© L. Sankar 1
Helicopter Aerodynamics and
Performance
Thrust
AeroelasticResponse 0 270 180 90 Dynamic Stall on Retreating Blade Blade-Tip Vortex interactions Unsteady Aerodynamics Transonic Flow on Advancing BladeMain Rotor / Tail Rotor / Fuselage Flow Interference
V
Noise WavesShock
Tip Vortices
© L. Sankar 3
A systematic Approach is
necessary
•
A variety of tools are needed to understand, and predict these phenomena.
•
Tools needed include
– Simple back-of-the envelop tools for sizing helicopters, selecting engines, laying
out configuration, and predicting performance
– Spreadsheets and MATLAB scripts for mapping out the blade loads over the
entire rotor disk
– High end CFD tools for modeling
• Airfoil and rotor aerodynamics and design • Rotor-airframe interactions
• Aeroacoustic analyses
– Elastic and multi-body dynamics modeling tools
– Trim analyses, Flight Simulation software
•
In this work, we will cover most of the tools that we need, except for elastic
analyses, multi-body dynamics analyses, and flight simulation software.
•
We will cover both the basics, and the applications.
•
We will assume familiarity with classical low speed and high speed
aerodynamics, but nothing more.
Plan for the Course
• PowerPoint presentations, interspersed with
numerical calculations and spreadsheet
applications.
• Part 1: Hover Prediction Methods
• Part 2: Forward Flight Prediction Methods
• Part 3: Helicopter Performance Prediction
Methods
• Part 4: Introduction to Comprehensive Codes
and CFD tools
• Part 5: Completion of CFD tools, Discussion of
Advanced Concepts
© L. Sankar 5
Text Books
• Wayne Johnson: Helicopter Theory, Dover
Publications,ISBN-0-486-68230-7
• References:
– Gordon Leishman: Principles of Helicopter
Aerodynamics, Cambridge Aerospace Series, ISBN
0-521-66060-2
– Prouty: Helicopter Performance, Stability, and
Control, Prindle, Weber & Schmidt, ISBN
0-534-06360-8
– Gessow and Myers
– Stepniewski & Keys
Grading
• 5 Homework Assignments (each worth 5%).
• Two quizzes (each worth 25%)
• One final examination (worth 25%)
• All quizzes and exams will be take-home type.
They will require use of an Excel spreadsheet
program, or optionally short computer programs
you will write.
© L. Sankar 7
Instructor Info.
• Lakshmi N. Sankar
• School of Aerospace Engineering, Georgia
Tech, Atlanta, GA 30332-0150, USA.
• Web site:
www.ae.gatech.
edu/~lsankar/AE6070.Fall2002
Earliest Helicopter..
Chinese Top
© L. Sankar 9
Leonardo da Vinci
(1480? 1493?)
Human Powered Flight?
HP
6.7
5.33/0.8
Merit
of
re
Power/Figu
Ideal
Power
Actual
33
.
5
A
2
W
W
Power
Ideal
slugs.
0.00238
Desnity
sq.ft
100
Area
Rotor
6ft
~
Radius
Rotor
160lbf
Weight
HP
© L. Sankar 11
D’AmeCourt (1863)
Paul Cornu (1907)
© L. Sankar 13
De La Cierva
Cierva introduced hinges at the root
that allowed blades to freely flap
Hinges
Only the lifts were transferred to the fuselage,
not unwanted moments.
In later models, lead-lag hinges were also used to
Alleviate root stresses from Coriolis forces
© L. Sankar 15
Igor Sikorsky
Started work in 1907, Patent in 1935
Used tail rotor to counter-act the reactive torque exerted by
the rotor on the vehicle.
© L. Sankar 17
Ways of countering the
Reactive Torque
© L. Sankar 19
Coaxial rotors
Kamov KA-52
© L. Sankar 21
© L. Sankar 23
Helicopters tend to grow in size..
16,027 lb (7270 kg) Lot
1 Weight
15,075 lb (6838 kg)
11,800 pounds Empty
Primary Mission Gross
Weight
17.15 ft (5.227 m)
17.15 ft (5.227 m)
Wing Span
13.30 ft (4.05 m)
15.24 ft (4.64 m)
Height
58.17 ft (17.73 m)
58.17 ft (17.73 m)
Length
AH-64D
AH-64A
© L. Sankar Helicopter Aerodynamics
25
147 kt (273 kph)
[Sea Level Standard Day]
149 kt (276 kph)
[Hot Day 2000 ft 70 F (21
C)]
150 kt (279 kph)
[Sea Level Standard Day]
153 kt (284 kph)
[Hot Day 2000 ft 70 F (21 C)]
Cruise Speed (MCP)
147 kt (273 kph)
[Sea Level Standard Day]
149 kt (276 kph)
[Hot Day 2000 ft 70 F (21
C)]
150 kt (279 kph)
[Sea Level Standard Day]
153 kt (284 kph)
[Hot Day 2000 ft 70 F (21 C)]
Maximum Level Flight
Speed
2,635 fpm (803 mpm)
[Sea Level Standard Day]
2,600 fpm (793 mpm)
[Hot Day 2000 ft 70 F (21
C)]
2,915 fpm (889 mpm)
[Sea Level Standard Day]
2,890 fpm (881 mpm)
[Hot Day 2000 ft 70 F (21 C)]
Maximum Rate of Climb
(IRP)
1,775 fpm (541 mpm)
[Sea Level Standard Day]
1,595 fpm (486 mpm)
[Hot Day 2000 ft 70 F (21
C)]
2,175 fpm (663 mpm)
[Sea Level Standard Day]
2,050 fpm (625 mpm)
[Hot Day 2000 ft 70 F (21 C)]
Vertical Rate of Climb
(MRP)
10,520 ft (3206 m)
[Standard Day]
9,050 ft (2759 m)
[Hot Day ISA + 15 C]
12,685 ft (3866 m)
[Sea Level Standard Day]
11,215 ft (3418 m)
[Hot Day 2000 ft 70 F (21 C)]
Hover Out-of-Ground Effect
(MRP)
14,650 ft (4465 m)
[Standard Day]
13,350 ft (4068 m)
[Hot Day ISA + 15 C]
15,895 ft (4845 m)
[Standard Day]
14,845 ft (4525 m)
[Hot Day ISA + 15C]
Hover In-Ground Effect
(MRP)
16,027 lb (7270 kg) Lot 1
Weight
15,075 lb (6838 kg)
11,800 pounds Empty
Primary Mission Gross
Weight
17.15 ft (5.227 m)
17.15 ft (5.227 m)
Wing Span
13.30 ft (4.05 m)
15.24 ft (4.64 m)
Height
58.17 ft (17.73 m)
58.17 ft (17.73 m)
Length
AH-64D
AH-64A
Power Plant Limitations
• Helicopters use turbo shaft engines.
• Power available is the principal factor.
• An adequate power plant is important for
carrying out the missions.
• We will look at ways of estimating power
requirements for a variety of operating
conditions.
© L. Sankar 27
High Speed
Forward Flight Limitations
• As the forward speed increases, advancing side
experiences shock effects, retreating side stalls.
This limits thrust available.
• Vibrations go up, because of the increased
dynamic pressure, and increased harmonic
content.
• Shock Noise goes up.
• Fuselage drag increases, and parasite power
consumption goes up as V3.
• We need to understand and accurately predict
the air loads in high speed forward flight.
Concluding Remarks
• Helicopter aerodynamics is an interesting area.
• There are a lot of problems, but there are also
opportunities for innovation.
• This course is intended to be a starting point for
engineers and researchers to explore efficient
(low power), safer, comfortable (low vibration),
environmentally friendly (low noise) helicopters.
© L. Sankar 29
Hover Performance
Prediction Methods
Background
• Developed for marine propellers by
Rankine (1865), Froude (1885).
• Extended to include swirl in the slipstream
by Betz (1920)
• This theory can predict performance in
hover, and climb.
• We will look at the general case of climb,
and extract hover as a special situation
with zero climb velocity.
© L. Sankar 31
Assumptions
• Momentum theory concerns itself with the
global balance of mass, momentum, and
energy.
• It does not concern itself with details of the
flow around the blades.
• It gives a good representation of what is
happening far away from the rotor.
• This theory makes a number of simplifying
assumptions.
Assumptions (Continued)
• Rotor is modeled as an actuator disk
which adds momentum and energy to the
flow.
• Flow is incompressible.
• Flow is steady, inviscid, irrotational.
• Flow is one-dimensional, and uniform
through the rotor disk, and in the far wake.
• There is no swirl in the wake.
© L. Sankar 33
Control Volume is a Cylinder
V
Disk area A
Total area S
Station1
2
3
4
V+v2
V+v3
V+v4
Conservation of Mass
4
4
4
1
)
(
A
-S
V
bottom
he
through t
Outflow
m
side
he
through t
Inflow
VS
top
he
through t
Inflow
A
v
V
© L. Sankar 35
Conservation of Mass through the
Rotor Disk
Flow through the rotor disk =
4
4
3
2
v
v
v
V
A
V
A
V
A
m
Thus v2=v3=v
There is no velocity jump across the rotor disk
Global Conservation of Momentum
4 4 4 4 4 2 4 2 4 4 4 1 2v
v
)
v
(
A
T
in
Rate
Momentum
-out
rate
Momentum
T
,
Thrust
.
boundaries
field
far
the
all
on
c
atmospheri
is
Pressure
v
A
-S
bottom
through
outflow
Momentum
v
A
V
m
side
he
through t
inflow
Momentum
V
op
through t
inflow
Momentum
m
V
A
V
V
V
S
Mass flow rate through the rotor disk times
Excess velocity between stations 1 and 4
© L. Sankar 37
Conservation of Momentum at the
Rotor Disk
V+v
V+v
p2
p
3
Due to conservation of mass across the
Rotor disk, there is no velocity jump.
Momentum inflow rate = Momentum outflow rate
Conservation of Energy
Consider a particle that traverses from
Station 1 to station 4
We can apply Bernoulli equation between
Stations 1 and 2, and between stations 3
and 4.
Recall assumptions that the flow is
steady, irrotational, inviscid.
1
2
3
4
V+v
V+v4
4 4 2 3 2 4 2 3 2 2 2v
2
v
v
2
1
v
2
1
2
1
v
2
1
V
p
p
V
p
V
p
V
p
V
p
© L. Sankar 39
4
4
2
3
4
4
2
3
v
2
v
v
2
v
#38,
slide
previous
the
From
V
A
p
p
A
T
V
p
p
From an earlier slide # 36, Thrust equals mass flow rate
through the rotor disk times excess velocity
between stations 1 and 4
v
v
4
A
V
T
Induced Velocities
V
V+v
V+2v
The excess velocity in the
Far wake is twice the induced
Velocity at the rotor disk.
To accommodate this excess
Velocity, the stream tube
© L. Sankar 41
Induced Velocity at the Rotor Disk
Now we can compute the induced velocity at the
rotor disk in terms of thrust T.
T = Mass flow
rate through the
rotor disk *
(Excess velocity
between 1 and
4).
T = 2 A (V+v) v
A
T
V
2
2
2
V
-v
2
There are two solutions. The – sign
Corresponds to a wind turbine, where energy
Is removed from the flow. v is negative.
The + sign corresponds to a rotor or
Propeller where energy is added to the flow.
In this case, v is positive.
Induced velocity at the rotor disk
A
T
A
T
V
2
v
0
V
velocity
climb
Hover,
In
2
2
2
V
-v
2
© L. Sankar 43
Ideal Power Consumed by the Rotor
A
T
V
V
T
V
T
V
m
m
m
P
2
2
2
v
v
v
2
V
2
1
2v
V
2
1
in
flow
Energy
-out
flow
Energy
2
2
2
In hover, ideal power
A
T
T
2
Summary
• According to momentum theory, the downwash
in the far wake is twice the induced velocity at
the rotor disk.
• Momentum theory gives an expression for
induced velocity at the rotor disk.
• It also gives an expression for ideal power
consumed by a rotor of specified dimensions.
• Actual power will be higher, because momentum
theory neglected many sources of losses-
viscous effects, compressibility (shocks), tip
losses, swirl, non-uniform flows, etc.
© L. Sankar 45
Figure of Merit
• Figure of merit is
defined as the ratio of
ideal power for a rotor
in
hover
obtained
from momentum
theory and the actual
power consumed by
the rotor.
• For most rotors, it is
between 0.7 and 0.8.
P
T
T
C
C
C
T
FM
2
P
v
Hover
in
Power
Actual
Hover
in
Power
Ideal
Some Observations on
Figure of Merit
• Because a helicopter spends considerable
portions of time in hover, designers
attempt to optimize the rotor for hover
(FM~0.8).
• We will discuss how to do this later.
• A rotor with a lower figure of merit (FM~0.
6) is not necessarily a bad rotor.
• It has simply been optimized for other
© L. Sankar 47
Example #1
• A tilt-rotor aircraft has a gross weight of
60,500 lb. (27500 kg).
• The rotor diameter is 38 feet (11.58 m).
• Assume FM=0.75, Transmission
losses=5%
• Compute power needed to hover at sea
level on a hot day.
Example #1 (Continued)
HP
11528
1.05
*
10980
shaft
the
to
engine
by the
supplied
Power
loss
ion
transmiss
5%
is
There
HP
10980
power
actual
total
rotors,
two
For the
HP
5490
power
Actual
4117/0.75
Power/FM
ideal
Power
Actual
HP
4117
Power
Ideal
ft/sec
lb
74.86
x
30250
Tv
Power
Ideal
!
ft/sec
150
far wake
in the
Downwash
ft/sec
86
.
74
v
A
2
T
v
velocity,
Induced
lbf
30250
T
rotors.
two
are
There
feet
c
slugs/cubi
0.00238
Density
feet
square
12
.
1134
19
A
Area
Disk
2
A
© L. Sankar 49
Alternate scenarios
• What happens on a hot day, and/or high
altitude?
– Induced velocity is higher.
– Power consumption is higher
• What happens if the rotor disk area A is
smaller?
– Induced velocity and power are higher.
• There are practical limits to how large A
can be.
Disk Loading
• The ratio T/A is called disk loading.
• The higher the disk loading, the higher the
induced velocity, and the higher the power.
• For helicopters, disk loading is between 5 and
10 lb/ft2 (24 to 48 kg/m2).
• Tilt-rotor vehicles tend to have a disk loading of
20 to 40 lbf/ft2. They are less efficient in hover.
• VTOL aircraft have very small fans, and have
© L. Sankar 51
Power Loading
• The ratio of thrust to power T/P is called
the Power Loading.
• Pure helicopters have a power loading
between 6 to 10 lb/HP.
• Tilt-rotors have lower power loading – 2 to
6 lb/HP.
• VTOL vehicles have the lowest power
loading – less than 2 lb/HP.
Non-Dimensional Forms
C
2 Q 3 P 2 TC
Q
P
Torque
locity x
Angualr ve
Power
hover,
In
R
AR
Q
t
Coefficien
Torque
C
R
A
P
t
Coefficien
Power
C
R
A
T
t
Coefficien
Thrust
C
form.
l
dimensiona
-non
in
expressed
usually
are
Power
and
Torque,
Thrust,
© L. Sankar 53
Non-dimensional forms..
P
T
T
i
C
C
C
T
FM
2
P
v
Hover
in
Power
Actual
Hover
in
Power
Ideal
2
C
A
2
T
R
1
R
v
inflow
Induced
T
Tip Losses
R
A portion of the rotor near the
Tip does not produce much lift
Due to leakage of air from
The bottom of the disk to the top.
One can crudely account for it by
Using a smaller, modified radius
BR, where
b
C
B
1
2
T
BR
B = Number of blades.
© L. Sankar 55
Power Consumption in Hover
Including Tip Losses..
2
1
1
T
T
P
C
C
B
FM
C
Hover Performance
Prediction Methods
© L. Sankar 57
Preliminary Remarks
• Momentum theory gives rapid,
back-of-the-envelope estimates of Power.
• This approach is sufficient to size a rotor
(i.e. select the disk area) for a given power
plant (engine), and a given gross weight.
• This approach is not adequate for
designing the rotor.
Drawbacks of Momentum Theory
• It does not take into account
– Number of blades
– Airfoil characteristics (lift, drag, angle of zero
lift)
– Blade planform (taper, sweep, root cut-out)
– Blade twist distribution
© L. Sankar 59
Blade Element Theory
• Blade Element Theory rectifies many of these
drawbacks. First proposed by Drzwiecki in 1892.
• It is a “strip” theory. The blade is divided into a
number of strips, of width r.
• The lift generated by that strip, and the power
consumed by that strip, are computed using 2-D
airfoil aerodynamics.
• The contributions from all the strips from all the
blades are summed up to get total thrust, and
total power.
Typical Blade Section (Strip)
R
dr
r
Tip
Out
Cut
Tip
Out
Cut
dP
b
P
dT
b
T
dT
Root Cut-out
© L. Sankar 61
Typical Airfoil Section
r
V
v
arctan
r
V+v
Line of Zero Lift
Sectional Forces
Once the effective angle of attack is known, we can look-up
the lift and drag coefficients for the airfoil section at that strip.
We can subsequently compute sectional lift and drag forces
per foot (or meter) of span.
T
P
d
l
P
T
cC
U
U
D
cC
U
U
L
2
1
2
1
2
2
2
2
These forces will be normal to and along
the total velocity vector.
UT=r
UP=V+v
© L. Sankar 63
Rotation of Forces
r
V+v
L
D
T
Fx
X x T l d P T x d l P TrdF
dF
U
dP
dr
C
C
c
U
U
dr
L
D
dF
dr
C
C
c
U
U
dr
D
L
dT
sin
cos
2
1
sin
cos
sin
cos
2
1
sin
cos
2 2 2 2Approximate Expressions
• The integration (or summation of forces)
can only be done numerically.
• A spreadsheet may be designed. A
sample spreadsheet is being provided as
part of the course notes.
• In some simple cases, analytical
expressions may be obtained.
© L. Sankar 65
Closed Form Integrations
•
The chord c is constant. Simple linear twist.
•
The inflow velocity v and climb velocity V are small.
Thus, << 1.
•
We can approximate cos( ) by unity, and approximate
sin( ) by ( ).
•
The lift coefficient is a linear function of the effective
angle of attack, that is, Cl=a() where a is the lift
curve slope.
•
For low speeds, a may be set equal to 5.7 per radian.
•
Cd is small. So, Cd sin() may be neglected.
•
The in-plane velocity r is much larger than the normal
component V+v over most of the rotor.
Closed Form Expressions
dr
r
C
r
r
V
r
r
V
cba
P
dr
r
r
r
V
cba
T
R r r d R r r 3 0 3 2 0 2v
v
2
1
v
2
1
© L. Sankar 67
Linearly Twisted Rotor: Thrust
Here, we assume that the pitch angle varies as
E
Fr
R
v
V
a
R
bc
where
a
R
abc
C
R
R
ca
b
R
R
v
V
FR
E
ca
b
T
R T R
Ratio
Inflow
)
2
(~
slope
Curve
Lift
/
DiskArea
BladeArea/
solidity
2
/
3
2
2
/
3
2
2
/
3
2
2
4
3
3
1
2
75 . 75 . 75 . 2 3 2
Linearly Twisted Rotor
Notice that the thrust coefficient is linearly proportional to the
pitch angle at the 75% Radius.
This is why the pitch angle is usually defined at 75% R
in industry.
The expression for power may be integrated in a similar
manner, if the drag coefficient Cd is assumed to be a
constant, equal to Cd0.
8
0
d
T
P
C
C
C
© L. Sankar 69
Closed Form Expressions for
Ideally Twisted Rotor
r
R
tip
tip
T
a
C
4
C
P
C
T
C
d
0
Figure of Merit according to Blade
Element Theory
Area
Area/Disk
Blade
Solidity
R
v)/
(V
Ratio
Inflow
,
8
/
0
where
C
C
C
FM
d
T
T
High solidity (lot of blades, wide-chord, large blade area) leads to higher
Power consumption, and lower figure of merit.
© L. Sankar 71
Average Lift Coefficient
• Let us assume that
every section of the
entire rotor is
operating at an
optimum lift
coefficient.
• Let us assume the
rotor is untapered.
T T RC
R
bc
R
R
T
C
R
bc
dr
r
c
b
T
6
C
6
C
6
C
6
C
C
2
1
C
t
Coefficien
Lift
Average
l l l 2 2 3 2 l l 0 2 l
Rotor will stall if average lift coefficient exceeds 1.2, or so.
Optimum Lift Coefficient in Hover
minimized.
is
/
C
if
maximized
is
FM
6
/
C
If
8
2
2
2
C
hover,
In
8
2
/
3
d0
T
0
2
/
3
2
/
3
T
0
l
l
d
T
T
d
T
T
C
C
C
C
C
FM
C
C
C
FM
© L. Sankar 73
Drawbacks of Blade Element Theory
• It does not handle tip losses.
– Solution: Numerically integrate thrust from the cutout
to BR, where B is the tip loss factor. Integrate torque
from cut-out all the way to the tip.
• It assumes that the induced velocity v is uniform.
• It does not account for swirl losses.
• The Predicted power is sometimes empirically
corrected for these losses.
15
.
1
8
0
T
d
P
C
C
C
Example
(From Leishman)
• Gross Weight = 16,000lb
• Main rotor radius = 27 ft
• Tail rotor radius 5.5 ft
• Chord=1.7 ft (main), Tail rotor chord=0.8 ft
• No. of blades =4 (Main rotor), 4 (tail rotor)
• Tip speed= 725 ft/s (main), 685 ft/s (tail)
• K=1.15, Cd0=0.008
• Available HP =3000Transmission losses=10%
• Estimate hover ceiling (as density altitude)
© L. Sankar 75
Step I
• Multiply 3000 HP by 550 ft.lb/sec.
• Divide this by 1.10 to account for available
power to the two rotors (10% transmission
loss).
• We will use non-dimensional form of power into
dimensional forms, as shown below:
• P=Tv+(R)3A [Cd0/8]
• Find an empirical fit for variation of with
altitude:
4.255316
.
288
00198
.
0
1
h
level sea
Step 2
• Assume an altitude, h. Compute density, .
• Do the following for main rotor:
– Find main rotor area A
– Find v as [T/(2A)]1/2 Note T= Vehicle weight in lbf.
– Insert supplied values of , Cd0, W to find main rotor P.
– Divide this power by angular velocity W to get main rotor
torque.
– Divide this by the distance between the two rotor shafts
to get tail rotor thrust.
• Now that the tail rotor thrust is known, find tail rotor
power in the same way as the main rotor.
• Add main rotor and tail rotor powers. Compare with
available power from step 1.
• Increase altitude, until required power = available
power.
© L. Sankar 77
Hover Performance
Prediction Methods
III. Combined Blade Element-Momentum
(BEM) Theory
Background
• Blade Element Theory has a number of
assumptions.
• The biggest (and worst) assumption is
that the inflow is uniform.
• In reality, the inflow is non-uniform.
• It may be shown from variational calculus
that uniform inflow yields the lowest
© L. Sankar 79
Consider an Annulus of the rotor
Disk
r
dr
Area = 2rdr
Mass flow rate =2rV+vdr
dT = (Mass flow rate) * (twice
the induced velocity at the
annulus)
Blade Elements Captured by the
Annulus
r
dr
Thrust generated by these
blade elements:
dr
r
v
V
r
abc
dr
C
c
r
b
dT
l
2 22
1
2
1
© L. Sankar 81
Equate the Thrust for the Elements
from the
Momentum and Blade Element
Approaches
R
v
,
0
8
8
2
V
R
V
where
R
r
a
a
c c
2
16
8
2
16
2 c ca
R
r
a
a
Total Inflow Velocity from Combined
Blade Element-Momentum Theory
Numerical Implementation of
Combined BEM Theory
• The numerical implementation is identical
to classical blade element theory.
• The only difference is the inflow is no
longer uniform. It is computed using the
formula given earlier, reproduced below:
2
16
8
2
16
2
c
c
a
R
r
a
a
© L. Sankar 83
Effect of Inflow on Power in Hover
thrust!
of
level
specified
a
for
power,
induced
least
produces
inflow
Uniform
constant.
a
be
must
that v
follows
it
),
multiplier
n
(Lagrangea
contant
a
is
Since
0
v
2
v
3
if
is
v
s
variation
possible
all
for
vanish
will
integral
he
only way t
The
0
vdr
v
2
v
3
4
0
v
4
v
4
0
T
-P
.
multiplier
Lagrangean
a
is
where
T
-P
minimize
we
Therefore,
T.
of
value
specified
a
for
power,
induced
minimize
wish to
We
v
4
dT
T
v
4
vdT
2 0 2 0 2 3 0 2 0 0 3 0
R R R R R R inducedr
dr
r
r
dr
r
dr
r
P
Variation of a functional
constraint
Ideal Rotor vs. Optimum Rotor
• Ideal rotor has a non-linear twist: = CR/r
• This rotor will, according to the BEM theory, have a
uniform inflow, and the
lowest induced power
possible.
• The rotor blade will have very high local pitch angles
near the root, which may cause the rotor to stall.
• Ideally Twisted rotor is also hard to manufacture.
• For these reasons, helicopter designers strive for
optimum rotors that minimize
total power
, and maximize
figure of merit.
• This is done by a combination of twist, and taper, and
the use of low drag airfoil sections.
© L. Sankar 85
Optimum Rotor
• We try to minimize total power (Induced power +
Profile Power) for a given T.
• In other words, an optimum rotor has the maximum
figure of merit.
• From earlier work (see slide 72), figure of merit is
maximized if is maximized.
• All the sections of the rotor will operate at the angle of
attack where this value of Cl and Cd are produced.
• We will call this Cl the optimum lift coefficient Cl,
optimum .
d l
C
C
32Optimum rotor (continued..)
twisted.
be
must
blade
the
how
determines
This
2
R
v
and
r
v
arctan
-from
find
we
selected,
is
attack
of
angle
Once
maximum.
is
C
C
at which
a
optimum
an
at
operate
will
stations
radial
All
d 2 3 l TC
© L. Sankar 87
Variation of Chord for the Optimum
Rotor
r
c
C
dr
b
dT
2
l
2
1
dT = (Mass flow rate) * (twice the induced velocity at the annulus)
= 4r(v)vdr
Compare these two. Note that Cl is a constant (the optimum value).
It follows that
r
Const
r
RC
R
bc
r
l
8v
21
2
Local solidity
Planform of Optimum Rotor
Root
Cut out
Tip
Chord is proportional to 1/r
Such planforms and twist distributions are hard to manufacture, and are optimum
only at one thrust setting.
Manufacturers therefore use a combination of linear twist, and linear variation
in chord (constant taper ratio) to achieve optimum performance.
© L. Sankar 89
Accounting for Tip Losses
• We have already accounted for two sources of
performance loss-non-uniform inflow, and blade
viscous drag.
• We can account for compressibility wave drag
effects and associated losses, during the table
look-up of drag coefficient.
• Two more sources of loss in performance are tip
losses, and swirl.
• An elegant theory is available for tip losses from
Prandtl.
Prandtl’s Tip Loss Model
Prandtl suggests that we multiply the sectional inflow by
a function F, which goes to zero at the tip, and unity in the interior.
r
b
f
where
e
arcCos
F
f
1
2
,
2
When there are infinite number of blades,
F approaches unity, there is no tip loss.
© L. Sankar 91
Incorporation of Tip Loss Model in
BEM
All we need to do is multiply the lift due to inflow by F.
r
dr
Thrust generated by the annulus:
dT =
Resulting Inflow (Hover)
1
32
1
16
16
8
16
2
R
r
a
F
F
a
F
a
R
r
F
a
F
a
© L. Sankar 93
Hover Performance
Prediction Methods
BACKGROUND
• Extension of Prandtl’s Lifting Line Theory
• Uses a combination of
– Kutta-Joukowski Theorem
– Biot-Savart Law
– Empirical Prescribed Wake or Free Wake Representation of Tip
Vortices and Inner Wake
• Robin Gray proposed the prescribed wake model in
1952.
• Landgrebe generalzied Gray’s model with extensive
experimental data.
• Vortex theory was the extensively used in the 1970s and
1980s for rotor performance calculations, and is slowly
giving way to CFD methods.
© L. Sankar 95
Background (Continued)
• Vortex theory addresses some of the drawbacks
of combined blade element-momentum theory
methods, at high thrust settings (high CT/).
• At these settings, the inflow velocity is affected
by the contraction of the wake.
• Near the tip, there can be an upward directed
inflow (rather than downward directed) due to
this contraction, which increases the tip loading,
and alters the tip power consumption.
Kutta-Joukowsky Theorem
r
V+v
T
Fx
T (r)
Fx= (V+v)
: Bound Circulation surrounding
the airfoil section.
This circulation is physically stored
As vorticity in the boundary Layer
over the airfoil
© L. Sankar 97
Representation of
Bound and Trailing Vorticies
Since vorticity can not abruptly increase in space, trailing
vortices develop. Some have clockwise rotation,
others have counterclockwise rotation.
Robin Gray’s Conceptual Model
Tip Vortex has a
Contraction that can
be fitted with
an exponential curve
fit.
Inner wake descends at a near
constant velocity. It descends
faster near the tip than at the
© L. Sankar 99
Landgrebe’s Curve Fit for the
Tip Vortex Contraction
Rv
v
2v
R
R
R
0
.
707
2
v
Radial Contraction
blade
the
from
measured
Filament
vortex
the
of
Position
Azimuthal
Age
Vortex
27
145
.
0
78
.
0
)
1
(
R
R
:
vortex
tip
the
of
position
Radial
v vortex v
TC
A
e
A
A
© L. Sankar 101
Vertical Descent Rate
v
Zv
Initial descent is slow
Descent is faster
After the first blade
Landgrebe’s Curve Fit for
Tip Vortex Descent Rate
degrees
twist,
2
degrees
twist,
1
2
1
1
01
.
0
001
.
0
25
.
0
2
2
k
2
2
0
T
T
T
V
V
V
V
V
V
C
C
k
C
k
b
b
b
k
R
z
b
k
R
z
twist,degrees: Blade twist=Tip Pitch angle – Root Pitch Angle
This quantity is usually negative.
© L. Sankar 103
Circulation Coupled Wake Model
• Landgrebe’s earlier curve fits (1972) were
based on the thrust coefficient, blade twist
(change in the pitch angle between tip and
root, usually negative).
• He subsequently found (1977) that better
curve fits are obtained if the tip vortex
trajectory is fitted on the basis of peak
bound circulation, rather than CT/.
Tip Vortex Representation in
Computational Analyses
• The tip vortex is a continuous helical structure.
• This continuous structure is broken into
piecewise straight line segments, each
representing 15 degrees to 30 degrees of vortex
age.
• The tip vortex strength is assumed to be the
maximum bound circulation. Some calculations
assume it to be 80% of the peak circulation.
• The vortex is assumed to have a small core of
an empirically prescribed radius, to keep
© L. Sankar 105
Tip Vortex Representation
Control Points on the Lifting Line where induced flow is calculated
15
degrees
The x,y,z positions of the
End points of each segment
Are computed using
Landgrebe’s
Prescribed Wake Model
Inner Wake
(Optional)
Biot-Savart Law
1
r
Segment
Control Point
2
r
© L. Sankar 107
Biot-Savart Law (Continued)
2
1
2
2
2
1
2
2
2
1
2
2
1
2
1
2
1
2
1
2
1
2
1
4
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
V
c
induced
Core radius used to keep
Overview of Vortex Theory Based
Computations (Code supplied)
• Compute inflow using BEM first, using Biot-Savart law
during subsequent iterations.
• Compute radial distribution of Loads.
• Convert these loads into circulation strengths. Compute
the peak circulation strength. This is the strength of the
tip vortex.
• Assume a prescribed vortex trajectory.
• Discard the induced velocities from BEM, use induced
velocities from Biot-Savart law.
• Repeat until everything converges. During each iteration,
adjust the blade pitch angle (trim it) if CT computed is
© L. Sankar 109
Free Wake Models
• These models remove the need for empirical
prescription of the tip vortex structure.
• We march in time, starting with an initial guess
for the wake.
• The end points of the segments are allowed to
freely move in space, convected the
self-induced velocity at these end points.
• Their positions are updated at the end of each
time step.
Free Wake Trajectories
(Calculations by Leishman)
© L. Sankar 111
Background
• We now discuss vertical descent operations,
with and without power.
• Accurate prediction of performance is not done.
(The engine selection is done for hover or climb
considerations. Descent requires less power
than these more demanding conditions).
• Discussions are qualitative.
• We may use momentum theory to guide the
analysis.
© L. Sankar 113
Phase I: Power Needed in
Climb and Hover
Climb Velocity, V
Power
A
T
V
V
T
V
T
P
2
2
2
v
2Descent
Non-Dimensional Form
It is convenient to non-dimensionalize these graphs, so that
universal behavior of a variety of rotors can be studied.
h
h
Tv
by
lized
dimensiona
-non
is
v)
T(V
Power
A
2
T
v
velocity
inflow
hover
by
lized
dimensiona
-non
is
locity
descent ve
or
Climb
© L. Sankar 115
Momentum Theory gives incorrect
Estimates of Power in Descent
V/vh
(V+v)/vh
Climb
Descent
No matter how fast we descend, positive power is
still required if we use the above formula.
This is incorrect!
0
2
2
2
v
2
A
T
V
V
T
V
T
P
The reason..
Climb or hover
Physically acceptable Flow
V is down
V+v is down
V+2v is down
V is down
V is down
V is up
V+v is down
V+2v is down
V is up
V is up
Descent: Everything inside
Slipstream is down
© L. Sankar 117
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