Nonlinear Boussinesq Equation Research Articles

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Fuzzy Finite Elements Solution Describing Recession Flow in Unconfined Aquifers

In this work, a novel fuzzy FEM (Finite Elements Method) numerical solution describing the recession flow in unconfined aquifers is proposed. In general, recession flow and drainage problems can be described by the nonlinear Boussinesq equation, while the introduced hydraulic parameters (Conductivity K and Porosity S) present significant uncertainties for various reasons (e.g., spatial distribution, human errors, etc.). Considering the general lack of in situ measurements for these parameters as well as the certain spatial variability that they present in field scales, a fuzzy approach was adopted to include the problem uncertainties and cover the disadvantage of ground truth missing data. The overall problem is encountered with a new approximate fuzzy FEM numerical solution, leading to a system of crisp boundary value problems. To prove the validity and efficiency of the new fuzzy FEM, a comparative analysis between the proposed approach and other well-known and tested approximations was carried out. According to the results, the proposed FEM numerical solution agrees with Karadinumerical method for the crisp case and is in close agreement with the original analytical solution proposed by Boussinesq in 1904 with the absolute reduced error to be 4.6‰. Additionally, the possibility theory is applied, enabling the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of hydraulic properties (e.g., water level, outflow volume) and make the right decisions for rational and productive engineering studies.

Estimate of Wave Overtopping Rate on Armoured Slope Structures Using FUNWAVE-TVD Model

In this study, the program was modified by adding the empirical formula of EurOtop (2018) to enable calculation of wave overtopping on armoured slope structures in the FUNWAVE-TVD model using the fully nonlinear Boussinesq equation. The validity of the modified numerical model was verified by comparing it with CLASH data and experiment data for the rubble mound structure. This model accurately reproduced the change in wave overtopping rate according to the difference in the roughness factor of the armoured block, and well reproduced the rate of decrease in wave overtopping rate due to the increase in relative freeboard. The overtopping rate of the armoured slope structures showed significant differences depending on the positioning condition of the armoured blocks. When Tetrapods were placed with regular positioning, the overtopping rate increased significantly compared to when they were placed with random positioning, and it was consistent with when they were placed with Rocks. Meanwhile, when rocks were placed in one row, the wave overtopping rate was greater than when rocks were placed in two rows, which is believed to be due to the influence of the roughness and permeability of the structure’s surface.

A set of fully nonlinear mild slope equations

A set of time-dependent fully nonlinear mild slope equations is established in terms of velocity vector at still water level and free surface elevation, which is derived from the Euler equations without approximation in nonlinearity and only with the potential flow assumption adopted for bottom slope terms. Apart from the fully nonlinear property, the model is also different from previous nonlinear mild slope equations by adopting the velocity expression accurate to the first-order in bottom slope, which allows the new expressions of higher-order bottom slope terms to be established and the fully nonlinear terms related to bottom slope to be included in the equations. The model can be reduced to the classical mild slope equation for linear waves and the fully nonlinear Boussinesq equations for nonlinear waves. The advantages of the present model are validated against the related laboratory experimental results and demonstrated by comparison with the numerical results of other relevant models.

Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative

Abstract A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time ( t ) \left(t) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.

Modeling of water table profile variations owing to stream-aquifer interaction.

Spatial and temporal variations of the water table could be explained by the one-dimensional Boussinesq equation by incorporating the variables of evapotranspiration and groundwater recharge with appropriate initial and boundary conditions. In this study, the stream-aquifer interaction has been investigated through a numerical example model with the implementations of Galerkin's method-based Finite Element Solution (FES), Hybrid Finite Analytic Solution (HFAS), Fully Implicit Finite Difference Solution (FIFDS) of one-dimensional nonlinear Boussinesq equation, and analytical solutions of the Boussinesq equation linearized by Baumann's transformation (AS I) as well as linearized by Werner's transformation (AS II). Considering HFAS as the benchmark solution, it was observed that in both recharging and discharging aquifers, water table profiles at 1 day and 5 days as obtained from FES followed by FIFDS were observed quite close to HFAS. Based on L2 and Tchebycheff norms, FES and FIFDS were ranked in first and second place, respectively. L2 and Tchebycheff norms could not consistently establish the performance ranking of analytical solutions but their performance ranking was certainly below the numerical solutions. The performance ranking of analytical solutions could not consistently be established using the L2 and Tchebycheff norms, but it was certainly below the numerical solutions.

TSUNAMI MODELLING IN A BUILT-IN COASTAL ENVIRONMENT WITH ADAPTIVE MESH RIFINEMENT AND VARYING BOTTOM FRICTION

The great Japan earthquake on March/11/2011provoked an extreme tsunami wave toward the northeastern Japanese coast. The tsunami inundation covered a wide range of territories stretching over 500 km2 and caused devastating influences that claimed more than 15,000 lives (Mori et al. (2011)). Miyagi Prefecture, the closest mainland to the epicentre, received massive tsunami waves that reached over 40 m in run-up height. Prasetyo et al. (2019) aimed to reproduce the tsunami wave inundation in Onagawa Town of Miyagi Prefecture by building a physical model using Hybrid Tsunami Open Flume in Ujigawa Open Laboratory (HyTOFU) at Kyoto University to study the inundation process by measuring the wave height and arrival time. For that purpose, two wave types were used, the (Pump-type) hydraulic bore, and the (Piston-type) solitary wave. Also, 2D and quasi-3D (Q3D) models were tested for their robustness and applied to emulate tsunami propagation in the complex coastal city model. In this study, we adopted adaptive mesh refinement (AMR) for its efficiency in tsunami modelling with Basilisk open-source flow solver for non-linear shallow water (Saint-Venant) equations and fully non-linear Boussinesq (Green-Naghdi) equations (Popinet (2015)). The numerical experiment was conducted to validate the model by comparing the wave height with the physical model experiment, the 2D, and Q3D model results by Prasetyo et al. (2019).

A Nonlinear Recession Model for Horizontal Aquifers

AbstractRecession analysis is a powerful tool for determining the hydraulic characteristics of a riparian aquifer. The basis of the method relies on two analytical solutions of the nonlinear Boussinesq equation: one applies to a uniform water table profile in a semi‐infinite aquifer while the other pertains to a hypothetical initial profile in a finite aquifer. Both solutions assume that the water depth in the adjoining stream is negligible. In the present work, a nonlinear solution of the one‐dimensional Boussinesq equation is derived using the traveling wave approximation. The solution better represents the field conditions as it pertains to a uniform water table profile in a finite aquifer that is adjoining a stream with a non‐zero water level. It allows the derivation of simple algebraic expressions for the recession of the water table, the flow rate, and the associated drainage volume. Approximations relating the discharge rate to the outflow volume were also obtained for practical implementation in the estimation of the hydraulic parameters of the aquifer. A comparison of the proposed methodology of parameter estimation with the standard method of recession analysis showed that the present approach is superior as it avoids the pitfalls associated with the classical method, especially when daily discharge values with inherent observational errors are used. Application of the recession model to real‐world cases demonstrates its effectiveness in the estimation of the catchment‐scale hydraulic conductivity and drainable porosity.

Propagation characteristics of tsunami waves in shear flow based on the Boussinesq model with constant vorticity

Propagation of tsunami wave in linear shear flow with constant vorticity is studied in this paper. The nonlinear and dispersive Boussinesq equations for water waves in shear flow are derived by introducing uniform vorticity into the Euler equations. Generally, the transoceanic propagation of tsunami can be well simulated by the weakly nonlinear Boussinesq model. Thus, the nonlinear and dispersive terms are retained to the first-order and second-order (ϵ,μ2) respectively in the Boussinesq equations with uniform vorticity in the present work, which satisfies the simulation accuracy especially the dispersion effects of the long-distance propagation of tsunamis and does not require huge computational cost for the simulation. The set of equations is solved with the total variation diminishing (TVD) Lax–Wendroff scheme which is second-order both in space and time. This TVD scheme is applicable to handle the dispersion effects of the N-waves and undular bores. Different patterns of tsunami wave in the water with presence of vorticity are simulated by the numerical model. Wave height and wavelength are affected by the current with vorticity. The pattern of solitary wave is significantly changed in upstream and downstream conditions. The wave deformation and propagation speed of N-waves and undular bores in the shear flow are also studied. The vorticity has impacts on the leading crest amplitude and the maximum wave appearance time for N-wave and changes the wave amplitude and the following water level for undular bores.

Searching For (2+1)-dimensional nonlinear Boussinesq equation from (1+1)-dimensional nonlinear Boussinesq equation

A novel (2+1)-dimensional nonlinear Boussinesq equation is derived from a (1+1)-dimensional Boussinesq equation in nonlinear Schrödinger type based on a deformation algorithm. The integrability of the obtained (2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the (1+1)-dimensional Boussinesq equation. Because of the effects of the deformation, the (2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multi-valued.

Modeling wave processes in a reef-lagoon-channel system based on a Boussinesq model

The reef-lagoon-channel system is very typical on the low-lying reef-lined coasts in tropical and subtropical areas. Comparing to the well-studied one-dimensional horizontal (1DH) fringing reefs, little is known about the alongshore wave runup variation and its implication for coastal flooding in such a two-dimensional horizontal (2DH) reef configuration. To better understand the wave processes in an idealized reef-lagoon-channel system, a numerical model is presented in this study, which is based on a set of fully nonlinear Boussinesq equations. Wave breaking is addressed by a shock-capturing scheme and bottom friction is formulated by the Manning frictional law. The adopted model is first validated by a published laboratory dataset in view of wave spectrum, wave height, wave setup, mean current as well as wave runup on the back-reef beach. Subsequently, the effects of hydrodynamic (incident wave height, incident wave period, reef-flat wave level) and reef morphological (fore-reef slope, cross-shore reef-flat width, channel width and frictional coefficient) parameters that are not fully considered by the laboratory experiments are investigated through the numerical simulations. Finally, the model is used to analyze the infragravity (IG) wave resonant modes in the system, and the effects of channel on such modes are also demonstrated.

Fractional Analysis of Nonlinear Boussinesq Equation under Atangana–Baleanu–Caputo Operator

This article proposed two novel techniques for solving the fractional-order Boussinesq equation. Several new approximate analytical solutions of the second- and fourth-order time-fractional Boussinesq equation are derived using the Laplace transform and the Atangana–Baleanu fractional derivative operator. We give some graphical and tabular representations of the exact and proposed method results, which strongly agree with each other, to demonstrate the trustworthiness of the suggested methods. In addition, the solutions we obtain by applying the proposed approaches at different fractional orders are compared, confirming that as the value trends from the fractional order to the integer order, the result gets closer to the exact solution. The current technique is interesting, and the basic methodology suggests that it might be used to solve various fractional-order nonlinear partial differential equations.

Fuzzy solution of nonlinear Boussinesq equation

Abstract In this paper, the solution of the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation is examined. The physical problem concerns unsteady flow in a semi-infinite unconfined aquifer bordering a lake. In the examined problem, there is a sudden rise and subsequent stabilization of the lake's water level, thus the aquifer is recharging from the lake. The aquifer boundary conditions are considered fuzzy and, therefore, ambiguities are created to the solution of the overall physical problem. Then, the fuzzy problem is translated to a system of crisp boundary value problems. By using a Boltzmann transformation, the crisp problem is transformed into an integro-differential equation and solved with the help of a special numerical method. This method has a simple iterative procedure, which converges rapidly and is proven very accurate in comparison with other analytical methods. Additionally, the algebraic equation estimates very close to the storage coefficient, the flux at the stream-aquifer origin, and the water stored volume with respect to other exact solutions. This method is very useful for practical cases (artificial recharge problems, irrigation, and drainage projects), giving the possibility to decision-makers to take the right decision.

Bright Soliton Behaviours of Fractal Fractional Nonlinear Good Boussinesq Equation with Nonsingular Kernels

In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo–Fabrizio and Atangana–Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study the convergence, existence, uniqueness, boundedness, and stability of the solution of the considered good BEQ under the aforementioned derivatives. The obtained solutions are presented with numerical illustrations considering a particular example by two cases based on both derivatives with suitable initial conditions. The results are illustrated graphically where good agreements are obtained. Our results show that fractal-fractional derivatives are a very effective tool for studying nonlinear systems. Furthermore, when t increases, the solitary waves of the system oscillate. As the fractional order α or fractal dimension β increases, the soliton solutions become coherently close to the exact solution. For compactness, an error analysis is performed. The absolute error reveals an approximate linear evolution in the soliton solutions as time increases and that the system does not blow up nonlinearly.

Numerical investigation of the effects of rigid emergent vegetation on wave runup and overtopping

The effects of rigid emergent vegetation on wave overtopping have been studied by numerical simulation in this study. The numerical model is based on second-order fully nonlinear Boussinesq equations and solved by a combination of finite difference and finite volume methods. Physical laboratory results of wave propagation in the vegetated area and wave overtopping on a bare slope were used to validate the capability of the model, and the numerical and experimental results agreed well, which indicated that the model can well predict wave overtopping on a vegetated slope. Then, based on the model, the effects of rigid emergent vegetation on wave overtopping were investigated. The results showed that the wave overtopping volume decreases exponentially with the increase of vegetation volume per unit water volume Vveg/Vw, and the wave overtopping volume can be decreased by 50% when Vveg/Vw equals 0.01.

Topographic influence on special modes with narrow amplification diagrams in harbor oscillations

In studies of harbor oscillations, modes with extremely-narrow resonant peaks and long response times (i.e., extreme modes) are common. With the advent of extreme modes, a nodal line appears at the entrance, which results in a very weak flow at the entrance and significantly reduces the exchange efficiency of wave energy between the harbor and the sea. However, previous studies are limited to harbors with a constant water depth, and the topographic influence on extreme modes is unclear.In this work, an extended mild-slope equation and a fully nonlinear Boussinesq equation are adopted to study the influence of the topography inside the harbor on extreme modes. The frequency-domain results indicate that the topographies of the symmetrical axes not passing through the entrance have a great influence on the angle between the nodal line and entrance, thereby leading to considerable changes in the resonant peak widths. However, the other topographies have little influence on this angle, resulting in negligible changes in the resonant peak widths. The time-domain results show that the wave nonlinearity can reduce the time required for the oscillation to reach a steady state, and various topographies result in different reduced times.

Analytical and numerical solutions to describe water table fluctuations due to canal seepage and time-varying recharge

Abstract Hybrid finite analytic solution (HFAS), Galerkin's method based finite element solution (FES) and fully implicit finite difference solution (FIFDS) of one dimensional nonlinear Boussinesq equation and Analytical solution of Boussinesq equation linearized by Baumann's transformation (analytical solution I, AS I) as well as linearized by Werner's transformation (analytical solution II, AS II) were employed to obtain water table rise in a horizontal unconfined aquifer lying between two canals located at finite distance having different elevations and subjected to various patterns of recharge, i.e. zero recharge, constant recharge, as well as time varying recharge. Considering HFAS as benchmark solution, water table in mid region as obtained from FES followed by FIFDS was observed quite close to that obtained from HFAS and as per L2 and Tchebycheff norms computation, it was ranked at first and second place, respectively. Both AS I and AS II predicted higher water table at t = 5 days but at t = 10 days, AS I predicted lower and AS II predicted higher water table at all distances due to linearization effect. So, analytical solutions of linearized Boussinesq equation were rated lower than numerical solutions of nonlinear Boussinesq equation.

Comparative study of bistatic scattering pattern of nearshore water waves impacted by underwater topography at UHF-band

The electromagnetic (EM) scattering from nearshore water waves impacted by gradually-varied underwater topography at UHF-band is numerically investigated using the second-order small slope approximation (SSA-II). Based on the fully nonlinear Boussinesq equation solver, the sea wave-like sinusoidal-type waves are generated and simulated in the shoaling effect-impacted nearshore region using the free surface wave modeling package COULWAVE. It is found that the wave profile would be obviously deformed due to the shoaling effect when the wave propagates inshore. The rapid decrease of water depth would intensify this wave deformation. Numerical results of the normalized radar cross section (NRCS) are presented in bistatic configuration at UHF-band. The comparison of the NRCS of waves in the region of different underwater topography shows that the presence and position of the high-order harmonic peaks would be significantly changed in different topography cases. Moreover, the wave height, as well as the wavelength of water waves, would lead to different scattering patterns. Numerical results provide a qualitative perspective to investigate the impact of the nonlinear effect between water waves on EM scattering of sea waves in the nearshore region.