## 1 Introduction
[PERSON] and [PERSON] [12] were among the first to experimentally investigate oscillatory flow in a pipe and discovered the so-called \"annular effect\", i.e., that the maximum velocity in an oscillatory flow occurs near the wall rather than at the center of the pipe, as is the case under steady flow. Experiments such as those of [PERSON] et al. [13], [PERSON] et al. [14], and [PERSON] and [PERSON] [15] showed that the appearance of turbulence (perturbation) in an alternating flow is different from that in a quasi-stationary, unidirectional flow. The onset of turbulence happens only in the early deceleration phase, whereas in the acceleration phase the flow regime is quasi-laminar. Other works, performed by [PERSON] et al. [16], [PERSON] et al. [17] and [PERSON] and [PERSON] [18], also show that the oscillating flow regime can be either laminar, disturbed laminar (small perturbations appearing during the acceleration phase of the cycle), highly turbulent for the entire cycle, or intermittently turbulent (at the beginning of the deceleration part of the cycle). Not only is there a fundamental difference between steady and oscillatory flow, but many experimental and analytical investigations have shown that under conditions of oscillating flow, the pressure drop is not always in phase with the velocity. Thus, predictingphase shift in alternating flow is crucially important for designing alternating flow-based devices. This is why the phase shift has received particular attention. For example, [PERSON] et al. [19] showed that the influence of the regenerator length on the phase shift between pressure and velocity is negligible compared to that of frequency. The authors prove that the pressure drop coefficient is difficult to determine due to the phase shift variation with velocity (up to 50\({}^{\circ}\)). [PERSON] et al. [20] used modified phase shift as a dimensionless parameter to predict the phase shift \(\phi_{\Delta P}\) between the pressure drop and the mass flow rate at the warm end of oscillating flow through a twill screen regenerator. They found that all their experimental cases can be expressed by a single relationship:  
\[\phi_{\Delta P}=1.32\times 10^{-8}(D_{h}/L)^{-2.62}\;Va, \tag{3}\]  
where \(Va=Re_{\omega}/4\) is the Valensi number (same as kinetic Reynolds number, ratio of the time scale of viscous penetration to the oscillation period [11]). Another analytical correlation was obtained by [PERSON] [21] to predict the phase shift between the velocity and the pressure of oscillating flow through a porous medium channel bounded by two impermeable parallel plates. He found a significant phase shift of 90\({}^{\circ}\) between the velocity and the pressure. His results showed that the phase shift can be predicted using the Stokes number, external diameter, and porous medium shape parameter. The annular effect reported by [PERSON] [21] could be responsible for this phase shift. [PERSON] et al. [22] observed significant phase differences between the movements of gas inside porous media and the movements of the piston. This phase shift increases with the increasing pressure drop of porous media. A combined theoretical and experimental study on fluid motion in the wire mesh regenerator of a Stirling engine in an oscillatory flow was conducted by [PERSON] et al. [23]. They found a small phase shift between the pressure drop and velocity. It was stated that the turbulent intensity in the regenerator exit flow field is stronger during the decelerating period than during the accelerating one. Friction factors defined by adjusting the phase shift between the pressure drops and velocity variations show that it appears bigger in the accelerating period than in the decelerating period. [PERSON] and [PERSON] [8] found that the phase shift between the pressure drop and the velocity strongly depends on the kinetic Reynolds number \(Re_{\omega}\) and less so on the dimensionless displacement of the fluid. [PERSON] et al.