DOCSLIB.ORG

Natural Convection Above Circular Disks of Evaporating Liquids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Natural Convection Above Circular Disks of Evaporating Liquids Natural convection above circular disks of evaporating liquids Benjamin Dollet1 and Fran¸coisBoulogne2 1Institut de Physique de Rennes, UMR 6251 CNRS and Universit´eRennes 1, Campus Beaulieu, B^atiment 11A, 35042 Rennes Cedex, France 2Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Universit´eParis-Saclay, Orsay 91405, France May 16, 2017 Abstract We investigate theoretically and experimentally the evaporation of liquid disks in the presence of natural convection due to a density difference between the vapor and the surrounding gas. From the analogy between thermal convection above a heated disk and our system, we derive scaling laws to describe the evaporation rate. The local evaporation rate depends on the presence of a boundary layer in the gas phase such that the total evaporation rate is given by a combination of different scaling contributions, which reflect the structure of the boundary layer. We compare our theoretical predictions to experiments performed with water in an environment controlled in humidity, which validate our approach. 1 Introduction where ρs and ρ are respectively the vapor density at saturation and at1 infinity, g the gravity constant, R the The evaporation of small liquid disks takes its roots in radius of the drop and ν the kinematic viscosity of air. botany, in particular with some studies on the transpi- For small Grashof numbers, evaporation is limited by dif- ration of plants published in the early twentieth century fusion while for large Grashof numbers, buoyancy creates [1, 2, 3, 4]. Beyond this original inspiration, evaporation a convective flow. is an ubiquitous phenomenon in nature as for the evapo- Additional experimental questioning [11] and evi- ration of liquids from water drops [5] to lakes or oceans [6] dences [12, 13, 14, 15, 16, 17] of the importance of con- and in the industry, in particular for coating processes. vective effects have been reported more recently in the Indeed, the transport of solutes is often driven by evap- literature. Direct visualizations of evaporating drops oration and dictates the self-organization of particles at have been achieved by X-ray imaging method [12], IR micro and macroscopic lengthscales [7]. Therefore, a cor- absorption [14], Schlieren technique [18] or by interfer- rect modeling of the evaporation dynamics is crucial for ometric measurements [19] and these studies concluded understanding evaporation kinetics and colloidal deposi- that evaporation can be enhanced by convection. The tion. transition between diffusive and convective evaporation When the evaporation is limited by the diffusion in the regimes has been reported and the convective evapo- β vapor phase [8], the derivation of the evaporative flux ration rate can be captured as Gr with β 0:20 ≈ shows that the total evaporation rate is linear with the [13, 14, 15]. radius of the liquid surface [9]. In 2006, Shahidzadeh- As we expect from the definition of the Grashof num- Bonn et al. reported experimental observations of evap- ber (Eq. (1)), the threshold for convective evaporation orating drops of water and hexane [10]. They observed is a function of the radial lengthscale of the evaporating arXiv:1704.03243v2 [physics.flu-dyn] 14 May 2017 that while a drop of hexane evaporates as predicted by a surface. Recently, Carrier et al. investigated the mutual diffusive model, a drop of water has an anomalous evap- influence of closely deposited drops on the evaporation oration rate above a certain radius. They attributed this rate [20] as it can be encountered for a sprayed liquid. If different behavior to the natural convection that takes the drops are separated by a distance comparable to their place, as water vapor is less dense than air. The di- radius, they found that a cooperative effect induces con- mensionless number representing the balance between vective evaporation. In addition, they studied the evap- the buoyant forces in favor of convection and the vis- oration of circular evaporating surfaces of different radii cous forces in favor of diffusion, is the Grashof number and they concluded that for large Grashof numbers, con- defined as vection dominates with an evaporation rate that scales 3 as R2. ρs ρ gR Gr = − 1 ; (1) ρ ν2 From the point of view of the governing equations, the 1 1 evaporation of a liquid is similar to the dissolution of a z liquid into another. Recent attention has been devoted to the sessile drop dissolution [21, 22, 23] and bubble ~g Water growth in supersaturated solutions [24] with a combina- tion of experimental, numerical and theoretical investiga- R tions. These studies show that natural convection is also r observed above dissolving drops. The main difference Ring between evaporation and dissolution is the magnitude of the Schmidt number Sc = ν= defined as the ratio of the kinematic viscosity ν of theD surrounding phase and Figure 1: Sketch presenting the notations for the convec- the diffusivity of the molecules. For evaporation, the tive evaporation above a circular disk of volatile liquid. Schmidt numberD is close to unity as diffusivity and kine- matic viscosity are similar for gases. In contrast, the constant, except in the terms where it acts as a driving Schmidt number is large for dissolution [22]. This differ- force for the flow; in particular, the flow is considered ence of Schmidt number is important for the structure of incompressible. the boundary layer of the convective flow [25]. Under these conditions, the velocity field of the vapor In this paper, we propose to derive scaling laws for a above the drop writes u = u e + w e and depends only flat circular evaporating surface in a regime dominated r z on the cylindrical coordinates r and z, with origin at by convection. We base our analysis on an analogy be- the center of the drop interface (Fig. 1). The continuity tween convective evaporation of flat circular surfaces and equation is the thermal convection that takes place above a heated 1 @(ru) @w disk. We show that our prediction is in agreement with + = 0: (2) r @r @z our measurements and with some empirical predictions available in the literature. With p the pressure field, c the mass concentration field, ρ(c) the gas density field and µ its dynamic viscosity, the radial and vertical components of the Navier-Stokes 2 Model equation are respectively @u @u @p As stated in the introduction, the evaporation of the liq- ρ u + w = + uid changes the composition of the surrounding atmo- 1 @r @z − @r sphere and thus, its local density. Therefore, we con- @2u 1 @u u @2u µ + + ; sider the natural convection that can take place above @r2 r @r − r2 @z2 a circular disk of liquid. In this Section, we establish (3a) a scaling law between the evaporation flux and the ra- @w @w @p dius of the flat drop. By analogy between heat and mass ρ u + w = ρ(c)g + 1 @r @z − − @z transfer, this derivation closely follows that of the heat @2w 1 @w @2w transfer above a horizontal heated surface, which is a µ + + : (3b) classical problem [26, 27, 28, 29]. Dehaeck et al. have @r2 r @r @z2 also performed a detailed study of the role of natural In these equations, according to the Boussinesq approx- convection on an evaporating pendant droplet, see the imation, the gas density is taken as constant and equal Supporting Information of [19]. In particular, they dis- to its value at infinity ρ , except in the driving force cussed the role of the varying slope of the droplet, and of term of natural convection,1 ρ(c)g in the equation (3b) of thermal Marangoni flows induced in the droplet by the vertical motion. From the ideal gas law, the gas density latent heat released during evaporation. However, these varies linearly with the vapor concentration: papers either considered a plate geometry instead of a disk [26, 27], or started from dimensionless or simplified c ρ(c) = ρ0 ∆ρ ; (4) equations [28, 29, 19], which is why we prefer to present − cs the derivation in details. where ρ0 is the density of dry air, ∆ρ = ρ0 ρs the den- sity difference between pure air and air saturated− with 2.1 Diffusion-convection equations vapor, and cs the saturation concentration of vapor in We establish the equations to describe the flow in the air. Finally, the diffusion-convection equation for the gas phase and our analysis requires the following as- concentration field is sumptions. (i) The liquid-vapor interface is horizontal. @c @c 1 @ @c @2c u + w = r + ; (5) (ii) The flow of vapor is axisymmetric and in a steady @r @z D r @r @r @z2 state. (iii) Thermal effects are negligible. (iv) We use the Boussinesq approximation: the air density is assumed where is the diffusion coefficient of vapor in air. D 2 Let p = pstat + $ with pstat the hydrostatic pressure, 2.0 1.0 such that ρ g @p =@z = 0. The Navier-Stokes stat 0.8 equations (3)− 1 become:− 1.5 0.6 ˜ ˜ c @u @u @$ z 1.0 ρ u + w = + 0.4 1 @r @z − @r 0.5 @2u 1 @u u @2u 0.2 µ + + ; @r2 r @r − r2 @z2 0.0 0.0 (6a) 2 1 0 1 2 − − r˜ @w @w @$ ρ u + w = [ρ(c) ρ ] g + 1 @r @z − − 1 − @z Figure 2: Dimensionless vapor concentration map above @2w 1 @w @2w an evaporating disk in the dimensionless space (~r; z~) ob- µ + + : (6b) @r2 r @r @z2 tained from equation (12). White and cyan lines are the oblate spheroidal coordinates (κ, σ), for respectively con- We now nondimensionalize equations (2), (5) and (6) stant κ and σ values.
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]
  • Laws of Similarity in Fluid Mechanics 21
    Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimen- sional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists.
    [Show full text]
  • Turbulent-Prandtl-Number.Pdf
    Atmospheric Research 216 (2019) 86–105 Contents lists available at ScienceDirect Atmospheric Research journal homepage: www.elsevier.com/locate/atmosres Invited review article Turbulent Prandtl number in the atmospheric boundary layer - where are we T now? ⁎ Dan Li Department of Earth and Environment, Boston University, Boston, MA 02215, USA ARTICLE INFO ABSTRACT Keywords: First-order turbulence closure schemes continue to be work-horse models for weather and climate simulations. Atmospheric boundary layer The turbulent Prandtl number, which represents the dissimilarity between turbulent transport of momentum and Cospectral budget model heat, is a key parameter in such schemes. This paper reviews recent advances in our understanding and modeling Thermal stratification of the turbulent Prandtl number in high-Reynolds number and thermally stratified atmospheric boundary layer Turbulent Prandtl number (ABL) flows. Multiple lines of evidence suggest that there are strong linkages between the mean flowproperties such as the turbulent Prandtl number in the atmospheric surface layer (ASL) and the energy spectra in the inertial subrange governed by the Kolmogorov theory. Such linkages are formalized by a recently developed cospectral budget model, which provides a unifying framework for the turbulent Prandtl number in the ASL. The model demonstrates that the stability-dependence of the turbulent Prandtl number can be essentially captured with only two phenomenological constants. The model further explains the stability- and scale-dependences
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • Analysis of Mass Transfer by Jet Impingement and Heat Transfer by Convection
    Analysis of Mass Transfer by Jet Impingement and Study of Heat Transfer in a Trapezoidal Microchannel by Ejiro Stephen Ojada A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Frank Pyrtle, III, Ph.D. Rasim Guldiken, Ph.D. Date of Approval: November 5, 2009 Keywords: Fully-Confined Fluid, Sherwood number, Rotating Disk, Gadolinium, Heat Sink ©Copyright 2009, Ejiro Stephen Ojada i TABLE OF CONTENTS LIST OF FIGURES ii LIST OF SYMBOLS v ABSTRACT viii CHAPTER 1: INTRODUCTION AND LITERATURE REVIEW 1 1.1 Introduction (Mass Transfer by Jet Impingement ) 1 1.2 Literature Review (Mass Transfer by Jet Impingement) 2 1.3 Introduction (Heat Transfer in a Microchannel) 7 1.4 Literature Review (Heat Transfer in a Microchannel) 8 CHAPTER 2: ANALYSIS OF MASS TRANSFER BY JET IMPINGEMENT 15 2.1 Mathematical Model 15 2.2 Numerical Simulation 19 2.3 Results and Discussion 21 CHAPTER 3: ANALYSIS OF HEAT TRANSFER IN A TRAPEZOIDAL MICROCHANNEL 42 3.1 Modeling and Simulation 42 3.2 Results and Discussion 48 CHAPTER 4: CONCLUSION 65 REFERENCES 67 APPENDICES 79 Appendix A: FIDAP Code for Analysis of Mass Transfer by Jet Impingement 80 Appendix B: FIDAP Code for Fluid Flow and Heat Transfer in a Composite Trapezoidal Microchannel 86 i LIST OF FIGURES Figure 2.1a Confined liquid jet impingement between a rotating disk and an impingement plate, two-dimensional schematic. 18 Figure 2.1b Confined liquid jet impingement between a rotating disk and an impingement plate, three-dimensional schematic.
    [Show full text]
  • Turbulence, Entropy and Dynamics
    TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions
    [Show full text]
  • Huq, P. and E.J. Stewart, 2008. Measurements and Analysis of the Turbulent Schmidt Number in Density
    GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L23604, doi:10.1029/2008GL036056, 2008 Measurements and analysis of the turbulent Schmidt number in density stratified turbulence Pablo Huq1 and Edward J. Stewart2 Received 18 September 2008; revised 27 October 2008; accepted 3 November 2008; published 3 December 2008. [1] Results are reported of measurements of the turbulent where rw is the buoyancy flux, uw is the Reynolds shear Schmidt number Sct in a stably stratified water tunnel stress. The ratio Km/Kr is termed the turbulent Schmidt experiment. Sct values varied with two parameters: number Sct (rather than turbulent Prandtl number) as Richardson number, Ri, and ratio of time scales, T*, of salinity gradients in water are used for stratification in the eddy turnover and eddy advection from the source of experiments. Prescription of a functional dependence of Sct turbulence generation. For large values (T* 10) values on Ri = N2/S2 is a turbulence closure scheme [Kantha and of Sct approach those of neutral stratification (Sct 1). In Clayson, 2000]. Here the buoyancy frequency N is defined 2 contrast for small values (T* 1) values of Sct increase by the gradient of density r as N =(Àg/ro)(dr/dz), and g is with Ri. The variation of Sct with T* explains the large the gravitational acceleration. S =dU/dz is the velocity scatter of Sct values observed in atmospheric and oceanic shear. Analyses of field, lab and numerical (RANS, DNS data sets as a range of values of T* occur at any given Ri. and LES) data sets have led to the consensus that Sct The dependence of Sct values on advective processes increases with Ri [see Esau and Grachev, 2007, and upstream of the measurement location complicates the references therein].
    [Show full text]
  • The Analogy Between Heat and Mass Transfer in Low Temperature Crossflow Evaporation
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AUT Scholarly Commons The analogy between heat and mass transfer in low temperature crossflow evaporation Reza Enayatollahi*, Roy Jonathan Nates, Timothy Anderson Department of Mechanical Engineering, Auckland University of Technology, Auckland, New Zealand. Corresponding Author: Reza Enayatollahi Email Address: [email protected] Postal Address: WD308, 19 St Paul Street, Auckland CBD, Auckland, New Zealand Phone Number: +64 9 921 9999 x8109 Abstract This study experimentally determines the relationship between the heat and mass transfer, in a crossflow configuration in which a ducted airflow passes through a planar water jet. An initial exploration using the Chilton-Colburn analogy resulted in a coefficient of determination of 0.72. On this basis, a re-examination of the heat and mass transfer processes by Buckingham’s-π theorem and a least square analysis led to the proposal of a new dimensionless number referred to as the Lewis Number of Evaporation. A modified version of the Chilton-Colburn analogy incorporating the Lewis Number of Evaporation was developed leading to a coefficient of determination of 0.96. 1. Introduction Heat and mass transfer devices involving a liquid interacting with a gas flow have a wide range of applications including distillation plants, cooling towers and aeration processes and desiccant drying [1-5]. Many studies have gone through characterising the heat and mass transfer in such configurations [6-9]. The mechanisms of heat and mass transfer are similar and analogical. Therefore, in some special cases where, either the heat or mass transfer data are not reliable or may not be available, the heat and mass transfer analogy can be used to determine the missing or unreliable set of data.
    [Show full text]
  • 1 Pe = Re× Sc Peclet Number
    Passive Scalar Diffusion in the Near Field Region of Turbulent Rectangular Submerged Free Jets Andrea Boghia, Ivan Di Venutab, Fabio Gorib,* a School of Water, Energy and Agrifood, Cranfield University, Cranfield, Bedfordshire MK43 0AL, United Kingdom, b Department of Industrial Engineering, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy. * Corresponding Author, [email protected] Abstract Jets are a common way to transfer mass among fluids, or from a fluid to a surface. At moderate Reynolds numbers and low turbulent intensities the jet exhibits a Near Field Region (NFR) several diameters long. Numerical results and a theoretical model are presented for the passive scalar diffusion in the NFR of a submerged free jet. Large Eddy Simulations (LES), in the Reynolds number 5000-40,000 and the Schmidt number range 1-100, are performed obtaining the passive scalar fields. Three mathematical models for the passive scalar diffusion are presented; the first one is valid in the NFR, specifically in the Undisturbed Region of Flow (URF), and the other two, obtained under the hypotheses of Tollmien and Görtler momentum spreadings, are valid in the Potential Core Region (PCR). The last two models employ a turbulent Schmidt number inversely proportional to the mean velocity gradient, conclusion obtained by the LES numerical results. The self-similar solutions of the passive scalar show good agreement with the LES results. The wide range of Reynolds and Schmidt numbers investigated gives generality to the results. Keywords: Submerged Rectangular Free Jet; Turbulent Flow; Near Field Region; Undisturbed Region of Flow; Passive Scalar; Large Eddy Simulation; Self-Similarity.
    [Show full text]
  • Compressible Effects Modelling in Turbulent Cavitating Flows Jean Decaix, Eric Goncalvès Da Silva
    Compressible effects modelling in turbulent cavitating flows Jean Decaix, Eric Goncalvès da Silva To cite this version: Jean Decaix, Eric Goncalvès da Silva. Compressible effects modelling in turbulent cav- itating flows. European Journal of Mechanics - B/Fluids, Elsevier, 2013, 39, pp.11-31. 10.1016/j.euromechflu.2012.12.001. hal-00768757 HAL Id: hal-00768757 https://hal.archives-ouvertes.fr/hal-00768757 Submitted on 12 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Compressible Effects Modelling in Turbulent Cavitating Flows Jean Decaix∗, Eric Goncalv`es∗ LEGI-Grenoble INP, 1025 rue de la Piscine, 38400 St Martin d’Heres, France Abstract A compressible, multiphase, one-fluid RANS solver has been developed to study turbulent cavitating flows. The interplay between turbulence and cav- itation regarding the unsteadiness and structure of the flow is complex and not well understood. This constitutes a determinant point to accurately sim- ulate the dynamic behaviour of sheet cavities. In the present study, different formulations including compressibility effects on turbulence are investigated. Numerical results are given for two partial cavities on Venturi geometries and comparisons are made with experimental data.
    [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]
  • Three-Dimensional Free Convection in Molten Gallium A
    Three-dimensional free convection in molten gallium A. Juel, T. Mullin, H. Ben Hadid, Daniel Henry To cite this version: A. Juel, T. Mullin, H. Ben Hadid, Daniel Henry. Three-dimensional free convection in molten gallium. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2001, 436, pp.267-281. 10.1017/S0022112001003937. hal-00140472 HAL Id: hal-00140472 https://hal.archives-ouvertes.fr/hal-00140472 Submitted on 6 Apr 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Fluid Mech. (2001), vol. 436, pp. 267–281. Printed in the United Kingdom 267 c 2001 Cambridge University Press Three-dimensional free convection in molten gallium By A. JUEL1, T. MULLIN1, H. BEN HADID2 AND D. HENRY2 1Schuster laboratory, The University of Manchester, Manchester M13 9PL, UK 2Laboratoire de Mecanique´ des Fluides et d’Acoustique UMR 5509, Ecole Centrale de Lyon/Universite´ Claude Bernard Lyon 1, ECL, BP 163, 69131 Ecully Cedex, France (Received 22 March 1999 and in revised form 11 December 2000) Convective flow of molten gallium is studied in a small-aspect-ratio rectangular, differentially heated enclosure. The three-dimensional nature of the steady flow is clearly demonstrated by quantitative comparison between experimental temperature measurements, which give an indication of the strength of the convective flow, and the results of numerical simulations.
    [Show full text]