HYDRAULICS OF H I G H - G R A D I E N T STREAMS Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. By Robert D . Jarrett,1 M. ASCE ABSTRACT: Onsite surveys and 75 measurements of discharge were made on 21 high-gradient streams (slopes greater than 0.002) for the purpose of computing the Manning roughness coefficient, n, and to provide data on the hydraulics of these streams. These data show that: (1) n varies inversely with depth; (2) n varies directly with slope; and (3) streams thought to be in the supercritical flow range were actually in the subcritical range. A simple and objective method was employed to develop an equation for predicting the n of high-gradient streams by using multiple-regression techniques and measurements of the slope and hydraulic radius. The average standard error of estimate of this prediction equation was 28% when tested with Colorado data. The equation was verified using other data available for high-gradient streams. Regimeflow equations for velocity and discharge also were developed. INTRODUCTION Hydraulic calculations of the flow in channels and overbank areas of flood plains require an evaluation of roughness characteristics. Most commonly, the Manning roughness coefficient, n, is used to describe the flow resistance or relative roughness of a channel or overbank areas. Term n appears in the general M a n n i n g equation for open-channel flow, i.e. V = 1.49R 2/3 S 1/2 (1) n in which V = the average cross-section velocity, in ft/sec; R = the hydraulic radius, in ft; S = the energy gradient or friction slope; a n d n = Manning's roughness coefficient. The M a n n i n g equation is often substituted into the continuity equation, i.e. Q = AV (2) in which Q = the discharge, in cu ft/sec; a n d A = the cross-sectional area, in sq ft. This substitution yields a variation of the Manning equation 1.49AR 2/3 S 1/2 Q = (3) n and the variables are as defined previously. Eqs. 1 and 3 were developed for conditions of uniform flow in which the water-surface slope, friction slope, and energy gradient are parallel to the streambed, a n d the area, hydraulic radius, and d e p t h remain relatively constant throughout the stream reach. The Manning equation has been u s e d extensively as an indirect method for computing discharges or d e p t h s of flow in natural channels. 'Hydro., U.S. Geological Survey, Lakewood, Colo. Note.—Discussion open until April 1, 1985. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on September 6, 1983. This paper is part of the Journal of Hydraulic Engineering, Vol. 110, No. 11, November, 1984. ©ASCE, ISSN 0733-9429/84/0011-1519/$01.00. Paper No. 19272. 1519 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. In this application it is assumed that the equation is also valid for the nonuniform reaches usually found in natural channels and flood plains, and that the velocity distribution is logarithmic (11). The Manning equation has provided reliable results when used within the range of verified channel-roughness data. The selection of appropriate n values, however, requires considerable experience, even though extensive guidelines are available. Many studies based on hydraulic theory pertaining to flow resistance have been made and are summarized by Chow (11), Limerinos (23), and Carter, et al. (10). A study by Bray (8), who evaluated a number of equations used to predict roughness coefficients of gravel-bed streams, found that the equations of Limerinos (23) were the most accurate. The results of the theoretical and laboratory studies relating roughness to relative smoothness (a depth parameter divided by a particle size) are not entirely consistent. Verification using field data has not always been conclusive (23). This is probably due to a combination of several factors: (1) The theoretical and laboratory-derived relations are based on uniform flow (a condition rare or absent in natural channels), on particle size and shape, and on distribution of particle size; (2) few natural-flow streams exist in which some other factors do not affect channel roughness; and (3) there is an unknown model-to-prototype error associated with the theoretical and laboratory-derived relations. In addition, although theoretical solutions provide a sound description of the processes involved in flow resistance, the solutions have been tested only generally with flume data or are too complex in terms of data requirements for practical application. Barnes (4) presented verified n-value data, color photographs, and descriptive data for 50 stream channels. These verified data are for nearbankfull discharges but do not provide information on the change of Manning's roughness coefficient with depth of flow. Limerinos (23) presented verified K-value data for 11 streams and a predictive equation for Manning's n for various depths of flow as a function of relative smoothness. The equation for n developed by Limerinos (23) is: (0.0926)£1/6 1.16+ 2.0 log — dm in which du = the intermediate particle diameter, in ft, that equals or exceeds that of 84% of the particle diameters determined by methods described by Wolman (33). The other variables are as described earlier. Methods such as Limerinos' require particle-size data. These data may not be available due to the hydraulic conditions of the stream (such as large depths of flow and high flow velocities), and economic or time constraints. Studies have shown that many factors influence flow resistance. Chow (11), Fasken (15), and Aldridge and Garrett (1), expanded on a practical technique developed by Cowan (12) to aid in evaluating the total flow resistance in a channel reach. Total flow-resistance factors include crosssection irregularities, channel shape, obstructions, vegetation, channel meandering, suspended material, bed load, and channel and flood-plain 1520 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. conditions in agricultural or urban areas. A detailed description of estimating total flow resistance for these conditions can be found in Chow (11), Hejl (17), and Ree and Palmer (25). In general, all factors that tend to cause turbulence and retardance of flow, and thus energy loss, increase the roughness coefficient; those that result in smoother flow conditions tend to decrease the roughness coefficient. Normally, one n value is selected for the entire range of depth of flow. If the ratio of the depth of flow to the size of the roughness element (relative smoothness) is low, roughness is not constant. Most relations between roughness and depth of flow are too technical for general use and often involve variables that are not usually measured onsite. On high-gradient streams (for the purpose of this study, slope greater than 0.002), channel roughness can vary markedly with depth of flow. The available guidelines for selecting roughness coefficients for highgradient streams are based on limited verified roughness data, and, equally important, are handicapped by a lack of easily applied methods for evaluating the changes of roughness with depth of flow. Streams flowing on higher gradients generally have shallower depths of flow and larger bed materials that affect the flow resistance more than the bed materials in flatter slope streams having larger flow depths. Only a few studies have been made of the hydraulics of high-gradient channels. Judd and Peterson (20), Bathurst, et al. (6), and Peterson and Mohanty (24) have been primarily theoretically-oriented or concerned with methods to evaluate average flow velocity. A critical need exists to obtain additional data on high-gradient streams over a range of flow depths and to provide practical guidelines for evaluating channel-flow resistance. The investigation described here was conducted to provide data and improved methods for estimating the Manning roughness coefficient as well as other aspects of the hydraulics of high-gradient streams. COLLECTION OF DATA Seventy-five current-meter measurements of discharge using a Price AA meter and appropriate field surveys were made at 21 high-gradient natural stream sites in the Rocky Mountains of Colorado for the purpose of computing channel roughness by the Manning formula, and to evaluate other aspects of the hydraulics of high-gradient streams. A typical high-gradient stream, Lake Creek (Table 1, site 10) is shown in Fig. 1. These sites were selected to represent a wide range of channel type, flow width and depth, channel slope and roughness, and bed-material size. The reaches were straight and uniform and had a connected water surface and a stable bed and banks with minimal vegetation. The roughness measured reflects primarily the bed and bank-material roughness. The discharges at these sites ranged from low to high flows, and recurrence intervals ranged up to about 25 yr. The following description of field methods is brief because standard U.S. Geological Survey procedures were used to measure streamflow (7,9,13,29). The well-known methods of Wolman (33) were used to measure particle size. At each site, 3-5 cross sections were established and marked with metal stakes to define the reach of the stream. The cross 1521 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. TABLE 1.—Summary of Basic Data and Results Site number (1) 1. Velocity, Discharge,3 in feet Area, in in cubic Friction square Width, per Froude feet per second number slope feet in feet second (5) (6) (7) (2) (4) (3) Arkansas River at Pine Creek School, above Buena Vista 925 1,450 2,120 2,760 4,530 2. 69 73 78 79 80 3.72 4.30 5.27 6.11 8.65 0.35 0.35 0.40 0.45 0.60 0.026 0.023 0.021 0.025 0.026 Clear Creek near Lawson (latitude 39°45'57", 53 214 360 765 3. 249 340 407 454 526 43 71 102 141 42 46 49 52 1.25 3.00 3.58 5.48 0.22 0.42 0.44 0.59 0.015 0.017 0.018 0.019 Cottonwood Creek below Hot Springs, near Buena Vista 31 115 281 465e 4. 21 36 43 67 24 29 30 33 1.48 3.24 6.61 6.98 0.28 0.51 0.97 0.86 0.030 0.034 0.033 0.030 Crystal River above Avalanche Creek, near Redstone 83 272 530 1,220 60 112 161 220 204 224 233 577 2,300 3,710 123 125 135 226 443 528 5. 82 88 95 94 1.40 2.44 3.32 5.58 0.29 0.38 0.45 0.65 0.003 0.004 0.004 0.004 Eagle River below Gypsum (latitude 39°38'58", 6. 101 92 94 112 125 129 1.66 1.89 1.82 2.60 5.19 7.04 0.27 0.29 0.28 0.33 0.48 0.61 0.003 0.004 0.004 0.004 0.004 0.004 Egeria Creek near Toponas (latitude 40°02'12", 14 26 111 14 19 42 26 27 36 7. 0.98 1.36 2.63 0.24 0.28 0.42 0.003 0.003 0.002 Elk River at Clark (latitude 40°43'03", 39 254 1,050 1,410 59 72 81 90 39 105 185 272 1.01 2.42 5.73 5.21 1522 J. Hydraul. Eng. 1984.110:1519-1539. 0.22 0.35 0.66 0.53 0.003 0.004 0.006 0.006 £ x> 2 o "5 •LUO 1 > Q3 CD 3 0) "o T3 CD ft CD CL — ' s 5! 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Copyright ASCE. For personal use only; all rights reserved. Q ~o o 3 CO -J? I 52 •o CD >.-o X o 3 CO X i n 0) IN rH TS 3 O CM LO J. Hydraul. Eng. 1984.110:1519-1539. lie s I 00) 00 ^ o o o o 00) 106 tude 00) agi 106 Ion 908 :ude -43 -44 -37 -32 -14 3.61 4.66 5.22 5.75 6.58 3.24 3.99 4.46 4.85 5.51 0.026 0.022 0.020 0.024 0.023 agi 105 Ion Alpha (11) 0.081 0.074 0.071 0.074 0.074 0.142 0.132 0.112 0.110 0.086 1.25" 1.34" 1.45" 1.43" | obs irve | val (equation 9) rcenl 14) (1 3) Manning's Water | slope (8) atio TABLE 1.— (1) (2) (3) Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. 8. 12 94 242 122 224 252 264 (6) (7) 0.88 2.73 5.06 0.23 0.46 0.73 0.011 0.016 0.014 53 78 82 84 4.05 6.26 6.36 6.94 0.47 0.66 0.65 0.70 0.019 0.014 0.014 0.014 Lake Creek above Twin Lakes Reservoir (latitude 39°03'47", 148 830 1,360 68 147 185 11. 53 64 68 2.21 5.70 7.41 0.35 0.67 0.79 0.019 0.023 0.024 M a d Creek near Steamboat Springs (latitude 40°33'56", 48 92 331 409 32 46 91 127 2,920 3,170 1 12. 54 56 61 63 1.53 2.03 3.72 3.27 0.35 0.39 0.54 0.41 0.026 0.026 0.025 0.021 Piedra River a t Piedra (latitude 37°13'20", 109 110 419 451 13. 6.97 7.03 0.63 0.61 0.004 0.004 Rio G r a n d e at W a g o n w h e e l G a p (latitude 37°46'01", 103 453 680 151 2,060 4,040 116 152 170 1.47 4.56 5.94 0.28 0.46 0.52 0.004 0.004 0.003 Roaring Fork River a t G l e n w o o d Springs (latitude 39°32'37", 571 650 1,170 3,260 245 256 366 559 15. 145 147 158 170 2.34 2.56 3.19 5.83 0.32 0.35 0.37 0.57 0.002 0.002 0.003 0.003 San J u a n River a t Pagosa Springs (latitude 37°15'58", 2,700 3,175 16. (5) Hermosa Creek near H e r m o s a (latitude 37°25'19", 493 1,380 1,580 1,800 14. 29 32 32 14 35 48 9. 10. (4) Halfmoon Creek near Malta (latitude 39°10'20", 396 434 119 126 6.84 7.34 0.66 0.70 0.008 0.007 South Fork Rio Grande at South Fork (latitude 37°39'25", 70 800 l,450 e . 17. 48 157 271 49 64 75 1.51 5.12 5.36 0.27 0.58 0.50 0.009 0.007 0.007 Trout Creek near O a k Creek (latitude 40°18'44", 13 29 57 164 190e 22 23 25 26 33 11 14 19 31 54 1.23 2.11 2.97 5.36 3.54 1524 J. Hydraul. Eng. 1984.110:1519-1539. 0.31 0.48 0.59 0.87 0.49 0.016 0.017 0.016 0.013 0.013 Continued Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. (8) (12) (9) (10) (11) longitude 106°23'19") (Gaging station 07083000) (13) (14) 0.079 0.080 0.072 -28 28 73 0.087 0.052 0.054 0.049 0.076 0.065 0.065 0.063 -13 26 20 28 longitude 107°50'40") (Gaging station 07084500) 2.00d 0.019 1.20 1.28 0.098 0.023 2.30 1.11" 0.062 2.12, 1.08b 0.024 2.72 0.056 2.53 0.084 0.083 0.082 -15 33 47 0.106 0.100 0.091 0.081 -10 -7 11 -22 0.039 0.040 15 9 0.049 0.039 0.036 -15 -3 1 0.037 0.036 0.037 0.037 -16 -11 -13 14 0.050 0.049 20 32 0.087 0.043 0.052 0.064 0.050 0.048 -26 17 -7 0.089 0.065 0.053 0.033 0.064 0.091 0.091 0.084 0.074 0.070 2 41 60 123 10 0.011 0.016 0.015 0.50 1.05 1.42 0.48 1.09 1.50 C 1.40" 1.68b 0.109 0.062 0.042 longitude 107°50'40") (Gaging station 09361000) 0.019 0.014 0.014 0.014 2.23 2.85 3.03 3.36 2.30 2.87 3.07 3.14 1.64b 1.27b 1.21" 1.27b longitude 106°53'19") (Miscellaneous site) 0.026 0.026 0.027 0.023 0.60 0.80 1.40 1.92 0.59 0.82 1.49 2.02 1.56d 1.40d 1.36b 1.90" 0.117 0.108 0.082 0.105 longitude 107°20'32") (Gaging station 09349500) 3.80 0.034 1.20" 0.004 3.84 0.037 4.03 1.16" 0.005 4.10 longitude 106°49'51") (Gaging station 08217500) 0.004 0.004 0.004 0.89 2.97 3.98 0.89 2.98 4.00 c 1.30d 1.15d 0.058 0.041 0.035 longitude 10719'44") (Gaging station 09085000) 0.003 0.003 0.003 0.004 1.73 1.80 2.32 3.29 1.69 1.74 2.32 3.29 1.14d 1.25d 1.15d 1.10d 0.044 0.041 0.043 0.032 longitude 107°00'37") (Gaging station 09342500) 0.008 0.007 3.34 3.43 3.33 3.44 1.23b 1.39" 0.042 0.038 longitude 106b38'55") (Gaging station 08219000) 0.009 0.006 0.007 0.98 2.44 3.52 0.98 2.45 3.61 < 1.78d 1.18d longitude 107°00,34") (Miscellaneous site) 0.016 0.018 0.016 0.015 0.014 0.50 0.60 0.80 1.13 1.57 0.5 0.61 0.76 1.19 1.64 1.00d 1.46d 1.42d 1.11" 1.16" 1525 J. Hydraul. Eng. 1984.110:1519-1539. TABLE 1.— (1) Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. 18. (2) (3) (5) (6) (7) 46 51 73 110 3.27 5.46 0.45 0.66 0.027 0.031 White River above Coal Creek, near Meeker (latitude 4 0 W 1 8 " , 358 1,350 1,740 20. 61 88 95 154 276 314 2.40 4.91 5.54 1 0.32 0.49 0.54 0.002 0.003 0.004 Yampa River at Steamboat Springs (latitude 40°29'01", 86 335 1,170 1,870 68 86 103 105 63 117 250 282 1.37 2.89 4.68 6.64 0.25 0.44 0.53 0.71 0.006 0.006 0.005 0.005 Yampa River near O a k Creek (latitude 40°16'47", 21. Range: Minimum Maximum 1 Walton Creek near Steamboat Springs (latitude 40°34'39", 234 590 19. (4) 51 119 135 29 44 50 38 42 42 1.85 2.74 2.72 0.38 0.48 0.44 0.004 0.004 0.004 12 4,530 11 680 22 170 0.88 8.65 0.22 0.97 0.002 0.034 "Discharge does not exactly equal the product of area and velocity as they are b A natural channel with bridge piers, abutments, or manmade obstructions c Not available. d A natural trapezoidal-shaped channel without overbank flow and no bridge e Not used to develop prediction equation due to extreme bank vegetation. FIG. 1.—Typical High-Gradient Stream. Upstream View on Lake Creek above Twin Lakes Reservoir, Colo. 1526 J. Hydraul. Eng. 1984.110:1519-1539. Continued (8) (9) (10) (12) (11) (13) (14) 0.091 0.095 -11 28 0.032 0.034 0.038 -18 0 8 0.074 0.047 0.041 0.032 0.056 0.052 0.046 0.045 -24 11 11 44 Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. longitude 106°47'11") (Gaging station 09238500) 0.103 1.59 1.40" 0.074 2.16 1.34" longitude 107°49'29") (Gaging station 09304200) 0.027 0.034 1.63 1.87 0.002 0.003 0.004 1.80 3.10 3.25 2.52 3.14 3.31 c . 1.17" 1.11" 0.039 0.034 0.035 longitude 106°49,54") (Gaging station 09239500) 0.006 0.006 0.005 0.006 0.90 1.30 2.40 2.66 0.93 1.36 2.43 2.66 1.38" 1.38" 1.37" 1.54" longitude 106°50'50" (Miscellan sous site 0.004 0.005 0.005 0.76 1.10 1.20 0.76 1.05 1.19 1.19" 1.35" 1.03 d 0.041 0.034 0.038 0.049 0.047 0.048 20 38 25 0.002 0.039 0.50 5.51 0.48 6.58 1.00 2.00 0.028 0.159 0.032 0.106 -44 123 averages for the reach; similarly, other values are averages for the reach, which may affect the flow pattern. piers or other manmade obstructions. sections were spaced approximately one channel width apart. W h e n a current-meter m e a s u r e m e n t of discharge w a s m a d e , concurrent watersurface elevations were measured from about one channel w i d t h u p stream to one channel w i d t h d o w n s t r e a m of the site on each bank, including the e n d s of each cross section to define t h e water-surface profile. The average maximum wave w a s h u p d u e to extreme turbulence w a s taken to represent the high-water elevation, as d o n e by Barnes (4) a n d Limerinos (23). W h e n streamflow w a s low, a transit-stadia survey w a s made of each reach to obtain cross sections. The size distribution of the b e d materials w a s determined by measuring the intermediate diameters of sampled particles during low flows. For each particle, a determination also w a s m a d e to which axis—short, intermediate, or long—was closest to being vertical (here called axis orientation). Finally, r o u n d n e s s of the particle w a s determined using guidelines provided by Krumbein (22) a n d Wolman (33). Channel evidence indicated that the streambed material did n o t move during high flows measured in this study. DATA ANALYSIS AND INTERPRETATION Streamflow Data.—The simple forms of the M a n n i n g equation s h o w n 1527 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. in Eqs. 1 and 3 (13) are used only for uniform flow—that is, flow in a channel whose cross-sectional area does not vary within the reach. The energy equation for a reach of nonuniform channel between two sections (1 and 2) is (h + Mi = (h + Kh + (fy)i.2 + KUivhz (5) in which h = elevation above a common datum of the water surface at the respective section; hv = velocity head at the respective section = aV2/ 2 ? ; a =• velocity-head coefficient which is considered to be 1.0 for a uniformly shaped cross section; g = acceleration due to gravity = 32.2 ft/ sec2 (9.81 m/s 2 ); hf = energy loss due to boundary friction in the reach; Ahv = upstream velocity head minus the downstream velocity head; k{Ahv) - energy loss due to acceleration of velocity or deceleration of velocity in a contracting or expanding reach, respectively; and k = a coefficient equal to 0 for contracting reaches and 0.5 for expanding reaches (4). The friction slope, S, to be used in the Manning equation is thus defined as: hf_Ah S =^ = L + Ahv - k(Ahv) (6) in which Ah = the difference in water-surface elevation at the two sections; and L = the length of the reach (13). The quantity (1.49/n)AR2/3 in the Manning equation is called the conveyance, K, and is computed for each cross section. The mean conveyance in the reach between any two sections is computed as the geometric mean of the conveyance of the two sections. The discharge equation in terms of conveyance is: Q = K&S (7) 0.15 1 COTTONWOOD 1 (SITE 3) 0.14 ARKANSAS RIVER (SITE 1) - CREEK " 0,1 1 0-10 0.13 0.13 0,09 ~ } \\ 0.08 0.07 0.06 \ - 1 \l\ t \ \ 0.05 RIO GRANDE (SITE 13) HYDRAULIC 0.04 RADIUS, IN FEET \f\S \, . \l/\ \ < > /TROUT CREEK V ( S I T E 17) HYDRAULIC FIG. 2.—Relation of Manning's Roughness Coefficient to Hydraulic Radius SOUTH FORK RIO GRANDE SITE 16 . - RADIUS, IN FEET FIG. 3.—Relation of Manning's Roughness Coefficient to Hydraulic Radius, Showing Effects of Streambank Vegetation 1528 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. in which S = the friction slope as previously defined. In this investigation the average value of the Manning n was computed for each reach from the k n o w n discharge, the water-surface profile, and the hydraulic properties of the reach as defined by the cross sections. The equation applicable to a multisection reach of M cross sections which are designated 1, 2, 3, . . . M - 1, M is: 1.486 (h + hv), -(h + hv)M - [(fcAfo)L2 + (fcAfo)2.3 + ... + (kAhv\M-lyM] ^1.2 ZjZ 2 1 ^2.3 \- ^ ^ ^ -|MM-1)-M Z2Z3 Z( M _DZ M in which Z = AR2/3 and other quantities are as previously defined (4). Although Manning's n w a s c o m p u t e d for each subreach or combination of cross sections within the reach, an average value of n for each reach was adopted to represent the average conditions at the site. The average TABLE 2.—Summary of Bed-Material Data for Colorado Streams Statistical Size Distribution of Intermediate Diameter of Bed Material, in Feet," Shown in Following Percentiles Site number in Table 1 (1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 16 25 50 ^16 ^25 ^50 (2) 0.5 0.1 0.2 0.2 0.1 (3) 1.0 0.2 0.2 0.3 0.2 (4) 1.4 0.6 0.5 0.4 0.4 0.05d 0.7 0.3 0.8 1.0 0.4 0.4 0.3 0.5 0.4 0.5 0.2 0.7 0.2 0.4 0.1 d c C 0.3 0.1 0.3 0.4 0.1 0.05 0.2 0.3 0.1 0.2 0.1 0.1 0.1 0.1 0.4 0.2 0.4 0.6 0.2 0.05 0.2 0.4 0.1 0.3 0.1 0.3 0.1 0.1 C C Range Minimum 0.05 Maximum 0.4 0.05 1.0 0.2 1.4 Average Krumbein roundness" (9) 0.5 0.6 0.4 0.4 0.5 75 84 90 95 dre (5) 2.0 1.3 1.2 0.6 0.7 du (6) 2.6 1.8 1.4 0.6 0.8 d9o (7) 3.2 2.3 2.1 0.7 1.0 dgs c c c (8) 4.0 2.8 2.4 0.8 1.2 c C 1.0 1.1 1.2 1.5 0.9 0.8 0.4 0.7 0.8 0.9 0.4 1.3 0.2 0.7 1.3 1.3 1.5 2.0 1.2 0.9 0.5 0.8 1.1 0.9 0.5 1.6 0.3 0.9 1.5 1.4 1.6 2.2 1.4 1.2 0.6 1.0 1.2 1.0 0.5 2.0 0.3 1.1 1.7 1.7 1.9 2.4 1.6 1.6 0.6 1.1 1.3 1.3 0.6 2.5 0.4 1.4 0.5 0.6 0.3 0.4 0.5 0.4 0.6 0.6 0.7 0.5 0.6 0.5 0.5 0.4 c c 0.2 2.0 0.3 2.6 c c 0.3 3.2 0.4 4.0 "Determined by using methods of Wolman (33). Determined by using methods of Krumbein (22). c Data not available. d Estimated. b 1529 J. Hydraul. Eng. 1984.110:1519-1539. c 0.3 0.7 Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. TABLE 3.—Correlation Coefficients for Selected Hydraulic Characteristics for Colorado Streams8 (1) n S Manning's Water Bed coeffiFriction slope, material cient, n slope, S size, dm Sm (2) (5) (3) (4) 1.00 bw dM R D Q a — — — — — — — 0.71 1.00 — — — — — — 0.68 0.99 1.00 — — — — — Hydraulic radius, R (6) Hydraulic depth, D (7) -0.09 0.02 -0.02 0.33 1.00 -0.04 0.07 0.04 0.39 0.99 1.00 0.64 0.66 0.62 1.00 — — — — — — — — — Discharge, Alpha," a Q (8) (9) -0.23 -0.12 -0.14 0.12 0.91 0.88 1.Q0 — 0.52 0.52 0.51 0.79 -0.24 -0.24 -0.34 1.00 a For untransformed data. b For a natural trapezoidal-shaped channel without overbank flow and no bridge piers or other manmade obstructions. hydraulic properties for the reach and computed values of the Manning coefficient, n, are given in Table 1. Occasional inconsistencies in the data are due to difficulties in data collection as a result of the extremely turbulent flow conditions. The minimum and maximum values of each variable are given at the end of Table 1. The data in Table 1 indicate the marked variation of Manning's roughness, n, with depth in terms of hydraulic radius. The relation of Manning's roughness coefficient to the hydraulic radius of the four streams shown in Fig. 2 is typical of the relations of all of the streams listed. Roughness decreases markedly as depth of flow increases. This change indicates the need for developing relations between roughness and depth of flow. On three streams—Cottonwood Creek, South Fork Rio Grande, and Trout Creek—flow was affected by bank vegetation at the highest discharge. Dense willows created additional turbulence and increased channel roughness markedly. The relation of Manning's roughness coefficient to the hydraulic radius of these three streams is shown in Fig. 3. This indicates that dense vegetation can have a marked effect on total flow resistance and should be accounted for. Particle-Size Data.—The intermediate-axis particle-size data on the bed material and the average Krumbein roundness are summarized in Table 2. Correlation coefficients showing the relation between selected hydraulic data and bed-material data are shown in Table 3. Data on axis orientation indicates that the particle offers the least resistance to flow; that is, when the short axis is vertical. PREDICTION EQUATION FOR MANNING'S ROUGHNESS COEFFICIENT Most equations used to predict channel roughness require streambed particle-size information (5,10,11,23,30). Studies by Golubtsov (16), Riggs (27), and Ayvazyan (3) indicate that channel roughness is directly related to channel gradient in natural stable channels. Ayvazyan (3) evaluated a number of formulas used worldwide to evaluate channel roughness in 1530 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. earthen canals and found that the equations yield basically equivalent results, but do not truly reflect the nature of hydraulic resistance. Ayvazyan (3) showed the reason for disagreement was the failure to allow for the effect of slope. This relation of resistance and slope is due, in part, to the interrelation between channel slope and particle size of the bed material. As slope increases, finer material is removed and larger particles remain in the channel. The effect of increased turbulence and resistance results in increased friction slope. The correlation coefficients for selected hydraulic characteristics of the data in Table 1 are shown in Table 3. The coefficient for Manning's n is higher for friction slope (0.71) than for dm particles size (0.64). This supports the idea that slope has a strong influence on roughness. For similar bed-material size, channels with low gradients have much lower n values than channels with high gradients. Values of n as small as 0.032 have been obtained for channels having very low gradients, shallow depths, and large boulders (4). This implies that the channel roughness associated with streambedmaterial size can be evaluated in terms of the more easily obtained friction slope. The relation of Manning's roughness coefficient to friction slope, which indicates that roughness increases with slope, is shown in Fig. 4. The scatter is due to the at-a-site decrease of roughness with increasing depth of flow. There also is much greater scatter, or change - . .. 1 : • •• • •. . ! t I o : • ! S.• - ' OBSERVATIONS . ONE A TWO a FOUR . 1 0.001 0.002 0.005 0.01 0.02 FRICTION SLOPE, S FIG. 4.—Relation of Manning's Roughness Coefficient to Friction Slope FRICTION SLOPE, FIG. 5.—Relation of Manning's Roughness Coefficient to Friction Slope and Hydraulic Radius 1531 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. in the roughness coefficient, on higher gradient streams. The method for predicting channel roughness uses multiple-regression analysis, which related Manning's roughness coefficient to the easily measured hydraulic characteristics shown in Tables 1 and 2. Multiple-regression analyses were performed using several different types of equations (arithmetic, polynomial, semilogarithmic, and logarithmic or power) to determine the best type of equation to estimate channel roughness. The three highest measurements for Cottonwood Creek, South Fork Rio Grande, and Trout Creek were not used to develop the equation because of the extreme effect of bank vegetation. The resulting equation from the multiple-regression analyses developed for predicting Manning's n in steep natural channels is n = 0.39 s03BR-°-16 (9) and is graphically depicted in Fig. 5. In Eq. 9, S = the friction slope. However, the data in Table 1 indicate that the water-slope values are about the same and could be used interchangeably for fairly uniform channels. Similarly, Table 1 indicates that the values of hydraulic radius and hydraulic depth were approximately the same and could be used interchangeably. The average standard error of estimate of Eq. 9 is 28% and ranges from -24-32% for the data in Table 1. Eq. 9 was used to predict n for the sites, and the percent deviation of computed from observed values is also shown in Table 1. The algebraic mean of percentage differences was 5.8 and ranged from -44-123%, indicating that Eq. 9 tends to slightly overestimate n. The standard deviation of the percentage differences was 31%. The w-values having the greatest error are typically low-flow measurements when the ratio of R-dso is less than 7 for the data in Table 1. The concept of flow resistance at low flows may be subject to question due to nonconnection of the water surface. Another explanation is that ongoing research indicates that the vertical-velocity profile is S-shaped rather than logarithmic with much lower bottom velocities and higher surface velocities in shallow, steep, cobble-and-boulder bed streams. Overall, the measured velocity is too small; thus, the n-value would be somewhat overpredicted. Roughness-prediction equations such as Eq. 4 were developed in terms of relative smoothness. However, the standard error of estimate for Eq. 4 and for a similarly developed relative smoothness type equation was considerably higher, biased, and did not fit the data as well as Eq. 9. Data from Barnes (4) and Limerinos (23) were used to determine whether Eq. 9 produced reasonable results and to determine the equation's range of applicability. These data are based on 59 observations of n in which slopes are greater than 0.002, and the hydraulic radii are less than 7 ft (2.1 m). The algebraic mean of the percentage differences in the results was —7.8. The standard deviation of the percentage differences was 23% and ranged from —44-50%. RESULTS Although Eq. 9 provides a good means of estimating channel roughness, there are several explanations for the error associated with the 1532 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. equation. Eq. 9 predicts the average roughness of the reach of a stream rather than of an individual subreach. The flow conditions in high-gradient streams are extremely turbulent and add an unknown component to the measurement error, which is not present in more tranquil streams. Although the channel reaches selected were primarily uniform to slightly contracting, there were some cases where expansion did occur and could not be avoided. Expanding reaches also affected the stream data collected by Barnes (4) and Limerinos (23) and probably most other natural channel data. Although a detailed investigation was not made, it was noted that in some cases the energy loss in terms of the variation of observed Manning's n was as much as 61% higher in expanding reaches than in contracting reaches. There were no measurable differences in bed material throughout each reach, indicating that the energy losses were due to channel expansion. These losses could pose serious problems in hydraulic studies of streams because these studies encompass many expanding reaches. REGIME EQUATIONS FOR VELOCITY AND DISCHARGE One of the reasons for developing an equation to predict Manning's roughness coefficient (Eq. 9) using friction slope and hydraulic radius as power variables was that the Manning equations (Eqs. 1 and 3) are in the form of power equations. Therefore, for the simple case of uniform flow and no other factors affecting bank roughness, Eq. 9 could be directly substituted into the Manning equations (Eqs. 1 and 3). Substituting Eq. 9 into Eqs. 1 and 3 results in regime equations for velocity: V = 3.81 R°-83S0-12 (10) and for discharge Q = 3.81 ARomS012 (11) in which the variables are the same as defined previously. Eq. 10 was also derived by multiple-regression techniques using velocity as the dependent variable, and Eq. 11 was derived by substituting Eq. 10 into the continuity equation, Eq. 2. These equations provide a means of solving directly for velocity and discharge in uniform natural channels without the need for subjectively evaluating channel roughness. LIMITATIONS OF PREDICTION AND REGIME EQUATIONS The following restrictions need to be observed when using the previously developed equations to predict the Manning's n (Eq. 9), the velocity (Eq. 10), and the discharge (Eq. 11) of high-gradient streams: 1. The equations are applicable to natural main channels having stable bed and bank materials (gravels, cobbles, and boulders) without backwater. 2. The equations can be used for slopes from 0.002-0.04 and for hydraulic radii from 0.5-7 ft (0.15-2.1 m). The upper limit on slope is due to a lack of verification data available for the slopes of high-gradient streams. Results of the regression analyses indicated that for hydraulic 1533 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. radius greater than 7 ft (2.1 m), n did not vary significantly with depth; thus extrapolation to larger flows should not be too much in error as long as the bed and bank material remain fairly stable. 3. The energy-loss coefficients were assigned the values 0 and 0.5. 4. Hydraulic radius does not include the wetted perimeter of bed particles. 5. These equations are applicable to streams having relatively small amounts of suspended sediment. FLOW REGIME IN STEEP CHANNELS Standard hydraulic theory and analysis indicate that when slope exceeds critical slope—that is, when the Froude number exceeds unity— higher velocities and supercritical flow result. Peterson and Mohanty (24) observed extended reaches of supercritical flow; however, these were observed in high-gradient flumes. The field data collected for this study (Table 1) included slopes as steep as 0.052 and indicate that the Froude numbers for flow in high-gradient streams are less than unity (4,5,8,20,21,23). The combined effects of channel and cross-section variations create extreme turbulence and energy losses that result in increased flow resistance. The characteristic turbulence of a high-gradient stream is shown in a photograph (Fig. 6) taken at the Arkansas River site (Table 1, site 1). Studies of the flow resistance of boulder-filled streams indicated that there is a spill-resistance component with increasing flow (18,26). Spill resistance is a result of increased turbulence or roughness resulting from the velocity of water striking the large area of protruding bedroughness elements and eddy currents set up behind the larger boulders. Aldridge and Garrett (1) believe the effect of the disturbance of water surrounding boulders and obstructions increases with velocity and may overlap with nearby obstruction disturbances and further increase FIG. 6.—Characteristic Turbulence. Upstream View at Arkansas River at Pine Creek School, above Buena Vista, Colo. (Discharge is 4,530 cu ft/sec = 128 m3/s) 1534 J. Hydraul. Eng. 1984.110:1519-1539. 1.0 1 ' 0.9 - 0.8 - Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. 1 1 9 a - 0 - s 9 e „ tr dQ ° - 1 e • 6 a * d a e 9 9 • a ft 0 a 0 s a . Q 0.4 tt 0.3 • 9 A • 0 0 " 0.1 0 0.000 A 9 " I A 9 9 A 0 . A 0.2 ' 9 0 0 OBSERVATIONS . * • ONE A TWO 1 0.005 0.010 0.015 FRICTION 0.020 0.025 0.030 0.035 SLOPE, S FIG. 7.—Observations of Froude Number and Various Friction Slopes turbulence and thus, roughness. Bathurst (5) noted supercritical flow over boulders in natural channels and hydraulic jumps occurring just downstream, but these were very limited in areal extent. Very localized areas of supercritical flow were observed during the collection of data for this study where flow went over boulders. Observations of Froude number and various friction slopes for the data in Table 1 are shown in Fig. 7. The Froude number is computed as: V (12) where F = the Froude number and the other variables are as previously defined. Froude numbers less than 1, indicating subcritical flow, were characteristic of all sites. There also does not appear to be any tendency for Froude numbers to increase with friction slope. The question remains as to whether flow becomes supercritical (Froude number exceeds 1) at higher flood discharges. At higher flows, channelbank roughness (vegetation or bank irregularities) can markedly increase total channel roughness as shown in Fig. 3. During larger floods, channel erosion is common. Usually when large floods occur in small steep basins, large amounts of channel erosion occur and sediment is subsequently transported. About 60,600 tons (55,000 metric tons) of sediment were eroded from the valley floor of Loveland Heights tributary to the Big Thompson River, Colo., during the 1976 flood (2). Additional energy is consumed in transporting the bed material. Flume studies by Bathurst et al. (6) indicate there is a sharp increase in resistance when bed material moves. Rubey (28) showed that energy is required to move sediment, and transport of fine-grained sediment tends to reduce turbulence and, thus, flow resistance. However, this decrease in resistance is normally offset by a much greater increase in flow resistance caused by the formation of dunes (32). The movement of the bed material probably 1535 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. makes the channel react like an alluvial channel in which bed forms produce standing waves and additional energy losses as a result of an increase in friction. Critical and supercritical flow can occur locally in these channels, in smooth bedrock channels, and in fine-grained alluvial channels. Dobbie and Wolf (14), Thompson and Campbell (31), and the writer believe that during large floods, n values are much higher than those normally selected and that flows in high-gradient natural channels containing cobbles and boulders generally approach, but do not exceed, critical flow. For these conditions of high-gradient streams and extreme flows, a limiting assumption of critical depth in subsequent hydraulic analyses appears reasonable. Chow (11) provides information on the critical-depth method for computing discharge. VELOCITY HEAD COEFFICIENT Eqs. 5, 6, and 8 require the computation of the velocity head V2/2g. Streambed roughness, cross-section irregularities, channel variations, obstructions, vegetation, channel meandering, and other factors cause velocity in a channel to vary from point to point. Because of this variation in velocity, velocity head is greater than the value computed from V2/2g. True velocity head is expressed as aVz/2g, where alpha is velocity head coefficient. Velocity head coefficient, or kinetic energy coefficient (11) is computed as: in which v = the measured velocity in an elementary area A A and the other variables are as previously defined. Alpha was computed from the discharge measurements made using Eq. 13 and the values are shown in Table 1. These values are based on the average velocity in the vertical subarea rather than on the vertical-velocity distribution (from multiplepoint velocity measurements) in each subarea because these data were not available. Hulsing and others (19) showed that the values of alpha computed from multiple-point velocity measurements were similar to the one-point (0.6 depth) and two-point (0.2 and 0.8 depth) velocity measurements. The two-point method of determining velocity was used for the majority of the measurements in this study. The values of alpha shown in Table 1, which ranged from 1.0-2.0, had the correlation coefficients shown in Table 3 indicate a slight tendency for alpha to decrease with discharge and depth. There also is a slight tendency for alpha to increase with channel roughness, slope, and particle size. Because the alpha values consist of two subsets, one including natural channels and another including channels having roanmade obstructions, subsequent analyses were made on each subset. Attempts made to develop relations between alpha and other hydraulic or bed-material properties were unsuccessful because of the complexity of the changes in alpha and these variables. The mean of all the alpha values and the means of the two subsets ranged from 1.33-1.34. Inspection of the values of alpha in Table 1 indicates the values are much greater than the value of 1 assumed in Eqs. 5, 6, and 9. However, 1536 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. the solution of Eq. 8 for a multisection reach involves an evaluation of the difference between the alpha coefficients of upstream and downstream sections. Therefore, although the value of alpha may be greater than 1.0, what is important and consequently what would affect the accuracy of the computed n value is the relative difference between alpha upstream and downstream. It would be nearly impossible to measure the alpha of all cross sections at high flows. The majority of the reaches used are basically uniform throughout, although slightly contracting, as indicated by the basic hydraulic properties. Therefore, the higher values of alpha should not introduce much error in the computed n values. However, those studies that evaluate the velocity head to determine mean channel velocity at one cross section, such as techniques using superelevation around bends, slope-conveyance, or dynamic uprise (velocity head buildup), could be in considerable error in assuming alpha equals 1 in a simple trapezoidal-shaped channel cross section. SUMMARY AND CONCLUSIONS Hydraulic calculations of the flow in channels and overbank areas of flood plains require an evaluation of roughness characteristics. Most commonly, Manning's roughness coefficient is used to describe the flow resistance or relative roughness of a channel or overbank area. Available guidelines for selecting the roughness coefficients of high-gradient streams have not been verified. Field surveys and 75 current-meter measurements of discharge were made at 21 high-gradient natural stream sites in the Rocky Mountains of Colorado for the purpose of computing channel roughness by the Manning formula and to evaluate other hydraulic characteristics of high-gradient streams. These data indicate that Manning's roughness coefficient decreases markedly with depth and increases with friction slope. Multiple-regression techniques were used to develop a method for predicting Manning's roughness coefficient from the easily measured friction (or water) slope and hydraulic radius (or hydraulic depth). The average standard error of the prediction equation is 28% and ranges from —2432%. The equation was verified using other available data on high-gradient streams. Regime equations were developed to estimate velocity and discharge without requiring a subjective estimate of channel roughness. The data collected for this and other studies indicate that Froude numbers for flow in high-gradient, natural mountain streams are generally less than unity. There is no tendency for Froude numbers to increase with slope. Subcritical flow is due to the combined effects of channel and cross-section variations, bank roughness, spill resistance, and to an increase in the effect of these factors with increasing discharge, which creates extreme turbulence and energy losses that result in increased flow resistance. During floods, energy is required to transport eroded material; thus, flow conditions may approach, but probably do not exceed, critical flow, except in very localized areas in the channel. Alpha, the velocity-head coefficient, was computed from discharge measurements and was found to range from 1.0-2.0 in these high-gradient streams. Alpha showed a slight tendency to increase with an in1537 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. crease in M a n n i n g ' s r o u g h n e s s coefficient, b u t this w a s n o t always the case. Streambed particle-size data were obtained a n d presented but not used directly in the analysis of flow resistance. The bed-material size (rf84) ranged from 0.3-2.6 ft (0.1 to 0.8 m). ACKNOWLEDGMENTS This study was authorized by a cooperative agreement between the U.S. Geological Survey a n d the State of Colorado, D e p a r t m e n t of Natural Resources, Colorado Water Conservation Board. I would like to thank my colleagues at the Colorado District for their help with field work a n d preparation of the data. APPENDIX.—REFERENCES 1. Aldridge, B. N., and Garrett, J. M., "Roughness Coefficients for Streams in Arizona," U.S. Geological Survey Open-File Report, Feb., 1973. 2. Andrews, E. D., and Costa, J. E., "Stream Channel Charges and Estimated Frequency of a Catastrophic Flood, Front Range of Colorado," Geological Society of America Abstracts with Programs, Vol. 11, No. 7, 1979, p. 379. 3. Ayvazyan, O. M., "Comparative Evaluation of Modern Formulas for Computing the Chezy Coefficient," Soviet Hydrology, Vol. 18, No. 3, 1979, pp. 244-248. 4. Barnes, H. H., Jr., "Roughness Characteristics of Natural Channels," USGeological Survey Water-Supply Paper 1849, 1967. 5. Bathurst, J. C , "Flow Resistance of Large-Scale Roughness," Journal of the Hydraulics Division, ASCE, No. HY12, Dec, 1978, pp. 1587-1603. 6. Bathurst, J. C , Li, R-M, and Simons, D. B., "Hydraulics of Mountain Rivers," Report No. CER78-79JCB-RML-DBS55, Civil Engineering Department, Colorado State University, Fort Collins, Colo., 1979. 7. Benson, M. A., and Dalrymple, T., "General Field and Office Procedures for Indirect Discharge Measurements," U.S. Geological Survey Techniques of WaterResources Investigations, Book 3, Chpt. A-l, 1967. 8. Bray, D. I., "Estimating Average Velocity in Gravel-Bed Rivers," Journal of the Hydraulics Division, ASCE, No. HY9, Sept., 1979, pp. 1103-1122. 9. Buchanan, T. J., and Somers, W. P., "Discharge Measurements at Gaging Stations," 17.5. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chpt. A-8, 1969. 10. Carter, R. W., et al., "Friction Factors in Open Channels," Journal of the Hydraulics Division, ASCE, No. HY2, Mar., 1963, pp. 97-143. 11. Chow, V. T., Open Channel Hydraulics, New York, McGraw-Hill, 1959. 12. Cowan, W. L., "Estimating Hydraulic Roughness Coefficients," Agricultural Engineering, Vol. 37, No. 7, July, 1956, pp. 473-475. 13. Dalrymple, T., and Benson, M. A., "Measurement of Peak Discharge by the Slope-Area Method," U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chpt. A-2, 1967. 14. Dobbie, G. H., and Wolf, P. O., "The Lynmouth Flood of August 1952," Proceedings of the Institute of Civil Engineers, Part 2, 1953, pp. 522-588. 15. Fasken, G. B., "Guide for Selecting Roughness Coefficient 'n' Values for Channels," Soil Conservation Service, U.S. Department of Agriculture, D e c , 1963. 16. Golubtsov, V. V., "Hydraulic Resistance and Formula for Computing the Average Flow Velocity of Mountain Rivers," Soviet Hydrology, No. 5, 1969, pp. 500-510. 17. Hejl, H. R., "A Method for Adjusting Values of Manning's Roughness Coef1538 J. Hydraul. Eng. 1984.110:1519-1539. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 08/21/13. Copyright ASCE. For personal use only; all rights reserved. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. ficients for Flooded Urban Areas," U.S. Geological Survey, Journal of Research, Vol. 5, No. 5, Sept.-Oct., 1977, pp. 541-545. Herbich, J. B., and Shulits, S., "Large-Scale Roughness in Open-Channel Flow," Journal of the Hydraulics Division, ASCE, No. HY6, Nov., 1964, pp. 203-230. Hulsing, H., Smith, W., and Cobb, E. D., "Velocity-Head Coefficients in Open Channels," U.S. Geological Survey Water-Supply Paper 1869-C, 1966. Judd, H. E., and Peterson, D. F., "Hydraulics of Large Bed Element Channels," Report No. PRWG17-6, Utah Water Research Laboratory, Utah State University, Logan, Utah, Aug., 1969. Kellerhals, R., "Stable Channels with Gravel-Paved Beds," Journal of the Waterways and Harbors Division, ASCE, No. WW1, Feb., 1967, pp. 63-84. Krumbein, W. C , "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology, Vol. 11, No. 2, Aug., 1941, pp. 64-72. Limerinos, J. T., "Determination of the Manning Coefficient From Measured Bed Roughness in Natural Channels," U.S. Geological Survey Water-Supply Paper 1898-B, 1970. Peterson, D. F., and Mohanty, P. K., "Flume Studies of Flow in Steep, Rough Channels," Journal of the Hydraulics Division, ASCE, No. HY9, Nov., 1960, pp. 55-76. Ree, W. O., and Palmer, V. J., "Flow of Water in Channels Protected by Vegetative Linings," U.S. Soil Conservation Service, Department of Agriculture, Technical Bulletin 967, 1949. Richards, K. S., "Hydraulic Geometry and Channel Roughness-—A Non-Linear System," American Journal of Science, Vol. 273, 1973, pp. 877-896. Riggs, H. C , "A Simplified Slope-Area Method for Estimating Flood Discharges in Natural Channels," Journal of Research, U.S. Geological Survey, Vol. 4, No. 3, May-June, 1976, pp. 285-291. Rubey, W. W., "Equilibrium Conditions in Debris-Laden Streams," Transactions, American Geophysical Union, June, 1933, pp. 497-505. Savini, J., and Bodhaine, G. L., "Analysis of Current-Meter Data at Columbia River Gaging Stations, Washington and Oregon," U.S. Geological Survey WaterSupply Paper 1869-F, 1971. Simons, D. B., and Senturk, F., "Sediment Transport Technology," Water Resources Publications, 1977. Thompson, S. M., and Campbell, P. L., "Hydraulics of a Large Channel Paved with Boulders," Journal of Hydraulic Research, Vol. 17, No. 4, 1979, pp. 341354. Vanoni, V. A., and Nomicos, G. N., "Resistance Properties of SedimentLaden Streams," Transactions, ASCE, Vol. 125, 1960, pp. 1140-1175. Wolman, M. G., "A Method of Sampling Coarse River-Bed Material," Transactions, American Geophysical Union, Vol. 35, No. 6, 1954, pp. 951-956. 1539 J. Hydraul. Eng. 1984.110:1519-1539.