B,30 • VELOCITY DISTRIBUTIONS IN JETS AND WAKES 



3. Modified theory: e^ = (e^)t7, where (€;,)t = constant eddy viscosity 

 in the turbulent region, y = intermittency factor: 



/i = 1.835 exp \ -UA^^ 



In 1, 2, and 3, /i and | are 



Ue - U /x - 



1 + 



3 V0.35/ 



(30-3) 



fi = 



/ x — a;o Y 



? 



y 



xo)d]^ 



Ue V d / ' " [(X 



Xo = virtual origin {xo/d = +90) 

 d = diameter of cylinder producing the wake 



In these cases 



= Du = 0.017^Ued 



(30-4) 



It is seen that mixing length theory makes the distribution too narrow 

 near the axis. The constant exchange coefficient fits in this region but 



2.0 



+ Mixing length 1.835 [1 - {^/0A8)^f 



• Constant shear coefficient 1 .835 e - (5/0.253)' 



O Modified theory 1 .835 e - '^-^^^n + i(|/o.35)^j 



Mean of observations 



fil.O 



Fig. B,30. Comparison of velocity distribution formulas 

 for plane wake, after Townsend [126]. 



makes the velocity difference approach zero too slowly in the outer region. 

 In reality e^ is not constant, and an all-over fit is obtained only by adjust- 

 ing e^, as in Eq. 30-3. 



The distribution of axial velocity across the round wake may also be 

 represented by a Gaussian error function. Such representations are char- 

 acteristically faulty near the outer edges. The round wake has not been 

 investigated so thoroughly as the plane wake. 



< 173 ) 



