end of the injector tube and moves downstream in a turbulent flow 

 its trajectory might appear as follows: 



Let the vertical velocity component of the particle at time t ■$ 

 after leaving the injector be v(t). Then the transverse displace-, 

 ment y(t) of the particle at time t is given by y(t) ■ f v(r)dt. 

 The mean square displacement is then given by 



_ }■ £ 



y z (t) - j f V(rjv(a) Jr J<r 



or, introducing the correlation coefficient 



RiVjV) = Vlv)v(<r)/ v'Wv'Ccr) J y l -^T^- , 



r t A 



y z (t) = ) o J o Y'Mv'MRfarfiJTjcr . 



it should be noted that the means are not time averages of the 

 functions v2(t) and y 2 (t), but are ensemble averages taken over 

 a large number of particles. From the last equation one easily 

 derives 



t i 



J- 7ft) = 2Y\i)+Uv\i)\ v'( (r )Rtt J <r)clo-+ZY'it)(v , (ff)jL'R(t<r)Jcr 



dt z ' TV Jo ° at 



and, setting t = 0, 



d 1 y^o) = Z V'to) 



One may recast this formula as follows: 



j£ yVo>^[2/^^/^J=^i^J^2/y^^^^ = 2v'?o> 



or, finally, 



^d-^yHo) ~ v ' { 



= v (o) 



dt 



If one now makes the assumption that the fluctuations of velocity 

 in the longitudinal direction are small compared with the free- 

 stream velocity U, one may write, approximately, X - Ut and write 

 the last equation above 



jLjy^fo) = Vrty/U . 



- 6 - 



