This paper examines the theory and computational methods behind predicting the linear unsteady motion of a ship with steady forward speed in waves. The focus is on the wave exciting force impulse-response function as computed via the transient free-surface Green function. The linear equation of motion for a ship in waves was first written in a rational form, using the concept of the impulse-response function, by Cummins (1962). Some years later King et al (1988) added the corresponding wave exciting force in its appropriate convolution form. We extend this work by clarifying the definition of the impulsive incident wave in following seas, and show it to be easily computable. Continuing truncated calculations towards infinite time becomes especially important in following waves, and the method suggested by Bingham et al (1994) is employed here. A novel filtering scheme is also introduced to prevent short wave contamination of the solution. These developments allow calculations in following waves to be presented for the first time using this approach. The integral equation formulation of the linear seakeeping problem is reviewed in some detail, and the relevant equations derived. Transient Haskind relations for bodies with forward speed are also derived although, like their frequency-domain counterparts, these are only approximate. Computed, first-order exciting forces and response-amplitude operators for real ship geometries, in head and following seas, are presented that demonstrate the usefulness of the transient approach for the diffraction problem.
Unsteady RANS simulation of three-stage centrifugal pumps with different impeller-diffuser gaps
