Let there be a figure of arbitrary shape with area ω in the plane Ol , inclined to the horizon at an angle α (Fig. 3.17).

For the convenience of deriving a formula for the fluid pressure force on the figure under consideration, we rotate the wall plane by 90 ° around the axis 01 and align it with the drawing plane. On the plane figure under consideration, we single out at a depth h from the free surface of the liquid to an elementary area d ω . Then the elementary force acting on the area d ω , will

Rice. 3.17.

Integrating the last relation, we obtain the total force of fluid pressure on a flat figure

Considering that , we get

The last integral is equal to the static moment of the platform with respect to the axis OU, those.

where l With – axle distance OU to the center of gravity of the figure. Then

Since then

those. the total force of pressure on a flat figure is equal to the product of the area of ​​the figure and the hydrostatic pressure at its center of gravity.

The point of application of the total pressure force (point d , see fig. 3.17) is called center of pressure. The center of pressure is below the center of gravity of a flat figure by an amount e. The sequence of determining the coordinates of the center of pressure and the magnitude of the eccentricity is described in paragraph 3.13.

In the particular case of a vertical rectangular wall, we get (Fig. 3.18)

Rice. 3.18.

In the case of a horizontal rectangular wall, we will have

hydrostatic paradox

The formula for the pressure force on a horizontal wall (3.31) shows that the total pressure on a flat figure is determined only by the depth of the center of gravity and the area of ​​the figure itself, but does not depend on the shape of the vessel in which the liquid is located. Therefore, if we take a number of vessels, different in shape, but having the same bottom area ω g and equal liquid levels H , then in all these vessels the total pressure on the bottom will be the same (Fig. 3.19). Hydrostatic pressure is due in this case to gravity, but the weight of the liquid in the vessels is different.

Rice. 3.19.

The question arises: how can different weights create the same pressure on the bottom? It is in this seeming contradiction that the so-called hydrostatic paradox. The disclosure of the paradox lies in the fact that the force of the weight of the liquid actually acts not only on the bottom, but also on other walls of the vessel.

In the case of a vessel expanding upward, it is obvious that the weight of the liquid is greater than the force acting on the bottom. However, in this case, part of the weight force acts on the inclined walls. This part is the weight of the pressure body.

In the case of a vessel tapering to the top, it suffices to recall that the weight of the pressure body G in this case is negative and acts upward on the vessel.

Center of pressure and determination of its coordinates

The point of application of the total pressure force is called the center of pressure. Determine the coordinates of the center of pressure l d and y d (Fig. 3.20). As is known from theoretical mechanics, at equilibrium, the moment of the resultant force F about some axis is equal to the sum of the moments of the constituent forces dF about the same axis.

Rice. 3.20.

Let's make the equation of the moments of forces F and dF about the axis OU:

Forces F and dF define by formulas

Center of pressure

the point at which the line of action of the resultant of the pressure forces of the environment (liquid, gas) applied to a resting or moving body intersects with some plane drawn in the body. For example, for an airplane wing ( rice. ) C. d. is defined as the point of intersection of the line of action of the aerodynamic force with the plane of the wing chords; for a body of revolution (body of a rocket, airship, mine, etc.) - as the point of intersection of the aerodynamic force with the plane of symmetry of the body, perpendicular to the plane passing through the axis of symmetry and the velocity vector of the center of gravity of the body.

The position of the center of gravity depends on the shape of the body, and for a moving body it can also depend on the direction of motion and on the properties of the environment (its compressibility). Thus, at the wing of an aircraft, depending on the shape of its airfoil, the position of the central airfoil may change with a change in the angle of attack α, or it may remain unchanged (“a profile with a constant central airfoil”); in the latter case x cd ≈ 0,25b (rice. ). When moving at supersonic speed, the center of gravity shifts significantly towards the tail due to the influence of air compressibility.

A change in the position of the central engine of moving objects (aircraft, rocket, mine, etc.) significantly affects the stability of their movement. In order for their movement to be stable in the event of a random change in the angle of attack a, the central air must shift so that the moment of aerodynamic force about the center of gravity causes the object to return to its original position (for example, with an increase in a, the central air must shift towards the tail). To ensure stability, the object is often equipped with an appropriate tail unit.

Lit.: Loitsyansky L. G., Mechanics of liquid and gas, 3rd ed., M., 1970; Golubev V.V., Lectures on the theory of the wing, M. - L., 1949.

The position of the center of flow pressure on the wing: b - chord; α - angle of attack; ν - flow velocity vector; x dc - distance of the center of pressure from the nose of the body.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what the "Center of Pressure" is in other dictionaries:

This is the point of the body at which they intersect: the line of action of the resultant forces of pressure on the body of the environment and some plane drawn in the body. The position of this point depends on the shape of the body, and for a moving body it also depends on the properties of the surrounding ... ... Wikipedia

A point at which the line of action of the resultant of the pressure forces of the environment (liquid, gas) applied to a body at rest or moving intersects with a certain plane drawn in the body. For example, for an airplane wing (Fig.) C. d. determine ... ... Physical Encyclopedia

The conditional point of application of the resultant aerodynamic forces acting in flight on an aircraft, projectile, etc. The position of the center of pressure depends mainly on the direction and speed of the oncoming air flow, as well as on the external ... ... Marine Dictionary

In hydroaeromechanics, the point of application of the resultant forces acting on a body moving or at rest in a liquid or gas. * * * CENTER OF PRESSURE CENTER OF PRESSURE, in hydroaeromechanics, the point of application of the resultant forces acting on the body, ... ... encyclopedic Dictionary

center of pressure- The point at which the resultant of pressure forces is applied, acting from the side of a liquid or gas on a body moving or resting in them. Engineering topics in general… Technical Translator's Handbook

In hydroaeromechanics, the point of application of the resultant forces acting on a body moving or at rest in a liquid or gas ... Big Encyclopedic Dictionary

The point of application of the resultant aerodynamic forces. The concept of C. D. is applicable to the profile, wing, aircraft. In the case of a flat system, when the lateral force (Z), transverse (Mx) and track (My) moments can be neglected (see Aerodynamic forces and ... ... Encyclopedia of technology

center of pressure- slėgimo centras statusas T sritis automatika atitikmenys: angl. center of pressure vok. Angriffsmittelpunkt, m; Druckmittelpunkt, m; Druckpunkt, m rus. center of pressure, m pranc. center de poussee, m … Automatikos terminų žodynas

center of pressure- slėgio centras statusas T sritis fizika atitikmenys: engl. center of pressure vok. Druckmittelpunkt, m rus. center of pressure, m pranc. center de pression, m … Fizikos terminų žodynas

center of pressure Encyclopedia "Aviation"

center of pressure- center of pressure - the point of application of the resultant aerodynamic forces. The concept of C. D. is applicable to the profile, wing, and aircraft. In the case of a flat system, when the lateral force (Z), transverse (Mx) and track (My) can be neglected ... ... Encyclopedia "Aviation"

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The task of determining the resulting force of hydrostatic pressure on a flat figure is reduced to finding the magnitude of this force and the point of its application or the center of pressure. Imagine a tank filled with liquid and having an inclined flat wall (Fig. 1.12).

On the wall of the tank, we outline some flat figure of any shape with area w . We choose the coordinate axes as indicated in the drawing. Axis z perpendicular to the drawing plane. In plane uz the figure under consideration is located, which is projected as a straight line, indicated by a thick line, this figure is shown on the right in combination with the plane uz.

In accordance with the 1st property of hydrostatic pressure, it can be argued that at all points of the area w, the fluid pressure is directed normal to the wall. Hence we conclude that the hydrostatic pressure force acting on an arbitrary flat figure is also directed normally to its surface.

Rice. 1.12. Fluid pressure on a flat wall

To determine the pressure force, we select an elementary (infinitely small) area d w. Force of pressure dP on an elementary platform, we define it as follows:

dp=pd w = (p 0 + r gh)d w,

where h- platform immersion depth d w .

As h = y sina , then dP=pd w = (p 0 + r gy sina) d w .

Pressure force on the entire area w:

The first integral is the area of ​​the figure w :

The second integral is the static moment of the area w about the axis X. As you know, the static moment of the figure about the axis X is equal to the product of the area of ​​\u200b\u200bthe figure w and the distance from the axis X to the center of gravity of the figure, i.e.

.

Substituting into equation (1.44) the values ​​of the integrals, we obtain

P=p o w + r g sina y c. t w.

But since y c.t. sina = h c.t - the depth of immersion of the center of gravity of the figure, then:

P=(p 0 + r gh c.t)w. (1.45)

The expression enclosed in brackets is the pressure at the center of gravity of the figure:

p 0 + r gh c.t. =p c.t.

Therefore, equation (1.45) can be written as

P=p c.t w . (1.46)

Thus, the force of hydrostatic pressure on a flat figure is equal to the hydrostatic pressure at its center of gravity, multiplied by the area of ​​this figure. Let us determine the center of pressure, i.e. pressure point R. Since the surface pressure, passing through the liquid, is uniformly distributed over the area under consideration, the point of application of the force w will coincide with the center of gravity of the figure. If the pressure above the free surface of the liquid is atmospheric ( p 0 =p atm), then it should not be taken into account.

The pressure due to the weight of the liquid is unevenly distributed over the area of ​​the figure: the deeper the point of the figure is, the more pressure it experiences. Therefore, the point of application of force
P= r gh c.t w will lie below the center of gravity of the figure. We denote the coordinate of this point y c.d. To find it, we use the well-known position of theoretical mechanics: the sum of the moments of the constituent elementary forces about the axis X equal to the moment of the resultant force R about the same axis X, i.e.

,

as dp= r ghd w = r gy sina d w , then

. (1.47)

Here the value of the integral is the moment of inertia of the figure about the axis X:

and strength .

Substituting these relations into equation (1.47), we obtain

y c.d = J x / y c.t w . (1.48)

Formula (1.48) can be transformed using the fact that the moment of inertia J x relative to an arbitrary axis X equals

J x = J 0 +y2 c.t w, (1.49)

where J 0 - moment of inertia of the area of ​​the figure about the axis passing through its center of gravity and parallel to the axis X; y ts.t - the coordinate of the center of gravity of the figure (i.e. the distance between the axes).

Taking into account formula (1.49), we obtain: . (1.50)

Equation (1.50) shows that the center of pressure, due to the weight pressure of the liquid, is always located below the center of gravity of the figure under consideration by an amount and is immersed to a depth

, (1.51)

where h c.d =y ts.d sina - immersion depth of the center of pressure.

We limited ourselves to defining only one coordinate of the center of pressure. This is sufficient if the figure is symmetrical about the axis at passing through the center of gravity. In the general case, the second coordinate must also be determined. The method of its determination is the same as in the case considered above.

Center of pressure of the wing called the point of intersection of the resultant of aerodynamic forces with the chord of the wing.

The position of the center of pressure is determined by its coordinate X D - distance from the leading edge of the wing, which can be expressed in fractions of the chord

Direction of force R determined by the angle  formed with the direction of the undisturbed air flow (Fig. 59, a). It can be seen from the figure that

where To - aerodynamic quality of the profile.

Rice. 59 The center of pressure of the wing and the change in its position depending on the angle of attack

The position of the center of pressure depends on the shape of the airfoil and the angle of attack. On Fig. 59, b shows how the position of the center of pressure changes depending on the angle of attack for the profiles of the Yak 52 and Yak-55 aircraft, curve 1 - for the Yak-55 aircraft, curve 2 - for the Yak-52 aircraft.

It can be seen from the graph that the position CD when changing the angle of attack, the symmetrical profile of the Yak-55 aircraft remains unchanged and is approximately 1/4 of the distance from the toe of the chord.

table 2

When the angle of attack changes, the pressure distribution along the wing profile changes, and therefore the center of pressure moves along the chord (for the Yak-52 asymmetric airfoil), as shown in Fig. 60. For example, with a negative angle of attack of the Yak 52 aircraft, approximately equal to -4 °, the pressure forces in the nose and tail sections of the profile are directed in opposite directions and are equal. This angle of attack is called the zero-lift angle of attack.

Rice. 60 Moving the center of pressure of the Yak-52 wing when changing the angle of attack

With a slightly larger angle of attack, the pressure forces directed upwards are greater than the forces directed downwards, their resultant Y will lie behind the greater force (II), i.e., the center of pressure will be located in the tail section of the airfoil. With a further increase in the angle of attack, the location of the maximum pressure difference moves closer and closer to the nose edge of the wing, which naturally causes movement CD along the chord to the leading edge of the wing (III, IV).

most forward position CD at critical angle of attack  cr = 18° (V).

AIRCRAFT POWER PLANTS

PURPOSE OF THE POWER PLANT AND GENERAL INFORMATION ABOUT PROPELLERS

The power plant is designed to create the thrust force necessary to overcome drag and ensure the forward movement of the aircraft.

The traction force is generated by an installation consisting of an engine, a propeller (a propeller, for example) and systems that ensure the operation of the propulsion system (fuel system, lubrication system, cooling system, etc.).

At present, turbojet and turboprop engines are widely used in transport and military aviation. In sports, agricultural and various purposes of auxiliary aviation, power plants with piston internal combustion aircraft engines are still used.

On Yak-52 and Yak-55 aircraft, the power plant consists of an M-14P piston engine and a V530TA-D35 variable-pitch propeller. The M-14P engine converts the thermal energy of the burning fuel into the rotational energy of the propeller.

Air propeller - a bladed unit rotated by the engine shaft, which creates thrust in the air, necessary for the movement of the aircraft.

The operation of a propeller is based on the same principles as an aircraft wing.

PROPELLER CLASSIFICATION

Screws are classified:

according to the number of blades - two-, three-, four- and multi-blade;

according to the material of manufacture - wooden, metal;

in the direction of rotation (view from the cockpit in the direction of flight) - left and right rotation;

by location relative to the engine - pulling, pushing;

according to the shape of the blades - ordinary, saber-shaped, spade-shaped;

by types - fixed, unchangeable and variable step.

The propeller consists of a hub, blades and is mounted on the engine shaft with a special bushing (Fig. 61).

Fixed pitch screw has blades that cannot rotate around their axes. The blades with the hub are made as a single unit.

fixed pitch screw has blades that are installed on the ground before flight at any angle to the plane of rotation and are fixed. In flight, the installation angle does not change.

variable pitch screw has blades that during operation can, by means of hydraulic or electric control or automatically rotate around their axes and be set at the desired angle to the plane of rotation.

Rice. 61 Fixed-pitch two-blade air propeller

Rice. 62 Propeller V530TA D35

According to the range of blade angles, propellers are divided into:

on conventional ones, in which the installation angle varies from 13 to 50 °, they are installed on light aircraft;

on weathercocks - the installation angle varies from 0 to 90 °;

on brake or reverse propellers, have a variable installation angle from -15 to +90 °, with such a propeller they create negative thrust and reduce the length of the aircraft run.

The propellers are subject to the following requirements:

the screw must be strong and weigh little;

must have weight, geometric and aerodynamic symmetry;

must develop the necessary thrust during various evolutions in flight;

should work with the highest efficiency.

On the Yak-52 and Yak-55 aircraft, a conventional paddle-shaped wooden two-bladed tractor propeller of left rotation, variable pitch with hydraulic control V530TA-D35 is installed (Fig. 62).

GEOMETRIC CHARACTERISTICS OF THE SCREW

The blades during rotation create the same aerodynamic forces as the wing. The geometric characteristics of the propeller affect its aerodynamics.

Consider the geometric characteristics of the screw.

Blade shape in plan- the most common symmetrical and saber.

Rice. 63. Forms of a propeller: a - blade profile, b - blade shapes in plan

Rice. 64 Diameter, radius, geometric pitch of the propeller

Rice. 65 Helix development

Sections of the working part of the blade have wing profiles. The blade profile is characterized by chord, relative thickness and relative curvature.

For greater strength, blades with variable thickness are used - a gradual thickening towards the root. The chords of the sections do not lie in the same plane, since the blade is made twisted. The edge of the blade that cuts through the air is called the leading edge, and the trailing edge is called the trailing edge. The plane perpendicular to the axis of rotation of the screw is called the plane of rotation of the screw (Fig. 63).

screw diameter called the diameter of the circle described by the ends of the blades when the propeller rotates. The diameter of modern propellers ranges from 2 to 5 m. The diameter of the V530TA-D35 propeller is 2.4 m.

Geometric screw pitch - this is the distance that a progressively moving screw must travel in one complete revolution if it were moving in air as in a solid medium (Fig. 64).

Propeller blade angle  - this is the angle of inclination of the blade section to the plane of rotation of the propeller (Fig. 65).

To determine what the pitch of the propeller is, imagine that the propeller moves in a cylinder whose radius r is equal to the distance from the center of rotation of the propeller to point B on the propeller blade. Then the section of the screw at this point will describe a helix on the surface of the cylinder. Let's expand the segment of the cylinder, equal to the pitch of the screw H along the BV line. You will get a rectangle in which the helix has turned into a diagonal of this rectangle of the Central Bank. This diagonal is inclined to the plane of rotation of the BC screw at an angle  . From the right-angled triangle TsVB we find what the screw pitch is equal to:

The pitch of the screw will be the greater, the greater the angle of installation of the blade  . Propellers are subdivided into propellers with a constant pitch along the blade (all sections have the same pitch), variable pitch (sections have a different pitch).

The V530TA-D35 propeller has a variable pitch along the blade, as it is beneficial from an aerodynamic point of view. All sections of the propeller blade run into the air flow at the same angle of attack.

If all sections of the propeller blade have a different pitch, then the pitch of the section located at a distance from the center of rotation equal to 0.75R, where R is the radius of the propeller, is considered to be the common pitch of the propeller. This step is called nominal, and the installation angle of this section- nominal installation angle .

The geometric pitch of the propeller differs from the pitch of the propeller by the amount of slip of the propeller in the air (see Fig. 64).

Propeller pitch - this is the actual distance that a progressively moving propeller moves in the air with the aircraft in one complete revolution. If the speed of the aircraft is expressed in km/h and the number of propeller revolutions per second, then the pitch of the propeller is H P can be found using the formula

The pitch of the screw is slightly less than the geometric pitch of the screw. This is explained by the fact that the screw, as it were, slips in the air during rotation due to its low density relative to a solid medium.

The difference between the value of the geometric pitch and the pitch of the propeller is called screw slip and is determined by the formula

S= H- H n . (3.3)

1. Methods for applying the laws of hydraulics

1. Analytical. The purpose of applying this method is to establish the relationship between the kinematic and dynamic characteristics of the fluid. For this purpose, the equations of mechanics are used; as a result, the equations of motion and equilibrium of the fluid are obtained.

For a simplified application of the equations of mechanics, model fluids are used: for example, a continuous fluid.

By definition, not a single parameter of this continuum (continuous fluid) can be discontinuous, including its derivative, and at each point, if there are no special conditions.

Such a hypothesis makes it possible to establish a picture of the mechanical motion and equilibrium of a fluid at each point of the space continuum. Another technique used to facilitate the solution of theoretical problems is the solution of the problem for the one-dimensional case with the following generalization for the three-dimensional one. The fact is that for such cases it is not so difficult to establish the average value of the parameter under study. After that, you can get other equations of hydraulics, the most commonly used.

However, this method, like theoretical hydromechanics, the essence of which is a strictly mathematical approach, does not always lead to the necessary theoretical mechanism for solving the problem, although it reveals its general nature of the problem quite well.

2. Experimental. The main technique, according to this method, is the use of models, according to the theory of similarities: in this case, the obtained data are applied in practical conditions and it becomes possible to refine the analytical results.

The best option is a combination of the above two methods.

It is difficult to imagine modern hydraulics without the use of modern design tools: these are high-speed local networks, an automated workplace for a designer, and so on.

Therefore, modern hydraulics is often called computational hydraulics.

Liquid Properties

Since gas is the next aggregate state of matter, these forms of matter have a property that is common to both aggregate states. This property fluidity.

Based on the properties of fluidity, having considered the liquid and gaseous state of aggregation of matter, we will see that the liquid is the state of matter in which it is no longer possible to compress it (or it can be compressed infinitely little). A gas is a state of the same substance in which it can be compressed, that is, a gas can be called a compressible liquid, just like a liquid can be called an incompressible gas.

In other words, there are no special fundamental differences, except for compressibility, between gas and liquid.

An incompressible fluid, the balance and movement of which is studied by hydraulics, is also called drip liquid.

2. Basic properties of the liquid

Liquid density.

If we consider an arbitrary volume of liquid W, then it has mass M.

If the liquid is homogeneous, that is, if its properties are the same in all directions, then density will be equal to

where M is the mass of the liquid.

If you need to know r at every point BUT volume W, then

where D– elementarity of the considered characteristics at the point BUT.

Compressibility.

Characterized by the coefficient of volumetric compression.

It can be seen from the formula that we are talking about the ability of liquids to reduce the volume with a single change in pressure: due to the decrease, there is a minus sign.

temperature expansion.

The essence of the phenomenon is that a layer with a lower speed "slows down" the neighboring one. As a result, a special state of the liquid appears, due to intermolecular bonds in neighboring layers. This state is called viscosity.

The ratio of dynamic viscosity to fluid density is called kinematic viscosity.

Surface tension: due to this property, the liquid tends to occupy the smallest volume, for example, drops in spherical shapes.

In conclusion, we give a brief list of the properties of liquids that were discussed above.

1. Fluidity.

2. Compressibility.

3. Density.

4. Volumetric compression.

5. Viscosity.

6. Thermal expansion.

7. Tensile strength.

8. The ability to dissolve gases.

9. Surface tension.

3. Forces acting in a liquid

Liquids are divided into resting and moving.

Here we consider the forces that act on the liquid and outside it in the general case.

These forces themselves can be divided into two groups.

1. The forces are massive. In another way, these forces are called forces distributed over the mass: for each particle with mass? M= ?W force acting? F, depending on its mass.

Let the volume? W contains a dot BUT. Then at the point BUT:

where FA is the force density in an elementary volume.

Is the mass force density a vector quantity related to a unit volume? W; it can be projected along the coordinate axes and get: Fx, Fy, Fz. That is, the mass force density behaves like a mass force.

Examples of these forces include gravity, inertia (Coriolis and portable inertia forces), electromagnetic forces.

However, in hydraulics, except for special cases, electromagnetic forces are not considered.

2. surface forces. What are called forces that act on an elementary surface? w, which can be both on the surface and inside the liquid; on a surface arbitrarily drawn inside the liquid.

Forces are considered as such: pressure forces that make up the normal to the surface; friction forces that are tangent to the surface.

If by analogy (1) to determine the density of these forces, then:

normal stress at point BUT:

shear stress at point BUT:

Both mass and surface forces can be external, which act from the outside and are attached to some particle or each element of the liquid; internal, which are paired and their sum is equal to zero.

4. Hydrostatic pressure and its properties

General differential equations of liquid equilibrium - L. Euler's equations for hydrostatics.

If we take a cylinder with a liquid (at rest) and draw a dividing line through it, we get a liquid in a cylinder of two parts. If we now apply some force to one part, then it will be transmitted to the other through the separating plane of the section of the cylinder: we denote this plane S= w.

If the force itself is designated as the interaction transmitted from one part to another through the section? w, and is the hydrostatic pressure.

If we estimate the average value of this force,

Considering the point BUT as an extreme case w, we define:

If we go to the limit, then? w goes to the point BUT.

So ?p x -> ?p n . End result px= pn, in the same way you can get py= p n , p z= p n.

Hence,

py= p n , p z= p n.

We have proved that in all three directions (we chose them arbitrarily) the scalar value of the forces is the same, that is, does not depend on the orientation of the section? w.

This scalar value of the applied forces is the hydrostatic pressure, which was discussed above: it is this value, the sum of all components, that is transmitted through? w.

Another thing is that in total ( px+ py+ pz) some component will be equal to zero.

As we will see later, under certain conditions, the hydrostatic pressure can still be different at different points of the same fluid at rest, i.e.

p= f(x, y, z).

Properties of hydrostatic pressure.

1. Hydrostatic pressure is always directed along the normal to the surface and its value does not depend on the orientation of the surface.

2. Inside a fluid at rest at any point, the hydrostatic pressure is directed along the internal normal to the area passing through this point.

And px= py= pz= p n.

3. For any two points of the same volume of a homogeneous incompressible fluid (? = const)

1 + ?P 1 = ? 2 + ?P 1

where? is the density of the liquid;

P 1 , P 2 is the value of the field of body forces at these points.

A surface for which the pressure is the same for any two points is called equal pressure surface.

5. Equilibrium of a homogeneous incompressible fluid under the influence of gravity

This equilibrium is described by an equation called the basic equation of hydrostatics.

For a unit mass of a fluid at rest

For any two points of the same volume, then

The resulting equations describe the distribution of pressure in a liquid that is in equilibrium. Of these, equation (2) is the main equation of hydrostatics.

For reservoirs of large volumes or surfaces, clarification is required: whether it is co-directed to the radius of the Earth at a given point; how horizontal the surface in question is.

From (2) follows

p= p 0 + ?g(z – z 0 ) , (4)

where z 1 = z; p 1 = p; z 2 = z 0 ; p 2 = p 0 .

p= p 0 + ?gh, (5)

where? gh- weight pressure, which corresponds to a unit height and a unit area.

Pressure R called absolute pressurep abs.

If a R> p abs, then p – p atm= p 0 + ?gh – p atm- he's called overpressure:

p meas= p< p 0 , (6)

if p< p atm, then we talk about the difference in the liquid

p wack= p atm – p, (7)

called vacuum pressure.

6. Pascal's laws. Pressure measuring instruments

What happens at other points in the fluid if we apply some force?p? If we select two points and apply a force?p1 to one of them, then according to the basic equation of hydrostatics, at the second point the pressure will change by?p2.

whence it is easy to conclude that, with the other terms being equal, there must be

P1 = ?p2. (2)

We have obtained the expression of Pascal's law, which says: the change in pressure at any point of the fluid in an equilibrium state is transmitted to all other points without change.

So far we have been assuming that = const. If you have a communicating vessel that is filled with two liquids with? one ? ? 2 , and the external pressure p 0 = p 1 = p atm, then according to (1):

1gh = ? 2gh, (3)

where h 1 , h 2 is the height from the section of the surface to the corresponding free surfaces.

Pressure is a physical quantity that characterizes the forces directed along the normal to the surface of one object from the side of another.

If the forces are distributed normally and uniformly, then the pressure

where – F is the total applied force;

S is the surface to which the force is applied.

If the forces are unevenly distributed, then they talk about the average pressure value or consider it at a single point: for example, in a viscous fluid.

Pressure measuring instruments

One of the instruments used to measure pressure is a manometer.

The disadvantage of pressure gauges is that they have a large measurement range: 1-10 kPa.

For this reason, liquids are used in pipes that "reduce" the height, such as mercury.

The next instrument for measuring pressure is a piezometer.

7. Analysis of the basic equation of hydrostatics

The height of the pressure is usually called the piezometric height, or pressure.

According to the basic equation of hydrostatics,

p 1 + ?gh A = p 2 + ?gh H ,

where? is the density of the liquid;

g is the free fall acceleration.

p2, as a rule, is given by p 2 \u003d p atm, therefore, knowing h A and h H, it is easy to determine the desired value.

2. p 1 \u003d p 2 \u003d p atm. It is quite obvious which of = const, g = const it follows that h А = h H . This fact is also called the law of communicating vessels.

3.p1< p 2 = p атм.

A vacuum is formed between the surface of the liquid in the pipe and its closed end. Such devices are called vacuum gauges; they are used to measure pressures that are less than atmospheric pressure.

Height, which is a characteristic of the change in vacuum:

Vacuum is measured in the same units as pressure.

Piezometric head

Let's return to the basic hydrostatic equation. Here z is the coordinate of the considered point, which is measured from the XOY plane. In hydraulics, the XOY plane is called the comparison plane.

The coordinate z counted from this plane is called differently: geometric height; position height; geometric head of point z.

In the same basic equation of hydrostatics, the magnitude of p/?gh is also the geometric height to which the liquid rises as a result of pressure p. p/?gh, like the geometric height, is measured in meters. If atmospheric pressure acts on the liquid through the other end of the pipe, then the liquid in the pipe rises to a height pex /?gh, which is called the vacuum height.

The height corresponding to the pressure pvac is called the vacuum height.

In the main equation of hydrostatics, the sum z + p /?gh is the hydrostatic head H, there is also a piezometric head H n, which corresponds to atmospheric pressure p atm /?gh:

8. Hydraulic press

The hydraulic press serves to accomplish more work on a short path. Consider the operation of a hydraulic press.

For this, in order for work to be done on the body, it is necessary to act on the piston with a certain pressure P. This pressure, like P 2, is created as follows.

When the piston of the pump with the bottom surface area S 2 rises, it closes the first valve and opens the second one. After filling the cylinder with water, the second valve closes, the first one opens.

As a result, water fills the cylinder through the pipe and presses on the piston using the lower section S 1 with pressure P 2.

This pressure, like the pressure P 1, compresses the body.

It is quite obvious that P 1 is the same pressure as P 2, the only difference is that they act on different areas S 2 and S 1.

In other words, pressure:

P 1 = pS 1 and P 2 = pS 2 . (one)

Expressing p = P 2 /S 2 and substituting in the first formula, we get:

An important conclusion follows from the obtained formula: a piston with a larger area S 1 from the side of a piston with a smaller area S 2 is transferred to a pressure as many times greater as the times S 1 > S 2 .

However, in practice, due to friction forces, up to 15% of this transmitted energy is lost: it is spent on overcoming the resistance of friction forces.

And yet, hydraulic presses have an efficiency of ? = 85% - a fairly high figure.

In hydraulics, formula (2) will be rewritten in the following form:

where P 1 is denoted as R;

hydraulic accumulator

The hydraulic accumulator serves to maintain the pressure in the system connected to it constant.

Achieving a constant pressure occurs as follows: on top of the piston, on its area?, the load P acts.

The pipe serves to transfer this pressure throughout the system.

If there is an excess of liquid in the system (mechanism, installation), then the excess enters the cylinder through the pipe, the piston rises.

With a lack of fluid, the piston descends, and the pressure p created in this case, according to Pascal's law, is transmitted to all parts of the system.

9. Determination of the pressure force of a fluid at rest on flat surfaces. Center of pressure

In order to determine the force of pressure, we will consider a fluid that is at rest relative to the Earth. If we choose an arbitrary horizontal area in the liquid?, then, provided that p atm = p 0 acts on the free surface, on? excess pressure is applied:

R iz = ?gh?. (one)

Since in (1) ?gh ? is nothing but mg, since h ? and? V = m, the excess pressure is equal to the weight of the liquid contained in the volume h ? . The line of action of this force passes through the center of the square? and is directed along the normal to the horizontal surface.

Formula (1) does not contain a single quantity that would characterize the shape of the vessel. Therefore, R izb does not depend on the shape of the vessel. Therefore, an extremely important conclusion follows from formula (1), the so-called hydraulic paradox- with different shapes of vessels, if the same p 0 appears on the free surface, then with equality of densities?, areas? and heights h, the pressure exerted on the horizontal bottom is the same.

When the bottom plane is inclined, wetting of the surface with an area of ​​\u200b\u200btakes place. Therefore, unlike the previous case, when the bottom lay in a horizontal plane, it cannot be said that the pressure is constant.

To determine it, we divide the area? on elementary areas d?, any of which is subject to pressure

By definition of pressure force,

and dP is directed along the normal to the site?.

Now, if we determine the total force that affects the area ?, then its value:

Having determined the second term in (3), we find Р abs.

Pabs \u003d? (p 0 + h c. e). (4)

We have obtained the desired expressions for determining the pressures acting on the horizontal and inclined

plane: R izb and R abs.

Consider one more point C, which belongs to the area?, more precisely, the point of the center of gravity of the wetted area?. At this point, the force P 0 = ? 0?.

The force acts at any other point that does not coincide with point C.

10. Determination of the pressure force in the calculations of hydraulic structures

When calculating in hydraulic engineering, the overpressure force P is of interest, at:

p 0 = p atm,

where p0 is the pressure applied to the center of gravity.

Speaking of force, we will mean the force applied at the center of pressure, although we will mean that this is the force of excess pressure.

To determine P abs, we use moment theorem, from theoretical mechanics: the moment of the resultant about an arbitrary axis is equal to the sum of the moments of the constituent forces about the same axis.

Now, according to this resultant moment theorem:

Since at р 0 = р atm, P = ?gh c. e.?, so dP = ?ghd ? = ?gsin?ld ? , therefore (hereinafter, for convenience, we will not distinguish between p el and p abs), taking into account P and dP from (2), and after transformations, it follows:

If we now transfer the axis of the moment of inertia, that is, the line of the liquid edge (axis O Y) to the center of gravity?, that is, to point C, then relative to this axis the moment of inertia of the center of pressure of point D will be J 0.

Therefore, the expression for the center of pressure (point D) without transferring the axis of the moment of inertia from the same edge line, coinciding with the axis O Y , will look like:

I y \u003d I 0 + ?l 2 c.t.

The final formula for determining the location of the center of pressure from the axis of the liquid edge:

l c. d. \u003d l c. + I 0 /S.

where S = ?l c.d. is a statistical moment.

The final formula for l c.d. allows you to determine the center of pressure in the calculations of hydraulic structures: for this, the section is divided into component sections, for each section l c.d. is found. relative to the line of intersection of this section (you can use the continuation of this line) with a free surface.

The centers of pressure of each of the sections are below the center of gravity of the wetted area along the inclined wall, more precisely along the axis of symmetry, at a distance I 0 /?l c.u.

11. General procedure for determining forces on curved surfaces

1. In general, this pressure is:

where Wg is the volume of the considered prism.

In a particular case, the directions of the lines of action of the force on the curvilinear surface of the body, the pressures depend on the direction cosines of the following form:

The pressure force on a cylindrical surface with a horizontal generatrix is ​​completely determined. In the case under consideration, the O Y axis is directed parallel to the horizontal generatrix.

2. Now consider a cylindrical surface with a vertical generatrix and direct the O Z axis parallel to this generatrix, what does it mean? z = 0.

Therefore, by analogy, as in the previous case,

where h "c.t. - the depth of the center of gravity of the projection under the piezometric plane;

h" c.t. - the same, only for? y .

Similarly, the direction is determined by the direction cosines

If we consider a cylindrical surface, more precisely, a volumetric sector, with a radius? and height h, with a vertical generatrix, then

h "c.t. \u003d 0.5h.

3. It remains to generalize the obtained formulas for the applied application of an arbitrary curvilinear surface:

12. Law of Archimedes. Conditions of buoyancy of submerged bodies

It is necessary to find out the conditions for the equilibrium of a body immersed in a liquid and the consequences that follow from these conditions.

The force acting on the submerged body is the resultant of the vertical components P z1 , P z2 , i.e. e.:

P z1 = P z1 – P z2 = ?gW T. (1)

where P z1 , P z2 - forces directed downwards and upwards.

This expression characterizes the force, which is commonly called the Archimedean force.

The Archimedean force is a force equal to the weight of an immersed body (or part of it): this force is applied to the center of gravity, directed upwards and quantitatively equal to the weight of the fluid displaced by the immersed body or part of it. We formulated the law of Archimedes.

Now let's deal with the basic conditions for the buoyancy of the body.

1. The volume of fluid displaced by the body is called volumetric displacement. The center of gravity of volumetric displacement coincides with the center of pressure: it is in the center of pressure that the resultant force is applied.

2. If the body is completely immersed, then the volume of the body W coincides with W T, if not, then W< W Т, то есть P z = ?gW.

3. The body will only float if the body weight

G T \u003d P z \u003d ?gW, (2)

i.e., equal to the Archimedean force.

4. Swimming:

1) underwater, that is, the body is completely submerged, if P = G t, which means (with the body homogeneous):

GW=? t gW T, whence

where?,? T is the density of the liquid and the body, respectively;

W - volumetric displacement;

W T is the volume of the submerged body itself;

2) surface, when the body is partially submerged; in this case, the depth of immersion of the lowest point of the wetted surface of the body is called the draft of the floating body.

The waterline is the line of intersection of the immersed body along the perimeter with the free surface of the liquid.

The area of ​​the waterline is the area of ​​the submerged part of the body bounded by the waterline.

The line that passes through the centers of gravity of the body and pressure is called the axis of navigation, which is vertical when the body is in equilibrium.

13. Metacenter and metacentric radius

The ability of a body to restore its original equilibrium state after the termination of an external influence is called stability.

According to the nature of the action, statistical and dynamic stability are distinguished.

Since we are in the framework of hydrostatics, we will deal with statistical stability.

If the roll formed after external influence is irreversible, then the stability is unstable.

In the case of conservation after the cessation of external influence, the balance is restored, then the stability is stable.

The condition for statistical stability is swimming.

If swimming is underwater, then the center of gravity should be located below the center of displacement on the axis of navigation. Then the body will float. If surfaced, then stability depends on what angle? body rotated around its longitudinal axis.

At?< 15 o , после прекращения внешнего воздействия равновесие тела восстанавливается; если? >= 15 o , then the roll is irreversible.

The point of intersection of the Archimedean force with the axis of navigation is called the metacenter: in this case, it also passes through the center of pressure.

The metacentric radius is the radius of the circle, part of which is the arc along which the center of pressure moves to the metacenter.

Designations are accepted: metacenter – M, metacentric radius – ? m.

At?< 15 о

where I 0 is the central moment of the plane relative to the longitudinal axis contained in the waterline.

After the introduction of the concept of “metacenter”, the stability conditions change somewhat: it was said above that for stable stability the center of gravity must be above the center of pressure on the axis of navigation. Now suppose that the center of gravity should not be above the metacenter. Otherwise, the forces and will increase the roll.

How obvious is the roll distance? between the center of gravity and the center of pressure varies within?< ? м.

In this case, the distance between the center of gravity and the metacenter is called the metacentric height, which, under condition (2), is positive. The greater the metacentric height, the less likely the floating body is to roll. The presence of stability relative to the longitudinal axis of the plane containing the waterline is a necessary and sufficient condition for stability relative to the transverse axis of the same plane.

14. Methods for determining the movement of a liquid

Hydrostatics is the study of a fluid in its equilibrium state.

Fluid kinematics studies a fluid in motion without considering the forces that generate or accompany this motion.

Hydrodynamics also studies the movement of a fluid, but depending on the effect of forces applied to the fluid.

In kinematics, a continuous model of a fluid is used: some of its continuum. According to the continuity hypothesis, the considered continuum is a liquid particle in which a huge number of molecules are constantly moving; it has no gaps or voids.

If in the previous questions, studying hydrostatics, a continuous medium was taken as a model for studying a fluid in equilibrium, then here, using the same model as an example, they will study a fluid in motion, studying the movement of its particles.

There are two ways to describe the motion of a particle, and through it a fluid.

1. Lagrange method. This method is not used in describing wave functions. The essence of the method is as follows: it is required to describe the motion of each particle.

The initial time t 0 corresponds to the initial coordinates x 0 , y 0 , z 0 .

However, by the time t they are already different. As you can see, we are talking about the movement of each particle. This motion can be considered definite if it is possible to indicate for each particle the coordinates x, y, z at an arbitrary time t as continuous functions of x 0 , y 0 , z 0 .

x = x(x 0 , y 0 , z 0 , t)

y \u003d y (x 0, y 0, z 0, t)

z = z(x 0 , y 0 , z 0 , t) (1)

Variables x 0 , y 0 , z 0 , t are called Lagrange variables.

2. Method for determining the motion of particles according to Euler. The movement of the fluid in this case occurs in some stationary area of ​​the fluid flow in which the particles are located. Points are arbitrarily chosen in the particles. The time t as a parameter is given at each time of the considered region, which has coordinates x, y, z.

The area under consideration, as is already known, is within the flow and is motionless. The speed of a fluid particle u in this area at each time t is called the instantaneous local speed.

The velocity field is the totality of all instantaneous velocities. Changing this field is described by the following system:

u x = u x (x,y,z,t)

u y = u y (x,y,z,t)

u z = u z (x, y, z, t)

The variables in (2) x, y, z, t are called Euler variables.

15. Basic concepts used in fluid kinematics

The essence of the above velocity field are vector lines, which are often called streamlines.

A streamline is such a curved line, for any point of which, at a selected moment of time, the local velocity vector is directed tangentially (we are not talking about the normal component of the velocity, since it is equal to zero).

Formula (1) is the differential equation of the streamline at time t. Therefore, by setting different ti according to the obtained i, where i = 1,2, 3, …, it is possible to construct a streamline: it will be the envelope of a broken line consisting of i.

Streamlines, as a rule, do not intersect due to the condition? 0 or? ?. But still, if these conditions are violated, then the streamlines intersect: the intersection point is called singular (or critical).

1. Unsteady motion, which is so called due to the fact that local velocities at the considered points of the selected area change with time. Such motion is completely described by a system of equations.

2. Steady motion: since with such motion the local speeds do not depend on time and are constant:

u x = u x (x,y,z)

u y = u y (x,y,z)

u z = u z (x, y, z)

The streamlines and particle trajectories coincide, and the differential equation for the streamline has the form:

The totality of all streamlines that pass through each point of the flow contour forms a surface, which is called a stream tube. Inside this tube moves the liquid contained in it, which is called a trickle.

A trickle is considered elementary if the contour under consideration is infinitesimal, and finite if the contour has a finite area.

The cross section of the trickle, which is normal at each of its points to the streamlines, is called the live cross section of the trickle. Depending on the finiteness or infinite smallness, the area of ​​the trickle is usually denoted, respectively, by ? and d?.

A certain volume of liquid that passes through the free section per unit time is called the flow rate of the trickle Q.

16. Vortex motion

Features of the types of motion considered in hydrodynamics.

The following types of movement can be distinguished.

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 motion, despite the 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.

17. Laminar motion

This motion is also called potential (irrotational) motion.

With such a motion, there is no rotation of particles around the instantaneous axes that pass through the poles of liquid particles. For this reason:

x=0; ? y=0; ? z = 0. (1)

X=? y=? z = 0.

It was noted above that when a fluid moves, not only the position of the particles in space changes, but also their deformation along linear parameters. If the vortex motion considered above is a consequence of a change in the spatial position of a liquid particle, then laminar (potential, or irrotational) motion is a consequence of deformation phenomena of linear parameters, for example, shape and volume.

The vortex motion was determined by the direction of the vortex vector

where? - angular velocity, which is a characteristic of angular deformations.

The deformation of this movement is characterized by the deformation of these components

But, since laminar motion? x=? y=? z = 0, then:

It can be seen from this formula: since there are partial derivatives related to each other in formula (4), then these partial derivatives belong to some function.

18. Velocity potential and acceleration in laminar motion

? = ?(x, y, z) (1)

Function? called the speed potential.

With that in mind, components? look like this:

Formula (1) describes the unsteady motion, since it contains the parameter t.

Acceleration in laminar motion

The acceleration of the motion of a liquid particle has the form:

where du/dt are total time derivatives.

The acceleration can be represented in this form, based on

Components of the desired acceleration

Formula (4) contains information about the total acceleration.

The terms ?u x /?t, ?u y /?t, ?u z /?t, are called local accelerators at the point under consideration, which characterize the laws of change in the velocity field.

If the motion is steady, then

The velocity field itself can be called convection. Therefore, the remaining parts of the sums corresponding to each row (4) are called convective accelerations. More precisely, projections of convective acceleration, which characterizes the inhomogeneity of the velocity field (or convection) at a particular time t.

The full acceleration itself can be called some substance, which is the sum of projections

dux/dt, duy/dt, duz/dt,

19. Fluid continuity equation

Quite often, when solving problems, you have to define unknown functions of the type:

1) p \u003d p (x, y, z, t) - pressure;

2) n x (x, y, z, t), ny(x, y, z, t), n z (x, y, z, t) are velocity projections on the coordinate axes x, y, z;

3) ? (x, y, z, t) is the density of the liquid.

These unknowns, there are five in total, are determined by the Euler system of equations.

There are only three Euler equations, and, as we see, there are five unknowns. Two more equations are missing in order to determine these unknowns. The continuity equation is one of the two missing equations. The equation of state of a continuum is used as the fifth equation.

Formula (1) is a continuity equation, that is, the desired equation for the general case. In the case of fluid incompressibility??/dt = 0, because? = const, so from (1) it follows:

since these terms, as is known from the course of higher mathematics, are the rate of change in the length of a unit vector in one of the directions X, Y, Z.

As for the whole sum in (2), it expresses the rate of relative volume change dV.

This volumetric change is called differently: volumetric expansion, divergence, divergence of the velocity vector.

For a trickle, the equation will look like:

where Q is the amount of liquid (flow rate);

? is the angular velocity of the jet;

L is the length of the elementary section of the considered trickle.

If the pressure is steady or the free area? = const, then?? /?t = 0, i.e. according to (3),

Q/?l = 0, therefore,

20. Fluid flow characteristics

In hydraulics, a flow is considered such a mass movement when this mass is limited:

1) hard surfaces;

2) surfaces that separate different liquids;

3) free surfaces.

Depending on what kind of surfaces or their combinations a moving fluid is limited to, the following types of flows are distinguished:

1) non-pressure, when the flow is limited by a combination of solid and free surfaces, for example, a river, a canal, a pipe with an incomplete section;

2) pressure, for example, a pipe with a full section;

3) hydraulic jets, which are limited to a liquid (as we will see later, such jets are called flooded) or gaseous medium.

Free section and hydraulic radius of the flow. Continuity equation in hydraulic form

The flow section from which all streamlines are normal (i.e., perpendicular) is called the live section.

The concept of the hydraulic radius is extremely important in hydraulics.

For a pressure flow with a circular free section, diameter d and radius r 0 , the hydraulic radius is expressed as

When deriving (2), we took into account

The flow rate is the amount of fluid that passes through the free section per unit of time.

For a flow consisting of elementary jets, the flow rate is:

where dQ = d? is the flow rate of the elementary flow;

U is the fluid velocity in the given section.

21. A kind of movement

Depending on the nature of the change in the velocity field, the following types of steady motion are distinguished:

1) uniform, when the main characteristics of the flow - the shape and area of ​​the free section, the average flow velocity, including along the length, depth of the flow (if the movement is free-flowing) - are constant, do not change; in addition, along the entire length of the stream along the streamline, local velocities are the same, and there are no accelerations at all;

2) uneven, when none of the factors listed for uniform motion is met, including the condition of parallelism of the current lines.

There is a smoothly varying movement, which is still considered uneven movement; with such a motion, it is assumed that the streamlines are approximately parallel, and all other changes occur smoothly. Therefore, when the direction of movement and the OX axis are co-directed, then some quantities are neglected

Ux? U; Uy = Uz = 0. (1)

Continuity equation (1) for smoothly changing motion has the form:

similar for other directions.

Therefore, this kind of movement is called uniform rectilinear;

3) if the movement is unsteady or unsteady, when local speeds change over time, then the following varieties are distinguished in such movement: rapidly changing movement, slowly changing movement, or, as it is often called, quasi-stationary.

Pressure is divided depending on the number of coordinates in the equations describing it, into: spatial, when the movement is three-dimensional; flat, when the motion is two-dimensional, i.e. Uх, Uy or Uz is equal to zero; one-dimensional, when the movement depends on only one of the coordinates.

In conclusion, we note the following continuity equation for a stream, provided that the fluid is incompressible, i.e., ?= const, for a flow this equation has the form:

Q=? one ? 1=? 2? 2 = … = ? i? i = idem, (3)

where? i? i are the speed and area of ​​the same section with number i.

Equation (3) is called the hydraulic continuity equation.

22. Differential equations of motion of an inviscid fluid

The Euler equation is one of the fundamental ones in hydraulics, along with the Bernoulli equation and some others.

The study of hydraulics as such practically begins with the Euler equation, which serves as a starting point for reaching other expressions.

Let's try to derive this equation. Let we have an infinitesimal parallelepiped with faces dxdydz in an inviscid fluid with density ?. It is filled with liquid and moves as part of the flow. What forces act on the selected object? These are mass forces and surface pressure forces that act on dV = dxdydz from the side of the liquid in which the selected dV is located. Just as mass forces are proportional to the mass, surface forces are proportional to the areas under pressure. These forces are directed to the faces inward along the normal. Let us define the mathematical expression of these forces.

Let us name, as in obtaining the continuity equation, the faces of the parallelepiped:

1, 2 – perpendicular to the ОХ axis and parallel to the ОY axis;

3, 4 - perpendicular to the O Y axis and parallel to the O X axis;

5, 6 - perpendicular to the O Z axis and parallel to the O X axis.

Now you need to determine what force is applied to the center of mass of the parallelepiped.

The force applied to the center of mass of the parallelepiped, which causes this fluid to move, is the sum of the forces found, that is

Divide (1) by mass?dxdydz:

The resulting system of equations (2) is the desired equation of motion of an inviscid fluid - the Euler equation.

Two more equations are added to the three equations (2), since there are five unknowns, and a system of five equations with five unknowns is solved: one of the two additional equations is the continuity equation. Another equation is the equation of state. For example, for an incompressible fluid, the equation of state can be the condition? = const.

The equation of state must be chosen in such a way that it contains at least one of the five unknowns.

23. Euler equation for different states

The Euler equation for different states has different forms of writing. Since the equation itself was obtained for the general case, we consider several cases:

1) the movement is unsteady.

2) liquid at rest. Therefore, Ux = Uy = Uz = 0.

In this case, the Euler equation turns into an equation for a uniform fluid. This equation is also differential and is a system of three equations;

3) the fluid is nonviscous. For such a fluid, the equation of motion has the form

where Fl is the projection of the distribution density of mass forces on the direction along which the tangent to the streamline is directed;

dU/dt – particle acceleration

Substituting U = dl/dt into (2) and taking into account that (?U/?l)U = 1/2(?U 2 /?l), we obtain the equation.

We have given three forms of the Euler equation for three particular cases. But this is not the limit. The main thing is to correctly determine the equation of state, which contained at least one unknown parameter.

Euler's equation, combined with the continuity equation, can be applied to any case.

The equation of state in general form:

Thus, the Euler equation, the continuity equation, and the equation of state are sufficient to solve many hydrodynamic problems.

With the help of five equations, five unknowns are easily found: p, Ux, Uy, Uz, ?.

An inviscid fluid can also be described by another equation

24. Gromeka Form of the Equation of Motion for an Inviscid Fluid

The Gromeka equations are simply a different, slightly modified form of the Euler equation.

For example, for the x coordinate

To convert it, use the equations of the components of the angular velocity for the vortex motion.

Transforming the y-th and z-th components in the same way, we finally arrive at the Gromeko form of the Euler equation

The Euler equation was obtained by the Russian scientist L. Euler in 1755, and transformed into the form (2) again by the Russian scientist I. S. Gromeka in 1881

Gromeko equation (under the influence of body forces on the liquid):

Insofar as

– dP = Fxdx + Fydy + Fzdz, (4)

then for the components Fy, Fz one can derive the same expressions as for Fx, and, substituting this into (2), arrive at (3).

25. Bernoulli equation

The Gromeka equation is suitable for describing the motion of a fluid if the components of the motion function contain some vortex quantity. For example, this vortex value is contained in the components?x,?y,?z of the angular velocity w.

The condition that the movement is steady is the absence of acceleration, that is, the condition that the partial derivatives of all velocity components are equal to zero:

Now if we fold

then we get

If we project the displacement by an infinitesimal value dl onto the coordinate axes, we get:

dx=Uxdt; dy = Uy dt; dz = Uzdt. (3)

Now we multiply each equation (3) by dx, dy, dz, respectively, and add them:

Assuming that the right side is equal to zero, and this is possible if the second or third rows are equal to zero, we get:

We have obtained the Bernoulli equation

26. Analysis of the Bernoulli equation

this equation is nothing but the equation of a streamline in steady motion.

From this follows the conclusions:

1) if the motion is steady, then the first and third rows in the Bernoulli equation are proportional.

2) rows 1 and 2 are proportional, i.e.

Equation (2) is the vortex line equation. The conclusions from (2) are similar to the conclusions from (1), only the streamlines replace the vortex lines. In a word, in this case condition (2) is satisfied for vortex lines;

3) the corresponding members of rows 2 and 3 are proportional, i.e.

where a is some constant value; if we substitute (3) into (2), then we obtain the streamline equation (1), since from (3) it follows:

X = aUx; ? y = aUy; ? z = aUz. (4)

Here follows an interesting conclusion that the vectors of linear velocity and angular velocity are co-directed, that is, parallel.

In a broader sense, one must imagine the following: since the motion under consideration is steady, it turns out that the particles of the liquid move in a spiral and their trajectories along the spiral form streamlines. Therefore, streamlines and particle trajectories are one and the same. This kind of movement is called screw.

4) the second row of the determinant (more precisely, the members of the second row) is equal to zero, i.e.

X=? y=? z = 0. (5)

But the absence of angular velocity is equivalent to the absence of vortex motion.

5) let line 3 be equal to zero, i.e.

Ux = Uy = Uz = 0.

But this, as we already know, is the condition for the equilibrium of the liquid.

The analysis of the Bernoulli equation is completed.

27. Application examples of the Bernoulli equation

In all cases, it is required to determine the mathematical formula of the potential function that enters the Bernoulli equation: but this function has different formulas in different situations. Its form depends on what body forces act on the liquid under consideration. So let's consider two situations.

One massive force

In this case, gravity is implied, which acts as the only mass force. Obviously, in this case, the Z axis and the distribution density Fz of the force P are oppositely directed, therefore,

Fx=Fy=0; Fz = -g.

Since - dP = Fxdx + Fydy + Fzdz, then - dP = Fzdz, finally dP = -gdz.

We integrate the resulting expression:

P \u003d -gz + C, (1)

where C is some constant.

Substituting (1) into the Bernoulli equation, we have an expression for the case of the action of only one mass force on the liquid:

If we divide equation (2) by g (because it is constant), then

We have received one of the most frequently used formulas in solving hydraulic problems, so you should remember it especially well.

If it is required to determine the location of the particle in two different positions, then the relation for the coordinates Z 1 and Z 2 characterizing these positions is fulfilled

We can rewrite (4) in another form

28. Cases when there are several mass forces

In this case, let's complicate the task. Let the following forces act on the particles of the liquid: gravity; centrifugal force of inertia (carries movement away from the center); Coriolis force of inertia, which causes the particles to rotate around the Z-axis with simultaneous translational motion.

In this case, we were able to imagine a screw motion. Rotation occurs with an angular velocity w. It is necessary to imagine a curvilinear section of a certain fluid flow, in this section the flow, as it were, rotates around a certain axis with an angular velocity.

A special case of such a flow can be considered a hydraulic jet. So let's consider an elementary stream of liquid and apply the Bernoulli equation in relation to it. To do this, we place an elementary hydraulic jet in the XYZ coordinate system in such a way that the YOX plane rotates around the O Z axis.

Fx 1 = Fy 1 = 0; Fz 1 = -g -

the components of gravity (that is, its projections on the coordinate axes), referred to a unit mass of fluid. A second force is applied to the same mass - the force of inertia? 2 r, where r is the distance from the particle to the axis of rotation of its component.

Fx2=? 2x; Fy 2 = ? 2y; Fz 2 = 0

due to the fact that the OZ axis "does not rotate".

The final Bernoulli equation. For the case in question:

Or, which is the same, after dividing by g

If we consider two sections of an elementary jet, then, using the above mechanism, it is easy to verify that

where z 1 , h 1 , U 1 , V 1 , z 2 , h 2 , U 2 , V 2 are the parameters of the corresponding sections

29. Energy meaning of the Bernoulli equation

Let now we have a steady motion of a fluid, which is inviscid, incompressible.

And let it be under the influence of gravity and pressure, then the Bernoulli equation has the form:

Now we need to identify each of the terms. The potential energy of the position Z is the height of the elementary stream above the horizontal comparison plane. A liquid with mass M at a height Z from the comparison plane has some potential energy MgZ. Then

This is the same potential energy per unit mass. Therefore, Z is called the specific potential energy of the position.

A moving particle with mass Mi and speed u has weight MG and kinematic energy U2/2g. If we correlate the kinematic energy with a unit mass, then

The resulting expression is nothing but the last, third term in the Bernoulli equation. Therefore, U 2 / 2 is the specific kinetic energy of the jet. Thus, the general energy meaning of the Bernoulli equation is as follows: the Bernoulli equation is a sum containing the total specific energy of the liquid cross section in the flow:

1) if the total energy is related to unit mass, then it is the sum gz + p/? + U 2 / 2;

2) if the total energy is related to a unit volume, then?gz + p + pU 2 / 2;

3) if the total energy is related to unit weight, then the total energy is the sum z + p/?g + U 2 / 2g. It should not be forgotten that the specific energy is determined relative to the comparison plane: this plane is chosen arbitrarily and horizontally. For any pair of points arbitrarily chosen from a flow in which the motion is steady and which moves in a potential vortex, and the fluid is inviscid-incompressible, the total and specific energies are the same, that is, they are distributed uniformly along the flow.

30. Geometric meaning of the Bernoulli equation

The basis of the theoretical part of such an interpretation is the hydraulic concept of pressure, which is usually denoted by the letter H, where

The hydrodynamic head H consists of the following types of heads, which are included in formula (198) as terms:

1) piezometric head, if in (198) p = p izg, or hydrostatic, if p ? p out;

2) U 2 /2g - velocity head.

All terms have a linear dimension, they can be considered heights. Let's call these heights:

1) z - geometric height, or height by position;

2) p/?g is the height corresponding to pressure p;

3) U 2 /2g - high-speed altitude corresponding to the speed.

The locus of the ends of the height H corresponds to a certain horizontal line, which is commonly called a pressure line or a line of specific energy.

In the same way (by analogy), the geometrical places of the ends of the piezometric pressure are usually called the piezometric line. The pressure and piezometric lines are located at a distance (height) p atm /?g from each other, since p \u003d p izg + pat, i.e.

Note that the horizontal plane containing the pressure line and located above the comparison plane is called the pressure plane. The characteristic of the plane during different movements is called the piezometric slope J p, which shows how the piezometric head (or piezometric line) changes per unit length:

The piezometric slope is considered positive if it decreases along the stream (or stream), hence the minus sign in formula (3) in front of the differential. For J p to remain positive, the condition must be satisfied

31. Equations of motion of a viscous fluid

To obtain the equation of motion for a viscous fluid, consider the same fluid volume dV = dxdydz, which belongs to the viscous fluid (Fig. 1).

The faces of this volume will be denoted as 1, 2, 3, 4, 5, 6.

Rice. 1. Forces acting on an elementary volume of a viscous fluid in a flow

xy=? yx; ? xz=? zx ; ? yz=? zy. (one)

Then only three of the six shear stresses remain, since they are equal in pairs. Therefore, only six independent components are sufficient to describe the motion of a viscous fluid:

p xx , p yy , p zz , ? xy (or? yx), ? xz(?zx), ? yz(?zy).

A similar equation can easily be obtained for the axes O Y and O Z ; by combining all three equations into a system, we obtain (after dividing by?)

The resulting system is called the equation of motion of a viscous fluid in stresses.

32. Deformation in a moving viscous fluid

In a viscous fluid, there are friction forces; therefore, when moving, one layer slows down the other. As a result, there is compression, deformation of the liquid. Because of this property, the liquid is called viscous.

If we recall Hooke's law from mechanics, then according to it, the stress that occurs in a solid body is proportional to the corresponding relative deformation. For a viscous fluid, the relative strain is replaced by the strain rate. We are talking about the angular velocity of deformation of a liquid particle d?/dt, which is otherwise called the shear strain rate. Even Isaac Newton established a regularity about the proportionality of the internal friction force, the area of ​​​​contact of the layers and the relative speed of the layers. They also installed

coefficient of proportionality of the dynamic viscosity of the liquid.

If we express the shear stress in terms of its components, then

And as for the normal stresses (? is the tangential component of the deformation), which are dependent on the direction of action, they also depend on the area to which they are applied. This property is called invariance.

Sum of normal stress values

To finally establish the dependency between pud?/dt via the dependency between normal

(p xx ,p yy , p zz) and tangents (? xy = ? yx ; ? yx = ? xy ; ? zx = ? xz), representing from (3)

pxx = -p + p? xx , (4)

where p? xx - additional normal stresses, which depend on the direction of action, according to

analogy with formula (4) we get:

Having done the same for the components p yy , p zz , we got the system.

33. Bernoulli's equation for the motion of a viscous fluid

Elementary trickle in the steady motion of a viscous fluid

The equation for this case has the form (we give it without derivation, since its derivation is associated with the use of some operations, the reduction of which would complicate the text)

The loss of pressure (or specific energy) h Пp is the result of the fact that part of the energy is converted from mechanical to thermal. Since the process is irreversible, there is a loss of pressure.

This process is called energy dissipation.

In other words, h Pp can be considered as the difference between the specific energy of two sections; when the fluid moves from one to the other, there is a loss of pressure. Specific energy is the energy contained in a unit mass.

A flow with a steady, smoothly varying motion. Specific kinematic energy coefficient X

In order to obtain the Bernoulli equation in this case, one has to proceed from equation (1), that is, one must move from a trickle to a stream. But for this you need to decide what the flow energy is (which consists of the sum of potential and kinematic energies) with a smoothly changing flow

Let's deal with potential energy: with a smooth change in motion, if the flow is steady

Finally, during the motion under consideration, the pressure over the living section is distributed according to the hydrostatic law, i.e.

where X is called the kinetic energy coefficient, or the Coriolis coefficient.

The coefficient X is always greater than 1. From (4) it follows:

34. Hydrodynamic impact. Hydro and piezo slopes

Due to the smoothness of fluid motion for any point of the free section, the potential energy is Ep = Z + p/?g. Specific kinetic Еk= X? 2/2g. Therefore, for the cross section 1–1, the total specific energy

The sum of the right side of (1) is also called the hydrodynamic head H. In the case of a nonviscous fluid, U 2 = x? 2. Now it remains to take into account the head loss h pr fluid when it moves to section 2–2 (or 3–3).

For example, for section 2–2:

It should be noted that the smooth variability condition must be satisfied only in sections 1–1 and 2–2 (only in the considered ones): between these sections, the smooth variability condition is not necessary.

In formula (2), the physical meaning of all quantities was given earlier.

Basically, everything is the same as in the case of a non-viscous liquid, the main difference is that now the pressure line E \u003d H \u003d Z + p /?g + X? 2 /2g is not parallel to the horizontal plane of comparison, because there are head losses

The degree of pressure loss hpr along the length is called the hydraulic slope J. If the loss of pressure hpr occurs evenly, then

The numerator in formula (3) can be considered as the increment of head dH over the length dl.

Therefore, in the general case

The minus sign in front of dH / dl is because the change in head along its course is negative.

If we consider the change in the piezometric head Z + p/?g, then the value (4) is called the piezometric slope.

The pressure line, also known as the specific energy line, is above the piezometric line by a height u 2 /2g: the same is here, but the difference between these lines is now x? 2/2g. This difference is also maintained in non-pressure motion. Only in this case does the piezometric line coincide with the free flow surface.

35. Bernoulli's equation for the unsteady motion of a viscous fluid

In order to obtain the Bernoulli equation, it will be necessary to determine it for an elementary trickle with an unsteady motion of a viscous fluid, and then extend it to the entire flow

First of all, let us recall the main difference between unsteady motion and steady motion. If in the first case, at any point in the flow, local velocities change with time, then in the second case, there are no such changes.

Here is the Bernoulli equation for an elementary trickle without derivation:

what is taken into account here? =Q; ?Q = m; m? = (KD) ? .

Just as in the case of specific kinetic energy, consider (KD) ? not so easy. To count, you need to associate it with (KD) ? . For this, the coefficient of momentum is used.

Coefficient a? also known as the Businesq coefficient. Taking into account a?, the average inertial head over the free section

Finally, the Bernoulli equation for the flow, the receipt of which was the task of the issue under consideration, has the following form:

As for (5), it is obtained from (4) taking into account the fact that dQ = wdu; substituting dQ into (4) and reducing ?, we arrive at (6).

The difference between hin and hpr is primarily that it is not irreversible. If the movement of the fluid is accelerated, which means d? / t\u003e 0, then h in\u003e 0. If the movement is slow, that is, du / t< 0, то h ин < 0.

Equation (5) relates the flow parameters only at a given time. For another moment, it may no longer be reliable.

36. Laminar and turbulent regimes of fluid motion. Reynolds number

As it was easy to see in the above experiment, if we fix two speeds in the forward and reverse transitions of motion to the laminar -> turbulent modes, then

where? 1 is the speed at which the transition from laminar to turbulent regime begins;

2 - the same for the reverse transition.

Usually, ? 2< ? 1 . Это можно понять из определения основных видов движения.

Laminar (from lat. lamina - layer) is such a movement when there is no mixing of liquid particles in the liquid; such changes will be called pulsations in what follows.

The movement of a fluid is turbulent (from Latin turbulentus - erratic) if the pulsation of local velocities leads to mixing of the fluid.

Transition speeds? one , ? 2 are called:

1 - the upper critical speed and denoted as? in. cr, this is the speed at which laminar motion turns into turbulent;

2 - lower critical speed and denoted as? n. cr, at this speed, the reverse transition from turbulent to laminar occurs.

Meaning? in. cr depends on external conditions (thermodynamic parameters, mechanical conditions), and the values? n. kr do not depend on external conditions and are constant.

It has been established empirically that:

where V is the kinematic viscosity of the liquid;

d is the pipe diameter;

R is the coefficient of proportionality.

In honor of the researcher of hydrodynamics in general and this issue in particular, the coefficient corresponding to un. cr is called the critical Reynolds number Re cr.

If you change V and d, then Re cr does not change and remains constant.

If Re< Re кр, то режим движения жидкости ламинарный, поскольку? < ? кр; если Re >Re kr, then the mode of motion is turbulent due to the fact that?> ? cr.

37. Average speeds. Ripple components

In the theory of turbulent motion, a lot is connected with the name of the researcher of this motion, Reynolds. Considering chaotic turbulent motion, he presented instantaneous velocities as some sums. These sums look like:

where u x , u y , u z are the instantaneous values ​​of velocity projections;

p, ? – the same, but for pressure and friction stresses;

the line at the top of the values ​​means that the parameter is averaged over time; for u? x, u? y, u? z, p?, ?? the overline means that the pulsation component of the corresponding parameter (“additive”) is meant.

Averaging of parameters over time is carried out according to the following formulas:

is the time interval during which the averaging is carried out.

From formulas (1) it follows that not only velocity projections pulsate, but also normal and tangent ones? voltage. The values ​​of time-averaged “additives” should be equal to zero: for example, for the x-th component:

The time interval T is determined to be sufficient so that when repeated averaging, the value of the “additive” (pulsating component) does not change.

Turbulent motion is considered to be unsteady motion. Despite the possible constancy of the averaged parameters, the instantaneous parameters still fluctuate. It should be remembered: averaged (in time and at a specific point) and average (in a specific live section) speeds are not the same thing:

Q is the flow rate of a fluid that flows at a speed? through w.

38. Standard deviation

A standard has been adopted, which is called the standard deviation. For x

To obtain a formula for any “additive” parameter from formula (1), it is enough to replace u x in (1) with the desired parameter.

The standard deviation can be related to the following speeds: the average local speed of a given point; vertical average; average living section; maximum speed.

Normally, maximum and average vertical speeds are not used; two of the above characteristic velocities are used. In addition to them, they also use dynamic speed

where R is the hydraulic radius;

J - hydraulic slope.

The standard deviation, referred to the average speed, is, for example, for the x-th component:

But the best results are obtained if the standard deviation is related to u x , i.e. dynamic speed, for example

Let us determine the degree (intensity) of turbulence, as the quantity e is called

However, the best results are obtained if the dynamic velocity u x is taken as the velocity scale (that is, the characteristic velocity).

Another property of turbulence is the frequency of velocity pulsations. Average pulsation frequency at a point with radius r from the flow axis:

where N is half of the extremum outside the curve of instantaneous velocities;

T is the averaging period;

T/N = 1/w is the pulsation period.

39. Distribution of speeds with uniform steady motion. Laminar film

Nevertheless, despite the above and other features that are not mentioned because of their lack of demand, the main feature of turbulent motion is the mixing of fluid particles.

It is customary to speak of this mixing from the point of view of quantity as the mixing of moles of liquid.

As we have seen above, the turbulence intensity does not increase with increasing Re number. Despite this, nevertheless, for example, at the inner surface of a pipe (or at any other solid wall) there is a certain layer within which all velocities, including pulsating "additives", are equal to zero: this is a very interesting phenomenon.

This layer is called the viscous flow sublayer.

Of course, at the boundary of contact with the main mass of the flow, this viscous sublayer still has some speed. Therefore, all changes in the main stream are transferred to the tie layer, but their value is very small. This makes it possible to consider the motion of the layer as laminar.

Previously, assuming that these transfers to the garter layer are absent, the layer was called a laminar film. Now it is easy to see that, from the point of view of modern hydraulics, the laminarity of movement in this layer is relative (intensity? in the tie layer (laminar film) can reach 0.3. For laminar movement, this is a fairly large value)

Garter layer? in a very thin compared to the main thread. It is the presence of this layer that generates pressure losses (specific energy).

What about the laminar film thickness? c, then it is inversely proportional to the number Re. This is more clearly seen from the following comparison of thicknesses in flow zones during turbulent motion.

Viscous (laminar) layer - 0< ua / V < 7.

Transition zone - 7< ua/V < 70.

Turbulent core - ua/V< 70.

In these relationships, u is the dynamic flow velocity, a is the distance from the solid wall, and V is the kinematic viscosity.

Let's delve a little into the history of the theory of turbulence: this theory includes a set of hypotheses, on the basis of which the dependences between the main parameters u i ,? turbulent flow.

Different researchers have different approaches to this issue. Among them are the German scientist L. Prandtl, the Soviet scientist L. Landau and many others.

If before the beginning of the XX century. the laminar layer, according to scientists, was a kind of dead layer, in the transition to which (or from which) there is a break in speeds, that is, the speed changes abruptly, in modern hydraulics there is a completely different point of view.

The flow is a "living" phenomenon: all transient processes in it are continuous.

40. The distribution of velocities in the "live" section of the flow

Modern hydrodynamics has managed to solve these problems by applying the method of statistical analysis. The main tool of this method is that the researcher goes beyond traditional approaches and uses for analysis some time-averaged flow characteristics.

Average speed

It is clear that at any point of the live section, any instantaneous speed and can be decomposed into u x , u y , u z components.

The instantaneous speed is determined by the formula:

The resulting speed can be called the time-averaged speed, or the average local speed; this speed u x is fictitiously constant and allows one to judge the flow characteristics.

Calculating u y ,u x you can get the average velocity vector

shear stresses? = ? +? ,

Let us also determine the total value of shear stress?. Since this stress arises due to the presence of internal friction forces, the fluid is considered Newtonian.

If we assume that the contact area is unity, then the resistance force

where? is the dynamic viscosity of the liquid;

d?/dy - speed change. This quantity is often referred to as the velocity gradient, or shear rate.

Currently guided by the expression obtained in the aforementioned Prandtl equation:

where? is the density of the liquid;

l is the length of the path on which the movement is considered.

Without derivation, we present the final formula for the pulsating "additive" of shear stress:

42. Flow parameters on which the pressure loss depends. Dimension method

An unknown type of dependence is determined by the method of dimensions. For this, there is a?-theorem: if some physical regularity is expressed by an equation containing k dimensional quantities, and it contains n quantities with independent dimension, then this equation can be transformed into an equation containing (k-n) independent, but already dimensionless complexes.

For what we will determine: what the pressure loss depends on during steady motion in the field of gravity.

These options.

1. Geometric dimensions of the flow:

1) characteristic dimensions of the open section l 1 l 2;

2) the length of the section under consideration l;

3) angles that complete the live section;

4) roughness properties: ? is the height of the protrusion and l? is the nature of the longitudinal size of the roughness protrusion.

2. Physical properties:

one) ? – density;

2) ? is the dynamic viscosity of the liquid;

3) ? is the force of surface tension;

4) Е f is the modulus of elasticity.

3. The degree of intensity of turbulence, the characteristic of which is the root mean square value of the fluctuation components?u.

Now let's apply the?-theorem.

Based on the above parameters, we have 10 different values:

l, l2, ?, l? , ?p, ?, ?, E f,? u, t.

In addition to these, we have three more independent parameters: l 1 , ?, ?. Let's add the fall acceleration g.

In total, we have k = 14 dimensional quantities, three of which are independent.

It is required to obtain (kkn) dimensionless complexes, or, as they are called?-terms.

To do this, any parameter from 11 that would not be part of the independent parameters (in this case, l 1 , ?, ?), denoted as N i , now you can determine the dimensionless complex, which is a characteristic of this parameter N i , that is, i- ty?-member:

Here are the dimension angles of the base quantities:

the general form of dependence for all 14 parameters is:

43. Uniform movement and coefficient of resistance along the length. Chezy formula. Average speed and flow rate

With laminar motion (if it is uniform), neither the free cross section, nor the average velocity, nor the velocity diagram along the length change with time.

With uniform motion, the piezometric slope

where l 1 is the flow length;

h l - pressure loss over the length L;

r 0 d are the radius and diameter of the pipe, respectively.

In formula (2) dimensionless coefficient? is called the coefficient of hydraulic friction or the Darcy coefficient.

If in (2) d is replaced by the hydraulic radius, then

We introduce the notation

then taking into account the fact that

hydraulic slope

This formula is called the Chezy formula.

is called the Chezy coefficient.

If the Darcy coefficient? – dimensionless value

naya, then the Chezy coefficient c has the dimension

Let's determine the flow rate with the participation of the coefficient

Officer Chezi:

We transform the Chezy formula into the following form:

the value

called dynamic speed

44. Hydraulic likeness

The concept of similarity. Hydrodynamic modeling

To study the issues of building hydroelectric power plants, the method of hydraulic similarities is used, the essence of which is that exactly the same conditions are simulated in laboratory conditions as in nature. This phenomenon is called physical modeling.

For example, for two streams to be similar, you need them:

1) geometric similarity, when

where the indices n, m respectively mean "nature" and "model".

However, the attitude

which means that the relative roughness in the model is the same as in nature;

2) kinematic similarity, when the trajectories of the corresponding particles, the corresponding streamlines are similar. In addition, if the corresponding parts have passed similar distances l n, l m, then the ratio of the corresponding times of movement is as follows

where M i is the time scale

The same similarity exists for speed (speed scale)

and acceleration (acceleration scale)

3) dynamic similarity, when it is required that the corresponding forces are similar, for example, the scale of forces

Thus, if fluid flows are mechanically similar, then they are hydraulically similar; coefficients M l , M t , M ? , M p and others are called scale factors.

45. Criteria for hydrodynamic similarity

The conditions of hydrodynamic similarity require the equality of all forces, but this is practically impossible.

For this reason, the similarity is established by one of these forces, which in this case prevails. In addition, uniqueness conditions are required, which include flow boundary conditions, basic physical characteristics, and initial conditions.

Let's consider a special case.

The influence of gravity prevails, for example, when flowing through holes or weirs

If we go to the relationship P n and P m and express it in scale factors, then

After the necessary transformation,

If we now make the transition from scale factors to the ratios themselves, then taking into account the fact that l is the characteristic size of the free section, then

In (4) complex? 2 /gl is called the Froudy criterion, which is formulated as follows: flows dominated by gravity are geometrically similar if

This is the second condition of hydrodynamic similarity.

We have obtained three criteria for hydrodynamic similarity

1. Newton's criterion (general criteria).

2. Froude's criterion.

3. Darcy criterion.

We only note that in special cases the hydrodynamic similarity can also be established from

where? is the absolute roughness;

R is the hydraulic radius;

J– hydraulic slope

46. ​​Distribution of shear stresses with uniform motion

With uniform motion, the head loss over the length l he is determined by:

where? - wetted perimeter,

w is the open area,

l he is the length of the flow path,

G is the density of the liquid and the acceleration due to gravity,

0 - shear stress near the inner walls of the pipe.

From where, taking into account

Based on the results obtained for? 0 , shear stress distribution? at an arbitrarily chosen point of the allocated volume, for example, at the point r 0 - r \u003d t, this distance is equal to:

thus, we introduce a shear stress t on the surface of the cylinder, acting on a point in r 0 - r= t.

From comparisons (4) and (3) it follows:

Substituting r= r 0 – t into (5), we get

1) with uniform motion, the distribution of shear stress along the radius of the pipe obeys a linear law;

2) on the pipe wall, the shear stress is maximum (when r 0 \u003d r, i.e., t \u003d 0), on the pipe axis it is zero (when r 0 \u003d t).

R is the hydraulic radius of the pipe, we get that

47. Turbulent uniform flow regime

If we consider plane motion (i.e., potential motion, when the trajectories of all particles are parallel to the same plane and are functions of two coordinates to it and if the motion is unsteady), which is simultaneously uniform turbulent in the XYZ coordinate system, when the streamlines are parallel to the OX axis, then

Average speed for highly turbulent motion.

This expression: the logarithmic law of the distribution of velocities for turbulent motion.

In a forced motion, the flow consists mainly of five areas:

1) laminar: paraxial region, where the local velocity is maximum, in this region? lam = f(Re), where the Reynolds number Re< 2300;

2) in the second region, the flow begins to change from laminar to turbulent, hence the Re number also increases;

3) here the flow is completely turbulent; in this area, the pipes are called hydraulically smooth (roughness? less than the thickness of the viscous layer? in, that is?< ? в).

In case when?> ? c, the pipe is considered "hydraulically rough".

Typically, what if for? lam = f(Re –1), then in this case? where = f(Re - 0.25);

4) this area is on the path of the flow transition to the garter layer: in this area? lam = (Re, ?/r0). As can be seen, the Darcy coefficient is already beginning to depend on the absolute roughness?;

5) this region is called the quadratic region (the Darcy coefficient does not depend on the Reynolds number, but is determined almost entirely by the shear stress) and is near-wall.

This region is called self-similar, i.e., independent of Re.

In the general case, as is well known, the Chezy coefficient

Pavlovsky's formula:

where n is the roughness coefficient;

R is the hydraulic radius.

At 0.1

moreover, for R< 1 м

48. Uneven motion: Weisbach's formula and its application

With uniform motion, the pressure loss is usually expressed by the formula

where the head loss h CR depends on the flow rate; it is constant because the motion is uniform.

Consequently, formula (1) has corresponding forms.

Indeed, if in the first case

then in the second case

As can be seen, formulas (2) and (3) differ only in the drag coefficient x.

Formula (3) is called the Weisbach formula. In both formulas, as in (1), the drag coefficient is a dimensionless quantity, and for practical purposes it is usually determined from tables.

To conduct an experiment to determine xm, the sequence of actions is as follows:

1) the uniformity of the flow in the structural element under study must be ensured. It is necessary to ensure sufficient distance from the entrance of the piezometers.

2) for the steady motion of a viscous incompressible fluid between two sections (in our case, this is an inlet with x 1 ? 1 and an outlet with x 2 ? 2), we apply the Bernoulli equation:

In the sections under consideration, the flow should be smoothly changing. Anything could happen between sections.

Since the total head loss

then we find the pressure loss in the same section;

3) according to formula (5) we find that h m \u003d h pr - h l, after that, according to formula (2), we find the desired coefficient

resistance

49. Local resistance

What happens after the flow has entered the pipeline with some pressure and speed.

It depends on the type of movement: if the flow is laminar, that is, its movement is described by a linear law, then its curve is a parabola. The pressure loss during such a movement reaches (0.2 x 0.4) x (? 2 / 2g).

During turbulent motion, when it is described by a logarithmic function, the head loss is (0.1 x 1.5) x (? 2 / 2g).

After such pressure losses, the flow movement stabilizes, that is, the laminar or turbulent flow is restored, which was the input.

The section where the above pressure losses occur is restored in nature, the previous movement is called the initial section.

And what is the length of the initial section l beg.

Turbulent flow recovers 5 times faster than laminar flow with the same hydraulic associated data.

Let us consider a special case when the flow does not narrow, as discussed above, but suddenly expands. Why do head losses occur with this flow geometry?

For the general case:

To determine the coefficients of local resistance, we transform (1) into the following form: dividing and multiplying by? 12

Define? 2/? 1 from the continuity equation

1 w 1 = ?2w2 how? 2/? 1 = w 1 / w 2 and substitute into (2):

It remains to conclude that

50. Calculation of pipelines

Problems of calculation of pipelines.

The following tasks are required:

1) it is required to determine the flow rate Q, while the pressure H is given; pipe length l; pipe roughness?; liquid density r; fluid viscosity V (kinematic);

2) it is required to determine the pressure H. The flow rate Q is given; pipeline parameters: length l; diameter d; roughness?; liquid parameters: ? density; viscosity V;

3) it is required to determine the required pipeline diameter d. The flow rate Q is given; head H; pipe length l; its roughness?; liquid density?; its viscosity V.

The methodology for solving problems is the same: the joint application of the Bernoulli equations and continuity.

The pressure is determined by the expression:

fluid consumption,

since J = H / l

An important characteristic of the pipeline is a value that combines some parameters of the pipeline, based on the diameter of the pipe (we consider simple pipes, where the diameter is constant along the entire length l). This parameter k is called the flow characteristic:

If we start observation from the very beginning of the pipeline, we will see: some part of the liquid, without changing, reaches the end of the pipeline in transit.

Let this amount be Q t (transit expense).

The liquid is partially distributed to consumers along the way: let's denote this part as Q p (travel expense).

Given these designations, at the beginning of the pipeline

Q \u003d Q t + Q p,

respectively, at the end of the flow rate

Q - Q p \u003d Q t.

As for the pressure in the pipeline, then:

51. Water hammer

The most common, that is, the most common type of unsteady motion is water hammer. This is a typical phenomenon during fast or gradual closing of valves (a sharp change in speeds in a certain flow section leads to water hammer). As a result, pressures arise that propagate throughout the pipeline in a wave.

This wave can be destructive if special measures are not taken: pipes can burst, pumping stations fail, saturated steams can arise with all destructive consequences, etc.

Water hammer can cause fluid breaks in the pipeline - this is no less serious accident than a pipe break.

The most common causes of water hammer are as follows: sudden closing (opening) of gates, sudden stop of pumps when filling pipelines with water, release of air through hydrants in the irrigation network, start-up of a pump with an open gate.

If this has already happened, then how does the water hammer proceed, what consequences does it cause?

It all depends on what caused the water hammer. Let's consider the main of these reasons. The mechanisms of occurrence and course for other reasons are similar.

Instant shutter closing

The water hammer that occurs in this case is an extremely interesting phenomenon.

Let we have an open reservoir, from which a hydraulic straight pipe is discharged; at some distance from the tank, the pipe has a shutter. What happens when it closes instantly?

First, let:

1) the reservoir is so large that the processes occurring in the pipeline are not reflected in the liquid (in the reservoir);

2) pressure loss before closing the shutter is negligible, therefore, the piezometric and horizontal lines coincide

3) the fluid pressure in the pipeline occurs with only one coordinate, the other two projections of local velocities are equal to zero; movement is determined only by the longitudinal coordinate.

Secondly, now let's suddenly close the shutter - at time t 0 ; two cases can happen:

1) if the walls of the pipeline are absolutely inelastic, i.e. E = ?, and the liquid is incompressible (E f = ?), then the movement of the fluid also suddenly stops, which leads to a sharp increase in pressure at the gate, the consequences can be devastating.

Pressure increment during hydraulic shock according to the Zhukovsky formula:

P = ?C? 0 + ?? 0 2 .

52. Water hammer wave velocity

In hydraulic calculations, of considerable interest is the velocity of propagation of the shock wave of a hydraulic shock, as well as the hydraulic shock itself. How to define it? To do this, consider a circular cross section in an elastic pipeline. If we consider a section with a length? l, then above this section during the time? t the liquid still moves with a speed? 0 , by the way, as before closing the shutter.

Therefore, in the corresponding length l, the volume? V ? liquid will enter Q = ? 0? 0 , i.e.

V? = Q?t = ? 0? 0?t, (1)

where is the circular cross-sectional area - the volume formed as a result of pressure increase and, as a consequence, due to stretching of the pipeline wall? V 1 . The volume that arose due to the increase in pressure on?p will be denoted as?V 2 . This means that the volume that arose after the hydraulic shock is

V = ?V 1 + ?V 2 , (2)

V? included in?V.

Let's decide now: what will be equal to? V 1 and? V 2.

As a result of pipe stretching, the pipe radius will increase by ?r, that is, the radius will become equal to r = r 0 + ?r. Because of this, the circular section of the cross section will increase by ?? = ?– ? 0 . All this will lead to an increase in volume by

V1 = (?– ? 0)?l = ???l. (3)

It should be borne in mind that index zero means that the parameter belongs to the initial state.

As for the liquid, its volume will decrease by? V 2 due to the increase in pressure by? p.

The desired formula for the propagation velocity of a hydraulic shock wave

where? is the density of the liquid;

D/l is a parameter characterizing the pipe wall thickness.

It is obvious that the greater D/l, the lower the propagation velocity of wave C. If the pipe is absolutely rigid, that is, E = ?, then, as follows from (4)

53. Differential equations of unsteady motion

In order to make an equation of any type of motion, you need to project all the acting forces on the system and equate their sum to zero. So let's do it.

Let us have a pressure pipeline of circular cross section, in which there is an unsteady movement of fluid.

The flow axis coincides with the l axis. If we single out the element dl on this axis, then, according to the above rule, we can compose the equation of motion

In the above equation, the projections of the four forces acting on the flow, more precisely, on?l, are equal to zero:

1) ?M - inertial forces acting on the element dl;

2) ?p – forces of hydrodynamic pressure;

3) ?T are tangential forces;

4) ?G - gravity forces: here, speaking of forces, we meant the projections of forces acting on the element?l.

Let's move on to formula (1), directly to the projections of the acting forces on the element? t, on the axis of motion.

1. Projections of surface forces:

1) for hydrodynamic forces?p the projection will be

2) for tangential forces?T

The projection of tangential forces has the form:

2. Projection of gravity? ?G per element? ?

3. Projection of inertial forces? ?M is

54. Outflow of liquid at constant pressure through a small hole

We will consider the outflow that occurs through a small unflooded hole. For a hole to be considered small, the following conditions must be met:

1) pressure at the center of gravity H >> d, where d is the hole height;

2) the pressure at any point of the hole is practically equal to the pressure at the center of gravity H.

As for flooding, it is considered to be outflow under the liquid level, provided that the following do not change with time: the position of the free surfaces before and after the holes, the pressure on the free surfaces before and after the holes, atmospheric pressure on both sides of the holes.

Thus, we have a reservoir with a liquid whose density is ?, from which an outflow occurs through a small hole under the level. The pressure H in the center of gravity of the hole is constant, which means that the outflow velocities are constant. Therefore, the movement is steady. The condition for the equality of velocities on opposite vertical boundaries of the holes is the condition d

It is clear that our task is to determine the velocity of the outflow and the flow rate of the liquid in it.

The jet section spaced from the inner wall of the tank at a distance of 0.5d is called the compressed jet section, which is characterized by the compression ratio

Formulas for determining the speed and flow rate:

where? 0 is called the speed factor.

Now let's complete the second task, determine the flow rate Q. By definition

Let's call it E? 0 = ? 0 where? 0 is the flow rate, then

There are the following types of compression:

1. Full compression is a compression that occurs around the entire perimeter of the hole, otherwise the compression is considered incomplete compression.

2. Perfect compression is one of two varieties of complete compression. This is such a compression when the curvature of the trajectory, and hence the degree of compression of the jet, is the greatest.

Summing up, we note that incomplete and imperfect forms of compression lead to an increase in the compression ratio. A characteristic feature of perfect compression is that, depending on the forces under the influence, the outflow occurs.

55. Outflow through a large hole

A hole is considered small when its vertical dimensions d< 0,1Н. Большим отверстием будем считать такое отверстие, для которого тот же d>0.1N.

Considering the outflow through a small hole, we practically neglected the difference in velocities at different points of the jet cross section. In this case, we cannot do the same.

The task is the same: to determine the flow rate and velocities in the compressed section.

Therefore, the flow rate is determined in the following way: an infinitely small horizontal height dz is selected. Thus, a horizontal strip with a variable length bz is obtained. Then, integrating over the length, we can find the elementary flow

where Z is a variable pressure along the height of the hole, the top of the selected strip is submerged to such a depth;

? - coefficient of flow through the hole;

b z - variable length (or width) of the strip.

Consumption Q (1) can determine if? = const and the formula b z = f(z) is known. In the general case, the flow rate is determined by the formula

If the shape of the hole is rectangular, then bz= b = const, integrating (2), we obtain:

where H 1, H 2 - heads at the levels, respectively, at the upper and lower edges of the hole;

Nts - pressure above the center of the hole;

d is the height of the rectangle.

Formula (3) has a more simplified form:

In the case of outflow through a round hole, the limits of integration in (2) are H 1 = H c - r; H 2 \u003d H c + r; Z \u003d H c - rcos?; dz = ?sin?d?; bz = 2r?sin?.

Avoiding mathematical excess, we give the final formula:

As can be seen from the comparison of the formulas, there is no particular difference in the formulas for the flow rate, only for large and small holes, the flow coefficients are different

56. System flow rate

It is required to clarify the issue of flow if the outflow occurs through pipes connected to one system, but having different geometric data. Here we need to consider each case separately. Let's take a look at some of them.

1. The outflow occurs between two tanks at a constant pressure through a system of pipes that have different diameters and lengths. In this case, at the output of the system E = 1, therefore, numerically? = ?, where E, ?, ? are the coefficients of compression, flow rate, and speed, respectively.

2. The outflow occurs through a pipe system with different? (cross-sectional area): in this case, the total resistance coefficient of the system is determined, which consists of the same coefficients, but for each section separately.

The outflow occurs into the atmosphere through an unflooded hole. In this case

where H = z = const - head; ?, ?– flow coefficient and cross-sectional area.

since in (2) the Coriolis coefficient (or kinetic energy) x is related to the outlet section, where, as a rule, x? one.

The same outflow occurs through a flooded orifice

in this case, the flow rate is determined by the formula (3), where? = ? syst, ? is the area of ​​the outlet section. In the absence or insignificance of velocity in the receiver or pipe, the flow coefficient is replaced by

You just need to keep in mind that with a flooded hole? vy = 1, and this? vy enters? syst.