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Fig. 6. Relative error of transverse (left) and longitudinal (right) pressure force on 1/8 of a sphere for patch (continuous) and higher-order method (broken line)

(19)

0 is the constant potential generated by the source distribution δ in the space surrounded by Sb, i.e. inside the body. δ is the ‘eigen potential' following from the homogeneous integral equation

(20)

Eq. (20) has a solution δ which has non-zero values everywhere on Sb. Eq. (20) is also desingularised:

(21)

The nonsingular integrals in (21) and (19) are evaluated by Simpson 's rule using 9-knot panels (8 on the circumference, one in the center), to obtain a system of linear algebraic equations for the potential at each knot. Numerical interpolation and differentiation over the panels gives velocities, velocity derivatives, and pressures on Sb.

For a sphere in uniform flow, Fig.6 compares relative errors of the pressure force on 1/8 sphere between the higher-order and the patch method. For both methods, the mesh consisted of quadrilaterals bounded by meridians and latitude circles with uniform angular spacing, poles being at the stagnation points. (Results for the patch method given in Fig.6 differ somewhat from Table III because there a mesh of nearly uniform, equilateral triangles was used.) The higher-order method is, roughly, 10 times more accurate than the patch method which is again much more accurate than an ordinary first-order panel method. For more complicated bodies, however, the difference is expected to be much smaller. Both the maximum error in (not shown) and the pressure force (Fig.6) converge with errors ~ h3.5 to h4, where h is the grid spacing.

References

1. Bertram, V., “Numerische Schiffshydro-dynamik in der Praxis”, rep. 545, 1994, Inst. für Schiffbau, Hamburg Univ., Germany

2. Bertram, V.; Jensen, G., “Recent applications of computational fluid dynamics”, Ship Techn. Res. 41/3, 1994, pp. 131–134

3. Hughes, M.; Bertram, V., “A higher-order panel method for 3-d free surface flows”, rep. 558, 1995, Inst. für Schiffbau, Hamburg Univ., Germany

4. Zou, Z.; Söding, H., “A panel method for lifting potential flows around a yawed ship in shallow water”, 20th Symp. on Ship Hydrodyn., 1994, Santa Barbara

5. Zou, Z., “Calculation of the three-dimensional free-surface flow about a yawed ship in shallow water”, Ship Techn. Res. 42/1, 1995, pp. 45–52

6. Bertram, V.; Yasukawa, H., “Rankine source methods for seakeeping problems”, Proc. Schiffbautechn. Gesellschaft, Springer, Berlin, Heidelberg, New York, Tokyo, 1996

7. Söding, H., “A method for accurate force calculations in potential flow”, Ship Techn. Res. 40/3, 1993, pp. 176–186

8. Hackbusch, W., Multi-grid methods and applications , Springer, Berlin, Heidelberg, New York, Tokyo, 1980

9. Bertram, V., “Ship motions by a Rankine source method”, Ship Techn. Res. 37/4, 1990, pp. 143–152

10. Hess, J.L., “A higher order panel method for three-dimensional potential flow”, NADV-Report, MDC J8519, 1979

11. Kouh, J.S.; Ho, C.H., “A high order panel method based on source distribution and Gaussian quadrature”, Ship Techn. Res. 43/1, 1996, pp. 38–47



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