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<br />730
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<br />HYDRAULIC ENGINEERING '94
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<br />10 either Q,., or 02; they just serve as convenient and easily defined index discharges,
<br />The data were arbitrarily subdivided into three groups based on the value of
<br />Q/Qm' Values of Q/Qm larger than 0,7 were assumed to be representative of flood
<br />flows; values less than 0.2 were considered ro represent low discharges. and values
<br />from 0.2 to 0.7 represent moderate discharges. Four equations were used to estimate n
<br />for the reaches in each of the three groups of QIQ,.., The tests were limited to reaches
<br />with S > 0,002; the data set has 342 measurements with S > 0,002, In addition,
<br />Janeu's equation was tested separately for reaches with R < 1.8 m because his
<br />equation was derived only for that condition. The errors in estimating n for the
<br />composite data set were tabulated separately for the 3 groups of QIQ,.. and are
<br />summarized in figure 1. Because Umerinos expressed a preference for the equation
<br />using d84' his equation using d50 was not tested,
<br />
<br />I- 40 .-
<br />~ 20 Q/Qm < 0,2 n
<br />
<br />
<br />; ~~~nr&8B ~- & frfj-H-
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<br />
<br />~ -40 0 O,2<Q/Qm<O,7 Q/Qm> 0,7
<br />w .60 '
<br />0/~~~~e\lo 0~#~'<Jo~'i~'f#~ ~,\tP\~~#~
<br />
<br />EQUATION
<br />Figure 1, Modified boxplO1 01 errors in estimales 01 n lor the equations
<br />tested, [Top 01 box = 75th percentile, bottom 01 box = 25th
<br />percentile. and the line through the box is the median error,]
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<br />Results and Conclusions
<br />
<br />All equations tested underestimate n for the lowest flow range tested
<br />(QIQ,. < 0,2), Griffiths' equation had the largest bias (median error = -34 percent); the
<br />biases of the other equations were comparable to one another with median errors
<br />ranging from -12 10 -18 percenL For moderate discharges (O,2SQ/Q,.sO,7),
<br />Griffiths' equation was biased IOward underestimation (median error = -15 percenl),
<br />and Janell's equation was slightly hiased toward overestimation (median error = 6
<br />percent), The other two equations were relatively free from bias for moderale
<br />discharges, It must be noted that Bray stated that his equation was applicable only for
<br />high, in-bank flows corresponding approximately 10 QIQ,.=1.
<br />For ftoot1 discharges (Q/Q,. > 0,7), jarrell's equation was biased lOWard
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<br />t
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<br />
<br />BIAS IN MANNING'S N
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<br />4
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<br />overprediction with a median error of 10 percent. The other equations were essentially
<br />unbiased, Because of the bias in Janell's equation for flood discharges in the
<br />composite data set, his equation was tested separately for this discharge range against
<br />his own data and against Barnes' data, The median error against each of those data
<br />sets was 11 percent.
<br />Janell noted that his equation applied for the range of R from 0.15-2,1 m (his
<br />maximum reported R was 1.1 m). To test an assumption that extrapolation to larger
<br />flows should not be too much in error, the equation was tested against the composite
<br />data sel for R > 1.8 m, Only 17 values qualified; even when the Irue values of R were
<br />replaced with a constant R of 1.8 m, all n values were underestimated, and the median
<br />error was -17 percent.
<br />The above results show that the equations that include a relative smoothness
<br />term are unbiased for discharges in the range of the median annual peak discharge,
<br />However, jarrell's equation based on S and R overpredicts n for thaI range of
<br />discharges, None of the equations are reliable for low discharges (QIQ,. < 0.2)
<br />without adjustment. Furthennore. Jarrett's equation is not reliable when extrapolated
<br />'0 reaches baving R > 1,8 m because the equation will significantly underestimate n,
<br />Exn-apolation of any of these equations 10 discharges greater than about 1.5 times
<br />the median annual peak discharge is presently unwarranted, In the absence of
<br />overhanging banks or vegetation. n generally decreases with increasing relative
<br />discharge (Q/Qm)' However, the paucity of verification data for large floods is shown
<br />by the disnibution of QIQ,. for the composite data set; only 12 (3,5 percent) of the
<br />verifications are for QIQ,..>I,5, and only 3 (0,9 percent) are forQlQ,..>2, Many of the
<br />applications of the Mannin8 equation for high-gradient mountain streams are for large
<br />floods, Yet, the data are presently not available 10 define and test such equations
<br />against floods greater than about the median annual peak discharge, This is a serious
<br />limi.ation and emphasizes the need for additional data collection 10 define the
<br />hydraulic processes for large discharges on high-gradient streams,
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<br />Rt':ft':rencell
<br />
<br />Barnes, H, H" "Roughness Characteristics of Natural Channels," U,S, Geological
<br />Survey Water-Supply Paper 1849, 1967, 213 p,
<br />Bray, D, I., "Estimating Average Velocity in Gravel-Bed Rivers," Journal of the
<br />Hydraulics Division, ASCE. Vol. 105, No, HY9, Sept, , 1979, pp, 1103-1122,
<br />Griffiths, G, A" "Flow Resistance in Course Gravel Bed Rivers," Journal of the
<br />, Hydraulics Division, ASCE, Vol. un, No, HY7, July. 1981, pp, 899-918,
<br />llocks, D, M" and Mason, p, 0" "Rougbuess Characteristics of New Zealand Rivers,"
<br />Woter Resources Survey, Kilbimie, Wellington, New Zealand, 1991,329 p.
<br />1am:1I, R, D" "Hydraulics of High-Gradient Streams," Journal of Hydraulic
<br />Engineering, ASCE, Vol. 110, No, 11, November 1984, pp, 1519-1539,
<br />Limerinns, J, T" "Determination of the Manning Coefficient From Measured Bed
<br />Roughness in Natural Channels," U,S, Geological Survey Water-Supply Paper
<br />1898,8, 1970,47 p,
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