Finite Amplitude Effects Research Articles

On the quasi-steady aerodynamics of normal hovering flight part I: the induced power factor

An analytical treatment to quantify the losses captured in the induced power factor, k, is provided for flapping wings in normal hover, including the effects of non-uniform downwash, tip losses and finite flapping amplitude. The method is based on a novel combination of actuator disc and lifting line blade theories that also takes into account the effect of advance ratio. The model has been evaluated against experimental results from the literature and qualitative agreement obtained for the effect of advance ratio on the lift coefficient of revolving wings. Comparison with quantitative experimental data for the circulation as a function of span for a fruitfly wing shows that the model is able to correctly predict the circulation shape of variation, including both the magnitude of the peak circulation and the rate of decay in circulation towards zero. An evaluation of the contributions to induced power factor in normal hover for eight insects is provided. It is also shown how Reynolds number can be accounted for in the induced power factor, and good agreement is obtained between predicted span efficiency as a function of Reynolds number and numerical results from the literature. Lastly, it is shown that for a flapping wing in hover k owing to the non-uniform downwash effect can be reduced to 1.02 using an arcsech chord distribution. For morphologically realistic wing shapes based on beta distributions, it is shown that a value of 1.07 can be achieved for a radius of first moment of wing area at 40% of wing length.

Simple models for linear and nonlinear modal vibration of circular cylindrical shells

Beginning shortly after the Second World War, ONR sponsored projects at Columbia University, New York, that analyzed vibration and shock response of shell structures. This work necessitated ingenuity because it preceded the advent of numerical modeling tools. A central theme was energy methods. This paper will review two works of this type. Baron and Bleich [J. Appl. Mech. 21, 178–184 (1954)] used the mechanical energies of the Flugge-Byrne-Lur’ye shell theory and a fundamental property of the modal energies to obtain accurate formulas for the three vibration branches and associated modal displacements of a simply supported cylindrical shell. This author [Ginsberg, J. Appl. Mech. 40, 471–477 (1973)] used these modes to analyze finite amplitude effects at resonances. Examination of the order of magnitude of terms in the mechanical energies led to identification of nonlinear mode coupling. The resulting differential equations for the modal coordinates were solved by a singular perturbation technique. The ou...

Up-gradient eddy fluxes of potential vorticity near the subtropical jet

The role of eddy fluxes in the general circulation is often approached by treating eddies as (macro)turbulence. In this approach, the eddies act to diffuse certain quasiconservative quantities, such as potential vorticity (PV), along isentropic surfaces in the free atmosphere. The eddy fluxes are determined primarily by the eddy diffusivities and are necessarily down-gradient of the basic state PV field. Support for the (macro)turbulence approach stems from the fact that the eddy fluxes of PV in the free atmosphere are generally down-gradient in the long-term mean. Here we call attention to a pronounced and significant region of upgradient eddy PV fluxes on the poleward flank of the jet core in both hemispheres. The region of up-gradient (i.e., notionally “antidiffusive”) eddy PV fluxes is most pronounced during the winter and spring seasons and partially contradicts the turbulence approach described above. Analyses of the PV variance (potential enstrophy) budget suggest that the up-gradient PV fluxes represent local wave decay and are maintained by poleward fluxes of PV variance. Finite-amplitude effects thus represent leading order contributions to the PV variance budget, whereas dissipation is only of secondary importance locally. The appearance of up-gradient PV fluxes in the long-term mean is associated with the poleward shift of the jet—and thus the region of wave decay relative to wave growth—following wave-breaking events.

Perturbation methods for the spectral analysis of a weakly nonlinear acoustic field generated by a transient insonation

In this study an infinite plane piston is considered which oscillates with finite amplitude in unbounded homogeneous fluid. To illustrate the shape of the weakly nonlinear acoustic field generated by a transient insonation, the function defined by Funch/Muller representing a damped sinusoid is used to simulate the temporal waveform of the piston vibration. The acoustic transient wave generates harmonic components as result of nonlinearities in the material properties of the fluid and in the convective terms of the propagation equation. The mathematical approach is based upon the generalized Burgers’ equation, which is a good approximation of the exact equation for the nonlinear propagation when diffraction effects are assumed to be negligible. The pressure amplitude of the fundamental is considered large enough to produce the second harmonic wave. Under the quasi-linear approximation, an analytical description of the fundamental and the second harmonic waves is elaborated. To simulate the spectrum of the weakly nonlinear acoustic field, the pressure field is written in a perturbation series where the first term is the linear acoustic field that results from an infinitesimal oscillation of the piston and the second term contains the first nonlinear contribution to the acoustic field due to the finite amplitude effects.

Rain-induced attenuation of deep-water waves

AbstractA laboratory investigation has been undertaken to quantify water wave attenuation rates as a function of rainfall rate. Vertical artificial rainfall is shown to generate weak near-surface velocity fluctuations that decline systematically away from the free surface and are independent of rainfall rate across the range of rainfall rates investigated (40–$170~\mathrm{mm} ~{\mathrm{h} }^{- 1} $). In the absence of rain, the observed attenuation of gravity waves is at levels consistent with classical viscous theory, but with a systematic finite-amplitude effect observed above a mean steepness of 0.10. Wave attenuation rates were found to be independent of the mean wave steepness and identical when artificial rainfall rates of 108 and$141~\mathrm{mm} ~{\mathrm{h} }^{- 1} $were applied. Reassessment of complementary theoretical and experimental studies of individual droplets impacting on undisturbed water surfaces indicates that above a weak threshold rainfall rate of$30~\mathrm{mm} ~{\mathrm{h} }^{- 1} $, the surface irradiation becomes so frequent that droplet-generated violent surface motions directly interact with the incoming droplets. Present evidence is that a matching of time scales develops between the incoming surface irradiation and surface water motions generated by antecedent droplets as the rainfall rate increases. Consequently, at high rainfall rates, a highly dissipative surface regime is created that transmits little of the incident rainfall kinetic energy to the aqueous layers below. Rainfall-induced wave attenuation rates are compared with measurements of other wave attenuation processes to obtain a hierarchy of strength in both the laboratory and the field. Comparison is also made with wave attenuation theories that incorporate momentum and energy flux considerations. Rain-induced wave attenuation rates are weak or very strong depending on whether they are expressed in terms of energy scaling obtained from above or below the surface respectively, due to the high dissipation rate that occurs in the vicinity of the interface.

Contrast enhancement of medical ultrasound imaging with reversal phase-inversion pulse technology

A new ultrasound contrast imaging technique was proposed for eliminating the harmonic components from the emission signal transmitted by the broadband ultrasonic system. Reversal phase-inversion pulse was used for the first time to separate the contrast harmonics from the harmonics in the emission signal to improve the detection of contrast micro-bubbles. Based on the nonlinear acoustic theory of finite-amplitude effects and the associated distortion of the propagating wave, the Bessel-Fubini series model was applied to describe the nonlinear propagation effects of the reversal phase-inversion pulse, and the Church’s equation for zero-thickness encapsulation model was used to produce the scattering-pulse of the bubble. For harmonic imaging, the experiment was performed using a 64-element linear array, which was simulated by Field II. The results show that the harmonic components from the emission signal can be completely cancelled, and the harmonics generated by the nonlinear propagation of the wave through the tissue, can be reduced by 15–30 dB. Compared with the short pulse, the reversal phase-inversion pulse can improve the contrast and definition of the harmonic image significantly.

Exact firing-rate response of the integrate-and-fire neuron receiving finite amplitude excitatory and inhibitory post-synaptic potentials

Neurons in active networks are subject to a fluctuating synaptic drive comprising excitatory and inhibitory post-synaptic potentials arriving at high rates relative to the integration time of the cell. The standard approach for treating stochastic synaptic currents and conductances has been to approximate them by a Gaussian white-noise or coloured-noise process in which the amplitudes of the underlying post-synaptic potentials are considered small. Importantly, this diffusion approximation has allowed for various firing-rate properties of populations of fluctuation-driven neurons to be calculated using Johanessma’s [1] and Ricciardi’s [2] Fokker-Planck framework. This framework in turn has proven to be a powerful theoretical tool for deriving the emergent states of coupled neuronal networks as a function of the properties of constituent neurons. The Gaussian/diffusion model of fluctuating synaptic conductance is in widespread use theoretically, computationally and experimentally and, in many cases, is an excellent approximation to the synaptic drive experienced by neurons in active networks in vitro or in vivo. The typical synaptic coupling between neurons in many brain regions, however, is often sufficiently strong such that relatively few synchronous synaptic events are required to bring a neuron from rest to the spiking threshold. For example, pairs of neocortical layer-5 pyramidal cells form synaptic connections with amplitudes in the range 0.5mV-6mV with a mean of 1.3mV [2] and, when correlations are taken into account, near-synchronous pre-synaptic activity will lead to even stronger aggregate EPSPs. Given that the difference between rest and threshold is around 10mV for this class of cell [3] it is likely that finite-amplitude effects missed by the standard Gaussian/diffusion approximation will play an important role in shaping the steady-state and response properties of neurons in vivo. Here we present a significant generalization of the standard Gaussian/diffusion approach to treating synaptic fluctuations by exactly solving the firing-rate response for a neuron receiving finite-amplitude Poissonian excitatory and inhibitory post-synaptic potentials [5]. The mathematical framework involves the solution of a master equation and can be applied to synaptic-amplitude distributions decomposable into combinations of exponential functions -similar to those seen in experiment. It will be shown that this more general mathematical description of the neuronal response to stochastic synaptic dynamics is significantly richer and qualitatively distinct from that predicted by the Gaussian/diffusion approximation. As well as providing analytical solutions for the firing-rate response of the leaky integrate-and-fire model, it will be demonstrated how an efficient numerical scheme can be used to calculate, numerically but exactly, the response properties of non-linear integrate-and-fire neurons and also to better capture the effects of synaptic reversal potentials.

Tidal Spectroscopy of the Lower St. Johns River from a High-Resolution Shallow Water Hydrodynamic Model

The nonlinear distortion of the astronomic tide in the Lower St. Johns River is investigated. Computed tidal elevations are analyzed at various locations within the Lower St. Johns River. The modeling approach first evaluates the boundary condition applied at the open ocean with regards to it providing a complete description of the tidal elevation, followed by numerical experimentation and a tidal constituent analysis that examines the effects of finite wave amplitude, advection, and bottom friction towards distorting the tide as it propagates upriver. The distortions caused by each nonlinear source are presented in both the time and frequency domains. Analysis at observation stations reveals a river tide with more coastal characteristics near the mouth and with more considerable distortion upriver. The spectroscopy of the astronomic tide for the Lower St. Johns River is established in terms of a custom set of tidal constituents.

Firing-Rate Response of a Neuron Receiving Excitatory and Inhibitory Synaptic Shot Noise

The synaptic coupling between neurons in neocortical networks is sufficiently strong so that relatively few synchronous synaptic pulses are required to bring a neuron from rest to the spiking threshold. However, such finite-amplitude effects of fluctuating synaptic drive are missed in the standard diffusion approximation. Here exact solutions for the firing-rate response to modulated presynaptic rates are derived for a neuron receiving additive excitatory and inhibitory synaptic shot noise with exponential amplitude distributions. The shot-noise description of the neuronal response to synaptic dynamics is shown to be richer and qualitatively distinct from that predicted by the diffusion approximation. It is also demonstrated how the framework developed here can be generalized to multiplicative shot noise so as to better capture effects of the inhibitory reversal potential.

Generalizations of the Wedderburn number: Parameterizing upwelling in stratified lakes

The effect of lake geometry on wind‐induced upwelling, in a two‐layer stratified lake with a variable bottom slope and generic planar shape is investigated. (1) The traditional linearized classification parameter for upwelling, the Wedderburn number, is extended to include finite amplitude effects in a rectangular basin; this extension is important when the Wedderburn number is < ~3. (2) The Wedderburn number is generalized to incorporate the influence of a bottom slope and a variable‐width basin for both the linear (large Wedderburn number) and nonlinear (small Wedderburn number) assumptions. Laboratory and numerical experiments of a two‐layer stratification were performed and used to validate these extensions. (3) Both extensions are combined to derive a methodology to estimate the interface displacement in real lakes where both geometry effects may act together, and this methodology is used to discuss the response of the thermocline to the onset of wind in Lake Como and Mundaring Weir reservoir. In most cases, symmetry of the basin geometry along the fetch is the key characteristic that defines tilting of the interface for real lakes. Asymmetries in terms of bottom slopes induce shortening or stretching of the effective internal fetch, while a larger excursion of the interface results in narrower upwind areas of the lake due to horizontal redistribution of displaced water volumes. This paper gives the correct definition of the length scales to define the Wedderburn number for any stratified basin.

Tissue non-linearity

The propagation of acoustic waves is a fundamentally non-linear process, and only waves with infinitesimally small amplitudes may be described by linear expressions. In practice, all ultrasound propagation is associated with a progressive distortion in the acoustic waveform and the generation of frequency harmonics. At the frequencies and amplitudes used for medical diagnostic scanning, the waveform distortion can result in the formation of acoustic shocks, excess deposition of energy, and acoustic saturation. These effects occur most strongly when ultrasound propagates within liquids with comparatively low acoustic attenuation, such as water, amniotic fluid, or urine. Attenuation by soft tissues limits but does not extinguish these non-linear effects. Harmonics may be used to create tissue harmonic images. These offer improvements over conventional B-mode images in spatial resolution and, more significantly, in the suppression of acoustic clutter and side-lobe artefacts. The quantity B/A has promise as a parameter for tissue characterization, but methods for imaging B/A have shown only limited success. Standard methods for the prediction of tissue in-situ exposure from acoustic measurements in water, whether for regulatory purposes, for safety assessment, or for planning therapeutic regimes, may be in error because of unaccounted non-linear losses. Biological effects mechanisms are altered by finite-amplitude effects.

Beyond ray tracing for internal waves. II. Finite-amplitude effects

We examine the transmission of small and moderately large internal gravity wavepackets through uniformly and nonuniformly stratified fluids using fully nonlinear numerical simulations. The simulations of finite-amplitude waves in a uniformly stratified fluid show that the weakly nonlinear theory developed for horizontally periodic wavepackets extends well to the dynamics of wavepackets with horizontal extent comparable to the horizontal wavelength. In simulations of small-amplitude wavepackets in a nonuniformly stratified fluid the transmission coefficient is found to be comparable to that computed analytically for horizontally periodic waves that radiate continuously (nontransiently) on a reflection level. Simulations of finite-amplitude waves in a nonuniformly stratified fluid show little dependence of transmission coefficient on the wavepacket extent. However, for a wide range of incident wave frequencies the simulations exhibit a monotonic increase in the transmission coefficient as a function of the incident wave amplitude.

Relevant parameters and finite amplitude effects in estuarine hydrodynamics

The propagation of the tidal wave in convergent estuaries is investigated through a one‐dimensional numerical model, which allows consideration of the role of finite amplitude effects. The relevant dimensionless parameters characterizing estuarine hydrodynamics are given in terms of independent quantities, namely the geometry of the channel and the tidal forcing. Hence a suitable scale of velocity is derived, which covers both the cases of convergent and dissipative estuaries. Furthermore, the marginal conditions for tidal wave amplification, resulting from a balance between the damping effect of friction and the amplification due to channel convergence, are investigated. We show that the marginal curves can be given the form of power laws whose coefficients depend on the dimensionless tidal amplitude; the linear behavior predicted by analytical solutions is only recovered for vanishing values of the ratio between the tidal amplitude and the tidally averaged water depth. Finally, the proposed scale of velocity and that derived from the model of H. H. G. Savenije and E. J. M. Veling (2005) are tested against numerical results for a wide range of parameters.

On Shallow-Water Bottom Reverberation Frequency Dependence

Determinations of bottom scattering strength in the decade below 1 kHz under downward refracting conditions have been made using acoustic reverberation and transmission data from the 2001 East China Sea Asian Seas International Acoustic Experiment (ASIAEX). The measurements were performed using explosive sources and receiving hydrophones in ship-suspended vertical-line arrays. The focus of this paper has been the dependence of bottom scattering strength on the frequency and characterization of the uncertainties associated with the extraction of scattering strength from reverberation. The derived bottom scattering strength gradually rises with frequency from 100-300 Hz and then more rapidly above 300 Hz. A potential explanation suggests that the frequency variation results from two scattering mechanisms, rough layer scattering at the low end of the band and sediment near-surface volume scattering at the high end. The spatial extrapolation of these results is explored by comparing them with similarly derived scattering strengths using data obtained under the Navy's Harsh Environments Program at a somewhat separated site (56 km) under environmental conditions similar to those during ASIAEX. In the ASIAEX analysis, it has been found that the largest source of uncertainty in the scattering-strength frequency dependence arises from persistence of finite-amplitude effects associated with the source signal

Temperatures near the interface between an ideal heat exchanger and a thermal buffer tube or pulse tube

A thermal buffer tube (or pulse tube) thermally isolates two heat exchangers at different temperatures in a thermoacoustic engine or refrigerator while allowing the flow of acoustic power. For many heat transport mechanisms, the quality of the thermal isolation depends on the time-averaged mean temperature distribution in the thermal buffer tube, which is determined by boundary conditions set up by the heat exchangers. However, finite-amplitude effects within one peak-to-peak gas displacement of the heat exchangers can lead to significant modification of the thermal boundary conditions and thus the heat transport. To explore these effects, measured mean temperature profiles in the vicinity of the interface between a heat exchanger and thermal buffer tube are reported for a broad range of acoustic and thermal conditions. A one-dimensional Lagrangian model is developed to predict the mean temperature distribution, and reasonable agreement between experimental data and model results is found for the majority of the acoustic conditions considered.

SU-FF-T-317: Options for SURLAS Design Modification Due to the Impact of Ultrasound Nonlinear Propagation

Purpose: To study the impact of acoustic nonlinear propagation on the operation of a Scanning Ultrasound Reflector — Linear Arrays System (SURLAS) and determine possible options to modify its existing design. Method and Materials: The SURLAS is a superficial ultrasound hyperthermia system for the delivery of simultaneous (and sequential) thermoradiotherapy. Heat is delivered by dual-frequency scanned planar ultrasound while radiation is delivered with external beam radiation. Compatibility between the SURLAS and a Linac requires an applicator design where ultrasound waves (1) are of higher intensity and (2) travel longer distances than usual. Hence, ultrasound waves, especially from a 4.9 MHz array may be affected by nonlinear propagation (i.e., finite amplitude effects) which causes a dramatic increase in ultrasonic attenuation in the water coupling medium inside the applicator. Ultrasound propagation for the SURLAS applicator was analyzed to identify possible options in design modification to minimize the impact of nonlinear propagation on acoustic output. Results: Results are given in terms of limits on maximum power output (MPO), maximum traveling distance (MTD), and maximum frequency (MF) that could be used for hyperthermia delivery. The effects of these parameters on the applicator's design and performance, along with options for adjusting its current design, are discussed. Conclusion: A compromise must be reached between MTD and MF for a given required MPO. For example, for a typical chest wall for a frequency of 4.9 MHz, the MTD must be < 11.2 cm. If the current distance of 16 cm is kept, the MF must be < 2.15 MHz. In sum, this work suggests potential solutions for the SURLAS applicator design for specific treatment sites in order to provide sufficient and sustainable heating of tissue during simultaneous thermoradiotherapy. Support: NIH grant R01-CA63121

Behavior of linear beam-plasma instabilities in the presence of finite amplitude circularly polarized waves

We review the effect of finite amplitude circularly polarized waves on the behavior of linear ion-beam plasma instabilities. It has been shown that left-hand polarized waves can stabilize linear right-handed instabilities [1]. It has also been shown that for beam velocities capable of destabilizing left-handed waves, left-hand polarized large amplitude waves can also stabilize these waves. On the other hand, when the large amplitude wave is right-hand polarized, they can either stabilize or destabilize right-handed instabilities depending on the wave frequency and beam speed [2]. Finally, we show that the presence of large amplitude left-hand polarized waves can also trigger electrostatic ion-acoustic instabilities by forcing the phase velocities of two ion acoutic waves to become equal, above a threshold amplitude value.

Eliassen–Palm Flux Diagnosis Based on Isentropic Representation

Abstract This paper demonstrates an isentropic representation of the Eliassen–Palm flux (EP flux) as a tool for diagnosis of the atmospheric zonal mean states. This is a full extension of the transformed Eulerian mean (TEM) to express lower boundary conditions and finite-amplitude effects of waves. The isentropic EP flux is diagnosed from the NCEP–NCAR reanalysis for 12 yr and compared with the TEM. In the free atmosphere, the two diagnoses show similarities in the mass streamfunction and in the EP flux except that the EP flux divergence is somewhat different near the polar vortex. The similarities reflect the equivalency between the two schemes under the quasigeostrophic assumptions. However, the lower boundary values are very different between the two schemes. In the TEM, the vertical EP flux inaccurately suggests a much larger momentum flux coming out from the surface. In the isentropic diagnosis, the lower boundary value of vertical EP flux, which expresses the momentum exchange with the earth's surfa...

A Geosolitary Wave Solution on anfPlane

Abstract This paper presents an exact solution for nonlinear shallow-water waves on an f plane. It is a long wave satisfying the hydrostatic balance. It is also a solitary wave maintained by the balance of a dispersion effect of Coriolis force and the nonlinear effect of advection and finite wave amplitude. It satisfies the conservation of potential vorticity. It has a negative wave height, a negative (positive) relative vorticity in the Northern (Southern) Hemisphere, and a propagation speed smaller than that of the shallow-water wave. It propagates upstream, as a consequence of the conservation of potential vorticity, with a speed of (1 − |a|/h)[gh(1 − |a|/h)]1/2, where a is the wave height (a < 0). The trough of the surface elevation is a singular point where the first-order derivative approaches infinity. However, the region of this singularity is very small for such a wave on earth. For instance, with a 1-m amplitude, the horizontal length scale of the wave is several hundred kilometers but the large...

Weakly nonlinear stability of Hartmann boundary layers

By means of a weakly nonlinear stability analysis it is shown that the Hartmann boundary layer presents subcritical instability in the proximity of the minimum linear critical Reynolds number. This gives further support to earlier speculations that finite amplitude effects account for the discrepancies between the results of the linear stability analysis and experimental evidence on laminarisation.