181 



FIGURE 17. Free-surface flow near the stern (MS-02) 



The quantity e, was first introduced by Stewartson 

 (1969) and e << 1 in case of a large Reynolds number. 



If the X3 coordinate can be assumed to be 

 linear, i.e. , 



h3 = 1, (29) 



the continuity equation and Reynolds equations are 

 written in streamline coordinates as follows. 



aq, aq2 3q3 



hi3xi h23x2 8x3 



Kjqj - K2q2 = 



^^ ^^ ^ -<^^^ - K2qiq2 . K,q^ 



hi 3x 



1 dxi 



8x 



3x3 



3 / P , „ • 2 , 3 



- — r ( hqi ) - - — 



h^dxj p h23x2 



ql<32 



(30) 



Subdivision of the Flow Field 



" a;:" 'isqi + 2K2q;q2 + Ki(q;2-q^2) 

 0X3 



It has been made clear by experimental studies 

 that the separated flow has at least two, quite dif- 

 ferent viscous regions where no single approximate 

 equation of Navier-Stokes equation seems to be 

 valid for both. It can be proposed to subdivide 

 the flow field near the stern into five regions as 

 shown in Figure 18; potential flow region, boundary 

 layer region, vorticity diffusion region, separated 

 retarding region, and viscous sublayer region. 



Their characteristics are as follows. 

 Potential flow region: 



The region where the viscous term can be wholly 

 neglected and only displacement effects should be 

 taken into account. 



Boundary layer region: 



The region where the boundary layer assumption 

 is valid and the backward influence of separation 

 can be neglected . 



Vorticity diffusion region: 



The region where the vorticity, which has been 

 generated in the boundary layer, is diffused con- 

 vectively and viscously. No vorticity is newly 

 generated in this region. Because the dividing 

 streamline is a kind of free-streamline, the pres- 

 sure on it might be constant. 



Separated retarding region: 



The region where the velocity is very small 

 and the turbulence is intensive. Because even a 

 recirculating flow can be observed, the governing 

 equation for this region should be an elliptic type. 

 Viscous sublayer region: 



This is the very thin layer region which just 

 adheres to the hull surface. The molecular viscos- 

 ity is predominant and the velocity profile should 

 satisfy the no-slip condition on the hull surface. 



CALCULATION OF VELOCITY DISTRIBUTIONS IN THE SHIP'S 

 WAKE 



Approximation of Navier-Stokes Equation by Local 

 Asymptotic Expansion 



In order to get appropriate approximations of the 

 Navier-Stokes equation for each region, local asymp- 

 totic expansions of relevant quantities are made, 

 using small parameter £ defined by 



R -1/E 

 e 



"2 I 3x3 ^ ^ 3x2 



qi 3q, , q2 3q2 



3q2 



hi 3x1 hj 3x 



+ rr -5^7 + <53^r- + K2qi - Kiqiq2 



^3x 



h23x2 p 



(£ + q2 ) 



d I I d 

 3x^ '^^qs - hi3xi 



qiq2 



K2 (qi - qa ) + 2Kiqlq2 



^"3 a 



— --— (hi^i) 



q, sq^ . 52, 3q3 



+ q3- 



3q3 



hj 3xi h2 3X2 3x3 8x3 p 



' (^+q3^) 



(31) 



(32) 



hj Sxj 



hih2 



qiq3 



h23X2 



iW-i + Kigjgs + K2q2q3 



3x 



(hiui) - T (h2U2) 



2 dx^ 



(33) 



SEPARATION 

 POSITION 



POTENTIAL FLOW REGION 

 BOUNDARY LAYER 

 VORTICITY DIFFUSION REGION 

 SEPARATED RETARDING REGION 

 VISCOUS SUBLAYER 



(28) 



FIGURE 18. Subdivision of separated flow field near 

 the stern. 



