# Derivation of Canopy Resistance in Turbulent Flow from First-Order Closure Models

^{1}

^{2}

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^{5}

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## Abstract

**:**

_{w})

^{1/3}describes the energy loss of flowing water caused by small-scale roughness elements characterized by size r (<<h

_{w}), where h

_{w}is the water depth. When the roughness obstacle size becomes large (but <h

_{w}) as may be encountered in flow within canopies covering wetlands or river ecosystem, the f becomes far more complicated. The presence of a canopy introduces additional length scales above and beyond r/h

_{w}such as canopy height h

_{v}, arrangement density m, frontal element width D, and an adjustment length scale that varies with the canopy drag coefficient C

_{d}. Linking those length scales to the friction factor f frames the scope of this work. By adopting a scaling analysis on the mean momentum equation and closing the turbulent stress with a first-order closure model, the mean velocity profile, its depth-integrated value defining the bulk velocity, as well as f can be determined. The work here showed that f varies with two dimensionless groups that depend on the canopy submergence depth and a canopy length scale. The relation between f and these two length scales was quantified using first-order closure models for a wide range of canopy and depth configurations that span much of the published experiments. Evaluation through experiments suggests that the proposed model can be imminently employed in eco-hydrology or eco-hydraulics when using the De Saint-Venant equations.

## 1. Introduction

_{f}necessitates a closure model, the subject of the work here.

_{f}. The most common ones rest on the assumption that the flow may be ‘locally’ steady and uniform [9,10] so that Manning’s formula $n={U}_{b}^{-1}{R}_{h}^{2/3}{S}_{f}^{1/2}$ can be used to link the bulk velocity for whole depth U

_{b}to S

_{f}, where n is Manning’s roughness coefficient, and R

_{h}is the hydraulic radius. To proceed further, a link between n and roughness element size r, given by n~r

^{1/6}, has been empirically proposed and is often referred to as the Strickler-scaling. This scaling describes several data sets reasonably as discussed elsewhere [11] but not over the entire range of ~r/h

_{w}[12], where h

_{w}is the water depth. Other versions replace Manning’s formula with a Chézy equation where a coefficient C is used to represent roughness effects on the flow. However, these equations can all be made identical by linking their roughness parameterizations via $C={R}_{h}^{1/6}/n={\left(8g/f\right)}^{1/2}$, where g is gravitational acceleration, f is the well-studied Darcy–Weisbach friction factor expressed as $f=8\tau /\left(\rho {U}_{b}^{2}\right)$, $\tau =\rho {u}_{*}^{2}$ denotes the turbulent shear stress acting on the interface between the flow and solid boundary, ρ is water density, and ${u}_{*}=\sqrt{g{S}_{f}{R}_{h}}$ represents the friction velocity. To summarize, a closure model for the friction slope S

_{f}can be inferred from the friction factor ($f=8g{S}_{f}{R}_{h}{U}_{b}^{-2}$) once the friction factor is determined from the flow conditions and the geometry of roughness elements.

_{w}<< 1) dominates the flow (left-side of Figure 1), traditional boundary layer theory leads to a mean velocity profile given as [13].

_{0}is the momentum roughness height that can be linked to n. Thus, the bulk velocity for flow over small-scale roughness (r/h

_{w}<< 1) at high Reynolds number (i.e., for thin buffer and wake regions when compared to the log-region) can be linked to the log-law via [14].

_{w}>> d, η = 1/7–1/6 thereby recovering the aforementioned Strickler scaling for a certain range of large r/h

_{w}[12] for r~z

_{o}. The power-law approximation to the log- function in Equation (2) only applies for a certain range of h

_{w}/z

_{o}>> 1 discussed and delineated elsewhere [12,14]. However, when the roughness height is comparable to the flow depth, as may be encountered in aquatic vegetation covering a wetland or a river channel (right-side of Figure 1), the flow region can be decomposed into three layers according to the dominant vortical structures. Moving from the bottom to the flow surface vertically, the turbulent mixing-length is dominated by three vortical flow types, namely, von Kármán street (being spawned from the canopy element wakes), mixing layer (being formed by the fast flow above the vegetation and the slow flow inside the vegetation), and attached eddies far from the vegetated surface as expected from canonical rough-wall boundary layer theory [15]. Support for this 3-layer representation is as follows: (1) In the deeper vegetation layer (z/h

_{v}< 1), spectral analysis reveals that the energetic motion is dominated by von Kármán vortex streets with size L

_{v}, which is proportional to the vegetation stem diameter D, where the mixing length is strictly dependent on the vegetation element diameter and independent of the local velocity and canopy density; (2) well above the canopy (z/h

_{v}>> 1), the flow resembles a canonical rough-wall boundary layer, where the characteristic vortex size L

_{CBL}scales with the distance from the boundary (or zero-plane displacement) as expected for attached eddies; (3) for the zone near the canopy top (z/h

_{v}≈ 1), the flow is dominated by mixing-layer eddies [16] but occasionally is disturbed by attached eddies, although these types of eddies do not coexist in space. Simulations and experiments provide substantial support for this hypothesis as reviewed elsewhere [17].

## 2. Theory

#### 2.1. Vegetation Resistance

_{d}(discussed in Section 2.1.1) and the choice of the reference velocity within the vegetated region of the flow U

_{v}(discussed in Section 2.1.2). Then the friction factor of vegetation is given in Section 2.1.3 based on the approach of Darcy-Weisbach friction using the results of Section 2.1.1 and Section 2.1.2. The novelty of the proposed approach here lies in their joint specification via two dimensionless groups that accommodate the vortical structures featured in Figure 1.

_{d}is the overall vegetation drag coefficient, which is a function of the flow state quantified by a Reynolds number (to be discussed later); m is the vegetation density determined by the number of stems per unit bed area; D is the frontal width of the canopy; and U

_{v}is a reference velocity representing the flow in the vegetation section. There are several plausible velocities to represent U

_{v}, and those are discussed later. Beyond the vegetation properties (m, D, h

_{v}), Equation (6) requires two flow-rate parameters describing the effects of vegetation elements on the flow: The drag coefficient C

_{d}and reference velocity of vegetated region U

_{v}.

#### 2.1.1. Drag Coefficient

_{d}, which codifies the interaction between the flow state and the morphology and density of the canopy, has been widely studied [28,29]. When neglecting the interaction between adjacent wake elements of a vegetation array, the standard expression [7,9,30] for the drag coefficient of an isolated cylinder-type canopy for the range of Re

_{d}between 0.02 and 2 × 10

^{5}is:

^{5}), Equations (7)–(9) predict a constant C

_{d,iso}≈ 1.2. In most cases, the interaction between wakes of elements cannot be overlooked; thus, the effect of the canopy array is not the same as that of an isolated one [9,31]. For this reason, the drag coefficient of a vegetation array has attracted significant research interest [14,30,32,33,34,35,36].

_{d}exhibits a parabolic-shape along with Reynolds number [9], indicating that the drag effect initially increases and then decreases along with the Reynolds number. When adding rainfall on nonuniform flow in laboratory experiments [7], the rainfall also changes the features of the cross-sectional drag coefficient, which changes the parabolic shape of the C

_{d}–Re relation into a monotonous one where the drag coefficient decreases along with increasing Reynolds number during large rainfall events.

_{d}–Re was proposed on the basis of measured profiles of Reynolds stress and the velocity from laboratory experiments. Naturally, this linear relation applies for a restricted range of water depth and Re

_{d}.

_{d}–Re. The monotonic decline trend was also found in other experiments. Cheng and Nguyen [38] assembled numerous experimental results [23,37,39,40,41,42,43] and reported a monotonic decline in drag coefficient with an increasing vegetation-related Reynolds number Re

_{v}. An expression that fits their data and incorporates the fractional volume concentration is given by:

_{d}(>2 × 10

^{5}) and small ϕ, Re

_{v}reduces to (π/4) Re

_{d}and C

_{d,array}≈ 0.8 (a constant).

#### 2.1.2. Reference Velocity

_{v}is flow rate for vegetation layer.

_{e}= B(1 − ϕ) is used instead of the maximum flow width B.

_{c}= B(1 − D/L

_{s,stag}), and L

_{s,stag}is the spacing distance defined by Etminan et al. [47]. The work of Etminan et al. [47] also showed that for very large Re

_{d}(>2 × 10

^{5}) and small ϕ, C

_{d}≈ 1 (a constant) for a staggered cylinder arrangement.

_{p}is a kinetic energy of the sub-grid scale for a Smagorinsky eddy-viscosity model.

_{v}= U

_{v,pore}) due to the diverse vegetation arrangements covered by the experiments analyzed here.

#### 2.1.3. A Friction Formula

_{c}is the adjustment length scale, which is expressed as:

_{v}. Define U

_{s}as the depth-averaged velocity for the free water layer overlying the vegetation, and ∆U = U

_{s}− U

_{v}is the bulk velocity difference between the surface and vegetation layers, then the depth-averaged velocity for the whole depth U

_{b}can be expressed as:

_{v}/U

_{b}in the friction factor of vegetation expression (Equation (17)) can be made as a function of submergence h

_{v}/h

_{w}and ∆U/U

_{v}via

_{v}indicates the blocking effects of the vegetation on the flow. Previous results [49] showed that f

_{v}can be linked to two dimensionless groups, namely, submergence and drag length scale, which are defined as

_{v}(Equation (17)) becomes a function of three terms: α, β and ∆U/U

_{v}. A formula linking ∆U/U

_{v}to α, β can be derived using a first-order closure model thereby allowing the friction factor to be uniquely varying as a function of α and β, i.e., f

_{v}= f

_{v}(α, β).

#### 2.2. First-Order Closure Models

_{o}= sinθ is the bed slope, g is the gravitational acceleration, z is the height above the bed. Term $\partial \overline{{u}^{\prime}{w}^{\prime}}/\partial z$ is the spatially averaged Reynolds stress whereas the term $\partial {\overline{u}}^{\u2033}{\overline{w}}^{\u2033}/\partial z$ is the dispersive stress arising from spatial correlations in the time-averaged velocity field. It has been shown by Poggi et al. [50] that the dispersive stress is less than 10% of the Reynolds stress for mDh

_{v}> 0.1. Term $\nu {\partial}^{2}u/\partial {z}^{2}$ is the viscous stress whereas term F

_{d}is vegetation drag, and δ = 1 in the vegetation layer (z/h

_{v}< 1) and δ = 0 for the surface layer (z/h

_{v}> 1). For planar homogeneous and steady-uniform turbulent flow, the dispersive ($\partial {\overline{u}}^{\u2033}{\overline{w}}^{\u2033}/\partial z$) and viscous ($\nu {\partial}^{2}u/\partial {z}^{2}$) stresses are neglected, and the mean pressure gradient ∂p/∂x = 0 for uniform flow (assuming the pressure is quasi-hydrostatic). For such simplifications, Equation (23) reduces to [15,49].

_{m}is the eddy diffusivity for momentum and is impacted by the vortical structure dominating the various layers as shown in Figure 1.

_{m}and a model for the drag coefficient C

_{d}are required to solve the above second-order ordinary differential equation.

_{eff}is the effective mixing length parameterized as a function of the established vortex sizes of the different layers described in Figure 1.

## 3. Laboratory Experiments

_{d}= U

_{b}D/ν) for these data ranged from 61 to 9936. The Froude number (Fr = U

_{b}/(gh

_{w})

^{1/2}) ranged from 0.0045 to 0.5649 (sub-critical for all runs). The vegetation drag coefficient (C

_{d,array}of Equation (11)) ranged from 0.84 to 6.35 with the averaged value 1.28 for all the data points here. A summary of the basic information for the experiments analyzed here is featured in Table 1.

_{d,array}(calculated by Equation (11)) was adopted in this model, and the value is shown in Table 1. The results show that using C

_{d,array}derived from numerous experiments reasonably predicts measured bulk velocity when employed with the effective mixing length featured in Figure 1. The modeled bulk velocity was determined from the first-order closure by using the computed mean velocity profile via:

_{v}.

## 4. First-Order Closure Model Runs

_{v}reflecting the blocking effects of vegetation on the flow is investigated. A link between ∆U/U

_{v}and the two dimensionless groups α and β is to be established using the first-order closure model results. In Section 4.1, a scaling analysis is first conducted between ∆U/U

_{v}and α while fixing β. In Section 4.2, a similar scaling analysis is to conducted between ∆U/U

_{v}and β while fixing α. Finally, In Section 4.3, a general relation between ∆U/U

_{v}and α, β is proposed by combining results from Section 4.1 and Section 4.2. Last, an empirical formula for ∆U/U

_{v}to be used in the determination of f is proposed and comparisons with bulk velocity and friction factor measurements are conducted.

#### 4.1. Scale Analysis with Submergence

_{v}must vary only with α. As customary with scaling analysis, it is assumed that:

_{s}= h

_{w}− h

_{v}= 0, indicating an emergent condition, a ∆U = 0 is recovered.

^{−2}. For vegetation height h

_{v}= 0.5 m, flow depth h

_{w}varied from 0.6 m to 4.0 m. Figure 3 shows the best fit curve to results obtained from the first-order closure model.

#### 4.2. Scale Analysis with Vegetation Attributes

_{v}also increases with vegetation drag coefficient originating from the frontal area per unit bed area. Thus, an empirical formula can be given as follows based on a scaling analysis.

_{v}equals zero), ∆U = 0 is recovered.

_{w}= 1 m, vegetation h

_{v}= 0.5 m, the number of vegetation stems per unit bed area m ranging from 100 stems/m

^{2}to 2000 stems/m

^{2}. The first-order closure model calculations, summarized in Figure 4, are used to determining the coefficients in Equation (35) using regression analysis.

#### 4.3. Expression for the Combined Influences of Submergence and Vegetation

_{v}and α, β is now proposed by combining Equations (34) and (35) to yield.

_{1}, c

_{2}, and c

_{3}are constant.

_{1}= 1.7237, c

_{2}= 0.8545, and c

_{3}= 0.4944. Hence, it follows that:

## 5. Model Validation

_{v}is used to model the friction factor f

_{v}and bulk velocity U

_{b}and then compare against laboratory experiments described in Table 1. The comparisons between model calculations (Equations (37)–(39)) and data are featured in Figure 6, Figure 7 and Figure 8. It is to be noted that α, β here are determined from the first-order closure model results and thus c

_{1}= 1.8629, c

_{2}= 0.7909, and c

_{3}= 0.5137 are simply a summary of those model calculations. No data were used in the determination of α, β or coefficients c

_{1}to c

_{3}. Hence, the comparisons with laboratory experiments in Figure 6, Figure 7 and Figure 8 can be treated as validation.

_{v}are presented in Figure 6. Almost all the data points collapsed on a 1-to-1 line except for six points when ∆U/U

_{v}was small (submergence was minimal), and they were affected by many other factors. The flow in the surface layer may not have been fully developed in those cases. From Figure 6, agreement between modeled and measured is acceptable for ∆U/U

_{v}> 0.3.

_{v}show scatter, the modeled bulk velocity reasonably predicts the measurements. The Mean Squared Error (MSE) is used here to evaluate deviations between modelled and measured results and is given as:

_{sample}is the number of the data sample.

_{v}, the comparison between modelled and measured variables yields MSE = 0.0894, and maximal departure from modelled to measured is 1.5565.

_{v}, the comparison between modelled and measurements yields MSE = 0.2706 and maximal departure of 2.8152.

_{b}, comparison between model and measurements yields MSE = 0.0041 m

^{2}/s

^{2}, with maximal departure between model and measured being 0.3581 m/s.

_{v}can be modeled when ∆U/U

_{v}> 0.3. Using Equations (37) and (38) that are derived from Equation (36), the friction factor and bulk velocity can be reliably determined and used for modeling overland flow via the SVEs.

## 6. Conclusions

_{w}such as canopy height, arrangement density, frontal element width, and drag coefficient. To link those length scales to the friction factor, scaling analysis aided by first-order closure model calculations were adopted. It was shown that the Darcy–Weisbach friction factor for canopies varies with two dimensionless groups, namely, canopy submergence and a dimensionless canopy length scale. These two dimensionless groups were then determined from a first-order closure model calculation that explicitly considered vortical sizes within and above the various canopy zones. Comparison between experiments and calculated bulk velocity and friction factor suggested that the proposed link between S

_{f}and bulk velocity employed in SVEs can be used for describing flow through vegetation.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Sketch of the roughness effect and canonical mean velocity profile shape for a small-scale obstacle (

**left**) and tall vegetation (

**right**).

**Figure 2.**Comparison between measured and modeled bulk velocities (m/s) using the first-order closure model along with C

_{d,array}and the mixing length model featured in Figure 1.

**Figure 5.**Comparing the first-order closure model results for ∆U/U

_{v}to the power-law expression in Equation (36) when using c

_{1}= 1.8629, c

_{2}= 0.7909, and c

_{3}= 0.5137.

Authors | Stem Shape | Flow Condition | Q (m^{3}/s) | B (m) | S_{o} (%) | h_{w} (m) | h_{v} (m) | D (m) | m (Stems/m^{2}) | C_{d,array} | Re_{d} | Fr |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Dunn [52] | cylindrical | uniform | 0.046–0.181 | 0.91 | 0.36–1.61 | 0.164–0.391 | 0.118 | 0.006 | 43–387 | 0.84–1.06 | 1891–5421 | 0.2057–0.5649 |

Ghisalberti and Nepf [53] | cylindrical | uniform | 0.002–0.014 | 0.38 | 0.0002–0.01 | 0.467 | 0.138–0.139 | 0.006 | 391–1250 | 1.66–6.35 | 61–516 | 0.0045–0.0377 |

Liu et al. [41] | cylindrical | uniform | 0.011 | 0.3 | 0.3 | 0.087–0.119 | 0.076 | 0.006 | 97–496 | 0.89–1.11 | 2028–2774 | 0.2957–0.4730 |

López and García [54] | cylindrical | uniform | 0.046–0.181 | 0.91 | 0.36–1.61 | 0.164–0.391 | 0.12 | 0.006 | 42–384 | 0.84–1.06 | 1906–5463 | 0.2057–0.5649 |

Meijer [55] | cylindrical | nonuniform | 0.866–8.98 | 3 | 0.055–0.205 | 0.990–2.500 | 0.45–1.5 | 0.008 | 64–256 | 0.89–1.13 | 1400–9936 | 0.0397–0.2767 |

Meijer and Velzen [56] | cylindrical | nonuniform | 3.557 | 3 | 0.138 | 2.08 | 0.9 | 0.008 | 256 | 1.08 | 4560 | 0.1263 |

Murphy et al. [57] | cylindrical | nonuniform | 0.002–0.014 | 0.38 | 0.0003–0.1340 | 0.088–0.467 | 0.070–0.140 | 0.006 | 417–1333 | 1.11–3.63 | 90–1060 | 0.0088–0.1546 |

Nezu and Sanjou [58] | flat strip | uniform | 0.003–0.008 | 0.4 | 0.0196–0.1553 | 0.063–0.200 | 0.05 | 0.008 | 947–3676 | 1.42–4.44 | 800–960 | 0.0714–0.1278 |

Poggi et al. [15] | cylindrical | nonuniform | 0.162 | 0.9 | 0.004–0.0320 | 0.6 | 0.12 | 0.004 | 67–1072 | 0.96–1.38 | 1200 | 0.1237 |

Shimizu et al. [59] | cylindrical | uniform | 0.002–0.016 | 0.40–0.50 | 0.0660–0.7000 | 0.050–0.106 | 0.041–0.046 | 0.01–0.02 | 2501–9995 | 1.09–1.83 | 65–496 | 0.0826–0.3529 |

Stone and Shen [36] | cylindrical | uniform | 0.002–0.065 | 0.45 | 0.009–4.400 | 0.151–0.314 | 0.124 | 0.003–0.013 | 166–692 | 0.91–1.60 | 126–5405 | 0.0279–0.4436 |

Yan [60] | cylindrical | uniform | 0.014–0.038 | 0.42 | 0.065–1.280 | 0.120–0.300 | 0.06 | 0.006 | 500–2000 | 1.09–1.87 | 1714–1845 | 0.1703–0.2763 |

Yang [61] | cylindrical | uniform | 0.008–0.011 | 0.45 | 0.141–0.269 | 0.075 | 0.035 | 0.002 | 1400 | 1.09–1.12 | 444–622 | 0.2592–0.3629 |

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**MDPI and ACS Style**

Wang, W.-J.; Peng, W.-Q.; Huai, W.-X.; Katul, G.; Liu, X.-B.; Dong, F.; Qu, X.-D.; Zhang, H.-P.
Derivation of Canopy Resistance in Turbulent Flow from First-Order Closure Models. *Water* **2018**, *10*, 1782.
https://doi.org/10.3390/w10121782

**AMA Style**

Wang W-J, Peng W-Q, Huai W-X, Katul G, Liu X-B, Dong F, Qu X-D, Zhang H-P.
Derivation of Canopy Resistance in Turbulent Flow from First-Order Closure Models. *Water*. 2018; 10(12):1782.
https://doi.org/10.3390/w10121782

**Chicago/Turabian Style**

Wang, Wei-Jie, Wen-Qi Peng, Wen-Xin Huai, Gabriel Katul, Xiao-Bo Liu, Fei Dong, Xiao-Dong Qu, and Hai-Ping Zhang.
2018. "Derivation of Canopy Resistance in Turbulent Flow from First-Order Closure Models" *Water* 10, no. 12: 1782.
https://doi.org/10.3390/w10121782