Issue 
La Houille Blanche
Number 3, Avril 1967



Page(s)  249  256  
DOI  https://doi.org/10.1051/lhb/1967016  
Published online  24 March 2010 
Loi hauteurdébit d'un déversoir en mince paroi fonction de la charge totale H et valable dans une large plage d'utilisation
The head/discharge relationship for a sharpedged weir, based on total head and with a wide range of application
^{1}
MaitreAssistant à la Faculté des Sciences de Toulouse.
^{2}
Institut de Mécanique des Fluides de Toulouse.
^{3}
Ingénieur en chef des Services Techniques de la Ville de Paris.
^{4}
Ingénieur des Ponts et Chaussées.
^{5}
Professeur à la Faculté des Sciences de Toulouse ; Directeur technique de l'Institut de Mécanique des Fluides de Toulouse.
The H/Q relationship for a sharpedged weir without lateral contractions is generally used in its form (1), where B is the channel and weir width (both the same), H is the depth of water on the sill, and m is a dimensionless factor depending on H and the weir characteristics by various relationships proposed by different authorities. The Q(H) formula referred to is obtained from the Weisbach and Francis formulae (18411852), which lead to expression (2). Expansion of the first bracket of the latter to the third term produces expression (3). If the dynamic head V20/2 g corresponding to the approach velocity V0 is small compared to H, this expansion reduces to its first term. In some cases, however, the dimensions of discharge measurement channels for a sharpedged weir give mean approach velocities for which the dynamic head can no longer he neglected. An attempts as been made, therefore, to establish a more general relationship allowing explicity for dynamic head. Systematic tests have been carried out in a glasswalled flume 8 metres long by 0.30 metre wide with a sharpedged weir perpendicular to the flow, for seven different well heights (P = 0.10 m, 0.15 m, 0.20 m, 0.25 m, 0.30 m, 0.40 m and 0.50 m) and values of the dimensionless parameter (H + Vo2/2 g) /P =H/P ranging from 0.03 to 2.5. Five different values of the discharge coefficient m were found for each test by entering the experimental values found for QE and HE into the relationships listed below. All but one of the relationships chosen considered dynamic head. These relationships were the following : a) The WeisbachFrancis relationships, i.e. formula (2) mentioned above, which gave m ; b) Relationship (3) giving m' ; c) The above relationship only taken as far as the second terme, i.e. (5). This gave m1 ; d) A relationship one might call the 'simplified weisbachFrancis relationship', which is obtained by ignoring the second term in the square brackets of formula (2) (V02/2 g)3/2, and which can therefore be written in the form (6). It gave m2 ; e) The usual relationship (1) referred to above, which gave m3. It is seen that this relationship ignores the firstpower V20/2 g term, so that it is less precise than (6) which neglects a Vo2/2g terme to the power of 3/2. The results are as follows : 1) There is no significant difference between m and m' from (a) and (b) ; 2) Relationship (d) gives extremely small differences with respect to formula (a) ( 0.5 % to  0.7 %) where H/P ≤ 1, and very small differences where H/P > 1 (1.6 % for H/P = 1.25 and  3 % for H/P = 2). It also requires the least calculation time ; 3) Relationship (e) gives differences of the same order as formula (d) and almost identical value for m, but it requires longer calculation ; 4) Relationship (e) gives major differences, from + 2.5 % for H/P = 0.45 to + 16 % for H/P = 2.5 ; 5) m, m1 or m2 vary little with test conditions, but m3 varies a lot (20 % relative error), showing the importance of the Vo2/2g in the overflow. Relationship (6) of paragraph (d) has therefore been retained for the calculations. Values of m were thus determined for a very considerable number of tests. 121 values of m = f (H,QE) were obtained, all in the 0.03 ≤ H/P ≤ 2.5 range. These experimental values are shown on the dimensionless, [m, H/P] plot on Plate l and are seen to lie in a cluster of points grouped close to an average straight line, which was determined by the least squares approximate interpolation method, and the equation of which is (8). The proposed relationship is then given by (11), which in no way replaces existing formulae but is a useful complement to them as its validity  which has been checked experimentallyextends to a much range, i.e. H/P ≤ 2.5. This formula was checked on four weirs with widely differing characteristics in order to establish what degree of approximation it could ensure. The discrepancy between calculated discharge and the very accurate calibration data available for these weirs was generally found to be less than 1 %. The results obtained for one of these weirs (B = 0.40m, P = 0.299 m) by Bazin's (1888), Rehbock's (1929), KindsvaterCarter's (1959) formulae, the proposed formula and by volumetric calibration are shown on Plate II. The proposed formula and Rehbock's show particularly close agreement with the experimental calibration data.
© Société Hydrotechnique de France, 1967
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