obtained simultaneously with the determination of the lift coefficient C^. 

 The method used to this effect consisted of measuring the differential 

 pressure in the fluid between two layers; one at the top of the spheres 

 and the other near their base. This pressure difference which is a mea- 

 sure of the lift force can be expressed as 



AP = C L p f % (4-5) 



The analysis of the experimental results revealed that Cl in the above 

 expression had a constant value Cl = 0.178 for a wide range of flow con- 

 ditions provided that the average velocity was measured at a distance 

 0.35 D from the theoretical bed. Since it is practically impossible to 

 determine the value Cl in an unsteady mean flow, it would be reasonable 

 to assume that it is about the same as in a steady stream. As a matter 

 of fact, as it will be shown shortly, the only assumption that is really 

 necessary is that Cl is constant throughout the entire cycle of oscilla- 

 tion. Before we proceed any further we wish to summarize the statements 

 adopted in this study which have their origin in El-Samni's experiments 

 with steady mean flows. 



u. The theoretical bed lies at a distance 0.2 D below the top 

 of the grains resting on the fixed bed. 



b. The lift force associated with the mean flow can be calculated 

 by using the velocity at a distance 0.35 D from the theoretical 

 bed. 



c. The lift coefficient associated with the mean flow has a constant 

 value independent of the Reynolds number. 



If it were not for turbulence it would be rather easy to establish a cri- 

 terion of stability similar to Jeffreys' in the form of the inequality 



L > W (4-6) 



(W is the submerged weight of the particle) 

 _2 

 or C L p f u_ A 2 D 2 > (p s - pf) gA 2 D 3 (4-7) 



where by u we mean the amplitude of the velocity calculated from equation 

 (3-24) with y = 0.35 D. 



In a turbulent stream, however, all the local flow parameters, and 

 consequently the local life as well, vary rapidly with time. An accurate 

 description of the temporal variation of the local life by analytical 

 methods would be possible only if sufficient information regarding the 

 structure of turbulence in the boundary layer were available. Considerable 

 amount of effort is made now by numerous investigators toward developing 

 new theories or toward improving existing ones on the subject. The fact 

 remains that even if we could wait until the theoretical work had been 

 sufficiently advanced to permit practical applications, still some 



13 



