# <span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a>

**Alternate powers in Serrin's swirling vortex solutions 2**

Douglas Dokken (University of St. Thomas)

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.**Super-convergence for the 3D Navier-Stokes**

Giordano Tierra Chica (University of Sevilla)

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.**Active and hibernating turbulence in channel flow of Newtonian and viscoelastic fluids**

Michael Graham (University of Wisconsin, Madison)

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.**Miscible and immiscible Rayleigh-Taylor turbulence**

Andrea Mazzino (Università di Genova)

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.**Low-dimensional models from upper bound and energy**

stability theory

Gregory Chini (University of New Hampshire)

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.**Lagrangian dynamics in stochastic inertia-gravity waves**

Wenbo Tang (Arizona State University)

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.**Fast chemical reactions in chaotic flows: Reaction rate**

and mixdown time

Yue-Kin Tsang (University of California)

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.**On the effect of initial velocity field and phase shifting of an initial**

binary perturbation for Rayleigh-Taylor instability

Bertrand Rollin (Los Alamos National Laboratory)

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.**Bridging the Boussinesq and primitive equations through spatio-temporal filtering**

Traian Iliescu (Virginia Polytechnic Institute and State University)

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.**Shear cell rupture of nematic droplets in viscous fluids**

Xiaofeng Yang (University of South Carolina)

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.**Estimating generalised Lyapunov exponents for random flows**

Jacques Vanneste (University of Edinburgh)

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.**A fast explicit operator splitting method for passive scalar advection**

Alina Chertock (North Carolina State University)Charles Doering (University of Michigan)Alexander Kurganov (Tulane University)

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.**Numerical studies in shallow moist convection**

Joerg Schumacher (Ilmenau University of Technology)

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.**Rayleigh-Taylor turbulence: a simple model for heat transfer in thermal convection**

Guido Boffetta (Università di Torino)

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.**A numerical study of the effect of diffusion on a fast chemical**

reaction in a two-dimensional turbulent flow

Farid Ait Chaalal (McGill University)

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.**The spectrally-hyperviscous Navier-Stokes equations**

Joel Avrin (University of North Carolina)

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 10^{7}) Reynolds numbers.**Chaotic granular mixing in quasi-two-dimensional tumblers: streamline jumping, piecewise isometries and strange eigenmodes**

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).**A class of Hamilton-Jacobi PDE in space of measures and its associated compressible Euler equations**

Jin Feng (University of Kansas)

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.**Homogenization and mixing measures for a replenishing passive scalar field**

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.