Ocean University of Qingdao, Qingdao, China

Fax: 865322032799; E-mail: bings@mail.ouqd.edu.cn

Sichuan University, Chengdu, China

Fax: 86285405148 ; E-mail: caosy@mail.sc.cninfo.net

China institute of Water Resources and Hydropower Research, Bejing, China

Fax: 861068412173; E-mail: gfli@iwhr.com

Abstract: Based on the vectorial analysis of forces acting on non-cohesive sediment particle to be on arbitrarily vegetated riverbed, the basic equation of coarse particle movement is derived, a vectorial bed-load formula is founded under non-equilibrium condition. To verify above-mentioned formulas, some flume experiments are done by means of a straight channel with different planting length and density. The observed data on channel width adjustment are compared with calculating results of channel processes using above vectorial formula.

 

Keywords: river width adjustment, vectorial equation, vegetation

A lot of methods are used to describe channel morphology and morphological adjustments for river engineering purposes. Available approach ranges from equations that predict the regime or graded morphology of equilibrium channels to mathematical models that simulate channel changes in time and space. Most of them, however, neglect time-dependent channel width adjustment and do not simulate processes of bank erosion or deposition (Thorne .C., Knight. D. W. et al., 1998^, Shuyou Cao, 1996). Kovacs and Parker (1994) derived a vectorial bed-load transport equation for coarse sediment transport that could be applied for slopes up to angle of repose both in the stream-wise and transverse direction; However, their vectorial bed-load equation and bank erosion models represented only considerable advances in modeling non-cohesive sediment transport in a non-vegetated riverbed. So far, none of available methods could be used simulation vegetated channel width adjustment.

The role of vegetation in affecting bank erosion and width adjustment is complex. Vegetation generally reduces soil erodibility, its impact on bank stability will be positive. Because it will produce a decrease in the bank-line shifting, and can play an important role in limiting the effectiveness of bank erosion by detachment and entrainment of individual grain or aggregate of bank material. Compared to non-vegetated banks, the erosion of well-vegetated banks is reduced by one to two orders of magnitude (Carson and Kirkby, 1972). As pointing out by Thorne and Osman (1988), a great deal of further research is very necessary before vegetation effect can be properly understood and incorporated into the technical description if bank material characteristics encountered along natural stream and waterways. Based on above research results and vegetation positive effect on the river width adjustment. A general vectorial equation of sediment transport on the vegetated bed surface under non-equilibrium condition is derived, and its mathematical model is developed in this paper.

Firstly built up a spatial coordinate system (see Fig.1), let stand for a unit normal vector of riverbed and -represent the unit downward vertical vector, and -can be decomposed into a component normal and a component tangential to the plane of riverbed, they are defined as

,                                         (1,2)

Assuming the unit vector in stream-wise direction, then

,                                        (3,4)

Generally speaking, and are not orthogonal to each other under the condition of non-vegetation. Let the bottom velocity of flow equal to , in this way, we may get some formulas as follows

(1) Average velocity decreased value owning to backwater caused by vegetation

                                        (5)

In which,; k₁ =;is an increment of water level by backwater; B represents width of water surface; A₀ denotes the wetted area of the cross section in non-vegetated channel.

Fig.1   Spatial coordinate system

(2) Average velocity decreased value owning to wake flow after vegetation

                                        (6.a)

In which, k is a row number of upstream vegetation planted; is velocity decay in ith row, which could be expressed as

                                        (6.b)

In which, B_i denotes a distance between two vegetations along cross-section direction; l_I is Prandtl length; and

According to this formula on bottom velocity

                                        (7)

The real bottom velocity decay of flow is

                                        (8)

The bottom velocity of flow in vegetated bed is . If we suppose sediment particle velocity to be , then the fluid velocity relative to the moving sediment particle is

                                        (9)

(3) Sediment movement basic equations

The forces acting on a coarse particle are respectively as drag force, uplift, immerse weight tangential to plane, immersed weight normal to plane, and Coulomb friction. According to Newton’s law, then：

                                       (10)

Equation (10) could be further expressed as：

Let ，then

In order to simplify above equations, let

,

n this way, we may get

                                        (11)

                                        (12)

Solve above basic equations, and then particle movement velocity could be obtained respectively.

(4) Bed-load transport rate and riverbed deformation equations

Suppose the bed-load layer thickness is z ，then the transport rates in , direction can be expressed as

                                        (13,14)

Exner’s sediment conservation equation is employed in this analysis，

                                        (15)

Let the angle betweenand equal to j ，then

                                        (16,17)

Substituting equation (16) and (17) into equation (15), then

                                        (18)

In which, q_x、q_y are unit transport rate in x、y direction respectively; Z is bed elevation；l _p is porosity of bed material.

Kovacs and Parker considered a small volume in the form of an elementary parallelepiped with height z normal to the bed. Here z corresponds to the thickness of the bed-load layer. Assuming sediment particle steady motion, the force balance per unit bed area acting on this infinitesimal box can be expressed as

                                        (19)

In which, , ,

The bed-load layer thickness is obtained by solving Equation (19)

                                        (20)

Let, We may get:

                                        (21)

                                        (22)

In which, R=，m _c is resistance factor, is shear stress of bed surface that differs from non-vegetated case， represents critical shear stress.

The Equation (18) can be further simplified for a straight channel with uniform vegetation in the stream direction x, a partial differential equation using the time evolution of the cross-sectional geometry of a straight channel (z =z (x, y)) is obtained

                                        (23)

In open channel, the mean turbulent flow of the viscous, incompressible fluid known as water is governed by the Reynolds and continuity equations:

                                        (24)

                                        (25)

In which, t is the time, is the ensemble mean part of the velocity field, p is the ensemble mean pressure, is the acceleration due to gravity, r is the fluid density and m is the dynamic viscosity of the fluid, is the Reynolds stress tensor. The origin of the secondary flow is embedded in the structure of turbulence in straight channels, as discussed by Demuren and Rodi (1984) and others. It is estimated that the magnitude of straight-channel secondary flow is approximately 1% of the mean flow, so that it could be negligible in straight channel. Otherwise it is assumed that the term is negligible. In this way, the bed geometry and flow field are taken to be independent of x. Applying for the Boussinesq eddy-viscosity concept to equation (24), the following momentum equation for the stream-wise mean velocity can be obtained (Kovacs and Parker, 1994),

                                        (26)

Where, denotes isotropic turbulent eddy diffusivity, the other two components of the Reynolds equation yield the condition that the pressure distribution is essentially hydrostatic. A deformation-calculating model with vegetation is mainly composed by above equation (11), (12), (24), (18), and (19).

In order to verify above equations, a mathematical model developed to simulate the time development of a straight self formed channel is calibrated against experimental results that had been done by authors with feather regarding as model “vegetation” in 1997, its calculating conditions for the experimental Run A2 are shown as Table 1. For a straight channel, assumed that bed-load transport rate is uniform in the stream direction, and took 0.03(as the value for.

In view of vegetation effects on river width adjustment, and combined with the analysis result of experimental data, a shear stress relation dealt with vegetation factors is founded (Shi Bing, 1998)

=                                        (27)

In which, is an average retarding water rate of vegetation in section; f_v represents volume content of vegetation in certain scope, k’ is a coefficient, is shear stress of bed face under non-vegetation condition. Solve Equations (23), (26), and (27), then the width adjustment processes with vegetation bed could be obtained. Fig.2 only gives out the comparison results when experimental time equals 1000 seconds and 8640 Seconds respectively. As shown in Fig.2, the predicted cross sections in width adjustment processes are in good agreement with the laboratory observed data. It should be noted in Fig.2 that there is a tread in width adjustment processes in which the cross-section develops a flat center bed region, and a deposited area will be formed at the front of vegetation zone. At the same time, the effect of vegetation on width adjustment in transverse direction is greatly lower than non-vegetation case, the larger density of vegetation is, and then the smaller the velocity of width adjustment and stable width in transverse direction is too.

Fig.2   Comparison predicted section with experimental Run A2

A vectorial bed-load formula using calculation the transport of coarse sediment in vegetated riverbed is derived on the basis of Kovacs and Parker's ideal in this paper. It is fully non-linear and vectorial equation, and its application scope becomes wider. Simplifying this formula, a force balance equation and Kovacs and Parker‘s bed-load formula that uplift acting on sediment particle had been negligible can be obtained respectively. To calibrate the approach, a series of flume experiments dealt with the effect of vegetation on width adjustment in a straight channel are conducted according to different layout and planting length. At the same time, a mathematical model is founded too. Compared the predicted cross sections in width adjustment processes with experimental data, we may find they are in good agreement with.

The effect of vegetation on width adjustment is positive and obvious; its final cross-sectional profile is characterized by a flat bed region near the center of the channel, similar to non-vegetation condition. However, it will take a much longer time to reach a stable state and its stable width obviously less than that on non-vegetated riverbed. The larger density of vegetation on the slope and bank of river is, then the smaller stable width is.

This research was supported by the National Science Fund (grant no.59679010), Open Fund of State Key Hydraulics Laboratory of High Speed Flows of Sichuan University (grant no.9902) as well as Visiting Scholar fund of Education Ministry of China (1999-153).

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