(Yalin, 1977; Ch. 3; Clift, Grace and Weber, 1978, pp. 113-116). Figures 4-9 

 and 4-10 display empirical results for the fall velocity of spheres (solid 

 curve) . 



More significant littoral processes are empirical results for the fall 

 velocity of common natural grains (Hallermier, 1981). The dashed curve in 

 Figure 4-9 displays these results as (V d / v) versus B = [(y /y -1) 



g d;.^ / V ] , where the grain diameter is measured by the median sieve size 



d^Q . For common grains, the three segments of the Figure 4-9 dashed curve 

 are given by 



Vj = (Yg/Y -1) g ^3^/18 V (B < 39) (4-7) 



Vf = [(Yg/Y -1) g]°*^ dJ^Vev^-^ (39 <B < 10^) (4-8) 



Vf = [(Yg/Y -1) g d3Q/0.91]°*^ (10^ < B) (4-9) 



Equation (4-8) is most useful because it provides the fall velocity in 

 water of common quartz grains described as fine to coarse on the Wentworth 

 scale (Fig. 4-7). Equation (4-6) is identical to results for spheres and 

 pertains to laminar fluid motion in settling of very fine grains. Equation 

 (4-9) pertains to turbulent fluid motion in settling of very heavy grains; 

 this dependence of fall velocity is identical to asymptotic results for 

 spheres, but for common grains fall velocity is lower and turbulent motion 

 occurs at lower values of the buoyancy index B . 



According to its definition, V^ is a measure of grain behavior in an 

 ideal situation. Actual fall velocity can be affected by several factors; 

 e.g., the terminal fall velocity is reduced somewhat in a turbulent fluid 

 (Murray, 1970; Nieraczynowicz, 1972). However, the most appreciable effect 

 seems to be that due to particle concentration or proximity, which can reduce 

 fall velocity by two orders of magnitude. In a concentrated suspension of 

 spheres, the fall velocity V^g^, is related to the fall velocity in isolation 

 Vf3 by 



^fsc=^fs (l-^>"' (^-10) 



Here c is the volumetric particle concentration (between and about 

 0.7), and the povrer n is the empirical function of the buoyancy index 

 displayed in the solid curve of Figure 4-10 (Richardson and Jeronimo, 1979). 



Although this concentration effect has not been defined empirically for 

 common natural grains, behavior analogous to that for spheres may be 

 expected. Presuming a smooth transition for the settling behavior of common 

 grains between the two Figure 4-9 asymptotes (rather than the approximation in 

 equation (4-8)), the dashed curve given in Figure 4-10 should be appropriate 

 for the power n in 



V = V^ (1-c)" (4-11) 



re r 



4-19 



