Analysis of Formulas of the Flow Rate Coefficient of a Broad-Crested Spillway
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Vol. 54, No. 3, September, 2020
ANALYSIS OF FORMULAS OF THE FLOW RATE COEFFICIENT OF A BROAD-CRESTED SPILLWAY A. R. Gur’ev1 and B. A. Khaek2 Translated from Gidrotekhnicheskoe Stroitel’stvo, No. 3, March 2020, pp. 41 – 48.
An analysis is conducted of the calculation of a broad-crested spillway and of the errors obtained using the equation of the energy of slowly varying movement of liquid in this calculation. Keywords: broad-crested spillway; equation of slowly-changing liquid flow; flow rate coefficient; curvature; current lines convergence.
ficient must equal m0 = 0.385. Figure 1, taken from R. R. Chugaev’s textbook, shows a schematic diagram [12]. The equation of continuity of the stream of thickness h at the threshold, which corresponds to k = h/H, was the initial one for calculating the flow density q under pressure at the threshold H:
The broad-crested spillway can be designated the leader in hydraulic engineering construction. This is used on lowhead hydraulic units, which number tens of thousands throughout the world. In modern hydraulics [1 – 8], two markers of a structure that passes water as a “broad-crested spillway” are defined: — existence of a horizontal threshold; — relative length of the threshold 2(3) < ä/H < 10(15). In the standards document [8], art. 3.6.32 provides the following definition for “a broad-crested spillway”: “A spillway and the conditions of water spillover, through which it is defined by the current along its horizontal surface. Spillways, the size of horizontal surface of which in the current direction, as a rule, is more than two and less than eight head over the crest, pertain to this type.” Consequently, the class of broad-crested spillways can include free-flow Venturi trays, and spillways with a trapezoidal cross-section, and Parshall-type trays, as well as other similar pass-through structures having the specified indicators. It follows from this, that prismatic broad-crested spillways are only a particular case of a wide class of spillways that can be considered as broad-crested spillways. Nevertheless, for the more than 170-year study of these spillways, scientists did not consider this fact, and explicitly studied only prismatic spillways. The first theoretical solution of the flow rate coefficient was given by Bélanger in 1818 [9 – 11] for a spillway with vertical parallel walls according to which the flow rate coef1 2
q = h 2 g( H - h ) = k 1 - k 2 gH 3 ;
(1)
Analysis of Eq. (1), done by Bélanger, showed that for k = 0 the specific flow rate q = 0, as for k = 1. Consequently, there is a depth h at which (1) has a maximum. Solving (1) to determine an extremum, Bélanger obtained k = 2/3, and with this value the flow rate coefficient m0 = k 1 - k
(2)
has a maximum m0 =
2 2 2 1- = = 0.385; q = 0.385 2 gH 3 . 3 3 3 3
Bélanger’s graph of the dependence q = f(k) is shown in Fig. 1 to the right of the spillway scheme. At the same time acceptance of laminar flow on the threshold contradicts the theory of non-uniform motion in a channel with a horizontal bottom. For e
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