Author | Prins, H. |

Title | Time-Domain Calculations of Drift Forces and Moments |

Conference/Journal | PhD-thesis, Delft University of Technology |

Month | March |

Year | 1995 |

Abstract (EN)

In this thesis we study the effect of the forward speed of a ship on the hydrodynamic forces, in particular the drift forces. Previous studies, restricted to zero forward speed, were based on the assumption of harmonic waves, thus reducing the time-dependent equations to frequency-dependent equations. In this study the actual time-dependent equations including forward speed, are solved, enabling to simulate non-harmonic waves. The physical problem of a ship sailing at full sea, has been translated into a mathematical model. Within this model, some non-linearities arise. Under the assumption that the motion of the ship and the waves are small, these non-linearities are removed. Based on this same assumption a perturbation series is introduced, used to derive formulae for the drift forces and moments. Furthermore we assume that the stationary waves generated by the ship do not influence the drift forces substantially. Thus we approximate the stationary fluid flow by the double-body flow, neglecting the stationary waves. To solve the mathematical model we developed a numerical algorithm. This algorithm is based on the boundary-element method. This method only considers values of the quantities on the boundary of the computational domain. Using this method, the governing differential equation is written as an integral equation, which is discretized assuming constant quantities over each element. Then it is shown that the algorithm suggested in literature is unstable when including forward speed. A new algorithm is developed, based on combining the integral equation with the boundary conditions. It is shown that this algorithm is stable for all relevant speeds and grid sizes. To test the numerical algorithm, two test cases are considered: a cylinder of infinite length, and a floating hemisphere. Results are obtained for the hydrodynamic coefficients and the drift forces using simulations of harmonic waves. The agreement between our results and results found in literature based on frequency-domain approaches, is very good. Convergence of the algorithm, both with respect to time and space discretization, is shown for the case of the cylinder of infinite length. Results are presented for a commercial supertanker, for speeds of 0, 5 and 10 knots. It appears that the main hydrodynamic coefficients are accurate; some of the coupling coefficients may lack accuracy. The maximum value of the horizontal drift force increases considerably with forward speed. For low speed the accuracy of the drift forces is satisfactory. For higher speeds the accuracy still leaves to be desired. The fact that we solve the equations in the time domain, allows us to study general waves. Calculations are shown for a signal consisting of several harmonic waves. From these calculations, the step-response function and hydrodynamic coefficients are determined. The results are compared with results obtained using one single harmonic wave; the greement is measurements appears to be better. Furthermore, solving the equations in the time domain makes it possible to calculate the slow-drift forces of an object, both in deep and shallow water. Calculations were performed for the case of the infinitely long cylinder. It appears that the finite depth has a large influence on the slow-drift forces. For future research the following recommendations are made:

- A better absorbing boundary condition has to be developed in order to reduce the grid size.

- The double-body potential can be calculated more accurately using a higher-order panel method.

- The mesh used in the calculations should be refined.

- The matrix equation can be solved more accurately using preconditioning or iterative techniques, or by double-precision calculations.

- For higher velocities, linearization should be performed around the stationary wave field, approximated by the wave height due to the

double-body potential.

In this thesis we study the effect of the forward speed of a ship on the hydrodynamic forces, in particular the drift forces. Previous studies, restricted to zero forward speed, were based on the assumption of harmonic waves, thus reducing the time-dependent equations to frequency-dependent equations. In this study the actual time-dependent equations including forward speed, are solved, enabling to simulate non-harmonic waves. The physical problem of a ship sailing at full sea, has been translated into a mathematical model. Within this model, some non-linearities arise. Under the assumption that the motion of the ship and the waves are small, these non-linearities are removed. Based on this same assumption a perturbation series is introduced, used to derive formulae for the drift forces and moments. Furthermore we assume that the stationary waves generated by the ship do not influence the drift forces substantially. Thus we approximate the stationary fluid flow by the double-body flow, neglecting the stationary waves. To solve the mathematical model we developed a numerical algorithm. This algorithm is based on the boundary-element method. This method only considers values of the quantities on the boundary of the computational domain. Using this method, the governing differential equation is written as an integral equation, which is discretized assuming constant quantities over each element. Then it is shown that the algorithm suggested in literature is unstable when including forward speed. A new algorithm is developed, based on combining the integral equation with the boundary conditions. It is shown that this algorithm is stable for all relevant speeds and grid sizes. To test the numerical algorithm, two test cases are considered: a cylinder of infinite length, and a floating hemisphere. Results are obtained for the hydrodynamic coefficients and the drift forces using simulations of harmonic waves. The agreement between our results and results found in literature based on frequency-domain approaches, is very good. Convergence of the algorithm, both with respect to time and space discretization, is shown for the case of the cylinder of infinite length. Results are presented for a commercial supertanker, for speeds of 0, 5 and 10 knots. It appears that the main hydrodynamic coefficients are accurate; some of the coupling coefficients may lack accuracy. The maximum value of the horizontal drift force increases considerably with forward speed. For low speed the accuracy of the drift forces is satisfactory. For higher speeds the accuracy still leaves to be desired. The fact that we solve the equations in the time domain, allows us to study general waves. Calculations are shown for a signal consisting of several harmonic waves. From these calculations, the step-response function and hydrodynamic coefficients are determined. The results are compared with results obtained using one single harmonic wave; the greement is measurements appears to be better. Furthermore, solving the equations in the time domain makes it possible to calculate the slow-drift forces of an object, both in deep and shallow water. Calculations were performed for the case of the infinitely long cylinder. It appears that the finite depth has a large influence on the slow-drift forces. For future research the following recommendations are made:

- A better absorbing boundary condition has to be developed in order to reduce the grid size.

- The double-body potential can be calculated more accurately using a higher-order panel method.

- The mesh used in the calculations should be refined.

- The matrix equation can be solved more accurately using preconditioning or iterative techniques, or by double-precision calculations.

- For higher velocities, linearization should be performed around the stationary wave field, approximated by the wave height due to the

double-body potential.