HELICOPTER AERODYNAMICS

BASIC DEFINITIONS Shaft Axis The shaft axis is the axis of the main rotor shaft and about which the blades are permitted to rotate . Axis of Rotation An imaginary line which passes thru a point about which a body rotates & which is normal to the plane of rotation. The axis of rotation is through the head of the main rotor shaft about which the blades actually rotate. In steady flight conditions , the axis of rotation coincides with the shaft axis; however, this will not always be the case since the rotor disc is permitted to tilt under certain conditions of flight . Plane of Rotation A plane formed by the average tip path of the rotor blades & which is normal to the axis of rotation. Tip path plane The tip path plane is the path described by blade tips during rotation, and is at right angles to the axis of rotation. The area contained within this path is referred to as the rotor disc .

Advancing blade : The blade of the rotor moving into the free stream air is known as the advancing blade. Retreating blade : The blade of the rotor moving away from the free stream air is known as the retreating blade. Pitch angle or pitch : The acute angle between the chord of a rotor blade & a reference surface on the rotor hub. This reference surface is normal to the drive shaft.

IN HPTRS ROTATION OF THE BLADES ABOUT THE ROTOR AXIS PRODUCES THE RELATIVE MOTION WITH AIR & THUS PRODUCE LIFT EVEN WITHOUT MOTION OF HPTR IN ANY DIRECTION. FLIGHT CONDITION WHERE IN THERE IS NO RELATIVE MOTION OF THE HPTR WITH RESPECT TO AIR IS CALLED HOVER. TWO THEORIES IN HPTR DYNAMICS FOR LIFT GENERATION NAMELY MOMENTUM THEORY & BLADE ELEMENT THEORY.

MOMENTUM OR ACTUATOR DISC THEORY BASIC CHRACTERISTICS OF THE ROTOR IN THE HOVER & AXIAL (VERTICAL) FLIGHT CAN BE OBTAINED FROM THE ACTUATOR DISC THEORY WHICH IS BASED ON MOMENTUM APPROACH. THE ASSUMPTIONS MADE ARE: ROTOR HAS INFINITE NO. OF BLADES.(UNIFORM SYSTEM) ROTOR IS MODELLED BY A CONSTANT PRESSURE DIFFERENCE ACROSS THE DISC. VELOCITY OF AIR DOWNWARD THRU THE DISC (DOWNWASH) IS CONSTANT ACROSS THE DISC. VERTICAL VELOCITY IS CONTINOUS THRU THE ROTOR DISC.

THERE IS NO SWIRL IN THE WAKE (NO ROTATIONAL VELOCITY IS IMPARTED TO THE AIR PASSING THRU THE DISC , ALL FLOW VELOCITIES ARE AXIAL). THE AIRFLOW IS DIVIDED INTO TWO DISTINCT REGIONS, NAMELY THAT WHICH PASSES THRU THE ROTOR DISC & THAT WHICH IS EXTERNAL TO THE DISC.THE DIVISION OF THESE TWO TYPES OF FLOW IS A STREAMTUBE WHICH PASSES THRU THE PERIMETER OF THE DISC. THE WAKE FORMS A UNIFORM JET.

Let A= disc area A2=wake area P =Atmospheric static pressure r=Atmospheric density Pu=Atmospheric pressure above the rotor disc Pl= Atmospheric pressure below the rotor disc Vi= Induced velocity induced at the rotor V2= Velocity in the wake far downstream

The flow in the wake is assumed to come from air ahead of the rotor at rest with free stream pressure & to return to pressure far downstream of the rotor. The rotor thrust is given by T= A ( Pl – Pu) (1) Applying Bernoulli’s equation above the rotor gives, P1= Pu + ½ rVi 2 (2) Applying Bernoulli’s equation below the rotor gives, P1+ ½ rV22= Pl + ½ rVi2 (3) Substituting ,we obtain, Pu + ½ rVi2 + ½ rV22= Pl + ½ rVi2 Pl – Pu = ½ rV22 From (1), we have, T = ½ rV22 A (4)

Thrust developed at the rotor disc imparts a downward velocity Vi to the air at the disc & is called the downwash velocity or induced velocity. Thrust at the rotor disc ( T) is equal to the mass of air passing through the disc in unit time into total change in velocity. Mflow = ρ A Vi where (5) A= Disc Area , ρ = Air Density & Vi = induced velocity. Rate of Change of Momentum = Mflow ( V2- 0 ) where V2 is the velocity far down stream of the disc. Thrust ( T )= ρ A Vi V2 (6) From eqns (4) & (6), we have, ½ ρAV22 = ρAViV2 V2 = 2Vi Substituting to eqn (4) & rearranging gives, T = ½ rA (4Vi2)= 2rAVi2 Vi = T 2 ρ A

The value of Vi obtained is called the ideal induced velocity & because it is uniform over the rotor disc, is the minimum value for the given rotor thrust. This velocity gives the power required to generate the rotor thrust. In hover, the greatest source of power is that required to maintain the rotor thrust T working against the induced velocity Vi. This is called thrust induced power & is given by, Pi = T Vi

INFERENCES Rotor Velocity Depends on the Thrust & Disc Area. Downstream Velocity is twice the Induced velocity.

Power Required to Produce Thrust T is the work done by the thrust due to the relative velocity v at the Disc and is P = T v. This power is called the induced Power and also corresponding to the minimum power required to produce the thrust. Power Requirement is based on Idealized Conditions & in reality the following exist. Induced Flow is uniform. There will be Additional Power Required for Blade Tip Effect, Rotational Slip Stream Flow & Profile Drag of the Blades. In General Total Unaccounted Loss is about 40% of the total Power Required.

ROTOR FIGURE OF MERIT Efficiency of Lifting Rotor System can be estimated by comparing the actual power required to produce a given thrust with the minimum possible power required to produce that thrust. Rotor Hovering Efficiency is called the Figure of Merit and is given by FOM= Minimum Possible Power Required to Hover Actual Power Required to Hover = T Vi / Pi = 1/√2 * T/Pi *√ T/ ρ A M generally varies between 0.6 to 0.7

LIMITATIONS OF MOMENTUM THEORY Flow analysis over the rotor blade was difficult to make. Provides only a good estimate of Induced Flow. Provides a means to Evaluate Rotor System Efficiency. Neither Provides Detailed Information about Flow Field or Exact Dependence of the Rotor Thrust on Rotor Speed or Collective Pitch setting.

BLADE ELEMENT THEORY

BLADE ELEMENT THEORY A Means to Remove Limitations of Momentum Theory. Considers Finite No of Blades and the Power loss due to Profile drag and to some extent the Non- uniform Flow. For Calculation of Thrust & Power of rotor , first forces Acting on an Element of Blade ( Airfoil) is Considered.

Tangential Velocity of the Airfoil is due to the Rotation of the Rotor and is Ωr The vertical Component corresponds to the induced Velocity (u). θ is the Blade Pitch angle measured from the rotor disc plane of rotation to the airfoil chord line. Net Angle of Attack of the airfoil is α = θ – tan-- ( u/ Ωr) for angles u ‹ ‹ Ωr where Ω the rotational velocity of rotor in rad / sec. φ = tan-- ( u/ Ωr) = u/ Ωr and α = θ - u/ Ωr = θ - φ Thrust Velocity V R = u2 + (Ωr)2 Lift & Drag Equations are given by dL = ½ ρ V 2R a α (b c dr) & dD = ½ ρ V 2R CD (b c dr) Where a the lift curve slope, b the No of Blades, bcdr the area for b no of blades having a thickness of c

Assuming Resultant Velocity V R = u2 + (Ωr)2 = Ωr Elemental Thrust dT act perpendicular to the plane of rotation and for small angles dT = dL = ½ ρ V 2R a α (b c dr) Assuming Resultant Velocity V R = u2 + (Ωr)2 = Ωr Drag of the Blade Consists of two Parts Profile Drag =½ ρ V 2R CD (b c dr) & the Induced drag due to Lift component and given by dL(φ ) The Torque is the sum of the above two Drag equations i e dQ = ½ bρ(Ωr)2c[CD + φa α] rdr dT & dQ equations give the thrust & torque for the blade element . For a Given pitch θ the only unknown parameters in the above equations is the Induced velocity u. This can be obtained by combining Blade Element & Momentum Theories.

COMBINED BLADE ELEMENT & MOMENTUM THEORY In this theory, an annular ring of rotor with radius r & width dr is considered. From blade element theory for b number of blades, the expression for thrust in the annular ring is given by, dT= ½ brac( Ωr)2(

COMBINATION OF MOMENTUM & BLADE ELEMENT THEORIES Thrust = CT* ρπ r 2*( Ωr)2 or Thrust Coefft = CT = T/ ρπ r 2* ( Ωr)2 Torque Coefft = CQ = Q/ ρ A * ( Ωr)2 Power Coefft = Cp = P/ ρ A * ( Ωr)3 With Respect To Pitch Angle & No of Blades Torque = ½ b A c ( Ωr)3* (θ/3 – λ/3 ) Thrust T= ½ ρ b Ω2 c A r3 (θ/3 – λ/2) Also Thrust = CT* ρπ r 2*( Ωr)2 or CT = ½ * bc/π r *a (θ/3 – λ/3 ) =1/2* σ a (θ/3 – λ/2) where σ = bc/π r = bcr /π r2 = Actual Blade area Disc area = Rotor Solidity Factor

λ = Inflow ratio due to induced Velocity = v/ Ωr µ = Advance ratio for AC Speed = V/ Ωr σ = Rotor Solidity factor = bc/ π r =Blade Area / Rotor Area θ = Pitch Angle & a - lift curve slope Lesser the no of Blades higher the losses and vice versa. However beyond a particular no blades configuration due to blade interference effect the losses tends to increase. Generally the ideal combination is four to six bladed rotor configurations.

Profile power = Cd /8 * σ *1/2* ρA (Ωr)3 FOM = 0.707 3/2 * CT3/2 / CP Figure of Merit (FOM) = Min Power Required to the actual power Reqd = Induced Power Induced Power + Profile power Induced power = T * v Profile power = Cd /8 * σ *1/2* ρA (Ωr)3 FOM = 0.707 3/2 * CT3/2 / CP Total rotor thrust in Hover = ρ Cl av * (RPM) 2* σ * R2 * A Lbs 230,000

FORCES ON A ROTOR BLADE The Various forces acting on a rotor blade are: Weight of the blade acting downwards Aerodynamic lift Centrifugal force in the plane of rotation Inertia Forces