SOME EXAMPLES OF NUMERICAL MODELING OF NON-STEADY LIQUID FLOWS IN SIMPLE AND BRANCHED PIPELINES
DOI:10.30825/5.EJPAU.194.2020.23.4

Vyacheslav Berman¹, Stepan Kril¹, Emil Bournaski^2
1 Institute of Hydromechanics at National Academy of Sciences of Ukraine, Kiev, Ukraine
² Climate, Atmosphere and Water Research Institute at Bulgarian Academy of Sciences, Sofia, Bulgaria

ABSTRACT

In this paper we suggest to develop the theoretical approach and numerical modeling for non-steady turbulent transport of liquid flows in simple and branched pipelines. The basic problems for non-steady flows, such as starting and stopping of hydraulic pipeline systems, hydraulic impact and other were studied. Some concrete numerical algorithms for simulation of such kind of non-steady flows were also proposed in this paper.

Key words: non-steady liquid flows, pipeline hydraulic systems, numerical modeling.

1. INTRODUCTION

Existing systems of simple and branched pipelines suggest the availability of reliable methods for the hydrodynamic calculation of such systems at the design stage and in the process of ensuring their stable operation and maintenance. Without concretization here the specifics and purposes of the above-mentioned hydraulic pipeline systems, we only note that it can be both traditional water conduits and pipelines for pumping oil and oil products, special pipelines, pipeline systems designed for irrigation, etc.

Moreover, according to the generally accepted approach, as a basic in the field of pipeline transport, primarily for homogeneous flows, priority is given, of course, to the study of steady modes of transportation. At the same time, the development of hydrodynamic calculation methods for the case of non-steady flows of a similar class is also of no less interest for practice. The importance of the problems of non-steady motion is due, first of all, to the high requirements for the reliability of pipeline systems, especially for systems with a complex structure (this applies especially to branched pipelines).

From the experience of operating such pipelines, it is known that during operation, constant changes in transportation conditions are possible, and this, in turn, can be the cause of sharp and often dangerous fluctuations in pressure and flow rate. Jung and Karney [6], for example, analyzed the practical differences and advantages of four transitional models – water hammer models, rigid water column analysis, quasi-stationary analysis and the so-called Joukowski approach [3] and gave guidelines for determining the degree of instability combined with an appropriate non-stationary model.

In this regard, already at the design stage it is necessary to have available convenient methods for calculating non-steady motion, which allow determining possible fluctuations in pressure and flow rate under various operating modes of transport systems. Based on these methods, it is possible to obtain the necessary information for the calculation and selection of system for protecting pipelines from excessively high pressures, as well as for the calculation and commissioning of automatic control and protection for such systems.

2. STUDY OF NUMERICAL ALGORITHMS FOR CALCULATION OF NON-STEADY FLOWS OF HOMOGENEOUS MEDIA IN PIPELINE

As noted above, during the operation of industrial and, in particular, complex main pipeline systems, sharp pressure fluctuations can practically constantly occur, which sometimes leads to undesirable emergency situations. Such transient (non-steady) operating modes make take place during starts and stops of the main and auxiliary pumps, during a planned or emergency power outage, as well as when switching on and off various control valves.

A full account of non-steady nature will ensure the stable operation of the entire transport system, increase the reliability and durability of pressure pipelines and all hydromechanical equipment. Currently, there are a sufficient number of papers [1–7, 10–14] related to the solution of some of the problems formulated above. The approaches proposed in these papers, as a rule, contain a number of assumptions, which in some cases can affect both the character and value of the determining the hydrodynamic parameters.

In its most general form, the problem of non-steady motion of both an incompressible and a compressible homogeneous medium within the frame of the one-dimensional and isothermal approximation should be solved on the basis of the general system of dynamic equations obtained, for example, in [8, 9]. For complete closure, it is necessary to add the corresponding initial and boundary conditions to this system. This area of research still remains relevant and a number of authors can be noted [5–7, 10, 14] who use a variety of approaches for the solution formulated above problem regarding of non-steady liquid flows in pipelines.

In this paper, to solve a number of problems of interest to us, several simplified approaches were used, which also successfully allow modeling the flows of the class in question. So the general problem of non-steady motion of a homogeneous medium, formulated within the frame of a one-dimensional problem, is a special case of the system [9] and reduces to solving a system of two partial differential equations:

	(1)

Here, V – is the average velocity of the carrier medium over the cross-sectional area, P – is the pressure in the system, ρ – the density of fluid the carrier medium, a – is the sound velocity, λ – is the hydraulic friction factor, D – is the diameter of the pipeline.

Like the more general system of equations of motion [2, 8, 9], the simplified system of equations (1) is of the hyperbolic type and the same analytical and numerical methods can be used for solving them. In other words, by setting the initial and boundary conditions (while also knowing the law of variation, ρ, λ and a), the system of equations (1) can be, in principle, solved for various operating modes of pressure pipeline systems.

At the same time, it should be noted that the correct setting of boundary conditions and the choice of a convenient method for solving system (1) are most often associated with certain difficulties. Let us dwell in more detail on the method of solving this system of equations.

As is known, both analytical and numerical methods for solving a system of hyperbolic equations are widely used in the literature. When using analytical methods of solution, due to the nonlinearity of system (1), it is supposed to simplify (or linearize) the initial equations in advance. Most authors, as applied to such problems, performed linearization of system (1) by replacing the term λ|V|V with its constant value equal to the averaged value in coordinate and time. In addition, when solving specific problems, it is usually taken ρ = const, which eliminates the possibility of phase transitions. The assumptions noted above lead to the fact that it is not always possible to achieve a satisfactory agreement between the calculated and experimental data.

In this regard, it becomes necessary to develop a universal method for solving system (1). It is obvious that in such a situation it’s possible to use only numerical methods of calculation. The method of characteristics is still considered as the most popular for solving system (1).

After analyzing various numerical algorithms, we took as the basis explicit Lax and Lax-Wendroff difference schemes, which have been proven themselves very well in the field of gas dynamics. In such formulation, we were able to use the initial system of equations (1) to solve a number of problems in the field of not only the motion of a homogeneous fluid, but also for hydrotransport of various solid dispersed materials in pipelines in the regime of developed turbulence [2, 8, 9]. It is clear that for the case of turbulent flows of suspensions in pipeline, this model involves taking into account the variable parameter values ρ, λ and a both in space and time.

The goal of this paper is, first of all, to demonstrate the possibility of using the proposed numerical calculation method for simulation a sufficiently wide variety of non-steady processes that can take place in hydrotransport systems of a homogeneous fluid in simple and complex pipelines. Before proceeding directly to a discussion of the features of using the proposed numerical algorithm for solving specific problems, we’ll clarify one fundamental issue, which until now has not been practically discussed in the literature.

In all numerical calculations for non-steady flows, even for homogeneous mixtures, it was usually assumed that the hydraulic friction factor λ included in the determining system of equations (1) is determined at each calculation step strictly from the conditions of the stationary problem. Separate research known from the literature, for example [1], have shown that such assumptions are, as a rule, not sufficiently justified.

In fact, the problem of determining the coefficient λ for conditions strictly corresponding to the real regimes of non-steady motion of one-phase mixtures is of independent interest and requires additional research. This paper is mainly devoted to assessing the influence of the non-stationary coefficient on solving concrete problems of fluid movement in pipeline.

In addition, as already noted, trunk and branched pipelines are a fairly complex network on which there are various local resistances losses (particularly, gate valves, control valves, diameter changes, etc.). Taking into account that local resistances losses are extremely important, especially when setting boundary conditions (opening or closing a valve, starting or stopping pumping equipment, triggering a check valve, etc.) in non-steady motion modes. Until recently, it was assumed that the value of local resistances losses, which is expressed in determining the coefficient of local resistances losses ξ, is usually taken also from the conditions of the stationary problem. An attempt to calculate the non-steady factor in determining the value ξ was also partially considered in this paper.

3. STUDY OF FRICTION HEAD LOSS FOR NON-STEADY FLOWS IN PIPELINE

A review of the literature data related to the non-steady motion of one-phase flows in a pipeline showed that there are virtually no strict recommendations on taking into account the non-steady character of hydraulic friction factor λ and local resistance losses ξ. In this regard, we have analyzed those publications that are at least indirectly related to the study of these issues. Such an analysis made it possible, to a certain extent, to judge the nature of the change in the main parameters for the class of currents studied by us. It should be noted that even for the case of a homogeneous fluid, basically, only a few papers are known in which an attempt was made to take into account the non-steady nature of only the hydraulic friction factor coefficient λ.

Of the few publications devoted mainly to the experimental study of the hydraulic friction factor coefficient, the papers [1, 11, 12] can be noted. In general, the known results of papers in this area are very contradictory, not only in quantitative but also in qualitative terms. Some authors [1, 12], for example, have found that the coefficient of hydraulic friction factor in accelerated flow can be greater than the corresponding value in steady flow. At the same time, the data of other authors [11] indicate that the coefficient of hydraulic friction factor can be either greater or less than the value corresponding to stationary motion. These contradictions, to a certain extent, apply to the mode of slow motion of the liquid, in addition, there is no unanimous opinion in the literature about the value of the hydraulic resistance factor itself as a dimensionless complex. Of the available publications, we dwell in more detail only on those where, along with the experimental studies performed, attempts are made to generalize the obtained results.

Among various types of non-steady motion, the accelerated fluid motion in pressure pipelines has been studied in most detail. It should be noted in this respect the already mentioned group of authors [1, 12]. In these papers, the acceleration motion of liquid under the influence of a stepwise changing pressure gradient in a cylindrical pipe was considered. In this case, the value of the hydraulic friction factor coefficient was determined from the first equation of system (1):

	(2)

In this case, the flow velocity was determined by an induction flow meter, and the pressure P was determined by pressure sensors.

For such conditions during laminar acceleration flow, it was found in the paper that the ratio λ_N / λ_W is uniquely related to the ratio of the instantaneous value of the Reynolds Re number to the value of this Reynolds number Re_∞, corresponding to the end of the acceleration process:

	(3)

Here λ_N and λ_ST are the hydraulic resistance factor for non-stationary and stationary motion, respectively, µ are the roots of the Bessel function of zero order; ν – kinematic viscosity of the liquid, Re_∞= V _∞ D / ν – the value of the Reynolds number at the end of the acceleration process, t – time.

The experiments [1, 12] performed on a series of large-scale stands with various liquids has been showed that dependence (3) is well satisfied in the initial period of the acceleration section corresponding to the laminar flow regime. For a flow in a turbulent region, the direct use of dependence (3) becomes problematic. Despite this, the authors of [1, 12] derived that this dependence after some assumptions can also be used to describe the initial acceleration period (even in the turbulent region) :

	(4)

Thus, using relation (3) or (4), it is possible to take into account the non-steady of the coefficient λ for accelerating fluid motion in the pipeline. The ratios thus obtained can be used in calculating the start-up (acceleration) modes of pumping units.

In the systems we are considering, in addition to accelerating motion, there can also be slow motion and a simultaneous combination of accelerated and slow motion (water hammer). Therefore, the study of such flow patterns is also of interest. In this regard, research devoted to the study of the parameter λ for the case of water hammer deserve the special attention. In this case, the pressure was measured by piezometric sensors, and special sensors of the original design were used to measure the average velocity [11]. Processing the results of these experiments showed that the data obtained can be well generalized on the base of the approach described in [11]. In this paper, the non-steady mode of motion was created with using an automatic slotted shutter installed at the end of the pipe. The study of the kinematic characteristics of the flows was carried out with using the photo-visualization method. An aluminum finely dispersed powder was used as an indicator. The velocity flow was fixed in a vertical plane along the longitudinal axis of the pipe in the alignment, where the flow was already completely stabilized.

For estimation the degree of non-steady, a dimensionless quantity was adopted as N – the non-stationary parameter, which was determined as follows:

	(5)

Here H is half the height of the horizontal rectangular pipe, V is the average flow velocity. The parameter N is often used to describe non-steady motion, and is some generalization of the Strouhal number. In paper [11] both accelerated and slowed down fluid motion were considered.

Using the values of tangential stresses τ found on the wall with using the kinematic characteristics, the values of the hydraulic friction factor coefficient λ for non-steady flows were determined:

	(6)

For control the obtained above characteristic λ, the value of this coefficient was calculated also according to the pressure drops obtained with using special inductive pressure sensors. In this case the value of the quantity λ was determined from equation (2).

The values calculated from the data of the oscillograms were slightly higher than the values obtained from the kinematic characteristics of the flow. The values obtained by the kinematic characteristics of the flow were taken as true.

Processing of the experimental material showed that for accelerated motion there is the increasing of parameter λ, and with slow motion there is the decreasing λ compared to this value for steady flow. For practical calculations, the experimental data obtained for accelerated (+) and slowed motion (–), respectively, maybe for given condition conveniently generalized by empirical relationships:

	(7)

For visibility, the character of these dependencies can be traced from Figure 1.

Fig.1. Dependence λN / λW on N for accelerated and slow motion modes.

Thus, as can be seen from the above reasoning, when calculating various non-steady processes in pipeline systems, it is desirable to additionally use either dependencies of the type (3)–(4) or dependence (7). Considering the above results, we now turn directly to the consideration of some problems associated with the non-steady motion of a homogeneous liquid in pressure pipeline systems. In this case, the main attention here will focus on the movement of liquid in simple pipelines, where the influence of local resistances is insignificant and, therefore, quite accurately, we can trace the effect of the non-steady of the coefficient λ on the main transportation parameters. Accelerated and slowed down fluid movements were chosen here as examples.

4. STUDY THE ACCELERATING OF ONE-PHASE FLOWS IN THE STARTUP MODE FOR PRESSURE PIPELINES

One of the characteristic and frequently encountered in practice is the problem of accelerating the movement of one-phase flows in a pipeline with an open or closed end (to an open or closed valve). At the same time, two types of conditions can be set at the beginning of the pipeline: either the valve is opened near the working pump, or the pump unit is started.

Consider the initially case of opening of a valve near the pump when a homogeneous fluid moves. For definiteness, we can assume that the velocity near the pump is changed according to the law:

	(8)

where V₀ – is the velocity corresponding to the steady state flow; th – hyperbolic tangent, A – coefficient characterizing the degree of valve opening; t – time.

Using this problem as an example, as noted earlier, it is convenient to verify the accuracy of the proposed finite-difference computational schemes. The verification performed with well-known solutions for non-steady and steady problems showed a rather high accuracy and mutual coherence of these schemes.

The greatest interest in the problem formulated above is the calculation of the pressure change in time near the pump. In addition, the question of the degree of influence of non-steady the coefficient λ on the value and character of the change the main parameters has also significant importance.

In Figure 2 the results of such calculations for two pipelines with lengths of 500 m and 5000 m, respectively are presented. The hydraulic friction factor coefficient for non-steady problem was determined according to (3–4). Moreover, for the joint solution of the first two equations of this system, the iteration method was used. As can be seen from the above dependences, the maximum dimensionless pressure (the pressure referred to the impact pressure according to the formula of N.E. Zhukovsky [3]) for the case of non-steady is turned out to be approximately 25–30% less than the corresponding pressure under steady mode. The obtained result confirmed the assumption that taking into account the non-steady of hydraulic resistance coefficient λ can affect on the value of the main hydrodynamic parameters. As for the value of the impact pressure, the following can be noted.

As can be seen from Figure 2, both an excess of the maximum permissible pressure and the formation of vacuum are possible near the pump, which can lead to the formation of discontinuities.

The last circumstance is especially dangerous, since it is known from practice that discontinuities maybe accompanied by significant pressure surges (several times higher than the pressure during hydraulic impact). A sufficient number of papers are currently devoted to the study of this question. Most authors of these papers do not yet have a unified point of view on the issue of increasing pressure connected with a discontinuity in flow continuity, and therefore the increase in impact pressure due to discontinuity is taken into account differently. There was no consensus concerning character of the pressure change. The explanation of the appearance of maximum pressures after continuity breaks by the effect of dead ends, wave interference or excess of the reverse flow velocity (into the vacuum zone) over the steady velocity before impact is discussed now as outdated and unreasonable.

A) B)

Fig. 2. The dependence of pressure on time for accelerating fluid flow a) l = 500 m, V0 = 0.5 m / sec; b) l = 1500 m, V0 = 1.5 m / sec;

More correct in this regard is the point of view held by the authors [1, 11, 12] that the physical nature of the phenomenon of water hammer during flow continuity breaks can be explained based on the cavitations effect. This point of view is confirmed by the results of experimental studies obtained in [3]. In this paper, pressure increase pulses were first recorded at the beginning of positive phases after periods with discontinuities, and it was shown that the magnitude of the pulse pressure significantly exceeds the pressure increment in the first phase of direct hydraulic impact, which begins with a pressure increase wave. In this case, the occurrence of the indicated pressure pulsations is explained by saltatory change in the physical properties of the substance during the “vapor-liquid” phase transition and the action of forces arising from the collapse of cavitations cavities.

Thus, as follows from Figures 2a and 2b, we have considered the simplest example of accelerating fluid movement, and we see that in this case there may be a double reason hazard in the initial section of the pipeline. Therefore, it becomes quite clear why, as a rule, it’s necessary to place safety devices near the pump. In addition, also as can be seen from Figure 2, to estimate the true pressure value (in this case, near the pump), it is desirable to take into account the non-steady nature of the hydraulic friction factor λ used in the calculations.

5. STUDY OF TRANSIENTS PROCESS DURING EMERGENCY STOPS OF PIPELINE SYSTEMS

Initially, we consider the problem of direct hydraulic impact during the motion of a homogeneous fluid in a simple pipeline. As an example of this task, one can, as in the previous case, additionally to check the influence of the non-steady of the coefficient λ on the main transportation parameters. In this case, it was assumed that in the pipeline, where uniform movement with a given velocity is realized, the valve at the free end of pipeline is closed instantly. For this class of problems, as boundary conditions, it is necessary to put a constant pressure at the beginning of the pipeline and zero velocity at the end of it.

In Figure 3, as an example, the results of calculating the change in pressure (referring to the pressure near the valve) for a pipeline with a length of 5000 m are given. In this Figure, curve 1 corresponds to a stationary value, curve 2 – taking into account the non-steady mode, according to (3) – (4), curve 3 – taking into account non-steady, according to formulas (7). As can be seen from this figure, taking into account the non-steady for the case of hydraulic impact leads to an overestimation the maximum pressure. A certain disparity in the calculated values of this pressure according to (3) – (4) and (7) is apparently due to the fact that these dependences were obtained for various types of non-steady flows. And as noted, for example in [11], the type of non-steady can be significantly affected on the character of the change in all hydrodynamic parameters.

Fig. 3. Dependence of pressure vs. time at hydraulic impact in simple pipeline

As can be seen from Figure 3, by analogy with Figure 2, taking into account the non-steady of hydraulic friction factor λ is not only desirable, but also necessary for estimation the real value of pressure and other hydrodynamic characteristics.

We now turn to the consideration of the problem for motion of a homogeneous liquid in a branched pipeline. A characteristic feature of such pipelines is the fact that these systems maybe consisted from pipes of different diameters and equipped with various control valves. It’s clear that in such systems, local resistance can play a significant role in the calculations. In this regard, the first attempt was made to take into account not just the influence of local resistances, but the non-stationary nature of these resistances.

6. STUDY OF LOCAL RESISTANCE IN NON-STEADY PRESSURE FLOWS

Now we turn to the consideration of the issue of local resistance losses coefficient ξ . As noted above, to date, the problem of taking into account the influence of non-steady of the process on the value of the parameter ξ has not been posed at all. Therefore, in all hydraulic calculations where local resistances are encountered, the value ξ is taken from the conditions of the stationary problem. It can be assumed that such an assumption can lead to significant errors. In this regard, it's of interest to discuss the effect of the non-steady of the coefficient ξ on the hydrodynamic processes under consideration.

In this regard at the Institute of Hydromechanics of National Academy of Sciences of Ukraine together with the National University of Water and Environmental Engineering (Rovno, Ukraine) the attempt to solve this problem for certain types of hydraulic devices was made. Given that branched pipeline systems contain various varieties of local resistances, it is of great interest to try to take into account the non-steady factor, at least for certain types of control valves.

Fig. 4 Dependence ξN / ξW on b / dSH – relative valve opening

In our preliminary experimental ones, we have considered so far only a valve of the Ludlow type, which is often used in various pipeline systems. During the experiments, pressure and velocity were measured before and after the valve. In this case, pressure and velocity sensors were placed, respectively, before and after the valve. The experiments were performed for various values of valve opening. In this case, the experiments were carried out in the valve opening diapason b / d_SH (Fig. 4) from 0.2 to 0.4. It is clear that these data cannot be generalized to other values b / d_SH. The complexity of setting up such experiments allows us to consider the obtained results as evaluative and preliminary. As follows from the analysis of these data (Fig. 4), the obtained values of the quantity ξ_N for the non-steady flow are, as a rule, larger than the corresponding value ξ_W for the stationary problem.

The above data concerning account the non- steady coefficients λ and ξ were used by us to solve one of the practical problems associated with the calculation the basic parameters for specific branched pipeline.

7. DEPENDENCE OF PRESSURE ON TIME DURING THE HYDRAULIC IMPACT IN A BRANCHED PIPELINE

Now we turn to the consideration of the issue of local resistance losses coefficient ξ . As noted above, to date, the problem of taking into account the influence of non-steady of the process on the value of the parameter ξ has not been posed at all. Therefore, in all hydraulic calculations where local resistances are encountered, the value ξ is taken from the conditions of the stationary problem. It can be assumed that such an assumption can lead to signific

As an example, we chose a branched pipeline system, which was studied in detail in [4]. In fig. 5 shows a diagram of this system, which was as mounted from steel pipes and is characterized by the following dimensions and parameters:

inner diameter

D₁ = D₄ = 311 mm; D₂ = D₃ = 259 mm; D₅ = 514 mm; D₆ = 233 mm; D₇ = 207 mm; D₈ = 614;

length of pipeline sections

l₁ = l₃ = l₅ = 920 m; l₂ = l₄ = l₆ = l₇ = 460 m; l₈ = 500 m;

Fig. 5. Layout of the branched water supply network

The problem is solved in the following statement: on a branched pipeline network, where CE, FG and FH are peak branches, and at the point A the valve is closing gradually. The water flow at point A is Q₀ = 0.1 m³ / sec, and the pressure P₀ = 0.65 Mpa.

It is required to determine the necessary valve closing time so that the maximum pressure near point A does not exceed 2 Mpa ≈ 20 atm.

The literature knows the calculation of the operation for such hydraulic network, performed by the traditional hydraulic methodic [4]. According to this approach, the required valve closing time is around 35 seconds.

We have accomplished a solution of this problem based on the presented in this paper recommendations. At the same time, the non-steady losses of friction pressure λ and local resistances losses ξ were also taken into account.

In Figure 6 comparison the calculated values of pressure near the valve obtained in this paper and according to [4] are presented. As can be seen from this Figure, the maximum pressure value differs by about 10%, which indicates that the valve closing time should be exceeded 35 seconds. Similar results were obtained when calculating the movement and for other types of simple and branched pipelines.

Fig. 6. Comparison of experimental and calculated values for impact pressure near the valve 1- basic calculation according to [4]; 2- calculation according to equations (1)

8. CONCLUSIONS

The general preliminary conclusion that can be made on the basis of even these some calculations is, first of all, the fact that when solving problems associated with the calculation of non-steady motion in pipes, it is necessary to take into account the non-steady nature of change in hydraulic friction factor λ and local resistances losses ξ . Therefore, when developing a common method for calculating of non-steady flows in pipeline systems, the presented in this paper results can be reflected in the form of additional calculation methods and subprograms.

REFERENCES

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Received: 10.11.2020
Revieved: 28.11.2020
Accepted: 1.12.2020

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Vyacheslav Berman
Institute of Hydromechanics at National Academy of Sciences of Ukraine, Kiev, Ukraine

email: slava_berman@yahoo.com

Stepan Kril
Institute of Hydromechanics at National Academy of Sciences of Ukraine, Kiev, Ukraine


Emil Bournaski
Climate, Atmosphere and Water Research Institute at Bulgarian Academy of Sciences, Sofia, Bulgaria

email: bournaski@aim.com

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