MC-F-002.Rotor Aerodynamics. Momentum Theory

Miguel A. Barcala Montejano Ángel A. Rodríguez Sevillano 1 HELICOPTERS

Professors: Miguel A. Barcala Montejano Ángel A. Rodríguez Sevillano ROTOR AERODYNAMICS Momentum Theory Vertial Climb Inital thoughts: Vertical climb flight is the easiest flight condition. The velocities in the rotor plane are symmetrical about the rotation axis. The aerodynamic forces on the blades are constant regardless of their angular position. The plane formed by the blade rotor tips is perpendicular to the drive shaft.

MT. VERTICAL CLIMBING FLIGHT Miguel A. Barcala Montejano Ángel A. Rodríguez Sevillano 4 Inital thoughts: Vertical climbing flight is the easiest flight condition. There are different theories for studying rotor aerodynamics. The momentum theory. The blade element theory. The vortex theory.

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 5 ROTOR AERODYNAMICS VERTICAL CLIMBING FLIGHT MOMENTUM THEORY

Thrust and Power Calculations. Hover flight. Velocity and Power ratios. Thurst and Power coefficients. Dimensionless expressions.

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 6 MOMENTUM THEORY INITIAL ASSUMPTIONS

High values of Re number flow. Replace the original rotor with blades that rotate for a totally porous disc of the same radius (R) as the rotor replaced . We assume the affected flow through the disc is defined by the streamtube. The fluid flow in the streamtube is considered to be unidimensional, steady and incompressible. The effects of the rotation of the slipstream and loses in the blade tips, are negleted.

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 7 MOMENTUM THEORY MATHEMATICAL MODEL

Vv The velocity of the upstream rotor fluid is the vertical velocity of the

rotor. (Vv). The fluid velocity in this section of the T disc is the climbing velocity of the rotor plus the induced velocity by the lifting disc. (Vv+ vi ). Vv + vi The velocity of the downstream rotor fluid is the vertical velocity of the rotor plus the induced velocity in the disc plane affected by a factor of A. (Vv+Avi ). Vv + Avi Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 8 MOMENTUM THEORY THRUST AND POWER CALCULATIONS V v r r r r F ex - ∫ AP ndA= G ( V s -V e )

T ρ ρπ 2 ( + ) G = VA = R V v vi ρ π 2 Vv + vi T = ( R )(V v+vi )Avi

Vv + Avi Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 9 MOMENTUM THEORY THRUST AND POWER CALCULATIONS

Vv Calculation of Parameter “A” Pa T = (P′- P)π R2

1 ρ 2 1 ρ 2 Pa + 2 V v= P+ 2 (V v+vi ) P′+ 1 ρ( + )2= Pa+ 1 ρ( + A )2 Vv + vi 2 V v vi 2 V v vi

V + Av 1 ρ π 2 v i T = 2 ( R )(2V v + Avi ) Avi P a A = 2

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 10 MOMENTUM THEORY THRUST AND POWER CALCULATIONS

THRUST ρ π 2 T = 2 ( R )vi (V v +vi ) POWER Pi = T(V v +vi )

ρ 1 P - Pa = - vi (V v + 2 vi ) ′ ρ 3 P - Pa = vi (V v + 2 vi )

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 11 MOMENTUM THEORY HOVER FLIGHT (THRUST AND POWER)

Flight condition VV=0

ρ π 2 2 T = 2 ( R )vio ρ π 2 3 Pio = 2 ( R )vio

T W v = = io 2ρ(πR 2 ) 2ρ(πR 2 )

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 12 MOMENTUM THEORY VELOCITY RATIO + W V v  vi  = 2 = ()+  V i   = 1 2 vio vi V v vi    2ρ()π R  vio  vio 

  2   vi 1  V v V v  =   +4 -   io 2   io   io  2 v  v v   vi   vi V v    +   -1= 0  vio   vio  vio   2  V v +vi V v  V v  = 1    +4 +  2      vio   vio   vio 

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 13 MOMENTUM THEORY POWER RATIO

 2  P T(V + v ) +     i = V i = V v vi 1  V v  V v  = 2   + 4 +  P Tv   io io vio   vio   vio  V + v 1 Pi = V i =  vi  Pio vio    vio 

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 14 MOMENTUM THEORY Power Ratio vs Vertical Velocity Ratio Induced Velocity Ratio vs Vertical Velocity Ratio. 4,5

3,5

3 P i/P io 2,5 V i/V io

1,5

0,5

0 0 1 2 3 4 V v/V io

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 15 MOMENTUM THEORY DIMENSIONLESS COEFFICIENTS F T = Thrust coefficient (Dimensionless) C N = CT 2 2 ρSV 2 ρ()π R ()ΩR

W Pi Power coefficient (Dimensionless) = CW = C Pi 3 ρSV 3 ρ(π R2 )( ΩR )

ρ (π 2) 2 2 2 2 R vio  vio  vi V v +vi  CT = = 2   C = 2   ρ ()πRSUP2 ()ΩR 2  ΩR  Pi ΩR  ΩR 

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 16 MOMENTUM THEORY DIMENSIONLESS EXPRESSIONS

 2   2  ΩR  ΩR   ΩR  vi 1  V v   V v  vi vi • 1   V v   V v    = = 2 +4 - = 2CT +  -   Ω   Ω   Ω  vio R vio  R vio   R vio  ΩR 2  ΩR ΩR        

 2  C Pi  V v   Vv  Pi C Pi V v +vio 1 1   = = = 2 2CT +  +  CT Ω Ω Pio C vio C   R   R  Pio Pio 2  

 2  C C C CP     Pi Pi Pio i 1  V v V v  = = 2CT +  +  CT 2 Ω Ω CT C CT   R   R  Pio  

Miguel A. Barcala Montejano MT. VERTICAL CLIMBING FLIGHT Ángel A. Rodríguez Sevillano 17