B,27 • LAWS OF MEAN SPREADING AND DECAY 

 Following a similar procedure for the round wake we obtain 



dAU 1 dirr) 



Round wake 



C/e 



dx rp dr 

 The equations for the conservation of momentum are: 



(27-6) 



Plane jet 



Round jet 

 Plane wake 



p / ' UHy = const 

 2rp / Vhdr = const 



(27-7) 



p f^^ U{U, - U)dy = const ^ pU. f^^ AUdy 



Round wake 



27rp fj U{U, - U)rdr = const ^ 27rpt/e fj MJrdr 



The method of employing the foregoing relations to find the laws of 

 spreading and decay will be illustrated by carrying through the steps for 

 the plane jet. Then the end results for all cases will be stated. If U^ is 

 the velocity at the center of the jet and h is any convenient measure of 

 the width (6 may be the distance from the center to where U is zero or 

 some fraction of JJ^, then similarity means 



and 





T 



f 



(27-8) 



(27-9) 



where / and g are any function whose form may remain unknown. We 

 now set b -^ a;™ and Uo '^ x~"'. Then the terms in the equation of motion 

 (Eq. 27-1) become of the following order in x: 



U 



dx 



V 



dy 



dr 

 dy 



In order that the equation shall be independent of x, we must have 

 2ri + 1 = 2n + w, or m = 1. The momentum relation 



Pl ^"dy 



const 



because of order 



and since this must be independent oi x, —2n-\-m = 0. Since w = 1, 

 n = I-. Thus it is found that the plane jet spreads hnearly with x and the 



< 161 ) 



