190 



INITIAL CONDITIONS ON 

 A CROSS SECTION l» ^ 



FREE SURFACE 



INITIAL CONDITIONS [i = 01 ON 

 THE PLANE OF SYMMETRY 



FIGURE 1. The nonorthogonal coordinate system and the 

 initial data lines for the ship hull. 



Continuity Equation 



9x 



(uh2sine) + hjsinew^ + — (vhih2sine) = (12 



3y 



x-Momentum Equation 



u 3u 8u „ 7 



Z— T— + V coteKiu^ 



hj dx 3y ^ 



"e -^Ug o „ 3 / 3u 



T— T Kiufcote + — - v-^ - u'v' 



hi 3x ^ e 3y \ 3y 



z-Momentum Equation 

 .,2 



(13) 



3w2 



1. T + 7~ + V T + K? 1 uw., 



hj 3x h2 3y •^^ 2 



^e 3"ze *'ze 



hi 3x ■ h2 + K2lUeW,g + g^^Vg^- (Wv'),)(14) 



3w, 



Where w^ = 3w/3z and (w'v')z = 3(w'v')/3z. These 

 equations are subject to the following boundary con- 

 ditions : 



flow is presumed (since it is unlikely that the flow 

 remains laminar after separation and reattachment 

 with high Reynolds number) . Generation of the 

 initial data for turbulent flows is much more in- 

 volved if there are no experimental data available. 

 It requires sound mathematical and physical judgment 

 and tedious trial-and-error efforts. We shall 

 discuss this aspect of the problem later in the 

 paper . 



Turbulence Model 



For turbulent flows, it is necessary to make closure 

 assumptions for the Reynolds stresses, -pu'v' and 

 -pv'w' . In our study, we satisfy the requirement 

 by using the eddy-viscosity concept and relate the 

 Reynolds stresses to the mean velocity profiles by 



-u'v' 



3u 

 "m 3y ' 



-v'w' = 



3w 



3y 



We use the eddy-viscosity formulation of Cebeci 

 (1974), and define Ej^ by two separate formulas. In 

 the inner region, e^j, is defined as 



(Em'i 



= t2 



3u\^ /3w\2 ^ ^ 



where 



L = 0.4 y [1 - exp(-y/A)] 



ay/ vlj) I '^^' 



(18a) 

 (18b) 



+ 2cose 



(18c) 



In the outer region e is defined by the following 

 formula 



e„ = 0.0168 

 m 



where 



(Uf 



u^)dy 



0: 



= 



(15a) 



"te 

 Ut 



(u| + w| + 2UgWgCOse) 

 (u^ + w^ + 2uwcos0) 



(19) 



(20a) 

 (20b) 



6: 



(15b) 



The other initial data should be selected near 

 the bow of the ship along the line perpendicular to 

 the 2 = const coordinate (see Figure 1) . However, 

 because of the variety of possible bow shapes , 

 approximations are necessary. For a simple, smooth 

 bow section, where curvatures are small and no 

 separation is expected, the flow along the initial 

 line can be successfully assumed to be two- 

 dimensional without pressure gradient, and the 

 governing two-dimensional equations for a flat 

 plate are solved. However, for most general mer- 

 chant ships , the bow section is complicated and 

 flow separation and reattachment are expected be- 

 cause of large curvature variations and adverse 

 pressure gradients; as a consequence, the boundary- 

 layer calculations can only be performed downstream 

 of the attachment line (or point) where turbulent 



The inner and outer regions are established by the 

 continuity of the eddy-viscosity formula. 



Transformation of the Basic Equations 



The boundary-layer equations can be solved either 

 in physical coordinates or in transformed coordinates. 

 Each coordinate system has its own advantage. In 

 three-dimensional flows, the computer time and 

 storage required is an important factor . The trans- 

 formed coordinates are then favored because the 

 coordinates allow larger steps to be taken in the 

 longitudinal and transverse directions. 



We define the transformed coordinates by 



/ "e \ r 



x=x, z=z, dn= [ dy, Si = / h,dx 



V vs 1 y ^ i 



(21) 



