The center for nonlinear studies: A personal history.

The Center for Nonlinear Studies (CNLS) was an integral part of my scientific career starting as a Postdoctoral Fellow in 1983 up to my tenure as CNLS Director from 2004 to 2015. As such, I experienced a number of scientific phases of CNLS through almost four decades of foundation, evolution, and transition. Throughout this entire interval, the inspiration and influence of David Campbell guided my way.

Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection.

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary.

Plume dynamics in Hele-Shaw porous media convection.

Mass transport in multi-species porous media is through molecular diffusion and plume dynamics. Predicting the rate of mass transport has application in determining the efficiency of the storage and sequestration of carbon dioxide. We study a water and propylene-glycol system enclosed in a Hele-Shaw cell with variable permeability that represents a laboratory analogue of the general properties of porous media convection.

Lagrangian statistics in weakly forced two-dimensional turbulence.

Measurements of Lagrangian single-point and multiple-point statistics in a quasi-two-dimensional stratified layer system are reported. The system consists of a layer of salt water over an immiscible layer of Fluorinert and is forced electromagnetically so that mean-squared vorticity is injected at a well-defined spatial scale ri. Simultaneous cascades develop in which enstrophy flows predominately to small scales whereas energy cascades, on average, to larger scales.

Stick-slip behavior in a continuum-granular experiment.

We report moment distribution results from a laboratory experiment, similar in character to an isolated strike-slip earthquake fault, consisting of sheared elastic plates separated by a narrow gap filled with a two-dimensional granular medium. Local measurement of strain displacements of the plates at 203 spatial points located adjacent to the gap allows direct determination of the event moments and their spatial and temporal distributions. We show that events consist of spatially coherent, larger motions and spatially extended (noncoherent), smaller events.

Chaos, patterns, coherent structures, and turbulence: Reflections on nonlinear science.

The paradigms of nonlinear science were succinctly articulated over 25 years ago as deterministic chaos, pattern formation, coherent structures, and adaptation/evolution/learning. For chaos, the main unifying concept was universal routes to chaos in general nonlinear dynamical systems, built upon a framework of bifurcation theory. Pattern formation focused on spatially extended nonlinear systems, taking advantage of symmetry properties to develop highly quantitative amplitude equations of the Ginzburg-Landau type to describe early nonlinear phenomena in the vicinity of critical points.

Heat transport in the geostrophic regime of rotating Rayleigh-Bénard convection.

We report experimental measurements of heat transport in rotating Rayleigh-Bénard convection in a cylindrical convection cell with an aspect ratio of Γ=1/2. The fluid is helium gas with a Prandtl number Pr=0.7.

Fragility and hysteretic creep in frictional granular jamming.

The granular jamming transition is experimentally investigated in a two-dimensional system of frictional, bidispersed disks subject to quasistatic, uniaxial compression without vibrational disturbances (zero granular temperature). Three primary results are presented in this experimental study. First, using disks with different static friction coefficients (μ), we experimentally verify numerical results that predict jamming onset at progressively lower packing fractions with increasing friction.

Local temperature measurements in turbulent rotating Rayleigh-Bénard convection.

We present local temperature measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with Prandtl number about 6, was confined in a cell with a square cross section of 7.3×7.

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.

Patterns in flowing sand: understanding the physics of granular flow.

Dense granular flows are often unstable and form inhomogeneous structures. Although significant advances have been recently made in understanding simple flows, instabilities of such flows are often not understood. We present experimental and numerical results that show the formation of longitudinal stripes that arise from instability of the uniform flowing state of granular media on a rough inclined plane.

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.

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.

Avalanche dynamics on a rough inclined plane.

The avalanche behavior of gravitationally forced granular layers on a rough inclined plane is investigated experimentally for different materials and for a variety of grain shapes ranging from spherical beads to highly anisotropic particles with dendritic shape. We measure the front velocity, area, and height of many avalanches and correlate the motion with the area and height. We also measure the avalanche profiles for several example cases.

Stretching fields and mixing near the transition to nonperiodic two-dimensional flow.

Although time-periodic fluid flows sometimes produce mixing via Lagrangian chaos, the additional contribution to mixing caused by nonperiodicity has not been quantified experimentally. Here, we do so for a quasi-two-dimensional flow generated by electromagnetic forcing. Several distinct measures of mixing are found to vary continuously with the Reynolds number, with no evident change in magnitude or slope at the onset of nonperiodicity.

Flow rule of dense granular flows down a rough incline.

We present experimental findings on the flow rule for granular flows on a rough inclined plane using various materials, including sand and glass beads of various sizes and four types of copper particles with different shapes. We characterize the materials by measuring hs (the thickness at which the flow subsides) as a function of the plane inclination theta on various surfaces. Measuring the surface velocity u of the flow as a function of flow thickness h, we find that for sand and glass beads the Pouliquen flow rule u/sqrt[gh] approximately betahhs provides reasonable but not perfect collapse of the u(h) curves measured for various theta and mean particle diameter d.

Scaling in laminar natural convection in laterally heated cavities: is turbulence essential in the classical scaling of heat transfer?

We analyze heat transfer and flow properties in laminar natural convection driven by a horizontal temperature gradient in a closed cavity and propose that for the classical scaling of heat transfer turbulence does not play a decisive role. Direct numerical simulations were performed with the Rayleigh number (Ra) from 1 to 10(8) and the Prandtl number Pr = 0.71.

Rapid granular flows on a rough incline: phase diagram, gas transition, and effects of air drag.

We report experiments on the overall phase diagram of granular flows on an incline with emphasis on high inclination angles where the mean layer velocity approaches the terminal velocity of a single particle free falling in air. The granular flow was characterized by measurements of the surface velocity, the average layer height, and the mean density of the layer as functions of the hopper opening, the plane inclination angle, and the downstream distance x of the flow. At high inclination angles the flow does not reach an x -invariant steady state over the length of the inclined plane.

Experimental characterization of vibrated granular rings.

We report an experimental study of the statistical properties of vibrated granular rings. In this system, a linked rod and bead metallic chain in the form of a ring is collisionally excited by a vertically oscillating plate. The dynamics are driven primarily by inelastic bead-plate collisions and are simultaneously constrained by the rings' physical connectedness.

Physical mechanism of the two-dimensional inverse energy cascade.

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments.

Pair dispersion and doubling time statistics in two-dimensional turbulence.

Experimental measurements of pair separation statistics in two-dimensional turbulence are reported for an electromagnetically forced stratified-layer system with simultaneous ranges of direct-enstrophy and inverse-energy transfer separated by a well-defined spatial injection scale. Data for pair separation as a function of time are analyzed to determine the dependence of separation statistics in both regimes. Using doubling-time statistics, we show how the measured scalings of the mean quantities are consistent with exponential behavior in the enstrophy range and power-law behavior in the inverse-energy range.

Two scenarios for avalanche dynamics in inclined granular layers.

We report experimental measurements of avalanche behavior of thin granular layers on an inclined plane for low volume flow rate. The dynamical properties of avalanches were quantitatively and qualitatively different for smooth glass beads compared to irregular granular materials such as sand. Two scenarios for granular avalanches on an incline are identified, and a theoretical explanation for these different scenarios is developed based on a depth-averaged approach that takes into account the differing rheologies of the granular materials.