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GRANT: CRITICAL FLOW CONSTRAINS FLOW HYDRAULiCS 353 Active bed - sand (Eq. 1~../"" .- / ........ ,../ ..............~uve bed - gravel (Eq. 12) / ,/ ...... /.... 1.5 m 1.0 .D E " Z <0 u " o U: 0.5 t 1<1 '.,.... '.' /'" , 1 0,0 0.00 0.02 ... Burrows et a1. 1979 t> Dinehart 1992 ... Emmett 1976 V Fahnestock 1963 o G.Goversrill(unpub.) 'f Grimm & Leupold 1939 A Inbar&Schick 1979 ........ -" o Threshold chanoels (Eq. 1:1) 0.04 0.06 Channel Gradient 0.08 @LarooneelaI1986 .. Milhous1973 . Pierson & Scott 1985 o PiUick 1992 . Ritter 1967 o Thompson & CampbeU 1979 . OR Beach l This sludV Ii Pasig-polreroR..J . Figure 4. Plot of (6), (12), and (14) and published daIa for the relation between channel gradient and Froude number. Froude numbers are calculated using (1) with a = 1.0. Means and ranges of Froude number and stream gradient for each data set are shown and, except for those of Fahnestock [1963], are based on individual measurements of mean cross. section velocity and depth under flow conditions where bed loau was aClivdy being transported. Points for Fahnestock are means and ranges of single.currcnt meter measurements of maximum velocity and depth where bed load was being sampled. d T:r {5= 1.65 S By selting D = DB< and substituting (5) into (3) and (2), we can calculate Froude number at incipient motion for a given slope as Fr = 2,18[ In (1.65 T;,) + 1.35 ]sos (6) Equation (6) therefore gives the Froude number at conditions of incipient motion for a specified slope and value of T~r' At T;, = 0,06, which has been argued as the "channel-forming discharge" for gravel-bed streams [i,e., Andrews, 1983, 1984; Parker, 1990J, (6) predicts that the Froude number will in- crease asymptotically toward 1.0 as friction slopes approach 0,02-0,03 (Figure 4), For lower values of T;, the Froude num- ber increases but remains less than 1; higher values of T:r predict supercritical flows at incipient motion. Because it is based on an assumption of steady, one- dimensional, uniform flow and relies on empirical flow resis- lun~1! relutlon!!, this approach can only approximate the nec~ l:ssary channel slope; near-critical flow conditions are typically nonuniform and unsteady. This approach also neglects the potentially large component of flow resistance associated with bed forms as well as grain roughness [Davies, 1980J and does nOl account for the increased flow resistance created by free stlrface instabilities and hydraulic jumps at low values of rela- tive submergence (i,e" dID" < 1.5) [Ashida and Bayazit, 1973; Bathurst et a/., 1979J. Both of these factors result in lower velocities for a given slope and would therefore tend to push the asymptote more to the right. An additional complicating fuChlf il) lhut now resistance itself varies with Froude number (5) [Bathurst et at., 1979; Rosso et at., 1990]. Flume experiments by Rosso et at. [1990}. however. demonstrate that in channels with gradients less than 0.05, only a 5-10% error in determining resistance is introduced by neglecting this effect. Despite these uncertainties, the derived curve for T;r = 0.06 is in reasonable agreement with published data for mobile-bed channels (Figure 4). This threshold curve represents a lower bound for competent flows; therefore all channels with active transport (T(/T<;r <:: 1) should plot above the line, but incom- petent flows (TrJT" < I) should plot below it, Most published hydraulic data for competent flows (Le., where bed load trans- port was actually observed) for a wide range of stream types and slopes, including sand-bed, gravel-bed, and boulder-bed channels, lie on or above Ihe predicted threshold (Figure 4). Froude numbers calculated from published values of depth and velocity may be somewhat lower than actual Froude num- bers because most data report mean cross-section values of d and v, which are typically less than flow-weighted or maximum values. Most published velocity data also assume a = I, al- though il may be 10-15% larger [Wahl, 1993], The highest reported and greatest range in Froude numbers are from point measurements made in glacial outwash streams where flow conditions were explicitly described as breaking antidunes [Fahnestock, 1963, p, A28J, As previously shown, point Froude numbers can be significantly greater than 1.0 eVen when the cross section as a whole is near-critical (Figure 2a). Although there is some scatter in Figure 4, the data points are arrayed in a fairly narrow range above the lower envelope curve, but less than or equal to the critical flow line. Above a slope of about 0.01, all but one of the data sets report mean Froude numbers of 1.0 :t: 0.2. The mean Froude number for Inbar and Schick's [1979] dala falls slightly outside lhis range;