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.

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.

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.

scholarly journals Turbulent drag reduction by anisotropic permeable substrates – analysis and direct numerical simulations

2019 ◽  
Vol 875 ◽  
pp. 124-172 ◽  
Author(s):  
G. Gómez-de-Segura ◽  
R. García-Mayoral
Keyword(s):  
Drag Reduction ◽  
Numerical Simulations ◽  
Critical Value ◽  
Direct Numerical Simulations ◽  
Mean Flow ◽  
Turbulent Drag Reduction ◽  
Turbulent Drag ◽  
Theoretical Predictions ◽  
The Mean ◽  
The Difference

We explore the ability of anisotropic permeable substrates to reduce turbulent skin friction, studying the influence that these substrates have on the overlying turbulence. For this, we perform direct numerical simulations of channel flows bounded by permeable substrates. The results confirm theoretical predictions, and the resulting drag curves are similar to those of riblets. For small permeabilities, the drag reduction is proportional to the difference between the streamwise and spanwise permeabilities. This linear regime breaks down for a critical value of the wall-normal permeability, beyond which the performance begins to degrade. We observe that the degradation is associated with the appearance of spanwise-coherent structures, attributed to a Kelvin–Helmholtz-like instability of the mean flow. This feature is common to a variety of obstructed flows, and linear stability analysis can be used to predict it. For large permeabilities, these structures become prevalent in the flow, outweighing the drag-reducing effect of slip and eventually leading to an increase of drag. For the substrate configurations considered, the largest drag reduction observed is ${\approx}$20–25 % at a friction Reynolds number $\unicode[STIX]{x1D6FF}^{+}=180$.

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.

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).

Fluid-loaded elastic plate with mean flow: point forcing and three-dimensional effects

2001 ◽  
Vol 428 ◽  
pp. 305-331
Author(s):  
M. R. GREEN ◽  
D. G. CRIGHTON
Keyword(s):  
Elastic Plate ◽  
Critical Angle ◽  
Single Frequency ◽  
Absolute Instability ◽  
Frequency Range ◽  
Mean Flow ◽  
Curvature Effects ◽  
Flow Speeds ◽  
Neutral Mode ◽  
Asymptotic Forms

The unsteady behaviour of an infinitely long fluid-loaded elastic plate subject to a single-frequency line forcing in the presence of a uniform mean flow is known to exhibit a number of interesting phenomena. These include the onset of absolute instability for non-dimensional flow speeds U in excess of some critical speed Uc, and various interesting propagation effects when U < Uc. In the latter respect Crighton & Oswell (1991) have shown that over a particular frequency range there exists an anomalous neutral mode with group velocity directed towards the driver, in violation of the usual Lighthill outgoing radiation condition. Similar results have been found by Peake (1997) when transverse curvature effects are included. In this paper we seek to extend these results and consider the substantially harder problem of a fluid-loaded elastic plate with uniform mean flow which is subject to a point forcing, thereby resulting in a two-dimensional structural problem. A systematic method for determining the absolute instability threshold is developed, and it is shown that the flow is absolutely unstable for flow speeds U > Uc, where Uc is the one-dimensional value found by Crighton & Oswell. At flow speeds U < Uc the flow is marginally stable and convective growth is found to occur downstream of the driver, over a particular frequency range depending on the transverse Fourier wavenumber ky, within a wedge-shaped region. Outside this wedge-shaped region there is only neutral mode behaviour. Asymptotic forms are found for the dominant large-distance causal flexion response downstream of the driver inside and outside the wedge region, and the appropriate critical angle for the wedge region is identified. Within the convective instability wedge the flexion and critical angle take two different forms depending on whether the frequency ω is greater or less than U2/√5. In addition to this interesting behaviour, the flow also exhibits the usual anomalous neutral mode behaviour and, as with Peake's problem, we also find an extra stability (hoop) resulting in neutral mode behaviour over a small frequency range. Asymptotic forms are also found for the threshold frequencies which divide up the various regions of stability of the system (neutral, neutral anomalous, convectively unstable), as a function of ky, and are compared with the results of both Crighton & Oswell and Peake.

Peristaltic Waves in Circular Cylindrical Tubes

1969 ◽  
Vol 36 (3) ◽  
pp. 579-587 ◽  
Author(s):  
F. Yin ◽  
Y. C. Fung
Keyword(s):  
Pressure Gradient ◽  
Wall Motion ◽  
Amplitude Ratio ◽  
Cylindrical Tube ◽  
Critical Value ◽  
Mean Flow ◽  
Navier Stokes Equation ◽  
Mean Pressure ◽  
The Mean ◽  
Set Up

Peristaltic pumping in a circular cylindrical tube is analyzed. The problem is a viscous fluid flow induced by an axisymmetric traveling sinusoidal wave of moderate amplitude imposed on the wall of a flexible tube. A perturbation method of solution is sought. The amplitude ratio (wave amplitude/tube radius) is chosen as a parameter. The nonlinear convective acceleration terms in the Navier-Stokes equation is retained. The governing equations are developed up to the second order in the amplitude ratio. The zeroth-order terms yield the classical Poiseuille flow, the first-order terms yield the Sommerfeld-Orr equation. If there is no pressure gradient in the absence of wall motion, the mean flow and mean pressure gradient (averaged over time) are both shown to be proportional to the square of the amplitude ratio. Numerical results are obtained for this simple case by approximating a complicated group of products of Bessel functions by a polynomial. The results show that the mean axial velocity is dominated by two terms. One term corresponds to a parabolic profile which is due to the mean pressure gradient set up by the wall motion. The other term arises from satisfying the no-slip boundary condition at the wavy wall rather than at the mean position of the wall. In addition, there are perturbations arising from the convective acceleration. If the mean pressure gradient set up by the wall motion itself reaches a certain positive critical value, the velocity becomes zero on the axis. Values of the mean pressure gradient larger than the critical value will induce backward flow in the fluid. Values of the critical pressure gradient for several cases are presented.

On the behaviour of a fluid-loaded cylindrical shell with mean flow

1997 ◽  
Vol 338 ◽  
pp. 387-410 ◽  
Author(s):  
N. PEAKE
Keyword(s):  
Flat Plate ◽  
Critical Radius ◽  
Radiation Condition ◽  
Negative Energy ◽  
Flow Speed ◽  
Single Frequency ◽  
Engineering Practice ◽  
Absolute Instability ◽  
Mean Flow ◽  
Flow Speeds

The unsteady behaviour of an infinitely long fluid-loaded elastic plate which is driven by a single-frequency point-force excitation in the presence of mean flow is known to exhibit a number of unexpected features, including absolute instability when the normalized flow speed, U, lies above some critical speed U0, and certain unusual propagation effects for U<U0. In the latter respect Crighton & Oswell (1991) have demonstrated most significantly that for a particular frequency range there exists an anomalous neutral (negative energy) mode which has group velocity pointing towards the driver, in violation of the usual radiation condition of outgoing waves at infinity. They show that the rate of working of the driver can be negative, due to the presence of other negative-energy waves, and can also become infinite at a critical frequency corresponding to a real modal coalescence. In this paper we attempt to extend these results by including, as is usually the case in a practical situation, plate curvature in the transverse direction, by considering a fluid-loaded cylinder with axial mean flow. In the limit of infinite normalized cylinder radius, a, Crighton & Oswell's results are regained, but for finite a very significant modifications are found. In particular, we demonstrate that the additional stiffness introduced by the curvature typically moves the absolute-instability boundary to a much higher flow speed than for the flat-plate case. Below this boundary we show that Crighton & Oswell's anomalous neutral mode can only occur for a>a1(U), but in practical situations it turns out that a1(U) is exceedingly large, and indeed seems much larger than radii of curvature achievable in engineering practice. Other negative-energy waves are seen to exist down to a smaller, but still very large, critical radius a2(U), while the existence of a real modal coalescence point, leading to a divergence in the driver admittance, occurs down to a slightly smaller critical radius a3(U). The transition through these various flow regimes as U and a vary is fully described by numerical investigation of the dispersion relation and by asymptotic analysis in the (realistic) limit of small U. The inclusion of plate dissipation is also considered, and, in common with Abrahams & Wickham (1994) for the flat plate, we show how the flow then becomes absolutely unstable at all flow speeds provided that a>a2(U).

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