PLANE RIVER RESISTANCE

Abstract: The study on the hydraulic resistance of open channel with the help of dimensionless Darcy-Weisbach resistance coefficient f is presented in this paper. The structure of Darcy-Weisbach coefficient is thoroughly examined in the case of the square law for resistance of the flow. As a result of the investigation, connection between the friction factor and the cross-sectional shape of the river was found. Introducing the coefficient of the river bed shape as a function of the wetted perimeter and the river width, we take into account the complicated cross-sectional shape of any nature stream. The resistance coefficient f calculated by using the suggested method agrees well with that measured in rivers and flumes.

Keywords: river, velocity profile, hydraulic friction, Darcy-Weisbach coefficient

1 INTRODUCTION

Prediction of discharge through channels with compound cross-sections (flood-plain channels) has proved to be difficult. The interaction of the river-bed with the flow under any given discharge of water and sediments results in definite values of width, depth and slope for the flow. Hydraulic resistance is one of the main parameters, which defines the equilibrium stage of the flow [3]. Although there exists considerable number of papers in which the flow resistance of rivers and channels with simple geometry was calculated, it is quite difficult to estimate the discharge capacity of open flows because of the variability of their geometrical parameters.

The estimation of resistance of nature flows is also difficult because of different roughness of the parts of the river-bed, for example for the main channel and the floodplain. Traditionally compound channels were treated by dividing the compound cross-section into relatively large homogeneous and easier to analyze sub-areas [9]. Typically, these divisions were made using straight vertical lines at the edges of the main channel. This treatment assumes that the shear stress along these internal boundaries vanishes and as a result, these surfaces are not included in the wetted perimeter. In this consideration one usually supposes that there is no interaction between the subdivided areas. In paper [7] it was shown that interaction of turbulent flows in the main channel and floodplain has considerable effect on the discharge of the flow.

The authors of [13] investigated the resistance of the flow together with the turbulent structure of the floodplain channel and found the subdivision surfaces of zero shear stress, which can be excluded from the wetted perimeter. The direction of division lines in this work depends on the ratio of widths of the main channel and the floodplain.

When we consider hydraulic friction of open channels, the most important problem is to find connection between the shear stress and parameters of the flow. This connection is usually described by the Chezy or Darcy-Weisbach formulas. As a rule, the former is used for open streams and the latter for flows in pipes. However, Darcy-Weisbach formula is more rigorous, and it contains the dimensionless friction factor f. Our first attempt to apply Darcy-Weisbach formula for evaluation of open flow hydraulic friction was made in work [10].

This paper describes calculation of the resistance of natural open channels by using of the dimensionless friction factor f , which has certain advantage in comparison with Chezy and Manning coefficients having fractional dimension. The coefficient f can be defined exactly for pipes, but its definition is more difficult for rivers because the shear stress on movable interface is not determined. First of all we consider the structure of shear stress for natural flows.

2 SHEAR STRESS

Let us consider an open channel in Cartesian coordinates x is the streamwise direction, z is the vertical and y is the transverse coordinates, B is the width of the flow (z=0 at the bottom). The mean shear stress on the bottom is:

where Dp is the pressure drop along the length l ; h(y) is the depth of the flow at a distance y from the bank;

is the cross-sectional area;

is the channel wetted perimeter; g is the gravitational acceleration; i is the longitudinal water surface slope; R=A/P is the hydraulic radius;

is the shear velocity,

, r is the fluid density. The pressure drop along the length l is:

were

is the average cross-sectional velocity squared and u is the local time averaged velocity of the flow,

is a characteristic dimension of the flow.

In fact, this is the definition for the coefficient of hydraulic resistance of the flow, which can be also obtained by dimensional analysis.

3 CROSS-SECTIONAL SHAPE

From (2) one can separate the coefficient which depends only on the geometry of the river bed. The cross-sectional shape can be taken into account with the help of coefficient m:

Then, for Darcy-Weisbach coefficient, we obtain instead of (2) the following expression

In other words, m is the coefficient of cross-sectional shape. The characteristic dimension for round pipes is diameter

=d=4R, yielding m=8. For open plane channel (B/h>>1) the characteristic dimension is its depth

=h =R, and we obtain m=2. If the ratio of the flow width B to the depth h is not big enough or the cross-sectional shape is compound, then it is necessary to calculate m for each of these cases separately.

To study the structure of the coefficient m for open channel, let us consider the friction factor for a given vertical line f(y). Let us assume that in expression (2)

=h(y) and instead of

we have the average squared velocity on the vertical line:

The expression for local hydraulic radius

was used in [3,9]. Then for a given vertical line the value of

can be obtained as:

The expression for the mean cross-sectional value of

can be obtained by substituting of averaged cross-sectional values into (4)

The mean cross-sectional shear stress

can be expressed as a function of the shear stress

at a distance y from the bank

Substituting mean shear stress

into (5) and comparing the obtained expression with (3) one derives the coefficient

, which introduces the dependence on the shape of cross-section

Now let us consider the flow with trapezoidal shape of cross-section. Let

be the maximum depth of the flow,

be the distances of the tops of the trapezoid from the bank and

be the width of the flow. Then we obtain for the wetted perimeter P

Precise calculation of the coefficient

requires the knowledge of the form of cross-section, otherwise we can estimate it as 2 for plane rivers and channels because in this case the ratio <<1 and we can neglect the contributions of order

in (6). Table 1 shows the geometry of cross-sections for rivers of different scales and for flows in laboratory flumes modeling nature stream with compound cross-section (in Table 1

is the depth averaged across the flow,

is the mean depth for the reach). The data of the Table 1 confirm the possibility of using the coefficient

for plane rivers. Expression (6) enables one to take into account the shape of cross-section for deep and narrow channels.

Crosssection		m		m
Kirhzach, =18 m, H=0.46 m
1	48.61	0.45	9.73	21.97	2.02
2	33.14	0.52	8.98	17.84	2.07
3	39.73	0.46	8.38	18.58	2.04
4	37.38	0.48	8.43	18.10	2.04
5	36.75	0.47	7.98	17.53	2.05
6	43.79	0.42	7.71	18.56	2.02
7	38.31	0.46	7.99	17.83	2.04
8	39.10	0.44	7.61	17.50	2.03
9	35.00	0.48	8.02	17.10	2.04
10	38.70	0.44	7.46	17.19	2.02
Moskva, =62 m, H=1.4 m
1	47.26	1.37	88.86	65.08	2.01
2	39.70	1.43	80.69	56.85	2.01
3	34.49	1.53	80.53	53.03	2.01
4	45.55	1.41	90.20	64.45	2.01
5	60.98	1.17	82.90	71.50	2.01
Volga
1	77.94	7.75	4680.5	605.5	2.01
2	85.06	7.12	4317.5	607.1	2.00
3	406.1	2.96	3546.3	1200.7	2.00
Flume [7]
1	58.48	0.11	0.723	6.60	2.03
Flume [9]
1	143.4	0.07	0.683	10.05	2.03

4 VELOCITY DISTRIBUTION AND SHEAR VELOCITY

To calculate the friction factor

with the help of (3), it is necessary to have information about the velocity field. In the absence of self-similarity on the global Reynolds number, and incomplete self-similarity on the local Reynolds number, the power law for the velocity distribution was obtained with the help of methods of dimensions [1].

The power law is known to describe the experimental velocity profiles very well [4, 5, 10, 12] the exponent n depending on the Reynolds number [11].

To describe the depth distribution of velocity let us use the power law in the form

where

is conventional surface velocity, calculated from experimental data. The exponent

is one of the most reliable parameters, which can be measured in nature streams. The results of measurements of

vary from 0.1 to 0.3 [3].

Although there is some variation of n for nature streams, the variation of

, where u=<u> is not so big: 0.38 ÷ 0.42. So we can use for <u> the velocity was measured on horizon

= 0.4, and hence the value of

in (3) can be obtained. Using

instead of

results in inaccuracy of order

. To calculate

with the help of (3) it is necessary to know

. There are several ways to define the shear velocity for an open flow which are summarized in [5]. For the steady open channel the shear velocity can be obtained by measuring the water surface slope, assuming that it is equal to the bottom slope, or from data for velocity profiles. The first method requires precise measurements of water surface slope and gives the averaged value of the shear velocity along the considerable length of channel. It is quite difficult to obtain the shear velocity from such measurements in nature. More accurate values of shear velocity for a concrete cross-section can be obtained from velocity profile. It is well known, that in the main part of the flow excluding the boundary layer, the velocity distribution can be described both by power law and logarithmic laws. The average velocity does not depend on the law. From logarithmic law we have

It can be shown that

. This constant, as

in (8), has the meaning of the surface velocity. Equating the expressions for C and

we find:

5 DARCY-WEISBACH RESISTANCE COEFFICIENT

Now we have the values for estimate the Darcy-Weisbach coefficient f at a given cross-section. Substituting (9) in (3) and replacing

one obtains the expression for friction factor

To analyze the expression (10) the friction factor was calculated for rivers listed in Table 1 and for rivers investigated in [8]. The results are shown in Table 2.

River		from (10)	from (3)
Kirhzach (1972)	2 ×	0.011	-
Kirhzach (1974)	3 ×	0.011	0.012
Atrisco canal, [8]	4 ×	0.009	0.011
Moskva	9 ×	0.007	-
Missouri, [8]	6 ×	0.006	0.006
Volga	1 ×	0.004	-
Mississippi, [8]	3 ×	0.003	0.003

For all rivers under investigation the ratio

is less than 0.03, so one can use

and formula (10) for calculation of

. The Reynolds numbers in the Table 2 were calculated by using mean cross-sectional velocity and mean depth of the rivers. The power exponent n was obtained from the measured profiles of velocity of the rivers. The friction factors obtained from (10) are the mean values for a cross-section or for several cross-sections, when there were enough data of measurements. This averaging was made to compare the results of calculations by (10) and by (3) in that cases, when the magnitude of the shear velocity was obtained by measuring the water surface slope. The comparison of the friction factors calculated by using (3) and (10) shows satisfactory agreement.

The variation of the Darcy-Weisbach coefficient on the Reynolds number is shown in Figure 1. This dependence can be described by

which is comparable with that obtained by Blasius [11] for pipes of circular cross-section

Taking into account the notations of this paper

. Analyzing Figure 1 one can conclude that (11) fits the data of measurements for Reynolds number two orders more than that used in pipe flows to obtain (12). The lack of data for natural streams prevents us from making a more general conclusion. To illustrate the tendency of the friction factor to decrease with the increase of the Reynolds number we present the dependence of

(Figure 1) obtained in trapezoidal flume [9] (

), which is described by equation

Fig.1 Dependence of the friction coefficient on Reynolds number:
data for different rivers (1) and for flumes [2] (2)

6 CONCLUSION

Investigation of shear stress in open streams shows that the use of Darcy-Weisbach coefficient for evaluation of hydraulic resistance of nature flows is quite adequate. Analysis of structure of the friction factor makes possible to take into account the cross-sectional shape of the flow. The expression for

enables one to calculate this coefficient for any complicated river bed shape. Parametric expressions for velocity distribution and shear velocity give a simple formula for calculation of Darcy-Weisbach coefficient for open streams. The values of friction factor calculated with this formula are in good correspondence with available values of

calculated by other methods. Because of insufficient data for hydraulic resistance of open channels it is difficult to make more precise test of suggested method, but it is worthwhile to note the restriction of it. The method under consideration is valid for estimation of hydraulic resistance in wide rivers for which pressure drop and water surface slope can not be neglected.

The author would like to thank Einar Tesaker for fruitful discussion and the comments that clarified the restriction the application of (6).

References

[1] Barenblatt G.(1978) Similarity, self-similarity, intermediate asymptotics. Leningrad.

[2] Cardoso A.H., Graf W.H., Gust G. (1989) J. Hydr. Res.v.27, No.5, pp. 603-610.

[4] Dolgopolova E.N.(1993) The dependence of Darcy-Weisbach resistance coefficient on the shape of river bed. Proc. of the Inter.Symp. “Runoff and sediment yield modeling”, Warsaw, Poland, Sept. 14-16, pp. 285-290.

[5] Griffith F.O. & C.M. Grimwood (1981) J.Hydr. Div. ASCE, v.107, No.3, p.345-361.

[8] McQuivey R.S.(1973) Summary of turbulence data from rivers conveyance channels and laboratory flumes. Geol. Surv. Prof. Paper No.802 B.

[10] Orlov A.S., Dolgopolova E.N., Debolsky V.K., Gubeladze D.O.(1988) Distribution of velocity and hydraulic resistance in open flows. J. Water resources, No.2, p. 80-86.

[11] Schlichting H. (1968) Boundary layer theory. McGraw Hill Book Co. New York.

[12] Tsujimoto T. & W.H. Graf (1988) Velocity distribution in a gravel bed flume. The 6-th Congress of IAHR, Kyoto, Japan.