Transient growth analysis of flow through a sudden expansion in a circular pipe

Results are presented from a numerical study of transient growth experienced by infinitesimal perturbations to flow in an axisymmetric pipe with a sudden 1-2 diametral expansion. First, the downstream reattachment point of the steady laminar flow is accurately determined as a function of Reynolds number and it is established that the flow is linearly stable at least up to Re=1400. A direct method is used to calculate the optimal transient energy growth for specified time horizon tau, Re up to 1200, and low-order azimuthal wavenumber m. The critical Re for the onset of growth with different m is determined. At each Re the maximum growth is found in azimuthal mode m=1 and this maximum is found to increase exponentially with Re. The time evolution of optimal perturbations is presented and shown to correspond to sinuous oscillations of the shear layer. Suboptimal perturbations are presented and discussed. Finally, direct numerical simulation in which the inflow is perturbed by Gaussian white noise confirms the presence of the structures determined by the transient growth analysis. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3313931]