Numerical modelling of water-wave evolution based on the Zakharov equation

2001 ◽  
Vol 449 ◽  
pp. 341-371 ◽  
Author(s):  
SERGEI YU. ANNENKOV ◽  
VICTOR I. SHRIRA
Keyword(s):  
Numerical Modelling ◽  
Canonical Transformation ◽  
Water Waves ◽  
Water Wave ◽  
Numerical Models ◽  
Broad Band ◽  
Normal Modes ◽  
Physical Space ◽  
Zakharov Equation ◽  
Wave Evolution

We develop a new approach to numerical modelling of water-wave evolution based on the Zakharov integrodifferential equation and outline its areas of application.The Zakharov equation is known to follow from the exact equations of potential water waves by the symmetry-preserving truncation at a certain order in wave steepness. This equation, being formulated in terms of nonlinear normal variables, has long been recognized as an indispensable tool for theoretical analysis of surface wave dynamics. However, its potential as the basis for the numerical modelling of wave evolution has not been adequately explored. We partly fill this gap by presenting a new algorithm for the numerical simulation of the evolution of surface waves, based on the Hamiltonian form of the Zakharov equation taking account of quintet interactions. Time integration is performed either by a symplectic scheme, devised as a canonical transformation of a given order on a timestep, or by the conventional Runge–Kutta algorithm. In the latter case, non-conservative effects, small enough to preserve the Hamiltonian structure of the equation to the required order, can be taken into account. The bulky coefficients of the equation are computed only once, by a preprocessing routine, and stored in a convenient way in order to make the subsequent operations vectorized.The advantages of the present method over conventional numerical models are most apparent when the triplet interactions are not important. Then, due to the removal of non-resonant interactions by means of a canonical transformation, there are incomparably fewer interactions to consider and the integration can be carried out on the slow time scale (O(ε2), where ε is a small parameter characterizing wave slope), leading to a substantial gain in computational efficiency. For instance, a simulation of the long-term evolution of 103 normal modes requires only moderate computational resources; a corresponding simulation in physical space would involve millions of degrees of freedom and much smaller integration timestep.A number of examples aimed at problems of independent physical interest, where the use of other existing methods would have been difficult or impossible, illustrates various aspects of the implementation of the approach. The specific problems include establishing the range of validity of the deterministic description of water wave evolution, the emergence of sporadic horseshoe patterns on the water surface, and the study of the coupled evolution of a steep wave and low-intensity broad-band noise.

scholarly journals Super compact equation for water waves

2017 ◽  
Vol 828 ◽  
pp. 661-679 ◽  
Author(s):  
A. I. Dyachenko ◽  
D. I. Kachulin ◽  
V. E. Zakharov
Keyword(s):  
Wave Equation ◽  
Water Waves ◽  
Wave Breaking ◽  
Water Wave ◽  
Rogue Waves ◽  
Resonant Interaction ◽  
Zakharov Equation ◽  
Advection Term ◽  
New Equation ◽  
The Many

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (DystheProc. R. Soc. Lond.A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

On higher order Bragg resonance of water waves by bottom corrugations

2010 ◽  
Vol 659 ◽  
pp. 484-504 ◽  
Author(s):  
JIE YU ◽  
LOUIS N. HOWARD
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Amplitude ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Higher Order ◽  
Bragg Resonance ◽  
Sinusoidal Modulation ◽  
Integer Multiple ◽  
Bragg Resonances

The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Shallow water wave generation by unsteady flow

1968 ◽  
Vol 31 (4) ◽  
pp. 779-788 ◽  
Author(s):  
J. E. Ffowcs Williams ◽  
D. L. Hawkings
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Small Amplitude ◽  
Point Of View ◽  
Sound Waves ◽  
Aerodynamic Sound ◽  
Shallow Layer ◽  
Sound Theory ◽  
Water Simulation

Small amplitude waves on a shallow layer of water are studied from the point of view used in aerodynamic sound theory. It is shown that many aspects of the generation and propagation of water waves are similar to those of sound waves in air. Certain differences are also discussed. It is concluded that shallow water simulation can be employed in the study of some aspects of aerodynamically generated sound.

scholarly journals A Spherical Hybrid Triboelectric Nanogenerator for Enhanced Water Wave Energy Harvesting

Micromachines ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 598 ◽  
Author(s):  
Kwangseok Lee ◽  
Jeong-won Lee ◽  
Kihwan Kim ◽  
Donghyeon Yoo ◽  
Dong Kim ◽  
...  
Keyword(s):  
Energy Harvesting ◽  
Water Waves ◽  
Water Wave ◽  
Degrees Of Freedom ◽  
Low Frequency ◽  
Random Motion ◽  
Triboelectric Nanogenerator ◽  
Hybrid Energy ◽  
Pollution Levels ◽  
Self Powered

Water waves are a continuously generated renewable source of energy. However, their random motion and low frequency pose significant challenges for harvesting their energy. Herein, we propose a spherical hybrid triboelectric nanogenerator (SH-TENG) that efficiently harvests the energy of low frequency, random water waves. The SH-TENG converts the kinetic energy of the water wave into solid–solid and solid–liquid triboelectric energy simultaneously using a single electrode. The electrical output of the SH-TENG for six degrees of freedom of motion in water was investigated. Further, in order to demonstrate hybrid energy harvesting from multiple energy sources using a single electrode on the SH-TENG, the charging performance of a capacitor was evaluated. The experimental results indicate that SH-TENGs have great potential for use in self-powered environmental monitoring systems that monitor factors such as water temperature, water wave height, and pollution levels in oceans.

scholarly journals Forecasting the impact of buildings with multi-storey underground parts on the displacement of subsoil using modern numerical tools

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hanna Michalak ◽  
Paweł Przybysz
Keyword(s):  
Numerical Modelling ◽  
Numerical Models ◽  
Deep Excavation ◽  
Design And Implementation ◽  
Numerical Tools ◽  
Excavation Support ◽  
Area Of Influence ◽  
The Impact ◽  
New Buildings ◽  
Ground Part

Abstract The paper will analyse and review the experience to date in determining the impact range of implementation of deeply founded structures on the displacement of the subsoil in the vicinity. With the background of these experiences, primarily empirical, the present possibilities of using numerical modelling to forecast the displacements of the terrain surface in various stages of works, that is, execution of deep excavation support systems, excavation-deepening phases with successive adding of struts, construction of underground levels and erection of the above-ground part of the building, will be presented. Based on the results of own research, conclusions on the use of 3D numerical models in spatial shaping and designing the structure of underground parts of new buildings erected in dense urban development will be presented. The characterised 3D numerical models were verified, taking into account the actual results of geodetic measurements of the completed buildings. Determining the range and forecasting the displacements of the subsoil are necessary for the design and implementation of investments due to the need to ensure the safety of erection and use of a new building and the buildings located within the area of influence.

scholarly journals THE USE OF ARRAY PROCESSORS FOR NUMERICAL MODELLING OF TIDAL ESTUARY DYNAMICS

1980 ◽  
Vol 1 (17) ◽  
pp. 142
Author(s):  
D. Prandle ◽  
E.R. Funke ◽  
N.L. Crookshank ◽  
R. Renner
Keyword(s):  
Numerical Model ◽  
Numerical Modelling ◽  
Hybrid Model ◽  
Numerical Models ◽  
Computer Time ◽  
Scale Model ◽  
Hybrid Modelling ◽  
Tidal Propagation ◽  
Physical Scale ◽  
Array Processors

The use of array processors for the numerical modelling of estuarine systems is discussed here in the context of "hybrid modelling", however, it is shown that array processors may be used to advantage in independent numerical simulations. Hybrid modelling of tidal estuaries was first introduced by fiolz (1977) and later by Funke and Crookshank (1978). In a hybrid model, tidal propagation in an estuary is simulated by dynamically linking an hydraulic (or physical) scale model of part of the estuary to a numerical model of the remaining part in a manner such that a free interchange of flow occurs at the interface(s). Typically, the elevation of the water surface at the boundary of the scale model is measured and transmitted to the numerical model. In return, the flow computed at the boundary of the numerical model is fed directly into the scale model. This approach enables the extent of the scale model to be limited to the area of immediate interest (or to that area where flow conditions are such that they can be most accurately simulated by a scale model). In addition, since the region simulated by the numerical model can be extended almost indefinitely, the problems of spurious reflections from downstream boundaries can be eliminated. In normal use, numerical models are evaluated on the basis of computing requirements, cost and accuracy. The computer time required to simulate one tide cycle is, in itself, seldom of interest except in so far as it affects the above criteria. However in hybrid modelling this parameter is often paramount since concurrent operation of the numerical and scale models requires that the former must keep pace with the latter. The earlier hybrid model of the St. Lawrence (Funke and Crookshank, 1978) involved a one-dimensional numerical model of the upstream regions of the river. However, future applications are likely to involve extensive two-dimensional numerical simulation.

Numerical solutions to wave interaction with rubblemound breakwaters

1990 ◽  
Vol 17 (2) ◽  
pp. 252-261 ◽  
Author(s):  
Kevin R. Hall
Keyword(s):  
Numerical Modelling ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Scale Effects ◽  
Numerical Models ◽  
Internal Flow ◽  
Wave Interaction ◽  
Theoretical Development ◽  
Porous Media Flow ◽  
Complex Flow

The interaction of a wave with a rubblemound breakwater results in a complex flow field which is both nonlinear and turbulent, particularly within a region close to the surface of the structure. Numerical models describing internal flow in a rubblemound breakwater are becoming increasingly important, particularly as the influence of scale effects on internal flow in physical hydraulic models are becoming understood as important. A number of numerical models to predict the internal breakwater flow kinematics have been produced in the past two decades. This paper provides a review of the state-of-the-art of numerical modelling of wave interaction with rubblemound breakwaters. Details of the theoretical development and the resulting numerical solution techniques are presented. Methods for incorporating secondary effects such as two-phase (air–water) flow, inertia, and unbalanced boundary conditions are discussed. Limitations of the models resulting from the validity of the assumptions made in order to effect a numerical solution are discussed. Key words: breakwaters, internal flow, porous media flow, numerical modelling, rubblemound breakwaters.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

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