**3.1.2 Liquid film falling down on inclined flat plate**

In the fluid mixing phenomena, bubble, bubble/slug, slug and slug/churn flow are important. However, in slug flow, liquid film is observed between slug bubble and wall. Then verification of the TPFIT must be performed for film flow.

The TPFIT code was applied to numerical simulation of liquid film falling down on inclined flat plate. Simulations were performed with the same conditions as the experiment by Moran et al. (2003) (see Fig.5). Physical properties of the liquid were as follows: kinematic viscosity, *l*=2×10-5 m2/s, density, *l*=960 kg/m3, and surface tension, =2.06×10-2 N/m. And air properties at 300K and atmospheric pressure were used as gas properties. On all walls, nonslip boundary condition was assigned, and inlet pressure was fixed at atmospheric pressure.

The flow conditions were summarized in Table 1. The analysis conditions were set up to compare the probability density function (PDF) of local film thickness with the experimental results. In the table 1, *<sup>N</sup>* represents Nusselt's mean film thickness (Nusselt, 1916), and is evaluated by the following equation:

$$\mathcal{S}\_N = \left(\frac{\Im \nu\_1 f}{\mathcal{g}\_z}\right)^{1/3} \tag{15}$$

In this equation, *gz* is flow direction acceleration by gravity force, and *J* is mass flow rate of the liquid.


Table 1. Numerical conditions.

Fig. 5. Analytical geometry of a liquid film.

Development of Two-Phase Flow Correlation

experiment.

experimental results.

**3.1.3 Bubbly and slug flow in square duct** 

1 Air-water at 0.1MPa and 300K

3 Steam-water at 7.2MPa and 560K

predict mixing coefficient with accuracy of less than 10%.

Case Fluid Inlet Inlet void fraction:

4 B 0.111 Table 2. Numerical condition for bubbly flow in square duct.

for Fluid Mixing Phenomena in Boiling Water Reactor 295

close to zero for greater thickness values, indicating existence of few waves. At low film Reynolds number, the position and height of the first peak agreed well in the analysis and

(a) Case 1 (b) Case 2 (c) Case 3

In the experimental results, at relatively high Reynolds number (*Ref*=106 and 220), additional smaller peaks (second peaks) appeared to the right of the main peaks. Because the sampling numbers (*ns*) used in the processing of the experimental data (*ns* = 60) were smaller than those in the numerical results (*ns* =1200), scattered results were observed in the experimental PDF distributions. As shown in Fig.8, the numerical result agreed well with the experimental result including existence of second peaks and these positions. The predicted values of minimum liquid film thickness by the numerical simulations were slightly smaller than those measured by the experiments without relying on the mass flow rate of the liquid. It is thought that because the predicted minimum liquid film thicknesses were thin, the average liquid film thicknesses became smaller in comparison with the

By two-phase correlations for fluid mixing phenomena, volume or mass cross flow rate or mixing coefficients are evaluated. Then volume and mass conservation of two-phase flow is important function. As mentioned above, volume conservation equations for both phases are not solved, and the TPFIT has no special treatment to keep volume conservation. Then we must check volume conservation of the TPFIT. In two-phase flow fluid mixing phenomena, the maximum value of mixing gas flow rate is around 10% of gas flow rate in flow channel. An error of the volume conservation must be done below 1% if we want to

A 0.307

A 0.307

0.5 m/s 2 B 0.111

*in* Inlet velocity: *win*

Fig. 7. Probability density function of liquid film thickness.

Fig. 6. Observed liquid film shapes.

Figure 6 shows snapshot of the numerical results of the Case 3. In this study, the calculated gas-liquid interfaces were defined as isosurface at a volume fraction of liquid of 0.5. In 0.02s, two-dimensional wave was observed near liquid inlet section, and this wave moves to the downstream section. From 0.04 seconds later, small three-dimensional waves occurred on the surface of the liquid film. After that, these small waves gradually becomes big until t=0.2s. At t=0.4s, the liquid film exhibited a smooth, flat gas-liquid interface upon immediate entrance to the test section, but after a short distance small, small ripples were observed at the interface. At approximately 200mm (about *z*=100 *<sup>N</sup>*) from the inlet, the small ripples developed into a three-dimensional structure characterized by large waves, and wave structures were almost developed at this point. In general, the degree of waviness increased with increasing film Reynolds number. The average local film thicknesses in the numerical result (*ave\_cal*) at *x*=20mm and *z*=175 *<sup>N</sup>* are shown in Table 1 and almost agreed with Nusselt's mean film thickness. However, *ave\_cal* were slightly smaller than the average film thickness in the experimental results (*ave\_exp*).

The probability density functions (PDF) of liquid film thickness at *x*=20mm and *z*=175 *N* were evaluated to compare numerical results with the experimental results quantitatively. Figure 7 shows the PDF of film thickness. At low Reynolds numbers (*Ref*=13), the PDF distributions showed a sharp peak (first peak) at about average film thickness, but remained

(a) t=0.00s (b) t=0.02s

(c) t=0.04s (d) t=0.08s

(e) t=0.2s (f) t=0.4s

Figure 6 shows snapshot of the numerical results of the Case 3. In this study, the calculated gas-liquid interfaces were defined as isosurface at a volume fraction of liquid of 0.5. In 0.02s, two-dimensional wave was observed near liquid inlet section, and this wave moves to the downstream section. From 0.04 seconds later, small three-dimensional waves occurred on the surface of the liquid film. After that, these small waves gradually becomes big until t=0.2s. At t=0.4s, the liquid film exhibited a smooth, flat gas-liquid interface upon immediate entrance to the test section, but after a short distance small, small ripples were observed at

developed into a three-dimensional structure characterized by large waves, and wave structures were almost developed at this point. In general, the degree of waviness increased with increasing film Reynolds number. The average local film thicknesses in the numerical

were evaluated to compare numerical results with the experimental results quantitatively. Figure 7 shows the PDF of film thickness. At low Reynolds numbers (*Ref*=13), the PDF distributions showed a sharp peak (first peak) at about average film thickness, but remained

*ave\_exp*). The probability density functions (PDF) of liquid film thickness at *x*=20mm and *z*=175

*<sup>N</sup>*) from the inlet, the small ripples

*N*

*<sup>N</sup>* are shown in Table 1 and almost agreed with

*ave\_cal* were slightly smaller than the average film

Fig. 6. Observed liquid film shapes.

result (

the interface. At approximately 200mm (about *z*=100

*ave\_cal*) at *x*=20mm and *z*=175

Nusselt's mean film thickness. However,

thickness in the experimental results (

close to zero for greater thickness values, indicating existence of few waves. At low film Reynolds number, the position and height of the first peak agreed well in the analysis and experiment.

Fig. 7. Probability density function of liquid film thickness.

In the experimental results, at relatively high Reynolds number (*Ref*=106 and 220), additional smaller peaks (second peaks) appeared to the right of the main peaks. Because the sampling numbers (*ns*) used in the processing of the experimental data (*ns* = 60) were smaller than those in the numerical results (*ns* =1200), scattered results were observed in the experimental PDF distributions. As shown in Fig.8, the numerical result agreed well with the experimental result including existence of second peaks and these positions. The predicted values of minimum liquid film thickness by the numerical simulations were slightly smaller than those measured by the experiments without relying on the mass flow rate of the liquid. It is thought that because the predicted minimum liquid film thicknesses were thin, the average liquid film thicknesses became smaller in comparison with the experimental results.
