## 3 Results and Discussions
### Investigation of the Effect of the Initial Fluid Temperature on Flow and Heat Transfer
The velocity amplitudes at \(\phi 6\) and \(\phi 16\) shown in Figure 10b represent the highest velocity amplitude during the first and second parts of the flow cycle. The velocity amplitude for the model initialised at 360 K is always higher than the other two models for all the locations along the heat exchanger plate. Figure 9: Temperature contours at phase \(\phi 1\) from the experiment (**a**) and three numerical models initialised with different temperature fields (**b–d**). Finally, the effect of initial temperature on the heat flux is shown in Figure 11. Here, the heat flux is spatially averaged over the length of the heat exchanger (the same way as in [8]), and is presented as a function of time (phase in the cycle):  
\[q_{h,c}=\frac{1}{\overline{I}}\int_{0}^{l}-k\frac{dT}{dy}\bigg{|}_{null}dx. \tag{8}\]  
Subscripts \(h\) and \(c\) refer to HHX and CHX, respectively. The results were taken at cycle 70 after the model reached a steady oscillatory condition. Overall, the space-averaged wall heat fluxes from CFD are in-phase with the experimental values at both heat exchangers. However the magnitude differs at the HHX, especially when the flow starts slowing down at \(\phi\)6 and changes direction. The numerical space-time-averaged wall heat flux is calculated to be 30% lower than the experimental values. However, the difference is within the range reported in other numerical investigations [26]. The wall heat flux predicted by the model initialised at 360 K seems closest to the experiment. Also, the predictions for CHX seem to follow the experiment better. The above discussion shows that the choice of initial temperature field is an important factor in the model, and it may lead to difficulties with convergence and accuracy. Figure 10: The effect of initial temperature on (**a**) the oscillating velocity at location 1 mm from the joint above the cold plate and (**b**) the velocity amplitude along the heat exchanger’s plate taken at the centre of the channel for \(\phi\)6 and \(\phi\) 16. Figure 12 shows the temperature profiles plotted along the vertical direction \(y_{r}\), between the bottom and top wall of the resonator, at locations in the open areas: 40 mm from the cold end and 38 mm from the hot end of the heat exchangers, that is where most of natural convection effects are identified experimentally. Clearly, initialising the temperature at 360 K (dotted lines) gives the wrong values of temperature especially at the lower area of the domain compared to the experiments. Initialising the temperature field as \(T(x,y_{r})\) (dashed lines), seems to give a closer match in this area. Finally, it is interesting to look at the development of the temperature field as a function of the number of cycles taken in the solution. Here the model initialised at 300 K is taken as an example; the temperature profiles are plotted in Figure 12 for increasing the total number of cycles as coloured lines between the dash-dotted black line (45 cycles) and solid black line (490 cycles). The temperature profiles seem to tend towards the experimental values. However, the changes become smaller and smaller (especially after about 70 cycles) and it is unlikely that the profiles would ever reach the experimental data. Following this observation, all the models used in the analysis reported in this paper were run for at least 70 cycles so that the steady oscillatory thermal condition is reached from a practical point of view, in addition to the steady oscillatory flow condition reported in Figure 4.  
Figure 11: The effect of initial temperature on the wall heat flux of the heat exchangers. Figure 12: Temperature profiles at the open area next to the (**a**) cold and (**b**) hot ends of the heat exchanger assembly—comparison between the experimental and numerical model with different approaches of temperature initialisation. Figure 13 shows the temperature field within and around the heat exchanger assembly for the initial temperature of 300 K after 490 cycles.