189 



sented in Cebeci (1974, 1975) demonstrated that the 

 present eddy-viscosity model allows excellent agree- 

 ment between measurements and calculations but did 

 not include comparison with the three-dimensional 

 boundary-layer measurements of Vermeulen (1971) . 

 Since this data includes a strongly adverse-pressure 

 gradient case which allows a stringent test of the 

 present model, corresponding calculations and com- 

 parisons are reported. 



The calculation method is described in detail 

 in the following section which states the three- 

 dimensional, boundary-layer equations in curvi- 

 linear, nonorthogonal coordinates and describes 

 and discusses the required initial conditions, 

 turbulence model, and transformations in separate 

 subsections. Section 3 is devoted to the coordinate 

 system which is an essential feature of the present 

 method. The numerical method is discussed briefly 

 in Section 4 and calculated results are presented 

 in Section 5 which includes comparisons with the 

 measurements of Vermeulen (1971) and demonstrations 

 of the ability of the method to represent the geom- 

 etry of different hull configurations and to result 

 in realistic velocity and drag characteristics. 

 Siraimary conclusions are presented in Section 5. 



2. BASIC EQUATIONS 

 Boundary-Layer Equations 



geodesic curvatures of the curves z = const and x 

 = const, respectively. They are given by 



Ki = 



K 



2 



The parameters K12 ^"d K21 are defined by 



1 



K 



12 



^21 



-'-l-^lf)^-«(-a.^|t 



K2 .^|i).cose(Ki .^^ 

 h2 3z y V hj dx 



(5) 



(6a) 



(6b) 



_1 



For an orthogonal system 9 = Tr/2 and the parameters 

 Kj , K2 , Kj2f and K2 j , reduce to 



3h 



hjliT 3z 



K2 



3h2 



hjhj 3x 



-■12 



K21 - - K2 



(7) 



(8) 



At the edge of the boundary layer, (2) and (3) reduce 

 to 



u 3u w 3u 



r^ -;— + -^ ^7-^ - Kiu^cote + Kpw^csce + Kiju w 

 hj 3x h2 3z '^ e ^ e ■'■' e e 



The governing boundary-layer equations for three- 

 dimensional incompressible laminar and turbulent 

 flows in a curvilinear nonorthogonal coordinate 

 system are given by : 



Continuity Equation 



P \ cot9csc9 3 / ^ 1 

 jx\ py h2 3zl, p ' ^ ' 



u 3w w 3w 



:; — T + ; — t— ^ - KpW^cotB + KiU^cscB + Kt 1 u w 



hi 3x h2 3z e ^ e ^^ e e 



-;r-(uh2sin9) + Tr-(whisin6) + -— (vh ihosinB) 

 ox dz 3y ' •^ 



x-Momentum Equation 



u 3u w_ _3u 3u 



h 1 3x h2 3z 3y 



Kiu^cot9 + K2W^csc9 + K12UW 



hi 3x \p 

 3 / 3u 



cot9csc9 3 



3z V P 



3y V 3y 



z-Momentum Equation 



u 3w w 3w 3w 



hi "3^ ■^h^Sz'^^ay 



(2) 



K2W^cot9 + Kiu^csc9 + K21UW 



cot9csc9 3 /p 

 hi 3x I p 



3w 

 3y \ 3y 



h2 3z \P 



(3) 



Here, hi and h2 are the metric coefficients and 

 they are, in general, functions of x and z; that is. 



hi = hi (x,z) : 



h2 = h2 {x,z) 



(4) 



Also, 9 represents the angle between the coordinates 

 X and z. The parameters Ki and K2 are known as the 



cot9csce 3_/P 

 3x Vp 



csc^9 3 / P 

 3z Vp 



(10) 



hi 3x \ p / h2 



The boundary conditions for Eqs. (1) to (3) are: 



y = 0: u,v,w = (11a) 



y=6: u = u(x,z),w=Wg(x,z) (lib) 



Initial Conditions 



The solution of the system given by (1) to (3) , 

 siibject to (11) , requires initial conditions on 

 two planes intersecting the body along coordinate 

 lines. In general, the construction of these 

 initial conditions for three-dimensional flows on 

 arbitrary bodies such as ship hulls is difficult 

 due to the variety of bow shapes , which may be ex- 

 tensive and complicated. For this reason, assump- 

 tions are necessary in order to start the calculations. 



In our study we choose the inviscid dividing 

 streamline on which 3p/3z =0 to be one of the 

 initial data line (see Figure 1) . In the case of 

 rectilinear motion of a ship, this streamline runs 

 along the plane of symmetry. Because of symmetry 

 conditions, w and 3p/3z are zero on this line causing 

 (3) to become singular. However, differentiation 

 with respect to z yields a nonsingular equation. 

 After performing the necessary differentiation for 

 the z-momentum equation and taking advantage of 

 appropriate symmetry conditions , we can write the 

 so-called longitudinal attachment-line equations as : 



