Determination of the manning coefficient from measured bed roughness in natural channels

This report presents the results of a study to test the hypothesis that basic values of the Manning roughness coefficient of stream channels may be related to (1) some characteristic size of the streambed particles and to (2) the distribution of particle size. These two elements involving particle size 'can be combined into a single element by weighting characteristic particle sizes. The investigation was confined to channels with coarse bed material to avoid the complication of bed-form roughness that is associated with alluvial channels composed of fine bed material. Fifty current-meter measurements of discharge and appropriate field surveys were made at 11 sites on California streams for the purpose of computing the roughness coefficient, n, by the Manning formula. The test sites were selected to give a wide range in average size of bed material, and the discharge measurements and surveys were made at such times as to provide data covering a suitable range in stream depth. The sites selected were relatively free of the extraneous flow-retarding effects associated with irregular channel conformation and streambank vegetation. The characteristic bed-particle sizes used in the analyses were the 16,50and 84-percentile sizes as obtained from a cumulative frequency distribution of the diameters of randomly sampled surficial bed material. Separate distributions were computed for the minimum and intermediate values of the three diameters of a particle. The minimum diameters of the streambed particles were used in the study because a particle at rest on the bed invariably has its minimum diameter in the vertical position; this diameter is, therefore, the most representative measure of roughness height. The intermediate diameter was also studied because this is the diameter most easily measurable either by sieve analysis or by photographic techniques and because it is the diameter that had been used in previous studies by other investigators. No significant difference in reliability was found between the results obtained using minimum diameters and those obtained using intermediate diameters. Bl B2 STUDIES OF FLOW IN ALLUVIAL CHANNELS In analyzing the field data, the roughness parameter, -£ (where R is /C/o R hydraulic radius), was related to relative smoothness, (where d is a chara acteristic, or weighted characteristic, particle size). The parameter-^-, rather /CTO than n, was used because it is directly proportional to the square root of the Darcy-Weisbach friction factor, /, which is more widely used in theoretical studies of hydraulic friction. If the transformation of -^7to V / is made, /f the relations obtained in this study are of a form that is identical with that of the theoretical friction equation obtained by several investigators and that derived from field data by Leopold and Wolman (1957). The constants in the equation vary, of course, with the characteristic particle size used. The relations best fitting the field data for this study were obtained by using either a characteristic particle diameter equal to the 84-percentile size (dg4, the size equal to, or exceeding, that of 84 percent of the streambed particles), or a diameter obtained by weighting three characteristic particle sizes (dw, the size obtained by assigning a weight of 0.1 to d16, a weight of 0.3 to dgo, and a weight of 0.6 to d84 ). The use of d84 alone gave slightly better results than the use of dw, and, in addition, the use of d84 alone is attractive from a standpoint of simplicity. It is difficult, however, to rationalize the use of d84 alone because of the implication that the distribution of sizes is irrelevant, and it matters not at all whether 84 percent of the bed material is sand or whether it is large cobbles, as long as 16 percent of the material is of greater size. Consequently, the author recommends the use of dw rather than <J84 , although there was no unanimity of opinion on this recommendation among his colleagues who reviewed this paper. The reader is free to make his own choice between the use of d84 and dw. As an adjunct to this study an attempt was made to relate the roughness parameter,-^-, to the ratio of velocities at two-tenths depth (V02y) and at /C7o eight-tenths depth (VOSv ). These are the two depths at which velocity is usually observed when current-meter measurements of discharge are made. Twenty-seven sets of such velocity data, obtained at seven sites, were available from the discharge measurements made in connection with this study of the effect of streambed-particle size. If a suitable relation were found, the myriad of current-meter measurements of discharge in U.S. Geological Survey files could be used to compile an extensive library of n values for streams in the United States. Unfortunately, the range in magnitude of the 27 values of -^7was small, and therefore the small standard error of estimate obtained /£% by correlating -^7with the velocity ratio had little significance by itself. #% The correlation coefficient, 0.41, was low, and it was therefore concluded that, on the basis of the available data, the velocity ratio is an unsatisfactory index of the roughness parameter. MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B3 INTRODUCTION The Chezy equation with the empirically determined resistance coefficient n, ascribed to Manning (1891), has been used by many engineers in design of waterways and in other hydraulic applications. Under steady-state, uniform-flow conditions, satisfactory results can be obtained by the equation S* n where t* V=mean velocity, -, radius, -^ , S=energy slope, w=Manning roughness coefficient, Q= total discharge at a cross section, A cross-sectional area of the channel, and P=wetted perimeter.1 Equation 1, commonly referred to as the "Manning formula," has been used extensively as an indirect method for computing flows in natural channels. Although this equation was theorized and developed for uniform-flow conditions, adjustments for assumed energy losses attributable to an expanding or contracting river reach permit its use in channels with gradually varied flow. These assumptions have led to relatively reliable results when used within the range of verified channel-roughness data. Confident selection of values of the Manning roughness coefficient, n, usually requires considerable experience. Tables of computed n values for various channel conditions, and illustrations and stereoscopic color slides of channels for which n has been verified, are available to help the practicing engineer select an appropriate n value. Barnes (1967) has presented color photographs and descriptive data for 50 stream channels, including the average values of the Manning roughness coefficient for each channel. These aids show various channel conditions and the combined effect of the several factors that influence channel resistance. However, the lack of complete similarity in channel conditions and geometry from stream to stream makes it difficult to estimate channel roughness from illustrations and stereoscopic slides. 1 In this paper a uniform set of symbols has been adopted. Symbols in formulas of other authors have been changed where necessary for consistency, but the definitions remain the same.