## 3 Results and Discussion
### Turbulent Flow Models
Figure 5:1 Comparison between velocity profiles from the experiment and the SST k-\(\omega\) model; for drive ratio of (**a**) 0.65% and (**b**) 0.45%. Location 10 mm from the joint inside the cold channel. Figure 5:2 Vorticity contours at the left (**a**) and right (**b**) ends of heat exchanger assembly calculated from the SST k-\(\omega\) model. The drive ratio is 0.83%. Arbitrary distance \(x\) is measured along the “viewing area”; vertical axis, \(y\), is omitted for simplicity; the ratio of \(xy\) is 1:1. For drive ratio of 0.45%, the transition-SST model provides velocity profiles that agree well with the experiment, with the maximum deviation of 5.2% (cf. Figure 15b). This may indicate that the flow at this drive ratio is already transitional in its nature. Generally, in oscillatory flow, turbulent transition is a much more complex process than in uni-directional flow. [PERSON] [1] summarizes that the transitional region exists between the \"weakly turbulent region\" (flow with turbulent bursts at the centre of the pipe with the boundary layer remaining laminar) and the \"conditionally turbulent region\" (where the flow changes between \"weakly turbulent\" and \"fully-turbulent\" flow within a cycle). Furthermore, an oscillatory flow may also experience \"relaminarization\" where a \"turbulent-like\" flow is followed by a \"laminar-like\" flow in one cycle (cf. [8,9]). As seen from Table 1, the drive ratios investigated here correspond to [PERSON] layer-based Reynolds number, \(Re_{\delta}\), well below 400, which would be the critical Reynolds number for an infinite pipe [7]. However, the heat exchanger assembly in the current study has a short length. The disturbances from the edge of plates and the vortices shed and subsequently re-entrained (cf. Figure 13) may act as flow instabilities triggering turbulent transition. This may explain the need for the turbulence/transition model even though the Reynolds number is seemingly below the widely-accepted critical value. As discussed above, with reference to Figure 3, the velocity amplitude at location \(m\) is not a suitable benchmark to judge the quality of CFD against experimental results. On the other hand, judging the appropriateness of the models from vorticity maps may not be entirely objective. It is possible to introduce various \"metrics\" to judge the quality of CFD solutions; one of the simplest resembles the concept of the standard deviation. When considering \(u\)-velocity profiles in the channel, at a selected axial location as an example, it can be defined as follows:  
Figure 5:3  
Figure 5:4 Comparison between velocity profiles from the experiment and the transition-SST model for drive ratio of (**a**) 0.65% and (**b**) 0.45%. Location 10 mm from the joint inside the cold channel. where \(i\) and \(j\) simply denote the discrete locations \(y\) between the wall and channel centreline (here, the maximum \(i\) is \(N=14\)) and the phase number in the cycle (here, the maximum \(j\) is \(M=20\)), respectively. It attempts to describe the deviation of the CFD-based profile from the experimental profile, or in other words, the \"goodness\" of predictions. The numerical values are appropriately interpolated so that \(y\)-coordinates of CFD points match with the experimental ones. Of course, the smaller the metric \(\sigma_{met}\), the better the match. Figure 16 shows the plot of metric \(\sigma_{met}\) for the application of three models: laminar, transition-SST and SST k-\(\omega\) at the drive ratios studied. It is clear that for the smallest drive ratio, all models give a very similar value of \(\sigma_{met}\). As the drive ratio increases, the laminar model gives the largest discrepancies (the metric \(\sigma_{met}\) grows almost linearly). Meanwhile, the transition-SST model seems to give the best predictions for drive ratios of 0.45% and 0.65%, while SST k-\(\omega\) gives the best prediction for the highest drive ratio of 0.83%. This is congruent with the previous discussions that identify the transition processes for drive ratios as low as 0.45%.