Pulsating laminar fully developed channel and pipe flows

Analytical investigations are carried out on pulsating laminar incompressible fully developed channel and pipe flows. An analytical solution of the velocity profile for arbitrary time-periodic pulsations is derived by approximating the pulsating flow variables by a Fourier series. The explicit interdependence between pulsations of velocity, mass-flow rate, pressure gradient, and wall shear stress are shown by using the proper dimensionless parameters that govern the flow. Utilizing the analytical results, the scaling laws for dimensionless pulsation amplitudes of the velocity, mass-flow rate, pressure gradient, and wall shear stress are analyzed as functions of the dimensionless pulsation frequency. Special attention has been given to the scaling laws describing the flow reversal phenomenon occurring in pulsating flows, such as the condition for flow reversal, the dependency of the reversal duration, and the amplitude. It is shown that two reversal locations away from the wall can occur in pulsating flows in pipes and channels and the reversed amount of mass per period reaches a maximum at a certain dimensionless frequency for a given amplitude of mass-flow rate fluctuations. These analyses are numerically conducted for pipe and channel flows over a large frequency range in a comparative manner.