Hydraulic jump on smooth and uneven bottom

The mechanism of absorption of excess power of the flow within hydraulic jump has been studied in the article based on theoretical manner. Mathematic model of hydraulic jump has been investigated by taking additional water body (mass) in hydraulic jump zone as basic. Theoretical research has shown that main part of excess power is discharge for rotation of additional water mass and a formula has been obtained to make calculation thereof. The article also has provided a formula for calculating the portion of flow energy needed for overcoming friction resistance emerged in bed bottom. Because of conducted studies, formulas have been suggested for calculating hydraulic jump length occurred in flat and uneven beds. Obtained formulas have been mutually analyzed with results found by other researchers.


INTRODUCTION
It is obvious that process of absorption of excess flow energy in tailraces of hydro-technical facilities happen by hydraulic jump. One of the main parameters when designing tailraces of hydro-technical facilities is accurate and proper calculation of length of emerged hydraulic jump. Study of energy losses in the jump zone bears great importance when analyzing flow structure. Number of studies have been devoted to the given problem Results of these studies indicate that intensive turbulent agitation takes place in the area of hydraulic jumping, which causes penetration of large vortex structures in the form of additional discrete liquid masses from the turbulent (stormy) zone into transit (tranquil) one. An analysis of the existing assignments on this issue suggests that most of them are devoted to a flow with a constant mass. Currently, several empirical formulas are used in practice to determine the length of hydraulic jump а) The formula of N.N.Pavlovsky: г) The formula of Bakhmetov-Matchick: L= 5 (h2-h1) Great majority of these formulas have been proposed considering analysis of the results of studies conducted in various laboratories globally that carry out hydraulic investigations. Results derived from the calculation formulas sometimes vary from each other up to 50-80%. Proper design of water stilling wells constructed in tailraces of hydrotechnical facilities depends on accurate calculation of hydraulic jump length. Studies regarding hydromechanical analysis of energy absorption within hydraulic jump have not been conducted in known formulas. All calculation formulas have been empirically suggested based on the results of laboratory tests performed within a given range. Theoretical research. Unlike existing tasks and works, we consider a hydraulic jump in which the motion of a liquid occurs with a variable mass with decreasing number of motions. With a sudden transition of the flow from a turbulent state to a calm one between sections I-I and II-II, a hydraulic jump is generated within which highly complex hydrodynamic process takes place ( Fig.  1). Both connection to the main flow (between sections I-I and K-K) and separation of the additional flow (between sections K-K and II-II) from it happens within the limits of hydraulic jump. In this case, specific water discharge in the range of I-I will be q0, it will increase in the range of K-K and become (q0+qd), where qd is the specific flow rate of the connected flow. Separation of connected discharge -qd from main stream takes place in the area between the ranges of K-K and II-II. As a result, specific discharges of main stream in sections of I-I and II-II are the same and equal to -q0.  (2) Supposing that energy loss during the jump is equal to difference in energy of Е1 and Е2 in sections I-I and II-II, instead of (2) we find out: .
(3) Equation (3) reminds the Bernoulli equation, but with new term on the right side. During flow movement with variable discharge along the path, we determine the pressure loss on the friction against the bottom and the side walls of the channel bed by the formula of the same kind as with constant discharge, i.e.: (4) where Сmedis average value of Chezy coefficient between the sections; Rmed-average value of hydraulic radius between the sections. Another integral in (3) expresses mainly the pressure loss caused by the variability of the flow discharge: , (5) where: -means the ratio of the projection of the velocity of the attached flow to the main one. We will assume in future studies that 1 2=1 and (meeting of two streams happens at an angle of 90 °). Taking into account assumptions and considering that b = 1.0 m, dQ = qxdx, we determine the value of the pressure loss by formula (5) as follows: . (6) and in expression (6) are variable values and depend on the length of the jump. We determine the value of qх, for running value of separable or connected value from the expression.
(7) Where: lx-is the length of connection and separation section; qd.is additional discharge. By integrating he dependence (6) within the boundaries of I-I and K-K sections, we determine pressure loss for mixing additional discharge with main discharge: , (8) We apply change of Хand hx in these sections according to straightforward principle: , (10) Considering (9) and (10), equation (8) obtains the following form: .
(11) For hydraulic jump area between I-I and K-K sections with water depth h1 and hcr, expression (11) can be presented in the following form after minor transformations: .
(12) By integrating and transforming expression (12), we find out dependence of pressure loss on the mixing of the additional discharge with the primary one during the jump in the form below: (13) By integrating the dependence (8) within the limits of K-K -II-II sections, we determine pressure loss on separation of the additional discharge from the primary one: , We accept the change of Хand according to straightforward principle in the following manner: (15) .
(16) Considering (15) and (16), equation (14) obtains the following form: , (17) Expanding the integral in expression (17) and conducting certain transformations, we get the formula for pressure loss on separation of additional discharge during the jump: .
(18) Thus, two expressions (13) and (18) were obtained for determining pressure loss during hydraulic jump on connection and separation of additional discharge. It should be mentioned that, according to the adopted scheme, the energy of the stream gets decreased before the critical section. Arriving at minimum value in the critical section, and further due to separation of additional discharge, the flow partially recovers its energy. This condition indicates that pressure restoration happens in the section K-K -II-II. Pressure restoration value is determined by the expression (18). The values qd. and hcr. can be determined from the following expression with known magnitudes of hydraulic parameters of the jump: , . (20) Being aware of parameters on hydraulic jump elements, it is possible to determine pressure loss due to the connection and separation of the additional discharge along the length of the jump according to formulas (13) and (18) with great accuracy. Proposed dependencies (13) and (18)  on overcoming the resistance of the bed bottom. Specific energies Е1, Е2 and Еc. in sections I-I, II-II and K-K of the hydraulic jump are identified by using hydraulic flow parameters. At the same time, energy losses between the sections I-I and K-K constitute 1=Е1-Еcr, and between the sections of К-К and II-II they become 2=Еcr-Е2. In all cases, terms of Еcr 2, and at the same time 2 are satisfied. Hence it is obvious that during flow movement within the boundaries between the sections K-K and II-II, the specific flow energy increases additionally from the minimum (section K-K) to Е2 (section II-II) by an amount of 2. While calculating for (13) and (18) of values 1 and 2 you can determine energy loss to overcome resistance along a segment of length L1 from the expression 1 2-1 and along the length of section L2 from the expression 2 2 2. It becomes obvious from the presented material that, in hydraulic jump, energy loss necessary for overcoming bottom resistance of the bed will be equal to 1 2 It should be mentioned that in order to determine hydraulic parameters of the stream and to find the magnitude of value of the pressure loss in the hydraulic jump according to (13), (18) (13) and (18) are respectively 1=1,01…36,0 сm and 2=0,85…3,87 сm. The total pressure loss along the length of the hydraulic jump varied from 0.15 to 11.81 cm. Furthermore, according to the data in the table, it is obvious that for all the experiments 2 is negative and the conditions of To determine the magnitude of pressure loss for overcoming the bottom resistance of the channel, bypassing formulas (13) and (18), after processing numerous data, a dependence was obtained in the form: , Hence, energy loss in in hydraulic jumps will be: , It is obvious from the equation (22) that for hydraulic jumps, following conditions must be necessarily fulfilled: .
(23) We use Darcy-Weisbach formula for critical section form to determine length of hydraulic jump: , (24) where: -is the energy for overcoming frictional resistance; р -coefficient of hydraulic friction from the slopeof the gradient of pressure f-coefficient of hydraulic friction from the slopeof the friction ; Ljmp.-length of jump; cr.-critical flow velocity; hcr.critical flow depth.
We define the following from (24) for length of hydraulic jump: (25) For determining р and f we processed data of laboratory studies by several authors Based on results of these studies an expression was found for determining р in the following form: (26) In this case, the value of f is determined both for smooth and uneven bed separately. Coefficient of hydraulic friction f for smooth bed is determined by the formula: Extensive research activities have been carried out under the leadership of academician M. Vyzgo regarding the impact of bed roughness on hydraulic jump length . The length of hydraulic jump has been studied under laboratory conditions within same hydraulic parameters in smooth and uneven beds. Results of carried studies are summarized and presented in Figure 2. As it is obvious from this graph, obtained results are subject to parabolic functioning appropriateness. The results of the calculation done by formulas (25), (29), (29) and (30) that we obtained through theoretical method for the length of hydraulic jump occurring in uneven beds have been compared to research results by academician M. Vyzgo (Figure 2). When comparing the parameters of the hydraulic jump according to the recommendations developed by us for the uneven bottom, the data of M.S. Vyzgo and Y.A. Kuzminova were used results of which are presented in Figure 2. According to data of this figure, hydraulic jump parameters determined according to our recommendations, as well as by M.S. Vyzgo expression L.frik.=L0 almost coincide, which is confirmed by graphics