## Potential Flow Panel Methods for the Calculation of Free-surface Flows with Lift

Janson, Carl-Erik.

PhD Thesis. Chalmers University of Technology, Göteborg, n.d. 1997.

### Abstract

Two non-linear Rankine-source panel methods are developed and implemented in the same computer code. The first method uses a four-point upwind operator on the free-surface to compute the velocity derivatives and to enforce the radiation condition while the second method uses an analytical expression for the velocity derivatives and a collocation point shift one panel upstream to prevent upstream waves. First and higher order panels can be used for both methods. Source panels raised a small distance above the free-surface collocation points can be used together with both the four-point operator and the analytical method. A small upstream shift of the free-surface collocation points is also introduced to further enforce the radiation condition for the four-point operator. Lifting surfaces can be used together with the non-linear methods. The lift force is introduced as a dipole distribution on the lifting surfaces and on the trailing wake together with a flow tangency condition at the trailing edge of the lifting surface. The methods are compared by numerical computations in three dimensions and by a two-dimensional Fourier analysis to compare dispersion and damping. The Fourier analysis shows that the analytical method with raised panels has the best dispersion and damping properties of the methods investigated. A combination of raised source panels and a local upstream collocation point shift improves the dispersion for the four-point operator. In the present implementation both methods perform well for the Series 60 hull, C.BETA.=0.6, but for the Dyne tanker, C.BETA.=0.85, which may be of more practical interest the four-point operator gives the best results. Good agreement with measurements of the wave pattern and a possibility to include the interaction between the free-surface and the lift produced close to the surface is demonstrated. The three-dimensional numerical computations show that the non-linear convergence is improved when raised panels are used. First and higher order free-surface panels perform equally well in the present computations. Non-linear and linear computations are compared. Both the non-linear wave amplitude and the phase agree better with measurements than linear computations. This shows that it is important to satisfy the free-surface boundary conditions on the wavy free-surface. The improved geometry description of the hull due to the intersection with the wavy free-surface may also play an important role for the wave prediction. A method for automatic shape optimization of ship hulls is also developed. The wave resistance is computed from a linear potential-flow method and the viscous resistance is computed from a boundary layer and a Navier-Stokes method. The shape optimization worked well from a computational point of view but limitations in the ability to predict the resistance are identified for the computational methods.