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4 METHODS TO CALCULATE THE 2D POTENTIAL COEFFICIENTS Objective of the Chapter: Presentation of methods to numerically solve the radiation two-dimensional boundary value problem. Two-dimensional radiation results include: Radiation potential in the fluid Velocity of the fluid Free surface elevation Hydrodynamic pressures Added masses and damping coefficients NUNO FONSECA - IST 4.1 – General Description of 2D Methods (A) Methods Based on the Conformal Mapping and Ursell Theory (a1) Circular Cylinder: Ursell (1949) solved the linear boundary value problem of the circular cylinder oscillating on the free surface. The velocity potential is represented as a sum of an infinite set of multi poles satisfying the free surface boundary condition and each being multiplied by a coefficient in order to satisfy the body boundary condition. Reference: Ursell [1949], On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid, Quarterly Journal of Mechanics and Applied Mathematics, Vol. II, 1949. (a2) Lewis transformation Several researchers combined the Lewis conformal mapping transformation with the Ursell method to obtain the solution for more realistic ship cross sections. Lewis transformation: Transforms the ship cross sections to the circle using a scale factor and two mapping coefficients The mapped cross section will maintain the draft, the breadth and the area of the ship cross section References: Tasai [1959], On the Damping Force and Added Mass of Ships Heaving and Pitching, Technical Report, Research Institute for Applied Mechanics, Kyushu University, Japan, 1959, Vol. VII, No 26. Tasai [1961], Hydrodynamic Force and Moment Produced by Swaying and Rolling Oscillation of Cylinders on the Free Surface, Technical Report, Research Institute for Applied Mechanics, Kyushu University, Japan, 1961, Vol. IX, No 35. (a3) Multi-parameter Conformal Mapping More than two mapping coefficients can be used to represent the ship cross sections more accurately. Example reference: Tasai [1960], Formula for Calculating Hydrodynamic Force on a Cylinder Heaving in the Free Surface, (N-Parameter Family), Technical Report, Research Institute for Applied Mechanics, Kyushu University, Japan, 1960, Vol. VIII, No 31. (B) Frank Close Fit Method • Solution is valid for arbitrary cross sections, partly or fully submerged • The velocity potential is represented by a distribution of sources over the mean submerged cross section • Green functions satisfying the linear free surface boundary condition are used to represent the velocity potential of unit strength sources • The density of the sources is an unknown function to be determined from integral equations obtained by applying the body boundary condition Reference: Frank [1967], Oscillation of Cylinders in or below the Free Surface of Deep Fluids, Technical Report 2375, 1967, Naval Ship Research and Development Centre, Washington DC, U.S.A. Numerical Solution for Frank close fit method Boundary value problem • ‘Q’ is a point on the surface of the body – fundamental source point or influencing point • ‘P’ is a point in the fluid domain – the influenced point Applying Green theorem to the fluid volume together with the boundary conditions one obtains the Fredholm integral equation of the second kind: Numerical Solution for Frank close fit method The boundary integral equation is solved numerically: • The contour of the cross section is divided into a finite number of segments, N • On each segment the distribution of sources have an unknown constant strength, while the normal velocity is known • The former equation reduces to a set of 2N linear algebraic equations • The major difficulty is on the evaluation of the Green function • The numerical solution of the Green function includes some instabilities at certain frequencies where disparate results arise (irregular frequencies) Cross section of a fishing vessel discretized for application of the Frank method 0.0 -1.0 -2.0 -3.0 -4.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Advantages and Disadvantages of Conformal Mapping Methods • Smooth solutions over all frequency range (no irregular frequencies) • Sharp corners are not well represented • Sections with very low sectional area coefficient may not be well represented • Do not deal with fully submerged cross sections (like bow bulbous) Advantages and Disadvantages of Frank Close-Fit Method • Applicable for arbitrary cross sections, including sections with sharp corners and fully submerged sections • Have numerical problems at certain frequencies where the solution diverges (irregular frequencies). There are ways of reducing this problem 4.0 3.0 Frank B33 2.0 Conformal mapping 1.0 0.0 0 5 10 15 ω Lpp / g Figure 3 Non-dimensional heave damping coefficient 20