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2016 | Buch

Numerical Modeling of Sea Waves

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Presenting a novel approach to wave theory, this book applies mathematical modeling to the investigation of sea waves. It presents problems, solutions and methods, and explores issues such as statistical properties of sea waves, generation of turbulence, Benjamin-Feir instability and the development of wave fields under the action of wind. Special attention is paid to the processes of dynamic wind-wave interaction, the formation of freak waves, as well as the role that sea waves play in the dynamic ocean/atmosphere system. It presents theoretical results which are followed by a description of the algorithms used in the development of wave forecasting models, and provides illustrations to assist understanding of the various models presented. This book provides an invaluable resource to oceanographers, specialists in fluid dynamics and advanced students interested in investigation of the widely known but poorly investigated phenomenon of sea waves.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction: Different Approaches to Numerical Modeling of Sea Waves—Specifics of Current Approach
Abstract
Development of the mathematical modeling of wave processes and the advantages of the conformal coordinates for investigation of a two-dimensional flow with free surface are discussed. Different approaches to the numerical modeling of a three-dimensional flow are briefly analyzed.
Dmitry V. Chalikov
Chapter 2. Two-Dimensional Wave Model
Abstract
A method for numerical investigation of the nonlinear wave dynamics based on direct hydrodynamical modeling of the 1-D potential periodic surface waves is described. The model is a part of an interactive windwave model. Using non-stationary conformal mapping, the principal equations are rewritten in a surface-following coordinate system and reduced to two simple evolutionary equations for elevation and velocity potential of surface. Fourier expansion is used to approximate these equations.
Dmitry V. Chalikov
Chapter 3. Stationary Solutions of Potential Equations
Abstract
For stationary equations, the proposed approach coincides with the conventional complex variable method. For this case, the numerical algorithms for solution of gravity (Stokes) and gravity-capillary wave equations are proposed, and the examples of numerical solutions are given. The results suggest that gravity-capillary waves do not approach Stokes waves as the capillarity coefficient decreases. Both stationary and non-stationary schemes use Fourier series representation for spatial approximation and the Fourier transform method to calculate the nonlinearities. The main properties of Stokes, gravity-capillary, and capillary waves for infinite depth are discussed. The properties of Stokes waves for finite depth are investigated.
Dmitry V. Chalikov
Chapter 4. Two-Dimensional Wave Modeling Based on Conformal Mapping
Abstract 
High accuracy was confirmed by validation of a non-stationary model against known solutions and by comparison between the results obtained with different resolutions in the horizontal. The method developed is applied for simulation of wave evolution with different initial conditions. The numerical experiments with the initially monochromatic waves of different steepness show that the model is able to simulate the breaking conditions when the surface becomes a multi-valued function of the horizontal coordinate. An estimate of the critical initial wave height that separates non-breaking and eventually breaking waves is obtained. Simulation of nonlinear evolution of a wave field represented initially by two modes with close wave numbers (amplitude modulation) and a wave field with a phase modulation is given. Both runs result in appearance of large and very steep waves. Both of them also break if the initial amplitudes are sufficiently large. The interaction of two monochromatic waves at water surface enters a different dynamic regime if their wave numbers become very close. In the course of evolution of two waves, downshifting of the initial wave energy and growth of the first mode occur depending on wave steepness and a relative distance between modes in Fourier space.
Dmitry V. Chalikov
Chapter 5. Statistical Properties of One-Dimensional Waves
Abstract
A numerical model for long-term simulation of gravity surface waves is described. The model is designed as a component of a coupled wave boundary layer/sea waves model for investigation of a small-scale dynamic and thermodynamic interaction between ocean and atmosphere. The statistical properties of a nonlinear wave field are investigated on the basis of direct hydrodynamical modeling of the 1-D potential periodic surface waves. The high accuracy was confirmed by validation of the non-stationary model against known solutions and by comparison between the results obtained with different resolution in the horizontal. It is shown that the scheme allows to reproduce propagation of a steep Stokes wave for thousands of periods with a very high accuracy. The method developed is applied for simulation of the evolution of wave fields with a large number of modes for many periods of dominant waves. The statistical characteristics of a nonlinear wave field for the waves with different steepness have been investigated: spectra, curtosis and skewness, dispersion relation, and lifetime. The main result that wave field can be presented as a superposition of linear waves is valid just for small amplitudes. It is shown that a nonlinear wave field is rather a superposition of Stokes waves than that of the linear waves.
Dmitry V. Chalikov
Chapter 6. Nonlinear Interaction in One-Dimensional Wave Field
Abstract
Full nonlinear equations for one-dimensional potential surface waves were used for investigation of evolution of the initially homogeneous train of exact Stokes waves with steepness \({AK} = 0.01 - 0.42\). The numerical algorithm for integration of non-stationary equations and calculation of exact Stokes waves is described. Since the instability of the exact Stokes waves develops very slowly, a random small-amplitude noise was introduced in the initial conditions. Development of instability occurs in two stages: In the first stage, the growth rate of disturbances was close to that established for small steepness by Benjamin and Feir in (1967) and for medium steepness—by McLean (1982). For any steepness, Stokes waves disintegrate and create a random superposition of waves. For \({AK} < 0.13\), waves do not show tendency for breaking which is recognized by surface approaching a non-single value shape. Sooner or later, if \({AK} > 0.13\), one of the waves increases its height and finally comes to a breaking point. For the large steepness \({AK} > 0.35\), the rate of growth is slower than for medium steepness, but it does not turn to zero, as it was predicted by McLean (J Fluid Mech 114:315–330, 1982) on the basis of linearized equations for disturbances. The data for spectral composition of disturbances and their frequencies are given. The model is used for investigation of evolution of the wave field initially assigned as a train of harmonic waves. It is shown that a harmonic wave of any amplitude quickly generates new modes which undergo complicated evolution. These modes cannot be referred to neither as bound waves nor as free waves. The results of numerical simulation of adiabatic evolution of the waves assigned in the initial condition with empirical spectrum are presented. It is shown that wave spectrum is subject to strong fluctuations. Most of such fluctuations are reversible; however, a residual effect of the fluctuations causes downshifting of the spectrum. The rate of downshifting depends on the nonlinearity.
Dmitry V. Chalikov
Chapter 7. Modeling of Extreme Waves
Abstract
This chapter describes the results of more than 4000 long-term (up to thousands of peak wave periods) numerical simulations of nonlinear gravity surface waves performed for the investigation of properties and estimation of statistics of extreme (‘freak’) waves. The method of solution of 2-D potential wave’s equations based on conformal mapping is applied to the simulation of wave behavior assigned by different initial conditions, defined by JONSWAP and Pierson–Moskowitz spectra. It is shown that nonlinear wave evolution sometimes results in the appearance of very big waves. There are no predictors for appearance of extreme waves; however, a height of dimensional waves is proportional to a significant wave height. The initial generation of extreme waves can occur simply as a result of linear group effects, but in some cases, the largest wave suddenly starts to grow. It is followed sometimes by a strong concentration of wave energy around a peak vertical. It is taking place typically for one peak wave period. It happens to an individual wave in a physical space, no energy exchange with surrounding waves taking place. Probability function for steep waves has been constructed. Such type function can be used for the development of operational forecast of freak waves based on a standard forecast provided by the 3-D-generation wave prediction model (WAVEWATCH or WAM).
Dmitry V. Chalikov
Chapter 8. Numerical Investigation of Wave Breaking
Abstract
Results of numerical investigations, based on full dynamic equations, are presented for wave breaking in one-dimensional environment with wave spectrum. The breaking is defined as a process of irreversible collapse of an individual wave in physical space, and the incipient breaker is a wave which reached a dynamic condition of the limiting stability where the collapse has not started yet, but is inevitable. Main attention is paid to documenting the evolution of different wave characteristics before the breaking commences. It is shown that the breaking is a localized process which rapidly develops in space and time. No characteristics such as wave steepness, wave height, and asymmetry can serve as a predictor of the incipient breaking. Process of breaking is intermittent; it happens spontaneously and is individually unpredictable. Evolution of geometric, kinematic, and dynamic characteristics of the breaking wave describes the process of breaking itself rather than indicating an imminent breaking. It is shown that the criterion of breaking, valid for the breaking due to modulation instability in one-dimensional wave trains, is not universal if applied to the conditions of spectral environment. In this context, more important is development of algorithms for parameterization of breaking for wave prediction models and for direct wave simulations. Prototype of such algorithm is proposed on the basis of the diffusion-type highly selective operator. It is suggested that the main parameter is differential steepness calculated over entire spectrum. Thousands of exact short-term simulations of evolution of two superposed wave trains with different steepness and wave numbers were performed to investigate the effect of wave crests’ merging. Nonlinear sharpening of the merging crests is demonstrated. It is suggested that such effect may be responsible for the appearance of the typical sharp crests of surface waves, as well as for wave breaking.
Dmitry V. Chalikov
Chapter 9. Numerical Modeling of Wind–Wave Interaction
Abstract
The description of a coupled wind and wave model in conformal coordinates is given. The wave model is based on potential equations for the flow with a free surface, extended with the algorithm of breaking dissipation. The wave boundary layer (WBL) model is based on the Reynolds equations with the Kε closure scheme with the solutions for air and water matched through the interface. The structure of the WBL and vertical profiles of the wave-produced momentum flux (WPMF) in a long-term simulation of the coupled dynamics is investigated and parameterized. The shape of the \(\beta\) function connecting elevation and surface pressure is studied up to high non-dimensional wave frequencies. The errors of a linear presentation of the surface pressure are estimated.
Dmitry V. Chalikov
Chapter 10. One-Dimensional Modeling of the WBL
Abstract
The β function and the universal shape of the WPMF profile obtained in coupled simulations allow a formulation of the one-dimensional theory of the WBL and the carrying out of a detailed study of the WBL structure including the dependence of the drag coefficient on the wind speed. It is shown that a wide scatter of the experimental data on the drag coefficient can be explained, taking into account the age of waves. It is suggested that a reduction of the drag coefficient at high wind speeds can be qualitatively explained by the high-frequency wave suppression. A direct wave model based on the one-dimensional nonlinear equations for potential waves is used for simulation of wave field development under the action of energy input, dissipation, and nonlinear wave–wave interaction. The equations are written in conformal surface-fitted non-stationary coordinate system. New schemes for calculating the input and dissipation of wave energy are implemented. The wind input is calculated on the basis of the parameterization developed through the coupled modeling of waves and turbulent boundary layer. The wave dissipation algorithm, introduced to prevent wave breaking instability, is based on highly selective smoothing of the wave surface and surface potential. The integration is performed in Fourier domain with the number of modes M = 2048, broad enough to reproduce the energy downshifting. As the initial conditions, the wave field is assigned as train of Stokes waves with steepness ak = 0.15 at non-dimensional wave number k = 512. Under the action of nonlinearity and energy input, the spectrum starts to grow. This growth is followed by the downshifting. The total time of integration is equal to 7203 initial wave periods. During this time, the energy increased by 1111 times. Peak of the spectrum gradually shifts from wave number non-dimensional k = 512 down to k = 10. Significant wave height increases 33 times, while the peak period increases 51 times. Rates of the peak downshift and wave energy evolution are in good agreement with the JONSWAP formulation.
Dmitry V. Chalikov
Chapter 11. Numerical Investigation of Turbulence Generation in Non-breaking Potential Waves
Abstract
Theoretically, potential waves cannot generate the vortex motion, but the scale considerations (Babanin 2006) indicate that if the steepness of waves is not too small, the Reynolds number can exceed the critical values. This means that in presence of initial non-potential disturbances, the orbital velocities can generate the vortex motion and turbulence. This problem has been investigated by means of linear instability theory (Benilov et al. 1993). It was shown that pure two-dimensional motion always remains potential because one-dimensional vortex (in vertical plane) does not interact with the orbital motion. The waves can generate the vortex in horizontal plane, and further development of vorticity occurs due to exchange of energy between the components of vorticity. Then, due to nonlinearity, motion at smaller scales and more or less developed turbulent regime arise. This problem was investigated numerically on basis of full two-dimensional (xz) equations of potential motion with the free surface in cylindrical conformal coordinates. It was assumed that all variables are a sum of the 2-D potential orbital velocities and 3-D non-potential disturbances. Because the energy of waves is much larger than energy of turbulence, currently it was assumed that only one-way interaction exists: Non-potential motion takes the energy from potential waves. The non-potential motion is described directly with 3-D Euler equations, with very high resolution. The interaction between potential orbital velocities and non-potential components is accounted through additional terms which include the components of vorticity. The effects of turbulence are incorporated with a use of subgrid turbulent energy evolution equation. The turbulent scale is assumed to be proportional to grid resolution (LES technique). For small waves, the approach turns into a direct simulation method. Numerical scheme is based on 2-D Fourier transform method in ‘horizontal’ (in conformal coordinates) plane and on second-order approximation in the ‘vertical’. The pressure is calculated by means of Poisson equation in cylindrical conformal coordinates derived through covariant components of velocity. Poisson equation was solved with three-diagonal matrix algorithm (TDMA). Initial conditions for the elevations and the surface potential for waves were assigned according to the linear theory, and 3-D non-potential velocity components were inserted as a small-amplitude noise. Long-term numerical integration of the system of equations was done for different wave steepness. The vorticity and turbulence usually occurred in vicinity of wave crests (where the velocity gradients reach their maximum) and then spreads over upwind slope and downward. Specific feature of the wave turbulence is its strong intermittency: The turbulent patches are mostly isolated, and intermittency grows with the decrease of the wave amplitude. The maximum values of energy of turbulence are in qualitative agreement with experimental data. The results suggest that even non-breaking potential waves can generate the turbulence, which thus enhance the turbulence created by the shear current. Further investigation of this process will include the effect of tangential stress on a sea surface and flux of turbulent energy from the surface generated by breaking waves.
Dmitry V. Chalikov
Chapter 12. Three-Dimensional Modeling of Potential Waves
Abstract
A simple and exact numerical scheme for long-term simulations of three-dimensional potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following non-orthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid, and the fourth-order Runge–Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands of degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear two-dimensional surface waves, generation of extreme waves, and direct calculations of nonlinear interactions.
Dmitry V. Chalikov
Backmatter
Metadaten
Titel
Numerical Modeling of Sea Waves
verfasst von
Dmitry V. Chalikov
Copyright-Jahr
2016
Electronic ISBN
978-3-319-32916-1
Print ISBN
978-3-319-32914-7
DOI
https://doi.org/10.1007/978-3-319-32916-1