185 



repeated iterations may bring forth a reasonable 

 solution. 



The flow in the C-region can be determined by 

 taking a new streamline coordinate system O-x-x^x,, 

 where the x -axis coincides with DSL and the x^- 

 axis is normal to the DSL surface. 



By the finite-difference scheme, Eqs. (48) and 

 (49) are transformed into tridiagonal linear equa- 

 tions for k: > 2; 



0)2(1, j,k-l) - 2C(i, j,k)a)2(i,k,k) + 0)2 (i- 3 A+D 



= A2(i,j,k) , (80) 



M3(i,j,k-1) - 2C(i, j,k)U3(i,j,k) + 013(1, j,k+l) 



= A3(i,j,k) , (81) 



where ti)2(iij,k) etc. denote those values at xi=xi-i^, 

 ^2=^2 j' ^"'^ ^3=^3k' 



C(i,j,k) 



1 + 



AC^ 



VgAS 



qi (i, j ,]<) , 



A2(i,j,k) = - M2(i-1, j,k-l) + 



2u2 (i-1 , j ,k) 



A3(i,j,k) = 

 20)3 (i-l» j »k) 



AC" 



qi(i-l,j,k) 



013(1-1, j,k-l) + 



A?^ 



VgAS 



qi{i-l,j,k) 



o)2(i-l,j,k+l) , 



o)3(i-l,j,k+l) 

 (82) 



and AC, At; are short segments in the xj , X3 directions. 



Equations (80) and (81) can be solved by the 

 forward marching procedure if the velocity profile 

 of q]^ is given at the separation position. Here the 

 value of vorticity at k=l, on DSL, is made equal to 

 that at k=2 . 



Once the vorticity distributions are obtained 

 throughout, the boundary layer and wake, say V, 



velocity distributions can be calculated as induced 

 velocity of vorticity by invoking Biot-Savart' s law; 



1 01 ' 0) i' 



a,(x,y,z) = V X -^ /// ( i-)dx'dy'dz' 



V 4Tr V r ri 



I 



where 01 j is the mirror image of oi' whose components 



are oij;, o) , -oi^ and 



r2 = (x-x')^ + (y-y')2 + (z-z')2 , 



r2 = (x-x')2 + (y-y')2 + (z+z')2 . 



(84) 



Because Eq. (83) gives the viscous component of 

 velocity, the potential component should be added 

 to qv 



In the present calculation, DSL is determined 

 from experiments for the first iteration; it con- 

 sists of line segments, departing at Xs=0.9 and 

 reattaching at Xj-=1.1 (see Figure 18). The stream- 

 wise velocity qj in Eq. (82) is given by a quadratic 

 function of ^ which is equal to Uj at the outer edge 

 and to 2/3 Uj on DSL. The integral intervals for 

 X and C are 0.005 and 0.0025 respectively. 



In order to obtain the velocity distributions at 

 x=1.025, the region covering from x=0.8 to x=1.4 

 is integrated in Eq. (83). Here, 300-times molecular 

 kinematic viscosity is used as Vg. 



The boundary layer and the potential flow calcu- 

 lations are carried out in the same manner as in 

 Section 3. 



In Figure 19, typical calculated results of the 

 first iteration for MS-02 are shown compared with 

 experiments. The ship speed is Fn=0.1525 and the 

 corresponding equivalent Reynolds number is about 

 8700. Here the calculations for the D- and E-regions 

 have not been carried out; therefore both regions 

 are excluded from the vorticity-integrating region V. 



Satisfactory results are obtained, as far as 

 C-region is concerned, especially in u and w. The 

 velocity v is always underestimated, in other words, 

 overestimated in the negative direction; this may 



u.v.w 







ASSUMED 



DSL 



/T. • 



Z°-C.015 -0.4 



♦ ♦ • 



Z»-0.03 -0.4 



-0.2 



-0.6 



■i:^^^- 



FIGURE 19. Velocity distribu- 

 tions in wake at (1/8) L AFT from 

 A. P. (MS-02). 



