Analytical solutions of incompressible laminar channel and pipe flows driven by in-plane wall oscillations

Abstract

Emerging flow control strategies have been proposed to tackle long-lasting problems, for instance, precise mixing of chemicals and turbulent drag reduction. Employing actuators imposing in-plane wall oscillations are particularly popular. This paper investigates incompressible laminar rectangular channel and circular pipe flows driven by uniform and traveling wave in-plane wall oscillations. A comprehensive set of exact analytical solutions are presented describing parallel and concentric flows. Dimensionless groups are identified, and it is described how they characterize the one- and two-dimensional time-dependent velocity and pressure fields. The solutions enable to compute the oscillating boundary layer thickness. It is demonstrated that the dimensionless groups and the boundary layer thickness narrows the region of interest within the parameter space. In particular, the oscillating boundary layer thickness obtained from these laminar flows estimates a “radius of action” within which flow features can be altered to boost mixing or reduce turbulent friction drag. The results are suitable for software validation and verification, may open the way to promising complex wall oscillations, and ease the optimization task that delays the industrial application of flow controls.