Revisiting the stability of pulsatile pipe flow

We revisit the problem of the stability of pulsatile pipe flow for axisymmetric perturbations. In contrast to the earlier approach based on the Chebyshev expansion for the spatial discretization [J. Appl. Mech. ASME 53 (1986) 187], we use the set of the eigenfunctions derived from the longwave limit of the Orr-Sommerfeld equation. We show that the Orr-Sommerfeld basis gives greater accuracy than the Chebyshev basis if fewer terms are used in the Galerkin expansion. For the time evolution of the flow perturbation, instead of the usual Floquet analysis, a different representation for the solution of the periodic system of linear differential equations is employed. We found that the flow structures corresponding to the largest energy growth are toroidal vortex tubes. They are stretched by the shear stress of the mean flow so that a maximum energy growth occurs. The flow perturbation subsequently decays due to viscous effects. The maximum energy growth is then evaluated over a range of Reynolds and Womersley numbers. Asymptotic solutions provided for the longwave limit as well as the limit of large Womersley numbers agree well with the numerical results, confirming the known linear stability of the flow.