184 



disappears but its effects are still existing in- 

 directly through turbulence. 



Equation (65) gives the so-called boundary layer 

 approximation. But because cross terms of fluctu- 

 ating components exist in Eqs. (63) and (64), they 

 do not always yield the same type as boundary layer 

 equations which can not predict the recirculating 

 flow observed in experiments. 



satisfied when governing equations are solved, 

 (i) for upstream; 



lim ~ ,r ~ Z. lim ~ ,z ~ ~^ 



g^_„ uo(C,n,?) = Ug, ^_^_^ vi{C,n,U = v^, 



lim - ,= - -, 

 |^-„ "1 (5-1,0 = Wg 



(73) 



Viscous Sublayer (E-region) 



In the viscous sublayer, E-region, the no-slip con- 

 dition must be satisfied on the hull surface. Here 

 the intensity of turbulence may be very small and 

 all the turbulent terms in the Reynolds equations 

 vanish inf initesimally . 



The following asymptotic expansions are assumed 

 from the Blasius solution. 



qi/Uo = eu* + £2u2 + . . . , 



q2/Uo = Ev* + e^v^ + . . . , 



qa/uo = E wi + e W2 + 



(66) 



where the orders of each term are all 0(1). 

 The derivatives are represented by 



hi 3x 



Idxi 



3 

 3x3 



1 _3_ 

 L £3^ 



1 _ 



h23x2 



1 _3_ 

 L £3; 



H 



iei) 



and their orders are 



where Ug, Vg, and Wg are the velocity components 

 in the boundary layer in the xi,X2,X3 directions 

 respectively. 



(ii) far from the hull surface; the solutions 

 should be matched to the solution of the A-region, 

 potential flow. 



(iii) between the C- and D-region; 



uo(C,n,o) = , u (C,n,o) 



lim 



ui (t,r),lj - ?-^ uo(5,n,0 



C=0 



lim 



vi(e,ri,0) = ^^ VI (C,n,C) 



wi {C,n,o) = , w2(C,n,o) 



lim 



wi{S,n,i;) - ij^i{l,~r\,b |~^Q 



(iv) between the D- and E-region; 

 lim 



ui (5,1,0) 



Ui(C,ri,C ) 



lim 



VI (S,n-0) = ^*_^ vi (C,n,i; 



(74) 

 (75) 



(76) 



(77) 

 (78) 



^ = 0(1) 



~ = 0(1) 



0(1) 



(68) 



Substituting the above assumptions into Eqs. 

 (31, 32, 33) , the leading terms are obtained as 

 follows in original variables; 



3q. 



qi 



_L 



3q, 



hi3xi 

 1 3P 



'^h23x2 



+ V- 



32qi 



p hjSxi ^3x2 



3qi 



^3X3 



(69) 



wi {?,n,0) = , w2(C,n,0) = 



lim 



wf (C,n,?*) - C*-L wi(5,n,a U_„ 



°K "="0 



(79) 



The governing equations for the D-region do not 

 close. Some auxiliary equations are required, but 

 this problem is left for future work. 



3qj 



Sq, 



'hi3xi 



h23x2 



3P 



+ v: 



3q2 



P h23x2 ' ''3x2 



3p 

 3x3 



= 



The continuity equation is 

 3qi 3q2 



hi3xi h23x2 



3q3 



3X3 



0. 



(70) 

 (71) 



(72) 



Here the quantities of 0(6^) are omitted. 



These are the boundary layer equations themselves. 

 They must be matched with the solution for the D- 

 region in quite the same manner as the conventional 

 method of boundary layer calculation. 



The following matching conditions should be 



Numerical Calculations for the C-Region 



To solve the derived equations analytically is al- 

 most impossible; this is because not only are the 

 equations non-linear but also the hull surface, 

 where the boundary conditions are prescribed, is 

 very complicated in geometry. Instead, they must 

 be solved numerically. But it may be still more 

 difficult because the calculation should be carried 

 out for all the regions at the same time in order 

 to satisfy the matching conditions. However, this 

 difficulty can be removed by an iteration method; 

 the surface consisting of dividing streamlines (DSL) 

 is given a priori in the beginning as the inter- 

 mediate region between C- and D-regions where the 

 matching is carried out. Of course the surface of 

 DSL can be obtained finally as a solution of the 

 flow field, but the assumption of DSL makes it pos- 

 sible to solve the governing equations in every 

 region almost independently and it is expected that 



