Potential Flow Panel Methods for the Calculation of Free-surface Flows with Lift
Doctoral thesis, 1997

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.


shape optimization

Rankine source





Dawson operator

boundary layer


Method of Moving Asymptotes


sailing yacht

four point operator

surface piercing wing


numerical method

lift force

raised panel method

discrete Fourier transform

collocation point shift

potential flow


free surface

induced drag




analytical method



Carl-Erik Janson

Chalmers, Department of Naval Architecture and Ocean Engineering

Subject Categories

Mechanical Engineering



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1277

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