Manning's Formula and Strickler's Scaling Explained by a Co-Spectral

J. Fluid Mech. (2017), vol. 812, pp. 1189–1212. c Cambridge University Press 2017 1189

doi:10.1017/jfm.2016.863 Manning’s formula and Strickler’s scaling explained 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, USA 2Institute of Environmental Engineering, ETH Zurich, 8093 Zurich, Switzerland 3Nicholas School of the Environment, Duke University, Durham, NC 27708, USA 4Department 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)

Manning’s empirical formula in conjunction with Strickler’s scaling is widely used to predict the bulk velocity V from the hydraulic radius Rh, 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/Rh, 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/Rh, this existing theory requires the validity of the Strickler scaling to cancel the dependence of these constants on r/Rh 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/Rh. 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

† Email address for correspondence: [email protected]

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 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 granular roughness.

Key words: channel ﬂow, hydraulics, turbulence theory

1. Introduction Originally derived for a uniform free-surface turbulent channel flow driven by gravitational acceleration, Manning’s empirical formula (Manning 1891) relates the time- and cross-sectional area-averaged velocity V to the hydraulic radius (Rh) and the energy grade line slope (S) through an empirical roughness coefficient n that only varies with surface roughness (figure1). This roughness coefficient was proposed well before the classical pipe flow resistance experiments by Nikuradse and Colebrook in the 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 (Gauckler 1867; Manning 1891; Powell 1960) Q 1 V R2/3S1/2, (1.1) = A = n h where Q is the volumetric flow rate, A is the cross-sectional area orthogonal to the flow direction, Rh A/Pw and Pw is the wetted perimeter. Because of the large corpus of supporting data,= Manning’s formula is widely used in hydrology, hydraulics, sanitary engineering, irrigation science and stream restoration (Chow 1959). As early as 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 operational hydraulics, no equation has been advanced to displace Manning’s formula to date despite all the progress made in turbulent boundary-layer theories. Thus, derivation of Manning’s equation from theories of turbulence remained lacking until recently (Gioia & Bombardelli 2002). Using a phenomenological theory of turbulence and scaling arguments about the dominant size of vortices responsible for much of the momentum transport near granular roughness elements of uniform size r, Gioia & Bombardelli (2002) (hereafter referred to as GB02) provided a theoretical derivation of Manning’s equation and a link between the Strickler scaling, namely n r1/6, and Kolmogorov’s 5/3 ∼ k− scaling for inertial subrange turbulence (where k is the wavenumber). However, several unresolved issues remain with the derivation proposed in GB02 which are incompatible with what is known about flow statistics within the roughness sublayer (hereafter referred to as RSL). These issues are to be discussed in §2 but are summarized here to clarify the aims, scope and objectives of the present work. To begin with, GB02 assumed that eddies whose size exceeds the roughness element r do not contribute appreciably to the vertical velocity component near roughness elements, which is not supported by theories for the RSL (Raupach, Antonia & Rajagopalan 1991). Furthermore, Kolmogorov’s scaling (Kolmogorov 1941) rarely holds in close proximity 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 stress which, although plausible, were never corroborated by experiments. Last but not least, GB02 assumed that the turbulent kinetic energy dissipation rate in the vicinity of 3 the roughness elements scales as V /Rh, and that the associated similarity constant

Flow direction z

h y h s b 1 x Outer region Co-spectral budget formulated at this Logarithmic region z plane (attached eddies)

Roughness sublayer Surface r x roughness

FIGURE 1. (Colour online) Schematic representation of turbulent ﬂow in a rectangular channel having a width b and a water depth h. Flow within a very-high-Reynolds-number turbulent boundary layer over roughness elements with undulation size r is characterized by three sublayers: (i) the roughness sublayer, where eddy sizes responsible for momentum transport generally scale with r instead of z (distance from boundary); (ii) an extensive logarithmic 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 scale with h. The CSB is formulated near the top of the roughness sublayer at z γ r (γ > 1) just below the logarithmic region. Roughness undulations are assumed to be= periodic in the y direction and are assumed not to induce wakes or ﬂow separation. Any secondary circulation that may form at the corners of the channel is ignored. The axes are aligned such 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 given what 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 of GB02 and a critical appraisal of their assumptions is provided. An approach based on a co-spectral budget (hereafter referred to as CSB) model is developed, and it is then shown that Manning’s formula can be recovered when assuming a simplified formulation for the vertical velocity spectra Eww(k) which accommodates deviations 5/3 from k− at low k. This simplifying hypothesis is discussed in §3, where a sensitivity analysis of the proposed model is performed by integrating the CSB budget equation over the entire flow domain and using more complex (and perhaps more realistic) shapes for Eww(k). It is shown that the CSB model, besides capturing the main features of mean velocity profiles, also reproduces Nikuradse’s data on friction factors in the fully rough regime. The results also confirm that Strickler’s scaling represents a reasonable but inaccurate fit of experimental data also established by the Colebrook equation. There is no reason why it should have a ‘clean’ theoretical origin as suggested by GB02. Finally, a comparison between literature values of Manning’s coefficient and those predicted from the CSB is presented over seven decades of r variation in §4. The results provide novel links between Manning’s roughness coefficient, velocity–pressure interactions and the vertical velocity spectrum highlighted in the discussion and conclusion (§5).

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1192 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul 2. Theory 2.1. Background In a steady and uniform open channel ﬂow within a rectangular section having a water depth h and a width b (ﬁgure1), let u denote the velocity component along the longitudinal direction x and w the velocity component in the bed-normal direction z (positive upward). The total shear stress τ0 uniformly distributed along Pw 2h b (i.e. grains of size r are uniformly distributed along the bed and channel sides)= + can be determined from a force balance, resulting in

τ 0 gR S, (2.1) ρ = h

where ρ is the ﬂuid density and g is the gravitational acceleration. In fully rough ﬂow regimes and at some small distance from the wall, τ0 mainly originates from turbulent shear stress contributions given as

τ0 2 u max( u0w0), (2.2) ρ = ∗ ∼ −

where u0w0 is the turbulent momentum flux with a maximum in magnitude adjacent to Pw, an overbar denotes time averaging, u0 and w0 are the turbulent longitudinal and bed-normal fluctuations from u and w respectively and u is the shear or ∗ friction velocity corresponding to the bed stress τ0. A drag coefficient or friction 2 2 factor given by fd u /V can be inserted into (2.2) and combined with (2.1) to = ∗ 1/2 recover the classical Chezy formula V fd− √gRhS derived in 1776. The term 1/2 = Cz fd− is the Chezy coefficient and was initially assumed to vary with r only. Here,= f can be related to the well-known Darcy–Weisbach friction factor f 8f , d = d as discussed elsewhere (Brown 2002). Upon equating the Chezy V Cz√gRhS to 1/6 1/2 1= Manning’s formula in (1.1), it becomes clear that C R g− n− . Dimensional z = h considerations alone suggest that Cz must vary with the ratio r/Rh if n only varies with r, as presumed in the original work of Manning. That is, Cz F(r/Rh), where F(.) is an unknown function which may be determined from incomplete= similarity (i.e. K (r/R )a o (r/R )a and K is a similarity constant). Incomplete similarity 1 h + [ h ] 1 is necessary to determine the form of F(.) because when r 0, Cz does not approach a constant as predicted by complete similarity (Barenblatt→ 1996). When r is sufficiently small to be entirely embedded within the viscous sublayer, Cz must approach its smooth-wall formulation expressed as a function of the bulk Reynolds number Re VR /ν, where ν is the kinematic viscosity. Stated differently, = h incomplete similarity predicts Cz F(r/Rh) G(Re), where G(Re) encodes the = + α smooth boundary-layer solution expressed as fd Re− , with α 1 for laminar flow and α 1/4 for turbulent flows at moderate∼ Re (i.e. Blasuis= scaling) as discussed elsewhere= (McKeon, Zagarola & Smits 2005). With increasing Re, G(Re) is asymptotically reduced relative to F(r/Rh) when r > 0 and Rh r. With such a 1/6 1 naive argument, it becomes evident that if Cz only varies with r/Rh and Cz Rh n− , 1 1/6 ∼ incomplete similarity must predict a 1/6, resulting in n− r− . Here, a 1/6 = − ∼ = − is the well-known Strickler scaling at very high Re (i.e. G(Re)/F(r/Rh) 1). This is one of the results featured in the dimensional consideration by GB02.

(a)(z Side view b) Top view

V Longitudinal eddies Lateral Lateral eddies eddies

h b

FIGURE 2. (Colour online) (a,b) The linkage between the local variables Γ and and the bulk variables V, Rh and the roughness size r for uniform flow in a channel with width b and water depth h. Above the roughness sublayer, the bulk velocity is assumed to be V Q/(bh). Within the roughness elements, the mean velocity u is assumed to be negligible.∼ Hence, defining the roughness sublayer thickness to be proportional to r results in Γ (V 0)/r. Because the mean flow kinetic energy is constant along x in uniform flows,∼ the − must be dissipated by the production of turbulent eddies. The inverse 1 time scale (TL− ) of the primary eddies is V/h for the longitudinal eddies and V/(b/2) for the lateral eddies, as discussed elsewhere (Gioia & Bombardelli 2002). (c) Schematic representation of a wide channel with eddies of different sizes.

2.2. Review and critical appraisal of GB02’s derivation The salient features of the derivation of Manning’s formula in GB02 are now provided for completeness. When analysing the dominant velocity scales contributing to max( u w ) at the top of the roughness elements, it was assumed by GB02 − 0 0 that u0 V. A plausibility argument in support of this assumption is provided by Gioia∼ and co-workers in a subsequent paper (Gioia et al. 2010) and goes as follows. At any elevation above the bed, the turbulent shear stress can be thought of as a product between the momentum contrast (i.e. ρsdu/dz, where s is the characteristic scale over which velocity gradients occur) and the rate of momentum transfer (i.e. ws, which indicates the velocity scale of an eddy of size s), and hence τ(z) ρu w ρ(sdu/dz)w . At the roughness tops, s r and du/dz (V 0)/r, = − 0 0 ∼ s ∼ ∼ − resulting in u0 V, as shown in figure2. With regards to w0, GB02 assumed that eddies of size∼ s r contribute appreciably to the turnover velocity acting on ∼ du/dz, so that w0 ws wr. Here, wr is associated with eddies of size r which ‘fill’ the roughness∼ elements,∼ as shown in figure1. To infer its magnitude, the 1/3 Kolmogorov scaling for the inertial subrange was assumed so that wr (r) , and the concomitant turbulent momentum flux becomes | | ∼

gR S max( u w ) Vw V(r)1/3, (2.3) h = − 0 0 ∼ r ∼ where is the mean turbulent kinetic energy dissipation rate to be determined. In principle, must be interpreted as the dissipation rate just above the roughness

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1194 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul elements. However, GB02 argued that such a local (z r) and bulk dissipation rate = (b) are proportional, thereby allowing them to assume b. To infer this b, it is noted that the mean flow kinetic energy is unaltered spatially∼ by frictional losses in a steady uniform flow (by definition). It follows that the total energy losses are due to 2 2 turbulence and scale with V , leading to b V /TL. Here, TL reflects the lifetime of turbulent eddies dissipating the work done∼ in moving the fluid against wall and bed friction (or wetted perimeter). As in GB02, figure2 shows that the eddy frequency 1/TL must include contributions from longitudinal and lateral eddies whose inverse 1 lifetimes scale as V/h and V/(b/2). It directly follows that T− V/R , leading to L = h V3 b , (2.4) ∼ ∼ Rh

1/3 so that wr (r/Rh) V. Combining (2.3) and (2.4), Manning’s formula comprehensive of Strickler’s∼ scaling was recovered by GB02 as

1/6 r − p V RhgS. (2.5) ∼ Rh The derivation proposed by GB02 contains some implicit assumptions which are now discussed. The first (labelled as assumption A1) deals with the estimation of u0 V, or implicitly Γ(z) du(z)/dz V/r, while the second (assumption A2) deals∼ with the estimation of (=z) near the∼ top of the roughness elements as a function of bulk variables. With regards to assumption A1, figure2 provides a plausibility argument that in the vicinity of the roughness elements, Γ may be estimated from V 0 Γ − , (2.6) | | ∼ r when u 0 within the roughness elements and the thickness of the roughness sublayer = scales with r. More significant in GB02 is that the proportionality constant α1 linking Γ to V/r (i.e. Γ α V/r) is assumed to be independent of r/R . The implicit = 1 h assumption about the independence of α1 from r/Rh, at the very least, is questionable in the RSL. To illustrate, let the plane of analysis be vertically displaced from the roughness tops to an elevation right at the beginning of the logarithmic layer (figure1), where it is possible to exploit the known scaling for energy dissipation and the mean velocity gradient. This is one of the novelties introduced by the CSB approach explained later, but it is also used here to critically analyse the GB02 approach. This switch in point of analysis is small and is typically a linear function of r. Detailed experiments on flow over granular roughness elements reported a log region commencing at approximately z γ r independent of Rh with γ 1.6 (Manes, Pokrajac & McEwan 2007). Therefore,= the scaling for both the mean= velocity gradient and the dissipation rate proposed by GB02 should not be affected by this switch in the plane of analysis provided that r/Rh is small. At this location (i.e. z r), the local mean velocity gradient can be assumed to scale as du/dz u /r. This∼ observation, when combined ∼ ∗ with (2.6), implies that α1 must scale as the square root of the Darcy–Weisbach friction factor (i.e. α f 1/2), which notoriously depends on the relative roughness 1 ∼ d r/Rh in fully rough turbulent flows. With regards to assumption A2, there are two separate issues to be discussed. The first issue is whether the bulk dissipation b itself is correct when assuming

it to be independent from r/Rh and only proportional to the bulk variables V and Rh, as shown by (2.4). The second issue is whether b represents the local (z r) near the roughness top. The first issue is inconsistent with simple thermodynamic≈ considerations applied to the bulk flow irrespective of whether the local at the top of the roughness elements can be equated or proportional to b as in GB02 (i.e. the second issue). To illustrate, the work We required to move a fluid mass over a distance L is We τ0AL τ0 , with A L the fluid volume. Upon inserting (2.1), = = = × 2 2 the work per unit mass to move the fluid is We0 We/(ρ ) gRhS u or We0 fdV . The bulk turbulence kinetic energy (TKE) dissipation= rate= is the= change∗ in= work 3 during a turnover time, resulting in b We0 /TL fdV /Rh. While the scaling in (2.4) is actually recovered (i.e. V3/R ),∼ it is evident∼ that cannot be independent of b ∼ h b r/Rh because fd varies with r/Rh, as already hinted at from the incomplete similarity argument in the previous section. The same result can be obtained even more rigorously by integrating the TKE balance equation across the whole flow domain presented elsewhere (Abe & Antonia 2016). With regards to the second issue, we assume that the local dissipation at the roughness tops can be quantified by means of bulk variables as follows:

α3V3 (z r) 2 , (2.7) = = Rh

where α2 is an ad hoc scaling function which allows for the transition from local to bulk variables at a given z and r/Rh. Now, in the RSL just below the log 3 layer, the local dissipation rate can be computed as (z r) u0w0du/dz (u )/r = = − ∼ ∗ because du/dz u /r. Inserting this result into (2.7) implies that α2 must scale as 1/3 1∼/2 ∗ α2 (Rh/r) fd . Repeating the derivation of GB02 while accounting for the correct scaling∼ functions for the mean velocity gradient and the local dissipation rate, one obtains 1/3 1/3 2 r gRhS V(r) α1α2V , (2.8) ∼ ∼ Rh

and ultimately

1/6 − 1/2 r p V (α1α2)− RhgS. (2.9) ∼ Rh

Strickler’s scaling in (2.9) is recovered only if the product α1α2 is constant. From the 1/3 previous analysis, it turns out that α1α2 (r/Rh)− fd, which becomes constant only ∼ 1/3 if the friction factor fd follows the Strickler scaling (i.e. fd (r/Rh) ). This ﬁnding leads to a circular argument. Manning’s formula and the Strickler∼ scaling can only be recovered if α1α2 is constant independent of r/Rh, but α1α2 is independent of r/Rh only when the Strickler scaling applies. These are the main reasons why the derivation of Manning’s formula is re-examined using an alternative approach based on the CSB model which eliminates the need to a priori link local to global variables.

2.3. Co-spectral budget model The focus here is on wide channels, so that R h, although many of the results h ≈ derived next hold when replacing h with Rh. The momentum turbulent ﬂux at any

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1196 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul height z > r from the surface (or boundary) can be linked to eddy sizes characterized by wavenumber k through the normalizing property of the co-spectrum, given by

Z ∞ u0w0 Fwu(k) dk, (2.10) = − 0

where k corresponds to an inverse eddy size, Fwu(k) is the co-spectrum between u0 and w0, and where u w (z) (ghS)(1 z/h), as expected in open channel flows for − 0 0 ≈ − z > r. The Fwu(k) can now be modelled through a budget formulated in the RSL above the roughness elements at z γ r with γ > 1 but below the logarithmic region characterizing u(z), as shown in figure= 1. As already discussed, this region defines the thickness of the RSL for momentum transport, which for granular roughness results in γ 1.6 (Manes et al. 2007). It is further assumed that γ r/h 1, so that at z γ r, ≈ 2 = u0w0(z) (ghS) u . This budget is given by (Panchev 1971; Katul et al. 2013; −Katul &≈ Manes 2014≈ )∗ ∂F (k) wu 2νk2F (k) G(k), (2.11) ∂t + wu = where G(k) P (k) T (k) π(k) (2.12) = wu + wu +

and Pwu(K) Γ Eww(k) is the covariance (or turbulent stress) production term, Eww(k) is the turbulent= energy spectrum of the vertical velocity, Γ is now interpreted as a local mean velocity gradient across the plane where the CSB is being formulated (i.e. z γ r), T (k) is the co-spectral ﬂux-transfer term and π(k) is the velocity–pressure = wu interaction term, which acts to decorrelate u0 and w0 (Pope 2000). If a Rotta model is further invoked for π(K) but adjusted for the isotropization of the production effect, then (Rotta 1962; Choi & Lumley 2000)

Fwu(k) π(k) CR CIPwu(k), (2.13) = − τr(k) −

1/3 2/3 where τr(k) − k− is a wavenumber-dependent relaxation time scale (Onsager 1949; Corrsin= 1964; Bos et al. 2004), assumed to vary only with k and local , C 1.8 and C 3/5 are the Rotta and isotropization of the production R ≈ I = constants (Launder, Reece & Rodi 1975; Pope 2000). The CI 3/5 was earlier predicted from rapid distortion theory, as discussed by Pope (=2000). Accounting for a slow part (i.e. CRFwu(k)/τr(k)) and a fast part (i.e. CIPwu(k)) appears to be sufﬁcient to capture π(−k) in many geophysical applications such as the atmospheric boundary layer (Mellor & Yamada 1982). To what extent such a representation can be extrapolated to the cross-over from the RSL to the log region may be a subject of some debate given the possible large strain rate in this layer (Choi & Lumley 2000). Nonlinear corrections to π(k) that satisfy realizability conditions (Schumann 1977) can be derived with the aid of the Cayley–Hamilton theorem using symmetry group rotations and shear deformation scale, as extensively discussed elsewhere (Pope 1975; Lumley 1978; Sarkar & Speziale 1990; Taulbee 1992; Gatski & Speziale 1993; Chung & Kim 1995; Shih, Zhu & Lumley 1995; Girimaji 1996; Johansson & Wallin 1996; Jongen & Gatski 1998; Rung, Thiele & Fu 1998; Choi & Lumley 2000; Gatski & Jongen 2000; Pope 2000; Jacob et al. 2004, 2008; Schmitt 2007), and are beyond the scope here. In general, such corrections are signiﬁcant when the mean

1 1/3 2/3 Fwu(k) Γ − Eww(k)k− , (2.14) = Aπ

where Aπ CR/(1 CI) 4.5. Equations (2.13) and (2.14) neglect the wall-blocking component= for the− pressure–strain≈ correlation parametrization. The implications of such an assumption were further investigated and appeared to be minor (not shown). Combining (2.10) and (2.14), the CSB at z γ r can be re-arranged as = Z 1/3 u0w0 ∞ 2/3 Γ − Aπ − , Ik k− Eww(k) dk. (2.15a,b) | | = Ik = 0

1/3 1 Interestingly, equation (2.15) can be formulated as u0w0 µtΓ , where µt − Aπ− Ik may be interpreted as an effective nonlinear eddy viscosity= − dictated by the= shape of 2/3 k− Eww(k) and the local (z), known to vary with Γ in the RSL. When the TKE budget is reduced to (z) u0w0Γ , µt varies with Γ , rendering µt analogous to a non-Newtonian eddy viscosity.= − To evaluate Ik, assumptions about the shape of Eww(k) are now required for the RSL. At z γ r, just below the logarithmic region, Eww(k) can be reasonably described by the= regimes shown in figure3. The first regime (I) is associated with low-wavenumber contributions and is delineated by k 0, Kc , where Kc is a 1 1 ∈ [ ] wavenumber commensurate with Rh− or h− . The second regime (II) is defined for k K , K , where K is a wavenumber associated with the roughness size r when ∈ [ c a] a z 6 γ r. The third regime (III) is delineated by k > Ka, where the classical inertial subrange exists (Kolmogorov 1941; Frisch 1995). The presence of regimes (II) and (III) has been confirmed by a number of field and laboratory experiments collected in the RSL above tall forested canopies (Katul et al. 1998) and in laboratory flume studies (Poggi et al. 2004). Further support for this shift in the spectral regime 1 at Ka (z)− is provided by a number of recent experiments described elsewhere (Kunkel∼ & Marusic 2006; Nickels et al. 2007; Zhao & Smits 2007; Manes, Poggi & Ridolfi 2011). A fourth regime also exists and is related to viscous corrections to the inertial subrange commonly formulated as an exponential cutoff (Pope 2000) which becomes significant as k becomes sufficiently large (i.e. k 1/η, where η is known as the Kolmogorov microscale). However, distortions to the∼ overall energy content

a b ( )(I II III) I II III 102

100 100 E1

10–2

10–5 100 10–5 100

(c)(102 d)

100 100 E2

10–2

10–5 100 10–5 100

(e)(102 f )

100 100 E3

10–2

10–5 100 10–5 100

FIGURE 3. (Colour online) (a,c,e) Schemes of the idealized vertical velocity spectra (E1, E2 and E3) assumed in the numerical experiments. The co-spectrum Fwu(k) computed in the roughness sublayer at z γ r is also shown for each case (b,d,f ). Different break = points Kc for cases E2 and E3 are also illustrated (c–f ). The inset in (b) shows the 3 5 1/2 measured shear-stress co-spectra Fuw∗ Fuw(k)/( Γ − ) (original data from Saddoughi & Veeravalli (1994), digitized from Pope= − (2000)).

R ∞ R ∞ (i.e. 0 Eww(k) dk) and turbulent stress (i.e. 0 Fuw(k) dk) arising from this fourth regime vis-a-vis simply extending the inertial subrange indefinitely for k > Ka are minor provided that the CSB is formulated sufficiently above the roughness elements but still below the logarithmic region, and the Reynolds number is sufficiently large to allow for large-scale separation between r and η. Moreover, bottleneck effects in Eww(k) at the cross-over from inertial to viscous range are also neglected here (Katul et al. 2015). 3/4 Integrating Eww(k) and Fuw(k) to k is akin to assuming η/r Re − 0, → ∞ ∼ ∗ → where Re u r/ν defines a roughness Reynolds number. For n or fd to achieve independence∗ = of∗ Re , Re itself must be very large but is generally finite. Upon ignoring viscous corrections∗ ∗ and assuming that Re is sufficiently large so that η/r 1, the idealized E (k) shown in figure3 can be∗ expressed as ww  p p Ekol(Ka)Kc− k if 0 6 k 6 Kc, Eww(k) Ekol(Ka) if Kc 6 k 6 Ka, (2.16) = Ekol(k) if k > Ka,

2/3 5/3 where E (k) C k− is the Kolmogorov spectrum (Kolmogorov 1941; Frisch kol = o 1995), Co (24/55)Ck0 is the Kolmogorov constant for the vertical velocity component = 1/3 7/3 and Ck0 1.5 (Pope 2000). When Eww(k) Ekol, Fwu(k) Γ k− , consistent with other= co-spectral theories (Lumley 1967=) and measurements∼ (see the inset in figure3 b, data from Saddoughi & Veeravalli (1994)) within the inertial subrange 0 reviewed elsewhere (Katul et al. 2013). When Eww(k) k for low wavenumbers at the cross-over from inertial to larger scales as shown≈ in figure3, the co-spectrum 2/3 1/3 2/3 scales as Fwu(k) k− when τr(k) − k− . Wind tunnel experiments at very high Reynolds number≈ in the logarithmic= region indicate a co-spectral scaling between 0 and 2/3 (Saddoughi & Veeravalli 1994; Pope 2000) for values of k smaller than those− associated with the inertial subrange (see the inset in figure3 b), but these wind tunnel experiments do not span kh < 1. Furthermore, these experiments report 1-D spectra along x which may be distorted by the use of Taylor’s frozen turbulence hypothesis needed to convert time to wavenumbers at small k as well as potential aliasing effects (Tennekes & Lumley 1972). This shortcoming requires further assumptions about the precise shape of Eww(k) in the very-low-wavenumber regime. Assuming a linear increase for low k (as illustrated in figure3, case E3), equation (2.15) becomes

1/3 Aπ τ/ρ Γ − . Z Kc Z Ka Z | | = Ekol(Ka) p 2/3 2/3 ∞ 2/3 p k k− dk Ekol(Ka) k− dk k− Ekol(k) dk Kc 0 + Kc + Ka (2.17)

As a first approximation, the scaling for k 0, Kc may be described by p 8/3 (i.e. close to the von Kármán spectrum where∈ [ p ]17/6). This choice is consistent∼ with the scaling p 2 suggested by Pope (2000=), but no data exist to support it. Hence, an alternative≈ is also explored here to assess the sensitivity of Manning’s formula to the choice of p and Ka. A constant value (i.e. p 0 and Kc 0) of the energy spectrum E (k) E (K ) for low wavenumbers k =0, K (see= case E1 in ww ≈ kol a ∈ [ a] figure3) to ‘boost’ the low-wavenumber eddies for Eww(k) will be considered. Using this simplified description of Eww(k) and recalling that at z γ r, τ/ρ τ0/ρ ghS, equation (2.17) can be rewritten as = = =

1/3 Aπ ghS Γ − . (2.18) Z Ka Z | | = 2/3 ∞ 2/3 Ekol(Ka) k− dk k− Ekol(k) dk 0 + Ka Upon simplifying,

1/3 4Aπ Γ 4/3 ghS. (2.19) | | = 15CoKa− Setting (z) Γ u w Γ(z)ghS(1 z/h), formulating an appropriate model for = − 0 0 = − Ka variations with respect to z and twice integrating (2.19) with respect to z does not recover Manning’s formula for V but results in analytical expressions that are lengthy and not insightful. However, they do link the CSB model to Manning’s formula without invoking any relation between local and bulk variables. Before exploring those solutions numerically (described later on), it is instructive to examine conditions that admit Manning’s formula to a leading order, as was the case with incomplete similarity theory. This expression is ﬁrst discussed by directly linking and Γ in the RSL to the global variables V, h and r, as conducted in GB02.

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1200 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul 2.4. Manning’s formula from the CSB model using GB02 bulk variables As in GB02, Manning’s equation can also be recovered from the CSB model when subjecting it to the same ‘bulk’ formulation for the mean velocity gradient and the dissipation rate. Towards this end, we consider Γ and at z/r γ 1.6 (instead of z/r 1 assumed in GB02) as a function of the same bulk (or= global)= variables as in GB02.= We recall that the bulk and local velocity gradients in the RSL can be related by

du(z) V 0 Γ α − . (2.20) | | = dz = 1 r Likewise, the TKE dissipation rate may be deﬁned as

α3V3 2 . (2.21) = h Before proceeding with the derivation further, it is interesting to note that at z/r γ , (z) Γ ghS, and (2.19) can be recast as = ≈ 3/4 4Aπ p Γ Ka ghS. (2.22) = 15Co

Using only Γ α V/r in this expression leads to = 1 3/4 4Aπ Kar p V ghS. (2.23) = 15Co α1

1 With Ka r− in the RSL, equation (2.23) results in a link between fd and α1 given as ∼

V2 1 4A 3/2 1 π , 2 2 (2.24) u = fd ∼ 15Co α1 ∗ 1/2 or α1 fd , which is consistent with what was discussed in § 2.2. Inserting∼ estimates for Γ and from (2.20)–(2.21) into the CSB model (2.19) results in V 4Aπ α VK α ghS. 1 a 2 1/3 4/3 (2.25) h = 15CoKa− This equation can be further arranged to yield

2 4Aπ 4/3 V 1/3 gh S, (2.26) = α1α215CoKa− whence s 4Aπ √g 2/3 1/2 V 1/6 h S . (2.27) = α1α215Co Ka−

1 Manning’s formula is recovered on noting that Ka r− in the RSL at z/r < γ 1.6, and the roughness coefﬁcient n is ∼ = s 15α1α2Co 1/2 1/6 n g− r . (2.28) ∼ 4Aπ

With respect to the GB02 approach, the CSB model allows for the derivation of Manning’s equation without invoking any scaling argument for the turbulent shear stress and without incurring the inconsistencies of the GB02 approach discussed in § 2.2. However, even the CSB approach fails in providing a theoretical explanation for the Strickler scaling, as it fails in linking it to a general property of turbulence 5/3 (i.e. k− ). This is because, in (2.28), Strickler’s scaling can be recovered only when the product α1α2 is independent of r, which, as discussed earlier on, is true only if Strickler’s scaling is assumed, leading once again to a circular argument. Hence, numerically integrating (2.19) with respect to z appears necessary so as to avoid a priori assumptions about relations between local and bulk variables. The numerical solution allows an examination of whether the Strickler scaling and a constant α1α2 are compatible with the CSB model assumptions for a wide range of r/h and plausible spectral shapes for Eww(k).

2.5. Co-spectral budget model assumptions and approximations In comparison to GB02, the CSB approach can accommodate additions of the flux-transfer term disturbing the balance between production and pressure–velocity interaction, the viscous effects responsible for further decorrelation between w0 and u0, and wall-blocking effects modifying π(k) in the Rotta model. The significance of Twu(k) and the use of the Rotta closure without any wall-blocking effects to close π(k), which may be problematic in the RSL, were analysed, and it appeared that ignoring the flux-transfer term and wall-blocking effect does not alter the final outcome of Manning’s formula (not shown here). Approximations that deal with the role of low-wavenumber eddies modifying E (k) for kh 1 and the relation between local and bulk variables encoded in the ww dependence (or independence as in GB02) of α1 and α2 on r/h are discussed in the following section. The evaluation is conducted by solving for Γ through numerical integration of the CSB model over all k followed by double integration in physical space (i.e. z) to arrive at V. In the derivation of Manning’s formula, low-wavenumber contributions associated with different scalings (i.e. p exponent) have been neglected. To evaluate the effect of low-wavenumber adjustments on u(z) and subsequently V to compute fd, the general equation (2.17) should be considered instead of (2.18). The terms in the denominator of (2.17) represent the three energy regimes in figure3: an energy increase at low k (regime I), the maximum energy at Kc < k < Ka (regime II) and the well-known Kolmogorov scaling for k > K (regime III). For K 0, equation (2.19) is recovered, a c = i.e. case E1. To test different approximations to the shape of Eww(k), equation (2.17) is numerically integrated considering different choices of Kc.

3. Numerical results

The full CSB model is now used to determine numerically u(z), α1 and α2 at z/r 2 2 = γ , the bulk velocity V and the friction factor fd u /V . For simplicity, the results ∼ ∗

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1202 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul are again limited to wide channels (i.e. R h). It is now necessary to model the h ≈ wavenumber Ka, which expresses the cross-over between the constant region in the modelled spectra and the inertial subrange. Since attached eddies are the dominant turbulence structures for the ﬂow above roughness elements (i.e. z > γ r), they impact Ka. However, Ka is also restricted by r for 1 6 z/r 6 γ . Hence, a model for the low-wavenumber spectral cross-over to the inertial subrange for the entire water depth range can be expressed as 1 1 K δ (1 δ ) , (3.1) a = z r + − z αz where δ is a step function deﬁned as ( 1 if z γ r, δ 6 (3.2) z = 0 if z > γ r.

1 This definition provides a constant Ka r− below the RSL and linear variations in Ka 1 = 1 with z− above. When γ 1.6, continuity in Ka with z at z γ r requires α (γ )− 0.625. = = = = In principle, the CSB budget is applicable only for the flow region z/r > γ . Such an integration requires the specification of a lower boundary condition for u(z) at z γ r, which is not readily available. Instead, equation (2.17) may be integrated using= (3.1) while imposing an ad hoc but plausible boundary condition for u(z) at z/r 1. One plausible boundary condition at the roughness top may be u(r)/u 8.5, as= derived for rough-wall boundary layers and discussed elsewhere (Brutsaert ∗1975= ; Pope 2000). In the numerical integration, a linear u0w0 profile that varies from 0 at the water surface (z/h=1) to its maximum value τ0/ρ ghS at z r is used. A description of the numerical procedure is provided next, while= the parameters= used in the numerical calculations are summarized in table1. The general form for Γ(z) at any z is given by

3/4 Aπ √ghS(1 z/h) Γ(z) − . (3.3) = C 5/3 1/3 3/4 o 15 4/3 1/3 5/3 Ka− Kc K− 3K K− 4 a − c a + 1/3 p + The Γ in (3.3) is numerically integrated along z assuming different shapes for the vertical velocity energy spectrum (ﬁgure3) at low k using u(r)/u 8.5. Upon further integration with respect to z, V can be obtained as ∗ =

3/4 Aπ √ghS V = Co h   Z h Z h √1 z/h   − dz dz. (3.4) ×  5/3 1/3 3/4  0  0 15 4/3 1/3 5/3 Ka− Kc  K− 3K K− 4 a − c a + 1/3 p +

The proportionality constants α1 and α2 can now be numerically computed at z/r γ by equating local and bulk variables, thereby allowing us to assess their dependence=

Symbol Description Value

Co Kolmogorov constant for Eww(k) 0.65 kv von Kármán constant 0.4 S Bed slope 0.0012 α Proportionality constant for Ka above the RSL 0.625 γ Proportionality constant for the RSL thickness 1.6 p Coefﬁcient in (2.16) 8/3 CR Rotta constant 1.8 CI Isotropization constant 3/5 r Roughness (mm) 0.1–50 h Channel depth (m) 1–20 TABLE 1. Model parameters and the range of values used in the numerical integration of the CSB model. Unless otherwise stated, the ﬂow is stationary and uniform at very high Reynolds number in a wide channel, so that R h. Combinations of slope, water depth h ≈ and roughness height were chosen to ensure a roughness Reynolds number Re∗ >200, so that the exponential cutoff in the viscous sublayer can be reasonably neglected in the CSB formulation.

on r/h from

du(z) u Γlocal Γ(z) z γ r ∗ α1VKa, (3.5) = = | = dz z γ r ∝ γ r = = 3 3 3 du(z) u α2 V local u0w0 ∗ . (3.6) = dz z γ r ∝ γ r = Rh = To explore the validity of Manning’s formula, deviations from the Strickler scaling in the CSB computed α1 and α2 as well as their product are also featured. To investigate low-wavenumber adjustments to Eww(k), several scenarios were considered (see ﬁgure3). First, the u(z) was evaluated assuming a constant energy spectrum (E1) for k < Ka (regimes I and II) and the Kolmogorov spectrum for k > Ka (regime III). Second, the Eww(k) was set to zero for k < Kc 1/(βh) and calculations were conducted for different values of β 0.9, 2.5, 100 (case= E2). Finally, = Eww(k) was assumed to vary linearly from 0 to Kc with a slope of p 8/3 (case E3). The numerical integrations were conducted for r varying between 0.1= mm and 50 mm, and h varying between 1 and 20 m for illustration. In all cases, the TKE budget is assumed to be in equilibrium throughout, so that production and dissipation of TKE are balanced when computing (z). While a TKE balance between only production and dissipation is unlikely in the entire RSL and in the outer layer of the channel (Pope 2000), it was shown elsewhere (Katul & Manes 2014; McColl et al. 2016) that the effect of this imbalance has minor impact on the shape of u(z).

3.1. Effect of low-wavenumber adjustments of Eww(k) on u(z) Mean velocity profiles obtained under different r and h scenarios are compared with the logarithmic law (figure4) to assess how low wavenumbers in Eww(k) shape u(z). The different low-wavenumber Eww(k) models are then assessed by the agreement between the modelled f versus r/Rh curves with the Strickler scaling (figure5), the Colebrook equation and the virtual Nikuradse (VN) formula (Yang & Joseph 2009).

log-law log-law

log-law (a)(100 b)(100 c) 100

10–1

10–1 10–1 10–2

10–3 10–2 10–2 30 30 30

10–4 20 20 20

10 10 10 100 103 100 103 100 103 10–3 10–3 10 15 20 25 30 10 15 20 25 30 10 15 20 25 30

FIGURE 4. (Colour online) Velocity proﬁles for cases (a) E1, (b) E2 and (c) E3. The 1 impact of different break points Kc (βh)− on u(z) is shown in (b) and (c) for cases E2 and E3 respectively. Case E1 is illustrated= for different roughness sizes r, while cases E2 and E3 are shown for r 20 mm. Results are shown for S 0.0012 and h 10 m. = = = The classical log law u/u 8.5 (1/kv) log(z/r) for a rough wall is also shown for comparison (dashed grey lines).∗ = +

For the flow region above the roughness elements, all of the assumed energy spectra result in plausible mean velocity profiles that display a logarithmic region, with an apparent increase in the wake effect as β decreases (i.e. higher break point Kc), as shown in figure4. This finding is consistent with results for smooth pipes, where the low-frequency correction to the Kolmogorov spectrum has a key influence on the magnitude of the outer wake in the mean velocity profile (Katul & Manes 2014). This analysis suggests that the choice of the spectral break point Kc influences u(z) more when compared with the precise scaling laws in Eww for K 0, Kc and that a significant contribution of low-wavenumber eddies is required∈ [ in the] spectra to reproduce the outer wake. In further support of this claim, by imposing 1 Kc Ka (αz)− (i.e. setting to zero all of the large-scale eddies energy), the wakes in= the computed= mean velocity profiles u(z) are completely off (not shown here). All of these findings confirm that application of the E1 spectral shape to derive Manning’s formula in (2.27) was a reasonable assumption. All of the energy spectral models result in fd exhibiting a Strickler scaling restricted only to an intermediate range of r/h values (figure5). This result is robust for any (sensible) choice of free parameters contained in the CSB model. Such deviations from the Strickler scaling predicted by the proposed CSB approach are to be expected and are physically plausible. To illustrate, friction factor calculations conducted using 1/2 Colebrook’s empirical formula f − 2 log r/(14.8Rh) as well as the VN model for granular beds at very high bulk Reynolds= − number[ ( f ]0.1781(r/2R )0.4678 0.0098) = h +

(a) (b)(c) Colebrook –1.3 –1.3 –1.3 VN Nikuradse data –1.5 –1.5 –1.5

–1.7 –1.7 –1.7

–1.9 –1.9 –1.9

–2.1 –2.1 –2.1 –5–4 –3 –2 –1 –5–4 –3 –2 –1 –5 –4 –3 –2 –1

FIGURE 5. (Colour online) Comparison between f 8f (circles) and the Strickler scaling = d (r/h)1/3 (grey line) for cases (a) E1, (b) E2 and (c) E3. The impact of different break 1 points Kc (βh)− on f is shown in (b) and (c) for cases E2 and E3 respectively. Results are shown= for r varying between 0.1 and 50 mm, h between 1 and 20 m, and bed slope S set to 0.0012. Numerical results are shown for r/h values that ensure Re > 200 and Fr < 1. Colebrook’s formula (blue solid line) and predictions from the virtual∗ Nikuradse (VN) model (green dashed line) are featured to illustrate the plausible range of deviations from Strickler scaling across numerous experiments. Nikuradse’s data for high Re where f is independent of Re are also shown for reference (orange squares, digitized from Gioia & Chakraborty (2006)) so as to illustrate the range in r/h considered in the CSB approach versus the original experiments by Nikuradse. The dotted grey lines indicate the smallest relative roughness explored by Nikuradse.

along with Nikuradse’s original data are featured in ﬁgure5. Encouragingly, the CSB model predictions of fd are close to the VN equation, which is, to date, the best ﬁt of Nikuradse’s data on pipes with walls roughened with granular roughness. The CSB model also predicts an approximate Strickler scaling in the range of r/h values originally explored by Nikuradse. This acceptable agreement between experimental and model results further corroborates the applicability of the proposed CSB model.

3.2. Relations between local and bulk variables The α and α evaluated at z γ r with γ 1.6 are presented in figure6 as a 1 2 = = function of r/h for cases E1, E2 and E3. In principle, to recover Strickler’s scaling 1/6 1/6 in (2.28), α (r/h) and α (r/h)− are necessary, so that their product 1 ∼ 2 ∼ becomes independent of r/Rh. Consistent with figure5, these scaling rules appear to hold for intermediate values of r/h. Despite these deviations, and while the model calculations suggest that α1 and α2 do not strictly obey the Strickler scaling, they follow opposite power-law behaviours, as they should. That is, α1α2 becomes only weakly dependent on r/h (figure6 c). Specifically, for case E1 in figure6( c), the product α1α2 ranges between 0.048 and 0.08, resulting in only a weak variation of √α1α2 0.22 0.28 with r/Rh. The results from the CSB presented in figures5 ∈ [ ; ] 1/6 1/3 and6 confirm that Strickler’s scaling (i.e. n r or f (r/R )− ) represents a ∼ d ∼ h reasonable but inaccurate scaling law relating friction factors with roughness size for a wide range of r/Rh.

(a) 0.5 (b) –0.6 (c) –1.0 0 –0.7 –1.1 –0.5 –0.8 –1.0 –0.9 –1.2

–1.5 E1 –1.0 –1.3 –2.0 –1.1 –2.5 –1.2 –1.4 –4–3 –2 –1 –4–3 –2 –1 –4 –3 –2 –1 (d) 0.5 (e) –0.6 ( f ) –1.0 0 –0.7 –1.1 –0.5 –0.8 –1.0 –0.9 –1.2 –1.5 –1.0 –1.3 –2.0 –1.1 –2.5 –1.2 –1.4 –4–3 –2 –1 –4–3 –2 –1 –4 –3 –2 –1

FIGURE 6. (Colour online) Variation of the proportionality constants (a,d) α2 and (b,e) α1 (at z γ r) as a function of r/h in cases E1, E2 and E3. The 1/6 and 1/6 Strickler scalings= are shown for comparison (grey lines in (a,d) and (b,e)).− (c,f ) Variation of the product α1α2 as a function of r/h. Results are shown for r varying between 0.1 and 50 mm, h between 1 and 20 m, and bed slope S set to 0.0012. Numerical results are shown for r/h values that ensure Re > 200 and Fr < 1. The dotted grey lines indicate the smallest relative roughness explored∗ by Nikuradse. The green dashed line in (c) represents 1.26 the mean value α α 10− for case E1 used in the evaluation of n in ﬁgure7. 1 2 =

4. Evaluation of Manning’s roughness coefficient for permeable and impermeable rough surfaces To further evaluate (2.28) for n where the roughness elements can deviate from their assumed shape in figure1, a comparison with literature values is provided in figure7. The data span seven decades of roughness values. For uniformity in characterizing r across various roughness types, the following published datasets for n as a function of momentum roughness height zo ( r) for channel and atmospheric flows are used: (i) rigid canopy elements (Konings,∼ Katul & Thompson (2012), original data from Dunn et al. (1996), Meijer & Van Velzen (1999), López & García (2001), Ghisalberti & Nepf (2004), Poggi et al. (2004), Murphy et al. (2007), Nezu & Sanjou (2008), Yang & Choi (2010), Cheng (2011)); (ii) data for many bluff-rough surface covers including both vegetated and granular surfaces from Katul et al. (2002) (original data from Chow (1959), Brutsaert (1982)); (iii) measurements from Katul et al. (2002) including data for shallow gravel streams by Hey (1979), Bathurst (1985), Colosimo et al. (1988) and laboratory flumes by Bathurst et al. (1981); (iv) laboratory flume experiments from Manes et al. (2007). Where not available, zo is estimated as (Katul et al. 2002) h z , (4.1) o = r8 exp 1 kv + f where h and f are measured or reported in the experiments. The data here include rigid but densely arrayed rods, permeable rough canopies and uniform gravel beds

0.750 0 0.500 –0.5

0.250 –1.0

Canopy flow –4 –3 –2 –1 0.100

n 0.060 0.050 0.040 0.030 Granular roughness 0.020 (Chow 1959)

0.010

0.005 10–7 10–6 10–5 10–4 10–3 10–2 10–1 100

FIGURE 7. (Colour online) Comparison between measured and modelled Manning’s roughness coefficient n using (2.28) and literature values. Results from (2.28) are shown 1.26 1 for α1α2 constant (solid line) and set equal to the mean values 10− (black) and 10− (blue) obtained from the E1 spectrum imposing u(r)/u∗=8.5 and 3.3 respectively (see the 1.26 10− line in figure6 c). Values of n assuming a variable shape of α1α2 (spectrum E3, β 0.9) are shown for qualitative (or trend) comparison (dashed lines, black and blue = for the two different boundary conditions). The model results for the variable α1α2 are computed assuming z0 r/2. Data for granular roughness are shown for deep and shallow stream flows (red and= yellow symbols respectively). Data from canopy flow experiments are also shown for comparison (grey symbols). For dense canopies, flume and wind tunnel experiments using rod canopies and field experiments in the RSL of tall forests suggest u/u 3.3 at the canopy top. The green dashed line shows the n 0.03 value for granular roughness∗ = given by Chow (1959). Symbols refer to different= studies: (u) Katul et al. (2002), (q) Poggi et al. (2004), ( ) López & García (2001), (6) Meijer & Van Velzen (1999), (D) Ghisalberti & Nepf (2004+ ), (B) Murphy, Ghisalberti & Nepf (2007), (A) Yang & Choi (2010), (C) Cheng (2011), (?) Dunn, López & García (1996), (E) Nezu & Sanjou (2008), (p) Manes et al. (2007), (f) Hey (1979), ( ) Colosimo, Copertino & Veltri (1988), (@) Bathurst, Simons & Li (1981), ( ) Bathurst (∗1985). ×

with various diameters and thicknesses. The Strickler scaling did not originally cover such roughness shapes and was restricted to single-sized grains. The derivation here makes it clear that the existence of an RSL with a thickness that scales with r (or zo if r is ambiguously defined) may still follow an approximate Strickler scaling for n, although several deviations for shallow flows and canopy elements are apparent. 1.26 Manning’s n is evaluated by means of (2.28) assuming a constant α1α2 10− (see figure6, E1 spectrum and boundary condition equal to u(r)/u =8.5) and compared with literature values (figure7, black solid line). Since the∗ = roughness height r is not known for all experiments, the comparison is performed assuming

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 1208 S. Bonetti, G. Manoli, C. Manes, A. M. Porporato and G. G. Katul z r (Raupach, Hughes & Cleugh 2006). It should be noted that the slope, water o ∼ level height and boundary conditions (all influencing the value of α1α2) vary for each lab/field study, and the comparison with (2.28) is mainly featured for qualitative assessment of trends. The acceptable agreement between modelled and measured roughness coefficients shows that the derivation of Manning’s formula using the CSB model provides acceptable values for n even when r varies by seven orders of magnitude. Furthermore, the comparison between modelled and measured n also demonstrates that framing the CSB in the RSL above the roughness elements but below the logarithmic region reduces the sensitivity of the flow statistics to the precise geometry of the roughness elements provided that it can be encoded in a single effective roughness length r zo. The model provides a reasonable estimate of n for granular roughness in deep∼ streams (i.e. the conditions under which the CSB model was framed), while deviations can be observed for vegetated surfaces and shallow flows (h/r < 15), as already well known from extensive literature on flow resistance in open channel flows (see, e.g. Ferguson (2010) and references therein). Considering different boundary conditions and the dependence of α1α2 on r may explain such deviations. In particular, imposing u(r)/u 3.3 (typical for dense canopy flows) provides Manning’s roughness coefficient n ∗in= better agreement with the measured values.

5. Discussion and conclusions GB02 proposed the first theoretical derivation for two well-known empirical laws in hydraulics: (i) the Manning equation and (ii) the associated Strickler scaling for the roughness coefficient n. The work by GB02 provided a new strategy to address theoretically what has always been an empirical matter. However, despite its significance, simplicity, elegance and appeal, their work required revisions and critical appraisal. In deriving Manning’s formula, GB02 adopted a series of assumptions which have been critically investigated here considering known physics of rough-wall turbulent flows. These include (i) assuming the applicability of Kolmogorov scaling for velocity fluctuations immediately adjacent to roughness elements, (ii) the hypothesis that τ ρVur, where ur represents the velocity scale of an eddy of size r, (iii) the hypothesis∼ that eddies of size r are the major players in momentum transfer≈ in the 3 near-wall region, (iv) the mean TKE dissipation rate scales as V /Rh with a scaling factor independent of r/h near the roughness elements. GB02 have also assumed, although implicitly (but explicitly in a subsequent paper), that the mean velocity gradient at the roughness tops scales as V/r and that the associated scaling factor is also independent of r/h. The present paper first demonstrated that assumption (iv) is incorrect and that the scaling functions pertaining to the TKE dissipation rate and the mean velocity gradient must depend on r/Rh. When this dependence is taken into account within the GB02 derivation, Manning’s equation can still be recovered but not the Strickler scaling as argued by GB02. The new CSB approach proposed herein allows for the relaxation of hypotheses (i), (ii) and (iii) as it moves the point of analysis at the cross-over between the RSL and logarithmic layer, where Kolmogorov scaling for inertial subrange turbulence is applicable, it computes the turbulent shear stress directly from the integration of the co-spectrum and it incorporates all of the population of eddies that contribute to momentum transfer in the RSL, including very large eddies (i.e. eddies that exceed Rh in size).

Downloaded from https:/www.cambridge.org/core. Duke University Libraries, on 30 Jan 2017 at 13:21:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2016.863 Manning equation 1209 While relaxing the hypotheses pertaining to GB02, the CSB approach introduces some new ones. These include neglecting wall-blocking and flux-transfer term effects in the CSB budget and assuming equilibrium between production and dissipation of TKE throughout the flow domain. The former was explored (not shown), and proved to be a more than reasonable assumption, whereas the latter was addressed in preceding papers (Katul & Manes 2014; McColl et al. 2016) and was also shown to be acceptable. Perhaps the most stringent hypothesis of the proposed CSB approach is on the shape of the vertical velocity spectrum. This is seldom available in the literature, and, when it is, it relates to 1-D spectra which suffer major distortions induced by the use of Taylor’s frozen turbulence hypothesis and aliasing effects. This lack of information required a speculative approach with assumed spectral shapes especially for what concerns the low-wavenumber range. We therefore assumed the existence of an inertial subrange characterized by a Kolmogorov spectrum at high wavenumbers and a flat spectrum for the lower wavenumbers. The cross-over between the two regimes is dictated by scales commensurate with the elevation z above the bed (but r in proximity of the wall). This simplified version of the spectrum allowed for analytical foresight of the problem, leading to Manning’s equation from the CSB and momentum budget equations. To evaluate this assumed choice for the vertical velocity spectrum, numerical calculations were performed to explore how the full CSB model responded to different and more complex spectral shapes. In particular, a spectral decay at the low-wavenumber end of the spectrum was introduced, and we varied extensively the size of the flat spectral region at low wavenumbers. The results show that the model is weakly sensitive to how the spectrum decays at low wavenumbers. To reproduce realistic mean velocity profiles in the outer layer, the flat region (i.e. the contribution of large eddies) must be significant in extent. Both results work in favour of the simplified version of the spectrum used to derive Manning’s equation. Even though none of the investigated spectral shapes allowed a full recovery of the Strickler scaling across all of the r/Rh ranges considered, the CSB predicted departures from Strickler scaling that are consistent with well-established equations such as Colebrook’s formula and the VN model.

Acknowledgements Support from the US National Science Foundation (FESD EAR-1338694, NSF- EAR-1344703, NSF-CBET-1033467, NSF-EAR-1331846, NSF-EAR-1316258, NSF- DGE-1068871 through Duke WISeNet) and the USDA Agricultural Research Service cooperative agreement (58-6408-3-027) is acknowledged. C.M. acknowledges support from Compagnia San Paolo.

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