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<br />WATER RESOURCES RESEARCH, VOL. 35, NO, 3, PAGES 903-905, MARCIl 1999
<br />
<br />Comment on "Critical flow constrains flow hydraulics in
<br />mobile-bed streams: A new hypothesis" by G. E. Grant
<br />
<br />Hubert Chanson
<br />
<br />Dcparlmcnt of Civil Engineering, University of Queensland, Brisbane, Queensland, Australia
<br />
<br />Historically critical flow conditions were defined as the flow
<br />propcrlil:s at tht.: singularity of the backwater equation {Be-
<br />lunger, IH2H; Razf", It)65]. In the general case of a channel of
<br />nOllConslanl shape and longitudinal bell slope, the ooc-
<br />JinH.:nsiunal form of the energy equation (i.e., the backwater
<br />cquatiun) becomes
<br />
<br />ilff ild . ae iJz" Q2 iJA
<br />-:-- - :::: :-- cas 8 - d Sill 8 -:----- + - - a ~ ~:::: -s
<br />iJx dx Ilx ax gA 3 itx f
<br />(I) 3. Near-Critical Flows and Bed Shear Stress
<br />
<br />1. Introduction
<br />
<br />This paper tJiscusscs the challenging ideas proposctl by
<br />Grill/{ I I lj(J7 I on steep ll1ov;lblc bed L:hannd llows. First the
<br />ddinilioll or criticalllow arl; rl;visill;U. Then the calculation of
<br />bl:d ~hl:ar strl;~S in ncar-critical !lows <Ire discussed. Ncw ex-
<br />pl:riml:l1tal data an; presellted, and they highlight the three-
<br />L1inH.:n~ional variations uf boundary sh~ar stress. Further, it is
<br />bdievl:L1 that thl: <lpplicability of Grant's hypothesis is rc-
<br />slrich::u hcci.lusc or the confusion between critical and near-
<br />crili..:al Bows.
<br />Thl: discusser wishl;s to congratulate Grallt [1997] for some
<br />inti.:rl:sting anll chalh.:nging ideas on steep movable bed chan-
<br />nel !lows. This paper comments on the definition of critical
<br />IhlW (based un Gml/t [1997, eqUation 1], the Calculation of bed
<br />~hl.::ar ~lrcss in ncar-critical flows, and the applicability of the
<br />Grant's hypothesis. I hope that the information will assist to
<br />n.:IiIU.': tht.: Grant\ dcvdopmcnts.
<br />
<br />2. Definition of Critical Flow Conditions
<br />
<br />where H is the mean total head (i.e., average over the flow
<br />cross-section area), x is the longitudinal coordinate in the flow
<br />direction, d is the flow depth measured normal to the channel
<br />bo(tom, fJ is the longitudinal bed slope, Zu is the bed elevation,
<br />a is the kinetic energy correction coefficient (i.e., Coriolis
<br />coefficient), Q is th~ water discharge, A is the cross-section
<br />area, B is the free-surface width, S,. is the friction slope,
<br />
<br />1 V'
<br />Sf ~ f - -
<br />DII2g
<br />
<br />f is the D..Ifcy friction factor, D H is the hydraulic diameter, and
<br />V is the ml.:an flow velocity (V = QIA). For a constant
<br />dwnncl shape, aA = Bad and (I) hecomes
<br />
<br />aB
<br />ad sinR-Sf+dsinfJjJ;
<br />
<br />ilx (j'B
<br />CosO-a---
<br />gAJ
<br />
<br />Copyright 1999 by the American Geophysical Union.
<br />
<br />Paper number 199XWR900054.
<br />OO-U-j ]lJ7il.)lJ/ I Y9l'lW R 900U54$09.00
<br />
<br />Morc recently [B(lkJuneleJJ: 1912, 1932; Hellllersoll, (l)hhl.
<br />critkal fluw conditions wefl~ dcfint;u as the now pmpcrtil:s
<br />(depth ami velucity) for which the mcun spedlic l,;nl.::rgy is
<br />minimum for a given flow ratc and channt.:l cross-scl.:liull
<br />shape. The mean specific energy is defined <IS
<br />
<br />V'
<br />E = d cos B + " 29
<br />
<br />(~)
<br />
<br />assuming a hydrostatic pressure distribution. E is similar to the
<br />energy per unit mass, measured with the channel bottum as thl:
<br />datum. In the general case of a nonrectangular channel [e.g.,
<br />Henderson, 1966, p. 51], the mean specific energy is minimum
<br />for
<br />
<br />oE Q'B
<br />-:--=cosfJ-a~-=O
<br />ad gA'
<br />
<br />t5)
<br />
<br />Today this second definition of critical flow (i.e., equation
<br />(3)) is commonly used, Equations (5) and (3) for B conslan!
<br />yield the criterion fur critical flow cunditiuns
<br />
<br />Q'B
<br />agcosOAJ
<br />
<br />(6)
<br />
<br />EqUation (6) is more general than Grant's [1997] equation (t)
<br />which did nut take into account the bed slope effect nor lhe
<br />cross-section shape. Here, a is larger than unity although
<br />rarely exceeds 1.15, and the ratio alcos 0 typically ranges be-
<br />tween 1 and 1.3 in natural streams.
<br />
<br />(2)
<br />
<br />Near~critical flows may be defined as flow situations charac.
<br />terized by the occurrence of critical or nearly critical flow
<br />conditions over a "reasonably long" distance and time perioo,
<br />The specific encrgy/ftow depth diagram shows that near tht:
<br />critical flow conditions, a very small change of energy (e.g.,
<br />caused by a bottom or sidewall irregularity) can induce a very
<br />large change of flow depth, Near,critical flows are indeed char,
<br />acterized by the development of large fret:-surface undulations
<br />[e.g., Imai and Nakagawa, 1992; Chanson, and MOflles, 1995;
<br />ChallSon, 1995, 1996; Montes and Chal/son, 1998], Experimen-
<br />tal observations showed further that "undular flows" may take
<br />place for 0.3 :s Fr ::os; 3, but this range of flow conditions coukl
<br />be broader depending upon the boundary conditions.
<br />Here I performed new experiments in a 20-1l1~long fixed-beu
<br />channel of rectangular cross section (W = 0.25 m) tu illVt:s-
<br />ligate the boundary shear stress under an undular hydraulic
<br />jump. The bed shear stress was measured with a Prandll~Pitut
<br />tube (0 = 3.3 mm) used as a Preston tube (see appendix).
<br />Bed shear stress measurements under an undular jump (in a
<br />fixed~bed channel) are presented in Figure l, in which the b~d
<br />shear stress at various positions across tht: channel (..:/JV = 0
<br />at the sidewall, z/W = 0.5 un the centerline) is plottl.::u as a
<br />
<br />(3)
<br />
<br />903
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