Mathematical and Numerical Modeling of Water Waves

A special issue of Fluids (ISSN 2311-5521). This special issue belongs to the section "Mathematical and Computational Fluid Mechanics".

Deadline for manuscript submissions: closed (1 May 2021) | Viewed by 46403

Special Issue Editors


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School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
Interests: ocean waves; rogue waves; wave–structure interaction; wave–ice interaction; ocean dynamics
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Guest Editor
Department of Mechanical Engineering and Product Design Engineering, Swinburne University of Technology, Melbourne, Victoria 3122, Australia
Interests: bubble acoustics; fluid dynamics; ocean wave-power; wave modelling and wave-induced processes; applied mathematics
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Special Issue Information

Dear Colleagues,

Water waves are commonly used as example of waves but “have all the complications that waves can have” as Prof Richard Feynman pointed out in his Lectures on Physics. Starting from the mid 19th century, fundamental wave theories for deep and shallow waters were formulated providing a mathematical framework for applications in physics and engineering. This Special Issue of Fluids collects reviews and original research on recent developments in the mathematical and numerical modeling of water waves phenomena. Specific topics may include wave breaking, nonlinear wave propagation, spectral wave modeling, wave turbulence, rogue waves, solitary waves, wave–current interaction, wave–structure interaction, and wave energy conversion.

Dr. Alberto Alberello
Prof. Dr. Richard Manasseh
Guest Editors

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Keywords

  • water waves
  • surface waves
  • ocean waves
  • wave modeling
  • CFD

Published Papers (17 papers)

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Research

22 pages, 8761 KiB  
Article
A Deep Learning Approach for Wave Forecasting Based on a Spatially Correlated Wind Feature, with a Case Study in the Java Sea, Indonesia
by Didit Adytia, Deni Saepudin, Sri Redjeki Pudjaprasetya, Semeidi Husrin and Ardhasena Sopaheluwakan
Fluids 2022, 7(1), 39; https://doi.org/10.3390/fluids7010039 - 17 Jan 2022
Cited by 16 | Viewed by 3131
Abstract
For safety and survival at sea and on the shore, wave predictions are essential for marine-related activities, such as harbor operations, naval navigation, and other coastal and offshore activities. In general, wave height predictions rely heavily on numerical simulations. The computational cost of [...] Read more.
For safety and survival at sea and on the shore, wave predictions are essential for marine-related activities, such as harbor operations, naval navigation, and other coastal and offshore activities. In general, wave height predictions rely heavily on numerical simulations. The computational cost of such a simulation can be very high (and it can be time-consuming), especially when considering a complex coastal area, since these simulations require high-resolution grids. This study utilized a deep learning technique called bidirectional long short-term memory (BiLSTM) for wave forecasting to save computing time and to produce accurate predictions. The deep learning method was trained using wave data obtained by a continuous numerical wave simulation using the SWAN wave model over a 20-year period with ECMWF ERA-5 wind data. We utilized highly spatially correlated wind as input for the deep learning method to select the best feature for wave forecasting. We chose an area with a complex geometry as the study case, an area in Indonesia’s Java Sea. We also compared the results of wave prediction using BiLSTM with those of other methods, i.e., LSTM, support vector regression (SVR), and a generalized regression neural network (GRNN). The forecasting results using the BiLSTM were the best, with a correlation coefficient of 0.96 and an RMSE value of 0.06. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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13 pages, 1363 KiB  
Article
Numerical Study for Unsteady Waves Generated by Flow over a Permeable Wavy Bed
by Ikha Magdalena and Kemal Firdaus
Fluids 2022, 7(1), 9; https://doi.org/10.3390/fluids7010009 - 27 Dec 2021
Cited by 2 | Viewed by 2427
Abstract
In this paper, we formulate a numerical model to study unsteady waves generated by fluid flow over a permeable wavy bed. The model is derived from boundary value problems using potential theory. We solve the model numerically using a finite difference method. As [...] Read more.
In this paper, we formulate a numerical model to study unsteady waves generated by fluid flow over a permeable wavy bed. The model is derived from boundary value problems using potential theory. We solve the model numerically using a finite difference method. As a result, we found that the flow over a porous layer generates wave disturbed by bumps on the porous layer. The simulation also showed that the wave profile shifts from the permeable bed. The results of this study can be incorporated into the design of submerged artificial and natural breakwaters. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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19 pages, 1005 KiB  
Article
Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas
by Agissilaos G. Athanassoulis and Odin Gramstad
Fluids 2021, 6(8), 291; https://doi.org/10.3390/fluids6080291 - 19 Aug 2021
Cited by 4 | Viewed by 2512
Abstract
The Alber equation is a phase-averaged second-moment model used to study the statistics of a sea state, which has recently been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear [...] Read more.
The Alber equation is a phase-averaged second-moment model used to study the statistics of a sea state, which has recently been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e., two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e., in general different directions and different moduli.) We also derive the associated two-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given nonparametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 nonparametric spectra provided by the Norwegian Meteorological Institute and find that the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel “proximity to instability” (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin–Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with the PTI (>85% Spearman rank correlation). Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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13 pages, 9600 KiB  
Article
Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation
by Chen Yue, Dianchen Lu and Mostafa M. A. Khater
Fluids 2021, 6(7), 235; https://doi.org/10.3390/fluids6070235 - 29 Jun 2021
Cited by 30 | Viewed by 2233
Abstract
This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to [...] Read more.
This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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15 pages, 5094 KiB  
Article
Generation of Gravity Waves by Pedal-Wavemakers
by Isis Vivanco, Bruce Cartwright, A. Ledesma Araujo, Leonardo Gordillo and Juan F. Marin
Fluids 2021, 6(6), 222; https://doi.org/10.3390/fluids6060222 - 13 Jun 2021
Cited by 3 | Viewed by 3233
Abstract
Experimental wave generation in channels is usually achieved through wavemakers (moving paddles) acting on the surface of the water. Although practical for engineering purposes, wavemakers have issues: they perform poorly in the generation of long waves and create evanescent waves in their vicinity. [...] Read more.
Experimental wave generation in channels is usually achieved through wavemakers (moving paddles) acting on the surface of the water. Although practical for engineering purposes, wavemakers have issues: they perform poorly in the generation of long waves and create evanescent waves in their vicinity. In this article, we introduce a framework for wave generation through the action of an underwater multipoint mechanism: the pedal-wavemaking method. Our multipoint action makes each point of the bottom move with a prescribed pedalling-like motion. We analyse the linear response of waves in a uniform channel in terms of the wavelength of the bottom action. The framework naturally solves the problem of the performance for long waves and replaces evanescent waves by a thin boundary layer at the bottom of the channel. We also show that proper synchronisation of the orbital motion on the bottom can produce waves that mimic deep water waves. This last feature has been proved to be useful to study fluid–structure interaction in simulations based on smoothed particle hydrodynamics. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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15 pages, 880 KiB  
Article
Resonant Periods of Seiches in Semi-Closed Basins with Complex Bottom Topography
by Ikha Magdalena, Nadhira Karima and Hany Qoshirotur Rif’atin
Fluids 2021, 6(5), 181; https://doi.org/10.3390/fluids6050181 - 9 May 2021
Cited by 13 | Viewed by 3123
Abstract
Seiches and resonances are two closely related phenomena that can cause damage to coastal areas. Seiches that occur in a basin at a distinct period named the resonant period may generate resonance when a wave induced by external forces enters the basin and [...] Read more.
Seiches and resonances are two closely related phenomena that can cause damage to coastal areas. Seiches that occur in a basin at a distinct period named the resonant period may generate resonance when a wave induced by external forces enters the basin and has the same period as the seiches. Studying this period has become essential if we want to understand the resonance better. Thus, in this paper, we derive the resonant period in various shapes of semi-closed basin using the shallow water equations. The equations are then solved analytically using the separation of variables method and numerically using the finite volume method on staggered grid to discover the resonant period for each basin. To validate the numerical scheme, we compare its results against the analytical resonant periods, resulting in a very small error for each basin, suggesting that the numerical model is quite reliable in the estimation of the analytical resonant period. Further, resonant wave profiles are also observed. It is revealed that, in the coupled rectangular basin, the maximum wave elevation is disproportionate to the ratio of the length of the basin, while, in the trapezoidal basin, the ratio of the depth of the basin has no significant impact on the maximum wave elevation. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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15 pages, 2457 KiB  
Article
A Computational Fluid Dynamics Model for the Small-Scale Dynamics of Wave, Ice Floe and Interstitial Grease Ice Interaction
by Rutger Marquart, Alfred Bogaers, Sebastian Skatulla, Alberto Alberello, Alessandro Toffoli, Carina Schwarz and Marcello Vichi
Fluids 2021, 6(5), 176; https://doi.org/10.3390/fluids6050176 - 29 Apr 2021
Cited by 10 | Viewed by 2735
Abstract
The marginal ice zone is a highly dynamical region where sea ice and ocean waves interact. Large-scale sea ice models only compute domain-averaged responses. As the majority of the marginal ice zone consists of mobile ice floes surrounded by grease ice, finer-scale modelling [...] Read more.
The marginal ice zone is a highly dynamical region where sea ice and ocean waves interact. Large-scale sea ice models only compute domain-averaged responses. As the majority of the marginal ice zone consists of mobile ice floes surrounded by grease ice, finer-scale modelling is needed to resolve variations of its mechanical properties, wave-induced pressure gradients and drag forces acting on the ice floes. A novel computational fluid dynamics approach is presented that considers the heterogeneous sea ice material composition and accounts for the wave-ice interaction dynamics. Results show, after comparing three realistic sea ice layouts with similar concentration and floe diameter, that the discrepancy between the domain-averaged temporal stress and strain rate evolutions increases for decreasing wave period. Furthermore, strain rate and viscosity are mostly affected by the variability of ice floe shape and diameter. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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18 pages, 1233 KiB  
Article
Performance of an U-Shaped Oscillating Water Column Wave Energy Converter Device under Oblique Incident Waves
by Kshma Trivedi, Santanu Koley and Kottala Panduranga
Fluids 2021, 6(4), 137; https://doi.org/10.3390/fluids6040137 - 1 Apr 2021
Cited by 13 | Viewed by 2249
Abstract
The present study deals with the performance of an U-shaped oscillating water column device under the action of oblique incident waves. To solve the associated boundary value problem, the dual boundary element method (DBEM) is used. Various physical parameters associated with the U-shaped [...] Read more.
The present study deals with the performance of an U-shaped oscillating water column device under the action of oblique incident waves. To solve the associated boundary value problem, the dual boundary element method (DBEM) is used. Various physical parameters associated with the U-shaped OWC device, such as the radiation susceptance and conductance coefficients, and the hydrodynamic efficiency, are analyzed for a wide range of wave and structural parameters. The study reveals that the resonance in the efficiency curve occurs for smaller values of wavenumber with an increase in chamber length, submergence depth of the front wall and opening duct, and width of the opening duct. It is observed that with appropriate combinations of the angle of incidence and incident wavenumber, more than 90% efficiency in the U-shaped OWC device can be achieved. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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15 pages, 617 KiB  
Article
Effect of Varying Bottom Topography on the Radiation of Water Waves by a Floating Rectangular Buoy
by Kshma Trivedi and Santanu Koley
Fluids 2021, 6(2), 59; https://doi.org/10.3390/fluids6020059 - 1 Feb 2021
Cited by 5 | Viewed by 1589
Abstract
In the present study, the effect of an undulated bottom topography on the radiation of water waves by a floating rectangular buoy is analyzed. Various physical quantities of interest such as the added mass and damping coefficients associated with the surge, heave, and [...] Read more.
In the present study, the effect of an undulated bottom topography on the radiation of water waves by a floating rectangular buoy is analyzed. Various physical quantities of interest such as the added mass and damping coefficients associated with the surge, heave, and pitch motions are analyzed for a variety of parameters associated with the incident waves and bottom undulations. The study reveals that the added mass and damping coefficients associated with the surge and pitch motions of the floating buoy vary in an oscillatory manner with the variation in wavenumber for a sinusoidally varying bottom topography. Moreover, the oscillation amplitude is higher around the primary Bragg value. Further, this oscillatory pattern and oscillation amplitude increase with an increase in the ripple amplitude and the number of ripples for a sinusoidally varying bottom. However, a reverse pattern is formed with an increase in the depth ratio. In the long-wave regime, the added mass and damping coefficient corresponding to the surge motion become higher for a protrusion-type bed profile and lower for a depression-type bed profile. However, a reverse pattern is observed in the intermediate- and short-wave regimes. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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19 pages, 2390 KiB  
Article
Physics-Capturing Discretizations for Spectral Wind-Wave Models
by Marcel Zijlema
Fluids 2021, 6(2), 52; https://doi.org/10.3390/fluids6020052 - 22 Jan 2021
Cited by 2 | Viewed by 1686
Abstract
This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind [...] Read more.
This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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13 pages, 2473 KiB  
Article
Time-Dependent Motion of a Floating Circular Elastic Plate
by Michael H. Meylan
Fluids 2021, 6(1), 29; https://doi.org/10.3390/fluids6010029 - 8 Jan 2021
Cited by 17 | Viewed by 2933
Abstract
The motion of a circular elastic plate floating on the surface is investigated in the time-domain. The solution is found from the single frequency solutions, and the method to solve for the circular plate is given using the eigenfunction matching method. Simple plane [...] Read more.
The motion of a circular elastic plate floating on the surface is investigated in the time-domain. The solution is found from the single frequency solutions, and the method to solve for the circular plate is given using the eigenfunction matching method. Simple plane incident waves with a Gaussian profile in wavenumber space are considered, and a more complex focused wave group is considered. Results are given for a range of plate and incident wave parameters. Code is provided to show how to simulate the complex motion. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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15 pages, 5287 KiB  
Article
A Momentum-Conserving Scheme for Flow Simulation in 1D Channel with Obstacle and Contraction
by Putu Veri Swastika, Sri Redjeki Pudjaprasetya, Leo Hari Wiryanto and Revi Nurfathhiyah Hadiarti
Fluids 2021, 6(1), 26; https://doi.org/10.3390/fluids6010026 - 6 Jan 2021
Cited by 6 | Viewed by 2710
Abstract
We consider the extension of the momentum conservative staggered-grid (MCS) scheme for flow simulation in channels with varying depth and width. The scheme is formulated using the conservative properties of the Saint-Venant equations. The proposed scheme was successful in handling various steady flows [...] Read more.
We consider the extension of the momentum conservative staggered-grid (MCS) scheme for flow simulation in channels with varying depth and width. The scheme is formulated using the conservative properties of the Saint-Venant equations. The proposed scheme was successful in handling various steady flows and achieved results that are in complete accordance with the analytical steady solutions. Different choices of boundary conditions have created steady solutions according to the mass and energy conservations. This assessment has served as a validation of the proposed numerical scheme. Further, in a channel with a contraction and a nonuniform bed, we simulate two cases of dam break. The simulation results show a good agreement with existing experimental data. Moreover, our scheme, that uses a quasi-1-dimensional approach, has shown some fair agreement with existing 2-dimensional numerical results. This evaluation demonstrates the merits of the MCS scheme for various flow simulations in channels of varying width and bathymetry, suitable for river flow modeling. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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28 pages, 3506 KiB  
Article
Intercomparison of Three Open-Source Numerical Flumes for the Surface Dynamics of Steep Focused Wave Groups
by Thomas Vyzikas, Dimitris Stagonas, Christophe Maisondieu and Deborah Greaves
Fluids 2021, 6(1), 9; https://doi.org/10.3390/fluids6010009 - 30 Dec 2020
Cited by 5 | Viewed by 2921
Abstract
NewWave-type focused wave groups are commonly used to simulate the design wave for a given sea state. These extreme wave events are challenging to reproduce numerically by the various Numerical Wave Tanks (NWTs), due to the high steepness of the wave group and [...] Read more.
NewWave-type focused wave groups are commonly used to simulate the design wave for a given sea state. These extreme wave events are challenging to reproduce numerically by the various Numerical Wave Tanks (NWTs), due to the high steepness of the wave group and the occurring wave-wave interactions. For such complex problems, the validation of NWTs against experimental results is vital for confirming the applicability of the models. Intercomparisons among different solvers are also important for selecting the most appropriate model in terms of balancing between accuracy and computational cost. The present study compares three open-source NWTs in OpenFOAM, SWASH and HOS-NWT, with experimental results for limiting breaking focused wave groups. The comparison is performed by analysing the propagation of steep wave groups and their extracted harmonics after employing an accurate focusing methodology. The scope is to investigate the capabilities of the solvers for simulating extreme NewWave-type groups, which can be used as the “design wave” for ocean and coastal engineering applications. The results demonstrate the very good performance of the numerical models and provide valuable insights to the design of the NWTs, while highlighting potential limitations in the reproduction of specific harmonics of the wave group. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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13 pages, 6983 KiB  
Article
Wave Modeling for the Establishment Potential Area of Offshore Aquaculture in Indonesia
by Muhammad Zikra, Haryo Dwito Armono and Fahrizal Pratama
Fluids 2020, 5(4), 229; https://doi.org/10.3390/fluids5040229 - 1 Dec 2020
Cited by 2 | Viewed by 2266
Abstract
Aquaculture is expected to further improve in the future and can provide 57 percent of fish for human consumption by 2025. In Indonesia, the aquaculture sector produced 5.77% of the world total production in 2014 and increases annually by, on average, 0.62%. Prigi [...] Read more.
Aquaculture is expected to further improve in the future and can provide 57 percent of fish for human consumption by 2025. In Indonesia, the aquaculture sector produced 5.77% of the world total production in 2014 and increases annually by, on average, 0.62%. Prigi Bay, located in the south of east Java, is one potential area to develop sustainable aquaculture in Indonesia. This study presents numerical wave modeling to investigate the potential area for offshore aquaculture in Prigi Bay. The method used Delft3D Flow and CG WAVE model to simulate wave and current. The superimposed analysis is used to select potential areas between the results of the model and the criteria of environmental parameters. The result shows that the location which meets the aquaculture criteria is located at coordinates 8.311° S–8.322° S and 111.734° E–111.747° E. This site has a depth of around 18–26 m with current velocity between 0.10 and 0.14 m/s and significant a wave height between 0.2–0.4 m. This location is the most suitable location for aquaculture in the Prigi Bay. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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25 pages, 5692 KiB  
Article
A Near-Shore Linear Wave Model with the Mixed Finite Volume and Finite Difference Unstructured Mesh Method
by Yong G. Lai and Han Sang Kim
Fluids 2020, 5(4), 199; https://doi.org/10.3390/fluids5040199 - 5 Nov 2020
Cited by 4 | Viewed by 2163
Abstract
The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models [...] Read more.
The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models have been developed to predict the wave and current state and their interactions. Most models, however, have relied on the two-model approach in which the wave model is developed independently of the current model and the two are coupled together through a separate steering module. In this study, a new wave model is developed and embedded in an existing two-dimensional (2D) depth-integrated current model, SRH-2D. The work leads to a new wave–current model based on the one-model approach. The physical processes of the new wave model are based on the latest third-generation formulation in which the spectral wave action balance equation is solved so that the spectrum shape is not pre-imposed and the non-linear effects are not parameterized. New contributions of the present study lie primarily in the numerical method adopted, which include: (a) a new operator-splitting method that allows an implicit solution of the wave action equation in the geographical space; (b) mixed finite volume and finite difference method; (c) unstructured polygonal mesh in the geographical space; and (d) a single mesh for both the wave and current models that paves the way for the use of the one-model approach. An advantage of the present model is that the propagation of waves from deep water to shallow water in near-shore and the interaction between waves and river inflows may be carried out seamlessly. Tedious interpolations and the so-called multi-model steering operation adopted by many existing models are avoided. As a result, the underlying interpolation errors and information loss due to matching between two meshes are avoided, leading to an increased computational efficiency and accuracy. The new wave model is developed and verified using a number of cases. The verified near-shore wave processes include wave shoaling, refraction, wave breaking and diffraction. The predicted model results compare well with the analytical solution or measured data for all cases. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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13 pages, 711 KiB  
Article
The Fast Discrete Interaction Approximation Concept
by Vladislav Polnikov
Fluids 2020, 5(4), 176; https://doi.org/10.3390/fluids5040176 - 10 Oct 2020
Cited by 2 | Viewed by 1618
Abstract
Hasselmann and coauthors proposed the discrete interaction approximation (DIA) as the best tool replacing the nonlinear evolution term in a numerical wind–wave model. Much later, Polnikov and Farina radically improved the original DIA by means of location all the interacting four wave vectors, [...] Read more.
Hasselmann and coauthors proposed the discrete interaction approximation (DIA) as the best tool replacing the nonlinear evolution term in a numerical wind–wave model. Much later, Polnikov and Farina radically improved the original DIA by means of location all the interacting four wave vectors, used in the DIA configuration, exactly at the nodes of the numerical frequency–angular grid. This provides a nearly two-times enhancement of the speed of numerical calculation for the nonlinear evolution term in a wind–wave model. For this reason, the proposed version of the DIA was called as the fast DIA (FDIA). In this paper, we demonstrate all details of the FDIA concept for several frequency–angular numerical grids of high-resolution with the aim of active implementation of the FDIA in modern versions of world-wide used wind–wave models. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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18 pages, 3349 KiB  
Article
Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows
by Novry Erwina, Didit Adytia, Sri Redjeki Pudjaprasetya and Toni Nuryaman
Fluids 2020, 5(3), 149; https://doi.org/10.3390/fluids5030149 - 31 Aug 2020
Cited by 6 | Viewed by 3298
Abstract
Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating [...] Read more.
Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al., 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow. Full article
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)
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