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<br />352
<br />
<br />GRANT: CRITICAL FLOW CONSTRAINS FLOW HYDRAULICS
<br />
<br />a
<br />
<br />I L i
<br /> L
<br />Two minute average for each station
<br />-r
<br />
<br />1,5
<br />
<br />~
<br />~ 1.0
<br />,
<br />Z
<br />~
<br />'0
<br />,
<br />J: 0.5
<br />
<br />0,0
<br />
<br />o
<br />
<br />5
<br />
<br />2 3 4
<br />Distance from left bank (m)
<br />
<br />1,5
<br />
<br />b
<br />
<br />
<br />0;
<br />D
<br />E
<br />,
<br />z
<br />~
<br />'0
<br />,
<br />J: 0.5
<br />
<br />0,0
<br />
<br />Cenlerolchannel,XS 1
<br />
<br />40 60
<br />Time (seconds)
<br />
<br />Figure 2. Variation of Froude number (a) over the cross
<br />section and (b) through time for Big Creek, Oregon, (a) Mean
<br />and range are shown for instantaneously measured values of
<br />Froude number at vertical sections sampled at 5-5 intervals
<br />over 2-3 min. Average Froude number for the entire cross-
<br />section is 0.87. Arrow denotes centcr of channel section used in
<br />Figure 2b. (b) Variation in Froude number over approximately
<br />2 min measured at center of channel; average Froude number
<br />is 1.00. Antidune and standing wave buildup accompany in-
<br />creasing Froude number; breaking waves and hydraulic jumps
<br />occur at maximum Froude number, and plane beds occur at
<br />lowest Froude numbers.
<br />
<br />o
<br />
<br />20
<br />
<br />ao
<br />
<br />100
<br />
<br />number. Based on the average depth and surface velocity,
<br />however, and assuming that the average cross~section velocity
<br />'~wa~ 80% of the surface velocity [Mathes, 1956], average Froude
<br />numher for the reach was 1.1.
<br />
<br />Froude Number of Mobile-Bed Streams:
<br />An Anatytical and Empirical Approach
<br />
<br />The physical mechanism that causes hydraulics to oscillate
<br />around critical flow requires that the channel slope be great
<br />enough for the flow to approach critical when sediment trans-
<br />port is actively occurring. At a minimum this requires that the
<br />incipient motion threshold for the bed be exceeded. The
<br />Froude number at this threshold condition can be determined
<br />analytically by simultaneously expressing a flow resistance
<br />equation and a sediment transport relation in terms of the
<br />relative submergence (d/DA4, where d is flow depth andDR4 is
<br />the 84th percentile grain size of tbe ebannel he d) and using
<br />these equatinns to solve (1) recast in tenns of the channel
<br />slnpc. Because the average shear velocity v. is equal to
<br />{ndS)1I.5. where S is the channel slope, (1) can be rewritten as
<br />
<br />v
<br />Fr = - SO.5
<br />v'
<br />
<br />(2)
<br />
<br />Flume experiments by Bayazi! (19831 have shown that in steep,
<br />hydraulically rough channels, the flow resistance is given by a
<br />Keulegan-type relation:
<br />
<br />vv. = 2.18 [ In (;J + 1.35]
<br />
<br />(3)
<br />
<br />Also, critical dimensionless shear stress,
<br />Shields relation [Shields, 1936J:
<br />
<br />. dS
<br />
<br />T" = [( ~:) _ 1] D
<br />
<br />.
<br />l'cr'
<br />
<br />is given by the
<br />
<br />(4)
<br />
<br />where 'Y~ and 'Yw are the specific weights of water and sediment
<br />respectively, and tan e = S = channel slope. Assuming 'Y~ =
<br />2.65, (4) can be rearranged as a relative roughness equation:
<br />
<br />~
<br />
<br />
<br />A
<br />
<br />-----
<br />
<br />
<br />B
<br />
<br />
<br />c
<br />
<br />~::::\:\\ ...........:....i.'...
<br />::..:;./...:.;?;.:,...: ;',i, \;?~.>\ :f~"~'~":'~'?'\"'!:<':;::::
<br />
<br />"
<br />.\<.;>///':~
<br />
<br />:':.;':';.::-;.'.:.'
<br />
<br />.'.',.'.,..,..:,,:.,-
<br />
<br />.,.\\'.\:.':,':::....g,.j.'.'...'.....;.....
<br />
<br />-----
<br />
<br />~
<br />fr<1
<br />
<br />'--~
<br />
<br />D
<br />
<br />...':<<"
<br />;"'::.',::
<br />:.': ,',:"':'.,:.
<br />.,.,:'
<br />
<br />..:.:::.::'.':."
<br />':";':":':':/
<br />
<br />............-......:,.
<br />
<br />~~:~~;:~;;/~~3{{;X~}!;
<br />
<br />.....:......,:.,:.'....':,'.
<br />
<br />::::~~()::;;:/,/?,:::: ".
<br />
<br />\'~W".~ ~.;.::':.:':.:.'::'.:.:'.~.:\.'.;;
<br />
<br />.'/.
<br />
<br />-----
<br />
<br />E=A
<br />
<br />Fr<1
<br />
<br />
<br />Figure 3, Cyclic sequence of surface wave and hed form de-
<br />formation in sand-bed channels near critical flow. (a) Plane.
<br />bed, subcritical flow. (b) Building antidunes, approximately
<br />critical flow, (c) Breaking antidunes, flow both suberitieal and
<br />supercritical. Note that upstream migration of antidune into
<br />wave trough induces hydraulic jump formation, causing stand-
<br />ing wave to break, (d) Upstream wave has broken, downstream
<br />wave is about to break. (e) Scour of antidunes returns channel
<br />to plane~bed condition.
<br />
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