SECT. 3] BEACH AND NEABSHOKE PROCESSES 517 



G. Work Done by the Fluid in Transporting a Suspended Load 



The fluid power expended in transporting a sediment load in suspension by 

 fluid turbulence can be found by reasoning similar to that leading to 

 relation (3). 



Suppose a mass nig of sediment grains is maintained supported by fluid 

 forces arising from the diffusion of upward eddy momentum components, and 

 is transported at a mean speed Us. Under steady transport conditions the 

 fluid must exert a normal stress [{ps —p)lps]gms cos ^ in order to support the 

 load. If the fall velocity of the grains is w relative to the fluid surrounding them, 

 the fluid must in unit time do 'work equal to [{ps —p)lps']gmsW cos /3 in raising 

 the grains at the velocity w to keep them statistically at a constant distance 

 from the bed. Substituting isjUs for [(ps — p)/ps]gws cos jS^the work done by 

 the fluid in unit time, i.e. the power expended, will be ivisjUg. But the existence 

 of an energy gradient in the direction of flow, owing to a gravity slope tan j3, _ 

 will reduce the work done by the fluid by an amount is sin ^.^[(f^. : m, L^ c^'f^ ^ 



Hence, the power expended by the fluid in transporting a suspended load at 

 the dynamic rate is must be 



is {^^ -tan ^ (8) 



The power expended in transporting the bed load being e^oj, the remaining 

 power, (1 — e^)aj, is notionally available in the form of a rate of supi)ly of 

 kinetic turbulent energy to maintain the suspended load. Writing €« as the 

 suspension efficiency, the total transport rate, in terms of i = ib + is, is ex- 

 pressible by 



i = ib + is = K> 



CO 



or 



where the dimensionless coefficient K = ilaj is the value of the group within 

 brackets on the right. 



It is not possible experimentally to make any separation of the measured 

 values of i into its elements ib and ig. So the relative factual values of e^ and eg 

 remain unknown. Fm-ther, our understanding of the precise mechanism of fluid 

 turbulence is insufficient to enable es to be deduced by theoretical reasoning. 

 However, on empirical evidence obtained from water-flume experiments, 

 augmented recently by unpublished evidence from a number of natural rivers 

 in the U.S.A., it appears, provided the gravity slope tan ^ is less than either 

 tan or wjUs, the overall coefficient K attains a maximum value which is a 

 constant for the system concerned. In other words, it appears that a critical 

 value of the power a* exists at which both €b and eg attain constant values, 

 and at which, therefore, the total sediment transport rate becomes proportional 

 to the power o). 



