Second-order interaction of irregular waves with a truncated column

A complete semi-analytical solution is obtained for second-order diffraction of plane bichromatic waves by a fixed truncated circular column. The fluid domain is divided into interior and exterior regions. In the exterior region, the second-order velocity potential is expressed in terms of ‘locked-wave’ and ‘free-wave’ components, both are solved using Fourier and eigenfunction expansions. The resulting ‘locked wave’ potential is expressed by one-dimensional Green's integrals with oscillating integrands. In order to increase computational efficiency, the far-field part of the integrals are carried out analytically. Solutions in both regions are matched on the interface by the potential and its normal derivative continuity conditions. Based on the present approach, the sum-and difference-frequency potentials are efficiently evaluated and are used to generate the quadratic transfer functions which correlates sum-frequency QTFs for a TLP column are present, which are compared for some frequency pairs with those from a fully numerical procedure. Satisfactory agreement has been obtained. QTF spectra for a case study TLP column, generated using the semi-analytical solution are presented. Also given are the results for nonlinear wave field around the column.     

Nonlinear free surface action with an array of vertical cylinders

Nonlinear diffraction of regular waves by an array of bottom-seated circular cylinders is investigated in frequency domain, based on a Stokes expansion approach. A complete semi-analytical solution is developed which allows an efficient evaluation of the second-order potentials in the entire fluid domain, and the wave forces on the structure. Expressions are derived for the second-order potential in the vicinity of individual cylinders. These expressions have a simple form, thus providing an effective means for investigating the wave enhancement due to nonlinear interactions with multiple cylinders. Based on the present method, the wave run-up and free-surface elevations around an array of two, three and four cylinders are investigated numerically.     

THE DIFFRACTION OF SECOND-ORDER BICHROMATIC WAVES BY A SEMI-IMMERSED HORIZONTAL RECTANGULAR CYLINDER

The diffraction of second-order bichromatic Stokes waves by a semi-immersed horizontal rectangular cylinder (prism) is investigated theoretically. The problem is assumed two-dimensional and the fluid domain is divided into three regions: upwave, beneath and downwave of the structure. Analytical expressions for the velocity potentials in each region at both first- and second-order are obtained by an eigenfunction expansion approach. The solutions in each fluid region are linked through matching conditions on the imaginary fluid interfaces between them. Semi-analytical expressions are derived for the sum-and difference-frequency hydrodynamic loads and the free-surface elevations upwave and downwave of the structure to second-order. Numerical results are presented which illustrate the influence of the different wave and structural parameters on these quantities at both first- and second-order.     

Regular wave integral approach to numerical simulation of radiation and diffraction of surface waves

A regular wave integral method is developed in the discretisation of a linear hydrodynamic problem on radiation and diffraction of surface waves by a floating or submerged body. The velocity potential of the problem is expressed as a solution of a body boundary integral equation involving the pulsating free surface Green function or pulsating free surface sources distributed on the body surface. With the use of a discretisation on the regular wave integral rather than discretisations on the singular wave integral of the Green function as in earlier investigations, the singular wave integral is approximated as an expansion of regular (or nonirregular) wave potentials. Influence coefficients between pulsating free surface source points are computed by the approximate expansion together with Hess–Smith panel integral formulas. Thus the velocity potential solution is evaluated by a boundary element algorithm. The numerical results produced from the proposed method agree well with semi-analytic solution results.     

On the interactions of surface waves with immersed structures

The fluid forces resulting from wave interaction with large submerged structures may be calculated using numerical procedures based on the solution of the associated boundary-value problem. In this paper, the analysis of wave interaction with a fixed submerged object of arbitrary cross-section and infinite length using a two-dimensional boundary value formation based on linear diffraction theory is summarized. Subsequently, the application of the boundary element method to obtain a solution is presented. The numerical considerations are emphasized with particular reference to computational efficiency. Numerical results are presented in the form of dimensionless wave force plots for various structural shapes. In the case of a bottom-seated half cylinder, for which there exists a closed-form solution, comparisons are made between results generated using both boundary element and equivalent finite element approaches. In the case of a submerged cylinder, comparisons are made between boundary element derived values and experimental results. The boundary element results compare well with both the closed-form solution and the experimental values.     

The application of finite element analysis to the solution of Stokes wave diffraction problems

A new finite-element based method of calculating non-linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface-piercing structures by solving a boundary value problem for the second-order velocity potential. Special attention has been given to the radiation condition for the second-order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second-order theory arising from inadequate radiation conditions have been overcome.     

Influence of an acoustic wave on a system of two spheres in a viscous fluid

In this article we formulate and solve the problem of the influence of radiation forces (forces created by the radiation pressure) on two spheres in a viscous fluid during the transmission of an acoustic wave. On the basis of these forces we investigate the nature of the interaction between the spheres as determined by the mutual disturbance of the flow fields around them as a result of interference between the primary and secondary waves reflected from the spheres. A previously proposed [2] approach is used in the investigations. The radiation force acting on one of the spheres is filtered by averaging the convolution of the stress tensor in the fluid with the unit normal to the surface of the sphere over a time interval and over the surface of the sphere. The stresses in the fluid are represented, to within second-order quantities in the parameters of the wave field, in terms of the velocity potentials obtained from the solution of the linear problem of the diffraction of the primary wave by the free spheres. The diffraction problem is formulated and solved within the framework of the theory of linear viscoelastic solids [6]. The case of an ideal fluid has been studied previously [3–5, 7]. Radiation forces are one of the causes of the relative drift of solid particles situated in a fluid in an acoustic field.S. P. Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 33–40, February, 1994.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

3-D inviscid self-excited vibrations of a blade row in the last stage turbine

Presented here is a three-dimensional (3-D) nonlinear time-marching method for the aeroelastic behaviour of an oscillating turbine blade row. The approach has been based in the solution of a coupled fluid–structure problem where the aerodynamic and structural dynamic equations are integrated simultaneously in time. This provides the correct formulation of a coupled problem, as the interblade phase angle (IBPA) at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity in the entire annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonic second-order accurate Godunov–Kolgan finite-volume scheme and a moving hybrid H–O (or H–H) grid. The fluid and the structural equations are solved using the modal superposition method. An aeroelasticity prediction of a turbine blade of 0.765 m is presented. The natural frequencies and modal shapes of the blade were calculated by using 3-D finite element models. The instability regions for five mode shapes and the distribution of the aerodamping coefficient along the blade length were shown for harmonic oscillations with an assumed IBPA. The coupled fluid–structure oscillations in which the IBPA is part of the solution are shown.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

The development of a Cartesian cut cell method for incompressible viscous flows

This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier–Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson‐type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual‐time iteration technique based on a slope‐limited, high‐order, Godunov‐type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh–Taylor instability problems. Finally, a test case for wave run‐up and overtopping over an impermeable sea dike is performed. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability of wave forms of motion of a viscous fluid layer under tangential stress

We study the stability of wave flow of a viscous incompressible fluid layer subjected to tangential stress and an inclined gravity force with respect to long-wave disturbances.An asymptotic solution is constructed for the equations of the disturbed motion and the problem is reduced to the study of a second-order ordinary differential equation. It is shown that after loss of stability by a Poiseuille flow the laminar nature of the flow is not destroyed, but the form of the free surface acquires a wave-like profile. The Poiseuille regime is stable for low Reynolds numbers. The critical Reynolds number for wave flow is found, and the stability and instability regions are determined.     

Regular wave integral approach to the prediction of hydrodynamic performance of submerged spheroid

A free surface Green function method is employed in numerical simulations of hydrodynamic performance of a submerged spheroid in a fluid of infinite depth. The free surface Green function consists of the Rankine source potential and a singular wave integral. The singularity of the wave integral is removed with the use of the Havelock regular wave integral. The finite boundary element method is applied in the discretisation of the fluid motion problem so that the panel integral of the Rankine source potential is evaluated by the Hess–Smith formula and the panel integral of the regular wave integral is evaluated in a straightforward way due to the regularity nature. Present method’s results are in good agreement with earlier numerical results.     

A second order stationary solution of the three dimensional wave diffraction potential

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solution of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been derived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is “free wave”, travelling independently of the first order wave system, the other is “phase locked waves”, which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical circular cylinder have been calculated, and the numerical results agree well with the experimental data.     

Computation of shock wave diffraction at a ninety degrees convex edge

This paper presents a numerical study of shock wave diffraction at a sharp ninety degrees edge, using an explicit second-order Godunov-type Euler scheme based upon the solution of a generalized Riemann problem (GRP). The Euler computations produce flow separation very close to the diffraction edge, leading to a realistic development of the separated shear layer and subsequent vortex roll-up. The diffracted shock wave, and the secondary shock wave, are both reproduced well. In addition a pair of vortex shocks are shown to form, extending well into the vortex core.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

On the effect of damping on dispersion curves in plates

This paper presents a study on quantitative prediction and understanding of time-harmonic wave characteristics in damped plates. Material dissipation is modelled by using complex-valued velocities of free dilatation and shear waves in an unbounded volume. As a numerical example, solution of the classical Rayleigh–Lamb problem for a viscoelastic plate is presented to illustrate and discuss the role of dissipation in the cut-off phenomenon and in the phenomenon of veering for dispersion curves. These phenomena are explained in more detail considering a simple model, which allows accurate asymptotic analysis of the perturbation of dispersion curves in the regions of cut-off and veering.     

Hydrodynamic forces and moments due to interaction of linear water waves with truncated partial-porous cylinders in finite depth

An exact analytical method is employed for studying the diffraction problems in an ocean due to the presence of a specific type of cylinders. In this current work, two models are studied: (i) a floating surface-piercing truncated partial-porous cylinder, (ii) a surface-piercing truncated partial-porous cylinder placed at the bottom. In both cases, the configuration of the composite cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. By using linear water wave theory, a three-dimensional representation of the problem is developed based on eigenfunction expansion method. The condition on the porous boundary is defined by applying Darcy’s law. Pressure and velocity satisfy continuity conditions across the linear interface between the adjacent fluid domains. Hydrodynamic force, moment and wave run-up are calculated by using the velocity potentials. Comparisons are carried out with results of wave diffraction by a floating and bottom-mounted compound cylinder, i.e., when the whole cylinder is non-porous. Handy agreements are observed from these comparisons. Through numerical tests, various experiments are carried out to investigate the impact of various parameters, such as porous coefficients, draft ratio, the ratio of inner and outer radii, the water depth etc., on hydrodynamic force, moment and wave run-up. The results clearly indicate that an appropriate optimal ratio for various parameters may be considered in designing practical ocean structures with minimum adverse hydrodynamic effect. The appearance of resonance in the results and role of porosity in mitigating resonance effect are explained. Proposal to select various appropriate parameters for the best possible effect is put forward.     

A WAVE EQUATION MODEL TO SOLVE THE MULTIDIMENSIONAL TRANSPORT EQUATION

The wave equation model, originally developed to solve the advection–diffusion equation, is extended to the multidimensional transport equation in which the advection velocities vary in space and time. The size of the advection term with respect to the diffusion term is arbitrary. An operator-splitting method is adopted to solve the transport equation. The advection and diffusion equations are solved separate ly at each time step. During the advection phase the advection equation is solved using the wave equation model. Consistency of the first-order advection equation and the second-order wave equation is established. A finite element method with mass lumping is employed to calculate the three-dimensional advection of both a Gaussian cylinder and sphere in both translational and rotational flow fields. The numerical solutions are accurate in comparison with the exact solutions. The numerical results indicate that (i) the wave equation model introduces minimal numerical oscillation, (ii) mass lumping reduces the computational costs and does not significantly degrade the numerical solutions and (iii) the solution accuracy is relatively independent of the Courant number provided that a stability constraint is satisfied. © 1997 by John Wiley & Sons, Ltd.