192 



y _ 



26 



(f") 

 ^ w' 



•■ref 





+ 2 ^f;;g;;cose (35) 



3. COORDINATE SYSTEM 



Since, in general, a ship hull is a complicated non- 

 developable surface, a Cartesian coordinate system 

 is not suitable for boundary-layer calculations. 

 Most existing merchant and naval vessels possess 

 the following features: a flat bottom [y = f {x,z) 

 is not a single-valued function] ; a bottom which is 

 not parallel to the water surface; and a bow which 

 has a submerged bulb extending toward the origin. 

 In addition, the problem is further complicated by 

 the existence of a free surface, corresponding to 

 the water level of a partly-submerged hull. The 

 chosen coordinate system must be sufficiently general 

 to allow these various features to be represented in 

 the boundary-layer calculations . 



The streamline coordinate system is superficially 

 attractive but the determination of the streamlines, 

 the orthogonal lines, and the associated geometrical 

 parameters requires considerable effort. They are 

 dependent on the Froude number, and also on the 

 Reynolds number if the displacement effect is taken 

 into account. Consequently, and in addition to 

 being hard to compute, this coordinate system be- 

 comes uneconomical to use when the effect of the 

 Froude number and the Reynolds number are to be 

 systematically examined. 



A desirable requirement of a coordinate system 

 for the boundary-layer calculations is that it be 

 calculated only once. Miloh and Patel (1972) pro- 

 posed an orthogonal coordinate system which depends 

 only on the body geometry and is calculated once 

 and for all. This coordinate system has been applied 

 by Chang and Patel (1975) to boundary-layer calcula- 

 tions on two simple ship hulls: ellipsoid and double 

 elliptic ship. One of the coordinates is taken as 

 lines of X = X = constant and the other as z(x,z) = 

 constant, which is orthogonal to x = constant lines 

 everywhere on the ship hull, and is obtained from 

 the solution of the differential equation 



dz 

 dx 



1 + f2 

 z 



(36) 



Here y = f{x,z) defines the ship hull, and (x,y,z) 

 denote the Cartesian coordinates. The major ad- 

 vantage of this coordinate system is its simplicity. 

 Because one of the coordinates is subject to the 

 condition (35) , there is no guarantee that the 

 boundaries of the ship hull are coincident with the 

 coordinate lines . Furthermore , for a ship with flat 

 bottom for which y = f(x,z) is not a single-valued 

 function, one of the coordinates cannot be calcu- 

 lated from (36). The coordinate system is limited, 

 therefore, to some special geometries only. 



In this study we adopt a nonorthogonal coordinate 

 system similar to that developed by Cebeci, Kaups , 

 and Ramsey (1977) for arbitrary wings. It is based 

 on body geometries only and, hence, it is calculated 

 once and for all. In addition, the system can deal 

 with the peculiar features of most merchant and 

 naval vessels discussed previously. The details 



of this coordinate system are described briefly 

 in the following paragraph. 



Now consider the ship hull as given in the usual 

 Cartesian coordinate system; that is, x along the 

 ship axis, y and z in the cross-plane (see Figure 

 1) . We select x = x = constant as one of the co- 

 ordinates and the other coordinate , z , lies in the 

 yz-plane. Because the coordinate system is non- 

 orthogonal, we are free to select the values of z 

 in the plane to satisfy the condition that the 

 boundary lines of the ship hull are coincident with 

 z = constant coordinate lines. There are several 

 ways of finding the z-values. Here z is determined 

 by mapping each yz crossplane into a half or hull 

 unit circle depending on whether the crossplane in- 

 tercepts the free surface or is completely submerged. 

 The polar angle, normalized by it or 2Tr on the unit 

 circle, is taken as z-values. The z-values then 

 range from to 1 on each crossplane. The advantage 

 of the mapping method is that equi-interval, z = 

 constant coordinate lines are automatically concen- 

 trated in the region of large curvature where the 

 boundary-layer characteristics are expected to vary 

 greatly. Hence the number of z = constant coordinate 

 lines can be reduced without loss of accuracy. 



There are several methods available for the map- 

 ping of an arbitrary body onto a unit circle. Here 

 we use the numerical mapping method developed by 

 Halsey (1977) . It makes full use of Fast Fourier 

 Transform techniques and has no restrictions on the 

 shape of the body to be mapped. To map a smooth 

 crossplane onto a unit circle, the procedure is 

 fairly easy. If there are inner corner points, or 

 trailing-edge and leading-edge corner points (see 

 Figure 2) caused by the reflection of the cross- 

 plane, they must be removed before mapping is per- 

 formed to improve numerical accuracy and to provide 

 rapid convergence. The inner corner points are 

 rounded off by using Fourier series expansion tech- 

 nique and the leading-edge and/or trailing-edge 

 corner points are removed by using the Karman- 

 Trefftz mapping. For details see Halsey (1977) . 



To use the mapping method to find the coordinate 

 system, it is only necessary to define the ship hull 

 as a family of points in the x = constant planes, 

 to locate the intersection of the ship hull and the 

 free surface, and to indicate whether corner points 

 exist. The data in each plane is then mapped into 

 a unit circle as y vs z and z vs z and interpolated 

 for constant values of z. Another set of spline 

 fits, in the planes z = constant for y vs x and z 

 vs X, completes the definition of the coordinate 



TRAILING EDGE / 

 CORNER POINT—' 



\ , REFLECTION OVER W.L. 



\ 



\ 



LEADING-EDGE 

 \\ CORNER POINT 

 \\. 



W.L. 



INNER CORNER POINT 



FIGURE 2. Notation of corner points used in the 

 mapping procedure. 



