## 3 Results and Discussion
### Laminar Flow Model  
According to [PERSON] and [PERSON] [7], an oscillatory flow is considered laminar as long as the Reynolds number (based on the thickness of Stokes' layer) is less than 400. This is defined as \(Re_{\delta}=2u_{m}/(v\omega)^{1/2}\), with \(u_{m}\), \(v\) and \(\omega\) representing the axial velocity amplitude, viscosity and angular velocity, respectively. The values of the Stokes Reynolds number for all drive ratios investigated by [PERSON] et al. [16; 17] are shown in Table 1. All of these are lower than the critical value suggested in [7],\(Re_{\delta}=Re_{c}=400\). Thus, at first, a laminar model was used to model the flow and heat transfer for the drive ratios (\(\text{Dr}=P_{a}/P_{o}\); \(P_{o}\) is the mean pressure in the resonator) in Table 1.  
Figure 3 shows the variation over the cycle of the axial velocity at the channel centre, 10 mm from the joint between the HHX and CHX plates, inside the cold channel. Results from the laminar model, for the four drive ratios, are plotted against the experimental results. For all drive ratios, the numerical and experimental values of velocity agree reasonably well, but with slight discrepancies at some phases of a flow cycle. Maximum deviation between numerical and experimental values for drive ratios of 0.30%, 0.45%, 0.65% and 0.83% are 8%, 4.7%, 10.1% and 4.6%, respectively. However, the centreline velocity alone is not sufficient to represent the flow, and therefore, the instantaneous velocity profiles across the channel need to be inspected. These are plotted in Figure 4 for the same axial location, from the wall surface (\(y=0\)) to the middle of the channel (\(y=3\) mm), for all 20 phases of a flow cycle, separately for each drive ratio. Clearly, the already shown mismatch between the CFD and experimental magnitudes of axial velocity at \(y=3\) mm can be seen also in Figure 4, but more importantly, the whole profiles show substantial discrepancies at other locations, especially near the wall. The over-prediction of velocity profiles near the wall as calculated by the laminar model hints at the presence of turbulence [25]. Furthermore, as the drive ratio increases, the discrepancies between velocity profiles from laminar CFD and experiments become larger, which indicates that the laminar model may not be sufficient to capture the flow behaviour. These discrepancies are also illustrated in Figure 5 as a vorticity contour map (drive ratio 0.83% selected as an example), which covers the area of the hot and cold channel. The numerical solutions appear to indicate the presence of very strong elongated secondary vortices (also referred to as a secondary shear layer), which are much less intense in reality, the best examples of such discrepancies being shown for \(\phi 3\). Furthermore, the laminar model also produces \"blobs\" of vorticity that travel within the channel between phases \(\phi 13\) and \(\phi 17\), and this also has no match in the experimental results. Many investigations (cf. [4; 5]) have shown that oscillatory flow past parallel plate structures creates strong vortex structures at the end of plates. The remains of these structures tend to be re-entrained into the channel as the flow changes direction. This suggests that there may be sufficient instabilities present in the flow to trigger turbulent transition. Furthermore, the critical Reynolds number defined in [7] was strictly for a flow within a long, ideally infinite, channel. Therefore, it is possible that transition may be triggered at much lower \(Re_{\delta}\) for flows past short plates. This is why a range of turbulent models has been tested as presented in the next section. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Drive Ratio,** & **Centreline Velocity Amplitude,** & **Stokes Reynolds Number,** \\
**Dr (\%)** & \(u_{m}\) **(m/s)** & \(Re_{\text{ffi}}\) \\ \hline
0.30 & 1.30 & 70.25 \\
0.45 & 1.90 & 101.49 \\
0.65 & 2.97 & 155.34 \\
0.83 & 3.84 & 202.33 \\ \hline \hline \end{tabular}
\end{table}