## 2 Computational Model
### CFD Model Validation  
The velocity amplitude obtained from the converged model that reaches a steady oscillatory state is compared to the experimental result to validate the model. The velocity amplitudes shown in  
Figure 6: Grid independency test for (**a**) axial velocity amplitude at the centre of the channel for phases \(\phi 6\) and \(\phi 16\) and (**b**) velocity profiles near the wall; both taken 15 mm from the joint above the cold plate. Figure 7a,b are for the location at the centre of the gap between the plates of the heat exchanger, 1 mm away from the joint--cf., Figure 2c--for all 20 phases in a cycle. In both graphs, the line denoted as \"Theory_dT = 0\" represents the theoretical/analytical solution for oscillatory velocity calculated from the general thermoacoustic theory as given by [1]:  
\[u_{1}=\frac{i}{\omega\rho_{m}}[1-h_{\
u}(y,z)]\frac{\partial p_{1}}{\partial x}, \tag{6}\]  
where \(\omega\), \(\rho_{m}\), and \(\partial p_{1}/\partial x\) are the angular velocity, mean density, and the pressure gradient of the oscillatory flow, respectively. The term \(h_{\
u}\) is a viscous shape factor which varies according to the geometry of the internal structure involved in the system. The shape factor for parallel plate geometry is given as [1]:  
\[h_{\kappa,\
u}=\frac{\cosh[(1\ +\ i)y/\delta_{\kappa,\
u}]}{\cosh[(1\ +\ i)y_{0}/ \delta_{\kappa,\
u}]}. \tag{7}\]  
The shape factor defines the viscous, \(\
u\), or thermal, \(\kappa\), effects depending on the definition of the penetration depths used. The terms: \(y\), \(y_{o}\), and \(\delta_{\kappa,\
u}\) are the distance from the centre of the gap to the wall of the plate, the centre of the gap, and the thermal (\(\delta_{\kappa}=\sqrt{2\kappa/\omega}\)) or viscous (\(\delta_{\
u}=\sqrt{2\
u/\omega}\)) penetration depth. Figure 7a, for the adiabatic case, compares the experimental results of [PERSON] et al. [11] (\"Experiment_dT = 0\") with the current numerical results (\"CFD_Laminar_dT = 0\"). The theoretical solution is calculated for the mean density taken at 300 K. A good match between the experiment, CFD, and analytical solution can be seen. Maximum discrepancies between the three methods appear at phases \(\phi\)6 and \(\phi\)16 representing the highest velocity values for the two parts of the flow cycle. Comparison between the theory and experiment shows that the theoretical value is 9.3% lower than the experimental value at \(\phi\)6, and 2.6% lower at \(\phi\)16. However, CFD results differ from the experimental results only by 0.3% and 1.8% for \(\phi\)6 and \(\phi\)16, respectively. Figure 7b shows the velocity oscillations obtained from the experiment and CFD for the case with imposed temperature difference (the theoretical adiabatic solution is only plotted for reference). Experimental and CFD results show that, with the imposed wall temperature condition, the flow in the second half of the cycle exhibits a larger amplitude of velocity compared to the first half of the cycle. This is counterintuitive since one would expect that a hotter fluid would travel with higher velocity in the first half of the cycle, while cold fluid would flow with a lower velocity in the second half of the cycle. However it is possible that there is a mean flow along the channels (streaming effect) due to the convective currents as discussed later in Section 3.2.2. Figure 7: Centreline velocity amplitude (1 mm from the joint, above the cold plate) obtained from the model with heat exchanger walls: (**a**) adiabatic and (**b**) with the temperature profile based on the experiment. Figure 8 shows the validation of temperature profiles for a location 15 mm from the joint--cf., Figure 2c--above both the cold and hot plates. The maximum relative discrepancies of 6% were found between the numerical and experimental values when using the absolute temperature scale (17 K on the HHX and 18 K on CHX).