Linear stability of spiral and annular Poiseuille flow for small radius ratio

David L. Cotrell, Arne J. Pearlstein

Research output: Contribution to journalArticlepeer-review

Abstract

For the radius ratio η ≡ Ri/Ro=0.1 and several rotation rate ratios μ ≡ Ωoi we consider the linear stability of spiral Poiseuille flow (SPF) up to Re=105, where Ri and Ro are the radii of the inner and outer cylinders, respectively, Re ≡ Vz (Ro-Ri/ν is the Reynolds number, Ωi and Ωo are the (signed) angular speeds of the inner and outer cylinders, respectively, ν is the kinematic viscosity, and Vz is the mean axial velocity. The Re range extends more than three orders of magnitude beyond that considered in the previous μ=0 work of Recktenwald et al. (Phys. Rev. E, vol. 48, 1993, p. 444). We show that in the non-rotating limit of annular Poiseuille flow, linear instability does not occur below a critical radius ratio ̂ηap;0.115. We also establish the connection of the linear stability of annular Poiseuille flow for 0<η≤̂η at all Re to the linear stability of circular Poiseuille flow (η=0) at all Re. For the rotating case, with μ=-1, -0.5, -0.25, 0 and 0.2 the stability boundaries, presented in terms of critical Taylor number Ta ≡Ωi(Ro-Ri)2/ν versus Re, show that the results are qualitatively different from those at larger η. For each η the centrifugal instability at small Re does not connect to a highRe Tollmien-Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for η<̂η. We find a range of Re for which disconnected neutral curves exist in the k-Ta plane, which for each non-zero μ considered, lead to a multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating (μ<0) case, there is a finite range of Re for which there exist three critical values of Ta, with the upper branch emanating from the Re instability of Couette flow. For the co-rotating (μ=0.2) case, there are two critical values of Ta for each Re in an apparently semi-infinite range of Re, with neither branch of the stability boundary intersecting the Re = 0 axis, consistent with the classical result of Synge that Couette flow is stable with respect to all small disturbances if μ > η2, and our earlier results for μ > η2 at larger η.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalJournal of Fluid Mechanics
Volume547
DOIs
StatePublished - Mar 21 2006

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

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