Stability Analysis of a Nonlinear Plate in Mean Flow

1997 ◽  
Vol 05 (02) ◽  
pp. 137-155 ◽  
Author(s):  
Sean F. Wu ◽  
Jinshuo Zhu
Keyword(s):  
Stability Analysis ◽  
Flexural Vibration ◽  
Flow Speed ◽  
Critical Values ◽  
Plate Vibration ◽  
Viscous Damping ◽  
Mean Flow ◽  
Nonlinear Plate ◽  
Structural Nonlinearities ◽  
The Mean

A stability analysis of a nonlinear plate clamped to an infinite baffle in mean flow is given. The effect of structural nonlinearities induced by in-plane forces and shearing forces due to stretching of plate bending motion, and that of viscous damping are taken into account in the derivation of the plate equation. The plate flexural displacement is obtained by modal expansion based on Galerkin's method. The critical mean flow speeds at which local instabilities may occur are determined by Routh algorithm. The mechanisms that trigger the local instabilities are uncovered. The effect of structural nonlinearities, and that of plate aspect and plate length/thickness ratios on local instabilities are examined. Numerical examples of the transition from stable to locally unstable vibration, as the mean flow speed exceeds the critical values, are demonstrated. The results show that while the overall amplitude of the plate flexural displacement may be bounded when the mean flow speed exceeds the critical values, plate vibration may be locally unstable, jumping from one equilibrium position to another. Furthermore, the jumping may be random, and the plate vibration may seem chaotic. The results also show that viscous damping may stabilize plate flexural vibration and settle the plate in one of its equilibria.

ABSOLUTE INSTABILITY OR CHAOS? — A TRIBUTE TO DR. DAVID G. CRIGHTON

2002 ◽  
Vol 10 (04) ◽  
pp. 407-419
Author(s):  
SEAN F. WU
Keyword(s):  
Elastic Plate ◽  
Equilibrium Position ◽  
Flexural Vibration ◽  
Critical Value ◽  
Flow Speed ◽  
Multiple Equilibrium ◽  
Absolute Instability ◽  
Mean Flow ◽  
Structural Nonlinearities ◽  
The Mean

The stabilities of an elastic plate clamped on an infinite, rigid baffle subject to any time dependent force excitation in the presence of mean flow are examined. The mechanisms that can cause plate flexural vibrations to be absolute unstable when the mean flow speed exceeds a critical value are revealed. Results show that the instabilities of an elastic plate are mainly caused by an added stiffness due to acoustic radiation in mean flow, but controlled by the structural nonlinearities. This added stiffness is shown to be negative and increase quadratically with the mean flow speed. Hence, as the mean flow speed approaches a critical value, the added stiffness may null the overall stiffness of the plate, leading to an unstable condition. Note that without the inclusion of the structural nonlinearities, the plate has only one equilibrium position, namely, its undeformed flat position. Under this condition, the amplitude of plate flexural vibration would grow exponentially in time everywhere, known as absolute instability. With the inclusion of structural nonlinearities, the plate may possess multiple equilibrium positions. When the mean flow speed exceeds the critical values, the plate may be unstable and jump from one equilibrium position to another. Since this jumping is random, the plate flexural vibration may seem chaotic.

scholarly journals Transitional Flow Dynamics Behind a Micro-Ramp

2019 ◽  
Vol 104 (2-3) ◽  
pp. 533-552
Author(s):  
J. Casacuberta ◽  
K. J. Groot ◽  
Q. Ye ◽  
S. Hickel
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Large Scale ◽  
Vortex Pair ◽  
Base Flow ◽  
Transitional Flow ◽  
Mean Flow ◽  
Primary Vortex ◽  
The Mean ◽  
Near Wall

AbstractMicro-ramps are popular passive flow control devices which can delay flow separation by re-energising the lower portion of the boundary layer. We compute the laminar base flow, the instantaneous transitional flow, and the mean flow around a micro-ramp immersed in a quasi-incompressible boundary layer at supercritical roughness Reynolds number. Results of our Direct Numerical Simulations (DNS) are compared with results of BiLocal stability analysis on the DNS base flow and independent tomographic Particle Image Velocimetry (tomo-PIV) experiments. We analyse relevant flow structures developing in the micro-ramp wake and assess their role in the micro-ramp functionality, i.e., in increasing the near-wall momentum. The main flow feature of the base flow is a pair of streamwise counter-rotating vortices induced by the micro-ramp, the so-called primary vortex pair. In the instantaneous transitional flow, the primary vortex pair breaks up into large-scale hairpin vortices, which arise due to linear varicose instability of the base flow, and unsteady secondary vortices develop. Instantaneous vortical structures obtained by DNS and experiments are in good agreement. Matching linear disturbance growth rates from DNS and linear stability analysis are obtained until eight micro-ramp heights downstream of the micro-ramp. For the setup considered in this article, we show that the working principle of the micro-ramp is different from that of classical vortex generators; we find that transitional perturbations are more efficient in increasing the near-wall momentum in the mean flow than the laminar primary vortices in the base flow.

scholarly journals Marginal Instability?

2009 ◽  
Vol 39 (9) ◽  
pp. 2373-2381 ◽  
Author(s):  
S. A. Thorpe ◽  
Zhiyu Liu
Keyword(s):  
Richardson Number ◽  
Maximum Growth ◽  
Flow Speed ◽  
Maximum Growth Rate ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Boundary Layer Flows ◽  
Naturally Occurring ◽  
Marginal State ◽  
The Mean

Abstract Some naturally occurring, continually forced, turbulent, stably stratified, mean shear flows are in a state close to that in which their stability changes, usually from being dynamically unstable to being stable: the time-averaged flows that are observed are in a state of marginal instability. By “marginal instability” the authors mean that a small fractional increase in the gradient Richardson number Ri of the mean flow produced by reducing the velocity and, hence, shear is sufficient to stabilize the flow: the increase makes Rimin, the minimum Ri in the flow, equal to Ric, the critical value of this minimum Richardson number. The value of Ric is determined by solving the Taylor–Goldstein equation using the observed buoyancy frequency and the modified velocity. Stability is quantified in terms of a factor, Φ, such that multiplying the flow speed by (1 + Φ) is just sufficient to stabilize it, or that Ric = Rimin/(1 + Φ)2. The hypothesis that stably stratified boundary layer flows are in a marginal state with Φ < 0 and with |Φ| small compared to unity is examined. Some dense water cascades are marginally unstable with small and negative Φ and with Ric substantially less than ¼. The mean flow in a mixed layer driven by wind stress on the water surface is, however, found to be relatively unstable, providing a counterexample that refutes the hypothesis. In several naturally occurring flows, the time for exponential growth of disturbances (the inverse of the maximum growth rate) is approximately equal to the average buoyancy period observed in the turbulent region.

An experimental investigation of pulsating turbulent water flow in a tube

1971 ◽  
Vol 46 (1) ◽  
pp. 43-64 ◽  
Author(s):  
J. H. Gerrard
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Water Flow ◽  
Cylindrical Tube ◽  
Flow Speed ◽  
Mean Flow ◽  
Mean Velocity ◽  
Central Plateau ◽  
Transition Range ◽  
The Mean

Experiments were made on a pulsating water flow at a mean flow Reynolds number of 3770 in a cylindrical tube of diameter 3·81 cm. Pulsations were produced by a piston oscillating in simple harmonic motion with a period of 12 s. Turbulence was made visible by means of a sheet of dye produced by electrolysis from a fine wire stretched across a diameter. The sheet of dye is contorted by the turbulent eddies, and ciné-photography was used to find the velocity of convection which was shown to be the flow speed except in certain circumstances which are discussed. By subtracting the mean flow velocity profile the profile of the component of the motion oscillating at the imposed frequency was determined.The Reynolds number of these experiments lies in the turbulent transition range, so that large effects of laminarization are observed. In the turbulent phase, the velocity profile was found to possess a central plateau as does the laminar oscillating profile. The level and radial extent of this were little different from the laminar ones. Near to the wall, the turbulent oscillating profile is well represented by the mean velocity power law relationship, u/U ∝ (y/a)1/n. In the laminarized phase, the turbulent intensity is considerably reduced at this Reynolds number. The velocity profile for the whole flow (mean plus oscillating) relaxes towards the laminar profile. Laminarization contributes appreciably to the oscillating component.Extrapolation of the results to higher Reynolds numbers and different frequencies of oscillation is suggested.

scholarly journals The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

2018 ◽  
Vol 48 (11) ◽  
pp. 2779-2797 ◽  
Author(s):  
Megan A. Stamper ◽  
John R. Taylor ◽  
Baylor Fox-Kemper
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Initial Conditions ◽  
Boussinesq Equations ◽  
Small Region ◽  
Computational Domain ◽  
Mean Flow ◽  
The Mean

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

The instability of the thin vortex ring of constant vorticity

1977 ◽  
Vol 287 (1344) ◽  
pp. 273-305 ◽  
Keyword(s):  
Stability Analysis ◽  
Vortex Ring ◽  
Mean Flow ◽  
Bending Waves ◽  
Constant Vorticity ◽  
Ring Radius ◽  
Amplification Rate ◽  
The Mean ◽  
The Stability ◽  
Good Agreement

A theoretical investigation of the instability of a vortex ring to short azimuthal bending waves is presented. The theory considers only the stability of a thin vortex ring with a core of constant vorticity (constant /r) in an ideal fluid. Both the mean flow and the disturbance flow are found as an asymptotic solution in e = a /R, the ratio of core radius to ring radius. Only terms linear in wave amplitude are retained in the stability analysis. The solution to 0 (e 2 ) is presented, although the details of the stability analysis are carried through completely only for a special class of bending waves that are known to be unstable on a line filament in the presence of strain (Tsai & Widnall 1976) and have been identified in the simple model of Widnall, Bliss & Tsai (1974) as a likely mode of instability for the vortex ring: these occur at certain critical wavenumbers for which waves on a line filament of the same vorticity distribution would not rotate (w 0 = 0). The ring is found to be always unstable for at least the lowest two critical wavenumbers ( ka = 2.5 and 4.35). The amplification rate and wavenumber predicted by the theory are found to be in good agreement with available experimental results.

EFFECT OF TURBULENCE ON SOUND RADIATION BY VORTEX–BODY INTERACTION

2009 ◽  
Vol 17 (01) ◽  
pp. 71-81
Author(s):  
TING-HUI ZHENG ◽  
GEORGIOS H. VATISTAS ◽  
S. K. TANG
Keyword(s):  
Flow Speed ◽  
Nonuniform Flow ◽  
Sound Waves ◽  
Mean Flow ◽  
Single Vortex ◽  
Body Interaction ◽  
Fundamental Properties ◽  
The Mean ◽  
Radiated Sound ◽  
Solid Bodies

This study examines the sound generated by the interaction between turbulent vortices and solid bodies, and its propagation in a nonuniform flow. Single vortex encounters with a flat plate and a nonrotating cylinder are considered. The solutions show that as the turbulence intensity increases, the sound radiated by the vortex–body interaction is strengthened while the effect of the mean flow speed on the sound waves weakens. The sound profile and sound directivity do not change with the Reynolds number. Neglecting turbulence in vortices will not affect the prediction of the fundamental properties of the radiated sound waves; however, it will underestimate the magnitude of the produced sound.

Hydraulic control of homogeneous shear flows

2003 ◽  
Vol 475 ◽  
pp. 163-172 ◽  
Author(s):  
CHRIS GARRETT ◽  
FRANK GERDES
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Water Waves ◽  
Flow Speed ◽  
Hydraulic Control ◽  
Mean Flow ◽  
Shallow Water Waves ◽  
Homogeneous Fluid ◽  
Apparent Paradox ◽  
The Mean

If a shear flow of a homogeneous fluid preserves the shape of its velocity profile, a standard formula for the condition for hydraulic control suggests that this is achieved when the depth-averaged flow speed is less than (gh)1/2. On the other hand, shallow-water waves have a speed relative to the mean flow of more than (gh)1/2, suggesting that information could propagate upstream. This apparent paradox is resolved by showing that the internal stress required to maintain a constant velocity profile depends on flow derivatives along the channel, thus altering the wave speed without introducing damping. By contrast, an inviscid shear flow does not maintain the same profile shape, but it can be shown that long waves are stationary at a position of hydraulic control.

The propagation of atmospheric Rossby gravity waves in latitudinally sheared zonal flows

1977 ◽  
Vol 287 (1340) ◽  
pp. 115-143 ◽  
Keyword(s):  
Gravity Waves ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Flow Speed ◽  
Wave Number ◽  
Mean Flow ◽  
Zonal Flows ◽  
Planetary Scale ◽  
The Mean ◽  
Critical Latitude

Using the B-plane approximation we formulate the equations which govern small perturbations in a rotating atmosphere and describe a wide class of possible wave motions, in the presence of a background zonal flow, ranging from ‘moderately high’ frequency acoustic-gravity-inertial waves to ‘low’ frequency planetary-scale (Rossby) waves. The discussion concentrates mainly on the propagation properties of Rossby waves in various types of latitudinally sheared zonal flows which occur at different heights and seasons in the earth’s atmosphere. However, it is first shown that gravity waves in a latitudinally sheared zonal flow exhibit critical latitude behaviour where the ‘intrinsic ’ wave frequency matches the Brunt-Vaisala frequency (in contrast to the case of gravity waves in a vertically sheared flow where a critical layer exists where the horizontal wave phase speed equals the flow speed) and that the wave behaviour near such a latitude is similar to that of Rossby waves in the vicinity of their critical latitudes which occur where the ‘intrinsic’ wave frequency approaches zero. In the absence of zonal flow in the atmosphere the geometry of the planetary wave dispersion equation (which is described by a highly elongated ellipsoid in wave-number vector space) implies that energy propagates almost parallel to the /--planes. This feature may provide a reason why there seems to be so little coupling between planetary scale motions in the lower and upper atmosphere. Planetary waves can be made to propagate eastward, as well as westward, if they are evanescent in the vertical direction. The W.K.B. approximation, which provides an approximate description of wave propagation in slowly varying zonal wind shears, shows that the distortion of the wave-number surface caused by the zonal flow controls the dependence of the wave amplitude on the zonal flow speed. In particular it follows that Rossby waves propagating into regions of strengthening westerlies are intensified in amplitude whereas those waves propagating into strengthening easterlies are diminished in amplitude. A classification of the various types of ray trajectories that arise in zonal flow profiles occurring in the Earth’s atmosphere, such as jet-like variations of westerly or easterly zonal flow or a belt of westerlies bounded by a belt of easterlies, is given, and provides the conditions giving rise to such phenomena as critical latitude behaviour and wave trapping. In a westerly flow there is a tendency for the combined effects on wave propagation of jet-like variations of B and zonal flow speed to counteract each other, whereas in an easterly flow such variations tend to reinforce each other. An examination of the reflexion and refraction of Rossby waves at a sharp jump in the zonal flow speed shows that under certain conditions wave amplification, or over-reflexion, can arise with the implication that the reflected wave can extract energy from the background streaming motion. On the other hand the wave behaviour near critical latitudes, which can be described in terms of a discontinuous jump in the ‘wave invariant’, shows that such latitudes can act as either wave absorbers (in which case the mean flow is accelerated there) or wave emitters (in which case the mean flow is decelerated there).

scholarly journals On the impact of swirl on the growth of coherent structures

2014 ◽  
Vol 741 ◽  
pp. 156-199 ◽  
Author(s):  
K. Oberleithner ◽  
C. O. Paschereit ◽  
I. Wygnanski
Keyword(s):  
Stability Analysis ◽  
Coherent Structures ◽  
Large Scale ◽  
Vortex Breakdown ◽  
Shear Mode ◽  
Mean Flow ◽  
Swirling Jet ◽  
The Mean ◽  
The Impact ◽  
Helical Mode

AbstractSpatial linear stability analysis is applied to the mean flow of a turbulent swirling jet at swirl intensities below the onset of vortex breakdown. The aim of this work is to predict the dominant coherent flow structure, their driving instabilities and how they are affected by swirl. At the nozzle exit, the swirling jet promotes shear instabilities and, less unstable, centrifugal instabilities. The latter stabilize shortly downstream of the nozzle, contributing very little to the formation of coherent structures. The shear mode remains unstable throughout generating coherent structures that scale with the axial shear-layer thickness. The most amplified mode in the nearfield is a co-winding double-helical mode rotating slowly in counter-direction to the swirl. This gives rise to the formation of slowly rotating and stationary large-scale coherent structures, which explains the asymmetries in the mean flows often encountered in swirling jet experiments. The co-winding single-helical mode at high rotation rate dominates the farfield of the swirling jet in replacement of the co- and counter-winding bending modes dominating the non-swirling jet. Moreover, swirl is found to significantly affect the streamwise phase velocity of the helical modes rendering this flow as highly dispersive and insensitive to intermodal interactions, which explains the absence of vortex pairing observed in previous investigations. The stability analysis is validated through hot-wire measurements of the flow excited at a single helical mode and of the flow perturbed by a time- and space-discrete pulse. The experimental results confirm the predicted mode selection and corresponding streamwise growth rates and phase velocities.

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