615 



APPENDIX : 



THE EFFECT OF NEGLECTING THE PRESSURE 



GRADIENT IN CALCULATIONS FOR THE CONVECTION 



PLUME IN A TRANSVERSE WIND 



J . P . Benque 



Electricity de France 



Chatou, France 



where the f is in no manner the same as the f in 

 Eq. (38) of Yih's paper, we have 



f" + 2nf' + af = 0, 

 g" + 2C,g' + bg = 0, 



(A. 5) 

 (A. 6) 



where 



In many previous studies on jets and plumes, the 

 pressure distribution in the jets or plumes is 

 assumed hydrostatic, so that if the body- force term 

 in the equation of motion is written in the form 

 -gAp , where Ap is the difference between the local 

 density and the ambient density, the pressure gradi- 

 ent can be neglected in the equations of motion. 

 If, further, the flow is two dimensional or axisym- 

 metric, only the equation of motion for the vertical 

 velocity component is then needed. After that 

 velocity component is determined, the equation of 

 continuity can be used to determine the other veloc- 

 ity component. 



In the preceding paper by Yih, the assumption 

 that the x component of the velocity is constant 

 leaves only two other velocity components to be 

 determined, and it is tempting to adopt the usual 

 procedure of neglecting the pressure gradient. Yih 

 has resisted that temptation. But it is useful to 

 see what effects such a neglect would have on the 

 flow and to determine whether in the problem treated 

 by Yih such a neglect is allowable. This Appendix 

 is devoted to this question. 



If the pressure distribution is assumed hydro- 

 static and the usual procedure is followed, one will 

 drop Eq. (1) and retain Eq. (2) , with the first 

 term on its right-hand side dropped. [Equation 

 numbers in Yih's paper are retained.] Equations 

 (3) to (7) will remain but (8) and (9) will not be 

 needed . 



Following Yih's development and using his nota- 

 tion, then, we have, as the dimensionless equations 

 to solve, (24) and 



(L - 3)W = -h + A (VW^ + WWjj) 



(A.l) 



Using the X-series (29) , we have again (33) for the 

 solution of hg . The equation for W„ , obtained from 

 (A.l), however, is now 



(L - 3)W„ 



(A. 2) 



The solution of this equation, satisfying all the 

 boundary conditions for W stated in Yih's paper, is 



3iT 



1 -r^ 1 -n^-E^ 



3tt 



(A. 3) 



Although it can be readily verified that Eq. 

 (A. 3) satisfies Eq. (A. 2), it is not obvious that 

 Eq. (A. 3) is the unique solution. We shall show 

 in the following that it indeed is the unique 

 solution. The complementary solution Wq^ of Eq. 

 (A. 2) satisfies 



(L - 3)Woc = (A. 4) 



and must be even in both n and C • Let 



W(j^ = f (n)g(C), 



Now let 



a + b = 1. 



f (n) = e "^ ^^aW 



Then Eq. (A. 5) becomes 



6" - (n^ + b)6 = 0. 

 Similarly, if we let 



gia = e ^ ■^2^(0. 



(A. 7) 



(A. 8) 



(A. 9) 



Then 



Y" - {i:^ + a)Y = 0. 



(A. 10) 



Because of Eq. (A. 7), a or b must be positive. Let 

 b be positive. (The argument is strictly similar 

 if a is positive.) Because of the symmetry with 

 respect to the C axis, 



6' (0) = 0. 



Then Eq. (A. 9) shows that 3 will approach infinity 

 as n approaches infinity, if B(0) is not zero. 

 [If B(0) =0 then 0=0 throughout.] To see how 

 f (n) behaves at infinity, it is necessary to see 

 how 3(ri) behaves asymptotically. A simple calcula- 

 tion shows that the two solutions of Eq. (A. 9) 

 behave, for large values of n , Ixke 



exp 



(n^ 



a - 2)^dn 



and exp 



(n^ - a)'*dn 



As we have seen, 6 must contain the second solution 

 since B approaches infinity as n ->- <". Using the 

 second solution as the dominant term (a constant 

 multiplier being understood) , and recalling that 



(n-^ 



a)- 



— +0 



2n 



we see from Eq. (A. 8) that for large n^ 



I -a/2 



f (n) 



n 



(A. 11) 



which can be seen to satisfy Eq. (A. 5) asymptotically. 

 If a is negative, (A. 11) shows that f(ri), and there- 

 fore Wqj,, cannot satisfy the condition on Wq at 

 infinity. If a is positive, it must be less than 

 1, because of (A. 7) and because b is positive. 

 Then if W. contains Wg^, 



