INTRODUCTION TO HELICOPTERS

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INTRODUCTION TO HELICOPTERS

• COURSE PRESENTATION
• Divahar J
• ME-AEROSPACE
• Indian Institute of Science, Bangalore,
• India

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Organization

• Extended Momentum theory
• Combined Blade element theory
• Tip effects
• Root cutout
• Inclusion of root Cutout and tip effects

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Extended Momentum Theory

• Includes the variation of induced velocity
• Key point ? the Conservation Laws
• Not just GLOBAL conservation

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Assumptions

• Disk is of zero thickness
• There is a well defined smooth slipstream
• Flow is incompressible.
• Flow is steady, inviscid, irrotational.
• Flow is one-dimensional, and NOT uniform through
  the rotor disk, and in the far wake.
• There is no swirl in the wake.

Assumptions
Extended Momentum Theory
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Conservation Laws
Conservation Laws
Extended Momentum Theory
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Model
Model
Extended Momentum Theory
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Model
S0-S1

• V Decreases
• ? p Increases
• Pressure increment ?p is Balanced

S0
Model
A
Extended Momentum Theory
S1
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Formulation

• Mass conservation
• Momentum Conservation

Formulation
Extended Momentum Theory
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Formulation

• Energy conservation

Formulation
Extended Momentum Theory
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Formulattion
KE added to flow
Work Done to move the air through Disk
Formulation
Extended Momentum Theory
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Optimum Induced Velocity Distribution

• Typical Optimization Problem
• Minimize
• Subject to
• Lagrangian fn

Optimum Induced Velocity Distribution
Extended Momentum Theory
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Optimum Induced Velocity Distribution Contd
Optimum Induced Velocity Distribution
Extended Momentum Theory
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Bernoullis Equation
Bernoullis Equation
Extended Momentum Theory
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Bernoullis Equation

• Through Bernoullis eqn...
• From station 0 ? 1
• From Station 2 ? 3

Bernoullis Equation
Extended Momentum Theory
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Uniformity

• W ?uniform, Disk loading ?uniform
• Assumptions of uniform ?p and w is validated but
  EMT does not say anything about distribution of v

Uniformity
Analogous to Trefftz plane analysis of fixed
wing, which shows that minimum drag is obtained
with uniform downwash in the far wake and
elliptical loading, but tells nothing about the
induced angle of attack at the wing
Extended Momentum Theory
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Conservation Laws in mean values

• For uniform ?p and w,
• Mass conservation
• Momentum Conservation
• Energy Conservation

Conservation Laws in mean values
Extended Momentum Theory
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Momentum Theory in Differential Form
Momentum Theory in Differential Form
Extended Momentum Theory
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Discussion on EMT

• No justification for this differential from.
• Assumes no mutual interference of the disk
  elements
• Key to the result is the assumption that v w/2
  is valid for individual streamlines
• Usefulness diff form of momentum theory can be
  applied to rotors with non uniform loading and
  inflow.

Discussion on EMT
Extended Momentum Theory
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Combined Blade Element Theory

• Blade element theory with the inclusion of the
  variation of inflow
• Inflow is given by differential momentum theory
• Assumption induced velocity at r is only due to
  elemental thrust dT at that station

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Formulation

• From Blade element theory,

Formulation
Combined Blade Element Theory
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Formulation

• Combining MT and BET Result for CT

Formulation
Combined Blade Element Theory
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Formulation

• Solving for ?,
• NOTE For const chord, uniform inflow is achieved
  if ?rconst

Formulation
Combined Blade Element Theory
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Tip Loses

• Lifting line theory is not strictly valid near
  tip
• Gives lift even at the tip
• Wing tip vortices spoil the lift

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Tip Loses

• Lift at the tip drops even faster than the fixed
  wings !!
• Leads to overestimation of lift
• Effect of tip is modeled rigorously by lifting
  surface or by vortex methos

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Accounting for tip loss
Lift
R
BR
B0.96 0.98
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Root Cutout
rRR
R
rR0.1 0.3
Dynamic pressure is low at root ? correction is
small
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Inclusion of Root Cutout and tip effects

• To include these effects,