1.3　Frictional characteristics of alluvial streams in the lower and upper regimes⁽⁴⁾

W. R. White^①, R. B ettes^②, Wang Shiqiang^③

（①Head of River Engineering, Hydraulics Research, Wallingford；

②Project Manager, River Engineering Department, Hydraulics Research；

③Lecturer, Department of Hydraulic Engineering, Tsinghua University, Beijing, China.）

Abstract: In 1980, White et al. proposed a method for predicting the frictional resistance of alluvial channels. The method established a relationship between the total shear and effective shear for different sediment sizes for flows with a Froude number less than 0.8. In this paper, it is shown that the previous method applied to lower regime flows in which the bed features are either plane-bed, ripples or dunes. A new relationship is proposed for upper regime flows in which the bed features are plane-bed or anti-dunes. A criterion to determine whether the flow is in upper or lower regime is described. This work greatly extends the range of applicability of the method for alluvial resistance prediction.

1.3.1　Introduction

1. The calculation of flow in an alluvial channel requires an estimate to be made of the frictional losses on the boundary of the channel. A method of determining the friction in an alluvial channel is also required for the determination of sediment transport rates related to the design of irrigation channels and river improvement works. The determination of these frictional losses is difficult as bed forms and hence frictional resistance vary with the flow and the sediment transport rate. Under certain conditions the bed may be plane, but as the flow varies, ripples, dunes or anti-dunes may develop, causing significant variations in the friction losses at the boundary.

2. In this paper, a method is proposed for predicting alluvial friction. It is essentially an extension of the approach described in White et al. Attention is confined to steady flows in which the friction is dependent only on the local values of the flow and sediment transport.

3. In the case, for example, of a laboratory channel with an initially plane, sediment bed, down which the discharge is gradually increased, the bed features will change as the discharge changes. Provided that the sediment is sufficiently fine, ripples will develop first. At a higher discharge, dunes will appear and increase in size. The dunes will then diminish as the discharge increases to be replaced by an approximately smooth bed. Antidunes will the develop.

4. Previous work, particularly by Engelund has suggested that there is one frictional relationship for dune beds, described as the lower flow regime and an essentially different relationship for plane beds, standing waves and anti-dunes, described as the upper flow regime. Engelund plotted the dimensionless bed shear due to skin friction against the total dimensionless bed shear for a sequence of flume data. The relationship he obtained and data from Guy et al. are shown in Fig. 1. The lower curve applies for dunes, ripples being excluded from Engelund's analysis, while the upper curve applies to flat beds, standing waves and anti-dunes.

5. In 1980, White et al.^[1] introduced a friction relationship which effectively described the friction associated with ripple and dune beds, corresponding to Engelund's lower flow regime^[1]. They showed that particle size is an additional important parameter not included in the analysis by Engelund. In this paper, the Authors extend that work by proposing a further relationship applicable to flat beds, standing waves, anti-dunes and chute and pools, corresponding to Engelund's upper flow regime.

1.3.2　Description of White, Paris, and Bettess theory

6. Following AckersandWhite, White et al.^[1] based their approach on a dimensionless sediment mobility F_gr, the ratio of the effective shear forces to the immersed weight of the particles. The mobility number was defined in such a way that only the relevant shear forces were used, that is, total shear for fine sediments, grain shear for coarse sediments and an intermediate value depending on the dimensionless grain size for the transitional sediments.

Notation

A　value of sediment mobility number F_gr at initial motion

a　constant

b　constant

d　mean depth of flow (m)

D　sediment diameter (m)

D₃₅　sediment diameter in meters for which 35% of the sample is finer

D_gr　dimensionless grain size, D[g (s-1)/ν²]^1/3

F　Froude number

F_cg　sediment mobility, coarse grains, V/

F_fg　sediment mobility, fine grains, V_∗/

F_gr　sediment mobility, general

g　acceleration due to gravity (m/s per s)

n　transition exponent dependent on particle size

R　hydraulic radius (m)

s　specific gravity of sediment ρ_s/ρ

S_f　friction slope, V²_∗/gD

U_E　dimensionless unit stream power, VS/[ (g ν)^1/3D_gr]

V_∗　shear velocity (m/s), V

V　mean velocity (m/s)

α　constant

ν　kinematic viscosity of fluid (m²/s)

ρ　density of fluid (kg/L)

ρ_s　density of sediment (kg/L)

ϕ　function

ψ　function

τ　shear stress (N/m²)

7. The dimensionless grain size D_gr was defined by)

where g is acceleration due to gravity, s is specific gravity of the sediment, ν is the kinematic viscosity and D is the sediment diameter. It was recommended that D₃₅ should be used as the representative diameter.

8. The dimensionless sediment mobility was defined by

where V_∗ is the shear velocity, V is the average flow velocity, d is the depth and n is the exponent which varies form 1.0 for fine sediments (D_gr=1.0) to 0.0 for coarse sediments (D_gr=60). Thus for fine sediments

and for coarse sediments

9. By considering flume data for which the Froude number was less than 0.8, White et al. found that there was a relationship between the total shear mobility F_fg and the effective shear mobility F_gr, the form of the relationship varying with sediment size D_gr (see Fig. 2). Sediment motion occurs only if F_gr exceeds the threshold value A, so that if F_gr is less than A then the friction will depend on the past flows as bed features will be present which were created by earlier, larger flows. For the region F_gr greater than A, it was found that there was a linear relationship between F_gr and F_fg for each value of D_gr. The lines fanned out from the point F_gr=F_fg=A, the steeper lines being associated with lager D_gr values. A relationship of the form

was postulated. Flume data were used to examine the form of the function ϕ, and the suggested equation was

This relationship was extensively tested on a selection of laboratory and field data. It performed significantly better than the methods proposed by Einstein and Barbarossa and Raudkivi, and marginally better than that attributable to Engelund.^[2] Of the predictions of friction factor, 48% were within 25% of the observed value and 89% within a factor of two.^[1]

1.3.3　Extension of White, Paris, and Bettess theory to upper regime

10. In their analysis, White et al.^[1] restricted their attention to data for which the Froude number was less than 0.8. This was done partly because their approach was based on that used by Ackers and White^[4], when the latter developed their sediment transport theory which was developed using data with Froude numbers less than 0.8, and also partly to avoid the problems of considering flows in the upper regime. This restriction has not prevented others from attempting to use the theory for Froude numbers lager than 0.8, with some success.

11. To further elucidate the problems of alluvial friction, a sequence of laboratory experiments was undertaken.These experiments indicated that if data for a particular sediment size for ripples, dunes, plane beds and anti-dunes were plotted on an F_gr/F_fg plot then two separate relationships became apparent, in a fashion similar to Engelund's analysis, ^[2] one for lower regime and one for upper regime. Data from Guy et al. and Gilbert were analysed and showed similar behaviour. These data from Guy et al. for three separate sediment sizes are plotted in Figs 3, 4 and 5. All demonstrate two distinct relationships, one for ripple and dune beds and another for transition, anti-dunes and chute and pool beds.

12. The lower regime relationship is that previously described by the White et al.^[1] equation (Eq. (6)).

13. The upper regime data show a consistent trend. Data for each D_gr value lie on a single curve, with different curves for different D_gr values. As for the lower regime, there is a clear progression away from the F_gr=F_fg line with increasing values of D_gr, the line through each data set converges towards the F_fg=F_gr line close to the value of the mobility number corresponding to the threshold of movement. For large values of F_fg, the curves bend away from the F_fg=F_gr line.

The lower regime case has been described by an equation of the form

where ϕ is a function of D_gr. To give the appropriate form for the upper regime, it was decided to fit an equation of the form.

The equation fitted to the available data was

14. As the data are not yet available to verify this equation for large sediment sizes, the Authors do not recommend its use for values of D_gr greater than 60. Unfortunately, little data is available for very high mobilities, particularly for coarse material. The Authors therefore consider that equation (9) may require modification when more data become available. The Authors would not expect, however, that the general behaviour would be different from that described by equation (9). The family of curves as predicted by equations (6) and (9) are given in Fig. 6 for graphical use.

1.3.4　Criterion for upper or lower regime

15. The use of two separate relationships for lower and upper regime creates two problems. The first is that of determining the appropriate regime to use in particular circumstances, and the second is that of determining the transition that must take place from one regime to the other. In the account so far the distinction between the two regimes has been provided by a description of the bed features associated with them. It seems reasonable, therefore, that the criterion used to define the upper and lower regime conditions should be related to those used to specify the occurrence of different bed forms. Simons and Richardson distinguished different features by plotting stream power, τ V, against median fall diameter (see Fig. 7). Further consideration of the problem has led the Authors to prefer the use of a non-dimensionalized unit stream power U_E, in the form

16. Figure 8 shows U_E plotted against D_gr for a range of data from Guy et al., ^[3] Gilbert^[8] and Wang Shiqiang.^[7]It can clearly be seen that different areas of the diagram correspond to different bed features. For values of U_E less than 0.00035 the bed is plane. Ripples occur for values of U_E between 0.00035 and 0.011 provided that the D_gr value is less than approximately 15. Otherwise, for values of U_E between 0.00035 and 0.011 the bed feature is predominantly dunes. The transition region is approximately for values of U_E between 0.011 and 0.02, while flat bed and anti-dunes occur for U_E values greater than 0.02. Therefore, the lower regime curves are appropriate if U_E is less than 0.011 and the upper regime curve if U_E is greater than 0.011.

17. Consideration will now be given to the way in which this method of determining the nature of the bed features can be used in the calculation of alluvial friction. If there is no a priori information as to the nature of the flow then it must be assumed, in turn, that the flow corresponds to lower and upper regime, whereupon the assumption which leads to consistent results must be determined. In the following, a superscript L will denote values of variables calculated using equation (6)—that is, assuming lower regime—and a superscript U will denote values of variables calculated using equation (9)—that is, assuming upper regime. If equations (6) and (9) are used to calculate the relevant variables and are determined, then, as >, three different cases must be considered.

(a) <0.011 and <0.011

In this case, as the use of the upper regime equation leads to a solution that implies lower regime conditions, the only consistent assumption is that the system is in lower regime.

(b) >0.011 and >0.011

Similarly, as the use of the lower regime equation leads to an inconsistency, the system must be in upper regime.

(c) <0.011 and >0.011

In this case, both assumptions lead to consistent results. Our interpretation is that either result represents a stable solution and that the form adopted in practice will depend on the previous history of the flows.

18. There is little or no experimental evidence available, but the Authors suggest that the system might display a hysteresis effect. If the system is in lower regime, it will remain in lower regime until =0.011, at which point the system will make the transition to upper regime. If the system is in upper regime then it will remain in the upper regime until =0.011, when it will make the transition to lower regime. There is not enough evidence for a complete interpretation of other people's experiments, but it seems likely that experimental results lying between the upper and lower regime curves do not represent stable, steady solutions, but are unsteady solutions in transition between the upper and lower regimes. A careful series of experiments is required to resolve these problems.

19. In the interim, an algorithm is required which can be used to determine the alluvial friction in an particular case. If information is available about the nature of the flow or its past history then this should be used to determine whether upper or lower regime conditions should be assumed. In the absence of any such information, the Authors suggest that the following criterion be used

+<0.022 use lower regime

+ 0.022 use upper regime

This is represented graphically in Fig. 9. Since is always greater than , only the area below the line OA is considered. Points to the left of the line ED represent lower regime；Points above the line EB represent upper regime. Two possible stable solutions exist for points in the region DEB. The above criterion is equivalent to assuming that points to the left and below the line EC are lower regime and points to the right and above the line EC are in upper regime.

1.3.5　Conclusions

20. An extension of the White et al.^[1] method for predicting alluvial friction has been proposed for upper regime flows. The method has been developed on flume experiments. This removes the restriction on the White et al. method to flows whose Froude number is less than 0.8. More data, however, are required to elucidate the behaviour of large sediment sizes under higher Froude number flows.

21. A criterion to determine when the flow in a channel is in upper or lower regime has been suggested. It has been further suggested that there is a range of conditions under which both the solutions in the upper and lower regime are stable. It is postulated that the solution that is achieved in practice depends upon the history of the flow and that the system may well exhibit hysteresis. Further experiments are required to investigate these suggestions.

22. The application of the extended method is described in Appendix 1.

Acknowledgements

23. The work described in this Paper was carried out while Wang Shiqiang was visiting Hydraulics Research, UK, funded by the Chinese government. The HR involvement in the work was funded by the Department of the Environment under Contract PECD 7/6/26-204/83, and the Paper is published with the agreement of the Department of the Environment.

Computational Procedure

24. It is assumed that D, D, g, s, ν and S_f are known

(a) Calculate

(b) Calculate

where D is D₃₅ (bed material)；then calculate the parameters n and A

(c) Calculate

(d) Calculate using

Calculate V^L using

Calculate using

(e) Calculate using

See paragraph 25.

Calculate V^U using

Calculate using

(f) If

If

25. Solution of equation (19)

If (-A) is denoted by x, then solving equation (19) is equivalent to solving the equation

where　α=0.07 and b=-(1.07-0.18lg D_gr) (F_fg-A).

26. The Authors recommend using the Newton-Raphson iteration method to solve this equation in practice. If x_n is an estimate of the solution of equation（24） then x_n+1 is a better estimate, where

References

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[2] Engelund F. Hydraulic resistance of alluvial streams[J]. J. Hydraul. Div. Am. Soc. Civ. Engrs, 1966, 92, HY2, 315-326；93, HY4, 287-296.

[3] Guy H. P. et al. Summary of alluvial channel data from flume experiments[R], 1956-1961. US Geological Survey, 1966, Professional Paper 462-I.

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[5] Einstein H A, Barbarossa N L. River channel roughness[J]. Trans. Am. Soc. Civ. Engrs, 1952, 117, 1121-1132.

[6] Raudkivi A J. Analysis of resistance in fluvial channels[J]. J. Hydraul. Div. Am. Soc. Civ. Engrs, 1967, 93, HY5, 73-84.

[7] Wang S Q et al. Experiments on alluvial friction[R]. Hydraulics Research, Wallingford, 1986, Report SR 83.

[8] Gilbert G K. The transportation of debris by running water[R]. US Geological Survey, 1914, Professional Paper 86.

[9] Simons D B, Richardson E V. A study of variables affecting flow characteristics and sediment transport in alluvial channels[C]. Proc of Federal Inter-Agency Sedimentation Conf, Washington, 1963.