Laminar flows starting-up from rest in round tubes are relevant to numerous industrial and biomedical applications. The two most common types are flows driven by an abruptly imposed constant pressure gradient or by an abruptly imposed constant volume flux. Analytical solutions are available for transient, fully developed flows wherein stream wise development over the entrance length is absent (Szymanski, 1932; Andersson & Tiseth 1988, respectively). They represent the transient responses of flows in tubes that are very long compared to the entrance length, a condition that is seldom satisfied.
This study establishes the entrance (development) length and development time of starting laminar flow in a round tube of finite length driven by a piston pump that produces a step change from zero flow to a constant volume flux for Reynolds numbers between 500 and 3000. The flows are examined experimentally, using stereographic particle image velocimetry (PIV) and computationally using computational fluid dynamics (CFD), and are then compared to the known analytical solutions for fully developed flow conditions in infinitely long tubes.
Results showed that step function volume flux start-up flows reach steady state and fully developed flow much more quickly than those driven by a step function pressure gradient. Based on these results, we present new, simple guidelines for achieving experimental flows that are fully developed in space and time in realistic (finite) tube geometries. To a first approximation, the time to achieve steady spatially developing flow is nearly equal to the time needed to achieve steady, fully developed flow. Conversely, the entrance length needed to achieve fully developed transient flow is approximately equal to the length needed to achieve fully developed steady flow. Beyond this level of description, the numerical results reveal interaction between the effects of space and time development and non-linear Reynolds number effects.
In this study, we examine the simultaneous entrance flow development and transient response of starting flow in a finite length tube when a constant volume flux at various Reynolds numbers is imposed suddenly on a resting flow. PIV measurements of the transient, spatially developing flow are used to verify the analytical solution for fully developed start-up and the numerical simulation of developing start-up flow. Numerical solutions at many Reynolds numbers are used to develop a simple, complete description of the simultaneous development of the flow in space and time. Based on these results, we formulate simple, approximate rules for achieving fully developed, steady state flow in starting flows driven by piston pumps.
We have manuscripts under review and in preparation discussing our methods and results in more detail. We are currently working on predicting input/output waveforms at downstream locations in phantom models.