Turbulence is a phenomenon observed in many flows of liquids and gases and consists in the fact that in these flows numerous vortices of various sizes are formed, as a result of which their hydrodynamic and thermodynamic characteristics (speed, pressure, temperature, density) experience chaotic fluctuations and therefore change in space and time irregularly.
A fluid flow in which turbulence is observed is called turbulent. With this flow, particles of liquid and gas undergo disordered, unsteady movements, which leads to their intense mixing.
In this way, turbulent flows differ from the so-called laminar flows, which have a regular character and can change in time only with a change in the acting forces or external conditions. In laminar flow, particles of liquid or gas move strictly in one direction in layers that do not mix with each other.
Due to the high intensity of chaotic mixing, turbulent flows have an increased ability to transfer heat and accelerate chemical reactions(for example, combustion), dispersion of sound and electromagnetic waves, as well as the transfer of momentum and, as a result, an increased force effect on the solid bodies flowing around them. Moreover, in turbulent flows, moving bodies experience significantly greater resistance, which leads to significant energy losses.
Turbulence occurs under certain conditions as a consequence of the hydrodynamic instability of laminar flows. Laminar flow loses stability and turns into turbulent flow when the ratio of inertial forces to viscous forces, the so-called Reynolds number (Re), exceeds a certain critical value characteristic of certain specific conditions.
The English physicist O. Reynolds (1842-1912) explained to his students the physical meaning of the criterion he discovered in the following way:
“A fluid can be likened to a squad of warriors, a laminar flow to a monolithic marching formation, a turbulent flow to disorderly movement. The speed of the fluid and the diameter of the pipe are the speed and size of the detachment, viscosity is discipline, and density is weapons. The larger the detachment, the faster its movement and the heavier the weapons, the sooner the formation breaks up. In the same way, turbulence arises in a liquid the faster, the higher its density, the lower the viscosity and the greater the speed of the liquid and the diameter of the pipe.”
The most thoroughly studied are turbulent flows in pipes, channels, boundary layers, around solid bodies flowing around a liquid or gas, and the so-called free turbulent flows - jets, wakes behind solid bodies moving relative to a liquid or gas, and mixing zones between flows of different velocities not separated by any or solid walls, etc., as well as the phenomenon of atmospheric turbulence.
Atmospheric turbulence plays big role In many atmospheric phenomena and processes - energy exchange between the atmosphere and the surface, transfer of heat and moisture, evaporation from earth's surface and reservoirs, diffusion of atmospheric pollution, generation of wind waves and wind currents in the sea, scattering of short radio waves in the atmosphere, etc.
In contrast to turbulence in artificial channels (pipes, jets, boundary layers, etc.), atmospheric turbulence has specific features: the range of scales of turbulent movements in the atmosphere is very wide - from several millimeters to thousands of kilometers; atmospheric turbulence develops in a space limited by one “wall” » - the surface of the Earth.
Of great practical interest is the question of energy losses during movement. solid in liquids and gas. The fact is that at low speeds, the resistance to movement increases in proportion to the speed. At the same time, as studies in the wind tunnel have shown, the moving flow remains laminar. With a further increase in speed, at some point turbulent vortices begin to form. From this moment, the resistance increases in proportion to the square of the speed, i.e., most of the energy is spent on the formation of vortices in the boundary layer and behind the moving body. Therefore, even a slight increase in speed requires large amounts of energy.
It was noticed that aquatic representatives of the animal world - dolphins - do not obey this pattern. They are known to reach speeds of up to 50 km/h and can easily maintain it for several hours. If we assume that the movement of a dolphin in water is similar to the movement of any solid body, then calculations show that the dolphin does not have enough muscular strength to do this (Gray's paradox).
A study of dolphins in a hydrodynamic tube showed that during movement the fluid flow around the dolphin's body remains laminar. Observations of the movements of dolphins in the aquarium led to the following results: when moving in the water, folds run through the thick elastic skin of the dolphin. They occur during critical flow regimes, when the speed increases so much that the flow is about to turn from laminar to turbulent. It is here that a “running wave” appears on the skin, which dampens the resulting turbulence, helping to maintain a constant laminar flow.
As soon as the secret of the dolphins' speed was revealed, engineers began to look for ways to use it. We made a “dolphin” skin for a steel torpedo. It consisted of several layers of rubber, the space between which was filled with silicone liquid, flowing through narrow tubes from one interlayer space to another. Of course, this was only a rough approximation, but it also made it possible to reduce the resistance to movement by 60% (when the torpedo was moving at a speed of 70 km/h).
Soft shells have found application not only in shipbuilding. Imagine thousands of kilometers of oil pipelines. Powerful pumping stations drive oil through them. The energy of these stations is also spent on overcoming turbulence and turbulent flows that arise in the pipes. If the pipes are covered from the inside with an elastic shell, the resistance will decrease due to laminarization of the oil flow, and therefore, as a result, energy consumption will be reduced.
· Stokes · Navier
Vortex street in flow around a cylinder
| Flow liquids and gas |
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| Creeping current | |
| Laminar flow | |
| Potential current | |
| Flow separation | |
| Vortex | |
| Instability | |
| Turbulence | |
| Convection | |
| Shock wave | |
| Supersonic flow | |
Turbulence, outdated turbulence(lat. turbulentus- stormy, disorderly) turbulent flow- the phenomenon lies in the fact that with an increase in the intensity of the flow of liquid or gas in a medium, numerous nonlinear fractal waves and ordinary, linear ones of various sizes spontaneously form, without the presence of external, random forces disturbing the medium and/or in their presence. To calculate such flows, various turbulence models have been created.
Turbulence was experimentally discovered by the English engineer Reynolds in 1883 while studying the flow of incompressible water in pipes.
In civil aviation, entering an area of high turbulence is called an air pocket.
Instantaneous flow parameters (speed, temperature, pressure, impurity concentration) fluctuate chaotically around average values. The square of the amplitude versus the oscillation frequency (or Fourier spectrum) is a continuous function.
For the occurrence of turbulence, a continuous medium is required, which obeys the Boltzmann or Navier-Stokes or boundary layer kinetic equation. The Navier-Stokes equation (it also includes the mass conservation equation or continuity equation) describes a variety of turbulent flows with sufficient accuracy for practice.
Typically, turbulence occurs when a certain critical Reynolds and/or Rayleigh number is exceeded (in the particular case of flow velocity at constant density and pipe diameter and/or temperature at the outer boundary of the medium).
In a particular case, it is observed in many flows of liquids and gases, multiphase flows, liquid crystals, quantum Bose and Fermi liquids, magnetic fluids, plasma and any continuous media (for example, sand, earth, metals). Turbulence is also observed during explosions of stars, in superfluid helium, in neutron stars, in human lungs, in the movement of blood in the heart, and during turbulent (so-called vibrational) combustion.
It occurs spontaneously when adjacent areas of the medium follow or penetrate one another, in the presence of a pressure difference or in the presence of gravity, or when areas of the medium flow around impervious surfaces.
It can arise in the presence of a compelling random force. Usually the external random force and the force of gravity act simultaneously. For example, during an earthquake or a gust of wind, an avalanche falls from a mountain within which the flow of snow is turbulent.
Turbulence, for example, can be created:
- by increasing the Reynolds number (increase the linear speed or angular velocity rotation of the flow, the size of the streamlined body, reduce the first or second coefficient of molecular viscosity, increase the density of the medium) and/or the Rayleigh number (heat the medium) and/or increase the Prandtl number (decrease the viscosity).
- and/or ask very complex look external force (examples: chaotic force, impact). The flow may not have fractal properties.
- and/or create complex boundary and/or initial conditions by specifying a boundary shape function. For example, they can be represented by a random function. For example: flow during the explosion of a vessel with gas. It is possible, for example, to inject gas into the medium and create a rough surface. Use the heat nozzle. Put the grid over. The flow may not have fractal properties.
- and/or create a quantum state. This condition applies only to helium isotopes 3 and 4. All other substances freeze, remaining in a normal, non-quantum state.
- irradiate the environment with high-intensity sound.
- through chemical reactions such as combustion. The shape of the flame, like the appearance of the waterfall, can be chaotic.
Theory
At large Reynolds numbers, flow velocities are weakly dependent on small changes at the boundary. Therefore, at different initial speeds of the ship, the same wave is formed in front of its bow when it moves at cruising speed. The nose of the rocket burns and the same picture of the peak is created, despite the different initial speed.
Fractal- means self-similar. For example, your hand has the same fractal dimension as your ancestors and descendants. A straight line has a fractal dimension of one. Y plane is equal to two. The ball has three. The river bed has a fractal dimension greater than 1, but less than two, when viewed from a satellite altitude. In plants, the fractal dimension grows from zero to a value greater than two. There is a unit of measurement geometric shapes, is called the fractal dimension. Our world cannot be represented in the form of many lines, triangles, squares, spheres and other simple figures. And the fractal dimension allows you to quickly characterize geometric bodies complex shape. For example, at a shell fragment.
Nonlinear wave- a wave that has nonlinear properties. Their amplitudes cannot be added during a collision. Their properties change greatly with small changes in parameters. Nonlinear waves are called dissipative structures. There are no linear processes of diffraction, interference, or polarization in them. But there are nonlinear processes, for example, self-focusing. At the same time, the diffusion coefficient of the medium, the transfer of energy and momentum, and the friction force to the surface sharply increase by orders of magnitude.
That is, in a particular case, in a pipe with absolutely smooth walls at a speed above a certain critical value, during any continuum, the temperature of which is constant, under the influence only of gravity, nonlinear self-similar waves and then turbulence are always spontaneously formed. In this case, there are no external disturbing forces. If you additionally create a disturbing random force or pits on the inner surface of the pipe, then turbulence will also appear.
In a particular case, nonlinear waves - vortices, tornadoes, solitons and other nonlinear phenomena (for example, waves in plasma - ordinary and ball lightning), occurring simultaneously with linear processes (for example, acoustic waves).
In mathematical language, turbulence means that the exact analytical solution of the partial differential equations of conservation of momentum and conservation of mass is Navier-Stokes (this is Newton's law with the addition of viscous forces and pressure forces in the medium and the equation of continuity or conservation of mass) and the energy equation is when exceeded some critical Reynolds number, a strange attractor. They represent nonlinear waves and have fractal, self-similar properties. But since the waves occupy a finite volume, some part of the flow region is laminar.
At a very small Reynolds number, these are well-known linear waves on water of small amplitude. At high speeds we observe nonlinear tsunami waves or breaking waves. For example, large waves behind a dam break up into smaller waves.
Due to nonlinear waves, any parameters of the medium: (speed, temperature, pressure, density) can experience chaotic fluctuations, changing from point to point and in time non-periodically. They are very sensitive to the slightest changes in environmental parameters. In a turbulent flow, the instantaneous parameters of the medium are distributed according to a random law. This is how turbulent flows differ from laminar flows. But by controlling the average parameters, we can control the turbulence. For example, by changing the diameter of the pipe, we control the Reynolds number, fuel consumption and filling rate of the rocket tank.
Literature
- Reynods O., An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. Roy. Soc., London, 1883, v.174
- Feigenbaum M., Journal Stat Physics, 1978, v.19, p.25
- Feigenbaum M., Journal Stat Physics, 1979, v.21, p.669
- Feigenbaum M., Uspekhi Fizicheskikh Nauk, 1983, v. 141, p. 343 [translation Los Alamos Science, 1980, v.1, p.4]
- Landau L.D., Lifshits E.M. Hydromechanics, - M.: Nauka, 1986. - 736 p.
- Monin A. S., Yaglom A. M., Statistical fluid mechanics. In 2 parts - St. Petersburg: Gidrometeoizdat, Part 1, 1992. - 695 p.; Moscow, Nauka Part 2, 1967. - 720 p.
- Obukhov A. M. Turbulence and atmospheric dynamics"Gidrometeoizdat" 414 pp. 1988 ISBN 5-286-00059-2
- Turbulence problems. Collection of translated articles, ed. M. A. Velikanov and N. T. Shveikovsky. M.-L., ONTI, 1936. - 332 p.
- D. I. Greenwald, V. I. Nikora, “River turbulence”, L., Gidrometeoizdat, 1988, 152 p.
- P. G. Frick. Turbulence: models and approaches. Lecture course. Part I. PSTU, Perm, 1998. - 108 p. Part II. - 136 s.
- P. Berger, I. Pomo, C. Vidal, Order in chaos, On a deterministic approach to turbulence, M, Mir, 1991, 368 pp.
- K.E. Gustafson, Introduction to partial differential equations and Hilbert space methods - 3rd ed., 1999 Encyclopedia of Technology
- (from the Latin turbulentus, stormy, disorderly), the flow of a liquid or gas, in which the particles of the liquid make disordered, chaotic movements along complex trajectories, and the speed, temperature, pressure and density of the medium experience chaotic... ... Big Encyclopedic Dictionary
Modern encyclopedia
TURBULENT FLOW, in physics, the movement of a fluid medium in which its particles move randomly. Characteristic of a liquid or gas with a high REYNOLDS NUMBER. see also LAMINAR FLOW... Scientific and technical encyclopedic Dictionary
turbulent flow- A flow in which gas particles move in a complex, disordered manner and transport processes occur at the macroscopic rather than at the molecular level. [GOST 23281 78] Topics: aerodynamics of aircraft Generalizing terms, types of flows... ... Technical Translator's Guide
Turbulent flow- (from the Latin turbulentus stormy, disorderly), the flow of a liquid or gas, in which the particles of the liquid make disordered, chaotic movements along complex trajectories, and the speed, temperature, pressure and density of the medium are experienced... ... Illustrated Encyclopedic Dictionary
- (from Latin turbulentus stormy, disorderly * a. turbulent flow; n. Wirbelstromung; f. ecoulement turbulent, ecoulement tourbillonnaire; i. flujo turbulento, corriente turbulenta) the movement of a liquid or gas, during which and ... ... Geological encyclopedia
turbulent flow- A form of flow of water or air in which its particles make disordered movements along complex trajectories, which leads to intense mixing. Syn.: turbulence… Dictionary of Geography
TURBULENT FLOW- a type of liquid (or gas) flow in which their small volumetric elements perform unsteady movements along complex random trajectories, which leads to intense mixing of layers of liquid (or gas). T. t. arises as a result... ... Big Polytechnic Encyclopedia
Structure of turbulent flow. A distinctive feature of turbulent fluid motion is the chaotic movement of particles in the flow. However, it is often possible to observe a certain pattern in this
movement. Using a thermohydrometer, a device that allows you to record changes in speed at a measurement point, you can take a speed curve. If you choose a time interval of sufficient length, it turns out that speed fluctuations are observed around a certain level and this level remains constant when choosing different time intervals. The magnitude of the speed at a given point in this moment time is called instantaneous speed. Graph of change in instantaneous speed over time u(t) presented in the figure. If you select a certain time interval on the speed curve and integrate the speed curve, and then find the average value, then this value is called the average speed
The difference between the instantaneous and average speed is called the pulsation speed And".
If the values of average velocities at different time intervals remain constant, then such turbulent fluid motion will be steady.
In unsteady turbulent motion 
fluid values of average velocities change over time
Fluid pulsation causes fluid mixing in the flow. The intensity of mixing depends, as is known, on the Reynolds number, i.e. while maintaining other conditions on the speed of fluid movement. Thus, in a specific thread
liquid (the viscosity of the liquid and the cross-sectional dimensions are determined by the primary conditions), the nature of its movement depends on the speed. For turbulent flow this is critical. So, in the peripheral layers of the liquid, the velocities will always be minimal, and the mode of motion in these layers will naturally be 
laminar. An increase in speed to a critical value will lead to a change in the fluid movement mode from laminar to turbulent mode. Those. In a real flow, both modes are present: laminar and turbulent.
Thus, the fluid flow consists of a laminar zone (at the channel wall) and a turbulent flow core (in the center) and, since the speed is towards the center of the turbulent flow,
current increases intensively, the thickness of the peripheral laminar layer is most often insignificant, and, naturally, the layer itself is called a laminar film, the thickness of which depends on the speed of fluid movement.
Hydraulically smooth and rough pipes. The condition of the pipe walls significantly influences the behavior of the liquid in a turbulent flow. So with laminar movement 
the liquid moves slowly and smoothly, calmly flowing around minor obstacles in its path. The local resistances arising in this case are so insignificant that their magnitude can be neglected. In a turbulent flow, such small obstacles serve as a source of vortex motion of the fluid, which leads to an increase in these small local hydraulic resistances, which we neglected in a laminar flow. Such small obstacles on the pipe wall are its irregularities. The absolute magnitude of such irregularities depends on the quality of pipe processing. In hydraulics, these irregularities are called roughness protrusions and are designated by the letter .
Depending on the ratio of the thickness of the laminar film and the size of the roughness protrusions, the nature of the movement of the liquid in the flow will change. In the case when the thickness of the laminar film is large compared to the size of the roughness protrusions ( , the roughness protrusions are immersed in the laminar film and they are inaccessible to the turbulent core of the flow (their presence does not affect the flow). Such pipes are called hydraulically smooth (scheme 1 in the figure). When the size of the roughness protrusions exceeds the thickness of the laminar film, then the film loses its continuity, and the roughness protrusions become a source of numerous vortices, which significantly affects the fluid flow as a whole. Such pipes are called hydraulically rough (or simply rough) (scheme 3 in the figure). Naturally, There is also an intermediate type of pipe wall roughness, when the roughness protrusions become commensurate with the thickness of the laminar film (Scheme 2 in the figure).
minary film can be estimated based on the empirical equation
Shear stresses in turbulent flow. In a turbulent flow, the magnitude of the tangential stresses should be greater than in a laminar flow, because To the tangential stresses determined when a viscous liquid moves along the pipe, additional tangential stresses caused by mixing of the liquid should be added.
Let's take a closer look at this process. In a turbulent flow, together with the movement of a liquid particle along the axis of the pipe at a speed And the same liquid particle is simultaneously transferred in a perpendicular direction from one layer of liquid to another with a speed equal to the pulsation speed And. Let's select an elementary platform dS, located parallel to the pipe axis. Through this platform, liquid will move from one layer to another at a pulsation speed, and the liquid flow rate will be:
Liquid mass dMr, moved across the platform in time dt will:
Due to the horizontal component of the pulsation speed their this mass will receive an increase in momentum in the new layer of liquid dM,
If the fluid flowed into a layer moving at a higher speed, then, consequently, the increment in the amount of motion will correspond to the force impulse dT, directed in the direction opposite to the movement of the fluid, i.e. speed their:
^ 
For average speed values:
It should be noted that when liquid particles move from one layer to another, they do not instantly acquire the speed of the new layer, but only after some time; During this time, the particles will have time to penetrate into the new layer at a certain distance /, called the length of the mixing path.
Now let's consider some liquid particle located at a point A Let this particle move to the adjacent layer of liquid and go deeper into it the length of the mixing path, i.e. got to the point IN. Then the distance between these points will be equal to /. If the fluid speed at a point A will be equal And, then the speed at the point
IN will be equal.
Let us make the assumption that the velocity pulsations are proportional to the increment in the velocity of the liquid volume. Then:
The resulting dependence is called the Prandtl formula and is a law in the theory of turbulent friction as well as the law of viscous friction for laminar fluid motion. , Let's rewrite the last dependence in the form:
Here the coefficient is called the turbulent exchange coefficient
plays the role of a dynamic viscosity coefficient, which emphasizes the commonality of the principles of the theory of Newton and Prandtl. Theoretically, the total shear stress should be equal to:
* 
"
but the first term on the right side of the equality is small compared to the second and its value can be neglected
Velocity distribution over the cross section of a turbulent flow. Observations of the values of average velocities in a turbulent fluid flow have shown that the diagram of average velocities in a turbulent flow is largely smoothed out and almost the velocities at different points of a living 
cross sections are equal to the average speed. Comparing the velocity diagrams of turbulent flow (diagram 1) and laminar flow allows us to conclude that there is an almost uniform distribution of velocities in the living section. Prandtl's work established that the law of change in shear stresses along the flow cross section is close to the logarithmic law. Under certain assumptions: flow along an infinite plane and equality of tangential stresses at all points on the surface
After integration: 
The last expression is converted to the following form:
Developing Prandtl's theory, Nikuradze and Reichardt proposed a similar relationship for round pipes. 
Head loss due to friction in a turbulent fluid flow. When studying the issue of determining the friction head loss coefficient in hydraulically smooth pipes, one can come to the conclusion that this coefficient depends entirely on the Reynolds number. There are known empirical formulas for determining the coefficient of friction; the Blasius formula is the most widely used:
According to numerous experiments, the Blasius formula is confirmed within the range of Reynolds numbers from up to 1-10 5. Another common empirical formula for determining the Darcy coefficient is P.K. Konakova:
Formula P.K. Konakova has a wider range of applications up to Reynolds numbers of several millions. G.K.’s formula has almost identical values in terms of accuracy and scope of application. Filonenko:
Study of the movement of fluid through rough pipes in an area where pressure losses are determined only by the roughness of the pipe walls and do not depend on speed
fluid movement, i.e. from Reynolds' number was carried out by Prandtl and Nikuradze. As a result of their experiments on models with artificial roughness, a relationship was established for the Darcy coefficient for this so-called quadratic region of fluid flow.
Turbulent flow is characterized by rapid and random fluctuations of speed, pressure and concentration around their average values. These fluctuations, as a rule, are of interest only in the statistical description of systems. Therefore, as a first step in the study of turbulent flow, equations for the average quantities believed to describe the flow are usually considered. In this case, for some average values we obtain differential equations, which include moments of higher orders. Thus, this method does not directly calculate any average value. The problem of turbulent flow has a direct analogy in the kinetic theory of gases, where the details of the random motion of molecules are unimportant, and only some average measurable quantities are of interest.
In many cases, it is possible to find a simple solution to the equation of motion (94-4), which describes laminar flow, but the observed flow is turbulent. This circumstance led to studies of the stability of laminar flow. The question of flow stability is formulated as follows: if the flow is disturbed indefinitely small amount, then will the disturbance increase in space and time or will it die out and the flow remain laminar? This issue is usually solved by linearizing the problem near the main, laminar solution. The results obtained are sometimes consistent with experimentally observed conditions for the transition to turbulence or to more complex laminar flow, as in the case of Taylor vortices in flow between rotating cylinders (Section 4). Sometimes there is
significant discrepancy with experiment, as in the case of Poiseuille flow in a pipe.
For turbulent flow, average values can be defined as time averages, e.g.
The time period U over which averaging is carried out must be large compared to the period of fluctuations, which can be estimated as 0.01 s.
For laminar flow, the stress is given by equation (94-1), which defines Newton's law for viscous flow. However, in turbulent flow there is an additional mechanism for momentum transfer due to the fact that random fluctuations in velocity tend to transfer momentum to a region with less momentum. Thus, the total average stress, or momentum flume, is equal to the sum of the viscous and turbulent momentum fluxes:
where the viscous momentum flow is determined by the time-averaged equation (94-1), and the turbulent momentum flow will be obtained in this section somewhat later.
Far from a solid wall, momentum transfer through the turbulent mechanism predominates. However, near a solid surface, turbulent fluctuations are damped, as a result of which viscous momentum transfer dominates. Therefore, the friction stress on the wall is still determined by the equality
relating to flow in a pipe of radius R. The damping of fluctuations near a solid surface is quite natural, since the liquid cannot cross the interface with a solid body.
The nature of the turbulent momentum flow can be determined by averaging the equation of motion (93-4) over time:
Here, the same stress tensor that was previously denoted by is denoted by . This tensor for Newtonian fluids is given by equality (94-1).
Let us introduce the deviation from the time-averaged values of velocity and pressure:
Let us call v the velocity fluctuation or the fluctuating part of the velocity. Several rules for time averaging follow directly from definition (98-1). Thus, the time average of the sum is equal to the sum of the time averages:
The average value of the derivative is equal to the derivative of the time average: . In general, the time average of the nonlinear term will yield more than one term. For example, Of course, the time average of the fluctuation is zero:
We assume that the characteristics of the fluid, e.g. In fact, a compressible laminar boundary layer may be more stable than an incompressible one. Taking these remarks into account, time averaging of the equation of motion (98-4) gives
The time-averaged continuity equation (93-3) has the form
The average viscous stress is found by averaging over time the equation (94-1):
These equations coincide with the equations that existed before averaging, with the exception that the term - appears in the equation of motion (98-6). If we express the turbulent momentum flux as
and write down the total average stress in accordance with equality (98-2), then the equation of motion takes the form
This equation is very similar to what it was before averaging.
These calculations illustrate the origin of the turbulent momentum flow or the so-called Reynolds stress, defined by equality (98-9). The turbulent mechanism of momentum transfer is to some extent similar to the mechanism of momentum transfer in gases, with the only difference being that in gases the transfer is carried out due to the random movement of molecules, and in liquids - due to the random movement of large molecular aggregates.
It can be seen that the averaging process does not reliably predict the Reynolds stress. Lacking a fundamental theory, many authors have written empirical expressions for , with varying degrees of success. It may be worth emphasizing that there is no simple relationship between turbulent stress and velocity derivatives, as is the case for viscous stress in a Newtonian fluid, where it is a characteristic of the state depending only on temperature, pressure and composition.
Many practical problems on turbulence involve a region near a solid surface, since in its meaning it is this region that serves as the origin of turbulence and since it is in this region that friction stresses and mass transfer rates need to be calculated. Many attempts have been made to study experimental data in order to generalize the properties different characteristics turbulent transport near the surface. Such characteristics include higher-order averages, such as the Reynolds stress, resulting from averaging the equations of motion and convective diffusion. This generalization has the form of a universal law of velocity distribution near the surface. The same result can be expressed using turbulent viscosity and turbulent kinematic viscosity- coefficients relating turbulent transport to velocity gradients. These coefficients depend significantly on the distance to the wall and therefore are not fundamental characteristics of the liquid. This kind of information is often obtained from studying fully developed flow in a pipe or some simple boundary layers.
When studying turbulent flow near the surface of a solid body, it was shown that the relation called the universal velocity profile is valid for the average tangential velocity, the dependence of which on the distance to the solid surface is shown in Fig. 98-1. This relationship describes a fully developed turbulent flow near a smooth surface.
walls and is valid both for flow in a pipe and for turbulent boundary layers. The expression for the turbulent velocity profile includes the friction stress on the wall:
Note that far from the wall the average speed changes linearly with the logarithm of the distance to the wall, and near it it increases linearly with the distance.
Rice. 98-1. Universal velocity profile in fully developed turbulent flow.
The main features of the curve are reproduced by the following approximate formulas:
In the logarithmic region
Here the term that includes the dependence of the velocity profile on y does not depend on the viscosity, which is included only in the additive constant.
From Fig. 98-1 shows that the Reynolds stress depends on the distance to the wall. Typically this dependence is expressed in terms of turbulent viscosity, defined by the relation
The introduction allows empirical data to be expressed in terms of turbulent viscosity. Since turbulent flow near a wall cannot be isotropic, a different turbulent viscosity is likely required to express other components of the Reynolds stress, even at the same distance to the wall.
Rice. 98-2. Turbulent viscosity as a universal function of distance to a solid surface.
The universal velocity profile (Fig. 98-1) appears to be valid only in the region near the wall, where the friction stress is essentially constant. This profile should break down near the center of the pipe, where the stress drops to zero. If we assume that the friction stress is constant over the entire region where the universal velocity profile is valid, then we can get an idea of the nature of the change with distance to the wall:
It is clear from this that the ratio must also be a universal function of the distance to the wall expressed in units. Rice. 98-2 was obtained by differentiating the universal velocity profile shown in Fig. 98-1. It is impossible to obtain accurate data for near the wall using this method,
possible, because in this area. However, this problem is not of particular importance, since the problems of hydrodynamics include only the sum
The universal velocity profile is one of the few conclusions obtained in the theory of turbulent flow near a wall. This profile is widely used in cases where experimental observations are not possible. Thus, the universal profile serves as the basis for a semi-empirical theory of turbulent flow, which is applied to the hydrodynamics of turbulent boundary layers, to mass transfer in turbulent boundary layers, and also in the inlet region in the case of fully developed pipe flow.
The chaotic, disordered movement of liquid particles significantly affects the characteristics of turbulent flows. These fluid flows are unsteady. Due to this, at each point in space the velocities change over time. The instantaneous speed value can be expressed:
(2.42)
where is the time-averaged speed in the direction x, – pulsation speed in the same direction. Typically, the averaged velocity maintains a constant value and direction over time, so such a flow must be taken as a steady-state average. When considering the velocity profile of a turbulent flow for any region, the profile of the averaged velocity is usually considered.
Let us consider the behavior of a turbulent fluid flow near a solid wall (Fig. 2.17).
Rice. 2.17. Velocity distribution near a solid wall
In the core of the flow, due to pulsating velocities, continuous mixing of the liquid occurs. Near solid walls, transverse movements of liquid particles are impossible.
Liquid flows laminarly near a solid wall.
There is a transition zone between the laminar boundary layer and the flow core.
The movement of a fluid in a turbulent regime is always accompanied by a significantly greater expenditure of energy than in a laminar regime. In laminar mode, energy is spent on viscous friction between layers of liquid; in a turbulent mode, in addition, a significant part of the energy is spent on the mixing process, which causes additional tangential stresses in the liquid.
To determine the stress of friction forces in a turbulent flow, the formula is used:
where is the viscous flow stress and is the turbulent stress caused by mixing. As is known, it is determined by Newton’s law of viscous friction:
|
(2.44)
Following Prandtl's semi-empirical theory of turbulence, assuming that the magnitude of transverse velocity pulsations is on average of the same order as longitudinal pulsations, we can write:
. (2.45)
Here r is the density of the liquid, l– mixing path length, – average velocity gradient.
Magnitude l, which characterizes the average path of fluid particles in the transverse direction, is caused by turbulent pulsations.
According to Prandtl's hypothesis, the length of the mixing path l is proportional to the distance of the particle from the wall:
where c is the universal Prandtl constant.
In a turbulent flow in a pipe, the thickness of the hydrodynamic boundary layer grows much faster than for a laminar one.
This leads to a decrease in the length of the initial section. In engineering practice it is usually accepted:
(2.47)
Therefore, quite often the influence of the initial section
the hydrodynamic characteristics of the flow are neglected.
Let us consider the distribution of averaged velocity over the pipe cross section. Let us assume constant tangential stress in a turbulent flow
and equal to the stress in the wall. Then after integrating equation (2.44) we obtain:
. (2.48)
Here is a quantity that has the dimension of speed, and is therefore called dynamic speed.
Expression (2.48) represents the logarithmic distribution law of averaged velocities for the core of a turbulent flow.
Through simple transformations, formula (2.48) can be given
to the following dimensionless form:
(2.49)
where is the dimensionless distance from the wall; M– constant.
As experiments show, c has the same value for all cases of turbulent flow. Meaning M was determined by Nikuradze's experiments: . So we have:
(2.50)
The following complex is used as a dimensionless parameter characterizing the thickness of the corresponding zones:
viscous laminar sublayer: 
,
transition zone: 
,
turbulent core: .
In turbulent mode, the ratio of the average speed
to the maximum axial is from 0.75 to 0.9.
Knowing the law of velocity distribution (Fig. 2.18), you can find the value of hydraulic resistance. However, to determine hydraulic resistance, you can use a simpler relationship, namely: the criterion equation of motion of a viscous fluid, obtained earlier in the first part of the discipline.
Rice. 2.18. Velocity distribution in the pipe
in laminar and turbulent modes
For a horizontal straight pipe in the case of pressure flow of a viscous fluid, the criterion equation has the form:
(2.51)
where are geometric complexes, is the Reynolds criterion, and is the Euler criterion. They are defined as:
where ∆ is the absolute roughness of the pipe, l– pipeline length,
d– internal diameter of the pipe. It is known from experience that pressure loss is directly proportional to . Therefore we can write:
(2.52)
Next we denote the unknown function 
, let's write down Euler's criterion. Then from equation (2.52) for pressure loss we obtain:
(2.53)
where l is the coefficient of hydraulic friction, w– average flow speed.
The resulting equation is called the Darcy–Weisbach equation. Equation (2.53) can be represented in terms of head loss:
(2.54)
Thus, the calculation of pressure loss or head is reduced to determining the coefficient of hydraulic friction l.
Nikuradze schedule
Among the numerous works on addiction research 
Let's choose Nikuradze's work. Nikuradze studied this dependence in detail for pipes with a uniformly grained surface created artificially (Fig. 2.19).
.
Rice. 2.19. Nikuradze schedule
The value of the coefficient is determined by empirical formulas obtained for various areas resistance according to Nikuradze curves.
1. For laminar flow regime, i.e. at , the coefficient l for all pipes, regardless of their roughness, is determined from the exact solution of the problem of laminar fluid flow in a straight round pipe using the Poiseuille formula:
2. In a narrow region, an abrupt increase in the resistance coefficient is observed. This region of transition from laminar to turbulent is characterized by an unstable flow. Here the turbulent regime is most likely in practice
and it is most correct to use the formulas for zone 3. You can also apply the empirical formula:
3. In the area of hydraulically smooth pipes with 
the thickness of the laminar layer at the wall d is greater than the absolute roughness of the walls D, the influence of the roughness protrusions washed by the continuous flow has practically no effect, and the drag coefficient is calculated here based on a generalization of experimental data
according to empirical relations, for example according to the Blausius formula:
4. In the range of Reynolds numbers 
there is a transition region from hydraulically smooth pipes to rough ones. In this area (partially rough pipes), when, i.e. roughness protrusions with a height less than the average value D continue to remain within the laminar layer, and protrusions with a height greater than the average find themselves in the turbulent region of the flow, and the braking effect of the roughness is manifested. In this case, the coefficient l is also calculated from empirical relations, for example
according to the Alshtuhl formula:
(2.58)
5. When the thickness of the laminar layer at the wall d reaches its minimum value, i.e. 
and doesn't change
with a further increase in the Re number. Therefore l does not depend on the number Re,
but depends only on e. In this area (rough pipes or the area of quadratic resistance), for example, the Shifrinson formula can be recommended to find the coefficient:
(2.59)
In this zone the value of l is within 
.
Studies have been carried out to determine l with natural roughness. For these pipes the second zone is not defined. For calculation
l The above formulas are usually suggested.
