impact of jet on vanes
Blade Element Theory
Prof. Dr. Mustafa Cavcar
Anadolu University, School of Civil Aviation
Eskisehir, Turkey mcavcar@anadolu.edu.tr The simple momentum theory provides an initial idea regarding the performance of a propeller but not sufficient information for the detailed design. Detailed information can be obtained through analysis of the forces acting on a blade element like it is a wing section. The forces acting on a small section of the blade are determined and then integrated over the propeller radius in order to predict the thrust, torque and power characteristics of the propeller.
R
ω
c r dr dT w
Ve
VR
V
dL
αi β α φ dF
ωr dD Figure 1. Propeller blade element with velocity and force diagram.
© Prof. Dr. Mustafa Cavcar, 2004.
1
A differential blade element of chord c and width dr , located at a radius r from the propeller axis, is shown in Figure 1. The element is shown acting under the influence of the rotational velocity, ωr , forward velocity of the airplane, V , and the induced velocity, w . Vector sum of these velocities produce
Ve = ω r + V + w
(1)
The section has a geometric pitch angle of its zero lift line of β . If it is assumed that V and ωr are known, then calculation of the induced velocity w is desired to find α i , and consequently the section angle of attack α . Knowing α , and the section type, C l and C d can be calculated, then the differential lift and drag of the section will follow. However, w depends on dL which in turn depends on w .
Thus the problem is closely related to the finite wing problem but is more complicated because of the helicoidal geometry of the propeller [1].
Combined Momentum – Blade Element Theory
A starting approximate value of w can be obtained by application of the momentum theory principles to an annulus of width dr and radius r (Figure 2).
In this case
dT = ρdA(V + w cos φ )2w cos φ
(2)
dr r R
Figure 2. Annulus to
Prof. Dr. Mustafa Cavcar
Anadolu University, School of Civil Aviation
Eskisehir, Turkey mcavcar@anadolu.edu.tr The simple momentum theory provides an initial idea regarding the performance of a propeller but not sufficient information for the detailed design. Detailed information can be obtained through analysis of the forces acting on a blade element like it is a wing section. The forces acting on a small section of the blade are determined and then integrated over the propeller radius in order to predict the thrust, torque and power characteristics of the propeller.
R
ω
c r dr dT w
Ve
VR
V
dL
αi β α φ dF
ωr dD Figure 1. Propeller blade element with velocity and force diagram.
© Prof. Dr. Mustafa Cavcar, 2004.
1
A differential blade element of chord c and width dr , located at a radius r from the propeller axis, is shown in Figure 1. The element is shown acting under the influence of the rotational velocity, ωr , forward velocity of the airplane, V , and the induced velocity, w . Vector sum of these velocities produce
Ve = ω r + V + w
(1)
The section has a geometric pitch angle of its zero lift line of β . If it is assumed that V and ωr are known, then calculation of the induced velocity w is desired to find α i , and consequently the section angle of attack α . Knowing α , and the section type, C l and C d can be calculated, then the differential lift and drag of the section will follow. However, w depends on dL which in turn depends on w .
Thus the problem is closely related to the finite wing problem but is more complicated because of the helicoidal geometry of the propeller [1].
Combined Momentum – Blade Element Theory
A starting approximate value of w can be obtained by application of the momentum theory principles to an annulus of width dr and radius r (Figure 2).
In this case
dT = ρdA(V + w cos φ )2w cos φ
(2)
dr r R
Figure 2. Annulus to
References: [1] McCormick, B.W., Aerodynamics of V/STOL Flight, Academic Press, Orlando, 1967. © Prof. Dr. Mustafa Cavcar, 2004. 5