613 



9Tr 



which is given in Table 1 also. The calculations 

 (52) for the second approximation have now been accom- 

 plished. 



where 



S(r) 



(e 



-r2 



- 1) + 



4r 



105283 2 

 278460 ^ 



8. ESTIMATE OF X 



The terras involving e in Eq. (2) have their origin 

 in the Reynolds stress terms 



, 114713 i| 3517 5 



1670760 



181 



68544 



247520 



10 



4320 



(53) 



|- (v'w') and |- (w'2), 



where the primes indicate turbulent quantities. 

 The terms were originally on the left-hand side of 

 Eq. (2) . The nonlinear terms on the left-hand 

 side of Eq. (2) can be written as 



The values of S(r) are tabulated in Table 1, from 

 which it can be seen that the maximum absolute 

 value of S occurs at about r = 0.95 and is about 

 0.847. Since S is negative throughout, inspection 

 of Eq. (52) shows that the maximum value of 5i is 

 at 



0.95, 



377 



4 ' 



and its minimum value (negative) at 



— - vw + ■— (W^ 

 3y 3z 



Thus the ratio of 



37 <"^' 

 is the ratio of 



3 

 and — {w'2) 



3z 



r = 0.95, 



^— - (w-^) and -Ew 

 dz ^^ 



The effect of S is to reduce the strenths of the 

 vorticity for the lower-half plane, but to augment 

 them in the upper-half plane, thus to raise the 

 eyes of the vortices . 



Finally, ^fj is to be found from 



^Irin + '^lliti 



Let 



3r 



1 3 



h — -^ + 



2 r 3r 



36 



and this ratio has the magnitude of 



-XW^/W^ 



The magnitude of Wg is 1/677, and the magnitude of 

 Wqj- is 0.267/377, which is the maximum value of Wqv- 

 along the r\ axis. Thus, approximately. 



0.267 (1277) 



(56) 



sin 26 „ , ^ 

 Vj = J — F(r) . 



977 



Then 



+ iF- 

 r 



— F = -S(r). 

 r 



(54) 



Two integrations by the method of variation of 

 parameters (since a complimentary solution of F is 

 simply r^) gives, with due regard for the boundary 

 conditions. 



r^Sdr 



dr 



r^Sdr - r"^ I r Sdr 



(55) 



where s is the square of w'/w. The convection in 

 the bent pliome is like the convection in a two- 

 dimensional plume, since the plume is bent by the 

 wind to a nearly horizontal position. The measure- 

 ments of Kotsovinos (1977) for the plane plume 

 give the value 0.2 to s. This is considered by 

 some people to be too high. But for the problem 

 under investigation s may be even higher, because 

 any swaying or deformation of the vortices would 

 contribute a good deal to turbulence. Thus using 

 0.2 for s in Eq. (56) would overesti/nate X. Using 

 0.2 for s, we obtain from Eq. (2) 



X 



48.5. 



This is probably too high. My estimate of X is 

 that it is somewhere in the range 



30 < A < 40. 



The value 30 for X corresponds to a value of 0.34 

 for s . 



Let us now see what errors would be committed 

 for h, y, and E, by stopping at the second approxi- 

 mation. For X = 30, the errors (in ratio of the 



