Since the time of Galileo's experiments on the Leaning Tower of Pisa, it has been known that all bodies fall in the field of gravity with the same acceleration. g.
However, everyday practice indicates otherwise: a light feather falls more slowly than a heavy metal ball. The reason for this is the air resistance.
Equations of motion. If we confine ourselves to the case forward movement non-rotating bodies in a stationary medium with resistance, then the resistance force will be directed against the speed. In vector form, it can be written as
where is the absolute value of this force, a is the modulus of the body's velocity. Taking into account the resistance of the medium changes the form of the equations of motion of a body thrown at an angle to the horizon:
The above equations also take into account the Archimedes buoyancy force acting on the body: free fall acceleration g replaced with a smaller one
where is the density of the medium (for air = 1.29 kg / m 3), and is the average density of the body.
Indeed, the weight of a body in a medium decreases by the value of the buoyant force of Archimedes
Expressing the volume of a body in terms of its average density
we arrive at the expression
In the presence of air resistance, the speed of a falling body cannot increase indefinitely. In the limit, it tends to some steady value, which depends on the characteristics of the body. If the body has reached a steady rate of fall, then it follows from the equations of motion that the resistance force is equal to the weight of the body (taking into account the Archimedean force):
The drag force, as we will soon see, is a function of the rate of fall. Therefore, the resulting expression for the resistance force is an equation for determining the steady rate of fall. It is clear that in the presence of a medium, the energy of the body is partially spent on overcoming its resistance.
Reynolds number. Of course, the equations of motion of a body in a fluid cannot even begin to be solved until we know nothing about the modulus of the drag force. The magnitude of this force essentially depends on the nature of the flow around the body with a counter flow of gas (or liquid). At low speeds, this flow is laminar(that is, layered). It can be imagined as a relative motion of layers of the medium that do not mix with each other.
The laminar flow of a liquid is demonstrated in the experiment shown in Fig. 13.
As already noted in Chapter 9.3, with the relative movement of layers of liquid or gas between these layers, forces of resistance to movement arise, which are called forces of internal friction. These forces are due to a special property of fluid bodies - viscosity, which is characterized numerically viscosity index. Here are the characteristic values for various substances: for air ( = 1.8 10 -5 Pa s), water ( = 10 -3 Pa s), glycerin ( = 0.85 Pa s). The equivalent designation of the units in which the viscosity coefficient is measured: Pa s = kg m -1 s -1.
Between the moving body and the medium there are always cohesive forces, so that immediately near the surface of the body the layer of gas (liquid) is completely delayed, as if "sticking" to it. It rubs against the next layer, which is slightly behind the body. That, in turn, experiences a friction force from an even more distant layer, and so on. Layers far from the body can be considered as resting. Theoretical calculation of internal friction for the movement of a ball with a diameter D leads to Stokes formula:
Substituting the Stokes formula into the expression for the resistance force in steady motion, we find the expression for the steady velocity of the ball falling in the medium:
It can be seen that the lighter the body, the less speed its fall in the atmosphere. The resulting equation explains why a piece of fluff falls more slowly than a steel ball.
When solving real problems, for example, calculating the steady speed of a parachutist falling during a long jump, one should not forget that the friction force is proportional to the speed of the body only for a relatively slow laminar counter air flow. With an increase in the speed of the body, air vortices arise around it, the layers are mixed, the movement at some point becomes turbulent, and the resistance force increases sharply. Internal friction (viscosity) ceases to play any significant role.
Rice. 9.15 Photograph of a liquid jet at the transition from laminar to turbulent flow (Reynolds number Re=250)
The emergence of a resistance force can then be imagined as follows. Let the body pass through the medium path. With a resistance force, work is expended on this
If the cross-sectional area of the body is equal to , then the body will "bump" into particles occupying the volume. Full mass particles in this volume is · Imagine that these particles are completely entrained by the body, acquiring speed . Then their kinetic energy becomes equal to
This energy did not appear from nowhere: it was created due to the work of external forces to overcome the force of resistance. That is, A=K, where
We see that now the resistance force is more dependent on the speed of movement, becoming proportional to its second degree (compare with the Stokes formula). In contrast to the forces of internal friction, it is often called dynamic drag .
However, the assumption that the particles of the medium are completely entrained by the moving body turns out to be too strong. In reality, any body is flowed around in one way or another, which reduces the resistance force. It is customary to use the so-called drag coefficient C, writing the drag force in the form:
With turbulent flow in a certain range of velocities C does not depend on the speed of the body, but depends on its shape: say, for a disk it is equal to one, and for a ball about 0.5.
Substituting the formula for the drag force into the expression for the drag force in steady motion, we arrive at an expression other than the previously obtained formula for the steady ball fall velocity (at C = 0,5):
Applying the found formula to the movement of a parachutist weighing 100 kg with a parachute transverse dimension of 10 m, we find
which corresponds to the landing speed when jumping without a parachute from a height of 2 m. It can be seen that the formula corresponding to a turbulent air flow is more suitable for describing the movement of a parachutist.
The expression for the drag force with the drag coefficient is convenient to use in the entire range of speeds. Since the drag regime changes at low velocities, the drag coefficient in the region of laminar flow and in the transition region to turbulent flow will depend on the velocity of the body. However, a direct relationship C from is impossible, since the drag coefficient is dimensionless. This means that it can only be a function of some dimensionless combination involving speed. Such a combination playing important role in hydro- and aerodynamics, is called Reynolds number(see topic 1.3).
The Reynolds number is a parameter that describes the regime change during the transition from laminar to turbulent flow. The ratio of the drag force to the internal friction force can serve as such a parameter. Substituting the expression for the cross-sectional area of the ball into the formula for the resistance force, we make sure that the magnitude of the drag force, up to now insignificant numerical factors, is determined by the expression
and the magnitude of the internal friction force - by the expression
The ratio of these two expressions is the Reynolds number:
If we are not talking about the movement of the ball, then under D the characteristic size of the system is understood (say, the diameter of a pipe in the problem of fluid flow). By the very meaning of the Reynolds number, it is clear that at its small values internal friction forces dominate: the viscosity is high and we are dealing with a laminar flow. At high Reynolds numbers, on the contrary, dynamic drag forces dominate and the flow becomes turbulent.
The Reynolds number is of great importance when modeling real processes on a smaller (laboratory) scale. If the Reynolds numbers are the same for two flows of different sizes, then such flows are similar, and the phenomena arising in them can be obtained one from the other by simply changing the scale of measuring coordinates and velocities. Therefore, for example, on a model aircraft or car in wind tunnel it is possible to predict and study the processes that will arise in the course of actual operation.
drag coefficient. So, the drag coefficient in the formula for the drag force depends on the Reynolds number:
This dependence has a complex character, shown (for a sphere) in Fig. 9.16. Theoretically, it is difficult to obtain this curve, and dependences experimentally measured for a given body are usually used. However, a qualitative interpretation is possible.
Rice. 9.16. The dependence of the drag coefficient on the Reynolds number (Roman numerals show the ranges of Re values; which correspond to various modes currents air flow)
Region I. Here the Reynolds number is very small (< 1) и течение потока ламинарно. Экспериментальная кривая описывается в этой области функцией
By substituting this value into the previously found formula for the resistance force and using the expression for the Reynolds number, we arrive at the Stokes formula. In this region, as already mentioned, resistance arises due to the viscosity of the medium.
Region II. Here the Reynolds number lies in the interval 1< < 2·10 4 . Данная область соответствует переходу от ламинарного к турбулентному течению. Экспериментальные данные свидетельствуют, что при увеличении числа Рейнольдса достигается некоторое его критическое значение, после которого стационарное ламинарное течение становится неустойчивым. Разумеется, это критическое значение не универсально и различается для different types currents. But its characteristic value is of the order of several tens.
At only slightly larger critical values, a non-stationary periodic motion of the flow appears, characterized by a certain frequency. With a further increase, the periodic motion becomes more complicated, and new and new frequencies appear in it. These frequencies correspond to periodic motions (vortices), the spatial scales of which become ever smaller. The movement becomes more complex and confusing - turbulence develops. In this region, the drag coefficient continues to fall with increasing , but more slowly. The minimum is reached at = (4–5) 10 3 , after which WITH rises somewhat.
Region III. This region corresponds to a developed turbulent flow around the ball, and we have already met with this regime above. The values of the Reynolds number characteristic here lie in the interval 2 10 4< < 2·10 5 .
When moving, the body leaves behind a turbulent wake, beyond which the flow is laminar. The vortex turbulent wake is easy to observe, for example, behind the stern of a ship. Part of the body surface is directly adjacent to the turbulent wake region, and its front part is adjacent to the laminar flow region. The boundary between them on the surface of the body is called the separation line. The physical cause of the resistance force is the difference in pressure on the front and back surfaces of the body. It turns out that the position of the separation line is determined by the properties of the boundary layer and does not depend on the Reynolds number. Therefore, the drag coefficient is approximately constant in this mode.
Region IV. However, such a flow around the body cannot be maintained up to arbitrarily large values of . At some point, the forward laminar boundary layer turbulizes, which pushes back the separation line. The turbulent wake behind the body narrows, which leads to a sharp (by a factor of 4–5) drop in the resistance of the medium. This phenomenon is called resistance crisis, occurs in a narrow range of values \u003d (2–2.5) 10 5 . Strictly speaking, the above theoretical considerations may change when the compressibility of the medium (air, in our case) is taken into account. However, this will manifest itself, as we have already discussed, at the speeds of objects comparable to the speed of sound.
http://ilib.mirror1.mccme.ru/djvu/bib-kvant/kvant_70.djvu - Stasenko A.L. Flight Physics, Quantum Library, issue 70 pp. 17–28 - aerodynamic forces acting on the wing.
http://d.theupload.info/down/8osiz73swyx22j1icv3641f3xxe8rtdp/butikov_e_i__kondratev_a_s__fizika_dlja_uglublennogo_izuchen.djvu - E.I. Butikov, A.S. Kondratiev, Tutorial; Book. 1, Mechanics, Fizmatlit, 2001 - chapter V - the movement of liquids and gases.
List of additional links
http://kvant.mirror1.mccme.ru/pdf/1998/02/kv0298fizfak.pdf - Kvant magazine - mathematical pendulum on inclined surfaces (P. Khadzhi, A. Mikhailenko).
http://kvant.mirror1.mccme.ru/1971/06/strannyj_mayatnik.htm - Kvant magazine - a mathematical pendulum with a moving suspension point (N. Mints);
http://edu.ioffe.ru/register/?doc=physica/lect4.ch1.tex - The lecture deals with harmonic oscillations, phase portrait of a pendulum, adiabatic invariants.
http://www.plib.ru/library/book/9969.html - E.I. Butikov, A.S. Kondratiev, Textbook; Book. 1, Mechanics, Fizmatlit, 2001 - pp. 279–295 (§§ 42,43) - damped oscillations in dry friction and natural oscillations in various physical systems are described.
http://mechanics.h1.ru/ - Mechanics at school, definitions of basic physical quantities, problem solving.
http://edu.ioffe.ru/register/?doc=mgivanov - A course of lectures on mechanics for a school of physics and technology (MG Ivanov).
http://ilib.mirror1.mccme.ru/djvu/bib-kvant/kvant63.djvu - Aslamazov L.G., Varlamov A.A. Amazing Physics, Quantum Library, issue 63, chapter 2 - simple physics of complex phenomena.
http://schools.keldysh.ru/sch1275/kross/ - Physical crosswords.
http://www.newsland.ru/News/Detail/id/211926/22 - The possibility of creating a sound and optical "cap of invisibility" is discussed.
http://ilib.mirror1.mccme.ru/djvu/bib-kvant/kvant_40.djvu - Khilkevich S.S., Physics around us, Kvant library, issue 40, chapter 1, § 5 - how vibration and what happens when you shake a bucket of potatoes.
Force and is always directed against the velocity vector of the body in the medium. Along with the lifting force, it is a component of the total aerodynamic force.
The drag force is usually represented as the sum of two components: drag at zero lift and induced drag. Each component is characterized by its own dimensionless drag coefficient and a certain dependence on the speed of movement.
Drag can contribute to both icing aircraft(at low temperatures air), and cause heating of the frontal surfaces of the aircraft at supersonic speeds by impact ionization.
| flow and form obstacles |
Resistance forms |
Influence friction viscosity |
|---|---|---|
| ~0,03 | ~100 % | |
 |
~0,01-0,1 | ~90 % |
 |
~0,3 | ~10 % |
 |
1,17 | ~5 % |
| hemisphere | 1,42 | ~10 |
Resistance at zero lift
This drag component does not depend on the magnitude of the created lift force and consists of the profile drag of the wing, the resistance of aircraft structural elements that do not contribute to lift, and wave drag. The latter is significant when moving at near- and supersonic speeds, and is caused by the formation of a shock wave that carries away a significant portion of the motion energy. Wave drag occurs when the aircraft reaches a speed corresponding to the critical Mach number, when part of the flow around the wing of the aircraft acquires supersonic speed. The critical number M is the greater, the greater the sweep angle of the wing, the more pointed the leading edge of the wing and the thinner it is.
The resistance force is directed against the speed of movement, its value is proportional to the characteristic area S, the density of the medium ρ and the square of the speed V:
X 0 = C x 0 ρ V 2 2 S (\displaystyle X_(0)=C_(x0)(\frac (\rho V^(2))(2))S) C x 0 (\displaystyle C_(x0))- dimensionless aerodynamic drag coefficient, obtained from similarity criteria, for example, Reynolds and Froude numbers in aerodynamics.The definition of the characteristic area depends on the shape of the body:
- in the simplest case (ball) - cross-sectional area;
- for wings and empennage - the area of the wing / empennage in plan;
- for propellers and rotors of helicopters - either the area of the blades or the swept area of the propeller;
- for streamlined underwater objects - wetted surface area;
- for oblong bodies of revolution oriented along flow (fuselage, airship shell) - reduced volumetric area equal to V 2/3, where V is the volume of the body.
The power required to overcome a given component of the drag force is proportional to the cube of the speed ( P = X 0 ⋅ V = C x 0 ρ V 3 2 S (\displaystyle P=X_(0)\cdot V=C_(x0)(\dfrac (\rho V^(3))(2))S)).
Inductive drag in aerodynamics
Inductive reactance(eng. lift-induced drag) is a consequence of the formation of lift on the wing of finite span. The asymmetric flow around the wing leads to the fact that the air flow escapes from the wing at an angle to the flow on the wing (the so-called flow bevel). Thus, during the movement of the wing, there is a constant acceleration of the mass of incoming air in a direction perpendicular to the direction of flight and directed downward. This acceleration, firstly, is accompanied by the formation of a lifting force, and secondly, it leads to the need to impart kinetic energy to the accelerating flow. The amount of kinetic energy required to communicate the speed to the flow, perpendicular to the direction of flight, will determine the value of the inductive resistance. The magnitude of the inductive drag is influenced not only by the magnitude of the lift (for example, in the case of a negative work of the lift, the direction of the vector of the inductive drag is opposite to the vector of the force due to tangential friction), but also by its distribution over the span of the wing. The minimum value of the inductive reactance is achieved with an elliptical distribution of the lifting force along the span. When designing a wing, this is achieved by the following methods:
- the choice of a rational wing shape in plan;
- the use of geometric and aerodynamic twist;
- installation of auxiliary surfaces - vertical wingtips.
Inductive reactance proportional to square lift force Y, and inversely wing area S, its elongation λ (\displaystyle \lambda ), medium density ρ and square speed V:
X i = C x i ρ V 2 2 S = C y 2 π λ ρ V 2 2 S = 1 π λ Y 2 ρ V 2 2 S (\displaystyle X_(i)=C_(xi)(\frac (\rho V^(2))(2))S=(\frac (C_(y)^(2))(\pi \lambda ))(\frac (\rho V^(2))(2))S= (\frac (1)(\pi \lambda ))(\frac (Y^(2))((\frac (\rho V^(2))(2))S)))Thus, inductive reactance makes a significant contribution when flying on low speed(and, as a result, at high angles of attack). It also increases as the weight of the aircraft increases.
Total resistance
It is the sum of all types of resistance forces:
X = X 0 + X i (\displaystyle X=X_(0)+X_(i))Since the resistance at zero lift is proportional to the square of the speed, and the inductive is inversely proportional to the square of the speed, they contribute differently at different speeds. With increasing speed X 0 (\displaystyle X_(0)) is growing and X i (\displaystyle X_(i))- falls, and the graph of the dependence of the total resistance X (\displaystyle X) on speed (“required thrust curve”) has a minimum at the point of intersection of the curves X 0 (\displaystyle X_(0)) And X i (\displaystyle X_(i)), at which both resistance forces are equal in magnitude. At this speed, the aircraft has the least resistance for a given lift (equal to weight), and therefore the highest
The drag coefficient makes it possible to take into account energy losses during body motion. Most often, two types of motion are considered: motion on a surface and motion in a substance (liquid or gas). If we consider the movement along the support, then we usually talk about the coefficient of friction. In the event that the motion of a body in a liquid or gas is considered, then the shape resistance coefficient is meant.
Determination of the coefficient of resistance (friction) of sliding
DEFINITION
Drag coefficient (friction) called the coefficient of proportionality connecting the friction force () and the force of normal pressure (N) of the body on the support. Usually this coefficient is denoted by the Greek letter. In this case, the friction coefficient is defined as:
We are talking about the coefficient of sliding friction, which depends on the combined properties of the rubbing surfaces and is a dimensionless quantity. The coefficient of friction depends on: the quality of surface treatment, rubbing bodies, the presence of dirt on them, the speed of movement of bodies relative to each other, etc. The coefficient of friction is determined empirically (experimentally).
Determination of the rolling resistance (friction) coefficient
DEFINITION
Rolling resistance (friction) coefficient usually denoted by the letter . It can be determined using the ratio of the moment of the rolling friction force () to the force with which the body is pressed against the support (N):
This coefficient has the dimension of length. Its basic unit in the SI system is the meter.
Determination of the shape resistance coefficient
DEFINITION
Shape drag coefficient- a physical quantity that determines the reaction of a substance to the movement of a body inside it. It can be said differently: it is a physical quantity that determines the body's response to movement in matter. This coefficient is determined empirically, its definition is the formula:
where is the resistance force, is the density of the substance, is the speed of the flow of the substance (or the speed of the body in the substance), the area of the projection of the body onto a plane perpendicular to the direction of motion (perpendicular to the flow).
Sometimes, if we consider the movement of an elongated body, then we consider:
where V is the volume of the body.
The drag coefficient under consideration is a dimensionless quantity. It does not take into account effects on the surface of bodies, so formula (3) may not be suitable if a substance with a high viscosity is considered. The drag coefficient (C) is constant as long as the Reynolds number (Re) is constant. IN general case.
If the body has sharp edges, then it is empirically obtained that for such bodies the drag coefficient remains constant in a wide range of Reynolds numbers. So it was experimentally obtained that for round plates placed across the air flow, at the values of the drag coefficient are in the range from 1.1 to 1.12. With a decrease in the Reynolds number (), the resistance law turns into the Stokes law, which for round plates has the form:
Ball drag has been investigated for a wide range of Reynolds numbers up to 
For received:
The handbooks present drag coefficients for round cylinders, balls and round plates depending on the Reynolds number.
In aviation technology, the problem of finding the shape of a body with minimal drag is of particular importance.
Examples of problem solving
EXAMPLE 1
| Exercise | The maximum speed of the car on a horizontal section of the road is equal to its maximum power equal to P. The drag coefficient of the car is C, and the largest cross-sectional area in the direction perpendicular to the speed S. The car has undergone reconstruction, the largest cross-sectional area in the direction perpendicular to the speed has been reduced to the value , leaving the coefficient resistance unchanged. Consider the force of friction on the road surface unchanged, find what is maximum power car, if its speed on a horizontal section of the road became equal to . The air density is . |
| Solution | Let's make a drawing. The power of the car is defined as: where is the traction force of the car. Assuming that the car on a horizontal section of the road moves with constant speed, we write Newton's second law in the form: In the projection onto the X axis (Fig. 1), we have: The resistance force experienced by the car, moving in the air, we express as:
Then the power of the car can be written:
Let us express from (1.5) the friction force of the car on the road:
Let's write the expression for the power, but with the parameters of the car changed according to the condition of the problem:
We take into account that the force of friction of the car on the road has not changed, and take into account the expression (1.6):
|
| Answer |
EXAMPLE 2
| Exercise | What is maximum speed a ball that freely falls in the air, if you know: the density of the ball (), air density (), mass of the ball (), drag coefficient C? |
| Solution | Let's make a drawing. We write Newton's second law for the free fall of a ball: |
In all real liquids, when some layers move relative to others, more or less significant friction forces arise.
From the side of the layer moving faster, the layer moving more slowly is affected by an accelerating force. From the side of the layer moving more slowly, the layer moving faster is affected by a retarding force. This internal friction is called the viscosity of the liquid or gas. These forces are directed tangentially to the surface of the layers. Let there be a liquid layer between two planes (Fig. 1); the upper plane moves relative to the lower one with a speed . Let us mentally divide the liquid into very thin layers by parallel planes spaced at a distance from each other. Layers of fluid that touch solids adhere to them. The intermediate layers have the distribution of velocities shown in fig. 1. Let the speed difference between adjacent layers be . The value of , which shows how quickly the speed changes when moving from layer to layer, is called the velocity gradient.
Calculations show that the force of internal friction between adjacent layers of liquid is the greater, the larger the area of the contact surface of the layers, and depends on the rate of change in velocity during the transition from layer to layer in the direction of the Ox axis, perpendicular to the velocity of the layers:
where S is the area of contact between the layers, is the coefficient of internal friction, or the viscosity of the liquid, is the velocity gradient.
Viscosity depends on temperature. As the temperature rises, the viscosity of the liquid decreases.
When a rigid body moves in a liquid or gas, a force of resistance to movement also arises, which is called the force viscous friction. But unlike dry friction, there is no static friction force in liquids and gases. The presence of a force of resistance to the motion of a body in a medium is explained by the existence of internal friction due to relative motion layers of liquid or gas.
It is established that the force of viscous friction depends on the velocity of the body. The dependence of the projection of the viscous friction force on the velocity is shown in Figure 2.
If the speed of the body is low, then the resistance force is directly proportional to the modulus of speed: , where k is the proportionality factor, which depends on the type of viscous medium, shape and size of the body. If the speed of the body increases, then the resistance force also increases:
With an increase in the speed of the body in a liquid or gas, vortices appear that slow down the movement: due to viscosity in the area adjacent to the surface of the body, a boundary layer of particles moving at lower speeds is formed. As a result of the decelerating action of this layer, the rotation of particles occurs, and the motion of the fluid in the boundary layer becomes vortex. If the body does not have a streamlined shape, then the boundary layer of the liquid is separated from the surface of the body. A fluid (gas) flow occurs behind the body, directed opposite to the oncoming flow. The detached boundary layer, following this flow, forms vortices rotating in opposite directions (Fig. 3, b). A fluid rotating in a vortex moves faster than a fluid in a stationary flow (Fig. 3, a). Therefore, on the rear side of the streamlined body, where vortices have formed, the pressure becomes less than on the front. The pressure difference in front of and behind the moving body creates resistance to the movement of the body. As a result, as the speed increases, the resistance force grows non-linearly (see Fig. 2).
The force of resistance depends on the shape of the body. Giving the body a specially designed streamlined shape significantly reduces the resistance force, since in this case the fluid is everywhere adjacent to its surface and is not swirled behind it (Fig. 3, c).
