ALBERT

Description: For spiral Poiseuille flow with radius ratios η ≡ Ri/Ro = 0.77 and 0.95, we have computed complete linear stability boundaries, where Ri and Ro are the inner and outer cylinder radii, respectively. The analysis accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Reynolds numbers Re for which the flow is stable for some range of Taylor number Ta, and extends previous work to several non-zero rotation rate ratios μ ≡ Ωo/Ωi, where Ωi and Ωo are the (signed) angular speeds. For each combination of μ and η, there is a wide range of Re for which the critical Ta is nearly independent of Re, followed by a precipitous drop to Ta = 0 at the Re at which non-rotating annular Poiseuille flow becomes unstable with respect to a Tollmien-Schlichting-like disturbance. Comparison is also made to a wealth of experimental data for the onset of instability. For Re 〉 0, we compute critical values of Ta for most of the μ = 0 data, and for all of the non-zero-μ data. For μ = 0 and η = 0.955, agreement with data from an annulus with aspect ratio (length divided by gap) greater than 570 is within 3.2% for Re ≤ 325 (based on the gap and mean axial speed), strongly suggesting that no finite-amplitude instability occurs over this range of Re. At higher Re, onset is delayed, with experimental values of Tacrit exceeding computed values. For μ = 0 and smaller η, comparison to experiment (with smaller aspect ratios) at low Re is slightly less good. For η = 0.77 and a range of μ, agreement with experiment is very good for Re 〈 135 except at the most positive or negative μ (where Taexptcrit 〉 Tacompcrit), whereas for Re ≥ 166, Taexptcrit 〉 Tacompcrit for all but the most positive μ. For η = 0.9497 and 0.959 and all but the most extreme values of μ, agreement is excellent (generally within 2%) up to the largest Re considered experimentally (200), again suggesting that finite-amplitude instability is unimportant. © 2004 Cambridge University Press.

Description: For spiral Poiseuille flow with radius ratio η ≡ Ri/Ro = 0.5, we have computed complete linear stability boundaries for several values of the rotation rate ratio μ ≡ Ωo/Ωi, where Ri and Ro are the inner and outer cylinder radii, respectively, and Ωi and Ωo are the corresponding (signed) angular speeds. The analysis extends the previous range of Reynolds number Re studied computationally by more than eightyfold, and accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Re for which spiral Poiseuille flow is stable for some range of the Taylor number Ta. We show how the centrifugally driven instability (beginning with steady or azimuthally travelling-wave bifurcation of circular Couette flow at Re = 0 when μ 〈 η2) connects, as conjectured by Reid (1961) in the narrow-gap limit, to a non-axisymmetric Tollmien-Schlichting-like instability of non-rotating annular Poiseuille flow at Ta = 0. For μ 〉 η2, we show that there is no instability for 0 ≤ Re ≤ Remin. For μ = 0.5, Remin corresponds to a turning point, beyond which exists a range of Re for which there are two critical values of Ta, with spiral Poiseuille flow being stable below the lower one and above the upper one, and unstable in between. For the special case μ = 1, with the two cylinders having the same angular velocity, Remin corresponds to a vertical asymptote smaller than found by Meseguer & Marques (2002), whose results for μ 〉 η2 fail to account for disturbances with a sufficiently wide range of azimuthal wavenumbers. © 2004 Cambridge University Press.