Effects of Desingularization and Collocation-Point Shift on Steady Waves with Forward Speed

Transcription

1 Effects of Desingularization and Collocation-Point Shift on Steady Waves with Forward Speed Yonghwan Ki* & Dick K.P. Yue** Massachusetts Institute of Technology, Departent of Ocean Engineering, Cabridge, MA, USA * yki@vfrl.it.edu, **yue@it.edu Introduction After the pioneering work of Dawson ([] for steady ship waves, any variations of panel ethod have been studied for steady ship wave probles. In the early stage of those studies, the application to wave-resistance coputation was of priary interest for practical reasons, and such studies contributed to the births of soe coputer progras, e.g. DAWSON, SHIPFLOW, SWIFT and RAPID. Sclavounos and Nakos ([] introduced a strong background of the Rankine panel ethod based on a rigorous stability analysis for steady wave probles, and also showed the advantage of the higher-order B-spline schee. Raven ([3] introduced the favorable properties of a raised-panel ethod using the siilar two-diensional stability analysis, and Bunnik ([] also included the effects of raising panel in his unsteady stability analysis. In the present study, theoretical and nuerical efforts are given to observe the effects of desingularization and collocation-point shift on the nuerical solution of panel ethod. We perfor a ore coplete three-diensional stability analysis, and a nuerical error is defined to copare the accuracy of different nuerical schees. For validating our theoretical analysis, the nuerical solutions of steady wave proble near a point singularity with forward speed are observed. Stability Analysis of Steady Rankine Panel Method The linear free surface boundary condition with a steady forward speed U is written as Φ Φ U + g =. ( z When the free surface is discretized into unifor rectangular panels of the size (, y, the disturbed freesurface flow (potential Φ or singularity strength σ in a discrete doain can be written as Φ π / π Rˆ i( ul + v y = e dudv. ( (π π / σ π W ˆ Wˆ includes the nuerical dispersion relation. At first, let s consider a constant source distribution ethod such that Figure. Definitions for stability analysis U σ + σ = l ds g l l ds (3 R z l R l where /R is the distance between the field and singularity points. Oitting the details, Wˆ corresponding equation (3 can be written as follows: I z l Wˆ = DI ˆ ˆ + k ˆ ( where k = g /U, and Iˆ πiu J d ˆ j ( j u = e i D (5 j= J p+ ( u / sin( v y / ( z ς u + + δ + v iu sin n q e = n u + v u / v y / n (

2 where u Iˆ p+ + + v + iu δ sin( u / sin( vn y / (7 q ( ς = z u z πe u / n / n v y = u + π /, = v + πn, and p=q=. ς ( = + y is the height of panel surface raised fro the v n free surface and δ ( = is the longitudinal shift of collocation points. Dˆ is the discrete Fourier transfor of finite difference, and d s are the corresponding coefficients. For eaple, the Dawson s -point difference is the case that j J=, J=, and d =/, -5/,, -/. The higher-order B-spline source distribution provides a siilar dispersion j relation, but Dˆ is not necessary and the order of the spline function can be controlled by p and q in equation ( and (7. For instance, p=q= indicates a bi-quadratic distribution. In addition, Î should be replaced to iu Iˆ. In the case of a potential-based ethod, the discrete Fourier transfor of the Green s nd identity should be observed and the desingularization is not applicable. For the potential-based ethod, the details are well described in []. Figure shows the nuerical dispersion and daping of the source ethods with different and, and three different schees are considered. The -ais in this figure is the eact wave nuber, and the y-aes are the wave nuber to appear in actual coputation and daping coefficient relative to the corresponding waves. Therefore, the schees of figure (a and (b can provide one or two nuerical wave coponents depending on, and no solution is also possible. u / π has a range of // in ost practical coputation, and the saw-tooth wave ay appear in this range. As epected, the nuerical daping increases when the collocation points are shifted to upstrea. The nuerical daping shown here is ore thoroughly studied by Ki and Yue ([5]. On the other hand, the desingularization induces longer wavelength than the conventional ethod. These results are consistent with the twodiensional cases of Raven ([3]. and do not affect uch the higher-order schee, ecept for a sall daping due to nonzero. In a practical point of view, u / π less than. is of great iportance u /π.3. eact solution =., =.. =.5, =. =., =.5 =.5, = u /π u /π π I (u / u π I (u / u π I (u / u u /π (= /λ eact u /π (= /λ eact u /π (= /λ eact (a -point conventional (b -point Dawson (c Bi-quadratic B-spline Figure. Nuerical dispersion and daping for different and : source ethod, =,.5, =,.5, = Nuerical Errors Two ajor sources of coputational error are dispersion error, and nuerical daping. The dispersion error causes the difference of wavelength, and the nuerical daping causes the difference of wave aplitude. To observe the overall accuracy of a certain nuerical schee, we define a paraeters such that E / u iu iu γ e e d (, N du γ = N / u u (

3 where u = u / π and u is the eact wave nuber. (, N E γ is an integral of weighted nuerical errors in a range of u γ over N wavelengths. u =/ is the aiu discrete wave nuber to possibly appear, and this wave is socalled saw-tooth wave which is not desirable in the viewpoint of nuerical stability. / u in ( is ultiplied to give ore weight when u is sall. It should be noted that the consistency of a nuerical schee iplies that the nuerator of ( should vanish as u approaches zero. (, N E γ reflects a liited daping effects at large u, but ay be a good inde to copare the overall errors in a specific range of u. Figure 3 copares E (, and (,. E for different nuerical schees. This result can be particularly iportant in a. practical perspective, since γ. is valid in ost nuerical coputations. According to this result, the desingularization ethod sees to reduce the overall nuerical error, and the optiu case depends on the nuerical schee. The 3-point and Dawson s difference have an optiu near.. On the other hand, the higher-order ethod does not have a significant benefit. The effect of collocation-point shift is not significant in this case. The - point conventional difference schee has soe reduction of error due to the nuerical daping copensating the negative daping in this range of γ (see Figure -(a. 5 5, E. (, E. (, 5 E. (,, E. (, E. (, E. (, (a = (b =,.5 Figure 3. E (, and (, E of four nuerical schees; source ethod; =.. E. (, 5 3-pt. conventional -pt. conventional pt. Dawson bi-quadratic B-spline E. (, Applications and Validation The steady wave elevations near a oving point singularity located at (,y,z=(,,-d are obtained using a few different nuerical schees to validate the stability and error analyses. In this coputation, U / gd =. is applied, and the coputational doain covers.5 (upstrea and 3.5 (downstrea ties of eact wavelength in longitudinal direction and twice in transverse direction. The nubers of applied grids are 5 ( u =/, 73 ( u =/5 and ( u =/. Figure copares the wave elevations at y/d=. for different and. As epected, shorter waves are observed in the conventional schee, i.e. when = and =. However, the wavelengths are longer in the case of nonzero. Also significant daping is shown for nonzero. Figure 5 is the elevation contour plots of three solutions, showing slightly different wave angles. A larger angle is found when = and =, but the solution obtained fro the desingularized schee shows an ecellent agreeent with the analytic solution. To understand this trend, we should observe both the longitudinal and transverse wave nubers, (u,v, as shown in Figure. Figure shows the three-diensional surfaces of ( u, u, v for the two cases of Figure 5. For a given eact wave nuber ( u and (e.g. bold line on botto, the conventional ethod induces saller u and larger v (contour on upper surface than the desingularized case (contour on lower surface. This causes shorter longitudinal waves and longer transverse wave, consequently the larger wave angle. Figure 7 copares the nuerical errors obtained fro the source ethod (, and E predicted by the stability analysis. The nuerical errors obtained fro the source ethod is defined as.5 follows:

4 =. =. analytic solution =. =. =. =. analytic solution =. =. =. =.3 /d Figure. Wave elevations at y/d=.; -pt. Dawson, 753 grids η η ds S ERSM = ( doain area ( S where η is the wave elevation of eact solution. Although the agnitudes of two errors are not sae (because of different definitions, they show a fair agreeent of the sensitivity on and. Siilar to Figure 3, the iniu errors are found near =. in both cases. analytic =. = v y / π /d.5..3 u y / π u y / π Figure. Wave contours near a oving sink: =., grids Figure. ( u, u, v surface for two cases of Figure E.5 (,.5 5 E RSM (, (a E (b Actual error.5 (, Figure 7. E and actual elevation error; 753 grids.5 References [] Dawson,, C.W. 977 Proc. nd Int. Conf. Nuerical Ship Hydrodynaics. [] Sclavounos, P.D. & Nakos, D.E. 99 Proc. 7 th Syp. Naval Hydrodynaics. [3] Raven, H.C. 99 PhD. Dissertation, Delft University of Technology. [] Bunnik, T. 999 PhD. Dissertation, Delft University of Technology. [5] Ki, Y. & Yue, D.K.P. 3 th Conf. Nuerical Ship Hydrodynaics (subitted

5 th IWWFB Le Croisic (France -9 April 3 Question by : X.B. Chen Thank you for your iportant work which helps e to understand the difficulty of R.P.M. in odelling surface waves. My question is whether you ve studied as well such effects on global values such as the wave resistance which are associated with wave patterns, and how uch are they? Author s reply: There is no doubt that coputational paraeters such as desingularization and collocation point shift affect the wave resistance predictions. The degree this is true depends on physical paraeters such as geoetry and speed. For eaple, Fig. A shows significant difference of wave contours around a Wigley hull obtained with and without desingularization and collocation-point shift. We are in the process of obtaining ore systeatic data on the wave resistance and look forward to publishing these when they are available. =., =. =., =.5. y/l.. /L Fig. A coparison of wave contours around Wigley Hull at Froude nuber