Optimal linear growth in spiral Poiseuille flow

Abstract

Computations are presented of the optimal linear growth in spiral Poiseuille flow(SPF). The aim is to complement a recent presentation of the complete neutral curves for this flow (Cotrell & Pearlstein, J. Fluid Mech. vol. 509, 2004, p. 331) with a study of the transient growth possible in the stable parameter regions. Maximum growth is computed over the full range of axial and azimuthal wavenumbers for the same three test cases as considered by Cotrell & Pearlstein: radius ratio η = −0.5 and rotation rate ratio μ= -0.5, 0 and 0.5. A connection is established between two regimes of optimal transients in spiral Poiseuille flow. The first occurs for axial Reynolds number Re≫1 and Taylor number Ta=O(1), with transient growth of streamwise disturbances analogous to that in non-swirling shear flows. In the second regime Ta≫1, and we find centrifugal transients of a different type. In this latter regime we obtain the first numerical verification of a recently conjectured scaling for centrifugal transient growth. Our results imply different transition scenarios, triggered by either transient growth or asymptotic instabilities, in the small-Re and large-Re regimes, consistent with previous experimental data. We also study a model for the secondary instability of the optimal transients, as a proposed explanation for the subcritical and delayed transition seen in experiments at moderately large Re. The model is found to favour delayed onset for smaller μ and subcritical onset for larger μ, in good qualitative agreement with the experimental data.