Wave forces on three-dimensional floating bodies with small forward speed

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves was put forward. The flow field was divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain was surrounded by a ship-wetted surface and a matching surface as well as part of the free surface. The free-surface condition for the inner domain was formulated by perturbation about the double-body (DB) flow assumption. The outer domain was surrounded by a matching surface and the rest had a free surface as well as an infinite far-field radiation boundary. The free-surface condition for the outer domain was formulated by perturbation of the uniform incoming flow. The simple Green function and transient free-surface Green function were used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. The Taylor expansion boundary element method (TEBEM) was adopted to solve the DB flow and inner-domain and outer-domain unsteady flow BIE. Matching conditions for the inner-domain flow and outer-domain flow were enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on the ship-wetted surface was applied to obtain the first- and second-order wave forces. The numerical prediction on the displacement, acceleration and added resistance of the 14000-TEU container ship at different forward speeds were investigated by the proposed TEBEM method. Reynolds-Averaged Navier–Stokes (RANS) equations based on Computational Fluid Dynamics (CFD) method were adopted to compare with TEBEM method. The physical tank experiment results also validated the accuracy of the numerical tank results. Compared with the experimental solutions, TEBEM obtained good agreement with the RANS CFD method. TEBEM, however, was much more efficient and robust.

We investigate the unsteady hydrodynamic force of solid objects vertically entering water with an air cavity behind the falling body. Physical models are proposed to represent the force components corresponding to the body acceleration, the gravity, and the velocity of the body and the fluid particles. The theoretical or numerical solutions of the physical models are presented to understand the evolution of the force components. The body-acceleration force component is expressed as the high-frequency added mass times the body acceleration. Near the undisturbed free surface, the added mass grows strongly with increasing the submerged depth. It tends to be steady after the submerged depth is greater than a few characteristic lengths. The gravity force component consists of an upward hydrostatic term and a downward dynamic term. Generally, the hydrostatic term, which is obtained by integrating the gravity term in Bernoulli’s equation over the wetted body surface, is much larger than the gravity force component. For the three-dimensional bodies, the gravity force component is found to vary as a power of the submerged depth, where the exponent is about 0.83. The velocity force component is represented as the drag coefficient defined by the V-squared law, which is characterized by the body geometry. The drag coefficient may experience three successive stages with increasing the submerged depth.

A Rankine source method is applied to predict linear and weakly nonlinear sectional loads of a modern container ship. The method uses solution in the frequency domain, linearised with respect to wave amplitude about the nonlinear steady flow due to forward speed, which accounts for the nonlinear free-surface conditions, ship wave and dynamic trim and sinkage. Weak nonlinearity of the sectional loads in waves (e.g. hogging-sagging asymmetry) is taken into account by pressure extrapolation and integration up to the estimated actual water line. The sectional forces obtained with this method are compared with the results of other methods, including (1) linear Rankine panel method, where flow due to waves is linearised about the double-body flow, (2) linear zero-speed Green function method with correction for forward speed, (3) fully nonlinear simulations based on field-based solution of Reynolds-averaged Navier-Stokes (RANS) equations and (4) model tests. Comparison with RANS solution and model tests shows, that the proposed method can accurately predict sectional loads for small to moderate wave heights.

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

The second-order wave excitation forces acting on offshore floating structures are much more sensitive to mesh quality than linear forces when using the boundary element method to solve potential flow problems. This paper demonstrates the importance of mesh refinement near the waterline and sharp corners for mean drift forces, as well as low-frequency components in bi-chromatic wave conditions. To achieve better local panel refinement, a stretching factor is introduced to the Cosine spacing formulation used in meshing for the potential flow solver. This approach yields an economical and effective mesh for second-order analyses. An optimal mesh for a bottom-fixed cylinder is identified by comparing the horizontal mean drift force from numerical results against analytical values. Off-body kinematics near the cylinder are also studied as a basis for the mid-field formulation. A truncated floating cylinder is used to test the mid-field formulation applied for the vertical drift force. It is found that the drift force calculated by the pressure integration method matches well with the result obtained by the mid-field formulation when the sharp corner at the bottom is properly refined. Two ship models are then used to test the meshing strategy towards the waterline for second-order forces using both the near-field and mid-field methods. Recommendations are provided for meshing strategies on wetted hulls for boundary element method codes applied to low frequency second-order problems.

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.

Wave interaction with a perforated wall breakwater with a submerged horizontal porous plate

The time-domain analysis is applied to study the added resistance of a restrained floating body moving with a steady forward speed in waves. It is treated as an initial-boundary value problem and solved by the time-domain Green function method. The body boundary condition is satisfied exactly on each instantaneous wetted body surface, and the linearized free surface condition is applied. The added resistance computation is carried out by integrating pressures on the wetted body surface. A submerged spheroid and a surface-piercing vertical circular cylinder are selected as numerical examples. Good agreements are obtained by comparing with the transformed results from the frequency-domain computation. Wave exciting force on Series 60 CB=0.7 is also computed to verify the computational method.

A method is presented to describe the behavior of an oscillating bubble in a fluid near a second elastic (biological) fluid. The elasticity of the second fluid is modeled through a pressure term at the interface between the two fluids. The Laplace equation is assumed to be valid in each of the fluids, and a difference in the respective densities is allowed. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be found. The boundary integral method is then used to solve for the unknown normal velocities at both the bubble interface and fluid-fluid interface. These said normal velocities are subsequently utilized to update the position of the interface(s) for the next time step. For bubbles oscillating near a second nonelastic fluid, the bubbles can develop a jet towards or away from the fluid-fluid interface (depending on the distance of the bubble from the fluid-fluid interface and the density ratios of the two fluids). This behavior can be greatly modified when the second fluid possesses some elastic properties. The elasticity causes a small perturbation to travel along the surface of the bubble. A complex interaction between this growing perturbation and the bubble in its collapse phase can lead to the bubble assuming a “mushroom” shape and/or even breakup into two smaller bubbles. These phenomena have not been observed when the elasticity is absent in the second fluid. Excellent agreement with experimental data was obtained for a wide range of parameters.

Results of numerical and experimental study are presented of the wave forces acting on a blunt ship advancing in oblique waves. A 3-D panel method using a new numerical scheme proposed one of the present authors for evaluating the Green function was applied to numerical estimation of wave forces in short waves provided that the line integral and the steady disturbance on the body surface can be neglected, and the numerical results are compared with the experimental ones in full detail.The effect of the number of discretized panels on the body surface is investigated by increasing the number of panels from about 400 to 1600 and it is confirmed that the number of panels affects significantly accuracy of computed second order steady forces but the effect is not so remarkable on the first order wave forces.Two different descriptions of the wave term of the Green function, one is the monopole type and the other is the panel type obtained by integrating analytically the monopole over the panel, are also investigated. We confirm that as large number of panels as the monopole type is necessary to obtain the accurate solutions even if the panel type estimation is adopted.The present study confirmed that 3-D panel method is useful to predict the wave forces comparative-ly in long waves, while in short waves it does not give such accurate solutions as be in good agreement with experiments even if the large number of panels is used.

A BEM-isogeometric method for the ship wave-resistance problem

Direct and boundary-only solutions of multilayer aquifer systems, Part A. Steady-state solution

Interaction of short-crested waves with cylinder and concentric segmented arc breakwaters

Time domain prediction of hydroelasticity of floating bodies

We study the impact of a line mass onto a liquid-gas interface. At early times we find a similarity solution for the interfacial deformation and show how the resulting surface tension force slows the fall of the mass. We compute the motion beyond early times using a boundary integral method, and find conditions on the weight and impact speed of the mass that determine whether it sinks or is trapped by the interface. We find that for given impact speed there is a critical weight above which the mass sinks, and we investigate the asymptotic behavior of this critical weight in the limits of small and large impact speeds. Below this critical weight, the mass is trapped by the interface and subsequently floats. We also compare our theoretical results with some simple tabletop experiments. Finally, we discuss the implications of our work for the vertical jumps of water-walking arthropods.