The efficiency with which an incompressible flow mixes a passive scalar field that is continuously replenished by a steady source-sink distribution has been quantified using the suppression of the mean scalar variance below the value it would attain in the absence of the stirring. We examine the relationship this mixing measure has to the effective diffusivity obtained from homogenization theory, particularly establishing precise connections in the case of a stirring velocity field that is periodic in space and time and varies on scales much smaller than that of the source. We explore theoretically and numerically via the Childress-Soward family of flows how the mixing measures lose their linkage to the homogenized diffusivity when the velocity and source field do not enjoy scale separation. Some implications for homogenization-based parameterizations of mixing by flows with finite scale separation are discussed.

Keywords: convection, heat transport, Nusselt number, Constantin-Doering-Hopf technique, long time statistical properties, numerical approximation, convergence of long time statistical properties

Abstract: We survey a few recent results on estimating the long time averaged rate of heat transport in the vertical direction (the Nusselt number) in Rayleigh-Bénard convection. In the first half of the talk, we recall rigorous upper bounds on the Nusselt number that are of the form of Ra^1/3 modulo logarithmic correction for both the infinite Prandtl number model and the classical Boussinesq model for convection with large but finite Prandtl number. The main technique is the Constantin-Doering-Hopf approach. We also discuss the infinite Prandtl number limit in the Boussinesq model for convection, and the formal infinite Rayleigh number limit within the infinite Prandtl number model for convection. In the second half of the talk, we discuss numerical schemes (time discretization) that are able to capture the long time statistical properties of the convection problems. We first recall that the maximum long time averaged rate of heat transport in the vertical direction (true maximum Nusselt number) is a long time statistical property of the convection system. We then show that appropriate time discretizations of the systems will be able to capture the true maximum Nusselt number asymptotically. Several specific schemes that satisfy the desired properties will be presented. This numerical approach complements the Constantin-Doering-Hopf approach in the sense that it provides a computational asymptotic lower bound. Noise effects will be mentioned.

Stratospheric Climate-Chemistry Models neglect the effects of sub-grid flow structures on chemistry. Several previous studies have pointed out that such unresolved small scales could significantly affect the chemistry . However this problem has not been thoroughly studied from a theoretical point of view. To fulfill this gap, we investigate the interactions between advection, diffusion and chemistry for a simple bimolecular reaction between two initially unmixed reactants, within the framework of two-dimensional isotropic and homogeneous turbulence. This is a highly simplified representation of quasi-isentropic mixing in the stratosphere. Our goal here is to describe and understand how the production rate is affected by the size of the smallest scales of the tracer field, as determined by the tracer diffusion. We focus on the case of an infinitely fast chemical reaction. Our results show a strong dependence of the total production on the diffusion coefficient. This production scales like the diffusion to the power of p(t), where p(t) is a positive decreasing function of time. This dependence is particularly important during an initial transient regime and is affected by the separation between the reactants at the initial time. This first regime is characterized by an exponential lengthening of the boundary between the reactants. The evolution of the tracer gradients along this interface explains the dependence of the chemistry on the diffusion. For larger times, our simulations suggest the appearance of an asymptotic strange eigenmode that controls the decay of the reactants.

We regularize the 3-D Navier-Stokes equations with hyperviscosity of degree alpha, but applied only to the high wavenumbers past a cutoff m; such a technique is also designed to approximate the subgrid-scale modeling effects of spectral eddy viscosity. Attractor estimates stay within the Landau-Lifschitz degrees-of-freedom estimates even for very large m. An inertial manifold exists for m large enough whenever alpha is at or above 3/2. Galerkin-convergence and inviscid-limit results are optimized for the high wavenumbers; the latter case is defined to mean that nu goes to zero while the spectral hyperviscous term stays fixed. Computational studies over many runs produce parameter choices that facilitate close-to-parallel agreement (over a good-sized portion of the inertial range) with the Kolmogorov energy-spectrum power law for high (up to 10⁷) Reynolds numbers.

I will discuss turbulent mixing within the framework of Rayleigh-Taylor geometry.

Large scale properties of mixing are described by a simple non-linear diffusion model, derived within the general framework of Prandtl mixing theory, which fits very well the evolution of turbulent profiles obtained from numerical simulations.

The effect of polymer additives is then discussed and on the basis of numerical simulations of complete viscoelastic models we obtain clear evidence that the heat transport is enhanced up to 50% with respect to the Newtonian case. This phenomenon is accompanied by a speed up of the mixing layer growth.

Keywords: Turbulent mixing, Rigorous bounds, stratified shear flows

Abstract: Parameterizing the mixing of a stratified fluid subject to shear is a fundamental challenge for models of environmental and industrial flows. In particular, it is of great value to parameterize the efficiency of turbulent mixing, in the sense of the proportion of the kinetic energy converted into potential energy (through irreversible mixing of fluid of different density) compared to the total amount converted to both potential energy and internal energy (through viscous dissipation). Various competing models have been presented to relate the mixing efficiency to bulk properties of the flow, especially through different Richardson numbers, which quantify the relative importance of buoyancy and shear within the flow. One promising approach is to construct rigorous bounds on the long-time average of the buoyancy flux (i.e. the mixing rate) within simple model stratified shear flows, imposing physically reasonable constraints on the model flow fields. In this talk, we apply this technique to stably stratified Couette flow. By identifying the stratification which leads to maximal buoyancy flux, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We discuss the implications of our results for various classical stratified shear turbulence models.

Joint work with W. Tang (Arizona State University) & R. R. Kerswell (University of Bristol).

Joint work with Alina Chertock, Charles R. Doering and Eugene Kashdan.

The dispersal and mixing of scalar quantities such as concentrations or thermal energy are often modeled by advection-diffusion equations. Such problems arise in a wide variety of engineering, ecological and geophysical applications. In these situations a quantity such as chemical or pollutant concentration or temperature variation diffuses while being transported by the governing flow. In the passive scalar case, this flow prescribed and unaffected by the scalar. Both steady laminar and complex (chaotic, turbulent or random) time-dependent flows are of interest and such systems naturally lead to questions about the effectiveness of the stirring to disperse and mix the scalar. The development of reliable numerical methods for advection-diffusion equations is crucial for understanding their properties, both physical and mathematical. In this work, we extend a fast explicit operator splitting method, recently proposed in [A. Chertock, A. Kurganov, and G. Petrova, International Journal for Numerical Methods in Fluids, 59 (2009), pp. 309-332] for solving deterministic convection-diffusion equations, to the problems with random velocity fields and singular source terms. A superb performance of the method is demonstrated on several two-dimensional examples.

Joint work with N. Dianati, Z. Zhang, and C. Doering.

A novel model reduction strategy for forced-dissipative infinite-dimensional nonlinear dynamical systems is described. Unlike popular but empirical methods (e.g. based on the Proper Orthogonal Decomposition), this new approach does not require extensive data sets from experiments or full PDE simulations. Instead, truly predictive reduced-order models are constructed via Galerkin projection of the governing PDEs onto a-priori basis functions. This basis set is obtained by solving a constrained eigenvalue problem drawn from energy stability and upper bound theory. Within the context of porous medium convection, we show that these eigenfunctions contain information about boundary layers and other complex dynamic features and, thus, are well suited for the low-order description of highly nonlinear phenomena. Crucially, our analysis reveals a gap in the eigenvalue spectrum that persists even for strongly supercritical forcing conditions, thereby enabling the identification of a rational truncation scheme. We demonstrate the efficacy of our approach via comparisons with Fourier--Galerkin approximations of various orders.

The singular limit of a vanishing flowing layer in a tumbled granular flow is studied numerically, analytically and experimentally. We formulate the no-shear-layer dynamical system as a piecewise isometry (PWI) and show that the mechanism of streamline jumping leads to mixing. In the special case of a half-full tumbler, chaotic behavior is shown to disappear completely in the singular limit. Experimental results are compared to the zeroth-order PWI model and a realistic continuum model, showing that the no-shear-layer dynamics form the skeleton of the granular flow. Even though, in this limit, stretching in the sense of shear strain is replaced by spreading due to streamline jumping, finite-time Lyapunov exponents and Lagrangian coherent structures still reveal the manifold structure of the flow. Finally, using the simplified mapping method, the asymptotic mixing pattern in a tumbled granular flows is decomposed into eigenmodes, showing a significant number of strange eigenmodes. These appear align with the unstable manifolds, which have been shown to outline the shape of segregation patterns in bidisperse granular mixing experiments in polygonal tumblers.

Co-authors: Julio M. Ottino (Northwestern Univeristy, http://mixing.chem-eng.northwestern.edu) and Richard M. Lueptow (Northwestern University, http://www.mech.northwestern.edu/lueptow).

Keywords: turbulent energy dissipation, eddy viscosity, effective diffusion, mixing efficiency

Abstract: Several open problems in the analysis of turbulent transport and mixing are described. Among these are the determination of physically relevant bounds on turbulent dissipation and eddy viscosities in a variety of flow configurations, the accurate estimation of active and passive scalar transport and effective diffusivites, and questions of effective mixing.

Joint work with Kurt Scholz, and Misha Shvartsman, University of St. Thomas, St. Paul, MN, USA.

Motivated by results of Cai(2005, Monthly Weather Review), we consider alternate power dependencies in Serrin's Swirling Vortex model. We also give a heuristic argument to justify Cai's power law for tornados.

We introduce a class of action integrals defined over probability measure-valued path space. We show that minimal action exists and satisfies a compressible Euler equation in weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton-Jacobi equation, in the space of probability measures, are well posed. There are two key arguments which involves relaxation and regularization in formulation of the problem. They are probabilistically motivated. This is a joint work with Truyen Nguyen.

Keywords: meshless methods, radial basis functions, geosciences

Abstract: The most classical approach for solving PDEs numerically is finite difference methods (FD). Although they are easy to implement, their accuracy is often low. In contrast, pseudospectral methods (PS) can give spectral (exponential) convergence but suffer from severe geometric restrictions. On the other hand, radial basis function methods (RBF), based on expansions of translates of a single radially symmetric function, combine algorithmic simplicity with spectral accuracy while generalizing to arbitrary node layouts. Since they do not depend on any grid, RBFs allow for high geometric flexibility, permitting local node refinements in critical areas. Only within the last few years have they been applied to non-trivial PDE systems, illustrating their potential in the geosciences. Here, the application of the RBF method to three common geoscience benchmark cases of increasing complexity will be discussed. The first case is idealized cyclogenesis, which models the wrap-up of vortices as they traverse the sphere. For this case, we will also show the simplicity of implementing local node refinement. The second case is unsteady nonlinear flows described by the shallow water equations. The third case is thermal convection in a 3-D spherical shell, a situation of interest in modeling the earth's mantle. Current research topics focus on developing fast and efficient RBF algorithms that are parallelizable.

Turbulent channel flow of drag-reducing polymer solutions is simulated in minimal flow geometries. Even in the Newtonian limit, we find intervals of ``hibernating'' turbulence that display many features of the universal maximum drag reduction (MDR) asymptote observed in polymer solutions: weak streamwise vortices, nearly nonexistent streamwise variations and a mean velocity gradient that quantitatively matches experiments. As viscoelasticity increases, the frequency of these intervals also increases, while the intervals themselves are unchanged, leading to flows that increasingly resemble MDR.

Keywords: Transit-time distribution (TTD); tracer transport rates and pathways

Abstract: I will explain the concept of transit-time distributions (TTDs) to diagnose passive tracer transport in geophysical fluids, like the Earth's ocean and atmosphere. The TTD is directly related to the Green's function to the advection/diffusion equation for the concentration of a dynamically-passive trace substance. Diagnosing and interpreting the TTD, rather than the passive tracer field itself, focuses attention on the advective/diffusive transport properties of the underlying flow, and removes the influence of the tracer sources and sinks. The TTD therefore quantifies fundamental transport rate and pathway information about the flow. Various basic mathematical constraints on the TTD function exist, which may illuminate traditional diagnostics of tracer stirring and mixing. Applications of the TTD approach are presented in ocean circulation models, and using ocean tracer measurements.

For many realistic geophysical flows, the numerical discretization of the Boussinesq equations yields a prohibitively high computational cost. Thus, a significant research effort has been directed at generating mathematical models that are more computationally efficient than the Boussinesq equations, yet are physically accurate. The tool of choice in generating these simplified models has been scaling.

In this note, we put forth spatio-temporal filtering as an alternative methodology for generating simplified mathematical models for the ocean and atmosphere. In particular, we show that spatio-temporal filtering represents a natural approach for bridging the Boussinesq equations and the primitive equations.

Keywords: Turbulent energy dissipation, eddy viscosity, stability

Abstract: Variational techniques have been successful in establishing limits on transport in turbulent flow systems (e.g heat in convection, mass in pressure-driven shear flow or momentum in boundary-driven flows). However, there is typically a significant discrepancy between the limit derived and observations as well as a disconnect between the theoretical `optimal flow' solution and what is actually seen. After giving examples of this, I will discuss some past work directed at closing this gap and motivate the use of plausible stability criteria.

Keywords: passive scalar, enhanced diffusion, mixing properties

Abstract: We consider passive scalar equation on a compact domain or manifold. The fluid flow can aid diffusion and increase the speed of convergence of the initial distribution to its average. We consider either stationary or time periodic flows, and derive a sharp characterization of flows that are particularly effective in enhancing the relaxation speed to mean value. The characterization links enhanced relaxation with spectral properties of the dynamical system generated by flow. The results also provide an indication that time dependence of the flow may improve relaxation enhancing properties. Methods used involve a mix of PDE techniques and functional analysis. A key role is played by estimates similar to ones used in quantum dynamics to measure the rate of wavepacket propagation. The talk is based on works joint with P. Constantin, L. Ryzhik, R. Shterenberg and A. Zlatos.

The Rayleigh–Taylor (RT) instability is a fluid-mixing mechanism occurring at the interface between two fluids of different density when subjected to an external acceleration. The relevance of this mixing mechanism embraces several phenomena occurring in different contexts: astrophysical supernovae and solar-flare development are some examples. Although this instability has been known since 1883, much remains unknown especially on the turbulent regime. A deeper understanding of the mechanism of flows driven by RT instability would thus shed light on the many processes that underpin fully developed turbulence. Along this direction, we performed 2D and 3D direct numerical simulations in order to investigate the statistical properties of turbulent mixing in both miscible and immiscible situations. An introduction to this instability will be provided and some of our numerical results discussed.

Keywords: mixing, shear dispersion, Taylor dispersion

Abstract: The evolution of a passive scalar diffusing in simple parallel shear flows is a problem with a long history. In 1953, GI Taylor showed theoretically and experimentally that on long times, the passive scalar experiences an enhanced diffusion in the longitudinal direction. On shorter times the scalar evolution is anomalous, characterized by second moments growing faster than linear in time as we show by analysis of the stochastic differential equations underlying the passive scalar equation. The spatial structures associated with this intermediate time evolution are predicted using WKBJ analysis of an associated non-self adjoint eigenvalue problem. This analysis predicts a sorting of wall modes and interior modes with specific predictions of the decay and propagation rates as a function of the Peclet number. Monte Carlo simulations demonstrate non-trivial skewness evolution, and skewness is studied in the new WKBJ modes. Time permitting, new behavior distinguishing channel from pipe flow will be presented along with comparisons between some of these predictions and experiments in the pipe geometry. This is joint work with Roberto Camassa, Zhi Lin, Keith Mertens, Nick Moore, and Claudio Viotti.

Keywords: mixing, mix-norm, optimal stirring, ergodicity

Abstract: I will discuss several issues in analyzing kinematics of a purely advective mixing process:

1) Determine whether the scalar quantity, such as dye, introduced into the flow field is - asymptotically - in time thoroughly mixed.

2) Determine how good is the mixture at any finite time.

3) Provide methods for open-loop optimization or feedback control of the mixture.

The concept of ergodic partition allows us to discuss 1) precisely, and I will discuss some new results that allow us to compute it effectively. Concerning 2), the problem of an effective norm for mixing has attracted a lot of work over the last decade. I will discuss one family of norms - the so called mix-norm that connects to negative Sobolev space norms - that allows us to pursue study of opimization and control, thus covering 3).

I will also discuss the question of ergodicity of a system and how to measure it. This is a departure from the standard off-on definition of ergodicity providing a measure of how close to ergodicity a system is. In all of the above, ergodic theory plays a prominent role.

Keywords: Shallow water, Layer model, hydraulic jumps, shocks, mixing, stability

Abstract: Shallow-fluid models are often the first step in modeling many geophysical flows. These models apply when the horizontal scales of motion are much larger than the vertical scales. For a single fluid layer with a free-surface, the shallow water approximation results in hyperbolic equations which are well understood and broadly applied. Generically, waves steepen and break creating hydraulic jumps which satisfy the PDEs in a weak sense. Physically one must ensure that certain conserved quantities - usually mass and momentum - are preserved across shocks. For the single layer case this results in a prediction of the small scale energy dissipation at the shock. In layered shallow water models the situation is complicated by at least two issues: that the flow may be shear unstable (the Kelvin-Helmholtz instability), and that breaking waves may mix the fluids. We shall discuss some physically motivated mathematical results on these issues.

Consider a Brownian particle in a deterministic time-independent incompressible flow in a bounded domain. We are interested how flow affects the expected exit time, the time the particle needs to reach the boundary of the domain. In particular, whether the presence of the flow decreases the maximum of this expected exit time. One would expect that any stirring improves mixing, thus decreasing the expected exit time. We will show that generally it is not true in two dimensions. This is a joint work with G.Iyer, L.Ryzhik, and A.Zlatos.

Keywords: Transit-time distribution (TTD); tracer transport; inverse problem; maximum-entropy deconvolution

Abstract: I will discuss the inverse problem of inferring the Green's function for advective-diffusive transport (also known as the transit-time distribution) from tracer observations. Tracers with different boundary conditions and/or different radio-active decay rates probe different transport pathways and timescales. Using multiple tracers in combination can therefore help constrain the full transit-time distribution (TTD). I will review two inversion methodologies applicable to ocean tracer measurements, one based on a parametric model for the TTD and one based on a more flexible maximum-entropy deconvolution approach. Because the oceanographic inverse problem is grossly under-determined an important focus of this talk will be on quantifying the uncertainty associated with the inversion results.

Keywords: jet, buoyancy, stratified flow, stability

Abstract: I discuss the flow structure and stability of a planar saline jet descending into a stable, density-stratified fluid. The jet retains its slender shape, largely due to the low salt diffusion. As the jet descends it entrains fresher water due to the relatively high mechanical viscous effects, when these are compared to inertial effects. This fresher water forms a recirculation cell. The jet exhibits a rapid acceleration on release, then deceleration, as it encounters the more dense surrounding fluid, and stops at a location much higher than the neutral buoyancy point.

I will recount preliminary work aimed at explaining the fluid dynamics of the jet: Stratification, mechanical diffusion and nonlinear inertial effects, as well as salt diffusion are all found to be crucial to the dynamics. I will also summarize our work on characterizing the basic instability modes of the jet by numerical means. We successfully captured the inception of the most salient symmetric and anti-symmetric instabilities and their dependence on the Reynolds number and the non-dimensional stratification gradient number. This jet, though deceptively simple, is far from well understood. I will enumerate key dynamic aspects that are beyond our present understanding and worthy of further study due to their relevance to other important physical phenomena.

This is joint work with Sam Schofield, Los Alamos National Laboratory, with contributions from Adriana Pesci and Raymond Goldstein, Cambridge University.

Starting (initial) conditions (ICs) can influence the development of hydrodynamic turbulence and material mixing in buoyancy driven flows. The overall goal of our research is to determine the extent to which starting conditions can be used to predict and design turbulent transport/material mixing. In particular, this work studies the effect of the initial velocity field and phase shifting on a binary initial perturbation. Results of an experimental investigation in which precisely defined initial conditions have been prescribed are presented. These experimental results serve as references that we try to match as closely as possible with numerical simulations. Our simulations show that the initial velocity field drives the growth of the initial perturbation in this experiment. Also, a “leaning” of the growing flow structures observed in the experiment is captured by the simulations, and linked to the phase shift.

Convective turbulence with phase changes and latent release is an important dynamical process in the atmosphere of the Earth which causes, e.g., the formation of clouds. Here we study moist convection in simplified setting - shallow and nonprecipitating moist Rayleigh-Benard convection with a piecewise linear thermodynamics on both sides of the phase boundary. The presented model is a first nontrivial extension of the classical dry Rayleigh-Benard convection. The equations of motion and the fully developed turbulent dynamics in very flat Cartesian cells are discussed.

Keywords: eddy diffusivity, conserved tracer, reactive tracer

Abstract: Eddies have an important role in transport and dynamical processes in the atmosphere and ocean. They influence the distribution of chemical species and are responsible for driving mean flows. I will discuss the quantification of eddy effects in the atmosphere and ocean. I will focus in particular on two problems. The first is how to quantify the geographic (latitude-longitude) variation of eddy diffusivity of a conserved tracer in such flows. I will describe two different techniques and discuss the implications of the results for an atmospheric and an oceanic case. The second is how to quantify the effects of eddies on the distribution of a reactive tracer. I will take the example of different tracers at the sea surface (temperature, salinity, chlorophyll, etc). Eddy stirring directly influences the distribution of such tracers, but small-scale eddies in the ocean can also influence air-sea interactions and I will describe how this latter effect may be quantified.

We perturb the analytic deterministic solution of inertia-gravity waves with stationary random noise and solve for the Fokker-Planck equation to study the evolution in time of the probability density function of passive tracers in such a flow. We find that at initial times the probability density closely follows the nonlinear background flow and non-trivial Stokes drift ensues as a result. Over finite time, we measure chaotic mixing based on the stochastic mean flow and identify non-trivial mixing structures of passive tracers, as compared to their absence in the deterministic flow. At later times, when the probability density field spreads out to sample larger regions, the mean Stokes drift approaches an asymptotic value, indicating suppression of Lagrangian mixing at long time-scales. However, the skewness of the probability density remains non-Gaussian even at large times.

Keywords: stirring, mixing, biomixing, Brownian motion.

Abstract: As fish or other bodies move through a fluid, they stir their surroundings. This can be beneficial to some fish, since the plankton they eat depends on a well-stirred medium to feed on nutrients. Bacterial colonies also stir their environment, and this is even more crucial for them since at small scales there is no turbulence to help mixing. It has even been suggested that the total biomass in the ocean makes a significant contribution to large-scale vertical transport, but this is still a contentious issue. We propose a simple model of the stirring action of moving bodies through both inviscid and viscous fluids. In the dilute limit, this model can be solved using Einstein and Taylor's formula for diffusion (Brownian motion). We compare to direct numerical simulations of objects moving through a fluid. This is joint work with Steve Childress and George Lin.

Keywords: Oldroyd-B, viscoelastic, instabilities, mixing

Abstract: To understand observations of low Reynolds number mixing and flow transitions in viscoelastic fluids, we study numerically the dynamics of the Oldroyd-B viscoelastic fluid model. The fluid is driven by a simple time-independent forcing that creates a cellular flow with extensional stagnation points. We find that at O(1) Weissenberg number these flows lose their slaving to the forcing geometry of the background force, become oscillatory with multiple frequencies, and show continual formation and destruction of small-scale vortices. This drives flow mixing. These new flow states are dominated by a single large vortex, which may be stationary or move persistently from cell to cell. Increasing the number of degrees of freedom by increasing the number of driving cells broadens the temporal frequency spectrum and yields richer dynamics with no persistent vortices and improved fluid mixing.

This work is devoted to study the stability and error estimates of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. Space is discretized by using the Finite Element Method, whereas time is discretized using the Finite Difference Method. We introduce an extension to mixed elliptic problems of the negative-norm estimates for uniformly elliptic problems. Using this extension, we prove some super-convergence results in space for velocity which have been observed in several computational experiments. Furthermore, we obtain some error estimates results for the pressure without restrictions relating time and space discrete parameters.

We study the effect of chaotic flows on the progress of fast bimolecular reactions. Simulations show that the reactant concentration decays exponentially and then crosses over to the algebraic law of chemical kinetics in the final stage of the reaction. By transforming the reactive mixing problem to an equivalent passive scalar problem, we make prediction to the crossover time and the overall reaction rate. Depending on the relative length scale between the velocity and the concentration fields, the overall reaction rate is either related to the distribution of the finite-time Lyapunov exponent or given in terms of an effective diffusivity. Preliminary results on a variation of this problem in which the reactants are initially isolated from one another is also presented. Here, we focus on the mixdown time, i.e. the time taken for the flow to bring the reactants into contact, and its dependence on the various length scales in the system.

Keywords: reactive flows, chaos, structure functions, Holder exponents, delay differential equations, Lyapunov exponents

Abstract: Motivated by the spatial heterogeneity observed in plankton distributions in the meso-scale ocean, we examine the stationary-state spatial structure of reacting tracer fields, for the case for which the reaction equations contain delay terms. The fields are advected by a flow that gives rise to chaotic parcel trajectories and the structures are maintained by a large-scale source. Previous theoretical investigations have shown that, in the absence of delay terms and in a regime where diffusion can be neglected (large Peclet number), the structures are filamental and characterized by a single scaling regime with a Holder exponent that depends on the rate of convergence of the reactive processes and the strength of the stirring measured by the average stretching rate (Lyapunov exponent). In the presence of delay terms, we show that for sufficiently small scales all interacting fields should share the same spatial structure, as found in the absence of delay terms. However, depending on the strength of the stirring and the magnitude of the delay time, two further scaling regimes that are unique to the delay system may appear at intermediate length-scales. An expression for the transition length-scale dividing small-scale and intermediate-scale regimes is obtained and the scaling behavior of the tracer field is explained. Finally, we discuss the dependence of the field's scaling exponents on the distribution of the stretching statistics.

Joint work with P. H. Haynes.

The generalised Lyapunov exponents (GLEs) quantify the growth of the separation between particles advected in fluid flows. They provide valuable information about mixing, in particular because, in some cases, the decay rate of passive scalars released a flow can be directly related to specific GLEs of this flow. Here we discuss some numerical and asymptotic methods for the estimation of the GLEs of random renewing flows (such as the alternating-sine flow) in which the particle separation is described by a product of random matrices. Specifically, we propose an importance-sampling Monte Carlo algorithm as a general purpose numerical method which is both efficient and easy to implement. We also discuss asymptotic approximations for the GLEs characterising extremes of stretching.

Keywords: microfluidics, acoustic mixing, surface waves, acoustic streaming

Abstract: Acoustic streaming, the generation of flow by dissipating acoustic waves, provides a promising method for flow pumping in microfluidic devices. In recent years, several groups have been experimenting with a acoustic streaming induced by leaky surface waves: (Rayleigh) surface waves excited in a piezoelectric solid interact with a small volume of fluid where they generate acoustic waves and, as result of the viscous dissipation of these waves, a mean flow. We discuss the basic mathematical model that has been employed in simulations of this type of acoustic streaming and reformulate it to account for the dynamical constraints imposed by vorticity conservation. The formulation proposed makes it clear that dissipative processes in the bulk of the fluid are essential to the streaming, and separates the Eulerian and Stokes contributions to the mean flow. Particular attention is paid to the thin boundary layer that forms at the solid/liquid interface, where both the acoustic waves and their streaming effect are best computed by asymptotic means. A simple two-dimensional model of mean-flow generation by surface acoustic waves is discussed as an illustration.

Joint work with Oliver Buhler (Courant).

Keywords: mixing reaction broadcast spawning coral scaling

Abstract: Coral and other marine organisms reproduce through the mechanism of broadcast spawning, where egg and sperm are released at separate locations and brought together by fluid mixing and transport. Coral fertilization is particularly important because corals are threatened by anthropogenic climate change. We idealize broadcast spawning as point sources of two reacting tracers separated by a neutral fluid. This represents a new class of problems, different from well-studied problems such as flame fronts, where two tracers fill the fluid and are separated by an interface. For the case of broadcast spawning within a vortex, we show that the vortex stirring leads to a self-similar solution with enhanced fertilization rates scaling as the Peclet number^(1/3) and reduced fertilization times scaling as the Peclet number^(-2/3).

We model the hydrodynamics of a two-phase system of a nematic liquid crystal drop in a viscous fluid using an energetic variational approach with phase-field methods cite{YFLS04}. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme following is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. We simulate the deformation and rupture of nematic droplets, identifying the formation of surface topological defects, and exploring the shear and normal stress distributions that accompany the evolution. A bipolar global defect structure, with two half-integer surface point defects called boojums, emerges in every daughter droplet when tangential anchoring conditions are imposed together with Oseen-Frank distortional bulk elasticity. The fate of the original mother drop is compared for the limiting case of an immiscible viscous drop versus strength of the liquid crystal interfacial and bulk potentials.

Keywords: Jet, layer, potential vorticity, beta-plane, stratified turbulence, Cahn-Hilliard equation, negative viscosity

Abstract: I'll discuss two examples of uniformly forced turbulent flows in which quasi-steady structures spontaneously form. This results in the intensity of turbulence becoming spatially inhomogeneous on length scales larger than those of the eddies i.e., as argued by Owen Phillips in 1972, a spatially homogeneous turbulent flow may be subject to a large-scale instability.

The first example is stirring a fluid with strong gravitational stability due to, for example, dissolved salt. The resulting stratified turbulence produces well-mixed layers with uniform density, separated by strongly stable steps in density. The second example is the formation of zonal jets in forced-dissipative beta-plane turbulence.

I'll discuss the prospects of understanding these systems using models related to the Cahn-Hilliard equation.