Nonlinear Effects in the Collapse of a Nearly Spherical Cavity in a Liquid

1972 ◽  
Vol 94 (1) ◽  
pp. 142-145 ◽  
Author(s):  
R. B. Chapman ◽  
M. S. Plesset
Keyword(s):  
Linear Theory ◽  
Spherical Cavity ◽  
Spherical Shape ◽  
Nonlinear Effects ◽  
Vapor Bubbles ◽  
Prolate Ellipsoid ◽  
Oblate Ellipsoid ◽  
Linearized Theory ◽  
Final Speed

The collapse of pure vapor bubbles in an infinite liquid initially perturbed from a spherical shape is simulated numerically for two cases. In Case A the bubble is initially close to a prolate ellipsoid, and in Case B it is initially close to an oblate ellipsoid. Nonlinear effects are determined by comparing the results with those predicted by the linearized theory of Plesset and Mitchell. These nonlinear effects are found to be important only in the final stages of collapse. In Case A a pair of inward moving jets develop and strike each other with a final speed which is roughly half that, predicted from the linear theory.

scholarly journals TSUNAMI GENERATION AND PROPAGATION

1972 ◽  
Vol 1 (13) ◽  
pp. 146
Author(s):  
Joseph L. Hammack ◽  
Frederic Raichlen
Keyword(s):  
Linear Theory ◽  
Source Region ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Its Region ◽  
Experimental Results ◽  
Specific Class ◽  
Two Dimensional ◽  
Three Dimensional Models

A linear theory is presented for waves generated by an arbitrary bed deformation {in space and time) for a two-dimensional and a three -dimensional fluid domain of uniform depth. The resulting wave profile near the source is computed for both the two and three-dimensional models for a specific class of bed deformations; experimental results are presented for the two-dimensional model. The growth of nonlinear effects during wave propagation in an ocean of uniform depth and the corresponding limitations of the linear theory are investigated. A strategy is presented for determining wave behavior at large distances from the source where linear and nonlinear effects are of equal magnitude. The strategy is based on a matching technique which employs the linear theory in its region of applicability and an equation similar to that of Korteweg and deVries (KdV) in the region where nonlinearities are equal in magnitude to frequency dispersion. Comparison of the theoretical computations with the experimental results indicates that an equation of the KdV type is the proper model of wave behavior at large distances from the source region.

Leading-Edge Corrections to the Linear Theory of Partially Cavitating Hydrofoils

1991 ◽  
Vol 35 (01) ◽  
pp. 15-27
Author(s):  
Spyros A. Kinnas
Keyword(s):  
Integral Equations ◽  
Linear Theory ◽  
Leading Edge ◽  
Foil Thickness ◽  
System Of Integral Equations ◽  
Cavitation Inception ◽  
Perturbation Velocity ◽  
Linearized Theory ◽  
Work First ◽  
Cavitating Hydrofoil

In this work, first, the partially cavitating hydrofoil problem is formulated in linear theory in terms of vorticity and source distributions on the projection of the hydrofoil to the free-stream direction. The resulting system of integral equations is inverted and the solution is expressed in terms of integrals of the horizontal perturbation velocity in fully wetted flow, multiplied by weighting functions that are independent of the shape of the hydrofoil. Second, the linearized dynamic boundary condition on the cavity is modified so that the total velocity on the cavity as predicted by applying Lighthill's leading-edge corrections is a constant. This results in a varying horizontal perturbation velocity on the cavity rather than a constant as required by conventional linear theory. The modified system of integral equations is inverted and the solution is expressed in terms of integrals of known quantities. The present linearized theory with the leading-edge corrections included, predicts a finite cavitation inception number as well as the correct effect of foil thickness on cavity size.

Laminar Swirling Pipe Flow

1954 ◽  
Vol 21 (1) ◽  
pp. 1-7
Author(s):  
L. Talbot
Keyword(s):  
Swirling Flow ◽  
Nonlinear Effects ◽  
Critical Conditions ◽  
Flow Conditions ◽  
Rotationally Symmetric ◽  
Rate Of Decay ◽  
Linearized Theory ◽  
The Stability ◽  
Experimental Values ◽  
Rates Of Decay

Abstract The problem of the decay of a rotationally symmetric steady swirl superimposed on Poiseuille flow in a round pipe was investigated theoretically and experimentally. The object was to determine the degree to which the rate of decay of the swirl as predicted by a linearized theory agreed with measured rates of decay at flow conditions near the critical conditions for swirl instability. The solution to the linearized equation of motion for the swirl was obtained. Swirling flow was produced experimentally by rotating a section of the test pipe. Swirl velocities were determined from motion-picture studies of colored oil droplets introduced in the flow. The stability of the swirl was investigated through visualization of a dye filament, and a critical curve for swirl instability was determined experimentally relating the angular velocity of the rotating section to the Reynolds number. The theoretical and experimental values for the decay parameter were found to agree closely, even at conditions of flow near the critical conditions for instability. It was concluded that in the problem under consideration the nonlinear effects are not appreciable for stable decay of the swirl.

Nonlinear effects in the squeezing spectrum of a parametric amplifier

1988 ◽  
Vol 66 (10) ◽  
pp. 854-858 ◽  
Author(s):  
H. R. Zaidi
Keyword(s):  
Linear Theory ◽  
Nonlinear Theory ◽  
Nonlinear Effects ◽  
Parametric Amplifier

A general nonlinear theory is formulated for the calculation of the squeezing spectrum of a parametric amplifier. The lowest order nonlinear corrections are calculated for both modes below the threshold, and the results are compared with linear theory. The corrections to the spectrum are found to be significant.

scholarly journals A note on tsunamis: their generation and propagation in an ocean of uniform depth

1973 ◽  
Vol 60 (4) ◽  
pp. 769-799 ◽  
Author(s):  
Joseph L. Hammack
Keyword(s):  
Linear Theory ◽  
Linear Approximation ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Solid Boundary ◽  
Far Field ◽  
Bed Deformation ◽  
Downstream Region ◽  
Generation Region ◽  
Block Section

The waves generated in a two-dimensional fluid domain of infinite lateral extent and uniform depth by a deformation of the bounding solid boundary are investigated both theoretically and experimentally. An integral solution is developed for an arbitrary bed displacement (in space and time) on the basis of a linear approximation of the complete (nonlinear) description of wave motion. Experimental and theoretical results are presented for two specific deformations of the bed; the spatial variation of each bed displacement consists of a block section of the bed moving vertically either up or down while the time-displacement history of the block section is varied. The presentation of results is divided into two sections based on two regions of the fluid domain: a generation region in which the bed deformation occurs and a downstream region where the bed position remains stationary for all time. The applicability of the linear approximation in the generation region is investigated both theoretically and experimentally; results are presented which enable certain gross features of the primary wave leaving this region to be determined when the magnitudes of parameters which characterize the bed displacement are known. The results indicate that the primary restriction on the applicability of the linear theory during the bed deformation is that the total amplitude of the bed displacement must remain small compared with the uniform water depth; even this restriction can be relaxed for one type of bed motion.Wave behaviour in the downstream region of the fluid domain is discussed with emphasis on the gradual growth of nonlinear effects relative to frequency dispersion duringpropagationand the subsequent breakdown of the linear theory. A method is presented for finding the wave behaviour in the far field of the downstream region, where the effects of nonlinearities and frequency dispersion have become about equal. This method is based on the use of a model equation in the far field (which includes both linear and nonlinear effects in an approximate manner) first used by Peregrine (1966) and morerecently advocated by Ben jamin, Bona & Mahony (1972) as a preferable model to the more commonly used equation of Korteweg & de Vries (1895). An input-output approach is illustrated for the numerical solution of this equation where the input is computed from the linear theory in its region of applicability. Computations are presented and compared with experiment for the case of a positive bed displacement where the net volume of the generated wave is finite and positive; the results demonstrate the evolution of a train of solitary waves (solitons) ordered by amplitude followed by a dispersive train of oscillatory waves. The case of a negative bed displacement in which the net wave volume is finite and negative (and the initial wave is negative almost everywhere) is also investigated; the results suggest that only a dispersive train of waves evolves (no solitons) for this case.

Nonlinear Ripple Formation in Sputter Erosion

1999 ◽  
Vol 585 ◽  
Author(s):  
A.-L. Barabási ◽  
B. Kahng ◽  
H. Jeong ◽  
S. Park
Keyword(s):  
Surface Morphology ◽  
Linear Theory ◽  
Characteristic Time ◽  
Nonlinear Effects ◽  
Morphological Features ◽  
Ripple Formation ◽  
Morphological Transitions ◽  
Sputter Erosion ◽  
Short Times

AbstractWe investigate the morphological features of sputter eroded surfaces, demonstrating that while at short times ripple formation is described by the linear theory, after a characteristic time the nonlinear terms determine the surface morphology. We also show that the morphological transitions induced by the nonlinear effects can be detected by monitoring the surface width and the erosion velocity.

Nonlinear thermal convection in an elasticoviscous layer heated from below

1977 ◽  
Vol 356 (1685) ◽  
pp. 161-176 ◽  
Keyword(s):  
Linear Theory ◽  
Nonlinear Effects ◽  
Horizontal Layer ◽  
Plane Poiseuille Flow ◽  
Elastic Parameters ◽  
Temperature Contrast ◽  
The Mean ◽  
Exchange Of Stability ◽  
Steady Convection ◽  
Adiabatic Temperature Gradient

The linear and nonlinear stabilities of a horizontal layer of an elasticoviscous fluid, whose stress-rate-of-strain relations are due to Oldroyd (1958), are studied. In the linear theory it is already shown that steady convection (the situation generally referred to as the exchange of stability) is preferred for all relevant values of the Prandtl number (which is the ratio of the kinematic viscosity to the thermal diffusivity). The study of nonlinear effects for slightly supercritical Rayleigh number (which measures the temperature contrast across the layer) shows that plane disturbances for the case where the exchange of stability is valid and plane or centred disturbances for the case of overstability are governed by equations similar to those derived by Hocking, Stewartson & Stuart (1972) for plane Poiseuille flow. The influence of elasticity is to give rise to a burst only when the principle of exchange of stability is valid and provided certain conditions relating to the elastic parameters of the fluid are satisfied. The effect of the adiabatic temperature gradient is also discussed. It is shown that it stabilizes the layer in the linear theory. However, in the nonlinear theory it can destabilize the layer if the ratio the mean temperature of the layer to the temperature difference across the layer is large enough. For most practical purposes it does not influence the conditions necessary for a burst to occur.

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