arXiv subjects

Richard M. McLaughlin

Publications and source records attributed to Richard M. McLaughlin.

14 recordsLinked to original sources

arXiv2024

Diffusion-limited settling of highly porous particles in density-stratified fluids

The vertical transport of solid material in a stratified medium is fundamental to a number of environmental applications, with implications for the carbon cycle and nutrient transport in marine ecosystems. In this work, we study the diffusion-limited settling of highly porous particles in a density-stratified fluid through a combination of experiment, analysis, and numerical simulation. By delineating and appealing to the diffusion-limited regime wherein buoyancy effects due to mass adaptation dominate hydrodynamic drag, we derive a simple expression for the steady settling velocity of a sphere as a function of the density, size, and diffusivity of the solid, as well as the density gradient of the background fluid. In this regime, smaller particles settle faster, in contrast with most conventional hydrodynamic drag mechanisms. Furthermore, we outline a general mathematical framework for computing the steady settling speed of a body of arbitrary shape in this regime and compute exact results for the case of general ellipsoids. Using hydrogels as a highly porous model system, we validate the predictions with laboratory experiments in linear stratification for a wide range of parameters. Lastly, we show how the predictions can be applied to arbitrary slowly varying background density profiles and demonstrate how a measured particle position over time can be used to reconstruct the background density profile.

physics.flu-dyn↗

arXiv2023

Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to general fluid flow, we restrict the examples of our theory to the tracer advected by flows that are induced by a periodically oscillating wall in a Newtonian fluid between two infinite parallel plates as well as flow in an infinitely long duct. We first derived the formula of the flow produced by the wall motions. Second, we calculate the second Aris moment for all time and its long-time limiting effective diffusivity as a function of the geometrical parameters, frequency, viscosity, and diffusivity. Using a new formalism based upon the Helmholtz operator we establish a new single series formula for the variance. We show that the viscous-dominated limit results in a linear shear layer for which the effective diffusivity is bounded. For finite viscosities, the enhanced diffusion diverges in the high-frequency limit. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exist for various parameter sweeps. Physical experiments are performed in water using particle tracking velocimetry to quantitatively measure the fluid flow. Using fluorescein dye as the passive tracer, we document that the theory is quantitatively accurate. Specifically, image analysis suggests that the distribution variance be measured using the full width at half maximum is robust to noise. Further, we show that the scalar skewness is zero for linear shear flows at all times, whereas for the nonlinear Stokes layer, the skewness sign can be controlled through the oscillating phase. Last, for single-frequency wall motion, the long-time skewness decays at the faster rate as compared with the case with steady flow.

physics.flu-dyn↗

arXiv2023

Tauberian identities and the connection to Wile E. Coyote physics

The application of the motion of a vertically suspended mass-spring system released under tension is studied focusing upon the delay timescale for the bottom mass as a function of the spring constants and masses. This ``hang-time", reminiscent of the Coyote and Road Runner cartoons, is quantified using the far-field asymptotic expansion of the bottom mass' Laplace transform. These asymptotics are connected to the short time mass dynamics through Tauberian identities and explicit residue calculations. It is shown, perhaps paradoxically, that this delay timescale is maximized in the large mass limit of the top ``boulder". Experiments are presented and compared with the theoretical predictions. This system is an exciting example for the teaching of mass-spring dynamics in classes on Ordinary Differential Equations, and does not require any normal mode calculations for these predictions.

math.DS↗

arXiv2023

Dispersion induced by unsteady diffusion-driven flow in parallel-plate channel

We investigate diffusion-driven flows in a parallel-plate channel domain with linear density stratification, which arise from the combined influence of gravity and diffusion in density-stratified fluids. We compute the time-dependent diffusion-driven flows and perturbed density field using eigenfunction expansions under the Boussinesq approximation. In channel domain, the unsteady flow converges to a steady-state solution either monotonically or non-monotonically (highly oscillatory), depending on the relation between the Schmidt number and the non-dimensionalized stratified scalar diffusivity, while the flow in the half-space inclined plane problem exhibits oscillatory convergence for all parameters. To validate the Boussinesq approximation, we propose the quasi-Boussinesq approximation, which includes transverse density variation in the inertial term. Numerical solutions show that the relative difference between the Boussinesq and quasi-Boussinesq approximations is uniformly small. We also study the mixing of a passive tracer induced by the advection of the unsteady diffusion-driven flow and present the series representation of the time-dependent effective diffusion coefficient. For small Schmidt numbers, the effective diffusion coefficient induced by the unsteady flow solution can oscillate with an amplitude larger than the effective diffusion coefficient induced by the long-time-limiting steady-state flow. Interestingly, the unsteady flow solution can reduce the time-dependent effective diffusion coefficient temporally in some parameter regimes, below even that produced by pure molecular diffusion in the absence of a flow. However, at long times, the effective diffusion is significantly enhanced for large Péclet numbers.

physics.flu-dyn↗

arXiv2023

Critical density triplets for the arrestment of a sphere falling in a sharply stratified fluid

We study the motion of a rigid sphere falling in a two-layer stratified fluid under the action of gravity in the potential flow regime. Experiments at a moderate Reynolds number of approximately 20 to 450 indicate that a sphere with the precise critical density, higher than the bottom layer density, can display behaviors such as bounce or arrestment after crossing the interface. We experimentally demonstrate that such a critical sphere density increases linearly as the bottom fluid density increases with a fixed top fluid density. Additionally, the critical density approaches the bottom layer fluid density as the thickness of density transition layer increases. We propose an estimation of the critical density based on the potential energy. With assuming the zero layer thickness, the estimation constitutes an upper bound of the critical density with less than 0.043 relative difference within the experimental density regime 0.997 $g/cm^{3}$ $\sim $ 1.11 $g/cm^{3}$ under the zero layer thickness assumption. By matching the experimental layer thickness, we obtain a critical density estimation with less than 0.01 relative difference within the same parameter regime.

physics.flu-dyn↗

arXiv2022

Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

physics.flu-dyn↗

arXiv2022

Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an $n$-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the $N$-point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.

math.AP↗

arXiv2021

Determinism and invariant measures for diffusing passive scalars advected by unsteady random shear flows

Here we study the long time behavior of an advection-diffusion equation with a general time varying (including random) shear flow imposing no-flux boundary conditions on channel walls. We derive the asymptotic approximation of the scalar field at long times by using center manifold theory. We carefully compare it with existing time varying homogenization theory as well as other existing center manifold based studies, and present conditions on the flows under which our new approximations give a substantial improvement to these existing theories. A recent study \cite{ding2020ergodicity} has shown that Gaussian random shear flows induce a deterministic effective diffusivity at long times, and explicitly calculated the invariant measure. Here, with our established asymptotic expansions, we not only concisely demonstrate those prior conclusions for Gaussian random shear flows, but also generalize the conclusions regarding determinism to a much broader class of random (non-Gaussian) shear flows. Such results are important ergodicity-like results in that they assure an experimentalist need only perform a single realization of a random flow to observe the ensemble moment predictions at long time. Monte-Carlo simulations are presented illustrating how the highly random behavior converges to the deterministic limit at long time. Counterintuitively, we present a case demonstrating that the random flow may not induce larger dispersion than its deterministic counterpart, and in turn present rigorous conditions under which a random renewing flow induces a stronger effective diffusivity.

physics.flu-dyn↗

arXiv2020

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan-Majda model and Taylor-Aris Dispersion

We study the long time behavior of an advection-diffusion equation with a random shear flow which depends on a stationary Ornstein-Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary $N$-point correlator using the ground state eigenvalue perturbation approach proposed in \cite{bronski1997scalar}. In turn, appealing to the conclusion of the Hausdorff moment problem \cite{shohat1943problem}, we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability distribution function at long times. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. This noteably contrasts the white noise case from the OU case. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar's long time limiting probability distribution function (PDF) for different types of initial conditions (e.g. deterministic and random). All results are verified by Monte-Carlo simulations.

math.AP↗

arXiv2019

Cluster formation and self-assembly in stratified fluids: a novel mechanism for particulate aggregation

An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Some examples include: sedimenting marine snow particles in lakes and oceans (central to carbon sequestration), dense microplastics in the oceans (which impact ocean ecology and the food chain), and even iron snow on Mercury (conjectured as its magnetic field source). These fluid systems all have stable density stratification, which is known to trap particulates through upper lightweight fluid coating the sinking particles, thus providing transient buoyancy. The current understanding of aggregation of such trapped matter involves collisions (due to Brownian motion, shear, and differential settling) and adhesion. Here, we observe and rationalize a new fundamental effective attractive mechanism by which particles suspended within stratification may self-assemble and form large aggregates without need for short range binding effects. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these toroidal flows yield attractive horizontal forces between particles. We observe that many particles demonstrate a collective motion revealing a system which self-assembles, appearing to solve jigsaw-like puzzles on its way to organizing into a disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments with two objects isolate the individual dynamics, which are quantitatively predicted through numerical integration of the underlying equations of motion. This new mechanism may be an important process in formation of marine snow aggregates and distribution of phytoplankton in lakes and oceans. Further, it potentially provides a new mechanism for general sorting and packing of layered material.

physics.flu-dyn↗

arXiv2018

Experimental investigation of nonlinear internal waves in deep water with miscible fluids

Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW) propagation in deep water configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representative of geophysical applications in the deep ocean where weakly nonlinear theories fail to capture the characteristics of large amplitude ISWs from field observations.

physics.flu-dyn↗

arXiv2018

On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the symmetry properties of the evolving point wise probability measure for the random scalar. We identify, for both point line source initial data, regions in the x-y plane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernels for the scalars statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the random Green's functions allowing for the fast construction of the PDF for single point statistics, and multi-point statistics natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on a channel and observe a dramatically different long time state resulting from the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states heavily leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.

physics.flu-dyn↗

arXiv2018

Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross-sections

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross-sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle, short time skewness is calculated exactly to be positive, while long-time asymptotics shows it to be negative. Monte-Carlo simulations confirm these predictions, and document the time scale for sign change. Interestingly, the equilateral triangle is the only regular polygon with this property, all other polygons possess positive skewness at all times, although this cannot cannot be proved on finite times due to the lack of closed form flow solutions for such geometries. Alternatively, closed form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (back) loading properties of the evolving tracer distribution along the pipe. Short and long time asymptotics confirm this condition, and Monte-Carlo simulations verify this at all times.

physics.flu-dyn↗

arXiv2008

Vortex Induced Oscillations of Cylinders

This article submitted to the APS-DFD 2008 conference, accompanies the fluid dynamics video depicting the various orientational dynamics of a hinged cylinder suspended in a flow tank. The different behaviors displayed by the cylinder range from steady orientation to periodic oscillation and even autorotation. We illustrate these features using a phase diagram which captures the observed phenomena as a function of Reynolds number and reduced inertia. A hydrogen bubble flow visualization technique is also used to show vortex shedding structure in the cylinder's wake which results in these oscillations.

physics.flu-dyn↗