On an analytic solution for general unsteady/transient turbulent pipe flow and starting turbulent flow

Abstract The literature already offers analytic solutions for steady-state laminar (Hagen–Poiseuille), transient laminar (Szymanski) and steady-state turbulent (Pai) incompressible pipe flows. However, no reference could be found for the most general case that encompasses all of them: the unsteady turbulent incompressible pipe flow. This study presents a General Analytic Solution (GAS) for the Reynolds–averaged Navier–Stokes equations corresponding to unsteady turbulent incompressible circular pipe flow, which is composed of the transient response of the initial mean velocity field to external interactions, the unsteady response to time-dependent pressure gradient, and the unsteady component due to the turbulence’s Reynolds stress. The particular cases of laminar, turbulent, steady-state and transient incompressible pipe flows emanate directly from such solution. A general method is proposed to obtain an analytical solution for any particular pipe flow problem for which some relevant function could be provided. That general method is exemplified in a test case, a much simplified version of the starting flow initiated in a circular pipe, after suddenly applying a constant pressure gradient. The time evolution for such starting flow, with a growing stage and a shrinking stage, is presented graphically and discussed.