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 liquid
, 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. measure of intensity vortex motion serves as the tension of the vortex cord .
. 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 . 
.
Features of the types of motion considered in hydrodynamics.
Can be distinguished the following types movement.
Unsteady, according to the behavior of speed, pressure, temperature, etc.; steady, according to the same parameters; uneven, depending on the behavior of the same parameters in a living section with an area; uniform, on the same grounds; pressure, when the movement occurs under pressure p > p atm, (for example, in pipelines); non-pressure, when the movement of fluid occurs only under the influence of gravity.
However, the main types of movement, despite a large number of their varieties are vortex and laminar motion.
The motion in which fluid particles rotate around instantaneous axes passing through their poles is called vortex motion.
This movement of a liquid particle is characterized by an angular velocity, the components (components), which are:
The angular velocity vector itself is always perpendicular to the plane in which the rotation occurs.
If we define the modulus of angular velocity, then
By doubling the projections onto the corresponding axis coordinates? x, ? y, ? z , we obtain the components of the vortex vector
The set of vortex vectors is called a vector field.
By analogy with the velocity field and the streamline, there is also a vortex line that characterizes the vector field.
This is such a line, in which for each point the angular velocity vector is co-directed with the tangent to this line.
The line is described by the following differential equation:
in which the time t is taken as a parameter.
Vortex lines behave in much the same way as streamlines.
Vortex motion is also called turbulent.
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 ω0 (∂ w/ ∂n0).
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 flow of the angular velocity vector in this moment 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 voltage)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 some contour is called computed along the contour curvilinear integral from the vector projection 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.
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.
Vortex is called the rotational motion of a particle around the axes passing through the particle.
The study of the vortex motion of liquid and gas in aerodynamics is of great importance. practical value. In particular, methods for determining the aerodynamic characteristics of wings of infinite and finite span are based on the vortex theory. When flowing around bodies with a real flow, flow separation can occur with the formation of vortices (Fig. 2.6).
The rotational motion of particles is characterized by angular velocities:
, 
,
.
That is, at each point in space, the rotation of liquid particles can be characterized by the angular velocity vector , the modulus of which is equal to 
. Each such vector characterizes the local rotation of the fluid.
In the study of angular velocity fields, concepts are usually introduced similar to those introduced in relation to the linear velocity field. To describe the field of angular velocities of rotation, the concept of vortex lines is introduced. The construction of vortex lines is similar to the construction of streamlines (Fig. 2.7).
Vortex line called a line drawn at a given time in a stream of liquid or gas, at each point of which the angular velocity vector is directed tangentially to it.
By analogy with streamlines, we can write the differential equations of vortex lines:
.
In addition to the concept of vortex lines, the concept of vortex tubes is introduced. Consider an arbitrary small closed contour that does not coincide with the vortex line, and draw a vortex line through each point of this contour (Fig. 2.8). The combination of these lines forms a vortex tube. The liquid or gas enclosed in it is called a vortex cord (vortex thread or vortex).
The side surfaces of the vortex tube are formed by vortex lines, and, consequently, the velocity vector vortex flow through side surface equals zero.
Because 
, then the vortex flow for any cross sections of the vortex tube (vortex intensity) is the same: 
. If for the cross section of the vortex tube 
, then the intensity of the vortex tube is constant:
.
Hence, the second Helmholtz theorem reads as follows:
The flow of the velocity vector vortex through an arbitrarily drawn cross section of the vortex tube at a given moment of time is the same along the entire tube.
From this theorem, we can conclude about the possible forms of existence of vortices:
1. The cross section of the vortex tube is nowhere equal to 0, since at a constant intensity of the vortex tube the angular velocity of rotation is , which is physically impossible.
2. Vortex tubes cannot end inside the liquid: they either close on themselves (vortex rings) or rest on a wall (solid surface) or on a free surface (interface between two media with different densities). Vortices can theoretically have an infinite extent, which is possible only in an ideal fluid. Under real conditions, under the action of viscous friction forces, the vortex is gradually destroyed. The value of the intensity (or stress) of the vortex is related to the circulation of the velocity vector around the vortex.
In the absence of vortex motion. If in this case the particle trajectories are closed curves, then such motion is a special case of a circulation flow (particles rotate about an axis that does not pass through it, and do not rotate about their own axes).
In aerohydromechanics important role plays concept circulation speed D. Let us single out an arbitrary closed contour in a moving fluid (Fig. 2.9). Let at some point of this contour the speed is equal to V, and its projection onto the tangent to a given point of the contour is equal to 
. Let's write down the product and take from it a curvilinear integral along the contour:
The quantity Γ determined in this way is called closed loop speed circulation. When calculating Г, the direction of traversal of the contour (the direction of integration) is considered positive if the area covered by the contour remains to the left.
Let us consider, as an example of a circulation flow, a plane-parallel flow around an asymmetric wing profile.
Let us assume that the medium flows around the wing, causing the appearance of lift. Then the flow velocity under the lower surface of the wing is less than the velocity of the undisturbed incoming flow, and above the upper surface it is greater. The nature of the disturbed flow near the wing can be determined by subtracting the velocities of the rectilinear translational flow from the local velocities. As a result, we obtain a perturbation flow, i.e., a movement that occurs in the medium from the presence of a wing. Because the influence 
of the wing is local, then the streamlines of the perturbation flow do not go to infinity, but must have a beginning and end on the wing surface or be closed. Such a flow with closed streamlines is called circulation flow. Thus, the flow near the wing can be represented as the sum of the translational undisturbed flow and the flow along closed trajectories (Fig. 2.10).
The intensity of the circulation flow near the wing is characterized by the value of the velocity circulation in a closed loop:
where is the element of the contour arc; is the projection of the velocity on the element . In the general case, an arbitrarily chosen contour may not coincide with the streamline of the circulation flow (Fig. 2.11). Thus, circulation is called movement, in which the circulation of speed ; if , then the motion of the medium occurs without circulation.
If the velocity circulation around the profile (wing) is equal to zero, then the profile (wing) does not create lift. If the magnitude of the lifting force is not equal to zero, then a circulation flow and circulation of speed are necessarily created near the profile.
Let us apply the concept of velocity circulation to a section of a vortex tube drawn along the normal to its axis. The vortex filament induces a velocity field around itself. At , the particle velocity at a distance from the vortex axis is defined as . Let us choose a closed contour enclosing the vortex in the form of a circle with radius . Then the circulation of the velocity vector along this contour will be equal to , where is the area covered by the circle. The resulting expression is nothing but the doubled intensity of the vortex tube.
Thus, we have considered methods for describing the motion of the medium, mathematical description motion of a liquid particle, motion without rotation of a particle, vortex motion. Further, the equations of motion of gas as a continuous medium will be considered.
test questions and tasks
1. Based on the analysis of the streamline equation 
show that an infinite number of streamlines can pass through the critical point.
2. At some point in the space of a moving fluid, the cross-sectional area of the current tube becomes equal to zero. What kinematic object is located at this point in space if the streamlines are directed towards it?
3. Why can only one streamline be drawn through each point of the stream? Isn't this provision in conflict with the kinematic image referred to in task 2?
4. What is the fundamental difference between the motion of a liquid particle and the motion of a solid body?
5. The velocity potential for a certain point in the space of a moving fluid is . Write down an expression for calculating the value of the flow rate through the potential.
DA is equal to .
In which of the options is there a circulation flow?
9. Explain why in fig. 2.11 the velocity vector at the bottom of the bypass loop is directed in this way.
10. Based on the position that the angular velocity of rotation cannot be equal to ¥, explain what will happen to the vortex rope formed in a certain place in space, and how it can behave.
