Richard M. McLaughlin

An internal splash: Levitation of falling spheres in stratified fluids

We experimentally explore the motion of falling spheres in strongly stratified fluids in which the fluid transitions from low density at the top to high density at the bottom and document an internal splash in which the falling sphere may reverse its direction of motion (from falling, to rising, to falling again) as it penetrates a region of strong density transition. We present measurements of the sphere’s velocity and exhibit nonmonotonic sphere velocity profiles connecting the maximum and minimum terminal velocities, matching earlier measurements [J. Fluid Mech. 381, 175 (1999)], but further exhibit the new levitation phenomenon. We give a physical explanation of this motion which necessarily couples the sphere motion with the stratified fluid, and vice versa, and supplement this with a simplified, reduced mathematical model involving a nonlinear system of ordinary differential equations which captures the nonmonotonic transition and agrees with the measured velocity profiles at all depths except those...

Rigorous Estimates of the Tails of the Probability Distribution Function for the Random Linear Shear Model

In previous work Majda and McLaughlin, and Majda computed explicit expressions for the 2Nth moments of a passive scalar advected by a linear shear flow in the form of an integral over RN. In this paper we first compute the asymptotics of these moments for large moment number. We are able to use this information about the large-N behavior of the moments, along with some basic facts about entire functions of finite order, to compute the asymptotics of the tails of the probability distribution function. We find that the probability distribution has Gaussian tails when the energy is concentrated in the largest scales. As the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer, and the distribution moves smoothly from Gaussian through exponential and “stretched exponential.” We also show that the derivatives of the scalar are increasingly intermittent, in agreement with experimental observations, and relate the exponents of the scalar derivative to the exponents of the scalar.

Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime

The behavior of settling particles in stratified fluid is important in a variety of applications, from environmental to medical. We document a phenomenon in which a sphere, when crossing density transitions, slows down substantially in comparison to its settling speed in the bottom denser layer, due to entrainment of buoyant fluid. We present results from an experimental study of the effects of the fluid interface on flight times as well as a theoretical model derived from first principles in the low Reynolds number regimes for stratified miscible fluids. Our work provides a new predictive tool and gives insight into the role of strong stratification in particle settling.

A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number

A sphere exhibits a prolonged residence time when settling through a stable stratification of miscible fluids due to the deformation of the fluid-density field. Using a Greens function formulation, a first-principles numerically assisted theoretical model for the sphere-fluid coupled dynamics at low Reynolds number is derived. Predictions of the model, which uses no adjustable parameters, are compared with data from an experimental investigation with spheres of varying sizes and densities settling in stratified corn syrup. The velocity of the sphere as well as the deformation of the density field are tracked using time-lapse images, then compared with the theoretical predictions. A settling rate comparison with spheres in dense homogeneous fluid additionally quantifies the effect of the enhanced residence time. Analysis of our theory identifies parametric trends, which are also partially explored in the experiments, further confirming the predictive capability of the theoretical model. The limit of infinite fluid domain is considered, showing evidence that the Stokes paradox of infinite fluid volume dragged by a moving sphere can be regularized by density stratifications. Comparisons with other possible models under a hierarchy of additional simplifying assumptions are also presented.

Subsurface Trapping of Oil Plumes in Stratification: Laboratory Investigations

Laboratory experiments demonstrating how the addition of surfactants creates the possibility of trapping buoyant immiscible fluids are presented. In particular, these experiments demonstrate that buoyant immiscible plumes like those which occurred during the Deepwater Horizon Gulf oil spill can be trapped as they rise through an ambient, stratified fluid. The addition of surfactants is an important mechanism by which trapping can occur. In this paper, we describe experiments and theory on trapping/escape of plumes containing an oil/water/surfactant mixture released into nonlinear stratification. We also present results on the timescale for trapping and for destabilization and release of trapped subsurface plumes. This timescale is shown to be a function of the oil to surfactant ratio.

The averaging of gravity currents in porous media

We explore the problem of a moving free surface in a water-saturated porous medium that has either a homogeneous or a periodically heterogeneous permeability field. We identify scaling relations and derive similarity solutions for the homogeneous, constant coefficient case in both a Cartesian and an axisymmetric, radial coordinate system. We utilize these similarity scalings to identify half-height slumping time scales as a rough guide for field groundwater cleanup strategies involving injected brines. We derive averaged solutions using homogenization for a vertically periodic, a horizontally periodic, and a two-dimensional periodic case—the solution of which requires solving a cell problem. Using effective coefficients, we connect the first two of these homogenized solutions to the similarity scaling solution derived for the homogeneous case. By simplifying to a thin limit, retaining variations of the porous media in the horizontal direction, we derive a homogenization solution in agreement with the general horizontally layered solution and an expression for the leading-order correction. Finally, we implement two numerical solution approaches and show that self-similar scaling and agreement with leading-order averaging emerge in finite time, and demonstrate the accuracy and convergence rate of the leading order correction for both the interior and the boundary of the domain.

The problem of moments and the Majda model for scalar intermittency

Abstract An enormous and important theoretical effort has been directed at studying the origin of broad-tailed probability distribution functions observed for numerous physical quantities measured in fluid turbulence. Despite the amount of attention this problem has received, there are still few rigorous results. One model which has been amenable to rigorous analysis is the Majda model for the diffusion of a passive scalar in the presence of a random, rapidly fluctuating linear shear layer, an anisotropic analog of the Kraichnan model. Previous work, by Majda, lead to explicit formulas for the moments of the distribution of the scalar. We examine this model, and construct the explicit large moment number asymptotics. Using properties of entire functions of finite order, we calculate the rigorous tail of the limiting probability distribution function for normalized scalar fluctuations. Through this process, we obtain an explicit relation between the limiting tail of the scalar probability distribution function and that of the scalar gradient. We additionally apply the method to moments derived asymptotically by Son, and those derived phenomenologically by She and Orszag.

Vortex Induced Oscillations of Cylinders at Low and Intermediate Reynolds Numbers

We study the orientational behavior of a hinged cylinder suspended in a water tunnel in the presence of an incompressible flow with Reynolds number (Re), based on particle dimensions, ranging between 100 and 6000 and non-dimensional inertia of the body(I *) in the range 0.1–0.6. The cylinder displays four unique features, which include: steady orientation, random oscillations, periodic oscillations and autorotation.We illustrate these features displayed by the cylinder using a phase diagram which captures the observed phenomena as a function of Re and I *. We identify critical Re and I * to distinguish the different behaviors of the cylinders. We used the hydrogen bubble flow visualization technique to show vortex shedding structure in the cylinder’s wake which results in these oscillations.

Trajectory and flow properties for a rod spinning in a viscous fluid. Part 1. An exact solution

An exact mathematical solution for the low-Reynolds-number quasi-steady hydrodynamic motion induced by a rod in the form of a prolate spheroid sweeping a symmetric double cone is developed, and the influence of the ensuing fluid motion upon passive particles is studied. The resulting fluid motion is fully three-dimensional and time varying. The advected particles are observed to admit slow orbits around the rotating rods and a fast epicyclic motion roughly commensurate with the rod rotation rate. The epicycle amplitudes, vertical fluctuations, arclengths and angle travelled per rotation are mapped as functions of their initial coordinates and rod geometry. These trajectories exhibit a rich spatial structure with greatly varying trajectory properties. Laboratory frame asymmetries of these properties are explored using integer time Poincare sections and far-field asymptotic analysis. This includes finding a small cone angle invariant in the limit of large spherical radius whereas an invariant for arbitrary cone angles is obtained in the limit of large cylindrical radius. The Eulerian and Lagrangian flow properties of the fluid flow are studied and shown to exhibit complex structures in both space and time. In particular, spatial regions of high speed and Lagrangian velocities possessing multiple extrema per rod rotation are observed. We establish the origin of these complexities via an auxiliary flow in a rotating frame, which provides a generator that defines the epicycles. Finally, an additional spin around the major spheroidal axis is included in the exact hydrodynamic solution resulting in enhanced vertical spatial fluctuation as compared to the spinless counterpart. The connection and relevance of these observations with recent developments in nano-scale fluidics is discussed, where similar epicycle behaviour has been observed. The present study is of direct use to nano-scale actuated fluidics.

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Peclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad-tailed distribution is similar for shear flows and time-varying nonsheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long-time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also...