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Modeling streamwise velocity and boundary shear stress of vegetation-covered flow
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Original Articles
Modeling streamwise velocity and boundary shear stress of vegetationcovered flow ⁎
Lijuan Hana, Yuhong Zenga, , Li Chena, Ming Lib a b
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, P.R. China State Key Laboratory of Ocean Engineering, School of Naval Architecture, Shanghai Jiao Tong University, Shanghai, P.R. China
A R T I C L E I N F O
A B S T R A C T
Keywords: Vegetation-covered flow Velocity distribution Two-power law expression Asymmetric flow Shear stress
Suspended vegetation floating in open channel alters the flow structure and generates vertically asymmetric flow because of the different roughnesses of the river bed and vegetation cover. Moreover, its typical profile of streamwise velocity can be used for a rough estimation of the fate of solute transportation. In this study, a twopower law expression was adopted to predict the vertical profile of streamwise velocity. The influence of roughness of the floating vegetation patches and channel bed was also analyzed. To verify the model, the vertical distribution of longitudinal velocity and shear stress of a vegetation covered flow was investigated by laboratory measurements. Results showed that the location of the maximum streamwise velocity was close to the smooth boundary (i.e., the vegetation cover). The vertical profile of shear stress indicated that the turbulent structure was intensively influenced by the presence of vegetation cover and its roughness characteristics. In addition, the turbulence intensity values were amplified in the vicinity of simulated vegetation cover but reduced near the channel bed. The location of the maximum values of shear stress was close to the vegetation cover. The large values of shear stress near the vegetation canopy indicate the high turbulent levels because of the perturbation resulting from vegetation drag and the canopy gap.
1. Introduction Floating vegetation is commonly seen in rivers, lakes, wetland marshes, and offshore environments (Bocchiola et al., 2002). Floating vegetation also is customarily planted in constructed floating wetlands (Rao et al., 2014). It can transfer biomass, nutrients, and energy across water (Dierssen et al., 2015), absorb nutrients and heavy metals to purify waste water (Hubbard et al., 2004), and finally achieve the goal of restoration of the river and lake ecosystems (Downing-Kunz and Stacey, 2012; C. Wang et al., 2015; C.Y. Wang et al., 2015). The existence of such rough covers (e.g., ice jam, floating vegetation, floating debris, and kelp beds) alters the flow structure and velocity distribution (Teal et al., 1994; Smith and Ettema, 1997). These changes derived from the vegetation cover not only influences sediment and contaminant transportation (Walker and Wang, 1997) but also affects phytoplankton and zooplankton biomass as well as the predation and habitat of fish communities (Adams et al., 2002; Padial et al., 2009). Therefore, investigating the flow structure of open channel covered with suspended vegetation is essential. Compared with submerged or emergent vegetation, suspended vegetation can be categorized into two groups (Folkard, 2011), namely,
⁎
rooted plants with floating leaves (e.g., water lilies, some pondweeds, and American lotus) and free-floating leaves with developed or underdeveloped roots (e.g., Eichhornia crassipes, duckweeds, and common bladderworts). Numerous studies regarding the open-channel flow with aquatic vegetation (submerged or emergent) have been carried out (Nepf, 1999; Stephan and Gutknecht, 2002; Maltese et al., 2007). However, the investigations on suspended vegetation flow were limited. Prior studies on floating vegetation have focused on vegetation that protrudes some distance into the flow (Plew et al., 2006; Huai et al., 2012; Plew et al., 2006, 2011; O’Donncha et al., 2015) rather than forming a thin layer on the surface (Downing-Kunz and Stacey, 2011, 2012). For example, Plew et al. (2006) found that the strong turbulence might enhance vertical exchange and turbulence dissipation within the suspended rigid cylinder array, and they further presented an analytical model with respect to the depth-averaged drag coefficient for open channel with rigid suspended canopy (Plew, 2011). O & Donncha et al. (2015) presented a three-dimensional hydrodynamic model by incorporating the influence of suspended canopy. Huai et al. (2012) adopted a three-layer model based on mixing length theory to simulate the flow structure of open channel flow covered by suspended rigid vegetation.
Corresponding author at: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, P.R. China. E-mail address: [email protected] (Y. Zeng).
http://dx.doi.org/10.1016/j.ecolind.2017.04.012 Received 31 December 2016; Received in revised form 14 March 2017; Accepted 6 April 2017 1470-160X/ © 2017 Published by Elsevier Ltd.
Please cite this article as: HAN, L., Ecological Indicators (2017), http://dx.doi.org/10.1016/j.ecolind.2017.04.012
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flow is commonly taken as a pressure flow or strong analogy to the pressure pipe flow and the drag force acts on the river bank. However, the vegetation cover flow is a free flow with a variable population density. The drag force also acts on water surface. As illustrated in Fig. 1, the vegetation covered flow can be divided into a bed zone and a vegetation zone by the horizontal plane of the maximum velocity umax; hb and hv are the distances from umax to the bed and vegetation cover, respectively; H is the flow depth; hτ is the distance between the bed and the plane of zero shear stress; τb and τv are shear stress associated with bed and vegetation boundaries, respectively. Previously, the logarithmic velocity expressions and the two-layer hypothesis that considered the plane of maximum velocity as a free surface were employed to establish the vertical profile of averaged streamwise velocity. However, this method showed some drawbacks as follows: the gradient of velocity at the location of umax is discontinuous, and the two-layer hypothesis was not suitable for icecovered bend flows (Urroz and Ettema, 1994), and it overestimated the streamwise velocity near the location of velocity maximum (Lau, 1982). Therefore, the vertical distribution of averaged streamwise velocity u is predicted by the two-power law expression (Tsai and Ettema, 1994; Uzuner, 1975), which is a continuous-gradient function and can avoid the drawbacks of discontinuous gradient caused by logarithmic velocity expressions and the two-layer hypothesis as follows:
For suspended vegetation floating on the water surface with undeveloped roots (Downing-Kunz and Stacey, 2012; Rao et al., 2014), the vegetation roughness and the channel bed roughness work together to the flow. The characteristics of flow with floating vegetation is akin to the ice-covered flow or duct flow, i.e., with a top rough cover, due to the additional drag force exerted by vegetation on the water surface and the bed friction on the bottom. The maximum velocity is located near the mid depth close to the smooth boundary rather than near the water surface for open water flow. For this reason, this covered flow has always been taken as the asymmetric flow (Hanjalic and Launder, 1972; Parthasarathy and Muste, 1994; Robert and Tran, 2012). To model the asymmetric ice-covered flow, Tsai and Ettema (1994) established a modified eddy viscosity model to predict the vertical profile of streamwise velocity to avoid the discontinuity of eddy viscosity. Robert and Tran (2012) found that the presence of floating cover can increase turbulent kinetic energy and Reynolds stress. In this study, to investigate the vertical profile of streamwise velocity and turbulent characteristics under the combined action of the channel bed and vegetation cover, laboratory experiments were conducted in a flume covered with artificial lotus leaf. The velocity was measured by acoustic Doppler velocimeter (ADV). The two-power law expression was adopted to characterize the vertical profile of streamwise velocity and the boundary shear stresses that were related to the roughness feature of bed and vegetation cover. The position of zero shear stress and the influence of gaps among lotus leaves have been discussed.
1m ⎛ z ⎞1 mb ⎛ z⎞ v u = K0 ⎜ ⎟ ⎜1 − ⎟ ⎝ ⎝H⎠ H⎠
(1)
where z is the distance from the boundary at which the streamwise velocity is u; K0 is a parameter related to the per-unit-width discharge; mb and mv are the parameters relevant to the frictional effects of vegetation and riverbed, respectively. When mv approaches infinity, Eq. (1) can be simplified to a single-power law expression for open channel z flow u = K0 ( H )1 mb , and where K0 becomes the umax. The theoretical position of the maximum velocity can be deduced by ∂u ∂z = 0 , and the maximum velocity is founded to be
2. Theoretical analysis 2.1. Two-power law expression for velocity profile The maximum velocity is located near the mid depth and deflects to the smooth boundary due to the presence of floating canopy (Fig. 1). Therefore, this covered flow has always been taken as an asymmetric flow with two different roughness boundaries. One is associated with the bed and the other with the floating vegetation. Under the combined action of the upper and lower boundaries, the velocity profiles of vegetation covered flow are analogous to that of ice-covered flow. Moreover, the velocity profile was affected by the combined action of surface and bottom boundary roughness. However, the distinction between each other exists in some ways as follows: (i) the ice cover is commonly seen as a consecutive slab in open channel, which cannot move with flow except broken up but the vegetation cover in channels may exhibit different distribution patterns related to the vegetation type and population density and may drift with flow. (ii) Ice-covered
u max = K0 (hb H )1 mb (1 − hb H )1 mv
(2)
and occurs at the position of hb, where
⎛z⎞ h mv = b = ⎜ ⎟ ⎝ H ⎠umax H m v + mb
(3)
A large boundary roughness exponent m indicates a smooth boundary and to which the maximum velocity umax appears close (Hanjalic and Launder, 1972). By integrating u by z, the depth-averaged velocity U can be obtained
Fig. 1. Sketch of vegetation-covered flow in laboratory flume.
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⎛ 8 ⎞1 2 mi = κ ⎜ ⎟ ⎝ fi ⎠
as
U = K0
∫0
1
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ ⎜1 − ⎟ d ⎜ ⎟ = K0 K1 ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠
The depth-averaged velocity of the bed zone is attained by integrating u from z = 0 to z = hb and can be expressed as
1 Ub = K0 hb
∫0
hb
fi =
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎜1 − ⎟ dz = K0 K2 ⎜ ⎟ ⎝ ⎝H⎠ H⎠
(5)
1 H − hb
H
∫H−h
b
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎜1 − ⎟ dz = K0 K3 ⎜ ⎟ ⎝ ⎝H⎠ H⎠
(6)
In these expressions, K1, K2, and K3 are the normalized depthaveraged streamwise velocity for the total, bed, and vegetation zone, respectively. By integration, K1 can be easily obtained:
K1 =
∫0
1
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ Γ (1 + 1 mb ) Γ (1 + 1 m v ) ⎜1 − ⎟ d ⎜ ⎟ = ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠ Γ (2 + 1 mb + 1 m v )
For fully developed turbulent flow, shear stress can be considered equivalent to Reynold shear stress, -ρu′w′(ρ = fluid’s density; u′, w′ = fluctuating velocity of turbulence motion in streamwise and vertical direction, respectively). The vertical gradient of shear stress is balanced by the longitudinal pressure gradient. For a constant pressure gradient along the main flow direction, the linear distribution of the shear stress is applicable
(7)
∞
mb + m v mv
∫0
mv (mb + mv )
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ ⎜1 − ⎟ d ⎜ ⎟ ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠
(8)
m + mb ⎛ mv ⎞ K3 = v ⎜K1 − K2 ⎟ mb ⎝ mb + m v ⎠
(11)
3. Shear stress
in which Γ (n ) = ∫ e−xx n −1dx . Based on the continuity equation and Eq. 0 (3), K2 and K3 can be transformed into the following expressions, which are merely related to mb and mv as follows:
K2 =
8 n i 2g Ro1 3
in which Ro is the hydraulic radius for open-water flow, which is equivalent to the water depth for a wide and shallow channel; g is the gravitational acceleration. In turn, the values of fv and fb can be defined by the corresponding streamwise boundary shear stresses τi and the mean velocity Ui, as fi = 8τi/Ui2 = 8(ui*/Ui)2(i = v,b), in which ui* is the shear velocity and mean velocity corresponding to the region i. The detailed expression about fi was deduced subsequently.
The depth-averaged velocity of the vegetation zone is obtained by integrating u from z = H-hb to z = H and can be stated as
Uv = K0
(10)
where κ = Kármán’s constant of 0.41. For a fully developed steady flow, combined with the Chezy–Manning formula, the relations between fi and Manning’s number ni can be described as
(4)
τ (z ) = τb − (τb − τv )
z H
(12)
As shown in Fig. 1, τ(z) is positive near the bed but negatively close to the suspended vegetation. The zero shear stress is located at
(9)
⎛z⎞ τb = ⎜ ⎟ ⎝ H ⎠τ =0 τb + τv
The values of K1 and K2 in terms of mb and mv are plotted in Fig. 2. 2.2. Determination of the parameters for two-power law expression
(13)
As power–law expression for vertical distribution of velocity does not have the information about boundary shear stress, to consider that the mean and maximum velocity from logarithmic law are equal to those expressed using the power–law expression is usual according to Sayre and Song (1979)
The two-power law expression contains three parameters: K0, mb, and mv. The shape of velocity profiles depends on the exponents mb and mv, whereas K0 is associated with the flow rate. Once the three parameters are determined, the vertical profile of velocity can be depicted, and the per-unit-width discharge can be attained by integrating u over z. The parameter mb does not vary observably from that in open water, even when the cover is present (Teal and Ettema, 1993). Thereby, mb can be estimated from open-water profiles. In two-power law expression, the exponents mb and mv can be described in terms of Darcy–Weisbach resistance coefficient fi (i = v, b) as
u − u max 1 z = ln , 0 < z < hi , i = v, b ui* κ hi Ui = − ui* =
ui* + u max , i = b , v κ
fi Ui, i = b, v 8
Fig. 2. Values of K1, K2 versus mb and mv.
3
(14) (15) (16)
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Table 2.
Then, the Darcy–Weisbach resistance coefficients for the bed and vegetation canopy (fb, fv) are obtained, respectively: 2 ⎛ 1 ⎞ fb = 8κ 2 ⎜1 − ⎟ ⎝ Cm K2 ⎠
⎛ 1 fv = 8κ 2 ⎜1 − ⎟ ⎝ Cm K3 ⎠
5.1. Vertical distribution of streamwise velocity (17)
Fig. 4 shows the vertical distribution of velocity measured and predicted for the six runs. As flow was hampered by the cover, streamwise velocity decreased beneath the cover and increased in the middle portion of the flow, which is similar to the findings of Plew (2011) and Robert and Tran (2012). The profiles of velocity for all runs were obviously asymmetric regarding the plane of the maximum velocity. The position of maximum velocity is deflected to the vegetation cover, or the vegetation canopy is smoother than the channel bed. Fig. 4 shows that the predicted velocity profiles are acceptable, and the performance of the two-power law expression is further evaluated by the averaged velocity bias of each runs. The velocity bias can be stated as
⎞2 (18)
where Cm = [(mb + m v ) m v ]1 mb [(mb + m v ) mb]1 mv . The correction coefficients cmb and cmv have been introduced because of the slight discrepancy of power–law expression and logarithmic law. Here, their values are both 1.05. 2 ⎛ c ⎞ fb = 8κ 2 ⎜1 − mb ⎟ ⎝ Cm K2 ⎠
⎛ c fv = 8κ 2 ⎜1 − mv ⎟ ⎝ Cm K3 ⎠
(19)
⎞2
ε = (up − u m )/ u m
(20)
where up is the predicted depth-averaged streamwise velocity and um is the measured one. Values of velocity bias and correlation coefficients of the six runs are listed in Table 3. From Fig. 4, we found that the velocities in vegetation region were well estimated but that in the bed region were underestimated obviously. Therefore, the velocity bias was almost always negative in Table 3. In addition, correlation coefficients for six runs ranged from 0.925 to 0.967, with an average value of 0.939. In conclusion, the performance of the two-power expression is satisfied.
Now, we can calibrate mb and mv deduced by the Darcy–Weisbach resistance coefficient for the bed fb and that for vegetation canopy fv. Substituting Eqs. (19) and (20) into this expression fi = 8τi/ Ui2 = 8(ui*/Ui)2(i = v,b), the shear stresses are obtained 2 ⎛ c ⎞ τb = ρκ 2 ⎜1 − mb ⎟ K0 2K2 2 ⎝ Cm K2 ⎠
⎛ c τv = −ρκ 2 ⎜1 − mv ⎟ K0 2K32 ⎝ Cm K3 ⎠
(23)
(21)
⎞2
(22)
5.2. Vertical distribution of shear stress The suspended leaves altered flow structures by generating turbulence at the canopy–water interface. The Reynolds shear stress (u′w′) has been normalized here by the shear velocity u*, u* = gRS , and the vertical profiles of that for six runs are presented in Fig. 5. For open water condition, the Reynolds shear stress was maximized near the channel bed. For the vegetation-covered condition, the vertical distribution of shear stress can be regarded as linearly distributed along the z direction (Fig. 1). The turbulence intensity values were amplified in the vicinity of simulated vegetation cover but reduced near the bed compared with the open water flow. The shear stress was increasingly negative in the vegetation zone with respect to z. The maximum values of Reynolds stress were present near the vegetation cover. The averaged shear stress bias of each run also was calculated as follows:
4. Laboratory experiments Experiments were conducted in a rectangular, circulating water flume (16 m long, 0.6 m wide, and 0.6 m deep). A stabilizer was installed at the entrance of the flume to promote a steady flow in a short distance. At the end of the channel, a variable tail gate was equipped to control the water depth. To imitate the irregular pattern of floating vegetation in nature, the biomimetic leafs have been adopted in our experiment. Twelve circular leaf pieces made by polyurethane, each with a diameter of 0.6 m, were connected with one another to simulate a continuous vegetation canopy (Fig. 3). The simulated vegetation cover was fixed over the flume using a fine cotton thread to ensure it will not move with the flow. A total of six runs were performed, and the relevant flow parameters are summarized in Table 1. Velocity data (u, v, and w) were collected by ADV at a sampling frequency of 50 Hz for 160 s, ensuring a reliable database for analyzing. A small hole was cut on the canopy cover to permit sampling by ADV. Vertical interval was almost set as 1 cm, except the two boundaries around where it was 0.5 cm. The measurement arrangements are shown in Fig. 3. The five test lines were performed at the central of the flume to study the change process of velocity along the main flow direction. Compared with the velocity and shear stress profiles of five lines, we found that the flow at cross-section L4 may be considered fully developed. Thus, we adopted the velocity data in L4 to verify the analytical model.
ε = (sp − sm )/ sm
(24)
where sp is the predicted depth-averaged shear stress and sm is the measured one. Values of shear stress bias and correlation coefficients of the six runs are listed in Table 4. Correlation coefficients for six runs ranged from 0.895 to 0.971, with an average value of 0.937. The shear stress fitted the linear distribution but the location of zero shear stress by analytical model has a deviation to the measured values, and it defected to the bed zone. 5.3. Locations of maximum velocity and zero shear stress The locations of maximum velocity and zero shear stress are not coincident (Table 5). Moreover, the location of maximum velocity is much closer to the smooth boundary (vegetation boundary) than that of zero shear stress herein, which is not in accordance with the results of Hanjalic and Launder & s (1972) for ice-covered flow. Nonetheless, the location of the zero shear stress remains inclined to the smooth side (i.e., hτ/H > 0.51).
5. Results The flume experiments showed that the vegetation cover presence increased flow depth, similar to the results of Tsai (1991), and resulted in corresponding reductions in the maximum values of depth-averaged streamwise velocity and bed shear stress. The results of measurements indicated that flow can be considered as fully developed after the test line 4. Thus, the measured velocity data at L4 were used for further analysis. For each run, parameters in Eq. (1) were obtained by Eqs. (10–11), and the predicted and measured values of hb are listed in
5.4. Parameter K0 To reveal how K0 varies with Q, the K0 values for six runs are plotted 4
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Fig. 3. Laboratory setup: (a) layout plan of measuring lines along the main flow direction; (b) vertical plan of measuring points; (c) simulated vegetation covering the flume.
6. Discussion
Table 1 Summary of experimental parameters. Runs
Q(l/s)
B(cm)
S
H(cm)
R(cm)
U(cm/s)
Re
1 2 3 4 5 6
10.31 11.94 14.81 16.40 18.04 20.43
60 60 60 60 60 60
0.0004 0.0005 0.0006 0.0008 0.0008 0.0008
18.30 18.95 19.60 20.02 20.53 20.96
9.15 9.48 9.80 10.01 10.27 10.48
9.38 10.51 12.59 13.66 14.65 16.24
8523 9894 12252 13578 14940 16901
6.1. mv and mb The shape of velocity profile was appreciably dependent upon the exponents mb and mv. The inflection point of asymmetric flow trends to the smooth boundary in terms of Eq. (3). The two parameters were different in the six runs, which indicated that the roughness of two boundaries varied under different flow rates. With the increase of flow rate, the values mv and mb decreased first in runs 1 and 2 and then increased in runs 3–6. Further laboratory investigation is needed to explain the rules. In our study, the values of mv and mb fell in the range of 2 < mb,v < 5 and coincided with the results of Teal et al. (1994), in which those fell in the range 1.5 < mb, mv < 8.5 with the preponderance of values between 3 and 4. However, values of mb for open channel flow (mb ≈ 7) were larger than the covered flow (Simth and Ettema, 1997), which is considerably interesting. As a consequence, to conclude that mv and mb for vegetation-covered flow were influenced by each other rather than independent parameters is reasonable.
Note: Q is the flow discharge; B is the channel width; S is the energy slope; R is the hydraulic radius (R = H/2); U is the depth-averaged streamwise velocity calculated by U = Q/HB; Re is the Reynolds number for each fully developed flow (Re = UH/ & , where & is the kinematic viscosity of water at 20 °C). Table 2 Parameters used in the two-power model and the comparison between the measured and predicted hb (hbm, hbp, respectively). Runs
K0 (cm/s)
1/mb
1/mv
hbm (cm)
hbp (cm)
1 2 3 4 5 6
19.1 22.5 25.8 27.1 28.4 30.5
0.3763 0.4209 0.3720 0.3707 0.3691 0.3219
0.233094 0.275672 0.256552 0.218294 0.199384 0.198692
11.30 11.45 11.60 12.60 13.33 12.96
11.25 11.40 11.54 12.62 13.30 12.91
6.2. Shear stress Detailed profile measurements of Reynolds stress indicated that the turbulence structure is strongly influenced by the presence of suspended vegetation and its roughness characteristics, as vegetation presence nearly doubles the wetted perimeter of a smooth wall river channel and the average shear stress also increases over its wetted perimeter. Shear stress on vegetation cover is larger than those at the channel bed due to the gap among leaves and the drag from the fixed chain even if the bed is rougher than the vegetation. This finding is similar to the results of Attar and Li. (2013) and Zare et al. (2016). The reason maybe is the intense turbulence caused by the vegetation gaps and the broken up ice. Large values of Reynolds stress reflected not only high turbulence levels but also increased transport of turbulent momentum. Additionally, the presence of vegetation cover made the shear stress near the bed decreased relative to those in equivalent open-water flow.
in Fig. 6. In this study, the Q values are in the range of 10.31–20.43 l/s, and one can find that K0 increases with increasing Q. A combination of the overall mv and mb values and typical K0 values with channel geometry makes it possible to estimate discharges for suspended vegetation flows. After being normalized by the depth-averaged velocity, the parameter K0 may be taken to be 2.0 for simplicity.
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Fig. 4. Vertical profiles of streamwise velocity.
Thereby, the erosion process of river bed will be retarded.
Table 3 Velocity bias. Runs
Mean bias (%)
Correlation coefficients
6.3. Gap among leaves
1 2 3 4 5 6
−1.37 1.25 −2.21 −1.35 −4.13 −0.02
0.967 0.934 0.935 0.937 0.935 0.925
From Fig. 5, the measured shear stress near the vegetation canopy (smooth side) is larger than that of the bed (rough side), which is different from that of the ice-covered flow. Here, the simulated floating vegetation is patched rather than continuously covered on the water surface. This discontinuity may have effects on the bulk hydraulics of aquatic vegetation canopies and details of advection, diffusion, and canopy–open water exchange in water vicinity. Previously, Folkard (2011) identify different flow regions of open channel flow with flexible 6
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Fig. 5. Spatially-averaged vertical profiles of shear stress for flume experiments. Table 4 Shear stress bias.
Table 5 Locations of maximum velocity and zero shear stress.
Runs
Mean bias (%)
Correlation coefficients
1 2 3 4 5 6
−0.06 −0.24 −0.17 0.22 0.17 0.56
0.971 0.955 0.935 0.895 0.922 0.946
Runs
Q(l/s)
hbm/H
hτ/H
1 2 3 4 5 6
10.31 11.94 14.81 16.40 18.04 20.43
0.61 0.60 0.59 0.63 0.65 0.62
0.55 0.52 0.55 0.59 0.51 0.52
To study the hydrodynamics of gaps, the quadrant analysis (Lu and Willmarth, 1973) have been made here (Fig. 7). It assigns each instantaneous measurement of turbulent velocity fluctuations u′, w′ to one of the four quadrants in the u′–w′ plane, denoted as Q1–4. In Q1 (outward interactions), u′ > 0 and w′ > 0; in Q2 (ejections), u′ < 0 and w′ > 0; in Q3 (inward interactions), u′ < 0 and w′ < 0; and in Q4
submerged vegetation patches and found that the gaps between vegetation patches determined the location of flow recirculation cells. Maltese et al. (2007) found a recirculation cell formed in the gap. The sweeps dominated the region near the top of the canopy for submerged vegetation flow. 7
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These results may be caused by the spatial heterogeneities in velocity and turbulence because of the shape of vegetation that may generate significant dispersive fluxes. To evaluate the values of dispersive fluxes caused by the gaps, the velocities (in front of a leaf, the centerline of the leaf, and cross of the leaf) have to be measured. The calculation method was shown in Poggi et al. (2004). Dispersive fluxes were only significant for sparse canopies (Poggi et al., 2004) similar to the finding of Hamagami et al. (2004) who demonstrated that when the gregarious density of floating plants became small, the values of turbulent characteristics variables in the upper layer increased. In our experiments, the dispersive fluxes are significant (> 10%) in the vicinity of the canopy gaps, especially at high Reynolds number. Therefore, we can conclude that the presence of gaps exhibits a meaningful influence to the dispersive fluxes and shear stress distribution. 7. Conclusions The present study confirms that the vegetation-covered flow structure is affected by the roughness characteristics of the channel bed and vegetation boundaries. The vertical distribution of velocity is asymmetric regarding the plane of the maximum velocity because of the different roughnesses of the two boundaries. The gaps between the simulated leaf patches make the vegetation-covered flow different from that of both open-channel and cover flows or the shear stress near the smooth vegetation cover is larger than that near the river bed. As the shear stress near the bed is reduced, the erosion process of river bed may be retarded. These findings help expand the understanding of river hydrodynamics during the vegetation cover existence. Parameters can also be utilized in numerical modeling of the vegetation covered flows. Several further studies are needed to be performed to fully understand the flow structure of the suspended vegetation flow. For example, the deduction of Darcy–Weisbach coefficients for two boundaries deserves considerable attention. Besides, the gap between canopies and the shape of leaves make the flow structure complicated and require more experiments to investigate. Moreover, further study is required for vegetation covered flow to elucidate the links between flow dynamics and aquatic ecosystems.
Fig. 6. K0 values varying with Q.
Acknowledgements
Fig. 7. Quadrant analysis (Run 5).
This work was supported in part by the National Natural Science Foundation of China (Nos. 51379154, 51439007,11472199 and 51622905).
(sweeps), u′ > 0 and w′ < 0. Si,0 is the contribution value of Reynolds stress in Qi. The figure shows that Q1 and Q3 are dominant quadrant in the vegetation zone but Q2 and Q4 dominate the bed zone. At the location of umax, no dominant quadrant is observed (i.e., the sum of Si,0 ≈ 0), which indicates that the shear stress near the location of umax is equivalent to zero. The maximum value of |S1,0| and |S3,0| near the vegetation cover is larger than that of |S2,0| and |S4,0|. In other words, the momentum exchanges are intense at the canopy–gap interface. In addition, Zare et al. (2016) confirmed that the shear stress varied dynamically with transformation of the ice cover, and the upper boundary shear stress is many times larger than the lower boundary shear stress during the solid ice cover to the ice cover removal, which is strongly similar to the gaps of vegetation cover. Therefore, the maximum of shear stress occurred at the water–vegetation interface, and the location of zero shear stress is close to the bed region in terms of the linear distribution. The water microspheres have to move downstream through the underneath of a leaf because of the resistance of leaves on the water surface. Once the gap occurs, the path of flow will change. In the gap region, water particles start to move upwards and leave the original position under the carry of high velocity cells, which also has been reflected by the quadrant analysis. The outward interaction Q1 is dominant at z/H > 0.9. Thus, the quadrant analysis can fully demonstrate the impact of gaps among leaves.
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L. Han et al. Huai, W.X., Hu, Y., Zeng, Y.H., Han, J., 2012. Velocity distribution for open channel flows with suspended vegetation. Adv. Water Resour. 49, 56–61. http://dx.doi.org/10. 1016/j.advwatres.2012.07.001. Hubbard, R.K., Gascho, G.J., Newton, G.L., 2004. Use of floating vegetation to remove nutrients from swine lagoon wastewater. T ASAE 47, 1963–1972. Lau, Y.L., 1982. Velocity distributions under floating covers. Can. J. Civ. Eng. 9 (1), 76–83. http://dx.doi.org/10.1139/l82-008. Lu, S.S., Willmarth, W.W., 1973. Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60 (3), 481–511. Maltese, A., Cox, E., Folkard, A.M., Ciraolo, G., La Loggia, G., Lombardo, G., 2007. Laboratory measurements of flow and turbulence in discontinuous distributions of ligulate seagrass. J. Hydraul. Eng.-ASCE 133, 750–760. http://dx.doi.org/10.1061/ (Asce)0733-9429(2007)133:7(750). Nepf, H.M., 1999. Drag, turbulence, and diffusion in flow through emergent vegetation. Water Resour. Res. 35, 479–489. http://dx.doi.org/10.1029/1998wr900069. O’Donncha, F., Hartnett, M., Plew, D.R., 2015. Parameterizing suspended canopy effects in a three-dimensional hydrodynamic model. J. Hydraul. Res. 53, 714–727. http:// dx.doi.org/10.1080/00221686.2015.1093036. Padial, A., Thomaz, S., Agostinho, A., 2009. Effects of structural heterogeneity provided by the floating macrophyte Eichhornia azurea on the predation efficiency and habitat use of the small Neotropical fish Moenkhausia sanctaefilomenae. Hydrobiologia 624, 161–170. Parthasarathy, R.N., Muste, M., 1994. Velocity-measurements In asymmetric turbulent channel flows. J. Hydraul. Eng.-ASCE 120, 1000–1020. http://dx.doi.org/10.1061/ (Asce)0733-9429(1994)120:9(1000). Plew, D.R., Spigel, R.H., Stevens, C.L., Nokes, R.I., Davidson, M.J., 2006. Stratified flow interactions with a suspended canopy. Environ. Fluid Mech. 6, 519–539. http://dx. doi.org/10.1007/s10652-006-9008-1. Plew, D.R., 2011. Depth-averaged drag coefficient for modeling flow through suspended canopies. J. Hydraul. Eng.-ASCE 137, 234–247. http://dx.doi.org/10.1061/(Asce) Hy.1943-7900.0000300. Poggi, D., Katul, G.G., Albertson, J.D., 2004. A note on the contribution of dispersive fluxes to momentum transfer within canopies. Boundary Layer Meteorol. 111 (3), 615–621. Rao, L., Qian, J., Ao, Y.H., 2014. Influence of artificial ecological floating beds on river hydraulic characteristics. J. Hydrodyn 26, 474–481. http://dx.doi.org/10.1016/ S1001-6058(14)60054-8. Robert, A., Tran, T., 2012. Mean and turbulent flow fields in a simulated ice-covered
channel with a gravel bed: some laboratory observations. Earth Surf. Proc. Land 37, 951–956. http://dx.doi.org/10.1002/esp.3211. Sayre, W.W., Song, F.B., 1979. Effects of Ice Cover on Alluvial Channel Flow and Sediment Transport Process, IIHR REPORT No.218. Iowa Institute of Hydraulic Research, the University of Iowa, Iowa City, Iowa. Smith, B.T., Ettema, R., 1997. Flow resistance in ice-covered alluvial channels. J. Hydraul. Eng.-ASCE 123, 592–599. http://dx.doi.org/10.1061/(Asce)07339429(1997)123:7(592). Stephan, U., Gutknecht, D., 2002. Hydraulic resistance of submerged flexible vegetation. J. Hydrol. 269, 27–43. http://dx.doi.org/10.1016/S0022-1694(02)00192-0. Teal, M.J., Ettema, R., 1993. Estimation of Mean Velocity for Ice-affected Stream Flow. IIHR Rep. No . 390. Iowa Inst. of Hydr. Res. (IIHR), Iowa City, Iowa. Teal, M.J., Ettema, R., Walker, J.F., 1994. Estimation of mean flow velocity In ice-covered channels. J. Hydraul. Eng.-ASCE 120, 1385–1400. http://dx.doi.org/10.1061/(Asce) 0733-9429(1994)120:12(1385). Tsai, W.F., Ettema, R., 1994. Modified eddy viscosity model in fully-developed asymmetric channel flows. J. Eng. Mech-Asce. 120, 720–732. http://dx.doi.org/10. 1061/(Asce)0733-9399(1994)120:4(720). Tsai, W.F., 1991. A Study in Ice-covered Bend Flow, PhD Thesis. University of Iowa, Iowa City, Iowa. Urroz, G.E., Ettema, R., 1994. Application of two-layer hypothesis to fully developed flow in ice-covered curved channels. Can. J. Civ. Eng. 21, 101–110. http://dx.doi.org/10. 1139/l94-010. Uzuner, M.S., 1975. The composite roughness of ice covered streams. J. Hydraul. Res. 13, 79–102. http://dx.doi.org/10.1080/00221687509499721. Walker, J.F., Wang, D.P., 1997. Measurement of flow under ice covers in North America. J. Hydraul. Eng.-ASCE 123, 1037–1040. http://dx.doi.org/10.1061/(Asce)07339429(1997)123:11(1037). Wang, C., Li, S.C., Lai, D.Y.F., Wang, W.Q., Ma, Y.Y., 2015. The effect of floating vegetation on CH4 and N2O emissions from subtropical paddy fields in China. Paddy Water Environ. 13, 425–431. http://dx.doi.org/10.1007/s10333-014-0459-6. Wang, C.Y., Sample, D.J., Day, S.D., Grizzard, T.J., 2015. Floating treatment wetland nutrient removal through vegetation harvest and observations from a field study. Ecol. Eng. 78, 15–26. http://dx.doi.org/10.1016/j.ecoleng.2014.05.018. Zare, S.G.A., Moore, S.A., Rennie, C.D., Seidou, O., Ahmari, H., Malenchak, J., 2016. Boundary shear stress in an ice-Covered river during Breakup. J. Hydraul. Eng. 142, 1943–7900. http://dx.doi.org/10.1061/(Asce)Hy.1943-7900.0001081.
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Contents lists available at ScienceDirect
Ecological Indicators journal homepage: www.elsevier.com/locate/ecolind
Original Articles
Modeling streamwise velocity and boundary shear stress of vegetationcovered flow ⁎
Lijuan Hana, Yuhong Zenga, , Li Chena, Ming Lib a b
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, P.R. China State Key Laboratory of Ocean Engineering, School of Naval Architecture, Shanghai Jiao Tong University, Shanghai, P.R. China
A R T I C L E I N F O
A B S T R A C T
Keywords: Vegetation-covered flow Velocity distribution Two-power law expression Asymmetric flow Shear stress
Suspended vegetation floating in open channel alters the flow structure and generates vertically asymmetric flow because of the different roughnesses of the river bed and vegetation cover. Moreover, its typical profile of streamwise velocity can be used for a rough estimation of the fate of solute transportation. In this study, a twopower law expression was adopted to predict the vertical profile of streamwise velocity. The influence of roughness of the floating vegetation patches and channel bed was also analyzed. To verify the model, the vertical distribution of longitudinal velocity and shear stress of a vegetation covered flow was investigated by laboratory measurements. Results showed that the location of the maximum streamwise velocity was close to the smooth boundary (i.e., the vegetation cover). The vertical profile of shear stress indicated that the turbulent structure was intensively influenced by the presence of vegetation cover and its roughness characteristics. In addition, the turbulence intensity values were amplified in the vicinity of simulated vegetation cover but reduced near the channel bed. The location of the maximum values of shear stress was close to the vegetation cover. The large values of shear stress near the vegetation canopy indicate the high turbulent levels because of the perturbation resulting from vegetation drag and the canopy gap.
1. Introduction Floating vegetation is commonly seen in rivers, lakes, wetland marshes, and offshore environments (Bocchiola et al., 2002). Floating vegetation also is customarily planted in constructed floating wetlands (Rao et al., 2014). It can transfer biomass, nutrients, and energy across water (Dierssen et al., 2015), absorb nutrients and heavy metals to purify waste water (Hubbard et al., 2004), and finally achieve the goal of restoration of the river and lake ecosystems (Downing-Kunz and Stacey, 2012; C. Wang et al., 2015; C.Y. Wang et al., 2015). The existence of such rough covers (e.g., ice jam, floating vegetation, floating debris, and kelp beds) alters the flow structure and velocity distribution (Teal et al., 1994; Smith and Ettema, 1997). These changes derived from the vegetation cover not only influences sediment and contaminant transportation (Walker and Wang, 1997) but also affects phytoplankton and zooplankton biomass as well as the predation and habitat of fish communities (Adams et al., 2002; Padial et al., 2009). Therefore, investigating the flow structure of open channel covered with suspended vegetation is essential. Compared with submerged or emergent vegetation, suspended vegetation can be categorized into two groups (Folkard, 2011), namely,
⁎
rooted plants with floating leaves (e.g., water lilies, some pondweeds, and American lotus) and free-floating leaves with developed or underdeveloped roots (e.g., Eichhornia crassipes, duckweeds, and common bladderworts). Numerous studies regarding the open-channel flow with aquatic vegetation (submerged or emergent) have been carried out (Nepf, 1999; Stephan and Gutknecht, 2002; Maltese et al., 2007). However, the investigations on suspended vegetation flow were limited. Prior studies on floating vegetation have focused on vegetation that protrudes some distance into the flow (Plew et al., 2006; Huai et al., 2012; Plew et al., 2006, 2011; O’Donncha et al., 2015) rather than forming a thin layer on the surface (Downing-Kunz and Stacey, 2011, 2012). For example, Plew et al. (2006) found that the strong turbulence might enhance vertical exchange and turbulence dissipation within the suspended rigid cylinder array, and they further presented an analytical model with respect to the depth-averaged drag coefficient for open channel with rigid suspended canopy (Plew, 2011). O & Donncha et al. (2015) presented a three-dimensional hydrodynamic model by incorporating the influence of suspended canopy. Huai et al. (2012) adopted a three-layer model based on mixing length theory to simulate the flow structure of open channel flow covered by suspended rigid vegetation.
Corresponding author at: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, P.R. China. E-mail address: [email protected] (Y. Zeng).
http://dx.doi.org/10.1016/j.ecolind.2017.04.012 Received 31 December 2016; Received in revised form 14 March 2017; Accepted 6 April 2017 1470-160X/ © 2017 Published by Elsevier Ltd.
Please cite this article as: HAN, L., Ecological Indicators (2017), http://dx.doi.org/10.1016/j.ecolind.2017.04.012
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flow is commonly taken as a pressure flow or strong analogy to the pressure pipe flow and the drag force acts on the river bank. However, the vegetation cover flow is a free flow with a variable population density. The drag force also acts on water surface. As illustrated in Fig. 1, the vegetation covered flow can be divided into a bed zone and a vegetation zone by the horizontal plane of the maximum velocity umax; hb and hv are the distances from umax to the bed and vegetation cover, respectively; H is the flow depth; hτ is the distance between the bed and the plane of zero shear stress; τb and τv are shear stress associated with bed and vegetation boundaries, respectively. Previously, the logarithmic velocity expressions and the two-layer hypothesis that considered the plane of maximum velocity as a free surface were employed to establish the vertical profile of averaged streamwise velocity. However, this method showed some drawbacks as follows: the gradient of velocity at the location of umax is discontinuous, and the two-layer hypothesis was not suitable for icecovered bend flows (Urroz and Ettema, 1994), and it overestimated the streamwise velocity near the location of velocity maximum (Lau, 1982). Therefore, the vertical distribution of averaged streamwise velocity u is predicted by the two-power law expression (Tsai and Ettema, 1994; Uzuner, 1975), which is a continuous-gradient function and can avoid the drawbacks of discontinuous gradient caused by logarithmic velocity expressions and the two-layer hypothesis as follows:
For suspended vegetation floating on the water surface with undeveloped roots (Downing-Kunz and Stacey, 2012; Rao et al., 2014), the vegetation roughness and the channel bed roughness work together to the flow. The characteristics of flow with floating vegetation is akin to the ice-covered flow or duct flow, i.e., with a top rough cover, due to the additional drag force exerted by vegetation on the water surface and the bed friction on the bottom. The maximum velocity is located near the mid depth close to the smooth boundary rather than near the water surface for open water flow. For this reason, this covered flow has always been taken as the asymmetric flow (Hanjalic and Launder, 1972; Parthasarathy and Muste, 1994; Robert and Tran, 2012). To model the asymmetric ice-covered flow, Tsai and Ettema (1994) established a modified eddy viscosity model to predict the vertical profile of streamwise velocity to avoid the discontinuity of eddy viscosity. Robert and Tran (2012) found that the presence of floating cover can increase turbulent kinetic energy and Reynolds stress. In this study, to investigate the vertical profile of streamwise velocity and turbulent characteristics under the combined action of the channel bed and vegetation cover, laboratory experiments were conducted in a flume covered with artificial lotus leaf. The velocity was measured by acoustic Doppler velocimeter (ADV). The two-power law expression was adopted to characterize the vertical profile of streamwise velocity and the boundary shear stresses that were related to the roughness feature of bed and vegetation cover. The position of zero shear stress and the influence of gaps among lotus leaves have been discussed.
1m ⎛ z ⎞1 mb ⎛ z⎞ v u = K0 ⎜ ⎟ ⎜1 − ⎟ ⎝ ⎝H⎠ H⎠
(1)
where z is the distance from the boundary at which the streamwise velocity is u; K0 is a parameter related to the per-unit-width discharge; mb and mv are the parameters relevant to the frictional effects of vegetation and riverbed, respectively. When mv approaches infinity, Eq. (1) can be simplified to a single-power law expression for open channel z flow u = K0 ( H )1 mb , and where K0 becomes the umax. The theoretical position of the maximum velocity can be deduced by ∂u ∂z = 0 , and the maximum velocity is founded to be
2. Theoretical analysis 2.1. Two-power law expression for velocity profile The maximum velocity is located near the mid depth and deflects to the smooth boundary due to the presence of floating canopy (Fig. 1). Therefore, this covered flow has always been taken as an asymmetric flow with two different roughness boundaries. One is associated with the bed and the other with the floating vegetation. Under the combined action of the upper and lower boundaries, the velocity profiles of vegetation covered flow are analogous to that of ice-covered flow. Moreover, the velocity profile was affected by the combined action of surface and bottom boundary roughness. However, the distinction between each other exists in some ways as follows: (i) the ice cover is commonly seen as a consecutive slab in open channel, which cannot move with flow except broken up but the vegetation cover in channels may exhibit different distribution patterns related to the vegetation type and population density and may drift with flow. (ii) Ice-covered
u max = K0 (hb H )1 mb (1 − hb H )1 mv
(2)
and occurs at the position of hb, where
⎛z⎞ h mv = b = ⎜ ⎟ ⎝ H ⎠umax H m v + mb
(3)
A large boundary roughness exponent m indicates a smooth boundary and to which the maximum velocity umax appears close (Hanjalic and Launder, 1972). By integrating u by z, the depth-averaged velocity U can be obtained
Fig. 1. Sketch of vegetation-covered flow in laboratory flume.
2
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⎛ 8 ⎞1 2 mi = κ ⎜ ⎟ ⎝ fi ⎠
as
U = K0
∫0
1
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ ⎜1 − ⎟ d ⎜ ⎟ = K0 K1 ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠
The depth-averaged velocity of the bed zone is attained by integrating u from z = 0 to z = hb and can be expressed as
1 Ub = K0 hb
∫0
hb
fi =
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎜1 − ⎟ dz = K0 K2 ⎜ ⎟ ⎝ ⎝H⎠ H⎠
(5)
1 H − hb
H
∫H−h
b
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎜1 − ⎟ dz = K0 K3 ⎜ ⎟ ⎝ ⎝H⎠ H⎠
(6)
In these expressions, K1, K2, and K3 are the normalized depthaveraged streamwise velocity for the total, bed, and vegetation zone, respectively. By integration, K1 can be easily obtained:
K1 =
∫0
1
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ Γ (1 + 1 mb ) Γ (1 + 1 m v ) ⎜1 − ⎟ d ⎜ ⎟ = ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠ Γ (2 + 1 mb + 1 m v )
For fully developed turbulent flow, shear stress can be considered equivalent to Reynold shear stress, -ρu′w′(ρ = fluid’s density; u′, w′ = fluctuating velocity of turbulence motion in streamwise and vertical direction, respectively). The vertical gradient of shear stress is balanced by the longitudinal pressure gradient. For a constant pressure gradient along the main flow direction, the linear distribution of the shear stress is applicable
(7)
∞
mb + m v mv
∫0
mv (mb + mv )
1m ⎛ z ⎞1 mb ⎛ z⎞ v ⎛z⎞ ⎜1 − ⎟ d ⎜ ⎟ ⎜ ⎟ ⎝H⎠ ⎝ ⎝H⎠ H⎠
(8)
m + mb ⎛ mv ⎞ K3 = v ⎜K1 − K2 ⎟ mb ⎝ mb + m v ⎠
(11)
3. Shear stress
in which Γ (n ) = ∫ e−xx n −1dx . Based on the continuity equation and Eq. 0 (3), K2 and K3 can be transformed into the following expressions, which are merely related to mb and mv as follows:
K2 =
8 n i 2g Ro1 3
in which Ro is the hydraulic radius for open-water flow, which is equivalent to the water depth for a wide and shallow channel; g is the gravitational acceleration. In turn, the values of fv and fb can be defined by the corresponding streamwise boundary shear stresses τi and the mean velocity Ui, as fi = 8τi/Ui2 = 8(ui*/Ui)2(i = v,b), in which ui* is the shear velocity and mean velocity corresponding to the region i. The detailed expression about fi was deduced subsequently.
The depth-averaged velocity of the vegetation zone is obtained by integrating u from z = H-hb to z = H and can be stated as
Uv = K0
(10)
where κ = Kármán’s constant of 0.41. For a fully developed steady flow, combined with the Chezy–Manning formula, the relations between fi and Manning’s number ni can be described as
(4)
τ (z ) = τb − (τb − τv )
z H
(12)
As shown in Fig. 1, τ(z) is positive near the bed but negatively close to the suspended vegetation. The zero shear stress is located at
(9)
⎛z⎞ τb = ⎜ ⎟ ⎝ H ⎠τ =0 τb + τv
The values of K1 and K2 in terms of mb and mv are plotted in Fig. 2. 2.2. Determination of the parameters for two-power law expression
(13)
As power–law expression for vertical distribution of velocity does not have the information about boundary shear stress, to consider that the mean and maximum velocity from logarithmic law are equal to those expressed using the power–law expression is usual according to Sayre and Song (1979)
The two-power law expression contains three parameters: K0, mb, and mv. The shape of velocity profiles depends on the exponents mb and mv, whereas K0 is associated with the flow rate. Once the three parameters are determined, the vertical profile of velocity can be depicted, and the per-unit-width discharge can be attained by integrating u over z. The parameter mb does not vary observably from that in open water, even when the cover is present (Teal and Ettema, 1993). Thereby, mb can be estimated from open-water profiles. In two-power law expression, the exponents mb and mv can be described in terms of Darcy–Weisbach resistance coefficient fi (i = v, b) as
u − u max 1 z = ln , 0 < z < hi , i = v, b ui* κ hi Ui = − ui* =
ui* + u max , i = b , v κ
fi Ui, i = b, v 8
Fig. 2. Values of K1, K2 versus mb and mv.
3
(14) (15) (16)
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Table 2.
Then, the Darcy–Weisbach resistance coefficients for the bed and vegetation canopy (fb, fv) are obtained, respectively: 2 ⎛ 1 ⎞ fb = 8κ 2 ⎜1 − ⎟ ⎝ Cm K2 ⎠
⎛ 1 fv = 8κ 2 ⎜1 − ⎟ ⎝ Cm K3 ⎠
5.1. Vertical distribution of streamwise velocity (17)
Fig. 4 shows the vertical distribution of velocity measured and predicted for the six runs. As flow was hampered by the cover, streamwise velocity decreased beneath the cover and increased in the middle portion of the flow, which is similar to the findings of Plew (2011) and Robert and Tran (2012). The profiles of velocity for all runs were obviously asymmetric regarding the plane of the maximum velocity. The position of maximum velocity is deflected to the vegetation cover, or the vegetation canopy is smoother than the channel bed. Fig. 4 shows that the predicted velocity profiles are acceptable, and the performance of the two-power law expression is further evaluated by the averaged velocity bias of each runs. The velocity bias can be stated as
⎞2 (18)
where Cm = [(mb + m v ) m v ]1 mb [(mb + m v ) mb]1 mv . The correction coefficients cmb and cmv have been introduced because of the slight discrepancy of power–law expression and logarithmic law. Here, their values are both 1.05. 2 ⎛ c ⎞ fb = 8κ 2 ⎜1 − mb ⎟ ⎝ Cm K2 ⎠
⎛ c fv = 8κ 2 ⎜1 − mv ⎟ ⎝ Cm K3 ⎠
(19)
⎞2
ε = (up − u m )/ u m
(20)
where up is the predicted depth-averaged streamwise velocity and um is the measured one. Values of velocity bias and correlation coefficients of the six runs are listed in Table 3. From Fig. 4, we found that the velocities in vegetation region were well estimated but that in the bed region were underestimated obviously. Therefore, the velocity bias was almost always negative in Table 3. In addition, correlation coefficients for six runs ranged from 0.925 to 0.967, with an average value of 0.939. In conclusion, the performance of the two-power expression is satisfied.
Now, we can calibrate mb and mv deduced by the Darcy–Weisbach resistance coefficient for the bed fb and that for vegetation canopy fv. Substituting Eqs. (19) and (20) into this expression fi = 8τi/ Ui2 = 8(ui*/Ui)2(i = v,b), the shear stresses are obtained 2 ⎛ c ⎞ τb = ρκ 2 ⎜1 − mb ⎟ K0 2K2 2 ⎝ Cm K2 ⎠
⎛ c τv = −ρκ 2 ⎜1 − mv ⎟ K0 2K32 ⎝ Cm K3 ⎠
(23)
(21)
⎞2
(22)
5.2. Vertical distribution of shear stress The suspended leaves altered flow structures by generating turbulence at the canopy–water interface. The Reynolds shear stress (u′w′) has been normalized here by the shear velocity u*, u* = gRS , and the vertical profiles of that for six runs are presented in Fig. 5. For open water condition, the Reynolds shear stress was maximized near the channel bed. For the vegetation-covered condition, the vertical distribution of shear stress can be regarded as linearly distributed along the z direction (Fig. 1). The turbulence intensity values were amplified in the vicinity of simulated vegetation cover but reduced near the bed compared with the open water flow. The shear stress was increasingly negative in the vegetation zone with respect to z. The maximum values of Reynolds stress were present near the vegetation cover. The averaged shear stress bias of each run also was calculated as follows:
4. Laboratory experiments Experiments were conducted in a rectangular, circulating water flume (16 m long, 0.6 m wide, and 0.6 m deep). A stabilizer was installed at the entrance of the flume to promote a steady flow in a short distance. At the end of the channel, a variable tail gate was equipped to control the water depth. To imitate the irregular pattern of floating vegetation in nature, the biomimetic leafs have been adopted in our experiment. Twelve circular leaf pieces made by polyurethane, each with a diameter of 0.6 m, were connected with one another to simulate a continuous vegetation canopy (Fig. 3). The simulated vegetation cover was fixed over the flume using a fine cotton thread to ensure it will not move with the flow. A total of six runs were performed, and the relevant flow parameters are summarized in Table 1. Velocity data (u, v, and w) were collected by ADV at a sampling frequency of 50 Hz for 160 s, ensuring a reliable database for analyzing. A small hole was cut on the canopy cover to permit sampling by ADV. Vertical interval was almost set as 1 cm, except the two boundaries around where it was 0.5 cm. The measurement arrangements are shown in Fig. 3. The five test lines were performed at the central of the flume to study the change process of velocity along the main flow direction. Compared with the velocity and shear stress profiles of five lines, we found that the flow at cross-section L4 may be considered fully developed. Thus, we adopted the velocity data in L4 to verify the analytical model.
ε = (sp − sm )/ sm
(24)
where sp is the predicted depth-averaged shear stress and sm is the measured one. Values of shear stress bias and correlation coefficients of the six runs are listed in Table 4. Correlation coefficients for six runs ranged from 0.895 to 0.971, with an average value of 0.937. The shear stress fitted the linear distribution but the location of zero shear stress by analytical model has a deviation to the measured values, and it defected to the bed zone. 5.3. Locations of maximum velocity and zero shear stress The locations of maximum velocity and zero shear stress are not coincident (Table 5). Moreover, the location of maximum velocity is much closer to the smooth boundary (vegetation boundary) than that of zero shear stress herein, which is not in accordance with the results of Hanjalic and Launder & s (1972) for ice-covered flow. Nonetheless, the location of the zero shear stress remains inclined to the smooth side (i.e., hτ/H > 0.51).
5. Results The flume experiments showed that the vegetation cover presence increased flow depth, similar to the results of Tsai (1991), and resulted in corresponding reductions in the maximum values of depth-averaged streamwise velocity and bed shear stress. The results of measurements indicated that flow can be considered as fully developed after the test line 4. Thus, the measured velocity data at L4 were used for further analysis. For each run, parameters in Eq. (1) were obtained by Eqs. (10–11), and the predicted and measured values of hb are listed in
5.4. Parameter K0 To reveal how K0 varies with Q, the K0 values for six runs are plotted 4
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Fig. 3. Laboratory setup: (a) layout plan of measuring lines along the main flow direction; (b) vertical plan of measuring points; (c) simulated vegetation covering the flume.
6. Discussion
Table 1 Summary of experimental parameters. Runs
Q(l/s)
B(cm)
S
H(cm)
R(cm)
U(cm/s)
Re
1 2 3 4 5 6
10.31 11.94 14.81 16.40 18.04 20.43
60 60 60 60 60 60
0.0004 0.0005 0.0006 0.0008 0.0008 0.0008
18.30 18.95 19.60 20.02 20.53 20.96
9.15 9.48 9.80 10.01 10.27 10.48
9.38 10.51 12.59 13.66 14.65 16.24
8523 9894 12252 13578 14940 16901
6.1. mv and mb The shape of velocity profile was appreciably dependent upon the exponents mb and mv. The inflection point of asymmetric flow trends to the smooth boundary in terms of Eq. (3). The two parameters were different in the six runs, which indicated that the roughness of two boundaries varied under different flow rates. With the increase of flow rate, the values mv and mb decreased first in runs 1 and 2 and then increased in runs 3–6. Further laboratory investigation is needed to explain the rules. In our study, the values of mv and mb fell in the range of 2 < mb,v < 5 and coincided with the results of Teal et al. (1994), in which those fell in the range 1.5 < mb, mv < 8.5 with the preponderance of values between 3 and 4. However, values of mb for open channel flow (mb ≈ 7) were larger than the covered flow (Simth and Ettema, 1997), which is considerably interesting. As a consequence, to conclude that mv and mb for vegetation-covered flow were influenced by each other rather than independent parameters is reasonable.
Note: Q is the flow discharge; B is the channel width; S is the energy slope; R is the hydraulic radius (R = H/2); U is the depth-averaged streamwise velocity calculated by U = Q/HB; Re is the Reynolds number for each fully developed flow (Re = UH/ & , where & is the kinematic viscosity of water at 20 °C). Table 2 Parameters used in the two-power model and the comparison between the measured and predicted hb (hbm, hbp, respectively). Runs
K0 (cm/s)
1/mb
1/mv
hbm (cm)
hbp (cm)
1 2 3 4 5 6
19.1 22.5 25.8 27.1 28.4 30.5
0.3763 0.4209 0.3720 0.3707 0.3691 0.3219
0.233094 0.275672 0.256552 0.218294 0.199384 0.198692
11.30 11.45 11.60 12.60 13.33 12.96
11.25 11.40 11.54 12.62 13.30 12.91
6.2. Shear stress Detailed profile measurements of Reynolds stress indicated that the turbulence structure is strongly influenced by the presence of suspended vegetation and its roughness characteristics, as vegetation presence nearly doubles the wetted perimeter of a smooth wall river channel and the average shear stress also increases over its wetted perimeter. Shear stress on vegetation cover is larger than those at the channel bed due to the gap among leaves and the drag from the fixed chain even if the bed is rougher than the vegetation. This finding is similar to the results of Attar and Li. (2013) and Zare et al. (2016). The reason maybe is the intense turbulence caused by the vegetation gaps and the broken up ice. Large values of Reynolds stress reflected not only high turbulence levels but also increased transport of turbulent momentum. Additionally, the presence of vegetation cover made the shear stress near the bed decreased relative to those in equivalent open-water flow.
in Fig. 6. In this study, the Q values are in the range of 10.31–20.43 l/s, and one can find that K0 increases with increasing Q. A combination of the overall mv and mb values and typical K0 values with channel geometry makes it possible to estimate discharges for suspended vegetation flows. After being normalized by the depth-averaged velocity, the parameter K0 may be taken to be 2.0 for simplicity.
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Fig. 4. Vertical profiles of streamwise velocity.
Thereby, the erosion process of river bed will be retarded.
Table 3 Velocity bias. Runs
Mean bias (%)
Correlation coefficients
6.3. Gap among leaves
1 2 3 4 5 6
−1.37 1.25 −2.21 −1.35 −4.13 −0.02
0.967 0.934 0.935 0.937 0.935 0.925
From Fig. 5, the measured shear stress near the vegetation canopy (smooth side) is larger than that of the bed (rough side), which is different from that of the ice-covered flow. Here, the simulated floating vegetation is patched rather than continuously covered on the water surface. This discontinuity may have effects on the bulk hydraulics of aquatic vegetation canopies and details of advection, diffusion, and canopy–open water exchange in water vicinity. Previously, Folkard (2011) identify different flow regions of open channel flow with flexible 6
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Fig. 5. Spatially-averaged vertical profiles of shear stress for flume experiments. Table 4 Shear stress bias.
Table 5 Locations of maximum velocity and zero shear stress.
Runs
Mean bias (%)
Correlation coefficients
1 2 3 4 5 6
−0.06 −0.24 −0.17 0.22 0.17 0.56
0.971 0.955 0.935 0.895 0.922 0.946
Runs
Q(l/s)
hbm/H
hτ/H
1 2 3 4 5 6
10.31 11.94 14.81 16.40 18.04 20.43
0.61 0.60 0.59 0.63 0.65 0.62
0.55 0.52 0.55 0.59 0.51 0.52
To study the hydrodynamics of gaps, the quadrant analysis (Lu and Willmarth, 1973) have been made here (Fig. 7). It assigns each instantaneous measurement of turbulent velocity fluctuations u′, w′ to one of the four quadrants in the u′–w′ plane, denoted as Q1–4. In Q1 (outward interactions), u′ > 0 and w′ > 0; in Q2 (ejections), u′ < 0 and w′ > 0; in Q3 (inward interactions), u′ < 0 and w′ < 0; and in Q4
submerged vegetation patches and found that the gaps between vegetation patches determined the location of flow recirculation cells. Maltese et al. (2007) found a recirculation cell formed in the gap. The sweeps dominated the region near the top of the canopy for submerged vegetation flow. 7
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These results may be caused by the spatial heterogeneities in velocity and turbulence because of the shape of vegetation that may generate significant dispersive fluxes. To evaluate the values of dispersive fluxes caused by the gaps, the velocities (in front of a leaf, the centerline of the leaf, and cross of the leaf) have to be measured. The calculation method was shown in Poggi et al. (2004). Dispersive fluxes were only significant for sparse canopies (Poggi et al., 2004) similar to the finding of Hamagami et al. (2004) who demonstrated that when the gregarious density of floating plants became small, the values of turbulent characteristics variables in the upper layer increased. In our experiments, the dispersive fluxes are significant (> 10%) in the vicinity of the canopy gaps, especially at high Reynolds number. Therefore, we can conclude that the presence of gaps exhibits a meaningful influence to the dispersive fluxes and shear stress distribution. 7. Conclusions The present study confirms that the vegetation-covered flow structure is affected by the roughness characteristics of the channel bed and vegetation boundaries. The vertical distribution of velocity is asymmetric regarding the plane of the maximum velocity because of the different roughnesses of the two boundaries. The gaps between the simulated leaf patches make the vegetation-covered flow different from that of both open-channel and cover flows or the shear stress near the smooth vegetation cover is larger than that near the river bed. As the shear stress near the bed is reduced, the erosion process of river bed may be retarded. These findings help expand the understanding of river hydrodynamics during the vegetation cover existence. Parameters can also be utilized in numerical modeling of the vegetation covered flows. Several further studies are needed to be performed to fully understand the flow structure of the suspended vegetation flow. For example, the deduction of Darcy–Weisbach coefficients for two boundaries deserves considerable attention. Besides, the gap between canopies and the shape of leaves make the flow structure complicated and require more experiments to investigate. Moreover, further study is required for vegetation covered flow to elucidate the links between flow dynamics and aquatic ecosystems.
Fig. 6. K0 values varying with Q.
Acknowledgements
Fig. 7. Quadrant analysis (Run 5).
This work was supported in part by the National Natural Science Foundation of China (Nos. 51379154, 51439007,11472199 and 51622905).
(sweeps), u′ > 0 and w′ < 0. Si,0 is the contribution value of Reynolds stress in Qi. The figure shows that Q1 and Q3 are dominant quadrant in the vegetation zone but Q2 and Q4 dominate the bed zone. At the location of umax, no dominant quadrant is observed (i.e., the sum of Si,0 ≈ 0), which indicates that the shear stress near the location of umax is equivalent to zero. The maximum value of |S1,0| and |S3,0| near the vegetation cover is larger than that of |S2,0| and |S4,0|. In other words, the momentum exchanges are intense at the canopy–gap interface. In addition, Zare et al. (2016) confirmed that the shear stress varied dynamically with transformation of the ice cover, and the upper boundary shear stress is many times larger than the lower boundary shear stress during the solid ice cover to the ice cover removal, which is strongly similar to the gaps of vegetation cover. Therefore, the maximum of shear stress occurred at the water–vegetation interface, and the location of zero shear stress is close to the bed region in terms of the linear distribution. The water microspheres have to move downstream through the underneath of a leaf because of the resistance of leaves on the water surface. Once the gap occurs, the path of flow will change. In the gap region, water particles start to move upwards and leave the original position under the carry of high velocity cells, which also has been reflected by the quadrant analysis. The outward interaction Q1 is dominant at z/H > 0.9. Thus, the quadrant analysis can fully demonstrate the impact of gaps among leaves.
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