1.2　Alluvial resistance in transition regime⁽³⁾

Wang Shiqiang^①, William Rodney White^②

（①Prof., Dept. of Hydr. Engrg., Tsinghua Univ., Beijing, China；

②Res. Dir., HR Wallingford, Howbery Park, Wallingford, OX10 8BA, UK.）

Abstract: The resistance of alluvial channels depends upon flow and sediment characteristics. This paper considers the bed features and alluvial resistance that occur at, and close to, the transition zone between the upper and lower flow regimes. New flume and river data are published and the results are analyzed to provide satisfactory predictive techniques covering the lower, transition, and upper regimes for use by river engineers. The bed features that occur in the transition regime are shown to be washed-out dunes that diminish rapidly in height and steepness with increasing shear stress. The increase in depth with increasing unit discharge is much reduced in the transition regime due to lower hydraulic resistance. In certain circumstances, depths in the transition region can decrease with increasing unit discharge. A criterion for the upper limit of the transition regime corresponding to a moving flat bed is proposed by the writers.

1.2.1　Introduction

The resistance of alluvial channels changes with flow and sediment conditions as bed forms change. The bed forms that occur in what is described as the lower regime are flat-bed, ripples, and dunes. The upper regime is used to describe moving flat beds, antidunes, and chute and pool flows. There exists a transition regime between these lower and upper regime flows in which the bed forms are characterized by washed-out dunes and ripples (Simons and Richardson 1961). Research into the prediction of alluvial resistance has shown that, in general, there are two different relationships describing lower and upper regimes (Engelund 1966；Brownlie 1983；Wang Shiqiang et al. 1986；Hayashi 1986；White et al. 1987).

Bed features in the transition regime occur in natural rivers, perhaps more frequently than many people suspect. For example, the alluvial resistance in the middle Yangtze and lower Yellow rivers are in the transition regime for much of the flood season. This regime is characterized by the resistance coefficient decreasing rapidly with increasing flow strength. Though knowledge of alluvial resistance in the transition regime is necessary to understand such cases, the subject has received little study and is not yet well understood.

In this paper, two sets of experiments are described that were designed to investigate the transition regime. Data from these experiments, together with other published data, have been analyzed to describe the variation of the resistance coefficient throughout the transition region. A method for predicting the resistance coefficient in this region and also in the lower and upper regimes is proposed.

1.2.2　Laboratory experiments on alluvial resistance

1. Experiments performed at Tsinghua University

A set of experiments was carried out at the Sediment Research Laboratory of Tsinghua University, Beijing, China by Wang Shiqiang et al. The experiments were performed in a recirculating, tilting flume 60m long and 1.2m wide. Both sides of the flume are made of glass, and so the flow and bed features can be observed along the whole length of the flume. The sediment used in the experiments was a fine sand with a D₅₀ size of 0.076mm and a value for D₈₄/D₁₆ of 1.44. Its relative density was 2.65. The length of the measurement section used was 50m. A sand layer 15cm thick was placed on the flume bed. Each test was run to equilibrium and had a duration of 4～20 hours.

The discharge was measured using an electromagnetic flowmeter. The water surface level was measured at 36 locations using an automatic water surface follower reading to an accuracy of 0.1mm, and the bed level was measured at the same locations using a ruler.

The results of these experiments are summarized in Table 1 where λ, Δ, and c are the average length, height, and speed of the sand waves, respectively. The discharge Q was 24～410L/s；the depth H, 0.100～0.365m；and the slope S, 0.01×10^-3～3.054×10^-3. Out of the total 64 tests, 30% were in the transition region. For the fine sediment used in this experiment, the washed-out ripples in the transition region are regular and relatively stable in time, see Fig. 1.

2. Experiments at Hydraulics Research, Wallingford

The second set of experiments was carried out at Hydraulics Research, Wallingford, England, in 1989 by Wang Shiqiang while he was visiting that establishment. The experiments were performed using a tilting, recirculating flume 24m long and 0.917m wide. The length of the measurement section was 10.7m. The sediment used in the flume was a medium-coarse sand with a D₅₀ size of 0.76mm and a value for D₈₄/D₁₆ of 1.29. A sand layer 15cm thick was placed on the flume bed. Each test was run to equilibrium and lasted 0.5～5.5 hours. The time taken to achieve equilibrium was related to dΔ/Y_∗ and L, where dΔ is the change in height of the sand waves, Y_∗ is dimensionless shear stress due to resistance on the bed, and L is the length of the sand bed in the flume.

A weir was used to measure the discharge in the flume. The water surface level was measured using five tapping points connected to stilling pots and vernier point gauges reading to an accuracy of 0.02mm. The bed level was measured using a ruler.

The experimental results are summarized in Table 2. The discharge was 22.5～135L/s；the depth, 0.080～0.143m；and the slope, 0.111×10^-3～1.073×10^-2. A total off 44 tests were performed out of which 20 were in the transition regime. The washed-out dunes in the transition region that occurred during these tests were unstable and irregular.

1.2.3　Change of bed form shape in transition regime

The bed forms in the transition regime are washed-out dunes or ripples of which the height, steepness, and particularly the slope of the downstream face decrease rapidly with increasing flow strength. This is associated with a corresponding reduction in the resistance coefficient f, where f=8（V_∗/V）². Fig. 2, derived from the data in Table 1, shows how the height and steepness of the ripples vary with the dimensionless shear stress due to grain resistance Y'_∗, which is defined by

where γ and γ_s=the relative density of water and sediment, respectively.

The hydraulic radius due to grain resistance R'_b can be obtained from the equation

where k'_s is the grain roughness height given by

It can be seen that for bed features in the transition region, the height and the steepness of the ripples decrease with increasing Y'_∗.

Fig. 1 shows the profile of the washed-out ripples taken during run 19 at Tsinghua University. It shows that the shape of the ripple is almost symmetric, the downstream slope being less than 10°. Analysis of the profiles of washed-out dunes observed in the Yangtze and Yellow rivers, reveals that the downstream slopes of the dunes are generally less than 5°. These are much lower than those observed for dunes or ripples in the lower regime, where the slopes are in general significantly higher.

Associated with this change in the downstream slope of the bed featureis a change in the flow eddy in the trough of the dune. It may be observed that as the downstream slope reduces, the strength of the eddy also reduces. In the past it has been considered that the energy loss associated with the bed features was due to the sudden flow expansion downstream of the crest. For bed-forms in transition. however, it might be argued that a more appropriate model for the energy loss due to the bed features is a gradual flow expansion. The reduction in the strength of the eddy generated in the trough results in the reduction in the resistance coefficient. The downstream slope of the washed-out dune or ripple is an important factor affecting the strength of the eddy and the resistance coefficient in the transition regime.

1.2.4　Characteristics of resistance parameters in transition regime

1. Relationship between Y_∗ and Y'_∗

Engelund (1966) developed a method for predicting the alluvial resistance of channels by plotting values of Y_∗ against Y'_∗. Using observations he demonstrated two separate relationships, one for the lower regime and one for upper regime. No attempt was made to describe the transition between these two regimes. Values of Y_∗ and Y'_∗ for data from Table 1, field data from the Yellow River and some canal data are plotted in Fig. 3, where Y_∗ is ［γ R_bS/(γ_s-γ)/D₅₀］ and R_b is the hydraulic radius due to the resistance on bed, see Einstein (1942). A similar plot for the data from Table 2 and Gilbert (1914) is shown in Fig. 4. In Fig. 5, data with a sediment size of 0.2mm from Yin Yi (1989), Guy et al. (1966), and the Yangtze River are plotted. From these figures it can be seen that, for a given sediment size, Y'_∗ is only a function of Y'_∗ for lower and upper regimes. In the transition regime, for a given sediment size, the relation between Y_∗ and Y'_∗, becomes a family of curves for different values of R_b/D. The relationship between Y_∗ and Y'_∗ varies as the sediment size varies for lower, transition, and upper regimes. This dependence upon sediment diameter was not noticeable in the original plots by Engelund as the data used was for a restricted range of sediment sizes.

2. Relationship between f and Y'_∗

Fig. 6 shows the relationship between f and Y'_∗. It can be seen that f increases with increasing Y'_∗ in both the upper and lower regimes. In the transition regime, however, the value of f decreases with increasing Y'_∗. It can also be seen that different curves are obtained for differing values of the parameter R_b/D₅₀. In the lower and transition regimes, for a given value of R_b/D₅₀ and Y'_∗, f decreases for increasing values of D₅₀, while in the upper regime, f increases with increasing D₅₀.

All the curves of f against Y'_∗ have similar characteristics in that all demonstrate a local maximum and minimum. The local minimum can be identified with the upper limit of the transition region, that is, the boundary between the upper and the transition regions. The local maximum corresponds to the lower limit of the transition region, that is, the boundary between the lower regime and the transition region. In practice, this boundary between the lower and transition regimes is difficult to determine by observation.

3. H and V as Function of g, S, and D_gr

The data from Table 1 and from the Yellow River have been plotted in Figs. 7 and 8 to demonstrate how H and V vary with unit discharge q. The water depth increases with increasing unit discharge more slowly in the transition regime than in the upper and lower regimes. In Fig. 7, it can be seen that the depth can even decrease with increasing unit discharge for fine sediments and small values of H/D. In the transition regime the velocity increases rapidly with increasing unit discharge.

1.2.5　Criterion for transition regime

When performing experiments it is relatively easy to recognize the moving flat bed that is the boundary between the transition regime and the upper regime. This boundary corresponds to the local minimum in the f-Y'_∗ relationship and the turning points in the Y_∗-Y'_∗ relationship.

It was observed during the experiments that the flow velocity near the bottom of the eddy in the trough was close to that for incipient motion of bed material that is related to the value of the variable . The flow velocity at the top of the eddy, or equivalently, in the bottom of the main current above the eddy, is related to V/(H/D) ^m or the shear velocity due to grain V'_∗, based on the equations of flow velocity distribution and average velocity. One, therefore, has a reason to assume that the size and strength of the eddy and the corresponding bed forms and resistance coefficient are related to the ratio of the top and bottom velocities of the eddy. This ratio is given by .

In Fig. 9V/ is plotted against R_b/D₅₀ for over 500 data in the transition and upper regimes including the field data from the Yangtze, Yellow, and Missouri rivers and the Rio Grande conveyance channel. The criterion to determine the upper limit of the transition regime is given by the relationship, where F_rd is grain Froude number

Analysis shows that this simple criterion can be used to predict accurately the transition region in most situations, though errors may be larger for finer sediment. A more accurate, but more complicated, expression for the upper limit of the transition is given by

where

If

and

or

and

then the critical F_rd does not change with the ratio H/D₃₅.

In Eq. (5), the kinematic viscosity of water ν in the expression for D_gr reflects the influence of water temperature on the bed forms in the transition regime.

1.2.6　Prediction of alluvial resistance

Analysis of Fig. 6 and other data shows that, for a given sediment size and a given Y'_∗, the value of f decreases logarithmically with increasing R_b/D₅₀ and is almost constant for R_b/D₅₀ larger than 10000.

A relationship between f and Y'_∗ for different sediment diameters was derived from data for which the ratio R_b/D₅₀ was larger than 10000 and D₅₀ smaller than 0.8mm. This relationship is of the form

where,

This alluvial resistance equation, shown in Fig. 6, is suitable for natural sand channels and can be used to predict the alluvial resistance coefficient in lower, transition, and upper regimes. The advantages of this formulation are its simplicity and continuity for the various flow conditions without the need for any supplementary criteria or equations.

Over 1000 data shown in Fig. 10 from flumes and natural channels (Gilbert 1914；Guy et al. 1966；Brownlie 1981；Willis et al. 1972；Yin Yi 1989) have also been analyzed in terms of Y^∗ and Y'_∗. The data represent a wide range of D₅₀, 0.018～28.65mm；and R_b/D₅₀, 10～120000. To avoid confusion, the data from flumes in the transition regime is not plotted on this figure. On the basis of these data, a friction relation is proposed as follows for the lower regime for different values of R_b/D₅₀ and for the transition regime for sufficiently large values of R_b/D₅₀：

where

Eq. (11) is only valid for Y'_∗>0.04. The corresponding relationship for the upper regime is

where

Curves given by (11) and (13) for various values of D_∗ are shown in Fig. 10. By analyzing the data shown in Figs. 3, 4, and 5 and other flume data for the transition regime, a friction relationship can be proposed in the form

where

Y'_∗c can be calculated from known values of D₅₀ and S, while R'_bc can be calculated using (1). V_c can be obtained from (4) or (5) and Y_∗c is determined from (13).

Eq. (15) is a family of parallel straight lines for varying values of R_b/D₅₀ and a given D_∗. The gradient of the lines decreases with increasing D_∗. For a given D_∗, the critical Y'_∗ in the transition regime increases with increasing values of the ratio R_b/D₅₀. Curves given by (15) for various values of R_b/D₅₀ are shown in Figs. 3, 4, and 5.

For natural channels in which the value of H/D is sufficiently large, either (11) and (13) or (9) can be used to predict the alluvial resistance for all flow conditions.

Table 3 shows the results of an error analysis of (11) and (13) for field data. It can be seen that the average errors in Y_∗ are quite small. The errors obtained by using (9) are similarly small.

Table 4 shows the comparison between the depths predicted using (11) and (13) and the observed depths. The predictions give good agreement with the observations. The numbers of predictions in the different regimes are shown in Tables 3 and 4. This result demonstrates the number of flow conditions and corresponding bed forms that occur in the transition regime in natural channels. The predictions for the lower Yellow River and Yangtze River reveal the reason why the measured slopes of the downstream face of dunes are so small in both these rivers. All of the predicted bed forms in the Missouri River shown in Table 4 are in the transition regime, but some of the observed bed forms were described as dunes (Shen et al. 1978). This may be due to the difficulty of observing the small changes in the size of dunes or ripples that takes place when moving from the lower regime to the transition regime.

1.2.7　Conclusions

(1) The bed features in the transition regime are washed-out dunes. Not only their height and steepness, but also the slope of the down-stream face, decrease rapidly with increasing dimensionless effective shear stress Y'_∗. This is associated with a corresponding reduction in the resistance coefficient f. The model of energy loss due to the washed-out dunes or ripples is one of a gradual flow expansion.

(2) In the lower and upper regimes, water depth increases rapidly with increasing unit discharge, but it increases much more slowly, or even decreases, in the transition regime for a given slope and sediment diameter. In the lower and upper regimes, the dimensionless total shear stress Y_∗ is the function of Y'_∗ and the dimensionless sediment diameter D_∗. In the transition regime, Y_∗ is also a function of the relative hydraulic radius R_b/D. In the upper and lower regimes, for a given value of D_∗ and R_b/D, the resistance coefficient f increases with increasing Y'_∗. while in the transition regime f decreases with increasing Y'_∗. For a given D_∗ and Y'_∗, the value of f also decreases with increasing R_b/D and is almost a constant for sufficiently large values of R_b/D. For a given value of Y'_∗ in natural channels in the lower or transition regimes, the value of f increases with decreasing D_∗, while in the upper regime, f decreases with decreasing D_∗.

(3) A criterion for the upper limit of the transition regime corresponding to a moving flat bed is proposed by the writers. The bed forms are in the upper regime when the grain Froude number V/ is larger than a critical value, which depends on R_b/D and D_∗.

Acknowledgements

The experimental work carried out at Tsinghua University was funded by the Ministry of Water Conservation, China. The writers would like to acknowledge the assistance of Liu Ji-Xiang, Zhai Xiu-Ling, Ren Yu-Min, and Guo Jia-Chao in carrying out the experiments. The work at HR Wallingford was funded by the Construction Directorate of the Department of the Environment, U. K., under contract PECD 7/6/160. Special thanks go to Zhang Ren and R. Bettess for their assistance throughout the work.

References

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Notation

The following symbols are used in this paper：

B=width of channel；

C=speed of bed features；

D=grain diameter for uniform sediments；

D_n=grain diameter for which n% of sample is finer；

D_∗, D_gr=dimensionless grain size；

dΔ=change in height of sand waves；

E=coefficient,

f=resistance coeffircient, 8（v_∗/V）²；

F_rd=grain Froude number, V/;

g=gravitational acceleration；

H=mean flow depth；

H_a, H_c=assumed and calculated water depths, respectively；

k, k₁, k₂,...=coefficients；

k'_s=grain roughness height；

L=length of sand bed in flume；

m₀, m=coefficients；

n₀, n=coefficients；

Q=water discharge；

q=unit discharge；Q/B；

R_b, R'_b=hydraulic radius due to resistance on bed and due to grain resistance, respectively；

R'_bc=critical R'_b for upper limit of transition regime；

S=energy slope；

T=temperature；

V=mean velocity of flow；

V_c=critical velocity；

V_∗, V'_∗=shear velocity and shear velocity due to grain resistance, respectively；

X=variable, log（Y'_∗/0.04）；

Y_∗=dimensionless shear stress due to resistance on bed；

Y'_∗, Y″_∗=dimensionless shear stress due to grain resistance and bed features, respectively；

Y'_∗=specified Y_∗；

Y'_∗L, Y'_∗c, =critical Y'_c for lower limit and upper limit of transition regime, respectively；

γ, γ_s=specified gravity of water and sediment, respectively；

Δ=height of dunes or ripples；

λ=length of sand waves；and

ν=kinematic viscosity of water.