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<br />GRANT: CRITICAL FLOW CONSTRAINS FLOW HYDRAULICS
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<br />355
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<br />flow conditions drop below those required to sustain boulder
<br />transport. The observed transition between step-pool channels
<br />with well-defined step bed forms and lower-gradient, plane-
<br />bed channels without distinct bed forms is at a slope of about
<br />0.03 [Mol/lgomeJY alld Buffillgtoll, 19931, The bypothesis pre-
<br />sented in this paper suggests that the morphology changes
<br />because slopes are insufficient to generate critical flow in these
<br />coarse-bedded channels (Figure 4). and bed form roughness is
<br />replaced by bar roughness.
<br />AJJilional support for this general hypothesis comes from a
<br />variety llf published and unpublished observations ur recon-
<br />stcU<.:tiuns of Haws in steep channels. Recently published data
<br />for steep, gauged channels show that Froude numbers tend to
<br />converge toward 1.0 as discharge increases [Wahl, 1993); this
<br />trend can also be abseIVed in MiIlIOIIS' (1973] data, which shaw
<br />Froude number increasing at a site with increasing discharge.
<br />Recenl flume work on rill development show that in rills freely
<br />formed at very high slopes (S > 0,05) in tilled silty loam of
<br />moderate cohesion, without layering or anisotropy, Froude
<br />number remains constant at La, independent of both slope and
<br />discharge (G. Gowrs, Leuven University, written communica-
<br />tion, 1995). PaleohyJraulic reconstructions of channel-forming
<br />Hoous through mountain channels also document flows at or
<br />near critical [Bowman, 1977; Grant et al., 1990; House and
<br />Pew11zree, 1995} as do direct measurements of flow hydraulics
<br />[i.e., Beuul1w/lt and Oberlander, 1971; Lt/cchitta and SUlIeson,
<br />I'mll,
<br />Thcrc is also some evidence that hydraulics of hyperconcen-
<br />trateu nows on steep slopes arc ncar-critical. Pierson und Scott
<br />l19H5] lks(;ribc lhe transition from lahars to hypcrconcen-
<br />trated streamflow in a chnnnel draining Mt. St. Helens, Wash-
<br />ington. During lhe latter phase, they note breaking standing
<br />waves and calculate Froude numbers of 0.9 to 1.1 at stream
<br />gradients of 0,003 to 0,1104 (Figute 4), Tbeir data suggest that
<br />hyperconcentrated flows may achieve critical flow at corre-
<br />spondingly lower slopes than for "normal" streamflows, per-
<br />haps owing to suppression of turbulence. The relatively high
<br />Froudc numbers measured in the Pasig-Potrero river (Figure
<br />4), where sediment concentrations were estimated to be 5% by
<br />volume and hence approached hyperconcentrated conditions,
<br />~uppur( Lhi~ lnrcrpretation and help explain why these two data
<br />points fall above the predicted active-bed transport curve (Fig-
<br />ure 4). In addition, assumptions regarding fluid density and
<br />resistance used to derive the envelope curves are not met for
<br />hyperconcentrated flows.
<br />Recent work on floods in steep bedrock channels suggests
<br />that even where the boundary is nonadjustable, critical flow
<br />may still constrain flow hydraulics. Paleohydraulic reconstruc-
<br />tion of flow bydtaulics associated with rapid release of water
<br />from Lake Bonncvilll~ and associated cataclysmic flooding
<br />demonstrate near-critical flow conditions over long (tens of
<br />kihJl1HHfI.lrH) nuu.:hlllfl! ur Uhlull\~l [a'CUljJlU~t, 1UU3, IJp. 3"...301
<br />1~lgurc 241. Flouds from the breakout of Lake Missoula were
<br />generally subcritical but locally achieved critical flow at con-
<br />strictions and wben channel slopes exceeded 0,01 [O'Collnor
<br />and Waitt, 19951. Constrictions of the Colorado River at allu-
<br />vial fans in the Grand Canyon adjust to maintain near-critical
<br />flows through rapids because of scour by upstream migrating
<br />hydraulic Jumps during competeIlE Ilows [Kieffer, 19851, AI,
<br />though in the case of bedrock rivers the bed is not adjusting to
<br />the How, irregularities in the bed surface and protrusions into
<br />the nuw may induce formation of hydraulic jumps, resulting in
<br />encrgy lusses. Flow nci.lf critical is poised in the sense that
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<br />Standing wave -...
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<br />-
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<br />
<br />(a)
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<br />Hydraulic Jump
<br />"
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<br />'V
<br />-'
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<br />
<br />(b)
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<br />Incipient PQ~I
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<br />Incipient step'
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<br />Figure 5. Model of step-pool formation in coarse-grained
<br />channels, based on flume experiments [Gram and A4izuyama.
<br />1991J, (a) Large clasts come to rest under standing waves,
<br />trapping smaller clasts and beginning step formation, (b) Flow
<br />over clast locally becomes supercritical, returning to subcritical
<br />just downstream and forming a hydraulic jump; scour under
<br />jump creates pool.
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<br />shifts in flow regime and large amplitude changes in the water
<br />surface elevation can be triggered by small irregularities in the
<br />bed [e.g., Henderson, 1966, p. 45}. Supercrilical flow I.:an only
<br />be maintained in steep, hydraulically smooth channels, such ns
<br />those lined with concrete described by Vaughn [19901, where
<br />Froude numbers typically range from 1.0 to as high as 2.5 or
<br />more over short (tens of meters) distances. Costa [1987] also
<br />reported indirect measurements of Froude numbers in excess
<br />of 1 for extremely high peak flows but noted that these were
<br />probably instantaneous values for short periods of time.
<br />Viewed from this perspective, the scatter of Froude numbers
<br />for adjustable channels in Figure 4 seems quite modest.
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<br />Discussion
<br />
<br />Critical flow is highly efficient for routing water' through
<br />channels. At critical flow, discharge per unit width is maxi-
<br />mized for the available specific energy (d + v2/2g), and
<br />specific energy is minimized for the available discharge [Hell-
<br />dersOll, 1966J, Tbat steep channels adjust their bed forms and
<br />energy expenditures to maintain critical flow is therefore con,
<br />sistent with the hypothesis of minimum rate of energy expen-
<br />diture and. under some constraints, maximum friction factor
<br />[Yanget a/.,1981; Davies and Slither/and, ]983J. This parsimony
<br />arises not only because of thermodynamic. principles, but also
<br />because of close coupling of the flow hydraulics with bed de-
<br />rul'ttUHh.1n un\J .uuln1"U\ \tMnlllmrt. (;onvectivI uc~.:lerations in
<br />the flow field are balanced by development of bet! tormlt unu
<br />increased flow resistance. Critical flow represents an energy
<br />threshold beyond wbich flow instability (hydraulic jumps) re-
<br />sults in rapid energy dissipation and morphologic change coun-
<br />teracting flow acceleration. The role played by hydraulic jumps
<br />is key; in channels ranging from sand to gravel to boulder bcd,
<br />development of highly erosive and turbulent hydraulic jumps is
<br />the hydraulic mechanism that applies the "brake" to flow ac-
<br />celeration. The net result is a tendency for critical flow to
<br />constrain adjustment of channel hydraulics and flow resistance
<br />at competent discharges.
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