Mean flow and turbulence in vegetated open channel flow

WATER RESOURCES RESEARCH, VOL. 41, W07006, doi:10.1029/2004WR003475, 2005

Mean flow and turbulence in vegetated open channel flow Andrea Defina and Anna Chiara Bixio Dipartimento di Ingegneria Idraulica, Marittima, Ambientale e Geotecnica (IMAGE), Universita` di Padova, Padua, Italy Received 8 July 2004; revised 18 February 2005; accepted 14 March 2005; published 8 July 2005.

[1] Vegetation affects the mean and turbulent flow structure in surface water bodies, thus

impacting the local transport processes of contaminants and sediments. The present paper explores the capability of two different mathematical models to predict fully developed one-dimensional open channel flow in the presence of rigid, complex-shaped vegetation with leaves, submerged or emergent. The flow is described by applying two different turbulence closure schemes, both of which are based on the Boussinesq eddy viscosity model: a suitably modified k e model and a two-layer model based on the mixing length approach. To describe the turbulence structure within and above the canopy a turbulent kinetic energy budget equation was added to the two-layer model. The results of the models were compared with experimental data where simple cylinders, plastic plant prototypes, or real plants, all arranged in a scattered pattern, were employed. Since good agreement between the results of the models and measurements was found in comparing velocity and turbulent shear stress, the models could potentially be used to assess vegetative resistance. Significant disagreement was found when comparing measured and computed eddy viscosity distributions, streamwise turbulence intensity, and most of the terms comprising the turbulent kinetic energy budget. Citation: Defina, A., and A. C. Bixio (2005), Mean flow and turbulence in vegetated open channel flow, Water Resour. Res., 41, W07006, doi:10.1029/2004WR003475.

1. Introduction [2] Vegetation plays an important role in influencing the hydrodynamic behavior, ecological equilibrium and environmental characteristics of water bodies. The knowledge of mean and turbulent flow structure in vegetated environments is of great importance for understanding and assessing the associated transport processes of sediments and contaminants. [3] Much research has been devoted to this topic in recent years. Flume experiments have been performed with natural vegetation [Gambi et al., 1990; Shi et al., 1995; Andersen et al., 1996; Meijer and Van Velzen, 1999; Nepf and Koch, 1999; Stephan and Gutknecht, 2002], plant prototypes [Nepf and Vivoni, 2000; Velasco et al., 2003; Baptist, 2003], or simple elements such as strips or cylinders [Tsujimoto and Kitamura, 1990; Shimizu and Tsujimoto, 1994; Dunn et al., 1996; Nepf, 1999; Righetti and Armanini, 2002]. Field measurements have also been collected [Ackerman and Okubo, 1993; Leonard and Luther, 1995; Sand-Jensen and Mebus, 1996; Sand-Jensen, 1998; Koch and Gust, 1999; Leonard and Reed, 2002]. Many theoretical and numerical investigations have been performed as well, focusing mainly on evaluating vertical velocity and shear stress profiles and characterizing mean turbulence [Klopstra et al., 1997; Shimizu and Tsujimoto, 1994; Nepf and Vivoni, 2000; Lopez and Garcia, 2001; Fisher-Antze et al., 2001; Righetti and Armanini, 2002]. [4] However, the wide variety of vegetation types and hydrodynamic conditions considered in these works make it Copyright 2005 by the American Geophysical Union. 0043-1397/05/2004WR003475

difficult to compare the individual results and draw general conclusions. One solution to this problem is to use mathematical models to describe flow-vegetation interactions and predict the flow field given the actual hydraulic conditions (flow rate, water depth, bottom slope, pressure gradient) and vegetation characteristics (height, biomass distribution, flexibility, density). In particular, models can be used to analyze the influence of the single parameters on the flow field. [5] A review of recent studies dealing with a one-dimensional flow through rigid vegetation shows that there are two different approaches to determining velocity profile through and above submerged vegetation: a two-layer approach, which separately describes flow in the vegetation layer and in the upper layer [Klopstra et al., 1997; Meijer and van Velzen, 1999; Righetti and Armanini, 2002], and a suitably modified turbulence k e model, in which the drag due to vegetation is taken into account not only in the momentum equation but also in the equations for k and e [Burke and Stolzenbach, 1983; Shimizu and Tsujimoto, 1994; Lopez and Garcia, 2001]. [6] Klopstra et al. [1997] and Meijer and Van Velzen [1999] developed and experimentally tested a method to analytically determine the velocity profile that divides the flow domain into two layers, one within the vegetation called ‘‘vegetation layer’’ and the other above it called the ‘‘upper layer’’ and solves the momentum equation in the vegetation layer while keeping a logarithmic profile in the upper layer. Matching boundary conditions at the interface ensures the continuity of velocity and shear stress between the two layers and makes it possible to determine the parameters of the log law. Since experimental calibration of the assumed characteristic length scale of turbulence

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is required, the model can be considered a semiempirical model. A similar model, with only different assumptions to define the mixing length was recently proposed by Righetti and Armanini [2002]. [7] Lopez and Garcia [2001] proposed a k e model to compute the mean velocity profile and turbulence characteristics in open channel vegetated flows. In this model the drag-related sink terms accounting for the presence of vegetation are rigorously derived. [8] Both the models of Lopez and Garcia [2001] and Klopstra et al. [1997] were originally developed for the simulation of submerged vegetation, but can be easily extended to emergent conditions. They both were formulated and tested only on artificial cylindrical vegetation stems characterized by constant with depth geometry and drag coefficient. In reality, variability along the depth of plant geometry and drag coefficient strongly affect the flow inducing higher velocities where the plant biomass is lower and vice versa [Petryik and Bosmajian, 1975; Leonard and Luther, 1995; Nepf and Vivoni, 2000]. [9] In the study described here, these two models were revised and extended to consider plant geometry and drag coefficient variable with depth. In order to give a complete description of turbulence structure within and above the canopy, a turbulent kinetic energy budget equation was added to the two-layer model. Numerical simulations were then performed with both models to reproduce the flow field in the presence of real and artificial vegetation. The results of these simulations were then compared with available experimental data.

[10] In this section the models proposed by Klopstra et al. [1997] and Lopez and Garcia [2001] are briefly described and discussed. In addition, the former is extended to include an equation for the turbulent kinetic energy budget. [11] Both models assume uniform flow conditions and neglect the correction to gravity term for water volume excluded by plants volume, which may be important for very high plant density [Nepf, 1999; Nepf and Vivoni, 2000; Stone and Shen, 2002]. Bed and wall drag are neglected because they are small compared to vegetative drag [Kadlec, 1990; Nepf and Vivoni, 2000; Stone and Shen, 2002]. [12] The momentum equation reads @u 1 @t ¼ gS0 þ fD @t r @z

ð1Þ

where u is the average flow velocity, t time, g gravity, S0 the bottom slope, r the water density, t the viscous and turbulent shear stress, z the vertical coordinate originating at the bed and fD the drag force per unit mass exerted by the vegetation. The drag force in (1) is given as

CD a u2 =2 z hp 0 z > hp

[13] The eddy viscosity model of Boussinesq is employed to describe the turbulent shear stresses which arise from double (i.e., temporal and spatial) averaging of NavierStokes equations [Raupach and Shaw, 1982; Lopez and Garcia, 2001]. Therefore the total shear stress in (1) can be expressed as t ¼ r ðn þ nt Þ

@u @z

ð3Þ

where n is fluid viscosity and nT the eddy viscosity. The latter requires a suitable closure model to be evaluated. Two different closure models are considered here, namely the k e model in the form proposed by Lopez and Garcia [2001] and the two-layer model based on mixing length approach suggested by Klopstra et al. [1997]. 2.1. The K E Model [14] The system of partial differential equations expressing the budget of turbulent kinetic energy k, and dissipation rate e respectively, is [Lopez and Garcia, 2001] @k @ ¼ @t @z @e @ ¼ @t @z

nT @k þn þ Pk e þ Cf k uf D sk @z

nT @e e þ ½C1 ðPk þ Cf e uf D Þ C2 e þn @z k se

ð4Þ

ð5Þ

where sx is the Prandtl-Schmidt number for any variable x, Pk = nT(@u/@z)2 is the shear production,

2. Mathematical Models

f D ðzÞ ¼

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; a ¼ Az m

nT ¼ C m

ð6Þ

and the drag related source terms cfkufD and cfeufD account for the presence of vegetation. The set of standard constants is Cm = 0.09, C1 = 1.44, C2 = 1.92, sk = 1.0, and se = 1.30 [Rodi, 1984]. Moreover, as suggested by Lopez and Garcia [2001], it is assumed that Cfk = 1 and Cfe = (C2/C1) Cfk = 1.33. [15] Because of the small values of the near-bed velocity and because wake turbulence production is much larger than bed shear production, the boundary conditions at the bed have an almost negligible influence on the solution even close to the bed itself [Lopez and Garcia, 2001]. Therefore, to simplify the problem, the following boundary conditions are imposed at the bottom in the k e model kjz¼0 ¼ ejz¼0 ¼ 0

ð7Þ

Within the framework of the above simplifications, slip boundary conditions are applied on the bed by assuming [Klopstra et al., 1997] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gS0 ujz¼0 ¼ ðaCD Þjz¼0

ð2Þ

where hp is plant height, CD the drag coefficient, a the vegetation density or projected plant area per unit volume, Az the frontal area of vegetation per unit depth, and m the number of stems per unit

k2 e

ð8Þ

which expresses the local equilibrium between gravity force and vegetation drag when bottom shear stress is neglected.

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Introducing the dummy variable z = u2/2g, equation (11) can be rearranged to read @ 2 V a CD S0 Vþ ¼0 @z2 a a

Figure 1. Velocity profile within and above vegetation, with notation. [16] At the surface the boundary conditions suggested by Lopez and Garcia [2001] are used, namely, @u @k ¼ ¼ 0; @z z¼H @z z¼H

ejz¼H ¼ k3=2 =be H

2.2. Two-Layer Model [18] A different model based on the mixing length approach was suggested by Klopstra et al. [1997]. In this model the flow depth is split into a lower layer containing vegetation (referred to as the vegetation layer or the roughness layer) and a surface layer that is above the vegetation layer and contains no part of the roughness (Figure 1). [19] Velocity profiles are described separately for the vegetation layer and the surface layer, reflecting the different physical phenomena acting in the two layers, and they are matched at the interface [Klopstra et al., 1997]. Eddy viscosity in the vegetation layer is assumed to be the product of the flow velocity u and a characteristic length scale a which, in the form proposed by Meijer and Van Velzen [1999], reads a=hp ¼ 0:0144

qffiffiffiffiffiffiffiffiffiffiffi H=hp

ð10Þ

Equation (10) is fully empirical and was obtained from an extensive series of flume tests in which steel bars were used to simulate vegetation. The height of steel bars, their density, the energy gradient in the flow direction and the flow depth were systematically varied in the tests [Meijer and Van Velzen, 1999]. Importantly, equation (10) was obtained by comparing measured and computed velocity profiles, while no turbulence characteristics were measured or considered in the analysis. [20] Utilizing equations (2) and (3), momentum equation (1) under steady flow conditions can be written as: au

@2u þa @z2

@u

If vegetation density a and drag coefficient CD are assumed to be constant along the depth, then equation (12) has a simple analytical solution [Klopstra et al., 1997]. On the contrary, the present paper addresses real vegetation and equation (12) is solved numerically to account for the variation in both CD and a along z. [21] At the bottom, boundary condition (8) is imposed, while at the interface (i.e., at z = hp) the shear stresses are matched giving @V @z

ð9Þ

where be is a model coefficient and H the flow depth. [17] The unsteady terms in equations (1), (4), and (5) are retained in the model only for computational purposes. In fact, the steady state solution is obtained as the asymptotic state of transient solutions with constant boundary conditions [Lopez and Garcia, 2001]. The system of p.d.e. that makes up the k e model is solved using MATLAB.

ð12Þ

S0 H hp ¼ a z¼hp

ð13Þ

The above equations make it possible to compute the velocity profile and shear stress distribution within the vegetation layer, and are solved using MATLAB. [22] The flow over the top of the submerged vegetation has been shown experimentally to follow a logarithmic profile [Christensen, 1985; Gambi et al., 1990; Shi et al., 1995; Nepf and Vivoni, 2000; Stephan and Wibmer, 2001] with the virtual zero point of the velocity profile located inside the canopy. Accordingly, for the surface layer, Klopstra et al. [1997] assumed u* zd ln u¼ z0 c

ð14Þ

where u* is the friction velocity, c Von Karman’s constant, d = hp hs the zero-plane displacement of the logarithmic profile, hs the distance between the top of vegetation and the virtual bed of the surface layer (see Figure 1), and z0 the equivalent bed roughness height. [23] Since the actual depth of the flow above the canopy is H-d, the friction velocity is given as [Klopstra et al., 1997; Nepf and Vivoni, 2000] u* ¼ ½gS0 ðH dÞ 1=2

ð15Þ

The unknown parameters hs and z0 can be computed by imposing that the velocity and vertical velocity gradient (or, equivalently, the total shear stress) match at the interface between the surface layer and the vegetation layer (i.e., at z = hp). By imposing these conditions one finds "

ffi# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

2 hs ¼ gS0 þ ðgS0 Þ þ 4 c@u=@zjz¼hp gS0 H hp

2 =2c2 @u=@zjz¼hp h i z0 ¼ hs exp cujz¼hp =u*

ð16Þ

ð17Þ

2

¼ aCD u2 =2 gS0

ð11Þ

where ujz=hp and @u/@zjz=hp are the velocity and the vertical velocity gradient at the interface, which are computed

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Table 1. Vegetation and Flow Characteristics for the Flume Experiments Considereda Flow Characteristics

Vegetation Characteristics Source

Experiment

Type

Az, m

hp, m

m, m2

CD

H, m

S0

Shimizu and Tsujimoto [1994] Shimizu and Tsujimoto [1994] Shimizu and Tsujimoto [1994] Shimizu and Tsujimoto [1994] Shi et al. [1995] Shi et al. [1995] Shi et al. [1995] Shi et al. [1995] Shi et al. [1995] Meijer and van Velzen [1999] Meijer and van Velzen [1999] Nepf and Vivoni [2000] Nepf and Vivoni [2000] Nepf and Vivoni [2000] Nepf and Vivoni [2000] Nepf and Vivoni [2000] Nepf and Vivoni [2000] Lopez and Garcia [2001] Lopez and Garcia [2001]

ST-R31 ST-R32 ST-A31 ST-A71 S-20 S-40 S-60 S-80 S-100 MV-22 MV-R6 NV-1.0 NV-1.25 NV-1.50 NV-1.75 NV-1.90 NV-2.75 LG-1 LG-9

rigid cylinders rigid cylinders rigid cylinders rigid cylinders Spartina anglica Spartina anglica Spartina anglica Spartina anglica Spartina anglica rigid cylinders reeds plastic plant prototype plastic plant prototype plastic plant prototype plastic plant prototype plastic plant prototype plastic plant prototype rigid cylinders rigid cylinders

0.001 0.001 0.0015 0.0015 0.004 0.028b 0.004 0.028b 0.004 0.028b 0.004 0.028b 0.004 0.028b 0.008 0.0057 0.005 0.017c 0.005 0.017c 0.005 0.017c 0.005 0.017c 0.005 0.017c 0.005 0.017c 0.0064 0.0064

0.041 0.041 0.046 0.046 0.06 0.12 0.18 0.24 0.3 0.9 1.58 0.16 0.16 0.16 0.16 0.16 0.16 0.12 0.12

10000 10000 2500 2500 350 350 350 350 350 256 256 330 330 330 330 330 330 170 384

1 1 1 1 1 2.5b 1 2.5b 1 2.5b 1 2.5b 1 2.5b 0.87 1.8 1 3c 1 3c 1 3c 1 3c 1 3c 1 3c 1.13 1.13

0.0631 0.0747 0.0936 0.0895 0.332 0.342 0.338 0.346 0.348 2.08 1.99 0.16 0.2 0.24 0.28 0.304 0.44 0.335 0.214

0.00164 0.0008 0.00089 0.00087 0.00003 0.00003 0.00007 0.0006 0.0006 0.00188 0.0019 0.00017 0.00025 0.00022 0.0005 0.00022 0.00017 0.0036 0.0036

a

Values in italics were lacking in the original works and estimated by the authors. See Figure 2. c See Nepf and Vivoni [2000, Figure 3]. b

using the velocity profile determined for the vegetation layer. [24] Equations (14) to (17) fully define the velocity profile within the surface layer. Moreover, it can be shown that the eddy viscosity and shear stress distributions are given by nT ¼ c

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi gS0 ðH zÞ ðz dÞ= ðH dÞ

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ hp aCD =a ¼

ð22Þ

Moreover, from equations (3) and (10) we have

ð18Þ t ¼ tmax

t ¼ r g S0 ðH zÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ahp CD

a=hp

h

i cosh b z=hp coshðbÞ

tmax ¼ tjz¼hp ¼ rgS0 H hp ð23Þ

ð19Þ

3. Numerical Simulations It is clear that the mixing length approach used to compute the eddy viscosity in the two-layer model only provides limited information on the turbulence structure within and above the canopy. To overcome this limitation, the two-layer model was improved by adding an equation expressing the budget of turbulent kinetic energy. From equation (6) we have e = Cmk2/nT, which is used to remove the dissipation rate e in equation (4). The latter can thus be written as @k @ ¼ @t @z

2 nT @k @u þ nT þn Cm k2 =nT þ Cf k uf D @z @z sk ð20Þ

The above equation is solved with the same boundary conditions used in the k e model, i.e., k = 0 at the bottom and @k/@z = 0 at the free surface. [25] It is worth pointing out that when a and CD are considered constant, equation (11) has the following analytical solution vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h

i 8 9 u sinh b z=h u p u 2gS0 < H 1= þ u¼t

1 coshðbÞ b; b a=hp : hp

ð21Þ

[26] The k e and two-layer models were compared with experimental data reported in the literature. These data were from laboratory experiments where vegetation was simulated using simple rigid cylinders [Lopez and Garcia, 2001; Shimizu and Tsujimoto, 1994; Meijer and Van Velzen, 1999], plastic plants [Nepf and Vivoni, 1999, 2000], and real vegetation [Shi et al., 1995; Meijer and Van Velzen, 1999]. [27] The experimental data are summarized in Table 1. Slope S0 is the bed slope when uniform flow conditions were attained in the experiments, otherwise S0 is the free surface slope. The data of Nepf and Vivoni [2000] and Lopez and Garcia [2001] were derived using a double averaging procedure where values measured over a number of positions in the horizontal are averaged first in time and then in space. Unfortunately, the experimental data for Spartina anglica reported by Shi et al. [1995] are not complete, i.e., the vertical distribution of the drag coefficient and frontal plant area are not available. To overcome this problem, the vertical distributions of a and CD were assumed to be similar to those adopted by Nepf and Vivoni [1999] (Figure 2). 3.1. Mean Flow Structure [28] Good agreement between the results of the models and experimental measurements was found for velocity and

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Figure 2. Assumed vegetation density a(z) and drag coefficient CD(z) for the Spartina anglica (S experiments). Figure 4. Comparison between computed and measured turbulent shear stress and velocity profiles for flume experiment NV-1.50.

shear stress distributions along the vertical. Some examples of the velocity and shear stress profiles obtained with the two models and the experimental data they were compared with are shown in Figures 3 – 6. [29] Interestingly, for the plant prototypes used by Nepf and Vivoni [2000] (Figure 4) and the Spartina anglica used by Shi et al. [1995] (Figure 6), the use of a variable with depth geometry a(z) and drag coefficient CD(z) made it possible to improve the prediction of the velocity and shear stress profiles inside the canopy. Furthermore, the s-shaped velocity profiles characterized by the local maximum and minimum velocities, which are caused by the plant biomass distribution along the vertical, were accurately reproduced. [30] In order to better understand the importance of considering vegetation density a and drag coefficient CD variable with depth, some additional simulations were performed assuming a constant depth averaged value for both a and CD. All the test cases by Nepf and Vivoni [2000], referred to in Table 1 as NV, were considered for this purpose. Since the results show similar behaviors for all of the test cases, only the results for test case NV-1.75 are presented and discussed here. Figure 7 compares the k e model predictions with depth-varying and constant parameters. In the latter case, the values a = 4.45 m1 and CD = 1.194 were used in the simulations. The differences in the velocity profiles extend over the entire flow depth, averaging about ±10% of the actual velocity. The assumption of

constant parameters does not produce an s-shaped velocity profile like the measured one. The shear stress profiles only differ in the vegetation layer, but the differences are rather significant. Finally, the eddy viscosity distribution, which is discussed in the next section, proved to be less sensitive to constant or variable parameters; the differences along the depth are negligibly small. Similar results were obtained with the two-layer model. [31] Further numerical simulations were performed to study the impact of CD values on the predicted flow field. The test case NV-1.75 was simulated after varying the drag coefficient CD(z) by ±20%. The results are reported in Figure 8, showing the comparison between the actual velocity, shear stress, and eddy viscosity profiles and those generated with increased and decreased values of the drag coefficient. As can be seen in Figure 8, imprecise knowledge of the actual drag coefficient has a large impact on velocity and eddy viscosity profiles but not on shear stress profile. This is because shear stress within the vegetation layer is controlled by shear stress above the canopy, which depends on free surface slope and submergence H-hp. The impact of drag coefficient on the velocity within the vegetation layer can be easily deduced from equation (1)

Figure 3. Comparison between computed and measured turbulent shear stress and velocity profiles for flume experiment ST-R32.

Figure 5. Comparison between computed and measured turbulent shear stress and velocity profiles for flume experiment LG-1.

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Figure 6. Comparison between computed and measured velocity profiles for flume experiments S-20, S-40, S-60, S-80 and S-100.

if, as a first approximation, the vertical variation in shear stress is assumed to be constant. In this case CDu2 is found to be constant. 3.2. Turbulence Structure 3.2.1. Eddy Viscosity [32] Both models correctly predict the depth-averaged eddy viscosity (Figure 9). However, differences between the two models can clearly be seen when comparing the vertical distributions of the eddy viscosity. The k e model predicts a convex-shaped behavior for nT in the vegetation layer (i.e., @ 2nT/@z2 > 0) and generally overpredicts eddy viscosity values within this region. On the contrary, the two-layer model predictions compare favorably well with the experimental concave-shaped eddy viscosity profiles, as can be seen in Figures 10 and 11. It is worth noting that eddy viscosity is quite sensitive to average drag coefficient (Figure 8), while it seems unaffected by the assumption of constant or variable parameters (Figure 7). 3.2.2. Turbulent Kinetic Energy Budget [33] The turbulent kinetic energy (TKE) budget illustrates the importance of the physical processes that govern turbulent fluid motions. The presence of vegetation adds a further dimension to the balance since new regions of turbulence production are created in the shear layer at the top of the canopy and in the wakes of the plant elements.

[34] The different length scales involved must be considered carefully. The wake-generated TKE has a length scale (Lw) proportional to the dimensions of the elements in the canopy (i.e., the stem diameter), which is generally much smaller than the scale of the shear-generated turbulence (Lk) determined by the plant height [Raupach et al., 1996; Nepf, 1999; Nepf and Vivoni, 2000]. Lopez and Garcia [2001] argued that when the scale of the wakegenerated turbulence is smaller than the Kolmogorov microscale, which is commonly the case in aquatic flows with relatively low plant densities, most of the mean flow energy extracted by the plant drag is quickly dissipated into spatial fluctuations. [35] All this suggests that a major fraction of Pw, depending on the value of the length scale ratio Lk/Lw, is promptly dissipated and can be left out of the TKE budget. In view of the above discussion the TKE equation (4) is rewritten as: @k ¼ Tt þ Pk e þ Pw @t

ð24Þ

where Pw = CfkufD is the wake production and Tt is the total (i.e., inertial and pressure) vertical transport described by the first term on the right hand side of equation (4). The total dissipation e is then split into a shear-scale dissipation ek and a wake-scale dissipation ew. The latter approximately corresponds to the dissipation of the fraction of the TKE

Figure 7. Comparison between velocity, shear stress, and eddy viscosity profiles computed with constant (dashed line) and depth varying (solid line) a and CD, for flume experiment NV-1.75. Plotted results were com using the k e model. 6 of 12

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Figure 8. Comparison between velocity, shear stress, and eddy viscosity profiles computed for flume experiment NV-1.75 after increasing (dotted line) and decreasing (dashed line) the original drag coefficient (solid line) by 20%. Plotted results were computed using the two-layer model. which arises from the spatial variation of the mean flow field [Lopez and Garcia, 2001]. It is expressed as ew = jPw, with j = j(Lk/Lw) < 1. Consequently, equation (24) is written as @k ¼ Tt þ Pk ek þ ð1 jÞPw @t

ð25Þ

If, as Lopez and Garcia [2001] and Nepf and Vivoni [2000] have suggested, cfk = 1 is used, the last term in equation (25) becomes (1 j)cfkufD = (1 j)ufD. This is like solving equation (4) after substituting cfk with an equivalent weighting coefficient cfkk = (1 j)cfk = (1 j). When j = 0.93, cfkk = 0.07, which is the value suggested by Shimizu and Tsujimoto [1994]. If we assume, in accordance with Lopez and Garcia [2001], that the weighting coefficient in the dissipation rate equation (5) can be expressed as cfe = 1.33cfkk, then cfe = 0.093. This value is not very different from the one, cfe = 0.16, found by Shimizu and Tsujimoto [1994]. [36] Figure 12 reports some terms of the TKE budget together with the normalized TKE profile as calculated by the two models for the experiment NV-2.75. The two different sets of weighting coefficients, i.e., Cfk = 1 and Cfe = 1.33 and Cfk = 0.07 and Cfe = 0.16, suggested by Lopez and Garcia [2001] and Shimizu and Tsujimoto [1994], respectively, were used in the calculations. We can observe that both models pffiffiffi overpredict the normalized turbulent kinetic energy k=u when the first set of weighting coefficients is used, particularly in the vegetation layer. Slightly better results are obtained when using the second set of weighting coefficients. Nonetheless, neither model is able to predict the concave-shaped profile seen in the experimental measurements when approaching the bottom. [37] The shear production rate Pk is predicted well by the k e model but it is always overestimated by the two-layer model. Both models correctly estimate zero shear production in the lower part of the canopy. 3.2.3. Turbulence Intensity [38] After the turbulent kinetic energy k and the dissipation rate e are computed, the streamwise turbulence intensity can be calculated using an algebraic stress model. This kind of model was first propo Rodi [1984] for unvegetated

flow, and then modified by Lopez and Garcia [1997] to account for the extra turbulence generated by plants. [39] Lopez and Garcia [2001] showed that the weighting coefficient cfk should be set to unity if the total turbulence intensity, including spatial and temporal velocity fluctuations, is to be modeled. They also observed that experimental values of the total turbulence intensity are difficult to measure because the spatial variation of the mean flow field makes it necessary to consider a large number of measurement locations. As a consequence, available turbulence intensity measurements within vegetation usually only consider the temporal fluctuation term averaged over a limited number of spatial positions. [40] This is why the use of cfk = 1 cfe = 1.33 generally leads to overprediction of turbulence intensity data, while smaller values of cfk and cfe give better results [Lopez and Garcia, 2001]. Nonetheless, both the proposed sets of weighting coefficients lead to unsatisfactory results from the two models. In fact, while most of the available experimental data clearly shows concave-shaped profiles

Figure 9. Comparison between computed and measured depth-averaged eddy viscosity values for NV flume experiments.

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Figure 10. Comparison between computed and measured eddy viscosity profiles for flume experiment NV-1.75. of streamwise turbulence intensity within the vegetation layer, the models results always show convex-shaped profiles. This discrepancy is probably due to the highly nonisotropic character of the flow, which cannot be described by present models. Cui and Neary [2002] recently presented the results of a numerical simulation of a fully developed three-dimensional flow through vegetation. They showed that the streamwise and vertical component of turbulence intensity are greatest at the top of the vegetation layer while the spanwise component grows monotonically from the bottom up to the free surface. Moreover, the streamwise turbulence intensity within the vegetation layer shows a clear concave-shaped behavior as measured in most experiments. These results confirm the highly nonisotropic vertical behavior of turbulent intensities which cannot be predicted by the algebraic stress model proposed by Lopez and Garcia [1997]. 3.2.4. Penetration Depth [41] Nepf and Vivoni [2000] observed that flow through a submerged canopy can be divided into two regions: the ‘‘vertical exchange zone’’ in the upper canopy, where the vertical turbulent exchange with the overlying water has a significant influence on the momentum balance, and the

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‘‘longitudinal exchange zone’’ near the bottom, where the vertical turbulent transport of momentum is negligible and the pressure gradient is balanced by vegetative drag. The distance from the bottom to the limit between the two zones is called penetration depth. Nepf and Vivoni [2000] suggested evaluating the penetration depth as the distance dt between the bottom and the point within the canopy at which the turbulent stress has decayed to 10% of its maximum value. [42] Nepf and Vivoni [2000] also observed that when expressing the velocity profile in the upper layer by the logarithmic law (equation (14)), the zero-plane displacement d is directly related to the penetration depth, since it corresponds to the average height of momentum absorption by the vegetation [Jackson, 1981; Raupach et al., 1991], and consequently measures the distance into the canopy interested by the vertical penetration of turbulent stress. [43] Both d and dt were calculated with the k e model and the two-layer model for some of the flume tests reported in Table 1, and compared with available experimental data. When assuming a and CD to be constant, a simple analytical solution is provided by the two-layer model: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt 1 1:1 þ tanh b tanh2 b 0:99 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 ‘n hp b 1:1 tanh b þ tanh2 b 0:99

ð26Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffi! d H 2 ¼1M 1 1þ 1þ hp hp M

ð27Þ

where b is given by equation (22) and H 1 þ b 1 tanh b hp 1 a M¼ 2 3 c hp H b2 1 hp

ð28Þ

Therefore, recalling equation (10) for a, both d/hp and dt/hp are functions of H/hp and a hp.

Figure 11. Comparison between computed and measured turbulent shear stress, velocity, and eddy viscosity profiles me experiment NV-2.75. 8 of 12

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Figure 12. Comparison between computed and measured TKE, shear production, total turbulent transport, and dissipation profiles for flume experiment NV-2.75. Two different sets of weighting coefficients were used in the k e model, i.e., (top) Cfk = 1, Cfe = 1.33, as suggested by Lopez and Garcia [2001], and (bottom) Cfk = 0.07, Cfe = 0.16, as suggested by Shimizu and Tsujimoto [1994].

[44] When assuming CD = 1, equations (26) and (27) are plotted in Figure 13 as a function of H/hp for different values of ahp. The dt/hp curves monotonically decrease with increasing water depth. The limiting emergent conditions H/hp = 1 are characterized by dt/hp values varying between 0.2 and 0.7, depending on ahp values. [45] The d/hp curves show a different behavior, rapidly growing with water depth at low submergence levels, reaching a maximum between H/hp = 1.5 and 2.5 and then decreasing at high submergence levels. The limiting emergent conditions are approached by negative d/hp values. [46] The theoretical curves in Figure 13 qualitatively agree with the observations of Nepf and Vivoni [2000] at high submergence, while at low submergence the theoretical values of dt/hp and d/hp do not approach 1 as the submergence decreases toward H/hp = 1. The reason for these results must be ascribed to the inadequacy of equation (10) at low submergence. More refined assessment of the length scale a characterizing turbulence within the vegetation layer would improve the two-layer model performance. Moreover, Nepf and Vivoni [2000] observed a strict relationship between d and dt, i.e., (1 d/hp) 0.5(1 dt/hp). The theoretical curves based on the two-layer model show a similar relationship between d and dt only at the higher values of ahp and H/hp. A number of computations were also performed with the k e model to com e dt/hp and d/hp curves in

Figure 13. The results for hp = 0.2 m and CD = 1 are shown in Figure 14. We can observe that both dt and d increase as the water depth decreases and approach emergent conditions, i.e., dt/hp = d/hp = 1, as found by Nepf and Vivoni [2000]. Furthermore, the above quoted relationship between d and dt is predicted fairly well. A comparison between the results of the two models and available experimental data is shown in Figure 15. It is clear that the k e model fits the measured values of dt more accurately than the two-layer model. In particular, for the set of data with a = 1.1 m1, the shear stress profile

Figure 13. Penetration depths dt/hp and d/hp, computed using the two-layer model, versus H/hp for different values of ahp.

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and Vivoni [2000], showing good agreement between dT and d.

4. Conclusions

Figure 14. Penetration depths dt/hp and d/hp, computed using the k e model, versus H/hp for different values of ahp.

within the vegetation layer predicted by the two-layer model shows values that are always larger than 10% of the maximum shear stress (Figure 5). This leads to dt = 0 and thus to a predicted vertical exchange zone extending through the whole vegetation layer. [47] Nepf and Vivoni [2000] also argued that penetration depth should increase as vegetation density a increases. However, as can be seen in Figure 15, the penetration depth is greater in experiments with a = 5.5 m1 than in those with a = 10 m1. On the contrary, when the parameter ahp is used to qualify vegetation density, as suggested by the analytical solution of the two-layer model, we can see that the penetration depth does increase as the vegetation density increases (Figure 15). [48] A further definition of the penetration depth can be based on the transport term Tt of the turbulent kinetic energy budget equation (24). This term represents the total vertical transport of turbulent kinetic energy, and is responsible for removing energy from the canopy top region and for redistributing it within the vegetation layer. Since negative Tt values in the upper part of the canopy characterize the vertical exchange zone, the penetration depths d and dt have been compared with the elevation above the bottom, dT, at which Tt jumps to zero before assuming small negative values. Figure 16 compares the results computed with the k e model with the experimental data of Nepf

[49] A number of numerical simulations were performed using the k e model and the two-layer model in order to test their ability to predict the flow field in the presence of rigid vegetation. A comparison between experimental data and the results of these simulations clearly demonstrates that both models can fairly accurately reproduce the vertical profiles of velocity and shear stress within and above vegetation. Therefore these models can be used to assess vegetative resistance to flow. Moreover, the results can be even more accurate when the plant geometry and drag coefficient variation along the vertical are taken into account. It has also been shown that the values assumed for the drag coefficient strongly affect the velocity profile but have a minor impact on shear stress. [50] Eddy viscosity profiles are fairly accurately predicted by the two-layer model. The k e model, on the other hand, does not accurately predict the behavior of eddy viscosity within the vegetation layer. In this layer, the model predicts a parabolic profile and generally overestimates the experimental values. [51] Turbulence characteristics are poorly predicted by both the models. This is mainly because the models are not able to effectively account for the presence of both the shear and wake turbulence length scales. However, there is also some uncertainty in the experimental data because measuring flow velocity in the presence of vegetation is quite difficult given that the spatial variation of the mean flow field makes it necessary to consider a large number of measurement locations. [52] The penetration depth of turbulent stress inside the canopy was estimated according to different criteria. These were based on the analysis of vertical profiles of Reynolds’ stress, velocity and the total transport of turbulent kinetic energy. The results of the two models confirmed the experimentally observed trend of dt/hp increasing as vegetation density increases if the product ahp is used instead of a to characterize the vegetation density. [53] Results obtained using the k e model showed that penetration depth is a decreasing function of depth ratio

Figure 15. Comparison between computed and measured penetration depth dt/hp. Original values of drag coefficient and vegetation density were used in the computations. For the NV case the computed penetration depth used variable CD and a; the value a = 5.5 m1 reported here is a representative vegetation densit Nepf and Vivoni, 2000, Figure 9]. 10 of 12

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n, nt viscosity and eddy viscosity, m2 s1. r water density, kg m3. t shear stress, N m2. [55] Acknowledgments. This work was supported by CO.RI.LA. under the research program ‘‘Dispersione intermareale, morfologia e processi morfodinamici a lungo termine nelle lagune’’ (linea 3.18). The anonymous AE is also kindly acknowledged.

References

Figure 16. Penetration depths dt/hp, d/hp and dT/hp as a function of submergence H/hp and comparison between the k e model and experimental results for the flume experiments of Nepf and Vivoni [2000]. H/hp, in agreement with experimental data. However, the two-layer model predicts unrealistic values for the penetration depth for limiting emergent conditions. [54] To sum up, both models proved to be effective in predicting velocity and shear stress, but not quantitative turbulence. Future research efforts should focus on modeling the turbulence generated by the interaction between flow and vegetation.

Notation a projected plant area per unit volume, m1. AZ frontal area of vegetation per unit depth, m. C1, C2, Cm numerical parameters of standard k e model. CD drag coefficient. Cfk, Cfe weighting coefficients of k e model. d zero-plane displacement of the logarithmic profile, m. dT elevation at which TT = 0, m. dt penetration depth, m. fD drag force per unit mass, m s2. g gravitational acceleration, m s2. H water depth, m. hp vegetation height, m. k turbulent kinetic energy, m2 s2. m number of stems per unit area, m2. Pk shear production, m2 s3. Pw wake production, m2 s3. S0 bottom slope. t time, s. Tt total transport of turbulent kinetic energy, m2 s3. u average flow velocity, m s1. u* shear velocity, m s1. z vertical coordinate, m. z0 equivalent bed roughness height, m. a characteristic length scale of the two-layer model, m. e dissipation rate, m2 s3. ek shear-scale dissipation rate, m2 s3. ew wake-scale dissipation rate, m2 s3. c von Karman’s constant. number for variable x. sx Prandtl-Sc

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A. C. Bixio and A. Defina, Dipartimento di Ingegneria Idraulica, Marittima, Ambientale e Geotecnica (IMAGE), Universita` di Padova, via Loredan 20, I-35131 Padova, Italy. ([email protected])

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