Nonlinear Boussinesq Equation Research Articles

Comparison Between Boussinesq and Shallow-Water Models in Predicting Solitary Wave Runup on Plane Beaches

The nonlinear shallow water equations and Boussinesq equations have been widely used for analyzing the solitary wave runup phenomenon. In order to quantitatively assess the merits and limitations of these two approaches, a shock-capturing scheme has been applied to numerically solve both sets of equations for predicting the runup processes over plane beaches. The analytical and experimental data available in the literature have been used as references in the assessment. In this study, the uniform sloping beach is preceded by a length of flat seafloor. When incident solitary waves are specified close to the slope, the two approaches are found to produce almost identical results for nonbreaking waves. For breaking waves, the Boussinesq equations give a better representation of the wave evolution prior to the breaking point, whereas they overestimate the short undulations accompanying the breaking process. If incoming solitary waves need to travel a long distance over a flat bed before reaching the slope, the shallow water equations cannot capture the correct waveform transformation, and the predicted runup depends heavily on the length of the flat-bed section. As this length approaches zero, however, the shallow water equations somehow give roughly the same maximum runup heights as those predicted by the Boussinesq model. The bed friction has little effect on the runup for small waves, but becomes important for large waves. The roughness coefficient needs to be calibrated to reproduce the measured runup heights of breaking waves.

An integral contravariant formulation of the fully non-linear Boussinesq equations

In this paper we propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. Following the approach proposed by Chen (2006), the motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock-capturing finite volume scheme in which an exact Riemann solver is involved; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is verified against several benchmark tests, and the results are compared with experimental, theoretical and alternative numerical solutions.

An exact series and improved numerical and approximate solutions for the Boussinesq equation

We derive a series solution for the nonlinear Boussinesq equation in terms of the similarity variable of the Boltzmann transformation in a semi‐infinite domain. The first few coefficients of the series have been known for a long time, having been obtained by a truncated inversion of the series solution of the Blasius equation, but no direct recurrence relation was known for the complete series representing the solution of the Boussinesq equation. The series turns out to have a finite radius of convergence, which we estimate with a numerical complex‐plane integration method that identifies the singularities of the solution when the equation is extended to the complex plane. The homogeneous condition at the origin produces a singularity which complicates numerical solutions with Runge‐Kutta methods. We present two variable transformations that circumvent the problem and that are best suited to the complex‐variable and the real‐variable versions of the equation, respectively. Using those tools, an approximate solution accurate to 1.75 × 10−10 and valid for the entire positive real axis is then developed by matching a Padé approximant of the exact series and an asymptotic solution (to overcome the restriction imposed by the finite radius of convergence of the series), along the same lines of the expression proposed by Hogarth and Parlange (1999). The accuracies of all of the existing and the newly proposed solutions are obtained.

Late‐time drainage from a sloping Boussinesq aquifer

Numerical solutions to the nonlinear Boussinesq equation, applied to a steeply sloping aquifer and assuming uniform hydraulic conductivity, indicate that late‐time recession discharge decreases nearly linearly in time. When recession discharge is characterized by −dQ/dt = aQb, this is equivalent to constant dQ/dt or b = 0. This result suggests that a previously reported exponential decrease with time (b = 1) of modeled recession discharge from a similar sloping aquifer represented by the same equation appears to be an artifact of the numerical solution scheme and its interpretation. Because the linearly decreasing recession discharge (b = 0) is not known from field studies, these findings challenge the application of a nonlinear Boussinesq framework assuming uniform conductivity and geometric similarity to infer hydraulic properties of sloping aquifers from observations of streamflow. This finding also questions the validity of the physical interpretation of the exponential decline in late time resulting from the commonly used linearized form of the Boussinesq equation, opposed to the full nonlinear equation, when applied under these conditions. For this reason, application of the linearized equation to infer hydraulic properties of sloping aquifers is also challenged, even if the observed recession is consistent with that of the linearized Boussinesq equation.

NUMERICAL SOLUTION OF BOUSSINESQ EQUATION ARISING IN ONE-DIMENSIONAL INFILTRATION PHENOMENON BY USING FINITE DIFFERENCE METHOD

Infiltration is a gradual flow or movement of groun dwater into and through the pores of an unsaturated porous medium (soil). The fluid infiltered in porous medium (unsaturated soil ), its velocity decreases as soil becomes saturated , and such phenomena is called infiltration. The present model deals with the filt ration of an incompressible fluid (typically, water ) through a porous stratum, the main problem in groundwater infiltration. The present model was developed first by Boussinesq in 1903 and is related to original motivation of Darcy. The mathematical formulation oinfiltration phenomenon leads to a non-linear Boussinesq equation. In the present paper, a numerical solution of Boussinesq e quation has been obtained by using finite differenc e method. The numerical results for a specific set of initial and boundary conditio ns are obtained for determining the height of the fsurface or water mound. The moment infiltrated water enters in unsaturated soil ; the infiltered water will start developing a curv e between saturated porous medium and unsaturated porous medium, which is call ed water table or water mound. Crank-Nicolson finite difference scheme has been applied to obtain the required results for var ious values of time. The obtained numerical results resemble well with the physical phenomena. When water is infiltered through the ve rtical permeable wall in unsaturated porous medium the height of the free surface steadily and uniformly decreases due to the saturat ion of infiltered water as time increases. Forward finite difference scheme is conditionally stable. In the present paper, the gra phical representation shows that Crank-Nicolson fin ite difference scheme is unconditionally stable. Numerical solution of the g overning equation and graphical presentation has be en obtained by using MATLAB coding.

Quintic B-spline for the numerical solution of the good Boussinesq equation

Abstract A numerical method is developed to solve the nonlinear Boussinesq equation using the quintic B-spline collocation method. Applying the Von Neumann stability analysis, the proposed method is shown to be unconditionally stable. An example has been considered to illustrate the efficiency of the method developed.

Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation

The present paper is devoted to the study of the global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. In the absence of horizontal dissipation, we establish a growth estimate on vertical component of velocity, that is, supp⩾2‖u2(t)‖Lpplogp which is close to ‖u2(t)‖L∞ and is bounded via the low-high decomposition technique. This together with the smoothing effect in vertical direction enables us to obtain the H1-estimate for velocity. Based on this, we prove the existence and uniqueness of classical solution without smallness assumptions. In addition, we also discuss the global well-posedness result for the rough initial data.

Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation

We consider the initial value problem associated to the nonlinear periodic sixth-order Boussinesq equation. For given initial data in Sobolev spaces Hs(T), the local well-posedness of this equation is established for s>−1/2.

Groundwater Flow in Sloping Aquifer under Localized Transient Recharge: Analytical Study

AbstractThis paper analyzes the groundwater flow system in an unconfined downward-sloping aquifer of semi-infinite extent in response to localized transient recharge. The aquifer is in contact with a water body of constant water level at one end and receives localized transient recharge from a recharge basin of finite width. The mathematical model is based on the Boussinesq equation with Dupuit-Forchheimer assumption, in which the spatial coordinate of the recharge basin is treated as an additional parameter. Analytical solutions of the linearized Boussinesq equation are obtained using the Laplace transform technique by dividing the aquifer in a three-zone system containing both Dirichlet and Neumann boundary conditions at the hypothetical interfaces. Upward- and zero-sloping cases are deduced from the main results by appropriately adjusting the slope parameter. To assess the validity and efficiency of the linearization method, the nonlinear Boussinesq equation is also solved using a fully explicit predic...

Internal generation of waves on an arced band

We develop a technique of generating waves internally on an arced band in a rectangular grid system using a Gaussian-shaped source function. Tests are made for four different types of wave generation layouts. Numerical experiments are conducted under the following conditions: the propagation of linear and cnoidal waves on a flat bottom, and the refraction and shoaling of cnoidal waves on a planar slope. Numerical experiments are conducted using the linear extended mild-slope equations of Suh et al. (1997) [Coastal Eng. 32, 91–117], and nonlinear extended Boussinesq equations of Nwogu (1993) [J. Waterw. Port Coastal Ocean Eng. 119, 618–638]. The third layout type consisting of two orthogonal bands connected to a quadrant band show the best solutions especially for a fine grid size. Use of the present method on a source band gives smaller errors than Lee and Yoon's (2007) [Coastal Eng. 54, 357–368], method on a source line for a coarse grid size but the two methods yield similar results for a fine grid size. Generation of cnoidal waves gives larger errors than linear waves. To reduce these errors, it is necessary to use a fine grid size and nonlinear wave numbers for incident waves.

Exact solutions of (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations

The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space—time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space—time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.

Wavelet-Petrov-Galerkin Method for Numerical Solution of Boussinesq Equation

In this paper, we employ the Wavelet-Petrov-Galerkin method to obtain the numerical solutions of the nonlinear Boussinesq equation. Boussinesq equation has braod application areas at different branches of engineering and science including chemistry and physics. We first discretize the Boussinesq equation in terms of wavelet coefficients and scaling functions, secondly multiply the discrete equation with wavelet basis functions. Using connection coefficients we express the resulting equation as a matrix equation. One of the significant advantages of the present method is that it does not require a quadrature formula.

Analytical solutions to the stream-aquifer interaction problem: a critical review

The aim of the present paper is to give a systematic and critical presentation of important existing analytical solutions for transient stream-aquifer interaction, which can be used to give answers to simple interaction problems or to verify mathematical models. Stream-aquifer interaction is the most common subject of papers discussing surface water-ground water interaction and a review of analytical solutions to the problem is lacking from the literature. The analytical solutions presented in the paper are firstly distinguished based on whether only the ground water flow equations or both the ground water and stream flow equations are solved for their derivation and secondly based on the type of aquifer (confined or unconfined) interacting with the stream and on the type of equations solved. The literature review showed that there is only a small number of publications, where the authors consider both the ground water and the stream flow equations for the development of the analytical solutions. The majority of the available analytical solutions of stream-aquifer interaction are derived by solving only the ground water flow equations, taking into account the stream water level as a boundary condition. For each analytical solution presented in the paper, its accuracy, its ease of application to simple interaction problems and its suitability for the verification of mathematical models are discussed in detail. Specifically for the case of predicting the water table level in unconfined aquifers interacting with streams, an analytical solution of the non-linear Boussinesq equation is compared to two analytical solutions of different linearized forms of the Boussinesq equation, in order to quantify the error in estimating the water table level when using a linear solution. Among the very few analytical solutions found in the literature, where the authors consider both the stream flow and ground water flow equations for their development, the most comprehensive one is chosen to give an application example, which can be used as a benchmark case for the verification of integrated stream-aquifer mathematical models.

Analytical research on the tidally-induced groundwater wave propagation mechanism in coastal aquifers

ABSTRACT Kong, J., Shen, C.J., Ye, R.H., 2013. Analytical research on the tidally-induced groundwater wave propagation mechanism in coastal aquifers. With increasingly more attention paid to coastal environment, research on the interaction between subsurface and surface water has become a hot subject. Being different from surface water, tidally-induced groundwater moves slower with special over-height far away from the coastal line, which is higher than the mean sea level. Such a special hydrodynamic characteristic hints the special propagation mechanism is different from traditional surface wave. In order to reveal the tidally-induced groundwater wave propagation mechanism and analyze the influence of latent coastal beach profile variation because of mean sea level rise, theoretical research is focused on a newly derived high-order analytical solution corresponding to non-linear Boussinesq equation applied on sloping beach. Research indicates that: Firstly, the tidal induced groundwater wave energy mainl...

Boussinesq modelling of a rip current at Haeundae Beach in South Korea

Bae, J.S., Yoon, S.B., Choi, J., 2013. Boussinesq Modelling of a Rip Current at Haeundae Beach.The rip current occurring at Haeundae beach in Busan, South Korea was numerically investigated under directional random wave environment. The numerical simulations were performed using a fully nonlinear Boussinesq equation model, FUNWAVE which is capable of resolving phase interaction and simulating nearshore circulation since it includes the effect of wave-induced momentum flux and horizontal turbulent mixing. To examine the modelling, the rip current occurred on the 12th of June, 2011 at Haeundae beach was simulated by using observations. The results of numerical simulation showed the time-dependent evolution of the wave-induced nearshore circulation that was caused by transformation of random waves due to the unique topography of Haeundae beach. And, it was found that the rip current was well generated and developed through the region where relatively lower wave heights were caused by refraction and diffraction of waves due to the topographical characteristics (i.e., submerged shoal). In addition, we found that a narrow-banded spectral wave (i.e., a swell spectrum) close to a regular wave generates stronger rip current than a broad-banded spectral wave under an equivalent wave height and period condition.

Exponential Attractor for Lattice System of Nonlinear Boussinesq Equation

We study the lattice dynamical system of a nonlinear Boussinesq equation. We first verify the Lipschitz continuity of the continuous semigroup associated with the system. Then, we provide an estimation of the tail of the difference between two solutions of the system. Finally, we obtain the existence of an exponential attractor of the system.

Nonlinear run-ups of regular waves on sloping structures

Abstract. For coastal risk mapping, it is extremely important to accurately predict wave run-ups since they influence overtopping calculations; however, nonlinear run-ups of regular waves on sloping structures are still not accurately modeled. We report the development of a high-order numerical model for regular waves based on the second-order nonlinear Boussinesq equations (BEs) derived by Wei et al. (1995). We calculated 160 cases of wave run-ups of nonlinear regular waves over various slope structures. Laboratory experiments were conducted in a wave flume for regular waves propagating over three plane slopes: tan α =1/5, 1/4, and 1/3. The numerical results, laboratory observations, as well as previous datasets were in good agreement. We have also proposed an empirical formula of the relative run-up in terms of two parameters: the Iribarren number ξ and sloping structures tan α. The prediction capability of the proposed formula was tested using previous data covering the range ξ ≤ 3 and 1/5 ≤ tan α ≤ 1/2 and found to be acceptable. Our study serves as a stepping stone to investigate run-up predictions for irregular waves and more complex geometries of coastal structures.

수치모의를 통한 해운대 이안류의 주요 메커니즘 연구: 파랑의 벌집구조

파향이 약간 다르게 진행하는 두 규칙파는 비선형적 상호작용에 기인하여 파봉선이 벌집구조와 유사한 모습을 갖는 현상이 발생하고, 벌집구조 모양의 파봉선 사이에 일정하게 파랑에너지가 매우 낮게 유지되는 노드선 영역이 생성된다. 이러한 파봉선 벌집구조 모양이 해변에 형성되면, 해안선 직각방향으로 생성되는 노드선 영역을 통하여 이안류가 발달하게 된다. 본 연구에서는 해변가에 형성되는 벌집구조 파봉선 현상이 해운대에서 발생하는 이안류의 주요한 메카니즘임을 확인하기 위하여, Boussinesq 파랑모형을 이용하여 일방향 규칙파에 의한 해운대 연안흐름의 수치해석을 수행하고 이를 분석하였다. 그 결과, 해운대 앞바다의 해저천퇴에 따른 파랑굴절로 파향이 서로 약간 다른 파랑들이 해안에 전파되고, 이에 따라 벌집구조가 형성되어 노드영역을 따라 이안류가 매우 잘 발달함을 확인할 수 있었다. 또한, 일정한 파고와 주기를 갖는 다양한 폭의 스펙트럼에 따른 불규칙파 수치모의를 수행하므로, 폭이 넓은 스펙트럼 파랑조건보다 규칙파에 가까운 폭이 좁은 스펙트럼의 파랑조건에서 이안류가 더 잘 발달하는 것을 확인하였다. Two regular progressive wave trains, the directions of which are slightly different from each other, develop a honeycomb pattern of wave crests due to their nonlinear interaction. In the honeycomb pattern of wave crest, the nodal line area, which has very low wave energy, is formed. When the honeycomb pattern is developed near the beach area, rip current evolves through the nodal line area formed in the cross shore direction. In this study, to confirm that the formation of honeycomb pattern of waves near the beach area is a dominant mechanism of rip current occurred at Haeundae beach, we performed a numerical simulation of nearshore circulation at Haeundae beach under an unidirectional and monochromatic wave condition by using a nonlinear Boussinesq equation model. As a result, wave refraction due to topographical characteristics (i.e., submerged shoal) of Haeundae gave rise to several wave trains propagating with slightly different directions toward the beach, and consequently rip currents well developed through the nodal line area of honeycomb patterns of wave crest. In addition, we found that a narrow-banded spectral wave condition (i.e., a swell spectrum) increases more likelihood of rip current than a broad-banded spectral wave condtion based on the simulations employing various wave spectra with an equivalent wave height and period.

관측자료 분석과 수치모의에 의한 해운대 이안류 발생 특성 연구

To investigate the characteristics of rip current occurring at Haeundae beach, observations obtained from a buoy and a CCTV were analyzed and numerical experiments were conducted. During observed rip-current events, the CCTV images showed that a couple of wave-trains, which are close to regular waves with slightly different directions, propagated to the beach, and wavelet analyses of data from the buoy showed very narrow-banded spectra with a peak frequency. From the evidences, it was inferred that a known mechanism of generating rip current due to the nodal line area of honeycomb-patterned wave crest was one of the significant factors of rip current occurrences of Haeundae beach. The mechanism has been explained by the following: When two wave-trains with slightly different directions propagate to a beach, wave crests of the incident wave-trains form honeycomb pattern due to nonlinear interaction. The nodal lines of honeycomb pattern are developed in the cross-shore direction. And longshore currents flow toward the nodal line area which has very low wave energy. Consequently their mass flux is expelled through the area toward the sea direction. To confirm the generation, numerical experiments were performed using a nonlinear Boussinesq equation model. In the cases with two incident wave-trains with slightly different directions and with a monochromatic wave propagating over a submerged shoal, it was seen that the honeycomb pattern of wave crests was well developed, and thus rip currents were evolved along the nodal lines.

Boussinesq modelling of nearshore waves under body fitted coordinate

A set of nonlinear Boussinesq equations with fully nonlinearity property is solved numerically in generalized coordinates, to develop a Boussinesq-type wave model in dealing with irregular computation boundaries in complex nearshore regions and to facilitate the grid refinements in simulations. The governing equations expressed in contravariant components of velocity vectors under curvilinear coordinates are derived and a high order finite difference scheme on a staggered grid is employed for the numerical implementation. The developed model is used to simulate nearshore wave propagations under curvilinear coordinates, the numerical results are compared against analytical or experimental data with a good agreement.