls. Seth et al. [13] delved into the 1 y Bx ( )BX combined free and forced convection Couette– 0 x y Hartmann flow of a viscous, incompressible, and electrically conducting fluid. The fluid was placed in a T T 2T 2T rotating channel with arbitrary conducting walls, and Cpu v k( ) (4) 2 2 flow was examined under the presence of the Hall x y x y current. Makinde et al. [14] investigated the steady flow and heat transfer of an electrically conducting 2 2 fluid with variable viscosity and electrical Bx Bx u u 1 Bx Bx (5) u v Bx By ( 2 2 ) conductivity between two parallel plates in the x y x y σμ0 x y presence of a transverse magnetic field. In most of the above-mentioned studies, 2 2 By By v v 1 By By (6) mathematical and numerical methods were intended to u v B B ( ) x y x x y y σμ x 2 y2 determine the influence of any combination of 0 Grashof, Reynolds, Prandtl, and Hartmann numbers on flow and heat transfer. A technique common to these studies was the simplification of the magnetic induction equation on the basis of low Reynolds approximation [15, 16]. The dimensionless magnetic Reynolds number ( Re m 0 ) of flow represents the ratio of advection to diffusion in a magnetic field. At Rem<<1, advection is relatively unimportant, and a magnetic field thus tends to relax toward a purely diffusive state. This relaxation is determined by boundary conditions rather than flow; consequently, fluid motion has no influence on magnetic field distribution. In this work, a simulation of magneto-convective flow was carried out inside a square cavity. The purpose of the current study was to numerically solve full magnetic induction, continuum, momentum, and Fig. 1. Geometry and coordinates of cavity configuration energy equations to investigate the effect of the with magnetic effect M. Pirmohammadi / JHMTR 4 (2017) 149-155 151 More details regarding Equations (5) and (6) can B* B* V V U y V y B* B* be found in [17]. In the equations above, u and v are X Y x X y Y (13) velocity components, p denotes the pressure, T is the Pr 2B* 2B* ( y y ) temperature, represents the density, g is the 2 2 Rem X Y gravitational acceleration, is the viscosity, is the coefficient of thermal expansion, stands for the Bx To calculate the magnetic field instead of solving magnetic field in the X direction, and B represents Equations (12) and (13), we can use the magnetic y the magnetic field in the Y direction. Moreover, k potential vector ( A ) as follows: pertains to the thermal conductivity, Cp is the specific B A (14) heat capacity, is the electrical conductivity, and 0 denotes the magnetic permeability. A z A z A z (15) 3. The dimensionless variables included in the Bx ,By 0,Bz 0 analysis are defined thus: Y X Z x y CpuH CpvH Thus, Equations (12) and (13) are converted into X ,Y , U , V , H H k k Equation (15) in the following manner: 2 2 2 2 A A Pr A A Cp pH T TC (16) , , Re m 0 , U V ( 2 2 ) P 2 X Y Re m X Y k TH TC * B B k x * y The boundary conditions are Bx , By , Pr (7) B0 B0 c p B y at X 0 (17) A Y,Bx 1, 0 3 X c p g(Th T )H , Ra C , Ha B H k 0 By at X 1 (18) A Y, Bx 1, 0 On the basis of these variables, the governing X B x at Y 0 (19) equations are expressed in dimensionless form as A 0,By 0, 0 follows: Y B A 1,B 0, x 0 at Y 1 (20) y Y U V U & V 0 at all walls (21) 0 (8) X Y (X 0, X 1, Y 0, Y 1) (0,Y ) 1, (1,Y ) 0, , (22) 0 0 Y Y U U P 2U 2U Y 0 Y 1 U V Pr( ) X Y X X2 Y2 (9) 2 * * 2 Pr * By Bx Ha By ( ) 4.
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]
  • Small-Scale Dynamo Action in Rotating Compressible Convection
    Small-scale dynamo action in rotating compressible convection B. Favier∗and P.J. Bushby School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK Abstract We study dynamo action in a convective layer of electrically-conducting, compressible fluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly- conducting boundary conditions are adopted for the magnetic field. Two different levels of thermal stratification are considered. If the magnetic diffusivity is sufficiently small, the con- vection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number RM that is based upon the horizontal integral scale and the horizontally-averaged velocity at the mid-layer of the domain, we find that rotation tends to reduce the critical value of RM above which dynamo action is observed. Increasing the level of thermal stratification within the layer does not significantly alter the critical value of RM in the rotating calculations, but it does lead to a reduction in this critical value in the non-rotating cases. At the highest computationally-accessible values of the magnetic Reynolds number, the saturation levels of the dynamo are similar in all cases, with the mean magnetic energy density somewhere be- tween 4 and 9% of the mean kinetic energy density. To gain further insights into the differences between rotating and non-rotating convection, we quantify the stretching properties of each flow by measuring Lyapunov exponents. Away from the boundaries, the rate of stretching due to the flow is much less dependent upon depth in the rotating cases than it is in the corre- sponding non-rotating calculations.
    [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]
  • Effects of a Uniform Magnetic Field on a Growing Or Collapsing Bubble in a Weakly Viscous Conducting Fluid K
    Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang, I. S. Kang, and C. M. Lee Citation: Phys. Fluids 14, 29 (2002); doi: 10.1063/1.1425410 View online: http://dx.doi.org/10.1063/1.1425410 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v14/i1 Published by the American Institute of Physics. Related Articles Dynamics of magnetic chains in a shear flow under the influence of a uniform magnetic field Phys. Fluids 24, 042001 (2012) Travelling waves in a cylindrical magnetohydrodynamically forced flow Phys. Fluids 24, 044101 (2012) Two-dimensional numerical analysis of electroconvection in a dielectric liquid subjected to strong unipolar injection Phys. Fluids 24, 037102 (2012) Properties of bubbled gases transportation in a bromothymol blue aqueous solution under gradient magnetic fields J. Appl. Phys. 111, 07B326 (2012) Magnetohydrodynamic flow of a binary electrolyte in a concentric annulus Phys. Fluids 24, 037101 (2012) Additional information on Phys. Fluids Journal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions PHYSICS OF FLUIDS VOLUME 14, NUMBER 1 JANUARY 2002 Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea I.
    [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]
  • Toward a Self-Generating Magnetic Dynamo: the Role of Turbulence
    PHYSICAL REVIEW E VOLUME 61, NUMBER 5 MAY 2000 Toward a self-generating magnetic dynamo: The role of turbulence Nicholas L. Peffley, A. B. Cawthorne, and Daniel P. Lathrop* Department of Physics, University of Maryland, College Park, Maryland 20742 ͑Received 6 July 1999͒ Turbulent flow of liquid sodium is driven toward the transition to self-generating magnetic fields. The approach toward the transition is monitored with decay measurements of pulsed magnetic fields. These mea- surements show significant fluctuations due to the underlying turbulent fluid flow field. This paper presents experimental characterizations of the fluctuations in the decay rates and induced magnetic fields. These fluc- tuations imply that the transition to self-generation, which should occur at larger magnetic Reynolds number, will exhibit intermittent bursts of magnetic fields. PACS number͑s͒: 47.27.Ϫi, 47.65.ϩa, 05.45.Ϫa, 91.25.Cw I. INTRODUCTION Reynolds number will be quite large for all flows attempting to self-generate ͑where Re ӷ1 yields Reӷ105)—implying The generation of magnetic fields from flowing liquid m turbulent flow. These turbulent flows will cause the transition metals is being pursued by a number of scientific research to self-generation to be intermittent, showing both growth groups in Europe and North America. Nuclear engineering and decay of magnetic fields irregularly in space and time. has facilitated the safe use of liquid sodium, which has con- This intermittency is not something addressed by kinematic tributed to this new generation of experiments. With the dynamo studies. The analysis in this paper focuses on three highest electrical conductivity of any liquid, sodium retains main points: we quantify the approach to self-generation and distorts magnetic fields maximally before they diffuse with increasing Rem , characterize the turbulence of induced away.
    [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]
  • Stability and Instability of Hydromagnetic Taylor-Couette Flows
    Physics reports Physics reports (2018) 1–110 DRAFT Stability and instability of hydromagnetic Taylor-Couette flows Gunther¨ Rudiger¨ a;∗, Marcus Gellerta, Rainer Hollerbachb, Manfred Schultza, Frank Stefanic aLeibniz-Institut f¨urAstrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany bDepartment of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom cHelmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany Abstract Decades ago S. Lundquist, S. Chandrasekhar, P. H. Roberts and R. J. Tayler first posed questions about the stability of Taylor- Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number Pm on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fieldspbecomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number Rm = Re · Rm (with Re the Reynolds number of rotation, Rm its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for Pm 1 while for Pm 1 the instability instead sets in for supercritical Rm at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting.
    [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]