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

Final Talk: Thursday, May 10. 10:00 AM

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

The Lagrangian evolution of the velocity gradient tensor depends upon the pressure Hessian and the viscous term to regularize the otherwise finite-time 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 two-time vorticity-strain 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 time-step, 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 tensor-based 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 two-time correlation function and observe a slight time-delay 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 finite-time 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

Haller and Beron-Vera (2012) have recently introduced a geodesic theory for the objective (i.e., frame-independent) identification of key material curves (transport barriers) that shape global mixing patterns in temporally-aperiodic two-dimensional flows defined over a finite-time interval, such as simulated or observed large-scale ocean flows. Seeking transport barriers as least-stretching 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 flux-gradient 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

We present a non-comprehensive survey of Eulerian-Lagrangian 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.
The survey includes:
(i) Soward's approach termed "HEL" (hybrid Eulerian-Lagrangian) which is a useful model in rotating magneto-hydrodynamics (MHD), at scales that are "large" as compared to dissipation and diffusion scales, but "short" as compared with the size of the physical system;
(ii) Holm's approach termed "Navier-Stokes-alpha" model, which is in the same spirit as the HEL approach, but whose equations are more tractable from the point of view of integrability and boundedness;
(iii) The so-called Clebsch-Weber approach, developed by Constantin and Ohkitani and later by Cartes, Bustamante and Brachet, which is actually not a model but an exact representation of 3D Euler/Navier-Stokes/MHD, that takes advantage of the Lagrangian nature of certain fields such as the magnetic induction field and the vorticity field.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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
References:
Liberzon, A., Guala, M., Kinzelbach, W., and Tsinober, A. (2006). On turbulent kinetic energy production and dissipation in dilute polymer solutions. Physics of Fluids, 18(12):125101.
Liberzon, A., Guala, M., Lüthi, B., Kinzelbach, W., and Tsinober, A. (2005). Turbulence in dilute polymer solutions. Physics of Fluids, 17(3):031707.
Tsinober, A. (2009). An informal conceptual introduction to turbulence. Springer.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 quasi-2D 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 Navier-Stokes 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 K41-prediction of the PDF of acceleration.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 three-dimensional 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 non-zero 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.

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)

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 Rayleigh-B´enard convection. Direct cascades in three dimensions will be analyzed by the statistics of the vorticity field, which is characterized by the presence of Burgers-like vortices. We will explicitly show that the statistics of the vorticity field is strongly non-Gaussian 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 like-signed vortices.
References:
M. Wilczek, R. Friedrich, Dynamical Origins for Non-Gaussian Vorticity Distributions in Turbulent Flows, Phys. Rev. E 80, 016316 (2009)
M. Wilczek, A. Daitche, R. Friedrich, Theory for the single-point velocity statistics of fully developed turbulence, EPL, 93, 34003 (2011)
M. Wilczek, A. Daitche, R. Friedrich, On the velocity distribution in homogeneous isotropic turbulence, J. Fluid Mech., first view article (2011)
J. Lûlff, M. Wilczek, R. Friedrich, Temperature Statistics in Turbulent Rayleigh Bénard convection, New J. Phys. 13, 015002 (2011)
R. Friedrich, M. Vosskuhle, O. Kamps, M. Wilczek, Two-Point Vorticity in the Inverse Turbulent Cascade, arXiv:1012.3356 (2010)
J. Friedrich, R. Friedrich, Vortex model for the inverse cascade of 2d-turbulence, arXiv:1111.5808

• Thematic program: Fusion and Turbulence (2012)
• Event: Meeting "Euler Lagrangian" (2012)