Rayleigh-B\'enard convection with a melting boundary

We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-B\'enard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appears as the depth -- and therefore the effective Rayleigh number of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.Comment: 36 pages, 16 figure

Inertial convection in a rotating narrow annulus: Asymptotic theory and numerical simulation

The following article appeared in Physics of Fluids, 2015, Volume 27, and may be found at http://scitation.aip.org/content/aip/journal/pof2/27/10/10.1063/1.4934527.An important way of breaking the rotational constraint in rotating convection is to invoke fast oscillation through strong inertial effects which, referring to as inertial convection, is physically realizable when the Prandtl number Pr of rotating fluids is sufficiently small.We investigate, via both analytical and numerical methods, inertial convection in a Boussinesq fluid contained in a narrow annulus rotating rapidly about a vertical symmetry axis and uniformly heated from below, which can be approximately realizable in laboratory experiments [R. P. Davies-Jones and P. A. Gilman, “Convection in a rotating annulus uniformly heated from below,” J. Fluid Mech. 46, 65-81 (1971)]. On the basis of an assumption that inertial convection at leading order is represented by a thermal inertial wave propagating in either prograde or retrograde direction and that buoyancy forces appear at the next order to maintain the wave against the effect of viscous damping, we derive an analytical solution that describes the onset of inertial convection with the non-slip velocity boundary condition. It is found that there always exist two oppositely traveling thermal inertial waves, sustained by convection, that have the same azimuthal wavenumber, the same size of the frequency, and the same critical Rayleigh number but different spatial structure. Linear numerical analysis using a Galerkin spectral method is also carried out, showing a quantitative agreement between the analytical and numerical solutions when the Ekman number is sufficiently small. Nonlinear properties of inertial convection are investigated through direct three-dimensional numerical simulation using a finite-difference method with the Chorin-type projection scheme, concentrating on the liquid metal gallium with the Prandtl number Pr = 0.023. It is found that the interaction of the two counter-traveling thermal inertial waves leads to a timedependent, spatially complicated, oscillatory convection even in the vicinity of the onset of inertial convection. The nonlinear properties are analyzed via making use of the mathematical completeness of inertial wave modes in a rotating narrow annulus, suggesting that the laminar to weakly turbulent transition is mainly caused by the nonlinear interaction of several inertial wave modes that are excited and maintained by thermal convection at moderately supercritical Rayleigh numbers.Leverhulme TrustMacau FDCTChinese Academy of Science

Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

We provide a new framework for the study of fluid flows presenting complex uncertain behaviour. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal field equations. By numerically solving these equations, we evolve in a fully coupled way the mean flow and the statistical and spatial characteristics of the stochastic fluctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that considerably reduce the computational cost. We analyse the transformation of energy from stochastic modes to mean dynamics, and vice versa, by deriving exact expressions that quantify the interaction among different components of the flow. The developed framework is illustrated through specific flows in unstable regimes. In particular, we consider the flow behind a disk and the Rayleigh–Bénard convection, for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different parameters associated with the considered flows. We reveal the low dimensionality of the underlying stochastic attractor.United States. Office of Naval Research (Grant N00014-08-1-1097 (ONR6.1))United States. Office of Naval Research (Grant N00014-08-1-0586 (QPE))United States. Office of Naval Research (Grant N00014-09-1-0676 (Science of Autonomy - A-MISSION))United States. Office of Naval Research (Grant N00014-12-1-0944 (ONR6.2))Natural Sciences and Engineering Research Council of Canad

1991 Summer Study Program in Geophysical Fluid Dynamics : patterns in fluid flow

The GFD program in 1991 focused on pattern forming processes in physics and geophysics. The pricipallecturer, Stephan Fauve, discussed a variety of systems, including our old favorite, Rayleigh-Bénard convection, but passing on to exotic examples such as vertically vibrated granular layers. Fauve's lectures emphasize a unified theoretical viewpoint based on symmetry arguments. Patterns produced by instabilties can be described by amplitude equations, whose form can be deduced by symmetry arguments, rather than the asymptotic expansions that have been the staple of past Summer GFD Programs. The amplitude equations are far simpler than the complete equations of motion, and symetry arguments are easier than asymptotic expansions. Symmetry arguments also explain why diverse systems are often described by the same amplitude equation. Even for granular layers, where there is not a universaly accepted continuum description, the appropnate amplitude equation can often be found using symmetry arguments and then compared with experiment. Our second speaker, Daniel Rothan, surveyed the state of the art in lattice gas computations. His lectures illustrate the great utility of these methods in simulating the flow of complex multiphase fluids, particularly at low Reynolds numbers. The lattice gas simulations reveal a complicated phenomenology much of which awaits analytic exploration. The fellowship lectures cover broad ground and reflect the interests of the staff members associated with the program. They range from the formation of sand dunes, though the theory of lattice gases, and on to two dimensional-turbulence and convection on planetary scales. Readers desiring to quote from these report should seek the permission of the authors (a partial list of electronic mail addresses is included on page v). As in previous years, these reports are extensively reworked for publication or appear as chapters in doctoral theses. The task of assembling the volume in 1991 was at first faciltated by our newly acquired computers, only to be complicated by hurricane Bob which severed electric power to Walsh Cottage in the final hectic days of the Summer.Funding was provided by the National Science Foundation through Grant No. OCE 8901012

Thermal convection experiments in liquid metal flows with and without magnetic field

The interaction of electrically conducting fluid flows with magnetic fields appears in numerous natural phenomena and technical applications. Since the relevant fluids – such as liquid metals and plasmas – are generally very hot, the flows are often accompanied or even driven by thermal convection. The study of this so-called magnetoconvection is thus of interest for a number of physical systems. Two aspects are investigated in this thesis. The first concerns the case when an imposed magnetic field does not alter the fluid flow. The second case explores the changes of the flow structure and global transport properties in the presence of strong magnetic fields. The first point is relevant for inductive measurement techniques, which are required to probe the flow without disturbing it. Here, the size of the fluid volume affected by a localised magnetic field is of major importance. This topic is investigated theoretically by deriving an algorithm to calculate the penetration depth of the magnetic field into the medium. This allows the prediction of a magnetic field strength, above which a flow is significantly disturbed. The theoretical results are verified for the measurement method of local Lorentz force velocimetry which is applied to a vertical convection flow. The second point is investigated experimentally for a Rayleigh-Bénard convection system that is subject to a homogeneous vertical magnetic field. The set-up consists of a cylindrical cell of aspect ratio one. The large-scale flow structure is monitored using temperature measurements and ultrasound Doppler velocimetry. The evolution of the flow with increasing magnetic field strength is classified into different regimes and compared with theoretical predictions, and numerical simulations. Global transport properties of the flow concerning its momentum, and the heat passing through the fluid are analysed and their behaviour is interpreted in light of the aforementioned flow regimes. Additionally, a new theoretical model is developed to predict the turbulent heat and momentum transfer in the fluid by extending the Grossmann-Lohse theory for the classical Rayleigh-Bénard convection setting by the effects of a vertical magnetic field. Experimental data of the present study and from literature are used to verify and enhance the model, and to identify relevant physical mechanisms responsible for the observed results.Die Wechselwirkung zwischen elektrisch leitfähigen Fluiden und Magnetfeldern tritt in zahlreichen natürlichen Phänomenen und technischen Anwendungen auf. Weil die dabei relevanten Medien - meist Flüssigmetalle oder Plasmen - im Allgemeinen sehr heiß sind, werden die Strömungen meist von thermischer Konvektion begleitet oder werden sogar von dieser getrieben. Das Phänomen der sogenannten Magnetokonvektion ist damit von Interesse für eine große Anzahl physikalischer Systeme. Die vorliegende Arbeit untersucht hierbei zwei Aspekte. Zum einen wird der Fall betrachtet, wenn ein aufgeprägtes Magnetfeld das Strömungsfeld nicht verändert. Zum anderen werden die Modifizierungen von Strömungsstruktur und globalen Transporteigenschaften durch starke Magnetfelder untersucht. Der erste Fall ist wichtig für induktive Messtechniken, welche die Bewegung eines Mediums untersuchen müssen, ohne dieses dabei zu stören. Die Größe des Fluidvolumens, welches von einem örtlich begrenzten Magnetfeld beeinflusst wird, ist hier ein äußerst wichtiger Faktor. Dieses Thema wird untersucht, indem die Eindringtiefe des Magnetfeldes in das Medium theoretisch hergeleitet wird. Das erlaubt die Vorhersage einer Magnetfeldstärke, oberhalb derer eine Strömung maßgeblich gestört wird. Die theoretischen Ergebnisse werden mittels experimenteller Messungen überprüft. Dazu wird die Messmethode der lokalen Lorentzkraft-Anemometrie auf eine vertikale Konvektionsströmung angewandt. Für den zweiten Fall wird das System der Rayleigh-Bénard Konvektion unter einem homogenen, vertikalen Magnetfeld experimentell untersucht. Der Aufbau besteht aus einer zylindrischen Zelle mit einem Aspektverhältnis von eins. Die großskalige Struktur der Strömung wird mittels Temperaturmessungen und Ultraschall Doppler Anemometrie überwacht. Die Entwicklung der Strömung mit ansteigender Magnetfeldstärke kann in verschiedene Regime kategorisiert und mit theoretischen Vorhersagen sowie numerischen Simulationen verglichen werden. Globale Transporteigenschaften des Systems bezüglich Impuls und übertragener Wärme werden analysiert und ihr Verhalten anhand der zuvor gefundenen Strömungsregime interpretiert. Zusätzlich wird ein theoretisches Modell entwickelt um den turbulenten Wärme- und Impulstransport vorherzusagen. Dazu wird die Großmann-Lohse Theorie für klassische Rayleigh-Bénard Konvektion durch den Effekt eines vertikalen Magnetfeldes erweitert. Die experimentellen Daten aus der vorliegenden Arbeit und aus der Literatur werden genutzt, um dieses Modell zu verifizieren und zu optimieren. Dabei werden physikalische Prozesse identifiziert, welche maßgeblich zu den beobachteten Ergebnissen beitragen

On the Sensitivity of 3-D Thermal Convection Codes to Numerical Discretization: A Model Intercomparison

Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how does the discretization of the governing equations affect the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth\u27s mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in development mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes: an established, community-developed, and benchmarked finite element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBF) are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ =4). How variations in the behavior of the cubic pattern to perturbations in the initial condition and Rayleigh number between the two numerical discrezations is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods converge first to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady state pattern is presented as a combination of order 3 and 4 spherical harmonics leading to a five cell or a hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes

Countertraveling waves in rotating Rayleigh-Bénard convection

Linear and nonlinear counter-traveling waves in a fluid-filled annular cylinder with realistic no-slip boundary conditions uniformly heated from below and rotating about a vertical axis are investigated. When the gap of the annular cylinder is moderate, there exist two three-dimensional traveling waves driven by convective instabilities: a retrograde mode localized near the outer sidewall and a prograde mode adjacent to the inner sidewall with a different wave number, frequency and critical Rayleigh number. It is found that the retrogradely propagating mode is always more unstable and is marked by a larger azimuthal wave number. When the Rayleigh number is sufficiently large, both the counter-traveling modes can be excited and nonlinearly interacting, leading to an unusual nonlinear phenomenon in rotating Rayleigh-Bénard convection