Session KJ: Convection and Buoyancy-Driven Flows VI

Heat Transfer in Turbulent Rotating Convection

Rayleigh-B\'{e}nard convection is a classical problem in which a fluid layer enclosed between two parallel horizontal walls is heated from below. In a rotating frame of reference the dynamics can change considerably through the fundamental involvement of a combination of buoyancy and Coriolis forces. The rotating Rayleigh-B\'{e}nard (RRB) setting is important for many applications, e.g., in engineering and climate modelling. Direct numerical simulation (DNS) is used to calculate the heat transfer at systematically varied rotation rates. The DNS code solves the incompressible Navier-Stokes equations in a rotating frame of reference, coupled to the heat equation within the Boussinesq approximation. Periodic boundary conditions are adopted in the horizontal directions and the vertical boundaries are treated as isothermal, no-slip walls. The velocity and temperature averages from this DNS will be compared to measurements in a water-filled cylindrical convection cell. Detailed velocity and temperature data will be obtained using stereoscopic particle image velocimetry and laser induced fluorescence, respectively.   

Modulated rotating convection: Quenching wall modes

The onset of thermal convection in an enclosed rotating cylinder when the rotation is Coriolis force dominated is to wall modes -- precessing hot/cold thermal plumes rising and descending in the sidewall layer of the cylinder. If the cylinder rotation is modulated with a frequency about twice its rotation frequency, an oscillatory large scale circulation is established which quenches the three-dimensional wall mode of convection, leaving an oscillatory axisymmetric convective state, even when the modulation amplitude is small. The associated physical mechanism leading to the quenching of the wall modes is explored numerically using the Navier-Stokes-Boussinesq equations.   

Rotating convection: Eckhaus-Benjamin-Feir instability

A numerical study of the onset of thermal convection in a rotating circular cylinder of depth-to-radius ratio equal to 4 is considered in a regime dominated by the Coriolis force where the onset is to wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and decending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, $m$. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by $m$, coexist and are stable. Their dyanmics are controlled by an Eckhaus-Benjamin-Feir instability, the most basic features of which are captured by a complex Ginsburg-Landau equation model. This instability in rotating convection has been analyzed for the first time using direct numerical simulations of the Navier-Stokes equations in the Boussinesq approximation. Several properties of the wall modes have been computed, extending the results to far beyond the onset of convection. Extensive favorable comparisons between our numerical results and previous experimental observations and complex Ginsburg-Landau model results of Liu and Ecke (PRL 1997, JFM 1999) are made.   

Measured spatial distribution of the local thermal dissipation rate in turbulent Rayleigh-B{\'e}nard convection

The timed-averaged local thermal dissipation rate $\epsilon_{N} ({\bf r})$ in turbulent convection is obtained from simultaneous measurements of three components of the temperature gradient vector in convection cells filled with water. It is found that the measured $\epsilon_{N}({\bf r})$ contains two contributions; one is generated by thermal plumes and occupies mainly in the plume-dominated bulk region. The other contribution comes from the mean temperature gradient and is concentrated in the thermal boundary layers. The experiment provides new insights into the mechanism of heat transport in turbulent convection. \newline \newline *Work supported by the Research Grants Council of Hong Kong SAR.   

Horizontal infinite-Prandtl-number convection

Differential heating of a horizontal surface in a finite fluid volume, leading to so-called horizontal convection, is a hot topic for both the oceans and glass furnaces. Here we investigate the infinite-Prandtl-number limit relevant to the latter, using numerics and scaling arguments. Two regimes are identified, and universal scaling functions are found for the transition between the regimes. Conclusive numerical evidence is presented for the Rossby boundary-layer scaling for both rigid and shear-free boundary conditions. The surprising lack of dependence on the boundary condition is explained by showing that the return flow comprises of a horizontal intrusion at the bottom of the horizontal boundary layer, and a weak downwelling plume that returns at depth. We also present results for highly temperature-dependent viscosity.   

Computational Homology in Rayleigh-Benard convection experiments

Computational homology is used to analyze spatial structures of spiral defect chaos (SDC) in Rayleigh-Benard convection (RBC) experiments. Geometric structures composed of hot (up) and cold (down) flows in SDC are visualized by a shadowgraph system producing images used to compute the homology of the rolls. The analysis of experimental data yields Betti numbers, which count the number of connected components and holes of the hot and cold flow regions in the images. The homology is used to detect symmetry breakings between up and down flows (Non-Boussinesq effects) in SDC. The probability distribution and sequence of Betti numbers at different parameter values are used for identifying and characterizing different attractors and states of SDC.   

Prediction of the center temperature for compressible non-Oberbeck-Boussinesq thermal convection

Thermal convection in gaseous ethane at high pressure is theoretically analyzed, focusing on non-Oberbeck-Boussinesq (NOB) effects. On the basis of boundary-layer equations with variable transport properties, it is shown that the top-down symmetry of the velocity, temperature, and density profiles is broken. In particular, we predict that the temperature $T_{c}$ in the center of the convection container is less than the mean temperature $T_{m} = (T_{t} + T_{b})/2$ between the top ($T_{t}$) and bottom ($T_{b}$) plates, in contrast to the corresponding NOB effect in liquids. We also characterize the temperature profile across the top boundary-layer, as $T_{t}$ approaches the gas-liquid coexistence curve and the corresponding thermal conductivity is enhanced.   

An experimental study of highly lazy plumes

We present results from an experimental study of highly lazy turbulent plumes, {\em{i.e.}} plumes with relatively low source momentum flux, or equivalently very large source Richardson numbers. Experimental observations indicate that the plumes contract as they move vertically away from the source and that the extent of the contraction is independent of the source Richardson number (consistent with previous experimental studies). Using the experimental technique developed by Baines (1983), we made volume flux measurements in the near source region of the plume. Our experimental results indicate that the volume flux increases linearly with distance from the source and scales with the source Richardson number to the one third power. This result is discussed in relation to existing entrainment models for forced plumes (low source Richardson number) and we demonstrate that these do not adequately describe the near source region of highly lazy plumes. It is also noted that the near source behaviour is similar to that of a line plume and a possible explanation for this behaviour is presented. \newline Baines, W.D. (1983), ``A technique for the direct measurement of volume flux of a plume,'' {\it{J. Fluid Mech.}} \textbf{132}, 247--256.