Using the friction factor 



2t 

 PU2 



(23) 



given as function of Ua/v and a/Dso in a plot by Jonsson (1966), Madsen 

 and Grant (1976) calculated values of ^ with data from five previous 

 studies and found that, plotted against r, trends were reasonably 

 compatible with the Shields curve. 



In Figure 13, the Shields curve and values of ^ from the data in 

 this study are plotted against r. To plot the data here, equations 

 (21), (22), and (23) have been used with f taken from a plot by 

 Kamphuis (1975). That plot gives f as a function of Ua/v and a/Dgo 

 from recent experiments using a sand-roughened shear plate. Values of 

 Dgo were taken from the curves in Figure 7. Values of t were found to 

 lie in the transition zones between rough turbulent, smooth turbulent 

 and laminar, and interpolations on the plot (which had first to be re- 

 drawn on a larger scale) were difficult and uncertain. The points for 

 the three sands are found to be considerably above the Shields curve 

 (as most of the trends shown by Madsen and Grant, 1976). The two points 

 for the compressed beds plot above others for the 0.55-millimeter sand. 

 There is a spread in the data for each sand, with a definite tendency 

 for m to decrease as T increases. It was found that, when recalculated 

 with f from the Jonsson (1966) plot, average values of ^ for the 

 0.55-, 0.21-, and 0.18-millimeter sands were raised by 25, 45, and 41 

 percent, and a tendency for ^ to decrease with increasing T was no 

 longer apparent. Thus, the use and interpretation of a Shields criterion 

 appears to depend significantly upon the calculation of f . 



Carstens, Neilson, and Altinbilek (1969) have offered a semi- 

 empirical derivation of a l'(r) based on the concept of a critical velocity 

 u at some elevation in the boundary layer such that the relevant force 

 on a surface grain is proportional to D^p^^^ times a (free fall) drag 

 coefficient which is a given function of u^D/v. From three observations 

 (Fig. 2 in this study) they found that "^ is best evaluated at an eleva- 

 tion 0.6 D where, for laminar (thick) and turb ulent (thin) boundary 

 layers, u^ is taken to be 0.6 Dxc/pv and 7.22/tc/p . On this basis, 

 Carstens, Neilson, and Albinbilek (1969) derive the curves for 4* 

 shown in Figure 21 of their study and again in Figure 13 of this 

 study. (Their equation 39 is clearly in error, but the curve so 

 labeled in their Figure 21 was computed correctly.) Their curve for 

 4'(r), which remains undefined in a transition zone between its two 

 branches, fits the finer sands of this study rather well (Fig. 13). 

 This is not surprising in view of the agreement between values of 

 (})(, from Carstens, Neilson, and Altinbilek (1969) and from this study 

 (Figs. 2 and 12) . 



42 



