Towards Predicting the Added Resistance of Slow Ships in Waves

The objective of this project was to develop a calculation tool for the added resistance of ships in ocean waves. To this end a linear potential flow time-domain numerical seakeeping solver has been developed. The solver is based on highorder finite-difference schemes on overlapping grids and has been implemented using the Overture framework for solving partial differential equations on overset, boundary-fitted grids.

This library includes support for parallel processing and a variety of direct and iterative system solvers. The non-linear water water wave problem is linearised about two base flows namely: the uniform stream, and the double body flow. The resulting linearised initial boundary value problem has been solved in the time domain.

In order to march the free surface in time, the fourth-order Runge-Kutta integration scheme has been used to integrate the kinematic and dynamic free-surface boundary conditions. The field continuity equation has been discretised by a centered fourth-order finite difference scheme which also includes ghost layers at the boundaries.

For the zero-speed hydrodynamic problem, the same centered scheme can be utilised to calculate the free-surface derivatives. In the case of the forward-speed problem however, the convective terms in the free-surface conditions have been calculated using an upwind biased scheme, where the stencil is weighted in the upwind direction.

As an alternative to using the biased scheme, a flexible filtering scheme has been implemented which can be applied to the solution after each time step. The filtering scheme can be used with the centered finite difference scheme. Both of these strategies introduce numerical diffusion into the model to ensure the stability in the case of the forward-speed hydrodynamic problems.

The developed computational strategy has been applied to solve three hydro-dynamic problems: the wave resistance problem, the radiation problem, and the diffraction problem. The main objective was to find the first-order velocity potentials, free-surface elevation and the body motions that are required to calculate the wave drift force or the added resistance.

Instead of solving the time-domain water wave problem by the impulse response function approach, a pseudo-impulsive Gaussian motion is used in this project. In the case of the diffraction problem the pseudo-impulse describes the elevation of the incident waves. In the radiation problem this is the displacement which will be applied to the body in the time-domain.

The time-domain solutions of the hydrodynamic problems are then Fourier transformed to get the frequency-domain solutions. In the case of the radiation problem these are the added mass and damping coefficients. For the diffraction problem we obtain the wave exciting forces in the frequency domain. By solving the equation of motion the response amplitude operators for six degrees of freedom are also calculated.

For each hydrodynamic problem, the free-surface elevation along the waterline, the velocity potential and its gradients on the body surface, are obtained in the frequency domain via Fourier transform of the transient solutions. All this frequency-domain data is then used to calculate the added resistance in the frequency domain.

This has been implemented using the near-field formulation. The solver has been validated against analytical solutions for simple exact geometries like a cylinder and a sphere. The solver is now ready to be exercised on real ship geometries.