Hydraulics of the subcritical Ventury channels

The purpose of the research is to develop and verify the method of hydraulic calculation of the Venturi channel in the flooded outflow mode. The criterion that separates the flooded and free flow regimes is the ratio of the flow depth in the throat of the Venturi channel hm to the critical depth hc exceeding one (hm/hc > 1). The article theoretically and experimentally substantiates a method for determining the flow rate in a flooded Venturi canal using two control cross-sections, one of which is in the upper basin of the canal, the other in its throat. The differential equation for an uneven flow in a non-prismatic channel was obtained in the form (13). The analysis showed that in a flooded Venturi channel with a subcritical flow at Froude numbers less than one (Fr <1), the denominator of equation (13) is always greater than zero; therefore, at the entrance to the channel in the region of constriction, when db/dl < 0, the depth is downstream decreases (dh/dl < 0), and in the lower diffuser, where db/dl > 0, the flow depth increases (dh/dl > 0). Thus, the curve of the flow surface has a dip in the throat of the flooded Venturi channel. Integration of differential equation (13) made it possible to obtain the function of changing the Froude numbers in the flow along the length of the flooded Venturi channel (16), and then, using relations (17), to determine the depths and flow velocities. The verification showed that the calculated data are in good agreement with the experimental ones. The deviations between calculation and experiment do not exceed ±3%. This makes it possible to recommend the outlined methods for application in engineering practice.


Introduction
Let us consider the method of hydraulic calculation of the Venturi flowmeter channel in a submerged water outflow (Figure 1), when the depth of the flow in the throat of the channel hm is greater than the critical hc. According to the current State Standard of the Russian Federation MИ 2406-97, subcritical flow regimes in water measuring channels are unacceptable. However, in practice, every year the conditions of free non-flooded outflow are violated more and more often [1]. This is directly related to global climate changes in the world with a tendency for the volume and intensity of floods and storm runoffs to increase, raising the level in front of water retention dams and in overflowing water bodies and rivers receiving extreme runoff.
The relevance of the issue is determined by the fact that if the free flow regimes in the Venturi flowmeter channel of critical depth are regulated by the State Standard of the Russian Federation MИ 2406-97, and the methods for calculating the flow in this case are well developed and have been tested by many years of practice [2 -10], then the flooded outflow regimes are not only not regulated by the IOP Publishing doi: 10.1088/1757-899X/1159/1/012014 2 State Standard, but practically not investigated and methods of their calculation are absent [1 and 11]. The incorrectness of the operational accounting of extreme flows at water measuring posts is often revealed with a significant delay, when, at best, the measurement results begin to raise doubts among the service personnel, at worst, when the water retaining structures lost their stability and begin to erode with all the ensuing, including catastrophic, consequences. The purposes of the research are to determine the criterion separating the free and flooded outflow regimes in the Venturi water-measuring channel of critical depth, to develop and verify the method of hydraulic calculation of the flow in the Venturi channel in the mode of submerged outflow.

Flow rate in the subcritical Venturi channel
We will assume that, due to the smooth configuration of the vertical walls in the Venturi channel, there are no vortex zones and, as a result, there are no local hydraulic losses, and, given the small length of the working section from upper pool (cross-section 0 in Figure 1) to the throat, where the flow depth hm is minimal (section 1), we assume that the hydraulic losses along the length are also practically negligible here. Overall, this makes it possible to consider the flow-measuring Venturi flume as an ideal channel not having hydraulic losses, and to write the Bernoulli equation for crosssections 0 and 1 in the form.
where V and H are the mean flow velocity and depth in the upper pool at cross-section 0; Vm and hm is the velocity and depth of the flow in the throat in cross-section 1, with hm greater than the critical  Figure 1); α is the Coriolis (Saint -Venant) coefficient; and g is the acceleration due to gravity, g = 9.81 m/sec 2 . But the flow velocities at cross-sections 0 and 1 are determined through the flow rate (Q) by the equalities here B and bm are the widths, respectively, of the upper pool and the throat of the Venturi channel. Then Bernoulli's equation (1), taking into account equalities (2), should be rewritten as as a result we get Formula (4) determines the flow rate of the flow following through the Venturi channel in both free and flooded outflows. To determine the flow rate through the channel, it is required to know the geometric dimensions of the channel (B and bm) and depths in the upper pool H and in the throat of the channel hm.
Formula (4) can be written in general form is a variable flow rate coefficient depending on the relations (3). The criterion for evaluating the separation of the free and flooded outflow modes in the Venturi water measuring channel is the ratio of the depth in the throat of the channel hm to the critical depth With a flooded outflow, this ratio should be greater than one 1 > In the latter case, formulas (4), (5) and (6) are reduced to formulas for calculating the flow rates of the Venturi channels of the critical depth: (15) from paper [9] or (9) and (10) from paper [10].

Flow motion in the subcritical Venturi channel
The differential equation of steady-state smoothly varying fluid motion in a non-prismatic Venturi channel with a horizontal bottom, vertical walls and negligible hydraulic losses has the form Since the depth-independent partial derivative of the function of changing the cross-sectional area of the flow for a channel of variable width b and with vertical walls is equal to then equation (9) can be rewritten in the form Let us now turn to an analysis of formula (10), which reflects the hydraulics of flows in nonprismatic channels with vertical walls and a horizontal bottom, to which the Venturi channels belong.
If the Froude number is written as then Let us bring equation (10) to the form The analysis shows: at the inlet portion of the Venturi channel, where there is a calm subcritical flow with Froude numbers less than unity (Fr < 1) the denominator of Eq. (13) is greater than zero, therefore, in this area the narrowing of the channel when db/dl < 0, causes a decrease in the depth of the stream (dh/dl < 0), a the expansion of the channel (db/dl > 0) causes an increase in the depth (dh/dl > 0); in the lower diffuser of the channel, where a calm subcritical flow with Froude numbers less than one (Fr < 1) is also observed, the denominator of the Eq. (13) is also greater than zero, the depth of such a flow increases (dh/dl <0) with expansion of the channel (db/dl > 0), and a decrease in depth (dh/dl > 0) occurs with its narrowing (db/dl < 0). Thus, the curve of the free surface of the flow has a dip in the throat of the flooded Venturi channel. In this case, in the section of the maximum dip of the free surface of the flow, its depth hm does not fall below the critical one hc according to (7) and (8).
Let's express the derivative dh/dl in terms of the Froude number. According to equalities (11) and Substituting the obtained value of the dh/dl derivative into expression (13), we come to equation Integrating (14), we find The constant of integration C is found from the following condition. Since within the throat the flow passes through the cross-section with minimum depth hm (cf. Figure 1), width bm, and area ω = bmhm, where, following (12) where the width b along the length of the Venturi channel varies from bm to B.
With respect to the sought value αFr, equality (15) is reduced to the cubic Cardano equation [12], which has a trigonometric solution satisfying the condition 0 < αFr <1 .
Having determined the Froude numbers depending on the variable width b of the Venturi channel, the depths h and the flow velocity V are then calculated Let us verification how adequately the computed formulas obtained correspond to actually observed flows. To do this, let's compare the computed values with the analogous measured characteristics of the model of Venturi channel shown in Figure 1.  All the equipment is certified consistent with the Russian Laws.

Methods and results of the experimental research
Before performing experimental studies the hydraulic flume НМ 162 was set in horizontal position with a bottom slope i = 0 and the model of Venturi channel НМ 162.51 placed in the middle part of the flume (cf. Figure 1). The digital level gauge НМ 162.91 was mounted on instrument carriage and its zero adjusted relative to the floor of the model НМ 162.51. The HM 162.12 specialized software package was loaded into the control computer in order to record the discharge, which was measured in course of the study by an electromagnetic flowmeter Promag 10 D.
On the control panel of the laboratory flume HM 162 operation mode or on the control computer, the flow rate was specified and the working pump SHS4 80-200/40/P of the flume was turned on.
Following stabilization of the operating mode of the flume (stabilization time 5 minutes), the flow rate values measured by the Promag 10D electromagnetic flow meter using the НМ 162.12 software were written to the disk of the control computer into the generated data file; the flow measurement time was set 200 seconds with an interval between measurements of 1 second; in the process of processing, the obtained data were transferred to an Excel file, in which the average value of the flow rate (mathematical expectation) was calculated where k is the sample size, k = 200; Qj is j-th element of the sample. The values of Q0, σ and flow depth in the upstream pool H (cf. Figure 1) are written in the title lines of Table 1. The same Table 1 Table 1.
Next, the discharge transmitted through the flume was changed with preliminarily selected step ∆Q. The total being investigated 9 regimes of transmission discharge Q0 from 110.37 to 69.92 m 2 /h. All measured values were written in units of dimensions of the measuring instruments. The results of comparing the computational methods described in subsections 2.1 and 2.2 are summarized in Table 2 and presented graphically in Figures 2 and 3. According to the data in Table 2, it can be seen that the discrepancies (∆) between the calculated (Q) and measured (Q0) in the process of hydraulic studies, the values of flow rates were within the limit of ±3%, and the normed standard deviation is σ = 0.01324. Figure 2 shows the graphs of changes in Froude numbers (αFr), as well as relative depths (h/H0) and velocities ( 0 2gH V ) in the flow along the length of the model Veturi channel when it is flooded in the downstream pool (h0). The regime with a flow rate Q0 = 110.37 m 3 /h at a tailwater level h0 = 218.26 mm is considered. Hydrodynamic head here was calculated as the sum When calculating the Froude numbers in the diffuser, hydraulic losses were taken into account.

Conclusion
According to the State Standard of the Russian Federation MИ 2406-97, flooded modes in water measuring channels are unacceptable. In practice, the conditions of free flow are often violated due to the increase in the intensity of effluents in the context of global climate change. The incorrectness of the operational accounting of extreme flows at water measuring posts is revealed with a significant delay. The purpose of the research is to determine the criterion separating the free and flooded outflow modes in the Venturi channel, and to develop and verify the method for its hydraulic calculation in case of flooded outflows. The criterion for separating the free and flooded outflow modes in the Venturi water-measuring channel is the ratio of the flow depth in the channel throat hm to the critical depth hc. With flooded outflow, this ratio should be greater than one (hm/hc > 1).
The method for determining the flow rate in the flooded Venturi canal using two control crosssections, one of which is located in the upper pool of the channel, the other in its throat, has been theoretically and experimentally substantiated.
A differential equation for an uneven flow in a non-prismatic channel is obtained in the form (13). An analysis of this equation showed that in a flooded Venturi channel with a subcritical flow at Froude numbers less than one (Fr < 1), the denominator of equation (13) is always greater than zero; therefore, at the channel inlet in the narrowing region, when db/dl < 0, the flow depth is down decreases downstream (dh/dl < 0), and in the downstream diffuser, where the channel expands (db/dl > 0), the flow depth increases (dh/dl > 0). Thus, the curve of the flow surface has a dip in the throat of the flooded Venturi channel. In this case, in the section of the maximum dip of the free surface, the flow depth hm does not fall below the critical value hc.
The integration of the differential equation (13) was carried out, which made it possible to obtain the function of changing the Froude numbers in the flow along the length of the flooded Venturi channel (16), and then, using (17), to determine the depths and flow velocities. When calculating the flow characteristics in the diffuser of channel, hydraulic losses must be taken into account.
The verification of the developed calculation methods showed that the data obtained by the calculation are in good agreement with the experimental ones. The deviations between calculation and experiment are less than 3%. This makes it possible to recommend the outlined methods for application in engineering practice.