The variables in Equation (35) and auxiliary formulas of Equation (37) are 

 defined in terms of the principal unknown momentum and displacement thick- 

 nesses 0,,, 0,2> ©21' *^22' ^1' ^'^'^ ^2 ^^ ^^® appendix. It is assumed that 

 the values of W are known at a given station (^ = ^ for all values of 6 and 

 that the boundary layer is to be computed between (j)_ and a final station <p 

 downstream of cf) . Because of symmetry, it is only necessary to solve 

 Equation (35) between 6 = - tt/2 and = 0. An (N+1) x (M+1) grid with 

 spacing Acf) and A0, respectively, is superimposed on the region [(<}), 6) | (})„< 

 (j)<(J) ,-TT/2<e<0]. For simplicity of notation, W"^-' will denote W((t)Q+iA(})Q, 



jA0). For i = 0, 1,...,N and j = 0, 1, . . . ,M. Also D^^ , B^^ , and C^^ will 

 denote the values of the matrices B, C, and D, respectively, at the point 

 ((()Q+iA(!),jA0). 



Equation (35) is hyperbolic if there is a nonzero crossflow. The 

 three characteristics at a point (4),0) lie between the angles a and a + B; 

 the equation is parabolic at a point if 3 is zero, as the characteristics 

 have the same tangent or, equivalently, the same direction. Along a line 

 of flow symmetry, such as at the keel or at the waterline on a double 

 model, the crossflow is zero and the governing Equation (35) is parabolic. 

 Consequently, in the present case of the double hull models. Equation (35) 

 is a mixed equation, that is hyperbolic and parabolic at different points 

 of the region of integration unless the crossflow is everywhere zero. In 

 this latter case. Equation (35) reduces to parabolic form. A solution 

 method which is applicable to both parabolic and hyperbolic equations must, 

 therefore, be used to solve Equation (35) for double hull models or hulls 



for which the crossflow is zero or very small everywhere. 



19 

 The O'Brien et al. , implicit finite difference scheme is used to 



solve Equation (35) . It is a stable scheme for any positive grid spacing 



ratio r = A0/A(}) and is applicable to both hyperbolic and parabolic 



equations. It consists of a one-step forward difference in the (})-direction 



and a central difference at the i + 1 step in the 0-direction. In this 



numerical integration scheme. Equation (35) is approximated by the equation 



^(^^)^lj^(^^^^^^#^)=c« 



21 



