Direct Resonance of Nonaxisymmetric Disturbances in Pipe Flow

Abstract

A mechanism which may lead to algebraic growth, followed by exponential decay, of small nonaxisymmetric disturbances in pipe flow is considered. The mechanism is interpreted as a direct resonance between the perturbations of the pressure and the streamwise velocity. The eigenvalue problems for the pressure and the velocity modes have been solved numerically for complex streamwise wave number, and 36 resonances have been investigated. A plot of the propagation speed versus the damping rate shows that the resonances follow certain sequences as the azimuthal wave number increases. The largest propagation speed is found to be ≈ 0.69 times the centerline velocity. No lowest speed is obtained, and as the azimuthal wave number increases the propagation speed decreases. The effects of changing the Reynolds number have also been investigated. It is found that the streamwise wave number and the damping rate are proportional to 1/R as R → ∞. The complex phase speed and the propagation speed become independent of R in the same limit.