5 Two-point statistics for turbulent relative dispersion
Time-dependent flow field
0 2 4 6 8 10 12 14 16 18
h/d 0
0.1 0.2 0.3 0.4
fH Ctd
˜t= 0
˜t= 33
˜t= 66
˜t= 98
(a) Time-averaged mean flow field
0 2 4 6 8 10
h/d 0
0.1 0.2 0.3 0.4
fH Ctd
˜t= 0
˜t= 33
˜t= 66
(e)
Time-dependent flow field
0 1 2 3 4 5 6 7 8 9
v/d 0
0.1 0.2 0.3 0.4
fV Ctd
˜t= 0
˜t= 33
˜t= 66
˜t= 98
(b) Time-averaged mean flow field
0 2 4 6 8 10
v/d 0
0.1 0.2 0.3 0.4
fV Ctd
˜t= 0
˜t= 33
˜t= 66
(f)
Time-dependent flow field
10−4 10−2 1 102 104
m
10−4 10−3 10−2 0.1 1
fM Ct
t˜= 0 ˜t= 33
˜t= 66 t˜= 98
(c) Time-averaged mean flow field
10−4 10−2 1 102 104
m
10−4 10−3 10−2 0.1 1
fM Ct
˜t= 0 ˜t= 33
˜t= 66
(g)
Time-dependent flow field
0 2 4 6 8 10 12 14 16 18
d/d 0
0.1 0.2 0.3
fD Ctd
˜t= 0 t˜= 33
˜t= 66
˜t= 98
(d)
Time-averaged mean flow field
0 2 4 6 8 10 12
d/d 0
0.1 0.2 0.3
fD Ctd
˜t= 0
˜t= 33
˜t= 66
(h)
Figure 5.9: Comparison between the time evolutions (the colour-scale used is the same to that used in figure 5.6) of the p.d.f.s in the case of the cluster of virtual particles initially seeded in the core of the time-dependent (a–d) (see figure 5.6c) and the time-averaged (e–h) (see figure 5.6f) velocity fields, for: (a,e) the dimensionless lateral distance between pairs of particles fCHtdcomputed using (5.10) and (5.11) with n= 3721 and N = 100; (b,f) the dimensionless streamwise distance between pairs of particlesfCVtdcomputed using (5.10) and (5.11) withn= 3721 andN = 100; (c,g) the ratio of the lateral distance to the streamwise distance between pairs of particles fCMt computed using (5.10) and (5.11) withn= 3721 and N = 100 (in a log–log plot); and (d,h) the dimensionless distance between pairs of particles fCDtdcomputed using (5.10) and (5.11) withn= 3721 and N = 100.
5.5 Discussion and Conclusion
consists of seeding and tracking clusters of virtual particles (or passive tracers) in experimentally-measured velocity fields. As we discussed in the previous chapter (see §4.1.2), there are numerous advantages to using this technique. The spatial and temporal resolutions are only limited by the resolution of the acquisition tech-nique used to measure the velocity field.1 Virtual particle tracking can potentially be applied to any laboratory flows, with a possible range of Reynolds numbers, Schmidt numbers or Prandtl numbers far exceeding the capabilities of numerical simulations. A large quantity of virtual particles can be seeded instantaneously in the flow field, with any arbitrary initial distribution, and then tracked over a spatial range only limited by the size of the measured velocity field. One could ar-gue that virtual particle tracking is not adapted to the study of three-dimensional flow fields. With only a two-dimensional velocity field of a three-dimensional flow field, it is true that virtual particle tracking cannot give meaningful information, because the trajectories of real Lagrangian particles are also three-dimensional.
We believe that the recent development of volumetric particle image velocimetry to measure the three components of the velocity in three-dimensional domains (see e.g. Kitzhofer et al., 2011; Cierpka & Kaehler, 2012, for recent reviews) can address this shortcoming.
The flow in quasi-two-dimensional jet is appropriate for the application of par-ticle tracking velocimetry because the three-dimensionality of the flow can be considered insignificant in the first order. In § 4.1.2 we report that the mean di-vergence of the flow is small compared with the mean vorticity. Moreover, Dracos et al. (1992) found that the flow of quasi-two-dimensional jets is primarily gov-erned by a two-dimensional inverse cascade of turbulence, except at scales of the order of (or less than) the gap width of the tankW. Therefore, we believe that par-ticle tracking velocimetry can give physically meaningful information about the dispersion in quasi-two-dimensional jets. However, the three-dimensional small-scale turbulence, typically of the order ofW = 1 cm or less, cannot be adequately resolved in this study, with only a two-dimensional velocity field.
Bearing in mind the limited spatial resolution of our data, we have probed the large-scale dispersion of the (large-scale) eddy and core structures of the flow.
1It can be noted that particle image velocimetry, a common technique to measure veloc-ity fields, is considered technically less demanding than experimental Lagrangian particles tracking techniques, such as particle tracking velocimetry or other optical particle tracking techniques (Kitzhofer, Nonn & Br¨ucker, 2011)
5 Two-point statistics for turbulent relative dispersion
The time evolution of the probability distributions of key two-point properties (such as the lateral distance, the streamwise distance, the Euclidean distance and the ratio of the lateral distance to the streamwise distance between two points) in the main structures of quasi-two-dimensional jets has shown different behaviours for the different parts of the flow. We compare the results of the two-point statis-tics obtained in the time-dependent velocity field with results obtained in the time-averaged velocity field of the same jet and results obtained with simple ge-ometrical distributions of points (a circle, an ellipse and a square). From the study of these simple geometrical distributions, we have been able to understand how the variation in time of general shape characteristics of the distribution af-fects the p.d.f.s of the two-point properties. In particular, we have been able to measure that, in the eddy, the distribution of particles disperses slowly and in a rather axisymmetric manner. At the interface between the core and the eddy, the distribution of particles stretches considerably in the streamwise direction at a high rate. This is accompanied by thinning of the particle cluster. In the core of the jet, the particle distribution disperses slowly in the cross-jet direction and splits along the jet axis. Finally, we believe that the comparison between the p.d.f.s for the time-averaged flow field and the p.d.f.s for the time-dependent flow field demonstrates the intense stirring (and potentially the resulting vigorous tur-bulent mixing) occurring within the eddy and, to some extent, at the interface between the eddy and the core. This aspect is revealed by the rapid displacement through time of the peaks in the distribution offEDt(t) (the time evolution of the p.d.f. of the distance between two particles initially seeded in the eddy) for the time-dependent velocity field of the eddy. The chaotic dynamics of the turbulent flow in the eddy strongly perturbs the distribution of the virtual particles, which manifests itself in the time evolution of the p.d.f. for the separation distance between particles.
Future research about the turbulent relative dispersion of the flow of quasi-two-dimensional jets could investigate the ideas of Richardson (1926) and Batchelor (1952) to describe the relative dispersion in the jet by a differential equation based on the p.d.f.s of two-point properties. In Chapter 3, we propose a model for the transport and streamwise dispersion in the jet, based on the Eulerian description of the flow. Forming the connection between the Eulerian and the Lagrangian descriptions of the turbulent dispersion could provide invaluable insight in the
5.5 Discussion and Conclusion
physics of anisotropic turbulent processes. One particular question of interest is to relate the streamwise turbulent eddy diffusivity KdM01/2z1/2 in the general effective advection–diffusion equation (3.15) (obtained using a mixing length hy-pothesis) to the p.d.f. of the streamwise distance between two points obtained directly from virtual particle tracking, in an effort to identify and parameterize the cumulative quantitative effect of the complex time-dependent flow on streamwise dispersion.
Another possible avenue of research would be to improve the spatial resolution of the velocity field, and perhaps to measure a truly three-dimensional velocity field of the flow. With a fully resolved velocity field in time (i.e. resolving Kol-mogorov time scaleτηK ≈40 ms) and in space (i.e. resolving Kolmogorov length scale ηK ≈ 0.2 mm), we could explore, for instance, the two-point dispersion model of Batchelor (1950). As Bourgoin et al. (2006) pointed out, there is a need for more experimental evidence. A comparison between the results of two-point statistics for the flow field of quasi-two-dimensional jets with the results for three-dimensional turbulent flows could shed new light on the physics of turbulent relative dispersion.