Multi-domain Boundary Element Method Research Articles

Novel analytical solutions without finding complex roots for oblique wave scattering by submerged porous/perforated structures

This study develops new analytical solutions to oblique wave scattering by submerged porous/perforated structures on the basis of linear potential theory. Two cases of a porous (rubble mound) breakwater and a horizontal perforated plate are considered. The new solutions have the novelty of using a contour integral technique to avoid finding the complex roots (wave numbers) of complex dispersion equations for water wave motion over porous/perforated structures. In the solution procedure, the linear equation systems for the expansion coefficients in velocity potentials are obtained by using the conventional mode-matching method. However, the matrix elements involving the complex wave numbers are recast by using the contour integral technique, and then the expansion coefficients are determined without knowing the explicit knowledge of complex wave numbers. As a result, the difficulties arising in solving the complex dispersion equations in the traditional solutions are avoided completely. The calculation results of the new solutions agree well with known results of analytical approaches with complex wave numbers, analytical solutions based on the velocity potential decomposition method, and numerical solutions based on the multi-domain boundary element method. This study gives a simple and elegant procedure to tackle water wave interactions with submerged porous/perforated structures, and the solutions can be used as a reliable alternative tool for fast engineering preliminary analysis.

Surface gravity wave scattering by multiple slatted screens placed near a caisson porous breakwater in the presence of seabed undulations

In the present study, the effectiveness of using multiple slatted screens placed in front of a caisson porous breakwater to dissipate the incident wave energy is analyzed. To model the water flow through the slatted screens, a quadratic (nonlinear) pressure drop condition which involves both the inertial and drag effects is considered. Further, for thick porous structure, the well known Sollitt and Cross (1972) model is used. To handle the nonlinear boundary conditions, an iterative multi-domain boundary element method (BEM) is used. The numerical convergence of the multi-domain BEM based solutions is presented. Energy identities for flow past a single slatted screen, multiple slatted screens and for the present problem are derived. The study shows that the minimum reflection coefficient (<1%) and maximum wave energy dissipation (>98%) can be obtained by using four slatted screens. Further, for moderate values of perforation-effect Keulegan-Carpenter number (KC), the reflection coefficient attains maximum for b1λ≈n2,n=1,2,3,⋯ (b1 is the distance between the rigid caisson and the front slatted screen, λ is the incident wavelength) irrespective of the variations in the number of slatted screens. Further, the transmission coefficient, horizontal and vertical wave forces acting on the rigid caisson attain maximum for b1/λ≈0.55n, and minimum occurs for 2n+14<b1λ<3n+26,n=0,1,2,3,⋯. The Bragg resonance in the reflection coefficient occurs at Bragg value ≈1.6 irrespective of the variations of KC number. Moreover, the reflection coefficient increases around the Bragg value for higher values of KC number without altering the peak value in the reflection coefficient.

Impact of sloping porous seabed on the efficiency of an OWC against oblique waves

The study examines the influence of a sloping porous bed on the efficiency of an oscillating water column (OWC) device facing oblique water waves. A vertical, surface piercing, thin plate near a rigid wall approximates the OWC. The system is simulated using a multi-domain boundary element method assuming the linear potential theory. The impact of varying sloping bed structural parameters and influence of the incident wave angle on the performance of the OWC is evaluated and discussed. The significance of this model to act as a breakwater protecting near-shore marine facilities is also highlighted. The OWC efficiency is found to be highly sensitive to the slope of the porous seabed, and seabed porosity is found to stabilize the resonant frequency against changes in water levels. OWCs are found to be more efficient over porous seabeds with higher frictional coefficients and lower values of porosity. The results indicate the optimum values of the sloping bottom inclination, the porosity and the frictional coefficient for the porous sloping bottom that can be used for the design and implementation of an efficient OWC device.

Analytical study for oblique wave interaction with a submerged horizontal perforated plate near a partially reflecting vertical wall

A novel iterative analytical solution is developed to study the oblique wave interaction with a horizontally submerged perforated plate near a partially reflecting vertical wall. The boundary condition on the perforated plate is considered as a quadratic pressure drop to incorporate the effect of wave height on the wave energy dissipation. The boundary value problem is analyzed analytically under the assumptions of small amplitude wave theory. To validate the analytical model, an iterative solution based on MBEM (multi-domain boundary element method) is independently developed, and the comparison between the analytical solution and the experimental data are given. The performance characteristics of a submerged perforated plate before a partially reflecting vertical wall is studied by analyzing the hydrodynamic coefficients of the system for the effects of plate porosity, relative plate width, spacing, depth of submergence, and the direction of wave attack. The proposed model is expected to act as an effective breakwater in the attenuation of wave energy for the creation of a tranquillity zone in the marine environment. Moreover, the analytical model developed in the present study can be effective and reliable for practical preliminary design.

Wave Trapping Efficiency of a Flexible Membrane Near a Partially Reflecting Seawall

Abstract The gravity wave interaction with a flexible membrane placed at a finite distance from the partially reflecting seawall is analyzed under the framework of linear water wave theory using the multi-domain boundary element method (BEM). The flow through a flexible membrane is assumed to follow Darcy’s law in addition to membrane displacements. As a viable alternative to the existing wave dampers, the flexible membrane is examined for the effective dampening of incident waves. The correctness of the numerical results is affirmed with the known results available in the literature. The effect of membrane tension, submergence depth, membrane width, porosity, angle of inclination, and confined chamber spacing on hydrodynamic coefficients is discussed as a function of dimensionless wavenumber. The partially reflecting harbor wall diminishes the wave reflection coefficient in the long-wave regime. The increase in the flexible membrane width does not necessarily ensure the ideal wave capturing performance. A shift in the peak of the maximum deflection is observed with the increase of membrane width while there is a shift in peak outward for the increase in the submergence depth. Moreover, the maximum deflection is found to decrease with the increase in porosity, and it is 62% reduction for membrane porosity b = 1 due to the significant wave damping. The wave run-up and the wall force coefficients are found to be minimum when the relative plate width is B/h = 1. The present study is expected to be useful for the design of cost-effective wave attenuating systems.

Boundary element method for wave trapping by a multi-layered trapezoidal breakwater near a sloping rigid wall

This study examines the multiple layers in a rubble mound breakwater and their effect on reflection and dissipation of incoming ocean waves. The numerical model is developed using multi-domain boundary element method for oblique water wave trapping near a sloping wall by a multi-layered trapezoidal porous structure, which is utilized to model armour, filter and core layers while examining the hydrodynamics in different configurations. Both, the constant element and linear element approaches to boundary element method are discussed. The cases of bottom-standing porous structures as being submerged and fully extended are considered. The wave hydrodynamics over the structure is described by the reflection and dissipation coefficients along with the forces acting on the sloping wall, and is influenced by wave and structural parametrics of the system. The influence of armour layer in different configurations is highlighted for various structural and wave parameters.

New analytical solutions of oblique wave scattering by submerged horizontal perforated plates using quadratic pressure drop condition

This study develops new analytical solutions for oblique wave scattering by horizontal perforated plates submerged below the water free surface using a quadratic pressure drop boundary condition. Both kinds of submerged plates are considered: a) A horizontal perforated plate as an offshore breakwater; b) A horizontal perforated plate attached to a vertical wall as a wave absorber. The velocity potential decomposition method combined with an efficient iterative algorithm is adopted to obtain the analytical solutions, and the hydrodynamic parameters of engineering interest for the reflection, transmission and energy-loss coefficients as well as the vertical wave force are considered. A new simple formula for predicting the discharge coefficient in the quadratic pressure drop condition, which is critical to make the analytical solutions to be practically applicable, is developed by fitting the analytical results and an amount of experimental data in literature. The present analytical solutions are in good agreement with previous analytical solution for special case and the numerical results of an independently developed multi-domain boundary element method solution. The reflection and transmission coefficients calculated by the present analytical solutions are also in good agreement with experimental data when adopting a well-predicted discharge coefficient. The present analytical solutions are valuable for fast assessing the hydrodynamic performance of horizontal perforated plates in the initial stage of engineering design. The present analytical model can also be readily extended to the multi-layer horizontal perforated plates and similar structures.

Wave scattering by inverted trapezoidal porous boxes using dual boundary element method

A numerical model is developed based on the dual boundary element method (DBEM) to solve the problem of surface gravity waves past multiple floating inverted trapezoidal porous boxes under the framework of linear potential flow theory. The continuity of velocity and semi-empirical quadratic pressure drop equation is adopted on the porous boundary to incorporate the effect of incident wave amplitude on the energy loss coefficient. Various results by an array of inverted trapezoidal porous boxes are discussed in the form of scattering coefficients (wave reflection, transmission, and energy loss coefficients), and force coefficients (horizontal force, vertical force, and moment). The verification of the numerical model is carried out by an independently developed multi-domain boundary element method (MBEM) code, and also with results available in the literature. The study reveals that, in most cases, the long waves exhibit insignificant hindrance whereas intermediate waves delineate the harmonic oscillations and deep water waves feel maximal hindrance by inverted trapezoidal porous boxes. The wave damping of KL≥0.95 is obtained for structural width B/λ≥0.66 by a pair of identical inverted trapezoidal porous boxes. The angle of porous wings is helpful to alter the scattering coefficients and also helps in the fine-tuning of force coefficients. The angle of partially open wings within 300≤θ≤600 is suggested along with structural porosity μ=0.25 for effective constriction of incident waves. The parametric analysis will be useful in the meticulous design of cost-effective breakwaters.

Bragg scattering of surface gravity waves by a submerged wavy porous plate

Bragg scattering of surface gravity waves by a slender pile-supported submerged wavy porous plate is studied under the assumption of small amplitude water wave theory. The waves past porous barriers are modeled using linearized pressure drop boundary condition known as Darcy law. Numerical solution for the boundary value problem is obtained using the Multi-Domain Boundary Element Method and the outcomes are validated with known results in the literature. For different wave and structural parameters such as the effect of the number of ripples of the wavy plate, effect of the relative ripple amplitude, relative plate length, plate porosities, and relative submergence depth; various physical quantities of interests like reflection and transmission coefficients, vertical force, and horizontal force coefficients are computed and analyzed. The study reveals that Bragg scattering occurs in the water of shallow and intermediate-depth for a wavy plate having more than two ripples and wave reflection increases with an increase in the amplitude of the wavy plate. For higher wavenumber or with an increase in plate length, Bragg scattering with harmonic and sub-harmonic peaks in the reflection coefficients are noticed in the case of wavy plate irrespective of permeability unlike the oscillatory decreasing pattern in the case of a flat plate. Wave forces exerted on the submerged plate decreases significantly with the introduction of structural porosity due to the dissipation of wave energy. Moreover, an increase in the number of ripples and ripple amplitude reduces the wave transmission; thus improving the overall performance for a wide range of frequencies. The study will be useful in the design of effective breakwaters for mitigating wave effects in the marine environment.

Wave interaction with multiple wavy porous barriers using dual boundary element method

The gravity wave interaction with horizontally staggered multiple thin wavy porous barriers is analyzed under the assumption of potential flow theory. The dual boundary element method (DBEM) is used to solve the boundary value problem. The accuracy and reliability of the DBEM solutions are ascertained by comparing with known results in the literature and with an independently developed multi-domain boundary element method (MBEM) code. The study is focused on the scattering coefficients and vertical forces on an array of wavy barriers. The numerical results demonstrate that wave transmission by a horizontally staggered porous barrier is much better compared to a single horizontal plate. It is recommended to use plate porosity of 0.05 to 0.20, if transmission coefficient of less than 0.50 is needed, especially for relative water depth k0h > 2. With an increase in the number of ripples and ripple amplitude on the porous plates, it is possible to further reduce wave transmission by 5% to 10%. Also, an increase in the inclination of the plate from 0 to 10 degrees can reduce the wave transmission and increases the wave reflection to 5% to 10%. Further, a reduction of about 10% to 30% in the vertical force on the wavy plate is possible for k0h ≥ 1 with an increase in the relative ripple amplitude, the number of ripples and increased plate rotation. This essentially will help in the reduction of pile penetration length into the sea bed soil. The results of the present study can be used for the optimal design of wavy plate as a wave barrier for offshore and coastal engineering applications.

Gravity wave interaction with multiple submerged artificial reefs

In the present study, gravity wave interaction with a series of submerged artificial permeable reefs is analysed within the framework of linearised water wave theory. For wave past porous walls of the artificial reefs, a quadratic pressure drop is assumed to account for the wave energy dissipation due to the changes in wave height. The physical problem is handled for a solution using a numerical model based on the iterative multi-domain boundary element method and the developed numerical model is validated with known results in the literature. The iterative model is compared with the numerical model based on linear pressure drop boundary condition (i.e., Darcy law). The study reveals that the wave transmission reduces with the increase in the number of reef units. It is demonstrated that the transmission coefficient can be reduced to less than 0.5 when the number of reef units is greater than or equal to three for a relative height greater than 0.7, reef porosities less than 20% and for 0.4&lt;k0h&lt;3.0. Bragg reflection is observed when the porosity is in the range of 0 to 10% and above which the recurrent nature of wave reflection gradually fades away due to dissipation effects. The number of peaks occurring in the shallow and intermediate water depths is equal to the number of reef units wherein the maxima occur at the first frequency. A relative increase in the base width improves the energy losses but the rate of increase in energy loss decreases. The scattering coefficient pattern is oscillatory when the relative spacing between the barriers is varied and their hydrodynamic performance is invariant for higher relative base width.

Scattering of gravity waves by a pontoon type breakwater with a series of pervious and impervious skirt walls

ABSTRACT The scattering of surface gravity waves by a skirt wall breakwater is analysed in two dimensions under the assumption of small-amplitude wave theory. The oscillatory wave past the porous skirt walls is assumed to follow Darcy’s law. The boundary value problem is solved using multi-domain boundary element method to obtain the numerical results for the different number of porous skirt walls, relative pontoon width, relative submergence and different porosity of the skirt walls. The study reveals that an increase in the number of skirt walls from one to four can reduce the wave transmission by 61% but at the expense of an increase in horizontal wave force by 90% and decrease in the vertical force by 31%. A minimum of two-skirt walls is needed for reducing wave transmission. Skirt wall porosity of 0.40 is recommended for field applications considering the trade-off between the wave transmission and wave forces.

Frontal collision of two nonlinear internal solitary waves in a stratified fluid

Head-on collision of two internal solitary waves (ISWs) in a two-layer inviscid fluid with fully nonlinear free-surface (FS) condition is investigated numerically in this paper. The collision process of and nonlinear interaction of internal solitary waves is studied based on nonlinear potential flow theory and a series of numerical simulations are conducted using a multi-domain boundary element method (MDBEM). The numerical model is validated by experiment for the simulation of ISWs propagating. The computation results show that the collision of ISWs is the asymmetric and nonlinear process in which the rundown motion causes the interface displacement penetrates the interface. The collision leaves imprints on small dispersive trailing waves which accounts for the attenuation with each departing ISW tilting slightly backward with respect to the direction of its propagation. In addition, the characteristic of collision process including run up, residence time, surface wave feature is discussed. During the collision, the wave spends more time falling than rising for all cases and the residence time is very long for waves of small amplitude, large depth ratios and density ratios. Considering the free surface of the top layer whether or not makes contributions to the run up and the residence time is larger if the top boundary is free surface.

Multi-Domain Boundary Element Method for Sound Scattering on a Partly Perturbed Water Surface

The problem is the scattering of a plane sound wave at a rough water-air interface. The purpose of this paper is to describe in detail the method and demonstrate its work with simple examples. The main advantage of this approach is that there are no limits on the relation between the shape of the surface and the incident wave, so we can consider large Rayleigh parameter, shading, multiple scattering. The solution of the Helmholtz equation in the form of an integral over the boundary is used only in the inner domain, in the outer domain the separation of variables is used to obtain a nonlocal integral boundary condition on the artificial boundary.

Gravity wave interaction with a submerged wavy porous plate

ABSTRACT A numerical model based on multi-domain boundary element method (BEM) is developed to study the gravity wave interaction with a horizontally submerged wavy porous plate. The boundary condition on the porous barrier is considered a quadratic pressure drop to incorporate the effect of wave height on the wave energy dissipation directly. The performance characteristics of the wavy porous plate are studied by analysing the hydrodynamic coefficients of the breakwater for different physical parameters. Numerical results demonstrate that wavy porous plate improves the energy dissipation and reduce wave transmission compared against the conventional flat plate. The increase in wave energy dissipation is governed by the ripple amplitude and the number of ripples. The proposed model is expected to act as an effective wave barrier in the attenuation of wave energy for the creation of a tranquillity zone in the marine environment for water depths ranging from intermediate to deep waters.

Gravity wave interaction with a wave attenuating system

In the present study, gravity wave interaction with a wave attenuating system (WAS) comprised of a series of surface piercing thin vertical porous plates fixed on a fully submerged rigid pontoon is analyzed. For the wave past porous barriers, a quadratic pressure drop is considered to account for the energy dissipation due to changes in wave steepness. A numerical model based on iterative multi-domain boundary element method is developed under the assumption of a small amplitude water wave theory. The scattering coefficients (reflection and transmission) and wave force coefficients (horizontal, vertical and moments) are computed for different wave and structural parameters such as the number of plates, relative plate width, the relative depth of submergence, plate porosities, and relative wave steepness. The results demonstrate that with the creation of trap zones through multiple porous plates on a submerged rigid pontoon, it is possible to reduce wave transmission by 60–70%. For optimum performance across all water depths, it is suggested for the WAS to have 1 ≤ B/h ≤ 1.5 and h1/h ≤ 0.3. Also, interestingly there is a 25–30% reduction in the vertical wave forces which reduces the load acting on the ancillary equipment. The WAS is found to perform well for porosity μ=20%, above which the wave transmission increases gradually. The study results can be useful in the cost-effective design of this type of special wave barrier for a range of coastal engineering field applications.

Oblique wave scattering by porous breakwaters/seawalls: Novel analytical solutions based on contour integral without finding complex roots

This article develops novel analytical solutions to oblique wave scattering by porous rubble-mound structures based on linear potential theory. The novelty of the solutions is that the usual complex root finding algorithm is avoided by applying a contour integral technique. In the solving procedure, the whole fluid domain is divided into multiple regions, and the series solution of the velocity potential in each region is obtained using the separation of variables. A linear equation system for the expansion coefficients is achieved by matching the interface conditions between adjacent regions. The matrix elements, which initially include the complex roots of complex dispersion relations, are evaluated by applying the contour integral technique. The evaluation process is free of complex roots, and thus the difficulties arising in solving the complex dispersion relations in traditional solutions are avoided. Case studies show that the series solution converge rapidly, and the calculation results of the new solutions agree well with those of previous analytical solutions with complex roots and multi-domain boundary element method (BEM) solutions. Besides providing a simple and elegant approach for analyzing the hydrodynamic performance of porous structures, the new solutions can be extended to water wave problems involving other dissipative coastal structures.

New interface integration BEM for solving multi-medium nonlinear heat transfer problems

In this paper, a new single interface integral equation method is established for solving non-linear multi-medium heat transfer problems with temperature- dependent thermal conductivity. At first, the boundary-domain integral equation for nonlinear heat transfer in single medium is established based on the fundamental solution of Laplace equation. Then, based on the variation feature of the material properties, a new single integral equation for material nonlinear multi-medium is established according to the degeneration rule from a domain integral to an interface integral. Next, the final system equations are solved by Newton-Raphson iterative method after the discretization. Comparing with conventional multi-domain boundary element method, the presented method is more efficient in computational time, data preparing and program coding. Three numerical examples are given to demonstrate the correctness and robustness of the method presented in the paper.

Analytical and experimental studies on Bragg scattering of water waves by multiple submerged perforated semi-circular breakwaters

This study develops analytical solutions for Bragg scattering of water waves propagating over a series of submerged perforated semi-circular breakwaters (bars) based on potential theory. The series solutions for velocity potentials in the external common fluid domain of the bars are expressed through multipole expansions, whilst those in the internal fluid domains are given using the separation of variables. The continuous condition of fluid velocity and a quadratic pressure drop condition on the perforated semi-circular arcs are applied to determine the unknowns in the series solutions. In the analytical solutions, obliquely and normally incident waves are independently considered. The calculation methods of the reflection, transmission and energy loss coefficients are given. Experimental tests are performed to measure the reflection and transmission coefficients of the breakwaters at different wave periods and water depths. The analytical solutions are in excellent agreement with the independently developed multi-domain BEM (boundary element method) solutions, and are also compared with the wide-spacing approximations and the experimental results. Calculation examples are presented to check the effects of various parameters (e.g., bar number, bar radius, bar spacing and incident wave angle) on the hydrodynamic quantities of the breakwaters. Main valuable results for engineering application are draw.

Iterative multipole solution for wave interaction with submerged partially perforated semi-circular breakwater

This study develops an iterative multipole solution for water wave interaction with a submerged partially perforated semi-circular breakwater based on the linear potential theory. Multipole expansion method and separation variable method are used for getting the velocity potential expressions of the flow fields, and the expansion coefficients in velocity potentials are determined by iteratively solving the non-linear pressure loss boundary condition on the partially perforated wall. The calculation results of the present multipole solution are in good agreement with the numerical results of multi-domain BEM (boundary element method) solution. The effects of the radius, porosity and perforated position on the reflection coefficient, transmission coefficient and wave force of breakwater are thoroughly discussed. Some meaningful suggestions are presented for engineering design.