UNSTEADY PRESSURE MOVEMENT OF A LIQUID AT VISCOUS RESISTANCES

ABSTRACT

All the important information necessary for an engineer of this profile about steady-state uneven laminar and turbulent flows of homogeneous media is concentrated instead of a scattered and insufficiently complete presentation of this information, common in educational literature. The construction of the section and its title are fully consistent with the systematic approach to the consideration of complex objects, according to which, in particular, each of its elements, along with a specific value, must serve as a component of a certain system - in this case, it is the technical mechanics of fluids. In the implementation of system-forming relationships between sections on uniform and uneven fluid movements, as well as in the analysis of specific types of deformable flows, paramount importance is given to the processes of hydrodynamic stabilization and destabilization of flows.

АННОТАЦИИ

Сосредоточивается вся необходимая инженеру данного профиля важная информация об установившемся неравно­мерном ламинарном и турбулентном течениях однородных сред вместо разрозненного и недостаточно полного изложения этих сведений, распространенного в учебной литературе. Построение раздела и его название полностью соответствуют системному подходу к рассмотрению сложных объектов, согласно которому, в частности, каждый его элемент наряду с конкретным значением должен выполнять роль компонента опреде­ленной системы в данном случае ею является техническая ме­ханика жидкости. В реализации системообразующих взаимосвязей между разделами о равномерном и неравномерном движениях жидко­сти, а также при анализе конкретных разновидностей деформируемых течений, важное значение отводится процессам гидродинамической стабилизации и дестабилизации потоков.

Keywords: differential equations, unsteady motion, friction slope, velocity pulsation, laminar flow, turbulence, kinetic energy.

Ключевые слова: дифференциальные уравнения, нестационарное движение, наклон трения, пульсация скорости, ламинарное течение, турбулентность, кинетическая энергия.

Method

During the study, scientific research methods were used using literature analysis, as well as scientific views on the laws adopted in hydraulics. The study of uneven pressure motion of a liquid, including dependencies for and other quantities, is carried out mainly by the differential method, by setting up appropriate the equation and processing measurement results.

Introduction

First of all, attention should be paid to the rigor of the derivation of one-dimensional differential equations of unsteady motion. Such equations exist in two forms in the form of equations of quantity of motion and kinetic energy. They are obtained as a result of integrating the differential equations of motion of a real liquid in the Cauchy form over the cross section of the flow. However, one-dimensional equations can only be obtained for a smoothly varying flow under constraints similar to those used in boundary layer theory, which make it possible to reduce the system of three equations of motion to one.

Integrating this equation over a finite flow section allows us to obtain two equalities, depending on whether it is integrated over a live section or pre-multiplied by the local velocity in order to obtain an energy dependence. In view of this, one-dimensional differential equations take the form of two equalities: the equation of quantity of motion and the equation of kinetic energy, the integral of which in the case of unsteady motion of water is sometimes called the generalized D. Bernoulli equation [1]. The above dependencies are approximate due to the fact that the differential equations of hydromechanics cannot be integrated in the general case.

It should be noted that the equation of the amount of motion has greater operational capabilities compared to the energy one. As a result, it is widely used in solving a number of practical problems. However, the kinetic energy equation contains a lot of information about the flow and is useful from the point of view of a deeper study of the phenomenon of non-stationarity.

Research results and analysis

It should be emphasized that there is a difference between the terms of these equations. The first one contains one correction of the amount of motion a_0, recorded with the convective acceleration term, at the same time as in the kinetic energy equation, along with , there is a correction of kinetic energy a. Both coefficients appear in the terms of local and convective accelerations and depend on the path and time [1, 2]. It is important to keep in mind the existing difference between the terms of these equations, which take into account the forces of resistance to flow movement.

In the equations, these terms are represented as friction slopes

Marked dependencies in the case of a live section  where s — the longitudinal coordinate will take the form:

If you subtract (2) from equation (1), then you can see the difference between the terms .

Really,

It follows from equality (3) that, contrary to the existing belief in uneven non-stationary flows  in case, if  do not depend on s and the values of the coefficients at the first, the third and fourth terms in dependence (3) are small. ( if you take everything marked with

Using examples of flat or axisymmetric flows, it can be shown that the friction slope in the equation of the amount of motion is due to the tangential stress at the wall

where R — is the hydraulic radius.

An analogous term in the kinetic energy equation is represented by the integral of energy dissipation and, for example, for a flat fluid flow has the form

Here q — is the volumetric flow rate per unit width of the flow; Н — is the height of the flow; u_s— is the longitudinal component of the local velocity; у — is the vertical coordinate. Believing the integral in equality (5) in parts and bearing in mind that yy, the vertical component of the local velocity, we obtain

Note that the name "friction slope" corresponds more to a uniform unsteady flow. In this case, equality (4) and (5) are valid. In case of uneven movement, the "friction slope", in addition to tangential stresses, must take into account the gradients of projections of normal stresses on the longitudinal axis, which do not vanish, as a result of which the slope is due to more than one friction resistance. Taking the coefficients  and  independent of s and s in nonstationary uneven flows and equal to one lead to the identification of two friction slopes due to tangential stress at the wall and energy dissipation.

Further, due to the greater simplicity of the equation of the amount of motion (1), we will focus on the derivation of dependencies for  (or )  in nonstationary motion.

The dependence for  for laminar uniform pressure motion can be obtained using the Navier-Stokes equation. The study of this issue, if the pressure gradient is given by a periodic function of the form , allows us to judge the nature of the distribution of   over the flow section and conclude that the coefficient , as well as  and C change over time, which characterize the dimensionless coefficients of resistance in the friction slopes  and  (fig. 1,3).

From the graph of the change of , shown in Fig.1, it follows that  at the wall in the case of accelerated motion is greater than the corresponding quasi-stationary motion. In this case, the convexity of the curve  is directed downstream. In the case of slow motion,  is, and the curve is convex against the flow (here * = is the angular frequency of oscillations).

In the problem, the pipeline was assumed to be infinitely long.

Subsequently, the formulation of the problem can be complicated and brought closer to the specific conditions of water movement in a finite-length pipeline connecting two reservoirs

The solution is also performed using the Navier—Stokes equation. Unlike the previous problem, the equation of flow constancy, initial and boundary conditions are added to it, which make it possible to close the system of equations. This formulation of the problem does not require setting the pressure gradient as a function of time, as it was taken when solving similar problems by a number of authors.

In conclusion, we present the result of solving the general problem of determining  in uniform motion, independent of the nature of the change in the pressure gradient over time.

The solution is implemented in order to determine the tangential friction stress at the wall in flat or cylindrical pressure pipelines. In the case of a flat problem, it will take the form

where v— average cross-section speed;  — a function of a certain type

It follows from equality (6) that in the case of a stationary flow, when   turns into a known dependence of the stationary flow.

If , and if , as noted earlier.

It also follows that  depends not only on v, but also on acceleration.

A complete solution to a particular problem is possible by combining equality (6) with a one-dimensional differential equation of the amount of motion. So, for example, in the case of uniform unsteady movement of water in a flat pressure stream (it should be noted that equality (6) is obtained from the condition of uniformity of flow), the one-dimensional equation of motion will take the form

Equation (7) must be implemented in a specific task together with the equation of continuity of motion, initial and boundary conditions.

Along with the above exact solution, it is possible to solve the problem in question approximately, based on the representation of   in the form of a polynomial written as a function of the transverse coordinate of the flow, with time-dependent coefficients. To determine the latter, boundary conditions and differential equations of motion and continuity are used. [1,2,4,5,6]

As a result, a dependence of the form (6) is obtained, but differing in a simple form of the integrand function  at . Figure 4 shows graphs of integral functions in cases of exact and approximate solutions. The proximity of these functions makes it possible to extend the approximate method to the case of unsteady turbulent flows, in which it is not possible to determine the motion parameters using precise methods. All of the above had to do with unsteadiness in laminar motion.

However, as is known, turbulent flows are more common in various fields of technology than laminar flows. The study of these currents, even stationary ones, is fraught with great difficulties. The cardinal solution of the issues of non-stationary turbulence is associated with solving the problem of turbulence in general. Therefore, for now we have to be content with approximate methods, using along with experimental data some analogy between laminar and turbulent flows [7].

Bearing in mind the above, the tangential friction stress  in turbulent flow was also expressed by a polynomial of the fifth degree of y with time-dependent coefficients. The coefficient of turbulent viscosity Ac was assumed to be approximately equal to its quasi—stationary value, similar to the proposal of V. M. Makkaveev adopted in the diffusion theory of turbulence — Ac - kv. In this case, the dependence for  is reduced to the form

where  С — the Shezi coefficient, т = 24

A similar problem is solved precisely under the previously accepted conditions.

Using a new variable

the solution is obtained in the form of

where  the roots of the equation

The parameter l according to the diffusion theory of turbulence depends on the roughness of the walls of the ceiling.

Just as in all previous cases, equations (8) and (9) in the case of flat uniform motion  turn into a known dependence for the tangential stress at the wall, in the case of stationary motion

To solve specific problems, the expressions obtained must be substituted into one-dimensional equations of the amount of motion.  can also be determined if the nature of the dependence  is known.

It should be noted that the solutions given (even the exact ones) will be approximate in nature due to the fact that the coefficient of turbulent mixing A is assumed to be quasi-stationary. Moreover, it reflects the unsteadiness of the flow by depending on the average velocity, which is a function of time. However, this representation of the coefficient A does not respond properly to the dynamic effects that accompany the inertial motion of the flow [1,2,3].

It is known that the turbulent regime is characterized by chaotic motion of moles of liquid, characterized by pulsation of velocities. The magnitude of these pulsations is not in phase with the average velocity of the liquid and may differ from the nature of its changes in the case of non-stationary motion. N. A. Panchurin hypothesized that in the case of accelerated motion, the level of RMS pulsation values lags, due to inertia, from the corresponding pulsations of the quasi-stationary flow and exceeds it in the case of deceleration movements. The noted circumstances are not taken into account by the accepted formula structure for A.

It can be shown that the coefficient of turbulent viscosity A in a nonstationary flow will also depend on acceleration.

In conclusion: it should be noted that no less important in the problem under consideration is the issue of determining the boundaries of the application of stricter dependencies of unsteady motion.

It is known that simplified equations of unsteady motion have long been used in engineering calculation practice, in which the resistance is quasi-stationary, and the coefficients and  and a are assumed to be constant and equal to one. An example is the tasks related to the calculation of open flows, hydraulic shock in pipes, the process of filling and emptying the lock chambers, hydraulic and pneumatic systems, etc. However, in some cases, the use of simplified equations does not adequately describe the processes taking place. Thus, the phenomenon of attenuation and transformation of a hydraulic shock wave, described without taking into account non-stationary friction, does not correspond to reality, the process of filling the airlock chamber from the point of view of wave formation in it leads to a significant difference between experiment and theory, wave propagation in channels and the process of their attenuation is also characterized by certain errors.

In this regard, it seems necessary to revise the existing equations of unsteady motion from the point of view of the possibility of their application with varying degrees of accuracy in specific tasks.

Here it seems useful to follow the path of defining criteria that characterize the degree of non-stationary and allow us to judge the boundaries of the appropriate application of strict dependencies in a specific engineering task.

References:

1. Bazarov D.R., Karimov R.M., Matyakubov R.M., Khidirov S.K. Hydraulics I (main course), Tashkent, Tashkent 2018, 537 b.;
2. Bazarov D.R., Karimov R.M., Matyakubov R.M., Khidirov S.K. Hydraulics II (main course), Tashkent, Tashkent 2018, 557 b.;
3. Bahodir, N., Javlonbek, R., & Zarina, A. (2023). FORMULAS FOR DETERMINING THE LOSS OF NAPOR BY THE ADDITION AND SEPARATION OF FLOWS IN PIPES.. Innovations in Technology and Science Education, 2(10), 460-467.
4. Bahodir, M. N., Raxmanov, J. D., & Maxmudov, A. J. (2023). ADDING A HYDROCYCLONE DEVICE WHEN TINTING DRINKING WATER. Educational Research in Universal Sciences, 2(3), 112-117.
5. Musulmanovich, N. B. (2021). ANALYSIS OF CHANGE OF FLOW DYNAMICS IN LOW BENEFITS OF WATER SUPPLY FACILITIES. Journal of Advanced Scientific Research (ISSN: 0976-9595), 1(1).
6. Bazarov, D., Shodiev, B., Norkulov, B., Kurbanova, U., & Ashirov, B. (2019). Aspects of the extension of forty exploitation of bulk reservoirs for irrigation and hydropower purposes. In E3S Web of Conferences (Vol. 97, p. 05008). EDP Sciences.
7. Isabaev, K., Berdiev, M., Norkulov, B., Tajieva, D., & Akhmadi, M. (2020, July). The dynamics of channel processes in the area of damless water intake. In IOP Conference Series: Materials Science and Engineering (Vol. 883, No. 1, p. 012033). IOP Publishing.х