(209i) 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

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_ijs_jk s_ki - 1/4 ω_iω_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.