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Topics: "Lagrange versus Euler for turbulent flows and/or vice versa, with some emphasis on the relation”  

Arkady Tsinober  WPI, Seminarroom C 714  Mon, 7. May 12, 10:00 
Introductory and Final Talk  
Introduction: "Lagrange versus Euler for turbulent flows and/or vice versa, with some emphasis on the relation”  
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Charles Meneveau  WPI, Seminarroom C 714  Mon, 7. May 12, 11:00 
Lagrangian dynamics of the velocity gradient tensor in isotropic turbulence  
The Lagrangian evolution of the velocity gradient tensor depends upon the pressure Hessian and the viscous term to regularize the otherwise finitetime singularity producing dynamics of the Restricted Euler system. We review the Recent Fluid Deformation closure (Chevillard & Meneveau, 2006) and summarize its predictions reproducing recent observations by Xu, Pumir & Bodenschatz (2010) on twotime vorticitystrain alignment statistics. We also describe a new tool associated with the public turbulence database cluster, namely the "getPosition function" that is particularly useful for studies of the Lagrangian dynamics of turbulence. Given an initial position, integration timestep, as well as an initial and end time, the getPosition function tracks arrays of fluid particles and returns particle locations at the end of the trajectory integration time. The getPosition function is tested by comparing with trajectories computed outside of the database. It is then applied to study Lagrangian velocity structure functions as well as tensorbased Lagrangian time correlation functions. The roles of pressure Hessian and viscous terms in the evolution of the symmetric and antisymmetric parts of the velocity gradient tensor are explored by comparing the time correlations with and without these terms. We also test the pressure Hessian model based on the Recent Fluid Deformation (RFD) approximation and examine the twotime correlation function and observe a slight timedelay between model and real pressure Hessian. Work performed in collaboration with Dr. Laurent Chevillard, Dr. Huidan Yu and the Turbulence Database Group at JHU, and supported by the US National Science Foundation.  
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JeanLuc Thiffeault  WPI, Seminarroom C 714  Mon, 7. May 12, 12:00 
Extracting flow information from sparse Lagrangian trajectories  
In many applications, particularly in geophysics, we often have fluid trajectory data from floats, but little or no information about the underlying velocity field. The standard techniques for finding transport barriers, based for example on finitetime Lyapunov exponents, are then inapplicable. However, if there are invariant regions in the flow this will be reflected by a `bunching up' of trajectories. We show that this can be detected by tools from topology. This is joint work with Michael Allshouse and Tom Peacock.  
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Francisco J. BeronVera  WPI, Seminarroom C 714  Mon, 7. May 12, 14:30 
Geodesic transport barriers in ocean flows  
Haller and BeronVera (2012) have recently introduced a geodesic theory for the objective (i.e., frameindependent) identification of key material curves (transport barriers) that shape global mixing patterns in temporallyaperiodic twodimensional flows defined over a finitetime interval, such as simulated or observed largescale ocean flows. Seeking transport barriers as leaststretching material curves, it is found that such transport barriers must be shadowed by (minimal) geodesics of the Cauchy–Green strain tensor. Three relevant types of transport barriers are identified: hyperbolic (generalized stable and unstable manifolds); elliptic (generalized KAM curves); and parabolic (generalized shear jets). In this talk the main elements of the geodesic theory will be described, and results from its application to ocean flows inferred using satellite altimetry measurements will be presented.  
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Charles R. Doering  WPI, Seminarroom C 714  Mon, 7. May 12, 16:00 
Measures of mixing in (turbulent) fluid flows  
There are a variety of commonly accepted quantitative measures of mixing in fluid dynamics. These include tracer particle and/or pair dispersion and passive scalar fluxgradient relations. Both lead to familiar notions of "effective diffusion" and "turbulent diffusivity" that presumably characterize transport properties of a flow. Other measures of mixing, and hence other notions of effective diffusion and turbulent diffusivity, arise in applications where sources and sinks sustain scalar inhomogeneities. For example overall scalar concentration variance reduction in a bounded domain naturally characterizes the effectiveness of stirring in the presence of sources and sinks. It turns out that these notions are not always compatible, i.e., effective diffusions decided by tracer dispersion may not agree with thase determined by concentration variance reduction for the same flow. We discuss why this is so, and contemplate the robustness of concepts like turbulent diffusivities.  
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Miguel D. Bustamante  WPI, Seminarroom C 714  Tue, 8. May 12, 10:00 
Dynamics of Vorticity Near the Position of its Maximum Modulus  
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Miguel D. Bustamante  WPI, Seminarroom C 714  Tue, 8. May 12, 11:30 
EulerianLagrangian methods in fluid mechanics, based on Hamilton's principle  
We present a noncomprehensive survey of EulerianLagrangian approaches in fluid mechanics, having as common starting point the Hamilton's principle for the 3D Euler equations. In the inviscid case, we present derivations of the methods from first principles. For the dissipative case, we show some applications of these methods in numerical simulations of relevant physical processes such as vortex reconnection and the magnetic dynamo, and also in turbulence closure models.  
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Alex Liberzon  WPI, Seminarroom C 714  Tue, 8. May 12, 16:00 
Lagrangian and Eulerian aspects of turbulent flows with dilute polymer solutions  
Flows with polymer solutions provide another important example where the Lagrangian approach is unavoidable at least for two additional reasons: 1) since the material elements (purely Lagrangian objects) in such flows are not passive and 2) there are no equations reliably describing flows of polymer solutions such as NSE for Newtonian fluids. So there is a need for Lagrangian experimentation with such turbulent flows in the first place. A similar statement is true of flows with any other active additives. However, Lagrangian methods alone are limited in several respects so that there is a necessity of using Eulerian approaches in parallel with the Lagrangian ones. We bring a number of examples demonstrating this point. The first concerns the fluid particle acceleration (a purely Lagrangian quantity) along with its various it Eulerian components which help to elucidate a number of issues. Similarly, though important issues in the evolution of small scales are addressed via Largangian approaches one is using such quantities as strain and vorticity in their Eulerian form. Similarly, even when the dominant role of Lagrangian approaches is clear when dealing with the issue of material elements one needs again Eulerian quantities such as strain and vorticity. On the other hand, Eulerian approaches are of utmost value dealing with such large scale issues as Reynolds stresses (RS) and TKE production. We bring a number of results on the above issues obtained by the Particle tracking technique with access to velocity derivatives and , possibly on the direct interaction of large and small scales and time evolution  
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Nick Ouellette  WPI, Seminarroom C 714  Wed, 9. May 12, 10:00 
Sweeping and the Cascade: Physical Transport of Spectral Properties  
We typically think about turbulence in two distinct ways: we study the dynamics in space, and we study the dynamics in scale. In both cases, the interaction of the large and small scales can be studied. In the spatial sense, this interaction takes the form of the sweeping of small eddies by the large scales. In the spectral case, it takes the form of the energy cascade: a net flux of energy from large to small scales. But how are these two pictures linked? I will discuss recent progress we have made in trying to understand the advection of the spectral properties of the flow in quasi2D laboratory weak turbulence. Using high resolution velocimetry and a filtering technique, we extract spatially resolving energy fluxes between scales in our flow. We then study the Lagrangian transport of these fluxes and connections to the transport of fluid elements.  
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Victor Lvov  WPI, Seminarroom C 714  Wed, 9. May 12, 11:30 
EulerianLagrangian bridge for the energy and dissipation spectra in homogeneous turbulence  
For homogeneous isotropic hydrodynamic turbulence we derived from the first principles equations that bridge the Eulerian and Lagrangian energy spectra, EE(k) and EL(ω), as well as the Eulerian and Lagrangian dissipation, εE(k) = 2νk2EE(k) and εL(ω). We demonstrate that both analytical relationships, EL(ω) ⇔ EE(k) and εL(ω) ⇔ εE(k), are in very good quantitative agreement with our DNS results, which show that not only EL(ω; t) but also the Lagrangian spectrum of the dissipation rate εL(ω; t) has its maximum at low frequencies (about the turnover frequency of energy containing eddies) and vanishes at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence.  
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Rainer Grauer  WPI, Seminarroom C 714  Wed, 9. May 12, 16:00 
Exact relation between Eulerian and Lagrangian velocity increment statistics  
We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems. In order to get insight into the role played by the dissipative structures we compare different turbulent systems e.g. 2D and 3D NavierStokes flows, 3D MHD flows and 2D electron MHD flows. In addition, we study the situation for compressible fluids where the density clustering has to be taken into account. If time allows we will present results on conditional Lagrangian statistics where we propose a novel condition for Lagrangian increments which is shown to reduce the flatness of the corresponding PDFs substantially and thus intermittency in the inertial range of scales. The conditioned PDF corresponding to the smallest increment considered is reasonably well described by the K41prediction of the PDF of acceleration.  
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Luca Biferale  WPI, Seminarroom C 714  Thu, 10. May 12, 10:00 
The Multifractal approach to Lagrangian and Eulerian statistics in homogeneous and isotropic turbulence: successes and pitfalls  
I review recent applications of the multifractal phenomenology to Eulerian and Lagrangian turbulence. In particular, I will stress the main ideas behind a bridge relation between the two ensembles. I will show benchmarks of such relation against experimental and numerical data and discuss opens problems, in particular concerning new questions arising when inertial particles are considered.  
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Eberhard Bodenschatz  WPI, Seminarroom C 714  Thu, 10. May 12, 11:00 
On Lagrangian particle statistics  
In turbulence, ideas of energy cascade and energy ux, substantiated by the exact Kolmogorov relation, lead to the determination of scaling laws for the velocity spatial correlation function. Here we ask whether similar ideas can be applied to temporal correlations. We critically review the relevant theoretical and experimental results concerning the velocity statistics of a single fluid particle in the inertial range of statistically homogeneous, stationary and isotropic turbulence. We stress that the widely used relations for the second structure function, D2(t) ≡ ‹[v(t)  v(0)]2 › ≈ εt, relies on dimensional arguments only: no relation of D2(t) to the energy cascade is known, neither in two nor in threedimensional turbulence. State of the art of the experimental and numerical results demonstrate that at high Reynolds numbers, the derivative dD2(t)/dt has a finite nonzero slope starting from t ≈ 2τη. The analysis of the acceleration spectrum ΦA(ω) indicates a possible small correction with respect to the dimensional expectation ΦA(ω) ~ ω0, but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics. If time permits we shall also discuss the multiple particle statistics.  
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Rudolf Friedrich  WPI, Seminarroom C 714  Thu, 10. May 12, 12:00 
Kinetic Equations for Turbulent Cascades  
The talk intends to provide an introduction to the application of kinetic equations for the statistics of turbulent flows. We will focus both on the inverse cascade in two dimensional flows as well as the direct cascade in homogeneous isotropic three dimensional turbulence. Furthermore, we discuss kinetic equations for the temperature statistics of RayleighB´enard convection. Direct cascades in three dimensions will be analyzed by the statistics of the vorticity field, which is characterized by the presence of Burgerslike vortices. We will explicitly show that the statistics of the vorticity field is strongly nonGaussian and we will trace this nonnormality back to the presence of strong vorticity events. We shall discuss how the wings of the vorticity probability distribution can be related to the properties of these coherent structures. Two dimensional cascades will be investigated on the basis of a generalized Onsager vortex model explicitly showing that the energy transfer from small to large scales arises due to a clustering of likesigned vortices.  
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