Laminar (unidirectional) air flow. Laminar and turbulent fluid movement

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
,

- head loss along the length.

At
, i.e. on the pipe axis
,

.

With laminar motion, the velocity diagram along the cross section of the pipe will have the shape of a quadratic parabola.

Turbulent mode of fluid movement

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 different kind(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 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 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 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).

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 flow of real liquid 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 n 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 factor ζ, thus, there is a ratio of lost head to velocity head.

Hydraulic losses are usually divided into local losses and friction losses along the length.

M 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.

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 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

,

Where - friction stress on the pipe wall.

If we take into account
, you can get

,

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

There are two different forms, two modes of fluid flow: laminar and turbulent flow. The flow is called laminar (layered) if along the flow each selected thin layer slides relative to its neighbors without mixing with them, and turbulent (vortex) if intense vortex formation and mixing of the liquid (gas) occurs along the flow.

Laminar the flow of liquid is observed at low speeds of its movement. In laminar flow, the trajectories of all particles are parallel and their shape follows the boundaries of the flow. In a round pipe, for example, the liquid moves in cylindrical layers, the generatrices of which are parallel to the walls and axis of the pipe. In a rectangular channel of infinite width, the liquid moves in layers parallel to its bottom. At each point in the flow, the speed remains constant in direction. If the speed does not change with time and magnitude, the motion is called steady. For laminar motion in a pipe, the velocity distribution diagram in the cross section has the form of a parabola with a maximum speed on the pipe axis and a zero value at the walls, where an adhering layer of liquid is formed. The outer layer of liquid adjacent to the surface of the pipe in which it flows adheres to it due to molecular adhesion forces and remains motionless. The greater the distance from the subsequent layers to the pipe surface, the greater the speed of subsequent layers, and the layer moving along the pipe axis has the highest speed. The profile of the average speed of a turbulent flow in pipes (Fig. 53) differs from the parabolic profile of the corresponding laminar flow by a more rapid increase in speed v.

Figure 9Profiles (diagrams) of laminar and turbulent fluid flows in pipes

The average value of the velocity in the cross section of a round pipe under steady laminar flow is determined by the Hagen-Poiseuille law:

(8)

where p 1 and p 2 are the pressure in two cross sections pipes spaced from each other at a distance Δx; r - pipe radius; η - viscosity coefficient.

The Hagen-Poiseuille law can be easily verified. It turns out that for ordinary liquids it is valid only at low flow rates or small pipe sizes. More precisely, the Hagen-Poiseuille law is satisfied only at small values ​​of the Reynolds number:

(9)

where υ is the average speed in the cross section of the pipe; l- characteristic size, in this case - pipe diameter; ν is the coefficient of kinematic viscosity.

The English scientist Osborne Reynolds (1842 - 1912) in 1883 carried out an experiment according to the following scheme: at the entrance to the pipe through which a steady flow of liquid flows, a thin tube was placed so that its opening was on the axis of the tube. Paint was supplied through a tube into the liquid stream. While laminar flow existed, the paint moved approximately along the axis of the pipe in the form of a thin, sharply limited strip. Then, starting from a certain speed value, which Reynolds called critical, wave-like disturbances and individual rapidly decaying vortices arose on the strip. As the speed increased, their number became larger and they began to develop. At a certain speed, the strip broke up into separate vortices, which spread throughout the entire thickness of the liquid flow, causing intense mixing and coloring of the entire liquid. This current was called turbulent .

Starting from a critical speed value, the Hagen-Poiseuille law was also violated. Repeating experiments with pipes of different diameters and with different liquids, Reynolds discovered that the critical speed at which the parallelism of the flow velocity vectors is broken varied depending on the size of the flow and the viscosity of the liquid, but always in such a way that the dimensionless number
took on a certain constant value in the region of transition from laminar to turbulent flow.

The English scientist O. Reynolds (1842 - 1912) proved that the nature of the flow depends on a dimensionless quantity called the Reynolds number:

(10)

where ν = η/ρ - kinematic viscosity, ρ - fluid density, υ av - average fluid velocity over the pipe cross-section, l- characteristic linear dimension, for example pipe diameter.

Thus, up to a certain value of the Re number there is a stable laminar flow, and then in a certain range of values ​​of this number the laminar flow ceases to be stable and individual, more or less quickly decaying disturbances arise in the flow. Reynolds called these numbers critical Re cr. As the Reynolds number increases further, the motion becomes turbulent. The region of critical Re values ​​usually lies between 1500-2500. It should be noted that the value of Re cr is influenced by the nature of the entrance to the pipe and the degree of roughness of its walls. With very smooth walls and a particularly smooth entrance into the pipe, the critical value of the Reynolds number could be raised to 20,000, and if the entrance to the pipe has sharp edges, burrs, etc. or the pipe walls are rough, the Re cr value can drop to 800-1000 .

In turbulent flow, fluid particles acquire velocity components perpendicular to the flow, so they can move from one layer to another. The velocity of liquid particles increases rapidly as they move away from the pipe surface, then changes quite slightly. Since liquid particles move from one layer to another, their speeds in different layers differ little. Due to the large velocity gradient near the pipe surface, vortices usually form.

Turbulent flow of liquids is most common in nature and technology. Air flow in. atmosphere, water in seas and rivers, in canals, in pipes is always turbulent. In nature, laminar movement occurs when water filters through the thin pores of fine-grained soils.

The study of turbulent flow and the construction of its theory is extremely complicated. The experimental and mathematical difficulties of these studies have so far been only partially overcome. Therefore, a number of practically important problems (water flow in canals and rivers, the movement of an aircraft of a given profile in the air, etc.) have to be either solved approximately or by testing the corresponding models in special hydrodynamic tubes. To move from the results obtained on the model to the phenomenon in nature, the so-called similarity theory is used. The Reynolds number is one of the main criteria for the similarity of the flow of a viscous fluid. Therefore, its definition is practically very important. In this work, a transition from laminar flow to turbulent flow is observed and several values ​​of the Reynolds number are determined: in the laminar flow region, in the transition region (critical flow) and in turbulent flow.

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 varies 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 jet 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 m is equal to the product of the second air flow 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 of the stream 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) this is static pressure and in the second case (when the air flow is mobile) this 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.

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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

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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

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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 L. t. is sustainable and practically implemented; meaning R e cr is usually determined experimentally. At R e> R e cr, by taking special measures to prevent random disturbances, it is also possible to obtain a linear t., but it will not be stable and, when disturbances arise, it will turn into disordered turbulent flow.Theoretically, L. t. are studied with the help Navier - Stokes equations movement of viscous fluid. Exact solutions to these equations can be obtained only in a few special cases, and usually when solving specific problems one or another approximate methods are used.