170 



were carried out from this standpoint. But they 

 are only the beginning of research on ship wakes 

 and many future problems were pointed out, especially 

 requirements for further experimental studies . 



The present authors are firmly convinced that, 

 for such viscous flow problems, marriages of experi- 

 mental and theoretical studies are primarily impor- 

 tant in order to make further progress. Because 

 of this, the present paper is divided into two parts; 

 experimental studies on ship boundary layers and 

 wakes (Section 2 and 4) , and theoretical studies 

 and numerical calculations (Section 3 and 5) . 



TABLE 1 PRINCIPAL DIMENSIONS OF MODELS 



Coordinate Systems and Models Used 



Two coordinate systems are employed throughout the 

 present paper. One is the right-hand linear coordi- 

 nate system, 0-xyz , whose origin is at midship and 

 on the waterplane and the oncoming flow, Ug , is in 

 the x-direction. The other is the streamline coor- 

 dinate, X1X2X3,- the curves of constant X2 coincide 

 with potential flow streamlines on hull surface and 

 X3 is normal direction to hull surface (Figure 1) . 



All quantities are dimensionless by half ship 

 length I (=L/2) , ship speed Uq , and fluid density p, 

 unless specified in another form. 



For the present research three ship models, 

 GBT-125, GBT-30, and MS-02 were used whose body 

 plans with potential streamlines and principal di- 

 mensions are shown in Figure 2 and Table 1. 



GBT-125 and GBT-30 are practical tanker ship 

 models, similar in geometry to each other. GBT-125 

 is a double model and was used under submerged con- 

 ditions for studies of boundary layer flow. MS-02, 

 which was used for the studies of wake flow, has a 

 rather simple stern form; the framelines are ellip- 

 tic and given by the equation. 



where 



X > a (a=0.4) 



yo = fciQ 



x-a 



1-a 



0.7 



(1) 



(2) 



and bo is the half breadth of the waterplane at 

 X = 0.4 (S.S.3) and d is the draft. The remainder, 

 (x < 0.4) , has a practical hull form. This is be- 

 cause the practical stern form produces a very com- 

 plicated stern flow, e.g., an intensive longitudinal 

 vortex, not suitable for the present investigations. 



Experiments were carried out in the circulating 

 water channel and the towing tank of Hiroshima 

 University . 



NOTATION 



L,S. ship model length and half length 



b ship model breadth 



Cj^ block coefficient of the ship model 



d ship model draft 



p density 



V kinematic coefficient of viscosity 



Vg eddy viscosity coefficient 



g gravity acceleration 



Uq velocity of oncoming flow, ship speed 



Fj^ Froude number =Uo//gL 



Re Reynolds number =UoL/v 



£ small parameter for asymptotic ex- 

 pansions = Rg~ /° 



x,y,z orthogonal linear coordinates 



xi,X2,X3 orthogonal curvilinear coordinates 



£,ri,C distances along xi,X2,X3 coordinates 

 hi,h2,h3 corresponding metric coefficients 

 Ki,K2 convergences defined by Kj = 



1 3h2 , „ ^ _ _1 3hi 



hih23xi ^ hih2 3x2 



q 



^v 

 u, v,w 



qi'q2'q3 



qi»q2/q3 



Ui,Vi,Wi 



^0'Yi'Yi'"i 



U^.V; ,Wi 



* * * 



U*,Vi,Wi 



(i=l,2 ) 



Ui,Vi,Wi 



(i=l,2,...) 



'^x'^y'^z 



^i ,102,103 



Mgi,io^i,a)j,i 



(i=l,2 ) 



FIGURE 1. Coordinate systems. 



normalized distances for vorticity 

 diffusion region, separated re- 

 tarding region, and viscous sub- 

 layer respectively 

 velocity vector 



viscous part of velocity vector 

 velocity components in x,y,z direc- 

 tions excluding uniform flow 

 mean velocity components in xi,X2,X3 

 directions 



fluctuating velocity components in 

 xi,X2,X3 directions 



resultant velocity at boundary layer 

 edge 



velocity components at boundary layer 

 edge in xi,X2,X3 directions 



asymptotic terms of normalized mean 

 velocity for vorticity diffusion 

 region, separated retarding region, 

 and viscous sublayer region 

 asymptotic terms of normalized fluc- 

 tuating velocity for separated re- 

 tarding region 

 vorticity vector 



vorticity components in x,y,z di- 

 rections 



vorticity components in xi,X2,X3 

 directions 



asymptotic terms of normalized vor- 

 ticity for vorticity 

 diffusion region 

 pressure 



