Nulidad absoluta y nulidad relativa del testamento

Chapter 5

Two-point statistics for turbulent

relative dispersion in

quasi-two-dimensional jets

5 Two-point statistics for turbulent relative dispersion

flow is turbulent everywhere in a quasi-two-dimensional jet. We believe that such distinctive Eulerian characteristics (of the flow in the core, in the eddies and at the interface between the two) also have distinctive dispersive and mixing properties.

From the interaction between these structures, in time and space, results the global, mean dispersion mechanism of quasi-two-dimensional jets, which we model in Chapter 3 along the streamwise direction.

Conversely, in this chapter, we adopt a Lagrangian approach to investigate the dispersion and mixing properties of the core and eddy structures of quasi-two-dimensional jets. In figure 4.4 presented in the previous chapter, we showed the evolution in time of clusters of virtual particles (or passive tracers) released in different parts of a quasi-two-dimensional jet: in an eddy (see figure 4.4a), between the eddy and the core (see figure 4.4b), and in the core (see figure 4.4c).

We qualitatively described how the clusters of particles disperse and mix, while being transported by the jet. The virtual particles seeded in the eddy travel significantly slower than the virtual particles seeded in the core. The virtual particles seeded in the eddy appear to experience more vigorous stirring than the virtual particles seeded in the core. We also noticed that the virtual particles seeded in the core disperse laterally as they are advected by the flow. On the other hand, the virtual-particle cluster seeded between the eddy and the core display intense streamwise stretching.

The aim of this study is to quantify these observations about the dispersion and mixing of the virtual particles in figure 4.4. We use statistical analysis to understand the underlying physical mechanisms. We study the probability dis-tribution of two-point properties, such as the lateral (or x-) distance between two points, the streamwise (or z-) distance between two points, the distance be-tween two points, and the ratio of the lateral distance to the streamwise distance between two points. We apply these probabilistic tools to clusters of virtual parti-cles released in quasi-two-dimensional jets (such as those in figure 4.4), where the two-point measurements are made between pairs of virtual particles. We compute the probability distribution of the two-point properties at each instant in time to obtain meaningful quantitative insight about the temporal and spatial dispersive dynamics of the jet structures.

The work of Richardson (1926) pioneers the use of two-point statistics to study diffusion in turbulent flows (see e.g. Sawford, 2001; Salazar & Collins, 2009, for

5.1 Introduction

recent reviews). Observing considerable discrepancies (by more than ten orders of magnitude) in the measurements of the atmospheric diffusivity, he argued that two-point statistics are more appropriate to explain diffusion in the atmosphere than single-point statistics (used previously to measure the diffusivity in the sense described by Fick’s law). Two-point statistics (such as the time average of the distance between two points) enable the study of the dispersion in the flow at each spatial scale (defined, for example, by the eddy size), without being influenced by the larger scales. From the probability density function of the distance between particles, Richardson derived his famous 4/3 law of diffusion. Batchelor (1952) developed a rigorous mathematical framework for the idea of Richardson (1926) to use two-point statistics in order to study turbulent relative dispersion. He applied two-point statistics to the diffusion of passive scalars in homogeneous isotropic turbulence.

The concept of two-point statistics has then been used to study turbulent dis-persion in the ocean and in the atmosphere (see e.g. Monin & Yaglom, 1975, pp. 556–567, for a review). Salazar & Collins (2009) and Yeung (2002) give a summary of experimental and numerical works investigating turbulent relative dispersion. In experimental turbulent flows, two-point statistics can be calculated by tracking Lagrangian particles. According to Toschi & Bodenschatz (2009), the most successful current technique to perform Lagrangian particle tracking is called particle tracking velocimetry. For example, Bourgoin et al. (2006) used particle tracking velocimetry to measure the mean square distance between particles in a turbulent flow (generated “between coaxial counter-rotating baffled disks in a closed chamber”). They confirmed the theoretical prediction of Batchelor (1950) that the temporal evolution of the distance between pairs of particles during the superdiffusion stage (i.e. the regime when the mean square distance between par-ticles increases in time like t^α with α > 1, Bourgoin et al., 2006) is influenced by the initial distance separation of the particles. Bourgoin et al. (2006) also commented on the scarcity of direct experimental evidence for turbulent relative dispersion. Toschi & Bodenschatz (2009) attributed the lack of experimental ev-idence to the technical difficulties of the implementation of Lagrangian particle tracking in fully turbulent flow.

We believe that applying two-point statistics to the turbulent flow of quasi-two-dimensional jets can give new insight about turbulent relative dispersion in

5 Two-point statistics for turbulent relative dispersion

the case of a non-homogeneous and anisotropic turbulent flow. We use what we believe to be a new method to calculate these two-point statistics, which we call virtual particle tracking (see § 4.1). The virtual-particle-tracking technique (which we use to produce the results shown in figure 4.4, mentioned above) con-sists of seeding and tracking virtual passive tracers in velocity fields measured using particle image velocimetry. The results presented in this study focus pri-marily on the dispersion properties of quasi-two-dimensional jets, but not directly on the transport or turbulent mixing properties. By definition, two-point statis-tics do not depend on mean transport motion, and thus cannot investigate it.

(The transport properties of the jet have actually been studied extensively in Chapters 2, 3 and 4.) On the other hand, we believe that mixing properties can-not be directly examined from the results we present in this thesis for technical reasons. The measurements of the velocity fields (performed using particle image velocimetry), though well-resolved in time (the time resolution is one order of magnitude smaller than the Kolmogorov time scale, τηK ≈ 40 ms), do not have the spatial accuracy necessary to investigate the finest scales of turbulence in our flow (the Kolmogorov length scale is of the order of ηK ≈ 0.2 mm, as discussed in§2.2.2). In Chapter 4, we quantify the mixing through the dilution of the dye concentration. Likewise, in this chapter, we infer indirectly the turbulent mixing processes from the dispersion, stretching and folding of our particle distributions.

In order to comprehend fully the temporal evolution of the probability distri-butions of two-point properties applied to virtual-particle clusters seeded in the different parts of the flow of quasi-two-dimensional jets, we compare our results with other flow fields. As a preliminary study, we apply our statistical tools to simple distributions of points (such as a circle, an ellipse and a square) evolving in diverging velocity fields. The purpose of this preliminary study is to understand how the probability distributions of two-point properties are related to a given initial distribution of particles, and how they evolve in time. Then, we compare the results for the time-dependent flow field of a quasi-two-dimensional jet with results obtained using the time-averaged flow field of the same jet. This compari-son allows us to identify some key dispersive mechanisms due to the core and eddy structures and emphasizes the importance of their time-dependent interactions.

The rest of this chapter is organized as follows. In § 5.2, we describe mathe-matically how to compute the probability distribution of the two-point properties

5.2 Mathematical definitions of two-point probability distributions

stated above, in both the continuous case and the discrete case. In § 5.3, we present a preliminary study of three analytical and numerical test cases: an ax-isymmetric expansion of a circular domain, a non-axax-isymmetric expansion of an elliptical domain, and a diffusive expansion of a square domain. In § 5.4, we present the results of the probability distributions for the three clusters of virtual particles seeded in the quasi-two-dimensional jet shown in figure 4.4. We compare these results with similar results obtained in the equivalent time-averaged velocity field of the jet. Finally, we draw our conclusions in § 5.5.

5.2 Mathematical definitions of two-point