642 research outputs found
Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation
No full textWe develop a Fourier-Chebyshev pseudospectral direct numerical simulation
(DNS) to examine a potentially singular solution of the radially bounded,
three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou,
Proc. Natl. Acad. Sci. USA, 111.36 (2014)]. We demonstrate that: (a) the time
of singularity is preceded, in any spectrally truncated DNS, by the formation
of oscillatory structures called tygers, first investigated in the
one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the
analyticity-strip method can be generalized to obtain an estimate for the
(potential) singularity time.Comment: 17 pages. 13 figure
Spaces, Places, Dwellings, and Beings : A Contribution to a Topoanalysis of Rahul Sankrityayan's Baisvim Sadi
No full textPandit Rahul Sankrityayan's creative writing Baisvim sadi (Twenty-second century) is an imaginary journey in time and space. Mobility is a crucial element in this story: it implies changes of space, but also physical and mental challenges. In light of topoanalysis - the term coined by French philosopher Gaston Bachelard to describe the detailing of intimate spaces - a better understanding of the self can be attained through a research of the places in which the subject has lived on account of the close link between self and place. Place attachment is in direct proportion to the integrity of identity. This paper aims to contribute to a revised topoanalysis of Baisvim sadi.</p
Universal properties of the two-dimensional Kuramoto-Sivashinsky equation
No full textJayaprakash, Hayot, and Pandit Reply: We disagree completely with the Comment that L’vov and Procaccia [1]make on our Letter 121, so we explain the coarse-graining procedure and the renormalization-group (RG) arguments we use in the light of their Comment
Multiscaling in the randomly forced and conventional Navier-Stokes equations
No full textWe present an overview of some results we have obtained recently (A. Sain, Manu and R. Pandit, Phys. Rev. Lett. 81 (1998) 4377) from a pseudospectral study of the randomly forced Navier–Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance ~k4−d−y, with k the wavevector and the dimension d=3. These include the multiscaling of velocity structure functions for y>4 and a demonstration that the multiscaling exponent ratios p/2 for y=4 are in agreement with those obtained for the Navier–Stokes equation forced at large spatial scales (3dNSE). We also study a coarse-graining procedure for the 3dNSE and examine why it does not lead to the RFNSE
Cataracts and Visual Decline – A Conversation with Rahul T. Pandit, MD, Blanton Eye institute, Houston Methodist Hospital
No full textCahn-Hilliard-Navier-Stokes Investigations of Binary-Fluid Turbulence and Droplet Dynamics
No full textThe study of finite-sized, deformable droplets adverted by turbulent flows is an active area of research. It spans many streams of sciences and engineering, which include chemical engineering, fluid mechanics, statistical physics, nonlinear dynamics, and also biology. Advances in experimental techniques and high-performance computing have made it possible to investigate the properties of turbulent fluids laden with droplets. The main focus of this thesis is to study the statistical properties of the dynamics of such finite-size droplets in turbulent flows by using direct numerical simulations (DNSs). The most important feature of the model we use is that the droplets have a back-reaction on the advecting fluid: the turbulent fluid affects the droplets and they, in turn, affect the turbulence of the fluid. Our study uncovers (a) statistical properties that characterize the spatiotemporal evolution of droplets in turbulent flows, which are statistically homogeneous and isotropic, and (b) the modification of the statistical properties of this turbulence by the droplets.
This thesis is divided into seven Chapters. Chapter 1 contains an introduction to the background material that is required for this thesis, especially the details about the equations we use; it also contains an outline of the problems we study in subsequent Chapters. Chapter 2 contains our study of “Droplets in Statistically Homogeneous Turbulence: From Many Droplets to a few Droplets”. Chapter 3 is devoted to our study of “Coalescence of Two Droplets”. Chapter 4 deals with “Binary-Fluid Turbulence: Signatures of Multifractal Droplet Dynamics and Dissipation Reduction”. Chapter 5 deals with “A BKM-type theorem and associated computations of solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations”. Chapter 6 is devoted to our study of “Turbulence-induced Suppression of Phase Separation in Binary-Fluid Mixtures”. Chapter 7 is devoted to our study of “Antibubbles: Insights from the Cahn-Hilliard-Navier-Stokes Equations”
Spiral-Wave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tissue
No full textThere is a growing consensus that life-threatening cardiac arrhythmias like ven- tricular tachycardia (VT) or ventricular fibrillation (VF) arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have shown that in homogeneities in cardiac tissue can have dramatic effects on such spiral waves.
In this thesis we focus on spiral-wave dynamics in mathematical models of human ventricular tissue which contain (a) conduction in homogeneities, (b) ionic in- homogeneities, (c) fibroblasts, (d) Purkinje fibers. We also study the effect of a periodic deformation of the simulation domain on spiral wave-dynamics. Chapter 2 contains our study of “Spiral-Wave Dynamics and Its Control in the Presence of In homogeneities in Two Mathematical Models for Human Cardiac Tissue”; this chapter follows closely parts of a paper we have published [1]. Chapter 3 contains our study of “Spiral-wave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Fibroblasts”; this chapter follows closely a paper that we have submitted for publication. Chapter 4 contains our study of “Spiral-wave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tis- sue: The Effects of Periodic Deformation”; this chapter follows closely a paper that we have submitted for publication. Chapter 5 contains our study of “Spiral-wave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Purkinje fibers”; this chapter follows closely a paper that we will submit for publication soon.
In chapter 2, we study systematically the AP morphology in a state-of-the-art mathematical model of human ventricular tissue due to ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model); we also look at the contribution of individual ionic currents to the AP by partially or completely blocking ion channels associated with the ionic currents. We then carry out systematic studies of plane- wave and circular-wave dynamics in the TNNP04 model for cardiac tissue model. We present a detailed and systematic study of spiral-wave turbulence and spa- tiotemporal chaos in two mathematical models for human cardiac tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model). In particular, we use extensive numerical simulations to elucidate the interaction of spiral waves in these models with conduction and ionic in homogeneities. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such in homogeneities. A major goal here is to develop low amplitude defibrillation schemes for the elimination of VT and VF, especially in the presence of in homogeneities that occur commonly in cardiac tissue. Therefore, we study a control scheme that has been suggested for the control of spiral turbulence, via low-amplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control scheme in the presence of in homogeneities in biophysical realistic models. We find that a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence than other control schemes. We discuss the theoretical and experimental implications of our study that have a direct bearing on defibrillation, the control of life-threatening cardiac arrhythmias such as ventricular fibrillation.
In chapter 3, we study the role of cardiac fibroblasts in ventricular tissue; we use the TNNP04 model for the myocyte cell, and the fibroblasts are modelled as passive cells. Cardiac fibroblasts, when coupled functionally with myocytes, can modulate their electrophysiological properties at both cellular and tissue levels. Therefore, it is important to study the effects of such fibroblasts when they are coupled with myocytes. Chapter 3 contains our detailed and systematic study of spiral-wave dynamics in the presence of fibroblasts in both homogeneous and inhomogeneous domains of the TNNP04 model for cardiac tissue. We carry out extensive numerical studies of such modulation of electrophysiological properties in mathematical models for (a) single myocyte fibroblast (MF) units and (b) two-dimensional (2D) arrays of such units; our models build on earlier ones and allow for no, one-way, or two-way MF couplings. Our studies of MF units elucidate the dependence of the action-potential (AP) morphology on parameters such as Ef , the fibroblast resting membrane potential, the fibroblast conductance Gf , and the MF gap-junctional coupling Ggap. Furthermore, we find that our MF composite can show autorhythmic and oscillatory behaviors in addition to an excitable response. Our 2D studies use
(a) both homogeneous and inhomogeneous distributions of fibroblasts, (b) various ranges for parameters such as Ggap, Gf , and Ef , and (c) intercellular couplings that can be no, one-way, and two-way connections of fibroblasts with myocytes. We show, in particular, that the plane-wave conduction velocity CV decreases as a function of Ggap, for no and one-way couplings; however, for two-sided coupling, CV decreases initially and then increases as a function of Ggap, and, eventually, we observe that conduction failure occurs for low values of Ggap. In our homogeneous studies, we find that the rotation speed and stability of a spiral wave can be controlled either by controlling Ggap or Ef . Our studies with fibroblast inhomogeneities show that a spiral wave can get anchored to a local fibroblast inhomogeneity. We also study the efficacy of a low-amplitude control scheme, which has been suggested for the control of spiral-wave turbulence in mathematical models for cardiac tissue, in our MF model both with and without heterogeneities.
In chapter 4, we carry out a detailed, systematic study of spiral-wave dynamics in the presence of periodic deformation (PD) in two state-of-the-art mathematical models of human ventricular tissue, namely, the TNNP04 model and the TP06 model. To the best of our knowledge, our work is the first, systematic study of the dynamics of spiral waves of electrical activation and their transitions, in the presence of PD, in such biophysically realistic mathematical models of cardiac tissue. In our studies, we use three types of initial conditions whose time evolutions lead to the following states in the absence of PD: (a) a single rotating spiral (RS),
(b) a spiral-turbulence (ST) state, with a single meandering spiral, and (c) an ST state with multiple broken spirals for both these models. We then show that the imposition of PD in these three cases leads to a rich variety of spatiotemporal pat- terns in the transmembrane potential including states with (a) an RS state with n-cycle temporal evolution (here n is a positive integer), (b) rotating-spiral states with quasiperiodic (QP) temporal evolution, (c) a state with a single meandering spiral MS, which displays spatiotemporal chaos, (d) an ST state, with multiple bro- ken spirals, and (e) a quiescent state in which all spirals are absorbed (SA). For all three initial conditions, precisely which one of the states is obtained depends on the amplitudes and the frequencies of the PD in the x and y directions. We also suggest specific experiments that can test the results of our simulations. We also study, in the presence of PD, the efficacy of a low-amplitude control scheme that has been suggested, hitherto only without PD, for the control of spiral-wave turbulence, via low-amplitude current pulses applied on a square mesh, in mathematical models for cardiac tissue. We also develop line-mesh and rectangular-mesh variants of this control scheme. We find that square- and line-mesh-based, low-amplitude control schemes suppress spiral-wave turbulence in both the TP06 and TNNP04 models in the absence of PD; however, we show that the line-based scheme works with PD only if the PD is applied along one spatial direction. We then demonstrate that a minor modification of our line-based control scheme can suppress spiral-wave turbulence: in particular, we introduce a rectangular-mesh-based control scheme, in which we add a few control lines perpendicular to the parallel lines of the line- based control scheme; this rectangular-mesh scheme is a significant improvement over the square-mesh scheme because it uses fewer control lines than the one based on a square mesh.
In chapter 5, we have carried out detailed numerical studies of (a) a single unit of an endocardial cell and Purkinje cell (EP) composite and (b) a two-dimensional bilayer, which contains such EP composites at each site. We have considered bio- physically realistic ionic models for human endocardial cells (Ecells) and Purkinje cells (Pcells) to model EP composites. Our study has been designed to elucidate the sensitive dependence, on parameters and initial conditions, of (a) the dynamics of EP composites and (b) the spatiotemporal evolution of spiral waves of electrical activation in EP-bilayer domains. We examine this dependence on myocyte parameters by using the three different parameter sets P1, P2, and P3; to elucidate the initial-condition dependence we vary the time at which we apply the S2 pulse in our S1-S2 protocol; we also investigate the dependence of the spatiotemporal dynamics of our system on the EP coupling Dgap, and on the number of Purkinje- ventricular junctions (PVJs), which are measured here by the ratio R, the ratio of the total number of sites to the number of PVJs in our simulation domain.
Our studies on EP composites show that the frequency of autorhythmic activity of a P cell depends on the diffusive gap-junctional conductance Dgap. We perform a set of simulations to understand the source-sink relation between the E and P cells in an EP composite; such a source-sink relation is an important determinant of wave dynamics at the tissue level. Furthermore, we have studied the restitution properties of an isolated E cell and a composite EP unit to uncover this effect on wave dynamics in 2D, bilayers of EP composites.
Autorhythmicity is an important property of Purkinje cell; it helps to carry electrical signals rapidly from bundle of His to the endocardium. Our investigation of an EP composite shows that the cycle length (CL) of autorhythmic activity decreases, compared to that of an uncoupled Purkinje cell. Furthermore, we find that the APD increases for an EP composite, compared to that of an uncoupled P cell. In our second set of simulations for an EP-composite unit, we have obtained the AP behaviors and the amount of flux that flows from the E to the P cell during the course of the AP. The direction of flow of this flux is an important quantity that identifies which one of these cells act as a source or a sink in this EP composite. We have found that the P cell in an EP composite acts as a stimulation-current source for the E cell in the depolarization phase of the AP, when the stimulus is applied to both cells or to the P cell only. However, the P cell behaves both as a source and a sink when the stimulus is applied to the E cell only. In our third set of simulations for an EP composite unit, we have calculated the restitution of the APD; this plays an important role in deciding the stability of spiral waves in mathematical models for cardiac tissue. Our simulation shows that, for the EP composite with high coupling (Dgap = Dmm~10), the APDR slope decreases, relative to its value for an isolated E cell, for parameter sets P1 and P2, and first increases (for 50 ≤ DI ≤ 100 ms) and then decreases for the parameter set P3 ; however, for low coupling (Dgap = Dmm~100), the variation of the AP D as function of DI, for an EP composite, shows biphasic behavior for all these three parameter sets. We found that the above dynamics in EP cable type domains, with EP composites, depends sensitively on R.
We hope our in silico studies of spiral-wave dynamics in a variety of state-of-the- art ionic models for ventricular tissue will stimulate more experimental studies that examine such dynamics
Particles and Fields in Partial-differential-equation Models for fluid and Superfluid Turbulence
No full textWe have shown that uniformly rotating vortex-containing gravitationally-bound
solutions of the GPPE can be generated by starting the evolution from initial data
obtained by integrating to convergence the (imaginary-time) ARGLPE (6.7). We
have built on this GPPE and introduced a minimal model, with a single, angular,
dynamical variable for a solid crust coupled with a rotating GPPE star. We
have demonstrated that this model exhibits stick-slip dynamics, whose statistical
properties we have characterized by computing the event-size and event-duration
CPDFs Q( Jc=Jc0) and Q(ted
0), which show power-law forms, and the waitingtime
CPDF Q(twt
0), which exhibits an exponential tail. These SOC-type desiderata
are in consonance with measurements on a class of pulsars [22].
We plan to study pulsar-glitch models that are more realistic than our minimal
model. Examples include models with (a) a solid crust with 6 degrees of freedom, 3
rotational and 3 translational, instead of only one angle of rotation, or (b) a super6.4.
Conclusions 129
conducting component with magnetic flux tubes. We expect that such generalizations
of our minimal model should help us to undertand all the types of statistical
properties that are displayed by pulsar glitches in different pulsar
Particles and Fields in Superfluid Turbulence : Numerical and Theoretical Studies
No full textIn this thesis we study a variety of problems in superfluid turbulence, princi-pally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these.
In Chapter 1, we provide an overview of several problems in superfluid turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give: (a) a brief introduction of fluid turbulence; (b) an overview of superfluidity and the phenomenological two-fluid model; (c) a brief overview of experiments on superfluid turbulence; (d) an introductory accounts of the phenomenological models used in the study of superfluid turbulence. We end with a summary of the problems we study in subsequent Chapters of this thesis.
In Chapter 2, we present a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power- law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc ; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other.
In Chapter 3, we present the first calculation of the mutual-friction coefficients α and α (which are parameters in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model that we study in chapter 5) as a function of temperature in a homogeneous Bose gas in two-dimensions by using the Galerkin-truncated Gross-Pitaevskii equation, with very special initial conditions, which we obtain by using the advective, real, Ginzburg-Landau equation (ARGLE) and an equilibration procedure that uses a stochastic Ginzburg-Landau equation (SGLE). We also calculate the normal-fluid density as a function of temperature.
In Chapter 4, we elucidate the interplay of particles and fields in superfluids, in both simple and turbulent flows. We carry out extensive direct numerical simulations (DNSs) of this interplay for the two-dimensional (2D) Gross-Pitaevskii (GP) equation. We obtain the following results: (1) the motion of a particle can be chaotic even if the superfluid shows no sign of turbulence; (2) vortex motion depends sensitively on particle charateristics; (3) there is an effective, superfluid-mediated, attractive interaction between particles; (4) we introduce a short-range repulsion between particles, with range rSR, and study two- and many-particle collisions; in the case of two-particle, head-on collisions, we find that, at low values of rSR, the particle collisions are inelastic with coefficient of restitution e = 0; and, as we in-crease rSR, e becomes nonzero at a critical point, and finally attains values close to 1; (5) assemblies of particles and vortices show rich, turbulent, spatio-temporal evolution.
In Chapter 5, we present results from our direct numerical simulations (DNSs) of the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) two-fluid model in two dimensions. We have designed these DNSs to study the statistical properties of inverse and forward cascades in the HVBK model. We obtain several interesting results that have not been anticipated hitherto: (1) Both normal-fluid and superfluid energy spectra, En(k) and Es(k), respectively, show inverse- and forward-cascade regimes; the former is characterized by a power law Es(k) En(k) kα whose exponent is consistent with α 5/3. (2) The forward-cascade power law depends on (a) the friction coefficient, as in 2D fluid turbulence, and, in addition, on (b) the coefficient B of mutual friction, which couples normal and superfluid compo-nents. (3) As B increases, the normal and superfluid velocities, un and us, re-spectively, get locked to each other, and, therefore, Es(k) En(k), especially in the inverse-cascade regime. (4) We quantify this locking tendency by calculating the probability distribution functions (PDFs) P(cos(θ)) and P(γ), where the angle θ ≡ (un • us)/( |un||us|) and the amplitude ratio γ = |un|/|us |; the former has a peak at cos(θ) = 1; and the latter exhibits a peak at γ = 1 and power-law tails on both sides of this peak. (4) This locking increases as we increase B, but the power-law exponents for the tails of P(γ) are universal, in so far as they do not depend on B, ρn/ρ, and the details of the energy-injection method. (5) We characterize the energy and enstrophy cascades by computing the energy and enstrophy fluxes and the mutual-friction transfer functions for all wave-number scales k.
In Chapter 6, we examine the multiscaling of structure functions in three-dimensional superfluid turbulence by using a shell-model for the three-dimensional HVBK equations. Our HVBK shell model is based on the GOY shell model. In particular, we examine the dependence of multiscaling on the normal-fluid fraction and the mutual-friction coefficients.
We hope our in silico studies of 2D and 3D superfluid turbulence will stimulate new experimental, numerical, and theoretical studies
Numerical Studies of Problems in Turbulence : 1) Fluid Films with Polymer Additives; 2) Fluid Films with Inertial and Elliptical Particles; 3) Scaled Vorticity Moments in Three- and Two-dimensional Turbulence
No full textIn this thesis we study a variety of problems in fluid turbulence, principally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these.
In Chapter 1, we provide an overview of several problems in turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give (a) natural and laboratory examples of fluid turbulence, (b) and introductory accounts of the equations of hydrodynamics, without and with polymer additives, Eulerian and Lagrangian frameworks, and the equations of motion of inertial particles in fluid flows. We end with a summary of the problems we study in subsequent Chapters of this thesis.
In Chapter 2, we carry out the most extensive and high-resolution direct numerical simulation, attempted so far, of homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental studies and propose new experiments.
In Chapter 3, we perform a direct numerical simulation (DNS) of the forced, incompressible two-dimensional Navier-Stokes equation coupled with the FENE-P equations for the polymer- conformation tensor. The forcing is such that, without polymers and at low Reynolds numbers Re, the lm attains a steady state that is a square lattice of vortices and anti-vortices. We nd that, as we increase the Weissenberg number (Wi), this lattice undergoes a series of nonequilibrium phase transitions, first to spatially distorted, but temporally steady, crystals and then to a sequence of crystals that oscillate in time, periodically, at low Wi, and quasiperiodically, for slightly larger Wi. Finally, the system becomes disordered and displays spatiotepmoral chaos and elastic turbulence. We then obtain the nonequilibrium phase diagram for this system, in the Wi − Re plane, and show that (a) the boundary between the crystalline and turbulent phases has a complicated, fractal-type character and (b) the Okubo-Weiss parameter provides us with a natural measure for characterizing the phases and transitions in this diagram.
In Chapter 4, our study is devoted to heavy, inertial particles in two-dimensional (2D) tur- bulent, but statistically steady, flows that are homogeneous and isotropic. The inertial particles are distributed uniformly in our simulation domain when St = 0; they start to cluster as St increases; this clustering tendency reaches a maximum at St 1 and decreases thereafter. We then obtain PDFs of and show that their left tails, which come from extensional regions, do not depend sensitively on St; in contrast, their right tails, from the vortical regions of the flow, are consistent with the exponential form ∼ exp ‰− + Ž; and we nd that the scale + decreases with St until St _0:1 and then saturates at a value _0:75. Our persistence-type studies yield the following results, when we consider forcing that leads to an energy spectrum that is dominated by a forward-cascade regime: In strain-dominated or extensional regions of the flow, wend that the cumulative PDF of the persistence time decays exponentially; this decay yields a time scale T−, which increases rapidly with St, at low values of St, but more slowly after St _0:75. By contrast, in vortical regions of the flow, this cumulative PDF displays a tail that has power-law and exponential parts; the power-law part yields the persistence exponent _ and the exponential tail gives a time scale T−; _ increases with St, whereas T− decreases with St; _ and T− reach saturation values as St increases. From the cumulative PDF of the particle mean-square displacement r2, we obtain the time scale Ttrans at which there is a crossover from ballistic to diffusive behavior; we _nd that Ttrans increases with St. The PDFs of v2, the square of the particle velocity, and v2 ejected, the square of the velocity of a particle just as it is ejected from a region with _ > 0 (vortical region) to one that has _ < 0 (extensional region), do not show a significant dependence on St; the tails of these PDFs are characterized by power-law decays with exponents _1 and _5~3, respectively. Our next set of results deal with statistical properties of special combinations of the acceleration a =dv~dt and the velocity v. For instance, the curvature of the trajectory is _ =aÙ~v2, where the subscript Ù denotes the component perpendicular to the particle trajectory; we obtain PDFs of _ and _nd there from that particles in regions of elongational flow have, on average, trajectories with a lower curvature than particles in vortical regions; this . We also determine how the number of number of points NI , at which a ×v changes sign along a particle trajectory, as time increases; we _nd that the increase of NI with time and decrease as St increases. Our ninth set of results show that the characteristic decay time T_ for decreases with St.
In Chapter 5, we study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For the large-scale forcing, the spatial distribution of particle orientations forms large- scale structures, which are absent for the intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For the intermediate- scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate decreases as the aspect ratio increases.
In Chapter 6, we study the issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box [0; L]3. This is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t) = where All four simulations unexpectedly show that the Dm are ordered for m =1 ….,9 such that Dm+1 <Dm. Moreover, the Dm squeeze together such that Dm+1/Dm 1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occuring which could be the cause of Navier{Stokes regularity. The first simulation, by R. Kerr, is of very anisotropic decaying turbulence ; the second and third, which have been carried out by me, are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number, respectively ; the fourth, by D. Donzis, is of very-high-Reynolds-number forced, stationary, isotropic turbulence at resolutions up to 40963 collocation points. For the sake of completeness and for a comparison of the data from all these four simulations, all the results are presented; however, in the Sections that deal with the simulations, I indicate who carried out the calculations reported there. I also present an extension of this work to two-dimensional fluid turbulence; this has not been submitted for publication so far.
We hope our in silico studies of 2D and 3D turbulence will stimulate new experimental, numerical, and theoretical studies
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