Robert E. Ecke

Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry

We consider experimentally the instability and mass transport of flow in a Hele-Shaw geometry. In an initially stable configuration, a lighter fluid (water) is located over a heavier fluid (propylene glycol). The fluids mix via diffusion with some regions of the resulting mixture being heavier than either pure fluid. Density-driven convection occurs with downward penetrating dense fingers that transport mass much more effectively than diffusion alone. We investigate the initial instability and the quasisteady state. The convective time and velocity scales, finger width, wave number selection, and normalized mass transport are determined for 6000<Ra<90,000. The results have important implications for determining the time scales and rates of dissolution trapping of carbon dioxide in brine aquifers proposed as possible geologic repositories for sequestering carbon dioxide.

Mechanisms of extensive spatiotemporal chaos in Rayleigh-Benard convection

Spatially extended dynamical systems exhibit complex behaviour in both space and time—spatiotemporal chaos. Analysis of dynamical quantities (such as fractal dimensions and Lyapunov exponents) has provided insights into low-dimensional systems; but it has proven more difficult to understand spatiotemporal chaos in high-dimensional systems, despite abundant data describing its statistical properties. Initial attempts have been made to extend the dynamical approach to higher-dimensional systems, demonstrating numerically that the spatiotemporal chaos in several simple models is extensive (the number of dynamical degrees of freedom scales with the system volume). Here we report a computational investigation of a phenomenon found in nature, ‘spiral defect’ chaos in Rayleigh–Bénard convection, in which we find that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom. By studying the detailed space–time evolution of the dynamical degrees of freedom, we find that the mechanism for the generation of chaotic disorder is spatially and temporally localized to events associated with the creation and annihilation of defects.

Rotating Rayleigh–Bénard convection: asymmetric modes and vortex states

We present optical shadowgraph flow visualization and heat transport measurements of Rayleigh–Benard convection with rotation about a vertical axis. The fluid, water with Prandtl number 6.4, is confined in a cylindrical convection cell with radius-to-height ratio Γ = 1. For dimensionless rotation rates 150 R c (Ω) much less than those predicted by linear stability analysis for a laterally infinite system and qualitatively consistent with finite-aspect-ratio, linear-stability calculations of Buell & Catton (1983). As in the calculations, the forward bifurcation at onset is to states of localized flow near the lateral walls with azimuthal periodicity of 3 m et al . (1992), with a frequency that is finite at onset but goes to zero as Ω goes to zero. At Ω = 2145 we find primary and secondary stability boundaries for states with m = 4, 5, 6, and 7. Further, we show that at higher Rayleigh number, there is a transition to a vortex state where the vortices form with the symmetry of the existing azimuthal periodicity of the sidewall state. Aperiodic, time-dependent heat transport begins for Rayleigh numbers at or slightly above the first appearance of vortices. Visualization of the formation and interactions of thermal vortices is presented, and the behaviour of the Nusselt number at high Rayleigh numbers is discussed.

Knots and Random Walks in Vibrated Granular Chains

We study experimentally statistical properties of the opening times of knots in vertically vibrated granular chains. Our measurements are in good qualitative and quantitative agreement with a theoretical model involving three random walks interacting via hard-core exclusion in one spatial dimension. In particular, the knot survival probability follows a universal scaling function which is independent of the chain length, with a corresponding diffusive characteristic time scale. Both the large-exit-time and the small-exit-time tails of the distribution are suppressed exponentially, and the corresponding decay coefficients are in excellent agreement with theoretical values.

Turbulent rotating convection: an experimental study

We present experimental measurements of velocity and temperature fields in horizontal planes crossing a cylindrical Rayleigh–Benard convection cell in steady rotation about its vertical axis. The range of dimensionless rotation rates Ω is from zero to 5×10 4 for a Rayleigh number R = 3.2×10 8 . The corresponding range of convective Rossby numbers is ∞ > Ro > 0.06. The patterns of velocity and temperature and the flow statistics characterize three basic flow regimes. For Ro [Gt ] 1, the flow is dominated by vortex sheets (plumes) typical of turbulent convection without rotation. The flow patterns for Ro ∼ 1 are cyclone-dominated, with anticyclonic vortices rare. As the Rossby number continues to decrease, the number of anticyclonic vortex structures begins to grow but the vorticity PDF in the vicinity of the top boundary layer still shows skewness favouring cyclonic vorticity. Velocity-averaging near the top of the cell suggests the existence of a global circulation pattern for Ro [Gt ] 1.

Soap film flows: Statistics of two-dimensional turbulence

Soap film flows provide a very convenient laboratory model for studies of two-dimensional (2-D) hydrodynamics including turbulence. For a gravity-driven soap film channel with a grid of equally spaced cylinders inserted in the flow, we have measured the simultaneous velocity and thickness fields in the irregular flow downstream from the cylinders. The velocity field is determined by a modified digital particle image velocimetry method and the thickness from the light scattered by the particles in the film. From these measurements, we compute the decay of mean energy, enstrophy, and thickness fluctuations with downstream distance, and the structure functions of velocity, vorticity, thickness fluctuation, and vorticity flux. From these quantities we determine the microscale Reynolds number of the flow Rλ≈100 and the integral and dissipation scales of 2D turbulence. We also obtain quantitative measures of the degree to which our flow can be considered incompressible and isotropic as a function of downstream ...

Heat transport measurements in turbulent rotating Rayleigh-Bénard convection

We present experimental heat transport measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with a Prandtl number (sigma) of about 6, was confined in a cell with a square cross section of 7.3 x 7.3 cm2 and a height of 9.4 cm. Heat transport was measured for Rayleigh numbers 2 x 10(5)<Ra<5 x 10(8) and Taylor numbers 0<Ta<5 x 10(9). We show the variation in normalized heat transport, the Nusselt number, at fixed dimensional rotation rate OmegaD, at fixed Ra varying Ta, at fixed Ta varying Ra, and at fixed Rossby number Ro. The scaling of heat transport in the range of 10(7) to about 10(9) is roughly 0.29 with a Ro-dependent coefficient or equivalently is also well fit by a combination of power laws of the form a Ra1/5+b Ra1/3. The range of Ra is not sufficient to differentiate single power law or combined power-law scaling. The data are roughly consistent with an assumption that the enhancement of heat transport owing to rotation is proportional to the number of vortical structures penetrating the boundary layer. We also compare indirect measures of thermal and Ekman boundary layer thicknesses to assess their potential role in controlling heat transport in different regimes of Ra and Ta.

Rotating Rayleigh-Be´nard convection: Ku¨ppers-Lortz transition

Abstract Rayleigh-Benard convection with rotation about a vertical axis is investigated for small dimensionless rotation rates 0

Fluid Mixing in Stratified Gravity Currents: The Prandtl Mixing Length

Shear-induced vertical mixing in a stratified flow is a key ingredient of thermohaline circulation. We experimentally determine the vertical flux of momentum and density of a forced gravity current using high-resolution velocity and density measurements. A constant eddy-viscosity model provides a poor description of the physics of mixing, but a Prandtl mixing length model relating momentum and density fluxes to mean velocity and density gradients works well. For the average gradient Richardson number Ri(g) approximately 0.08 and a Taylor Reynolds number Re(lambda) approximately 100, the mixing lengths are fairly constant, about the same magnitude, comparable to the turbulent shear length.

Mode-locking and chaos in Rayleigh—Benard convection

Abstract We have experimentally studied the transition to chaos in a thermally convecting dilute solution of 3 He in superfluid 4 He. Two natural oscillatory instabilities are found in our geometry. Mode-locking is observed over a range of Rayleigh and Prandtl number. The mode-locking steps have a measured fractal dimension in good agreement with theoretical predictions and the larger steps show Farey tree ordering. Far below chaos many features of the measurements are suggestive of the circle map. We produce a one-dimensional map showing a tangent bifurcation to a mode-locked state. An effective phase angle change across a mode-locked step is calculated from the data. Close to chaos, hysteresis and other effects are suggestive of a two-dimensional map. The fractal dimension of the attractor is measured over a range of Rayleigh number close to the chaotic onset. Our Poincare sections show the fractalization of the torus and are characterized by extremely high resolution.