We discussed earlier that an initial disturbed interface is introduced in 3-D model by a user defined subroutine in ANSYS Fluent. As the time progresses, interfacial waves grow and eventually reach saturation. The first six pictures shown in Figure 96 illustrate the evolution of the interfacial waves for the first 0.5 seconds. Waves are still evolving. As the time progresses, waves eventually reach saturation at around 1.5 seconds. The last three pictures of Figure 96 show the wave shape after 1.5 seconds of simulation time where changes in amplitudes of the wave are diminishing over time. More details on saturated wave and its natures are described in a later section of this chapter.
144
Figure 96: Evolution of the interfacial wave in three dimensions at different times.
Time = 0.04 sec Time = 0.14 sec Time = 0.26 sec
Time = 0.36 sec Time = 0.43 sec
Time = 0.49 sec
Time = 1.55 sec
145
Once the wave reaches saturation, the saturated wave shape resembles an axisymmetric wave. We must ensure that the resultant waves are indeed axisymmetric. In order to do this, volumetric fraction of oil and water for X-Y and Z-X plane are drawn in Figure 97 for an instant of time. The blue color represents oil and red color represents water. The interface is green where the volumetric fraction is 0.5. A quick examination of Figure 97 reveals that wave shape at the interface on X-Y plane is symmetric to axis of the pipe. The interface waves are designated as upper wave and lower wave for X-Y plane. Similarly, the interface line in Z-X plane is also symmetric with respect to centerline and they are labeled as left and right waves.
Figure 97: Volume fraction of water showing blue as oil and red as water. Contour plot of volumetric fraction showing the saturated wave shape at the interface between the oil and water.
To examine whether the waves are truly axisymmetric, the upper and lower interfacial waves are plotted in X-Y plane in Figure 98. Similarly, left and right waves are also plotted in Z-X plane. Then the X-Y plot and Z-X plot are superimposed. Superposition of the upper wave with the left wave coincides exactly. On the other hand, superposition of lower and right waves perfectly matches with each other. Therefore, the saturated wave is indeed an
Upper
Lower Left
146
axisymmetric wave. The three-dimensional model of the core-annular flow confirms that the wave generated by the specific parameters for this benchmark case are truly axisymmetric.
Figure 98: Superposition of the wave shapes. Upper wave is superimposed with left wave and lower wave is superimposed with right wave. They perfectly match.
In addition to the superposition of the wave shapes, another approach is discussed here to ensure that the waves generated at each instance of time shown in Figure 96 are truly axisymmetric. In Figure 99, volume fraction contour of oil and water is drawn at an arbitrary cross section inside the domain. If we examine the cross section from the front view or Z-Y plane, we notice that cross sections are circular but the area changes with time. Therefore, if we plot the centroidal coordinates of the cross section, namely 𝑍𝑚 and 𝑌𝑚 ,
Upper and Lower Wave
Left and Right Wave Superposition of Upper and Left
and lower and right waves
Upper
Lower Left
147
for axisymmetric wave, the centroidal coordinates (𝑍𝑚, 𝑌𝑚) at every instance of time will coincide with each other and the plot will appear as the point plot shown in Figure 100.
Therefore, we can conclude that for axisymmetric wave or Bamboo wave there is no movement of the centroidal coordinates from the center or axis of the pipe and as a result a point plot is obtained.
Figure 99: Cross section of the domain at different times showing the volume fraction of core with blue color and annulus with red color.
Figure 100: The centroid of the core is plotted in Y-Z plane. It is a point plot.
𝑇𝑖𝑚𝑒 = 𝑡1
𝑇𝑖𝑚𝑒 = 𝑡2
𝑌𝑚= Coordinate of 𝑌 at Centroid of Core (Oil)
𝑍𝑚= Coordinate of 𝑍 at Centroid of Core (Water)
𝑍𝑚
𝑌𝑚
t0, t1, t2, t3
148
From 3-D simulation, centroidal coordinates of the oil volumetric fractions are drawn as a function of time. They are shown in Figure 101. In Figure 101(a), centroidal coordinates of 𝑌𝑚 and 𝑍𝑚 constitute a point graph in Y-Z plane. On the other hand, 𝑍𝑚 and 𝑌𝑚 vs. time plot, Figure 101(b), shows a straight line which suggests that as the wave
travels, the cross section of the core fluid changes at the arbitrary location but the centroid position does not change with time. It remains in the center.
Figure 101: Centroidal coordinate 𝑍𝑚 vs. 𝑌𝑚 at different simulation times of 0.5 to 1.5 seconds. (a)
Coordinate 𝑍𝑚 and 𝑌𝑚 over time (b) Centroidal coordinate location (𝑍𝑚 and 𝑌𝑚 ) over time in a 3-
D plot.
From all of the discussions above, it is clear that the centroid of the core fluid at any cross section of the pipe flow remains at the center over time and the wave is axisymmetric. The trajectories of the centroidal coordinate (𝑍𝑚, 𝑌𝑚)remains at a fixed point at the center of the pipe. Therefore, the wave will remain axisymmetric at all times.
Another very important finding of 3-D simulation is the confirmation of the wave shape. In Figure 102, saturation wave shapes are plotted at two different times. Notice
0.00200 0.00100 0.00000 0.00100 0.00200 0.00200 0.00100 0.00000 0.00100 0.00200 Re 3.73: Bamboo Waves Zm Vs. Ym Over Time Ym Zm Ym Zm, , T_Ym ( )
149
that the interface wave plotted here is for three wave lengths. The waves are pointed at the peak and wider at the trough and symmetric in both side of the peak and of course axisymmetric. Therefore, it is true that for the set of given parameters 𝑎 = 1.61, 𝑚 = 0.00166,𝜁 = 1.1, 𝐽 = 0.0633, and ℝ = 3.737, the saturated wave evolves into a bamboo wave and it is corroborated by full blown 3-D and 2-D axisymmetric model and by experimental work.
Figure 102: Wave shape from 3-D simulation. Interface waves of three consecutive domains (three wave lengths) show the wave shape for the benchmark flow parameters. They constitute the shape of a bamboo stem.
150