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<br />WATER RESOURCES RESEARCH, VOL. 35, NO, 3, PAGE 907, MARCH 1999
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<br />Reply
<br />
<br />Gordon E. Grant
<br />USDA Forest Service, Pacific Northwest Research Station, Forestry Sciences Laboratory, Corvallis, Oregon
<br />
<br />Chanson (this issue] raises some interesting points, and I
<br />appreciate the opportunity to clarify and elaborate on the
<br />conclusions from Grant [1997], The three issues considered by
<br />Chanson are the definition of critical flow, the nonuniform
<br />shear stress distribution associated with hydraulic jumps and
<br />standing waves, and the application of the conclusion that
<br />critical flow is a limiting condition for steep, mobile-bed chan-
<br />nels.
<br />As Chanson [this issue] and others [e,g" Henderson, 1966,
<br />equation 2,21, p, 51] point out, Chanson's equation (1) pro-
<br />vides a more general definition of critical flow for nonrectan.
<br />gular channels; it equals the square of the Proude number.
<br />While explicit inclusion of the slope fJ is theoretically correct,
<br />the error introduced in calculating the Froude number without
<br />it is of the order of <1 % for channel slopes as high as 0,10, For
<br />this reason, most formulations exclude the slope correction.
<br />Slope was explicitly included in the recast Froude number
<br />equation [Grant, 1997, equation 2],
<br />The point that the shear stress distribution is nonuniform in
<br />both the longitudinal and cross-channel directions is nicely
<br />shown by Chanson's [this issue] detailed flow observations, It is
<br />precisely this non uniformity and consequent bed deformation
<br />that drives the cyclical growth and collapse of bedforms in
<br />sand-bed channels, which, in turn, maintains near-critical flow
<br />conditions, as shown by Grant [1997, Figure 3J, Moreover,
<br />minimum shear stresses below the crests of undular waves with
<br />maximums in the troughs tend to localize deposition of clasts,
<br />which may lead to step formation in coarse-grained channels
<br />[Grant, 1997, Figure 5]. The nonuniform cross-channel shear
<br />stress distribution (Chanson [this issue, Figure 1]) closely par.
<br />allels the nonuniform cross-channel Froude number distribu-
<br />tion, with lowest Froude numbers near the boundary [Grant,
<br />1997, Figure 2a], With high sediment transport rates and
<br />Froude numbers both concentrated in the center of the chan-
<br />nel, free-surface undulations should be greatest there, as has
<br />'been noted by others [Tinkler, 1997aJ,
<br />No claim is made that the tendency toward critical flow in
<br />steep channels represents an "ultimate simplification," at least
<br />in terms of fully predicting the complex, three-dimensional
<br />flow field of near-critical flow. ~ both Chanson [this issue] and
<br />I point out, near-critical flows are typically both unsteady and
<br />rapidly varying conditions that nullify most hydraulic simplifi-
<br />cations and for which few models, empirical or otheIWise,
<br />
<br />This paper is not subject to U.S. copyright. Published in 1999
<br />by the American Geophysical Union.
<br />Paper number 1998WR900055,
<br />
<br />apply, As noted by Grant [1997, p, 353], the use of empirical
<br />flow resistance correlations neglects form drag, free-surface
<br />instabilities, and hydraulic jumps as sources of energy loss; a
<br />comprehensive theory of energy losses in steep channels awaits
<br />further work, Chanson's equation (2) is similarly hampered by
<br />use of an empirical resistance coefficient. My point was that
<br />within the uncertainties introduced by use of empirical coeffi-
<br />cients and an assumption of uniform boundary shear stress, a
<br />general trend toward critical flow showing good agreement
<br />with field data can be observed. This agreement may result, in
<br />part, because the velocity measurements used to calculate
<br />Froude numbers are typically both time- and space-averaged,
<br />thereby integrating some of the unsteady and nonuniform as-
<br />pects of the flow,
<br />None of the Chanson's [this issue} comments are directed at
<br />the central tenet of the paper: to wit that critical flow repre-
<br />sents a boundary condition for steep, mobile, and potentially
<br />even immobile [i.e" Tink/er, 1997a, b] bed channels, main-
<br />tained by complex interactions between the free surface and
<br />the bed, The points raised do highlight the nature of some of
<br />those complexities and underscore the importance of recog-
<br />nizing near-critical flow as a "state" in its own right, with a
<br />distinctive suite of hydraulic and sedimentalogical features and
<br />behaviors. Further efforts should be directed toward testing
<br />the applicability of this concept across a range of channel types
<br />and flows.
<br />
<br />References
<br />
<br />Chanson, H., Comment on "Critical flow constrains flow hydraulics in
<br />mobile-bed streams: A new hypothesis" by G. E. Grant, Water Re-
<br />sour. Res., this issue.
<br />Grant, G. E., Critical flow constrains flow hydraulics in mobile-bed
<br />streams: A new hypothesis, Water Resour. Res., 33, 349-358, 1997.
<br />Henderson, F. M., Open Channel Flow, MacMillan, New York, 1966.
<br />Tinkler, K.. 1., Critical flow in rockbed streams with estimated values
<br />for Manning's n, Geomorphology, 20(1-2), 147-164, 19970,
<br />Tinkler, K.. J., Indirect velocity measurement from standing waves in
<br />rockbed streams, 1. Hydrau/. Eng" /23(10), 918-921, 1997b,
<br />
<br />G. E. Grant, USDA Forest Service, Pacific Northwest Research $ta.
<br />tion, Forestry Sciences Laboratory, 3200 SW Jefferson Way, Corvallis,
<br />OR 97331. (e-mail: grant@diceo.unifi.it)
<br />(Received July 20, 1998; revised October 5, 1998;
<br />accepted October 13, 1998.)
<br />
<br />OM
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