SHEAR STRESS DISTRIBUTION IN RECTANGULAR CHANNELS

 

MARIA OLIVERO N. , JULIAN AGUIRRE-PE and ALIX MONCADA

 

Centro de Investigaciones Hidráulicas y de Mecánica de los Fluidos-CHIDRA

Universidad de Los Andes - Mérida -Venezuela - Fax: 58-74-402812

marial@ing.ula.ve; aguirrej@ing.ula.ve; alix@ing.ula.ve

 

 

ABSTRACT

With the purpose of obtaining shear stress distributions at the walls and at the bed of an open channel, experimental data available in literature, for smooth open channels and smooth rectangular ducts, are analysed and confronted. Similitude criteria between a free surface flow and a close non-pressurized flow are given. The influence of the aspect ratio on shear distribution is determined and different functional relationships are obtained. Mean shear stresses at the walls and at the bed are analysed and empirical relationships are obtained. Experimental equations show satisfactory correlation coefficients.

 

Keywords: Shear distribution, shear stresses, aspect ratio, rectangular channels, rectangular ducts.

 

INTRODUCTION

Einstein (1942) developed the first method to estimate mean shear stresses at the bed and at the walls in an open channel. Meyer-Peter and Müller (1948) presented a similar method to Einstein's, without making any reference to it. Taylor (1961) concluded that Einstein's method was appropriate to evaluate friction with an aspect ratio smaller than 0.5. Johnson (1942) admitted the convenience of using the friction logarithmic law with Einstein's method. Vanoni and Brooks (1957) refined Johnson's method and explained how to separate bed and wall shear stresses. The ASCE (1975) recommended the use of Vanoni and Brooks´s method, warning about possible deficiencies in the estimation of the friction factor. The ASCE advises about making direct measurements to obtain true values of shear stress at the bed and at the walls since there is not any experimental support in Vanoni and Brooks's method.

 

Olivero et al. (1992a) presented a model that, while keeping Einstein's hypothesis, may be solved in a general form for any type of rugosities at the bed and at the walls. Additionally, Olivero et al (1992b) developed a model which takes into account the velocity profile and allows to obtain a curve which separates the bed and wall corresponding areas. Even through promising results were obtained, considered aspect ratios were rather limited.

 

Different researches have considered the shear distribution problem by obtaining local shear stress as a fraction of the total shear stress and as a function of the aspect ratio B/y, where B is the channel width and y is the depth. (Rajaratnam and Muralidhar, 1969, Ghosh and Roy, 1971, y 1972, Knight and Macdonald, 1979a and 1979b, Knight, 1981, Knight and Demetriou, 1984, and Knight and Patel, 1985). These authors studied smooth channels, separatedly from rough channels.

In this paper, it is intended to verify that experimental data related to shear stress show similar results for channel and rectangular ducts if they keep geometric similitude. It is also intended to find the influence of the aspect ratio on shear stresses at the boundaries.

 

EXPERIMENTAL PROCEDURE AND ANALYSIS

It may be thought that a liquid flow in an open channel is similar to the flow in a duct of the same width and twice the depth of the channel, as may be observed in Fig. 1. Open channel free surface is related to the symmetry surface of the close conduit in which shear stress is zero. To obtain experimental information on velocity and shear stress distributions an air duct was designed whose rectangular section had a width B = 0.80 m and a high 2y = 0.20 m. A control gate was located upstream of the 10 m long conduit. The working section was located downstream, where the boundary layer was totally developed. Shear stresses were determined from experimental velocity profiles obtained from 297 data at each section. To extend experimental information, data by Knight and Patel (1985) were included. They worked on a smooth rectangular duct of variable section and 9.5 m long.

 

For evaluation of channel flow, data by Knight and Demetriou (1984) and by Rajaratnam and Muralidhar (1969) were employed. They work on rectangular channels as indicated in Fig. 1.

 

RESULTS

 

BED AND WALL SHEAR STRESSES RATIOS FOR SMOOTH CHANNEL

Rajaratnam and Muralidhar (1969) found it possible to obtain a function relating shear stress ratios and aspect ratios B/y in open channels. In Fig. 2 shear stress at the wall t_w divided by shear stress at the bed t_b as a function of the aspect ratio, for channels and ducts is given. It is observed that all data may be represented by the curve

 

(1)

 

 

It is evident that the aspect ratio determines the ratio between shear stresses at the walls and shear stresses at the bed, both in open channels and in ducts in a similar way.

 

RELATIONSHIP BETWEEN MEAN BED AND WALL SHEAR STREESSES FOR SMOOTH CHANNELS

Models to estimate bed and wall shear stresses assume some basic hypothesis. It must be taken into account that the mean channel shear stress is related to the bed and wall shear stresses properly applied to the wet perimeter.

 

Figs. 3 and 4 show the ratio between wall shear stresses and bed shear stresses, related to t_m, or mean shear value in open channels and ducts, It is found that the best fit equations, are.

 

(2)

 

(3)

 

An inspection of Figs. 3 and 4 indicates that t_w approaches t_m for B/y smaller than 2 and that t_B approaches t_m for B/y larger than 2. Here again, open channels and ducts of similar geometry present similar shear distributions, even though there is larger dispersion for data corresponding to open channels. Aspect ratio clearly determines shear stress ratios for open channels and ducts.

 

Fig. 3. Wall and mean shear stress ratio for smooth open channels and ducts.

Fig. 4. Bed and mean shear stress ratio for smooth open channels and ducts.

 

SHEAR STRESS DISTRIBUTION IN SMOOTH CHANNELS

Several authors have expressed shear stresses through corresponding mean shear forces of the channel and their shear forces SP_w and SP_b, for the walls and the bottom respectively, as a function of the aspect ratio B/y. Knight and Demetriou (1984), based on a previous analysis by Knight (1981), proposed experimental relationships for shear stresses distributions. But after confrontation with open channel and duct data by other authors, it results evident that their relationships are adequate only for their data.

 

Figs. 5 and 6 show dimensionless best fit logarithmic equations for the shear stresses at the walls and at the bed, where the depth is used for the mean shear stress gyS. Fig. 5 shows that experimental data fall below the best fit equation, for B/y<1 and B/y>10, and above the best fit equation for 1<B/y<10. It may be observed that dimensionless bed shear stress exhibits two tails that seem to approach 0 to the left and 1 to the right. Indeed, if the aspect ratio approaches 0 then their shear stress must also approach 0. For aspect ratios larger than 20 the bed shear stress approaches the values corresponding to a two dimensional channel.

 

(4)

 

(5)

 

Figs. 7 and 8 show the bed and wall dimensionless shear stresses, here the hydraulic radius is used for the mean shear stress gRS. The corresponding logarithmic relationships are given as

 

(6)

 

(7)

 

From Eqs. 6 and 7, it is evident that the wall shear stress becomes equal to the bed shear stress for and aspect ratio B/y=0

 

 

SHEAR STRESS DISTRIBUTION IN CHANNELS WITH ROUGH BED AND SMOOTH WALLS

Figs. 9 and 10 show bed and wall shear stresses in dimensionless form, by the use of gyS, and Figs. 11 and 12 show dimensionless stresses by the use gRS, with the data from Knight et al. (1981) and Ghosh and Roy (1972). Dimensionless shear stress equations for smooth channel have been included in Figs. 9 to 12. Smooth channels curve is an upper limit to experimental shear stresses at the bed and it is lower limit to experimental shear stresses at the bed. Relative high dispersion is observed for rough bed channels.

 

 

CONCLUSIONS

In order to determine shear stresses at the bed and at the walls of smooth rectangular channels, experimental data available in the literature have been analysed both for smooth channels and for ducts. Similitude between free surface and close duct flows has been established. No clear trend in shear distribution has been observed for rough bed channels.

 

Functional relationships between shear stress at the walls and at the bed have been analysed as a function of mean shear stresses. Power expressions with correlation degrees higher than 0.90 have been obtained.

Logarithmic experimental relationships for a wide data field are given in Eqs. 4, 5, 6 and 7 for the dimensionless shear stresses. Their correlation degrees are higher than 0.80 both for ducts and for open channel flow.

 

ACKNOWLEDGEMENTS

The authors acknowledge the support given by the Consejo de Desarrollo Científico, Humanístico y Tecnológico, Universidad de Los Andes, through Project I-563-96-02-B.

 

REFERENCES

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