By analyzing PSD results at different concentration in both channels, we try to investigate corrugation structure at the free surface in shear flow of concentrated suspension. Our objective is also to study the effect of concentration, effect of particle size, and effect of suspending fluid viscosity on the surface roughness of fluctuations near the free surface which is required to characterize the flow structures in free surface flow of suspensions.
Moreover apparent wall slip co-efficient measurement is of fundamental importance to application to material processing as the surface effect is particularly manifest at the interface where the transport of mass, momentum and energy are affected. We present here the comparison of power spectral densities of image intensity for both temporal and spatial directions in both plane and serrated channels. Though the exact relation between the surface curvature and intensity of the image is not known, such spectra do provide a qualitative estimate of the length and time scale of the surface roughness. Such analysis along with the PIV data can increase the understanding of the free-surface flow, its scales and the nature of the flow structures. The near surface particle fluctuation causes deformation of the interface.
When the deformation relaxes, the energy contained within it is released back into the fluid.
This may further affect the surface topography elsewhere. The length scale associated with the flow structures can be estimated by using Taylor hypothesis, i.e. multiplying the time scale with the local average velocity at that position (Taylor, 1938). In the turbulence literature, this energy cascade is revealed by the turbulent kinetic energy spectra. However the correct analysis of the kinetic energy spectra requires continues velocity measurements at small time intervals. Since this was not possible from our velocity measurements, we have studied the spectra of the image intensity. For temporal measurements photographs were taken at a frame rate of 19 frames/s. The power spectrum densities (PSD) of image intensity values were computed using the fast Fourier transform similar to Singh et al. (2006). A total of 444 frames were taken to get the PSD. To study the effect of particle size we have prepared suspension with different particle sizes (80 µm, 250 µm and 500 µm mean diameter). The viscosity of the suspending fluid was 2.05 cP for all the three suspensions.
The corresponding power spectra for these suspensions are shown in Fig. 4.3 through Fig.4.5.
To study the effect of viscosity of suspending fluid, we have prepared suspensions with 500 µm particles mixed in three different suspending fluids of viscosity 2.05 cP, 98 cP, and 204 cP. The corresponding spectra for these cases are shown in the Fig.4.5 through Fig.4.7.
Figure 4.8 shows the power spectra for suspension of 200 µm particles in suspending fluid of
viscosity 19 cP. All these plots show large peaks at regular intervals indicating the presence of flow structures at multiples time scale. The frequencies of the peaks are nearly same for both plane and serrated channels. In addition to this, the nature of the plots are same for both channels at all concentrations that we studied. There is no marked difference in the spectral distribution for all particle size and suspending fluid viscosity. This could be due to the fact that spectra were studied in the plane of free surface and it is possible that the spread of corrugation structure in the plane of free surface is similar for all the suspensions. However, it is desired to study the corrugation in velocity-vorticity plane (perpendicular to the plane of free surface) to quantify the corrugation and its dependence on particle size and viscosity of the suspending fluid. This is done in the next chapter. For a given suspension, the comparison of spectra from plane channel and rough channel gives an idea about the effect of wall slip on surface corrugation. In all the cases it was observed that the natures of these plots are nearly same. This indicates that wall slip may not have any apparent effect on the surface corrugations due to particle size, suspending fluid viscosity, particle fractions and plane or serrated wall.
Figure 4.3: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 80 µm and the viscosity of suspending fluid was 2.04 cP.
Figure 4.4: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 250 µm in this case. The suspending fluid was same as in fig. 4.3.
Figure 4.5: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 500 µm in this case. The suspending fluid was same as in fig. 4.3.
Figure 4.6: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 500 µm and the viscosity of suspending fluid was 98 cP.
Figure 4.7: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 500 µm and the fluid viscosity was 204 cP in this case.
Figure 4.8: Temporal PSD at different particle concentrations for (a) plane and (b) serrated channel.
The particle size was 200 µm and the fluid viscosity was 19 cP.
As mentioned previously, the autocorrelation function represents how the flow structure at a particular location is correlated with that at the neighboring downstream locations. A faster decay of autocorrelation with spatial dimension indicates larger fluctuation at the free- surface. On the other hand, if the flow structure at the interface is largely moving with the mean flow this would result into a slower decay of autocorrelation function. In Fig.4.9 through Fig.4.14, we have compared spatial autocorrelation in x and y directions for suspensions of various particle size (80 µm, 250 µm and 500 µm) in 2.05 cP of suspending fluid at various particle fraction. The similar plots for various suspension of 500 µm particles in different suspending fluids are shown in Fig.4.15 through Fig.4.18. Figures. 4.19 and 4.20 are for suspension of 200 µm particles in suspending fluid of viscosity 19 cP. A common observation in all the plots is that after sharp decay the curve flattens for all the concentrations. The mean level of the plateau portion of the curve is expected to have no relation to corrugation estimation. It could be that the overall intensity of the cold light source in the experiments for plane and serrated channel was not same. The plots for different concentrations reveal that the flow structures uncorrelated faster for particle concentration of 0.40 and 0.45 (it can be observed from the rate of decay of autocorrelation function in the inset of the plots). However, with particle concentration of 0.50, we observe that the autocorrelation decay is relatively slow. The behavior is similar for both plane and serrated channels and the initial rate of decay was found to be nearly equal. The results are similar for all suspensions irrespective of the size and viscosity of suspending fluid. This observation is in agreement with the earlier findings of Loimer et al. (2002) and Singh et al. (2006) who reported maximum disorder at particle concentrations between 0.40 and 0.45. The rate of decay of autocorrelation function is related to the flow structures. At low concentration, the particle fluctuations are not strong enough and the flow structures convect with the mean flow. On the other extreme, i.e., when the particle concentration approaches to maximum flowing fraction, the particles again follow ordered motion due to crowding effect (Ackerson, 1990; Ovarlez et al., 2006). There exists a critical particle fraction value when the fluctuations are strong enough to produce wide spectrum of uncorrelated flow structures.
Therefore, our study shows that wall slip does not change the nature of surface corrugation patterns observed during free surface flow of concentrated suspensions.
Figure 4.9: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 80 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.10: Normalized spatial auto-correlation in y (flow) direction for suspension of 80 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.11: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 250 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.12: Normalized spatial auto-correlation in y (flow) direction for suspension of 250 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.13: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 500 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.14: Normalized spatial auto-correlation in y (flow) direction for suspension of 500 µm particles in fluid of viscosity 2.05 cP. (a) plane and (b) serrated channel.
Figure 4.15: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 500 µm particles in fluid of viscosity 98 cP. (a) plane and (b) serrated channel.
Figure 4.16: Normalized spatial auto-correlation in y (flow) direction for suspension of 500 µm particles in suspending fluid of viscosity 98 cP. (a) plane and (b) serrated channel.
Figure 4.17: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 500 µm particles in suspending fluid of viscosity 204 cP. (a) plane and (b) serrated channel.
Figure 4.18: Normalized spatial auto-correlation in y (flow) direction for suspension of 500 µm particles in suspending fluid of viscosity 204 cP. (a) plane and (b) serrated channel.
Figure 4.19: Normalized spatial auto-correlation in x (spanwise) direction for suspension of 200 µm particles in fluid of viscosity 19 cP. (a) plane and (b) serrated channel.
Figure 4.20: Normalized spatial auto-correlation in y (flow) direction for suspension of 200 µm particles in fluid of viscosity 19 cP. (a) plane and (b) serrated channel.