Re-analysis and Correction of Bedload Relation of Meyer-peter and Müller Using Their Own Database

The pioneering predictor of fluvial bedload transport rate proposed by Meyer-Peter and Müller in 1948 is still extensively used in basic research and engineering applications. A review of the basis for its formulation reveals, however, that an unnecessary bed roughness correction was applied to cases of plane-bed morphodynamic equilibrium. Its inclusion followed a flow resistance parameterization in terms of the Nikuradse roughness height, which has been shown (well after the publication of their work) to be inappropriate for the characterization of mobile bed rough conditions in rivers. Removing the unnecessary correction and incorporating an improved correction of the boundary shear stress due to sidewall effects allow elucidation of the most parsimonious form of the bedload relation of Meyer-Peter and Müller that is dictated by their own data set. The new predictor is presented in terms of two alternative power law forms. These amended forms show that, in the case of lower-regime plane bed equilibrium transport of uniform bed sediment, the new estimates of volume bedload transport rates are less than or equal to half the values that would be obtained with the original relation of Meyer-Peter and Müller in the absence of the unnecessary bed roughness correction. The meticulous database and clear analysis of the original work of Meyer-Peter and Müller greatly aided the present authors in their re-analysis, which liberally uses the hindsight offered by 57 years of subsequent research. CE Database subject headings: Bed load, Boundary shear, Flow resistance, Wall friction, Bedforms, Flumes 1 Graduate student, St. Anthony Falls Laboratory, University of Minnesota, #2 3 Ave SE, Minneapolis, MN 55414, Phone: (612)-624-4675, Fax: (612)-624-4398, e-mail: [email protected] 2 Professor, Ven Te Chow Hydrosystems Laboratory, University of Illinois, 205 N Mathews Ave, Urbana, IL 61801, Phone: (217)-244-5159, Fax: (217)-333-0687, e-mail: [email protected] Manuscript number HY/2004/023810 Wong and Parker (2005) C:\Old Data\mwe\research topic on tracers\meyer-peter\SECOND revised draft 02-asce jhe.doc Page 2 Introduction A focus of current research on sediment transport in rivers, particularly in the case of gravel-bed streams, is on developing more accurate predictors of the bedload transport rate. Estimating this rate usually involves relating it to mean characteristics of the driving force of the flow and the corresponding reach-averaged resistance properties of the bed. More than a century has passed since the introduction of the first mechanistic relation of this type by Du Boys in 1879 (Ettema and Mutel 2004), but still no formulation can be claimed to be of universal applicability. There is a lively debate, for instance, concerning the effects of turbulence and bed heterogeneities at micro and macro scales on bedload transport rates, but no agreement has been reached (see e.g., Ashworth et al. 1996; Cao and Carling 2002; Wilcock 2004). One of the formulae most widely used in laboratory and field investigations as well as in numerical simulations of bedload transport is the empirical relation proposed by Meyer-Peter and Müller (1948; abbreviated to “MPM” below). This relation was derived from experiments carried out during a period of sixteen years in the flume facilities of the Laboratory for Hydraulic Research and Soil Mechanics of the Swiss Federal Institute of Technology (ETH) at Zürich, Switzerland. It allows estimation of the bedload transport rate in an open-channel, as a function of the excess bed shear stress applied by the flowing water. MPM worked out their relation from a comprehensive experimental data set for equilibrium bedload transport under steady, uniform flow that included 135 runs from ETH and 116 more runs from the set due to Gilbert (1914; abbreviated as “GIL” below). The data pertain to both sediment of uniform size and size mixtures, various values of sediment specific gravity, and cases both with and without the presence of bedforms. The original data of MPM were included in an internal report of ETH, but were not published until Smart and Jäggi’s paper (1983) on bedload transport on steep slopes. Review of the experimental techniques and data analysis carried out by MPM reveals a meticulous attention to detail. The bedload transport relation of MPM has been used extensively for almost six decades. With only very few exceptions, this usage has not been accompanied by detailed re-analysis of the formula itself, the data on which it is based, and its range of validity. Re-analysis to date has concentrated on extension of the formula to: (i) channels that are steeper than those of the experiments by MPM (Smart and Jäggi 1983; Smart 1984), and (ii) poorly-sorted sediment mixtures (Hunziker 1995; Hunziker and Jäggi 2002). The simplest and most common case to which the relation of MPM is typically applied, i.e. the transport of uniform sediment over a flat bed with a slope not exceeding 0.02, however, does not appear to have been revisited in the sole context of the original data used by Manuscript number HY/2004/023810 Wong and Parker (2005) C:\Old Data\mwe\research topic on tracers\meyer-peter\SECOND revised draft 02-asce jhe.doc Page 3 MPM. Hence the question arises: Does the relation of MPM in fact fit the data used in its derivation? Answering this is of significance not only because the MPM bedload transport predictor is commonly used for comparative purposes in basic research (see e.g., Abdel-Fattah et al. 2004; Barry et al. 2004; Bettess and Frangipane 2003; Bolla Pittaluga et al. 2003; Bravo-Espinosa et al. 2003; Defina 2003; Gaudio and Marion 2003; Knappen and Hulscher 2003; Martin 2003; Mikoš et al. 2003; Nielsen and Callaghan 2003; Ota and Nalluri 2003; Singh et al. 2004; Yang 2005), but also because it is frequently used in engineering applications (it is one of the relationships available in HEC-6, a computer software used for sediment transport calculations that is very popular in the USA). This paper is devoted to a thorough review of the basis for the formulation of the MPM relation, with special attention to the procedures by which bedform and sidewall corrections were applied by MPM. Hindsight offered by the results of more current research on sediment transport and flow resistance in rivers reveals that a) the form drag correction used by MPM in analyzing their data for plane-bed conditions is unnecessary, and b) if this unnecessary correction is omitted, then the MPM bedload transport relation needs to be modified in order to accurately reproduce the experimental observations used to derive it. The analysis presented here culminates in an amended form of the MPM relation for the case of lower-regime plane-bed equilibrium transport conditions. It is worth clarifying here that the goal of this paper is not to propose a new or improved universal predictor of the bedload transport rate, but to correct the data analysis and results of MPM for the case of plane-bed bedload transport in light of research results that have become available since the publication of their work. It is equally important to point out that the present work is not the first one to conclude that the relation of MPM overpredicts bedload transport under plane-bed conditions in the absence of a form drag correction. Credit must go to Hunziker, Jäggi, and Smart for first recognizing this (Hunziker 1995; Hunziker and Jäggi 2002; Smart and Jäggi 1983; Smart 1984). The database used in their analysis is, however, larger than that used by MPM alone. As a result, it is not readily apparent that the relation of MPM significantly overpredicts (in the absence of a form drag correction) when applied solely to the data (for plane-bed transport) originally used in its derivation. Nor is it apparent that the form drag correction is not necessary for the plane-bed data of MPM. In this paper: (a) the overprediction in the absence of a form drag correction is demonstrated in the narrow context of the plane-bed data used by MPM, (b) the fact that a form drag correction is unnecessary for the plane-bed data used by MPM is made evident, and (c) simpler modified forms of MPM which are faithful to the original data set are presented. Manuscript number HY/2004/023810 Wong and Parker (2005) C:\Old Data\mwe\research topic on tracers\meyer-peter\SECOND revised draft 02-asce jhe.doc Page 4 Fourteen references which use the original form of the MPM bedload transport equation without the form drag correction of MPM are presented in the third paragraph of this Introduction. All of these references were published subsequently to the works of Hunziker, Jäggi and Smart aforementioned. It is the hope of the authors that the present work, combined with the contributions cited above, will finally lead to the recognition that a) the MPM bedload equation without the MPM form drag correction overpredicts plane-bed bedload transport by a factor of about 2, b) the form drag correction of MPM is unnecessary for plane-bed conditions, and c) a modified form of the MPM bedload relation (with no form drag correction) that predicts significantly lower transport rates than the original MPM bedload relation (when used with no form drag correction) should be used in the future for plane-bed conditions. Empirical bedload transport relation Three data sets were used by MPM for the derivation of their bedload transport relation; they are named here ETH-up, GIL-up, and ETH-nup. ETH-up (“up” is an abbreviation for “uniform sediment, plane bed”) consists of the results for 52 runs, all pertaining to plane-bed transport of material of uniform size (32 runs with Dm = 28.65 mm, and 20 runs with Dm = 5.21 mm, where Dm = arithmetic mean diameter of the sediment) and with a constant value of submerged specific gravity of the sediment, R, of 1.68 (R = ρs/ρ – 1, where ρs = density of the sediment, and ρ = density of water). GIL-up similarly includes the results of 116 runs, all corresponding to plane-bed transport of material of uniform size (27 runs with Dm = 7.01 mm, 69 runs with Dm = 4.94 mm, and 20 runs with Dm = 3.17 mm), with a constant value of R of 1.65. ETH-nup (“nup” is an abbreviation for “not necessarily both uniform material and plane bed”) comprises the results of 83 runs, in many of which bedforms were present (channel aspect ratio, B/H, varied between 1.7 and 72.2, where B = channel width, and H = water depth), the sediment consisted of size mixtures (Dm ranged between 0.38 and 5.21 mm, and D90/Dm varied between 1.00 and 2.52, where D90 = particle size for which 90% of the sediment is finer by weight), and the value of R differed from 1.68 (R ranged between 0.25 and 3.22). The descriptor “not necessarily” is motivated by the observation that although Meyer-Peter and Müller (1948) indicated that bedforms were present in many of the runs in this set ETH-nup, they do not specify which of the runs had bedforms and which did not. Since the focus of the present work is on the transport of uniform material over a plane bed, most of the data analysis herein is performed using the sets ETH-up and GIL-up. Manuscript number HY/2004/023810 Wong and Parker (2005) C:\Old Data\mwe\research topic on tracers\meyer-peter\SECOND revised draft 02-asce jhe.doc Page 5 The relation proposed by MPM estimates the rate of bedload transport in a river as a function of the shear force exerted by the flowing water over the sediment bed. This relation evolved over time. A first form of the relation was presented by Meyer-Peter et al. (1934); it was based solely on the sets ETH-up and GIL-up, for which bedforms were not observed and the sediment could be approximated as uniform in size. An empirical analysis of these data resulted in the following equation: