On The Physics Of Flight

ON THE PHYSICS OF FLIGHT Emil Marinchev*

Dian Geshev

Ivan Dimitrov

Stoil Donev**

Ivaylo Nedyalkov*

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

Department of Air Transport, Technical University-Sofia, 1000, Bulgaria *

Department of Applied Physics, Technical University-Sofia, 1000, Bulgaria

**

Institute for Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, Sofia, 1784, Bulgaria

Abstract A universal explanation of the physics of flight, regarding it as a case of reactive motion, caused by the interaction of flying objects and airflow, is presented. The results obtained are exact and valid for all kinds of flying objects in real 3-dimensional fluid. They constitute, on one hand, a good complement to the Kutta-Joukowski theorem and, on the other hand, a generalization of this theorem is derived for low-viscosity real fluid. Key words: physics of flight, lift, drag, Magnus effect, Kutta-Joukowski theorem.

1. Introduction For more than 100 years the Kutta-Joukowski theorem is used successfully by aeronautical engineers for subsonic flights [1, 4, 7, 8, 9, 15]. A basic disadvantage of using this theorem is that it considers the case of ideal fluid, while any flight is performed in a real fluid. The aim here is to reconsider the lift problem in the case of a real fluid making use mainly of the conservation laws. The following theorems from Mechanics are used: - The momentum of a mechanical system p is equal to the momentum of the center of masses pc

p = p c = mv c ,

where m is the mass of the system, and vc - the velocity of the mass center. - The reactive force R is determined by the action of momentum flow mu ( m is mass flow, and u is the change of its velocity) R = mu .

2. Physics of flight Air flow, interacting with the wing is diverted at an effective angle 11, 12, 13]. It is equal to the effective angle of attack for the entire wing.

α

(Fig. 1) [5, 6, 10,

Fig. 1 The mass flow interacting with the flying object is denoted by | m |. The diverted mass flow, leaving the wing is m < 0 , and the incoming flow is −m > 0 . As the air flow follows the

shape (curvature) of the wing, the mass flow m is proportional to the streamlined surface of the wing S w , the velocity of the airflow v ( v ≡ vc ), and air density ρ - m ∼ ρ v Sw . If S w is

expressed by the wing planform area S , we get m = Cρ v S , where C is a coefficient of proportionality. The resulting aerodynamic force R acting on the wing, is due to the joint action of the incoming and diverted momentum flows: (1)

R = −mv + m( v + u ) = mu .

Consider a flight with a constant velocity v in a low-viscosity real fluid. Neglecting losses of energy leads to v + u ≈ v , and u = 2 vsin ( α 2 ) (Fig. 1). Than from expression (1) it follows (2)

R = Cρ v S 2 v sin ( α 2 ) = 4C sin ( α 2 ) .S

ρ v2 . 2

The vector u is represented by its normal and parallel components with respect to the incoming flow: u⊥ = vsin α

u = v− v cos α = v(1 − cos α) = 2 vsin 2

α v 2 ≈ α. 2 2

For the lift L and the drag D (Fig. 1) and their dimensionless coefficients C L and C D we derive L = mu⊥ = 2C sin α S (2')

D = mu ≈ C α 2 S

ρ v2 2

ρ v2 2

L = 2C sin α S ρ v2 / 2 (2′') D CD = ≈ Cα 2 2 Sρ v / 2 CL =

Taking into account the loss of energy leads to: first, v ' =| v + u | < v (Fig. 2), second, the angle between R and L increases (α′ > α/2):