The laminar decay of suddenly blocked channel and pipe flows

Abstract

This paper is a theoretical investigation of the stable laminar decay of a fully established channel or pipe flow following a sudden blockage such as would be caused by the rapid closure of a valve or imposition of an end wall or gate. The development of the subsequent velocity and pressure fields is examined from the instant the initial pressure wave passes until the final decay of all motion. Three time scales of hydrodynamic interest are identified and the relevant solutions are obtained. The time scales are as follows: (i) a very short time characteristic of the passage of the pressure wave during which the velocity field adjusts inviscidly to the new boundary conditions imposed by the presence of the end wall, (ii) a short diffusion time during which the displacement interaction generated by the diffusion of the primary Rayleigh layer induces a substantial secondary motion with distinct side-wall boundary layers and an inviscid core and (iii) a long diffusion time during which the boundary layers fill the entire channel or pipe and the residual motion then dies out. The secondary flow for short diffusion times is of special interest in that it is an example of an unsteady boundary layer where the external pressure gradient and inviscid outer flow are unknown and determined by the integrated time history of the combined mass flow displacement generated by the primary- and secondary-flow boundary layers. The paper closes with some preliminary comments and experimental observations on decelerating pipe flows.