193 



system. The lines formed by the intersection of 

 the planes x = constant and z = constant with the 

 hull constitute the nonorthogonal coordinate net 

 on the surface, and the third boundary-layer coor- 

 dinate is taken as the distance normal to the surface 

 in accordance with first-order boundary-layer approx- 

 imation. 



Since the spline-fitting also yields derivatives, 

 the metric coefficient and the geodesic curvatures 

 of the coordinate lines can be calculated from the 

 formulas given below. 



The metric coefficients: 



2 

 hi 



= 1 



The angle between the coordinate lines: 



hih 



1"2 



3y 



{37a) 



{37b) 



{38) 



The geodesic curvature of the z = constant line: 



{39) 



The geodesic curvature of the x = constant line: 



K2 



hihlisine 



'9z 9y By 9z 

 Sz 8z2 3z 3z2 



(40) 



The other parameters K12 ^nd K2 1 are calculated from 

 {6) . It may be noted that Kj and K2 can also be 

 obtained from (5) . This provides a check on the ex- 

 pressions given by (39) and (40) . 



In the boundary-layer calculations, we need the 

 invisid velocity components along the surface 

 coordinates. Let V be the total velocity vector 

 on the hull, (u,v,w) the corresponding velocity com- 

 ponents in the Cartesian coordinates, and {ug,we) 

 in the adopted surface coordinates . As can be seen 

 from Figure 3 , 



(41) 



(42) 



Here tj and t2 are the unit tangent vectors along 

 X and z coordinates and are given by 



ti = r- 



t2 



h2 



3y 



J + 



3z 



(43) 

 (44) 



FIGURE 3. Resolution of the velocity components. 



With the definition of V and with the use of (43) 

 and (44), Eqs . (41) and (42) can be written as 



„+vl^) +w^ 



v^^^ .wfP 



\3z 



3z 



(45) 



Bin^e hj 



V m + w ^ 



3Y 



3x 



3z 



(46) 



zj 



4. 



NUMERICAL METHOD 



We use the Box method to solve the boundary-layer 

 equations given in Section 2. This is a two-point 

 finite-difference method developed by Keller and 

 Cebeci . This method has been applied to two- 

 dimensional flows as well as three-dimensional 

 flows and has been found to be efficient and accu- 

 rate . Descriptions of this method have been pre- 

 sented in a series of papers and reports and a 

 detailed presentation is contained in a recent 

 book by Cebeci and Bradshaw (1977) . 



In using this numerical method, or any other 

 method, care must be taken in obtaining solutions 

 of the equations when the transverse velocity com- 

 ponent, w, contains regions of flow reversal. Such 

 changes in w-profiles will lead to numerical in- 

 stabilities resulting from integration opposed to 

 the flow direction unless appropriate changes are 

 made in the integration procedure . Here we use the 

 procedure developed by Cebeci and Stewartson (1977) . 

 In this new and very powerful procedure, which fol- 

 lows the characteristics of the locally plane flow, 

 the direction of w at each grid point across the 

 boundary layer is checked and difference equations 

 are written accordingly. At each point to be calcu- 

 lated, the backward characteristics which determine 

 the domain of dependence, are computed from the 

 local values of the velocity. Since the character- 

 istics must be determined as part of the solution 

 a Newton iteration process is used in the calcula- 

 tion procedure to correctly determine the exact 

 shape of the domain of dependence. 



To illustrate the basic numerical method, we shall. 



