Effects of velocity skewness on the linear stability of the oscillatory Stokes layer

AbstractQuantitative results for the linear stability of planar Stokes layers subject to small, high-frequency perturbations are obtained for both a narrow channel and a flow approximating the classical semi-infinite Stokes layer. Previous theoretical and experimental predictions of the critical Reynolds number for the classical flat Stokes layer have differed widely with the former exceeding the latter by a factor of two or three. Here it is demonstrated that only a 1 % perturbation, at an appropriate frequency, to the nominal sinusoidal wall motion is enough to result in a reduction of the theoretical critical Reynolds number of as much as 60 %, bringing the theoretical conditions much more in line with the experimentally reported values. Furthermore, within the various experimental observations there is a wide variation in reported critical conditions and the results presented here may provide a new explanation for this behaviour.

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Linear stability of plane Poiseuille flow over a generalized Stokes layer

Journal of Physics Conference Series ◽

10.1088/1742-6596/318/2/022033 ◽

2011 ◽

Vol 318 (2) ◽

pp. 022033

Author(s):

Maurizio Quadrio ◽

Fulvio Martinelli ◽

Peter J Schmid

Keyword(s):

Linear Stability ◽

Poiseuille Flow ◽

Plane Poiseuille Flow ◽

Stokes Layer

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On the linear stability of Stokes layers

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2008.0055 ◽

2008 ◽

Vol 366 (1876) ◽

pp. 2685-2697 ◽

Cited By ~ 8

Author(s):

P.J Blennerhassett ◽

Andrew P Bassom

Keyword(s):

Reynolds Number ◽

Linear Stability ◽

Practical Interest ◽

Transition To Turbulence ◽

Large Reynolds Number ◽

Poor Agreement ◽

Oscillatory Flows ◽

Stokes Layer ◽

The Stability ◽

Time Periodic

Oscillatory flows occur naturally, with applications ranging across many disciplines from engineering to physiology. Transition to turbulence in such flows is a topic of practical interest and this article discusses some recent work that has furthered our understanding of the stability of a class of time-periodic fluid motions. Our study starts with an examination of the linear stability of a classical flat Stokes layer. Although experiments conducted over many years have demonstrated conclusively that this layer is unstable at a sufficiently large Reynolds number, it has only been relatively recently that rigorous theoretical confirmation of this behaviour has been obtained. The analysis and numerical calculations for the planar Stokes layer were subsequently extended to flows in channels and pipes and for the flow within a torsionally oscillating circular cylinder. We discuss why our predictions for the onset of instability in these geometries are in disappointingly poor agreement with experimental results. Finally, some suggestions for future experimental work are given and some areas for future theoretical analysis outlined.

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The linear stability of flat Stokes layers

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1978.0037 ◽

1978 ◽

Vol 359 (1697) ◽

pp. 151-166 ◽

Cited By ~ 68

Keyword(s):

Reynolds Number ◽

Continuous Spectrum ◽

Linear Stability ◽

Parameter Range ◽

Discrete Eigenvalue ◽

Stokes Layer

The linear stability of a flat Stokes layer is investigated. The results obtained show that, in the parameter range investigated, the flow is stable. It is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number. In addition, a set of discrete eigenvalues exists for certain values of the Reynolds number. The eigenfunctions associated with this set are confined to the Stokes layer while those corresponding to the continuous spectrum persist outside the layer. The effect of introducing a second boundary a long way from the Stokes layer is also considered. It is shown that the least stable disturbance of this flow does not correspond to the least stable discrete eigenvalue of the infinite Stokes layer when this boundary tends to infinity.

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The linear stability of flat Stokes layers

Journal of Fluid Mechanics ◽

10.1017/s0022112002001052 ◽

2002 ◽

Vol 464 ◽

pp. 393-410 ◽

Cited By ~ 49

Author(s):

P. J. BLENNERHASSETT ◽

ANDREW P. BASSOM

Keyword(s):

Growth Rate ◽

Flat Plate ◽

Linear Stability ◽

Floquet Theory ◽

Neutral Curve ◽

Basic Flow ◽

Wall Velocity ◽

Stokes Layer ◽

Complicated Geometry ◽

Narrow Bands

The linear stability of the Stokes layer generated by an oscillating flat plate is investigated using Floquet theory. The results obtained include the behaviour of the growth rate of the disturbances, part of the corresponding neutral curve and the structure of neutrally stable disturbances. Previously unknown properties of the growth rate cause the neutral curve to have a complicated geometry: the majority of the marginal curve is defined by waves propagating relative to the basic flow and the curve is smooth in character, but for certain very narrow bands of wavenumbers it was found that stationary modes are the first to become unstable. This phenomenon has the consequence that the underlying smooth neutral curve is punctuated by thin finger-like features. The structure of the eigenfunctions showed that the neutrally stable disturbances tend to grow most rapidly just after the wall velocity passes through zero.

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The linear stability of high-frequency flow in a torsionally oscillating cylinder

Journal of Fluid Mechanics ◽

10.1017/s0022112007004831 ◽

2007 ◽

Vol 576 ◽

pp. 491-505 ◽

Cited By ~ 11

Author(s):

P. J. BLENNERHASSETT ◽

ANDREW P. BASSOM

Keyword(s):

Linear Stability ◽

High Frequency ◽

Longitudinal Axis ◽

Critical Conditions ◽

Neutral Stability ◽

Entry Length ◽

Stokes Layer ◽

Shear Modes ◽

The Stability ◽

Pseudo Spectral

The linear stability of the Stokes layer induced in a fluid contained within a long cylinder oscillating at high frequency about its longitudinal axis is investigated. The disturbance equations are derived using Floquet theory and the resulting system solved using pseudo-spectral methods. Both shear modes and axially periodic centripetal disturbance modes are examined and neutral stability curves and corresponding critical conditions for instability identified. For sufficiently small cylinder radius it is verified that the centripetal perturbations limit the stability of the motion but that in larger-radius configurations the shear modes associated with the Stokes layer take over this role. These results suggest a possible design, free of entry-length effects, for experiments intended to examine the breakdown of oscillatory boundary layers.

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The linear stability of a Stokes layer with an imposed axial magnetic field

Journal of Fluid Mechanics ◽

10.1017/s0022112010004210 ◽

2010 ◽

Vol 662 ◽

pp. 320-328 ◽

Cited By ~ 2

Author(s):

CHRISTIAN THOMAS ◽

ANDREW P. BASSOM ◽

CHRISTOPHER DAVIES

Keyword(s):

Magnetic Field ◽

Linear Stability ◽

Floquet Theory ◽

Axial Magnetic Field ◽

Direct Numerical Simulations ◽

Critical Parameters ◽

Neutral Stability ◽

Steady Flows ◽

Stokes Layer ◽

Boundary Wall

The effects of a uniform axial magnetic field directed towards an oscillating wall in a semi-infinite viscous fluid (or Stokes layer) is investigated. The linear stability and disturbance characteristics are determined using both Floquet theory and via direct numerical simulations. Neutral stability curves and critical parameters for instability are presented for a range of magnetic field strengths. Results indicate that a magnetic field directed towards the boundary wall is stabilizing, which is consistent with that found in many steady flows.

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