Studying the properties of liquid and gas flows is very important for industry and utilities. Laminar and turbulent flow affects the speed of transportation of water, oil, and natural gas through pipelines for various purposes and affects other parameters. The science of hydrodynamics deals with these problems.
Classification
In the scientific community, the flow regimes of liquids and gases are divided into two completely different classes:
- laminar (jet);
- turbulent.
A transition stage is also distinguished. By the way, the term “liquid” has a broad meaning: it can be incompressible (this is actually a liquid), compressible (gas), conducting, etc.
Background
Back in 1880, Mendeleev expressed the idea of the existence of two opposite flow regimes. British physicist and engineer Osborne Reynolds studied this issue in more detail, completing his research in 1883. First practically, and then using formulas, he established that at low flow speeds, the movement of liquids takes on a laminar form: layers (particle flows) hardly mix and move along parallel trajectories. However, after overcoming a certain critical value (it is different for different conditions), called the Reynolds number, the fluid flow regimes change: the jet flow becomes chaotic, vortex - that is, turbulent. As it turned out, these parameters are also characteristic of gases to a certain extent.
Practical calculations of the English scientist showed that the behavior of, for example, water strongly depends on the shape and size of the reservoir (pipe, channel, capillary, etc.) through which it flows. Pipes with a circular cross-section (such as are used for the installation of pressure pipelines) have their own Reynolds number - the formula is described as follows: Re = 2300. For flow along an open channel, it is different: Re = 900. At lower values of Re, the flow will be ordered, at higher values - chaotic .
Laminar flow
The difference between laminar flow and turbulent flow is the nature and direction of water (gas) flows. They move in layers, without mixing and without pulsations. In other words, the movement occurs evenly, without random jumps in pressure, direction and speed.
Laminar flow of liquid is formed, for example, in narrow living beings, capillaries of plants and, under comparable conditions, during the flow of very viscous liquids (fuel oil through a pipeline). To clearly see the jet flow, just open the water tap slightly - the water will flow calmly, evenly, without mixing. If the tap is turned off all the way, the pressure in the system will increase and the flow will become chaotic.
Turbulent flow
Unlike laminar flow, in which nearby particles move along almost parallel trajectories, turbulent fluid flow is disordered. If we use the Lagrange approach, then the trajectories of particles can intersect arbitrarily and behave quite unpredictably. The movements of liquids and gases under these conditions are always nonstationary, and the parameters of these nonstationarities can have a very wide range.
How the laminar regime of gas flow turns into turbulent can be traced using the example of a stream of smoke from a burning cigarette in still air. Initially, the particles move almost parallel along trajectories that do not change over time. The smoke seems motionless. Then, in some place, large vortices suddenly appear and move completely chaotically. These vortices break up into smaller ones, those into even smaller ones, and so on. Eventually, the smoke practically mixes with the surrounding air.
Turbulence cycles
The example described above is textbook, and from its observation, scientists have drawn the following conclusions:
- Laminar and turbulent flow are probabilistic in nature: the transition from one regime to another does not occur in a precisely specified place, but in a rather arbitrary, random place.
- First, large vortices appear, the size of which is larger than the size of a stream of smoke. The movement becomes unsteady and highly anisotropic. Large flows lose stability and break up into smaller and smaller ones. Thus, a whole hierarchy of vortices arises. The energy of their movement is transferred from large to small, and at the end of this process disappears - energy dissipation occurs at small scales.
- The turbulent flow regime is random in nature: one or another vortex can end up in a completely arbitrary, unpredictable place.
- Mixing of smoke with the surrounding air practically does not occur in laminar conditions, but in turbulent conditions it is very intense.
- Despite the fact that the boundary conditions are stationary, the turbulence itself has a pronounced non-stationary character - all gas-dynamic parameters change over time.
There is another important property of turbulence: it is always three-dimensional. Even if we consider a one-dimensional flow in a pipe or a two-dimensional boundary layer, the movement of turbulent vortices still occurs in the directions of all three coordinate axes.
Reynolds number: formula
The transition from laminarity to turbulence is characterized by the so-called critical Reynolds number:
Re cr = (ρuL/µ) cr,
where ρ is the flow density, u is the characteristic flow speed; L is the characteristic size of the flow, µ is the coefficient cr - flow through a pipe with a circular cross-section.
For example, for a flow with speed u in a pipe, L is used as Osborne Reynolds showed that in this case 2300 A similar result is obtained in the boundary layer on the plate. The distance from the leading edge of the plate is taken as a characteristic size, and then: 3 × 10 5 Laminar and turbulent fluid flow, and, accordingly, the critical value of the Reynolds number (Re) depend on a large number of factors: pressure gradient, height of roughness tubercles, intensity of turbulence in the external flow, temperature difference, etc. For convenience, these total factors are also called velocity disturbance , since they have a certain effect on the flow rate. If this disturbance is small, it can be extinguished by viscous forces tending to level the velocity field. With large disturbances, the flow may lose stability and turbulence occurs. Considering that the physical meaning of the Reynolds number is the ratio of inertial forces and viscous forces, the disturbance of flows falls under the formula: Re = ρuL/µ = ρu 2 /(µ×(u/L)). The numerator contains double the velocity pressure, and the denominator contains a quantity of the order of friction stress if the thickness of the boundary layer is taken as L. The high-speed pressure tends to destroy the balance, but this is counteracted. However, it is not clear why (or the velocity pressure) leads to changes only when they are 1000 times greater than the viscous forces. It would probably be more convenient to use the velocity disturbance rather than the absolute flow velocity u as the characteristic velocity in Recr. In this case, the critical Reynolds number will be of the order of 10, that is, when the disturbance of the velocity pressure exceeds the viscous stresses by 5 times, the laminar flow of the fluid becomes turbulent. This definition of Re, according to a number of scientists, well explains the following experimentally confirmed facts. For an ideally uniform velocity profile on an ideally smooth surface, the traditionally determined number Re cr tends to infinity, that is, the transition to turbulence is actually not observed. But the Reynolds number, determined by the magnitude of the speed disturbance, is less than the critical one, which is equal to 10. In the presence of artificial turbulators that cause a burst of speed comparable to the main speed, the flow becomes turbulent at much lower values of the Reynolds number than Re cr determined from the absolute value of the speed. This makes it possible to use the value of the coefficient Re cr = 10, where the absolute value of the speed disturbance caused by the above reasons is used as the characteristic speed. Laminar and turbulent flow is characteristic of all types of liquids and gases under different conditions. In nature, laminar flows are rare and are characteristic, for example, of narrow underground flows in flat conditions. This issue worries scientists much more in the context of practical applications for transporting water, oil, gas and other technical liquids through pipelines. The issue of laminar flow stability is closely related to the study of the perturbed motion of the main flow. It has been established that it is exposed to so-called small disturbances. Depending on whether they fade or grow over time, the main flow is considered stable or unstable. One of the factors influencing the laminar and turbulent flow of a fluid is its compressibility. This property of a liquid is especially important when studying the stability of unsteady processes with a rapid change in the main flow. Research shows that laminar flow of incompressible fluid in pipes of cylindrical cross-section is resistant to relatively small axisymmetric and non-axisymmetric disturbances in time and space. Recently, calculations have been carried out on the influence of axisymmetric disturbances on the stability of the flow in the inlet part of a cylindrical pipe, where the main flow depends on two coordinates. In this case, the coordinate along the pipe axis is considered as a parameter on which the velocity profile along the pipe radius of the main flow depends. Despite centuries of study, it cannot be said that both laminar and turbulent flow have been thoroughly studied. Experimental studies at the micro level raise new questions that require reasoned computational justification. The nature of the research also has practical benefits: thousands of kilometers of water, oil, gas, and product pipelines have been laid throughout the world. The more technical solutions are implemented to reduce turbulence during transportation, the more effective it will be. Laminar flow liquid is called layered flow without mixing of liquid particles and without pulsations of speed and pressure. The law of velocity distribution over the cross section of a round pipe in a laminar mode of motion, established by the English physicist J. Stokes, has the form Where At With laminar motion, the velocity diagram along the cross section of the pipe will have the shape of a quadratic parabola. Turbulent called a flow accompanied by intense mixing of the liquid and pulsations of speeds and pressures. As a result of the presence of vortices and intense mixing of liquid particles, at any point in the turbulent flow at a given moment in time there is an instantaneous local velocity of its own in value and direction u, and the trajectory of particles passing through this point has a different appearance (they occupy different positions in space and have different shapes). Such a fluctuation in time of instantaneous local speed is called speed pulsation. The same thing happens with pressure. Thus, turbulent motion is unsteady. Average
local speed ū
– fictitious average speed at a given point of the flow for a sufficiently long period of time, which, despite significant fluctuations in instantaneous speeds, remains almost constant in value and parallel to the flow axis P The laminar sublayer, located directly at the pipe walls, has a very small thickness δ
, which can be determined by the formula In the transition layer, the laminar flow is already disrupted by the transverse movement of particles, and the further the point is located from the pipe wall, the higher the intensity of particle mixing. The thickness of this layer is also small, but it is difficult to establish a clear boundary. The main part of the live cross-section of the flow is occupied by the core of the flow, in which intense mixing of particles is observed, therefore it is this that characterizes the turbulent movement of the flow as a whole. THE CONCEPT OF HYDRAULICALLY SMOOTH AND ROUGH PIPES P Depending on the ratio of the thickness of the laminar sublayer δ
and the heights of roughness protrusions Δ are distinguished hydraulically smooth And rough pipes. If the laminar sublayer completely covers all protrusions on the pipe walls, i.e. δ>Δ, pipes are considered hydraulically smooth. At δ<Δ трубы считаются
гидравлически шероховатыми. Так как
значение δ зависит от Re,
то одна и та же труба может быть в одних
и тех же условиях гидравлически гладкой
(при малых Re),
а в других – шероховатой (при больших
Re). Lecture No. 9
HYDRAULIC LOSSES GENERAL INFORMATION. When a real fluid flow moves, pressure losses occur, since part of the specific energy of the flow is spent on overcoming various hydraulic resistances. Quantitative determination of head loss h P
is one of the most important problems of hydrodynamics, without solving which the practical use of Bernoulli’s equation is not possible: Where α
–
kinetic energy coefficient equal to 1.13 for turbulent flow, and 2 for laminar flow; v-average flow speed; h- a decrease in the specific mechanical energy of the flow in the area between sections 1 and 2, occurring as a result of internal friction forces. Loss of specific energy (pressure), or, as they are often called, hydraulic losses, depend on the shape, size of the channel, flow speed and viscosity of the liquid, and sometimes on the absolute pressure in it. The viscosity of the liquid, although it is the root cause of all hydraulic losses, does not always have a significant effect on their magnitude. As experiments show, in many, but not all cases, hydraulic losses are approximately proportional to the fluid flow velocity to the second power, therefore in hydraulics the following general method of expressing hydraulic losses of total head in linear units is accepted: or in pressure units This expression is convenient because it includes the dimensionless proportionality coefficient ζ
called loss factor, or the resistance coefficient, the value of which for a given channel is constant in the first rough approximation. Loss ratio ζ,
thus, there is a ratio of the lost head to the velocity head. Hydraulic losses are usually divided into local losses and friction losses along the length. M Local pressure losses are determined using the Weisbach formula as follows: or in pressure units Where v- average cross-sectional speed in the pipe in which this local resistance is installed. If the diameter of the pipe and, consequently, the speed in it varies along the length, then it is more convenient to take the larger of the speeds as the design speed, i.e. the one that corresponds to the smaller pipe diameter. Each local resistance is characterized by its own resistance coefficient value ζ
, which in many cases can be approximately considered constant for a given form of local resistance. Friction losses along the length are energy losses that occur in their pure form in straight pipes of constant cross-section, i.e. with uniform flow, and increase in proportion to the length of the pipe. The losses under consideration are due to internal losses in the liquid, and therefore occur not only in rough, but also in smooth pipes. Friction head losses can be expressed using the general formula for hydraulic losses, i.e. however, the coefficient is more convenient ζ
connect with relative long pipe l/
d.
Let us take a section of a round pipe with a length equal to its diameter and denote its loss coefficient by λ
. Then for the entire long pipe l
and diameter d.
the loss factor will be in l/
d
times more: Then the pressure loss due to friction is determined by the Weisbach-Darcy formula: or in pressure units Dimensionless coefficient λ
called friction loss coefficient along the length, or Darcy coefficient. It can be considered as a coefficient of proportionality between the loss of pressure due to friction and the product of the relative length of the pipe and the velocity pressure. N Where
If consider those. coefficient λ
is a value proportional to the ratio of the friction stress on the pipe wall to the dynamic pressure determined by the average speed. Due to the constant volume flow of incompressible fluid along a pipe of constant cross-section, the speed and specific kinetic energy also remain constant, despite the presence of hydraulic resistance and pressure losses. The pressure loss in this case is determined by the difference in the readings of two piezometers. Lecture No. 10
The movement of fluid observed at low speeds, in which individual streams of fluid move parallel to each other and the flow axis, is called laminar fluid movement. A very clear idea of the laminar regime of fluid movement can be obtained from Reynolds' experiment. Detailed description . The liquid flows out of the tank through a transparent pipe and goes through the tap to the drain. Thus, the liquid flows at a certain small and constant flow rate. At the entrance to the pipe there is a thin tube through which a colored medium enters the central part of the flow. When paint enters a flow of liquid moving at low speed, the red paint will move in an even stream. From this experiment we can conclude that the liquid flows in a layered manner, without mixing and vortex formation. This mode of fluid flow is usually called laminar. Let us consider the basic laws of the laminar regime with uniform movement in round pipes, limiting ourselves to cases where the pipe axis is horizontal. In this case, we will consider an already formed flow, i.e. flow in a section, the beginning of which is located from the inlet section of the pipe at a distance that provides the final stable form of velocity distribution over the flow section. Bearing in mind that the laminar flow regime has a layered (jet) character and occurs without mixing of particles, it should be assumed that in a laminar flow there will only be velocities parallel to the pipe axis, while transverse velocities will be absent. One can imagine that in this case the moving liquid seems to be divided into an infinitely large number of infinitely thin cylindrical layers, parallel to the axis of the pipeline and moving one inside the other at different speeds, increasing in the direction from the walls to the axis of the pipe. In this case, the velocity in the layer directly in contact with the walls due to the adhesion effect is zero and reaches its maximum value in the layer moving along the axis of the pipe. The accepted motion scheme and the assumptions introduced above make it possible to theoretically establish the law of velocity distribution in the cross section of the flow in laminar mode. To do this, we will do the following. Let us denote the internal radius of the pipe by r and choose the origin of coordinates at the center of its cross section O, directing the x axis along the axis of the pipe, and the z axis vertically. Now let’s select a volume of liquid inside the pipe in the form of a cylinder of a certain radius y and length L and apply Bernoulli’s equation to it. Since due to the horizontal axis of the pipe z1=z2=0, then where R is the hydraulic radius of the section of the selected cylindrical volume = y/2 τ – unit friction force = - μ * dυ/dy Substituting the values of R and τ into the original equation we get By specifying different values of the y coordinate, you can calculate the velocities at any point in the section. The maximum speed will obviously be at y=0, i.e. on the axis of the pipe. In order to represent this equation graphically, it is necessary to plot the velocity on a certain scale from some arbitrary straight line AA in the form of segments directed along the fluid flow, and connect the ends of the segments with a smooth curve. The resulting curve will represent the velocity distribution curve in the cross section of the flow. The graph of changes in friction force τ across a cross section looks completely different. Thus, in a laminar mode in a cylindrical pipe, the velocities in the cross section of the flow change according to a parabolic law, and the tangential stresses change according to a linear law. The results obtained are valid for pipe sections with fully developed laminar flow. In fact, the liquid that enters the pipe must pass a certain section from the inlet section before a parabolic velocity distribution law corresponding to the laminar regime is established in the pipe. The development of a laminar regime in a pipe can be imagined as follows. Let, for example, liquid enter a pipe from a large reservoir, the edges of the inlet hole of which are well rounded. In this case, the velocities at all points of the inlet cross section will be almost the same, with the exception of a very thin, so-called wall layer (layer near the walls), in which, due to the adhesion of the liquid to the walls, an almost sudden drop in speed to zero occurs. Therefore, the velocity curve in the inlet section can be represented quite accurately in the form of a straight line segment. As we move away from the entrance, due to friction at the walls, the layers of liquid adjacent to the boundary layer begin to slow down, the thickness of this layer gradually increases, and the movement in it, on the contrary, slows down. The central part of the flow (the core of the flow), not yet captured by friction, continues to move as one whole, with approximately the same speed for all layers, and the slowdown of movement in the near-wall layer inevitably causes an increase in the speed in the core. Thus, in the middle of the pipe, in the core, the flow velocity increases all the time, and near the walls, in the growing boundary layer, it decreases. This occurs until the boundary layer covers the entire flow cross section and the core is reduced to zero. At this point, the formation of the flow ends, and the velocity curve takes on the parabolic shape usual for the laminar regime. Under certain conditions, laminar fluid flow can become turbulent. As the speed of the flow increases, the layered structure of the flow begins to collapse, waves and vortices appear, the propagation of which in the flow indicates increasing disturbance. Gradually, the number of vortices begins to increase, and increases until the stream breaks into many smaller streams mixing with each other. The chaotic movement of such small streams suggests the beginning of the transition from laminar flow to turbulent. As the speed increases, the laminar flow loses its stability, and any random small disturbances that previously caused only small fluctuations begin to develop rapidly.
In everyday life, the transition from one flow regime to another can be traced using the example of a stream of smoke. At first, the particles move almost parallel along time-invariant trajectories. The smoke is practically motionless. Over time, large vortices suddenly appear in some places and move along chaotic trajectories. These vortices break up into smaller ones, those into even smaller ones, and so on. Eventually, the smoke practically mixes with the surrounding air. Laminar is an air flow in which air streams move in one direction and are parallel to each other. When the speed increases to a certain value, the streams of air flow, in addition to translational speed, also acquire rapidly changing speeds perpendicular to the direction of translational movement. A flow is formed, which is called turbulent, i.e. disorderly. Boundary layer The boundary layer is a layer in which the air speed changes from zero to a value close to the local air flow speed. When an air flow flows around a body (Fig. 5), air particles do not slide over the surface of the body, but are slowed down, and the air speed at the surface of the body becomes zero. When moving away from the surface of the body, the air speed increases from zero to the speed of the air flow. The thickness of the boundary layer is measured in millimeters and depends on the viscosity and pressure of the air, the profile of the body, the state of its surface and the position of the body in the air flow. The thickness of the boundary layer gradually increases from the leading to the trailing edge. In the boundary layer, the nature of the movement of air particles differs from the nature of the movement outside it. Let's consider an air particle A (Fig. 6), which is located between streams of air with velocities U1 and U2, due to the difference in these velocities applied to opposite points of the particle, it rotates, and the closer this particle is to the surface of the body, the more it rotates (where the difference speeds are highest). When moving away from the surface of the body, the rotational motion of the particle slows down and becomes equal to zero due to the equality of the air flow speed and the air speed of the boundary layer. Behind the body, the boundary layer turns into a cocurrent jet, which blurs out and disappears as it moves away from the body. The turbulence in the wake falls on the tail of the aircraft and reduces its efficiency and causes shaking (buffeting phenomenon). The boundary layer is divided into laminar and turbulent (Fig. 7). In a steady laminar flow of the boundary layer, only internal friction forces due to the viscosity of the air appear, so the air resistance in the laminar layer is low. Rice. 5 Rice. 6 Air flow around a body - deceleration of the flow in the boundary layer
Rice. 7 In a turbulent boundary layer, there is a continuous movement of air streams in all directions, which requires more energy to maintain a random vortex motion and, as a consequence of this, creates a greater resistance to the air flow to the moving body. To determine the nature of the boundary layer, the coefficient Cf is used. A body of a certain configuration has its own coefficient. So, for example, for a flat plate the resistance coefficient of the laminar boundary layer is equal to: for a turbulent layer where Re is the Reynolds number, expressing the ratio of inertial forces to frictional forces and determining the ratio of two components - profile resistance (shape resistance) and friction resistance. Reynolds number Re is determined by the formula: where V is the air flow speed, I - nature of body size, kinetic coefficient of viscosity of air friction forces. When an air flow flows around a body, at a certain point the boundary layer transitions from laminar to turbulent. This point is called the transition point. Its location on the surface of the body profile depends on the viscosity and pressure of the air, the speed of the air streams, the shape of the body and its position in the air flow, as well as the surface roughness. When creating wing profiles, designers strive to place this point as far as possible from the leading edge of the profile, thereby reducing friction drag. For this purpose, special laminated profiles are used to increase the smoothness of the wing surface and a number of other measures. When the speed of the air flow increases or the angle of position of the body relative to the air flow increases to a certain value, at a certain point the boundary layer is separated from the surface, and the pressure behind this point sharply decreases. As a result of the fact that at the trailing edge of the body the pressure is greater than behind the separation point, a reverse flow of air occurs from a zone of higher pressure to a zone of lower pressure to the separation point, which entails separation of the air flow from the surface of the body (Fig. 8). A laminar boundary layer comes off more easily from the surface of a body than a turbulent boundary layer. Air flow continuity equation The equation of continuity of a jet of air flow (constancy of air flow) is an equation of aerodynamics that follows from the basic laws of physics - conservation of mass and inertia - and establishes the relationship between the density, speed and cross-sectional area of a jet of air flow. Rice. 8 Rice. 9 When considering it, the condition is accepted that the air under study does not have the property of compressibility (Fig. 9). In a stream of variable cross-section, a second volume of air flows through section I over a certain period of time; this volume is equal to the product of the air flow velocity and the cross section F. The second mass air flow rate m is equal to the product of the second air flow rate and the density p of the air flow of the stream. According to the law of conservation of energy, the mass of the air flow m1 flowing through section I (F1) is equal to the mass m2 of the given flow flowing through section II (F2), provided that the air flow is steady: m1=m2=const, (1.7) m1F1V1=m2F2V2=const. (1.8) This expression is called the equation of continuity of a stream of air flow of a stream. F1V1=F2V2= const. (1.9) So, from the formula it is clear that the same volume of air passes through different sections of the stream in a certain unit of time (second), but at different speeds. Let us write equation (1.9) in the following form: The formula shows that the speed of the air flow of the jet is inversely proportional to the cross-sectional area of the jet and vice versa. Thus, the air flow continuity equation establishes the relationship between the cross section of the jet and the speed, provided that the air flow of the jet is steady. Static pressure and velocity head Bernoulli equation air plane aerodynamics An airplane located in a stationary or moving air flow relative to it experiences pressure from the latter, in the first case (when the air flow is stationary) it is static pressure and in the second case (when the air flow is moving) it is dynamic pressure, it is more often called high-speed pressure. The static pressure in the stream is similar to the pressure of a liquid at rest (water, gas). For example: water in a pipe, it can be at rest or in motion, in both cases the walls of the pipe are under pressure from the water. In the case of water movement, the pressure will be slightly less, since a high-speed pressure has appeared. According to the law of conservation of energy, the energy of a stream of air flow in various sections of a stream of air is the sum of the kinetic energy of the flow, the potential energy of pressure forces, the internal energy of the flow and the energy of the body position. This amount is a constant value: Ekin+Er+Evn+En=sopst (1.10) Kinetic energy (Ekin) is the ability of a moving air flow to do work. It is equal where m is air mass, kgf s2m; V-air flow speed, m/s. If we substitute air mass density p instead of mass m, we obtain a formula for determining the velocity pressure q (in kgf/m2) Potential energy Ep is the ability of an air flow to do work under the influence of static pressure forces. It is equal (in kgf-m) where P is air pressure, kgf/m2; F is the cross-sectional area of the air stream, m2; S is the path traveled by 1 kg of air through a given section, m; the product SF is called the specific volume and is denoted by v. Substituting the value of the specific volume of air into formula (1.13), we obtain Internal energy Evn is the ability of a gas to do work when its temperature changes: where Cv is the heat capacity of air at a constant volume, cal/kg-deg; T-temperature on the Kelvin scale, K; A is the thermal equivalent of mechanical work (cal-kg-m). From the equation it is clear that the internal energy of the air flow is directly proportional to its temperature. Position energy En is the ability of air to do work when the position of the center of gravity of a given mass of air changes when rising to a certain height and is equal to where h is the change in height, m. Due to the minutely small values of the separation of the centers of gravity of air masses along the height in a stream of air flow, this energy is neglected in aerodynamics. Considering all types of energy in relation to certain conditions, we can formulate Bernoulli’s law, which establishes a connection between the static pressure in a stream of air flow and the speed pressure. Let's consider a pipe (Fig. 10) of variable diameter (1, 2, 3) in which the air flow moves. Pressure gauges are used to measure pressure in the sections under consideration. Analyzing the readings of pressure gauges, we can conclude that the lowest dynamic pressure is shown by a pressure gauge with cross section 3-3. This means that as the pipe narrows, the air flow speed increases and the pressure drops. Rice. 10 The reason for the pressure drop is that the air flow does not produce any work (friction is not taken into account) and therefore the total energy of the air flow remains constant. If we consider the temperature, density and volume of air flow in different sections to be constant (T1=T2=T3;р1=р2=р3, V1=V2=V3), then the internal energy can be ignored. This means that in this case it is possible for the kinetic energy of the air flow to transform into potential energy and vice versa. When the speed of the air flow increases, the speed pressure and, accordingly, the kinetic energy of this air flow also increases. Let us substitute the values from formulas (1.11), (1.12), (1.13), (1.14), (1.15) into formula (1.10), taking into account that we neglect the internal energy and position energy, transforming equation (1.10), we obtain This equation for any cross section of a stream of air is written as follows: This type of equation is the simplest mathematical Bernoulli equation and shows that the sum of static and dynamic pressures for any section of a stream of steady air flow is a constant value. Compressibility is not taken into account in this case. When taking compressibility into account, appropriate corrections are made. To illustrate Bernoulli's law, you can conduct an experiment. Take two sheets of paper, holding them parallel to each other at a short distance, and blow into the gap between them. Rice. eleven The sheets are getting closer. The reason for their convergence is that on the outside of the sheets the pressure is atmospheric, and in the interval between them, due to the presence of high-speed air pressure, the pressure decreased and became less than atmospheric. Under the influence of pressure differences, sheets of paper bend inward. Wind tunnels An experimental setup for studying the phenomena and processes accompanying the flow of gas around bodies is called a wind tunnel. The principle of operation of wind tunnels is based on Galileo's principle of relativity: instead of the movement of a body in a stationary medium, the flow of gas around a stationary body is studied. In wind tunnels, the aerodynamic forces and moments acting on the aircraft are experimentally determined, the distribution of pressure and temperature over its surface is studied, the pattern of flow around the body is observed, and aeroelasticity is studied. etc. Wind tunnels, depending on the range of Mach numbers M, are divided into subsonic (M = 0.15-0.7), transonic (M = 0.7-1 3), supersonic (M = 1.3-5) and hypersonic (M = 5-25), according to the principle of operation - into compressor (continuous action), in which the air flow is created by a special compressor, and balloons with increased pressure, according to the circuit layout - into closed and open. Compressor pipes have high efficiency, they are convenient to use, but they require the creation of unique compressors with high gas flow rates and high power. Balloon wind tunnels are less economical than compressor wind tunnels, since some energy is lost when throttling the gas. In addition, the duration of operation of balloon wind tunnels is limited by the gas reserves in the tanks and ranges from tens of seconds to several minutes for various wind tunnels. The widespread use of balloon wind tunnels is due to the fact that they are simpler in design and the compressor power required to fill the balloons is relatively small. Closed-loop wind tunnels utilize a significant portion of the kinetic energy remaining in the gas stream after it passes through the work area, increasing the efficiency of the tube. In this case, however, it is necessary to increase the overall dimensions of the installation. In subsonic wind tunnels, the aerodynamic characteristics of subsonic helicopter aircraft are studied, as well as the characteristics of supersonic aircraft in takeoff and landing modes. In addition, they are used to study the flow around cars and other ground vehicles, buildings, monuments, bridges and other objects. Figure shows a diagram of a subsonic closed-loop wind tunnel. Rice. 12 1 - honeycomb 2 - grids 3 - prechamber 4 - confuser 5 - flow direction 6 - working part with model 7 - diffuser, 8 - elbow with rotating blades, 9 - compressor 10 - air cooler Rice. 13 1 - honeycomb 2 - grids 3 - pre-chamber 4 confuser 5 perforated working part with model 6 ejector 7 diffuser 8 elbow with guide vanes 9 air exhaust 10 - air supply from cylinders Rice. 14 1 - compressed air cylinder 2 - pipeline 3 - regulating throttle 4 - leveling grids 5 - honeycomb 6 - deturbulizing grids 7 - prechamber 8 - confuser 9 - supersonic nozzle 10 - working part with model 11 - supersonic diffuser 12 - subsonic diffuser 13 - atmospheric release Rice. 15 1 - high pressure cylinder 2 - pipeline 3 - control throttle 4 - heater 5 - pre-chamber with honeycomb and grids 6 - hypersonic axisymmetric nozzle 7 - working part with model 8 - hypersonic axisymmetric diffuser 9 - air cooler 10 - flow direction 11 - air supply into ejectors 12 - ejectors 13 - shutters 14 - vacuum tank 15 - subsonic diffuser LAMINAR FLOW(from Latin lamina - plate) - an ordered flow regime of a viscous liquid (or gas), characterized by the absence of mixing between adjacent layers of liquid. The conditions under which stable, i.e., not disturbed by random disturbances, L. t. can occur depend on the value of the dimensionless Reynolds number Re. For each type of flow there is such a number R e Kr, called lower critical Reynolds number, which for any Re An idea of the features of linear motion is given by the well-studied case of motion in a round cylindrical. pipe For this current R e Kr 2200, where Re=
(
- average fluid velocity, d- pipe diameter,
- kinematic coefficient viscosity, - dynamic coefficient viscosity, - fluid density). Thus, practically stable laser flow can occur either with a relatively slow flow of a sufficiently viscous liquid or in very thin (capillary) tubes. For example, for water (= 10 -6 m 2 / s at 20 ° C) stable L. t. s = 1 m / s is possible only in tubes with a diameter of no more than 2.2 mm. With LP in an infinitely long pipe, the speed in any section of the pipe changes according to the law -(1 - - r 2 /A 2), where A- pipe radius, r- distance from the axis, - axial (numerically maximum) flow velocity; the corresponding parabolic. the velocity profile is shown in Fig. A. The friction stress varies along the radius according to a linear law where = is the friction stress on the pipe wall. To overcome the forces of viscous friction in a pipe with uniform motion, there must be a longitudinal pressure drop, usually expressed by the equality P 1 -P 2 Velocity distribution over the pipe cross section: A- with laminar flow; b- in turbulent flow. When the flow becomes turbulent, the flow structure and velocity profile change significantly (Fig. 6
) and the law of resistance, i.e. dependence on Re(cm. Hydrodynamic resistance). In addition to pipes, lubrication occurs in the lubrication layer in bearings, near the surface of bodies flowing around a low-viscosity fluid (see Fig. Boundary layer), when a very viscous liquid flows slowly around small bodies (see, in particular, Stokes formula). The theory of laser theory is also used in viscometry, in the study of heat transfer in a moving viscous fluid, in the study of the movement of drops and bubbles in a liquid medium, in the consideration of flows in thin films of liquid, and in solving a number of other problems in physics and physical science. chemistry. Lit.: Landau L.D., Lifshits E.M., Mechanics of Continuous Media, 2nd ed., M., 1954; Loytsyansky L.G., Mechanics of liquid and gas, 6th ed., M., 1987; Targ S.M., Basic problems of the theory of laminar flows, M.-L., 1951; Slezkin N.A., Dynamics of a viscous incompressible fluid, M., 1955, ch. 4 - 11. S. M. Targ.Concept of speed disturbance
Calculations and facts
Stability of laminar flow in a pipeline
Flow of compressible and non-compressible fluids
Conclusion
,
,
- head loss along the length.
, i.e. on the pipe axis 
,
.Turbulent mode of fluid movement
.
o Prandtl turbulent flow consists of two regions: laminar sublayer And turbulent core flow, between which there is another area - transition layer. The combination of a laminar sublayer and a transition layer in hydrodynamics is usually called boundary layer.
.
the surface of the walls of pipes, channels, trays have one or another roughness. Let us denote the height of the roughness protrusions by the letter Δ. The quantity Δ is called absolute roughness, and its ratio to the pipe diameter (Δ/d) - relative roughness; the reciprocal value of the relative roughness is called relative smoothness(d/Δ).
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natural losses energy is caused by the so-called local hydraulic resistance, i.e. local changes in the shape and size of the channel, causing deformation of the flow. When a fluid flows through local resistances, its speed changes and large vortices usually appear. The latter are formed behind the place where the flow separates from the walls and represent areas in which fluid particles move mainly along closed curves or trajectories close to them.
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It is difficult to find out the physical meaning of the coefficient λ
, if we consider the condition of uniform motion in a pipe of cylindrical volume with length l and diameter d, i.e. the equality to zero of the sum of forces acting on the volume: pressure forces and friction forces. This equality has the form
,
- friction stress on the pipe wall.
, you can get
,Laminar motion mode in experiments
Laminar flow formula
Development of laminar regime in a pipe
Transition from laminar to turbulent flow
Video about laminar flow
Where p 1 And p 2- pressure in the kn. two cross sections located at a distance l from each other - coefficient. resistance, depending on for L. t. The second flow rate of liquid in a pipe at L.t. is determined by Poiseuille's law. In pipes of finite length, the described L. t. is not established immediately and at the beginning of the pipe there is a so-called. the entrance section, where the velocity profile gradually transforms into parabolic. Approximate length of the input section 