J. Fluid Mech. (2017), vol. 812, pp. 11891212. c Cambridge University Press 2017doi:10.1017/jfm.2016.863

1189

Mannings formula and Stricklers scalingexplained by a co-spectral budget model

S. Bonetti1, G. Manoli2,3, C. Manes4,, A. Porporato1,3 and G. G. Katul3,1

1Pratt School of Engineering, Duke University, Durham, NC 27708, USA2Institute of Environmental Engineering, ETH Zurich, 8093 Zurich, Switzerland

3Nicholas School of the Environment, Duke University, Durham, NC 27708, USA4Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino,

10129 Torino, Italy

(Received 21 January 2016; revised 14 December 2016; accepted 14 December 2016;first published online 12 January 2017)

Mannings empirical formula in conjunction with Stricklers scaling is widely usedto predict the bulk velocity V from the hydraulic radius Rh, the roughness size rand the slope of the energy grade line S in uniform channel and pipe flows at highbulk Reynolds numbers. Despite their importance in science and engineering, bothMannings and Stricklers formulations have waited for decades before finding atheoretical explanation. This was provided, for the first time, by Gioia & Bombardelli(Phys. Rev. Lett., vol. 88, 2002, 014501), labelled as GB02, using phenomenologicalarguments. Perhaps their most remarkable finding was the link between the Stricklerand the Kolmogorov scaling exponents, the latter pertaining to velocity fluctuationsin the inertial subrange of the turbulence spectrum and presumed to be universal. Inthis work, the GB02 analysis is first revisited, showing that GB02 employed severalad hoc scaling assumptions for the turbulent kinetic energy dissipation rate and,although implicitly, for the mean velocity gradient adjacent to the roughness elements.The similarity constants arising from the GB02 scaling assumptions were presumedto be independent of r/Rh, which is inconsistent with well-known flow propertiesin the near-wall region of turbulent wall flows. Because of the dependence of thesesimilarity constants on r/Rh, this existing theory requires the validity of the Stricklerscaling to cancel the dependence of these constants on r/Rh so as to arrive at theStrickler scaling and Mannings formula. Here, the GB02 approach is corroboratedusing a co-spectral budget (CSB) model for the wall shear stress formulated at thecross-over between the roughness sublayer and the log region. Assuming a simplifiedshape for the spectrum of the vertical velocity w, the proposed CSB model (subjectto another simplifying assumption that production is balanced by pressurevelocityinteraction) allows Mannings formula to be derived. To substantiate this approach,numerical solutions to the CSB over the entire flow depth using different spectralshapes for w are carried out for a wide range of r/Rh. The results from this analysissupport the simplifying hypotheses used to derive Mannings equation. The derivedequation provides a formulation for n that agrees with reported values in the literatureover seven decades of r variations. While none of the investigated spectral shapesallows the recovery of the Strickler scaling, the numerical solutions of the CSBreproduce the Nikuradse data in the fully rough regime, thereby confirming that the

Email address for correspondence: costantino.manes@polito.it

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1190 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul

Strickler scaling represents only an approximate fit for the friction factor for granularroughness.

Key words: channel flow, hydraulics, turbulence theory

1. IntroductionOriginally derived for a uniform free-surface turbulent channel flow driven by

gravitational acceleration, Mannings empirical formula (Manning 1891) relates thetime- and cross-sectional area-averaged velocity V to the hydraulic radius (Rh) andthe energy grade line slope (S) through an empirical roughness coefficient n that onlyvaries with surface roughness (figure 1). This roughness coefficient was proposed wellbefore the classical pipe flow resistance experiments by Nikuradse and Colebrook inthe 1930s. First presented by Gauckler (in 1867) for S< 0.0007 and later by Manning(in 1891) for open channels and pipes, the formula is given in SI units as (Gauckler1867; Manning 1891; Powell 1960)

V = QA= 1

nR2/3h S

1/2, (1.1)

where Q is the volumetric flow rate, A is the cross-sectional area orthogonal tothe flow direction, Rh = A/Pw and Pw is the wetted perimeter. Because of the largecorpus of supporting data, Mannings formula is widely used in hydrology, hydraulics,sanitary engineering, irrigation science and stream restoration (Chow 1959). As earlyas 1899, it was marked as one of best formulas of the day (Willcocks & Holt 1899;Powell 1960), and this assessment remains unchanged (Brutsaert 2005). In operationalhydraulics, no equation has been advanced to displace Mannings formula to datedespite all the progress made in turbulent boundary-layer theories. Thus, derivation ofMannings equation from theories of turbulence remained lacking until recently (Gioia& Bombardelli 2002). Using a phenomenological theory of turbulence and scalingarguments about the dominant size of vortices responsible for much of the momentumtransport near granular roughness elements of uniform size r, Gioia & Bombardelli(2002) (hereafter referred to as GB02) provided a theoretical derivation of Manningsequation and a link between the Strickler scaling, namely n r1/6, and Kolmogorovsk5/3 scaling for inertial subrange turbulence (where k is the wavenumber).

However, several unresolved issues remain with the derivation proposed in GB02which are incompatible with what is known about flow statistics within the roughnesssublayer (hereafter referred to as RSL). These issues are to be discussed in 2 butare summarized here to clarify the aims, scope and objectives of the present work. Tobegin with, GB02 assumed that eddies whose size exceeds the roughness element r donot contribute appreciably to the vertical velocity component near roughness elements,which is not supported by theories for the RSL (Raupach, Antonia & Rajagopalan1991). Furthermore, Kolmogorovs scaling (Kolmogorov 1941) rarely holds in closeproximity to the wall, whether it is rough or smooth (see, e.g. Poggi, Porporato &Ridolfi 2002; McKeon & Morrison 2007), in contrast to the assumption by GB02.GB02 also employed scaling arguments to estimate the near-wall turbulent shear stresswhich, although plausible, were never corroborated by experiments. Last but not least,GB02 assumed that the turbulent kinetic energy dissipation rate in the vicinity ofthe roughness elements scales as V3/Rh, and that the associated similarity constant

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Manning equation 1191

x

x

y

b

r

z

z

hh

s

Outer region

Logarithmic region

Roughness sublayerSurface

roughness

Co-spectral budgetformulated at this

plane (attached eddies)

Flow direction

1

FIGURE 1. (Colour online) Schematic representation of turbulent flow in a rectangularchannel having a width b and a water depth h. Flow within a very-high-Reynolds-numberturbulent boundary layer over roughness elements with undulation size r is characterizedby three sublayers: (i) the roughness sublayer, where eddy sizes responsible for momentumtransport generally scale with r instead of z (distance from boundary); (ii) an extensivelogarithmic region, where eddy sizes responsible for momentum transport scale with z(attached eddies); (iii) the outer region, where the eddy sizes tend to be large and scalewith h. The CSB is formulated near the top of the roughness sublayer at z= r ( > 1)just below the logarithmic region. Roughness undulations are assumed to be periodic inthe y direction and are assumed not to induce wakes or flow separation. Any secondarycirculation that may form at the corners of the channel is ignored. The axes are alignedsuch that x and z are the longitudinal and vertical directions respectively.

is independent of r/Rh, which, as will be discussed later on, cannot be correct givenwhat is known about the flow properties in the RSL (Raupach 1981; Raupach et al.1991).

The present work addresses these issues as follows. In 2, a brief overview ofGB02 and a critical appraisal of their assumptions is provided. An approach basedon a co-spectral budget (hereafter referred to as CSB) model is developed, and it isthen shown that Mannings formula can be recovered when assuming a simplifiedformulation for the vertical velocity spectra Eww(k)