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 Movement of the center of pressure of the wing of the Yak-52 aircraft with a change in 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 It 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)
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
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Center of pressure atmospheric pressure forces pOS will be in the center of gravity of the site, since atmospheric pressure is transmitted equally to all points of the liquid. The center of pressure of the fluid itself on the site can be determined from the theorem on the moment of the resultant force. resultant moment
forces about the axis OH will be equal to the sum of the moments of the component forces about the same axis.
Where 
where: - position of the center of excess pressure on the vertical axis, - moment of inertia of the site S about the axis OH.
The center of pressure (the point of application of the resultant force of excess pressure) is always located below the center of gravity of the platform. In cases where the external acting force on the free surface of the liquid is the force of atmospheric pressure, then two forces of equal magnitude and opposite in direction due to atmospheric pressure (on the inner and outer sides of the wall) will simultaneously act on the vessel wall. For this reason, the real operating unbalanced force remains the overpressure force.
| Previous materials: |
The point of application of the total pressure force is called the center of pressure. Determine the coordinates of the center of pressure and (Fig. 3.20). As is known from theoretical mechanics, at equilibrium, the moment of the resultant F relative to some axis is equal to the sum of the moments of the component forces dF about the same axis.
Let's make the equation of the moments of forces F and dF about the 0y axis.
Forces F and dF define by formulas
Reducing the expression by g and sin a, we get
where is the moment of inertia of the area of the figure relative to the axis 0 y.
Replacing according to the formula known from theoretical mechanics, where J c - moment of inertia of the area of the figure about the axis parallel to 0 y and passing through the center of gravity, we get
From this formula it follows that the center of pressure is always located below the center of gravity of the figure at a distance. This distance is called the eccentricity and is denoted by the letter e.
Coordinate y d is found from similar considerations
where is the centrifugal moment of inertia of the same area about the axes y and l. If the figure is symmetrical about an axis parallel to axis 0 l(Fig. 3.20), then, obviously, , where y c - coordinate of the center of gravity of the figure.
§ 3.16. Simple hydraulic machines.
Hydraulic Press
The hydraulic press is used to obtain high forces, which are necessary, for example, for pressing or stamping metal products.
A schematic diagram of a hydraulic press is shown in fig. 3.21. It consists of 2 cylinders - large and small, interconnected by a tube. The small cylinder has a piston with a diameter d, which is actuated by a lever with shoulders a and b. When the small piston moves down, it exerts pressure on the liquid p, which, according to Pascal's law, is transferred to a piston with a diameter D located in a large cylinder.
When moving up, the piston of the large cylinder presses the part with a force F 2 Define strength F 2 if the strength is known F 1 and press sizes d, D, as well as lever arms a and b. Let's first define the force F acting on a small piston with a diameter d. Consider the balance of the press lever. Let us compose the equation of moments relative to the center of rotation of the lever 0
where is the reaction of the piston to the lever.
where is the cross-sectional area of the small piston.
According to Pascal's law, pressure in a fluid is transmitted in all directions without change. Therefore, the pressure of the liquid under the large piston will also be equal to p well. Hence, the force acting on the large piston from the side of the liquid will be
where is the cross-sectional area of the large piston.
Substituting into the last formula p and taking into account that , we get
To take into account friction in the cuffs of the press, sealing the gaps, the efficiency of the press h is introduced<1. В итоге расчетная формула примет вид
hydraulic accumulator
The hydraulic accumulator serves for accumulation - accumulation of energy. It is used in cases where it is necessary to perform short-term large work, for example, when opening and closing lock gates, when operating a hydraulic press, hydraulic lift, etc.
A schematic diagram of the hydraulic accumulator is shown in Fig. 3.22. It consists of a cylinder A in which the piston is placed B connected to the loaded frame C to which loads are suspended D.
With the help of a pump, liquid is pumped into the cylinder until it is completely filled, while the loads rise and thereby energy is accumulated. To raise the piston H, it is necessary to pump a volume of liquid into the cylinder
where S- sectional area of the piston.
If the size of the loads is G, then the pressure of the piston on the liquid is determined by the ratio of the weight force G to the cross-sectional area of the piston, i.e.
Expressing from here G, we get
Work L, spent on lifting the load, will be equal to the product of the force G for the path length H
Law of Archimedes
Archimedes' law is formulated as the following statement - a body immersed in a liquid is subjected to a buoyant force directed upwards and equal to the weight of the liquid displaced by it. This force is called sustaining. It is the resultant of the pressure forces with which a fluid at rest acts on a body at rest in it.
To prove the law, we single out in the body an elementary vertical prism with bases d w n1 and d w n2 (Fig. 3.23). The vertical projection of the elemental force acting on the upper base of the prism will be
where p 1 - pressure on the base of the prism d w n1 ; n 1 - normal to the surface d w n1 .
where d w z - area of the prism in the section perpendicular to the axis z, then
Hence, taking into account that according to the hydrostatic pressure formula, we obtain
Similarly, the vertical projection of the elemental force acting on the lower base of the prism is found by the formula
The total vertical elemental force acting on the prism will be
Integrating this expression for , we obtain
Where is the volume of the body immersed in the liquid, where h T is the height of the submerged part of the body on the given vertical.
Hence for the buoyant force F z we get the formula
Selecting elementary horizontal prisms in the body and making similar calculations, we obtain , .
where G is the weight of the fluid displaced by the body. Thus, the buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by the body, which was to be proved.
It follows from the law of Archimedes that two forces ultimately act on a body immersed in a liquid (Fig. 3.24).
1. Gravity - body weight.
2. Supporting (buoyant) force, where g 1 - specific weight of the body; g 2 - specific gravity of the liquid.
In this case, the following main cases may occur:
1. The specific gravity of the body and liquid are the same. In this case , the resultant , and the body will be in a state of indifferent equilibrium, i.e. being submerged to any depth, it will neither rise nor sink.
2. For g 1 > g 2 , . The resultant is directed downward, and the body will sink.
3. For g 1< g 2 . Равнодействующая направлена вверх, и тело будет всплывать. Всплытие тела будет продолжаться до тех пор, пока выталкивающая сила не уменьшится настолько, что сделается равной силе веса, т.е. пока не будет . После этого тело будет плавать на поверхности.
§ 3.19. Conditions of buoyancy and stability of bodies,
partially immersed in liquid
The presence of a condition is necessary for the equilibrium of a body immersed in a liquid, but it is still not enough. For the balance of the body, in addition to equality, it is also necessary that the lines of these forces be directed along one straight line, i.e. matched (Fig. 3.25 a).
If the body is homogeneous, then the points of application of the indicated forces always coincide and are directed along one straight line. If the body is inhomogeneous, then the points of application of these forces will not coincide and the forces G and F z form a pair of forces (see Fig. 3.25 b, c). Under the action of this pair of forces, the body will rotate in the fluid until the points of application of forces G and F z will not be on the same vertical, i.e. the moment of the pair of forces will be equal to zero (Fig. 3.26).
Of greatest practical interest is the study of equilibrium conditions for bodies partially immersed in a liquid, i.e. when swimming tel.
The ability of a floating body, taken out of equilibrium, to return to this state again is called stability.
Consider the conditions under which a body floating on the surface of a liquid is stable.
On fig. 3.27 (a, b) C- center of gravity (point of application of the resultant forces of weight g);
D- point of application of the resultant buoyant forces F z M- metacenter (point of intersection of the resultant buoyant forces with the navigation axis 00).
Let's give some definitions.
The weight of a fluid displaced by a body immersed in it is called displacement.
The point of application of the resultant buoyant forces is called the center of displacement (point D).
Distance MC between the metacenter and the center of displacement is called the metacentric radius.
Thus, a floating body has three characteristic points:
1. Center of gravity C, which does not change its position during a roll.
2. Displacement center D, which moves when the body rolls, since the outlines of the volume displaced in the liquid change in this case.
3. Metacenter M, which also changes its position during roll.
When swimming the body, the following 3 main cases may present themselves, depending on the relative location of the center of gravity C and metacenter M.
1. The case of stable equilibrium. In this case, the metacenter lies above the center of gravity (Fig. 3.27, a) and when the pair of forces rolls G and F z tends to return the body to its original state (the body rotates counterclockwise).
2. The case of indifferent equilibrium. In this case, the metacenter and the center of gravity coincide, and the body, taken out of equilibrium, remains motionless.
3. The case of unstable equilibrium. Here, the metacenter lies below the center of gravity (Fig. 3.27, b) and the pair of forces formed during the roll causes the body to rotate clockwise, which can lead to capsizing of the floating vehicle.
Task 1. Direct-acting steam pump delivers liquid F to the height H(Fig. 3.28). Find the working steam pressure with the following initial data: ; ; . Liquid - water (). Find also the force acting on the small and large pistons.
Decision. Find the pressure on the small piston
The force acting on the small piston will be
The same force acts on the large piston, i.e.
Task 2. Determine the pressing force developed by a hydraulic press, which has a large piston diameter, and a small piston, with the following initial data (Fig. 3.29):
Decision. Find the force acting on the small piston. To do this, we compose the equilibrium condition for the press lever
The fluid pressure under the small piston will be
Fluid pressure under large piston
According to Pascal's law, pressure in a fluid is transmitted in all directions without change. From here or
Hydrodynamics
The branch of hydraulics that studies the laws of fluid motion is called hydrodynamics. When studying the motion of liquids, two main problems are considered.
1. The hydrodynamic characteristics of the flow (velocity and pressure) are given; it is required to determine the forces acting on the fluid.
2. The forces acting on the liquid are given; it is required to determine the hydrodynamic characteristics of the flow.
As applied to an ideal fluid, hydrodynamic pressure has the same properties and the same meaning as hydrostatic pressure. When analyzing the motion of a viscous fluid, it turns out that
where are the real normal stresses at the point under consideration, related to three mutually orthogonal areas arbitrarily marked at this point. The hydrodynamic pressure at a point is considered the value
It is assumed that the value p does not depend on the orientation of mutually orthogonal areas.
In the future, the problem of determining the velocity and pressure for known forces acting on the fluid will be considered. It should be noted that the velocity and pressure for different points of the fluid will have different values and, in addition, for a given point in space, they may change in time.
To determine the velocity components along the coordinate axes , , and pressure p in hydraulics, the following equations are considered.
1. The equation of incompressibility and continuity of a moving fluid (the equation for the balance of fluid flow).
2. Differential equations of motion (Euler equations).
3. Balance equation for the specific energy of the flow (Bernoulli equation).
All these equations, which form the theoretical basis of hydrodynamics, will be given below, with preliminary explanations of some of the initial provisions from the field of fluid kinematics.
§ 4.1. BASIC KINEMATIC CONCEPTS AND DEFINITIONS.
TWO METHODS FOR STUDYING LIQUID MOVEMENT
When studying the motion of a fluid, two research methods can be used. The first method, developed by Lagrange and called the substantive one, is that the motion of the entire fluid is studied by studying the motion of its separate individual particles.
The second method, developed by Euler and called local, is that the motion of the entire fluid is studied by studying the motion at individual fixed points through which the fluid flows.
Both of these methods are used in hydrodynamics. However, the Euler method is more common due to its simplicity. According to the Lagrange method at the initial moment of time t 0, certain particles are noted in the liquid and then the movement of each marked particle and its kinematic characteristics are monitored in time. The position of each fluid particle at a time t 0 is determined by three coordinates in a fixed coordinate system, i.e. three equations
where X, at, z- particle coordinates; t- time.
To compose equations that characterize the motion of various flow particles, it is necessary to take into account the position of the particles at the initial moment of time, i.e. the initial coordinates of the particles.
For example, dot M(Fig. 4.1) at the time t= 0 has coordinates a, b, with. Relations (4.1), taking into account a, b, with take the form
In relations (4.2), the initial coordinates a, b, with can be considered as independent variables (parameters). Therefore, the current coordinates x, y, z some moving particle are functions of variables a, b, c, t, which are called Lagrange variables.
For known relations (4.2), the fluid motion is completely determined. Indeed, the velocity projections on the coordinate axes are determined by the relations (as the first derivatives of the coordinates with respect to time)
The acceleration projections are found as the second derivatives of the coordinates (the first derivatives of the velocity) with respect to time (relations 4.5).
The trajectory of any particle is determined directly from equations (4.1) by finding the coordinates x, y, z selected liquid particle for a number of time points.
According to the Euler method, the study of fluid motion consists in: a) the study of changes in time of vector and scalar quantities at some fixed point in space; b) in the study of changes in these quantities during the transition from one point in space to another.
Thus, in the Euler method, the subject of study is the fields of various vector or scalar quantities. A field of some magnitude, as is known, is a part of space, at each point of which there is a certain value of this magnitude.
Mathematically, a field, such as a velocity field, is described by the following equations
those. speed
is a function of coordinates and time.
Variables x, y, z, t are called Euler variables.
Thus, in the Euler method, the fluid motion is characterized by the construction of the velocity field, i.e. patterns of motion at different points in space at any given moment in time. In this case, the velocities at all points are determined in the form of functions (4.4).
The Euler method and the Lagrange method are mathematically related. For example, in the Euler method, partly using the Lagrange method, one can follow the motion of a particle not during time t(as it follows according to Lagrange), and in the course of an elementary interval of time dt, during which a given fluid particle passes through the considered point in space. In this case, relations (4.3) can be used to determine the velocity projections on the coordinate axes.
From (4.2) it follows that the coordinates x, y, z are functions of time. Then there will be complex functions of time. By the rule of differentiation of complex functions, we have
where are the projections of the acceleration of the moving particle onto the corresponding coordinate axes.
Since for a moving particle
Partial derivatives
are called projections of local (local) acceleration.
Kind sums
are called projections of convective acceleration.
total derivatives
are also called substantive or individual derivatives.
Local acceleration determines the change in time of speed at a given point in space. Convective acceleration determines the change in velocity along the coordinates, i.e. when moving from one point in space to another.
§ 4.2. Particle Trajectories and Streamlines
The trajectory of a moving fluid particle is the path of the same particle traced in time. The study of particle trajectories underlies the Lagrange method. When studying the movement of a fluid using the Euler method, a general idea of the movement of a fluid can be drawn up by constructing streamlines (Fig. 4.2, 4.3). A streamline is such a line, at each point of which at a given time t the velocity vectors are tangent to this line.
| Fig.4.2. | Fig.4.3. |
In steady motion (see §4.3), when the liquid level in the tank does not change (see Fig. 4.2), the particle trajectories and streamlines coincide. In the case of unsteady motion (see Fig. 4.3), the particle trajectories and streamlines do not coincide.
The difference between the particle trajectory and the streamline should be emphasized. The trajectory refers to only one particular particle, studied during a certain period of time. The streamline refers to a certain collection of different particles considered at one instant
(at the current time).
STEADY MOVEMENT
The concept of steady motion is introduced only when studying the motion of a fluid in Euler variables.
Steady-state is the movement of a fluid, in which all the elements characterizing the movement of a fluid at any point in space do not change in time (see Fig. 4.2). For example, for the velocity components we will have
Since the magnitude and direction of the speed of movement at any point in space do not change during steady motion, then the streamlines will not change in time. It follows from this (as already noted in § 4.2) that, under steady motion, the particle trajectories and streamlines coincide.
A motion in which all the elements characterizing the motion of a fluid change in time at any point in space is called unsteady (, Fig. 4.3).
§ 4.4. JETTING MODEL OF LIQUID MOTION.
CURRENT PIPE. FLUID CONSUMPTION
Consider the current line 1-2 (Fig. 4.4). Let's draw a plane at point 1 perpendicular to the velocity vector u 1 . Take in this plane an elementary closed contour l covering the site d w. We draw streamlines through all points of this contour. A set of streamlines drawn through any circuit in a liquid form a surface called a stream tube.
| Rice. 4.4 | Rice. 4.5 |
The set of streamlines drawn through all points of the elementary area d w, constitutes an elementary trickle. In hydraulics, the so-called jet model of fluid movement is used. The fluid flow is considered as consisting of individual elementary jets.
Consider the fluid flow shown in Figure 4.5. The volumetric flow rate of a liquid through a surface is the volume of liquid flowing per unit time through a given surface.
Obviously, the elementary cost will be
where n is the direction of the normal to the surface.
Full consumption
If we draw a surface A through any point of the stream orthogonal to the streamlines, then . The surface, which is the locus of fluid particles whose velocities are perpendicular to the corresponding elements of this surface, is called the free flow section and is denoted by w. Then for an elementary stream we have
and for flow
This expression is called the volumetric flow rate of liquid through the living section of the flow.
Examples.
The average speed in the flow section is the same speed for all points of the section, at which the same flow occurs, which actually takes place at actual velocities that are different for different points of the section. For example, in a round pipe, the distribution of velocities in a laminar fluid flow is shown in Fig. 4.9. Here is the actual velocity profile in laminar flow.
The average speed is half the maximum speed (see § 6.5)
§ 4.6. CONTINUITY EQUATION IN EULER VARIABLES
IN CARTSIAN COORDINATE SYSTEM
The equation of continuity (continuity) expresses the law of conservation of mass and the continuity of the flow. To derive the equation, we select an elementary parallelepiped with ribs in the liquid mass dx, dz, dz(Fig. 4.10).
Let the point m with coordinates x, y, z is in the center of this parallelepiped. Liquid density at a point m will .
Let us calculate the mass of fluid flowing into and out of the parallelepiped through opposite faces during the time dt. The mass of fluid flowing through the left side in time dt in axis direction x, is equal to
where r 1 and (u x) 1 - density and velocity projection on the axis x at point 1.
The function is a continuous function of the coordinate x. Expanding this function in a neighborhood of the point m into the Taylor series up to infinitesimals of the first order, for points 1 and 2 on the faces of the parallelepiped we obtain the following values
those. the average flow velocities are inversely proportional to the areas of the living sections of the flow (Fig. 4.11). Volume flow Q incompressible fluid remains constant along the channel.
§ 4.7. DIFFERENTIAL EQUATIONS OF MOTION OF AN IDEAL
(NON-VISCOUS) LIQUIDS (EULER EQUATIONS)
An inviscid or ideal fluid is a fluid whose particles have absolute mobility. Such a fluid is unable to resist shear forces and, therefore, shear stresses will be absent in it. Of the surface forces, only normal forces will act in it.
in a moving fluid is called hydrodynamic pressure. Hydrodynamic pressure has the following properties.
1. It always acts along the internal normal (compressive force).
2. The value of hydrodynamic pressure does not depend on the orientation of the site (which is proved similarly to the second property of hydrostatic pressure).
Based on these properties, we can assume that . Thus, the properties of hydrodynamic pressure in a nonviscous fluid are identical to those of hydrostatic pressure. However, the magnitude of the hydrodynamic pressure is determined by equations different from the equations of hydrostatics.
To derive the equations of fluid motion, we select an elementary parallelepiped in the fluid mass with ribs dx, dy, dz(Fig. 4.12). Let the point m with coordinates x,y,z is in the center of this parallelepiped. Point pressure m will . Let the components of the mass forces per unit mass be X,Y,Z.
Let us write the condition for the equilibrium of forces acting on an elementary parallelepiped in the projection onto the axis x
, (4.9)
where F1 and F2– forces of hydrostatic pressure; Fm is the resultant of the mass forces of gravity; F and - resultant of inertia forces.
Of great practical interest is the location of the point of application of the force of the total hydrostatic pressure. This point is called center of pressure.
In accordance with the basic equation of hydrostatics, the pressure force F 0 =p 0 · ω , acting on the surface of the liquid, is evenly distributed over the entire site, as a result of which the point of application of the total surface pressure force coincides with the center of gravity of the site. The place of application of the total force of excess hydrostatic pressure, which is unevenly distributed over the area, will not coincide with the center of gravity of the site.
At R 0 =p atm the position of the center of pressure depends only on the magnitude of the excess pressure force, so the position (ordinate) of the center of pressure will be determined taking into account only this force. To do this, we use the moment theorem: the moment of the resultant force about an arbitrary axis is equal to the sum of the moments of its constituent forces about the same axis. For the axis of moments, we take the line of the edge of the liquid OH(Figure 1.14).
Let us compose the equilibrium equation for the moment of the resultant force F and moments of the constituent forces dF, i.e. M p =M ss:
M p \u003d F y cd; dM cc=dF y. (1.45)
In formulas (1.45)
where is the moment of inertia of the platform about the axis X.
Then the moment of the constituent forces
M ss =γ· sin α I x.
Equating the values of the moments of forces M p and M ss, we get
,
Moment of inertia I x can be determined by the formula
Ix=I 0 +ω· , (1.49)
where I 0 is the moment of inertia of the wetted figure, calculated relative to the axis passing through its center of gravity.
Substituting value I x into formula (1.48) we obtain
. (1.50)
Consequently, the center of excess hydrostatic pressure is located below the center of gravity of the area under consideration by the value .
Let us explain the use of the dependencies obtained above with the following example. Let on a flat rectangular vertical wall with a height h and width b a fluid acts, the depth of which in front of the wall is equal to h.
