The lifting force of an airplane wing when flying horizontally at high speeds. Explanation of the vortex trail.


 In this paper, an explanation is given of the lift force of an airplane during horizontal flight. It is shown that during a flight, five vertical forces act on the airplane: gravity; pressure gradient with a minus sign; Archimedes force; potential force and the vortex force obtained from the action minimum. The first three forces were known before. The potential force was also known from the Bernoulli equation, but its effect on the airfoil from the air had not previously been taken into account. The vortex force obtained from the minimum action in the application to a continuous medium was not taken into account in aerodynamics. In horizontal flight the vortex force is directed upwards, it compensates for the gravity of the airplane at high speed commensurate with the speed of sound. The paper provides an explanation of the vortex trail behind the airplane, mentioned in the Millennium problem Navier-Stokes equation.


Introduction
The idea to write this paper came after journal «Scientific American» published the paper: "No one can explain why planes stay in the air" [1]. In [1], the diagram of the pressure and speed distribution around the airfoil, which occurs during horizontal flight at zero angle of attack, is justified (Fig. 1). The diagram in figure 1 is well known and it is used in many training manuals to explain the lift of an airplane wing, see for example [2,3]. under the airfoil and this explains the lift of the wing of the airplane, shown in Fig.1. However, there is no evidence that the air speed is larger over the wing of the airplane. As it turned out, this problem is more than a hundred years old.
If you look at the movement of air around the airfoil in horizontal flight, without paying attention to the inscriptions on Fig. 1, it becomes obvious that the air speed above the wing of the airplane is less than the air speed under the wing of the airplane.
Indeed, figure 1 clearly shows that the wing of the airplane is beating the air from the top of the airfoil, like a hill in an open field. It is obvious that the wind behind the hill is less, therefore, the air speed above the airfoil is less than below the airfoil in horizontal flight. This is how all airplane wings are designed, they have a thickening in front of the wing above the horizon line.
In addition, the air really presses on the airplane wing from above, based on the design of the aerofoil, since the tangent to the aerofoil has an acute angle with the horizon, and from below the air passes parallel to the aerofoil and does not exert additional pressure (Fig. 1). The acute angle between the tangent to the profile and the horizon line at the top of the aerofoil in fact, means that the profile of the airplane wing creates an angle of attack from above for oncoming air flow, when the wing is horizontal (with zero angle of attack under the wing of the airplane).
As it is known [2][3][4], Zhukovsky used the fact that the air flow velocity is larger over the wing of the airplane when deducing the lift force ( Fig. 1). Therefore, he received a lifting force. However, if we take into account the real situation that the speed of air flow under the wing is greater than the speed over the wing of the airplane in horizontal flight, then the Zhukovsky force is directed downward, along with the air pressure that presses on the wing of the airplane from above.
Indeed, the analogy between the airfoil with a rotating cylinder in an oncoming air flow, which Zhukovsky used, does not stand up to criticism. Firstly a symmetrical cylinder is not equivalent to an asymmetric airfoil so they will behave differently in oncoming air flow.
Secondly, even if we assume that there is some correlation in their behavior, the asymmetric airfoil at zero angle of attack slows down the oncoming air flow from the top of the airplane wing, as it creates a barrier to the air flow from above. Therefore, an asymmetric profile at zero angle of attack will correspond to a cylinder rotating counterclockwise in the oncoming flow, so that the air velocity is less on the top of the cylinder. As is known [4], a downward force will act on such a rotating cylinder in the oncoming flow.
The viscosity in the air is low, it obviously will not help to keep the airplane in the air. However, Prandtl took into account the air viscosity in the boundary layer when describing the lift force of an airplane [4]. But he also assumed that the air velocity is larger over the airfoil (Fig. 1), since the strength of the viscosity depends on the speed. Therefore, if we now recalculate the forces received by Prandtl, they will also be directed downward.
Thus, the viscosity and similarity principle, which is used in describing the motion of a viscous fluid [4], will not help us to explain the lift force of an airplane. The airfoil sets the pressure gradient and speed gradient along the vertical axis. Therefore, if Zhukovsky and Prandtl obtained upward forces based on the distribution (Fig. 1), then now, when recalculating, given that the air velocity is greater under the wing of the airplane, and the air pressure is greater above the wing of the airplane, these forces will be directed downward.
Then the question arises, how to explain the lift of the airplane in horizontal flight and why the airplane does not fall?
The purpose of this paper is to describe the lift of an airplane in horizontal flight and answer the question why the airplane remains in the air [1].

The equation of motion of the air and equilibrium of the airplane.
To solve this problem, we will use the equations of air movement obtained from the minimum action [5], which have the form: (1) Here j x P    is the pressure gradient with a minus sign from Newton's equations; centrally symmetric force obtained from the action minimum; vortex force obtained from the minimum action; cspeed of sound; j air speed;  -air density; P -pressure.
In (1), the air viscosity is not taken into account, since it is not essential to explain the lift of the airfoil. Therefore, in the future we will not appeal to the Navier-Stokes equations, but to Euler's equations of hydrodynamics.
As is known [4], the classical Euler equations of hydrodynamics were obtained by differentiating the velocity field as a complex function and have the form: It is easy to see that equations (1) and (4') coincide for the case of stationary one-dimensional Bernoulli motion [4]: This is where the coincidences between the equations of motion (1) and (4') -end.
As it was shown in [5] and [6], the problem of the general description of turbulence is connected with the incorrectness of Euler's equations. For example, the problem of describing cyclones and anticyclonesthe largest vortex movements on Earth is not related to the complexity of solutions to the Euler or Navier-Stokes hydrodynamic equations, as mechanics believe, but to their incorrectness. This was proved in p. 3 [6]. In p. 2 paper [6] show that equations (4) prohibit the very existence of anticyclones, and for cyclones give an incorrect ratio between the kinetic energy of air and pressure when they change, which means that the Euler and Navier-Stokes equations are invalid. Therefore, in this paper, we will not repeat the reasoning of the paper [4], but will solve the problem of describing the lift of an airplane in horizontal motion, based on the equations of motion (1) obtained from the minimum action. Especially since this problem is known [1] and obvious (Fig. 1).
The logic of the diagram shown in figure 1 is understandable. The airplane is flying horizontally and does not fall, which means that its wings are affected by the resulting upward force in the form of a pressure difference multiplied by the area of the wings. According to Bernoulli's law, if the air pressure is greater below, under the airfoil, then the air speed must be greater above the airfoil.
However, there is an error in this reasoning. It consists in identifying the air pressure gradient and the force acting on the airfoil from the air. After all, several forces can act on the airfoil from the air, not just the pressure gradient. For example, the wing of an airplane and its fuselage are affected by the Archimedean force directed upwards from the air. It is equal to the density of air multiplied by the volume of the airplane and the acceleration of free fall. The Archimedes force is small and, in the first approximation, it can be ignored, but it is not related to the air pressure gradient that is formed around the airfoil when the airplane moves.
In addition, from figure 1 it is clearly seen that not only is the air speed less above the wing of the airplane, but also the air pressure is greater above the wing of the airplane. Indeed, the oncoming air flow literally presses the airplane down. This is due to the special shape of the airfoil, which has an acute angle between the tangent to the wing profile and the horizon above, and below the wing, the air flow passes smoothly, almost without exerting pressure on the airplane. The acute angle between the tangent to the airfoil and the horizon, in fact, means the angle of attack from above, which creates the airfoil. Thus, it is obvious that the air pressure is greater above the wing of the airplane when flying horizontally ( fig. 2). Note that we did not use Bernoulli's law in these arguments. The fact that the Bernoulli equations are fulfilled for the air around the airplane during stationary horizontal flight is not obvious, since the air, for example, is strongly compressed and heated when it collides with the airplane. However, in the first approximation, we can assume that the air blowing around the wings of the airplane satisfies the potential movement of Bernoulli (5). Therefore, in the future, for simplicity of reasoning, we will use the expression (5), just as it is used in [1] and [2,3].
In our opinion, the desire to prove the incorrect statement that the air speed is larger above the wing of the airplane (Fig. 1) [1,2] is due to a wrong understanding of equation (5), from which the Bernoulli equation follows. The fact is that during a stationary one-dimensional potential movement of air, the sum of the forces acting on the air is zero (5). Equation (5) follows from equation (1). Thus, according to (5), two forces act on the air: the pressure gradient with a minus sign j x P    , and the potential force , and this is well known.
However, in the scientific literature on mechanics, the potential force is not considered a force. This may be due to the fact that it was obtained from the convective derivative of the velocity field: [4]. Obviously, it is impossible to obtain the force acting in a continuous medium by differentiating the velocity field. In [6] it was proved that the convective derivative of the velocity field is identically equal to zero. On the other hand, the was obtained as a summand of the centrally symmetric force (2) from the action minimum [5]. Therefore, of course has the Newton dimension, it stands in the equations of motion (1) and (4') as a force, so of course it is a force.
Why did the aerodynamicists decide that if the sum of all forces acting on a unit volume of air is zero (5), then only the air pressure gradient acts on the wing of the airplane, and the kinetic energy gradient of the air does not act in the form of: -the question is rhetorical. Most likely, because they wanted to explain the lift force of the airplane (Fig. 1), but how can it be explained if with a potential movement of air that satisfies the Bernoulli equation, the sum of the forces acting on the wing of the airplane from the air is zero (5).
The problem with Fig. 1 is that it was necessary to explain the lift of the airplane, and there were no other forces other than the pressure gradient. In this case, the air pressure gradient was identified with the air pressure on the wing of the airplane, which is not the same thing. Therefore, the task was to show that the pressure in the air during horizontal flight is greater under the wing of the airplane [1][2][3], although it is obvious to everyone that this is not the case (Fig. 2).
was not considered a force, but was considered a certain "ploy" associated with the replacement of Lagrange coordinates with Euler coordinates. As shown in [6] this is not true. The potential force exists and must be taken into account when moving a continuous medium. It is responsible for the flat rotation of air in the form of cyclones and anticyclones when the pressure in the atmosphere changes [6].
If we take into account the potential force, then for a potential stationary air flow it does not matter where the pressure is greater below the airfoil or above the airfoil. Since according to the Bernoulli equations, which are referred to in [1] and [2,3], the sum of the forces acting on the wing of the airplane from the air at its potential flow is zero (5).
Thus, to describe the lifting force of the airplane, we have only one vortex force: , since the Archimedean force, equal to the weight of the displaced air, is small, and it is usually not taken into account [2,3].
Note that the second term of the vortex force: this force is directed upwards, which was necessary to prove.
Thus, the sum of the forces acting on the airplane in the projection on the vertical axis, in the first approximation, without the Archimedean force, has the form Now it becomes clear why you need to thicken the top of the wing of the airplane (Fig. 2), so that the pressure is larger above, and not below the wing of the airplane. When constructing an airfoil, developers achieve potential airflow around the wing of the airplane without vortex movement around the wing of the airplane. Then the vortex force j x P c   2 2  compensates for the force of gravity and keeps the airplane in the air at high speeds.

The turbulent trace of the plane.
The problem of describing the lift force of an airplane wing is directly related to the problem of describing the vortex trail of an airplane. When the plane flies, a vortex movement of air forms behind its wings (Fig. 3), Q.E.D. Indeed, it is the wings that prevent the occurrence of vortex air movement as a result of the action of vortex force (3).
While the plane is flying, the vortex force component j x P c   2 2  compensates for the airplane's gravity and does not act on the air. It's like hiding under the wing of an airplane. As soon as the airplane has passed, the vortex force (3) acts on the air, it accelerates it, and together with the central force (2) makes air to move in the form of vortices, as shown in (Fig. 3).
Note that the Euler's equations (4) and, consequently, the Navier-Stokes equations cannot describe vortex motion (Fig.3). The invalidity of the Navier-Stokes equations is reflected in the Millennium problem: http://www.claymath.org/millennium-problems . Let's prove this statement.
Let an airplane fly in the direction of the axis Obviously, the viscosity in the Navier-Stokes equations is not able to correct this situation, since the viscosity in the air is very small. In addition, due to the isotropy of the air, the viscosity can't explain the resulting anisotropic pattern of the airplane trace (Fig. 3). The invalidity of the Euler and Navier-Stokes equations is that they can't describe the vortex trace of an airplane (Fig.3), as well as plane vortex movements [6].
On the other hand, the potential force acts on the air perpendicular to its movement, which explains the vortex trail of the airplane (Fig. 3). As it was shown in [4], this is and the problem of describing plane vortex movements of air in the atmosphere.
However, as shown in this paper, descriptions of the continuous medium movement in (1) and (4) was deeper and it is related to relativism. Therefore we will devote the rest of this paper to discussing relativism and the vortex force that keeps an airplane in the air.

Lifting force of the airplane in horizontal flight.
We will arrange, as they say in such cases, a debriefing. As a result of taking into account the vortex force (3)  Note that this result does not depend on the assumption that the movement of air around the plane satisfies the Bernoulli equation. When the speed of the airplane is nearby to the speed of sound, the pressure gradient is annulated in the equations of motion for air (1) when the plane is moving horizontally. Therefore, the wings of all airplanes are arranged so that when flying horizontally, the air speed is greater under the wing of an airplane (Fig. 2), otherwise, the airplane simply can not be in the air when flying horizontally.
In this paper, we used the Bernoulli equation for methodological reasons to explain clearly the action of the vortex force: . It is obvious that in general, it is necessary to set boundary conditions and solve the problem of equilibrium for the airplane and for the air when the airplane is flying horizontally.
However, schematically, in the first approximation, we can assume that the air near the wing of the airplane moves potentially, without vortices, and the condition (5)  The fact that the vortex force (3) is the lifting force was known from [5], since it is responsible for the lifting force in tornadoes.
In tornadoes, the lifting force is equal to  is the horizontal vortex velocity in the tornado, and the pressure gradient 3 x P   is directed along the vertical axis upwards towards the expansion of the funnel of the tornado. At the same time, in [5] it was assumed just as for an airplane, that the gradient of air pressure along the vertical axis is compensated by the gradient of the kinetic energy of air rotation in the tornado, since the tornado, in the first approximation, can be considered a stable vortex movement of air.
Thus, the lift in a tornado is the same as the lift in an airplane. The only difference is that in a tornado the air moves in a circle in a horizontal plane, and the airplane flies straight and itself creates forces acting on it from the air. Therefore, it was not at all obvious that the lifting force for an airplane could be a vortex force (3). This problem was solved thanks to the statement of the question in the paper [1].
On the other hand, everything is less obvious in a tornado than in a situation with an airplane, since it is difficult to check something in a tornado. This is the main problem associated with studying tornadoes, since the tornado breaks all the instruments before they have time to measure anything.
There is a scale of intensity of tornadoes it is called The Fujita scale [7]. The Fujita scale clearly shows that the speed at which a large passenger plane fly up h km/ 270 and the wind speed in a tornado F3, at which it is able to lift large cars h km/ 254 , are of the same order. From this we can conclude that the lifting force in tornadoes is of the same nature as the lifting force of an airplane.
It is obvious that the lift effect in (6)  We must pay tribute to the fact that thanks to the efforts of Zhukovsky and Prandtl, the lifting force was written from dimensional considerations in the form C is a dimensionless coefficient [2][3][4]. As noted above, this force was derived from the incorrect assumption that the speed is greater over the wing of the airplane (Fig. 1).
However, it was guessed almost correctly. Indeed, if we put that 2 c P   , then, in the first approximation, the lifting force in (6) can be written as: . Here  is the density above the wing of the airplane, and the density   is equal to the usual air density  taken into account above.
It is obvious that at high speeds the counter flow of air is compressed. Therefore the density of air above the airfoil is greater than the density of air under the airfoil, where the air flow passes without resistance at zero angle of attack. Therefore, if the density increases by ten percent over the wing of the airplane (7) due to the thickening of the wing from above, we will get the lifting coefficient that corresponds to the lifting coefficient of the airplane at zero angle of attack 2 . 0  C [2,3]. This result was obtained under the assumption that the Bernoulli equations for air flowing around the airfoil (6) (1) and it is possible to calculate the coefficient C for the lifting force without referring on the Bernoulli equations.
Indeed, let the speed value be above than the airplane wing  and under than the airplane wing   , then the lifting force after integration can be written as As is known [2][3][4], the lifting coefficient C at zero angle of attack has a value of order 2 . 0 .
Therefore, to get such the lifting coefficient, the airfoil must slow down the speed of the air flow above the wing of the airplane by ten percent, which is actually happening.
Thus, it becomes clear that it is the force (7) Zhukovsky and Prandtl tried to obtain. However, they acted out of misconceptions about the movement of air around the wing of the airplane (Fig.  1). Therefore, even if the relativistic force (7) was guessed correctly, it does not give a complete understanding of the movement of the airplane in the air, since only horizontal flight is considered here.
Moreover, the Zhukovsky and Prandtl forces are global (integral) forces [4], not local (differential) forces (1). The five components of the volume force (1) can't be obtained for dimensional reasons. Forces (2, 3) [5] were derived from translational symmetry [8]. The absence of at least one component of these forces will not allow you to correctly describe the flight of the airplane. In papers [5,6] it was shown what each component of the forces (2, 3) is responsible for. Therefore to describe the movement of the airplane it is necessary to take into account all the forces acting in the air (1-3), obtained from the minimum action.

Explanation and conclusions.
In conclusion, let us recall how the forces (2) and (3) acting in a continuous medium appeared and why they are relativisticthey have a relative multiplier 2 2 c  . First, we constructed a minimal interaction [8] induced by a subgroup of translations in the form of an expanded derivative, in accordance with the Pauli principle [9]. It is known that the electrodynamics described by Pauli is a relativistic theory [10]. After constructing the minimal interaction induced by the translation subgroup [7], the forces of this interaction were calculated from the action minimum in the same way as the Coulomb and Lorentz forces are calculated in the field theory [9].
However, the first relativism in this theory [5,8] was introduced by Kadic-Edelen [11]. By analogy with electrodynamics they wrote down the pseudo-Lorentz calibration condition, with the only difference that c -here is the speed of sound, not the speed of light. The pseudo-Lorentz calibration condition is physical, for example, it follows that mechanical waves in a continuous medium propagate at the speed of sound [5].  [5]. Note that the first term is the potential force which had been known since the time of Bernoulli and Euler, and the other terms are relativistic forces, all of which contain the speed of sound squared in the denominator.
At high speeds comparable to the speed of the sound at which large airplane is fly, the vortex force component is not small. It is the lifting force for the airplane in horizontal flight. To increase lift, you need to increase the speed of the airplane or increase the air pressure gradient. Given that in the first approximation, it can be assumed that the air moves potentially in the area of the wing of the airplane (5), an increase in the pressure gradient in the air can be achieved by increasing the air velocity gradient.
Thus, it is necessary to make such a design of the airfoil that the speed above the wing is less than below the wing of the airplane (Fig. 2). Our airplane designers successfully coped with this task, without theoretical aerodynamics [1][2][3]. Now that there is an explanation for such a wing design in an airplane, it will be possible to design new airplane models using computer calculations, without resorting to expensive and not always possible experiments using wind tunnels.
Computer simulation of the airplane flight will be possible due to the knowledge of local forces (1) that act on the airplane from the air at each point of its surface. To date, only the pressure gradient j x P    and the integral lifting force S C F 2   obtained from heuristic dimensional considerations are taken into account in aerodynamics from forces (1). Obviously, these two forces can't replace the five components of forces acting in a continuous medium (1).