Analytic solution of liquid-drop impact problems

1981 ◽  
Vol 377 (1770) ◽  
pp. 289-308 ◽  
Keyword(s):  
Linear Theory ◽  
Wave Speed ◽  
Liquid Drop ◽  
Contact Region ◽  
Nonlinear Effects ◽  
Numerical Calculations ◽  
Escape Time ◽  
Good Representation ◽  
Time Step ◽  
Geometrical Acoustics

The pressure in an impacting liquid drop against both a rigid and an elastic target is calculated for the period when the contact region is expanding faster than the wave speed in the liquid. For very low speed impact a geometrical-acoustics model is shown to give a good representation of the solution, until the edge speed approaches the wave speed. A self-similar solution, that takes account of nonlinear effects, is used in the neighbourhood of the contact edge. Comparisons are made with linear theory and numerical calculations. It is shown that linear theory is totally inadequate in predicting the escape of the shock system from the contact edge and that numerical calculations have used too large a time step to calculate the time of escape correctly. The delay in escape time from the previous theoretical predictions of Heymann (1969) is attributed to the elasticity of the target, an effect that is taken into account in the present work.

On the instability of inextensible elastic bodies: nonlinear evolution of non-neutral, neutral and near-neutral modes

1993 ◽  
Vol 443 (1917) ◽  
pp. 59-82 ◽  
Keyword(s):  
Linear Theory ◽  
Wave Speed ◽  
Nonlinear Evolution ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Effects ◽  
Elastic Bodies ◽  
Zero Growth ◽  
Main Action ◽  
Amplitude Growth

For inextensible elastic bodies, linear theory predicts that if the reaction stress is compressive and sufficiently large, a transverse progressive wave travelling in the direction of inextensibility may have an imaginary wave speed and grow without bound as a standing wave (Chen & Gurtin 1974). The development of these growing standing waves under the influence of nonlinearity is considered in this paper. Attention is focused on the case for which the negative reaction stress deviates by a small amount from the value corresponding to the zero wave speed, so that the question addressed is how the evolution of the near-neutral waves is (slowly) modulated by nonlinear effects. It is shown, both numerically and analytically, that depending on initial conditions, nonlinearity can make a near-neutral wave grow, decay or have constant amplitude (growth occurs even in the neutral case for which linear theory predicts zero growth), but in every case its main action is to distort the wave profile and make it evolve into a shock within a finite time. It is found that the evolution of some near-neutral waves (corresponding to certain initial conditions) is governed by analytical solutions, with the aid of which we can show that any shock, once it has formed, will eventually decay to zero algebraically. For general initial conditions, the further evolution of the shock cannot be determined from the present analysis, but we may conjecture that the shock thus formed will also decay to zero. Hence nonlinearity stabilizes near-neutral waves through the formation of shocks. However, an important result found for near-neutral waves is that corresponding to some initial conditions, high values of strain (and thus stress) may obtain just before the shock forms, so that there is the possibility that the elastic body may fracture before the decay of shock amplitude occurs. The effects of nonlinearity on non-neutral travelling waves are also studied and it is shown that nonlinearity also makes non-neutral travelling waves evolve into shocks, but in contrast with the situation for near-neutral waves it does not change their amplitudes as time evolves. The present analysis is also applicable to surface waves in pre-stressed materials where zero wave speed may be induced by large enough pre-stresses.

scholarly journals A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

2021 ◽  
Author(s):  
Chennakesava Kadapa
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Critical Time ◽  
Incompressible Material ◽  
Implicit Scheme ◽  
Bulk Wave ◽  
Lumped Mass ◽  
Time Step ◽  
Material Models ◽  
Shear Wave Speed

AbstractThis paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

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.

Time Simulation of Water Shipping for a Ship Advancing in Large Longitudinal Waves

1993 ◽  
Vol 37 (02) ◽  
pp. 126-137
Author(s):  
Ming-Chung Fang ◽  
Ming-Ling Lee ◽  
Chwang-Kuo Lee
Keyword(s):  
Nonlinear Effects ◽  
Ship Motion ◽  
Irregular Waves ◽  
Wave System ◽  
Time Step ◽  
Large Wave ◽  
Time Simulation ◽  
Head Waves ◽  
Speed Effect

The technique of time-domain numerical simulation for the occurrence of water shipping on board in head waves is presented. The nonlinear effects of the large-amplitude motion are treated. These nonlinear factors include the effect of large wave amplitude, large ship motion, the change of hull configuration below the free surface and the nonlinear resultant wave. Therefore, the variation of the potentials and the hydrodynamic coefficients for a ship at each time step must be carefully treated. While handling the determination of the instantaneous wave surface around the ship hull, the complete incident, diffracted, and radiated wave system is used rather than the incident wave only. The complexity of the ship speed effect on the related terms is also treated at each time step, especially for the radiation problems. An experimental setup is also designed to measure the motion response and the relative motion, and comparisons are made. The results show excellent agreement and the validity of the theory is confirmed. The successful development of the present technique can be extended to analyze the dynamic stability, capsize phenomena, and ship motion in irregular waves

Efficient Computations of Fully-Nonlinear Wave Interactions With Floating Structures

2010 ◽  
Author(s):  
Hongmei Yan ◽  
Yuming Liu
Keyword(s):  
Boundary Element ◽  
High Efficiency ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Computational Effort ◽  
Boundary Element Methods ◽  
Computational Method ◽  
Wave Interactions ◽  
Time Step ◽  
Fully Nonlinear

We consider the problem of fully nonlinear three-dimensional wave interactions with floating bodies with or without a forward speed. A highly efficient time-domain computational method is developed in the context of potential flow formulation using the pre-corrected Fast Fourier Transform (PFFT) algorithm based on a high-order boundary element method. The method reduces the computational effort in solving the boundary-value problem at each time step to O(NlnN) from O(N2∼3) of the classical boundary element methods, where N is the total number of unknowns. The high efficiency of this method allows accurate computations of fully-nonlinear hydrodynamic loads, wave runups, and motions of surface vessels and marine structures in rough seas. We apply this method to study the hydrodynamics of floating objects with a focus on the understanding of fully nonlinear effects in the presence of extreme waves and large-amplitude body motions.

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.

Assessing the impact of the development of an urban district on shallow groundwater using the integrated urban hydrological model URBS

2020 ◽  
Author(s):  
emmanuel berthier ◽  
jérémie sage ◽  
emmanuel dumont ◽  
marie-laure mosini ◽  
fabrice rodriguez ◽  
...  
Keyword(s):  
Water Table ◽  
Water Levels ◽  
Integrated Modelling ◽  
Model Parameters ◽  
Good Representation ◽  
Surprising Result ◽  
Time Step ◽  
Urban District ◽  
The Impact ◽  
Underground Constructions

<p>The urbanisation leads to modifications in the water budget, not only at the surface but in groundwater as well. Few urban modelling studies deal with this topic, due to the lack of appropriate models. The URBS (Urban Runoff Branching Structure) model has been developed since several decades to simulate water transfers at the scale of an urban district. An integrated modelling approach is deliberately adopted to account for the numerous elements that influence urban hydrology: the spatial distribution of the sealed surfaces, interactions between the urban soil and water networks or underground, sustainable drainage systems…. In URBS, the spatial discretization of a catchment is based on Urban Hydrologic Elements (UHE) constituted by cadastral parcels and the adjacent streets, connected to the drainage network. URBS is able to perform continuous and long-period simulations (typically several years) of water fluxes in urban districts for small time-steps (typically few-minutes), with rainfall and potential evapotranspiration as input data.</p><p>The URBS model is adopted to study the hydrological impact of the Moulon district layout, a 200 ha development operation of the Paris-Saclay Cluster (currently underway). The project should result in an increase of sealed surfaces from 14% to 35% and a densification of underground constructions such as networks and basements. A shallow unconfined aquifer extends on the whole area. The fluctuations of ground-water levels have been monitored at an hourly time-step with 8 piezometers since 2012. Water-table levels exhibit significant variations, with near-saturation levels during winter and several meters depths during summer, although the piezometers do not all exhibit the same dynamics.</p><p>A calibration of the URBS model is first conducted for a 2-year period using only piezometric data and no flowrate data. The calibration is solely performed for the parameters influencing the soil compartment: soil permeability and parameters of the sewer infiltration process. Model performances are rather satisfactory with good representation of the observed levels for several piezometers, despite some difficulties for two piezometers exhibiting atypical variations. Once the URBS model is calibrated for the initial situation, simulations are conducted for the project layout (accounting for land-use modification and underground constructions) so as to evaluate the hydrological impacts of the development. Simulation results suggest that an increase of water table levels might be expected after the development of the district (this somehow surprising result may partly originate from the decrease of evapotranspiration fluxes associated with the increased of sealed surfaces).</p><p>The analysis of these first simulations also suggests that large uncertainties might be expected regarding the water levels computed by URBS. A simplified uncertainty analysis (based on Monte-Carlo simulations) is thus conducted to evaluate and distinguish uncertainties associated with model parameters and the total uncertainties in model outputs. While the results clearly evidence the importance of total uncertainties (although the uncertainties due to the model parameters remain low), they also confirm that groundwater depths could be reduced by the construction of the Moulon district.</p>

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