Air-Liquid Flows
6.1.3. Effect of Bubble Coalescence and Break-up in Drag Reduction
In the light of the aforementioned results, one should notice that a prescribed bubble diameter has been specified throughout all numerical simulations for Inhomogeneous two- fluid model. Although encouraging results have been presented in previous sections, the value of bubble diameter (i.e. 500 µm) unfortunately at best served as a fair engineering estimation which is calibrated against experimental data based on trail-and-error without solid physical interpretations. As discussed before, the constant diameter assumption may introduce numerical errors if the bubble coalescence and break-up become dominant in the problem, especially when the air injection rate is considerably high. In attempting to overcome this problem, the MUSIG model is introduced into the simulation allowing bubble diameter to be evaluated mechanistically using the coalescence and breakage kernels.
Figure 6.8a shows the numerical comparison against its experimental counterpart for a free stream velocity of 9.6m/s, the MUSIG model in addition to Inhomogeneous model is used for comparison. Herein the MUSIG model had been specified 10 groups of bubbles, diameters ranging from 100µm-1000µm. As depicted in the figure, at high flow rates (i.e. Q4 and Q5), MUSIG model gives the best agreement while the inhomogeneous model tends to slightly over-predict the skin friction coefficients. However, using the default coefficients for
break-up and coalescence models serious under-prediction has been observed for the MUSIG model for low gas flow rates (i.e. Q1-Q3). Figure 6.8b shows the similar comparison of experimental data with higher free-stream velocity of 14.2 m/s. Analogy to the previous result, predictions of the MUSIG model appear marginally superior to the inhomogeneous at high gas injection rates (i.e. Q4 & Q5), while considerably under predictions have shown for lower gas injection rates.
0.50 0.60 0.70 0.80 0.90 1.00 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 Expt Two-Fluid-Inhomogenous MUSIG C f /C fo
Flow rate of Air (Qa)
9.6m/s
Figure 6.8.a. Comparison of computed skin-friction co-efficient Inhomogeneous & MUSIG models U∞= 9.6m/s. U=14.2m/s 0.5 0.6 0.7 0.8 0.9 1.0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Expt Two-Fluid-Inhomogenous MUSIG
Flow rate of Air (Qa)
C
f
/C
fo
Figure 6.8.b. Comparison of computed plate drag co-efficient Inhomogeneous & MUSIG models U∞=
One possible reason attributed to the under-prediction of the MUSIG model for low gas injection rates could be the over-estimation of bubble break-up rate which sequentially introduced more small bubbles into calculations. These additional small bubbles were thereby dispersed with the boundary layer caused greater drag reduction on the surface. It should be emphasized that default model parameters of the MUSIG have been adopted directly in the above numerical investigation. These parameters were calibrated with bubbly flow condition where isotopic turbulence was assumed. At high air injection rate, such assumption may be quite close the physical behaviour as higher turbulence modulation has been introduced by the presence of bubbles. However, it may become invalid for low flow rates. In essence, it is well known that these model parameters may vary from cases to cases which should be re- calibrated for particular flow condition.
0.50 0.60 0.70 0.80 0.90 1.00 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 Expt
MUSIG Breakup co-efficient=1.0 MUSIG Breakup co-efficient=0.05
C
f
/C
fo
Flow rate of Air (Qa)
9.6m/s
Figure 6.9.a. Comparison of skin friction co-efficients with different break-up co-efficients for U∞=
9.6m/s.
Based on the above argument, also serves as a confirmation of the above observations, another set of simulations have been carried out with the break up coefficient deliberately decreased 20 times to minimize the resultant bubble break-up rate. Figure 6.9a shows the corresponding predictions of the skin-friction coefficient plots for the free stream velocity of 9.6m/s. Compared with the default MUSIG model, the predicted results were generally in satisfactory agreement with measurements across varying flow rates, while a considerably improvements have been obtained for the lower air injection rates. Similar observations can also be found for the freestream velocity of 14.2m/s showing in Figure 6.9b.
The effect of the reduced break-up rate can be exemplified by a closer visualization of the predicted bubble size distributions of the two numerical results. Figure 6.10a shows the predicted bubble size distribution at the outlet obtained from the default and the re-calibrated MUSIG models. With the default break-up coefficient, in both free-stream velocities, the relatively high volume fraction of small bubble clearly demonstrated that the default model
tends to create additional small bubble via the dominating break-up mechanism. In contrast, by limiting the break-up rate, bubble coalescence overcomes the break-up mode forming relatively higher volume fraction for the larger bubbles.
0.5 0.6 0.7 0.8 0.9 1.0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 Expt
MUSIG breakup co-efficient=1.0 MUSIG breakup co-efficient=0.05
Flow rate of Air (Qa)
C
f
/C
fo
14.2 m/s
Figure 6.9.b. Comparison of skin friction co-efficients with different break-up co-efficients for U∞=
14.2m/s. 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 100 200 300 400 500 600 700 800 900 1000
MUSIG breakup co-efficient=0.05 MUSIG breakup co-efficient=1.0
d (µm) V o lu m e f ra c tio n
In the present study, the deduction of break-up coefficient arose only as an engineering estimation. In fact, as the air injection rates are considerably low, interactions between bubbles are relatively insignificant compared with that for the high injection rates. It is thereby unsurprising to re-calibrate the model constants for obtaining a “better” comparison. Although revealing the short-comings of the current kernels is certainly one of the findings, this drawback of the model should be circumvented by refining the model assumption and the mechanism which unfortunately is left far beyond the focus of this paper.
Directing back to the theme of current work, one could easily state that good predictions of the skin-friction coefficients can be obtained by specifying a proper bubble size for each simulation. Nevertheless, one should also be reminded that bubble sizes may change significantly which is impossible to be represented by a fixed average value. This problem is further exacerbated; if rigorous bubble interactions are involved at high air injection rate. In practical micro-bubble problems, it is more easily to acquire the range of bubble size rather than exact bubble diameter. The MUSIG model which tailored to resolve the bubble size distribution mechanistically within a given range of bubble size appeared as the best candidate to resolve the physics embedded in micro-bubble drag reduction problems.
0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 100 200 300 400 500 600 700 800 900 1000
MUSIG breakup co-efficient=0.05 MUSIG breakup co-efficient=1.0
d (µm) V o lu m e fr a c ti o n
Figure 6.10.b. Bubble diameter distribution function for Q2-V14.2.
Conclusion
In this article outlined above, a lot of work was undertaken numerically to study the behavior of two-phase turbulent flows of varying density regimes viz., Gas-Particle, Liquid- Particle and Liquid-Air flows. In addition to the carrier and the dispersed phases mean and turbulent behavior, Turbulence Modulation (TM) has also been investigated. It is given by the
change in the carrier phase amidst the dispersed phase or the effect of the dispersed phase on to the carrier phase at the turbulence level.
For the Gas-Particle flow, the particle-turbulence two-phase flow interaction has been successfully investigated with the Eulerian two-fluid model. The numerical code has been validated against the experimental results of Fessler and Eaton (1999) for mean velocity and the fluctuating velocities for both carrier and the dispersed phases. Two classes of particles sharing the same Stokes number, but different particle Reynolds number has also been investigated in this study. The majority of the results agree well with the experimental data; however there have been some minor discrepancies felt at the proximity of the experimental results.
The Turbulence Modulation (TM) of the carrier phase for these two classes of particles have been studied along the three sections that is near the inlet (x/h=2), in the mid-section (x/h=7) and just aft of the exit (x/h=14). It can be concluded that even though the 70μm copper and 150μm glass particles share the same Stokes number, their behavior seems to be quite different, which suggests that Stokes number alone does not characterize the particle behavior, thereby making particle Reynolds number an important parameter in classifying the way the particles behave.
Particles response to turbulent GP (Gas-Particle) and LP (Liquid-Particle) flow, behind a turbulent backward-facing step geometry have also been successfully analysed and simulated numerically using an Eulerian two-fluid model. A significant amount of work was undertaken to provide an in-depth understanding of the particle response, amidst turbulent flow conditions for two different carrier phases namely the gas and the liquid (diesel oil). From the two sets of experimental data, at the mean velocity level, the particles seem to ‘lead’ and later ‘catch up’ with the carrier phase for the LP flow, whereas they ‘lag’ behind and later ‘lead’ for the GP flow. While at the turbulence level, the particles seem to ‘lag’ and then ‘catch up’ for the LP flow while they ‘lag’ and phenomenally ‘lead’ for the GP flow. The detailed study and also the numerical diagnosis undertaken for turbulent particulate flows with two different carrier phases, to study the particle response both at the mean velocity and at the turbulence level, behind a shear flow sudden expansion geometry is quite unique and one of its kind, as there is no current published work dealing with the analysis and numerical validation of the same.
Numerically the code was validated against the benchmark experimental data of Fessler and Eaton (1997) for GP and the experimental data of Founti & Klipfel (1998) for the LP flows. The numerical results revealed good agreement with the experimental data. From there the code was further used to investigate Stokes number effect on the two different carrier phases both at the mean velocity and at the turbulence level, for this exercise the experimental geometry of the GP flow and the inlet conditions of the LP flow were used. In order to present the results in a more methodical manner, 12 points consisting of a matrix of three sections along the length of the step and four along the height of the step were used to study the Stokes number effect on the two types of flows.
At the mean velocity level the particles seem to move faster that the carrier phases for both the GP and the LP flow. However, at the particle fluctuation level, although the GP flow show an escalation with the increase in the Stokes number, the same feature seem to be absent in the LP flow, wherein the particle fluctuation seem to decrease and almost flatten out with the increase in its Stokes number. The main reason for this behaviour is the difference in the physical characteristics of the carrier phase namely the liquid, which is far denser than the
gas, this eventually changes the cross-stream and the mean gradient behaviour, which is shown to cause elevated particle fluctuations in the GP flow.
For the Liquid-Air flows, the turbulent micro-bubble laden flow has been investigated with the help of two numerical models namely the two-fluid Inhomogeneous and MUSIG models, for two different free-stream velocities. Inhomogeneous model, which uses a fixed bubble diameter, shows a very good comparison of the skin-friction co-efficients with the experiment. This model is further probed to study the various physical phenomenon’s causing the drag reduction along the boundary layer, firstly it was observed that there is drop in the mean streamwise water velocities with a subsequent increase in the normal along varying gas injection rates. Secondly, the presence of the micro-bubbles caused turbulence attenuation for some distance along the boundary layer and later an augmentation was felt due to the shedding of the vortices behind the bubbles. Thirdly, the peak of the void fractions seem to differ in relation to the degree of drag reduction along the two free-stream velocities considered in our study.
However, with respect to the drag reduction caused due to the presence of micro-bubbles in the turbulent boundary layer MUSIG model seem to show good predictions for higher gas flow rates while under predicting for lower gas flow rates. This poor prediction of the model at low flow rates was investigated to be the dominating break-up phenomenon that was taking place within the flow. This was done in order to represent the actual flow condition, where by groups of bubbles of varying bubble sizes are found within the boundary layer. Thereby allowing the MUSIG model to resolve the bubble size distribution mechanistically within a given range of bubble size and feeding it back to the Inhomogeneous model appeared as the best candidate to resolve the hidden physics in micro-bubble induced drag reduction problems.
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