B • TURBULENT FLOW 



constant density. If heat is added or is generated by friction, or if another 

 gas is added, it is assumed that the amounts are too small to affect the 

 dynamical problem. The type of problem considered is that of fully devel- 

 oped rectilinear flow such as applies to the j et in a stationary surrounding 

 medium and a wake at sufficient distance from a body. 



The Reynolds equations in simple form become acceptable approxi- 

 mations under the conditions that (1) the viscous stresses may be neg- 

 lected compared to the turbulent stresses, and (2) the mean pressure is 

 so nearly constant that the gradients have a negligible effect on the axial 

 motion and momentum. With regard to condition 2, it should be pointed 

 out that the pressure in jets is slightly different from the ambient pres- 

 sure [111], but this may be disregarded as far as our present interests are 

 concerned. 



Let X be measured along the axis of mean flow from some suitable 

 origin, and U denote the mean velocity in the x direction. Let y be the 

 lateral coordinate for two-dimensional flow and r be the radial coordinate 

 for flow symmetrical about the x axis, and let V represent the lateral or 

 radial component of mean velocity in each case. Then for steady mean 

 flow the equations of motion and continuity are respectively: 



[f7^+7^ = i|^ (27-1) 



Plane jet and J ^^ ^V P ^V 



mixing zone | /irr /^t/ 



^9U SU^l_a^ (27-3) 



ax or rp dr 



^ + ^ = (27-4) 



dx dr 



Here t is the shear stress. 



For wakes, equations corresponding to Eq. 27-1 and 27-3 may be 

 further reduced because of conditions which apply at the great distances 

 from the object necessary for similarity to exist. These are that V has 

 become neghgible, and U is nowhere much less than the free stream ve- 

 locity Ue. If we express the velocity reduction by 



^U = U.- U 



and substitute in Eq. 27-1, at the same time dropping the term VdU/dy, 



we obtain 



,j^ . ^^. dAU 1 dr 



-{Ue - AU) -r— = - — 



dx p dy 



To a sufficient degree of approximation this may be written 



Plane wake - U, ^ = - ^ (27-5) 



dx p dy 



< 160 ) 



Round jet 



