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2014,26(3):467-473 DOI: 10.1016/S1001-6058(14)60053-6

Numerical investigation of flow through vegetated multi-stage compound cha- nnel* WANG Wen (王雯), HUAI Wen-xin (槐文信), GAO Meng (高猛) State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: wangwen@whu.edu.cn (Received February 15, 2014, Revised May 5, 2014) Abstract: This paper addresses the problem of the renormalization group k turbulence modeling of a vegetated multi-stage compound channel. Results from Micro acoustic Doppler velocimeter (ADV) tests are used with time and spatial averaging (double- averaging method) in the analysis of the flow field and the characterization. Comparisons of the mean velocity, the Reynolds stress, and the turbulent energy distribution show the validity of the computational method. The mean velocity profile sees an obvious de- celeration in the terraces because of vegetation. Secondary flow exists mainly at the junction of the main channel and the vegetation region on the first terrace. The bed shear stress in the main channel is much greater than that in the terraces. The difference of the bed shear stress between two terraces is insignificant, and the presence of vegetation can effectively reduce the bed shear stress. Key words: compound channel, open channel flow, vegetation, double-averaging concept, bed shear stress

Introduction A compound cross section is often formed in na-

tural rivers under the action of different dominant dis- charge rates. When the water level rises, the water sur- face extends from the main channel to the floodplain. Because of the increase both of the width of the free surface area and of the depth of the water, the area of the flow section increases, and this leads to an increa- se of the discharge capacity of the river channel[1]. In many cities, to increase the stability of river slopes and to facilitate constructions, river channels are often designed as multi-stage compound channels that can adapt to different flow rates. Vegetation may be pla- nted on the river terraces to beautify the environment. In the design of the cross-sectional shape of a channel,

* Project supported by the National Natural Science Foun- dation of China (Grant No. 11372232), the Specialized Resea- rch Fund for the Doctoral Program of Higher Education (Grant No. 20130141110016) and the State Water Pollution Control and Management of Major Special Science and Technology (Grant No. 2012ZX07205-005-03). Biography: WANG Wen (1986-), Male, Ph. D. Candidate Corresponding author: HUAI Wen-xin, E-mail: wxhuai@whu.edu.cn

one would normally consider the hindering effect of the river terrace vegetation on the flow, but ignore the reinforcing effect its roots have on the bank soil, which can reduce the amount of sand introduced into the atmosphere[2,3].

Under the action of a compound river channel boundary and the secondary flow, the velocity distri- bution within the river channel becomes extremely complex. Wu et al.[4,5] developed a depth-averaged 2- D numerical model to simulate the flow through eme- rgent and submerged rigid vegetations. Rameshwaran and Shiono[6] studied the depth-averaged velocity dis- tribution along a cross section. Hamidifar and Omid[7] experimented with rigid cylindrical rods to study the flow velocity and secondary velocity distributions within a compound open-channel. Chen et al.[8] obtai- ned the analytical solution of the flow velocity in a lateral distribution by considering the effects of the se- condary flow. Wu et al.[9] studied the vegetation effe- cts on the drag and inertia forces in the momentum equations and proposed a vertical two-dimensional model in the simulation of the wave propagation th- rough vegetated waterbodies.

When a river flow is hindered by vegetation, it is very important to understand the redistribution of the flow velocity, which can have a crucial effect on the

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discharge capacity. However, the Reynolds stress and the turbulent energy distribution also play important roles in determining the flow characteristics. Typically, as the distance from the vegetation varies, the spatial characteristics of the flow differ. For a macroscopic river modeling, the hydrodynamic variables generally represent a stationary and ergodic random field. When the time averaging is performed over a very long pe- riod, and the space averaging is performed over a re- presentative elementary volume, the averaged flow characteristics can be used for macro conditions. Based on this consideration, Souliotis and Prinos[10] studied the effects of a vegetation patch on the flow and turbulence characteristics. Elghanduri[11] and Prinos et al.[12] focused their attention on the turbulent flow over and within a porous bed.

The above studies concern with the flow chara- cteristics of a single-stage compound channel, and the plants on the river terrace are considered as non-sub- merged. This paper carries out experimental and nu- merical simulations to study the flow characteristics of a two-stage compound channel with both submerged and non-submerged plants. 1. Model description 1.1 Physical model[13]

The experiments are performed using the labora- tory flume at Wuhan University. The flume is 20 m long, 1 m wide, and 0.5 m deep, and the outlet and in- let structures of the flume are connected to a hydraulic circuit that allows a continuous recirculation of the stable discharge at a rate of 0.135 m3/s.

Fig.1 Schematic diagram of the two-stage compound open cha-

nnel

The flume bed slope 0( )S is set to 0.0004. Figure

1 shows the cross-sectional shape of the flume with the following geometrical parameters: =mB 0.5 m,

1 =B f 0.25 m, 1 =d 0.106 m, 2 =B f 0.25 m, and 2 =d

0.163 m. To model the vegetation, rigid cylinders are placed in the first and second terraces along the cen- tral portion of the flume of 8 m in length. The diame- ter and the height of the cylinders are 0.006 m and 0.14 m, respectively. The cylinders are arranged in

lines 0.05 m apart and with lengthwise spacing of 0.20 m. The water depth is adjusted by a gate at the end of the flume to form a water surface slope parallel the bed slope in the test area. This is referred to as uniform flow conditions. The water depth ( )H is

0.405 m. The 3-D water velocity and turbulence are mea-

sured with the SonTek/YSI 16MHz Micro acoustic Doppler velocimeter (ADV). Each measurement point with a sampling volume of 9×10–8 m3 works at a sam- pling frequency of 50 Hz for 60 s of sampling time. The test data with a signal-tonoise ratio less than 15 and a correlation coefficient less than 70% are re- moved from the final analysis. The measured data are processed using the space-averaging method.

Fig.2 Plan view of the test points and the weighting scheme

Figure 2 is the plan view of the spatial sampling points in the test and the weighting scheme. The cha- nnel is divided into 20 regions in the transverse dire- ction, and 12 measuring points are set within each re- gion, five measurement points behind the vegetation and seven within the vegetation. The weighted-avera- ge method is used for each measured value. The wei- ghting factor of each measurement point is the perce- ntage of its area over the total weighting area, as shown in Fig.2. 1.2 Mathematical model

Using the double-averaging method[14] for a gra- vity-driven, open-channel, free surface flow, the conti- nuity equation and the momentum conservation equa- tion can be written as:

+ = 0i

i

u

t x

(1)

1+ =

i i i j

j ij i j

u u p u uu g

t x x x

+ + +i i

ii j F V

j j j

uu u f f

x x x

(2)

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where iu is the component of the time-averaged ve-

locity in ix direction (the overbar indicates time ave-

raging), is the fluid density, p is the average pre-

ssure, represents the kinematic viscosity, iFf and

iVf represent the form and viscous drag forces on a

unit mass of water within the averaging volume V , and i ju u is the Reynolds stress due to the temporally

fluctuating velocities. The extra Reynolds stress

i ju u is a dispersive covariance, which means a

covariance that arises from the spatial correlation of quantities averaged in time, but carrying with the po- sition[15].

The Reynolds decomposition of the instanta- neous velocity component into = +i i iu U u , with iU

being the time-averaged velocity for the velocity com- ponent in direction i . The time fluctuating part is iu .

In the following equations, the plane-averaged quanti- ties are denoted using angled brackets . As highli-

ghted by Nikora et al.[14], two different spatial avera- ges are considered. The first is the so-called intrinsic spatial average, which when applied to iU is expre-

ssed as follows

1( , ) = ( + , )d

fi iA

f

U t U t SA x x r (3)

The second average is the so-called superficial spatial average[14]

0

1( , ) = ( + , )d

fi is A

U t U t SA x x r (4)

where fA is the area occupied by the fluid at a given

altitude z within 0A , the total averaging plane area

at this given altitude, r is the vector describing fA

around the location point x , dS is an infinitesimal area element, and cz is the vegetation height. Above

cz , fA is equal to 0A and =i isU U . Below cz ,

the roughness geometry function can be written as = ( )i is

U z U . Based on the intrinsic average, the

following decomposition of the local mean velocity component is introduced

= +i i iU U u (5)

where iu is the deviation of the local time-averaged

velocity from the double-averaged velocity ( )iU z .

By definition, =iu 0. In an extension, any flow qua-

ntity can be decomposed as

= + (6)

The renormalization group (RNG) k models are more accurate and reliable for a wider class of flows than the standard k model. They provide an improved performance for several types of flows, in- cluding the flows with separation zones and those around curved geometries. The features of the flow in open-channel sharp bends, such as the flow separation and the secondary flow are associated with a complex mean strain and are sensitive to the anisotropy of the turbulence[16].

The main difference between the RNG and the standard k models lies in the additional term in the equation. An expression for R , is based on the

additional expansion parameter , which is the ratio

of the turbulent to the mean strain time scale, and the resulting expression is

3

0

3

1

=1+

T S

R

(7)

where = /Sk , 1/ 2= (2 )i j i jS S S and 0 4.38.

The constant is chosen such that = 0.012,

which results in a value for the von Karman constant of 0.4, a recommended value for turbulent channel flows. In the /R k turbulence model, proposed

by the present authors, the R term is adopted in the

standard k formulation. The equation for the turbulence kinetic energy

( )k is

= + +i Tj i j

j i j k j

Uk kU

x x x x

(8)

where iU is the mean velocity, ix is the position ve-

ctor, i j is the Reynolds stress, T is the eddy visco-

sity, k is a closure coefficient with a value of unity,

and is the dissipation rate. The equation for the dis- sipation rate of the turbulence kinetic energy ( ) is

2

1 2= +ij i j

j j

UU C C

x k x k

3

0

3

1

+1+

T

T

j k j

S

x x

(9)

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The eddy viscosity concept is used for the treat- ment of the Reynolds stress. The Reynolds stress with time and space averaging can be written as follows

= + + +ji i

i j T Tj i j

uu uu u

x x x

2

3j

i ji

uk

x

(10)

In Eq.(10), the eddy viscosity is determined by the Prandtl-Kolmogorov relationship, such that

2

=T

c k

(11)

The model constants 1C , 2C , C and are

1.42, 1.68, 0.0845, and 1.3, respectively, i j is the

Kronecker delta, k is the turbulence kinetic energy, and is the turbulence dissipation rate. By ignoring the spatial fluctuation items, the Reynolds stresses can be expressed as

2=

3ji

i j T i jj i

uuu u k

x x

(12)

Fig.3 Schematic diagram of the computational model

Fig.4 Schematic diagram of the mesh around the vegetation

Equations (1) and (2) are discretized using the fi- nite volume method and the structured mesh is used in

the computational model. Figure 3 shows the compu- tational domain. The cylindrical surface of the vegeta- tion is simplified by using a 32-sided polygon, as shown in Fig.4, the smallest grid length is 5.910−4 m. The total grid number is approximately 7.5106.

The volume of fluid method is used for the treat- ment of the free surface, and the mass flow inlet and pressure outlet conditions are used for the boundary conditions. The total length of the computational model is 7 m. The pressure outlet boundary condition is used for the air outlet. A standard wall function and the no-slip wall boundary conditions are used for the treatment of the solid walls, such as the channel bed and the cylinders.

In this study, the pressure-velocity coupling is achieved using the SIMPLE algorithm. The body force weighting is used for the discretized pressure. For the momentum, the turbulence kinetic energy, and the turbulence dissipation, the Third-Order MUSCL scheme is used. The typical values of the under-re- laxation factors used in the study are 0.3 for the pre- ssure, 0.7 for the velocities, and 0.8 for other turbule- nce quantities. The convergence is reached when the residuals are less than 10−5 for all equations.

The cases studied are based on experimental arra- ngements[13]. The computational domain is the vegeta- tion zone in the center of the flume. The flow rate, the water level, and the vegetation arrangement are the same as in the test. The spatially averaged quantities are calculated on various planes, parallel to the cha- nnel bed.

Fig.5 Mainstream velocity contours from the test (m/s)

Fig.6 Mainstream velocity contours from the computation (m/s)

2. Analyses of the results

2.1 Mainstream velocity distribution Figures 5 and 6 show the mainstream velocity

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distribution. In these figures, the velocities are norma- lized by the friction velocity U . It can be seen that owing to the presence of the vegetation, the hindering effect on the velocity in the first and second terraces is significant. The velocity difference between the seco- nd terraces and the main river channel is clearly seen, and a clear deceleration zone can be seen around the vegetation on the first terrace. The flow velocity de- creases from the main river channel to the second te- rrace, reaching a relatively small value inside the ve- getation on both terraces. In the experiment, the maxi- mum velocity takes near the left-side wall a value of 0.55 m/s. The flow velocity on the second terrace is less than 0.25 m/s. The calculated velocity distribution is more uniform; the maximum velocity is about 0.53 m/s, and less than 0.3 m/s on the second terrace. The calculated flow velocity on the first terrace above the vegetation is slightly greater than that of the tested data.

Fig.7 Tested secondary flow distribution

Fig.8 Calculated secondary flow distribution 2.2 Secondary flow distribution

Figures 7 and 8 show the distribution of the se- condary flow in the test and by the calculation. The test results show that a clockwise low-intensity seco- ndary flow exists on the second terrace. There is a large circulation at the top of the first terrace vegeta- tion region and the main river junction. Significant momentum is transported from the main channel to the first terrace. Under the hindering effect of the ve- getation on the second terrace, the secondary current cannot reach the inside of the second terrace, and therefore, the flow descends along the slope of the se- cond terrace, forming a clockwise secondary flow on the top of the vegetation on the first terrace.

Similar distributions can be seen in the calculated results. A counterclockwise secondary flow appears in the junction between the main channel and the vegeta- tion on the first terrace. There is a counter-clockwise secondary flow at the top of the main channel near the left-side wall. Under its effect, the contour of the mai- nstream velocity budges towards the side wall, as can be seen in Fig.6.

The hindering effect of the vegetation on the te- rraces breaks down the large-scale vortex into smaller scale vortices. The high-energy vortex transfers mo- mentum to the terraces. The intensity of the secondary flow on the first terrace is significantly greater than that on the second terrace, which indicates that the momentum exchange occurs mainly on the first terra- ce. The Reynolds stress distribution also indicates that the shear stress in this region is relatively high.

The maximum value of the secondary flow velo- city in the test results is about 0.025 m/s, which is about 4% of the mainstream flow velocity. The calcu- lated maximum value of the secondary flow is 0.022 m/s, which is about 3% of the mainstream flow velocity, slightly smaller than the experimental values.

Fig.9 Tested Reynolds stress contours (Pa)

Fig.10 Calculated Reynolds stress contours (Pa)

2.3 Reynolds stress distribution The tested Reynolds stress distribution is shown

in Fig.9 and the calculated results are in Fig.10. It can be seen in both figures that the peak value of the Reynolds stress appears at the top of the vegetation on the first terrace, near the main river channel, where a significant momentum exchange occurs. In the main river channel, because there is no vegetation, the Reynolds stress near the wall is larger than that in the mainstream flow.

The calculated value of the Reynolds stress is re- latively large at the surface of the water, especially near the second terrace. This is due to the fact that the

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vegetation on the second terraces is non-submerged, i.e., the pressure and the velocity difference generated by the flow around such a cylinder produces a large shear stress. This result is similar to the experimental results obtained by Nezu and Sanjou[17]. The calcula- ted pattern of the surface water flow is shown in Fig.11.

Fig.11 Calculated free-surface flow pattern

Fig.12 Tested turbulent energy contours (m2/s2)

Fig.13 Calculated turbulent energy contours (m2/s2) 2.4 Turbulent energy distribution

Figures 12 and 13 show the distribution of the turbulent energy in the test and by the calculation, re- spectively. It is clear that there is a large magnitude turbulent energy in the middle region of the compou- nd channel and the turbulent energy takes smaller ma- gnitudes near the left-side wall. The maximum turbu- lent energy appears at the junction of the main channel and the vegetation on the first terrace. The turbulent energy in the main channel and on the second stage are relatively small, mainly because in the main cha- nnel the velocity has a high magnitude but a small gradient. The turbulent energy on the second terrace is small because both the velocity and the velocity gra- dient are small. There is a significant increase of the turbulent energy at the interface of the vegetation and surrounding flow. The maximum value of the turbule-

nt energy appears at the top of the vegetation on the first terrace near the main channel position, which in- dicates the place where there is a high shear velocity gradient.

2.5 Bed shear stress The boundary shear stress distribution can be in-

fluenced by many factors, such as the vegetation den- sity, the cross-sectional shape, and the water depth[18,19]. The presence of vegetation can effectively reduce the flow velocity through the resistance it ge- nerates. The interface of the vegetation and the flow can absorb large amounts of momentum, thereby re- ducing the bed shear stress. Thus, the presence of ve- getation can effectively reduce the flow velocity, pro- mpting sedimentation, and reducing erosion.

Fig.14 Distribution of bed shear stress

Figure 14 shows the variation of the bed shear st- ress along the bottom of the river. In the main channel, because there is no vegetation, the bed shear stress in- creases gradually from the main channel to the terra- ces, and then drops sharply at the slope interface. The main channel bed shear stress is much larger than that of the first and secondary terraces. The bed shear st- ress on the first terrace is large near the main river channel, and then decreases sharply inside the vegeta- tion region, where it is stabilized in the internal area of the vegetation. The bed shear stress is greater between two rods than behind a rod.

3. Conclusions With the use of an RNG k turbulence model

and the adoption of the double-averaging method, the turbulence characteristics of flow through a vegetated multi-stage compound channel are analyzed, and the calculated data are compared with laboratory measu- rements using ADV techniques. The main findings are:

(1) The double-averaging method can effectively predict the flow characteristics, such as the mainst- ream velocity and the Reynolds stress distribution. The secondary flow in the multi-stage compound cha- nnel is well captured by the RNG k turbulent model.

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(2) In the two-stage compound channel, the ma- ximum values of the Reynolds stress and the turbulent energy appear in the vegetation on the first stage.

(3) A counterclockwise circulation of the seco- ndary flow can be seen in the main channel near the first terrace, the presence of which indicates a mome- ntum exchange. The magnitude of the secondary flow in the test is slightly larger than that calculated.

(4) The bed shear stress results show that shear stress at the bottom of the main channel is larger than that on the vegetated terraces. There is no significant difference in the bed shear stress between the first and the second terraces. The presence of vegetation can ef- fectively reduce the shear stress near the riverbed, ac- hieving the effects of siltation and reducing erosion. References [1] ABRIL J. B., KNIGHT D. W. Stage-discharge predi-

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