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As a general comment, the present results by Dr. Zou seem to confirm the main conclusions of the paper by Dr. Landrini and myself which was presented at the Workshop on Water Waves and Floating Bodies (10th WWWFB) held in Oxford last year. There and in a subsequent paper [1] we showed some results concerning the steady drift and turning motion of a surface-piercing flat plate described by a linearized free-surface problem according to the Dawson procedure. In particular, by adopting a simplified but still nonlinear wake model, the relevant role of the keel vortex in determining the force coefficients was there emphasized.

I have a few specific comments on the paper included in the proceedings:

  • Concerning both trailing edge and tip vortices, we had used an iterative solution to adjust the direction of the trailing vortices with the “local” velocity. Have you already tried to follow that procedure? If not, in what sense is the model you considered nonlinear?

  • Dealing with the double body linearization, we experienced very sharp gradients of the second derivatives of the double body solution near the intersection between the leading edge of the plate and the undisturbed water plane. Although not adopting a low-pass filter, as suggested by Nakos and Sclavounos [2], we obtain satisfactory values for the convergence. What is your experience on that problem?

  • In the computation, we have found it useful to neglect the influence of the (perturbation) trailing vortex nearest to the free surface. This is because that vortex may generate a jerky behavior of the solution, being too close to the free surface collocation points. How do you handle this problem?

  • In our computations, increasing discrepancies between experiments and numerics were observed in the higher Froude range (see, for example, figure 9 in [3]). Could you please comment on this?


[1] M.Landrini, E.F.Campana, “Steady Waves and Forces About a Yawing Flat Plate,” Journal of Ship Research, 1996. In press.

[2] D.E.Nakos, P.D.Sclavounos, “Kelvin Wakes and Wave Resistance of Cruiser and Transom-Ships, Journal of Ship Research, Vol. 38 , No. 1, pp. 9–29, 1994.

[3] A.DiMascio, M.Landrini, E.F.Campana, “On the Modeling of the Flow past a Free-Surface Piercing Flat Plate, ” 21st Symposium on Naval Hydrodynamics, Trondheim, Norway, this volume.

  • The vortex model used in the present work is an extension of Bolley 's model which is nonlinear in the sense that the trailing vortices leave the plate at some angle to the plane of the plate and thus form a vortex sheet which does not lie in that plane. As the numerical results indicated, this vortex model is efficient to predict the nonlinear dependance of the hydrodynamic forces on drift angle and yaw rate.

  • For the linearized boundary condition on the free surface, derivatives of the double-body velocity potential up to the second order need to be calculated. In my calculation, for each drift angle or yaw rate only one free-surface grid is used. The inmost collocation points on the free surface lie at a small distance away from the intersection of the plate and the undisturbed free surface. I did not investigate the details of the flow near the leading edge.

  • The vortex distribution on the plate is discretized by a quasi-continuous method, resulting in a system of horseshoe vortices. The uppermost horseshoe vortices are between the inmost collocation points on both sides of the free surface and are included in the calculation.

  • In a potential-flow method for lifting problem, a Kutta condition is imposed at the trailing edge to fix the value of lift force which is ultimately due to viscosity. For a surface-piercing plate, the experimentally observed jump in the free-surface elevation across the wake just behind the trailing edge above some critical Froude number is contrary to the Kutta condition of pressure continuity. There, it cannot be expected that the potential-flow method would predict correctly the lift force at range of higher Froude number.

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