An analytical model for the oscillatory flow in constant cross-section ducts

In this paper, an analytical model for the incompressible laminar mean flow induced by an oscillatory pressure gradient is proposed for arbitrary constant cross-section, straight ducts of very large length. With the hydraulic diameter as the characteristic length scale and just one parameter, namely the cross-section loss-coefficient for steady flow, the proposed model provides the modulus and phase of the dimensionless mean flow and volumetric flow rate as a function of the Reynolds frequency number. The model, which is derived from the momentum integral equation using complex algebra, is compared to exact analytical solutions for laminar oscillatory flows. Results for circular, squared, rectangular and triangular cross-sections show that the model accurately reproduces the asymptotic behavior at low and high frequencies, with a reasonable approximation in the intermediate frequency Reynolds number range (1,100), where largest errors occur. The proposed model can be of interest in applied computer programs for engineering, environmental and biological systems that are subject to periodic flows, such as the human circulatory system or the design of hydraulic dampers. • The model gives the magnitude and phase of the oscillatory flow rate as a function of frequency Reynolds number. • The model length scale is the hydraulic diameter. • The only model input is the section friction coefficient in steady flow. • The method is applied to ducts of circular, squared, rectangular and triangular cross-sections.