As experiments show, two modes of flow of liquids and gases are possible: laminar and turbulent.
Laminar is a complex flow without mixing of fluid particles and without pulsations of speeds and pressures. With laminar movement of liquid in a straight pipe of constant cross-section, all flow lines are directed parallel to the axis of the pipes, there is no transverse movement of the liquid. However, laminar motion cannot be considered irrotational, since although there are no visible vortices in it, at the same time as the translational motion there is an ordered rotational movement individual fluid particles around their instantaneous centers with certain angular velocities.
Turbulent flow is a flow accompanied by intense mixing of liquid and pulsations of speeds and pressures. In turbulent flow, along with the main longitudinal movement of the liquid, transverse movements and rotational movement of individual volumes of liquid occur.
A change in the flow regime occurs at a certain ratio between the speed V, diameter d, and viscosity υ. These three factors are included in the formula of the dimensionless Reynolds criterion R e = V d /υ, therefore it is quite natural that it is the number R e that is the criterion that determines the flow regime in pipes.
The number Re at which laminar motion becomes turbulent is called critical Recr.
As experiments show, for round pipes Recr = 2300, that is, at Re< Reкр течение является ламинарным, а при Rе >Recr - turbulent. More precisely, a fully developed turbulent flow in pipes is established only at Re = 4000, and at Re = 2300 - 4000 a transitional critical region occurs.
The change in flow regime when Re cr is reached is due to the fact that one flow loses stability, and the other gains stability.
Let us consider laminar flow in more detail.
One of the most simple types movement of a viscous fluid is laminar movement in a cylindrical pipe, and in particular its special case- steady uniform motion. The theory of laminar fluid motion is based on Newton's law of friction. This friction between layers of moving fluid is the only source of energy loss.
Let us consider the established laminar flow of liquid in a straight pipe with d = 2 r 0
To eliminate the influence of gravity and thereby simplify the conclusion, let us assume that the pipe is located horizontally.
Let the pressure in section 1-1 be equal to P 1 and in section 2-2 - P 2.
Due to the constant pipe diameter V = const, £ = const, then the Bernoulli equation for the selected sections will take the form:
Hence, this is what piezometers installed in sections will show.
Let us select a cylindrical volume in the liquid flow.
Let us write down the equation of uniform motion of a selected volume of liquid, that is, the equality 0 of the sum of forces acting on the volume.
It follows that the tangential stresses in the cross section of the pipe vary linearly depending on the radius.
If we express the shear stress t according to Newton’s law, we will have
The minus sign is due to the fact that the direction of reference r (from the axis to the wall) is opposite the direction of reference y (from the wall)
And substituting the value of t into the previous equation, we get
From here we find the speed increment.
After performing the integration we get:
We find the integration constant from the condition for r = r 0; V=0
The speed in a circle of radius r is equal to
This expression is the law of velocity distribution over the cross section of a round pipe in laminar flow. The curve depicting the velocity diagram is a parabola of the second degree. The maximum speed occurring at the center of the section at r = 0 is
Let us apply the resulting velocity distribution law to calculate the flow rate.
It is advisable to take the area dS in the form of a ring with radius r and width dr
Then 
After integration over the entire cross-sectional area, that is, from r = 0, to r = r 0
To obtain the law of resistance, we express; (via previous flow formula)
(
µ=υρ r 0 = d/2 γ = ρg. Then we obtain Poireille's law;
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.
Laminar motion mode in experiments
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 experience we can conclude that the fluid 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 motion in round pipes, limited 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.
Laminar flow formula
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 asking different meanings coordinates y, 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.
Development of laminar regime in a 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.
Transition from laminar to turbulent flow
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.
Video about laminar flow
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.
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 isolated 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 of the pipe, spaced apart 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 hole 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 surface of the pipe, 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.
When fluid particles move without intersecting each other's trajectories, and the velocity vector becomes tangent to the trajectory, then such a flow is called directed. When it occurs, layers of liquid, as a rule, slide relative to each other. This flow is known as laminar flow. An important condition Its existence is due to the relatively small movement of particles.
In laminar flow, the layer that is in contact with fixed surface, has zero speed. In the direction perpendicular to the surface, the speed of the layers gradually increases. In addition, pressure, density and other dynamic properties of the fluid remain unchanged at each point in space inside the flow.
The Reynolds number is a quantitative indicator of the nature of fluid flow. When it is small (less than 1000) the flow is laminar. In this case, the interaction occurs through the force of inertia. For values between 1000 and 2000, the flow is neither turbulent nor laminar. In other words, there is a transition from one type of movement to another. The Reynolds number is a dimensionless quantity.
What is turbulent flow?
When the properties of a fluid in a flow change rapidly over time, it is called turbulent. Speed, pressure, density and other indicators, in this case, take completely random values.
A fluid moving in a uniform cylindrical tube of finite length, also known as a Poiseuille tube, will be turbulent when the Reynolds number reaches a critical value (about 2000). However, the flow cannot be explicitly turbulent when the Reynolds number is greater than 10,000.
Turbulent flow is characterized by the random nature of its characteristics, diffusion and eddies. The only method for studying them is experiment.
What is the difference between laminar and turbulent flow?
In laminar flow, flow occurs at low speeds with a low Reynolds number, and it becomes turbulent at high speeds and large numbers Reynolds.
In laminar flow, fluid parameters are predictable and practically do not change. In this case, there are no disturbances in the movement of layers and their mixing. In a turbulent flow, the flow pattern is chaotic. There are eddies, whirlpools, and cross currents.
Within laminar flow, the properties of the fluid at any point in space remain unchanged over time. In the case of turbulent flow they are stochastic.
) moves as if in layers parallel to the direction of the flow. L. t. is observed either in very viscous liquids, or in flows occurring at fairly low speeds, as well as in the slow flow of liquid around small bodies. In particular, luminescent processes take place in narrow (capillary) tubes, in a lubricant layer in bearings, in a thin boundary layer formed near the surface of bodies when liquid or gas flows around them, etc. With an increase in the speed of movement of a given liquid, luminous flows occur. . at some point turns into . At the same time, all its properties change significantly, in particular the flow structure, velocity profile, and the law of resistance. The fluid flow regime is characterized by the Reynolds number Re. When the Re value is less than critical. number Recr, L. t. liquid takes place; if Re > Recr, the flow becomes turbulent. The value of Recr depends on the type of flow under consideration. Thus, for flow in round pipes ReKp » 2300 (if the characteristic speed is considered to be the average over the cross section, and the characteristic size is the diameter of the pipe). At Recr
Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1983 .
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. with = 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 
Where p 1 And p 2 - pressure in Ph.D. two cross sections located at a distance l from each other - coefficient. resistance, depending on for L. t. The second of liquid in a pipe at L. t. determines 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 inlet section, where the velocity profile gradually transforms into parabolic. Approximate length of the input section
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 fluid 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 liquid, 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.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .
See what "LAMINAR FLOW" is in other dictionaries:
Modern encyclopedia
Laminar flow- (from the Latin lamina plate, strip), an ordered flow of liquid or gas, in which the liquid (gas) moves in layers parallel to the direction of flow. Laminar flow is observed either in flows occurring with... ... Illustrated Encyclopedic Dictionary
- (from the Latin lamina plate strip), a flow in which a liquid (or gas) moves in layers without mixing. The existence of laminar flow is possible only up to a certain point, the so-called. critical, Reynolds number Recr. When Re,... ... Big Encyclopedic Dictionary
- (from lat. lamina plate, strip * a. laminar flow; n. Laminarstromung, laminare Stromung; f. ecoulement laminaire, courant laminaire; i. corriente laminar, torrente laminar) ordered flow of liquid or gas, with liquid... ... Geological encyclopedia
- (from the Latin lamina plate, strip) a viscous fluid flow in which the particles of the medium move in an orderly manner through the layers and the processes of transfer of mass, momentum and energy between the layers occur at the molecular level. A typical example L.t.... ... Encyclopedia of technology
LAMINAR FLOW, calm flow of liquid or gas without mixing. A liquid or gas moves in layers that slide past each other. As the speed of movement of the layers increases, or as the viscosity decreases... ... Scientific and technical encyclopedic dictionary - the movement of a viscous liquid (or gas), in which the liquid (or gas) moves in separate parallel layers without turbulence and mixing with each other (as opposed to turbulent (see)). As a result (for example, in a pipe) these layers have... ... Big Polytechnic Encyclopedia
laminar flow- The calm, orderly movement of water or air moving parallel to the direction of the current, as opposed to a turbulent current... Dictionary of Geography
