Wave-structure Interaction Problems Research Articles

Wave structure interaction problems for two-layer fluids in three dimensions

The effect of compressive force on flexural gravity waves in two-layer fluids is analysed. Wave characteristics for surface and interfacial modes in the cases of deep and shallow water are studied and the effect of compression on these modes is analysed in special cases from the general problem. Generalized expansion formulae and associated orthogonal mode-coupling relations are derived for the velocity potentials to deal with wave structure interaction problems in three dimensions in both the cases of finite and infinite water depths in channels of finite and semi-infinite widths. Several characteristics of the eigenfunctions are derived in specific cases. As an application of the expansion formulae, wave scattering due to partially frozen cracks in floating ice sheet is analysed in a channel of finite width and depth in the presence of compressive force in two-layer fluids.

Three-dimensional interaction of waves and porous coastal structures: Part I: Numerical model formulation

This paper and its companion paper (Lara et al. (2012)) describe the capability of a new model, called IH-3VOF, to simulate wave–structure interaction problems using a three-dimensional approach, when porous structures are present. The lack of a universal approach for the formulation of porous media flow equations has motivated a new derivation in the present work. Applications dealing with heterogeneous media, where porosity varies along the porous body, such as the study of multilayered rubble-mound breakwaters, are the final objective of the study. In this first paper, a new derivation of the equations, eliminating the limitations imposed by previous approaches is presented. The model integrates a new set of equations which covers physical processes associated with flow interaction with porous structures. The model considers the multiphase VARANS equations, a volume-averaged version of the traditional RANS (Reynolds-Averaged Navier–Stokes) equations. Turbulence is modeled using a k–ε approach, not only at the clear fluid region but also inside the porous media. A VOF technique is used to track the free surface. In this first paper, the model has been validated using laboratory data of a two-dimensional flow. In the companion paper the model is further validated with new experimental data sets, considering porous and solid structures as well as the presence of air. The model predictions present an excellent agreement with the laboratory measurements.

Investigation of a Stochastic Inverse Method to Estimate an External Force: Applications to a Wave‐Structure Interaction

The determination of an external force is a very important task for the purpose of control, monitoring, and analysis of damages on structural system. This paper studies a stochastic inverse method that can be used for determining external forces acting on a nonlinear vibrating system. For the purpose of estimation, a stochastic inverse function is formulated to link an unknown external force to an observable quantity. The external force is then estimated from measurements of dynamic responses through the formulated stochastic inverse model. The applicability of the proposed method was verified with numerical examples and laboratory tests concerning the wave‐structure interaction problem. The results showed that the proposed method is reliable to estimate the external force acting on a nonlinear system.

Expansion formulae for wave structure interaction problems in three dimensions

In the present paper, expansion formulae for the velocity potentials and associated orthogonal modecoupling relations for 3D Laplace equation satisfying higher order boundary conditions arising in fluid structure interaction problems are derived. Characteristics of the eigensystems are derived in terms of several identities and the convergence of the developed expansion formulae is demonstrated. As an application of the expansion formulae, flexural gravity wave scattering due to a crack in a floating ice sheet is investigated. The present method of solution can be easily applied to a large class of problems in the area of wave structure interaction arising in ocean engineering and various branches of applied mathematics, mathematical physics and engineering.

IH-3VOF: A THREE-DIMENSIONAL NAVIER-STOKES MODEL FOR WAVE AND STRUCTURE INTERACTION

This paper describes the capability of a new model, called IH-3VOF to simulate wave-structure interaction problems using a three-dimensional approach. The model is able to deal with physical processes associated with wave interaction with porous structures. The model considers the VARANS equations, a volume-averaged version of the traditional RANS (Reynolds Averaged Navier-Stokes) equations. Turbulence is modeled using a k-ï¥ approach, not only at the clear fluid region (outside the porous media) but also inside the porous media. The model has been validated using laboratory data of free surface time evolution in a fish tank containing a porous dam. Numerical simulations were calibrated by adjusting the porous flow empirical coefficients for two granular material characteristics. Sensitivity analysis of porous parameters has also been performed. The model is proven to reproduce with a high degree of agreement the free surface evolution during the seeping process. Simulations of a three- dimensional porous dam breaking problem has been studied, showing the excellent performance of the model in reproducing fluid patterns around a porous structure. The model is powerful tool to examine the near-field flow characteristics around porous structures in three dimensional flow conditions.

Surface gravity wave interaction with elastic bottom

In the present paper, a hydroelastic model is developed to deal with surface gravity wave interaction with an elastic bed based on the small amplitude water wave theory and plate deflection in finite water depth. The elastic bottom bed is modelled as a thin elastic plate and is based on the Euler–Bernoulli beam equation. The wave characteristics in the presence of the elastic bed is analyzed in both the cases of deep and shallow water waves. Further, the linearized long wave equation is generalized to include bottom flexibility. A generalized expansion formula for the velocity potential is derived to deal with the boundary value problems associated with surface gravity waves having an elastic bed. The utility of the expansion formula is illustrated by demonstrating specific physical problems which will play significant role in the analysis of wave structure interaction problems. Behavior of the wave spectra are discussed in the case of closed basin having a free surface and an elastic bottom topography.

High-order finite difference solution for 3D nonlinear wave-structure interaction

This contribution presents our recent progress on developing an efficient fully-nonlinear potential flow model for simulating 3D wave-wave and wave-structure interaction over arbitrary depths (i.e. in coastal and offshore environment). The model is based on a high-order finite difference scheme OceanWave3D presented in [1, 2]. A nonlinear decomposition of the solution into incident and scattered fields is used to increase the efficiency of the wave-structure interaction problem resolution. Application of the method to the diffraction of nonlinear waves around a fixed, bottom mounted circular cylinder are presented and compared to the fully nonlinear potential code XWAVE as well as to experiments.

Scattering of membrane coupled gravity waves by partial vertical barriers

Scattering of membrane coupled gravity waves in deep water by partial vertical barriers is investigated by the recently developed expansion formulae for wave structure interaction problems. The horizontal thin membrane is considered to be under uniform tension and is covering the free surface. The analysis is based on the linearized theory of water waves, and by combining the kinematic and dynamic conditions at the membrane covered surface, one may derive a not so well-posed mixed boundary value problem for Laplace's equation with third order boundary condition. The flexible membrane is attached by a spring to the surface piercing barrier, giving suitable edge conditions for the unique solution. The boundary value problem has been converted into dual integral equations with kernels composed of trigonometric functions, which are then solved analytically. The important physical quantities such as reflection and transmission coefficients for both cases of submerged and surface piercing barriers are obtained analytically in terms of modified Bessel functions. It is found that complete reflection or transmission is possible at certain resonant frequencies for the incident membrane coupled waves. Numerical results are plotted and discussed for different values of the nondimensional membrane tension parameter. doi:10.1017/S1446181110000064

Second-order wave diffraction by a circular cylinder using scaled boundary finite element method

The scaled boundary finite element method (SBFEM) has achieved remarkable success in structural mechanics and fluid mechanics, combing the advantage of both FEM and BEM. Most of the previous works focus on linear problems, in which superposition principle is applicable. However, many physical problems in the real world are nonlinear and are described by nonlinear equations, challenging the application of the existing SBFEM model. A popular idea to solve a nonlinear problem is decomposing the nonlinear equation to a number of linear equations, and then solves them individually. In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. By splitting the forcing term into two parts, the physical problem is described as two second-order boundary-value problems with different asymptotic behaviour at infinity. Expressing the velocity potentials as a series of depth-eigenfunctions, both of the 3D boundary-value problems are decomposed to a number of 2D boundary-value sub-problems, which are solved semi-analytically by SBFEM. Only the cylinder boundary is discretised with 1D curved finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The method can be extended to solve more complex wave-structure interaction problems resulting in direct engineering applications.

Probability distributions of wave run-up on a TLP model

Wave run-up on multi-column compliant and rigid platforms both on the weather-side and beneath the platform deck is a complex wave–structure interaction problem. Predictions based upon higher order hydrodynamic design codes have been improved our understanding of this phenomena but have not appreciably reduced the reliance on model tests and statistical analyses. In this study a three parameter distribution model is developed based upon the inclusion of linear, second-order, and mean contributions to the wave run-up utilizing the method of L-moments. Explicit expressions are derived relating the L-moments and the input from experimental data. The sample L-moments are developed from a linear combination of ordered sequence of the data and consequently the high order L-moments, i.e. L-skewness and L-kurtosis, are less biased than the corresponding ordinary moments. The L-moment definitions of the variance, skewness and kurtosis are numerically compared with values obtained using the more standard definitions for these parameters. Rigid hull measurements are compared with the compliant hull configuration for a mini-TLP model. The sea-states investigated include the relatively benign sea conditions off of West Africa and the more extreme sea environments of Gulf of Mexico.

SCATTERING OF MEMBRANE COUPLED GRAVITY WAVES BY PARTIAL VERTICAL BARRIERS

AbstractScattering of membrane coupled gravity waves in deep water by partial vertical barriers is investigated by the recently developed expansion formulae for wave structure interaction problems. The horizontal thin membrane is considered to be under uniform tension and is covering the free surface. The analysis is based on the linearized theory of water waves, and by combining the kinematic and dynamic conditions at the membrane covered surface, one may derive a not so well-posed mixed boundary value problem for Laplace’s equation with third-order boundary condition. The flexible membrane is attached by a spring to the surface piercing barrier, giving suitable edge conditions for the unique solution. The boundary value problem has been converted into dual integral equations with kernels composed of trigonometric functions, which are then solved analytically. The important physical quantities such as reflection and transmission coefficients for both cases of submerged and surface piercing barriers are obtained analytically in terms of modified Bessel functions. It is found that complete reflection or transmission is possible at certain resonant frequencies for the incident membrane coupled waves. Numerical results are plotted and discussed for different values of the nondimensional membrane tension parameter.

Wave interaction with multiple articulated floating elastic plates

Flexural gravity wave scattering by multiple articulated floating elastic plates is investigated in the three cases for water of finite depth, infinite depth and shallow water approximation under the assumptions of two-dimensional linearized theory of water waves. The elastic plates are joined through connectors, which act as articulated joints. In the case when two semi-infinite plates are connected through a single articulation, using the symmetric characteristic of the plate geometry and the expansion formulae for wave-structure interaction problem, the velocity potentials are obtained in closed forms in the case of finite and infinite water depths. On the other hand, in the case of shallow water approximation, the continuity of energy and mass flux are used to obtain a system of equations for the determination of the full velocity potentials for wave scattering by multiple articulations. Further, using the results for single articulation and assuming that the articulated joints are wide apart, the wide-spacing approximation method is used to obtain the reflection coefficient for wave scattering due to multiple articulated floating elastic plates. The effects of the stiffness of the connectors, length of the elastic plates and water depth on the propagation of flexural gravity waves are investigated by analysing the reflection coefficient.

Solving water wave diffraction by an elliptic cylinder using scaled boundary finite element method

The scaled boundary finite element method (SBFEM), a novel semi-analytical mathematical method, is modified to solve the water wave interaction with an elliptic cylinder. By introducing a virtual circular cylinder surrounding the elliptic cylinder, the whole fluid domain is divided into one unbounded subdomain and several bounded subdomains. The corresponding boundary value problems in bounded and unbounded domains are solved by the SBFEM semi-analytically. Comparisons to the previous numerical solutions demonstrate excellent computational accuracy and efficiency of the present SBFEM approach, as well as the benefit of not suffering from the difficulties of irregular frequency, which are often encountered by the boundary element method. The method can be extended to solve more complex wave structure interaction problems resulting in direct engineering applications. References Mei, C. C., The applied dynamics of ocean surface waves, World Scientific, Singapore, 1989. Tao, L., Song, H. and Chakrabarti, S., Scaled boundary fem solution of short-crested wave diffraction by a vertical cylinder, Comput. Method Appl. M., 197, 2007, 232--242. doi:10.1016/j.cma.2007.07.025. Williams, A. N., Wave forces on an elliptic cylinder, J. Waterw. Port. C. ASCE, 111, 1985, 433--449. http://cedb.asce.org/cgi/WWWdisplay.cgi?8500462. Wolf, J. P., The scaled boundary finite element method, John Wiley and Sons Ltd, Chichester, England, 2003. Zhu, S. and Moule G., Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section, Ocean Eng., 21, 1994, 645--662. Au, M. C. and Brebbia, C. A., Diffraction of water waves for vertical cylinders using boundary elements, Appl. Math. Model., 7, 1983, 106--114. doi:10.1016/0307-904X(83)90120-8. Bettess, J. A. and Bettess, P., A new mapped infinite wave element for genearal wave diffraction problems and its validation on the ellipse diffraction problem, Comput. Method Appl. M., 164, 1998, 17--48. doi:10.1016/S0045-7825(98)00045-0. Chen, H. S. and Mei, C. C., Wave forces on a stationary platform of elliptical shape, J. Ship Res., 17, 1973, 61--71. Li, B., Cheng, L., Deeks, A. J. and Zhao, M., A semi-analytical solution method for two-dimensional Helmholtz equation, Appl. Ocean Res., 28, 2006, 193--207. doi:10.1016/j.apor.2006.06.003.

Gravity wave interaction with floating membrane due to abrupt change in water depth

Using the recently developed expansion formulae for wave structure interaction problems, the scattering of surface water waves by a semi-infinite floating membrane due to abrupt change in bottom topography is analyzed. Both the cases of finite and infinite steps are analyzed. In the present paper, the analysis is based on the linearized theory of water waves and small amplitude membrane response. Combining the linearized kinematic and dynamic surface conditions on the water surface with the dynamic pressure condition on the membrane, a third order differential equation is derived to describe the membrane covered free surface condition. General wave energy relation for wave scattering by floating horizontal membrane is derived by the application of law of conservation of energy flux and alternately by the direct application of Green's second identity. In the floating membrane covered region, the wave energy density is a combination of the kinetic and potential energy density due to the surface gravity waves, and the surface energy density which is due to the existence of the floating membrane on the free surface. Gravity wave transformations due to an abrupt change in bottom topography in the presence of a floating membrane in finite water depth are analyzed based on shallow water approximation. Numerical results are computed and analyzed to understand the wave transformation due to the floating membrane when there is an abrupt change in topography in different cases.

Scaled boundary FEM solution of short-crested wave diffraction by a vertical cylinder

The scaled boundary finite-element method (SBFEM) is a novel semi-analytical method developed in the elasto-statics and elasto-dynamics areas that has the advantages of combining the finite-element method with the boundary-element method. The SBFEM method weakens the governing differential equation in the circumferential direction and solves the weakened equation analytically in the radial direction. It has the inherent advantage of solving the unbounded fluid dynamic problem. In this paper, the boundary-value problem composed of short-crested waves diffracted by a vertical circular cylinder is solved by SBFEM. Only the cylinder boundary is discretized with curved surface finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The computation of the diffraction force based on the present SBFEM solution demonstrates a high accuracy achieved with a small number of surface finite elements. The method can be extended to solve more complex wave–structure interaction problems resulting in direct engineering applications.

Expansion formulae for wave structure interaction problems with applications in hydroelasticity

Alternate derivations of the expansion formulae for wave structure interaction problems are obtained in case of water of infinite depth and utilized to analyze the hydroelastic behavior of large floating structures. Considering the boundary value problem associated with Laplace equation having higher order boundary condition on the horizontal boundary and a Dirichlet type boundary condition on the vertical boundary in a quarter plane, Fourier sine transform is applied in the horizontal direction to convert the problem to a Sturm–Liouville type boundary value problem associated with non-homogeneous ordinary differential equation (ODE) in the transformed variable. Finally, inverting the transformed functions and applying the regularity criterion of the transformed function, the required expansion formula is derived. The expansion formula thus derived is extended to deal with similar boundary value problems having Neumann type boundary condition. The expansion formulae are applied to (i) analyze oblique scattering of flexural gravity waves by an articulated floating elastic plate and (ii) study the effect of compression on the oblique scattering of flexural gravity waves by a line discontinuity in a large floating ice sheet in water of infinite depth, which find applications in marine technology and arctic engineering, respectively. The present derivations of the expansion formulae are very simple and straightforward and can be easily used to study a large class of problems in the area of fluids and structures in mathematical physics and engineering.

Instability of nonlinear standing waves in front of a vertical wall

The standing waves formed in front of a vertical breakwater in Gdansk North Port Harbour are examined. To simplify the wave-structure interaction problem, a laboratory experiment and mathematical model were designed. Standing waves in a limited space were generated under resonance conditions between a vertical wall and wave generator. Such slowly growing standing waves eventually become unstable and, consequently, create an impact impulse on the vertical wall. The dynamics of the vertical wall and hydrodynamics of the standing wave were measured and compared with a numerical model derived by variational calculus. The phenomenon of standing wave instability observed in nature was reproduced by laboratory experimentation and numerical simulation. The presented mechanism of breaking standing waves is more complicated in reality due to wave randomness and the multidirectional wave field.

Expansion formulae in wave structure interaction problems

A large class of problems in the field of fluid–structure interaction involves higher-order boundary conditions for the governing partial differential equation and the eigenfunctions associated with these problems are not orthogonal in the usual sense. In the present study, mode-coupling relations are derived by utilizing the Fourier integral theorem for the solutions of the Laplace equation with higher-order derivatives in the boundary conditions in both the cases of a semi-infinite strip and a semi-infinite domain in two dimensions. The expansion for the velocity potential is derived in terms of the corresponding eigenfunctions of the boundary-value problem. Utilizing such an expansion of the velocity potential, the symmetric wave source potentials or the so-called Green's function for the boundary-value problem of the flexural gravity wave maker is derived. Alternatively, utilizing the integral form of the wave source potential, the expansion formulae for the velocity potentials are recovered, which justifies the completeness of the eigenfunctions involved. As an application of the wave maker problem, oblique water wave scattering caused by cracks in a floating ice-sheet is analysed in the case of infinite depth.

Stability and accuracy of periodic flow solutions obtained by a POD-penalty method

We develop a new penalty method to derive low-dimensional Galerkin models for fluid flows with time-dependent boundary conditions. We then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter(s) that yield asymptotically stable periodic solutions of the highest possible accuracy. We illustrate this new approach by studying flow past a circular cylinder using direct numerical simulation (DNS) data, and a wave-structure interaction problem using particle image velocimetry (PIV) data.

Diffraction–radiation of multiple floating structures in directional waves

The dynamics of multiple floating structures have been studied using the finite element method. The emphasis is on the hydrodynamic behaviour of multiple bodies under a multi-directional wave field. A two-dimensional numerical model has been adopted to evaluate hydrodynamic coefficients and forces in an oblique wave field. The responses in sway, heave and roll modes are reported. The linear filter technique is then used to extrapolate the responses under directional waves. The effect of mean wave direction and directional homogeneity on the hydrodynamic behaviour of the structure is studied. Based on the present study, it is found that the two-dimensional model is applicable to investigate the wave-structure interaction problems of the type herein considered.