(1)
Salah
Kouchakzadeh and
Alireza Vatankhah
Respectively, Assistant Prof. and former graduate student, Irrigation and Reclamation Engineering Dept., University of Tehran
P.O. Box 4111, Karaj, Iran, 31587-11167.
E-mail: skzadeh@chamran.ut.ac.ir
Abstract: The perturbation solution was previously applied for solving the ordinary non-linear differential equations of the spatially varied flows with increasing discharge. The momentum correction coefficient, b, was taken as unity in previous studies which result in highly approximate solutions. In this study, the momentum correction coefficient is considered in the governing equation and a modified perturbation solution for the spatially varied flow iss obtained. Comparison between computed and observed water surface profiles indicated that a mean value of b =1.52 could be obtained for this flow condition. The computed water surface profiles using the proposed mean value reasonably fits the observed data. Consequently, the use of the proposed mean value is recommended for the design purposes.
Flows having varying discharge in the direction of the flows are classified as spatially varied flows. According to the increase or decrease of the discharge along the channel, spatially varied flows with increasing or decreasing discharge are recognized, respectively. This type of flow is generally observed in many circumstances in natural flows and in many man-made hydraulic structures (Figure 1). It seems that Hinds (1926) was the first who introduced a reliable form of the governing equation for the spatially varied flow with increasing discharge based on the main physical principles.
Following Hinds original work, many researchers introduced modified forms of the governing equation (Li, 1955; Meyer-Peter & Faver, 1934). Applying the momentum principles in the direction of the flow yields the governing deferential equation of the water surface profile for the spatially varied flow. The governing equation is given (Chow, 1959; Henderson, 1966)
(1)
where dy/dx=slope of the water surface, So=longitudinal bed slope, Sf=friction slope, Q=discharge at a specified point x from the upstream end (figure 2), q*= the discharge of unit length of the channel and is given by q*=dQ/dx=Qo/L, Qo=total discharge, g=acceleration due to gravity, L=total length of the channel, A= crass-sectional area at the specified point x, T= channel width at the water surface, b =momentum correction factor due to non-uniformity in flow velocity distribution in the cross-section, D=A/T=hydraulic depth at specified point x.
The design of a channel having spatially varied flow with increasing discharge depends on the water surface profile within the channel, i.e. accurate determination of the water surface profile aids channel design and guarantees the anticipated hydraulic performance of the side weir. Different solution methods were proposed for the governing dynamic equation, the perturbation solution applied by Gill (1977) will be considered in this paper. Because of the high non-uniformity in the flow velocity distribution across the channel, researchers emphasized on applying the momentum correction factor for practical purposes regardless of the solution techniques used. Data indicating the value of b in spatially varied flow with increasing discharge are scarce, hence, engineers b =1 in the design process which can result in high approximation in the prediction of the water surface profile within the channel. In this paper, by incorporating b in the governing equation, a modified perturbation solution was obtained. Also based on Gill’s data the coefficient b was so determined that the best concordance between the observed and computed water surface profile is obtained for each set of data.
The governing equation was derived based on the assumption that the flow interring the flume produces no momentum component in the flow direction. Otherwise an additional term would be appear in the governing equation due to the mentioned momentum component (Fox and Goodwill, 1970). Also, the friction slope is the indication of the frictional head loss. In this paper the frictional head loss in a wide rectangular channel was evaluated using Darcy-weisbach formula. In order to present the governing equation in non‑dimensional form, the parameters of equation (1) were normalized as follows:
where Fo=Froud number at the down stream end of the channel, b=channel width, yo=flow depth at the down stream end. Using above parameters equation (1) is presented as:
(2)
According to Darcy-weisbach
formula, Sf in a wide
rectangular channel is evaluated by
which
when substituted in equation (2) yields:
(3)
where Gf=fL/yo and f= Darcy weisbach coefficient.
Solution to the governing equation for subcritical flows
The subscript (*) in equation (3) indicates non-dimensional values. For the sake of abbreviation the subscript will be dropped from now on, therefore, equation (3) becomes:
(4)
Considering Fo2 as a perturbation parameter, Equation (4) could be solved using the following series (Gill, 1977):
where y1, y2, y3,… are functions of x and are to be evaluated.
Applying the boundary values y1=1 (i.e., y*=y/yo=1) at x=1 (i.e. x*=x/L=1) yields y1=y2=y3=…=0. For the first order approximation, the summation of all F02 terms (i.e. terms having Fo with the power of 3 and more are ignored) yields:
Considering k=G/Fo2 which will temporarily enter the mathematical operation, hence,
substituting the above values in equation (4) and applying the boundary values yields:
(5)
similarly, summing all the Fo4 terms (i.e. terms with value of 5 and higher are ignored) yields:
substituting the above values in equation (4) and letting dy1/dx=k-2bx-Gf x2/8 yields:
substituting
y1 and dy1/dx
values in the above equation, integrating it and evaluating the constant of the
integration at the boundary conditions y*=1,
x=1 which yields y2=0, we obtain equation (6)
(6)
Therefore, based on the second order approximation, the flow depth at each point could be determined by substituting equations (5) and (6) in the following equation.
letting x=0, the relative depth yu/yo at the upstream end could be evaluated by the following equation:
(7)
or
(8)
where
,
,
Providing that flow depths at both upstream and down stream ends are available for different discharges, the coefficient b could be evaluated. Note that for evaluating Gf it is necessary that the Darcy-weisbach coefficient, f, be known for the range of steady uniform flows. Gill evaluated the Darcy-weisbach coefficient and obtained the value 0.028. It is evident that the value of f might be considered constant only in hydraulically rough flow conditions. In smooth flow condition the value of f depends on the Reynolds number. In this study the value of f was considered constant and this assumption will be valid as long as the variation of f is insignificant. It is worth mentioning that some of the differences between observed and computed values might be attributed to the above assumption.
Gill’s data were obtained in a rectangular tilting laboratory flume, 5000 mm long, 76.2 mm wide, and 250 mm deep. Further information regarding the experimental setup was presented in Gill (1977). Nineteen sets of Gill’s experimental data were used to evaluate b in equation (8). Observed water surface profiles along with the computed ones using three values of b, i.e., b =1, bavg=1.52, and the best value of b that fit each individual set of data were depicted in Figure (3). It is clearly observed that the accuracy of the perturbation solution results, like other solution methods, depends on the selected values of b. As it indicated oFigure (3), applying b =1 yields water surface profiles with significant differences with the observed ones. As it emphasized by researchers, to obtain satisfactory results for practical purposes, it is necessary to apply reasonable value for the momentum correction factor.
The observed variations of b for these set of data were in the range 1.19 and 2.72. Although the variation range seems rather wide, careful attention reveals that about 90% of the computed values are bellow 1.66, which seems a reasonable value. Incorporating b in the water surface profile calculation, produced results with no appreciable differences with the observed values. Therefore, it could be concluded that an average value of 1.5 might be used in design purposes.
The velocity distribution is highly non-uniform in spatially varied flows with increasing discharge, hence it is necessary to evaluate the momentum correction coefficient and incorporate it in the solution of the governing equation in order to obtain more realistic results. In this paper by considering b in the governing equation, the perturbation solution method was modified so that a new relation for evaluating the momentum correction factor was obtained. With the aid of the mentioned relation and Gill’s experimental data, the value of b for each test was obtained. Comparison between observed and computed water surface profiles indicated that applying b =1 results in values lower than the observed ones. This results show that it is necessary to incorporate the momentum correction factor in the perturbation solution, like other solution methods. Although it seems that the range of the computed b values is rather wide, applying the mean values resulted in satisfactory result, i.e. bavg good agreement between the computed and the observed results. Therefore, it is recommended that this mean value (b =1.5) be used in the design purposes.
Acknowledgement
The authors wold like to thank the vice chancellor for research of Tehran University for supporting the research financially.
References
Camp, T. R. (1940). “Lateral spillway channels.” Trans., ASCE, 105, 606-607.
Chow, V.T. (1959). “Open channel hydraulics.” McGraw-Hill Co., New York.
Fox, J.A. and Goodwill I.M.(1970). “Spatially varied flow in open channel.” Proc. Inst. Civ. Engrs., London, 46, Paper No 7289, 311-325.
Gill, M. A.(1977). “Perturbation solution of spatially varied flow in open channels.” J. Hydr. Res., IAHR, 15(4)4, 337-350.
Henderson, F.M. (1966). “Open channel flow.” MacMillan Co., New York.
Hinds, J.(1926). “Side channel spillways.” Transactions, ASCE, 89, 881-938.
LI, Wen-Hsiung (1955). “Open channels with non-uniform discharge.” Trans., ASCE, 120, 255-280.
Meyer-Peter, E. and Favre H.(1934). “Analysis of Boulder dam spillways made by Swiss laboratory.” Engineering News-Record, 113(17), 520-522.
Fig. 2 Comparison between results of equations (2) and (12) for prismatic channel
Fig. 3 Comparison between results of equations (2) and (12) for non-prismatic channel