## 1 Introduction  
The alternating or oscillating flows within a regenerator are of scientific and industrial interest. They are frequently found in many industrial applications, such as Stirling machines, magneto-caloric coolers with active regeneration, heat exchangers, compressors, and internal combustion engines. These machines operate according to thermodynamic cycles requiring the use of an alternating fluid flow. The characterization of regenerators operating in an alternating regime constitutes an essential need for the thermal or thermo-mechanical modeling of these systems. Experimental and theoretical works are being devoted to the thermo-fluidics of alternating flows within a regenerator. The first regenerator was developed in 1816 by [PERSON] for the improvement of his hot air engine. The regenerator or economizer of a Stirling machine increases the efficiency of the thermodynamic cycle and it is located between the hot and cold heat exchangers. It often consists of either a porous material or an assembly of small grids of fine wire cloth, plates, or balls, through which the working gas flows in an alternating fashion. Various authors, such as [PERSON] [1], [PERSON] [2], and [PERSON] et al. [3], have proposed correlationsfor the pressure drop (or the Darcy friction factor) in a regenerator based on unidirectional steady flow experiments. In previous works, the Darcy friction factor \(f\) has been calculated using the following equation ([PERSON] et al. [4]):  
\[f=\frac{\Delta PD_{h}}{\frac{1}{2}\rho u^{2}L}, \tag{1}\]  
where \(\Delta P\) is the pressure drop, \(D_{h}\) is the hydraulic diameter, \(\rho\) is the fluid density, \(L\) is the length of the regenerator matrix, and \(u\) is the velocity of the fluid flow. Previous experimental and analytical investigations showed that the steady flow condition is not exactly the same as that of the oscillating flow. Several authors ([PERSON] and [PERSON] [5], [PERSON] et al. [4], [PERSON] and [PERSON] [6], and [PERSON] and [PERSON] [7]) conducted experiments that reveal that pressure drops across the regenerator are different under oscillating flow conditions than under steady flow at the same Reynolds number. [PERSON] and [PERSON] [8] investigated the oscillatory pressure drop characteristics in packed columns (composed of three different sizes of woven screen) subjected to a periodically reversing flow of air. They found that the cycle-averaged pressure drop of the oscillatory flow in a packed column is four to six times higher than that of a steady flow. In an experimental work, [PERSON] et al. [9] presented two test rigs that could generate steady and oscillating flows through a woven screen regenerator under similar conditions. They measured velocity at both ends of the regenerator. This is different from theoretical velocity based on piston velocity, which should begin at a \(0^{\circ}\) crank angle of the piston (bottom dead center) and end at \(180^{\circ}\) (top dead center). It was also shown that the effect of gas compression cannot be ignored for steady and oscillating flows through a regenerator when mass flow increases. They stated that the theoretical velocity could be used only when the effect of gas compression is negligible. By studying the pressure drop for a smooth tube with tri-orifice baffle inserts under steady and oscillatory flow conditions, [PERSON] et al. [10] showed that the pressure drop coefficient is identical to the steady flow one at a very low kinetic Reynolds number (\(Re_{\omega}<30\)). The kinetic Reynolds number \(Re_{\omega}\) is a dimensionless expression for the oscillating inertial forces in relation to viscous forces ([PERSON] and [PERSON] [11]):  
\[Re_{\omega}=\frac{\rho\omega D_{h}^{2}}{\mu}, \tag{2}\]  
where \(\rho\) is the density, \(\omega\) is the angular frequency, and \(\mu\) is the dynamic viscosity. The kinetic Reynolds number is also called the oscillating Reynolds number. They also observed that the pressure drop signal under oscillatory flow is not a purely sinusoidal wave.