Entry Length Requirements for Two- and Three-Dimensional Laminar Couette–Poiseuille Flows

In this study, we examine the development length requirements for laminar Couette–Poiseuille flows in a two-dimensional (2D) channel as well as in the three-dimensional (3D) case of flow through a square duct, using a combination of numerical and experimental approaches. The parameter space investigated covers wall to bulk velocity ratios, r, spanning from 0 (purely pressure-driven flow) to 2 (purely wall driven-flow; 4 in the case of a square duct) and a wide range of Reynolds numbers (Re). The results indicate an increase in the development length (L) with r. Consistent with the findings of Durst et al. (2005, “The Development Lengths of Laminar Pipe and Channel Flows,” ASME J. Fluids Eng., 127(6), pp. 1154–1160), L was observed to be of the order of the channel height in the limit as $Re→0$⁠, irrespective of the condition at the inlet. This, however, changes at high Reynolds numbers, with L increasing linearly with Re. In all the cases considered, a uniform velocity profile at the inlet was found to result in longer entry lengths than in a flow developing from a parabolic inlet profile. We show that this inlet effect becomes less important as the limit of purely wall-driven flow is approached. Finally, we develop correlations for predicting L in these flows and, for the first time, also present laser Doppler velocimetry (LDV) measurements of the developing as well as fully-developed velocity profiles, and observe good agreement between experiment, analytical solution, and numerical simulation results in the 3D case.