Analytic method of hydraulic calculation of eddy shaft spillways

Abstract

An eddy shaft spillway is one of the common types of high-head hydraulic structures. Many such spillways have been constructed in Italy [11]. In the USSR the spillway of the Medeo mudflow-protection dam is also an eddy one [1].An eddy shaft spillway ( Fig. 1) consists of an inlet channel, spiral chamber having a box section [2], vertical shaft, bend, and outlet tunnel. The flow, entering the shaft from the spiral chamber, acquires rotation, as a result of which it is pressed against the shaft walls. Here occurs an excess pressure preventing the development of cavitation. Despite the fact that the diameter of the vertical shaft (2R) of the eddy spillway is greater than that of a circular shaft spillway with the same discharge capacity and the design of the portal is more complex, their hydraulic advantages --stable operation in the entire range of variation of the discharge from zero to the maximum --make it possible in a number of cases to give preference to a spillway of this type [2].Several methods of hydraulic calculation of eddy shaft spillways, empirical [2, 3, 12] and analytic [13, 14, 15], have been developed. Without dwelling on the first, since they have a limited applicability, we will compare the proposed analytic method examined below with the known ones and with results of experimental investigations [4, 11, 12].To determine the discharge capacity of an eddy shaft spillway, we will write the system of equations of conservation of energy, momentum, and continuity for sections I--I and II--II in Fig. 1. We will neglect the hydraulic losses due to friction in this zone. The distribution of the characteristics of the flow in the cross sections is assumed axisymmetric and in conformity with the laws of conservation of specific energy and momentum [2, 13, 15] re p where p is the water density; z 0 is the length of the zone in which equalization of the velocity distributions of the flow after entering the vertical shaft occurs and according to the data of [5] is assumed equal to 4(R --re); r e is the radius of the air core in section II--II; h is the depth of the flow in section I--I; 6 is the height of the edge at the inlet to the vertical shaft; v is the flow velocity in section I--I; Q is the discharge; v z, v r are the vertical and circumferential components of the flow velocity in section II--II, ~ is the hydraulic resistance coefficient of the edge at the inlet to the shaft, ~ = 0.11 [5]; P is the pressure in the flow in section II--II as a function of r, P = pf(vr2/r ) dr, A is the distance between the axes of the shaft and inlet channel; b is re the width of the inlet channel; p is the discharge coefficient of the eddy shaft spillway; e is the coefficient of contraction of the now (Zig. 2a [51).System (1) has the solution V ~ 4t?

Hydrodynamics and stability of swirling flows in channels with solid walls

“…The values of the coefficients in expressions ( 9) -( 10) are equal to 1, 0, 0.419, 0, 14, 0.75. The finite-difference iterative method was used to solve the initial boundary value problem ( 1)-( 4), ( 5)- (10). The direct method of incomplete reduction was applied to solve the Poisson equation (1).…”

“…Swirling flows are widely used in hydraulic engineering construction. In particular, these are vortex spillways, flow energy damping devices [10], counter vortex aerators for oxygenation of ponds [11,12]. A large number of technical devices using the properties of oppositely rotating flows are considered in [13].…”