Interaction Of Solitary Waves Research Articles

Interaction of Schamel type solitary waves in an electronegative plasma with trapped non-extensive electrons

Interaction of Schamel type solitary waves in an electronegative plasma with trapped non-extensive electrons

Predictions of the elastic modulus of trabecular bone in the femoral head and the intertrochanter: a solitary wave-based approach.

This paper deals with the numerical prediction of the elastic modulus of trabecular bone in the femoral head (FH) and the intertrochanteric (IT) region via site-specific bone quality assessment using solitary waves in a one-dimensional granular chain. For accurate evaluation of bone quality, high-resolution finite element models of bone microstructures in both FH and IT are generated using a topology optimization-based bone microstructure reconstruction scheme. A hybrid discrete element/finite element (DE/FE) model is then developed to study the interaction of highly nonlinear solitary waves in a granular chain with the generated bone microstructures. For more robust and reliable prediction of the bone's mechanical properties, a face sheet is placed at the interface between the last chain particle and thebone microstructure, allowing more bone volume to be engaged in the dynamic deformation during interaction with the solitary wave. The hybrid DE/FE model was used to predict the elastic modulus of the IT and FH by analysing the characteristic features of the two primary reflected solitary waves. It was found that the solitary wave interaction is highly sensitive to the elastic modulus of the bone microstructure and can be used to identify differences in bone density. Moreover, it was found that the use of a relatively stiff face sheet significantly reduces the sensitivity of the wave interaction to local stiffness variations across the test surface of the bone, thereby enhancing the robustness and reliability of the proposed method. We also studied the effect of the face sheet thickness on the characteristics of the reflected solitary waves and found that the optimal thickness that minimizes the error in the modulus predictions is 4mm for the FH and 2mm for the IT, if the primary reflected solitary wave is considered in the evaluation process. We envisage that the proposed diagnostic scheme, in conjunction with 3D-printed high-resolution bone models of an actual patient, could provide a viable solution to current limitations in site-specific bone quality assessment.

Experimental research on the mechanisms of condensation induced water hammer in a natural circulation system

Natural circulation systems (NCSs) are extensively applied in nuclear power plants because of their simplicity and inherent safety features. For some passive natural circulation systems in floating nuclear power plants (FNPPs), the ocean is commonly used as the heat sink. Condensation induced water hammer (CIWH) events may appear as the steam directly contacts the subcooled seawater, which seriously threatens the safe operation and integrity of the NCSs. Nevertheless, the research on the formation mechanisms of CIWH is insufficient, especially in NCSs. In this paper, the characteristics of flow rate and fluid temperature are emphatically analyzed. Then the formation types of CIWH are identified by visualization method. The experimental results reveal that due to the different size and formation periods of steam slugs, the flow rate presents continuous and irregular oscillation. The fluid in the horizontal hot pipe section near the water tank is always subcooled due to the reverse flow phenomenon. Moreover, the transition from stratified flow to slug flow can cause CIWH and enhance flow instability. Three types of formation mechanisms of CIWH, including the Kelvin-Helmholtz instability, the interaction of solitary wave and interface wave, and the pressure wave induced by CIWH, are obtained by identifying 67 CIWH events.

Kink–antikink interaction forces and bound states in a ϕ 4 model with quadratic and quartic dispersion

We consider the interaction of solitary waves in a model involving the well-known ϕ 4 Klein–Gordon theory, but now bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponential interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with sufficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corresponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink–antikink collisions and the corresponding single- and multi-bounce windows.

Analytical modelling of solitary wave diffraction from a V-shaped breakwater

This analytical study presents the solitary wave interaction with a bottom-mounted surface-piercing V-shaped breakwater. The breakwater is impermeable, thin, and rigidly fixed on the bottom. As a key element, an imaginary closed cylinder with an arm’s length radius was introduced so that the fluid domain was divided into three subdomains by an imaginary interface, within which the solutions were described by eigenfunctions. Furthermore, by satisfying the boundary and matching conditions in the subdomains, the velocity potential at any point in the fluid domain could be determined. The numerical results obtained for the limiting cases were verified by a comparison with previous predictions for a straight breakwater. The effects of the wave incident angle, opening angle of the breakwater, and wave-effect parameter on wave loads and wave run-up were studied. Furthermore, comparison and analysis of the hydrodynamic performance of the V-shaped and arc-shaped breakwaters with a similar structure showed that they had a similar wave attenuation effect. However, the V-shaped breakwater was subject to smaller wave loads. Moreover, for solitary wave diffraction caused by the breakwater, the Airy wave model might significantly underestimate wave run-up.

Using SAR imagery to survey internal solitary wave interactions: A case study off the Western Iberian shelf

Physical oceanography is increasingly relying on satellite remote sensing to survey the perpetually undersampled ocean, whereas the latest Synthetic Aperture Radars (SARs) are moving forward to provide a more continuous monitoring of the ocean. In this study we use a collection of SAR images to document the two-dimensional horizontal structure of Internal Solitary Waves (ISWs) propagating between two large submarine canyons off the Western Iberian Peninsula (between May and October 2018), which are observed to intersect approximately along the mid-shelf and originate a naturally-occurring interaction hotspot between different ISW packets. ISW interactions are well documented in theory and in laboratorial and numerical studies, but their observations in the real ocean are limited to airborne observations over the Strait of Georgia. The frequent SAR imagery of interacting ISWs in this region provides additional case studies to the literature, and we investigate if an energy proxy taken from their sea surface signatures can be used as an indicator for high-energy interaction events (e.g. when comparing with a non-interacting background). In particular, a quasi-synergetic event captured both in SAR and in a moored thermistor chain reveals that the often used weakly nonlinear theory for small-amplitude waves may underestimate the amplitudes measured in the waves’ interacting sections. ISWs provide the largest vertical displacements and velocities in the ocean. Understating how their vertical structure changes during wave-wave interactions may have important implications in the broader spectrum of ocean sciences, and SARs are shown in this study to be a first-approach tool to survey this frequent phenomenon in coastal regions.

Chaotic Dynamics of the Interface between Dielectric Liquids at the Regime of Stabilized Kelvin-Helmholtz Instability by a Tangential Electric Field

The nonlinear dynamics of the interface between two immiscible dielectric liquids at the regime of suppressed Kelvin-Helmholtz instability by external horizontal electric field is studied theoretically. The initial equations of the fluids motion are reduced to a single weakly nonlinear integro-differential equation that describes the interaction of solitary waves (rational solitons) propagating along the interface. The dynamics of two interacting solitons is regular and integrable; they can combine into a stable wave packet (breather). It is shown that the interaction of three solitons becomes complex and, for a wide rang of initial conditions, chaotic. The numerically obtained Poincaré sections demonstrate the destruction of toroidal trajectories in the phase space during the transition of the system to a chaotic regime of fluid motion. Such a behaviour is consistent with the Kolmogorov-Arnold-Moser theory describing quasi-periodic chaotic motion in Hamiltonian systems. At the developed chaotic state, the system fast loses the information on its initial state; the corresponding estimate for Lyapunov exponent is obtained. From the physical point of view, the chaotic behavior of the system is related with structural instability of the soliton triplet. The triplet can decay into a solitary wave and stable breather consisting of two interacting solitons.

Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

Numerical investigation of solitary wave interaction with a flapper wave energy converter using incompressible SPH method

In this research, mesh-free incompressible smoothed particle hydrodynamics (ISPH) was used to study the oscillatory behavior of a flapper wave energy converter located at the end of a numerical tank under the impact of solitary waves. A two-step projection approach including prediction and correction steps was employed to conduct temporal discretization of the governing equations for inviscid fluid flows. At different times, the pressure fields were implicitly calculated by the ISPH method to solve a pressure Poisson equation (PPE) derived upon the enforcement of the incompressibility conditions. To the authors’ knowledge, not many mesh-less numerical studies have been accomplished addressing different aspects of absorption of wave power using oscillating flappers. In this study, effects of different parameters, including solitary wave height, flapper spring stiffness, moment of inertia of the wall withstanding wave loads, initial angle of the flapper and initial longitudinal wave crest–flapper distance for a pair of solitary waves, were examined to analyze the wave energy absorption performance of the flapper. The results showed the necessity of adopting optimum spring stiffness to maximize the wave power absorption. Otherwise, wave overtopping may occur for sub-optimal spring stiffness values, while the absorbed wave power was lower than maximal for too rigid flappers. It was further found that the flapper oscillation period decreased with increasing the spring stiffness, with the other parameters imposing no significant effect on the period. As another finding, it was figured out that the moment of inertia of the downstream wall must be optimal to capture maximum wave power. The initial angle of the flapper was also found to be an important factor contributing to wave power absorption. The maximal absorbed wave power was achieved when a pair of solitary waves was flowing toward the flapper in such a way that the second wave hit the flapper at the end of the flapper period, causing some resonance phenomenon. Damping process of the oscillatory motion due to the flow viscosity and flapper friction was also considered. The results of the present work can be used to maximize the wave power absorption by adjusting the wave converter design considering the wave height, spring stiffness, moment of inertia and longitudinal distance of wave crest between two successive solitary waves.

Quintic B-spline collocation method for the numerical solution of the Bona–Smith family of Boussinesq equation type

Abstract Our main purpose in this work is to investigate a new solution that represents a numerical behavior for one well-known nonlinear wave equation, which describes the Bona–Smith family of Boussinesq type. A numerical solution has been obtained according to the quintic B-spline collocation method. The method is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The stability of the proposed method has been discussed and presented to be unconditionally stable. The efficiency of the proposed method has been demonstrated by studying a solitary wave motion and interaction of two and three solitary waves. The results are found to be in good agreement with the analytic solution of the system. We demonstrated the physical interpretation of some obtained results graphically with symbolic computation.

Interaction of internal solitary waves with long periodic waves within the rotation modified Benjamin–Ono equation

The interaction of a solitary wave with a long background periodic wave in a deep ocean is studied within the framework of the rotation modified Benjamin–Ono equation. Using a multiple scales analysis, we find stationary and nonstationary solutions for a Benjamin–Ono soliton trapped within a long sinusoidal wave. We show that the radiation losses experienced by the soliton and caused by the Earth’s rotation can be compensated by energy pumping from the background wave. However, the feedback of the soliton on the background wave eventually leads to the destruction of the coherent structure and energy dispersion in a quasi-random wave field. Estimates of the characteristic parameters of the soliton dynamics are presented for real oceanic conditions.

Trapped solitary-wave interaction for Euler equations with low-pressure region

Trapped solitary-wave interaction is studied under the full Euler equations in the presence of a variable pressure distribution along the free surface. The physical domain is flattened conformally onto a strip and the computations are performed in the canonical domain. Computer simulations display solitary waves that remain trapped in a low-pressure region. In terms of confinement, we observe that these waves are stable for small perturbations of either their amplitudes or the pressure forcing term. Furthermore, multiple solitary waves are considered within the low-pressure region without escaping the low-pressure region. We identify regimes in which multiple solitary waves remain trapped after several collisions. In particular, we display a regime where three solitary waves are trapped and collide several times, before one escapes at a time. The remaining solitary waves stay trapped in the low-pressure region.

Solitary Wave Diffraction with a Single and Two Vertical Circular Cylinders

This study investigates the three-dimensional (3-D) solitary wave interaction with two cylinders in tandem and side-by-side arrangements for two wave heights. The solitary wave generation and propagation are predicted using the volume of fluid method (VOF) coupled with the NavierStokes transport equations. The PHOENICS code is used to solve these transport equations. The solitary wave generation based on the source line developed by Hafsia et al. (2009) is extended in three-dimensional wave flow and is firstly validated for solitary waves propagating on a flat bottom. The comparison between numerical results and analytical solution for small wave height H / h = 0.1 and 0.2 shows good agreements. The wave crest and the pseudo-wavelength are well reproduced. Excellent agreements were found in terms of maximum run-up and wave forces by comparison with the present model and analytical studies. The present model can be tested for the extreme solitary wave to extend its application to a more realistic case study as the solitary wave diffraction with an offshore oil platform.

Matrix Spectral Problems and Integrability Aspects of the Błaszak-Marciniak Lattice Equations

A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

Numerical modelling of solitary wave and structure interactions using level-set and immersed boundary methods by adopting adequate inlet boundary conditions

ABSTRACT Solitary waves are often used to replicate tsunami waves and, therefore, the study of their propagation and interactions with structures could help to understand the impacts of tsunami waves on infrastructures. This paper presents a numerical model that uses the level-set framework for simulating the propagation of solitary waves and their interactions with structures using the immersed boundary method. The model imposes a set of appropriate boundary conditions to prescribe the incoming solitary waves within the sharp-interface immersed boundary method. Turbulence is modelled using the large-eddy simulation. Experimental tests are conducted to measure the free-surface elevation and velocity field evolution of solitary waves, as they propagate downwave interacting with two different wall-mounted structures. Analysis of the computed and experimental results revealed interesting dynamics of primary vortex-tubes forming near the sharp edges of structures. It is shown that the primary vortices eventually split to form pairs of asymmetrical doublets.

Numerical solutions of Boussinesq equation using Galerkin finite element method

AbstractIn this study, Galerkin finite element method has been applied to good Boussinesq (GBq) and bad Boussinesq (BBq) equations which are examples of Boussinesq type equations. To apply the method, cubic B‐spline basis functions are taken as element and weight functions. The solutions of the numerical schemes have been obtained using the fourth order Runge–Kutta method. The error norms L2 and L∞ have been used to test how compatible they obtained numerical solutions with those of exact ones. As numerical examples, solitary wave motion and interaction of solitary waves have been investigated for both GBq and BBq equation. Also blow‐up solutions related to the interaction of two solitary waves are considered for GBq equation.

Numerical investigation of solitary waves interaction with an emerged composite structure

The present study describes a numerical investigation on the hydrodynamic characteristics of the impinging and overtopping by solitary waves on an emerged impermeable trapezoidal seawall on a sloping beach. A flow solver of Navier-Stokes equations is employed with the constrained interpolation profile method and a volume-of-fluid type method to capture the strong nonlinear free surface. Three typical solitary waves are simulated to validate the capability of the numerical model. A favorable agreement between experimental and numerical results reveals that this model can reproduce the nonlinear hydrodynamic features of solitary wave impinging and overtopping involving vortex, breaking and impacting. Furthermore, parametric studies are conducted by numerical modeling to examine the effects of wave parameters including water level and wave nonlinearity and leeward slope ratio of the seawall on hydrodynamic stability including sliding and overturning stability and erosion potential. Vorticity and pressure fields are exhibited to show the dynamic properties of flow separation, air entrainment and breaking. It is found that the wave parameters and leeward slope ration influence the hydrodynamic stability of the seawall and erosion of the leeward slope in a different way.

Frontal collision of two nonlinear internal solitary waves in a stratified fluid

Head-on collision of two internal solitary waves (ISWs) in a two-layer inviscid fluid with fully nonlinear free-surface (FS) condition is investigated numerically in this paper. The collision process of and nonlinear interaction of internal solitary waves is studied based on nonlinear potential flow theory and a series of numerical simulations are conducted using a multi-domain boundary element method (MDBEM). The numerical model is validated by experiment for the simulation of ISWs propagating. The computation results show that the collision of ISWs is the asymmetric and nonlinear process in which the rundown motion causes the interface displacement penetrates the interface. The collision leaves imprints on small dispersive trailing waves which accounts for the attenuation with each departing ISW tilting slightly backward with respect to the direction of its propagation. In addition, the characteristic of collision process including run up, residence time, surface wave feature is discussed. During the collision, the wave spends more time falling than rising for all cases and the residence time is very long for waves of small amplitude, large depth ratios and density ratios. Considering the free surface of the top layer whether or not makes contributions to the run up and the residence time is larger if the top boundary is free surface.

Interaction of solitary wave with submerged breakwater by smoothed particle hydrodynamics

This study investigates the interaction between solitary wave and submerged breakwater based on smoothed particle hydrodynamics. The Goring wave making method is applied. The interactions between a solitary wave and a submerged breakwater with rectangular and semi-circular section are investigated to verify the ability of the presented numerical model. A composite submerged breakwater is proposed by combining a quarter circle and a square. The different arrangements of submerged breakwater are defined as Profile Type I (PT-I) and Profile Type II (PT-II). The forces of the semi-circular of these two submerged breakwaters are calculated and compared. In order to evaluate the wave dissipation effect of these two types of breakwaters, we propose an empirical formula of incident wave energy and a novel concept of an energy transmission coefficient. It is shown that potential energy is transmitted more easily through a submerged breakwater than kinetic energy. The results reveal that the different arrangements of PT-I and PT-II have no obvious effect on wave dissipation, whereas the depth of the submerged breakwater has a significant influence.

Propagation and interaction of dust ion acoustic solitary waves(DIASWs) for the damped forced modified Korteweg-de-Vries-Burger equation at some critical composition of parameters

The analytical solution of dust-ion-acoustic-solitary waves (DIASWs)with the influence of external periodic force are reported in a dusty plasma model which consists of dust-ion collisions. Using reduction perturbation technique (RPT) damped modified forced Korteweg-de-Vries-Burger (DMFKdVB)equation in some critical composition is obtained. Various physical parameters i.e. dust ion collision frequency (νid0), the entropic index (q), the velocity of travelling wave (M0) ,the ratio of the densities between electrons and ions (μ), viscosity coefficient (η), frequency (w) of the external perturbation and the strength(f0) of the perturbation are observed. This study may be helpful to interpret the characteristics of DIASWs in astrophysical rings, interstellar clouds and comet tails.