approximations would be required since most of the studies are concerned 

 with steady mean flows and smooth boundaries. In the meantime and due to 

 the lack of more reliable information we are forced to depend again on 

 experimental evidence. A possible criterion of stability in a turbulent 

 stream could be set as 



+ L > W 



(4-7a) 



where L' is the turbulent component of the lift force. The study of the 

 variation of L' with time constituted the last phase of the experimental 

 work conducted by El-Samni. By measuring the instantaneous values of 

 the lift force exerted by a steady stream of water on the plastic spheres 

 mentioned above, he was able to show that L* behaves like a random vari- 

 able having a normal distribution with mean zero and standard deviation 



a = L T|o =p — •<- . Assuming that the behavior of L' in an oscillatory mean 

 flow will be similar although with a different numerical value of 0" we 

 could proceed as follows: Let p be defined as the probability that a 

 particle resting at a certain location in the bed becomes just ready to 

 move; this implies that 



p = Pr 



{» 



or p = Pr 1 Z > 



Let Y 



V P f 



Pf 



+ L > W 



. I = Pr \1L > JL - _i_ \ 



L LTU LT1 h Tl J 



2g(P s -p f ) A 2 D 



C L p f D 2 u 2 Al \ 



Dg 

 -ZTT 



L J 



Ho 



(4-8) 



(4-9) 



(4-10) 



and B^ 



2 A, 



C L T1oAl 



Then p = Prjz>YB,-i- X 



(4-11) 



(4-12) 



L» 



and since Z = — — has a normal distribution with mean zero and standard 



L Tlo 

 deviation a - 1 



B *Y 



_z 2 /2 

 e dz 



1 



(4-13) 



Tlo 



where z of course is a dummy variable. 



14 



