Experimental Analysis of Aggregate Breakage In Turbulent Flow by 3D-PTV

It is absolutely vital to have thorough access to the properties of the turbulent flow in the close proximity of the aggregate before, during and after its disintegration to explore the breakage dynamics which is up to date poorly understood according to Soos et al. (2008). Therefore the main goal of this experimental effort is to investigate the underlying physics of breakup mechanism of dynamically grown aggregate in turbulent flow by three dimensional particle tracking velocimetry (3D-PTV), a whole field non intrusive flow diagnostic technique described in details by Lüthi et al. (2005). Before investigating the breakage phenomena in more realistic turbulent flow environment, we start with some important synopsis of the results obtained from an orifice producing extensional flow shown in Fig.1a.

Figure 1: a) Sketch of the orifice setup for extensional flow b) Sketch of the forcing device for turbulent flow

Fig.2a shows three representative breakage events, where the sharp drop in aggregate size (quantified by the number of pixels) pinpoints the breakage position on the horizontal x-y plane. It can be seen that in all three cases the aggregate slpit into two parts of comparable size. Due to the geometry of the orifice it there existed simple shear close to the solid wall and elongational shear because of the converging nature of the flow. Fig.2b classifies the breakage events according to the experienced shear.  The intermediate eigen value of the rate of strain tensor, Λ₂, which is negative, almost equates the compressive one, Λ₃, in magnitude and elongational shear drives the breakage upstream of the orifice (the orifice is located at 24.5 mm). But as the flow converges towards the contraction, the magnitude of Λ₂ attenuates and finally vanishes whereas the compressive eigen value, Λ_3, starts to dominate downstream of the orifice to more upstream producing simple shear and henceforth dominates the breakage.

Aiming towards observing agglomerate breakage in more realistic conditions, turbulent flow is generated within a glass aquarium, 120x120x140 mm³, by a device having two sets of four counter rotating disks with baffles driven by a conventional servo motor.  A detailed description of the set up in Fig.1b can be found in Liberzon et al. (2005). The baffled disks' rotation speed was set to 800 rpm and the characteristics of the resulting quasi-isotropic turbulent flow field are given in table 1.

Table 1: Some turbulent flow properties as measured from experiment

urms	L	ε	Reλ	η	τη	G	Δ/η
0.12 ms-1	25 mm	0.026 m2s-3	180	0.08 mm	0.006 s	160 s-1	120

Figure 2: a) Reconstructed trajectories showing the breakage locations (marked by red circle) as well as the path lines of the fragments b) Streamwise breakage location as a function of the ratio between intermediate Λ₂ and most compressive Λ₃ eigen value of the rate of strain tensor.

The integral scale and the mean rate of dissipation were obtained by fitting the parameterization of Borgas and Yeung (2004) to the measured longitudinal second order velocity structure function. The instantaneous spatial gradients of the velocity were extracted from the volumetric 3D-PTV data using a filtering scale of Δ=10 mm. Employing the approximation of Lüthi et al. (2007) the coarse grained mean strain <2s²> is estimated to be about 800s^-2, which is in good agreement with the measured strain and ensthophy PDFs and their respective means shown in Fig.3b.  With the dissipation rate thus reliably determined at ε ~ 0.026 m²s^-3 we obtain a typical shear rate of G=1/τ_η=160 s^-1. For turbulent flow characterization it has proven useful to study the so called Q-R plane (e.g. Cantwell (1992)). Q is the second invariant, Q = 1/4(2ω² - 2s²), of the velocity gradient tensor A_ij and R is its third invariant, R = -1/3 s_ij s_jk s_ki - 1/4 ω_i ω_j s_ij. In joint PDF plots of Q versus R a qualitatively identical “tear drop” shape (see Fig.3a) for different kinds of turbulent flows was found by a number of investigators.

Figure 3: a) Qualitative universal feature of turbulent flows, joint PDF of second and third invariant of the velocity gradients Q and R, showing the typical tear-drop shape. b) PDF of the first moment of the coarsed grained strain and enstrophy distribution. c) Reflection of the self-amplifying nature of turbulence, PDF of the cosine between vorticity ω and the vorticity stretching vector W = ω_j s_ij.  d) PDF of the shape of the coarse grained rate of strain tensor s_ij. Vertical dashed lines indicate the respective mean values.

Chacin (2000) argue that the shape is a universal characteristic of the small-scale motions of turbulence. Recently, very similar dynamics have also been observed for the coarse grained velocity gradients; see e.g. Lüthi et al. (2007). The most important underlying features of the self-amplifying dynamics of the velocity gradients are illustrated in Fig.3c, d. In Fig.3c the measured alignment between vorticity and the so-called vorticity stretching vector is shown along with its positive mean value (black vertical line). In Fig3d the reason for the self-amplification of strain is shown as the positive intermediate eigenvector Λ₂ of the rate of strain tensor.

In the next stage of this work, agglomerates of scale O(100-1000) microns will be released into this turbulent flow and PTV measurements will be used to simultaneously measure the field of coarse velocity gradients and to track breaking agglomerates and their products.

References:

Borgas M. S., Yeung P.K., Relative dispersion in isotropic turbulence. Part 2. A new stochastic model with Reynolds-number dependence, J. Fluid Mech,Vol. 503, pp. 125-160, 2004

Cantwell B.J., Exact solution of a restricted euler equation for the velocity-gradient tensor, Physics of Fluids, Vol. 4, pp.782-793, 1992

Chacin J. M., Cantwell B. J., Dynamics of a low Reynolds number turbulent boundary layer, J. Fluid Mech,Vol. 404, pp. 87-115, 2000

Liberzon A., Guala M., Lüthi B., Kinzelbach W., Turbulence in dilute polymer solutions, Physics of Fluids, Vol. 17, pp. 031707, 2005.

Lüthi B., Tsinober A., Kinzelbach W., Lagrangian measurement of vorticity dynamics in turbulent flow, J. Fluid Mech, Vol. 528, pp. 87-118, 2005.

Lüthi B., Ott S.,Berg J., Mann J., Lagrangian multi-particle statistics, Journal of Turbulence, Vol. 7, 2008.

Soos M., Moussa AS., Ehrl L., Sefcik J., Wu H., Morbidelli M., Effect of shear rate on the aggregates size and morphology investigated under turbulent conditions in stirred tank, Journal of Colloid and Interface Science, Vol. 319, pp. 577-589, March 2008.