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Workshop Programme

for period 29 September - 26 October 2008

Inertial-Range Dynamics and Mixing

29 September - 26 October 2008


Monday 29 September
08:30-09:55 Registration
09:55-10:00 David Wallace - welcome
Chair: C Doering
10:00-10:30 Leonard, A (Caltech)
  Curvature of vortex tubes and sheets in turbulence Sem 1

Weak structures of vorticity are studied as they evolve essentially passively as a result of the induced velocity field of the large-scale turbulence. Such objects are possibly an important component of the inertial and dissipation ranges of turbulence. Earlier work [1} along these lines concentrated on the effects of accumulated strain, apropos of Lagrangian chaos, on localized packets of vorticity. In particular it was shown that time-averaging the self-energy spectrum of such a structure leads to a fractional power law for the energy spectrum. In this paper we consider the effects of strain on the geometry of vortex tubes and sheets and concentrate on regions of high curvature, as an extension of other recent work [2]. In the case of a vortex tube, we find that a region of high curvature on the tube is likely to develop around a point where the vorticity vector is orthogonal to the principal axis of maximum strain. For vortex sheets, a region of high curvature is likely to occur where the above-mentioned principal axis is normal to the sheet. The strain tensor in question is that corresponding to the future deformation of a material element. We explore the notion that such high-curvature structures play a role in establishing the energy spectrum in inertial-range turbulence. References 1. A. Leonard 2002," Interaction of localized packets of vorticity with turbulence" in Proc. IUTAM Symp. on Tubes, Sheets, and Singularities in Fluid Dynamics (Kluwer) 201-210. 2. A. Leonard 2008, "The universal structure of high curvature regions of material lines in chaotic flows", submitted.

10:30-11:00 Falkovich, G (Weizmann Institute of Science)
  Emerging symmetries and condensates in inverse cascades Sem 1

I shall review symmetry aspects of turbulent inverse cascades in fluid mechanics, optics and plasma physics. Inverse cascades are generally scale invariant in distinction from most direct cascades. It has been recently discovered that in many cases scale scale invariance can be promoted to a more general and powerful conformal invariance. Namely, two-dimensional isolines of different fields (vorticity, temperature) belong to the remarkable class of curves called Schramm-Loewner evolution (SLE). I shall briefly review the properties of such curves. That discovery brings unexpected connections between fluid mechanics, mathematics (SLE was a subject of the recent Fields medal), theory of critical phenomena and quantum field theory. I then describe the appearance of spectral condensates (modes coherent across the whole system) due to inverse cascades. I briefly discuss the possible role of condensates in atmospheric physics, including the controversy on the energy flux at meso-scales. I shall describe initial findings on how condensates break symmetries established in inverse cascades.

11:00-11:30 Coffee
Chair: T Gotoh
11:30-12:00 Chen, H (Exa Corporation)
  Kinetic theory representation for turbulence modeling and computation Sem 1

One of the most common approximations in turbulence for the averaged effect of small scales is by the so called eddy viscosity modeling. That is, one approximates the Reynolds stress as a linear function of the local rate of strain of the averaged flow field. The proportionality constant is referred to as an eddy viscosity. This concept was first proposed over a century ago. It stems from an analogy for small eddy interactions with collisions of molecules resulting in Newtonian fluid constitutive relations. This approximation has made enormous impact particularly in computational fluid dynamics for turbulent flows. Many theoretical works were also developed since then, with various successes, in order to analytically derive such a functional relationship. However, unlike molecular interactions in a fluid, one of the apparent criticisms or difficulties in this analogy is the lack of scale separations between averaged fluid motions and fluctuating eddies. In this presentation, the speaker will give a somewhat provocative argument in favor of such analogy, provided that this concept be expanded in a generalized kinetic theory framework. In such an expanded framework, the analogy between eddy interactions and molecular collisions has a broader physical validity, while the eddy viscosity approximation is its consequence in the very long wave length limit. The kinetic theory representation itself needs not depend on scale separations. Using such an expanded analogy, one can also draw similarities between turbulent flow phenomena to that of non-Newtonian fluid flows in micro/nano scales. Nevertheless, other than phenomenological argument, as far as the speaker is aware, so far there have been little theoretical attempts to produce such a kinetic theory for turbulence on a concrete footing via first principle, if its physical soundness is acceptable.

12:00-12:30 Poster adverts
12:30-13:30 Lunch at Wolfson Court
Chair: H Chen
14:00-14:30 Nazarenko, SV (Warwick)
  Bottleneck crossover between classical and quantum superfluid turbulence Sem 1

We consider superfluid turbulence near absolute zero, which consists of a polarized tangle of mutually interacting vortex filaments with quantized vorticity. This system exhibits a classical K41-type cascade at the scales greater than the intervortex separation l, and Kelvin wave turbulence at the scales

14:30-15:00 Pandit, R (Indian Institute of Science)
  Homogeneous isotropic turbulence with polymer additives Sem 1

We investigate the effects of polymer additives on flows that display homogeneous, isotropic turbulence by extensive direct numerical simulations (DNS) of (a) a shell model and (b) the Navier-Stokes equation coupled to an equation for the polymer-conformation tensor (the FENE-P model). Our simulations show that the addition of polymers to such flows leads to dissipation reduction in both decaying and statistically steady turbulence; this dissipation reduction is the analogue of drag reduction in wall-bounded flows. Our numerical results agree well with recent experimental results. In particular, we find that polymers decrease the energy of the turbulent fluid at intermediate length scales but increase it at small length scales; a scale-dependent viscosity provides a natural means of understanding our results. Preliminary studies of the multiscaling of structure functions, in the presence of polymer additives, and shock-capturing schemes for this problem are also discussed.

15:00-15:30 Tea and Posters
Chair: R Pandit
15:30-16:00 Tsubota, M (Osaka City)
  Quantum turbulence -from superfluid helium to atomic Bose-Einstein condensates- Sem 1

Quantum turbulence (QT) was discovered in superfluid in the 1950s, and the research has tended toward a new direction since the mid 90s. The similarities and differences between quantum and classical turbulence have become an important area of research in low temperature physics. QT is comprised of quantized vortices that are definite topological defects, being expected to yield a model of turbulence that is much simpler than the classical model. The general introduction of the issue is followed by a description of the dynamics of quantized vortices. After reviewing the modern research trends on QT, we focus on the energy spectrum of QT at very low temperatures. The numerical simulation of QT by the Gross-Pitaevskii model shows that energy is transferred through the Richardson cascade of quantized vortices and the spectrum obeys the Kolmogorov law. Then we discuss QT in trapped atomic Bose-Einstein condensates (BECs). Under the combined rotations around two axes, a BEC cloud develops to a turbulent state and the energy spectrum obeys the Kolmogorov law. Ref. M. Tsubota, cond-mat/ 08062737

16:00-16:30 Roche, PE (CNRS)
  The spectrum of the vortex line density in a turbulent superfluid Sem 1

Over the last twenty years, experiments and numerical simulations revealed strong similarities between Superfluid turbulence and Navier-Stokes turbulence. In particular, most measurements of integral quantities -such as the head loss in a pipe- or fluctuating ones -such as the velocity in the inertial range- could be interpretated by analogy with classical turbulence. Recently, measurements of the local fluctuations of the vortex lines density in superfluid 4He near 1.5 K have been reported. Over more than one decade of inertial range, the power spectrum was fitted with a f^(-5/3) power law. Remarkably, such a scaling differs from the one found in Navier-Stokes turbulence for the enstrophy, or for the absolute value of vorticity, which are often considered as classical counterparts of the vortex line density. We argue that this observation could be interpreted as the signature of a passive advection of the superfluid vortex lines by the largest scales of the flow.

more info on :
Roche P.-E. & Barenghi C. , EPL 81: 36002 (2008)

Chair: J Jimenez
17:00-18:00 Sreenivasan, KR (ICTP)
  Cryogenic turbulence Sem 1

An overview of our knowledge of turbulence in both helium 4 and superfluid helium will be presented. The results will be cast in historical perspective and supplemented by numerical and theoretical ideas of the last fifty years.

18:00-18:30 Welcome Wine Reception
18:45-19:30 Dinner at Wolfson Court (Residents Only)
Tuesday 30 September
Chair: GL Eyink
09:30-10:00 Tsuji, Y (Nagoya)
  Anisotropic pressure and acceleration spectra in uniform shear flow Sem 1

According to the local isotropic hypothesis, the small scale physical quantities such as velocity, temperature and pressure fluctuations are to be universal in any kind of turbulent flow. At this stage, the question is not whether this assumption is correct or not, but seems to be how the large scale anisotropy lost its information as the scale becomes small. In this talk, the anisotropic effect on inertial-range quantities are directly checked following the formula presented by Ishihara, Yoshida and Kaneda (P.R.L.,vol.88,154501,2002), in which the velocity correlation spectrum was uniquely determined by the rate of strain tensor of mean flow, the energy dissipation of per unit mass, and the two-non dimensional constants. This idea is applied to the pressure field in the uniform shear flow, and the shear effect on pressure and pressure gradient (acceleration) is studied experimentally up to the Reynolds number based on Taylor micro scale is 800. The results show the excellent agreement with the prediction by theory, and the universal trend of anisotropic spectra was observed.

10:00-10:30 Jimenez, J (Aeronautics, UP Madrid)
  Intermittency in imperfect multiplicative cascades Sem 1

The standard multifractal cascade model assumes both a Markovian self-similar multiplicative cascade, and locality, in the sense that point properties depend only on their immediate neighbourhoods. Relaxing the second condition leads to more general cascades in which a point property v_{n+1} depends both on the local previous cascade step v_n, and on the global variance v'_n. The first contribution models a local breakdown process, while the second represents the effect of the background perturbations. There are two stochastic multipliers, one for each term, and they are characterised empirically for experimental high-Reynolds number experimental turbulence. General conditions are derived for such an imperfect multiplicative cascade to be intermittent, in the sense of creating unbounded high-order flatness factors after many steps. The experimental values are such as to be most likely intermittent, but they may not reach true multifractal distributions, and power laws for the structure functions, until extremely large Reynolds numbers.

10:30-11:00 Peinke, J (Oldenburg)
  New insights into turbulence Sem 1

We present a more complete analysis of measurement data of fully developed, local isotropic turbulence by means of the estimations of Kramers- Moyal coefficients, which provide access to the joint probability density function of increments for n- scales \cite{JFM}. In this contribution we report on new findings based on this technique and based on the investigation of many different flow data over a big range of Re numbers.

In particular we show:

- An improved method to reconstruct from given data the underlying stochastic process in form of a Fokker-Planck equation, which includes intermittency effects, will be shown.

- It is shown that a new length scale, for turbulence can be defined, which corresponds to a memory effect in the cascade dynamics. This coherence length can be seen as analogue to the mean free path length of a Brownian Motion. For length scales larger than this coherence length the complexity of turbulence can be treated as a Markov process. We show that this Einstein- Markovian coherence length is closely related to the Taylor micro-scale.

- It is shown that the stochastic process of a cascade will change with the Re-number and its universal or non-universal behavior with changing large scale boundary conditions will be discussed.

- For longitudinal and transversal velocity increments we present the reconstruction of the two dimensional stochastic process equations, which shows that the cascade evolves differently for the longitudinal and transversal increments. A different "speed" of the cascade for these two components can explain the reported difference for these components. The rescaling symmetry is compatible with the Kolmogorov constants and the Karman equation and give new insight into the use of extended self similarity (ESS) for transverse increments.

- A method is presented which allows to reconstruct time series from a estimated stochastic process evolving in scale. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The original and reconstructed time series coincide with respect to the unconditional and conditional probability densities. Therefore the method proposed here is able to generate artificial time series with correct n-point statistics.

11:00-11:30 Coffee and Posters
Chair: J Peinke
11:30-12:00 Arimitsu, T (Tsukuba)
  On extension of the formalism MPDFA and its application to the analyses of DNS 4096$^3$ conducted by Kaneda and Ishihara Sem 1

Our original theoretical framework, named Multi-fractal Probability Density Function Analysis (MPDFA), has been extended successfully in order to make it possible to analyze a series of probability density functions (PDFs), extracted from experiments and numerical simulations, with arbitrarily changed measuring areas or distances for various physical quantities representing intermittent behavior characterizing fully developed turbulence.

MPDFA is a unified self-consistent approach for the systems with large deviations, which has been constructed based on the Tsallis-type distribution function following the assumption raised by Frisch and Parisi that the singularities due to the scale invariance of the Navier-Stokes equation for high Reynolds number distribute themselves multifractal way in real physical space. MPDFA can be said as a generalization of the log-normal model. It was shown that MPDFA derives the log-normal model when one starts with the Boltzmann-Gibbs distribution function instead of Tsallis-type distribution function.

As a test of the validity of the extension, we analyzed the PDFs for energy transfer rates and for energy dissipation rates extracted by Kaneda and Ishihara group at Nagoya University from their DNS 4096$^3$.

In this talk, we will present mainly on the theoretical extension of MPDFA and its validity. The detailed analyses of PDFs out of DNS 4096$^3$ and the physical outcomes from them will be given at our poster presentation of this workshop.

Related Links
  • - Journal of Physics: Conference Series {\bf 7} (2005) 101--120.
  • - Anomalous Fluctuation Phenomena in Complex Systems: Plasma Physics, Bio-Science and Econophysics (Special Review Book for Research Signpost), eds. C.~Riccardi and H.E.~Roman (Transworld Research Network, Kerala, India, 2008) in press.

12:00-12:30 Danaila, L (CORIA)
  The structure of the velocity and passive scalar mixing in a multiple opposed jets reactor Sem 1

We document an experimental investigation of a confined chamber in which fluid is injected through two sets of 16 opposed jets that issue from top/bottom boundary porous planes. The investigated Reynolds numbers, based on injection velocity and jet diameter are up to 28,000. The high Reynolds numbers and impinging configuration of the flow produce very intense turbulence levels and a turbulence with zero mean velocity in the central region of the reactor. The analysis is done for basically two geometries: opposed jets with strong backflow, and opposed jets with very slow backflow. Particles Image Velocimetry (PIV) measurements in different planes allowed for a characterization of the mean and fluctuating velocity fields. Fluid recirculation in the reactor creates annular shear layers. The central region of the reactor includes stagnation regions, where mean vertical velocity gradients are very strong with low local mean velocity values, leading to high rms-to-mean velocity ratios. Such gradients are responsible for considerable kinetic-energy production, that sharply peaks in the central region. A particular attention is paid to the determination of the small-scales characteristics (energy dissipation rate) in different points of the flow, which is done using both PIV (using indirect methods, i.e. inertial-range information) and LDV (Laser Doppler Anemometry). Inertial-range behaviour is discussed, in terms of second and third-order structure functions, and a critical comparison with classical (forced) flows is done.

A passive scalar (Sc=1.3) is injected in the flow, in an alternate sequence (Z=0 and Z=1) among each two opposed jets and measured using Laser-Induced Fluorescence. Scalar structure functions are investigated, as well as the extent to which isotropy and homogeneity are adequate approximations for this flow. Flow visualisations exhibit very sharp scalar gradients at the frontier among two opposed jets. The dynamics of these regions closely follows that dictated by the back-and-forth motion developed in the central region, due to the opposed jets instabilities. Thus, the scalar is directly injected at the level of small scales, whereas the velocity field itself is injected over a whole range of scales. This directly leads to a very effective mixing in the stagnation region of the opposed jets.

12:30-13:30 Lunch at Wolfson Court
Chair: M Farge
14:00-14:30 Schumacher, J (TU Ilmenau)
  Statistics and growth rates of high-amplitude vorticity events in turbulence Sem 1

Fluid turbulence is often characterized as a tangle of many intermittent vortices embedded in regions of straining motion. Although there have been many experimental and numerical studies on the evolution of isolated intense vortices, pairs of them or on the kinematics of ensembles of randomly distributed vortices, not much is known about their dynamic evolution in a fully turbulent flow. We present a high-resolution numerical simulation that monitors the formation and time evolution of high-amplitude vorticity regions. In order to identify and follow these events, we track the turbulence fields in local Cartesian frames of reference which move with Lagrangian tracers through the fluid. The local enstrophy -a measure of vorticity- shows temporal growth compatible on average with a classical prediction by Howarth and von Karman (1938). It remains well below a recently predicted rigorous upper bound for the enstrophy growth (Lu and Doering, 2008). However locally, enstrophy growth rates are detected that go beyond the mean trend and approach the predicted global bound. Related Links * - Homepage of Theoretical Fluid Mechanics Group

14:30-15:00 Ishihara, T (Nagoya)
  Statistics of two-point velocity difference in high-resolution direct numerical simulations of turbulence in a periodic box Sem 1

Statistics of two-point velocity difference are studied by analyzing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to $4096^3$ grid points. The DNS consist of two series of runs; one is with $k_{max}\eta\sim 1$ (Series 1) and the other is with $k_{max}\eta\sim 2$ (Series 2), where $k_{max}$ is the maximum wavenumber and $\eta$ the Kolmogorov length scale. The maximum, time-averaged, Taylor-microscale Reynolds number $R_\lambda$ in Series 1 is about 1145, and it is about 680 in Series 2. Particular attention is paid to the possible Reynolds number ($Re$) and $r$ dependence of the statistics, where $r$ is the distance between two points. The statistics include the probability distribution functions (PDFs) of velocity differences and the longitudinal and transversal structure functions. DNS data suggest that the PDFs of the longitudinal velocity difference at different values of Re but the same values of $r/L$, where $L$ is the integral length scale, overlap well with each other when r is in the inertial subrange and when using the same method of forcing at large scales. The similar is also the case for the transversal velocity difference. The tails of the PDFs of normalized velocity differences ($X$'s) are well approximated by such a function as $\exp(-A|X|^a)$, where $a$ and $A$ depend on $r$, and where $a$ becomes $\approx 1$ in the inertial subrange. Analysis shows that the scaling exponents of the $n$th-order longitudinal and transversal structure functions are not sensitive to $Re$ but sensitive to the large-scale anisotropy and non-stationarity, and suggests that nevertheless their difference is a decreasing function of $Re$.

15:00-15:30 Tea and Posters
Chair: A Leonard
15:30-16:00 Farge, M (LMD-CNRS-IPSL)
  CVS filtering to study turbulent mixing Sem 1

Coherent Vortex Simulation (CVS) is based on the wavelet filtered Navier-Stokes equations, where at each instant the turbulent flow is split into two orthogonal contributions: the coherent flow made of vortices which is kept, and the incoherent flow made of the background fluctuations which is discarded. The CVS filter is based on an orthogonal wavelet decomposition of the vorticity field where only the wavelet coefficients whose modulus is larger than a given threshold are kept. The value of the threshold depends only on the total enstrophy and on the numerical resolution used to represent the flow. The CVS filter has already been applied to 2D [Farge, Schneider and Kevlahan in Phys. Fluids 11(8) 1999] and 3D [Farge, Pellegrino and Schneider in Phys. Rev. Lett. 87(55) 2001, Farge et al. in Phys. Fluids 15(10) 2003, Okamoto et al. in Phys. Fluids 19 2007] turbulent flows, where it has been shown that only few wavelet coefficients (from 0.7% for 256^2 up to 2.6% for 2046^3 resolution) are sufficient to represent the coherent flow which preserves the vorticity and velocity PDFs, the energy spectrum and the nonlinear transfers all along the inertial range. We will analyze the time evolution of a decaying homogeneous isotropic turbulent flow, by applying the CVS filter at each time step of a Direct Numerical Simulation (DNS). We will compare the Eulerian and Lagrangian mixing properties of the total, coherent and incoherent flows by studying how they advect a passive tracer and many particles during several eddy turn-over times. We will quantify the mixing properties of coherent and incoherent flows and show that efficient mixing is due to the transport by vortices, while the incoherent contribution is much weaker and only diffusive. Related Links * - Web

16:00-16:30 Domaradzki, JA (Southern California)
  Energy cascade in turbulent flows: quantifying effects of Reynolds number and local and nonlocal interactions Sem 1

The classical Kolmogorov theory of three-dimensional turbulence is based on the concept of the energy transfer from larger to progressively smaller scales of motion. The theory postulates that bulk of the energy transfer in the inertial range of turbulence occurs between scales of similar size, a process known as the local energy cascade. The locality allows to postulate that after multiple cascade steps the small scale dynamics become universal, i.e., independent of particulars of large scales that are determined by geometry, boundary conditions, and forces causing a flow. Yet despite its central role in the Kolmogorov theory the locality assumption cannot be easily verified, neither analytically nor experimentally. This is because the energy transfer is a result of interactions among different scales of motion originating from the nonlinear term in the Navier-Stokes equation that couples all scales. Relevant questions have been productively addressed for the first time using databases generated in large scale numerical simulations. We revisit and extend previous work and use such databases to compute detailed energy exchanges between scales of motion obtained by decomposing numerical velocity fields using banded filters, and investigate how the detailed transfers contribute to the global quantities such as the classical energy transfer, the energy flux, and the subgrid-scale transfer. We address two questions in detail. First, for the purposes of quantitative analyzes, various definitions of scales of motion can be used. This non-uniqueness leads to the possibility, raised in the literature on the subject, that properties of the energy transfer deduced from such analyzes can be qualitatively affected by the employed scale definitions. We address this question by computing detailed energy exchanges between different scales of motion defined by decomposing velocity fields using three specific filters: sharp spectral, Gaussian, and tangent hyperbolic. Second, we quantify the locality of the energy transfer and address a persistent controversy concerning the role of nonlocal interactions in the energy transfer process, i.e., the role of much larger scales than those transferring energy. The analysis of detailed interactions reveals that the individual nonlocal contributions are always large but significant cancellations lead to the global quantities asymptotically dominated by the local interactions. The detailed locality functions are computed and their behavior compared with the asymptotic scaling laws valid for infinite Reynolds numbers turbulence. Apart from an intellectual challenge of clarifying these issues, obtained results have bearing on practical questions of turbulence modeling that will also be addressed in the talk.

16:30-17:00 Chen, S (PKU & JHU)
  Physical-space decimation and constrained large Eddy simulation Sem 1

Traditional decimation theory of fluid turbulence was proposed by Kraichnan and the analysis was carried out in the Fourier space. It has been shown that the low-order decimation theory leads to the direct-interaction-approximation, while the high-order decimation theory can include the effect of intermittency. In this talk, we propose a physical-space decimation method which can be used for large-eddy-simulation. In particular, we propose to impose physical constraints in the dynamic procedure of the dynamic subgrid-scale (SGS) stress model in large eddy simulation, and to calculate the SGS model coefficients using a constrained variation. One simple constraint for fluid turbulence in both physical and Fourier space decimation models is the conservation of energy across the inertial range. Numerical simulations of forced and decaying isotropic turbulence demonstrate that the constrained dynamic mixed model predicts the energy evolution and the SGS energy dissipation well. The constrained SGS model also shows a strong correlation with the real stress and is able to capture the energy backscatter, manifesting a desirable feature of combining the advantages of dynamics Smagorinsky and mixed models. It should be mentioned that all previous LES models do not satisfy underlying physical constraints. We have also extended the constrained LES to helical, passive-scalar and intermittent systems.

Chair: JA Domaradzki
17:10-17:40 Horiuti, K (Tokyo Institute of Technology)
  Extraction of the hierarchical energy spectrum in forced turbulence Sem 1

Properties of the energy spectrum in turbulent flows are studied using the DNS data for forced homogeneous isotropic and shear turbulence. The perturbation expansion about the Kolmogorov -5/3 energy spectrum yields the hierarchical spectrum, in which the -7/3 spectrum induced by the fluctuation of the dissipation rate is added. Averaging conditioned on the temporal variations of dissipation rate is applied to the ensemble of the energy spectra. The base steady spectrum fits -5/3 power, but its deviatoric part exhibits a fitting with -7/3 power. The role of the -7/3 power spectrum in generation of energy cascade is elucidated by examining the temporal variations of the energy spectrum and transfer function. The cascade process is divided into two phases. The energy contained in the low-wavenumber range in Phase 1 is transferred to the high wavenumbers in Phase 2 with switchover of the sign in the -7/3 power component. In Phase 1, a very large gain in the energy transfer function occurs at the scale corresponding to the integral length. The vortex sheet whose lateral length is comparable to the scale of this input is created, and many Mode 3 or 2 spiral vortices (LSV) (Horiuti & Fujisawa 2008) are detected in Phase 1. These LSVs are converted to Mode 1 in Phase 2. The statistics are compared in Phases 1 and 2. The turbulent energy is larger in Phase 1 than in Phase 2. The moderately large dissipation rate dominates in Phase 2, but the dissipation field is more intermittent in Phase 1. Averaging conditioned on the dissipation and enstrophy indicates that the regions of extreme dissipation and enstrophy possess a significant degree of overlap in space in Phase 1. These extreme events occur along the spiral sheet of Mode 3 LSV which is strained and stretched by the tube in the core.

17:40-18:00 Li, Y (Sheffield)
  Lagrangian evolution of non-Gaussianity and material deformation in restricted Euler dynamics Sem 1

Small-scale intermittency in fluid turbulence refers to the infrequent but strong bursts in the signals of small scale parameters. These bursts display highly non-Gaussian statistics, and its prediction poses serious challenges to turbulence research. Based on the restricted Euler approximation, and following the recent idea of tetrad dynamics, we derive a simple system of equations for the short-time Lagrangian evolution of velocity and passive scalar increments. The system reproduces several important intermittency trends observed in turbulence. A generalization to rotating turbulence shows that system captures some qualitative effects of rotation. An analytic solution to the material deformation in restricted Euler dynamics, obtained following the same idea, is also presented.

18:45-19:30 Dinner at Wolfson Court (Residents Only)
Wednesday 1 October
Chair: G Falkovich
09:00-09:30 Davidson, PA (Cambridge)
  On the large-scale structure of two-dimensional turbulence Sem 1

We consider freely-decaying, two-dimensional, isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k->0)=Ik^3 , where I is the two-dimensional version of Loitsyansky’s integral. However, a second possibility is E(k->0)=Lk , where the pre-factor, L, is the two-dimensional analogue of Saffman’s integral. We show that, as in three dimensions, L is an invariant and that E~Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k->0)=Jk^-1, where J, like L, is an invariant. However, while E~Jk^-1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E~Jk^-1 spectra. The constraint imposed by the conservation of energy is particularly important,ruling out E~Jk^-1 spectra for certain classes of initial conditions. We illustrate these ideas with some direct numerical simulations.

09:30-10:00 Kerr, RM (Warwick)
  Resolving the cascade bottleneck in vortex-line turbulence Sem 1

Both in many superfluid experimental situations and simulations of a 3D hard-core interaction model, it is found that the vortex line length in superfluid turbulence decays in a manner consistent with classical turbulence. Two decay mechanisms have been proposed, Kelvin wave emission along lines and phonon radiation at small scales. It has been suggested that both would require a Kelvin wave cascade, which theory says cannot reach the smallest scales due to a bottleneck. In this presentation we will discuss a new approach using a recent quaterionic formulation of the Euler equations, coupled with the local induction approximation. Without the extra quaterionic terms It can be shown that if there are sharp reconnections, the above scenario occurs. But with the extra terms, the direction of propagation of nonlinear waves is reversed, there is a cascade to the smallest scales that could create phonons, and the paradox can be resolved.

10:00-10:30 Pouquet, A (NCAR)
  The absence of bottleneck in the Lagrangian-averaged model for incompressible magnetohydrodynamics Sem 1

In order to better understand the small scale dynamics of geophysical and astrophysical flows with huge Reynolds numbers, numerical modeling is an invaluable tool but it needs to be assessed against experimental and observational data as well as direct numerical simulations (DNS) at high resolution. In this context, we study the properties of the Lagrangian-averaged magnetohydrodynamics (MHD) $\alpha-$model, LAMHD hereafter; this model can be viewed as a norm-preserving filtering of the primitive MHD equations. Among its advantages is the fact that the LAMHD formulation preserves the basic properties of MHD, e.g. the Alfv\'en theorem of flux conservation, and invariants such as the total energy, the cross-correlation between the velocity and magnetic field and magnetic helicity, albeit in a modified (H_1) form. LAMHD has been tested in two and three space dimensions and is found to behave satisfactorily, for example reproducing the threshold for dynamo action at moderately low magnetic Prandtl numbers P_M, as encountered in the liquid core of the Earth, the solar convection zone or in liquid metals in the laboratory (where P_M is the ratio of viscosity to magnetic diffusivity). Here we demonstrate that, for the case when there is initially quasi-equipartition between the velocity and the magnetic field and with a magnetic Prandtl number equal to unity, the model reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes $\alpha-$model is somewhat more limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. The LAMHD model is thus shown to be capable of leading to large reductions in required numerical degrees of freedom for a given set of kinetic and magnetic Reynolds number. Specifically, we find a reduction factor of approx 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Taylor Reynolds number approx 1700. The DNS having been stopped at the peak of dissipation of total energy, the run was pursued using LAMHD. We thus also report on preliminary explorations of the decaying dynamics of that high Reynolds number MHD flow at late times using the LAMHD model.

10:30-11:00 Schekochihin, A (Imperial College London)
  Kinetic turbulence: a nonlinear route to dissipation through phase space Sem 1

This talk will describe a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (possibly arbitrarily small) degree of collisionality. The smallness of the collision rate is compensated by the emergence of small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring in the gyrocentre space and in velocity space. Scaling relations for the corresponding fluctuation spectra are derived. An estimate for the collisional cutoff is provided. The relevance of these results to understanding the dissipation-range turbulence in the solar wind and the electrostatic microturbulence in fusion plasmas is discussed. Related Links * - preprint

11:00-11:30 Coffee and Posters
Chair: S Nazarenko
11:30-11:50 Mazzucato, AL (Penn State)
  On enstrophy dissipation in 2D turbulence Sem 1

We consider dissipation of enstrophy, one half the integral of squared vorticity, in 2D incompressible, turbulent flows at very high Reynolds number. We prove rigorously that, if fully developed turbulence is to be modeled mathematically by irregular (weak) solutions of the 2D Euler equations in the limit of vanishing viscosity, then there is no dissipation as long as the initial enstrophy is finite. We also provide examples of dissipative flows when the initial enstrophy is infinite. Our analysis is inspired by work of G. Eyink. This is joint work with Helena and Milton Lopes.

11:50-12:10 Turner, M (Exeter)
  Thresholds for the formation of satellites in two--dimensional vortices Sem 1

We examine the evolution of a two--dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m=2 added to it. If the perturbation is weak then the vortex returns to an axisymmetric state and the non--zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.

We consider the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine--scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical--layer equations numerically. These calculations are supported by simulations of the full Navier--Stokes equations using a family of profiles based on the tanh function.

12:10-12:30 Bustamante, MD (Warwick)
  Capturing reconnection in Navier-Stokes and resistive MHD dynamics Sem 1

In this work, the phenomena of vortex reconnection in Navier-Stokes (and magnetic reconnection in MHD), of importance in fully developed turbulence, are studied from the point of view of the Eulerian-Lagrangian representation. This representation is interpreted as a full characterization of fluid motion using only particle description. New generalized equations of motion for the Weber-Clebsch potentials associated to this representation are derived. We perform direct numerical simulations in order to confirm the validity of the paradigm proposed by Constantin where particles will diffuse anomalously in the space -and time- vicinity of reconnection events. For Navier-Stokes, the generalized formalism captures the intense reconnection of vortices of the Boratav, Pelz and Zabusky flow, in agreement with the previous study by Ohkitani and Constantin. For MHD, the new formalism is used to detect magnetic reconnection in several flows: the 3D Arnold, Beltrami and Childress (ABC) flow and the (2D and 3D) Orszag-Tang vortex. It is concluded that periods of intense activity in the magnetic enstrophy are correlated with periods of increasingly frequent resettings. Finally, the positive correlation between the sharpness of the increase in resetting frequency and the spatial localization of the reconnection region is discussed. Related Links * - ArXiv Preprint of Paper

12:30-13:30 Lunch at Wolfson Court
Chair: J Schumacher
14:00-14:30 Vassilicos, JC (Imperial College London)
  Non-universality of the turbulence dissipation constant in homogeneous isotropic turbulence and the universal relations which account for it Sem 1

The dimensionless dissipation constant of homogeneous isotropic turbulence is equal to the third power of a number which reflects the number of large-scale eddies multiplied by a function of Reynolds number. This function of Reynolds number may tend to a constant in the limit of very high Reynolds number as a result of an eventual balance between a slow growth of the range of viscous length-scales and the increasing non-Gaussianity of the small scales. However, when the turbulence is generated by fractal grids this function of Reynolds number is inversely proportional to the Reynolds number for a very wide range of Taylor length-based Reynolds number up to about 1000 even though the turbulence energy spectrum has a well-defined -5/3 range. See Mazellier, N. & Vassilicos, J.C. 2008 The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology. Phys. Fluids 20, 015101 Seoud, R.E. & Vassilicos, J.C. 2007 Dissipation and decay of fractal-generated turbulence. Phys. Fluid 19, 105108

14:30-15:00 Oberlack, M (Fachgebiet für Strömungsdynamik)
  Scaling law of fractal-generated turbulence and its derivation from a new scaling group of the multi-point correlation equation Sem 1

Investigating the multi-point correlation equations for the velocity and pressure fluctuations in the limit of homogeneous turbulence a new scaling symmetry has been discovered. Interesting enought this property is not shared with the Euler or Navier-Stokes equations from which the multi-point correlation equations have orginally emerged. This was first observed for parallel wall-bounded shear flows (see Khujadze, Oberlack 1994, TCFD (18)) though there this property only holds true for the two-point equation. Hence, in a strict sense there it is broken for higher order correlation equations. Presently using this extended set of symmetry groups a much wider class of invariant solutions or turbulent scaling laws is derived for homogeneous turbulence. In particular, we show that the experimentally observed specific scaling properties of fractal-generated turbulence (see Vassilicos etal.) fall into this new class of solutions. This is in particular a constant integral and Taylor length scale downstream of the fractal grid and the exponential decay of the turbulent kinetic energy along the same axis. These particular properties can only be conceived from multi-point equations using the new scaling symmetry since the two classical scaling groups of space and time are broken for this specific case. Hence, extended statistical scaling properties going beyond the Euler and Navier-Stokes have been clearly observed in experiments for the first time.

15:00-15:30 Tea and Posters
Chair: S Chen
15:30-16:00 Sreenivasan, KR (ICTP)
  How deep should one go to get the inertial range right? Sem 1

Theoretical and empirical arguments will be presented to show that the grid resolution in direct numerical simulations ought to be finer than previously thought. Error estimates for poorer resolutions will be presented. By going deep in the dissipation region, it may be possible to recover inertial range properties at finite Reynolds numbers.

16:00-16:30 Tsinober, A (Imperial College London)
  Kolmogorov 4/5 law, nonlocality and sweeping decorrelation hypothesis Sem 1

Results of experiments at high Reynolds numbers - both field an airborne - are used to validate an equivalent form of the 4/5 Kolmogorov, which demonstrate one of important aspects of non-locality of turbulent flows in the inertial range and stand in contradiction with the sweeping decorrelation hypothesis understood as statistical independence between large and small scales. This is supported by a set of exact purely kinematic relations also validated experimentally.

16:30-17:00 Tatsumi, T (Kyoto)
  Statistical mechanics of fluid turbulence based on the cross-independence closure hypothesis Sem 1

Statistical theory of turbulence is presented, which deals with homogeneous isotropic turbulence and inhomogeneous turbulent flows, their large-scale structures and small-scale similarities on an equal footing, using the "cross-independence closure hypothesis" proposed by Tatsumi(2001) for closing the Lundgren-Monin equations(1967) for the multi-point velocity distributions. Homogeneous isotropic turbulence at large Reynolds numbers is shown to be governed by the closed set of the one- and two-point velocity distributions. The distributions are expressed as the universal inertial-normal distributions associated with their own energy-dissipation rates as only parameters. Only exception from this universality is the longitudinal velocity-difference distribution, which is given by the local non-normal distributions in the inertial and viscous subranges. These theoretical results are discussed in comparison with the existing experimental and numerical results. Inhomogeneous turbulent flows at large Reynolds numbers are shown to be governed by the closed set of equations for the mean velocity and the one- and two-point velocity distributions. These equations have eminent feature that the effect of the mean flow is limited to the lage-scale components of turbulence. This feature is expected to largely simplify the formalism of shear-flow turbulence just like the 'boundary layer' in laminar flows.

Chair: M Oberlack
17:10-17:40 Cambon, C (Ecole Centrale de Lyon)
  Modelling two-time velocity correlations for prediction of both Lagrangian and Eulerian statistics Sem 1

More information on two-point two-time velocity correlations are needed for a better prediction of turbulent dispersion as well as radiated noise using an acoustic analogy. Conceptual aspects will be emphasized and not applications. Only isotropic turbulence will be considered, although many applications are developped in our team towards strongly anisotropic turbulence, mainly in rotating, stably stratified and/or MHD flows.

A simple synthetic model of isotropic turbulence is firstly considered, using a random superposition of Fourier modes : This is the KS (Kinematic simulation) following Kraichnan and Fung et al. Unsteadiness of velocity field realizations is mimicked using temporal frequencies, which are expressed in term of a prescribed energy spectrum and the wavenumber. Even if the orientation of the wavevector is randomly chosen, the link of the temporal frequency to the wavenumber is deterministic in the simpler version of the KS model. Although such a model was relevant for several applications, it is dramatically questioned for the evaluation of two-time velocity correlations. It is shown that spurious oscillations are generated, and that it is needed to model the temporal frequencies as random Gaussian variables with a standard deviation of the same order of magnitude as their mean value. Further applications to noise radiation are touched upon, in order to illustrate dominant (Lagrangian) `straining' or dominant (Eulerian) `sweeping' effects, according to the scale under consideration.

The role of a typical time-scale for the decorrelation of triple velocity correlations is then recalled and discussed in the classical `triadic closures' from the Orszag and Kraichnan's legacy, such as EDQNM, DIA and semi-Lagrangian more sophisticated variants.

Finally, these different concepts (diffusive and/or dispersive eddy dampings, straining or sweeping processes) are applied to a recent closure theory of weakly compressible isotropic turbulence. A Gaussian kernel for the decorrelation of triple velocity correlations was shown to give much better results than the classical exponential kernel inherited from EDQNM in the incompressible case. A new explanation is given in accordance with the renormalization of the acoustic wave frequency by a pure random term with zero mean but with a standard deviation of the same order of the eddy damping term formerly used in EDQNM. This analysis can be related to the concept of Kraichnan's random oscillator, recently revisited by Kaneda (2007), with a connection to the much simpler KS problem firstly presented (see also the monograph `homogeneous turbulence dynamics' by Pierre Sagaut and Claude Cambon, just published in Cambridge University Press.)

19:30-23:00 Conference Dinner at St Catharine's College
Thursday 2 October
Chair: JF Pinton
09:00-09:30 Biferale, L (Tor Vergata)
  Lagrangian velocity statistics in turbulence: theory, experiments and numerics Sem 1

A detailed comparison between experimental and numerical data of Lagrangian velocity structure functions in turbulent flows is presented. Thanks to the integration of information coming from experimental and numerical data, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers can be achieved. Intermittency changes if measured close to the Kolmogorov time scales or at larger time lags. A quantitative comparison with prediction from multifractal theory for Lagrangian turbulence will also be presented. These results shed some new insight on the relevance of vortex filaments for the statistics of tracers and/or heavy/light particles in turbulence.

09:30-10:00 Bodenschatz, E (MPIDS)
  Experimental results on the dynamics of tracers and inertial particles in highly turbulent flows Sem 1

We report measurements on the statistics of two particle dispersion, acceleration, and velocity structure functions for tracers and inertial particles. We will especially discuss large, neutrally buoyant and small, heavy particles. The experiments are conducted in the center of von Karmann mixing flows at high Reynolds numbers using direct Lagrangian particle tracking. We will show that single time, single particle statistics are not sensitive to particle inertial for both particles; however, we observed an inertial effect on the two-time, or the two-particle statistics. We will also show that the preferential concentration (increase of the radial distribution function) measured in the center of the apparatus is caused by a decrease of the average particle number density as a function of the distance from the center of the apparatus, which is the statistical stationary point of the flow field. The work has been conducted with Mathieu Gibert and Haitao Xu.

10:00-10:30 Doering, CR
  Statistically stationary stirring of a scalar sustained by steady sources and sinks Sem 1

Stirring generally enhances mixing, aiding molecular diffusivity by amplifying scalar gradients. Scalars sustained by steady sources and sinks, however, may best be mixed by flows with optimal transport properties. In this talk we describe differences between transient and steady state mixing and discuss implications for the concept of effective (eddy) diffusion.

10:30-11:00 Eyink, GL (Johns Hopkins)
  Turbulent Lagrangian dynamics of vortex and magnetic-field line Sem 1

We do not understand the laws of motion of vortex and magnetic-field lines in high-Reynolds-number turbulent flows. The current lore is self-contradictory. On the one hand, vortex/magnetic-field lines are often assumed to wander and elongate nearly as material lines in the limit of small viscosity/resistivity, and thus also to intensify, as a consequence of the Kelvin/Alfvén theorems. On the other hand, the topology of the lines is assumed to be continuously altered by viscous/resistive reconnection, implying breakdown of those same theorems. We discuss experimental and numerical evidence that these laws are both sometimes observed and sometimes violated in high-Reynolds-number turbulence. Unfortunately, we have no rational criterion to say when the Kelvin/Alfvén theorems or the Helmholtz laws of ``frozen-in'' motion should hold and when they should not. The problem has grown more perplexing with the theoretical discovery of "spontaneous stochasticity" for Lagrangian particle evolution in a Kolmogorov inertial range. As a consequence of the forgetting of initial separations in Richardson pair-diffusion, Lagrangian trajectories are not unique and must be replaced with random distributions of trajectories in the limit of small viscosity. This result presents a major crisis to our understanding of the turbulent dynamics of vortex/magnetic-field lines. As a possible resolution, we discuss a conjectured generalization of the Kelvin/Alfvén theorems, namely, that they survive as "backward martingales" of the spontaneous stochastic flows at high Reynolds-number. This conjectured relation provides a precise mathematical framework for the theory of turbulent reconnection. We discuss current rigorous results related to the conjecture and also important questions for investigation by experiment and simulation.

11:00-11:30 Coffee and Posters
Chair: L Biferale
11:30-12:00 Gotoh, T (Nagoya Institute of Technology)
  Intermittency and scaling of passive scalar convected by isotropic steady turbulence under the uniform mean scalar gradient Sem 1

It has been more convincing that passive scalar in turbulence is more intermittent than the turbulent velocity field itself, implying that the small scales of the passive scalar are more affected by the large scale conditions. In order to get more precise knowledge about the scaling behavior of the passive scalar for various Reynolds (Peclet) numbers and large scale conditions, we have performed very high resolution direct numerical simulations (DNSs) of the passive scalar turbulence with or without uniform mean scalar gradient up to $2048^3$ grid points and $R_\lambda\approx 600$, and analysed the various statistical functions. Turbulent velocity field was statistically in a steady and isotropic state by Gaussian random force applied at large scales. Fundamental statistics such as the spectra of the kinentic energy, pressure, scalar variance, scalar-velocity flux were examined, especially in their scaling behavior. It is found that although curves of the kinetic energy and scalar spectra are well collapsed onto a single curve when the Kolmogorov variables are used, while the others are not as well as the former, suggesting need of more elaborated scaling. The scaling of the velocity structure functions is consistent with the existing data of experiments and DNSs, while the scaling of the passive scalar is not convincing and difficult to reach definite conclusion. When the isotropic random injection for the passive scalar is applied at large scales (Case R), each curve of the local scaling exponent at a given order has one local minimum and maximum point, unlike the velocity case, and plateau is not wide enough to precisely determine the scaling exponents. On the other hand, when the uniform mean scalar gradient is applied (Case G), the curves of the local scaling exponents of the isotropic sector are found to have well developed plateau, and their plateau levels are smaller than those of Case R, meaning stronger intermittency for the Case of G. Crossover of the velocity and scalar structure functions is also examined. The crossover of the transverse velocity structure functions is found to be very similar to that of the passive scalar. We seek the reason for the above differences and similarities.

12:00-12:30 Yeung, PK (Georgia Institute of Technology)
  Local flow structure and Reynolds number dependence of Lagrangian statistics in direct numerical simulations of homogeneous turbulence Sem 1

Reynolds number dependence including the effects of intermittency is a crucial issue in the study of Lagrangian statistics and in how information from direct numerical simulations can be useful for stochastic modeling. Intermittency in the form of localized regions of intense straining or rotation is, for example, expected to influence how rapidly a fluid particle undergoes acceleration, and how multiple diffusing fluid particles move apart from each other. An effective approach to delineate such effects is to compute Lagrangian statistics conditioned on energy dissipation rate, enstrophy, or pseudo-dissipation following fluid particle trajectories. In this talk we shall illustrate these issues via recent results from simulations of isotropic turbulence at Reynolds numbers sufficiently high for observing inertial range behavior in the Eulerian (but not necessarily Lagrangian) frame. We also discuss research directions in the near future, including flows of greater complexity, and the promise of simulations at ever-increasing grid resolution that rapid advances in computing power are expected to make feasible.

12:30-13:30 Lunch at Wolfson Court
Chair: E Bodenschatz
14:00-14:30 Sawford, B (Monash)
  Relative dispersion and Richardson’s constant Sem 1

This talk will describe some very recent analysis of Direct Numerical Simulation results for turbulent relative dispersion over a wide range of Reynolds numbers. We will start with some background discussion of the nature and significance of relative dispersion and of the role of Kolmogorov’s similarity theory, leading to the introduction of Richardson’s constant as a fundamental parameter of relative dispersion. Although it is of great fundamental and practical significance, Richardson’s constant has not been well-quantified, and model estimates for it range from 0.01 to 4. We will describe first a traditional analysis of relative dispersion data, concluding that this approach does not yield a good estimate for Richardson’s constant even at the highest Reynolds number currently available. We then use a modified version of a new approach developed by Ott & Mann (JFM, 422, 207, (2000)) to show that a well-defined Richardson scaling range exists in our data. We estimate Richardson’s constant over a range of Reynolds numbers showing that it decreases weakly with Reynolds number to an asymptotic value at large Reynolds number of 0.55 - 0.57.

14:30-15:00 Pinton, JF (ENS-Lyon)
  Dynamics of inertial particles in fully developed turbulence Sem 1

We focus on acceleration of inertial particles, using experimatal measurements and numerical studies. In the experiment, particles are optically tracked in a turbulent flow of water using an Extended Laser Doppler Velocimetry technique. The probability density functions (PDF) of particle accelerations and their auto-correlation in time are computed. Numerical results are obtained from a direct numerical simulation in which a suspension of passive pointwise particles is tracked, with the same finite density and the same response time as in the experiment. We observe that many effects cannot be accounted for by point-particle models. We show that a much better description is achieved when one includes Faxen corrections in the particle's equation of evolution.

15:00-15:30 Tea and Posters
Chair: PK Yeung
15:30-16:00 Youngs, D (AWE)
  Mixing due to Rayleigh-Taylor instability Sem 1

Rayleigh-Taylor instability occurs when a dense fluid rests on top of a light fluid in a gravitational field. It also occurs in an equivalent situation, in the absence gravity, where there is a pressure gradient normal to a interface between fluids of different density such that the direction of acceleration is from the light to the heavy fluid. This situation occurs in Inertial Confinement Fusion Implosions (ICF), see for exapmle Amemdt et al [1].

There have been a number of successful experiments on mixing due to Rayleigh-Taylor instability, for example Dimonte [2] and Dalziel [3]. However, it is impractical to perform the "perfect" experiment and experimental diagnostics are necessarily limited. High-resolution Large Eddy Simulation (LES) can now be used to greatly add to our understanding of the mixing processes and this is the subject of the talk. The numerical technique used,the TURMOIL code, was first used for Rayleigh-Taylor mixing by Youngs[4]. A MILES approach is used because of the need to treat discontinuities in the flow e.g. the initilal density discontinuity and shock waves (in some applications). The high Reynolds case is of most interest where it is assumed that the bulk properties of the turbulent zone are independent of the Reynolds number. It is argued that LES (rather that DNS) is then an appropriate technique. Mesh convergence, or near-mesh convergence, will be demonstrated for key statistical averages.

Results are discussed for a range of situations-(a) Rayleigh-Taylor mixing at a plane boundary, (b) three-layer Raleigh-Taylor mixing and (c) mixing in a spherical implosion (a simplified version of an ICF implosion).The three cases are illustrated in figs 1,2 and 3 in the attached file. Two main aspects of the mixing process will be discussed. Firstly the influence of initial conditions. It is argued that loss of memory of initial conditions is unlikely to occur in experimental situations. The initial conditions have a significant effect on the overall width of the mixing zone - an important issue for engineering models. It would very difficult to obtain corresponding results experimentally because of the lack of control and the difficulty in measuring initial conditions. Secondly, the internal structure of the turbulent mixing zone will also be discussed, in particular the dissipation of both turbulence kinetic energy and of density fluctuations. For the internal structure results are more universal and less dependent on the initial conditions.

1. P. Amendt et al., “Indirect-drive noncryogenic double-shell ignition targets for the National Ignition Facility: Design and analysis”, Physics of Plasmas, 9, p2221, (2002). 2. G Dimonte & M Schneider, “Density ratio dependence of Rayleigh-Taylor mixing for sustained and impulsive acceleration histories”, Physics of Fluids, 12, p304 (2000) 3. S.B.Dalziel, "Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability", J. Fluid Mech. 399, p1,

16:00-16:30 Schneider, K (Universite de Provence, Marseille)
  Lagrangian acceleration in confined 2d turbulent flow Sem 1

A Lagrangian study of two-dimensional turbulence for two different geometries, a periodic and a confined circular geometry, is presented to investigate the influence of solid boundaries on the Lagrangian dynamics. It is found that the Lagrangian acceleration is even more intermittent in the confined domain than in the periodic domain. The flatness of the Lagrangian acceleration as a function of the radius shows that the influence of the wall on the Lagrangian dynamics becomes negligible in the center of the domain and it also reveals that the wall is responsible for the increased intermittency. The transition in the Lagrangian statistics between this region, not directly influenced by the walls, and a critical radius which defines a Lagrangian boundary layer, is shown to be very sharp with a sudden increase of the acceleration flatness from about 5 to about 20. Related Links * - PDF file of a preprint to appear in PRL

16:30-17:00 Casciola, CM (Sapienza Università di Roma)
  Clustering of inertial particles in shear flows Sem 1

Recently, clustering of inertial particles in turbulence has been thoroughly analyzed for statistically homogeneous and isotropic flows. The most striking result concerns the singular behavior exhibited by the radial distribution function under proper resonance conditions, showing clustering below the Kolmogorov scale. Since anisotropy is strongly depleted through the inertial range, the advecting field anisotropy may be expected in-influential for the small scale features of particle configurations. By addressing direct numerical simulations (DNS) of a statistically steady particle-laden homogeneous shear flow, we find instead that the small scales of the particle distribution are strongly affected by the geometry of velocity fluctuations at large scales. The proper statistical tool is the angular distribution function of particle pairs (ADF). Its anisotropic component may develop a singularity whose strength quantifies the anisotropy of the small scale clustering. The data provide evidence that the process is essentially anisotropic, even in the range of scales where isotropization of velocity statistics already occurred. Possible implications and connections of the above findings for turbophoresis in wall bounded shear flows will be briefly outlined using DNS data of particle laden turbulent pipe flows as example.

Chair: B Sawford
17:10-17:40 Thomson, D (Met Office)
  The behaviour of particle pairs in kinematic simulations Sem 1

The way pairs of particles separate is an important aspect of turbulent mixing which has often been explored using the technique of kinematic simulation. However kinematic simulation is not like real turbulnce in that the Fourier modes are independent and the smaller eddies are not advected (or 'swept') by the large eddies. Our aim here is to explore this aspect of kinematic simulation both theoretically and numerically. The fact that the small eddies are not swept by the large eddies, but the particles in the flow are so swept, means that particle pairs are swept through the smaller eddies by the large eddies. This is expected to alter the time scale on which the relative velocity of the particles fluctuates. A simple argument then shows that the mean square separation of pairs is expected to grow, not as t cubed as expected following Richardson, but as t to the sixth power. This is confirmed in numerical simulations where we add a mean flow to the kinematic flow field to exaggerate the problem caused by lack of sweeping (with the eddies not being advected by the mean flow). Without the mean flow the situation is more complex with a significant contribution to the separation process from locations where the velocity is small and where there is no sweeping issue. This leads to a separation growing like t to the power 9/2. The time dependence of the kinematic flow field can also lead to a wider range of behaviours. The work described here is not especially new (we published the main idea in 2005) but it remains controversial and we hope the talk will generate some discussion of the ideas involved.

17:40-18:10 Rubinstein, R (NASA Langley)
  Closure theories for inhomogeneous turbulence Sem 1

Although Kraichnan formulated the Direct Interaction Approximation and the Test-Field model for general problems of inhomogeneous turbulence, the resulting equations, requiring repeated multiple integrations over the flow domain, are both difficult to understand and difficult to apply in practice; in the homogeneous case, triad interactions provide the key to unraveling the physics of the approximation. The goal of this work is to formulate the closure theory of some special inhomogeneous problems with comparable simplicity. It is done by decomposing the inhomogeneous problem into a set of coupled quasi-homogeneous problems, each of which admits a simple formulation. The formalism will be applied to the problem of weakly inhomogeneous turbulence, where previous heuristic theories have been incomplete. The same formalism applies to problems admitting scaling transformations; it will be applied to give a simple formulation of the problem of turbulence in a half-space.

18:45-19:30 Dinner at Wolfson Court (Residents Only)
Friday 3 October
Chair: JM Chomaz
09:40-10:10 Riley, JJ (Washington)
  Stratified turbulence: a possible interpretation of some geophysical turbulence measurements Sem 1

Several existing sets of smaller-scale ocean and atmospheric data appear to display Kolmogorov-Obukov-Corrsin inertial ranges in horizontal spectra for length scales up to at least a few hundred meters. It is argued here that these data are inconsistent with the assumptions for these inertial range theories. Instead, it is hypothesized that the dynamics of stratified turbulence explain these data. If valid, these dynamics may also explain the behavior of strongly stratified flows in similar dynamic ranges of other geophysical flows.

10:10-10:40 Lindborg, E (KTH)
  Vertical dispersion by stratified turbulence Sem 1

An analytical relation is derived for the growth of the vertical mean square displacement of fluid particles in stratified turbulence. A number of numerical simulations are carried out to test the analytical relation. The comparison shows good agreement between the analytical and the numerical results.

10:40-11:00 Brethouwer, G (KTH)
  Divergent-rotational modes and passive scalars in stratified turbulence Sem 1

Strongly stratified turbulent flows are anisotropic but have three-dimensional dynamics with a forward energy cascade as shown by Lindborg (2006). Simulations with hyperviscosity also revealed an inertial range with a horizontal k^-5/3-spectrum. We have continued this work and examined the features of divergent and rotational modes, and of passive scalars in the inertial range of stratified turbulence.

The Helmholtz decomposition of the velocity into a rotational and divergent part have been used to study vortices and internal waves in stratified flows. In flows with mainly vertically oriented vortices the rotational part dominates while the divergent part dominates when there are mainly internal waves. The timescale ratio of the 'fast' waves and the 'slow' horizontal vortical motions can be estimated as the vertical Froude number F_v = u/Nl_v, where u is a horizontal velocity scale, l_v a verti cal length scale, N the Brunt-Vaisala frequency. It is often assumed that F_v = 0 in strongly stratified flows suggesting separate time scales of vortices and waves, and consequently weak interactions. However, scaling analysis suggests l_v = u/N in stratified turbulence, i.e. F_v = 1, which was supported by DNS (Brethouwer et al. 2007). In stratified turbulence the divergent and rotational modes may thus have similar timescales leading to strong nonlinear interactions between rotational and dive rgent modes. This implies that forcing in either divergent or rotational modes may have minor effects on the inertial range dynamics of stratified turbulence. Results of simulations with different stratification and resolution show that the energy of divergent and rotational modes hav e the same magnitude in the inertial range when large-scale rotational modes are forced, see Lindborg & Brethouwer (2007). In the simulations with forcing of divergent modes, small horizontal wave numbers and one vertical wave number are forced which introduces a vertical length scale. Results show that the inertial range spectra are very similar in simulations with forcing of rotational modes and divergent modes, suggesting similar dynamics, if in the latter simulations the large-scale dynamics o beys F_v = 1, but deviations are found when this condition is not fulfilled. The next subject is a passive scalar in stratified turbulence. The Obukhov-Corrsin theory for the one-dimensional spectrum of the variance of a passive scalar in the Kolmogorov inertial range of turbulence predicts a k^-5/3 slope. The second-order structure function according to the same theory has a r^2/3 power-law range. Measurements of horizontal spectra and structure functions of passive scalars (e.g. ozone) in the mesoscale range of the middle atmosphere are consistent with this theory. This is remarkable because the mesoscale range is strongly stratified and does not resemble Kolmogorov turbulence. We have carried out numerical simulations of stratified turbulence with large-scale forcing, hyperviscosity

11:00-11:30 Coffee
Chair: J Riley
11:30-12:00 Kimura, Y (Nagoya)
  Transition in energy spectrum for forced stratified turbulence Sem 1

Energy spectrum for forced stably stratified turbulence is investigated numerically by solving the 3D Navier-Stokes equations under the Boussinesq approximation with stochastic forcing applied to the largest velocity scales. Using pseudo-spectral simulations with 1024^3 grid points, we could verify the transition in the vortex (horizontal) spectrum (as a function of horizontal wave number) from $k_{\perp}^{-3}$ to $k_{\perp}^{-5/3}$. Meanwhile the wave spectra shows $k_{\perp}^{-2}$ for the large scales, and $k_{\perp}^{-5/3}$ for the small scales. According to Carnevale {\it et.~al.}, the transition wave number is understood as the Ozmidov scale with a correction by the coefficients of the buoyancy spectrum, $E(k) =\alpha N^2k^{-3}$, and the Kolmogorov spectrum, $E(k)=C_K\epsilon^{2/3} k^{-5/3}$. By equating these spectra, $k_b \sim (\alpha/C_K)^{3/4}\sqrt {N^3/ \epsilon}$ is obtained for the transition wavenumber. Our calculation shows, however, that the vortex spectra at large scales seem to have the same slope irrespective of stratification, which implies a possibility of a different mechanism for producing the $k_{\perp}^{-3}$ spectrum. We will discuss possibility that the spectrum corresponds to two-dimensional turbulence. Referece: Carnevale,G.F. {\it et.~al}: 2001 J.~Fluid Mech. {\bf 427} 205--239.

12:00-12:30 Lohse, D (Twente)
  Non-Oberbeck-Boussinesq effects in Rayleigh-Benard convection Sem 1

The problem of Rayleigh-Benard convection is commonly analyzed within the so-called Oberbeck-Boussinesq (OB) approximation, in which the fluid properties are assumed to be temperature independent, apart from the density for which a linear temperature dependence is assumed. Under normal conditions, i.e., small temperature differences between the bottom and top plate, this approximation is rather good. However, in order to achieve ever larger Rayleigh numbers for given cell height and fluid properties the temperature difference is quite frequently increased to such an extent that the OB approximation has to be expected to fail. Non-Oberbeck-Boussinesq (NOB) effects on the mean center cell temperature, the Nusselt number Nu, and the Reynolds number Re then have to be expected at the largest Rayleigh numbers. We report on our recent experimental, theoretical, and numerical results on these NOB corrections. For water and glycerol they are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient: With increasing NOBness, for water and glycerol Nu goes down and the center temperature goes up, whereas for ethane gas in general Nu goes up and the center temperature goes down. However, for ethane close to the critical point the main origin of NOB corrections lies in the strong temperature dependence of the isobaric thermal expansion coefficient, namely in the nonlinear temperature dependence of the density, leading to NOB corrections which presently cannot be described by our extended Prandtl-Blasius boundary layer theory. Related Links * - Web page Phyiscs of Fluids group Twente

12:30-13:30 Lunch at Wolfson Court
Chair: C Cambon
14:00-14:30 Zhou, Y (LLNL)
  Scaling criteria for high Reynolds and Peclet number turbulent flow, scalar transport, mixing, and heat transfer Sem 1

Very high Reynolds (Re) and Peclet (Pe) number turbulent flows are commonly encountered in engineering, geophysical and astrophysical applications. In comprehensive statistical flow experiments or corresponding direct numerical simulations of high Re and Pe number turbulent flow, scalar transport, mixing, and heat transfer the energetic excitation influences of the entire range of dynamic spatial scales combining both velocity fluctuations and passive scalar variances must be considered together. However, direct computational simulations or experiments directed to the very high Re and Pe flows of practical interest commonly exceed the resolution possible using current or even foreseeable future super computer capability or spatial, temporal and diagnostic technique limitations of current laboratory facilities. Pragmatic considerations and practical needs promote use of statistical flow data bases developed from direct numerical simulations or experiments at the highest Re and Pe levels achievable within the currently available facility limitations. Unfortunately the obtainable levels are lower than those associated with the flows of practical interest. Moreover, at present, there is no metric to indicate whether and how much of the fully resolved physics of the flow of interest has been captured within the facilities available to the investigator. This talk presents metric criteria based on establishing a smaller subset of the total range of dynamic scale interactions that will still faithfully reproduce all of the essential, theoretically significant, influences of the complete range of scale interactions associated with the flows of practical interest. The present work leads to the identification of the minimum significant Re flow and Pe field that a researcher must attain in direct simulation or experiment (hereafter called the minimum state). These threshold criteria levels are minimum values to be attained in experiments or direct simulations which assure that the energy-containing scales of the flows ? and scalar fields under investigation are not contaminated by the (non-universal) velocity dissipation and scalar diffusivity inertial range scale limits.

14:30-15:00 Matsumoto, T (Kyoto)
  Numerical study of 3D Rayleigh-Taylor turbulence Sem 1

The Rayleigh-Taylor turbulence in 3D space is numerically studied via the Boussinesq approximation along the same line as the 2D numerical study by Celani et al. (2006 Phys.Rev.Lett. 96 134504). A comparison with the phenomenology proposed by Chertkov (2003 Phys.Rev.Lett. 91 115001), in particular deviation from the phenomenology (intermittency), will be presented.

15:00-15:30 Tea
Chair: E Lindborg
15:30-16:00 Benzi, R (U. Roma "Tor Vergata")
  Some remarks on the dissipative properties of homogenous and isotropic turbulence Sem 1

Dissipation of kinetic energy plays a key role in understanding the statistical properties of turbulence. In this talk, we shall review some recent results obtained by performing high resolution numerical simulations of homogenous and isotropic turbulence. In particular, we present an exhaustive investigation of the statistics of velocity gradients along the trajectories of neutral tracers and of heavy/light particles advected by an homogeneous and isotropic turbulent flow. We propose a Lagrangian rephrasing of the Refined Kolmogorov Similarity Hypothesis (RKSH) and test its validity along the particle trajectories. We also show that for homogenous and isotropic compressible turbulence, there is no statistical differences in the statistical properties of inertial range intermittency due either to the slight compressibility or to the different dissipative mechanism.

16:00-16:30 Barenghi, CF (Newcastle)
  Reconnection of superfluid vortex bundles Sem 1

Using the vortex filament model and the Gross Pitaevskii nonlinear Schroedinger equation, we show that bundles of quantised vortex lines in helium~II are structurally robust and can reconnect with each other maintaining their identity. We discuss vortex stretching in superfluid turbulence and show that, during the bundle reconnection process, a large amount of Kelvin waves is generated, in agreement with the finding that helicity is produced by nearly singular vortex interactions in classical Euler flows.

16:30-16:35 Closing Remarks
18:45-19:30 Dinner at Wolfson Court (Residents Only)
Sunday 26 October
Chair: M Oberlack
17:40-18:10 McComb, D (Edinburgh)
  Scale-invariance in three-dimensional isotropic turbulence Sem 1

We present a critical review of the Kolmogorov (1941) theory of isotropic turbulence, with particular reference to the `2/3' power-law for the second-order structure function (and the corresponding `-5/3' law for the energy spectrum). We begin by noting that the recent resolution of an associated paradox allows the inertial range to be defined in terms of the scale-invariance of the energy flux (David McComb, J. Phys. A: Math. Theor. 41, 075501 (2008)), thus permitting the Kolmogorov arguments to be presented independently of concepts such as localness which are themselves counter-intuitive when interpreted in terms of vortex-stretching. If this approach is pursued further, then a simple phenomenological analysis shows that we can regard turbulence as a statistical field theory possessing one nontrivial fixed point (corresponding to the top of the inertial range) and two trivial fixed points at the origin and infinity, respectively, in wavenumber space. Using this framework, various schools of criticism, ranging from the original criticism by Landau (1959), through `intermittency corrections' to present-day analogies with the theory of critical phenomena, with the introduction of `anomalous exponents', are analysed. In particular, we examine the conflict between the recent work of Lundgren (2002), which uses mathematical arguments to show that the `2/3' law must be asymptotically true in the limit of infinite Reynolds numbers; and that of Falcovich which uses mathematical arguments to show that the `2/3' law is incompatible with the observed values for higher-order moments. We conclude by attempting to put forward a picture in which various long-standing contentious issues may be seen as either resolved or at least potentially resolvable.


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