Fluid movement in pipeline with one inflow

. In engineering practice, we often encounter issues related to the connection of flows. In which, in the butt places, vortex zones are observed. The article considers the problem of fluid motion in a channel with one inflow. The scientific papers devoted to the connection of flows are studied. The pressure and non-pressure movement of fluid in the channels, as well as the influence of the structural elements of the system on the hydraulic parameters of the flow, are considered. In the article, the functions describing the main parameters of the channel and inflow are constructed using the formulas of the theory of jets, which made it possible to determine the boundary lines of the vortex zone, the angle of expansion, and the inflow with a uniform connection of the flow. The results obtained show that for a uniform connection of the flow along the channel and along the lateral inflow, it is necessary to take (cid:2)(cid:3) ≈ 30° − 45° at (cid:5)(cid:3) ≈ 0° − 2° , while the vortex boundary line is very close to the circular arc.


Introduction
Much work has been done on the movement of liquids in pressure systems.In the works of A. Arifzhanov, K. Rakhimov, and others [1] consider the theoretical foundations of fluid movement in a cylindrical pipeline.The influence of the slope of the pipeline on the flow rate of the liquid is substantiated.In the work of A. Arifzhanov, K. Rakhimov, and others [2], the issue of determining pressure losses in pipes and the coefficient of hydraulic friction is considered.In this paper, the authors explore the two-phase flow.In the work, the influence of river sediments on the determination of the coefficient of hydraulic friction was considered.In hydraulic calculations during the movement of inhomogeneous liquids, which contain mechanical elements, particularly river sediments, then in such cases, the determination of the coefficients of hydraulic resistance is very difficult and, in most cases, is empirical in nature.Problems dedicated to merging and splitting threads have been studied in many works.Merging streams are an important element of any drainage network.
Rhoads' writings say that when water passes through a drainage network, it is forced to converge at confluences [3].The flows at the confluence of the channels, which was presented by Best [4], due to the increase in water flow and the collision of converging flows near the junction, a complex and active turbulent movement arises.Therefore, deep leaching holes and dotted strips formed in this area, which led to a change in the morphology of the rivers.
Virakun and Tamia [5], on a rectangular and trapezoidal channel, modeled the flow structure, applied a turbulence model with two equations and used parabolic processing.In this work, the authors of [6] used a fully elliptical scheme to study asymmetric fusion at an angle of 60 degrees.They found that the estimated length of the downstream recirculation zone was about 30% shorter.Bradbrook et al. [7,8], using the theory of turbulence, studied the influence of the velocity ratio, merging angle, merging asymmetry level, and the degree of mismatch on the streamline curvature and the degree of topographic influence.In works [9][10][11], and [12], pressure reduction zones were studied using the Kaskaskia River as an example.Biron et al. [13] used a 3D model to study mixing processes immediately after the confluence and downstream of the main flow.Simulations are presented of laboratory connection and fusion fields for low and high flow conditions.
Huang et al. [14] validated the 3D model using experimental data on the flow at the 90° joint.Gobadian [15] conducted extensive investigations of the washing hole at the confluence of the river.All his tests were carried out at the junction of sharp edges.His results showed that as the downstream Froude number, discharge coefficient, and confluence angle of rivers increase, the scouring depth also increases.In addition, as the channel width factor increased or the deposit size increased, the erosion depth decreased.Borgay and Sahebari [16] studied the influence of the angle between the two channels, the flow rate, and the ratio of the inflow's width to the channel's lower branches on the maximum washout depth.In [17], the flow structure was studied in highly curved turns at 90-degree turns of an open channel.Their results showed that the cross slope of the water surface in the bends was not linear.Liu et al. [18] conducted an experimental study of the flow structure and sediment transport at the confluence of an open channel at an angle of 90 degrees with different ratios of inflow to total flow.The paper presents an experimental study of local erosion at the curved edge of the joints of an open channel.Their results showed that as the radius of curvature increases, the maximum local erosion decreases significantly, and the location of the maximum depth of the displacement flows downstream and towards the center of the main channel.
In a study by Rasool Ghobadian Mahsa Basiri, a 3D program was used to calculate local erosion and deposits at a 60-degree confluence of channels.In the works of Kai Bao Amgad Salam Shuyu Sun, a numerical study of the branching of the flow of two-phase immiscible fluids in a Y-shaped planning channel is carried out by solving the coupled Cahn-Hilliard and Navier-Stokes system by the finite element method.In this system, the horizontal channel branches into two identical and symmetrical branches with channel walls assigned several different wettability values.The studies of which the above analyzes are given is devoted to the study of erosion and other parameters at the confluence of flows; in this paper, we consider a channel with a diffuser section in the presence of one lateral inflow there is a vortex zone inside the channel (Fig. 1), that is, the flow area in this case various.The problem is solved based on a model of an ideal incompressible fluid; functions are constructed that allow one to determine the main parameters of the channel and inflow.

Materials and Methods
Let us consider a plane stationary flow of an ideal incompressible fluid in a channel with a diffuser section in the presence of one lateral inflow.There is a point vortex inside the channel (Fig. 1).It follows that the streamlines сonst \ in this section are circles centered at the point M 2 1 im m m t (Fig. 1 and 2).On a circle of radius R at each point, the velocity is the same is circulation of a point vortex.),i.e., a particle of a liquid makes a circular motion at a constant speed.In addition, it is assumed that the vertex E of the polygonal flow boundary is its critical point.Points B and F are also critical [1].
rate of the mixture at the end of the channel.
Further, using the Christopheles-Schwartz formula [2] 2 -are angles at the vertices Therefore, these angles are equal 0 , , , 0 ; 1 The correspondence between the points of the flow region z G the upper half-plane t G can be seen in Fig. 1 and 2.
Then formula (2) takes the form Using (1) and (3), we find the complex flow velocity

Results and Discussions
Thus, we obtain the following ordinary differential equations (5) In this problem, it suffices to determine the shape of the contour BN.Therefore, by integrating (5) in the limit At the confluence, a vortex is formed near the wall in the flow confluence zone on the right side of the channel.Formulas (1), (3), and (4) contain unknown parameters of the problem.To define them, let's do this: Using the condition of equality of the amount of flow at the beginning of the channel and in the inflow, from (1), we find the flow rate in the inflow, referred to as the total flow in the main channel According to formula (4), in the limit at f o t and 0 o t it is not difficult to obtain the current velocities at the beginning of the channel and in the inflow, respectively: The condition of constant flow rate leads , based on (9), to the relation C V H , are width at the end of the channel and its flow velocity.
From Fig. 1, one more condition follows for determining the problem's parameters.
The position of the point vortex M is determined based on the condition that the force acting on it from the liquid is equal to zero.Then the first of the Chaplygin-Blasius formulas [1] for such a force gives there J is a circle of small radius centered at the point   From here, having separated the real and imaginary parts, after simplification, we get the following:    16), we proceed as follows: Thus, the integral ( 16), taking into account (17), will take the form:   Thus, to determine the unknown parameters of the problem, using formulas ( 7)-( 9), ( 14), (15), and (18), a system of transcendental equations is obtained.Having solved this system for a given 2 0

S DS d
, according to the formula (6), the shape of the vortex zone interface is determined.In Fig. 3. and 4 the graphs of changes in this boundary are given (solid lines) and compared with an arc of a circle (dashed lines).

Conclusions
Numerical calculations show that for a uniform connection of the flow along the channel and along the lateral inflow, it is necessary to take

Fig. 2 .
Fig. 2. Canonical area To obtain a general solution to the problem of such a flow, we construct a function dt dw

.
The integral is calculated by subtracting the function at the point m

.
Therefore, using formula (4), we have one more condition for determining the unknown parameters of the problem

.
we use the formula for the difference of inverse trigonometric functions arguments of the sine and cosine in (18) can be simplified.

Fig. 3 . 2022 Fig. 4 .
Fig. 3. Graph of the change in the BN line from the angle of the tap.
line BN is very close to the arc of a circle.