Session AJ: Convection and Buoyancy-Driven Flows I

A geophysical scale model of turbulent natural convection next to a vertical isothermal surface

Modelling of many geophysical systems requires accurate prediction of heat transfer from large vertical surfaces. For example, the melt rate of a polar ice shelf submerged several hundred metres into the ocean is controlled in part by the convective heat flux to its surface. We use scaling ideas to develop a model of a vertical natural convection boundary layer with an inner laminar layer coupled to an outer turbulent region. Two distinct dynamical regimes are observed. At small heights the laminar layer is buoyancy driven resulting in a Nusselt number-Rayleigh number correlation of $Nu \propto Ra_x^{1/3}$ consistent with laboratory experiment. At larger heights the dominant buoyant forcing is in the outer turbulent region, which exerts a shear to drive the laminar sublayer. This regime is likely to be the one relevant to most geophysical situations, and yields $Nu \propto Ra_x^{1/2}$. Our scaling analyses are consistent with the ideas that the width of the laminar sublayer is determined by a buoyancy-driven instability in the first regime but by a shear-driven instability in the second.   

Morphological Evolution of Thermal Plumes in Turbulent Convection

An experimental study of the morphological evolution of thermal plumes in turbulent thermal convection is presented. Both visualization and quantitative velocity and temperature measurement techniques are used in the study. When viewed from the top the plumes look sheet-like and their lengths have an approximate log-normal distribution. When the sheet-like plumes curl-up or cluster together to form mushroom-like objects strong vertical vorticity fluctuations are generated. The fluctuating vorticity is found to have an exponential distribution and correlates strongly with temperature fluctuations. Moreover, the rms values of vorticity and temperature are found to exhibit similar scaling behavior with Ra.   

Forced convection in a mushy layer

The effect of an external shear flow on the convective instabilities inherent to the directional solidification of a dendritic mushy layer is investigated using a linear stability analysis. The external flow is coupled to advective perturbations in the liquid and to flow in the mush through a perturbed mush-liquid interface. A complete numerical solution of the stability of the system is performed and we find the critical porous medium Rayleigh number as a function of both the external flow speed and the wavenumber of the interfacial perturbations. By neglecting convection in the liquid and solving only for the pressure perturbations on the corrugated mush-liquid interface induced by the external flow a reduced model can be constructed and solved semi-analytically. These theoretical results are compared with experimental results obtained in a laboratory flume in which an ammonium-chloride solution was solidified from below.   

Percolation in the Rayleigh-Benard convection

The divergence of the viscosity coefficient and the heat conductivity in the Rayleigh-Benard convection was found in the numerical calculation from 2004 to 2005[1-3]. The interpretation for this phenomenon from the viewpoint of physics is incomplete. The present author proposes a physical interpretation introducing the percolation theory[4]. The temperature difference region where the divergence of the viscosity coefficient occurs depends on the system size length. This system size dependency gives us an insight for the physics in the divergence of the viscosity coefficient in the Rayleigh-Benard convection. \newline \newline [1] H. Shibata, \textit{Momentum flux in Rayleigh-Benard convection}, Physica A \textbf{333}, 71-86(2004). \newline [2] H. Shibata, \textit{Momentum flux in Rayleigh-Benard convection II}, Physica A \textbf{345}, 448-456(2005). \newline [3] H. Shibata, \textit{Heat flux in Rayleigh-Benard convection}, Physica A \textbf{352}, 335-346(2005). \newline [4] D. Stauffer, \textit{Introduction to Percolation Theory} (Taylor {\&} Francis, London, 1985).   

Localisation of mushy-layer convection by background flow

Under certain conditions, the directional solidification of a binary alloy leads to the development of a dendritic mushy region. Convective instability in the mush initiates the formation of chimneys (freckles). We examine the r\^ole of a weak and slowly varying background flow in the localisation of mushy-layer convection. Both two and three dimensional perturbations to a two dimensional background flow are considered in the near-eutectic limit, where the solid fraction in the mush is small. We find that three-dimensional disturbances are localised at places where there is upflow in the mush. We present amplitude equations to describe the evolution of both the 2D and 3D disturbances.   

Steady-state mushy layers: Experiments and theory

A new facility has been developed to investigate mushy layers formed during the steady directional solidification of transparent aqueous solutions in a quasi-two-dimensional system. Experiments have been conducted on NaCl--H20 solutions by translating a Hele-Shaw cell at prescribed rates between fixed heat exchangers providing a temperature gradient of approximately 1\,$^0$C/mm. Ice formed the primary solid phase and the dense residual fluid ponded within the mushy layer at the base of the system. Mathematical predictions of the steady-state temperature profile and mushy layer thickness as functions of freezing rate are in excellent agreement with experimental results. Experiments have also been performed on aqueous NH4Cl solutions, with the salt forming the primary solid phase, yielding buoyancy-driven convection in the mushy layer and the development of chimneys. The lifetime of the chimneys increased with decreasing freezing rate; however, no steady-state chimneys have been observed. Rather, a convecting chimney appears to deplete the surrounding solution and is eventually extinguished. At freezing rates larger than about 5.5\,$\mu$m/s a uniform mushy layer develops with no chimneys. However, at rates larger than about 5\,$\mu$m/s a second mode of behaviour is observed in which the mushy layer is thin and there is significant supercooling and nucleation above it. There is hysteresis between the two modes.   

Effect of the Earth's Coriolis force on the large-scale circulation of turbulent Rayleigh-B{\'e}nard convection

We present measurements of the large-scale circulation (LSC) of turbulent Rayleigh-B{\'e}nard convection in water-filled cylindrical samples of heights equal to their diameters. The LSC consists of a single convection cell circulating in a near- vertical plane. Its azimuthal orientation $\theta_0(t)$ had an irregular time dependence, but revealed a net azimuthal rotation with an average period of about 3 days for Rayleigh numbers $R \stackrel{>}{_\sim} 10^{10}$. On average there was also a tendency for the LSC to be aligned with upflow to the west and downflow to the east, even after physically rotating the apparatus in the laboratory. The rate of azimuthal rotation and the probability distribution $P(\theta_0)$ could be calculated from a model of diffusive azimuthal meandering in a potential due to the Earth's Coriolis force.   

The Hopping Dynamics of a Drifting Heat Blanket in Turbulent Thermal Convection

We report an experimental study in turbulent thermal convection that has an open upper surface. The geometry of the convective system is annular with aspect ratio (girth/height) on the order of 10 and with periodic boundary conditions. We investigate the intriguing interaction between the convective flow and a freely moving, floating boundary that partially covers the open fluid surface. The floating boundary position and the corresponding convective pattern are simultaneously recorded and correlated to reveal the coupled dynamics. We observe robust hopping behavior of the floating boundary as it exerts a thermal blanketing effect (reducing the local heat loss from the bulk fluid), which constantly modifies the bulk convective pattern.   

3D chaotic mixing promoted with time-dependent natural convection

Mixing with natural convection inside a three-dimensional cubic cavity can be achieved if the motion of a fluid is generated by imposing alternating hot and cold temperatures on opposite walls. In this work we study the flow produced inside a cubic prism with sections of its upper and lower horizontal walls cooled and heated in a periodic manner. These boundary conditions generate a vortex of time dependent intensity which moves around the geometrical center of the cavity. The three-dimensional trajectory followed by the vortex and its intensity are described. The governing equations of natural convection are solved numerically using the control volume method and equations are decoupled following the SIMPLEC (Semi-Implicit Method for Pressure Linked Equations) integration strategy. The mixing properties of the flow are described by Lagrangian tracking of collections of massless points. It is demonstrated that efficient mixing can be achieved with natural covection.   

Extensive Chaos in Rayleigh-Benard Convection

Spatiotemporal chaos is studied using large-scale numerical simulations of Rayleigh-Benard convection in cylindrical domains with experimentally realistic boundary conditions. The Lyapunov exponents and fractal dimension are calculated over a range of system sizes as given by aspect ratios of the cylindrical convection domain between 5 and 15. It is found that the chaos is extensive over this range of system size as illustrated by a linear dependence of the fractal dimension with the square of the aspect ratio. The chaos is extensive even though the convection pattern is found to transition from boundary to bulk dominated dynamics as the system size is increased. An analysis of the Lyapunov vectors is used to yield quantitative information describing the location of the largest growing perturbations. It is found that for small aspect ratios the largest perturbations are near the lateral sidewalls and as the aspect ratio increases the largest perturbations are found in the center of the convection domain. In all cases the largest perturbations are localized and correlate with defect structures in the fluid flow field.   

Spatio-temporal behavior of natural convection in low Prandtl number fluid

Natural convection appearing in liquid gallium layer, which has Prandtl number of 0.03, is investigated experimentally at the range of the Rayleigh number $R$, $10^3 < R < 10^5$. Spatio- temporal velocity distribution in the rectangular fluid layer measured by ultrasonic velocity profiler represents two kinds of periodic motion of convection roll; periodic movement of the roll and periodic variation of the size of the roll. Furthermore, simultaneous measurement of two velocity profiles clarifies existence of the wavy motion of the roll for the axial direction, which has been predicted by a stability analysis. Spatial distribution of the frequency of the periodic motion is investigated by frequency analysis of the obtained spatio- temporal velocity distribution. Variation of the period with respect to Rayleigh number is determined by frequency analysis of the temperature fluctuation measured by a thermistor. The frequency component cannot be detected at smaller Rayleigh number, under $R = 10^4$ and the frequency increases proportional to the Rayleigh number power 0.38.   

Why is there dispersion in the measured scaling exponent of the Reynolds number in turbulent Rayleigh-B\'{e}nard convection?

A riddle in turbulent thermal convection is the apparent dispersion from 0.42 to 0.5 in the value of the scaling exponent $\gamma $ of experimentally measured Reynolds number $\Re$ $\sim$ Ra$^{\gamma}$, where Ra is the Rayleigh number. The measured Re may be divided into two groups: one based on the circulation frequency of the mean wind and the other based on a directly measured velocity. With new experimental results we show that in frequency measurements the dispersion in $\gamma $ is a result of the evolution in the circulation path of the wind, and that in the velocity measurements it is caused by the inclusion of a counter-flow in the mean velocity. When these factors are properly accounted for both groups give $\gamma $ = 0.5, which may imply that a single mechanism is driving the flow for both low and high values of Ra.