Fluid movement in a flat pipe with a break

. A new task is set and solved about the flow of fluid in the canal with a break and considering the vortex and cavitation zones. Systematic calculations of stationary flow characteristics and cavity for cavitation flow in the channel.


Introduction
The classical theory of developed cavitation contains the following assumptions: 1) the fluid is considered perfect, incompressible, and weightless; 2) the pressure in the cavity is accepted constant; 3) The current is considered stationary.
In light of the accumulated experimental material and the latest theoretical developments, we consider consistently adopted assumptions, the limits of their applicability, and the possibility of their modification.
The difference between the speed of the unperturbed fluid flow and the rates of liquid particles on the walls of the cavity is small.Therefore, the effect of viscosity should be insignificant.In revenge, the viscosity can noticeably impact the cavitation flow if the cavity is formed on an elongated body with smooth circles (for example, on ellipsoids ).In this case, it becomes an uncertain point of separation of liquid jets.It may influence its position like the flow in the border layer.Pulsation phenomena in the boundary layer are transmitted downstream, and wave perturbations are generated on the surface of the cavity.The flows described relate to developed cavitation trends with a smooth gap.The uncertainty with the position of the point of separation is absent in the bodies with sharp edges, forging the forming (for example, in disks, cones, bodies with ledge, etc.).In this case, they talk about the flows with a fixed margin.The flux breaks through the body of the body surface.Bodies with sharp edges in an isolated form or installed on other bodies with smooth outlines are called caution-forming elements or just cavitators.Curiously, with a fixed separation, the viscosity (Reynolds number) practically does not affect the nature of the developed cavitation flow.Therefore, the model of the ideal fluid, in this case, becomes justified.This is all the more valid for the lamellar cavitators established across the flow since friction forces arising in a viscous fluid will not contribute to the overall resistance force.As part of the ideal flow fluid in most cases, it is logical to take potential.The exception is the tasks of the interaction of several bodies when the incident flow can be viewed for individual bodies.
The assumption of liquid incompressibility is justified by the fact that the cavitation barrier (the moment of cavitation) occurs significantly earlier than the effects of water compressibility begin to manifest.However, this restriction of the classical theory of developed cavitation with ultra-high speeds of motion, comparable to the speed of sound in water, becomes unfair, and therefore new models of cavitation flows are created taking into account the compressibility of water (Gurevich M.I.).
The extremely difficult way is the weight of water.In fact, a course of developed cavitation occurs with high speeds and, therefore, with large numbers of Froude ‫ܨ(‬ > 10).Therefore, the assumption of the classical theory of neglecting the weightiness of water in cavitation flows is generally justified.However, with a more thorough analysis, it is found that the effect of water weight is manifested not only through the number of frauds.In Kaverns with a vertical axis, the influence of the weight is significant, and the weight ability can generate a peculiar configuration of the cavity, missing in weight liquid.The problem of taking into account water weight is dictated by another circumstance.The characteristics of cavitation flows are studied in hydrodynamic laboratories at reduced speeds, and the necessary cavitation numbers are achieved by imaging gas.Under these conditions, the number of fruits turns out to be smaller, and it is necessary to be able to calculate the amendments for water weight in the experimental data.The classical theory developed sufficiently in this direction (G.V. Logvinovich, L.A. Epstein, A. F. Bolotin, etc.).
The most significant assumption about the constancy of pressure in the cavity.Careful experiments on pressure measurement confirm this position.Minor deviations are noted in the tail part of the cavity, but this is, due to the nonstationary flow in this area, accompanying the destruction of the walls of the cavity.
The proposal to constant pressure based on the Bernoulli equation immediately leads to the conclusion about the constancy of speed at the border of the cavity.In fact, ‫‬ At the turn of the channels or pipe, we get a curvature by the current line (Fig. 1).On the particles of the liquid moving along the curved current lines, the centrifugal force of inertia acts [5].Due to this force, the hydrodynamic pressure (and, consequently, the potential energy) in the revenge of the turn at the outer wall of the pipe increases and at the inner -decreases.The same circumstance causes a decrease in the speed pressure at the outer wall and increases it at the inner wall.Thus, at the turn, there is a redistribution of velocities along live sections and deformation of the speeds along the flow of m, e; there is a cavitation current at the turn of the pipe.Studies of twisted and cavitation currents in the channels that are elements of turbo machines have acquired relevant importance.Examples of such channels are inlet and exhaust pipes, intervenes channels of turbines, compressors, and fans.
This paper presents some results of a theoretical study of cavitation flow fluid in a canal with a break.Depending on the hydrodynamic and geometric parameters.
The distribution of parameters in the radial section with a radial input of the stream corresponds to the distribution of parameters when the potential vortex twist is shrilling.The course in the cylindrical ring channel occupies an intermediate position between the potentially swirling flow and the flow by the law of the twist of the solid body.In addition, the geometry of the radial-axial channel is characterized in that along the external circuit breaker, the turn turns smoothly, and along the internal -jump, as the straight angle is flowing.The flow around the convex angle leads to a local decrease in the static pressure in the neighborhood of the angle of rotation, and the flow around the concave angle leads to the local increase in pressure in the area of the angle.
Consider the flow of fluid in the floor of an infinite flat pipe rotated at an angle where 0 < ߙ ≤ 1 2 ൗ , in the cross-section of ВК (fig.1).The existing experimental material shows that in the case of the flow around the straight contours at the inner corner of the break, less π, a cavity is formed.The cavitation zone occupies a significant area of the flow; therefore, it cannot but affect the fluid in the mainstream.It causes damage to the solid walls of the channels, pulling out the material from their surface.
Therefore, when designing channels with breaks, it is very important to determine the cavitation zone and describe its character at various stages, depending on the flow parameters and channel geometry.

Methods
To study the cavitation flow in a tube with a break, we use the Epros scheme [1][2][3][4], with the formation of a cavity at point B.
Despite some mathematical complexity, the specified flow diagram of fluid is closer to the real current scheme [5,6].
Problems of fluid flow in channels with variable cross sections are considered experimentally and theoretically, with various hydraulic approximations by the authors of work [7][8][9].In such one-dimensional and approximate productions, it is difficult to determine the relationship between the geometric and hydrodynamic parameters of the channels (for example, the mutual relationship between the outflow width and flow rate between the shapes of the channel and cavitation, etc.).To account for these factors, it is necessary to consider the task of at least a two-dimensional formulation, applying the methods of jet theory.In all cases, conformal mappings of some canonical region were built [10][11][12] ‫ݑ(‬ = ߦ + ݅߫; usually, such areas are the half-plane, square, circle, strip, sector, etc.) to the region of the complex potential ‫ܩ‬ ௪ (where ‫ݓ‬ = ߮ + ݅߰ complex potential) and the flow area ‫ܩ‬ ௭ ‫ݖ(‬ = ‫ݔ‬ + ‫,)ݕ݅‬ or the area of the speed of the speed‫ܩ‬ : where ܸ is a velocity module on a free surface or to the area of change of the function of Zhukovsky‫ܩ‬ ௪ [13][14][15] To solve the problem, we will show the flow area in the physical plane by a quarter of the circle of a single degree (fig. 1

Results and Discussion
Knowing all zeros and poles function ௗ௪ బ
where N is permanent.To find this permanent, you need to integrate (1) at a quarter of the circumference of an infinite small radius (Fig. 2) Here is the Q-consumption of fluid in section AA.
With the help of the method of singular points, it is easy to build a complex speed at zeros and features [13]: where ܸ is the stationary speed on the free boundary BC (Fig. 2).
From formula (3), we obtain the expression for the speed in the section AA: Then the number of cavitation is determined by the formula Denote the width of the C at the point with through = ೖ .
The main task facing us is to find the hydrodynamic parameters of the cavitation flow depending on the dimensionless values specified ೖ , భ , ߙ where ‫ܮ‬ and ‫ܮ‬ ଵ are Channel widths in the section AA and ‫,ܦܦ‬ respectively.First, we find ‫ݖ݀‬ ‫ݑ݀‬ ൗ and z as a function from.From ( 1) and ( 3) implies that DD width is defined as the increment of the function ‫)ݑ(ݖ‬ when around the point ‫ݑ‬ = ݅݀: Exception from ( 2) and ( 7) constant N gives Taking the integral from the function (1) on the infinitely small semi-rapidity with the center at the point ‫ݑ‬ = ݅, we find the flow consumption in the jet Recalculating the width of the stream through ߜ = ೖ , from (9), we get From formula (3), it suggests that the point ‫ݑ‬ = ݁ ఈ బ గ , in which the cavity tangent to the boundary parallel to the BD wall is determined from the equation , the solution to this task is simplified.In particular, from equation (11), neglecting small higher orders, we define a point in which the cavity tangent is parallel to the BD wall [32]: Then from equation ( 6), you can determine the width of the cavity: Similarly, we obtain expressions to determine the distance from the start of the coordinates to the point H, where the speed is zero: From fig. 1.It can be seen that or (10) and (15), it follows that Equality ( 4), (8), and ( 16) constitute a closed system of equations relative to unknown k, d, h.This system can be solved only numerically.

Conclusions
The speed distribution along the KD channel wall at different values of the raid flow rate is shown in Fig. 4.From the graph; it follows that at sea zoom in the − У increases, with − У = 0.8 ÷ 1.5 reaches its greatest value and then, with increasing − У gradually decreases.
Ascending and reducing the speed value is explained by changing the living section of the flow.In Fig. 5, the relative width of the

ு
cavity was given on the ratio of the velocity ೖ at different values of the angle of the fault of the flat pipe απ.This result indicates that with the value of the velocity ೖ ∶ 0,83 ÷ 1, the width of the cavity remains unchanged, which is explained by sufficiently pushing the flow to the external lining of the flat pipe.

Fig. 4 .
Fig. 4. Speed distribution along the wall channel KD with different values of the raid flow rate.

Fig. 5 .
Fig. 5.The dependence of the relative width of the cavity from various.