## Appendix A Dimensional Analysis  
The dimensional analysis required in Section 3.4 is based on the analysis of the transport equations that govern the flow. The analysis is carried out using a two-dimensional Navier-Stokes equation for compressible flow. In this study, the normalisation parameters are introduced as:\[u^{*}=\frac{u}{\sigma u_{c}};\,\,v^{*}=\frac{v}{\sigma u_{c}};\,\,x^{*}=\frac{x}{ d_{c}};\,\,y^{*}=\frac{y}{d_{c}};\,\,t^{*}=t\omega;\,\,p^{*}=\frac{p}{\rho_{c} \sigma^{2}u_{c}^{2}}; \tag{11}\] \[\rho^{*}=\frac{\rho}{\rho_{c}};\,\,g^{*}=\frac{\delta}{g_{c}};\,\, \beta^{*}=\frac{\beta}{\beta_{c}};\,\,\mu^{*}=\frac{\mu}{\mu_{c}};\,\,T^{*}= \frac{T-T_{c}}{T_{H}-T_{c}};\,\,c_{p}^{*}=\frac{c_{p}}{c_{p}c};\,\,k^{*}=\frac{ k}{k_{c}}.\]  
The superscript \({}^{*}\) describes the dimensionless variable. The subscript c refers to a reference value, preferably selected in the open area a distance away from the stack/regenerator where the properties and behaviour of the fluid are not affected much by the presence of the object. The velocity is normalised by the velocity at the reference point multiplied by the porosity, \(\sigma\). The dimensions \(x\) and \(y\) are both normalised by the characteristic dimension, \(d_{c}\)--here the length of the heat exchanger plate (35 mm) was used. The temperatures \(T_{H}\) and \(T_{\rm C}\) selected represent the hot and cold plate/fluid temperatures, respectively.  
The resulting dimensionless equations for the two-dimensional Navier-Stokes equations are:  
Continuity equation  
\[Re_{\omega}\frac{\partial\rho^{*}}{\partial t^{*}}+\sigma Re_{d}\frac{ \partial(\rho^{*}u^{*})}{\partial x^{*}}+\sigma Re_{d}\frac{\partial(\rho^{*}v ^{*})}{\partial y^{*}}=0, \tag{12}\]  
\(x\)-momentum equation  
\[\begin{split}& Re_{\omega}\frac{\partial}{\partial t^{*}}(\rho^{*}u ^{*})+\sigma Re_{d}\frac{\partial}{\partial x^{*}}\big{(}\rho^{*}u^{*2}\big{)} +\sigma Re_{d}\frac{\partial}{\partial y^{*}}\big{(}\rho^{*}u^{*}v^{*})+ \sigma Re_{d}\frac{\partial p^{*}}{\partial x^{*}}=\\ &\frac{Gr}{\sigma Re_{d}}\rho^{*}g^{*}g^{*}T^{*}+\frac{\partial}{ \partial x^{*}}\bigg{[}2\mu^{*}\frac{\partial u^{*}}{\partial x^{*}}-\frac{2}{ 3}\mu^{*}\bigg{(}\frac{\partial u^{*}}{\partial x^{*}}+\frac{\partial v^{*}}{ \partial y^{*}}\bigg{)}\bigg{]}+\frac{\partial}{\partial y^{*}}\bigg{[}\mu^{* }\bigg{(}\frac{\partial v^{*}}{\partial x^{*}}+\frac{\partial u^{*}}{ \partial y^{*}}\bigg{)}\bigg{]},\end{split} \tag{13}\]  
\(y\)-momentum equation  
\[\begin{split}& Re_{\omega}\frac{\partial}{\partial t^{*}}\big{(} \rho^{*}v^{*}\big{)}+\sigma Re_{d}\frac{\partial}{\partial x^{*}}\big{(}\rho^ {*}u^{*}v^{*}\big{)}+\sigma Re_{d}\frac{\partial}{\partial y^{*}}\big{(}\rho^ {*}v^{*2}\big{)}+\sigma Re_{d}\frac{\partial p^{*}}{\partial y^{*}}=\\ &\frac{Gr}{\sigma Re_{d}}\rho^{*}g^  
flow. Depending on the definition of the reference temperature used, this dimensionless number may represent the effect of heat accumulation within the investigated area. The effect can be seen in the transient and convection part of the energy equation. This indicates that heat accumulation changes over time and is very much dependent on the amplitude of the flow. The Eckert number, _Ec_, expresses the relationship between the flow kinetic energy and enthalpy, and is used to characterise dissipation. The Prandtl number, _Pr_, describes the property of fluids, which is useful when comparing cases with different fluid media.