1974 MAY
University
State
.
GI-418340 No. Grant Under
Oregon
(RANN) Needs National to Applied Research
Foundation, Science National the by Supported
1-
Lissaman S. B. Peter And Wilson E. Robert
MACHINES
POWER WIND
Of
AERODYNAMICS
APPLIED
1974 July,
California Pasadena,
Inc. Aerovironment,
Lissaman S. B. Peter
and
97331 Oregon Corvallis,
University State Oregon
Wilson E. Robert
by
MACHINES POWER WIND
OF
AERODYNAMICS APPLIED
79
74
73
73
69
64
61
Relations Approximate
Gradient Wind Vertical of Effects The
Introduction
GRADIENT WIND VERTICAL TO DUE MOMENTS AND FORCES 6, CHAPTER
Rotor Circular The
Rotor Darrieus
Wake the of Modeling Vortex
Theory Vortex
Glauert Rotor; Optimum The
Models Loss Tip
Equations Flow Annulus
Wake the of Representation Vortex
59
54
49
44
39
MACHINES AXIS CROSS-WIND 5, CHAPTER
61
39
34
25
20
17
17
12
11
11
8
7
6
6
THEORY VORTEX/STRIP ROTORS; AXIS WIND 4, CHAPTER
Actuators Ducted
States Flow Multiple of Model Simple
Rotation Wake of Effect
Theory Rankine-Froude
THEORY MOMENTUM GENERAL ROTORS; AXIS WIND 3, CHAPTER
Translators Lifting
Translators Drag
MACHINES POWER WIND TRANSLATING 2, CHAPTER
Rotor Circulation-Controlled
Design Smith-Putnam
Rotor Ducted
Machines Axis - Wind
Power Wind in Aerodynamics of Role
2
Rotor Darrieus
Rotor Madaras
Rotor Savonius
Machines Cross-Wind-Axis
3
5
4
3
1
INTRODUCTION
1,
CHAPTER
page
CONTENTS OF TABLE
107
FACTORS "F" THE OF USE II, APPENDIX
REFERENCES
85
SYMBOLS
87
INSTRUCTIONS OPERATION PROGRAM I, APPENDIX
89
page
(cont'd.) Contents of Table
State Brake Propeller
26
Modes Operation Rotor
28
Modes Various for Force Blades
29
3.8
Coordinates Element Blade
30
3.9
Angles Pitch Blade Various for States Element Blade
32
Geometry Windmill Ducted
36
4.1
Disc Actuator
40
4.2
Shown) Are Blades Two (Only
Rotor Multi-Bladed a for System Lattice Vortex
41
Rotor Two-Bladed a of System Vortex of Idealization
43
Element Blade Rotor
45
Element Blade Rotor a for Diagram Velocity
45
4.5
4.4
4.3
3.10
3.7
3.6
3.5
Rotor Savonius
3
Effect Magnus
4
Rotor Darrieus
5
1.4
Rotor Ducted Enfield-Andreau
6
1.5
Turbine Wind Smith-Putnam
8
1.6
Machines Power Wind of Performance Typical
10
Device Drag Translating
11
2.2
Airfoil Translating
13
2.3
Ratio Lift-Drag vs Airfoil Translating a From Power
14
2.4
Wind Relative with Airfoil Translating
15
3.1
Turbine Wind a Past Flow One-Dimensional
17
3.2
Geometry Streamtube
21
3.3
Wake Vortex Rankine a with Rotor a for
Ratio Speed Tip vs Coefficient Power Maximum
Wake Inotational an with Flow for Velocities
Induced the on Ratio Speed Tip of Effect
25
24
3.4
2.1
1.3
1.2
1.1
page
FIGURES OF LIST
Diagram Velocity Blade
Gradient Wind a in Rotor
76
Gradient Wind To
Due Output Power In Reduction Percent
84
75
6.3
6.2
6.1
Diagram Velocity
55
5.1
Actuator Axis Cross-Wind of Shedding Vortex
62
5.2
Actuator Axis Crosswind Bladed Single of System Vortex
63
Length Equal of Caternary and Circle Troposkien,
Turbine Crosswind-Axis a for System Flow
65
70
53
00
5.4
5.3
4.9
= Op Turbine Wind
4.6
Rotor a of State Working
48
4.7
Turbine Wind Smith-Putnam the of Performance Calculated
52
4.8
Smith-Putnam the for Speed Wind Versus Output Power
page
(cont'd.) Figures of List
Disk Actuator Optimum the for Conditions Flow
56
Disk Actuator Optimum the for Parameters Blade
Disk Actuator Optimum the for X vs Cp
57
Sums Trigonometric
78
59
6.1
4.3
4.2
4.1
page
TABLES OF LIST
APPLIED AEROOY:\iAMICS OF WIND lvlAGIINES
CHAPTER 1
INTRODUCTI ON
Recent interest in wind machines has resulted in the reinvention and
analysis of many of the wind power machines developed over the past centuries.
Because of the considerable time period since the last large scale interest
in this country, which occurred over twenty-five years ago (1) a considerable
amount of information that was published is out of print or not generally
available.
An
excellent bibliography of the work published prior to 1945 was
collected by the War Production Board in a report issued by
sity (2).
~ew
York Univer-
Golding's work (3) published in 1955 also contains an extensive
bibliography and covers the work done in England in the 1950's.
It is the
purpose of this paper to review the aerodynamics of various types of wind
power machines and to indicate advantages and disadvantages of various schemes
for obtaining power from the wind.
The advent of the digital computer makes the task of preparing general
performance plots for wind machines quite easy.
Simple, one-dimensional
models for various power producing machines are given along with their performance characteristics and presented as a function of their elementary aerodynamic and kinematic characteristics.
Propeller type wind turb ine theory is
reviewed to level of strip theory including both induced axial and tangential
velocities.
It is intended that this publication be of use in rapid eval-
uation and comparative analys is of the aerodynamic performance of wind
power machines.
1.1
Role of Aerodynamics in Wind Power
The success of wind power as an alternate energy sources is obviously a
direct function of the economics of production of wind power machines.
In
this regard, the role of improved power output through the development of
better aerodynamic performance offers some potential return, however, the
focus is on the cost of the entire system of which, the air-to-mechanicalenergy transducer is but one part.
The technology and methodology used to
develop present day fixed and rotating-wing aircraft appears to be adequate
to develop wind power.
One of the key areas associated with future development of wind power is
rotor dynamics.
The interaction of inertial, elastic and aerodynamic forces
will have a direct bearing on the manufacture, life and operation of wind
power systems while at the same time have a minor effect on the power output.
Thus the aerodynamics of performance prediction, quasi-static in nature,
is deemed adequately developed while the subject of aeroelasticity remains to
be transferred from aircraft applications to wind power applications.
1.2
Wind Power Machines
Since 1920 there have been numerous attempts in designing feasible wind-
mills for large scale power generation in accordance with modern theories.
This section describes representative types of these designs.
It is convenient to classify wind-driven machines by the direction of
their axis of rotation relative to wind direction as follows:
1.
Wind-Axis Machines; machines whose axis of rotation is parall el
to the direction of the wind.
2.
Cross Wind-Axis Machines; machines whose axis or rotation is
perpendicular to the direction of the wind.
2
CROSS-WIND-AXIS MACHINES
SAVONIUS ROTOR
The Savonius Rotor in its most simplified form appears as a vertical
cylinder sliced in half from top to bottom; the two halves being displaced as
shown in Figure 1.1.
It appears to work on the same principle as a cup ane-
mometer with the addition that wind can pass between the bent sheets.
In
this manner torque is produced by the pressure difference between the concave
and convex surfaces of the half facing the wind and also by recirculation effects on the convex surface that comes backwards upwind.
The Savonius design
was fairly efficient, reaching a maximum of around 31%, but it was very inefficient with respect to the weight per unit power output since its construction results in all the area that is swept out being occupied by metal.
A
Savonius rotor requires 30 times more surface for the same power as a conventional rotor blade wind-turbine.
Therefore it is only useful and economical
for small power requirements.
Figure 1.1
Savonius Rotor
3
4
full-sized single A design. this with done being little very in resulted locations, top mountain
on use for unsuitability its with coupled losses, electrical and losses, mechanical design,
aerodynamic poor system's the However axles. car the to connected generators drive and track
the around cars the propel would effect Magnus The motor. electric an by driven and diameter,
in feet 18 high, feet 90 been have to was cylinder Each move. would flat-cars, on vertically
mounted cylinders, rotating which around track circular a construct to proposed Madaras
Effect Magnus 1.2 Figure
produced. be to cylinder the of
axis the and flow the to perpendicular force lift a causing induced is circulation Thus developed.
partly only is separation side lower the On eliminated. completely is separation direction,
same the in moving are cylinder the and flow the when surface, cylinder the of half upper
the On flow. the to angle right a at stream a in placed cylinder rotating a about exists which
pattern flow the shows 1.2 Figure cylinder. a of rotating the involves effect Magnus the achieve
to way simplest The boundary. solid the and fluid the between velocity relative the of reduction
by formation layer boundary suppress to attempts which technique control layer boundary
a involves it essence In effect. Magnus the of principle the on works Rotor Madaras The
ROTOR MADARAS
5
diameter.
in feet 20 than less are built rotors Darrieus largest the date To uncertain. remains more, or kw
100 production, power of levels higher to rOtor type Darrieus the scale to ability The production.
power economical for candidate promising a it make production cost low for potential associated
and design of simplicity The starting. for input power requires and rotor propeller-type
a of that near performance has rotor Darrieus The Ottawa. in Canada of Council Research
National the of (20) Rangi & South by investigated been recently has Rotor Darrieus The
Rotor Darrieus 1.3 Figure
below. 1.3 Figure in sketched
rotor axis vertical a for 1926 in patent States United a filed Paris of Darrieus Georges
ROTOR DARRIEUS
since. done
been has development further no but testing for Jersey New Burlington, in built was cylinder
6
they that so designed are blades The mph. 60 to 30 of speeds wind at effective were and pitch
blade the vary to used were motors hydraulic speed rotor constant maintain to order In
Rotor Ducted Enfield-Andreau 1.4 Figure
GENERATOR
INTAKES AIR
TUR8INE
VENT AIR
PROPELLER
VENT AIR
rpm. 95 at wind mph
35 a in kilowatts 100 producing of capable is and diameter in feet 80 was blade The generator.
a drives that turbine a through passes it tower the through flows air As tower. high foot
100 the in created semi-vacuum the through air up draws also pedestal blade the and rotor the
of tip the between difference pressure the time same the At tips. blade the through tower hollow
the from air pulls force centrifugal wind, the by turned are blades the As generator. the and
propeller the between coupling no has it machines conventional Unlike 1.4. Figure in shown as
blades airplane-type hollow two with windmill experimental an built British the 1954 In
ROTOR DUCTED
MACHINES WIND-AXIS
7
turbines. wind scale large by generation power electrical of possibilities
the illustrated design Smith-Putnam the blade, the of failure structural the of spite In
design. this with experimentation ended and off flew blades the of one 1945 In years. two
for secured be not could replacement a and failed bearing main inch 24 the when 1943 February
until unit test a as operated and 1941 M erected was 1.5, Figure in shown The:turbine,
kilowatts. 1,000 generate
to intended was turbine wind The edge. leading the on ice of inches six with mph 100 and
mph 120 to up wind withstand to designed was plant power The cylinders. oil-filled by damped
itself was coning The speed. constant reasonably a maintain still and gusts sudden against
guard to 20° to up cone to ability an with designed were blades The edgewise. turning by feather
to began blades the increased, velocity wind the As above. and mph 18 of velocities wind
at rpm 28.7 of speed constant a at blades the keeping automatic, was control pitch The
tower. foot 100 a by supported
were and tons 250 about weighed generator and rotor The sections. airfoil 4418 NACA
using blades steel stainless two of consisted and feet 175 was diameter rotor The constructed.
ever largest the was Vermont in Knob Grandpa's at built windmill Smith-Putnam The
DESIGN SMITH-PUTNAM
aloft. supported not is equipment generating power
the that is system this of advantage main The system. operated power a by controlled and aided
is wind the into face to rotor the of motion The gusts. heavy of pressure wind under flap can
8
lift. generate which of all surface, lower the on suction in decrease a and surface upper the
on suction in increase an is there and blowing by induced is circulation time same the At drag.
viscous in increase accompanying an is there but reduced is drag pressure the Consequently,
cylinder. the on back further separation of point the moving thereby cylinder the of
surface upper the of layer boundary energy low the re-energizes Blowing separation. flow delay
to technique control layer boundary a is briefly, principle, This slots. small from blades the
of surfaces upper the around tangentially air of sheets blowing by generated is lift rotor, the of
blades cylindrical the rotating of Instead 1920's. the of Rotor Flettner the and Rotor Madams the
of that to similar quite is Turbine Wind Rotor Circulation-Controlled the of concept The
ROTOR CIRCULATION-CONTROLLED
Turbine Wind Smith-Putnam 1.5 Figure
9
constructed. been have that rotors
of types various the of comparison performance a gives page following the on 1.6 Figure
rotor.
circular a of simplicity structural the offset to enough large was area) disk the by. divided area
projected (rotor solidity rotor required the ratios speed low-tip at while rotor, the from output
power than greater was jet the drive to power compressor the ratios speed high-tip at that found
was it and University State Oregon at made was design this of investigation analytical An
environment. operating its with cope to structure
rigid very a and control, construction, easy for provides design This controls. pitch complicated
foregoing thereby slot, the of location the adjusting by achieved is coefficient lift constant
spanwise The stiff. very be to it causes blade of type this of cross-section cylindrical a of inertia
of moment large The fastened. solidly was that blade propeller conventional a of difficulties
the without hub the to rigidly mounted be can blade the because needed is coning or flapping no
Second, gusts. sudden with up speed to tend not would rotor the therefore gusts, to insensitive
is cylinder the lift zero at First, advantages. of number a possesses design This
10
Machines. Power Wind of Performance Typical 1.6 Figure
SPEED SPEED/STREAM
6
5
4
PER
3
=TIP RATIO SPEED
2
Type Arm Four Dutch
Rotor Darrieus
Rotor Sayonius
Type Blade - Multi American
Type Blade Two Speed High
0
Windmills Type
Propeller for Efficiency Ideal
0.6
(2-1)
D
Sv C v)2
11
(Ko
p (1/2) = Dv = P
by expressed is power the that so v
-
V.Q
velocity relative a sees device drag The
velocity. translation the and drag the of product the is P, extracted, power the device a such For
Device Drag Translating 2.1 Figure
device. drag elementary the of action the illustrates
2.1
Figure drag. by driven be to machine the consider First, machine. rotating of element blade
instantaneous an as considered be can translation the since machines rotary various examining in
illustrative be can devices drag-driven and lift-driven translating of Analysis extraction. power
than rather propulsion for used been have devices translating wind-driven Historically,
wind. the of action under line
straight a in loves that device the is machine power wind of type simple most the Perhaps
TRANSLATORS DRAG
2.1
MACHINES POWER WIND TRANSLATING
2 CHAPTER
16
direction. wind in changes to sensitivity and ground
the to proximity are translators of disadvantages Other land. and machines in investment capital
extensive mean speeds large as disadvantage chief the are rates extraction power high achieve
to translator the for required speeds large The speeds. low at velocity translator of independent
relatively is translator a by produced force the contrast In velocity. translation the with
linearly vary to seen is translator a of output power the velocity, wind the below speeds At wind.
17
Turbine Wind a Past Flow One-Dimensional 3.1 Figure
motion.
rotational no with axial entirely be to assumed is flow the if power and thrust the determine may
we flow the to energy and momentum, continuity, Applying device. wind a by slowed is which
V. is wind stream free The
below. shown as turbine wind a past flow consider (5,6) Froude
E. R. and W. and (4) Rankine by originated theory momentum axial the with Starting
THEORY RANKINE-FROUDE
3.1
output. power to geometry
blade linking approach detailed more
a.
to proceed then and turbine wind a of output the of
analysis one-dimensional a consider will we step first a As production. power wind scale large
for candidate leading the and machine efficient most the (1), 1940 in as today, remains turbine
wind or windmill type propeller The turbines. wind to attention our turn us let Now
THEORY MOMENTUM GENERAL ROTORS; AXIS WIND
3
CHAPTER
+umv.
18
Or
2
c°
(v. upA
2
}=
2
pAu = P
1y,20
p. = pi with flow, isothermal assuming thermodynamics, of law first the From
is. power
the however, importance; great of immediately not is thrust The disc. the at change velocity
the twice is ui,
- V. change velocity wake final the , 2aV. Ui= - V. that note aV.,
u-
denote we If velocities. final and initial the of average the is disc the at velocity the i.e.,
2
(3-4)
=U
ui +
obtain we expression momentum the with together
2
(3-3)
p
T
(V02,,
that so wake the and turbine the of side downwind the between again and
turbine the of side upwind the and stream free between used be may Equation Bernoulli the Now
-p- =p Ap
(3-2)
,
where
AAp = T
machine wind the by caused drop pressure the of consideration from Second
-ui) (Voo pAu )=
(3-1)
=T
(Vo(,
theorem momentum the from First, obtained. be may thrust the for expressions Two
19
energy, kinetic translational to addition in wake the in energy kinetic rotational is there If sense.
opposite the in rotates wake the (windmill), device extracting energy an of case the in propeller,
the of direction the in rotates wake the propeller, a of case the In rotate. to wake the cause will
machine wind rotating a with interaction rotational, not is stream initial the As rotation. wake of
consideration additional the with accomplished be can modeling one-dimensional Further
88.8%. is coefficient power maximum at efficiency
The 0.5. of coefficient power a yields which 1/2 = a at 100% is efficiency maximum The
2
(3-7)
a) 441
pAu
= V2
by given is available power by divided output power efficiency, the Hence Au. but AV., not
is disc the through rate flow mass the since efficiency maximum the represent not does however,
(3-6), Equation rotor. the by out swept that to equivalent area an in contained wind the of energy
kinetic the is denominator the that noted be may it (3-6) equation examining When
.
1/2 =
axam
obtain we 2a), - (1 V., = ui as
so zero, is velocity wake final minimum The air. the on turbine the of influence the of measure a
is and factor interference axial the as known is (a) term The defined. is power maximum a Thus
(3-6)
Pmax
16
pAV.3 1/2
27
1/3 = a when maximum a has which
(3-5)
)
441 =
pAV:: 1/2
20
bounds. establishing in results the to utility affords which model
simple a yielding inserted core rotational a and out subtracted be may velocities angular high
of regions the of contribution the however axis, rotation the near velocities rotational unrealistic
produces irrotational, be to assumed if model, flow wake The estimated. be may removal
power on rotation wake of effect the turbines, wind of analysis the to notation his Adopting
propellers. of analysis the in rotation wake of effect the considered (7) Joukowski
ROTATION WAKE OF EFFECT
3.2
torque. low and velocity angular
,
high means which low is ER2 when occur will extraction power greatest the ETi energy
initial of amount given a for so, energy, kinetic rotational wake greater thereby and momentum
angular wake greater produces torque increased that note velocity), (angular x (torque) = P as
ER2
T2
E
ET1
=P
thermodynamics From
Translation
Rotation
ER2
+
T2
Extracted Power
P=
Energy Kinetic Final
E =
ETi = Energy Kinetic Initial
velocity.
angular rotor to energy kinetic rotational wake relate will example simple following The
translation. only having wake the of
case the in than extraction power lower expect may we considerations thermodynamic from then
(3-9)
21
ri2(01
=
r2CO
Momentum of Moment
(3-8)
uiridr = urdr
Continuity
are: equations resulting The
geometry Streamtube 3.2 Figure
Wake
Rotor
streamtube. the illustrates below 3.2 Figure
wake. the in and rotor the at velocities induced
the of values relative the on rotation of effects the of measure a is approach this of outcome main
The obtained. be can coefficient power the for expression an cases, special certain for addition,
In rotor. the at velocities corresponding the and rotational, and axial both velocities, wake the
between relation the express that written be can equations analysis, streamtube a Using
22
obtained. be also may element annular an for thrust and torque the for Expressions
zero. be to assumed been has drag Fluid
rotor. the at discontinuously changes rotor
the of rotation of direction the opposite is which fluid, the of velocity angular The
radially. vary velocities axial wake and rotor The
velocity. rotational the to due wake the across varies pressure The
noted. be may flow the of features Several
equation. Euler's and equation
Bernoulli's are used equation momentum the of fauns particular The solution. a obtain to order
in
co,
say variables, the of one of specification needs one and obtained be cannot Closure wake.
the in flow and torque thrust, between relations the obtain to used be may equations four These
1) 1 (3-
2
\ dr].
) (a)? d wi)
(c2
2
2
\
dr1
( d
obtained. be may velocity
axial in gradient radial the for expression an Finally rotor. the of velocity angular the is 5-2 where
U
(3-10)
ulahri2
2
U1
2
+
(01
(0"
=
+
00
2
/
(u1
)2
Energy
equation, energy an obtain
may we addition, In fluid. the of velocities angular wake and rotor the are
coi
and
co
where
(1
a)2
b
a
b2
23
A 0/003 y2
power
cp
and
a),
(3-12)
(13
4X2
42 (1
1
2
b'
=a
obtain may We
a)
(1 Voo
a u
b) (1 V.,
Defining
constant. is velocity axial wake the constant
is r2o) when that seen be may it gradient, velocity radial wake the for expression the From
r2codA
+
p(S)
22-2)
= dT
THRUST
pur2o3dA
= dQ
TORQUE
(3-13)
N)b] (1 +
24
a
a)2[2Na
b
= Cp
11(1
obtain we N::----0/coma, Letting wake. vortex
Rankine a substitute may we wake, vortex irrotational an of lieu In axis. the near velocities
infinite produces constant = r2e) since modification some requires coefficient power The
2. above ratios speed
tip for wake
in value the 1/2 approximately always is disc the at change velocity axial the that observed be
may It X. and a of function a as a/b ratio the of variation the illustrates above 3.3 Figure
Wake. Irrotational
an with Flow for Velocities Induced the on Ratio Speed Tip of Effect 3.3 Figure
-u
0.5
0.4
a:
0.2
0.3
0
0.1
0.2
0.4
III
8
0.6
0.8
I.0
25
the continue to simple quite is it 0 < a For 1. < <a 0 is that device, extraction power draglike
a as operating is device the that assumed tacitly been has it analysis previous the In
STATES FLOW MULTIPLE OF MODEL SIMPLE
3.3
turbines. wind of theories element blade in rotation
wake neglecting of effect the into insight some affords it model, this with associated difficulties
the of spite In model. flow this criticized recently has (8) Goorjian tubes. steam interacting
non- annular, in occur to flow the requires results these at arrive to used model flow The
Wake. Vortex Rankine
a with Rotor a for Ratio Speed Tip vs Coefficient Power Maximum 3.4 Figure
RD.
5
4
2
3
0
I
0
DISTRIBUTION VELOCITY
ROTATIONAL WAKE
0.2
0.3
8
n>
0.4
0.5
0.6
WMAX
least. the are rotation wake the consequently and torque the where ratios speed
tip high at occur coefficient power of values highest the expected be would As 3.4. Figure in
below shown is wake vortex Rankine a with rotor a for coefficient power maximum The
(3-16)
26
11al 4a = CT
form the in written be can cases three all Thus,
(3-15)
a) (1 4a
(3-14)
pAV:, X
= ,
= CT
Force
(1.
4a = Cp
that
case this in find we 3.1 Section in as approach same the using analysis the Continuing
State Brake Propeller 3.5 Figure
below, shown is pattern streamline idealized An state. brake or thrust, reverse the in propeller
a is that flow, forward a induces it that so adjusted pitch its with propeller powered a considering
by modeled physically be may This
1.
a> for occurs case interesting particularly A
propeller. a of type that of typical is regime flow This flow. wake the
to energy adding and thrust producing propulsion a as act will device the that show to analysis
.5
1
1
28
Modes Operation Rotor 3.6 Figure
a
i
i
0.5
1.0
0
-0.5
1
Flow To
Added Power
_ 0.6
0
op
0.6
F/ow From
n
Extracted Power
f
I
I
0.5
1.0
1.5
I
I
I
I
a
0
1
-0.5
_ -1.0
1
THRUST
.---v-A"
Propeller
Windmill
0
Theory
Momentum
Brake ---4.--
_ Propeller
//
(emperical)
Glauert
A
_ 1.0
//
1
t DRAG
I
2.0
I
29
Modes Various For Force Blades 3.7 Figure
Direction Wind
a>I
Brake Propellor
0<a<I/2
Windmill
0=a
Slip Zero
a<0
Propellor
Vco
r.r1
a
9
aVo
S7:7=1V.,
./1
Force
Blade
v/
rf2
Rotation of Plane
30
Coordinates Element Blade 3.8 Figure
a)V.
(1
Rotation of
Direction
L'
Direction
Wind
3.8. Figure in sketched is model This itself. disk the at that twice
is flow induced wake the assuming and terms swirl ignoring theory element blade conventional
use and (propeller) rotor axis wind the example, an as consider, will we model simple our For
themselves. elements blade the on forces lifting by represented as that to momentum wake by
represented as force the connect must we modes the of possibility the establish to order In
wake. the in flow the considering by and itself disc
the at flow the considering by both states these of models physical construct can we Thus,
here. down break will model simplified
our since
1
= a of vicinity close the in regimes flow the avoided-discussing have We blade. the
on torque and force the of sense the illustrate also which 3-7, Figure in sketched are states These
1.
a> state, brake propeller the enters rotor the negative, increasingly becomes now 0 As
rotation. of plane the to relative angle zero at be to blade the
for is, that 0; = 0 for exist still can state windmill the that noted be should It 1/3. = a at occurring
(3-23)
31
ila 4a 01= sin x
a)cos
K1 47cxa
a all for write can we Thus cdrircrdr.
=
a as a solidity local a define can We
previously. as result same the by given is force blade the
,
while a), - -4a(1 = CT theory momentum by get we
1
a> state propeller-brake the For
rON.. ratio speed tip local the is x where
(3-22)
0]
xsin
0
a)cos
(
XC
= CT
OrFdr = dT
(3-21)
by given is annulus the on force the Thus
(3-20)
0] sin Or
0 a)cos
(1 pnc[Vca =
a, 27csin = CL with and WcCL/2 = F
from circulation the get we itself element blade the at flow considering Now
a) 441
(3-19)
= CT
by given is coefficient thrust local the and
Ircrdr a)2a
(3-18)
(1 pV02 = dT
by given is annulus the on force
the theory, momentum By
1.
< a state windmill or propeller the in are we Assuming
32
than larger much becomes constant) = 8 (following disk the on drag-force the now increased,
slightly is flow induced the a, B, at that Assume follows. as is unstable is B state why of
explanation simplified A approached. is state the how upon depending occur and stable both are
C and A that and unstable is B as shown point the that appears it case, mode triple the For
ratio. speed tip
and solidity of fiuiction a is lines force blade the. of slope the that note to interest of is It
Angles Pitch Blade Various For States Element Blade 3.9 Figure
1.5
1.0
0.5
a
-0.5
AMMO
Brake
Propeller
mode. brake propeller one and modes windmill two
states, equilibrium possible three exhibit 0 > 0 02> range intermediate the in angles The occurs.
mode brake propeller the <82 0 for while occurs, propeller thrusting powered simple the <81 0
for that Note 3.9. Figure from seen be easily most can equation this to solutions of nature The
33
(16)]. [Shapiro, occur states ring vortex the and wake, turbulent the brake, parachute
the of states anomalous the where rotor descending lifting a with associated that is region this
analysis helicopter In theory. rotor helicopter with connection in problem this on done been has
research extensive quite although region, this in flow for exist theories satisfactory No
1.
= a of vicinity the in inadequate
quite considered be must analysis brake propeller simple the Again unacceptable. physically is
which configuration streamline a infinity, downstream and disc actuator the between somewhere
velocity wake zero and wake far- the in reversal flow implies here developed model dimensional
one- simple the
1
<
<a 0.5 for example, For
0.5.
a> for valid considered be not should thus
model, perturbation small a considered be should analysis present the that here note We
characteristic. the of region X low the at occurs which stall,
blade as same the not is behavior this that Note reduced. is a which in technique feather blade
normal the from distinct quite is which control speed of method a is This energy. dump to state
brake propeller the of flow confused the use to possible be may it loads, torque shaft reduced or
winds incoming high to due speed over rotor prevent to necessary is it where cases in However,
windmill a of range design major the in occur not should 0.5 a> of states that note We
continuity. flow violate somehow will 0.5 < a of value the with annuli outer and inner has
which annulus an on
1
< a giving model a that seen be can it example, For inconsistencies. flow
involves inevitably it that and model idealized an is above the that stressed be should It
occur. can
1
< <a 1/2
with solutions no that is theory element blade in assumption working a Thus arguments.
these to according stable are C and A hand, other the On 1.0. = a towards moves system
the and reduced, further is momentum wake this thus momentum, wake the by represented that
34
pressure; static freestream at exit duct the leaves flow the that assume to usual is it performance
propeller ducted of calculations In technically. effective not is propeller shrouded the account,
into taken are moments pitching duct other and drag duct when and increased, is speed forward
the as reduces shroud the of effectiveness the friction, skin ignoring even that, shown be can
It speed. forward zero at least at ratio, thrust/power its increase thus can and propeller thrusting
a of contraction slipstream the reduce effectively quite can duct a that established well been has
It propellers. powered of thrust static the increase to used frequently are ducts or Shrouds
ACTUATORS DUCTED
3.4
disc. the for a mean the is given perturbation axial of value the and a, uniform
non- to subjected is rotor the fact in thus remainder, the driving is rotor the of portion one
state autorotative the in that noted be should It state. autorotative the terminology, helicopter in
or 0, = Cp to correspond tests these running, free were these Since 3.9. and 3.6 Figures in shown
was curve This tests. windmill free-running of series a from data using and concepts these
using (10) Glauert by constructed was a versus CT of curve performance generalized A
(18). Townend and Bateman, Lock, by given is problem this
to approach classical A flow. axial for occur which singularities the of some eliminates freedom
of degree additional this of introduction The axis. rotor the to angle small some at yawed
but axial, precisely not is flow freestream the that assuming by removed be can 1.0 = a near
flow of anomalies the of many that Wolkovitch, by described as note, to interest of is It
(17).
Wolkovitch by discussed is a of range entire the for performance Rotor occur. well may 0.5
a> with solutions spurious conditions off-design for However,
0.5.
a> for conditions analyze to
necessary seldom is it thus 0.5; than less is a modes operating windmill normal most For
35
as extracted power the write immediately can we , rh is system the through
flow mass the Assuming system. windmill ducted typical a show we 3.10 Figure In
justify. to hard quite are
which assumptions without device this on made be cannot analysis type momentum a rotor, free
the unlike that shown is it which in follows, this of analysis simple A capacity. extraction power
its increased and rotor the through drawn be to flow more caused has duct the effect In 0.593. of
limit rotor free the exceed will coefficient power the area, rotor on based then, velocity, stream
free the thirds two of level optimal the at flow induced axial wake the keeping still white this,
than larger be to section cross wake optimal the cause to possible is it if Thus, disc. windmill the
of that twice be should section cross wake optimal the windmill free a for that showed have We
expansion. wake the increase to be will duct a of effect the viewpoint, physical a From
(19). Rainbird and Lilley 'y given is windmills ducted of analysis comprehensive
A performance. superior have might windmill ducted a that suggested been frequently
has it duct, a to due performance propeller thrusting in improvement the of Because
there.
pressures low the to due duct the of area entry and edge leading the on force the is contribution
this of part major a and duct, the on thrust forward a by represented is force increased
the that noted be should It contraction. slipstream no been lave will there thrust, the increase will
propeller the as section cross same the of duct cylindrical a even that observe to interest of is It
system. the of ratio thrust/power the increase will this than less be to contraction slipstream final
the causes which duct any Thus area. propeller the half one is slipstream ultimate the propeller,
free static a of case the In section. cross wake ultimate the as taken be may area exit duct the
calculation, of purposes for and velocity slipstream in change further no is there consequently,
36
required. is assumption additional one theory
momentum using Thus, edge. leading the to close very region the for except calculated, be can
interior duct the on pressure the theory one-dimensional By known. not is duct the of outside
the on field pressure the since theory momentum simple by determined be cannot force duct
the but AAp, by given is force propeller the that true still is it duct, the with case, our For
case. propeller free the for a) - pAVoo(1 = th result the gives immediately
This area. actuator the is A where AAH = AAp = T by given is which actuator the on force the is
system the on force the that stating by obtained readily is equation remaining the theory actuator
free For th. for expression an have not do we that in closed not are equations these that note We
(3-25)
2a riiVao = T
as written be may duct) and (rotor system entire the on force This
(3-24)
a) (1 2a MY!
= MAR = P
Geometry Windmill Ducted 3.10 Figure
vs.
37
higher and flow mass higher a with duct the of downstream expansion wake gives case, the be
must which static, freestream than lower is pressure exit the assuming that, seen be will It
(3-29)
b)
(3-28)
2a
b)
b)(2 (2a
41 (1 4a =
=
C;
Cp
us give which
C*p
coefficient pressure exit
duct the and area) exit on (based coefficient power the writing by expressed be can This duct.
the for condition flow exit of assumption the by determined are characteristics performance all
analysis of level this At alone. rotor the of that exceed will area, rotor on based system, ducted
the of coefficient power the then 1.54 exceed ratio area rotor to duct the if that observe and
(3-27)
/A 2a)Ae
41 (1 4a =
Cp
get we then area, rotor of terms in coefficient
power write we If 0.211. = a at 0.385 =
(3-26)
2a)
Cp..
41
give to maximumized be can expression This
4a = Cp
obtain we
A, area exit duct the on coefficient power the basing Then, a). 2
- AepV.,(1
Iii= as determined
be can flow mass the and 2a = b get we then static, freestream is exit duct the at pressure
the that theory, propeller ducted powered in done is as assume, we If b). - V.,,(1 as shown
is which exit, duct the at velocity the assuming by satisfied be to this consider can We
38
loadings. rotor different
at system given a for even and system, rotor duct every for notably vary will which quantity
this on dependent directly is result the since approximation; poor a is coefficient pressure exit the
assuming- that appears It flow. entire the of modeling a upon depends prediction performance
proper a and method simple any by analyzed be cannot windmills ducted Thus
wake. the surrounding tube vortex associated the with energy, different of wake
a implies disc actuator the of addition essential the since flow homoenergetic uniform a in wing
ring a as simply treated be cannot duct the that note We analysis. this into enter must geometry
pressure ensuring and
duct the of details the Evidently surface. bounding wake the on continuity
flows, external and internal computing contour, a assuming techniques, flow potential by shape
wake the compute to principle in possible is it sections, previous the in described As
pressure. exit duct assumed the of terms in plotted
power
is performance the noted be must it paper Rainbird's and Lilley studying In coefficient.
39
the that Assume process. shedding ring vortex the of model a create can we porosity, infinite to
zero from switched be arbitrarily may .which disc a actuator, prototype the consider we If
flow. freestream the to parallel is tube vortex
wake the that assumed is it where disc, actuator the for used are assumptions Analogous
deformation. wake for account to necessary is it that
wings loaded highly very of cases in only is it However, situation. inconsistent kinematically
a wake, the through flow downwash be will there that implies This flow. freestream
the to parallel wing the leaves wake vortex the assuming by removed usually is problem the
of nature interactive the however, theory; wing ordinary in occurs situation similar A consistent.
are fields pressure and streamline wake the assure to checked flow induced the and assumed
is strength and geometry initial the where process, iterative an by determined be generally
must strength and shape wake the words other In flow. the of dynamics the and kinematics
the both involve must wake inviscid the of solution proper a Thus equilibrium. in is it whether
determine to wake the on fields flow and pressure the compute to possible then is It wake.
this of flow induced the calculate to Law Biot-Savart the use then and distribution vorticity
wake the stipulate usually we models wake advanced more In flow. the of models developing
in assistance great of be can geometry, its and vorticity, this of generation of mode The vorticity.
of sheet a by represented be may head in discontinuity the flow, inviscid an For mainstream.
the from head total different of flow a of consists system windmill a of wake The
WAKE THE OF REPRESENTATION VORTEX
4.1
THEORY VORTEX/STRIP ROTORS: AXIS WIND
4 CHAPTER
40
low. fact in is torque the power given a for that so large be must ratio
speed tip the windmill, type propeller a of model approximate an be to this For torque. no is
there hence and wake the in velocity tangential no has system this that observed be will It
disc. the at that twice is wake downstream the in flow induced the
that analysis, momentum the from obtained already result the demonstrating of way another is
This tube. semi-infinite a of end the at that twice becomes flow axial the and removed are flows
radial singular the then system, this to added were strength appropriate of tube vortex another f I
edge. tube the at infinite are flows radial the although section, cross the over flow axial induced
uniform a gives solution the state to except here, these into go not will We end. its at tube
vortex semi-infinite a of flow induced the compute to available are methods Standard
disc. actuator the as diameter same the have will tube vortex the loads, light of hypothesis
an using and developed, is strength constant of tube vortex a assume we if limit, the In
Disc Actuator 4.1 Figure
4.1. Figure in shown as freestream the with downstream convected be will which vortices ring
of series a shed now will disc The motion. rearward the during porous fully and mainstream)
the (against motion forward the during solid is and backwards and forwards oscillated is disc
41
inside located be could induction different completely of annulus Another systems. tube vortex
circular two superimposing by seen be can as annuli, other affect not does system a such of flow
induced the words, other In cylinder. annular this to confined entirely perturbations tangential
and axial with model self-consistent fully a is this mainstream, the to parallel cylinders circular
right considered be may boundaries wake the which in system loaded lightly a For
Shown) Are Blades Two (Only Rotor Multi-Bladed a for System Lattice Vortex 4.2 Figure
itself. disc the at vorticity radial by
connected geometry, similar of lines vortex spanwise and ring of consists tube vortex inner the of
surface the Then annulus. an is disc actuator the that assume to is model this of variant A
4.2). (Figure
cylinder wake the along vorticity streamwise distributed with strength finite of vortex central
a implies this that see we lines, vortex of kinematics the on Laws Helmholtz's satisfy to order In
circulation. blade constant of system many-bladed a representing disc, the of plane the in lines
vorticity radial of number large a with this model can we theory, disc actuator with Consistent
torque. introduces which one is model simple this to add to refinement next The
42
realistic. more becomes sheet vortex bounding continuous a of idealization
the that so packed, densely more be to tern sys helix finite the cause will blades of number large
a or ratio, tip-speed large a that also note We 4.3. Figure of system helix the of components
as considered be could. systems vortex streamwise and ring the where 4.2, Figure of wake
the of that to structure similar somewhat contains model bladed finite this that note We
ratio. tip-speed the to related directly
is vortices the of angle helix The 4.3. Figure in sketched as becomes geometry vortex wake the
then flow, local the to parallel lies vortex the that and tips and root the at only occurs shedding
vortex that Assuming occurs. situation different somewhat a velocity finite a at rotating
and blades of number finite a having system rotor realistic more a consider now we If
models. simple in assumed as head, total reduced of be to flow wake the
permits it Thus sheet. the by separated flows the between head total in differences permits rings
vortex of composed sheet vortex bounding continuous a of concept the that also note We
large. be should ratio speed tip the
and blades of number the of product the that requires valid be to theory element blade interactive
non- for Thus annulus. an of induction the isolates effectively which tube wake vortex the
on vorticity streamwise continuous of idealization the is it that apparent is It stations. other all
at flows induced affect will- station spanwise one at geometry in changes where theory, wing in
situation the from different is this that note We interact. not do annuli that assumes which theory
element blade of basis the is independence annular of result interesting This flows. induced
adjacent affecting without changed be can annuli neighboring in blade the of chord or attack
of angle the and annulus that in geometry blade the of only function a is annulus each of system
flow induced the Thus, first. the in flows induced the affecting without one this outside or
43
here. it
discuss not will we and most than complicated more is model elegant This Model. Goldstein the
termed generally is sheets vortex helical of system a sheds blades the which in model The
wing. ratio high-aspect rolling a as modeled is rotor the where (21), Kuchemann
by given is ratio advance low very at system rotor two-bladed a for analysis An
ratios. advance low at blades few with rotor a for approximation an considered be must theory
element blade that so sections neighboring influence does section spanwise each on load the that
sense the in interaction spanwise is there model bladed finite the for
that noted be will It blade.
the from distance short a adequate quite be may vortex tip strength finite a of idealization the that
so tip the near concentrated is voracity this Generally, edge. trailing the
from shed is voracity
of sheet continuous a and wing yawing a of that to similar quite becomes situation local the Thus
there. washes induced infinite imply would this since tips, the from shed be cannot strength
finite of vortex a Consequently, wing. ratio aspect high a of that to similar is system vortex
and flow the blade each near that illustrate will 4.3 Figure of examination However,
Rotor Two-Bladed a of System Vortex of Idealization 4.3 Figure
(4-4)
44
a. attack of angle local the and W velocity
relative local the upon based coefficients drag and lift sectional the- are
(I)
COS
CD
and CL where
CD sin(I) CL = Cx
ID a' + 1 =
(4-2)
sing) CD ± 4) cos CL
(4-3)
Vco
a
(4-1)
(13,
Cy
tan
1
(I)
=
a
verified be may relations trigonometric following the diagram, the From
direction. downwind the in blade the to normal is y-direction the and x-direction the in
is rotation of plane The
a. attack, of angle local the and 0 angle pitch blade local the to relation
in shown is W, wind, relative the Here 4.5. Figure in rotation of axis the towards looking
tip the from viewed is 4.4 Figure in illustrated rotor turbine wind a of element The
follows. as proceeds station, radial
each at flow 2-D locally assume to seen be can which method, The theory. strip and theory
vortex theory, element blade theory, element blade modified including names of variety a by
called been has velocities induced the with conjunction in used when method This tubes. stream
circular non-interacting in occurs rotor the through flow the that assume to is rotors helicopter
and propellers for calculations performance for method accurate and used frequently A
EQUATIONS FLOW ANNULUS
4.2
45
Element Blade Rotor a for Diagram Velocity 4.5 Figure
Element Blade Rotor 4.4 Figure
vc.
(4-9)
Cdr
2
46
w2
pBc = dQ
is torque blade-produced The disk. rotor the at velocity
angular the twice be to assumed been has stream slip the to imparted velocity angular the where
(4-8)
(2an) (27Erdr)ur p = dQ
obtains one theorem momentum of moment the From
tube. stream differential annular an in element blade the from developed torque the to equated
is considerations momentum angular from determined torque the manner, similar a In
11)
sin2
87cr
(4-7)
1-a
a
BcCY
obtains one
2a, = b factor interference axial wake local the that assuming and expressions two these Equating
(4-6)
Bc = dTB
pW2Cydr
c, chord local having each blades B for while
1
(4-5)
)
u
(Ka
u (27crdr) p
dTm
momentum From considerations.
aerodynamic element blade from predicted force axial the to considerations momentum by
dr thickness of element annular an in generated dT force axial the equating by obtained be may
blade the on developed forces the and a factor interference axial the between relation A
47
A valid. not is analysis above the therefore, (4-10), and (4-7) equations to leading assumptions
the with consistent not is pattern flow a Such occur. may flow recirculating that seen
be may It
4-6.
Figure in below illustrated patterns flow the note procedure, above the qualify
to First, qualification. and modifications some required far so developed expressions The
readily. quite accommodated
be may taper blade and twist changes, section airfoil Blade rotor. the of force axial
and torque overall the determine to integrated be may force axial and torque to contributions
local the and known are properties flow sectional the converges iteration above the Once
repeat and B step to back Go
Cy and Cx
4-4) and (4-3
Calculate
a
(4-7)
Calculate
a'
(4-10)
CD and CL
Calculate
Calculate
Calculate
(4-1)
(4-2)
(i)
Calculate
start) to acceptable is 0 = a' = (a a' and a Guess
S2
V,
0, CD(cc),
CL(a),.
c, r, Given
procedure: following the by determined be may r station radial given
a at conditions flow local the then available, is data performance sectional airfoil suitable If
.2(1) sin 47cr
BcCx
(4-10)
a' 1+
a'
relations these Combining
48
ring vortex (e) windmill; turbulent (d)
windmill; (c) zero-thrust; (b) propeller; (a) Rotor: a of States Working 4.6 Figure
(e)
---s"--"----_,__-------
)d(
Cc)
(b)
c) (
rotor. a of states
ring vortex and windmill turbulent the quantify to results experimental used (10) Glauert 4-6.
Figure in illustrated states various the through pass can descent autorotational to ascent vertical
from going in helicopter, A section. next the in modified be will consideration This occur.
can recirculation , 'A
a> for hence 2a), -
V00(1
= u1 wake the in velocity The considerations.
momentum wake from obtained be may flow recirculating of onset the determining for criterion
49
follows.
as proceeds velocities induced for equation into factor loss tip the of incorporation The
used. be
can solution simple a yields which model Prandtl the blades, more or three involving situations
for results in difference little is there Since use. to difficult more is it functions, Bessel modified
of series infinite an involves model flow Goldstein's since however, accurate more the is
model Goldstein the models two the Of position. radial and blades of number ratio, speed tip of
function a is F factor The blades. of number infinite an with rotor a for circulation corresponding
the is Fc, and r station radial a at circulation the is F blades, of number the is B where
(4-11)
that such F, factor circulation-reduction is approach Goldstein's and
Prandtl's of result The tips. the at interaction wake to due circulation of reduction the for models
developed and contraction) wake (negligible propellers lightly-loaded about flow analyzed
have (12) Goldstein and (11) Prandtl radius. actual the of 97% of order the on characteristically
radius, actual the of fraction some to radius rotor maximum the reduce to being these
of simplest the approaches, of variety a in treated been have losses tip So-called neglected. be
cannot effects wake the however, machines; power wind most for neglected be can acceleration
radial of effects The pattern. flow assumed the alter can tip the at flow wake-induced
and acceleration Radial dimensional. two be to assumed been has position radial any at flow The
vorticity. shed of pattern the of because modification some requires analysis previous The
MODELS LOSS TIP
4.3
50
> aF becomes flow recirculating for criteria the
that noted be may It convergence. for required iterations of number the in reduction significant
and performance higher slightly yields (4-12) of lieu in (4-14) equation of use The
II. Appendix in given is (4-14) and (4-13) equations of derivation The
(I)
(4-14)
S1I12 8
=
aCY
aF)aF
(1
quadratic a becomes (4-12) equation however, same, the remains (4-13) Equation
aF). - V.(1 = u by given then is rotor the through velocity flow average The velocity. wake the
as manner same the in vary to assumed is (4-5) equation in disk rotor the through u velocity flow
axial the in made be can analysis the of refinement further A
a is
.
7tr
a by given solidity local
a quantity the and factor correction tip Prandtl the or correction tip Goldstein the is F where
(I)
(4-13)
cos 4) sin 8F
aCx
(I)
(4-12)
1+a'
a'
a
aC
-
sin2 8F
a
1
become then (4-10)
and (4-7) Equations 2a'FQ. written is change velocity angular the manner, similar in and 2aFV.
becomes 2aV. change velocity the Thus velocity. this of F fraction a only is change velocity
average the and sheets vortex the on only occurs slipstream the in 2aV., or ul) - (V. velocity,
axial of change maximum the that virtually is correction tip the of meaning physical The
51
speed tip low At machines. wind concerning generalizations some illustrate to used be can
also 4-7 Figure ratios. speed tip low at available power the increase can but power maximum
the reduces pitch Increased 4-7. Figure in seen be can angle pitch of effect The
curve. the develop to used was correction tip Goldstein The chord. 11'4" an with
feet 175 was diameter turbine The length. foot 65 a along 50 twisted discontinuously has which
airfoil 4418 NACA an employed turbine Smith-Putnam The ratio. speed tip of function a as
(1) Turbine Wind Smith-Putnam the of coefficient power calculated the gives 4-7 Figure
noted. be should analysis previous the of nature approximate the blade, the along
circulation of distribution optimum with rotor frictionless a for derived been has F factor the As
4)
(4-16)
sin R 2
Rr
=B
f
where
7C
(4-15)
Lexp
r -1
A
cos
2
=F
by
given is factor F Prandtl's calculations. the in factor dominant a being instead correction, loss tip
a be to ceases approach this cases such In blade. entire the over appreciable is loss tip the ratios
speed tip low at that noted be may it consideration practical a As determined. be can F(p,,g,,a)
= F factor correction tip Goldstein the thus and RpIr =
The
cot.*
IA
tiao
is ratio speed tip corresponding
relation the via ratio speed local a defines
4)
angle The /R). r (q), F = F
that so (I), of value local the on F of calculation the bases approach Lock's (13). Lock of method
the following determined be may rotor loaded heavily a for correction tip Goldstein The
12
52
(4-13) and (4-12) equations 00, = °pitch
(4-13) and (4-12) equations 5°, = Opitch
(4-14) and (4-13) equations 00, = °pitch
Turbine. Wind Smith-Putnam the of Performance Calculated 4.7 Figure
R
l0
4
8
0.1
0.2
cq
o
0_
itl>8
0.3
0.4
0.5
stability. the greater the curve, the steeper The speed. original its to
rotor the return will turn in which output power positive in result will speed rotational decreased
velocity, wind constant for since stable, be will (feathered) output power zero at operation
rotor the negative, is 0 = Cp at curve power the of slope the If zero. become will output power
net the ratio speed tip large some at Finally velocity. angular increasing with power in decrease
rapid a in result will coefficient drag high A dominant. becomes drag of effect the coefficient,
power peak the above ratios speed tip At speed. design the below operating when stalled
be can stations, inboard the partieularly rotor, the of much and ratios speed tip low at large is
(1)
angle The coefficient. lift maximum the by influenced strongly is coefficient power the ratios,
53
-= Op Turbine
Wind Smith-Putnam the For Speed Wind Versus Output Power 4.8 Figure
Rn
V.
0.5
0,4
0.2
0.3
0.1
0a.
1.1J
cc
0
N)
a.
In
0
111
10
12
4-7. Figure in shown calculations turbine wind Smith-Putnam the illustrates
4.8 Figure I/X. by given is velocity while Cp/X3 to proportional directly is power the variables,
our as X and Cp Retaining velocity. wind versus power of plot a is performance rotor illustrates
that display of type Another velocity. wind given a for speed rotation and efficiency, output,
power concerning infollnation yields ratio speed tip versus coefficient power of plot A
8) 1 (4-
54
a(1a) = a'(1a')x2
equation energy the is This
required. is relation another variables, dependent two involves power the for integral the Since
Vco
u
o,
V
=a
,
Voo
=
= a' and
RS)
Vco
rQ
=x
where
(4-17)
x2 =
°
dx 3
ix
pVco37CR2 Y2
8
= CP
is coefficient power the for relation The
ratio. speed tip given a for output power maximum the yielding results the constraint; energy
an to subject stationary made is integral power The power. the for integral an up set and blades)
of number infinite an with rotor a to corresponds (i.e., disk actuator rotating a as rotor the treat to
is used approach The windmill. optimum the for model simple a developed has Glauert
GLAUERT ROTOR; OPTIMUM THE
4.4
fluctuations. speed wind to sensitivity
greatest the hence and velocity wind in increase given a for power in increase greatest the give
will coefficient power maximum of point the near Operation coefficient. power maximum of
location the is importance considerable of point Another velocity. starting the controls drag and
output power maximum the controls stall the angle, pitch and RPM constant at that Note
(4-18)
55
a) (1 a = a')x2 a'(1+
that So
(4-19)
a'rf)
a')rS2 (1+ = tan
4\7,0 (1
are These velocity. relative the to normal is velocity induced total the that condition
the under developed be may
tan for expressions Two velocity. relative the to perpendicular
hence and lift to due be must rotor the at induced velocity the drag, of absence .the In
Diagram Velocity 4.9 Figure
Rotation of Plane
Direction Wind
assumed. be may flow two-dimensional conditions, these Under variations.
circumferential no with streamtubes annular in uniform be to assumed is flow The plane.
rotor the at velocities the consider to is relation this illustrating of way unique most the Perhaps
56
afx2
x
0
0
.0584
0.157
.1136
0.374
.1656
0.753
0.031
.2144
2.630
0
.2222
00
0.292
0.812
2.375
00
a'
1/3
.33
.31
.29
.27
.25
a
Disk Actuator Optimum The For Conditions Flow 4.1 Table
rotor ideal an of most that seen be can it more, or 7 of ratios speed tip reach easily rotors speed
high Since 4.1. Table in given are x and atx2, a', a, in variation The 1/4.
1 4a = a,
(4-21)
1) a)(4a (1 = atx2
(4-22)
.
._
a
. ._
1/3 Hence,
that so
3a
yields solution The
a) - a(1
= =
at)x2 + a'(1 0 x) a', G(a, With
"
P
JO =
C
x)dx a' Fla rX
(4-20)
posed now is problem variational The
57
lift. is blade the on acts that
force only the zero, is drag the that assumed have we since course, Of configuration. blade the
to relation in forces and velocities the illustrate to used be may 4.5 Figure determined. be may
(I)
angle the and velocity relative the position, radial each for known are a' and a quantities the
As theory. element blade the using model this from obtained be may infoimation Further
0.5
.288
1.0
.416
1.5
.480
2.0
.512
2.5
.532
5.0
.570
7.5
.582
10.0
.593
voo
Cp
RS-2
Disk Actuator Optimum The For X vs Cp 4.2 Table
1.6. Figure in illustrated is
ratio speed tip with Cp of variation The zero. approaches rotation wake the and 0.593 approaches
coefficient power the ratios, speed tip large At wake. the in energy kinetic rotational large the
of because low is coefficient power the ratios speed tip low At 4.2. Table in given is ratios speed
tip various for coefficient power The
2V
t
rV
2V
02
(2/9)
-->
cor2
a'x2
Go, -->
x as i.e. vortex
irrotational an of form the in distributed velocity rotational the and 1/3 = a with operate will
CL
.7.
58
xa-.=
at maximum a approaches that chord a have will
constant and X given a for blade optimum an that noted be may It results. the gives 4.3 Table
determined. be may blades the of shape the that so x of function a as known are a' and a Now
(4-28)
4)cos4) sin 8nr
sin BcCL
(I)
(4-27)
(I) sin2 87cr
cos BcCL
a' 1+
a'
a
-
1
a'
if,
that So
(4-26)
(4-25)
=
a)a'dr (a 47cr3pVc00 dQ
a)adr (1
4npA72
= dT
2a) = b (assuming yield expressions momentum The
(4-24)
dr
(I)
sin CL 2 rpW
2
Bc
= dQ
and
2
(4-23)
= dT
dr q) cos pW2CL
Bc
by given are c chord
having each blades B containing annulus an on acting torque and thrust incremental The
59
as Now Law. Biot-Savart the of integration direct a is method straightforward most the but
equations, differential partial solving by obtained been have element blade a at volocity induced
the for solutions Several blade. the of element each at velocities induce filaments vortex trailing
the while blade the on lift local the produce to serves voracity bound the scheme, this In
accuracy. of loss without neglected be may flow in variation chordwise the solidity, low
very have rotors windmill most Since line. vortex a of lieu in sheet vortex bound a by replace be
should blade the flow, the represent fully to order In section. blade the over chordwise vary will
flow induced the that note may one However, Law. Biot-Savart the via determined be to section
each at flow induced the enables scheme simple This line. vortex bound a as modeled is rotor
the of blade each wings, to applied as theory vortex of concepts the Following lift. represent
to vortices bound of use the involves model frequently-used A disk. actuator optimum the
describe to used model flow the from respects many in differs real-rotors over flow The
THEORY VORTEX
0.35
.497
1.00
.536
1.73
.418
2.43
.329
10
3.73
.228
7
5.39
.161
5
7.60
.116
4.5
15
20
30
50
27tVcc
(i)
Bcf/CL
Disk Actuator Optimum The For Parameters Blade 4.3 Table
60
propeller. optimum
the define to required techniques analytical cover (26) Weinig and (25), Lerbs (24), Theodoresen
Betz, of writings The surface. screw rigid a as back move to wake the requires criteria
This used. be may (23) criteria Betz the theory vortex using rotor optimum an For
velocities. induced calculate and geometry wake assume we here
forces calculates
and geometry deformation a assumes one where materials of strength elementary in used that
to likened be may approach This wake). the in up rolls also (which sheet vortex a as considered
be also may propeller a by shed wake the wing), the of downstream distance snort a up
roll to happens (which sheet vortex a is wing a from wake the as Just answers. acceptable very
gives theory Prandtl's of results the wake, sheet vortex the thru flow which velocities impossible
physically calculate can one although and wing the of plane the in lie to assumed is wake the
Theory, Lifting-Line Prandtl in Now coupled. are vortices the all unless established be cannot
configuration) (or trajectory vortex the other, each on acting each filaments, vortex of array
large a of superposition the of consist will wake the Since wake. the in filaments vortex the of
trajectory the of knowledge the input an as requires it appear, may method this as straightforward
61
increased. is number blade and ratio speed tip the as cancel will component vorticity streamwise
the is, that system; type ring simple a to converge will sides the on system crisscross the
that Note 5.2. Figure in shown as appears system wake final the Thus occurs. situation similar
a portion forward the across passage On path. the of portion upper the over lift zero assumes
it as vortex bound its sheds finally and shown, as pair trailing a and vortex starting a sheds it
sector, leeward the up moves blade a As 5.1. Figure in shown is model This path. the to relative
attack of angle zero at and velocity constant at path square a follow to constrained are but axis,
the about path circular a in move not do these assume simplicity for and blades, lifting slender
having machine axis crosswind a consider model, realistic more somewhat a construct To
system. ratio speed
tip high many-bladed a for model acceptable an be may this that see we Again develops. disc
actuator rotor the of that to similar system wake a and shed are device the as section cross of disc
actuator oscillating an as simply modeled be to device the assume we if that first note We
literature. the in discussed been not has this since actuator, axis crosswind a of system
vortex the construct to interest of is it 4.1, Section in discussed approach the Continuing
WAKE THE OF MODELING VORTEX
5.1
MACHINES AXIS CROSS-WIND
5
CHAPTER
62
Actuator Axis Cross-Wind of Shedding Vortex 5.1 Figure
ION ROTAT
OF
AXIS
V..
WIND
63
uniform. not is flow axial induced the
since system, axis wind a of coefficient power ideal the achieve cannot machine cross-axis ideal
an even that appears it Thus uniform. not is flow axial internal induced the that and tube the
across distributed linearly is vorticity spanwise this that shown be easily can It tube. the within
vorticity spanwise internal is there since case previous the from different importantly is This
5.2. Figure in sketched as system vortex wake a obtain now we ratios, advance high assuming
and system path square the for used those to similar arguments Adopting changing. continually
is vorticity starting and trailing shed the consequently and lift the then axis, fixed a about rotate
blades the where system axis crosswind a of case realistic more the consider now we If
flow. external zero and
flow, axial internal uniform a section, cross circular of one of that as same the is section cross
arbitrary of tube vortex infinite an of induction the for solution the that shown be can It
Actuator Axis Crosswind Bladed Single of System Vortex 5.2 Figure
,
;-1\
1
.--
,
1
1
,
WIND
64
R
a sin 2rc
--=
c
CL
0 = CD
0=p
assumptions, following the adopt shall we analysis the simplify to order In
5.3. Figure in illustrated
system flow the giving a), - V.0(1 velocity has flow incoming the itself device the at then
2aV,o, is perturbation windwise wake the if that obtain we Thus wake. the in value their half
one are device the at flows induced the assume will we theory, vortex with Consistent
side. the to deflected be can wake the that so force, cross-wind
a as well as windwise a experience can devices these general, In forces. wake the to equated
be may station each at device the on forces the that sense the in quasi-independently behaves
station axis) the to (parallel spanwise each that assume we considered device the For
flows. induced the determine to equations
sufficient obtains one forces wing and wake the equating By flows. induced unknown contains
forces these for expression The itself. surface lifting the at theory airfoil an by as well as
wake the of analysis momentum a by system the on forces the express to is which theory, wing
of approach standard the adopt we device crosswind-axis Darrieus-type a analyze To
ROTOR DARRIEUS
5.2
65
Turbine Crosswind-Axis a for System Flow 5.5 Figure
Wind
66
in dO/Q of increment time a spend will airfoil the period, time this Of 27c/Q. is which period one
be shall analysis our of interval time the so revolution every itself repeat will process The
(5-5)
I
0 sin Rd01 =
by dO to related is dx width The dO. + 0 position to
0 position angular from goes airfoil the when dx width of be streamtube the Let occupies. airfoil
the which streamtube the in lost momentum the to airfoil the on force the equate can we Now
(5-4)
ij 0)
cos 0 sin vavt + 0 sin
(v ej sin2 vavt
one
obtain we
p*xf = P
(5-3)
as expressed be can airfoil the on force the Since
a sin 7tcW =
(5-2)
2
WCL
=F
that so
(5-1)
= pWF = L
pW2cCL
write can we law, Kutta-Joukowski the with starting
and assumptions above the Using stall. near aerodynamics nonlinear the model to analysis
numerical requires performance ratio speed tip low The small. is a attack of angle maximum
the where ratios speed tip high at analysis inviscid an to limited be then will results Our
(5-10)
0 sin2 a)2
is blades B with rotor a for torque average The
K20
pncR = Q
by given is torque The
components. radial and torque into resolved be may force blade the defined, is a that Now
(5-9)
Kt, 2R
sinOI RS2 Bc = a
blades B for or
(5-8)
I
0 sin
c
RS2
2R
Vco
a=
blade one for a factor interference axial the for expression
an yields RO = Vt and a) - V.0(1 = Va that assumption the under forces two these Equating
(5-7)
20n
a 2K0 a)Ko
(1
I
sin pRd0 =
0
I
dFmomentum
as streamtube the in force the yields equation momentum the Now
(5-6)
dO
0 sin2 2pncVtVa = dFblade
as 2760 period time the for equation force blade the write may we 0 ± angles the to respect with
symmetrical be to seen is (5-4) equation from contribution force streamwise the Since wake.
the of portion rear the in
dO/S2
of increment time another and streamtube the of portion front the
then is torque drag The R.Q.
68
W is velocity local the that assuming by approximated
simple be may losses drag the speeds, tip low at operate not will rotor of type this Since
3. about of ratio speed
tip starting a have would radius foot 20 a and chord foot one a with rotor three-bladed a that so
Bc
(5-14)
1+2
Xstart
4
obtain we max 14° =
a. Using
tip starting the express to (5-13) equation rearrange may we am to equal set is
(5-13)
2R
Bc
X
X
am
1
ratio. speed
a When
a tan
where
Tr/2
=0
point the at approximately occurs attack of angle maximum The attack. of angle maximum
and drag of consideration with made be can refinements Further 0.401.
am= = BcX/2R
quantity the when 0.554 of coefficient power maximum a yields expression This
32
R2
B2c2X2 3
(5-12)
R 37r
BcX 4
Bc
=7rX Cp
by given is coefficient power sectional corresponding the and
)
(5-11)
32R
(BcX)2
3
R
37T
BcX 4
[pnBcRY,20]
= Q
69
rotation. of axis
the and tangent blade the between 7 angle local the and curve troposkien the illustrates below
5.4 Figure reduced. are lift the of component usable the and speed rotational the both since
substantial is rotation of axis the to closer blades the bringing by caused performance on effect
The parabola. or curve sine a by approximated is and integrals elliptic by described is curve
The troposkien. name the given and (22) Blackwell by investigated been has shape required The
tensile. entirely are loads the that so catemary the to similar shape a in blade the deploying by
removed be may loads bending These blades. the in loads bending substantial in result and large
are loads inertial the rotor, Darrieus-type the for required speeds rotational high the At
ROTOR CIRCULAR THE
2
D
(5-16)
62
\ CD
4
13n
3
C
3
=C
5.3
Cp that seen be may it and
noX = C
a3X3 + 2 X 2 16
becomes coefficient power The diameter.
disc to circumference blade of ratio the Bc/2R, =
2R
(5-15)
X3
CDBc
-=
a as defined be may solidity a
p
point, this At
AC
is coefficient power the to contribution the and
2
(5-14)
BCX3
pVcc,
D
3
C
= BcR f2 pR
2
2
2
CD
QD
(5-18)
y
8
3o-2X2CS2
+ y cos aX
70
32
% y cos 2pnBcVco =
dz
dQ
is axis rotor the along dz slice a by generated torque The R. of independent is product this
since
Rmax
= R point the at ratio speed tip and solidity the as taken be may crX product the where
0 sin y cos aX = a
(5-17)
becomes a for expression the rotor, the of height unit a analyze
we If 5.2. Section in as manner same the in proceeds rotor curved the of analysis The
Length Equal of Caternary and Circle Troposkien, 5.4 Figure
R/Rm
0.5
1.0
1.5
,
0.5
Circle
Catenary
Troposkien
1.0
71
rotor. wind-axis the for developed that to similar theory element blade a of development
by model above the in included be can stall and drag of effects The 0.461. = amax = erX
at occurs 0.536 of coefficient power maximum a which for blade circular the is case simple
one accomplished; be may geometry arbitrary an for (5-19) equation of integration The
(5-19)
y
COS2
8
a2X2
+ y cos GX
3n
72--8
y cos R
A
4rcaX
dzy2pVA
2
ciQ
dz
p
dC
is coefficient power incremental the and
U
73
height with relation law
power a by approximated frequently is terrain rough over flow mean the contrast, By length.
Monin-Obukhov the parameter, stability a and roughness surface the velocity, friction surface the
parameters, three involved relation Their stratification. unstable and neutral stable, encompasses
that flow mean the for relation a developed have (14) Obukhov and Monin terrain flat
over layer surface atmospheric the in structure wind of made been have studies Extensive
complete. more much is terrain
flat over flow of knowledge the contrast By terrain. rough over layer surface atmospheric the
in turning wind considerable of presence the reported has (13) Slade turning. wind and spectra
turbulence on data no or little is there terrain, rough over flow for exists data gradient wind
some While bather. considerable a represents terrain rough of regions in particularly conditions,
flow local of knowledge of lack The machines wind to adapted be must variations flow
the of effects the of magnitude the predict to techniques secondly, known; be must conditions
flow local the First, turbines. wind of operation and design the to difficulties double-edged
present all elevation with turning and-wind gustiness gradient, wind Vertical unidirectional.
nor steady uniform, neither be will flows Real occur. can motion rotor and irregularities
flow both reality, In rotor. the of rotation of plane the to perpendicular be to and parallel and
unifoim be to assumed was wind relative the previously, covered rotors of analysis the In
INTRODUCTION
6.1
GRADIENT WIND VERTICAL TO DUE MOMENTS AND FORCES
6 CHAPTER
74
coordinates.
(t,n) the now are 4 Chapter of (x,y) the Thus XYZ. coordinates the with confusion avoid to order
in (t) tangential and (n) normal designated been have element blade a on forces incremental the
noted be should it further, proceeding Before significant. is that rotor the of bottom and top the
between difference velocity the is it since scale, upon dependent is moments these of magnitude
the expected be would As moment. pitching the and variation torque the are these of largest
The 6-1. Figure in illustrated as moments and forces induce will gradient wind vertical A
GRADIENT WIND VERTICAL OF EFFECTS THE
6.2
I. Appendix in described program
the in included are forces and moments flapping (15), Ribner of analysis the using treated
be can yaw Rotor moments. and forces large induce can flapping and yaw rotor Both
gradient. wind to due loads aerodynamic the
with only deal will section This present. are moments and forces mechanical certain moments
and forces aerodynamic the to addition In turbine. wind a of operation and design the both hence
and dynamics system overall the effect will moments and forces These occur. can moments
pitching and forces side dependent time torque, in variations periodic however, output, mean
turbine the on effects order first no have previously studied flow the from departures The
relation. above the use shall we turbines wind for required is feet)
200 to 100
range limited a over variation wind the that fact the and parameters, fewer requires
it since (6-1), equation of simplicity the of Because one. than less is
-
\hJ
(6-1)
11
coefficient the where
VR
(Z
75
steady- the from generated be can moments and forces order second and first the for Expressions
rotation. of angle the with df',1 and dFt force the in variations cause
of angle in variation
The attack. of angle increased an be
increased of effect principle the that seen be may
will turn in attack
will gust) or gradient
it ratios
speed tip high
to (due velocity
At a. attack of angle
the and w velocity resultant the both increases Vw in increase The disk. half lower the in blade
a for than higher be
will velocity axial
the disk, rotor the of half upper the in blade a For
V. velocity wind local the with
replaced been has V. velocity freestream the that in illustrations previous from differs diagram
velocity This diagram. velocity blade the illustrates page following the on 6.2 Figure
Gradient. Wind a in Rotor 6.1 Figure
MOMENT PITCHING
AG
CI:1)R
X
TORQUE
(6-3)
2
Z2 A
76
T2 + Z TlA + =1T dFt
form the in (6-2) equation Rewriting blade. ith the of rotation of angle the to
refers
Ai
where 0i, rsin = AZ distance The elevation. with velocity wind in variation the to due 0)
= (Z values hub their from change forces These dFn. for obtained be may expression similar A
Vref
=u
and
2
au
dFt
,
2
zz
'AZ
u +dF, u2
+ AZ
[dF,
z
Uz dFtu
+
0
au = u
dFt
aFt
a2Ft
2
(6-2)
dFt =
where
dFt
that so hub rotor the about Series Taylor a of terms in expressed be may element
rotor a on force differential The manner following the in aerodynamics performance state
Diagram Velocity Blade
-a)
I
6.2
Figure
( Vw
t
77
NI. of evaluation by approach the
indicate us let straightforward, is differentiation of method the As moments. and forces the of
magnitude the determine to order in evaluated be to remain
T2
and
T1 N2, N1, terms
The
blades.
of numbers various for summations the of values the gives page next the on 6.1 Table
moment. yawing order second or first no induces gradient
wind that noted be may it and evaluated been has blades spaced equally B over summation The
odd is
B unless
(6-6)
jdOi
dr
2
+
fr3N2
i=1
Oi
2N1drIsin2
MOMENT PITCHING
i=1
(6-5)
s
r3T2dr
sin2
E fr2Tidr + drirTo = Q
TORQUE
2.
B where blades,
B over summing and blade the over integrating by moments and forces following the obtain we
2
(6-4)
+N2 N1AZ + dF=N0
z2
manner same the in dFr, expressing and
(6-9)
AZ
az
Z=0
4)
au
(I))
78
cos
sin2 + CD
sin++ cos4)+CL
a
CL pVRcv
2
= N1
is result final The
all
(6-8)
13
=
(I)
La
c
al
a+
aa
acL
a4)
au
acL
VR
As comment. some requires CL of ation differenti The-
Z=0
Z
(6-7)
sinliloauzi
CD
cos++
)(CL v2
sin3Qt
0
+
f(u2
v where
rS,
and cc +
constant remains
(z). vw
Z=0
au
a pA/;'c
= AZ
= u and
3/2
2
30t cos ---
0
i3
VAT
az au =
aU a(dFn)
N1
i=1
4
0
3
Oi
sin3
i=1
4
3 =B
4 =B
0
O1
cos
Oi
sin2
i=1
cos2Qt
0; sin2
1
2=B
Sums Trigonometric
6.1
Table
79
obtained. be may expressions following
the that shown be may it drag neglecting by and
(1)
situ')
ratios speed tip high At
RELATIONS APPROXIMATE
6.3
obtained. be can approximations form
closed case, this In blades. the of portions outer the over constant are c and CL of value the and
small is (I) angle the if made be can simplification considerable (6-12), and (6-11) equations from
seen be can As moments. and forces other the by obtained are integrals type Similar
hub 's
(6-12)
R
tr)4
d(-1.)
R
R
)(1.
TC
a' + sind)1(1 CD
(I)
sin CLa
sin24))CL
+ Lk1 r
ft
=
X
12
and
R,RR1
(6-11)
N
7
C
(f3 ar9\
"-hub
'it
cc
(1)CD
cos3 2
(I)CL
cos 2 (I)+ sin
cl))CL COS2
as evaluated are
Z=0
VRdZ2
(6-10)
Z=0)
+12
dZ VR
12
1.k2 ,j
ft
=
1
and I integrals the where
= CQ A, = Y2pVinR3
QA
dVw
d2Vw R2
analysis preceding the from evaluated be may coefficient torque in variation The
=120' hh
Z=0
,
0.14
80
2 dZ VR
d2Vw
1
hh
0.104
Z=0
dZ VR
dVw
1
yield data wind term long The turbine.
wind Smith-Putnam the from data the using by obtained be may values representative Some
4B
a)
OaCLa
5B
(1
crXCL
12
and
2B
2
a
(1
GCL
where
i
2
Z=0
Z=0
i=1
(6-14)
0
BE
VR(I7,2
S111
dZ VR
+12
R2d2Vw (
2
\
1
=
3
TCR pV12(.'
RdVw
QA
CHANGE TORQUE
i=1 Z=0
(6-13)
Oi
sin2
dZ VR
RdVw a)
4
La
2
pV127ER
M
crXC
MOMENT PITCHING
P
0.4 = C
81
X =Q
C
,
pVircR3
cP
3
2
CQ = Q
=
pV0TR2 Y2 CT D
CT
torque. the to moments the and turbine wind
the of drag the to forces the reference us let Accordingly, judge. to difficult are numbers The
(6-16)
Mt) (1cos
0.00153 =
TcR3 Y2pV12:
QA
VARIATION TORQUE
(6-15)
20t) (1cos
0.0173 =
Y2pV12(.7cR3
MY
MOMENT PITCHING
machine, Smith-Putnam the of motion flapping the neglecting obtain, we
0.33 = a
5.5 =
0.6 = CL
a
CL
0.083 = a
ft. 87.5 = R
6 =X
2=B
below listed values the using and
82
0.17.
r yields terrain smooth over
flow for evidence Experimental 1/6. = ii when value maximum a has expression This
3.
to equal approximately is (6-19) equation in expression bracketed the that noted be may It
appreciable. be will
scale large for variation torque The
C. low and X large have will point design the below speeds
wind at RPM constant at operating rotor a example For (6-19). equation from estimated maybe
load and solidity ratio, speed tip scale, of effects The hub. rotor the of height the is h where
(6-19)
ri
sin2
+111
a)CL 5(1
4XCL
}
+ 2,92
h
4Cp
CQ
(112 a)2CLa Xo.(1
obtain we (6-1) equation by given profile aw
1
CQ A
power the Adopting
change. torque percentage the express to modified be may which change
torque the for expression an gives (6-14) Equation discussion. more requires itself variation
torque The power. also hence and torque in decrease 4.6% a to up be to seen is variation
torque The torque. the of value the of 52% of variation a to amounting appreciable, quite
be to seen is moment pitching the while insignificant quite are forces drag and yaw The
(6-18)
200
cos
(1 =O.023 Q A
VARIATION TORQUE
200 cos (1 0.259
(6-17)
MOMENT PITCHING
83
4.8. Figure with along 6.3 Figure of results the using
by obtained be may gradient to due variation power the of magnitude absolute The example. this
in included not was blades the of motion Flapping turbine. wind Smith-Putnam the for gradient
wind to due output turbine mean in decrease percentage the illustrates page following the on 6.3
Figure zero. approaches output turbine net the as greatly increases gust) (or gradient wind to due
output turbine in variation percentage the that so speed tip with appreciably changes however
turbine wind a of output net The ratios. speed tip of range wide a over constant approximately
is gradient wind the to due torque in variation The gradient. wind to due output power in change
the evaluate to used be also may variation torque the for developed expressions The
14
84
Gradient Wind To Due Output Power In Reduction Percent 6.3 Figure
Vco
RS/
12
10
RATIO, SPEED TIP
6
8
4
0
2
4
6
8
10
12
120)
z16
(si
)1
vi
Velocity Wind
- 14
Neglected Flapping Blade 813=0°
_
Turbine Wind Smith-Putnam
16
85
1972. No.3, 17, Vol. Soc., Helicopter Amer. J. Descents. Steep
in Helicopters for Boundaries Ring Vortex of Prediction Analytical J., Wolkovitch,
1955. York, New McGraw-Hill, Engineering, Helicopter of Principles J., Shapiro,
1948. D.C., Washington, 820, Report NACA Yaw, in Propellers S., N. Ribner,
163-187. pp 1954, 24, No. 151, Vol. Trudy, Institut, Geofizicheskii
Leningrad, SSSR, Nauk Akademiia from translated Atmosphere, the of
Laver Ground the In Mixing Turbulent of Laws Basic M., A. Obukhov, and S. A. Monin,
293-7 p. 1969, April 8, Vol. Meteorology, Applied of Journal H., D. Slade,
1929. 440, p.
123, (A) Proc. Soc. Roy. Propellors, Screw of Theory Vortex the On S., Goldstein,
1919. 193-217, p. Nachr. Gottinger
Betz, A. by Energieverlust, gerngstein mit Schraubenpropellor to Appendix L., Prandtl,
1926. 1026, M & R Br, Airscrew, an of States Ring
Vortex and Brake Windmill the in Results Experimental of Analysis The H., Glauert,
6,
,1889. 390 p. 30, Vol. Architects, Naval of Institute Transactions, R.E., Froude,
7.
1918. Moscou, de Technique Superieure
l'Ecole de Aeronatuiques Essais et Calculs des Bureau du Travanx E., N. Joukowski,
8
543-4. p. 1972, April, No.4, 10, Vol. Journal, AIAA M., P Goorjian,
9
1935. Berlin Springer, Julius 324. p.
L, Division 6, Vol. Editor-in-chief), Durand, F. (W. Theory Aerodynamic H., Glauert,
1878. 47, p. 19, Vol. Architects, Naval of Institute Transactions, W., Froude,
1865. 13, p. 6, Vol. Architects, of-Naval Institute Transactions, J., W. Rankine,
1956. York, New
Library, Philosophical Power, Wind by Electricity of Generation The W., E. Golding,
1946.
31, January D.C., Washington, PB25370, Board, Production War Development,
and Research Production, of Office Turbine. Wind the on Report Final N.Y.U.,
1948. York, New Inc. Company, VanNostrand Wind, the From Power C. P. Putnam,
References
86
Command. Atl. Air Div. Documents
Air Ohio: Dayton, Author. by revised and Luftschraube der Aerodynamik
,
book, German 1939 from trans. Propeller, the of Aerodynamics 1939/1948,
Barth. J.A.
Leipzig: Turbomaschinen von Schaufeln die um Stromung Die 1935, F., Weinig,
73-117. 60, Engrs., Marine
and Architects Naval Soc. Trans. Circulation," of Distribution Arbitrary an
and Blades of Number Finite a with Propellers Loaded "Moderately 1952, W., H. Lerbs,
1948.
York, New Inc., Co.; Book McGraw-Hill Propellers, of Theory Theodore, Theodorsen,
68-92. pp. 1943), Bros., Edwards Arbor: Ann (reprint 1927 Gottingen, Betz,
A. and Prandtl L. by Aerodynamik und Hydrodynamik zur Abhandlungen Vier in
reprinted 193-217; pp. Klasse, Math.-Phys. Gottingen, zu Wiss. der Gesellschaft
Kgl. der Nach. Energieverlust," Geringstem mit ler "Schraubenpropel 1919, A., z, Bet
Mexico. New Albuquerque,
1974, April, SLA-74-0154, Report Energy Laboratories Sandia Turbine, Wind
Vertical-Axis of Type Troposkien a for Shape Blade E., G. Reis, and F., B. Blackwell,
1953.
York, New McGraw-Hill, Propulsion, of Aerodynamics Weber, J. and D., Kuchemann,
1973. 10-12, October Alberta, Calgary, Engineers, Agricultural of Society
American the of Region Northwest Pacific the of Meeting Annual 1973 the at
Presented Canada, Ottawa, Council, Research National the at Developed Turbine
Wind Vertical-Axis the of Economics and Performance The R., Rangi, and P. South,
1956. 102, No. Report CoA Cranfield, Windmills. Ducted of
Performance and Design the on Report Preliminary A Rainbird, J. W. and M. G. Lilley,
1925. 1014. M & R. A.R.C., Br. Results. Experimental
Including Pitch, Small of Airscrews in Applications with Airscrews of
Theory Vortex the of Extension An Townend, H. C. H. and Bateman, H. H., N. C. Lock,
rotor the at velocity flow Axial
area surface projected translator or Rotor
radius rotor Local
radius Rotor
Pressure
air the from extracted Power
L'
length unit per force lift Sectional
th
rate flow Mass
LID ratio, drag to Lift
CL
coefficient lift Sectional
CD
coefficient drag Sectional
Cx
rotation of direction the in coefficient force Sectional
Cy
rotation' of plane the to normal coefficient force Section
Cp
or 13/(l/2)pAK3 coefficient, Power
D'
length unit per force drag Sectional
PA1/2)pSVG,3
chord Blade
1-u1N.,,
b wake the in factor interference Axial
blades of Number
A
area disc rotor Projected
a
a rotor the at factor interference Axial
a'
co/20 -= a' rotor the at factor interference Tangential
u/Voc,
-
1
Symbols
88
rotor the of downwind velocity angular Fluid
velocity angular Rotor
density Fluid
6 5, Chapters angle, rotation blade 4; 3, Chapters angle,
pitch Blade
velocity Translator
(i)
velocity relative the and rotation of plane the between Angle
11
exponent relation Wind-height
5,6. Chapter angle,
pitch blade 2; Chapter velocity, translation the to normal the and wind the between Angle
attack of Angle
Greek
rON., ratio, speed Local
R.Q/Vc. ratio, speed Tip
X
element rotor the to relative velocity Resultant
velocity wind stream Free
wake the in velocity flow Axial
I XICENHddV
89
specified. conditions Operating
radius.
percent of function a as twist and chord i.e., specified, be must geometry Blade
0012. and 4418, NACA than other
if used, section airfoil to conform to rewritten be must NACAXX Subroutine
made. changes
and/or inputted be must infointation following the geometry propeller given a For
changes. program other without placed be can section airfoil any for
curvefit a which for subroutine empty an is NACAXX 4418; profile NACA for coefficients drag
and lift sectional determines NACA44 subroutine 0012; profile NACA for coefficients drag and
lift sectional determines NACA00 subroutines functions; Bessel modified calculates BESSEL
subroutine factor; hub-loss the arid factor loss tip the calculates TIPLOS subroutine parameters;
related and factors interference angular and axial determines CALC subroutine station;
given a at angle twist and chord calculates SEARCH subroutine output; for titles and listing
input prints TITLES subroutine functions; output and input integration, performs PROP program
main the subroutines: eight with program main a of consists package program The
respectively. punch card and
printer, line reader, card the to refer program the in 62 and 61, 60, numbers unit Logical
systems. other on implementation for differences
small some be may there therefore University, State Oregon at package operating OS-3
the under 3300 CDC a using written was It integration. numerical of method pass Rule/three
Simpson's a utilizes and language Fortran the in written is package program The
PROGRAM LOADS AND PERFORMANCE TURBINE WIND
90
degrees - THETI(I)
- HL
Prandt - 1
None 0-
1
stations for angle Twist
ft (CI(I)) stations dor Chord
(I) RR stations for radius Percent
- GO
None 2Goldstein 1
Prandtl - 0
NACAXX subroutine in be to curvefit Profile - 9999
0012 NACA - 0012
4418 NACA - 4418 -
controller mode loss Hub
controller model loss Tip
NPROF Profile NACA
NF specification geometry blade for stations inputted of Number
degrees - SI
angle Coning
ft - H level sea above Altitude
10
original
tl
seeAppendixII Wilson,
AMOD
Code Model Interference Axial
X ratio speed Tip
mph - V velocity Wind
B blades of Number
degrees - THETP
DR
angle Pitch
incrementation) integration for radius of (percent Percentage Incremental
ft - HB
ft - R
radius Hub
blade of Radius
are: inputted be to parameters The
91
program. the to changes
without profiles other use to user the enables This (degrees). attack of angle of function a as
coefficients drag and lift sectional for fit curve a add must one subroutine a is NACA)0( used.
be to is NACAXX subroutine means 9999 of input an 4418; profile NACA the for data drag and
lift sectional calculate to used be to is NACA44 subroutine means 4418 of input an 0012; profile
NACA the for data drag and lift sectional calculates that subroutine a is which used, be to is
NACA00 subroutine means 0012 of input An used. be to is method Prandtl's means 1.0 of input
an while used, be will model loss hub no means 0.0 of input An model. loss hub the controls
it HL; is controller third The used. be to is model loss tip no means 2.0 of input an and model,
Prandtl's choose will program the then two, than greater is blades of number the unless used, be
to is model Goldstein's means 1.0 input the used, be to is model Prandtl's means 0.0 of input An
model. loss tip the determines GO used. be will form root square the means 1.0 of input an while
used, be also will form Glauert the means 0.0 of input An used. be will factors interference
angular and axial of calculation for method which determines AMOD NPROF. and HL,
,
GO, AMOD are These -inputted. be must that controllers operation several are There
width. character-field 140 of basis the on printed be will Output
(I) THETI
CI(I)
I) RR(
NF) Card
4 5 (Card
V
81
AMOD
X
NPROF
GO
(21-22) NF
H
HL
4 Card
3
Card
B
2 Card
TBETP
HB
DR
R
1
Card
41-50
31-40
21-30
11-20
1-10
Columns
format: following the in be should Input
92
form. binary
to program the convert to desirable be would it desired, are runs many if Therefore, 3300. CDC
the on compiled be to seconds ten approximately takes program the that noted, be also should It
10
(4F10.3) FORMAT
R,DR,HB,THETP (60,10) READ
10
(4F10.3) FORMAT
B,V,X,AMOD (60,10) READ
30
(2F10.3,12,8X,F10.2,14) FORMAT
H,SI,NF,GO,NPROF (60,30) READ
40
(F10.3) FORMAT
HL (60,40) READ
20
(F5.1,5X,F10.5,F10.5) FORMAT
(RR(I),THETI(I),I=1,NF) (60,20) READ
used. statements Format and Read input the are following The
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)al'Id1WSIAIOH):1'VD31A10109/IS'WOOLAJVidlaHl'X'AIEVEIH1210121 NOIA11610D
(SZ)I131-11'(SZ)IDi(SZ)1111 NOISNANIO
Indlno
JAIldIliDSgO
5170T
6L1
8LT
D
D
LLT
9L1
SLT
17L1
NVQ:11110A LSO
97
END
RETURN
C
95
1XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX#)
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX FORMAT(//#
58
70
59
FORMAT(//6X,#MY#,16X,#W#,15X,#Q#,11X,#CT#,13X,#T#,9X,#CP#,13X,
1CX=,9X,#CY=,9X,#FX#,8X,*FY#,12X,MX#)
FORMAT(///4X,#R#,10X,#PC-R#,8X,#F,7X,#ALPHA,9X,#CL=,9X,=CD#,9X,#
FT=,18X,#FN#) FORMAT(///9X,=A#,14X,#AP*,26X,
200
779
778
659
101
100
320
310
=#,I4) PROFILE #,F20.5,10X,=NACA = RPM FORMAT(//#
USED#) MODEL HUBLOSS NO FORMAT(//#
PRANDTL#) BY MODELED HUBLOSSES FORMAT(//#
USED#) MODEL LOSS TIP NO FORMAT(//#
*) FORMULA GOLDSTEINS BY MODELED LOSSES TIP FORMAT(//#
#) FORMULA PRANDTLS BY MODELED LOSSES TIP FORMAT(//#
USED#) METHOD INTERFERENCE AXIAL STANOARO FORMAT(//#
USED#) METHOD INTERFERENCE AXIAL WILSON FORMAT(//#
NO#) PHI#,10X,=RE 1#P#,9X,
1HA#)
ALP
DEGREES -- ATTACK OF ANGLE F#/ -- FACTOR LOSS TIP FORMAT(#
-- NUMBER REYNOLDS #/# PHI -- DEGREES -- PHI ANGLE FORMAT(#
NC:1#) RE
P#) -- KILOWATTS -- POWER CP# -- COEFFICIENT POWER FORMAT(#
--T#) LB -- THRUST CT-#/ -- COEFFICIENT THRUST FORMAT(#
Q#) -- FT-LB -- TORQUE --W#/# FT/SEC -- VELOCITY RELATIVEC FORMAT(=
MY#) -- FT-LB 1-MOMENT-X-DIR/BLADE FORMAT(=
-
MOMENT-Y-DIR/BLAOE MX#/# -- FT-LB
LB
-
CY#)
71
69
68
67
66
65
1--FY#)
FORCE-Y-DIR/BLADE FX=/# -- LB -- FORCE-X-DIR/BLAOE FORMAT(=
FOKCE-X-DIR OF COEF FORMAT(#
FORCE-Y-DIR OF COEF CX4=
00#) -- COEFFICIENT DRAG CL=/= -- COEFFICIENT LIFT FORM4T(#
--FT#) FORCE TANGENTIAL FN4= -- FORCE NORMAL FORMAT(#
AP#)
FA
64
63
62
169
1CTOR
INTERFERENCE ANGULAR A4# -- FACTOR INTERFERENCE AXIAL FORMAT(#
PC-R*) -- RADIUS PERCENT R#/# -- FT -- RADIUS FORMAT(#
RECORD4///) OUTPUT DATA FORMAT(///////#
FORMAT(//5X,F5.1,8X,F10.5,10X,F10.5)
1)
ANGLE-DEGREES# RADIUS#,5X,=CHORD-FT=,5X,#TWIST PERCENT FORMAT(///#
1)
==,F7.3,5X,#NUMBER 1ANGLE-DEGREES
=#,I2
61
60
57
56
55
SPAN ALONG STATIONS DATA OF
LEVEL-FT=#,F10.2,5X,#CONING SEA ABOVE SITE OF ALTITUDE FORMAT(//#
=#,F6.3) RATIO SPEED 110X,#TIP
=#,F7.2, MPH - VELOCITY =#,F3.0,10X,#WIND BLADES OF NO. FORMAT(/
=,F7.4) DEGREES - ANGLE RADIUS-FT=#,F5.2,10X,=PITCH 110X,A-IUB
=#,F6.4, PERCENTAGE RADIUS-FT,F7.2,10X,ANCREMENTAL FORMAT(///#
#) RECORD INPUT DATA FORMAT(///#
1INE#)
1045 05/17/74
54
53
52
51
279
278
277
276
275
274
273
272
271
270
269
268
267
266
265
264
263
262
261
260
259
258
257
256
255
254
253
252
251
250
249
248
247
246
245
244
243
242
241
240
239
238
237
236
235
234
233
TITLES 3.12 VERSION FORTRAN 0S3
CIN3
'081/Id*I3H1=1.3H1
(dN)II3H1=13H1
CAN)ID=D
0.17
01 09
017
OE
(1-C)I12H1+((T-C)I13H1-(Z-C)I13HIMI3d=13H1
(T-0I3+C(T-0I0-(Z-C)ID)*?Ad=3
UT-01121-(Z-02111)/00T-02111-A21):1)=112d
T+I=C
311NIINOD
OE
OT
01 09
01 09 (AN'bI)AI
s3znun II
CINV CidOHD
(0,01:11:1'39*Al:W)AI
3110INI-1331 N0I1V10dli3INI
NVdS 3H1 9NO1V srucivi NDAI9 V
SBNIIAI213130 - HDI:IVDS
3HI
ZIE
i1/LT/S0
ZOE
TOE
E6Z
Z6Z
16Z
L8Z
98Z
S8Z
3178Z
3E8Z
3
3Nun0bens
N0IS:13A
00E
66Z
86Z
L6Z
96Z
S6Z
-176Z
OT
OZ
06Z
68Z
88Z
'001*((IS)S0D*X21)/Th=A111
AN'T=T OZ 00
X2I'Id'1WSIA'01-1):11V9DIA10109/IS'WCIOIAIV`d13HI'X'A'819W1101?1 NO141403
(SZ)II3H1j(SZ)I31(SZ)211:1 NOISN3LAII0
?3VDNI1 V
IV 319NV ISIMI 3H1
(131-11101ANIIIAH1'I3'Til1iTh)H321V3S
SNIT
Z8Z
T8Z
08Z
NV2111:10A ESO
)39011d BM_ NI C131;101S
66
(IIVA-(II-IcI)SOD*(IHd)NIS*A)/AVA=dV
('elAA+Z**(IHd)NIS*A)/1113A=V
X3*DIS*SZT'0=1:1VA
19E
09E
6SE
SSE
((A*A-1-219A)*.Z)/((NVD)111OS-A.21E1A*.Z)=V
LSE
9SE
(11VA'T)/2/VA=dV
85E
A3*DIS*5ZT'0=):18A
089
01
SLS
OD
Li7E
(I1-1d)SO3*CID-(IHd)NIS*13=X3
8.17E
0'0=NV3 (0'0'11'NV3)1I
(A--T)*A*V3A*17+A*A=NVD
C(IHd)SO3*(IHd)NIS*A)/(XD*DIS*SZT.0)=IiVA
(Z**(IHd)NIS)/(A3*DIS*SZT.0)=213A
SLS 01 OD (O'bTODWV)AI
(121*Id)/(3*3)=DIS
(IS)SO3*(IHd)NIS'C13+(IS)SO3*(IHd)S03*13 =AD
{7SE
ESE
ZSL
ISE
OSE
617E
917E
(1-11:11IHd'Thili'Id1H10948/A101XXI1XX)S01dI1 11VD
S3SSO1 enH C1NV
dIl AO
Si7E
1717E
3
NOI1V111D1V3
0.0=13 (CZT/Id*7).3D'(VHd1V)SEIV)AI
(OCI3'O13IOVHd1VIIS'XIIIII)XXV3VN 11VD
(CI3i13IVHd1V'IS'XI:11111)XXVDVN 11VD
008 01 OD
(CICI3'O13'CIVHd1V)00VDVN 11V3
(03IDIVHd1V)00VDVN 11V3
008 01 OD
(CICIDIC1101CIVHd1V)1717V3VN 11VD
(C1D1D'VHd1V)1717VDVN 11VD
(.8.I1717 VDVN 3sn -ram VIVI:1901d 31-11
ION 31IAM:Id VDVN
V CIBIAIDBdS AAVH
noA
)1VIts1110A
008
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6£E
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0017
(0091T9)31I2IM
OSS 01 OD (66662:19.A01:1dN)AI
009 01 00 (ZTOO'bTA01:1dN)AI
01 OD (8T171:7'b3.A01:1dN)AI
6ZE
8ZE
LZE
9ZE
0017
SIN3I3IAA303
IAI1 1VNOI1D3S
AO NOI1V111D1VD
D
SZE
17ZE
.
EZE
ZZE
TZE
TO(T0+VHd1V=CIVHd1V
d13H1-131-11-IHd=VHd1V
IHA=21IHd
Th/11*1XX=01XX
(VVIHd)NISAVVIHd)SOD=1XX
(IHd)S9V=VVIHd
((lX*(AV-1--T))/(IS)SOD*(V-'T))NVIV=IHd
6TE
81E
LIE
171E
0171T-C OT OC1
STE
dV=V1-130
V=V1AS
91E
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31-1=1-111
A/V93140*121=1X
81`LI'91'S111711EliZIll'A0ThldN'MT
)(21'Id'11-eSIA101-1211VD31n10'0D'IS'WCIONY'd13HI'X'A'8'3W2ICI'll
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((IHA)NIS*V1D+(II-Id)NIS*OD-1D*(Z**(IHA)NIS+ I )))+ 81= 81
7/1:10*(D*E**11:1*Z**(V-T)*(V10*(IHA)SOD'ZT
+CID*E**(IHA)SOD*.Z-(IHA)NIS*1D*(Z**(IHd)SOD-7)))+LI=L1
1117
9117
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£117
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1:1DACI*1A+ Ti= T1
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0117
6017
8017
1DACI*1:1DA*(V-. T)*NA+171=171
21DACI*):1Dc:1*(V-I1)*1A+91=91
liDdOIDA*1A+S1=S1
(L'OZAZ`X017)1V141:10d
669
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£017
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?:IDA*AID*(V-"T)*((IHcOANV1VicIXD+XD*7)=1A
XD-(I1-1d)NIS*VCID-1-(IHA)SOD*V13=dAD
AD-1-(IHA)SODANCID-(1-1c1)NIS*V1D=c1X0
1OO*0/(0D-OOD)=-VCID
66£
86£
L6£
Z6£
16E
06£
68E
(8H-1?:1)*AAA'''dAdlnlX
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176£
AD*11D=d1
96£
56£
TOO' 0/(1D-C110)=V1D
X21/0=1:ID
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(M*M)*(D*El*OHI:1*S*0)=T1D
(E1H-111)*AXA=AXAINX
AD*1SNOD=AAA
XD*1SNOD=AXA
(D*(Z**NO*OH11*S.0)=1SNIOD
SIA/3*M*OH21.--91:1
D
51101DVA 3DNa213A1:131NDIV1119NV
OS
((IHA)NIS)/((IS)SOD*A*(V-. T))=M
88£
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CINV 1VIXV NOdfl 11\1gC1N3d30 SNOLLONfld AO NOI1V11131VD
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69£
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31V3 ZI'E N0IS2GA Nt0:11):10A ESO
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'T+1A1V=IAIV
(onv/n)*(2- (on*on-vT)/(mv*on*on))+ziAins=ztAins
00=3 (0.-1.19*LAIV)AI
Z
(S8Z.T**OCI)/T£0.0=3
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01 OD (01.31\l'InIV)AI
T
(899.**011)/860'0-=3
T
ZAT1D -
01 OD (otyaNlAw)i
zninm+vms=vms
(n*n±-r)/(n*n)=nAd
(n*n-FT)/(n*n)-al
9L17
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(OT**(11*11+'T))/T
(OT**11*'9£-8**11*.0917-4-9**11*"590T-17**11*.£09+1411**SG--T)*fl*114:179=9-1.
(c**(n*n+-T))/(9**n*'i7 - v**n*-Tz+n*n*17-r--r)*n*n*'9I=i71
v**(n*n-F-T))*n>kn,o7-zi
S9.17
17917
£917
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Z917
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ZAT1- aId*A*S.)NIS*.Z)/(IV*Id*A)=ZAT1D
UZA-C1)*(ZA-D)*(ZA-43)*(ZA-V))/(17**ZZ)+((ZA-D)T
8S-17
SS17
"9*9=0
1S17
9S17
*(ZA-9)*(ZA-V))/(E**ZZ)+((ZA-8)*(ZA-V))/(ZZ*ZZ)+(ZA-V)/ZZ=ZAII
'17,07=8
17S17
£S17
(SD2)di
6t717
Z*Z=ZZ
OS-17
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(OIVIA/OZ)13SS3E1 11VD
(IViA1Z)13SS2E1 11VD
ZS-17
TS17
A*A=ZA
aoHlaw
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Si717
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(T-Fwv*z)=A
9t+
EH'
SNI31SC11OD
0
Z1717
T1717
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SOT
01
01
OD
OUT
OD
00?
996
311NI1NO3
171717
T =A
SOT 01 OD
(a(T000-+ Z**(IHd)NIS)121bS*121*.Z)/((121-'d)*b)-)dX3)ASODV*(IdtZ)=A
HIV
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8£17
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wo=nAtris
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0.0=1A1V
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"sassoi ON AO BSVD 31-11 210d 210
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EinH C1NV dI1 3H1 S3NI1421319C1
SOldIl
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sassoi
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ZI*£ N0IS213A NV211210d ES°
17L/LT/S0
CINB
Nntru..]
Id*d=d
((T000'T
+Z**(IHd)NIS)121bS*H1:1*.Z)MHI:1-120*b).-)dX3)dSODV*(IdtZ)=Id
006
SOT
006 01 OD
0-1=U
(0'Tb3-1H)dI
005
01
OD
T617
osv
6817
88-17.
005
3
SNOIlVinDivD ssmanH
L817
9817
S817
17817
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Di13,K((no)/(n*n+-0)=d
(-r+kivz)hiv=kmv
VaLT/S0 SOldIl ZT*£ NOIS'tlgA Ntql1?:10d
z8i7
T817
ost.
zinins*(idtz)-D=DIED
6LV
vms*((id*Id)/13)-(n*n+t)/(n*n)=9
SVOT
8Li7
OT
Z.L17
ESO
END
RETURN
AI=((.5*Z)**V)*S
CONTINUE
C=AK*C
S +
30
AK=AK+1.
S=B/(C*E)
E=P
40
5 TO GO
D=D-2.
P=D*TK*P
40 TO GO IF(TK.LE.0.0)
TK=D-1.
P=1.
D=V+AK
5
B=(.25*Z*Z)**AK
K=1,10 30 DO
1. C=
AK=0.0
S=0.0
C
MODEL LOSS TIP
GOLDSTEIN THE FOR FUNCTIONS BESSEL CALCULATES BESSEL
C
C
BESSEL(Z,V,AI) SUBROUTINE
1045 05/17/74
516
515
514
513
512
511
510
509
508
507
506
505
504
503
502
501
500
499
498
497
496
495
494
493
492
3.12 VERSION FORTRAN 053
END
RETURN
25
CD=1.0
25 TO GO IF(CD.LE.1.0)
+SD5*(A-AMAX)**4 CD=SD3+SD4*(A-AMAX)**2
CL=0.0
61 TO GO IF(CL.GE.0.0)
CL=(AO*A)-(A2*(A-AMAX)**2)
25 TO GO
CD=SD0+(SD1+A)+(SD2*A*A)
CL=A0*A
24 TO GO IF(A.GT.AMAX)
A=ALPHA
AMAX=0.218
SD5=12570.
SD4=0.0006
SD3=0.0168
SD2=.130
SD1-0.0006
SDO=0.0058
A2=7.*A0
A0=5.73
529. 669,PAGE NO. REPORT NACA IN PUBLISHED DATA NACA OF CURVEFIT
POLYNOMIAL ORTHOGNAL A BY OBTAINED WERE EQUATIONS THE
AIRFOIL. 0012 NACA A FOR ALPHA; ATTACK, OF ANGLE GIVEN A AT
DRAG AND LIFT OF COEFFICIENTS THE DETERMINES - NACA
C
C
C
C
C
C
NACA00(ALPHA,CL,CD) SU8ROUTINE
1045 05/17/74
545
544
543
542
541
540
539
538
537
536
535
534
533
532
531
530
529
528
527
526
525
524
523
522
521
520
519
518
517
3.12 VERSION FORTRAN 0S3
0.1851985
0.0248
-
-
END
RETURN
-
100
50
10 TO GO
3.23 + -0.11*ALP CL=
60 TO GO IF(ALP.GE,16.0)
10 TO GO
60
CD=+0.0131686686494*ALF
+1.934 -0.029*ALP CL=
0.00738*ALP**2
-
0.214377*ALP CL=
50 TO GO GE.15.5) IF(ALP.
30
10 TO GO
0.6078 + 0.0731*ALP CL=
30 TO GO IF(ALP.GE.12.0)
100 TO GO
0.00629752 1+
0.0000023229*ALP**3 + 0.00002732*ALP**2 + CD=0.0001695356*ALP
100 TO GO
0.000000704*ALP**3
20
10
1+0.00661
0.000028188623*ALP**2 + CD=0.00001644&ALP
10 TO GO IF(ALF.6G.+2.0)
0.3975 + CL=0.099375*ALP
20 TO GO IF(ALP.GE.8.0)
ALP=ALPHA*180./3.141593
401. PAGE 824, NO. REPORT NACA IN PUBLISHED DATA NACA OF CURVEFIT
POLYNOMIAL ORTHOGNAL A BY OBTAINED WERE EQUATIONS THE
AIRFOIL. 4418 NACA A FOR ALFHA; ATTACK, OF ANGLE GIVEN A AT
DRAG AND LIFT OF COEFFICIENTS THE DETERMINES - NACA
C
C
C
C
C
C
(ALPHA,CL,CD) NACA44 SUBROUTINE
575
574
573
572
571
570
569
568
567
566
565
564
563
562
661
660
559
558
557
556
555
554
553
552
551
550
549
548
547
546
3.12 VERSION FORTRAN 053
1045 05/17/74
00026
SUBPROGRAM. OF LENGTH
NACAXX FOR ERRORS NO
END
RETURN
CD AND CL FOR PROGRAM FIT CURVE ADD
C
ALP=ALPHA*188./3.141593
DEGREES. IN ATTACK OF ANGLE
OF FUNCTION A AS COEFFICIENTS DRAG AND LIFT SCCTIONAL FOR
EQUATICNS FIT CURVE INSERT MUST ONE STORED. PREVIOUSLY NOT
PROFILE A FOR USE FOR SUBROUTINE EMPTY AN IS NACAXX
C
C
C
C
C
C
NACAXX(ALPHA,CL,CD) SUBROUTINE
1045 05/17/74
597
585
584
583
582
581
580
579
578
577
576
3.12 VERSION FORTRAN 0S3
II
xiaNadav
107
W CL
2
= Frft
Kutta-Joukowski from time, same this At
blade per
=
a'rS2
total
4nrF
F
above calculated circulation the of function a is F circulation the Hence,
between. in decreasing and blades the near increasing 0, of function a is a' blades finite For
1
a'r0 47r
FtotalB
constant. is a' i.e., (0), a' # a' blades of number infinite an For
velocity
tangent
wake
= "total
{2a1rf2}rd0
2n
path circular a Using
= 'total
ds
Hence
blades the by "generated"
circulation the to equal be must wake the in circulation the conserved, is vorticity Since
FACTORS "F" THE OF USE THE
y
dr c C 2 pW
then also, disk rotor the at localized be to "a" Consider
-B
= a)2a\T,27urdr (1 pVco
element
dTblade
dTmomentum
by determined is thrust Now
(I)
cos sirul) 8F
aCx
a' 1+
1
Bc
CLW Bc
W
ID
1+a'
(I)
cos
dp
a since
sin CL
Cx
airS2 47trF
Combining
= dFx
cCxdr pW2
,
pcVxfdr
=
(ixd)
=
d-C)x
Since
sin(I) CL
rT
NOW
is FL to rtotal of ratio the circulation generated drag the included rtotal
a)2
F)
-
(1.
109
2(S+F2)
(1 4SF
+
VF2
(1)
S a.'
2
F = 2S
8sin2 =
aCY
a=
aF)aF (1