Small-scale anisotropy in Lagrangian turbulence

We report measurements of the second-order Lagrangian structure function and the Lagrangian velocity spectrum in an intensely turbulent laboratory flow. We find that the asymmetries of the large-scale flow are reflected in the small-scale statistics. In addition, we present new measurements of the Lagrangian structure function scaling constantC0, which is of central importance to stochastic turbulence models as well as to the understanding of turbulent pair dispersion and scalar mixing. The scaling of C0 with the turbulence level is also investigated, and found to be in agreement with an existing model. Turbulence governs the vast majority of fluid flows in nature and in industrial applications, including the dynamics of weather systems and clouds, the spread of odour plumes and pollutants, and mixing in chemical reactors. Despite the importance of turbulence, however, our fundamental understanding of the subject remains poor. Indeed, Feynman called turbulence one of the last great unsolved problems of classical physics [1]. Due to the complexity of the fluid equations of motion, we are forced to turn to phenomenological modelling to gain insight into the behaviour of turbulent flows. Statistical turbulence modelling has been dominated by the ideas of Kolmogorov [2], whose 1941 hypotheses have so influenced the field that they are simply known as the ‘K41’ model. Taken together, the K41 hypotheses assume that, in intense turbulence and well away from any boundaries or singularities, the statistics of turbulent flow should be universal at length 4 Author to whom any correspondence should be addressed. New Journal of Physics 8 (2006) 102 PII: S1367-2630(06)16684-7 1367-2630/06/010102+10$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Figure 1. Sketch of the experimental apparatus. The trajectories of tracer particles were recorded by three high-speed cameras in a 5 × 5 × 5 cm3 subvolume in the centre of the tank. The tracers were illuminated by two pulsed Nd :YAG lasers with a combined power of roughly 150 W. The cameras were arranged in a single plane in the forward scattering direction from the lasers with an angular separation of roughly 45◦. The discs rotated about the z -axis. and timescales that are small compared with the injection of energy into the flow. If these small-scale statistics are to be universal, they must be independent of the large-scale flow structure. In particular, K41 predicts that at small scales the turbulence should ‘forget’ any preferred directions of the large-scale flow and that the small-scale fluctuations should be statistically homogeneous and isotropic. Models and simulations of turbulence therefore commonly assume isotropic flow. Real flows, however, are never homogeneous and isotropic at large scales. Careful study of the effects of large-scale anisotropy on the small-scale turbulent fluctuations is therefore very important for understanding the behaviour of turbulent flows in nature. In addition, such study is necessary in order to relate current turbulence theory, modelling, and simulation to practical applications. We have investigated the K41 hypothesis of local isotropy in an optical three-dimensional (3D) particle tracking experiment. Our experimental facility consists of a closed cylindrical chamber where turbulence is generated between counter-rotating discs, as sketched in figure 1. The tank has a diameter of 48.3 cm, and the discs are separated by 43.9 cm. The flow is seeded with polystyrene tracer particles with a diameter of 25μm and a density 1.06 times that of water, which have been shown to act as passive tracers in this flow [3]. The particles are illuminated with two pulsed Nd :YAG lasers with a combined power of roughly 150 W, and their motion is followed using three Phantom v7.1 CMOS cameras from Vision Research. These cameras are capable of recording 27 000 images per second at a resolution of 256 × 256 pixels. Tracer particle tracks are found from the image sequences using particle tracking algorithms [4], and their velocities are calculated by convolving the particle tracks with a Gaussian smoothing and differentiating kernel [5]. New Journal of Physics 8 (2006) 102 (http://www.njp.org/) 3 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Because of the cylindrical symmetry of our apparatus, the large-scale flow is axisymmetric. We investigate the effects of this large-scale anisotropy in the context of the variance of the temporal increments of the turbulent velocity δui(τ) = ui(t + τ)− ui(t), known as the secondorder Lagrangian structure function DL ij(τ). Axisymmetric turbulence has been the subject of prior theoretical work [6]–[8], but has not yielded any experimentally verifiable predictions similar to those made by the K41 model. K41 theory predicts that the structure function should scale as DL ij(τ) = 〈δui(τ)δuj(τ)〉 = C0 τδij in the so-called inertial range where the only relevant flow parameter is the rate of energy dissipation per unit mass . According to K41, the structure function should be isotropic and C0 should have a universal value for all turbulent flows. It is an important parameter in stochastic models of turbulent transport and dispersion [9]–[11] and is, remarkably, also connected both to the Richardson constant governing the separation of fluid element pairs, assuming that the covariance of the relative acceleration of the pair is stationary, and to the structure functions of the fluctuations of a scalar field passively advected by the turbulence [12]. Previously measured values of C0 range from 2.1 to 7.0 [13], in part because Lagrangian experiments, where the trajectories of individual fluid particles are followed, have historically been very difficult. Here, we report new, better-resolved measurements of C0. Despite recent experimental and numerical studies of Lagrangian turbulence [3], [14]–[18], most of our understanding of turbulence still comes from Eulerian measurements, where probes are fixed with respect to some laboratory reference frame. For example, while the value of the Lagrangian constant C0 is very uncertain, the corresponding Eulerian constant C2 has a wellmeasured value of 2.13 ± 0.22 [19]. Lagrangian statistics also seem to require higher turbulence levels to observe K41 scaling [18]. The turbulence level is quantified by the Reynolds number, which measures the relative importance of the nonlinear inertial terms and the linear viscous terms in the Navier–Stokes equations. In the present work, we report the Reynolds number based on the Taylor microscale, Rλ ≡ u′λ/ν, where u′ is the root mean square velocity of the turbulent fluctuations and ν is the kinematic viscosity. λ, the Taylor microscale, is defined to be √ 15u′2ν/ . To define our Reynolds numbers, we measure from the secondand third-order Eulerian structure functions, which give us the spherically averaged dissipation rate. Since, as mentioned above, the large-scale velocity is different in the axial direction and the radial directions, we define the Reynolds number based on the radial root mean square velocity. Yeung [18] has suggested that a Reynolds number of at least Rλ = 600–700 is required to observe K41 scaling of Lagrangian quantities, a range difficult to achieve both in experiments and simulations. In this work, we report measurements at Reynolds numbers up to Rλ = 815. Figure 2 shows a single component of our measured Lagrangian structure functions compensated by τ at Rλ = 200, 350, and 815. Plotting the structure function in this fashion should display a plateau in the inertial range with value C0. A well-developed inertial range requires a large scale separation between the Kolmogorov timescale τη and the Lagrangian integral time TL. TL is usually measured from the Lagrangian velocity autocorrelation function, which decays approximately exponentially. We show the three diagonal components of the autocorrelation tensor R(τ) measured at Rλ = 815 in figure 3 along with fits of the function R(τ) = TLe −τ/TL − T2e−τ/T2 TL − T2 (1) proposed by Sawford [10] to account for the finite slope of the autocorrelation function at the origin. T2 here is related to the Kolmogorov time τη. We fit (1) between 0 and 40τη, and find an New Journal of Physics 8 (2006) 102 (http://www.njp.org/) 4 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT