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Linear stability of spiral and annular Poiseuille flow for small radius ratio
- DAVID L. COTRELL, ARNE J. PEARLSTEIN
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- Journal:
- Journal of Fluid Mechanics / Volume 547 / 25 January 2006
- Published online by Cambridge University Press:
- 11 January 2006, pp. 1-20
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For the radius ratio $\eta\,{\equiv}\,R_i/R_o\,{=}\,0.1$ and several rotation rate ratios $\mu\,{\equiv}\,\Omega_o/\Omega_i$, we consider the linear stability of spiral Poiseuille flow (SPF) up to ${\hbox {\it Re}}\,{=}\,10^5$, where $R_i$ and $R_o$ are the radii of the inner and outer cylinders, respectively, ${\hbox {\it Re}}\,{\equiv}\,\overline V_Z(R_o\,{-}R_i)/\nu$ is the Reynolds number, $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds of the inner and outer cylinders, respectively, $\nu$ is the kinematic viscosity, and $\overline V_Z$ is the mean axial velocity. The Re range extends more than three orders of magnitude beyond that considered in the previous $\mu\,{=}\,0$ 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 $\hat\eta\,{\approx}\,0.115$. We also establish the connection of the linear stability of annular Poiseuille flow for $0\,{<}\,\eta\,{\leq}\,\hat\eta$ at all Re to the linear stability of circular Poiseuille flow ($\eta\,{=}\,0$) at all Re. For the rotating case, with $\mu\,{=}\,{-}1$, ${-}\,0.5$, ${-}\,0.25$, 0 and 0.2, the stability boundaries, presented in terms of critical Taylor number ${\hbox {\it Ta}}\,{\equiv}\,\Omega_i(R_o\,{-}R_i)^2/\nu$ versus Re, show that the results are qualitatively different from those at larger $\eta$. For each $\mu$, the centrifugal instability at small Re does not connect to a high-Re Tollmien–Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for $\eta\,{<}\,\hat\eta$. We find a range of Re for which disconnected neutral curves exist in the $k$–Ta plane, which for each non-zero $\mu$ considered, lead to a multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating ($\mu\,{<}\,0$) case, there is a finite range of Re for which there exist three critical values of Ta, with the upper branch emanating from the ${\hbox {\it Re}}\,{=}\,0$ instability of Couette flow. For the co-rotating ($\mu\,{=}\,0.2$) 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 $\mu\,{>}\,\eta^2$, and our earlier results for $\mu\,{>}\,\eta^2$ at larger $\eta$.
The connection between centrifugal instability and Tollmien–Schlichting-like instability for spiral Poiseuille flow
- DAVID L. COTRELL, ARNE J. PEARLSTEIN
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- Journal:
- Journal of Fluid Mechanics / Volume 509 / 25 June 2004
- Published online by Cambridge University Press:
- 07 June 2004, pp. 331-351
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For spiral Poiseuille flow with radius ratio $\eta\,{\equiv}\,R_i\slash R_o\,{=}\,0.5$, we have computed complete linear stability boundaries for several values of the rotation rate ratio $\mu\equiv{\it\Omega}_o\slash{\it\Omega}_i$}, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively, and ${\it \Omega}_i$ and ${\it \Omega}_o$ 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 $\mu\,{<}\,\eta^2$) 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 $\mu\,{>}\,\eta^2$, we show that there is no instability for $0\,{\leq}\,Re\,{\leq}\,Re_{min}$. For $\mu\,{=}\,0.5$, $Re_{min}$ 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 $\mu\,{=}\,1$, with the two cylinders having the same angular velocity, $Re_{min}$ corresponds to a vertical asymptote smaller than found by Meseguer & Marques (2002), whose results for $\mu\,{>}\,\eta^2$ fail to account for disturbances with a sufficiently wide range of azimuthal wavenumbers.
Computational assessment of subcritical and delayed onset in spiral Poiseuille flow experiments
- DAVID L. COTRELL, SARMA L. RANI, ARNE J. PEARLSTEIN
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- Journal:
- Journal of Fluid Mechanics / Volume 509 / 25 June 2004
- Published online by Cambridge University Press:
- 07 June 2004, pp. 353-378
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For spiral Poiseuille flow with radius ratios $\eta \equiv R_i/R_o = 0.77$ and 0.95, we have computed complete linear stability boundaries, where $R_i$ and $R_o$ 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 $\mu \equiv \Omega_o/\Omega_i$, where $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds. For each combination of $\mu$ and $\eta$, 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 $\mu = 0$ data, and for all of the non-zero-$\mu$ data. For $\mu = 0$ and $\eta = 0.955$, agreement with data from an annulus with aspect ratio (length divided by gap) greater than 570 is within 3.2% for $Re \leq 325$ (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 $Ta_{\hbox{\scriptsize{\it crit}}}$ exceeding computed values. For $\mu = 0$ and smaller $\eta$, comparison to experiment (with smaller aspect ratios) at low $Re$ is slightly less good. For $\eta = 0.77$ and a range of $\mu$, agreement with experiment is very good for $Re < 135$ except at the most positive or negative $\mu$ (where $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$), whereas for $Re \geq 166$, $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$ for all but the most positive $\mu$. For $\eta = 0.9497$ and 0.959 and all but the most extreme values of $\mu$, agreement is excellent (generally within 2%) up to the largest $Re$ considered experimentally (200), again suggesting that finite-amplitude instability is unimportant.