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Manning’s formula and Strickler’s scaling explained by a co-spectral budget model

  • S. Bonetti (a1), G. Manoli (a2) (a3), C. Manes (a4), A. Porporato (a1) (a3) and G. G. Katul (a1) (a3)...

Manning’s empirical formula in conjunction with Strickler’s scaling is widely used to predict the bulk velocity $V$ from the hydraulic radius $R_{h}$ , the roughness size $r$ and the slope of the energy grade line $S$ in uniform channel and pipe flows at high bulk Reynolds numbers. Despite their importance in science and engineering, both Manning’s and Strickler’s formulations have waited for decades before finding a theoretical explanation. This was provided, for the first time, by Gioia & Bombardelli (Phys. Rev. Lett., vol. 88, 2002, 014501), labelled as GB02, using phenomenological arguments. Perhaps their most remarkable finding was the link between the Strickler and the Kolmogorov scaling exponents, the latter pertaining to velocity fluctuations in the inertial subrange of the turbulence spectrum and presumed to be universal. In this work, the GB02 analysis is first revisited, showing that GB02 employed several ad 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 presumed to be independent of $r/R_{h}$ , which is inconsistent with well-known flow properties in the near-wall region of turbulent wall flows. Because of the dependence of these similarity constants on $r/R_{h}$ , this existing theory requires the validity of the Strickler scaling to cancel the dependence of these constants on $r/R_{h}$ so as to arrive at the Strickler scaling and Manning’s formula. Here, the GB02 approach is corroborated using a co-spectral budget (CSB) model for the wall shear stress formulated at the cross-over between the roughness sublayer and the log region. Assuming a simplified shape for the spectrum of the vertical velocity $w$ , the proposed CSB model (subject to another simplifying assumption that production is balanced by pressure–velocity interaction) allows Manning’s formula to be derived. To substantiate this approach, numerical solutions to the CSB over the entire flow depth using different spectral shapes for $w$ are carried out for a wide range of $r/R_{h}$ . The results from this analysis support the simplifying hypotheses used to derive Manning’s equation. The derived equation provides a formulation for $n$ that agrees with reported values in the literature over seven decades of $r$ variations. While none of the investigated spectral shapes allows the recovery of the Strickler scaling, the numerical solutions of the CSB reproduce the Nikuradse data in the fully rough regime, thereby confirming that the Strickler scaling represents only an approximate fit for the friction factor for granular roughness.

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G. I. Barenblatt 1996 Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, vol. 14. Cambridge University Press.

J. C. Bathurst 1985 Flow resistance estimation in mountain rivers. ASCE J. Hydraul. Engng 111 (4), 625643.

W. J. T. Bos , H. Touil , L. Shao  & J. P. Bertogli 2004 On the behavior of the velocity–scalar cross correlation spectrum in the inertial range. Phys. Fluids 16, 38183923.

W. Brutsaert 1975 The roughness length for water vapor sensible heat, and other scalars. J. Atmos. Sci. 32 (10), 20282031.

W. B. Brutsaert 1982 Evaporation into the Atmosphere: Theory, History and Applications. Springer.

W. B. Brutsaert (Ed.) 2005 Hydrology: An Introduction. Cambridge University Press.

N.-S. Cheng 2011 Representative roughness height of submerged vegetation. Water Resour. Res. 47 (8), W08517.

M. K. Chung  & S. K. Kim 1995 A nonlinear return-to-isotropy model with Reynolds number and anisotropy dependency. Phys. Fluids 7 (6), 14251437.

C. Colosimo , V. A. Copertino  & M. Veltri 1988 Friction factor evaluation in gravel-bed rivers. ASCE J. Hydraul. Engng 114 (8), 861876.

S. Corrsin 1964 Further generalization of Onsager’s cascade model for turbulent spectra. Phys. Fluids 7, 11561159.

R. Ferguson 2010 Time to abandon the Manning equation? Earth Surf. Process. Landf. 35 (15), 18731876.

U. Frisch (Ed.) 1995 Turbulence. Cambridge University Press.

T. B. Gatski  & T. Jongen 2000 Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog. Aerosp. Sci. 36 (8), 655682.

T. B. Gatski  & C. G. Speziale 1993 On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech. 254, 5978.

M. Ghisalberti  & H. M. Nepf 2004 The limited growth of vegetated shear layers. Water Resour. Res. 40 (7), W07502.

G. Gioia  & P. Chakraborty 2006 Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory. Phys. Rev. Lett. 96, 044502.

G. Gioia , N. Guttenberg , N. Goldenfeld  & P. Chakraborty 2010 Spectral theory of the turbulent mean-velocity profile. Phys. Rev. Lett. 105 (18), 184501.

S. S. Girimaji 1996 Fully explicit and self-consistent algebraic Reynolds stress model. Theor. Comput. Fluid Dyn. 8 (6), 387402.

B. Jacob , L. Biferale , G. Iuso  & C. M. Casciola 2004 Anisotropic fluctuations in turbulent shear flows. Phys. Fluids 16 (11), 41354142.

B. Jacob , C. M. Casciola , A. Talamelli  & P. H. Alfredsson 2008 Scaling of mixed structure functions in turbulent boundary layers. Phys. Fluids 20 (4), 045101.

A. V. Johansson  & S. Wallin 1996 A new explicit algebraic Reynolds stress model. In Advances in Turbulence VI, pp. 3134. Springer.

T. Jongen  & T. B. Gatski 1998 General explicit algebraic stress relations and best approximation for three-dimensional flows. Intl J. Engng Sci. 36 (7), 739763.

G. G. Katul , C. D. Geron , C.-I. Hsieh , B. Vidakovic  & A. B. Guenther 1998 Active turbulence and scalar transport near the forest–atmosphere interface. J. Appl. Meteorol. 37 (12), 15331546.

G. Katul , P. Wiberg , J. D. Albertson  & G. Hornberger 2002 A mixing layer theory for flow resistance in shallow streams. Water Resour. Res. 38, 1250.

G. G. Katul , A. Porporato , C. Manes  & C. Meneveau 2013 Co-spectrum and mean velocity in turbulent boundary layers. Phys. Fluids 25 (9), 091702.

A. G. Konings , G. G. Katul  & S. E. Thompson 2012 A phenomenological model for the flow resistance over submerged vegetation. Water Resour. Res. 48 (2), W02522.

G. J. Kunkel  & I. Marusic 2006 Study of the near-wall-turbulent region of the high-Reynolds-number boundary layer using an atmospheric flow. ASCE J. Fluid Mech. 548, 375402.

B. E. Launder , G. J. Reece  & W. Rodi 1975 Progress in the development of Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537566.

F. López  & M. H. García 2001 Mean flow and turbulence structure of open-channel flow through non-emergent vegetation. ASCE J. Hydraul. Engng 127 (5), 392402.

J. L. Lumley 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855858.

C. Manes , D. Pokrajac  & I. McEwan 2007 Double-averaged open-channel flows with small relative submergence. ASCE J. Hydraul. Engng 133, 896904.

K. A. McColl , G. G. Katul , P. Gentine  & D. Entekhabi 2016 Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage. Phys. Fluids 28 (3), 035107.

B. J. McKeon  & J. F. Morrison 2007 Asymptotic scaling in turbulent pipe flow. Phil. Trans. R. Soc. Lond. A 365, 771787.

G. L. Mellor  & T. Yamada 1982 Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. 20 (4), 851875.

E. Murphy , M. Ghisalberti  & H. Nepf 2007 Model and laboratory study of dispersion in flows with submerged vegetation. Water Resour. Res. 43 (5), W05438.

I. Nezu  & M. Sanjou 2008 Turburence structure and coherent motion in vegetated canopy open-channel flows. J. Hydro-environment Res. 2 (2), 6290.

T. B. Nickels , I. Marusic , S. Hafez , N. Hutchins  & M. S. Chong 2007 Some predictions of the attached eddy model for a high Reynolds number boundary layer. Phil. Trans. R. Soc. Lond. A 365 (1852), 807822.

L. Onsager 1949 Statistical hydrodynamics. Il Nuovo Cimento 6, 279287.

S. Panchev (Ed.) 1971 Random Functions and Turbulence. Pergamon.

D. Poggi , A. Porporato  & L. Ridolfi 2002 An experimental contribution to near-wall measurements by means of a special laser Doppler anemometry technique. Exp. Fluids 32, 366375.

D. Poggi , A. Porporato , L. Ridolfi , J. D. Albertson  & G. G. Katul 2004 The effect of vegetation density on canopy sub-layer turbulence. Boundary-Layer Meteorol. 111 (3), 565587.

S. B. Pope 1975 A more general effective-viscosity hypothesis. J. Fluid Mech. 72 (02), 331340.

S. B. Pope (Ed.) 2000 Turbulent Flows. Cambridge Univeristy Press.

R. W. Powell 1960 History of Manning’s formula. J. Geophys. Res. 65, 13101311.

M. R. Raupach 1981 Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech. 108, 363382.

M. R. Raupach , R. A. Antonia  & S. Rajagopalan 1991 Rough-wall turbulent boundary layers. Appl. Mech. Rev. 44 (1), 125.

M. R. Raupach , D. E. Hughes  & H. A. Cleugh 2006 Momentum absorption in rough-wall boundary layers with sparse roughness elements in random and clustered distributions. Boundary-Layer Meteorol. 120 (2), 201218.

J. C. Rotta 1962 Turbulent boundary layers in incompressible flow. Prog. Aerosp. Sci. 2 (1), 195.

T. Rung , F. Thiele  & S. Fu 1998 On the realizability of nonlinear stress–strain relationships for Reynolds stress closures. Flow Turbul. Combust. 60 (4), 333359.

S. Saddoughi  & S. Veeravalli 1994 Local isotropy in turbulent boundary layers at high Reynolds number flow. J. Fluid Mech. 268, 333372.

S. Sarkar  & C. G. Speziale 1990 A simple nonlinear model for the return to isotropy in turbulence. Phys. Fluids A 2 (1), 8493.

F. G. Schmitt 2007 Direct test of a nonlinear constitutive equation for simple turbulent shear flows using DNS data. Commun. Nonlinear Sci. Numer. Simul. 12 (7), 12511264.

U. Schumann 1977 Realizability of Reynolds-stress turbulence models. Phys. Fluids 20 (5), 721725.

T.-H. Shih , J. Zhu  & J. L. Lumley 1995 A new Reynolds stress algebraic equation model. Comput. Meth. Appl. Mech. Engng 125 (1), 287302.

D. B. Taulbee 1992 An improved algebraic Reynolds stress model and corresponding nonlinear stress model. Phys. Fluids A 4 (11), 25552561.

W. Yang  & S.-U. Choi 2010 A two-layer approach for depth-limited open-channel flows with submerged vegetation. J. Hydraul. Res. 48 (4), 466475.

R. Zhao  & A. J. Smits 2007 Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow. J. Fluid Mech. 576, 457473.

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