The linear stability of flat Stokes layers

1978 ◽  
Vol 359 (1697) ◽  
pp. 151-166 ◽  
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.

scholarly journals The linear stability of a Stokes layer subjected to high-frequency perturbations

2014 ◽  
Vol 764 ◽  
pp. 193-218 ◽  
Author(s):  
Christian Thomas ◽  
P. J. Blennerhassett ◽  
Andrew P. Bassom ◽  
Christopher Davies
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
High Frequency ◽  
Wall Motion ◽  
Critical Reynolds Number ◽  
Narrow Channel ◽  
Critical Conditions ◽  
Stokes Layer ◽  
Quantitative Results ◽  
Sinusoidal Wall

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.

On the linear stability of Stokes layers

2008 ◽  
Vol 366 (1876) ◽  
pp. 2685-2697 ◽  
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.

scholarly journals Linear Stability of a Steady Convective Flow between Permeable Cylinders

Fluids ◽  
2021 ◽  
Vol 6 (10) ◽  
pp. 342
Author(s):  
Maksims Zigunovs ◽  
Andrei Kolyshkin ◽  
Ilmars Iltins
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Chemical Reaction ◽  
Linear Stability Analysis ◽  
Convective Flow ◽  
Base Flow ◽  
Marginal Stability ◽  
Heat Sources ◽  
Rate Of Increase

Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

Stability of Horizontal Boundary Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Mathematical Proof ◽  
Vertical Component ◽  
Ekman Layer ◽  
Numerical Results ◽  
Energy Estimates ◽  
Navier Stokes ◽  
The 1960S ◽  
The Stability

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

scholarly journals Towards the Understanding of Humpback Whale Tubercles: Linear Stability Analysis of a Wavy Flat Plate

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 212
Author(s):  
Miles Owen ◽  
Abdelkader Frendi
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Flat Plate ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Global Analysis ◽  
Leading Edge ◽  
Local Analysis ◽  
Mean Flow

The results from a temporal linear stability analysis of a subsonic boundary layer over a flat plate with a straight and wavy leading edge are presented in this paper for a swept and un-swept plate. For the wavy leading-edge case, an extensive study on the effects of the amplitude and wavelength of the waviness was performed. Our results show that the wavy leading edge increases the critical Reynolds number for both swept and un-swept plates. For the un-swept plate, increasing the leading-edge amplitude increased the critical Reynolds number, while changing the leading-edge wavelength had no effect on the mean flow and hence the flow stability. For the swept plate, a local analysis at the leading-edge peak showed that increasing the leading-edge amplitude increased the critical Reynolds number asymptotically, while the leading-edge wavelength required optimization. A global analysis was subsequently performed across the span of the swept plate, where smaller leading-edge wavelengths produced relatively constant critical Reynolds number profiles that were larger than those of the straight leading edge, while larger leading-edge wavelengths produced oscillating critical Reynolds number profiles. It was also found that the most amplified wavenumber was not affected by the wavy leading-edge geometry and hence independent of the waviness.

Linear stability of rotating Hagen-Poiseuille flow

1981 ◽  
Vol 108 ◽  
pp. 101-125 ◽  
Author(s):  
Fredrick W. Cotton ◽  
Harold Salwen
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Neutral Stability ◽  
Unstable Region ◽  
Azimuthal Index ◽  
Expansion Technique ◽  
Simple Scaling

Linear stability of rotating Hagen-Poiseuille flow has been investigated by an orthonormal expansion technique, confirming results by Pedley and Mackrodt and extending those results to higher values of the wavenumber |α|, the Reynolds number R, and the azimuthal index n. For |α| [gsim ] 2, the unstable region is pushed to considerably higher values of R and the angular velocity, Ω. In this region, the neutral stability curves obey a simple scaling, consistent with the unstable modes being centre modes. For n = 1, individual neutral stability curves have been calculated for several of the low-lying eigenmodes, revealing a complicated coupling between modes which manifests itself in kinks, cusps and loops in the neutral stability curves; points of degeneracy in the R, Ω plane; and branching behaviour on curves which circle a point of degeneracy.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Curvature- and rotation-induced instabilities in channel flow

1990 ◽  
Vol 210 ◽  
pp. 537-563 ◽  
Author(s):  
O. John E. Matsson ◽  
P. Henrik Alfredsson
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Channel Flow ◽  
Stability Theory ◽  
Critical Reynolds Number ◽  
Secondary Instability ◽  
Linear Stability Theory ◽  
Curved Channel ◽  
Dean Vortices ◽  
Coriolis Effects

In a curved channel streamwise vortices, often called Dean vortices, may develop above a critical Reynolds number owing to centrifugal effects. Similar vortices can occur in a rotating plane channel due to Coriolis effects if the axis of rotation is normal to the mean flow velocity and parallel to the walls. In this paper the flow in a curved rotating channel is considered. It is shown from linear stability theory that there is a region for which centrifugal effects and Coriolis effects almost cancel each other, which increases the critical Reynolds number substantially. The flow visualization experiments carried out show that a complete cancellation of Dean vortices can be obtained for low Reynolds number. The rotation rate for which this occurs is in close agreement with predictions from linear stability theory. For curved channel flow a secondary instability of travelling wave type is found at a Reynolds number about three times higher than the critical one for the primary instability. It is shown that rotation can completely cancel the secondary instability.

The development of a two-dimensional wavepacket in a growing boundary layer

1982 ◽  
Vol 384 (1787) ◽  
pp. 317-332 ◽  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Linear Stability ◽  
Stability Theory ◽  
Linear Stability Theory ◽  
Two Dimensional ◽  
Displacement Thickness ◽  
Power Comparisons ◽  
The One ◽  
Half Power

The evolution of a two-dimensional wavepacket in a growing boundary layer is discussed in terms of linear stability theory. The wavepacket is represented by an integral of periodic wavetrains, each of which is defined as a series in terms of the inverse of the local displacement thickness Reynolds number to the one half power. Comparisons are made between the waveforms computed directly from the integral, a steepest-descent expansion of the integral, and a global expansion about the peak of the wavepacket.

Unconditional Linear Stability of Plane Couette-Poiseuille Flow in Presence of Cross Flow

2010 ◽  
Author(s):  
Anirban Guha ◽  
Ian A. Frigaard
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Energy Analysis ◽  
Cross Flow ◽  
Base Flow ◽  
Lower Range ◽  
Long Wavelength ◽  
Flow Reynolds Number

We have investigated the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, Ri and the dimensionless wall velocity, k. Corresponding to each k ∈ [0,1], we have observed two ranges of Ri for which the flow is unconditionally linearly stable. In the lower range, we have a stabilisation of long wavelengths leading to a cut-off Ri. In this range, cross-flow stabilisation and Couette stabilisation appear to act via very similar mechanisms in this range, leading to the potential for robust compensatory design of flow stabilisation using either mechanism. As Ri is increased, we see first destabilisation and then stabilisation at very large Ri. The instability is again a long wavelength mechanism. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality.

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