If in the space occupied by the liquid there are regions in which ω  0, i.e., inside them, the rotation of the particles of the liquid takes place, then the movement in such regions is called eddy(for example, in the region of the boundary layer formed around solid body, streamlined by a viscous fluid flow). In the boundary layer in the direction of the normal to the body surface, the velocity increases sharply, and therefore in it ω0 (∂ w/ ∂n0).

The line is called vortex, when at each of its points the tangent coincides with the direction of the angular velocity vector ω. The differential equation of the vortex line is obtained from the relation ω dl= 0 and has the form

vortex tube is formed if through all points of a closed curve C(which is not a vortex line) to draw vortex lines. From the definition of a vortex line and a vortex surface, it follows that at any point of such lines and surfaces, the normal component of the angular velocity is equal to zero.

Angular velocity vector flow J through the surface  is called the integral:

where ω n is the projection of the angular velocity of rotation on the normal to the surface .

Another Helmholtz theorem is about vortices: the flux of the angular velocity vector through a closed surface is always zero. Let's prove it.

Indeed, by direct calculations from formulas (1.11) we obtain, on the one hand, that

a

on the other hand, that if the surface  is closed, then, according to the Ostrogradsky theorem (on the transformation of a volume integral into a surface one),

where V is the volume bounded by the surface .

But then, according to (1.18), we find that

Rice. 3. Vortex tube

Formula (1.19) implies an important property of vortex tubes. Let us single out in the vortex tube some closed surface (Fig. 3) formed by any two cross sections ( 1 and  2) and the side surface. Since the flow of the angular velocity vector along the lateral surface is equal to zero, then, according to (1.19):

Hence, due to the arbitrary choice of sections  1 and  2 , we obtain that the flux of the angular velocity vector at a given moment of time along the length of the elementary vortex tube does not change. Therefore, this flow is a quantity characteristic of the entire vortex tube, and it (the quantity) is called intensity(or tension)vortex tube.

If the magnitude of the angular velocity vector is constant over the cross section of the vortex tube, then from (1.20) we obtain

ω 1 n 1 \u003d ω 2 n 2 \u003d ω in  i= const.

On the basis of this, we draw the following conclusion: the cross section of the vortex tube is not equal to zero, since in such a case ω , which is physically incorrect. Thus, the vortex tube does not break inside the medium. But, however, only four types of vortex tubes can be distinguished, i.e., when the "vortex cord" (vortex tube): 1) begins and ends on the free surface of the liquid; 2) begins on the free surface of the liquid, and ends on a solid wall; 3) starts and ends on a solid wall; 4) is closed.

In an ideal fluid, vortices cannot change their intensity, they are, as it were, “doomed” to exist forever, unable to arise and degenerate. In a real fluid (due to friction), vortices are generated and then diffuse, i.e., degenerate.

The intensity of the tube, like the vortex of velocity, cannot be measured directly. It is relatively easy to determine the velocities of fluid particles. Therefore, the question arises of establishing a relationship between the intensity of the vortex tube and the distribution of velocities in the liquid. For solutions this issue Let us introduce the characteristic value for the velocity field: velocity circulation along some line.

Vector circulation along a certain contour is called the curvilinear integral calculated along the contour of the projection of the vector onto the tangent to the contour:

Then the relationship between the intensity of the vortex tube and the distribution of velocities is given by the well-known Stokes theorem: the intensity of the vortex tube is equal to the velocity circulation in a closed loop,once encircling the vortex tube:

Stokes' theorem reduces the quantitative determination of the intensity of a vortex tube to the calculation of the circulation velocity. Direct measurement of the velocity with special instruments is not difficult, and the summation of the terms included in the closed-loop integral is an operation more accurate than differentiation of the velocity distribution (necessary to calculate rot w) and subsequent summation.

An important corollary follows from this theorem: if in some region the flow is irrotational ( w= 0, rot w\u003d 0), i.e., potential, then the velocity circulation along any closed loop drawn in this area is zero (G \u003d 0). It also follows from the considered theorem that the finite circulation of the velocity determines whirlwind effect on the velocity field in a fluid flow.

Fluid streamlines and vortex lines. Smooth and abruptly changing movement

If we take an infinitely small closed loop in a moving fluid and draw streamlines through all its points, then a tubular surface is formed, called a stream tube. The part of the flow contained within the current tube is called the elementary trickle. As the transverse dimensions of the jet tend to zero, it contracts to the limit into the streamline.

At any point of the stream tube, i.e., the lateral surface of the stream, the velocity vectors are directed tangentially, and there are no velocity components normal to this surface, therefore, during steady motion, not a single particle of the liquid, at any point of the stream tube, can penetrate inside the stream or go outside. Thus, the current tube is like an impenetrable wall, and the elementary trickle is an independent elementary flow.

Fig 1.12 Fig 1.3

Streamlines Streamtube

We will first consider flows of finite sizes as a set of elementary jets, i.e., we will assume that the flow is jet. Due to the difference in velocities, adjacent streams will slide one over the other, but will not mix with each other. A living section, or simply a section of a stream, is generally called a surface within a stream, drawn normally to the streamlines. Further, we will consider in the flows such sections in which the filaments can be considered parallel and, consequently, the living sections are flat.

Distinguish between pressure and non-pressure fluid flows. Pressure flows are called flows in closed channels without a free surface, and pressureless flows are flows with a free surface. With pressure flows, the pressure along the flow is usually variable, with free-flow - constant (on the free surface) and most often atmospheric. Flows in pipelines with increased (or reduced) pressure, in hydraulic machines or other hydraulic units can serve as examples of pressure flow. The currents in the rivers are free-flowing, open channels and trays.

expense called the amount of liquid flowing through the living current of the stream (trickle) per unit time. This quantity can be measured in units of volume, in weight units or in units of mass, in connection with which there are volumetric Q, weight Q G and mass Q m flow rates.

For an elementary stream, which has infinitely small cross-sectional areas, we can assume that the true velocity is the same at all points of each section. Therefore, for this stream, volumetric (m 3 / s), weight (N / s) and mass (kg / s) flow

;

For a flow of finite dimensions, in the general case, the velocity has different meaning in different points section, so the flow rate must be determined as the sum of the elementary flow rates of the jets.

Usually, the cross-sectional average velocity is taken into consideration.

v cf =Q/S, whence Q= v cf S.

Based on the law of conservation of matter, on the assumption of continuity (continuity) of the flow, and on the above property of the stream tube, which consists in its "impenetrability", for a steady flow of an incompressible liquid, it can be argued that the volume flow in all sections of an elementary stream is the same:

dQ=v 1 dS 1 =v 2 dS 2 =const (along the trickle)

This equation is called the volumetric flow equation for an elementary trickle.

A similar equation can be drawn up for a flow of finite dimensions, limited by impermeable walls, but instead of true velocities, average velocities should be introduced.

Methods for studying fluid motions

a) Euler (local) - at a fixed point

b) Lagrange (substantial) - change in parameters when moving from the beginning. fixed floor. points

The internal task is the distribution of the state parameters of gases in a moving medium.

External task - explores the force interaction of a moving medium with a body in it.

Velocity field, types of flow.

Stationary, non-stationary.

One-dimensional, two-dimensional (flat), three-dimensional (spatial). The velocity vector field is a region of the space of a moving fluid at each point of which the velocity vector is uniquely determined. Streamline - a line tangent to which at any point coincides with the direction of the velocity vector at the point of contact. In a stationary flow, the streamline coincides with the trajectory of motion. The surface formed by a continuous set of streamlines is the stream surface. A part of the liquid enclosed within the current surface, drawn through all points of a certain closed contour in the flow - by the current tube. In the stationary case, the current surface is not permeable to the flow. A trickle is a streamline in a stationary flow. A trickle is called elementary if its transverse dimensions are small and the velocity does not change along the section.

Consumption and average speed

The cross section of the stream is the live section. Elementary weight consumption - . Email wt. consumption - . Email volume. consumption - . email square, specific gravity. V is the speed. The fluid flow rate is the amount of fluid flowing per unit of time through a fixed surface. ( ) average speed- this is a conditionally constant velocity over the flow section, which ensures the fluid flow rate equal to the true flow rate through the same section. For an incompressible fluid .

4. Differential equations of continuity

5. Total energy of the particles of the flowing fluid , Specific energy

6. Bernoulli's equation for a trickle

Differential equations of the dynamics of an inviscid fluid in the Euler form

Forces: pressure, mass, inertial.

Bernoulli integral

Multiplying the Euler equations by dx... we get , U(x, y, z) - the potential of body forces. .

9. Angular velocities of particles. . . The rotational motion of fluid particles is called vortex.

Vortex line, vortex tube, vortex cord.

The region of the space of a rotating fluid, at each point of which the vector is uniquely defined, is called the vortex field. A set of vortex lines penetrating a closed circuit is called a vortex tube, and the liquid filling it is called a vortex cord. The intensity of the vortex cord serves as a measure of the intensity of the vortex motion.

. An infinitely thin vortex cord is a vortex line.

Speed ​​circulation

Elementary Speed ​​Circulation - . , Г>0, if the "wind" is in the back, and vice versa.

Stokes' theorem

The velocity circulation along any closed contour that does not go beyond the limits of the liquid is equal to the sum of the stresses of all the vortices penetrating the surface based on this contour.

Remarks: a) if then , b) If , then . .

We have already written out the general equations for the flow of an incompressible fluid in the presence of vorticity:

The physical content of these equations was verbally described by Helmholtz in three theorems. First of all, imagine that instead of flow lines we have drawn wihray lines. By vortex lines, we mean field lines that have the direction of the vector Ω, and their density in any region is proportional to the value of Ω. From equation (II) the divergence Ω always is equal to zero [remember Ch.3, § 7 (issue 5): the divergence of the rotor is always zero]. Thus, the vortex lines are like the lines of the field B: they end nowhere and begin nowhere, and always tend to close. Formula (III) Helmholtz described in words: vortex lines moving togetherwith liquid. This means that if you were to mark liquid particles located on some vortex line, for example, by coloring them with ink, then in the process of fluid movement and transfer of these particles, they would always mark a new position of the vortex line. No matter how the atoms of the liquid move, the vortex lines move with them. This is one way of describing laws. It also contains a method for solving any problems. Given the initial flow, say v everywhere, you can calculate Ω. Knowing v, you can also tell where the vortex lines will be a little later: they move at a speed v. And with a new value of Ω, you can use equations (I) and (II) and find a new value v. (Just like in the problem of finding the field B given the currents.) If we are given the type of flow at any one moment, then in principle we can calculate it at all subsequent moments. We get common decision inviscid flow.

I would like to show you how (at least in part) one can understand Helmholtz's statement, and hence formula (III). In fact, this is simply the law of conservation of angular momentum applied to a liquid. Imagine a small liquid cylinder whose axis is parallel to the vortex lines (Fig. 40.13a). Some time later, the samemost the volume of liquid will be somewhere else. Generally speaking, it will be shaped like a cylinder with a different diameter and in a different location. It may also have a different orientation (Fig. 40.13b). But if the diameter changes, then the length must also change so that the volume remains constant (since we consider the fluid to be incompressible). Also, since the vortex lines are coupled with matter, their density increases in inverse proportion to the decrease in the cross-sectional area of ​​the cylinder. The product of Ω and the area of ​​a cylinder BUT will remain constant, so according to Helmholtz

Now note that at zero viscosity, all forces on the surface of the cylindrical volume (or any volume in this substance) are perpendicular to the surface. Pressure forces can make it change shape, but without tangentsocial forces momentum of a fluidinside cannot change. Angular momentum of a fluid inside a small cylinder is equal to the product its moment of inertia / on the angular velocity of the fluid, which is proportional to the vorticity Ω. The moment of inertia of the cylinder is proportional to tr 2 . Therefore, from conservation of angular momentum, we would conclude that

But the mass will be the same (M 1 = M2), and the area is proportional R 2 , so we again get just equation (40.21). Helmholtz's statement, which is equivalent to formula (III), is simply a consequence of the fact that in the absence of viscosity, the angular momentum of a fluid element cannot change.

There is good way demonstrate a moving vortex using the apparatus shown in Fig. 40.14. This is a "drum" with a diameter and length of about 60 cm, consisting of a cylindrical box with a open base thick rubber sheet. The drum stands on its side, and in the center of its solid bottom a hole with a diameter of about 8 cm. If you sharply hit the rubber diaphragm with your hand, then an annular vortex flies out of the hole. Although this vortex cannot be seen, we can safely say that it exists, since it extinguishes the flame of a candle standing at 3-6 m from the drum. By the delay of this effect, you can tell that "something" is propagating at a finite speed. You can better see what flies out by first blowing smoke into the drum. Then you will see whirlwinds in the form of amazingly beautiful rings of “tobacco smoke”.

The smoke rings (Fig. 40.15, a) are just a donut of vortex lines. Since Ω=Vx v, these vortex lines also describe the circulation v (Fig. 40.15,b). In order to explain why the ring moves forward (i.e., in the direction constituting the right screw with the direction Ω), one can argue as follows: the circulation velocity increases towards insideearly surface of the ring, and the speed inside the ring is directed forward. Since the lines Ω are carried along with the fluid, they also move forward with a speed v. (Of course, the higher speed on the inside of the ring is responsible for the forward movement of the vortex lines on the outside.)

Here it is necessary to point out one serious difficulty. As we have already noted, equation (40.90) says that if the vorticity Ω was originally equal to zero, then it will always remain equal to zero. This result is a collapse of the theory of "dry" water, because it means that if at some point the value of Ω is equal to zero, then it always will be zero, and under no circumstances create twist is not possible. However, in our simple experiment with the drum, we could create vortex rings in the air, which until then was at rest. (It is clear that until we hit the drum, v = 0 and Ω = 0 inside it.) Everyone knows that by rowing with an oar, you can create whirlwinds in the water. Undoubtedly, in order to fully understand the behavior of a liquid, it is necessary to move on to the theory of "wet" water.

Another incorrect statement in the theory of "dry" water is the assumption that we made when considering the flow at the boundary between it and the surface of a solid object. When we discussed the flow around a cylinder (for example, Fig. 40.11), we assumed that the liquid slides over the surface of a solid body. In our theory, the velocity on the surface of a solid could have any value depending on how the motion started, and we did not take into account any "friction" between the liquid and the solid. However, the fact that the velocity of a real liquid should go to zero on the surface of a solid body is an experimental fact. Consequently, our solutions for the cylinder both with and without circulation are wrong, as is the result about creating a vortex. I will tell you about more correct theories in the next chapter.