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.

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

2007 ◽  
Vol 576 ◽  
pp. 491-505 ◽  
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.

scholarly journals Suppression of the Kapitza instability in confined falling liquid films

2018 ◽  
Vol 860 ◽  
pp. 608-639 ◽  
Author(s):  
Gianluca Lavalle ◽  
Yiqin Li ◽  
Sophie Mergui ◽  
Nicolas Grenier ◽  
Georg F. Dietze
Keyword(s):  
Linear Stability ◽  
Pressure Difference ◽  
Film Surface ◽  
Water Film ◽  
Narrow Channel ◽  
Liquid Films ◽  
Interfacial Instability ◽  
Critical Conditions ◽  
Driving Mechanism ◽  
Strong Confinement

We revisit the linear stability of a falling liquid film flowing through an inclined narrow channel in interaction with a gas phase. We focus on a particular region of parameter space, small inclination and very strong confinement, where we have found the gas to strongly stabilize the film, up to the point of fully suppressing the long-wave interfacial instability attributed to Kapitza (Zh. Eksp. Teor. Fiz., vol. 18 (1), 1948, pp. 3–28). The stabilization occurs both when the gas is merely subject to an aerostatic pressure difference, i.e. when the pressure difference balances the weight of the gas column, and when it flows counter-currently. In the latter case, the degree of stabilization increases with the gas velocity. Our investigation is based on a numerical solution of the Orr–Sommerfeld temporal linear stability problem as well as stability experiments that clearly confirm the observed effect. We quantify the degree of stabilization by comparing the linear stability threshold with its passive-gas limit, and perform a parametric study, varying the relative confinement, the Reynolds number, the inclination angle and the Kapitza number. For example, we find a 30 % reduction of the cutoff wavenumber of the instability for a water film in contact with air, flowing through a channel inclined at $3^{\circ }$ and of height 2.8 times the film thickness. We also identify the critical conditions for the full suppression of the instability in terms of the governing parameters. The stabilization is caused by the strong confinement of the gas, which produces perturbations of the adverse interfacial tangential shear stress that are shifted by half a wavelength with respect to the wavy film surface. This tends to reduce flow-rate variations within the film, thus attenuating the inertia-based driving mechanism of the Kapitza instability.

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.

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.

How rapidly oscillating collapsible tubes extract energy from a viscous mean flow

2008 ◽  
Vol 601 ◽  
pp. 199-227 ◽  
Author(s):  
MATTHIAS HEIL ◽  
SARAH L. WATERS
Keyword(s):  
Reynolds Number ◽  
High Frequency ◽  
Computational Analysis ◽  
Functional Relationship ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Mean Flow ◽  
Collapsible Tubes ◽  
High Frequency Oscillations ◽  
Reynolds Number Flows

We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, Recrit, at which the wall begins to extract energy from the flow. We conjecture that Recrit corresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between Recrit and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

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.

Stability of obliquely driven cavity flow

2021 ◽  
Vol 928 ◽  
Author(s):  
Pierre-Emmanuel des Boscs ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Couette Flow ◽  
Critical Reynolds Number ◽  
Cavity Flow ◽  
Basic Flow ◽  
Driven Cavity ◽  
Aspect Ratios ◽  
Driven Cavity Flow ◽  
Yaw Angle

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part

2001 ◽  
Vol 434 ◽  
pp. 355-369 ◽  
Author(s):  
J. MIZUSHIMA ◽  
Y. SHIOTANI
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Nonlinear Stability ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Critical Conditions ◽  
Stable Steady State ◽  
Weakly Nonlinear ◽  
Symmetric Channel ◽  
Steady State Solutions

Transitions and instabilities of two-dimensional flow in a symmetric channel with a suddenly expanded and contracted part are investigated numerically by three different methods, i.e. the time marching method for dynamical equations, the SOR iterative method and the finite-element method for steady-state equations. Linear and weakly nonlinear stability theories are applied to the flow. The transitions are confirmed experimentally by flow visualizations. It is known that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number, regains the symmetry at another critical Reynolds number and becomes oscillatory at very large Reynolds numbers. Multiple stable steady-state solutions are found in some cases, which lead to a hysteresis. The critical conditions for the existence of the multiple stable steady-state solutions are determined numerically and compared with the results of the linear and weakly nonlinear stability analyses. An exchange of modes for oscillatory instabilities is found to occur in the flow as the aspect ratio, the ratio of the length of the expanded part to its width, is varied, and its relation with the impinging free-shear-layer instability (IFLSI) is discussed.

scholarly journals Linear stability of Hunt's flow

2010 ◽  
Vol 649 ◽  
pp. 115-134 ◽  
Author(s):  
JĀNIS PRIEDE ◽  
SVETLANA ALEKSANDROVA ◽  
SERGEI MOLOKOV
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Maximal Velocity ◽  
Square Duct ◽  
Stream Function Formulation ◽  
The Magnetic Field

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha ≈ 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec growing approximately as Ha. For Ha > 40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ≈ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

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