Enstrophy Production Research Articles

High-order WENO simulations of three-dimensional reshocked Richtmyer–Meshkov instability to late times: dynamics, dependence on initial conditions, and comparisons to experimental data

The dynamics of the reshocked multi-mode Richtmyer–Meshkov instability is investigated using 513 × 2572 three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations. A two-mode initial perturbation with superposed random noise is used to model the Mach 1:5 air/SF6 Vetter–Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces, and density cross-sections are utilized to show the detailed ow structure before, during, and after reshock. It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock. The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model. A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented. Finally, the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra, as well as the evolution of the baroclinic enstrophy production, buoyancy production, and shear production terms in the enstrophy and turbulent kinetic transport equations.

Expanding theQ–Rspace to three dimensions

The two-dimensional space spanned by the velocity gradient invariantsQandRis expanded to three dimensions by the decomposition ofRinto its strain production −1/3sijsjkskiand enstrophy production 1/4ωiωjsijterms. The {Q;R} space is a planar projection of the new three-dimensional representation. In the {Q; −sss; ωωs} space the Lagrangian evolution of the velocity gradient tensorAijis studied via conditional mean trajectories (CMTs) as introduced by Martínet al. (Phys. Fluids, vol. 10, 1998, p. 2012). From an analysis of a numerical data set for isotropic turbulence ofReλ~ 434, taken from the Johns Hopkins University (JHU) turbulence database, we observe a pronounced cyclic evolution that is almost perpendicular to theQ–Rplane. The relatively weak cyclic evolution in theQ–Rspace is thus only a projection of a much stronger cycle in the {Q; −sss; ωωs} space. Further, we find that the restricted Euler (RE) dynamics are primarily counteracted by the deviatoric non-local part of the pressure Hessian and not by the viscous term. The contribution of the Laplacian ofAij, on the other hand, seems the main responsible for intermittently alternating between low and high intensityAijstates.

Energy spectra power laws and structures

Direct numerical simulations (DNS) of two inviscid flows, the Taylor–Green flow and two orthogonal interacting Lamb dipoles, together with the DNS of forced isotropic turbulence, were performed to generate data for a comparative study. The isotropic turbulent field was considered after the transient and, in particular, when the velocity derivative skewness oscillates around −0.5. At this time, Rλ ≈ 257 and a one decade wide k−5/3 range was present in the energy spectrum. For the inviscid flows the fields were considered when a wide k−3 range was achieved. This power law spectral decay corresponds to infinite enstrophy and is considered one of the requirements to demonstrate that the Euler equations lead to a finite time singularity (FTS). Flow visualizations and statistics of the strain rate tensor and vorticity components in the principal axes of the strain rate tensor (λ, λ) were used to classify structures. The key role of the intermediate component 2 is demonstrated by its good correlation with enstrophy production. Filtering of the fields shows that the slope of the power law is directly connected to self-similar structures, whose radius of curvature is smaller the steeper the spectrum.

Entropy, Closures and Subgrid Modeling

Maximum entropy states or statistical mechanical equilibrium solutions have played an important role in the development of a fundamental understanding of turbulence and its role in geophysical flows. In modern general circulation models of the earth’s atmosphere and oceans most parameterizations of the subgrid-scale energy and enstrophy transfers are based on ad hoc methods or ideas developed from equilibrium statistical mechanics or entropy production hypotheses. In this paper we review recent developments in nonequilibrium statistical dynamical closure theory, its application to subgrid-scale modeling of eddy-eddy, eddy-mean field and eddy-topographic interactions and the relationship to minimum enstrophy, maximum entropy and entropy production arguments.

Modeling the pressure Hessian and viscous Laplacian in turbulence: Comparisons with direct numerical simulation and implications on velocity gradient dynamics

Modeling the velocity gradient tensor A=∇u along Lagrangian trajectories in turbulent flow requires closures for the pressure Hessian and viscous Laplacian of A. Based on an Eulerian–Lagrangian change in variables and the so-called recent fluid deformation closure, such models were proposed recently [Chevillard and Meneveau, Phys. Rev. Lett. 97, 174501 (2006)]. The resulting stochastic model was shown to reproduce many geometric and anomalous scaling properties of turbulence. In this work, direct comparisons between model predictions and direct numerical simulation (DNS) data are presented. First, statistical properties of A are described using conditional averages of strain skewness, enstrophy production, energy transfer, and vorticity alignments, conditioned upon invariants of the velocity gradient. These conditionally averaged quantities are found to be described accurately by the stochastic model. More detailed comparisons that focus directly on the terms being modeled in the closures are also presented. Specifically, conditional statistics associated with the pressure Hessian and the viscous Laplacian are measured from the model and are compared with DNS. Good agreement is found in strain-dominated regions. However, some features of the pressure Hessian linked to rotation-dominated regions are not reproduced accurately by the model. Geometric properties such as vorticity alignment with respect to principal axes of the pressure Hessian are mostly predicted well. In particular, the model predicts that an eigenvector of the rate of strain will be also an eigenvector of the pressure Hessian, in accord with basic properties of the Euler equations. The analysis identifies under what conditions the Eulerian–Lagrangian change in variables with the recent fluid deformation closure works well, and in which flow regimes it requires further improvements.

A Lagrangian investigation of the small-scale features of turbulent entrainment through particle tracking and direct numerical simulation

We report an analysis of small-scale enstrophy ω2and rate of strains2dynamics in the proximity of the turbulent/non-turbulent interface in a flow without strong mean shear. The techniques used are three-dimensional particle tracking (3D-PTV), allowing the field of velocity derivatives to be measured and followed in a Lagrangian manner, and direct numerical simulations (DNS). In both experiment and simulation the Taylor-microscale Reynolds number isReλ= 50. The results are based on the Lagrangian viewpoint with the main focus on flow particle tracers crossing the turbulent/non-turbulent interface. This approach allowed a direct investigation of the key physical processes underlying the entrainment phenomenon and revealed the role of small-scale non-local, inviscid and viscous processes. We found that the entrainment mechanism is initiated by self-amplification ofs2through the combined effect of strain production and pressure--strain interaction. This process is followed by a sharp change of ω2induced mostly by production due to viscous effects. The influence of inviscid production is initially small but gradually increasing, whereas viscous production changes abruptly towards the destruction of ω2. Finally, shortly after the crossing of the turbulent/non-turbulent interface, production and dissipation of both enstrophy and strain reach a balance. The characteristic time scale of the described processes is the Kolmogorov time scale, τη. Locally, the characteristic velocity of the fluid relative to the turbulent/non-turbulent interface is the Kolmogorov velocity,uη.

Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer. Part 1. Facilities, methods and some general results

This is a report on a field experiment in an atmospheric surface layer at heights between 0.8 and 10m with the Taylor micro-scale Reynolds number in the rangeReλ = 1.6−6.6 ×103. Explicit information is obtained on the full set of velocity and temperature derivatives both spatial and temporal, i.e. no use of Taylor hypothesis is made. The report consists of three parts. Part 1 is devoted to the description of facilities, methods and some general results. Certain results are similar to those reported before and give us confidence in both old and new data, since this is the first repetition of this kind of experiment at better data quality. Other results were not obtained before, the typical example being the so-called tear-dropR-Qplot and several others. Part 2 concerns accelerations and related matters. Part 3 is devoted to issues concerning temperature, with the emphasis on joint statistics of temperature and velocity derivatives. The results obtained in this work are similar to those obtained in experiments in laboratory turbulent grid flow and in direct numerical simulations of Navier–Stokes equations at much smaller Reynolds numbersReλ ~ 102, and this similarity is not only qualitative, but to a large extent quantitative. This is true of such basic processes as enstrophy and strain production, geometrical statistics, the role of concentrated vorticity and strain, reduction of nonlinearity and non-local effects. The present experiments went far beyond the previous ones in two main respects. (i) All the data were obtained without invoking the Taylor hypothesis, and therefore a variety of results on fluid particle accelerations became possible. (ii) Simultaneous measurements of temperature and its gradients with the emphasis on joint statistics of temperature and velocity derivatives. These are reported in Parts 2 and 3.

Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability

The ninth-order weighted essentially nonoscillatory (WENO) shock-capturing method is used to investigate the physics of reshock and mixing in two-dimensional single-mode Richtmyer-Meshkov instability to late times. The initial conditions and computational domain were adapted from the Mach 1.21 air (acetone)/ SF6 shock tube experiment of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)]: the growth of the bubble and spike amplitudes from fifth- and ninth-order WENO simulations of this experiment were compared to the predictions of linear and nonlinear amplitude growth models, and were shown to be in very good agreement with the experimental data prior to reshock by Latini, Schilling, and Don [Phys. Fluids 19, 024104 (2007)]. In the present investigation, the density, vorticity, baroclinic vorticity production, and simulated density Schlieren fields are first presented to qualitatively describe the reshock process. The baroclinic circulation deposition on the interface is shown to agree with the predictions of the Samtaney-Zabusky model and with linear instability theory. The time evolution of the positive and negative circulation on the interface is considered before and after reshock: it is shown that the magnitudes of the circulations are equal before as well as after reshock, until the interaction of the reflected rarefaction with the layer induces flow symmetry breaking and different evolutions of the magnitude of the positive and negative circulation. The post-reshock mixing layer growth is shown to be in generally good agreement with three models predicting linear growth for a short time following reshock. Next, a comprehensive investigation of local and global mixing properties as a function of time is performed. The distribution and amount of mixed fluid along the shock propagation direction is characterized using averaged mole fraction profiles, a fast kinetic reaction model, and mixing fractions. The modal distribution of energy in the mixing layer is quantified using the spectra of the fluctuating kinetic energy, fluctuating enstrophy, pressure variance, density variance, and baroclinic vorticity production variance. It is shown that a broad range of scales already exists prior to reshock, indicating that the single-mode Richtmyer-Meshkov instability develops nontrivial spectral content from its inception. The comparison of the spectra to the predictions of classical inertial subrange scalings in two-dimensional turbulence shows that the post-reshock spectra may be consistent with many of these scalings over wave number ranges less than a decade. At reshock, fluctuations in all fields (except for the density) are amplified across all scales. Reshock strongly amplifies the circulation, profiles, and mixing fractions, as well as the energy spectra and statistics, leading to enhanced mixing followed by a decay. The mole and mixing fraction profiles become nearly self-similar at late times following reshock; the mixing fraction exhibits an approach toward unity across the layer at the latest time, signifying nearly complete mixing of the gases. To directly quantify the amplification of fluctuations by reshock, the previously considered quantities are compared immediately after and before reshock. Finally, to investigate the decay of fluctuations in the absence of additional waves interacting with the mixing layer following reshock, the boundary condition at the end of the computational domain is changed from reflecting to outflow to allow the reflected rarefaction wave to exit the domain. It is demonstrated that the reflected rarefaction has an important role in breaking symmetry and achieving late-time statistical isotropy of the velocity field.

An experimental investigation on Lagrangian correlations of small-scale turbulence at low Reynolds number

Lagrangian auto- and cross-correlation functions of the rate of strains2, enstrophy ω2, their respective production terms −sijsjkskiand ωiωjsij, and material derivatives, Ds2/Dtand Dω2/Dtare estimated using experimental results obtained through three-dimensional particle tracking velocimetry (three-dimensional-PTV) in homogeneous turbulence atReλ=50. The autocorrelation functions are used to estimate the Lagrangian time scales of different quantities, while the cross-correlation functions are used to clarify some aspects of the interaction mechanisms between vorticity ω and the rate of strain tensorsij, that are responsible for the statistically stationary, in the Eulerian sense, levels of enstrophy and rate of strain in homogeneous turbulent flow. Results show that at the Reynolds number of the experiment these quantities exhibit different time scales, varying from the relatively long time scale of ω2to the relatively shorter time scales ofs2, ωiωjsijand −sijsjkski. Cross-correlation functions suggest that the dynamics of enstrophy and strain, in this flow, is driven by a set of different-time-scale processes that depend on the local magnitudes ofs2and ω2. In particular, there are indications that, in a statistical sense, (i) strain production anticipates enstrophy production in low-strain–low-enstrophy regions (ii) strain production and enstrophy production display high correlation in high-strain–high-enstrophy regions, (iii) vorticity dampening in high-enstrophy regions is associated with weak correlations between −sijsjkskiands2and between −sijsjkskiand Ds2/Dt, in addition to a marked anti-correlation between ωiωjsijand Ds2/Dt. Vorticity dampening in high-enstrophy regions is thus related to the decay ofs2and its production term, −sijsjkski.

Topographically-induced enstrophy production/dissipation in coastal models

Vorticity and enstrophy production and dissipation are studied for both wave-averaged and wave-resolving (Boussinesq-type) models of wave-induced near shore circulation. Quadratic flow properties of fundamental importance for shallow-water turbulence, i.e., energy and enstrophy, whose sources/sinks are clearly identifiable by positive/negative-definite contributions in the appropriate transport equations, are taken as the most suitable indicators for assessing model performance in describing flows characterized by large-scale vortices. Two state-of-the-art models, SHORECIRC and FUNWAVE2D, have been evaluated in detail. Suitable transport equations for enstrophy are derived and analyzed to get a clear insight into the mechanisms of generation/dissipation of this quantity in both models. Analytical results show that steep gradients of the total flow depth act as sinks as well as sources for vorticity and entrophy, similar to the results of Brocchini and Colombini [M. Brocchini and M. Colombini, Phys. Fluids 16, 2469 (2004)]. Predictive estimates have been given for the rate of change of circulation for waves breaking over a bar or breakwater and the vorticity source and sink terms have been numerically analyzed. The comparison between numerical results obtained using the two different circulation models reveals that while wave-resolving computations give well-structured rip currents, the wave-averaged model predicts less organized flows, given the different structure of the circulation forcing terms. The analysis of equivalent enstrophy-forcing terms characterizing the two models shows that they are all proportional to depth gradients in the case of wave-resolving models while their intensity is mainly due to the gradients of the wave-induced velocity for wave-averaged models. Energetic considerations are also given in support of the proposed vorticity/enstrophy generation mechanisms. Wave-averaged computations clearly show that, apart from bottom friction, the most intense dissipation mechanism is due to classic viscous effects [−νT(∇ω̃)2] while depth gradients weakly contribute. Rather surprisingly this also occurs for the wave-resolving model.

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition \cite{Kerr93}[Phys. Fluids {\bf 5} (1993), 1725]. We use a pseudo-spectral method with resolution up to $1536\times 1024 \times 3072$ to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than double exponential in time, up to $t=19$, beyond the singularity time $t=18.7$ predicted by Kerr's computations \cite{Kerr93,Kerr04}. The velocity, the enstrophy and enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.

Dual-plane stereo particle image velocimetry measurements of velocity gradient tensor fields in turbulent shear flow. II. Experimental results

Results are presented from highly resolved dual-plane stereo particle image velocimetry (DSPIV) measurements for the structure, statistics, similarity, and scaling of all nine simultaneous components of the velocity gradient tensor fields ∂ui∕∂xj on the quasi-universal intermediate and small scales of turbulent shear flows. Measurements were obtained at three combinations of the outer-scale Reynolds number Reδ and the local mean shear rate S in the fully developed self-similar far field of a turbulent jet, and thus reflect the combined effects of the large-scale structure, spatial inhomogeneities, and anisotropies inherent in such a flow. Conditions addressed in this study correspond to local outer-scale Reynolds numbers Reδ=6,000 and 30,000 and local mean shear values Sδ∕uc=0 and 1.7, corresponding to Taylor-scale Reynolds numbers Reλ≈44 and 113 and shear rates Sk∕ε=0 and 2.1. Gradient fields investigated here include the individual velocity gradient component fields, the strain rate component fields and the associated principal strain rates, the vorticity component fields and their orientations with respect to the principal strain axes, the enstrophy and enstrophy production rate fields, and the true kinetic energy dissipation rate field. Results normalized on both inner- and outer-scale variables are presented to allow interpretation relative to the similarity and scaling implied by classical turbulence theory. For both Reδ values at S=0, results show that most aspects of these gradient fields are essentially in agreement with the predictions from homogeneous isotropic turbulence, while for S≠0 there are significant and consistent departures from isotropy. Results also provide direct measurements of the exponential scaling factors in the left and right tails of the velocity gradient distributions, as well as quantification of the inner (viscous) length scales in the enstrophy and dissipation rate fields. In addition, strong evidence for multifractal scale similarity at length scales greater than about twice the viscous length λν is found in both the enstrophy and dissipation rate fields.

Structural studies of decaying fluid turbulence: Effect of initial conditions

We present results from a systematic numerical study of structural properties of an unforced, incompressible, homogeneous, and isotropic three-dimensional turbulent fluid with an initial energy spectrum that develops a cascade of kinetic energy to large wave numbers. The results are compared with those from a recently studied set of power-law initial energy spectra [C. Kalelkar and R. Pandit, Phys. Rev. E 69, 046304 (2004)] which do not exhibit such a cascade. Differences are exhibited in plots of vorticity isosurfaces, the temporal evolution of the kinetic energy-dissipation rate, and the rates of production of the mean enstrophy along the principal axes of the strain-rate tensor. A crossover between "non-cascade-type" and "cascade-type" behavior is shown numerically for a specific set of initial energy spectra.

The Rayleigh–Taylor instability of two-dimensional high-density vortices

We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that high-density vortices are subject to a Rayleigh–Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex centre.

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.

Lagrangian measurement of vorticity dynamics in turbulent flow

The full set of velocity derivatives, -alignment of vorticity. Lagrangian measurements of enstrophy budget terms suggest that there is no pointwise balancing of production and viscous reduction of enstrophy and that the role played by viscosity is of great importance.

The effect of subgrid-scale models on the vortices computed from large-eddy simulations

Direct numerical and large-eddy simulations (DNS/LES) of temporal plane jets are carried out in order to analyze the effect of the subgrid-scale (SGS) models on the vortices obtained from LES. The dynamics of the filtered vorticity norm (or filtered enstrophy) is analyzed through the application of a box filter to temporal DNS of turbulent plane jets (Reλ≈100), using a methodology similar to da Silva and Métais [J. Fluid Mech. 473, 103 (2002)]. Special emphasis is placed on the enstrophy SGS dissipation term, which represents the effect of the SGS models on the vortices computed from LES. When the filter is placed in the inertial range region the evolution of the vorticity norm is governed by the enstrophy production and enstrophy SGS dissipation, which represents, in the mean, a sink of resolved enstrophy. Thus the coherent vortices obtained from LES are subjected to an additional (nonviscous) dissipation mechanism. Locally, however, the enstrophy SGS dissipation can be either a sink or a source of resolved vorticity (forward/backward enstrophy cascade), but the forward cascade dominates, in analogy with what happens with the resolved kinetic energy equation. A priori tests are conducted using several SGS models in order to analyze their ability to represent the enstrophy SGS dissipation. The models analyzed are the Smagorinsky, structure function, filtered structure function, dynamic Smagorinsky, gradient, scale similarity, and mixed. It turns out that in terms of spatial localization all the models lead to a good correlation between the “real” and “modeled” enstrophy SGS dissipation. Moreover, all the SGS models, even of eddy-viscosity type, are able to provide enstrophy SGS backscatter. However, in terms of statistical behavior the eddy-viscosity models do not provide enough enstrophy backscatter as the non-eddy-viscosity models. LES are carried out and show that the Smagorinsky, structure function, and mixed models cause excessive vorticity dissipation compared to the other models, and although the enstrophy SGS dissipation affects mainly the smallest resolved scales, it may affect also some low-wave numbers. An estimation of the “vorticity error” and its wave number dependence is given, for each SGS model. Both a priori tests and LES show that the dynamic Smagorinsky and filtered structure function models seem to be the best suited to a correct prediction of the resolved vorticity field.

A simple model for self-sustained oscillations in homogeneous shear flow

Generation of the large-scale coherent vortical structures in homogeneous shear flow couples dynamical processes of energy and enstrophy production. In the large rate of strain limit, the simple estimates of the contributions to the energy and enstrophy equations result in a dynamical system, describing experimentally and numerically observed self-sustained nonlinear oscillations of energy and enstrophy. It is shown that the period of these oscillations is independent of the box size and the energy and enstrophy fluctuations are strongly correlated.

Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow

The budgets of the Reynolds stress, turbulent kinetic energy and streamwise enstrophy are evaluated through direct numerical simulations for the turbulent channel flow of a viscoelastic polymer solution modeled with the Finitely Extensible Nonlinear Elastic with the Peterlin approximation (FENE-P) constitutive equation. The influence of viscoelasticity on the budgets is examined through a comparison of the Newtonian and the viscoelastic budgets obtained for the same constant pressure drop across the channel. It is observed that as the extensional viscosity of the polymer solution increases there is a consistent decrease in the production of Reynolds stress in all components, as well as in the other terms in the budgets. In particular, the effect of the flow elasticity, which is associated with the reduction in the intensity of the velocity-pressure gradient correlations, potentially leads to a redistribution of the turbulent kinetic energy among the streamwise, the wall-normal and the spanwise directions. In this work, we also show that in the presence of viscoelasticity there is a significant reduction in all components of the production of streamwise enstrophy. This is consistent with a proposed mechanism for polymer-induced drag reduction through the inhibition of vortex stretching by the high extensional viscosity of the polymer solution.

Velocity derivatives in the atmospheric surface layer at Reλ=104

We report first results of an experiment in which explicit information is obtained on the field of velocity derivatives (all the nine components of the tensor ∂ui/∂xj) along with all the three components of velocity fluctuations at Reynolds number as high as Reλ∼104. This includes information on such basic processes as enstrophy and strain production, geometrical statistics, the role of concentrated vorticity and strain, and the reduction of nonlinearity.