944 945
In this section, both thek- model and the RSM optimized in the previous section of the paper 946
are used to evaluate the accuracy of different drag models. In Figure 10, liquid and air 947
velocities, void fraction, liquid streamwise velocity fluctuations (or turbulence kinetic energy) 948
and relative velocity radial profiles calculated with the k- model are compared against 949
experiments S2 and H1. Results from the drag model of Wang (1994), Tomiyama et al. 950
(1998) and Tomiyama et al. (2002a) combined with the Welleck et al. (1966) correlation for 951
the bubble aspect ratio are included. Experiment S2 is particularly relevant, since it is the only 952
case were a distinct dip in the liquid velocity was found using the k- model towards the 953
centre of the pipe. In Figure 10 a-d, no significant differences are found beween Wang (1994) 954
and Tomiyama et al. (1998), for which radial proflies are similar for all the physical quantities 955
considered. In addition, the introduction of the void fraction correction in Eq. (11) produced 956
negligible differences. Instead, significant differences are visible with the correlation of 957
Tomiyama et al. (2002a), used together with the Welleck et al. (1966) correlation for the 958
bubble aspect ratio. In addition to the higher drag coefficient that causes a lower relative 959
velocity in the pipe centre, the drag coefficient further increases near the wall, generating a 960
reduction in the relative velocity. Even if good quantitative agreement is not obtained, the 961
relative velocity reduction in the near-wall region is in qualitative agreement with the 962
experiment (Figure 10d). In the centre of the pipe, however, the accuracy of the prediction 963
worsens. Changes in the relative velocity are also reflected in the void fraction and velocity 964
profiles (Figure 10 a-b). For the velocity in particular, no dip towards the pipe centre is 965
observed and the predictions are significantly improved, in particular for the liquid (Figure 966
10a). Similar results are found for experiment H1 (Figure 10 e-h). In this case, quantitatively 967
good agreement is also obtained with Tomiyama et al. (2002a) for the relative velocity near 968
the wall (Figure 10h). Unfortunately, no measurements of the relative velocity are available 969
near the pipe centre. With respect to the S2 experiments, no significant differences occur with 970
the change of the drag model in the liquid velocity and the void fraction, which remain close 971
to those obtained with the Wang (1994) drag model, and in agreement with experiments 972
(Figure 10 e-f). In regards to experiment H1, the drag correction due to Legendre and 973
Magnaudet (1998) was also tested. As shown in Figure 10h, this model did not give 974
satisfactory agreement with data and its predictions were generally characterized by an 975
excessive correction, which implies a limit to its effect is necessary. In view of these results, it 976
was not adopted in successive simulations. 977
Comparisons were repeated for the Wang (1994) and Tomiyama et al. (2002a) 978
approaches with the RSM (Figure 11). Similarly to the previous comparisons, the drag 979
correlation of Tomiyama et al. (2002a) improves the relative velocity predictions near the 980
wall for S2 (Figure 11d). This allows more accurate estimations of the water and air velocity 981
profiles (Figure 11a). Even if the dip towards the pipe centre is still present, its magnitude is 982
reduced with respect to the Wang (1994) model predictions. However, the accuracy in the 983
pipe centre is low. Differences in the void fraction and the velocity fluctuations are lower, and 984
both models are in reasonable agreement with the experiments (Figure 11 b-c). Considering 985
experiment H1, similar predictions are obtained for the liquid velocity, the void fraction and 986
the r.m.s. of velocity fluctuations (Figure 11 e-g). In agreement with the k- comparisons in 987
Figure 10, the relative velocity is also improved near the wall, where the Tomiyama et al. 988
(2002a) predictions are in agreement with the data (Figure 11h). 989
In summary, and despite the deteriorating accuracy of predictions in the pipe centre, the 990
improvements obtained in the near-wall region encourage use of the Tomiyama et al. (2002a) 991
correlation to account for bubble aspect ratio in the drag model. Further validation with 992
additional experimental data is required, in particular to confirm the lower accuracy in the 993
pipe core region. If the latter is confirmed, additional work would be desirable to further 994
improve the drag model, maintaining the same accuracy in the near-wall region without 995
deteriorating it in the pipe centre. Finally, it must be pointed out that changes in the drag 996
coefficent near the wall had a large impact on the magnitude of the lift and wall forces, being 997
both functions of the relative velocity. Lift and wall forces essentially determine the void 998
fraction radial profile in these kinds of flow, therefore, to maintain the same accuracy in the 999
void fraction radial distribution, it was necessarry to re-optimize the wall force coefficients to 1000
Cw1 = -0.4 andCw2= 0.3 for the k- model, andCw1= -0.65 andCw2= 0.45 for the RSM. The 1001
differences between these two sets of values can be attributed to the differing interactions 1002
between the velocity, lift and wall forces and the turbulence field. A summary of the 1003
combination of models and model coefficients, together with the data used for validation, can 1004
be found in Table 3. 1005
Table 3. Models and model coefficients tested and data used for validation. 1006
Turbulence Drag Lift Wall Data
k- , RSM Wang (1994) CL= 0.1 WP CL= -0.05 CP Cw,1= -0.055 Cw,2= 0.09 Whole Database k- Tomiyama et al. (2002a) CL= 0.1 WP CL= -0.05 CP Cw,1= -0.4 Cw,2= 0.3 H1, S2, W4, K1, K2, K3, K4 RSM Tomiyama et al. (2002a) CL= 0.1 WP CL= -0.05 CP Cw,1= -0.65 Cw,2= 0.45 H1, S2, W4, K1, K2, K3, K4
WP wall-peaked void profiles 1007
CP core-peaked void profiles 1008
1010
1011
1012
1013
Figure 10. Radial profiles of predicted water (and air for S2) velocity, void fraction, 1014
streamwise r.m.s. of velocity fluctuations (turbulence kinetic energy for H1) and relative 1015
velocity fromk- and different drag models compared against experiments S2 and H1 (from 1016
left to right). TW: Tomiyama et al. (2002a) + Welleck et al. (1966); Wang: Wang (1994); 1017
Tom: Tomiyama et al. (1998); TLM: Tomiyama et al. (1998) + Legendre and Magnaudet 1018
(1998) 1019
1020
1021
1022
1023 1024
Figure 11 Radial profiles of predicted water (and air for S2) velocity, void fraction, 1025
streamwise r.m.s. of velocity fluctuations and relative velocity from RSM and different drag 1026
models compared against experiments S2 and H1 (from left to right). TW: Tomiyama et al. 1027
(2002a) + Welleck et al. (1966); Wang: Wang (1994). 1028
1029 1030