Temporal stability of eccentric Taylor–Couette–Poiseuille flow

Leclercq, Colin; Pier, Benoı̂t; Scott, Julian F.

doi:10.1017/jfm.2013.437

Cited by 9 publications

(18 citation statements)

References 64 publications

(100 reference statements)

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“…This case presents similarity with the eccentric Taylor-Couette instability problem (see [27][28][29] and references therein). Indeed, in those problems, the axial wavelength of the critical perturbations is always of the same order of magnitude of the gap.…”

Section: Results Of the Numerical Simulationmentioning

confidence: 62%

Centrifugal instability of Stokes layers in crossflow: the case of a forced cylinder wake

2015

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The wake flow around a circular cylinder at Re ≈ 100 performing rotatory oscillations has been thoroughly discussed in the literature, mostly focusing on the modifications to the natural Bénard-von Kármán vortex street that result from the forced shedding modes locked to the rotatory oscillation frequency. The usual experimental and theoretical frameworks at these Reynolds numbers are quasi-two-dimensional, because the secondary instabilities bringing a threedimensional structure to the cylinder wake flow occur only at higher Reynolds numbers. In this paper, we show that a three-dimensional structure can appear below the usual three-dimensionalization threshold, when forcing with frequencies lower than the natural vortex shedding frequency, at high amplitudes, as a result of a previously unreported mechanism: a pulsed centrifugal instability of the oscillating Stokes layer at the wall of the cylinder. The present numerical investigation lets us in this way propose a physical explanation for the turbulence-like features reported in the recent experimental study by the present authors.

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Section: Results Of the Numerical Simulationmentioning

confidence: 62%

Centrifugal instability of Stokes layers in crossflow: the case of a forced cylinder wake

2015

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“…Substituting the normal mode form into the Navier-Stokes equations (2.6)-(2.11) linearized about Q leads to a generalized eigenvalue problem in ω and q ′ . Following Leclercq et al (2013), this problem was reduced to a much smaller standard eigenvalue problem by eliminating the pressure, boundary points, and one velocity component (w ′ if m = 0 andṽ ′ otherwise) and then solved using standard LAPACK and ARPACK++ routines. The solver was validated against figure 6 of Avila et al (2008) with perfect agreement between our computed values of critical Re and values digitized from the plot for Γ = 3, 4, 6 and the various m shown.…”

Section: Linear Stabilitymentioning

confidence: 99%

Using stratification to mitigate end effects in quasi-Keplerian Taylor–Couette flow

et al. 2016

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Efforts to model accretion disks in the laboratory using Taylor–Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{3/2}$, where ${\it\Omega}_{o}/{\it\Omega}_{i}$ is the angular velocity at the outer/inner boundary and $r_{i}/r_{o}$ is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{1/2}$), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number $Re=O(10^{4})$. This observation is consistent with most recent computational (Ostilla-Mónico, et al.J. Fluid Mech., vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high $Re$, and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics.

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“…In this modified bipolar coordinate system, a FourierChebyshev pseudospectral projection method is implemented, with N φ = 2K φ + 1 Fourier modes, and N ξ Gauss-Lobatto collocation points. For more details on the numerical procedure, the reader is referred to Leclercq et al (2013).…”

Section: Basic Flowmentioning

confidence: 99%

“…On the other hand, convective instabilities correspond to wavepackets propagating only in the downstream direction: in the absence of forcing, the system eventually relaxes to its initial state at any fixed location, after perturbations have been 'blown away' from the source. The most temporally amplified perturbations are given by a classical temporal stability analysis, and such a study was recently carried out for this flow (Leclercq, Pier & Scott 2013). It was shown that the physics is essentially similar to the axisymmetric case (Takeuchi & Jankowski 1981;Ng & Turner 1982), with propagating toroidal vortices replaced by helical structures of increasing azimuthal complexity as Re z is increased.…”

Section: Introductionmentioning

confidence: 99%

“…The only known series of experiments were performed by Coney & Mobbs (1969), Coney (1971), Younes (1972), Younes, Mobbs & Coney (1972), Mobbs & Younes (1974), Coney & Atkinson (1978) and show good agreement with our a posteriori predictions, despite small discrepancies attributed to finite-length effects. For a brief review of other theoretical and experimental results on eccentric Taylor-Couette flow on the one hand, and axisymmetric Taylor-CouettePoiseuille flow on the other hand, we refer to Leclercq et al (2013).…”

Section: Introductionmentioning

confidence: 99%

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Absolute instabilities in eccentric Taylor–Couette–Poiseuille flow

2014

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The effect of eccentricity on absolute instabilities (AI) in the Taylor-Couette system with pressure-driven axial flow and fixed outer cylinder is investigated. Five modes of instability are considered, characterized by a pseudo-angular order m, with here |m| 2. These modes correspond to toroidal (m = 0) and helical structures (m = 0) deformed by the eccentricity. Throughout the parameter range, the mode with the largest absolute growth rate is always the Taylor-like vortex flow corresponding to m = 0. Axial advection, characterized by a Reynolds number Re z , carries perturbations downstream, and has a strong stabilizing effect on AI. On the other hand, the effect of the eccentricity e is complex: increasing e generally delays AI, except for a range of moderate eccentricites 0.3 e 0.6, where it favours AI for large enough Re z . This striking behaviour is in contrast with temporal instability, always inhibited by eccentricity, and where left-handed helical modes of increasing |m| dominate for larger Re z . The instability mechanism of AI is clearly centrifugal, even for the larger values of Re z considered, as indicated by an energy analysis. For large enough Re z , critical modes localize in the wide gap for low e, but their energy distribution is shifted towards the diverging section of the annulus for moderate e. For highly eccentric geometries, AI are controlled by the minimal annular clearance, and the critical modes are confined to the vicinity of the inner cylinder. Untangling the AI properties of each m requires consideration of multiple pinch points.

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Temporal stability of eccentric Taylor–Couette–Poiseuille flow

Cited by 9 publications

References 64 publications

Centrifugal instability of Stokes layers in crossflow: the case of a forced cylinder wake

Centrifugal instability of Stokes layers in crossflow: the case of a forced cylinder wake

Using stratification to mitigate end effects in quasi-Keplerian Taylor–Couette flow

Absolute instabilities in eccentric Taylor–Couette–Poiseuille flow

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