Total Velocity Potential Research Articles

Interaction of a Flexural-Gravity Wave with a Vertical Rigid Plate Built in a Floating Elastic Plate

The linear two-dimensional problem of interaction between an hydroelastic wave propagating along an elastic floating ice plate with built-in vertical rigid plate is studied. The fluid under the ice is inviscid and incompressible. The fluid depth is finite. The deflection of the ice plate is described by the linear theory of thin elastic plates. The flow under the ice is potential. The total velocity potential is decomposed into the potential of the incident wave, even potential caused by the vertical motion of the rigid plate, and an odd potential caused by the rotation of the rigid plate. The vertical mode method is used. The third potential is obtained by solving a mixed boundary-value problem numerically using Chebyshev polynomials. The solution is validated by analysis of its convergence. The first and second potentials, and the corresponding deflections and strains of the ice plate, are obtained analytically. The motions of the rigid plate, as well as deflection and strains in the floating plate, are numerically analyzed. It is shown that the rotation of the rigid plate due to the incident wave is the main factor of increasing strains in the ice plate.

Analytical study on an oscillating buoy integrated with a perforated structure with a quadratic pressure drop condition: A case study on a WEC-perforated breakwater integration system

A general analytical method for solving the linear wave interactions with the integration systems comprising an oscillating buoy and perforated structure is presented, with a quadratic pressure drop condition on the perforated surface. Using the method of variable separation and eigenfunction expansion, the motion response of the buoy and total velocity potential can be solved using a two-loop iterative method. The method is validated against the published results. The integration system with a heaving WEC arranged inside a perforated breakwater was investigated particularly. Under the quadratic pressure drop condition, the symmetry of the hydrodynamic coefficients and Haskind relation, are found no longer satisfied, and the dissipation through the perforated wall also contribute to the damping coefficient. Effects of the damping of the power take-off system, front wall porosity, wave amplitude, and position of the WEC are investigated in particular. It was found that the horizontal and vertical wave forces on the WEC increase significantly as wave amplitude increases; however, the hydrodynamic efficiency decreases. When the WEC is near the solid rear wall, the horizontal wave forces on the WEC and solid rear wall increase significantly at the frequencies of piston mode gap resonance and heave motion resonance.

Free vibration and dynamic response analysis of liquid in a rectangular rigid container with an elastic baffle

In practical applications, undesirable deformation and failure of the container may appear due to intense liquid oscillation. Anti-slosh baffle is extensively applied to the reduction of sloshing response. Thus, a semi-analytical mathematical model is proposed to extract the vibratory characteristics of liquid interacting with elastic baffle in a rectangular rigid container. The superposition method and the separation of variables are used to analytically formulate the velocity potentials of sub-domains. The baffle deflection is expanded using its dry-modal functions. A linear finite-dimension equation system is derived to treat the eigenvalue problem of liquid considering hydroelasticity through implementation of sloshing condition, continuity and compatibility conditions. Then, the total velocity potential is comprised of rigid and perturbed components for laterally excited container. The coupled modal orthogonality is demonstrated by use of Hamilton's principle. The system response equation is established by combination of wave equations and baffle equation. Convergence study is performed. The present results are compared with the theoretical and numerical solutions from the reported literatures. Good agreements are achieved. The effects of baffle parameters on the natural frequencies, mode shapes and dynamic responses are evaluated in detail. It is found that the thin baffle may increase the sloshing frequencies to higher values.

Numerical researches of three-dimensional nonlinear sloshing in shallow-water rectangular tank

The Boussinesq-type equations in terms of velocity potential are adopted for the solutions of three-dimensional nonlinear sloshing motions in a shallow-water rectangular tank. The total velocity potential is divided into two parts: a particular solution and the rest which is calculated by the Boussinesq-type equations. Particular solutions of six degrees of freedom motions are constructed. The fully nonlinear free surface boundary conditions are satisfied and finite difference scheme is adopted for the spatial derivatives. The free surface profiles in tank with external excitation frequencies close to the first natural frequency are presented. Numerical results demonstrate that the Boussinesq-type equations with particular solutions can accurately model nonlinear sloshing waves in shallow-water rectangular tank.

Green's function approach for the transmission loss of concentrically multi-layered circular dissipative chamber.

An analytical method has been proposed by using Green's function analysis to find the transmission loss (TL) of a concentrically multi-layered circular dissipative chamber. Each layer of the chamber is filled with a porous acoustic absorptive material and is separated from the adjacent one by a thin perforated screen. A tailored Green's function for this configuration, in the absence of mean flow, is expressed as the summation of eigenfunctions of the central duct. In the analysis, the walls of the chamber are assumed to be acoustically rigid. The cumulative effect of the layers has been incorporated in terms of the reflection coefficient in the eigenfunctions of the central duct. By using the piston analogy approach at the inlet and outlet ports of the chamber, the total velocity potential generated inside the central duct is estimated. Thence, the transfer matrix is evaluated to predict the TL. The results obtained from the current method are in good agreement with the numerical models and available literature. A parametric study has been conducted to investigate the effect of the number of layers, and their arrangement with respect to thickness and flow resistivity, on the acoustic performance of the chamber.

Wave action by arrays of vertical cylinders with arbitrary smooth cross-section

This paper presents an analytical model to solve the linear wave diffraction problem by arrays of bottom-mounted cylinders with arbitrary smooth cross-section. Based on the assumption of ideal fluid and potential theory, the unknown coefficients of total velocity potential can be solved by system of linear equations, which are obtained from the boundary conditions. The accuracy of the present method is verified by comparing it with the numerical tool in terms of the wave force and wave run-up. Multiple cylinders with different configurations are tested, the cross-section of which is circular with cosine perturbation. The results show that the proposed method could obtain an accurate prediction of the wave action with multiple cylinder problems. Finally, the diffraction wave is investigated on arrays of bottom-mounted cylinders with different cross-section and layout. The near-trapping problem with effects of the multi-body interaction are also investigated.

Efficient calculation of the hydrodynamic coefficients and dynamic stiffness of an air-spring type vibration absorber

Integration of an air-spring device into an offshore platform can be an effective solution to modify the system properties, such as the restoring stiffness. In this study, the dynamic stiffness due to the linear vibration of the air and fluid entrapped inside an air-spring type vibration absorber is investigated. The device is modelled as a hollow column with a closed top and floats vertically in a water of constant depth. At the initial time, the free surfaces inside and outside the device are at the same level and there is an amount of air entrapped above the inner water surface. Within the device, the variation to the air pressure is assumed to occur adiabatically and the free-surface boundary condition is derived based on the ideal gas state equation. Energy dissipation is introduced into the region within the device by adding a resistance force on the inner free surface. The wave radiation due to the heave motion of the device is then concerned and an analytical model is developed to evaluate the associated hydrodynamic coefficients and the induced dynamic stiffness by using the method of separation of variables in conjunction with the matching technology. An alternative solution of the dynamic stiffness has also been developed based on a decomposition of the total velocity potential into two parts which depend purely on the oscillation of the device and the dynamic air pressure applied on the inner free surface, respectively. Based on the developed model, detailed numerical analysis is performed. The relative phases between the body motion and the dynamic stiffness are discussed for different wave conditions and geometric parameters.

Prediction of non-linear wave loads on large floating structures using a 3D numerical wave tank approach

In the present work, non-linear wave loads acting on large floating structures are computed using a 3D numerical wave tank (NWT) approach in an approximate manner. The hydrodynamic initial boundary value problem is solved following a Rankine panel-based boundary element method in conjunction with time integration of free-surface constraints and body-motion equations. To enable long-duration simulation for practical offshore configurations, total velocity potential is split into incident and perturbation part, and the latter effects are linearized. In our earlier works (Ganesan and Sen in J Ocean Eng Mar Energy 1:299–324, 2015; Ganesan and Sen in Appl Ocean Res 51:153–170, 2015), the 2nd order contributions from the linearized perturbation potential were neglected. In this work, we propose a modified formulation for the external load in which the full non-linear loads from the incident wave and up to 2nd order loads from the linear perturbation potential are considered. After presenting some results to validate the present method for steady drift force computations, non-linear loads from the modified formulation are compared with the unmodified form of the method and also with widely used 3D frequency-domain Green-function-based method. Comparative results between these three computations are presented for different geometries and the results are discussed to bring out the relative advantage of the modified formulation in predicting non-linear loads across the range of frequency and wave steepness.

Shallow-water sloshing motions in rectangular tank in general motions based on Boussinesq-type equations

Based on the highly accurate Boussinesq-type equations in terms of velocity potential, the shallow-water sloshing in a two-dimensional rectangular tank is studied. The rectangular tank in harmonic sway, heave and roll motions with small excitation amplitudes is considered. The total velocity potential is divided into two parts: the particular solution and the remaining part to be determined by the Boussinesq-type equations. The Stokes-Joukowski potential is adopted in the particular solution for the roll excitation motion. The comparisons of the numerical results indicate that the shallow-water sloshing motions in a rectangular tank can be predicted well based on the Boussinesq-type equations.

Investigation of linear wave action around a truncated cylinder with non-circular cross section

This paper presents an analytical model to solve the linear wave diffraction problem using a truncated cylinder with an arbitrary smooth cross section. Using the method of separation of variables, the boundary condition equations for the surface of a truncated cylinder are derived. Based on the condition that the radius function of a cylinder surface can be expanded into a Fourier series, the unknown coefficients of the diffraction potential can be obtained from a linear system equation. The surface elevation and wave force are calculated and analyzed in terms of the total velocity potential for different cases. The truncated error introduced in the solving process is discussed. Then, the accuracy of this method is verified by comparison with the results of the boundary element method. Finally, this method is further extended to a practical engineering application with a quasi-ellipse caisson for different drafts and exposures.

Numerical model of sloshing in rectangular tank based on Boussinesq-type equations

The highly accurate Boussinesq-type equations in terms of velocity potential are adopted for the simulation of sloshing phenomena in a two-dimensional rectangular tank. The fully nonlinear free surface conditions are used and linearized energy dissipation term is added into the dynamic free surface condition. The total velocity potential is divided into two parts: the particular solution and the rest which is calculated by the Boussinesq-type model. Different filling levels of liquid in the tank are considered for the validation of the numerical model, including shallow water, intermediate water depth and finite water depth cases. The linear theory based on the superposition of eigen-modes corresponding to natural sloshing frequencies is introduced. Compared with results calculated by linear theory and results from literatures, the numerical model based on Boussinesq-type equations presents good performances in predicting the sloshing motions in rectangular tank with small excitation amplitude.

Resonance of a tension leg platform exited by third-harmonic force in nonlinear regular waves.

The resonance of a floating tension leg platform (TLP) excited by the third-harmonic force of a regular wave is investigated based on fully nonlinear theory with a higher order boundary element method (BEM). The total wave elevation and the total velocity potential are separated into two parts, based on the incoming wave from infinity and the disturbed potential by the body. A numerical radiation condition is then applied at the far field to absorb the disturbed potential without affecting the incident potential. The BEM mesh on the free surface is generated only once at the initial time and the element nodes are rearranged subsequently without changing their connectivity by using a spring analysis method. Through some auxiliary functions, the mutual dependence of fluid/structure motions is decoupled, which allows the body acceleration to be obtained without the knowledge of the pressure distribution. Numerical simulation is carried out for the interaction of a floating TLP with waves. The focus is on the motion principally excited by higher harmonic wave forces. In particular, the resonance of the ISSC TLP generated by the third-order force at the triple wave frequency in regular waves is investigated, together with the tensions of the tendons.

Fully nonlinear wave interaction with freely floating non-wall-sided structures

A fully nonlinear numerical model for a floating body in the open sea has been developed based on velocity potential together with a higher-order boundary element method (BEM). The total wave elevation and the total velocity potential are separated into two parts, based on the incoming wave from infinity and the disturbed potential by the body. The mesh is generated only once at the initial time and the element nodes are rearranged subsequently without changing their connectivity by using a spring analogy method. Through some auxiliary functions, the mutual dependence of fluid/structure motions are decoupled, which allows the body acceleration to be obtained without the knowledge of the pressure distribution. Numerical results are provided for forces and run-ups of a fixed cylinder with flare and the comparison is made with the second order theory in the frequency domain. Simulations are also made for a freely floating body responding to wave excitation. Resonance related to ringing excited by the high order force at the triple wave frequency is discussed. Further results are provided for motions, forces and run-ups of a floating cylinder with flare. Comparison with the results for the fixed body and body in single degree of freedom is made.

Three dimensional wave scattering by arrays of elliptical and circular cylinders

Recently, Chatjigeorgiou and Mavrakos (2009, 2010a) provided an analytic solution for the three dimensional wave scattering by arrays of elliptical cylinders. The present paper extends the contents of the existing study to tackle the problem of the hydrodynamic interactions between elliptical and circular cylinders. The main task is to derive an analytic solution for the total velocity potential for an arbitrary body of the array and accordingly, to express the hydrodynamic pressure, the exciting forces and the wave elevation in compact analytic closed-forms. The solution method is rather complicated as it considers the circular cylinders as different geometries and not special cases of elliptical cylinders with zero elliptic eccentricity. Nevertheless, the adopted procedure enhances the mathematical reconstruction of the physical subject as it requires the derivation and the employment of addition theorems that transform expressions from elliptic to polar coordinate systems in all four possible combinations.

Response Amplitude and Hydrodynamic Force for a Buoy over a Convex

A buoy as an offshore structure is often placed over a convex such as a caisson or a submerged island. The hydrodynamic fluid/solid interaction becomes more complex due to the convex compared with that on the flat. Both the buoy and the convex are idealized as vertical cylinders. Linear potential theory is used to investigate the response amplitude and the hydrodynamic force for a buoy over a convex due to diffraction and radiation in water of finite depth. These are derived from the total velocity potential. A set of theoretical added mass, damping coefficient, and exciting force expressions have been proposed. Analytical results of the response amplitude and hydrodynamic force are given. Finally, the numerical results show that the effect of the convex on the response amplitude and hydrodynamic force for the buoy is ignored if the size of the convex is relatively smaller.

A Mathematical Note: The Dipole Potential for the Swaying and Rolling Cylinder

In this study, the total velocity potential, for the swaying and rolling cylinder expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin for incom- pressible, inviscid and irrotational flow. The non-dimensional expression of the dipole potential and its conjugated stream functions are re-obtained by using the mathematical expressions with complete alge- braic manipulations. The total velocity potential UT ðx; y; tÞ, for the swaying and rolling cylinder which is a periodic harmonic function of time t, may be expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin (1),(2). The x-axis is horizontal which coincides with the free surface of the fluid and is perpendicular to the axis of the cylinder and the y-axis is vertical, positive downward and going through the mean position of the axis of the cylinder. De Jong (1) formulated the problem as follows: In a fluid of infinite depth a cylinder is considered which is oscil- lating one-dimensionally and harmonically with a frequency of x while the mean posi- tion of its axis is assumed to lie in the free surface of the undisturbed fluid. De Jong assumed the amplitude of the oscillation to be small with respect to the diam- eter of the cylinder and the length of the waves generated by the oscillation, so that in the linearized approximation, the values of all physical quantities could be referred to the mean position of the cylinders. Taking the cylinder very long with respect to the breadth or enclosing the cylinder at both ends between two infinitely long walls per- pendicular to the axis of the cylinder, De Jong neglected the velocity components par- allel to the axis of the cylinder and consequently the motion is two-dimensional. The determination of the motions of the fluid under influence of the harmonic oscillation of the cylinder could be reduced to the solution of a boundary-value prob- lem from the linear potential theory. So, the velocity potential UT ðx; y; tÞ is also a periodic harmonic function of time. Therefore, the potential expressed is given in the following form, using complex notation, UTðx; y; t Þ¼� i/ T ðx; yÞe ixt : ð1Þ

On scattering and radiation problem for a cylinder in water of finite depth

Wave loadings due to scattering and radiation for a floating vertical circular cylinder in water of finite depth are derived. These are derived from the total velocity potential which can be decomposed as four velocity potentials; one due to scattering in the presence of an incident wave on fixed structure (diffraction problem), and the other three due to radiation respectively by surge, heave and pitch motion on calm water (radiation problem). For each case, the velocity potential is derived by considering two regions, namely, interior region and exterior region. The complex matrix equations can be solved numerically to determine the unknown coefficients to compute the wave loads. Numerical results can be obtained for different depth to radius and draft to radius ratios.

Wave loadings on a vertical cylinder due to heave motion

Wave forces and moments due to scattering and radiation for a vertical circular cylinder heaving in water of finite depth are derived analytically. These are derived from the total velocity potential which can be decomposed as two velocity potentials; one due to scattering in the presence of an incident wave on fixed structure (diffraction problem), and the other due to radiation by the heave motion on calm water (radiation problem). For each part, the velocity potential is derived by considering two regions, namely, interior region and exterior region. The complex matrix equations are solved numerically to determine the unknown coefficients to compute the wave loads. Some numerical results are presented for different depth to radius and draft to radius ratios.

Comments on Jacobs and Tsakonas: “Propeller Loading-Induced Velocity Field by Means of Unsteady Lifting Surface Theory”

In the cited paper (2) a formula is given for the lth Fourier component of the velocity potential of an N-bladed propeller [equations (9) and (10) of the paper], (2). The total velocity potential is then, of course, given by the sum of all the components.

Scattering of a surface wave by a submerged sphere

The problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential. These equations are easily solved numerically, so long as the sphere is not too close to the surface. The approach has been to formulate the problem as an integral equation, expand the Green's function, the velocity potential of the incident wave, and the total velocity potential in spherical harmonics, impose the boundary condition at the surface of the sphere, and carry out the integrations. The scattering cross section has been evaluated numerically and is shown to peak for values of the product of radius and wave number somewhat less than unity. Also, the Born approximation to the cross section is obtained in closed form.