APPLIED APPLIED
AERODYNAMICS AERODYNAMICS
Of Of
POWER POWER WIND WIND
MACHINES MACHINES
Lissaman Lissaman And And Wilson Wilson Peter Peter S. S.
B. B. E. E. Robert Robert 1- 1-
the the Science Supported Supported National National Foundation, Foundation, by by
(RANN) (RANN) Applied Applied Research Research National National Needs to to
Oregon Oregon
GI-418340 GI-418340 No. No. Under Under Grant Grant State State
University University . .
MAY MAY 1974 1974
AERODYNAMICS AERODYNAMICS APPLIED APPLIED
OF OF
POWER POWER MACHINES MACHINES WIND WIND
by by
Wilson Wilson Robert Robert E. E.
University University Oregon Oregon State State
Corvallis, Corvallis, 97331 97331 Oregon Oregon
and and
Lissaman Lissaman Peter Peter B. B. S. S.
Aerovironment, Aerovironment, Inc. Inc.
California California Pasadena, Pasadena,
July, July, 1974 1974
CONTENTS CONTENTS OF OF TABLE TABLE
page page
CHAPTER CHAPTER INTRODUCTION INTRODUCTION 1, 1, 1 1
of of Wind Wind in in Power Aerodynamics Aerodynamics Role Role 2 2
Cross-Wind-Axis Cross-Wind-Axis Machines 3 3
Savonius Savonius Rotor Rotor 3 3
4 4 Madaras Madaras Rotor Rotor
Darrieus Darrieus Rotor Rotor 5 5
Wind Wind Machines Axis Axis 6 6
- -
Ducted Ducted Rotor Rotor 6 6
Design Design Smith-Putnam Smith-Putnam 7 7
Rotor Rotor Circulation-Controlled Circulation-Controlled 8 8
CHAPTER CHAPTER POWER POWER MACHINES MACHINES TRANSLATING TRANSLATING WIND WIND 2, 2, 11 11
Translators Translators Drag Drag 11 11
12 12 Lifting Lifting Translators Translators
MOMENTUM MOMENTUM THEORY GENERAL GENERAL ROTORS; ROTORS; AXIS AXIS CHAPTER CHAPTER WIND WIND 17 17 3, 3,
Theory Theory Rankine-Froude Rankine-Froude 17 17
of of Rotation Rotation 20 20 Wake Wake Effect Effect
of of Flow Flow Multiple Multiple Model Model States States 25 25 Simple Simple
34 34 Ducted Ducted Actuators Actuators
VORTEX/STRIP VORTEX/STRIP THEORY THEORY ROTORS; ROTORS; 39 39 AXIS AXIS WIND WIND 4, 4, CHAPTER CHAPTER
of of 39 39 Wake Wake the the Representation Representation Vortex Vortex
44 44 Equations Equations Flow Flow Annulus Annulus
49 49 Models Models Loss Loss Tip Tip
Glauert Glauert 54 54 Optimum Optimum The The Rotor; Rotor;
59 59 Vortex Vortex Theory Theory
MACHINES MACHINES 61 61 CROSS-WIND CROSS-WIND AXIS AXIS CHAPTER CHAPTER 5, 5,
of of Wake Wake 61 61 Modeling Modeling the Vortex Vortex
Rotor Rotor 64 64 Darrieus Darrieus
69 69 Rotor Rotor Circular Circular The The
VERTICAL VERTICAL MOMENTS MOMENTS WIND GRADIENT GRADIENT DUE DUE TO TO 73 73 CHAPTER CHAPTER FORCES FORCES AND 6, 6,
73 73 Introduction Introduction
74 74 of of Vertical Vertical Wind Gradient Gradient The The Effects
79 79 Relations Relations Approximate Approximate
page page (cont'd.) (cont'd.) of of Table Table Contents Contents
85 85 REFERENCES REFERENCES
87 87 SYMBOLS SYMBOLS
89 89
INSTRUCTIONS INSTRUCTIONS OPERATION OPERATION PROGRAM PROGRAM APPENDIX APPENDIX I, I,
107 107 FACTORS FACTORS "F" "F" THE THE USE USE OF OF APPENDIX APPENDIX II, II,
FIGURES FIGURES OF OF LIST LIST
page page
3 3 Savonius Savonius Rotor Rotor 1.1 1.1
4 4 Effect Effect Magnus Magnus 1.2 1.2
5 5 Rotor Rotor Darrieus Darrieus 1.3 1.3
6 6 Enfield-Andreau Enfield-Andreau Ducted Rotor 1.4 1.4
Wind Wind Turbine Turbine 8 8 Smith-Putnam Smith-Putnam 1.5 1.5
of of Machines Machines Wind Wind 10 10 Power Power Performance Performance Typical Typical 1.6 1.6
11 11 Device Device Translating Translating Drag Drag 2.1 2.1
13 13 Translating Translating Airfoil 2.2 2.2
Ratio Ratio 14 14 Lift-Drag Lift-Drag Airfoil Airfoil Translating Translating Power Power From From 2.3 2.3 vs vs a a
with with Wind Wind Relative Relative 15 15 Translating Translating Airfoil 2.4 2.4
Turbine Turbine Wind Wind 17 17 One-Dimensional One-Dimensional Flow Past Past 3.1 3.1 a a
21 21 Geometry Geometry Streamtube Streamtube 3.2 3.2
Induced Induced of of Ratio Ratio Speed Speed the the Effect Effect Tip Tip 3.3 3.3 on on
Inotational Inotational Wake Wake 24 24 with with Velocities Velocities Flow for for an an
Speed Speed Ratio Ratio Tip Tip Coefficient Coefficient Maximum Maximum Power Power 3.4 3.4 vs vs
25 25 Wake Wake Rankine Rankine Vortex Vortex with with for for Rotor Rotor a a a a
26 26 Brake Brake Propeller Propeller State State 3.5 3.5
28 28 Operation Operation Modes Modes Rotor Rotor 3.6 3.6
29 29 Various Various Modes Blades Blades for for Force Force 3.7 3.7
30 30 Coordinates Coordinates Element Element Blade Blade 3.8 3.8
Angles Angles Pitch Pitch 32 32 Various Various Blade for for Element Element Blade Blade States States 3.9 3.9
36 36 Ducted Ducted Windmill Geometry Geometry 3.10 3.10
40 40 Disc Disc Actuator Actuator 4.1 4.1
Multi-Bladed Multi-Bladed for for Rotor Lattice Lattice Vortex Vortex System System 4.2 4.2 a a
41 41 Blades Blades Shown) Shown) Are Are (Only (Only Two Two
Two-Bladed Two-Bladed 43 43 of of Rotor Rotor of of Idealization Idealization System System Vortex Vortex 4.3 4.3 a a
45 45 Element Element Blade Blade Rotor Rotor 4.4 4.4
45 45 Element Element Blade Blade for for Rotor Rotor Velocity Velocity Diagram 4.5 4.5 a a
of of (cont'd.) (cont'd.) Figures Figures List List page page
48 48 of of Rotor Rotor State State Working Working 4.6 4.6 a a
Wind Wind Turbine Turbine 52 52 Smith-Putnam Smith-Putnam of of the the Calculated Calculated Performance 4.7 4.7
Smith-Putnam Smith-Putnam for for the the Speed Speed Wind Wind Output Output Versus Versus Power Power 4.8 4.8
53 53 Turbine Turbine Wind Wind 00 00 Op Op = =
55 55 Velocity Velocity Diagram 4.9 4.9
62 62 Axis Axis Cross-Wind Cross-Wind of of Actuator Actuator Shedding Shedding Vortex Vortex 5.1 5.1
Axis Axis Actuator Crosswind Crosswind 63 63 of of Bladed Bladed Single Single System System Vortex Vortex 5.2 5.2
65 65
Turbine Turbine Crosswind-Axis Crosswind-Axis for for Flow Flow System 5.3 5.3 a a
70 70 Length Length of of Equal Equal Caternary Caternary and and Circle Circle Troposkien, Troposkien, 5.4 5.4
75 75
Wind Wind Gradient Gradient in in Rotor Rotor 6.1 6.1 a a
76 76 Velocity Velocity Diagram Blade Blade 6.2 6.2
Due Due In In Reduction Reduction Power Power Output Output Percent Percent 6.3 6.3
84 84 Gradient Gradient Wind Wind To To
TABLES TABLES LIST LIST OF OF
page page
Disk Disk 56 56 Conditions Conditions Actuator Actuator Optimum Optimum the the Flow Flow for for 4.1 4.1
57 57 Disk Disk
Actuator Actuator Optimum Optimum the the for for X X 4.2 4.2 Cp Cp vs vs
Disk Disk the the Optimum Optimum Actuator 59 59 for for Blade Blade 4.3 4.3 Parameters Parameters
78 78 Trigonometric Trigonometric Sums Sums 6.1 6.1 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 ~ew York Univer sity (2). 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 per formance characteristics and presented as a function of their elementary aero dynamic 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-mechanical energy 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 out put. 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 ef- fects on the convex surface that comes backwards upwind. The Savonius design was fairly efficient, reaching a maximum of around 31%, but it was very inef- ficient with respect to the weight per unit power output since its construc- tion 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 conven- tional rotor blade wind-turbine. Therefore it is only useful and economical for small power requirements.
-
Figure 1.1 Savonius Rotor
3
ROTOR ROTOR MADARAS MADARAS
of of it it the the Madaras Madaras principle involves works works effect. In Magnus Magnus The The the the Rotor Rotor on on essence essence a a
which which boundary layer layer formation boundary boundary technique technique by layer layer control control attempts attempts to to suppress suppress
of of between between the the fluid fluid and and the solid boundary. boundary. The reduction reduction the the relative velocity velocity simplest simplest to to way way
of of the the rotating rotating Figure Figure effect effect cylinder. cylinder. involves involves flow achieve achieve the the the shows shows Magnus Magnus 1.2 1.2 pattern pattern a a
which which in in placed placed cylinder cylinder right right angle angle about about rotating the the the exists exists flow. flow. On On stream stream at at to to a a a a a a
the the flow flow cylinder the the the surface, surface, and of of half half in in when when moving moving cylinder cylinder the the upper upper are are same same
completely completely direction, direction, lower separation separation partly eliminated. eliminated. the side side separation separation is On On only only is is
perpendicular perpendicular circulation circulation lift lift is is induced causing developed. developed. force force flow flow the the and Thus Thus the axis to to a a
of of the the cylinder cylinder produced. be be to to
Effect Effect Figure Figure Magnus Magnus 1.2 1.2
Madaras Madaras proposed proposed circular circular track track around around which which rotating rotating cylinders, cylinders, mounted mounted construct construct to to a a
vertically vertically flat-cars, flat-cars, cylinder cylinder would would Each Each have have been been feet feet high, feet feet 90 90 in in to to 18 18 on on was was move. move.
diameter, diameter,
driven driven and and by by electric electric The effect effect the Magnus Magnus would propel propel the the around around motor. motor. an an cars cars
track track
drive drive and and connected connected the the the axles. axles. However However system's system's aerodynamic aerodynamic generators generators to to car car poor poor
design, design, mechanical mechanical
electrical electrical
and and losses, losses, coupled coupled unsuitability with with its losses, losses, for for on on use use
mountain mountain resulted resulted locations, locations, in in being being little little done with with this this design. design. single single top top full-sized full-sized A A very very 4 4
testing testing in in cylinder cylinder Burlington, further development but but for for built built Jersey Jersey been has has New New no no was was
done done since. since.
DARRIEUS DARRIEUS ROTOR ROTOR
of of in in filed filed United United Darrieus Darrieus Paris axis States States for Georges Georges vertical 1926 1926 patent patent rotor rotor a a a a
below. below. sketched sketched in in Figure Figure 1.3 1.3
Darrieus Darrieus Figure Figure 1.3 1.3 Rotor Rotor
The The Darrieus been been recently recently investigated has has Rotor Rotor of of by by Rangi South South & & (20) the the National
of of Research Research Council Council in in
Canada Canada The Darrieus has has Ottawa. Ottawa. of of performance performance that that rotor rotor near near a a
propeller-type propeller-type and and requires requires input input for for of of The The simplicity starting. starting. and and design design rotor rotor power power
associated associated for for production production potential potential low low
make make it it promising promising candidate candidate economical economical for for cost cost a a power power
production. production.
ability ability The The Darrieus Darrieus the the scale scale of of higher higher levels levels to to production, production, 100 100 type type to to rOtor rOtor power power
kw kw
remains remains uncertain. uncertain. the the Darrieus largest largest date date built built To To less less than than in in feet feet 20 20 rotors rotors more, more, or or are are
diameter. diameter. 5 5
MACHINES MACHINES WIND-AXIS WIND-AXIS
ROTOR ROTOR DUCTED DUCTED
blades blades airplane-type airplane-type
with with hollow windmill windmill experimental experimental built built two two the the British 1954 1954 In In an an
propeller propeller between between the the machines machines has has coupling conventional conventional it it Unlike Unlike
in in Figure Figure 1.4. 1.4. shown shown no no as as
from from the the
air air pulls pulls the the centrifugal centrifugal force wind, wind, by by turned turned blades blades the the and and the As As generator. generator. are are
of of
the the tip difference difference the the time time between the the blade blade At the the tips. tips. through through hollow hollow tower tower pressure pressure same same
the the
100 100 through through the the in semi-vacuum semi-vacuum created air air pedestal pedestal draws draws also also blade blade the the and and the the rotor rotor up up
through through that turbine turbine drives drives it it through through the the air air flows flows tower tower high high As As foot foot a a tower. tower. a a passes passes
in in of of
producing producing capable capable kilowatts is is and and 100 100 35 35 in in diameter diameter feet feet blade blade 80 80 The The a a generator. generator. was was
mph mph wind wind
95 95 at at rpm. rpm.
AIR AIR VENT VENT
PROPELLER PROPELLER
AIR AIR VENT VENT
TUR8INE TUR8INE
INTAKES INTAKES AIR AIR
GENERATOR GENERATOR
Ducted Ducted Enfield-Andreau Enfield-Andreau Rotor Rotor Figure Figure 1.4 1.4
blade blade used used hydraulic hydraulic the the speed speed maintain maintain order order In In motors motors to to constant constant rotor rotor to to vary vary were were
that that
of of The The designed they blades blades mph. mph. pitch pitch speeds speeds 60 60 wind wind 30 30 effective effective and and to to at at are are so so were were 6 6
wind wind of of is is motion motion The The face the the into the the of of heavy heavy under under wind wind flap flap rotor rotor to to gusts. gusts. pressure pressure can can
of of
main main that operated operated is advantage advantage this the The The controlled controlled by aided aided and and system system system. system. power power a a
supported supported aloft. aloft. equipment equipment generating generating is is not not power power
DESIGN DESIGN SMITH-PUTNAM SMITH-PUTNAM
in in largest largest Grandpa's Grandpa's Knob Knob the the Vermont Vermont built built windmill windmill Smith-Putnam Smith-Putnam The The at at ever ever was was
using using of of blades blades steel steel consisted consisted and and stainless stainless diameter diameter feet feet constructed. constructed. 175 175 The The two two rotor rotor was was
weighed weighed about about and and and and 250 250 The The sections. sections. NACA NACA airfoil airfoil 4418 4418 generator generator tons rotor rotor were were
by by supported supported foot 100 100 tower. tower. a a
of of keeping keeping blades blades the the speed speed automatic, automatic, 28.7 28.7 pitch pitch control control The The constant constant at at at at rpm rpm a a was was
wind wind increased, increased, blades velocity velocity began the the the the of of velocities velocities and and above. As As mph mph wind wind to to 18 18
guard guard with with designed designed ability ability 20° blades blades by by edgewise. edgewise. The The turning turning feather feather to to to to to to were were an cone cone up up
itself itself maintain maintain reasonably reasonably coning coning The The speed. speed. still still sudden sudden and and against against constant constant gusts gusts was was a a
plant plant withstand withstand mph mph designed designed wind The The 120 120 cylinders. cylinders. by by oil-filled oil-filled damped damped to to to to up up was was power power
of of leading leading wind ice ice turbine the the intended edge. edge. The mph mph with and and inches inches six six 100 100 to to was was on on
kilowatts. kilowatts. 1,000 1,000 generate generate
unit unit erected erected and and operated until until in in Figure Figure M M 1941 1941 shown shown The:turbine, The:turbine, 1.5, 1.5, test test was was a a as as
and and replacement failed failed bearing bearing could main main inch inch be be secured for when when February February the the 24 1943 1943 not not a a
off off of of with with ended ended experimentation blades blades and this design. design. the the flew In In 1945 1945 two two one one years. years.
of of of of the the Smith-Putnam Smith-Putnam design illustrated the the blade, blade, the the failure failure spite spite structural the In In
by by turbines. turbines. wind wind large large generation generation scale of of possibilities possibilities electrical electrical power power 7 7
Wind Wind Turbine Turbine Figure Figure Smith-Putnam 1.5 1.5
ROTOR ROTOR CIRCULATION-CONTROLLED CIRCULATION-CONTROLLED
Circulation-Controlled Circulation-Controlled of of of that that Rotor Rotor the the Wind is is similar Turbine Turbine The The quite concept concept to to
of of of of and and Rotor 1920's. the the Flettner Instead rotating the the Madams the the cylindrical blades blades Rotor Rotor
of of of of of of the the lift lift air tangentially generated generated by blowing blowing around the the sheets sheets is is surfaces surfaces rotor, rotor, upper upper
boundary boundary layer This This briefly, is blades blades principle, slots. slots. from from control technique the the small small to to a a
low low layer layer of of delay delay flow flow Blowing re-energizes boundary the the separation. separation. the surface surface energy energy upper upper
of of of the the thereby moving moving further back cylinder cylinder point separation the the cylinder. cylinder. the the on on
the the reduced reduced there there but but is is is is in in Consequently, Consequently, accompanying accompanying increase increase drag drag viscous viscous pressure pressure an an
blowing blowing time time the the by circulation circulation is and and there induced induced drag. drag. is is At At suction suction increase increase in in an an on on same same
suction suction lower lower of of the the the the surface decrease surface, surface, and and in in which which all all lift. lift. generate generate upper upper on on a a 8 8
of of lift lift cylinder cylinder advantages. advantages. the First, First, number number is is design design This This at at zero zero possesses possesses a a
sudden sudden would would with tend tend speed speed Second, Second, the the insensitive insensitive therefore not not to to gusts. rotor rotor to to gusts, gusts, up up
rigidly rigidly blade blade mounted without the the be be the the needed needed the the hub because because coning coning is is flapping flapping to to can can or or no no
of of propeller propeller solidly solidly fastened. fastened. that that The The large large of of blade blade conventional conventional difficulties difficulties moment moment was was a a
of of this this blade blade it be spanwise of of of cylindrical cylindrical The The stiff. stiff. cross-section cross-section inertia inertia to to type type causes causes very a a
of of achieved achieved adjusting adjusting location foregoing foregoing the the the by by thereby lift lift slot, slot, coefficient coefficient is is constant constant
and and for for provides provides construction, construction, rigid rigid design design controls. controls. control, This This pitch pitch complicated complicated easy easy a a very very
with with operating operating environment. environment. its its structure structure to to cope cope
it it made made design design University this this of of investigation investigation Oregon Oregon analytical analytical and and State State An An at at was was was was
jet jet the the than drive drive ratios ratios the the that that speed speed found found high-tip to to greater greater at at compressor compressor power power power power was was
required required low-tip low-tip ratios projected the the speed speed solidity the the (rotor from from while rotor rotor output output rotor, rotor, at
enough enough of of disk disk the the large large offset simplicity structural structural divided divided the the area) circular by. by. to to a a area area was was
rotor. rotor.
of of of of performance performance comparison comparison Figure Figure following gives gives the the the the various various 1.6 1.6 types types page page a a on on
that that been been have have constructed. rotors rotors 9 9
0.6 0.6
for for Propeller Propeller Efficiency Efficiency Ideal Ideal
Windmills Windmills Type Type
0 0
Type Type Blade Blade High High Speed Speed Two Two
Blade Blade Type Type Multi Multi
American American - -
Rotor Rotor Sayonius Sayonius
Darrieus Darrieus Rotor Rotor
Type Type Four Four Dutch Dutch Arm Arm
4 4 5 5 6 3 3 2 2
SPEED SPEED SPEED/STREAM SPEED/STREAM =TIP =TIP PER PER RATIO RATIO SPEED SPEED
of of Wind Wind Machines. Machines. Performance Performance Typical Typical Power Power 1.6 1.6 Figure Figure 10 10
CHAPTER CHAPTER 2 2
POWER POWER MACHINES MACHINES TRANSLATING TRANSLATING WIND WIND
TRANSLATORS TRANSLATORS DRAG DRAG 2.1 2.1
straight straight that that in device device loves is is of of wind wind machine the simple simple the the Perhaps Perhaps most most type type power power a a
of of action action the the wind. wind. under under line line
than than rather rather for for propulsion devices devices used been been have have wind-driven wind-driven translating Historically, Historically,
illustrative illustrative
lift-driven lift-driven drag-driven drag-driven devices devices be and and Analysis Analysis translating of of extraction. extraction. can can power power
considered considered instantaneous instantaneous be be translation translation the the since since machines machines examining examining various various in in rotary rotary can can an as as
2.1 2.1 be be driven Figure machine machine by by drag. drag. consider consider the the of of rotating rotating machine. machine. First, element element blade blade to to
device. device. of of drag drag the the elementary action action illustrates illustrates the
Translating Translating Device Drag Drag Figure Figure 2.1 2.1
of of velocity. velocity. the the product product translation translation and and the the drag the the is is extracted, extracted, the the device device such such P, P, For For power power a a
that that the expressed expressed by by device device velocity velocity is is
relative relative drag drag The The V.Q V.Q
power power so so v v sees sees a a - -
(Ko (Ko v)2 v)2 (1/2) (1/2)
(2-1) (2-1) Sv Sv C C Dv Dv P P
p p = =
= = D D 11 11
the the wind. wind.
wind wind speeds speeds of of At At below the the velocity, translator translator linearly linearly is is output output to to power power a a seen seen vary vary
with with translation translation the the velocity. produced produced force force the the by by relatively translator translator In In is is contrast contrast a a
independent independent of of velocity velocity translator translator low low The The speeds. speeds. large required required for the the speeds speeds translator translator at at to to
chief chief high high extraction extraction achieve achieve the the disadvantage disadvantage large large speeds speeds extensive extensive rates rates power power are are as as mean mean
investment investment in
capital capital disadvantages disadvantages machines machines of of land. land. Other Other and and translators translators proximity proximity the the to to are are
sensitivity sensitivity ground ground and and in in wind changes changes direction. to to 16 16
CHAPTER CHAPTER 3 3
MOMENTUM MOMENTUM ROTORS; ROTORS; GENERAL THEORY THEORY WIND WIND AXIS
wind wind propeller wind The The windmill attention attention turbines. let let Now Now type type turn turn to to our our or or us us
the the in in efficient efficient leading remains remains the machine candidate and and turbine turbine today, today, for for 1940 1940 (1), (1), most most as as
will will first first one-dimensional consider consider production. production. large large analysis wind wind scale scale As As step step a a power power we we a a
of of then then of of wind wind approach turbine turbine proceed detailed linking and and blade the the
output output to to more more a a a. a.
output. output. geometry geometry to to power power
RANKINE-FROUDE RANKINE-FROUDE THEORY THEORY 3.1 3.1
originated originated theory theory axial axial Rankine by by and and and with with the the Starting Starting W. W. (4) (4) E. E. R. R. momentum momentum
V. V. wind wind Froude Froude wind turbine shown free free below. below. The The flow flow is is consider consider (5,6) (5,6) past past stream stream a a as as
which which wind wind by by Applying is is slowed slowed continuity, device. device. and and the flow momentum, momentum, to to energy energy we we a a
if if the the flow assumed is is determine determine the and entirely axial axial be be thrust thrust with with rotational rotational to to may may power no no
motion. motion.
One-Dimensional One-Dimensional Wind Wind Flow Flow Figure Figure Past 3.1 3.1 Turbine Turbine a a 17 17
theorem theorem the the First, First, obtained. obtained. from be be thrust thrust the the for for expressions expressions momentum momentum Two Two may may
-ui) -ui) )= )=
(Voo (Voo (Vo(, (Vo(,
(3-1) (3-1) pAu pAu T T = =
the the wind wind machine machine caused caused by by drop drop the the of of consideration consideration from from Second Second pressure pressure
-p- -p- (3-2) (3-2) where where Ap Ap AAp AAp T T
=p =p
= = , ,
of of upwind upwind turbine turbine the the side side the the free free and and between between used used be be Equation Equation Bernoulli Bernoulli the the stream stream Now Now may may
the the that that wake wake and and of of turbine turbine downwind downwind side side the
the the between between again again and and so so
(V02,, (V02,,
(3-3) (3-3)
T T
p p
2 2
obtain obtain expression expression with with the the together together momentum momentum we we
+ +
ui ui
(3-4) (3-4)
U U = =
2 2
If If of of initial initial final velocities. denote and and the the the the is is disc disc the the the velocity velocity i.e., i.e., at at we we average average
V. V.
the the the the twice final final wake velocity change change is is V. V. 2aV. 2aV. that that aV., aV.,
note note ui, ui, Ui= Ui= - - u u - -
- - , ,
of of immediately immediately importance; importance; however, however, the thrust thrust is is The The disc. disc. the the velocity velocity change great great not not at at
with with isothermal isothermal flow, flow, of of assuming assuming law law thermodynamics, thermodynamics, first first the the is. is. From From p. p. pi pi = power power
}= }=
1y,20 1y,20
pA pA
u u +umv. +umv.
(v. (v.
pAu pAu P P
= =
c° c° 2 2 2 2 2
Or Or 18 18
441 441
) ) (3-5) (3-5)
= =
pAV:: pAV:: 1/2 1/2
which which has when when 1/3 1/3 maximum maximum
a a = = a a
Pmax Pmax 16 16
(3-6) (3-6)
pAV.3 pAV.3 27 27 1/2 1/2
known known axial axial interference is is the the factor The The defined. defined. and is is maximum maximum (a) (a) is is Thus Thus term term as as power power a a
velocity velocity of of turbine turbine minimum the the final wake wake The The air. air. the the is is of of the the influence influence zero, zero, on on so so measure measure a a
obtain obtain
1/2 1/2 2a), 2a),
V., V.,
(1 (1
= =
we we ui ui = = am am as as
- - ax ax . .
it it noted noted denominator kinetic kinetic that that is is the the the the be be When When equation examining examining (3-6) (3-6) may may
the the equivalent equivalent that by Equation of of (3-6), (3-6), wind wind in in the the contained out out rotor. rotor. swept swept to to an an area area energy energy
is is the the disc disc maximum maximum flow flow through however, however, efficiency efficiency since since the the the the does does rate rate not not represent represent mass mass
by by by divided divided available available given is is Au. Au. the the efficiency, efficiency, but but AV., AV., Hence Hence output output not not power power power power
441 441
a) a) (3-7) (3-7)
= =
V2 V2
pAu pAu
2 2
of of which which coefficient coefficient yields yields is is The The 0.5. 0.5. 100% 100% maximum maximum efficiency efficiency 1/2 1/2 The The at at a a power power = = a a
coefficient coefficient is is 88.8%. 88.8%. maximum maximum efficiency efficiency at at power power
modeling modeling with accomplished accomplished the additional additional one-dimensional one-dimensional consideration be be Further Further can can
with with interaction interaction rotating wind wind rotational, rotational, machine initial initial is is rotation. rotation. the wake wake of of As As not not stream stream a a
of of of the the in in the propeller, propeller, wake direction the the the the wake wake the the will will In In rotates rotates to to rotate. rotate. a a cause cause case
of of device device wake wake extracting extracting (windmill), (windmill), in in the the the opposite opposite in in the the propeller, propeller, rotates rotates energy energy case case an
in in rotational rotational the wake wake addition translational kinetic kinetic kinetic in in If If there there is is to to energy energy energy, energy, sense. sense. 19 19
the the than than extraction extraction in in lower lower considerations considerations thermodynamic thermodynamic from from expect expect then then case case power power may may we we
translation. translation. having having only only wake wake of of the the
kinetic kinetic angular relate relate wake rotational will will example example simple simple following following to to rotor The The energy energy
velocity. velocity.
Kinetic Kinetic Initial Initial Energy Energy ETi ETi = =
Extracted Extracted P P Power Power = =
Final Final Kinetic E E
Energy Energy ER2 ER2 + + = =
T2 T2
Rotation Rotation Translation Translation
thermodynamics thermodynamics From From
E E ER2 ER2 P P ET1 ET1
= = T2 T2
produces produces wake wake angular
increased increased
that that (angular (angular velocity), greater greater (torque) (torque) torque torque P P note note x x = = as as
of of given given initial for for kinetic kinetic rotational rotational wake wake thereby thereby and and amount amount greater greater momentum momentum energy, energy, so, so, a a
high high when when low low which will will
is is extraction extraction ER2 ER2
the the ETi ETi greatest greatest occur occur means means power power energy energy
, ,
low low velocity velocity angular angular and torque. torque.
ROTATION ROTATION EFFECT EFFECT WAKE WAKE OF OF 3.2 3.2
of of analysis analysis propellers. of of in in effect effect rotation the considered considered wake the the (7) (7) Joukowski Joukowski
of of rotation rotation the the effect of of wake wake turbines, turbines, wind wind analysis analysis Adopting Adopting his notation the the to to on on power
if if assumed assumed be be flow flow model, model, irrotational, wake wake produces be be estimated. estimated. The removal removal to to may may
however however of of of of rotation rotation the the contribution the regions regions axis, axis, the the rotational rotational unrealistic unrealistic velocities velocities near near
rotational rotational inserted yielding simple simple subtracted subtracted and be be high high angular angular velocities velocities out out a a core core a a may may
establishing establishing bounds. bounds. utility utility in in results results model model affords affords the which which to to 20 20
between between relation relation the the that that written written
be be equations equations analysis, analysis, streamtube streamtube Using Using express express can can a a
In In velocities velocities the corresponding corresponding rotational, rotational, and and the the rotor. rotor. and and at axial axial both both velocities, velocities, wake wake the the
obtained. obtained. The
coefficient coefficient be be
the the expression expression for for special special for for certain addition, addition, can can power power cases, cases, an an
of of the the values values
relative relative of of the the
rotation rotation of of effects effects the the approach approach of of is is this this main main on on outcome outcome measure measure a a
wake. wake. in in the the and and velocities velocities the the induced induced rotor rotor at at
streamtube. streamtube. illustrates illustrates the below below Figure Figure 3.2 3.2
Rotor Rotor Wake Wake
Streamtube Streamtube Figure Figure 3.2 3.2 geometry geometry
equations equations resulting resulting The The are: are:
Continuity Continuity
(3-8) (3-8) uiridr uiridr
urdr urdr = =
of of Moment Moment Momentum
ri2(01 ri2(01 r2CO r2CO (3-9) (3-9)
= = 21 21
fluid. fluid. of of velocities velocities the the addition, addition, angular angular In wake wake the the and and and
where where
rotor rotor
we we may may are are coi coi co co
equation, equation, obtain obtain an an energy
Energy Energy
(0" (0"
(01 (01
+ + + +
2 2 2
ulahri2 ulahri2 )2 )2 (u1 (u1 (3-10) (3-10)
= =
/ / 2 2
00 00
U U U1 U1
in in of of Finally Finally for for gradient radial radial axial axial expression expression the the the the velocity velocity angular angular the the is is where where rotor. rotor. 5-2 5-2 an an
be be velocity velocity obtained. obtained. may may
( (
2 2 2 d d (a)? (a)? d d
) )
wi) wi) (c2 (c2
(3- (3- 1) 1) 1 1
\ \
dr]. dr]. 2 2 dr1 dr1
\ \
between between relations relations thrust, flow the the and and the the in in obtain obtain used used equations equations be be These These four four torque torque to to may may
of of of the the obtained obtained specification specification in needs needs variables, and and
be be wake. wake. Closure cannot cannot one one say say one one co, co,
of of Bernoulli's Bernoulli's equation equation used used the the fauns fauns particular particular The The solution. solution. obtain obtain momentum momentum order order to to are are a a
equation. equation. Euler's Euler's and and equation equation
of of flow flow be be noted. noted. the the features features Several Several may may
rotational rotational wake wake the the velocity. due due the the varies varies The The to to across across pressure pressure
velocities velocities radially. wake wake and and axial The The rotor rotor vary vary
of of the the direction direction of of of rotation rotation velocity velocity which the the is is opposite opposite the fluid, fluid, angular angular The The
the the discontinuously discontinuously changes changes at at rotor. rotor. rotor rotor
has has assumed assumed be be been been Fluid Fluid drag drag to to zero. zero.
element element also also obtained. the the thrust annular annular be and and for for for for Expressions Expressions torque torque may may an an 22 22
TORQUE TORQUE
pur2o3dA pur2o3dA
dQ dQ = =
THRUST THRUST
p(S) p(S) 22-2) 22-2)
r2codA r2codA
dT dT
+ + = =
r2o) r2o)
it it that that when gradient, gradient, is is velocity velocity be for for radial radial wake wake the the expression expression the the From From may may seen
velocity velocity is is the the wake wake axial constant. constant. constant constant
Defining Defining
(1 (1
b) b) V., V.,
a) a) (1 (1
Voo Voo u u a a
obtain obtain We We may may
42 42 (1 (1 b' b'
(3-12) (3-12)
1 1
a a
= =
4X2 4X2
2 2 a), a), (13 (13
and and
a)2 a)2 b2 b2 (1 (1
power power
cp cp
0/003 0/003
b b y2 y2 A A a a 23 23
I.0 I.0
0.8 0.8
0.6 0.6
8 8
III III
0.4 0.4
0.2 0.2
0.2 0.2 0.3 0.3 0.5 0.5 0 0 0.4 0.4 0.1 0.1
-u -u
a: a:
Induced Induced Flow with Velocities Velocities for for Ratio Ratio of of the the Speed Speed Tip Tip Effect Effect Figure Figure 3.3 3.3 an an on on
Wake. Wake. Irrotational Irrotational
of of
It It function function and and of of ratio ratio a/b a/b variation variation X. X. the the the the illustrates illustrates Figure Figure above above 3.3 3.3 may may a a a a as as
value value in in the the 1/2 1/2 approximately approximately always always disc disc the the is is velocity velocity change change axial axial the the that that observed observed at at be be
above above ratios ratios speed speed tip tip for for 2. 2. wake wake
r2e) r2e)
produces produces infinite since since modification modification
coefficient coefficient requires constant constant The The = = some some power power
Rankine Rankine substitute substitute wake, wake, irrotational irrotational
of of the the lieu axis. axis. vortex vortex In In velocities velocities a a may may we we an an near near
obtain obtain N::----0/coma, N::----0/coma, wake. wake. Letting vortex vortex we we
a)2[2Na a)2[2Na
11(1 11(1
(1 (1
N)b] N)b]
(3-13) (3-13)
Cp Cp + +
= =
b b a a 24 24
below below
wake wake shown shown is is
with with Rankine Rankine
vortex vortex coefficient coefficient for for rotor rotor maximum maximum The The a a a a power power
tip tip high high coefficient coefficient of of values values highest highest at at expected expected the the would would be be Figure Figure As occur occur in in 3.4. 3.4. power power
the the least. least. rotation rotation wake wake
consequently consequently the and and
where where the the speed speed ratios are are torque torque
WMAX WMAX
0.6 0.6
0.5 0.5
0.4 0.4
8 8 n> n>
0.3 0.3
0.2 0.2
ROTATIONAL ROTATIONAL WAKE WAKE
DISTRIBUTION DISTRIBUTION VELOCITY VELOCITY
0 0 I I
5 5 4 4 3 3 2 2 0 0
RD. RD.
with with Ratio Ratio
for for Rotor Rotor Speed Speed Tip Tip Coefficient Coefficient Maximum Maximum Power Power Figure Figure 3.4 3.4 a a a a vs vs
Wake. Wake. Vortex Vortex Rankine Rankine
in in annular, annular, flow flow the the requires requires results results these these used used arrive arrive to to model model non- non- flow flow The The at to to occur occur
of of the the model. model. spite spite flow flow this this criticized criticized In In recently recently has has Goorjian Goorjian (8) (8) tubes. tubes. interacting interacting steam steam
of of
neglecting neglecting wake wake effect effect the the insight insight into into affords affords it it model, model, with with this associated associated difficulties difficulties some some
turbines. turbines. of of wind wind theories theories blade blade element element rotation rotation in in
STATES STATES FLOW FLOW MULTIPLE MULTIPLE MODEL MODEL OF OF SIMPLE SIMPLE 3.3 3.3
is is operating device device that that assumed assumed the tacitly tacitly been been it it has has analysis analysis previous previous the the In In a a as as
the the continue continue simple simple quite quite it it is is 0 0 For For that that
to to device, device, is is 1. 1. extraction extraction 0 0 < < propulsion propulsion and producing producing adding adding thrust thrust will will that that analysis analysis show show the device to to act act to to a a energy energy as as of of of of that that propeller. propeller. typical typical This This is the the wake wake flow regime regime flow. flow. type type a a modeled modeled for for physically by by This This be be particularly particularly interesting interesting A A 1. 1. a> a> occurs occurs may case case induces induces it it that that powered powered with pitch propeller propeller adjusted forward forward that considering considering flow, flow, is is its its a a so so a a a a An An idealized idealized brake brake shown streamline streamline below, below, propeller propeller the is is in in thrust, thrust, pattern pattern state. state. reverse reverse or Brake Brake Propeller Propeller Figure Figure State State 3.5 3.5 find find in in in in this approach approach Section the the the the Continuing Continuing using analysis analysis 3.1 3.1 we we same same as as case case that that 4a 4a (1. (1. (3-14) (3-14) Cp Cp = = Force Force (1 (1 a) a) 4a 4a (3-15) (3-15) CT CT = = = = , , X X pAV:, pAV:, in in three three written form the the Thus, Thus, be be all all can can cases cases 11al 11al 4a 4a (3-16) (3-16) CT CT = = 26 26 2.0 2.0 A A I I I I / / t t DRAG DRAG / / Glauert Glauert 1 1 / / (emperical) (emperical) / / 1.0 1.0 _ _ _ _ Propeller Propeller ---4.-- ---4.-- Brake Brake Momentum Momentum Theory Theory 0 0 Windmill Windmill .---v-A" .---v-A" Propeller Propeller THRUST THRUST -1.0 -1.0 _ _ I I I I 1 1 1 1 0.5 0.5 0 0 -0.5 -0.5 1.5 1.5 1.0 1.0 a a I I I I I I I I f f Extracted Extracted Power Power n n From From F/ow F/ow 0.6 0.6 op op 0 0 Power Power Added Added 0.6 0.6 _ _ Flow Flow To To i i i i 1 1 1 1 -0.5 -0.5 0 0 0.5 0.5 1.0 1.0 .5 .5 1 1 a a Operation Operation Figure Figure Modes Rotor Rotor 3.6 3.6 28 28 Rotation Rotation of of Plane Plane ./1 ./1 rf2 rf2 S7:7=1V., S7:7=1V., v/ v/ Propellor Propellor aVo aVo a<0 a<0 9 9 Blade Blade a a Force Force r.r1 r.r1 Slip Slip Zero Zero Vco Vco 0 0 a a = = Windmill Windmill 0 Brake Brake Propellor Propellor a>I a>I Direction Direction Wind Wind Various Various Modes Modes For For Blades Blades Force Force Figure Figure 3.7 3.7 29 29 occurring occurring the the that that noted noted should should be be 1/3. 1/3. windmill windmill It It still still exist exist at at for for that that for for state state is, is, 0; 0; 0 0 a a = = can can = = blade blade the the relative relative be be of of the the angle angle plane plane to to rotation. rotation. at at to to zero zero increasingly increasingly becomes becomes 0 0 As As negative, negative, the the the the propeller propeller brake brake rotor rotor enters enters 1. 1. now now a> a> state, state, These These in in Figure Figure sketched sketched which which illustrate illustrate also also 3-7, 3-7, states states of of the the the the force force and and torque torque are are sense sense on on We We avoided-discussing the the have have blade. blade. the the flow flow in in regimes regimes the the vicinity vicinity of of close close since since 1 1 our our = = a a simplified simplified model model break down down will will here. here. Thus, Thus, physical physical of of these these models models both both by by construct construct considering the the flow flow states states the the can can we we at at disc disc itself itself by by considering considering and and flow flow the the in in the the wake. wake. order order In In possibility possibility establish establish the the of of the the modes modes the the force force to to must must connect connect represented we we as as wake wake by by that that represented represented by lifting lifting forces forces the the blade blade momentum momentum elements elements to to themselves. themselves. as as on on simple simple For For model model will will consider, the the example, example, wind axis (propeller) (propeller) and and rotor rotor our our we we an an as as use use conventional conventional blade blade element element theory ignoring ignoring swirl swirl and and assuming assuming the the wake induced induced flow terms terms is is that that twice twice the the disk disk itself. itself. model model This This is is in in sketched sketched Figure Figure at at 3.8. 3.8. Wind Wind Direction Direction L' L' Direction Direction of of Rotation Rotation a)V. a)V. (1 (1 Figure Figure Blade Blade 3.8 3.8 Element Element Coordinates Coordinates 30 30 the the theory, theory, By By windmill windmill 1. 1. momentum momentum propeller propeller the the state state < < in in Assuming Assuming a a or or are are we we by by given given annulus annulus is is the the force force on on (1 (1 a)2a a)2a (3-18) (3-18) pV02 pV02 Ircrdr Ircrdr dT dT = = by by given given is is coefficient coefficient thrust thrust local local the the and and 441 441 a) a) (3-19) (3-19) CT CT = = from from circulation circulation the the itself itself element element blade blade get get the the flow flow considering considering at at Now Now we we 27csin 27csin and and with WcCL/2 WcCL/2 CL CL F F a, a, = = = = (1 (1 0] 0] a)cos a)cos pnc[Vca pnc[Vca (3-20) (3-20) sin sin Or Or 0 0 = = by by given given is is annulus annulus the the force force Thus Thus the the on on (3-21) (3-21) OrFdr OrFdr dT dT = = 0 0 XC XC 0] 0] a)cos a)cos ( ( (3-22) (3-22) xsin xsin CT CT = = rON.. rON.. speed speed ratio ratio local local tip tip the the where where is is x x while while a), a), theory theory -4a(1 -4a(1 CT CT by by momentum momentum propeller-brake propeller-brake get get a> a> the the state 1 1 = = For For we we - - , , solidity solidity local local define define We We previously. previously. a a result result a a by by the = = as as force force given blade blade a a is is the the can can as as same same for for write write all all Thus Thus cdrircrdr. cdrircrdr. a a can can we we ila ila 01= 01= a)cos a)cos (3-23) (3-23) K1 K1 4a 4a sin sin 47cxa 47cxa x x 31 31 be be of of The The this this solutions solutions equation equation easily easily from from Figure Figure nature nature that that to to 3.9. 3.9. most most Note Note for for seen seen can can <81 <81 the the simple 0 0 propeller propeller powered powered thrusting while while for for <82 <82 propeller propeller the the brake brake 0 0 mode mode occurs, occurs, intermediate intermediate angles angles The The 02> 02> the the in in exhibit exhibit 0 0 possible possible three three 0 0 equilibrium equilibrium > > occurs. occurs. states, states, range range windmill windmill modes modes propeller propeller brake brake and and mode. mode. two two one one Propeller Propeller Brake Brake AMMO AMMO -0.5 -0.5 0.5 0.5 1.0 1.0 1.5 1.5 a a Figure Figure Blade Blade Element Element 3.9 3.9 For For Various Various States States Blade Blade Pitch Pitch Angles Angles of of interest interest that that It It is is the the slope slope of of blade blade the. the. lines lines force force note note fiuiction fiuiction to to is is of of solidity solidity and and a a speed speed tip tip ratio. ratio. triple triple the the For For mode it it that that the the point point shown shown is is unstable unstable B B that that and and and and A A C C case, case, appears appears as as both both stable stable and and depending depending how how the the is is approached. approached. simplified simplified A A explanation explanation state state are are occur occur upon upon of of why why is is unstable unstable is B B follows. follows. that that Assume Assume state state B, B, induced induced the the flow flow slightly slightly is is at at as as a, a, increased, increased, the the drag-force disk disk the the (following 8 8 becomes becomes constant) constant) much much larger larger than than now now on on = = 32 32 this this further further the the reduced, reduced, wake wake and is is thus thus by by wake wake that that represented the the momentum momentum momentum, momentum, and and stable stable other other hand, hand, according according these these the the C C A A towards towards On On 1.0. 1.0. to to system system are are a a = = moves moves with with theory theory element element solutions in in blade that assumption assumption is is Thus Thus working arguments. arguments. no no a a 1/2 1/2 it it idealized idealized inevitably inevitably involves model model that that above above and and is is that that It It should the be be stressed an an which which that that annulus annulus model model giving it it flow flow inconsistencies. inconsistencies. example, be be For For < < 1 1 an an seen seen on on a a a a can can of of flow flow with with somehow somehow violate will will continuity. inner inner and annuli the has has value 0.5 0.5 outer outer < < a a the the design windmill windmill major major in in of of should should of of that that We We 0.5 0.5 a> a> not not note note states states a a occur occur range incoming incoming high high speed speed winds due due it it in in is is where where However, However, prevent prevent rotor rotor to to to over over necessary necessary cases cases of of propeller propeller the the flow flow possible possible confused confused brake the the reduced reduced shaft shaft loads, it be to to torque torque use use may may or or of of which which distinct the control control from normal speed speed quite quite is is method method dump dump This is is to to state state energy. energy. a a this this that that behavior behavior blade the the reduced. reduced. is is in in feather feather which blade blade Note technique technique is is not not same same a a as as characteristic. characteristic. of of which which low low X the the the region stall, stall, at at occurs occurs perturbation perturbation analysis analysis considered should should be be small model, that that the the here here We We note note present a a For For valid valid should should considered simple thus thus example, the be be for for 0.5. 0.5. 0.5 0.5 a> not not 1 1 < < one- one- in in reversal reversal velocity velocity dimensional dimensional model flow developed developed here wake wake implies implies the and far- far- zero zero between between somewhere somewhere downstream infinity, which the the streamline configuration and and disc disc actuator actuator a a propeller propeller physically physically Again Again brake considered unacceptable. unacceptable. the simple analysis analysis be quite quite is is must must of of inadequate inadequate vicinity the the in in 1. 1. a a = = flow flow satisfactory satisfactory for extensive research theories theories exist region, although quite in in this this No No been been with with helicopter helicopter done done problem connection theory. theory. analysis has has In In in in this this rotor rotor on on of of where where descending descending anomalous associated associated the the lifting lifting that that region region this this with with is is the the rotor rotor states states a a wake, wake, the the turbulent turbulent parachute parachute the brake, brake, ring and and [Shapiro, (16)]. (16)]. states states vortex vortex occur occur 33 33 seldom seldom is is than than it it thus thus less less 0.5; 0.5; is is windmill windmill modes modes operating operating normal normal For For necessary necessary most most a a with with solutions solutions conditions conditions spurious spurious a> a> off-design off-design for for However, However, for for conditions conditions 0.5. 0.5. analyze analyze a> a> to to Wolkovitch Wolkovitch of of discussed discussed by is is for for the the entire entire performance performance well well Rotor Rotor 0.5 0.5 a a range range occur. occur. may may (17). (17). of of flow flow the the anomalies anomalies of of that that Wolkovitch, Wolkovitch, by by described described of of interest interest is is It It note, note, many many to to as as axial, axial, precisely precisely but but the the flow freestream freestream is that that assuming assuming not not removed removed by be be 1.0 1.0 can can near near = = a a of of of of introduction introduction additional additional this this degree degree The The the the axis. axis. small small angle angle yawed yawed rotor rotor to to at at some some approach approach classical classical for for which which A A axial axial flow. flow. singularities singularities to to the the of of eliminates eliminates freedom freedom occur occur some some Townend Townend and and (18). (18). given given by by Lock, Bateman, Bateman, is is problem problem this this using using constructed constructed by Glauert Glauert (10) (10) of of CT CT generalized generalized performance A A versus versus a a was was curve curve windmill windmill of of free-running free-running This This series series from from using using and and data tests. tests. these these was was concepts concepts curve curve a a these these running, running, correspond 0, 0, free free Cp Cp to to these these Since Since tests tests and and 3.9. 3.9. Figures Figures shown shown in in 3.6 3.6 or or = = were were noted noted that the the autorotative in in be be should should It It autorotative autorotative state state the the terminology, terminology, helicopter helicopter in in state. state. is is subjected in in fact fact the the thus thus remainder, remainder, driving driving the to to rotor rotor the the is is of of portion portion rotor rotor non- non- one one the the disc. disc. for for the the is is given given perturbation perturbation of of axial axial and and uniform uniform the the value value mean mean a a a, a, ACTUATORS ACTUATORS DUCTED DUCTED 3.4 3.4 of of propellers. propellers. powered powered It It thrust thrust the the increase increase static static frequently frequently used to to Shrouds Shrouds ducts ducts are are or or of of contraction contraction the the reduce reduce effectively effectively slipstream quite quite that that established established duct duct well well been been has has a a can can a a forward forward speed. speed. thrust/power thrust/power least least ratio, ratio, It It its its increase increase propeller propeller thus thus and and at at thrusting thrusting zero zero can can reduces reduces the the of of the the effectiveness effectiveness shroud shroud friction, friction, the skin skin ignoring ignoring be be that, that, shown shown as as even even can can taken taken into into pitching pitching duct duct other other drag drag and and duct duct when when moments moments and and increased, increased, forward forward speed is is are are of of ducted ducted propeller calculations calculations technically. technically. effective effective In In propeller propeller is is shrouded shrouded the the not not account, account, static static exit exit freestream freestream duct duct leaves leaves the the flow flow that that the the usual usual it it at at is is performance performance to to pressure; pressure; assume assume 34 34 of of velocity velocity calculation, calculation, and and for for slipstream slipstream in in further further change consequently, consequently, there is purposes purposes no no of of the the wake wake free free static static section. section. ultimate ultimate In In the the taken taken the the duct exit be be case case cross cross a a as as area area may may the the half half the the propeller propeller which which Thus Thus duct duct slipstream slipstream is is propeller, propeller, ultimate the the causes causes any any one one area. area. of of thrust/power thrust/power will will increase the the the ratio ratio than than this this slipstream slipstream contraction contraction final final less less be be system. system. to to of of section section propeller the the duct duct the cylindrical cylindrical of of that that interest interest observe observe It It is is to to cross cross even even same same a a as as that that noted noted the the slipstream slipstream contraction. contraction. should be been been will will thrust, there It It increase increase the lave lave will will no no of of major major the the this this duct, duct, and represented represented forward thrust by by increased increased force is is part part on on a a a a of of low low duct duct the and and the the due force force the the contribution contribution leading leading edge edge is is the the to to entry entry area area pressures pressures on on there. there. propeller propeller due due duct, duct, has has it it of of in in performance performance improvement improvement thrusting the the Because Because to to a a might might windmill windmill superior superior performance. A A that that ducted ducted suggested suggested have have been been frequently frequently a a Rainbird Rainbird of of windmills windmills given Lilley analysis analysis ducted and and 'y 'y (19). is is comprehensive comprehensive wake wake will will duct duct the of of increase increase be be expansion. physical physical the viewpoint, viewpoint, effect effect From From to to a a a a of of be be windmill windmill twice that wake wake section should optimal optimal that that for for the the free free showed showed have have We We cross cross a a be be if if than than optimal optimal section section larger wake wake the the possible possible the the windmill it it is Thus, Thus, disc. disc. to to to to cross cross cause cause of of the the flow flow the level level axial axial induced thirds optimal optimal wake wake free free this, this, white white keeping the still still two two at at coefficient coefficient exceed exceed limit the the the will will free velocity, velocity, then, based rotor rotor rotor rotor stream stream power power on on area, area, increased increased its its the the through through and caused caused flow drawn be be the the duct has has of of effect effect 0.593. 0.593. In In rotor rotor to to more more of of unlike unlike the that that which which it this this in is is simple simple follows, follows, shown analysis analysis extraction extraction capacity. capacity. A A power power assumptions assumptions which which without without made made this this device be be analysis analysis free free cannot cannot momentum momentum rotor, rotor, type on on a a hard hard justify. justify. quite quite to to are are windmill windmill flow flow Assuming Assuming typical typical ducted ducted the the Figure Figure show show 3.10 3.10 In In system. system. mass mass we we a a the the extracted immediately immediately write write through through the the is is rh rh system system power power can can as as we we , , 35 35 vs. vs. Windmill Windmill Ducted Ducted Geometry Geometry Figure Figure 3.10 3.10 (1 (1 a) a) MY! MY! (3-24) (3-24) 2a 2a MAR MAR P P = = = = written written be be duct) duct) and and entire entire (rotor (rotor This This force force the system system may may as as on on 2a 2a (3-25) (3-25) T T riiVao riiVao = = free free expression expression have have for th. th. For For that that do do in in closed closed equations equations that that not not these these We We not not note note an an we we are are the the by by the the that that obtained obtained readily readily stating stating force force remaining remaining equation equation is is system system theory theory the the actuator actuator on on This This the the where where is A A by by AAH AAH AAp AAp is is given given T T which which force force the the the actuator actuator is is actuator actuator = = area. area. = = on on for for the the free propeller a) a) pAVoo(1 pAVoo(1 result result the the gives gives th th immediately immediately case. case. = = - - the the by by but the the propeller force force given that that is is still still AAp, AAp, with with it the the is is duct, duct, For For true true case, case, our our field field the the the theory theory since since simple simple determined determined by by be be duct duct momentum force force cannot cannot pressure pressure on on interior interior theory theory the the the one-dimensional one-dimensional duct duct of of the the By By outside outside is is known. duct duct not not pressure pressure on on Thus, Thus, using using leading leading the the edge. edge. close close for for region region the the momentum momentum calculated, calculated, be be to to except except very very can can assumption assumption required. required. is is additional additional theory theory one one 36 36 be be the velocity velocity by by assuming assuming consider consider this satisfied satisfied which which the the duct duct exit, We We is is to to at at can can If If in in powered powered ducted propeller that that shown shown done done theory, theory, is is b). b). the the V.,,(1 V.,,(1 assume, assume, as as we we as as - - freestream freestream exit exit static, static, then then and and the the the duct b b is is 2a 2a flow flow be be get get at at pressure pressure mass mass we we = = can can determined determined coefficient coefficient basing basing the the Then, Then, Iii= Iii= the the duct duct AepV.,(1 AepV.,(1 exit exit A, A, 2 2 a). a). power power on on as as area area - - obtain obtain we we 41 41 2a) 2a) 4a 4a (3-26) (3-26) Cp Cp = = Cp.. Cp.. This This expression maximumized maximumized give give be be If If write write 0.385 0.385 0.211. 0.211. to to at at = = can can a a = = power power we we of of in in then coefficient coefficient get get rotor rotor terms terms area, area, we we 41 41 (1 (1 2a)Ae 2a)Ae /A /A 4a 4a Cp Cp (3-27) (3-27) = = if if that that of of and and ratio observe observe then then the the the the duct exceed coefficient the 1.54 1.54 to to rotor rotor area area power power of of that that based based ducted ducted exceed exceed the will will of of alone. alone. At At this this level level analysis rotor rotor rotor system, system, on on area, area, of of characteristics characteristics all all by by performance performance assumption determined determined the exit flow flow condition condition the for for are are coefficient coefficient expressed expressed writing writing the duct. duct. This duct duct by by (based exit the be be and and area) area) power power can can on on C*p C*p which which exit exit coefficient coefficient give give pressure pressure us us 41 41 (1 (1 b) b) 4a 4a Cp Cp (3-28) (3-28) = = b)(2 b)(2 C; C; b) b) (2a (2a 2a 2a (3-29) (3-29) = = be be than than will will exit exit assuming assuming freestream the the lower that, that, is is which which It It static, static, must must pressure pressure seen seen be be with with the the wake wake the of of expansion expansion downstream downstream higher duct duct gives gives flow flow higher higher and and case, case, a a mass mass 37 37 is is performance performance the the be be noted it it Rainbird's Rainbird's must must and and Lilley Lilley studying studying In In coefficient. coefficient. paper paper power power exit exit duct duct assumed assumed of of the the plotted plotted in in pressure. pressure. terms terms wake wake principle principle the the possible possible in in is is it it sections, sections, compute to to previous previous in in the described described As As external external flows, flows, internal internal and and computing computing assuming assuming techniques, techniques, flow flow contour, contour, potential potential by by shape shape a a duct duct of of details details the the Evidently Evidently the the surface. surface. bounding bounding wake wake the the continuity continuity ensuring ensuring and and on on pressure pressure ring ring simply simply treated treated be be that that duct duct the the cannot cannot analysis. analysis. We We this this note note into into a a as as enter enter must must geometry geometry implies implies disc disc of of the the addition addition essential essential the the actuator since since flow flow homoenergetic homoenergetic a a uniform uniform in in wing wing a a wake. wake. the the surrounding surrounding tube tube associated associated with with vortex the the different different of of wake wake energy, energy, and and method method simple simple analyzed analyzed by be be windmills windmills ducted ducted proper proper cannot cannot a a Thus Thus any any assuming- assuming- that that of of It It the the entire entire flow. flow. modeling modeling depends depends prediction prediction performance performance appears appears upon upon a a this this dependent dependent directly directly result result is is the the since since approximation; approximation; is is coefficient coefficient on on exit exit the the poor poor a a pressure pressure given given for for and and at at system system duct duct for for notably notably system, system, rotor rotor will will which which a a quantity quantity even even every every vary vary loadings. loadings. different different rotor rotor 38 38 CHAPTER CHAPTER 4 4 VORTEX/STRIP VORTEX/STRIP ROTORS: ROTORS: WIND WIND AXIS THEORY THEORY VORTEX VORTEX REPRESENTATION REPRESENTATION 4.1 4.1 WAKE WAKE THE THE OF OF of of The The windmill wake wake of of of flow flow consists consists different different total total head from from the the system system a a a a mainstream. mainstream. inviscid inviscid For For in in discontinuity discontinuity head the the flow, flow, be be represented represented by sheet sheet of of an an may may a a of of The The vorticity. vorticity. this this of of mode mode vorticity, generation generation of of and and its its be be assistance assistance in in geometry, geometry, great great can can developing developing of of models models advanced advanced flow. flow. the the wake wake In In models models usually usually stipulate stipulate the the wake wake more more we we and and vorticity vorticity distribution then then Law Law Biot-Savart Biot-Savart the the calculate induced induced the the of of flow flow this this to to use use wake. wake. then then is is It It possible possible flow flow the the and fields fields the the wake wake determine to to compute compute to to pressure pressure on on whether whether it it is is in in equilibrium. equilibrium. of of solution solution the the Thus Thus inviscid inviscid wake wake involve both both the the must must proper proper a a kinematics kinematics and and dynamics dynamics the the of of the the other other words words In In flow. flow. the the wake wake shape shape and and strength strength must must generally generally be be by by determined determined iterative iterative where where the the initial initial and and strength strength is is geometry geometry an an process, process, assumed assumed and and induced the the flow flow checked checked the the wake wake streamline streamline and and fields fields to to assure assure pressure pressure are are consistent. consistent. similar similar A A situation situation ordinary ordinary wing in in theory; theory; however, however, the the of of interactive interactive nature nature occurs occurs problem problem the the is is usually usually removed removed by by the assuming assuming wake wake leaves leaves the the parallel parallel wing wing vortex vortex the the to to freestream freestream flow. flow. implies implies This This that that downwash there there will will be be flow flow through through the wake, wake, a a kinematically kinematically inconsistent inconsistent situation. situation. However, However, it it is only only in of of highly highly loaded loaded wings wings very very cases cases that that it it is is for for deformation. deformation. wake wake to to account account necessary necessary Analogous Analogous used used for assumptions assumptions the the disc, disc, where where it it is is assumed assumed that actuator actuator the the wake wake are are tube tube is is the the parallel parallel freestream vortex vortex flow. flow. to to If If the the consider consider disc disc .which .which prototype prototype arbitrarily actuator, actuator, be switched switched from from we we a a may may zero zero infinite infinite porosity, porosity, of of model model the the to to ring ring shedding shedding create create that that Assume Assume vortex vortex the the we we can can a a process. process. 39 39 the the (against (against motion motion forward forward the the during during solid solid is is backwards backwards and and forwards forwards and and oscillated oscillated is is disc disc of of series series shed shed will will disc disc rearward rearward motion. motion. The The the the during during fully fully and and a a mainstream) mainstream) now now porous porous in in Figure Figure 4.1. 4.1. shown shown the the with with freestream freestream downstream downstream convected convected will will be which which vortices vortices ring ring as as Disc Disc Actuator Actuator Figure Figure 4.1 4.1 and and using using developed, developed, strength strength is is of of tube tube if if constant constant limit, limit, vortex vortex the the an an In In assume assume a a we we disc. disc. the the diameter diameter have have the the will will tube tube actuator the the loads, loads, light light of of hypothesis hypothesis vortex vortex as as same same semi-infinite semi-infinite of of induced induced flow flow the the vortex vortex available available methods methods to to compute Standard Standard a a are are uniform uniform solution solution gives gives the the here, here, these these into into to to will will state state except except end. end. tube tube We We its its a a not not at at go go tube tube edge. edge. infinite infinite the the radial radial flows flows the the although although section, section, at at the the axial axial flow induced induced are are cross cross over over radial radial singular singular then then the the added added this this strength strength appropriate appropriate f f of of system, system, to to tube tube another another I I vortex vortex were were This This semi-infinite semi-infinite tube. tube. of of the the end end that that twice twice becomes becomes axial axial flow the the and and at at removed removed flows flows a a are are that that analysis, analysis, obtained obtained the the result result already already from from momentum momentum demonstrating demonstrating the the of of another another is is way way that that twice twice the the disc. disc. is is wake wake at at flow flow in the induced induced downstream the the there there wake wake and and hence in in velocity velocity the tangential tangential has has that that this this will will be observed observed It It system system no no speed speed windmill, windmill, the the of of tip tip propeller propeller model model approximate approximate type type be be this this For For is is to to torque. torque. a a an an no no in in fact fact low. is is the the that that for for given given be be large large ratio ratio torque must must power power a a so so 40 40 refinement refinement The The this this add add simple simple model model which which is is next next introduces introduces to to to to torque. torque. one one Consistent Consistent with with disc disc model model theory, theory, this this with with large large of of number number actuator actuator radial radial vorticity vorticity we we a a can can in in lines lines of of the the plane plane the the disc, disc, many-bladed many-bladed representing representing of of blade blade circulation. circulation. system system constant constant a a In In order order satisfy Helmholtz's Helmholtz's the the kinematics of of Laws Laws lines, lines, to to this this that that implies implies vortex vortex on on we we see see a a of of central central finite finite with with strength strength distributed distributed vortex vortex vorticity streamwise streamwise along along the the wake wake cylinder (Figure (Figure 4.2). 4.2). of of variant variant A A this this model that that is is the the disc is is to to annulus. annulus. Then Then the the actuator actuator surface surface assume assume an an of of the the inner consists consists tube tube of of ring ring and and spanwise of of lines lines vortex vortex similar similar vortex vortex connected connected geometry, geometry, by by radial radial vorticity vorticity the the disc disc itself. itself. at at Figure Figure 4.2 4.2 Vortex Vortex Lattice Lattice for for System System Multi-Bladed Multi-Bladed Rotor Rotor (Only (Only Two Two Blades Blades Shown) Shown) Are Are a a lightly lightly For For loaded loaded in in boundaries boundaries which which the the wake wake system system be be considered considered right right a a may may circular circular cylinders cylinders parallel parallel the mainstream, mainstream, this this is is fully fully self-consistent self-consistent to to model model with with axial axial and and a a tangential tangential perturbations perturbations entirely entirely confined this this annular annular cylinder. cylinder. to to other other In In words, words, the the induced induced flow flow of of such such does does affect other other annuli, annuli, system system not not by by be be superimposing superimposing a a circular circular two two can can as as seen seen tube tube Another Another vortex vortex of of annulus annulus completely completely systems. systems. different different induction induction could could inside inside be be located 41 41 flow flow induced induced the the Thus, Thus, induced induced flows flows the the first. first. in in the the affecting affecting without without this this outside outside one one or or of of angle angle annulus annulus the the and and that that in in blade blade the the of of only only function function geometry geometry annulus annulus is is each each of of system system a a adjacent adjacent without without changed changed affecting affecting annuli annuli be be neighboring neighboring blade blade of of in in chord chord the the attack attack can can or or of of element element independence independence basis blade the the is is of of annular annular result result interesting interesting This This induced induced flows. flows. situation situation the the different different from from that that is is this this We We interact. interact. annuli annuli do that that note note which which not not theory theory assumes assumes induced induced flows station station will- affect affect spanwise spanwise at at in in changes changes where where wing wing theory, theory, at at in in geometry geometry one one vorticity vorticity streamwise streamwise continuous continuous of of idealization idealization it it that that the the is is other other stations. stations. It is is apparent apparent all all on on of of for for Thus Thus annulus. annulus. induction induction the the isolates isolates effectively effectively which which tube tube wake wake the the non- non- an an vortex vortex blades blades of of of of number number and and product product the the that that the the valid valid requires requires be be theory theory blade blade element element interactive interactive to to be be should should large. large. tip tip ratio ratio speed speed the the of of composed composed bounding bounding sheet of of continuous continuous vortex the the vortex vortex that that also also concept concept We We note note a a it it permits permits by by separated separated the the Thus Thus sheet. sheet. flows flows head head between between the the total total in in differences differences permits permits rings rings models. models. in in assumed assumed simple simple total total head, head, reduced reduced of of wake wake flow be be the the to to as as and and of of blades blades finite finite having having number number realistic realistic If If consider consider system system rotor rotor a a more more a a now now we we Assuming Assuming that that situation situation different different somewhat somewhat velocity velocity vortex vortex finite finite rotating rotating at at occurs. occurs. a a a a flow, flow, parallel parallel then then local local that that the the the and and lies lies and and tips the the shedding shedding only only vortex vortex to root root at at occurs occurs of of is is helix helix Figure Figure sketched sketched the vortices in in The angle angle 4.3. 4.3. becomes becomes the the wake wake geometry geometry vortex vortex as as ratio. ratio. tip-speed tip-speed related related the the directly directly to to of of model model somewhat that similar similar contains contains the bladed bladed this this finite that that structure structure We We to to note note be be streamwise streamwise considered could. could. and and ring ring the the of of where where wake wake Figure vortex systems systems 4.2, 4.2, as as that that ratio, ratio, large large tip-speed tip-speed also also of of Figure Figure We We of of 4.3. 4.3. the the helix note note components components system system a a a a or or be be that that densely densely the the packed, packed, helix helix the the finite finite of of number number large large will blades blades tern tern to to more more so so cause cause sys sys realistic. realistic. becomes becomes bounding bounding sheet sheet of of idealization idealization continuous continuous vortex vortex more more a a 42 42 Figure Figure 4.3 4.3 Idealization Idealization of of of of Vortex Vortex System System Two-Bladed Two-Bladed Rotor Rotor a a However, However, of of examination examination Figure will will illustrate illustrate that that 4.3 4.3 each each blade blade the the flow flow and and near near is is of of similar similar that that high high vortex vortex ratio ratio system system wing. wing. to to Consequently, Consequently, aspect aspect of of finite finite vortex vortex a a a a strength strength be be shed from the the tips, tips, since since cannot cannot this would imply imply infinite infinite induced induced washes washes there. there. Thus Thus the the local local situation situation becomes becomes quite similar similar that that of of yawing yawing wing wing and and to to continuous continuous of of sheet sheet a a a a voracity voracity is is shed shed from from trailing trailing the the edge. edge. Generally, Generally, this this voracity voracity is is concentrated concentrated tip tip the the near near so so that that the the idealization idealization of of finite finite tip tip strength strength be be quite quite adequate adequate vortex vortex short short distance distance from from the the a a may may a a blade. blade. It It will will be be noted noted that bladed bladed for for the the finite model model there there is is spanwise spanwise interaction interaction the the in in sense sense that that load the the each each spanwise spanwise section does does influence influence neighboring neighboring sections sections that that on on blade blade element element so so theory theory be be considered considered must must approximation approximation for for with with few few blades blades low low rotor rotor advance advance an an ratios. ratios. at at a a An An analysis analysis two-bladed two-bladed for for low low advance advance rotor rotor ratio ratio system system at at is is given by by a a very very Kuchemann Kuchemann (21), (21), where where the the is is modeled modeled rolling rolling high-aspect high-aspect rotor rotor ratio ratio wing. wing. a a as as The The model model in in which which the the blades sheds sheds of of helical helical system system sheets sheets is is generally generally termed termed vortex vortex a a Goldstein Goldstein the the Model. Model. elegant elegant This This model model is is complicated complicated than than and and will will discuss discuss most most more more not not we we here. here. it it 43 43 EQUATIONS EQUATIONS FLOW FLOW ANNULUS ANNULUS 4.2 4.2 and and propellers propellers calculations calculations for for performance performance method method for and and used used frequently frequently accurate accurate A A circular circular non-interacting non-interacting in in the the through through flow flow the the rotor rotor that that is is helicopter helicopter occurs occurs rotors rotors to assume assume called called has has been been induced induced velocities velocities with with the the conjunction conjunction used used in in method method when when This This tubes. tubes. stream stream theory, theory, element element blade blade vortex vortex theory, theory, blade blade element modified modified including including of of variety variety by by names names a a each each flow flow locally locally 2-D 2-D at at which which be be method, method, to to The The theory. theory. strip strip and and theory theory assume assume seen seen can can follows. follows. proceeds proceeds station, station, radial radial as as tip tip from from the viewed viewed is is Figure Figure 4.4 in in illustrated illustrated wind wind turbine turbine of of rotor rotor element element The The a a in in shown shown is is wind, wind, relative relative W, W, the the Here Here 4.5. 4.5. Figure Figure in in of of rotation rotation the the axis axis towards towards looking looking is is rotation rotation of of plane plane attack, attack, The The of of angle angle local local the the and and 0 0 angle angle pitch pitch blade blade local local the the relation relation a. a. to to direction. direction. the the downwind downwind blade blade in the the normal normal y-direction y-direction is is and and the the to to x-direction x-direction in in the the relations relations verified verified be be trigonometric trigonometric following following the the diagram, diagram, the the From From may may (4-1) (4-1) (I) (I) a a = = Vco Vco 1 1 a a (4-2) (4-2) tan tan (13, (13, = = a' a' ID ID 1 1 + + (4-3) (4-3) sing) sing) CD CD CL CL Cy Cy 4) 4) ± ± cos cos (4-4) (4-4) sin(I) sin(I) CD CD Cx Cx CL CL (I) (I) COS COS = = relative relative local local the the coefficients coefficients based based and and drag drag lift lift sectional sectional and and the- the- where where CL CL CD CD upon upon are are attack attack of of angle angle local local the the velocity velocity and and W W a. a. 44 44 vc. vc. Rotor Rotor Element Figure Figure Blade Blade 4.4 4.4 Velocity Velocity Figure Figure for for Diagram Diagram 4.5 4.5 Blade Blade Element Element Rotor Rotor a a 45 45 developed developed the blade blade the the factor factor and forces interference interference axial axial the the relation relation between between A A on on a a of of thickness thickness dr dr in in element element generated generated annular dT dT axial axial force force the the obtained obtained by equating equating be be an an may may aerodynamic aerodynamic predicted predicted from element element blade blade force force axial axial the the considerations considerations by by to to momentum momentum considerations. considerations. From From momentum momentum ) ) (Ka (Ka (27crdr) (27crdr) (4-5) (4-5) dTm dTm u u u u p p 1 1 local local chord having having blades blades each while while for for B B c, c, pW2Cydr pW2Cydr (4-6) (4-6) Bc Bc dTB dTB = = local local wake interference factor factor axial axial 2a, 2a, that that b b assuming assuming the the expressions expressions and and these these Equating Equating two two = = obtains obtains one one BcCY BcCY a a (4-7) (4-7) sin2 sin2 1-a 1-a 87cr 87cr 11) 11) angular angular considerations considerations is is from from determined determined the the similar similar In In momentum momentum torque torque manner, manner, a a in in differential differential tube. tube. blade blade element element annular annular the the from from developed developed the the equated equated stream stream torque torque to to an an obtains obtains theorem theorem of of the the From From momentum momentum moment moment one one (2an) (2an) (27Erdr)ur (27Erdr)ur (4-8) (4-8) dQ dQ p p = = the the been been has has angular angular assumed assumed twice be be slip slip angular angular velocity velocity imparted the the where where the the to to stream stream to to is is blade-produced blade-produced The The the the velocity velocity disk. disk. torque torque rotor rotor at at w2 w2 Cdr Cdr pBc pBc dQ dQ (4-9) (4-9) = = 2 2 46 46 relations relations Combining Combining these a' a' BcCx BcCx (4-10) (4-10) a' a' sin sin 47cr 47cr 1+ 1+ .2(1) .2(1) conditions conditions flow flow local local then then the the at at available, available, is is data data performance performance sectional sectional If If a a suitable suitable airfoil airfoil procedure: procedure: following following the the by by determined determined be be station station radial radial given given r r may may CL(a),. CL(a),. V, V, Given Given S2 S2 CD(cc), CD(cc), 0, 0, c, c, r, r, start) start) acceptable acceptable is is 0 0 to to and and a' a' (a (a a' a' Guess Guess = = = = a a (4-2) (4-2) Calculate Calculate (i) (i) (4-1) (4-1) Calculate Calculate and and Calculate Calculate CD CD CL CL and and 4-4) (4-3 (4-3 and and Cy Cy Cx Cx Calculate Calculate (4-7) (4-7) Calculate Calculate a a a' a' Calculate Calculate (4-10) (4-10) and and back back repeat repeat B B Go Go step step to to the the local local and and properties properties known known flow flow sectional sectional the the iteration iteration above above the the Once Once are are converges converges overall overall and and the the determine determine integrated integrated torque torque be be axial axial force force and and to to contributions contributions torque torque to to may may be be blade blade and and twist twist changes, changes, section section taper taper airfoil airfoil Blade Blade of of the the axial axial force force may may rotor. rotor. readily. readily. quite quite accommodated accommodated First, First, qualification. qualification. and and modifications modifications required required to to far far developed developed expressions expressions The The some some so so be be in in It It Figure Figure illustrated illustrated below flow flow the the 4-6. 4-6. procedure, procedure, qualify qualify patterns patterns the the above above may may note note the the with with consistent consistent flow flow is is Such Such flow flow not pattern pattern recirculating recirculating that that a a occur. occur. may may seen seen valid. valid. A A analysis analysis is is the the above therefore, therefore, not not and and (4-10), (4-10), equations equations (4-7) (4-7) leading leading assumptions assumptions to to 47 47 for for criterion criterion determining the the of of recirculating recirculating flow flow obtained obtained be be from from onset onset wake wake may may momentum momentum considerations. considerations. velocity velocity The The in in wake wake the the hence hence for for 2a), 2a), recirculation recirculation V00(1 V00(1 'A 'A a> a> = = u1 u1 can can - - , , This This consideration consideration will will be be modified modified in the the helicopter, helicopter, section. section. A A in in going going next next from from occur. occur. vertical vertical autorotational autorotational descent through through the the to to ascent ascent illustrated illustrated various various in in Figure Figure states states pass pass can can Glauert Glauert 4-6. 4-6. used used experimental experimental (10) (10) results results quantify the the turbulent turbulent windmill to to and and ring ring vortex vortex of of states states rotor. rotor. a a c) c) ( ( (b) (b) Cc) Cc) ( ( ) d d ---s"--"----_,__------s"--"----_,__------ (e) (e) Figure Figure of of Working Working 4.6 4.6 States States propeller; propeller; (a) (a) Rotor: Rotor: zero-thrust; zero-thrust; (b) (b) (c) (c) windmill; windmill; a a turbulent turbulent windmill; (d) (d) ring ring (e) (e) vortex vortex 48 48 MODELS MODELS LOSS LOSS TIP TIP 4.3 4.3 vorticity. vorticity. of of of of shed shed modification modification because because the analysis analysis requires pattern pattern previous previous The The some some acceleration acceleration and and Radial Radial dimensional. dimensional. be be assumed assumed been been has has position position radial radial flow flow two to to The The at at any any radial radial of of flow flow assumed assumed effects effects The The the the alter alter tip tip the the wake-induced wake-induced flow flow pattern. pattern. at at can can effects effects machines; machines; however, the wake wake wind wind neglected neglected cannot cannot for for acceleration acceleration be be most most power power can can of of the the simplest simplest approaches, approaches, of of in in treated treated variety variety been been have have tip tip So-called So-called losses losses neglected. neglected. be be a a of of radius radius fraction fraction actual the the radius, maximum maximum the the being being these these reduce to to rotor rotor to to some some Goldstein Goldstein have have of of actual actual radius. radius. Prandtl and and (11) (11) (12) the the of of order order 97% 97% the the characteristically characteristically on on (negligible (negligible wake developed contraction) contraction) and and propellers propellers lightly-loaded lightly-loaded flow flow about analyzed analyzed Prandtl's Prandtl's of of interaction interaction result wake wake the the The The tips. tips. due due of of circulation circulation reduction reduction for for models models the at at to to factor factor that circulation-reduction circulation-reduction such F, F, is is Goldstein's Goldstein's approach approach and and (4-11) (4-11) circulation circulation is is and and the the station station radial radial of of the the is is number number F the the blades, blades, Fc, Fc, is is where where B B at at a a r r of of factor factor with with function infinite infinite number The blades. blades. is is F F for for circulation circulation corresponding corresponding rotor rotor a a an an a a Goldstein Goldstein Of Of of of radial radial the models the blades blades position. model and and of of ratio, ratio, number number speed speed tip tip two two of of Goldstein's Goldstein's model model infinite flow flow involves series series since since the the however, however, is is accurate accurate an an more more difficult difficult results results for for there there in little little difference Since Since is is it it is is modified modified Bessel Bessel functions, functions, to to more more use. use. Prandtl Prandtl model which which simple yields yields solution the the blades, blades, three three involving involving situations situations can can a a more more or or be be used. used. for for factor factor induced velocities equation equation proceeds proceeds into into of of loss loss the the tip incorporation incorporation The The as as follows. follows. 49 49 of of maximum maximum axial axial that that virtually virtually correction correction the the change is is of of tip tip meaning meaning the physical physical The The the the only only slipstream slipstream and and the sheets sheets in in the (V. (V. 2aV., 2aV., velocity, velocity, ul) ul) vortex vortex on on average average occurs occurs or or - - 2aV. 2aV. of of velocity. velocity. velocity this this becomes becomes the the Thus Thus change velocity velocity fraction fraction change change only is is F F a a velocity velocity change change written Equations Equations and (4-7) (4-7) is is 2a'FQ. 2a'FQ. angular angular the the in in and and similar similar 2aFV. 2aFV. manner, manner, then then become (4-10) (4-10) aC aC a a (4-12) (4-12) sin2 sin2 8F 8F 1 1 (I) (I) a a - - a' a' aCx aCx (4-13) (4-13) 1+a' 1+a' sin sin 8F 8F 4) 4) (I) (I) cos cos the the Prandtl Prandtl correction correction correction factor quantity tip tip and is is Goldstein Goldstein tip tip the is is where where the the F F a a or or a a of of the the in in refinement refinement further further analysis analysis be be axial by by solidity solidity given A A the local local made made can can a a . . 7tr 7tr equation equation (4-5) (4-5) in in is is assumed the the disk disk in in velocity velocity through flow flow the the to to rotor rotor vary vary manner manner same same u u as as through through then then given given by velocity velocity the the V.(1 V.(1 is is the the wake wake flow velocity. velocity. The The aF). aF). rotor rotor = = u u average average - - equation equation Equation Equation however, becomes becomes quadratic remains remains the (4-12) (4-12) (4-13) (4-13) a a same, same, (1 (1 aF)aF aF)aF aCY aCY (4-14) = = S1I12 S1I12 8 8 (I) (I) of of derivation derivation given in Appendix is is equations equations The The and (4-14) (4-14) (4-13) (4-13) II. II. of of yields yields in in higher higher lieu lieu The The slightly and and of of equation equation (4-12) (4-12) performance (4-14) (4-14) use use required required iterations iterations of of in in be be that reduction reduction number significant significant noted the the for for It It convergence. convergence. may may the the recirculating criteria criteria becomes becomes flow flow aF aF for for > > 50 50 be be loaded loaded heavily heavily determined determined following the the tip tip correction for for Goldstein Goldstein The The rotor rotor may may a a of of of of calculation calculation local local the the that of of the the value value Lock's Lock's approach approach bases bases method method F F Lock Lock (13). (13). (I), (I), on on so so cot.* cot.* relation relation ratio ratio via via local local speed The The the the defines defines angle angle /R). /R). The F F F F (q), (q), 4) 4) a a r r = = IA IA Goldstein Goldstein tip correction correction the the thus thus RpIr RpIr ratio ratio factor speed speed and and is is F corresponding corresponding tip tip = = = = tiao tiao be be noted that low it it consideration consideration tip tip speed practical practical As As determined. determined. be be F(p,,g,,a) F(p,,g,,a) at at may may can can a a this this be be approach approach blade. blade. such In In the the entire entire loss loss tip tip the the ratios ratios appreciable appreciable is is to to a a cases cases ceases ceases over over Prandtl's Prandtl's in in the factor factor given calculations. calculations. being being dominant factor is is instead instead loss loss F correction, correction, tip tip a a by by 2 2 A A r r -1 -1 (4-15) (4-15) F F cos cos Lexp Lexp = = 7C 7C where where Rr Rr =B =B f f (4-16) (4-16) R R 2 2 sin 4) 4) factor factor with of of the the optimum frictionless frictionless derived derived for for been been distribution distribution circulation circulation As As has F F rotor rotor a a previous previous of of the the along along blade, the the approximate approximate the the analysis should be be noted. noted. nature nature Figure Figure of of the the calculated calculated gives gives coefficient Smith-Putnam Smith-Putnam Wind the the 4-7 4-7 Turbine Turbine (1) (1) power power function function of of Smith-Putnam Smith-Putnam speed speed turbine turbine employed employed The The ratio. ratio. tip tip NACA NACA 4418 4418 airfoil airfoil a a as as an an has has foot which which twisted discontinuously discontinuously length. length. turbine turbine along along diameter diameter The The feet feet 65 65 50 50 175 175 was was a a with with tip correction used Goldstein Goldstein develop develop The The 11'4" 11'4" chord. chord. the the to to an an was was curve. curve. in in pitch pitch of of Figure Figure effect effect The The angle angle be be pitch pitch Increased Increased reduces reduces the 4-7. 4-7. seen seen can can maximum maximum low low increase increase speed the the available ratios. ratios. but but tip Figure Figure also also 4-7 4-7 at at power power power power can used used wind wind machines. machines. illustrate illustrate generalizations concerning be be tip tip low low At At speed speed to to some some can can 51 51 lift lift by by the maximum influenced influenced coefficient. coefficient. The The strongly strongly angle angle is is the the coefficient ratios, ratios, (1) (1) power power of of partieularly partieularly inboard inboard the the the the stations, ratios ratios much be and and tip tip low low speed speed large large is is rotor, rotor, at at can can tip tip speed. speed. peak peak At At design design speed ratios the the above above below below when when the the operating operating stalled stalled power power high high drag drag coefficient dominant. dominant. of of A A result will will rapid in in drag drag becomes coefficient, coefficient, the effect effect a a Finally Finally large large with with increasing increasing the angular angular velocity. velocity. speed ratio tip tip decrease decrease in net net at at some some power power If If of of the the the the slope is is become become the the will will negative, 0 0 Cp Cp at at rotor rotor output output power power power zero. zero. curve = = will will for (feathered) (feathered) be be since stable, stable, operation operation wind wind velocity, velocity, output output constant constant at at power power zero zero in in in in positive positive which which speed speed result decreased decreased will the the will will rotational rotational output output turn turn return return rotor power power the the stability. stability. the the original original the the its its The The speed. speed. greater greater steeper steeper to to curve, curve, 0.5 0.5 0.4 0.4 0.3 0.3 itl>8 itl>8 0.2 0.2 0_ 0_ cq cq o o 0.1 0.1 4 4 8 8 l0 l0 12 12 R R of of Calculated Calculated Figure Figure Performance Smith-Putnam the the Wind Wind Turbine. Turbine. 4.7 4.7 equations equations and and 00, 00, °pitch °pitch (4-13) (4-13) (4-14) = = equations equations and and Opitch Opitch (4-12) (4-12) (4-13) (4-13) 5°, 5°, = = equations equations and and °pitch °pitch (4-12) (4-12) 00, 00, (4-13) (4-13) = = 52 52 infollnation infollnation concerning concerning ratio ratio yields yields speed speed tip tip coefficient coefficient of of plot plot power power A A versus versus power power of of that that display display Another Another wind wind velocity. velocity. given given speed speed for for rotation rotation type type and and efficiency, efficiency, output, output, a a velocity. velocity. Retaining and and wind wind X X Cp Cp of of plot plot is is performance performance illustrates illustrates our our as as rotor rotor power power versus versus a a Cp/X3 Cp/X3 Figure Figure 4.8 4.8 velocity velocity given given I/X. I/X. by by is is while while proportional proportional directly directly is is the the variables, variables, to to power power in in Figure Figure 4-7. 4-7. shown shown calculations calculations wind wind turbine turbine Smith-Putnam Smith-Putnam the the illustrates illustrates 12 12 10 10 111 111 0 0 In In a. a. N) N) 0 0 cc cc 1.1J 1.1J 0 0 a. a. 0.2 0.2 0.3 0.3 0,4 0,4 0.1 0.1 0.5 0.5 V. V. Rn Rn the the Wind Wind Smith-Putnam Smith-Putnam Speed Speed For For Wind Wind Output Output Versus Figure Figure Power Power 4.8 4.8 Turbine Turbine Op Op -= -= 53 53 Note Note that and and RPM RPM pitch pitch angle, angle, the the stall stall controls the the maximum maximum constant constant at at output output power power and and drag drag controls controls the the starting velocity. Another Another point of of considerable considerable importance importance the the is is location location of of maximum maximum Operation Operation coefficient. coefficient. of of point point the the maximum maximum coefficient coefficient will will power power near near power power the the give give in in increase increase increase increase given given for for in in wind wind greatest greatest velocity hence hence and and the power power greatest greatest a a sensitivity sensitivity wind wind speed speed fluctuations. fluctuations. to to 4.4 4.4 THE THE OPTIMUM OPTIMUM ROTOR; ROTOR; GLAUERT Glauert Glauert has has developed developed simple simple model model the for for optimum optimum windmill. windmill. The The approach approach used used is is a a the the rotating rotating disk disk (i.e., (i.e., treat treat corresponds corresponds to to rotor rotor with with infinite infinite actuator actuator number number of of rotor rotor to to a a as as a a an an blades) blades) and and for for integral integral the the The The integral integral is is made made set set stationary stationary subject subject an an up up power power power. power. to to an an constraint; constraint; the the results results yielding yielding the the maximum maximum for for given given tip tip output output speed speed ratio. ratio. energy energy power power a a The The relation relation for for the the coefficient coefficient is is power power 8 8 ix ix 3 3 dx dx CP CP (4-17) (4-17) = = = = x2 x2 pVco37CR2 pVco37CR2 Y2 Y2 ° ° where where rQ rQ V V RS) RS) u u a' a' and and o, o, x x = a a ====== , , Vco Vco Voo Voo Vco Vco Since Since integral integral the the for for the the involves involves dependent dependent variables, variables, another another relation relation two two is is required. required. power power This This is is the the equation equation energy energy a(1a) a(1a) a'(1a')x2 a'(1a')x2 (4- (4- 8) 8) 1 1 = = 54 54 of of this this relation illustrating illustrating is is velocities unique unique consider the the Perhaps Perhaps the rotor rotor to to at at most most way way in in with with annular annular circumferential streamtubes streamtubes flow flow uniform uniform The The assumed assumed be be plane. plane. is is to to no no two-dimensional two-dimensional flow these these conditions, variations. variations. assumed. assumed. be be Under Under may may Direction Direction Wind Wind Plane Plane Rotation of of Velocity Velocity Diagram Figure Figure 4.9 4.9 of of velocity velocity the the induced induced .the .the In In the the absence absence be be due due drag, drag, lift lift hence and and at at rotor rotor must must to to perpendicular perpendicular expressions expressions the the relative velocity. developed developed Two Two be for for under under the the tan tan to to may may that that velocity velocity condition condition normal the the total total induced the the is is relative relative velocity. velocity. These These to to are are (1 (1 4\7,0 4\7,0 a'rf) a'rf) (4-19) (4-19) tan tan = = a')rS2 a')rS2 (1+ (1+ that that So So (1 (1 a'(1+ a'(1+ a')x2 a')x2 a) a) (4-18) (4-18) a a = = 55 55 posed posed is is problem problem variational variational The The now now rX rX Fla Fla x)dx x)dx a' a' " " (4-20) C C = = JO JO P P at)x2 at)x2 a) a) a(1 a(1 a'(1 a'(1 With With G(a, G(a, a', x) 0 0 + + = = = = - - solution solution yields yields The The 3a 3a a, a, (4-21) (4-21) = = 1 1 4a 4a that that so so 1) 1) (1 (1 a)(4a a)(4a atx2 atx2 (4-22) (4-22) = = atx2, atx2, in in high Since Since in in variation variation given given Table a', a', and and 4.1. 4.1. The The 1/4. 1/4. 1/3 1/3 Hence, Hence, x x are are a, a, a a . . ._ ._ ._ ._ . . ideal ideal of of it it that that of of ratios ratios be be reach reach easily easily tip tip speed speed speed speed 7 7 rotor rotor most most rotors rotors seen seen more, more, or or an an can can Disk Disk The The Optimum Optimum Actuator Actuator Conditions Conditions Table Table Flow Flow For 4.1 4.1 afx2 afx2 a' a' x x a a 0 0 0 0 .25 .25 00 00 0.157 0.157 2.375 2.375 .0584 .27 .27 0.374 0.374 .1136 .1136 0.812 0.812 .29 .29 0.753 0.753 .1656 .1656 0.292 0.292 .31 .31 2.630 2.630 .2144 .2144 0.031 0.031 .33 .33 .2222 .2222 1/3 1/3 0 0 00 00 56 56 will will with with distributed distributed rotational rotational velocity and and the in in of of 1/3 1/3 form form the the irrotational operate operate a a = = an an 02 02 cor2 cor2 rV rV a'x2 a'x2 t t The The (2/9) (2/9) i.e. i.e. coefficient coefficient for for various tip tip vortex vortex as as x x power power Go, Go, --> --> --> --> 2V 2V 2V 2V speed speed ratios ratios ratios in in low tip given given Table Table is is At At low low coefficient coefficient speed speed the 4.2. 4.2. is is of of because because power power the the large in in rotational rotational kinetic the the At At ratios, large large wake. wake. tip speed the the coefficient coefficient energy energy power power rotation rotation the the approaches approaches wake and and of of variation variation approaches approaches The The 0.593 0.593 with with tip tip speed speed ratio Cp Cp zero. zero. is is illustrated illustrated in in Figure Figure 1.6. 1.6. Table Table The The Disk Disk X X For For Optimum Actuator 4.2 4.2 Cp Cp vs vs RS-2 RS-2 Cp Cp voo voo 0.5 0.5 .288 .288 1.0 1.0 .416 .416 1.5 1.5 .480 .480 2.0 2.0 .512 .512 2.5 2.5 .532 .532 5.0 5.0 .570 .570 7.5 7.5 .582 .582 10.0 10.0 .593 .593 infoimation infoimation Further Further obtained obtained from from be be this this model model using the the element element blade blade theory. theory. As As may may the the and and known known radial quantities quantities each for for position, position, a' a' the the relative relative velocity velocity the the and and angle a a are are (I) (I) be be determined. determined. Figure Figure used illustrate be be velocities velocities 4.5 4.5 the the and and forces forces relation relation in in to to may may to to may may blade blade the the configuration. configuration. Of Of since since have assumed assumed that that the drag drag is is the the only only force force course, course, we we zero, zero, that that the the blade blade is is lift. lift. acts acts on on 57 57 having having each each containing containing blades blades annulus annulus acting acting B B and and thrust thrust incremental incremental The The torque torque an an on on by by given given chord chord are are c c Bc Bc pW2CL pW2CL (4-23) (4-23) dr dr dT dT q) q) cos cos = = 2 2 and and Bc Bc 2 2 (4-24) (4-24) dr dr rpW rpW CL CL dQ dQ (I) (I) sin sin = = 2 2 (assuming (assuming b b 2a) yield yield expressions expressions The The momentum momentum = = (1 (1 a)adr a)adr 4npA72 4npA72 (4-25) (4-25) dT dT = = a)a'dr a)a'dr (a (a 47cr3pVc00 47cr3pVc00 (4-26) (4-26) dQ dQ = = that that So So a' a' BcCL BcCL if, if, cos cos (4-27) (4-27) sin2 sin2 87cr 87cr 1 1 (I) (I) a a - - a' a' sin sin BcCL BcCL (I) (I) (4-28) (4-28) a' a' sin sin 4)cos4) 4)cos4) 1+ 1+ 8nr of of blades blades be be the the determined. determined. that that the the shape shape of of function function known known and and Now Now a' a' may may a a so so x x as as are are a a given given optimum optimum blade blade for and and noted noted X that that CL CL be be results. results. constant constant the the It Table Table gives 4.3 4.3 an an a may may maximum maximum .7. .7. approaches approaches that that chord chord will will have have at at xa-.= xa-.= a a a a 58 58 Disk Disk Actuator Actuator Optimum Optimum The The For For Blade Blade Parameters Parameters Table Table 4.3 4.3 Bcf/CL Bcf/CL (i) (i) 27tVcc 27tVcc .497 .497 0.35 0.35 50 50 .536 .536 1.00 1.00 30 30 .418 .418 1.73 1.73 20 20 2.43 2.43 .329 15 15 .228 .228 3.73 3.73 10 10 5.39 5.39 .161 .161 7 7 7.60 7.60 .116 .116 5 5 VORTEX VORTEX THEORY THEORY 4.5 4.5 from from the used flow flow model describe describe differs differs in in flow flow real-rotors real-rotors The The respects respects to to many many over over of of model model the bound bound involves involves vortices vortices frequently-used frequently-used the the optimum A disk. disk. to to actuator actuator use use of of of of applied applied blade each each the wings, wings, theory theory Following Following the the lift. lift. to to vortex vortex concepts concepts represent represent as as induced induced flow enables enables simple simple the each scheme scheme line. line. This bound bound modeled modeled is is at at rotor rotor vortex vortex a a as as that that the the induced flow Biot-Savart Biot-Savart However, However, Law. Law. via via the section section determined determined be be note note to to may may one one blade blade should should fully fully order order the the the flow, flow, section. section. the the blade blade will will chordwise In In represent represent to to vary vary over over have have of of Since Since line. line. windmill windmill lieu lieu be be replace replace bound in sheet sheet by by most most vortex vortex rotors rotors vortex vortex a a very very a a neglected neglected of of without without flow flow be be loss in in variation variation the the chordwise chordwise solidity, solidity, low low may may accuracy. accuracy. the the produce produce local blade blade lift lift while while voracity voracity bound bound the the the the this this scheme, scheme, In In to to serves serves on on of of the the element element each each blade. the Several Several solutions for for trailing trailing induce filaments filaments velocities vortex vortex at at solving solving have have partial partial element element obtained been been by differential equations, induced induced volocity volocity blade blade at at a a of of method method integration Biot-Savart the the direct direct Now Now but but is is straightforward straightforward Law. Law. the the most most as as a a 59 59 straightforward straightforward method requires requires this this input the it it knowledge knowledge of of trajectory trajectory the the as as appear, appear, may may an an as as of of the the the the will of of filaments filaments Since wake consist consist in in wake. wake. the the of of superposition superposition large large vortex vortex a a of of the the each each trajectory other, other, filaments, filaments, each acting configuration) configuration) (or (or vortex vortex vortex vortex array array on vortices vortices established established unless coupled. coupled. all all the be be in Now Now Prandtl Prandtl Lifting-Line Lifting-Line Theory, Theory, cannot cannot are are of of plane plane and and wake wake the the wing in in is is the the assumed assumed although the the lie physically physically calculate calculate to to can can one one impossible impossible sheet sheet flow flow thru the the which which of of velocities velocities wake, wake, Prandtl's Prandtl's results results theory theory gives gives vortex vortex wing wing the the from acceptable acceptable wake sheet sheet is is Just Just happens happens (which (which roll roll vortex vortex to to a a very very a a answers. answers. as as of of the the by the the downstream downstream wake distance distance wing), wing), shed shed propeller propeller be be also also snort snort a a a a up up may may the the considered considered (which (which also sheet sheet rolls in wake). wake). This This approach approach likened be be vortex vortex to to up up a a as as may may materials materials of of in in that that where where used used elementary elementary deformation strength strength and and geometry geometry one one assumes assumes a a calculates calculates wake here here and and forces forces calculate calculate induced induced velocities. velocities. geometry geometry we we assume assume using using the the theory theory criteria optimum optimum Betz This This For For (23) (23) be be used. used. rotor rotor vortex vortex an an may may criteria criteria back back rigid requires requires of of the the wake wake writings The The surface. surface. Betz, Betz, to to a a screw as as move move Theodoresen Theodoresen Lerbs Lerbs Weinig Weinig and and (26) (26) techniques techniques analytical analytical (24), (24), (25), (25), required required the define define to to cover cover optimum optimum propeller. propeller. 60 60 CHAPTER CHAPTER 5 5 MACHINES MACHINES CROSS-WIND CROSS-WIND AXIS AXIS WAKE WAKE MODELING MODELING THE THE OF OF VORTEX VORTEX 5.1 5.1 of of it it in in interest interest Section Section is is Continuing Continuing approach discussed the the the 4.1, 4.1, construct construct vortex to to been been in in the discussed discussed has has this this literature. of of crosswind crosswind axis since since not not actuator, actuator, system system a a if if modeled modeled device device be be the the simply simply oscillating oscillating that that first first We We to to actuator actuator note note assume assume an as as we we that that of of the the wake wake shed shed similar of of and and the the section section disc disc device device rotor rotor system system to actuator actuator a a are are cross cross as as model model acceptable acceptable Again Again that many-bladed high tip develops. develops. be disc disc this for for may may an an a a we we see see speed speed ratio system. system. consider consider model, model, crosswind crosswind having somewhat somewhat machine machine realistic realistic axis axis To To construct construct a a more more a a these these in simplicity simplicity do do and and path path circular circular about lifting lifting for the the slender slender blades, not not move move assume assume a a of of path path velocity attack attack angle angle constrained constrained follow and and axis, axis, but constant constant at at at at to to square square are are zero zero a a the the model in relative relative This As shown shown Figure the leeward blade blade path. path. is is 5.1. 5.1. to to sector, sector, a a moves moves up up pair pair and it it trailing trailing finally bound starting starting and and it it shown, shown, sheds its sheds sheds vortex vortex vortex vortex as as a a a a as as of of the the On portion portion the the the forward lift lift path. portion portion upper upper passage passage zero zero assumes assumes over over across a a in in that situation situation shown shown similar similar Figure Figure Thus Thus the final wake Note Note 5.2. 5.2. system system appears appears occurs. occurs. as as the the the simple ring ring crisscross crisscross will sides sides that that the is, is, type type to to system system system; system; converge converge a a on on vorticity vorticity speed blade blade number streamwise streamwise the tip ratio and will will cancel increased. increased. is is component component as as 61 61 WIND WIND AXIS AXIS OF OF V.. V.. ROTAT ROTAT ION ION of of Actuator Actuator Cross-Wind Cross-Wind Shedding Shedding Axis Vortex Vortex Figure Figure 5.1 5.1 62 62 WIND WIND , , ;-1\ ;-1\ , , .-- .-- 1 1 1 , 1 1 of of Figure Figure Bladed Bladed Vortex Vortex System System Single Single 5.2 5.2 Crosswind Crosswind Axis Axis Actuator Actuator that that the the shown shown solution It It be be induction induction for for the the of of infinite infinite tube tube arbitrary arbitrary of of vortex vortex can can an an of of of of section section the the is is that that circular circular section, section, uniform uniform internal internal axial axial flow, flow, cross cross same same as as cross cross one one a a and and external flow. flow. zero zero If If realistic realistic consider consider the of of crosswind crosswind axis axis where where the the blades blades system system now now we we more more case case a a fixed fixed then then about about axis, axis, the the and and lift lift the the consequently consequently rotate rotate shed shed trailing trailing and and starting starting vorticity vorticity is is a a continually continually changing. changing. Adopting Adopting similar similar those those used used the the for for path path arguments arguments to to and and system system square square high high assuming assuming ratios, advance advance wake wake obtain obtain sketched sketched in in Figure Figure vortex vortex 5.2. 5.2. system system now now we we a a as as importantly importantly This This is is different different from from the the previous previous since since is is there there internal internal vorticity vorticity spanwise spanwise case case within within the the tube. tube. be be shown shown easily easily that that It It this vorticity vorticity spanwise spanwise linearly linearly is is distributed distributed can can across across tube tube the the and and that that internal internal the the induced flow flow is axial axial uniform. it it Thus Thus that that not not appears appears an an even even ideal ideal cross-axis cross-axis ideal ideal machine machine achieve achieve the coefficient coefficient of of wind wind cannot cannot axis since since system, system, power power a a induced induced the the axial axial flow uniform. uniform. is is not not 63 63 ROTOR ROTOR DARRIEUS DARRIEUS 5.2 5.2 of of standard standard approach crosswind-axis crosswind-axis adopt adopt device device the Darrieus-type Darrieus-type analyze analyze To To we we a a of of wake wake the the by by analysis analysis the the which which the the is is forces forces wing wing theory, theory, momentum momentum system system to to on on a a express express expression expression these these forces forces for for itself. itself. surface surface The lifting lifting theory theory the well well airfoil airfoil by by at at an an as as as sufficient sufficient obtains obtains wing wing forces the the wake wake equating equating and and induced induced flows. flows. By By contains contains unknown unknown one one induced induced flows. flows. the the determine determine equations equations to to station station (parallel (parallel the the axis) axis) each each spanwise spanwise that that considered considered the the device device to to For For assume assume we we each each station station be device device the the in in the that the forces quasi-independently quasi-independently behaves behaves at on on may may sense sense well well windwise windwise experience experience devices devices these these general, general, In In forces. forces. equated equated the wake wake to to a a a as as as as can can deflected deflected the be be side. side. the the wake wake that that cross-wind cross-wind to force, force, can can so so induced induced device device the the flows flows the the will will theory, theory, with with Consistent Consistent at at vortex vortex are are one one assume assume we we perturbation perturbation is is if if that that windwise windwise wake wake obtain obtain 2aV,o, 2aV,o, the the Thus Thus half half their their in in the the value value wake. we we giving giving the flow flow velocity velocity a), a), V.0(1 V.0(1 flow flow has has then then the the incoming incoming itself itself device device the the system at at - - in in illustrated illustrated Figure Figure 5.3. 5.3. following following assumptions, assumptions, the the adopt adopt shall shall analysis analysis simplify simplify the the order order In In to to we we p p 0 0 = = 0 0 CD CD = = sin sin CL CL 2rc a a --= --= R R c c 64 64 Wind Wind Crosswind-Axis Crosswind-Axis for for Flow Flow Figure Figure Turbine Turbine System System 5.5 5.5 a a 65 65 inviscid inviscid limited limited analysis analysis then then high will will results results tip tip be be ratios ratios Our Our the the speed speed where to to at at an an of of maximum maximum low low tip attack attack speed speed ratio angle angle The is is small. small. performance performance requires requires numerical numerical a a nonlinear nonlinear analysis analysis model aerodynamics aerodynamics Using stall. stall. the the the the above above assumptions assumptions and and to to near near write write Kutta-Joukowski Kutta-Joukowski starting starting with the the law, law, can can we we pW2cCL pW2cCL pWF pWF L L (5-1) (5-1) = = = = that that so so sin sin F F WCL WCL 7tcW 7tcW (5-2) (5-2) a a = = = = 2 2 Since Since force force the expressed expressed the the airfoil be be on on can can as as p*xf p*xf P P (5-3) (5-3) = = obtain obtain we we (v (v ij ij 0) 0) ej ej sin2 sin2 vavt vavt sin sin vavt vavt 0 0 sin 0 0 (5-4) (5-4) + + one one cos cos the the the the airfoil airfoil Now Now lost lost in the the force force the the which the streamtube streamtube momentum momentum equate equate to to on on we we can airfoil airfoil of of width width the the be be streamtube streamtube the airfoil occupies. occupies. dx dx when when Let Let position from from angular 0 0 goes goes position position by width width related related dx dx The The 0 0 is dO. dO. dO dO to to to + + Rd01 Rd01 I I sin sin 0 0 (5-5) (5-5) = = The The itself itself of of will will revolution interval interval time time the the analysis shall shall be repeat repeat process process every every so so our our period period Of Of which which the the airfoil this this time time period, is is will will spend spend of of 27c/Q. 27c/Q. increment increment time time in in dO/Q dO/Q one one a a 66 66 of of portion portion time front front the the and another another increment the the streamtube of of of of in in the portion the the dO/S2 dO/S2 rear rear Since Since wake. wake. contribution contribution from streamwise streamwise equation the the force force (5-4) (5-4) is is symmetrical symmetrical be be to to seen seen with with write write the the the angles angles blade equation equation 0 0 force force the the for for time 2760 2760 period period respect respect to to ± ± may may we we as as dO dO sin2 sin2 2pncVtVa 2pncVtVa 0 0 dFblade dFblade (5-6) (5-6) = = Now Now in equation equation the the the the yields the streamtube force force momentum momentum as as 20n 20n (1 (1 a)Ko a)Ko I I sin sin pRd0 pRd0 I 0 0 2K0 2K0 dFmomentum dFmomentum (5-7) (5-7) a a = = Equating Equating that that these these assumption assumption forces forces under the the and and RO RO yields yields a) a) Va Va Vt Vt V.0(1 V.0(1 two two = = = = an an - - for for expression expression axial axial for for blade blade interference interference factor the the a a one one RS2 RS2 c c I I sin sin 0 0 (5-8) (5-8) a= a= 2R 2R Vco Vco for for blades blades B B or or Bc Bc RS2 RS2 sinOI sinOI (5-9) (5-9) a a = = 2R 2R Kt, Kt, the the that that blade blade resolved resolved defined, defined, force force be into into is is Now Now radial radial and and torque torque components. components. a a may may given given The The is is by by torque torque a)2 a)2 sin2 sin2 K20 K20 pncR pncR Q Q 0 0 (5-10) (5-10) = = The The with with blades blades for for is is B B rotor rotor torque torque average average a a (BcX)2 (BcX)2 BcX BcX 4 4 3 3 [pnBcRY,20] [pnBcRY,20] (5-11) (5-11) Q Q = = ) ) 32R 32R R R 37T 37T by by is is given coefficient coefficient sectional sectional corresponding corresponding the the and and power power B2c2X2 B2c2X2 BcX BcX 4 4 Bc Bc 3 3 (5-12) (5-12) =7rX =7rX Cp Cp R2 R2 R R 32 32 37r 37r of of maximum maximum coefficient coefficient when quantity quantity the the 0.554 0.554 yields yields expression expression This This power power a a of of be be with with made made consideration refinements refinements drag and and Further Further BcX/2R BcX/2R 0.401. 0.401. can can am= am= = = point point approximately approximately of of maximum maximum attack attack the the angle angle maximum maximum The The attack. attack. of of angle angle at at occurs occurs where where Tr/2 Tr/2 0 0 = = Bc Bc 1 1 am am (5-13) (5-13) tan tan a a X X 2R 2R X X the the equation equation tip tip starting starting (5-13) (5-13) equal equal When When is is to to to to set set a a rearrange rearrange express express may may we we am am a. a. obtain obtain Using Using speed speed ratio. ratio. 14° 14° max max we we = = 4 4 (5-14) (5-14) Xstart Xstart Bc Bc 1+2 1+2 would would foot foot radius have chord chord and tip starting starting with with foot foot 20 20 that that three-bladed three-bladed rotor rotor a a a a a a one one a a so so of of ratio ratio speed speed about about 3. 3. simple simple low low tip the losses losses speeds, speeds, drag be be of of will will this this Since Since operate operate at at rotor rotor not not type type may may then then drag drag is is velocity velocity local local is is W W The The that that approximated approximated the assuming assuming by by R.Q. R.Q. torque torque 68 68 BCX3 BCX3 C C CD CD 2 2 3 3 D D 2 2 BcR BcR (5-14) (5-14) pR pR f2 f2 QD QD pVcc, pVcc, = = 2 2 2 2 coefficient coefficient is is contribution contribution the the the and and to to power power CDBc CDBc X3 X3 (5-15) (5-15) AC AC - - = = p p 2R 2R blade blade Bc/2R, Bc/2R, ratio of of the the point, point, this this solidity defined circumference disc be be At At to to a a = = as as a a may may coefficient coefficient becomes diameter. diameter. The The power power \ \ 13n 13n CD CD 16 16 a3X3 a3X3 2 2 2 noX noX X X (5-16) (5-16) C C + + = = 62 62 4 4 3 3 C C D D =C =C be be that that it it and and Cp Cp may may seen seen 3 3 2 2 ROTOR ROTOR THE THE CIRCULAR 5.3 5.3 Darrieus-type Darrieus-type inertial inertial required required for the the the the rotational high high speeds speeds the loads At At rotor, rotor, are are bending bending loads loads the in in blades. These in in result result large large bending and and substantial removed be be loads loads may may loads loads the the the the that similar similar entirely entirely by by deploying blade in the the shape tensile. tensile. catemary catemary to to so so a a are given given by by the investigated investigated Blackwell and and (22) (22) been been required required troposkien. troposkien. shape shape has has The The The The name name approximated approximated by elliptic elliptic integrals and and sine is is by by described described parabola. parabola. The The is is curve curve or or a a curve curve the the of of performance performance bringing blades rotation effect effect closer caused caused by the axis substantial is is to to on on of of usable usable lift lift both both the Figure Figure rotational rotational the the the the and since since reduced. reduced. speed speed 5.4 5.4 component component are are between between troposkien troposkien the local local angle blade and and the the the the below below illustrates illustrates and the tangent tangent curve curve 7 7 of of rotation. rotation. axis axis 69 69 1.0 1.0 Troposkien Troposkien Catenary Catenary Circle Circle 0.5 0.5 , , 0.5 0.5 1.0 1.0 1.5 1.5 R/Rm R/Rm of of Equal Equal Caternary Caternary Length Circle Circle and and Figure Figure Troposkien, Troposkien, 5.4 5.4 If If proceeds proceeds in the in Section Section of of curved curved 5.2. 5.2. the the analysis analysis The The rotor rotor manner manner same same we we as as expression expression becomes for for the the the of of unit unit height analyze analyze rotor, rotor, a a a a aX aX sin sin 0 0 (5-17) cos cos a a y y = = point point and and the speed speed the the tip ratio solidity solidity since since taken taken R R the the be be Rmax Rmax product product where where at at crX crX as as = = may may by by of of generated generated along axis axis slice slice dz is is independent independent The the product product this this R. R. is is rotor rotor torque torque a a 3o-2X2CS2 3o-2X2CS2 dQ dQ 2 2 % % (5-18) (5-18) aX aX pnBcVco pnBcVco + + cos cos cos cos y y y y y y = = 32 32 dz dz 8 8 70 70 incremental incremental the the coefficient coefficient and and is is power power dC dC 72--8 72--8 4rcaX 4rcaX ciQ ciQ a2X2 a2X2 p p COS2 COS2 R R GX GX (5-19) (5-19) + + cos cos cos cos y y y y y y dzy2pVA dzy2pVA 2 2 dz dz A A 3n 3n 8 8 of of integration integration The The equation equation arbitrary arbitrary (5-19) (5-19) for be be accomplished; accomplished; geometry geometry an an may may one one simple simple which which circular circular is is maximum maximum the the blade of of for for coefficient coefficient 0.536 0.536 at at case case power power a a occurs occurs of of included included be be drag drag and effects effects stall The The 0.461. 0.461. erX erX in in above above the the model model by = = can can = = amax amax of of development development that that element element theory blade blade developed developed similar similar for for the the wind-axis to to rotor. rotor. a a 71 71 U U 6 6 CHAPTER CHAPTER GRADIENT GRADIENT WIND WIND VERTICAL VERTICAL DUE DUE TO TO MOMENTS MOMENTS AND AND FORCES FORCES INTRODUCTION INTRODUCTION 6.1 6.1 unifoim unifoim assumed assumed be be wind wind relative relative previously, previously, the to to covered covered of of analysis analysis the the was was In In rotors both both flow flow reality, reality, of of the the In of of rotation rotation plane plane the the rotor. rotor. perpendicular perpendicular be be and and parallel parallel to to and and to to steady steady uniform, uniform, neither neither will will be be Real Real flows motion motion nor nor and and irregularities irregularities rotor rotor can can occur. occur. with with elevation elevation all all turning turning and-wind and-wind present present gustiness gustiness gradient, gradient, wind wind Vertical Vertical unidirectional. unidirectional. the the flow flow local local First, First, of of turbines. turbines. wind wind operation operation and and design design the the difficulties difficulties double-edged double-edged to to of of the the of of the the effects effects magnitude magnitude predict predict the the techniques techniques secondly, secondly, to to known; known; be be conditions conditions must must of of local local flow flow of of knowledge knowledge lack lack The The machines machines wind wind adapted adapted be be variations variations flow flow to to must must While While bather. bather. considerable considerable terrain terrain rough rough of of represents represents regions regions some some in in particularly particularly conditions, conditions, a a turbulence turbulence data data little little is is there there terrain, terrain, rough rough flow flow for for exists exists gradient gradient on on data data wind wind or or no over over in in turning turning wind wind considerable considerable of of reported reported the the has has (13) (13) Slade Slade turning. turning. wind wind and and presence presence spectra spectra flat flat of of flow flow knowledge knowledge the the terrain. terrain. By rough rough contrast contrast layer layer over over atmospheric atmospheric surface surface the the over over complete. complete. much much terrain terrain is is more more layer layer surface surface atmospheric atmospheric the the in in wind wind of of made made been been structure structure over have have studies studies Extensive Extensive that that flow flow the the relation relation developed developed for for Obukhov Obukhov have (14) (14) and and Monin Monin mean mean terrain terrain flat flat a a involved involved three relation relation Their Their parameters, parameters, stratification. stratification. unstable unstable neutral neutral and and stable, stable, encompasses encompasses Monin-Obukhov Monin-Obukhov the the stability stability and and roughness roughness parameter, parameter, the the surface surface velocity, velocity, friction friction surface surface the the a a by by approximated approximated terrain terrain frequently is is rough rough flow flow power power the the a a length. length. By By contrast, contrast, over over mean mean height height relation relation with with law law 73 73 (Z (Z - - (6-1) (6-1) \hJ \hJ VR VR where where the the coefficient coefficient is is than than less less of of Because Because the the of of simplicity simplicity equation equation (6-1), (6-1), it it since since one. one. 11 11 requires requires fewer fewer that that and and fact fact the the the the wind wind variation variation parameters, parameters, limited 100 100 200 200 to to over over a a range range feet) feet) is is required required wind wind for for turbines turbines the the shall shall above above relation. we we use use The The departures departures from from the the flow studied studied have have previously previously first first order effects effects the the turbine turbine no no on on however, however, periodic periodic variations variations in in output, output, time time dependent dependent side side forces forces torque, torque, pitching pitching and and mean mean These These forces forces and and moments moments will will effect effect the the overall overall moments moments dynamics dynamics and and can can occur. occur. system system hence hence both both the the of of and and design design operation operation wind wind turbine. turbine. addition addition In In the the aerodynamic aerodynamic forces forces and and to to a a certain certain mechanical mechanical forces forces and and moments moments This This section section moments moments will will only only deal deal with with present. present. are are the the aerodynamic aerodynamic loads loads due wind wind gradient. gradient. to to Both Both and and flapping flapping induce induce large large forces forces rotor rotor and and Rotor Rotor be be yaw yaw moments. moments. can can yaw yaw can can treated treated using using of of the the analysis analysis Ribner Ribner flapping flapping (15), (15), and and forces included included moments moments in in the the are are described described Appendix Appendix in in I. I. program program 6.2 6.2 EFFECTS EFFECTS THE THE VERTICAL VERTICAL OF OF WIND WIND GRADIENT GRADIENT A A vertical wind wind gradient gradient induce induce will will forces forces and and illustrated illustrated in in Figure Figure moments moments The The 6-1. 6-1. as as of of largest largest these these the the variation variation the the and and pitching pitching torque torque be be would would As As expected expected moment. moment. the the are are magnitude magnitude of of these these is is dependent dependent scale, scale, it it since since the the moments moments velocity velocity is is difference difference between between upon upon the the and and bottom bottom of of that that top top the the significant. significant. is is Before Before rotor rotor proceeding proceeding further, further, it it should should noted noted be be incremental incremental the the forces forces blade blade element element been been have have designated designated normal normal and and (n) (n) in in tangential tangential on on (t) (t) a a order order avoid avoid confusion confusion with with the the coordinates coordinates to to the the Thus Thus XYZ. XYZ. (x,y) (x,y) of of Chapter Chapter the the 4 4 (t,n) (t,n) are are now now coordinates. coordinates. 74 74 TORQUE TORQUE CI:1)R CI:1)R X X AG AG MOMENT MOMENT PITCHING PITCHING Gradient. Gradient. in in Wind Wind Rotor Rotor Figure Figure 6.1 6.1 a a velocity velocity velocity velocity the the This blade blade diagram. diagram. illustrates illustrates following following the the Figure Figure 6.2 6.2 page page on on V. V. velocity velocity replaced replaced in in been been has has that that the the freestream freestream illustrations illustrations from from previous previous differs differs diagram diagram V. V. velocity velocity wind wind local local with with the the will will for for velocity velocity higher higher than than be be disk, disk, axial of of half half the the the the in in blade blade the the rotor rotor For For a a upper upper a a velocity velocity and and the the both both resultant resultant in in the the increases increases half half The The increase increase Vw Vw disk. disk. in in lower lower blade blade the w w of of principle principle increased increased effect effect it it that that the the be be tip tip high high ratios ratios speed speed of of At At attack attack angle angle may may seen seen a. a. of of of of in in variation variation angle angle The The will will angle angle attack. increased increased be be gradient gradient velocity velocity gust) gust) (due (due to to an an or or of of with with rotation. rotation. angle angle the the will will and and dFt dFt force force in in the the df',1 df',1 variations variations in in attack attack turn turn cause cause from from steady- steady- the the generated generated and and be be first first forces forces for for order order second second and and moments moments Expressions Expressions the the can can 75 75 t t -a) -a) ( ( Vw Vw I I Figure Figure Blade Blade Velocity Velocity Diagram Diagram 6.2 6.2 performance performance in in aerodynamics aerodynamics the the following following differential differential state state The The force force rotor rotor manner manner a a on on element element of of be be expressed expressed in in Taylor Taylor the about about Series Series hub hub that that terms terms rotor rotor may may a a so so 2 2 'AZ 'AZ [dF, [dF, u2 u2 dFt dFt dFt dFt dFtu dFtu +dF, +dF, AZ AZ (6-2) (6-2) + + + + u u Uz Uz = = 0 0 zz zz z z 2 2 a2Ft a2Ft aFt aFt where where dFt dFt and and dFt dFt = = u u = = , , 2 2 au au au au u u Vref Vref similar similar A A expression expression obtained obtained be be for for forces forces These These change change dFn. dFn. from from their their hub hub values values (Z (Z may may = = the the due due in in 0) 0) variation variation wind velocity velocity with elevation. elevation. distance distance The The to to rsin rsin where where AZ AZ refers refers 0i, 0i, Ai Ai = = of of the the of of rotation rotation ith the the angle angle Rewriting Rewriting blade. blade. in in equation equation to to the the form form (6-2) (6-2) Z2 Z2 A A dFt dFt TlA TlA =1T =1T Z Z T2 T2 (6-3) (6-3) + + + + 2 2 76 76 the the expressing expressing in in dFr, dFr, and and manner manner same same z2 z2 (6-4) (6-4) dF=N0 dF=N0 +N2 +N2 N1AZ N1AZ + + 2 2 summing summing B B blade blade and and integrating integrating the the by by and and following following forces moments moments the the obtain obtain over over over over we we where where blades, blades, 2. 2. B B TORQUE TORQUE E E fr2Tidr fr2Tidr irTo irTo r3T2dr r3T2dr sin2 sin2 (6-5) (6-5) dr dr Q Q + + s s = = i=1 i=1 MOMENT MOMENT PITCHING PITCHING fr3N2 fr3N2 jdOi jdOi 2N1drIsin2 2N1drIsin2 (6-6) (6-6) dr dr Oi Oi + + 2 2 i=1 i=1 unless unless B B odd odd is is wind wind noted noted that it it and and evaluated evaluated be be been been has has spaced spaced blades equally equally summation summation The The B B may may over over yawing yawing order order second second first first moment. moment. induces induces gradient gradient or or no no of of various various numbers of of summations summations for for the the the values values gives gives the the Table Table 6.1 6.1 next next page page on on blades. blades. magnitude magnitude the the determine determine evaluated evaluated in order order be be remain remain and and to to to to The The N2, N2, N1, N1, T2 T2 T1 T1 terms terms straightforward, straightforward, indicate indicate let let of of differentiation differentiation is is method method the the and and As As forces forces of of the the us us moments. moments. of of evaluation evaluation by by approach approach NI. NI. the the 77 77 Table Table Trigonometric Trigonometric Sums Sums 6.1 6.1 2 2 B B B B 3 3 B B 4 4 ====== sin2 sin2 0; 0; cos2Qt cos2Qt 3/2 3/2 1 1 2 2 i=1 i=1 sin2 sin2 Oi Oi O1 O1 30t 30t 0 0 cos cos 0 0 cos cos ------ 4 4 i=1 i=1 3 3 sin3 sin3 Oi Oi 0 0 sin3Qt sin3Qt 0 0 4 4 i=1 i=1 a(dFn) a(dFn) sinliloauzi sinliloauzi a a aU aU f(u2 f(u2 )(CL )(CL pA/;'c pA/;'c v2 v2 N1 N1 AZ AZ cos++ cos++ CD CD + + Z Z = = (6-7) (6-7) = = az az au au au au Z=0 Z=0 Z=0 Z=0 (z). (z). vw vw rS, rS, where where and and VAT VAT differenti differenti The- The- ation ation of of requires requires CL CL v v u u As As = = comment. comment. (I) (I) some some 13 13 = = VR VR and and remains remains + + i3 i3 constant constant cc cc acL acL acL acL a4) a4) c c (6-8) (6-8) La La al al aa aa au au a+ a+ all all final final The The result result is is (I)) (I)) sin2 sin2 CD CD au au + + 2 2 sin++ sin++ pVRcv pVRcv N1 N1 cos4)+CL cos4)+CL CL CL AZ AZ = = (6-9) (6-9) a a az az 4) 4) cos cos Z=0 Z=0 78 78 analysis analysis preceding preceding from from evaluated evaluated the the be be coefficient coefficient in in variation variation The The torque torque may may d2Vw d2Vw R2 R2 dVw dVw Q Q A A (6-10) (6-10) +12 +12 CQ CQ A, A, = = = = VRdZ2 VRdZ2 Y2pVinR3 Y2pVinR3 dZ dZ VR VR Z=0) Z=0) Z=0 Z=0 evaluated evaluated and and integrals integrals the the where where I I 12 12 as as are are (f3 (f3 7 7 N N ft ft cl))CL cl))CL 1 1 C C \ \ ar9 ar9 , , cos3 cos3 COS2 COS2 (6-11) (6-11) (1)CD (1)CD sin sin (I)CL (I)CL (I)+ (I)+ 2 2 2 2 1.k2 1.k2 cos cos j j = = R,RR1 R,RR1 cc cc 'it 'it "-hub "-hub and and tr)4 tr)4 d(-1.) d(-1.) sind)1(1 sind)1(1 X X sin24))CL sin24))CL ft ft )(1. )(1. r r a' a' sin sin (6-12) (6-12) CD CD CLa CLa (I) (I) + + + + 12 12 Lk1 Lk1 = = R R R R R R TC TC hub hub 's 's be be and and As As other other forces forces by by the the obtained obtained moments. integrals integrals Similar Similar seen seen can can type type are are if if small small is is angle angle made made the be be simplification simplification considerable considerable (I) (I) (6-12), (6-12), and and (6-11) (6-11) equations equations from from can can closed closed this this of of blades. blades. the the In In portions portions the the of of outer outer and and value value constant constant the the CL CL and and case, case, over over are are c c obtained. obtained. be be approximations approximations form form can can RELATIONS RELATIONS APPROXIMATE APPROXIMATE 6.3 6.3 that that the the be be shown shown drag drag it it neglecting neglecting by by and and ratios ratios situ') situ') speed speed high high tip tip At At (1) (1) may may obtained. obtained. be be expressions expressions following following may may 79 79 PITCHING PITCHING MOMENT M M crXC crXC sin2 sin2 RdVw RdVw Oi Oi La La a) a) (6-13) (6-13) pV127ER pV127ER 2 2 dZ dZ 4 4 VR VR i=1 i=1 Z=0 Z=0 TORQUE TORQUE CHANGE CHANGE ( ( 2 2 B R2d2Vw R2d2Vw 2 2 RdVw RdVw Q Q 0 0 A A S111 S111 E E i i +12 +12 (6-14) (6-14) = = 1 1 VR(I7,2 VR(I7,2 pV12(.' pV12(.' 3 3 dZ dZ VR VR TCR TCR \ \ i=1 i=1 Z=0 Z=0 Z=0 Z=0 where where (1 (1 2 2 GCL GCL a a 2B 2B and and (1 (1 OaCLa OaCLa crXCL crXCL 12 12 a) a) 4B 4B 5B 5B be be values values representative representative Some Some using using by by obtained obtained the the data from from the the wind wind Smith-Putnam Smith-Putnam may may turbine. turbine. data data long long wind The The yield yield term term d2Vw d2Vw dVw dVw 0.104 0.104 0.14 0.14 1 1 1 1 =120' =120' hh hh , , 2 2 dZ dZ hh hh VR VR dZ dZ VR VR Z=0 Z=0 Z=0 Z=0 80 80 below below listed listed values values using using the the and and ft. ft. R R 87.5 87.5 0.083 0.083 = = a a = = 2 2 B B 6 6 X X = = = = 0.6 0.6 CL CL 5.5 5.5 CL CL = = = = a a 0.33 0.33 = = a a of of machine, machine, Smith-Putnam Smith-Putnam the the motion motion flapping flapping neglecting neglecting the the obtain, obtain, we we MOMENT MOMENT PITCHING PITCHING MY MY (1cos (1cos 20t) 20t) (6-15) (6-15) 0.0173 0.0173 = = Y2pV12(.7cR3 Y2pV12(.7cR3 VARIATION VARIATION TORQUE TORQUE Q Q A A Mt) Mt) (1cos (1cos (6-16) (6-16) 0.00153 0.00153 = = Y2pV12: Y2pV12: TcR3 of of the the drag drag the the forces forces the reference reference let let Accordingly, Accordingly, judge. judge. difficult difficult to to numbers numbers The The to to us us are are the the the the turbine turbine and and wind wind torque. torque. moments moments to to 2 2 pV0TR2 pV0TR2 CT CT Y2 Y2 D D CT CT = = 3 3 cP cP pVircR3 pVircR3 0.4 0.4 C C C C CQ CQ Q Q = = = = = = Q Q P P , , X X 81 81 PITCHING PITCHING MOMENT MOMENT (1 (1 200 200 0.259 0.259 (6-17) (6-17) cos cos TORQUE TORQUE VARIATION VARIATION Q Q A A (1 (1 200 200 =O.023 =O.023 (6-18) (6-18) cos cos The The insignificant insignificant drag drag forces forces and and quite quite while while the the pitching pitching is is be be moment moment yaw yaw to to are are seen seen quite quite of of of variation variation appreciable, appreciable, amounting the the of of value value 52% 52% the the The The to to torque. torque. torque a a variation variation be be is is decrease decrease in in 4.6% 4.6% hence hence and and also also to to to to The The torque torque torque torque seen seen a a up up power. power. itself itself variation variation requires requires discussion. discussion. Equation Equation (6-14) (6-14) gives gives expression expression for for the the torque torque more more an an change change which which be be modified modified the the to to change. percentage percentage torque torque may may express express Adopting Adopting profile profile the the by by given given equation equation obtain obtain (6-1) (6-1) 1 1 power power aw aw we we Xo.(1 Xo.(1 a)2CLa a)2CLa } } (112 (112 A A CQ sin2 sin2 4XCL 4XCL 2,92 2,92 +111 +111 (6-19) (6-19) + + ri ri 5(1 5(1 h h 4Cp 4Cp CQ CQ a)CL a)CL h h of of where where is is of of the the height the the hub. effects The The scale, scale, tip tip ratio, ratio, speed speed and and rotor rotor solidity load load maybe maybe from from estimated estimated example example equation equation (6-19). (6-19). operating operating For For wind wind rotor rotor RPM RPM constant constant at at at at a a below below will will speeds speeds design point the the low low have have C. C. and and large large X X The The for for variation variation large large scale scale torque torque appreciable. appreciable. will will be be noted noted that It It be be expression expression bracketed bracketed the the in in equation equation is (6-19) (6-19) approximately approximately equal equal may may to to 3. 3. This This expression expression maximum maximum value value has has when when Experimental Experimental evidence evidence 1/6. 1/6. flow flow for for a a ii ii = = smooth smooth terrain terrain yields 0.17. 0.17. over over r r 82 82 variation variation used used evaluate evaluate the the be be also also developed developed the for for expressions expressions torque torque to The The may may gradient gradient is is in in wind The The variation the the gradient. gradient. due due wind wind due due torque torque in in change change to to output output to to power power of of of wind wind turbine turbine speed speed ratios. tip tip The wide wide approximately approximately net net output output constant constant a a range range over over a a that that variation variation turbine the the in in speed speed with with appreciably appreciably tip tip percentage percentage however however output changes changes so so turbine turbine Figure approaches approaches greatly greatly the the increases increases gust) gust) wind wind gradient gradient output output (or (or net net due due to to zero. zero. as as in in wind due due decrease decrease turbine illustrates illustrates the the to to following following output output the the percentage percentage 6.3 6.3 mean mean page page on on of of in in Flapping Flapping the the included motion motion blades blades wind wind turbine. turbine. for for the the Smith-Putnam Smith-Putnam gradient gradient not not was was be be of of variation variation obtained by gradient gradient due due the the magnitude magnitude absolute absolute this this example. example. The The to to may may power power with with Figure Figure 4.8. 4.8. along along Figure Figure of of 6.3 6.3 using using results the the 83 83 16 16 Turbine Turbine Smith-Putnam Smith-Putnam Wind Wind _ _ Flapping Flapping Neglected Neglected Blade Blade 813=0° 813=0° z16 z16 )1 )1 ( ( 14 14 - - si si Velocity Velocity vi vi Wind Wind 120) 120) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 4 4 8 8 6 6 10 10 12 12 14 14 RATIO, RATIO, TIP TIP SPEED SPEED RS/ RS/ Vco Vco Power Power Figure Figure Reduction Percent Percent Wind Gradient In In Due Output Output To 6.3 6.3 84 84 References References the the Putnam, Putnam, Wind, Power Power VanNostrand VanNostrand From From Company, P. P. C. C. Inc. Inc. New New York, York, 1948. 1948. Final Final Wind Wind N.Y.U., N.Y.U., the Turbine. Turbine. of of Report Report Office Office Production, Production, Research Research and and on on Production Production Development, Development, War War Board, Washington, Washington, PB25370, PB25370, January January D.C., D.C., 31, 31, 1946. 1946. of of Golding, Golding, Electricity Electricity by Wind Generation Generation The The W., W., E. E. Power, Power, Philosophical Philosophical Library, Library, York, York, New New 1956. 1956. Rankine, Rankine, J., J., Institute of-Naval Transactions, Transactions, Architects, W. W. Vol. Vol. 6, 6, 1865. 1865. 13, 13, p. p. Froude, Froude, Transactions, of of Naval Naval Institute Institute W., W., Architects, Architects, Vol. Vol. 1878. 1878. 19, 19, 47, 47, p. p. of of Naval Naval Institute Institute Froude, Froude, Transactions, Architects, Architects, R.E., R.E., 6, 6, Vol. Vol. 390 390 30, 30, ,1889. ,1889. p. p. Joukowski, Joukowski, du du N. N. Travanx Travanx des des Calculs E., E., 7. 7. Essais Essais Bureau Bureau Aeronatuiques Aeronatuiques l'Ecole l'Ecole de de et et Superieure Superieure Technique de Moscou, Moscou, 1918. 1918. Goorjian, Goorjian, AIAA AIAA Journal, Journal, Vol. No.4, No.4, April, April, P P 8 8 M., M., 1972, 10, 10, 543-4. 543-4. p. p. Glauert, Glauert, Theory Theory Aerodynamic Aerodynamic H., H., 9 9 Editor-in-chief), Editor-in-chief), Durand, Durand, (W. (W. F. F. Division Division Vol. Vol. 6, 6, L, L, Berlin Berlin Julius Julius Springer, Springer, 324. 324. 1935. 1935. p. p. of of Glauert, Glauert, Experimental Experimental Analysis Analysis H., H., The The Results Results in in the the Windmill Windmill and and Brake Brake Vortex Vortex of of Ring Ring Airscrew, States States R R M M Br, Br, 1026, 1926. 1926. & & an an Appendix Appendix Prandtl, Prandtl, Schraubenpropellor mit mit L., L., gerngstein gerngstein Energieverlust, Energieverlust, to to by by Betz, Betz, A. A. Gottinger Gottinger Nachr. 193-217, 193-217, 1919. 1919. p. p. of of the the Theory Theory Goldstein, Goldstein, On S., S., Screw Screw Vortex Vortex Propellors, Propellors, Roy. Roy. Soc. Soc. Proc. Proc. (A) (A) 123, 123, 440, 440, 1929. 1929. p. p. of of Slade, Slade, Applied Applied Journal Journal D. D. Meteorology, Meteorology, H., H., April April Vol. Vol. 1969, 1969, 293-7 293-7 8, 8, p. p. Monin, Monin, and and Obukhov, Obukhov, of of Basic Basic A. A. Mixing Mixing Turbulent Turbulent Laws Laws A. A. S. S. M., M., the the In In Ground Laver Laver of of the the Atmosphere, from from translated translated Akademiia Nauk Nauk SSSR, SSSR, Leningrad, Leningrad, Geofizicheskii Geofizicheskii Institut, Institut, Trudy, Trudy, Vol. Vol. No. No. 151, 151, 1954, 1954, 24, 24, 163-187. 163-187. pp pp Ribner, Ribner, in in Propellers Propellers N. N. S., S., NACA NACA Yaw, Yaw, Report Report Washington, 820, 820, D.C., D.C., 1948. 1948. of of Shapiro, Shapiro, Principles Principles Helicopter Helicopter Engineering, Engineering, J., J., McGraw-Hill, McGraw-Hill, New New York, York, 1955. 1955. Wolkovitch, Wolkovitch, Analytical Analytical of of Prediction Prediction Vortex Vortex J., J., Ring Ring for for Boundaries Boundaries Helicopters Helicopters in in Helicopter Helicopter Descents. Descents. Amer. Amer. Steep Steep Soc., J. J. Vol. Vol. No.3, No.3, 17, 17, 1972. 1972. 85 85 Theory Theory of of Extension Extension the the An An Vortex Townend, Townend, H. H. and and H. H. C. C. Bateman, Bateman, N. N. H. H. H., H., Lock, Lock, C. C. Including Including Small Small Pitch, Pitch, of of Airscrews Airscrews in in with with Applications Applications of of Airscrews Airscrews 1925. 1925. 1014. 1014. A.R.C., A.R.C., M & & R. R. Results. Results. Br. Br. Experimental Experimental Performance Performance Design Design and and the the Preliminary Preliminary Report Report A A Rainbird, Rainbird, and and J. J. W. W. Lilley, Lilley, M. M. G. G. on on 1956. 1956. 102, 102, CoA CoA Cranfield, Cranfield, No. No. Report Report Windmills. Windmills. of of Ducted Ducted Wind Wind Economics Economics Vertical-Axis of of the the and and Performance Performance The The Rangi, Rangi, and and R., R., P. P. South, South, Presented Presented Research Research Ottawa, Ottawa, Council, Council, Canada, National National the the Developed Developed Turbine Turbine at at American American of of Pacific Pacific Region the Northwest Northwest of of the the Meeting Meeting Annual Annual the the 1973 1973 at at October October 1973. 1973. 10-12, 10-12, Alberta, Alberta, Calgary, Calgary, Agricultural Agricultural Engineers, Engineers, of of Society Society York, York, McGraw-Hill, McGraw-Hill, New New of of Propulsion, Propulsion, Aerodynamics Aerodynamics Weber, Weber, and and Kuchemann, Kuchemann, J. J. D., D., 1953. 1953. Vertical-Axis Vertical-Axis of of Type Type Troposkien Troposkien for for Blade Blade Shape Shape Reis, Reis, and and E., E., G. G. Blackwell, Blackwell, F., F., B. B. a a SLA-74-0154, SLA-74-0154, April, April, 1974, 1974, Energy Energy Laboratories Laboratories Report Sandia Sandia Turbine, Turbine, Wind Wind Mexico. Mexico. New New Albuquerque, Albuquerque, Kgl. Kgl. Nach. Nach. der der Geringstem Geringstem Energieverlust," mit mit ler ler "Schraubenpropel "Schraubenpropel 1919, 1919, Bet Bet A., A., z, z, reprinted reprinted 193-217; 193-217; Klasse, Klasse, Math.-Phys. Math.-Phys. Gottingen, Gottingen, Wiss. Wiss. der der Gesellschaft Gesellschaft pp. pp. zu zu and and Prandtl Prandtl by by Aerodynamik Aerodynamik A. A. Hydrodynamik Hydrodynamik und L. L. Abhandlungen Abhandlungen Vier Vier in in zur zur 68-92. 68-92. Bros., Bros., Edwards Edwards 1943), 1943), Arbor: Arbor: Ann Ann (reprint (reprint Gottingen, Gottingen, 1927 1927 Betz, Betz, pp. pp. York, York, McGraw-Hill McGraw-Hill New New Book Book Inc., Inc., Co.; Co.; of of Propellers, Propellers, Theory Theory Theodore, Theodore, Theodorsen, Theodorsen, 1948. 1948. of of Blades Blades and Number Number Finite Finite with with Propellers Propellers Loaded Loaded "Moderately "Moderately 1952, 1952, Lerbs, Lerbs, W., W., H. H. a a Architects Architects and and Naval Naval Circulation," Circulation," Soc. Soc. Trans. Trans. of of Distribution Distribution Arbitrary Arbitrary an an 73-117. 73-117. Marine Marine 60, 60, Engrs., Engrs., Leipzig: Leipzig: Turbomaschinen Turbomaschinen Schaufeln Schaufeln die die Die Die Stromung Stromung 1935, 1935, Weinig, Weinig, F., F., von von um um Barth. Barth. J.A. J.A. book, book, from from German German 1939 1939 Propeller, Propeller, of of the the Aerodynamics Aerodynamics trans. trans. 1939/1948, 1939/1948, , , Air Air Ohio: Ohio: Author. Author. and and revised revised by Dayton, Dayton, Luftschraube Luftschraube Aerodynamik Aerodynamik der der Command. Command. Air Air Atl. Atl. Div. Div. Documents Documents 86 86 Symbols Symbols Projected Projected disc disc A A rotor rotor area area Axial Axial interference interference factor the the u/Voc, u/Voc, at at rotor rotor 1 1 a a a a - - the the Tangential Tangential interference factor co/20 co/20 a' a' a' a' rotor rotor at at -= -= of of Number Number blades blades Axial Axial in in factor factor the interference interference wake wake b b 1-u1N.,, 1-u1N.,, chord chord Blade Blade lift lift coefficient coefficient Sectional Sectional CL CL coefficient coefficient Sectional Sectional drag drag CD CD of of in in rotation direction direction force force coefficient the Sectional Sectional Cx Cx of of plane plane Section Section coefficient coefficient normal the rotation' rotation' force force Cy Cy to to 13/(l/2)pAK3 13/(l/2)pAK3 PA1/2)pSVG,3 PA1/2)pSVG,3 Power Power coefficient, coefficient, Cp Cp or or drag drag length length unit unit Sectional Sectional D' D' force force per per drag drag Lift Lift ratio, LID LID to to Sectional Sectional unit unit lift lift length length L' L' force force per per flow flow Mass Mass th th rate rate Power Power extracted extracted from the the air Pressure Pressure radius radius Rotor Rotor radius radius Local Local rotor rotor projected projected Rotor Rotor translator surface surface area area or or flow flow Axial Axial velocity velocity the rotor rotor at at wake wake the the velocity velocity in in flow flow Axial Axial velocity velocity wind wind Free Free stream stream element element relative relative the velocity velocity Resultant Resultant rotor rotor to to speed speed ratio, ratio, R.Q/Vc. R.Q/Vc. Tip Tip X X rON., rON., Local Local ratio, ratio, speed speed Greek Greek of of attack attack Angle Angle pitch pitch blade blade translation translation velocity, velocity, Chapter Chapter 2; 2; the the and and normal wind wind the the the the to to between between Angle Angle Chapter Chapter 5,6. 5,6. angle, angle, relative relative velocity velocity of of rotation rotation the and and plane plane the the between between Angle Angle (i) (i) relation relation Wind-height Wind-height exponent exponent 11 11 velocity velocity Translator Translator Chapters Chapters angle, angle, 6 6 rotation rotation blade blade 5, 5, Chapters Chapters angle, angle, pitch pitch 3, 3, Blade Blade 4; 4; density density Fluid Fluid velocity velocity angular angular Rotor Rotor of of downwind downwind the the velocity velocity rotor rotor Fluid Fluid angular 88 88 XICENHddV XICENHddV I PROGRAM PROGRAM LOADS LOADS PERFORMANCE PERFORMANCE AND TURBINE TURBINE WIND WIND utilizes utilizes Simpson's Simpson's the the language and in in written written Fortran package package is is The The a a program program the the using using written written integration. integration. under 3300 3300 numerical numerical It CDC CDC of of Rule/three Rule/three method method a a was was pass pass small small therefore therefore University, University, there be be package package operating operating State State Oregon Oregon OS-3 OS-3 at at some some may may other other implementation implementation for for differences differences systems. systems. on on reader, reader, card card printer, printer, refer refer the the line line in in the the and and 62 62 Logical Logical unit numbers 60, 60, 61, to to program program respectively. respectively. punch punch and and card with with eight eight the main subroutines: subroutines: main main of of consists consists package package The The program program a a program program prints prints input input subroutine subroutine functions; functions; TITLES TITLES and and input input integration, integration, performs performs output PROP PROP program program twist twist and and given given angle angle calculates calculates chord chord SEARCH SEARCH subroutine subroutine listing listing for for and and titles titles at at output; output; a a related related and and angular angular factors interference interference axial axial and and determines determines subroutine subroutine station; station; CALC hub-loss hub-loss subroutine subroutine arid arid factor; the the loss loss factor factor the the tip tip calculates calculates subroutine subroutine TIPLOS TIPLOS parameters; parameters; lift lift sectional sectional subroutines subroutines determines NACA00 NACA00 functions; functions; modified modified Bessel Bessel calculates calculates BESSEL BESSEL lift lift determines determines sectional sectional and and subroutine subroutine NACA44 profile profile coefficients coefficients 0012; 0012; NACA NACA and and drag for for which which for for subroutine subroutine curvefit is is NACAXX NACAXX profile profile for for NACA coefficients coefficients 4418; 4418; empty empty drag drag a a an an other other changes. without without placed placed airfoil airfoil section be be for for program program can can any any inputted inputted and/or and/or be be infointation infointation following following the the propeller propeller must must given given For For geometry geometry a a made. made. changes changes if if airfoil airfoil used, section section conform conform rewritten rewritten be be NACAXX NACAXX Subroutine Subroutine to to must must to to and and than than NACA NACA 4418, 4418, 0012. 0012. other other of of function function twist twist chord chord and and specified, specified, i.e., be be percent percent Blade Blade must must geometry geometry a a as as radius. radius. specified. specified. conditions conditions Operating Operating 89 89 inputted inputted be be The The parameters parameters to to are: are: ft ft of of Radius Radius blade R R - - ft ft Hub Hub radius radius HB HB - - integration integration radius radius incrementation) incrementation) for for of of DR DR (percent (percent Incremental Incremental Percentage Percentage degrees degrees Pitch Pitch THETP THETP angle angle - - B B of of Number Number blades blades mph mph Wind Wind velocity velocity V V - - X X speed speed ratio ratio Tip Tip Model Model AMOD Code Code Axial Axial Interference Interference 10 10 original original tl tl seeAppendixII seeAppendixII Wilson, Wilson, ft ft level level H H above above Altitude Altitude sea sea - - degrees degrees angle angle Coning Coning SI SI - - specification specification NF blade blade stations stations for for inputted inputted of of Number Number geometry geometry NPROF NPROF NACA NACA 4418 4418 4418 4418 Profile Profile NACA NACA - - - - NACA NACA 0012 0012 0012 0012 - - Profile Profile subroutine be be NACAXX NACAXX curvefit curvefit in 9999 9999 to to - - Prandtl Prandtl controller controller 0 0 model model loss loss Tip Tip GO GO - - - Goldstein Goldstein 1 1 2- 2- None None None None 0- 0- controller controller mode mode HL Hub Hub loss loss - - Prandt Prandt 1 1 1 1 - - stations stations RR RR (I) (I) for for radius radius Percent Percent ft ft stations stations (CI(I)) (CI(I)) Chord Chord dor dor - - degrees degrees THETI(I) THETI(I) Twist Twist for for stations stations angle angle - - 90 90 in in the the following should should be be format: format: Input Input 4 4 (Card (Card NF) Card Card Card Card Card 4 4 Columns Columns Card Card Card 2 2 3 3 5 5 1 1 RR( RR( HL HL I) I) R R H H 1-10 1-10 B B CI(I) CI(I) V V DR DR 11-20 11-20 81 81 (21-22) (21-22) (I) (I) THETI THETI NF NF X X 21-30 21-30 HB HB AMOD AMOD GO GO 31-40 31-40 TBETP TBETP NPROF NPROF 41-50 41-50 character-field character-field of of basis basis width. width. the the printed printed 140 140 be be will will Output Output on on that that be be These -inputted. -inputted. controllers controllers operation operation AMOD AMOD several several GO, GO, There There must must are are are are , , of of axial axial method method for calculation calculation and which which angular determines determines AMOD AMOD and and NPROF. HL, HL, of of the the form form will input input Glauert Glauert be be also also 0.0 0.0 used, used, used. used. An An will will interference interference factors be be means means model. model. used. used. the the form form be determines determines will will tip tip loss GO GO of of the the while while input input root root 1.0 1.0 square square means means an an Goldstein's Goldstein's the the input input used, used, model model is is of of model model be be Prandtl's Prandtl's is is input input 1.0 1.0 An An 0.0 0.0 to to to to means means means means then then Prandtl's than than the the will will of of choose choose is is number number blades be be unless used, used, the two, two, greater greater program program model model controller controller third third it it is is is is tip tip be be The The used. used. HL; of of loss loss input input model, model, and and 2.0 2.0 to to means means no no an an model model will hub hub while be be loss loss used, of of input input model. model. 0.0 0.0 An An loss loss controls controls the hub means means an an no no used. used. input subroutine of of NACA00 NACA00 is is be Prandtl's Prandtl's 0012 0012 method method An of of input input 1.0 1.0 to to means means means means NACA NACA for for lift lift that that calculates calculates sectional sectional and data the drag drag subroutine subroutine which which is is used, be be is is to to a a lift lift sectional sectional used used calculate calculate be be is is of of NACA44 NACA44 subroutine subroutine profile profile input input 4418 4418 0012; 0012; to to to to means means an an is is be of of input input subroutine subroutine NACAXX NACAXX profile profile 9999 9999 NACA NACA drag drag and and data the for for 4418; 4418; to to means means an an lift lift sectional sectional coefficients coefficients and and fit fit for drag drag add add used. used. is NACA)0( NACA)0( subroutine subroutine must must curve curve a a one one a a profiles profiles without without This This the the of of of other other enables enables function function attack attack angle angle (degrees). (degrees). to to user user use as as a a changes changes the to to program. program. 91 91 used. used. the the Format Format input input Read and and following following statements statements The The are are R,DR,HB,THETP R,DR,HB,THETP (60,10) (60,10) READ READ (4F10.3) (4F10.3) FORMAT FORMAT 10 10 B,V,X,AMOD B,V,X,AMOD READ READ (60,10) (4F10.3) (4F10.3) FORMAT FORMAT 10 10 H,SI,NF,GO,NPROF H,SI,NF,GO,NPROF (60,30) (60,30) READ READ (2F10.3,12,8X,F10.2,14) (2F10.3,12,8X,F10.2,14) FORMAT FORMAT 30 30 HL HL (60,40) (60,40) READ READ FORMAT FORMAT (F10.3) (F10.3) 40 40 (RR(I),THETI(I),I=1,NF) (RR(I),THETI(I),I=1,NF) (60,20) (60,20) READ READ (F5.1,5X,F10.5,F10.5) (F5.1,5X,F10.5,F10.5) FORMAT FORMAT 20 20 approximately approximately compiled compiled seconds seconds the be be takes takes that that the the should should noted, noted, be be also also to to ten ten It It on on program program the the would would desirable if if desired, desired, it be to to convert convert Therefore, Therefore, to to CDC CDC 3300. 3300. program program are are many many runs runs binary binary form. form. 92 92 17L/LT/S0 17L/LT/S0 NOISA NOISA ZIE ZIE S17OT S17OT NV2111:10A NV2111:10A ESO ESO dOlid dOlid 1A1V1D021d 1A1V1D021d 1A1V21901id 1A1V21901id NIVKI NIVKI AO AO V V 30NV14111(0421ad 30NV14111(0421ad santinDivo santinDivo S113131A1V21Vd S113131A1V21Vd 1V3Ila2:102H1 1V3Ila2:102H1 d021d d021d 31-11 31-11 =nun =nun II II V V 31111:1-S#NOSdIA/IS gdyll gdyll ONIM ONIM .9NI8lif1.1 0 0 '83113dC:ftld '83113dC:ftld TdonnvinN TdonnvinN NOLLV1931NI NOLLV1931NI 211bINH331 211bINH331 SSVd SSVd AO AO 2g?11-11 2g?11-11 / / 3 3 00H131/1 00H131/1 9 9 (SZ)I131-11i(SZ)I01(SZ)2121 (SZ)I131-11i(SZ)I01(SZ)2121 FJOJNNI FJOJNNI 3 3 8 8 'XI:l'IdilWSIA10H13`V931,40109'IS11-11C101A/Vid12H1'X'A'9'91-1'1:10'd 'XI:l'IdilWSIA10H13`V931,40109'IS11-11C101A/Vid12H1'X'A'9'91-1'1:10'd 6 6 NO1A11400 NO1A11400 8S'Ll'911S1'1711E11Z11TI'ACMdN'M1 8S'Ll'911S1'1711E11Z11TI'ACMdN'M1 OT OT TT TT 111dNI 111dNI V1VC1 V1VC1 3 3 C1V321 C1V321 T T ET ET dlgHl'8H1210121(0T'09)C1V221 dlgHl'8H1210121(0T'09)C1V221 i7T i7T C10141V'X'AIE1(0T109)C1V3 C10141V'X'AIE1(0T109)C1V3 ST ST dOlidWODIAN'IS/H(OE'09)C1V311 dOlidWODIAN'IS/H(OE'09)C1V311 91 91 1H(017109)C1V311 1H(017109)C1V311 LT LT (ANIT=Ii(I)I13H11 (ANIT=Ii(I)I13H11 81 81 (DICY(I)1111)(0Z109)CIV911 9ES9Z6ST17T'E=Id 9ES9Z6ST17T'E=Id 61 61 (Id*'09)P0i79Z*1:1/X*A=V91A10 (Id*'09)P0i79Z*1:1/X*A=V91A10 OZ OZ TZ TZ JUNI JUNI 111c11110 111c11110 S11LL S11LL 1NII:Id 1NII:Id C1NV C1NV :100A :100A 3 3 ZZ ZZ EZ EZ (AWI131-111I01 (AWI131-111I01 11)S311I1 11)S311I1 11VD 11VD 17Z 17Z SZ SZ 11B131AIVIIVd1NIV1SNO3 11B131AIVIIVd1NIV1SNO3 SNOI1V111D1VD SNOI1V111D1VD NOLLVZI1VIlINI NOLLVZI1VIlINI 3 3 9Z 9Z C1NV C1NV LZ LZ 0=1011 0=1011 8Z 8Z 0.0=T1 0.0=T1 6Z 6Z 00=Z1. 00=Z1. OE OE 00=E1 00=E1 TE TE 00=171 00=171 ZE ZE 0'0=S_L 0'0=S_L EE EE 00=91 00=91 17E 17E 0.0=L1 0.0=L1 SE SE 00=81 00=81 9E 9E 0-0=A?) 0-0=A?) LE LE 00=A1 00=A1 8E 8E 00=Ad 00=Ad 6E 6E 0.0=Xb 0.0=Xb 017 017 0.0=X1 0.0=X1 11' 11' 0'0=AXI4X 0'0=AXI4X Z17 Z17 00=.1.MAX 00=.1.MAX E17 E17 00=XXV1X 00=XXV1X -1717 -1717 Si' Si' 0'0=XALA/X 0'0=XALA/X 00=d01SV 00=d01SV 917 917 90=V 90=V L17 L17 009ET08Z9*A=A 009ET08Z9*A=A 817 817 '081/Id*IS=IS '081/Id*IS=IS 617 617 08T/Id*d13H1=d131A1 08T/Id*d13H1=d131A1 OS OS 08T/Id*O1d1V=01d1V 08T/Id*O1d1V=01d1V TS TS (.00001/H*L6Z'O-)dX2*66T69LEZ000=0H21 (.00001/H*L6Z'O-)dX2*66T69LEZ000=0H21 ZS ZS 000T/H*170Z000000000 000T/H*170Z000000000 6TLE000000'0=SIA 6TLE000000'0=SIA - - ES ES 11/X*A=VOgIAIO 11/X*A=VOgIAIO 17S 17S T+ T+ 21C1/(8H-1:1)=NN 21C1/(8H-1:1)=NN SS SS 11=X21 11=X21 95 95 XII*('JCI-1T)=8111 XII*('JCI-1T)=8111 LS LS 85 85 (IS)SO3*(EITh-X11)=2113 (IS)SO3*(EITh-X11)=2113 65 65 (IS)SOD*21=11 £6 £6 17L/LI/S0 17L/LI/S0 S170T S170T dOlid dOlid ZT.£ ZT.£ NOIS):IAA NOIS):IAA NNY11):10d NNY11):10d ESO ESO (IS)SOD*81-1=9H (IS)SOD*81-1=9H 09 09 19 19 1:1=111 1:1=111 Z9 Z9 dill/101d dill/101d 01 01 enH enH 3 3 £9 £9 NOI1V1:19A1NI SSVd SSVd - 1VDD:131/411N 1VDD:131/411N 9A211-11 9A211-11 179 179 NN'T=1 NN'T=1 0C1 0C1 S9 S9 OOT OOT 01 01 090:1Cra9.(8H-11:1))AI 090:1Cra9.(8H-11:1))AI OS OS 99 99 +dOlSV=d01SV +dOlSV=d01SV 19 19 09 09 ('Z'A9'd01SV)dI ('Z'A9'd01SV)dI 01 01 £6 £6 89 89 69 69 (8H-11:1)=1:la (8H-11:1)=1:la *Z/210=n1C1 *Z/210=n1C1 OL OL OS OS TI TI =910 =910 '9/):10 '9/):10 ZZ ZZ =>IV =>IV EL EL (13H1'0'dN'I1DHl'ID12111'111)HMIV3S11VD (13H1'0'dN'I1DHl'ID12111'111)HMIV3S11VD 171 171 'Td'IdX1'IdXbIdXAVJX/IdXXIAIX'IdXAd'IdXXdilBH1'011*8)01VD 'Td'IdX1'IdXbIdXAVJX/IdXXIAIX'IdXAd'IdXXdilBH1'011*8)01VD 11VD 11VD SL SL (1`VHd1V511/11X/dVIVIA3'XD/C10101?:IIHd1 (1`VHd1V511/11X/dVIVIA3'XD/C10101?:IIHd1 91 91 LL LL Z110-121=Til Z110-121=Til 0'0=NV 0'0=NV 8/ 8/ 61 61 11VD I1DITIIHd'31:1`dX1'dXb'dXAIA/X'dXXVIX'AAA'AX`13H113'111)01VD11VD I1DITIIHd'31:1`dX1'dXb'dXAIA/X'dXXVIX'AAA'AX`13H113'111)01VD11VD 08 08 (d'VHd11/51e1X'dV'V'AD'XDICIDT (d'VHd11/51e1X'dV'V'AD'XDICIDT T8 T8 (dXb+TdX24.17+Xb)*910=Xylb (dXb+TdX24.17+Xb)*910=Xylb Z8 Z8 xib xib Ab=1.0 Ab=1.0 £8 £8 (dX1+ (dX1+ TdX1*17+X1)*91C1-FX1=A1 TdX1*17+X1)*91C1-FX1=A1 178 178 XAb*V93140+Ad=Ad XAb*V93140+Ad=Ad S8 S8 (dXXI4X-1-TdXXIA/X*17+XXIAIX)*91C1+AXIAX=AX161X (dXXI4X-1-TdXXIA/X*17+XXIAIX)*91C1+AXIAX=AX161X 98 98 (dXAlts/X+TdXMA1X*''17+XAIA1X)*9113-FAAIA,IX=AAVIX (dXAlts/X+TdXMA1X*''17+XAIA1X)*9113-FAAIA,IX=AAVIX 18 18 dX?)=X?) dX?)=X?) 88 88 dX1=X1 dX1=X1 68 68 06 06 dXXIAX=XXIA1X dXXIAX=XXIA1X dXMA1X=XAIA1X dXMA1X=XAIA1X 16 16 (Z**X1:1*Id*Z**A*01-121*S.)/A1=A1D (Z**X1:1*Id*Z**A*01-121*S.)/A1=A1D Z6 Z6 (Z**X21*Id*E**A*OH*d*SO/Ad=AdD (Z**X21*Id*E**A*OH*d*SO/Ad=AdD £6 £6 9.1E1/A.d=d1 9.1E1/A.d=d1 176 176 S6 S6 Idt081*V1-1d1V=VHd1V Idt081*V1-1d1V=VHd1V 96 96 ((IS)SOD*X10/111=)3d ((IS)SOD*X10/111=)3d 86 86 indino indino _UMW _UMW 3 3 66 66 OOT OOT I+101I=101I I+101I=101I TOT TOT (1b1.101DAI (1b1.101DAI 09 09 01 01 E17 E17 ZOI ZOI 01 01 H7 H7 09 09 LOT LOT (S6/T9)B1I1:1M (S6/T9)B1I1:1M 170T 170T £17 (OL'I9)31I11/1/1 (OL'I9)31I11/1/1 SOT SOT (8S'T9)31IIIM (8S'T9)31IIIM 90T 90T (6SIT9)31II:IM (6SIT9)31II:IM LOT LOT (S61I9)91II:IM (S61I9)91II:IM 801 801 0=101I 0=101I 60T 60T 311NTINO0 311NTINO0 OTT OTT 1717 1717 AXIA1X'AA41AXA'AD'XD'OD'1D'VHd1Vld'21d1121(TOVI9)31I2IM AXIA1X'AA41AXA'AD'XD'OD'1D'VHd1Vld'21d1121(TOVI9)31I2IM TI TI T T 31:110IHdid1'AdD'A1'AID'AnIA'AAIAIX(00n9)31flIM 31:110IHdid1'AdD'A1'AID'AnIA'AAIAIX(00n9)31flIM ZI ZI T T BflNI1NOD BflNI1NOD LIT LIT OOT OOT Id/Il*A/X21=3CINAd Id/Il*A/X21=3CINAd £6 £6 T T 17T 17T I I Id/Z1-=9CINAA Id/Z1-=9CINAA ST ST Id/£1*Ah0:1-=3C1>IDIAIX Id/£1*Ah0:1-=3C1>IDIAIX 9T 9T Id/t71=9CINDIA1X Id/t71=9CINDIA1X ITT ITT Id Id /£1*A/Xli /£1*A/Xli 811 811 =3CINZIA1X =3CINZIA1X Nt0:1D:10d Nt0:1D:10d ESO ESO d01d d01d N0ISII3A N0ISII3A 171/11/50 ZI*5 ZI*5 91701 91701 611 611 Id/171-=9CINZIAIX Id/171-=9CINZIAIX Id/SI*A/X1-=DCINbD Id/SI*A/X1-=DCINbD OZT OZT Id/91= Id/91= TZT TZT DC1NOD DC1NOD ZZT ZZT (S**)0:1*Id)/L1=131SId (S**)0:1*Id)/L1=131SId EZT EZT (9**X11*Id)/X*81=131SZId (9**X11*Id)/X*81=131SZId DCIN.1.(008'19)31111M DCIN.1.(008'19)31111M 17ZT 17ZT I I SZ SZ 9)31n1M 9)31n1M 081T 081T DCINAA(T DCINAA(T 9Z1 9Z1 ACINDIAIX(Z08'T9)3111:1M ACINDIAIX(Z08'T9)3111:1M DCIN3IAIX(C08'19)31Il1M DCIN3IAIX(C08'19)31Il1M LZT LZT SZT SZT ACINZIAIX(1709`19)31IIIM ACINZIAIX(1709`19)31IIIM 6Z1 6Z1 9)31D:1M DONZIA1X(SO8'T DONZIA1X(SO8'T BONbD(908'19)AIRIM BONbD(908'19)AIRIM OET OET 1ST 1ST 9CINOD(LOWT9)31111/14 131Sid(80819)31n1M 131Sid(80819)31n1M ZET ZET 131SZId(60819)31ThM 131SZId(60819)31ThM CET CET 1751 1751 SIV14210d SIV14210d SET SET JildIflO JildIflO D D 1fldNI 1fldNI SIN31431VIS SIN31431VIS ):10d ):10d CINV CINV 951 951 D D 01 01 LET LET (EO1d17)1VW110d (EO1d17)1VW110d (S'O1d'S'O1d'XSIT'5d)IVIAl210d (S'O1d'S'O1d'XSIT'5d)IVIAl210d SET SET OZ OZ 651 651 (17I1ZO1diX81ZVE*OHZ)IVIA11:10d (17I1ZO1diX81ZVE*OHZ)IVIA11:10d OE OE 017T 017T (E'OTA)IVIA11:10d (E'OTA)IVIA11:10d 017 017 56 56 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1171 1171 #11)1VInftl #11)1VInftl OA OA (#XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXT (#XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXT Zi7T Zi7T (#1\1d#IX9Ti#1d#IX9e#dV#'X17II#V#iX6///)1VIA110A (#1\1d#IX9Ti#1d#IX9e#dV#'X17II#V#iX6///)1VIA110A £171 £171 OL OL 17171 17171 85 85 #1X6i#CID#1X61#1D#1X6'n/Hd1tWXL1#A#'X8/#11-Dd#IXOTI#11#`Xl///)IVIArtI0A #1X6i#CID#1X61#1D#1X6'n/Hd1tWXL1#A#'X8/#11-Dd#IXOTI#11#`Xl///)IVIArtI0A 5171 5171 (#XIAWXZTI#Ad'X81#Xd#1X61#AD#IX6i#XOT (#XIAWXZTI#Ad'X81#Xd#1X61#AD#IX6i#XOT 65 65 NV'X911#Alnl#1X9//)1V141110d NV'X911#Alnl#1X9//)1V141110d 9171 9171 'XET'#c1D#IX6'#1#'XCI'#1.D#')(IT'#0#'XST' 'XET'#c1D#IX6'#1#'XCI'#1.D#')(IT'#0#'XST' (#n (#n 1?1#1XOTI#IHd#iX6'#d#T 1?1#1XOTI#IHd#iX6'#d#T 1171 1171 N N 'XS117.9d'XS117.9d'XS'17.9d'X5117'Ld'X5'V9A'X5117.9diX5'Z'LA/MIVIANO1 'XS117.9d'XS117.9d'XS'17.9d'X5117'Ld'X5'V9A'X5117.9diX5'Z'LA/MIVIANO1 TO? TO? 8171 8171 (Z.TId'XS'Z'Ld'XS'Z'Zd'XSIE'SdT (Z.TId'XS'Z'Ld'XS'Z'Zd'XSIE'SdT 6171 6171 00? 00? TA//)IVP:10d TA//)IVP:10d OS OS Id'XS'Z'T Id'XS'Z'T T T E'01A`XSiV9A4X51ZT E'01A`XSiV9A4X51ZT TA'XS'Z'T TA'XS'Z'T TAIXS1179d'XS'Z'T (MC' (MC' 9A/XS'T 9A/XS'T ZT21XS'Z' TS TS 1 1 / / 3n0 3n0 008 008 cu. cu. S3D110d S3D110d DNINOD DNINOD MVA)#/)1V1A1110d MVA)#/)1V1A1110d / / ZS ZS 1#= 1#= (>1131-11)NIS) (>1131-11)NIS) (3C1V19 (3C1V19 EST EST (-VOTJT (-VOTJT 3na 3na 108 108 MVA)#/)1Vkl210A MVA)#/)1Vkl210A S2D):10d S2D):10d 17S1 17S1 / / / / 01 01 (301/19 (301/19 (Z**(N131-11)NIS) (Z**(N131-11)NIS) liV3HS liV3HS SST SST (17'0TAT (17'0TAT 951 951 013na 013na ?OR ?OR / / imaktokt imaktokt DNIddVld DNIddVld (B0V19 (B0V19 9NIN0D)#/)1V1Al210d 9NIN0D)#/)1V1Al210d (.17.0Id'#= (.17.0Id'#= C09 C09 LS LS / / / 01 01 3n0 3n0 T T ((.131-11)111IS) ((.131-11)111IS) liVAHS liVAHS 91\11N0D)#/)1VV11:10d 91\11N0D)#/)1VV11:10d (3C1V19 11\131401A1 11\131401A1 1#= 1#= 951 951 (17.01J1 (17.01J1 651 651 IN214014 IN214014 9NIMVA)#/)1V141:10d 9NIMVA)#/)1V141:10d 01 01 1709 1709 / / 3n0 3n0 / / SNIddVid SNIddVid SOD) SOD) 1#= 1#= (N121-11) (N121-11) (ACIV19 (ACIV19 091 091 (17O1J1 508 508 9NIMVA)#/)1VIAlli0d 9NIMVA)#/)1VIAlli0d 191 191 / / 3na 3na 01 / / (30V19 (30V19 3HI)SOD*(N131-11)NIS) 3HI)SOD*(N131-11)NIS) IN314014 IN314014 1:11/31-1S 1:11/31-1S (17.01d= (17.01d= ((NIT ((NIT Z91 Z91 3nbwi1W)Ivkiliod 3nbwi1W)Ivkiliod 908 908 £91 £91 NOTIVIIIVA NOTIVIIIVA 01 01 / / 3n0 3n0 9NINOD 9NINOD (3C1V19 (3C1V19 (17-0TAT (17-0TAT 1791 1791 591 591 3nbli0i)/h1vm0d 3nbli0i)/h1vm0d / / 01 01 3n0 3n0 / / IVHS IVHS 108 108 ((1131-11)NIS) ((1131-11)NIS) (BOV19 (BOV19 NOTIVIIIVA NOTIVIIIVA *=. *=. 991 991 (17.0TJ'T (17.0TJ'T 2nNi0i 2nNi0i 808 808 191 191 #/)IVIAI210d #/)IVIAI210d 01 01 3na 3na 3AIIVAD20 3AIIVAD20 = = DINVNAC10113V DINVNAC10113V C1NOD3S C1NOD3S NOLLVD:IVA NOLLVD:IVA 891 891 (17.5-idi (17.5-idi 691 691 608 01 01 noioi#/)..L.vmod noioi#/)..L.vmod Bna Bna NOI1VDJVA NOI1VDJVA liV31-1S liV31-1S 0N0D3S 0N0D3S 3AIIVAD:130 3AIIVAD:130 = = OLT OLT (17'STA'1 (17'STA'1 111 111 D D OIS OIS d d ?LT ?LT ELT ELT C1N3 C1N3 S6 S6 ZT£ ZT£ N0IS213A N0IS213A -17L/LT/S0 -17L/LT/S0 5170T 5170T LSO LSO NVQ:11110A NVQ:11110A (dN'I1AH1'I3W5311I1 (dN'I1AH1'I3W5311I1 gancmans gancmans 17L1 17L1 SLT SLT Viva Viva In° In° JAIldIliDSgO JAIldIliDSgO NI NI V V 51Nnid 51Nnid SB11I1 SB11I1 - - 11.1dNI 11.1dNI 9L1 9L1 D D Indlno Indlno SN0I1dIliDS3O SN0I1dIliDS3O S311I1/51091A1AS S311I1/51091A1AS dO dO 1:10d 1:10d 51ND:Id 51ND:Id 114210d 114210d CINV CINV LLT LLT D D 8LT 8LT (SZ)I131-11'(SZ)IDi(SZ)1111 (SZ)I131-11'(SZ)IDi(SZ)1111 NOISNANIO NOISNANIO 6L1 6L1 )al'Id1WSIAIOH):1'VD31A10109/IS'WOOLAJVidlaHl'X'AIEVEIH1210121 )al'Id1WSIAIOH):1'VD31A10109/IS'WOOLAJVidlaHl'X'AIEVEIH1210121 NOIA11610D NOIA11610D 081 081 A0):IdN'M'T A0):IdN'M'T 181 181 (0SiT9)31DIM (0SiT9)31DIM Z8T Z8T (TS1T9)31IIIM (TS1T9)31IIIM £81 £81 d13H1j8H12101):1 d13H1j8H12101):1 (ZS1T9)31fliM (ZS1T9)31fliM 1781 1781 X'AiS(ES1T9)31D3M X'AiS(ES1T9)31D3M 581 581 (17SiT9)B1DiN1 (17SiT9)B1DiN1 ANITS'H 981 981 d0lJdNiV93110(00Z/T9)31IIIM d0lJdNiV93110(00Z/T9)31IIIM L81 L81 09 09 01 01 00£ 00£ 881 881 (0"0"bg*GOIAIV)dI (0"0"bg*GOIAIV)dI (OIC19)91I2JAA (OIC19)91I2JAA 681 681 01 01 OD OD 017£ 017£ 061 061 (OZE'T9)31IIIM (OZE'T9)31IIIM 00E 00E 161 161 01 01 (0"Z1D8)AI (0"Z1D8)AI 559 559 OD OD MT MT 017£ 017£ 01 01 (0"0"b9"09)dI (0"0"b9"09)dI OD OD LOT LOT £61 £61 01 01 959 959 OD OD Z)TOD)dI Z)TOD)dI (0"T (0"T 1761 1761 01 01 (0"rbg"00)AI (0"rbg"00)AI LS9 LS9 OD OD S6T S6T 01 01 859 859 961 961 OD OD 01 01 £59 £59 OD OD (0"Z"OT0D)dI (0"Z"OT0D)dI 559 559 L61 L61 LOT LOT 01 01 OD OD 861 861 (TOT/T9)31ITJM (TOT/T9)31ITJM 959 959 661 661 01 01 859 OD OD 00? 00? (6S9/T9)31n1M (6S9/T9)31n1M L59 L59 TO? TO? 859 859 09 09 01 01 ZOZ ZOZ (00T1T9)31fliM (00T1T9)31fliM HZ HZ LOT LOT 01 01 LLL LLL 09 09 859 859 (0"0"N"1H)AI 170Z 170Z (8LL1T9)31DIM (8LL1T9)31DIM SO? SO? 90Z 90Z 09 09 01 01 999 999 LOZ LOZ (6LLIT9)31IIIM (6LLIT9)31IIIM La, La, 80? 80? (SSIT9)91DIAA (SSIT9)91DIAA 999 999 60? 60? (AN'T=Ii(I)I1BH1'(I)ID1(I)Ift1)(9S/T9)3111:1M (AN'T=Ii(I)I1BH1'(I)ID1(I)Ift1)(9S/T9)3111:1M OTZ OTZ (LS'T9)31DIM (LS'T9)31DIM TT? TT? (T9'T9)J1I21M (T9'T9)J1I21M ZTZ ZTZ (69I'19)31I2IM (69I'19)31I2IM ET? ET? (TLiT9)31flIM (TLiT9)31flIM 17TZ 17TZ (09iI9)21TIJAA (09iI9)21TIJAA ST? ST? (Z919)31IIIM (Z919)31IIIM 91Z 91Z (£9'19)2111:1M (£9'19)2111:1M LTZ LTZ (179'19)B1I2IM (179'19)B1I2IM 81Z 81Z (5919)31I21M (5919)31I21M 61Z 61Z (991T9)21DIM (991T9)21DIM OZZ OZZ (L9'T9)31I1-1M (L9'T9)31I1-1M TZZ TZZ (89iT9)21IIIM (89iT9)21IIIM ZZZ ZZZ (691T9)31DIM (691T9)31DIM EZZ EZZ (S6'T9)31IIIM (S6'T9)31IIIM 17ZZ 17ZZ (OL`T9)1IIIM (OL`T9)1IIIM SZZ SZZ (851T9)1n1M (851T9)1n1M 9ZZ 9ZZ (6S'T9)31IIIM (6S'T9)31IIIM LZZ LZZ (S6119)21I21/1A (S6119)21I21/1A 8ZZ 8ZZ 6ZZ 6ZZ S1N31431V15 S1N31431V15 indino indino 0d 0d 51VIAl230d 51VIAl230d 0 0 0£Z 0£Z 'HZ 'HZ ONIM ONIM 3dA1 3dA1 11311i:10"dd 11311i:10"dd dO dO V V 1VDI13103H1 1VDI13103H1 3DNVIA11:104213d 3DNVIA11:104213d #////)1V%11:10d #////)1V%11:10d ZEZ ZEZ OS 05/17/74 05/17/74 TITLES TITLES 1045 1045 3.12 3.12 VERSION VERSION FORTRAN FORTRAN 0S3 0S3 1INE#) 1INE#) 233 233 #) #) FORMAT(///# FORMAT(///# RECORD RECORD INPUT INPUT 234 234 DATA DATA 51 51 PERCENTAGE PERCENTAGE RADIUS-FT,F7.2,10X,ANCREMENTAL RADIUS-FT,F7.2,10X,ANCREMENTAL =#,F6.4, =#,F6.4, FORMAT(///# FORMAT(///# 52 52 235 235 =,F7.4) =,F7.4) RADIUS-FT=#,F5.2,10X,=PITCH RADIUS-FT=#,F5.2,10X,=PITCH ANGLE ANGLE DEGREES DEGREES 110X,A-IUB 110X,A-IUB 236 236 - - =#,F7.2, =#,F7.2, =#,F3.0,10X,#WIND =#,F3.0,10X,#WIND VELOCITY VELOCITY FORMAT(/ FORMAT(/ MPH MPH BLADES BLADES OF OF NO. NO. 237 237 53 53 - - =#,F6.3) =#,F6.3) RATIO RATIO 110X,#TIP 110X,#TIP SPEED SPEED 238 238 LEVEL-FT=#,F10.2,5X,#CONING LEVEL-FT=#,F10.2,5X,#CONING SEA SEA ALTITUDE ALTITUDE SITE SITE FORMAT(//# FORMAT(//# ABOVE ABOVE OF OF 239 239 54 54 =#,I2 =#,I2 ==,F7.3,5X,#NUMBER ==,F7.3,5X,#NUMBER STATIONS STATIONS ALONG ALONG OF OF SPAN SPAN DATA DATA 1ANGLE-DEGREES 1ANGLE-DEGREES 240 240 1) 1) 241 241 ANGLE-DEGREES# ANGLE-DEGREES# RADIUS#,5X,=CHORD-FT=,5X,#TWIST RADIUS#,5X,=CHORD-FT=,5X,#TWIST FORMAT(///# FORMAT(///# 242 242 PERCENT PERCENT 55 55 1) 1) 243 243 FORMAT(//5X,F5.1,8X,F10.5,10X,F10.5) FORMAT(//5X,F5.1,8X,F10.5,10X,F10.5) 56 56 244 244 RECORD4///) RECORD4///) FORMAT(///////# FORMAT(///////# OUTPUT OUTPUT DATA DATA 245 245 57 57 PC-R*) PC-R*) R#/# R#/# PERCENT PERCENT RADIUS RADIUS FORMAT(# FORMAT(# RADIUS RADIUS FT FT 246 246 60 60 ------ A4# A4# INTERFERENCE INTERFERENCE ANGULAR INTERFERENCE FACTOR FACTOR AXIAL AXIAL FA FA FORMAT(# FORMAT(# 247 247 61 61 -- -- AP#) AP#) 1CTOR 1CTOR 248 248 --FT#) --FT#) FN4= FN4= TANGENTIAL TANGENTIAL FORCE FORCE FORMAT(# FORMAT(# NORMAL FORCE FORCE 169 169 249 249 -- -- 00#) 00#) CL=/= CL=/= COEFFICIENT COEFFICIENT COEFFICIENT COEFFICIENT LIFT LIFT DRAG DRAG FORM4T(# FORM4T(# 62 62 250 250 ------ CX4= CX4= FORCE-Y-DIR FORCE-Y-DIR CY#) CY#) COEF COEF FOKCE-X-DIR FOKCE-X-DIR OF OF OF OF FORMAT(# FORMAT(# COEF COEF 251 251 63 63 FX=/# FX=/# FORCE-Y-DIR/BLADE FORCE-Y-DIR/BLADE LB LB FORCE-X-DIR/BLAOE FORCE-X-DIR/BLAOE LB LB FORMAT(= FORMAT(= 64 64 252 252 ------ 1--FY#) 1--FY#) 253 253 MOMENT-Y-DIR/BLAOE MOMENT-Y-DIR/BLAOE MX#/# MX#/# MOMENT-X-DIR/BLADE MOMENT-X-DIR/BLADE FT-LB FT-LB FORMAT(= FORMAT(= 65 65 254 254 -- -- MY#) MY#) FT-LB FT-LB 1-- 1-- 255 255 -- -- Q#) Q#) --W#/# --W#/# FT-LB FT-LB VELOCITY VELOCITY FT/SEC FT/SEC TORQUE FORMAT(= FORMAT(= RELATIVEC RELATIVEC 66 66 256 256 ------ -- -- --T#) --T#) THRUST THRUST CT-#/ CT-#/ THRUST THRUST COEFFICIENT LB LB FORMAT(# FORMAT(# 67 67 257 257 ------ KILOWATTS KILOWATTS POWER POWER P#) CP# CP# COEFFICIENT COEFFICIENT POWER POWER FORMAT(# FORMAT(# 258 258 68 68 ------ #/# #/# NUMBER NUMBER NC:1#) NC:1#) RE RE PHI PHI REYNOLDS REYNOLDS DEGREES DEGREES FORMAT(# FORMAT(# PHI PHI ANGLE ANGLE 259 259 69 69 ------ -- -- DEGREES DEGREES F#/ F#/ FACTOR FACTOR ALP ATTACK ATTACK ANGLE ANGLE TIP TIP OF OF FORMAT(# FORMAT(# LOSS LOSS 260 260 71 71 ------ 1HA#) 1HA#) 261 261 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX FORMAT(//# FORMAT(//# 262 262 95 95 1XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX#) 1XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX#) 263 263 FT=,18X,#FN#) FT=,18X,#FN#) FORMAT(///9X,=A#,14X,#AP*,26X, FORMAT(///9X,=A#,14X,#AP*,26X, 70 70 264 264 FORMAT(///4X,#R#,10X,#PC-R#,8X,#F,7X,#ALPHA,9X,#CL=,9X,=CD#,9X,# FORMAT(///4X,#R#,10X,#PC-R#,8X,#F,7X,#ALPHA,9X,#CL=,9X,=CD#,9X,# 265 265 58 58 1CX=,9X,#CY=,9X,#FX#,8X,*FY#,12X,MX#) 1CX=,9X,#CY=,9X,#FX#,8X,*FY#,12X,MX#) 266 266 FORMAT(//6X,#MY#,16X,#W#,15X,#Q#,11X,#CT#,13X,#T#,9X,#CP#,13X, FORMAT(//6X,#MY#,16X,#W#,15X,#Q#,11X,#CT#,13X,#T#,9X,#CP#,13X, 267 267 59 59 NO#) NO#) PHI#,10X,=RE PHI#,10X,=RE 1#P#,9X, 1#P#,9X, 268 268 USED#) USED#) METHOD METHOD INTERFERENCE INTERFERENCE AXIAL AXIAL FORMAT(//# FORMAT(//# 310 310 WILSON WILSON 269 269 METHOD METHOD USED#) USED#) INTERFERENCE INTERFERENCE STANOARO STANOARO AXIAL AXIAL FORMAT(//# FORMAT(//# 270 270 320 320 FORMULA FORMULA #) #) PRANDTLS PRANDTLS FORMAT(//# FORMAT(//# MODELED MODELED BY BY TIP TIP LOSSES LOSSES 100 100 271 271 *) *) GOLDSTEINS GOLDSTEINS FORMULA FORMULA TIP TIP FORMAT(//# FORMAT(//# MODELED MODELED LOSSES LOSSES BY BY 101 101 272 272 USED#) USED#) MODEL MODEL FORMAT(//# FORMAT(//# LOSS LOSS TIP TIP NO NO 659 659 273 273 PRANDTL#) PRANDTL#) BY BY MODELED MODELED HUBLOSSES HUBLOSSES FORMAT(//# FORMAT(//# 778 778 274 274 USED#) USED#) FORMAT(//# FORMAT(//# MODEL MODEL HUBLOSS HUBLOSS NO NO 275 275 779 =#,I4) =#,I4) #,F20.5,10X,=NACA #,F20.5,10X,=NACA PROFILE PROFILE FORMAT(//# FORMAT(//# 276 276 RPM RPM 200 200 = = 277 277 C C RETURN RETURN 278 278 END END 279 279 97 97 ZIE ZIE i1/LT/S0 i1/LT/S0 N0IS:13A N0IS:13A SNIT SNIT ESO ESO NV2111:10A NV2111:10A (131-11101ANIIIAH1'I3'Til1iTh)H321V3S (131-11101ANIIIAH1'I3'Til1iTh)H321V3S 3Nun0bens 3Nun0bens 08Z 08Z T8Z T8Z ISIMI ISIMI IV IV 319NV 319NV 3HI 3HI 3H1 3H1 CidOHD CidOHD - - SBNIIAI213130 HDI:IVDS HDI:IVDS CINV CINV 3 3 Z8Z Z8Z s3znun s3znun II II 3E8Z 3E8Z NVdS NVdS ?3VDNI1 ?3VDNI1 srucivi srucivi V V 3H1 3H1 9NO1V 9NO1V NDAI9 NDAI9 V V 3178Z 3178Z 3110INI-1331 3110INI-1331 N0I1V10dli3INI N0I1V10dli3INI S8Z S8Z (SZ)II3H1j(SZ)I31(SZ)211:1 (SZ)II3H1j(SZ)I31(SZ)211:1 NOISN3LAII0 NOISN3LAII0 98Z 98Z X2I'Id'1WSIA'01-1):11V9DIA10109/IS'WCIOIAIV`d13HI'X'A'819W1101?1 X2I'Id'1WSIA'01-1):11V9DIA10109/IS'WCIOIAIV`d13HI'X'A'819W1101?1 L8Z L8Z NO141403 00 00 88Z 88Z AN'T=T OZ OZ '001*((IS)S0D*X21)/Th=A111 '001*((IS)S0D*X21)/Th=A111 68Z 68Z 09 09 01 01 06Z 06Z (0,01:11:1'39*Al:W)AI (0,01:11:1'39*Al:W)AI OT OT 09 09 01 01 (AN'bI)AI (AN'bI)AI OE OE 16Z 16Z 311NIINOD 311NIINOD Z6Z Z6Z OZ OZ T+I=C T+I=C E6Z E6Z OT OT UT-01121-(Z-02111)/00T-02111-A21):1)=112d UT-01121-(Z-02111)/00T-02111-A21):1)=112d -176Z -176Z (T-0I3+C(T-0I0-(Z-C)ID)*?Ad=3 (T-0I3+C(T-0I0-(Z-C)ID)*?Ad=3 S6Z S6Z (1-C)I12H1+((T-C)I13H1-(Z-C)I13HIMI3d=13H1 (1-C)I12H1+((T-C)I13H1-(Z-C)I13HIMI3d=13H1 96Z 96Z 09 09 01 01 L6Z L6Z 0.17 0.17 CAN)ID=D CAN)ID=D OE OE 86Z 86Z (dN)II3H1=13H1 (dN)II3H1=13H1 66Z 66Z '081/Id*I3H1=1.3H1 '081/Id*I3H1=1.3H1 00E 00E 017 017 TOE TOE CIN3 CIN3 ZOE ZOE t7L/LT/S0 t7L/LT/S0 St7OT St7OT NOISIGA NOISIGA NV211d0A NV211d0A ESO ESO £ £ Z 3Nun0bens 3Nun0bens zoz zoz 113121IHcli3biAl'AbiAAAIADCAXALAIXIAAA'AXA'13HI'D'111)D1V3 113121IHcli3biAl'AbiAAAIADCAXALAIXIAAA'AXA'13HI'D'111)D1V3 (A'VHd1VINV11XidV/VIAD'XD'CIDT (A'VHd1VINV11XidV/VIAD'XD'CIDT 170E 170E SOE SOE 33N3):13AII3INI 33N3):13AII3INI 1VIXV 1VIXV 90E 90E D1VD D1VD - - S3NI1Al21319C1 S3NI1Al21319C1 1:1V111DNV 1:1V111DNV 31-11 31-11 D D C1NV IV IV sniavb sniavb NJAID NJAID 3 3 SNOTIDN11A SNOTIDN11A IN3C1N3d3C1 S1101DVA S1101DVA V CINV CINV LOE LOE S3NI141:1313C1 S3NI141:1313C1 3S31-11 3S31-11 'S213131/s1V1iVd NOdf) NOdf) 80E 80E D D 60E 60E OIL OIL )(21'Id'11-eSIA101-1211VD31n10'0D'IS'WCIONY'd13HI'X'A'8'3W2ICI'll )(21'Id'11-eSIA101-1211VD31n10'0D'IS'WCIONY'd13HI'X'A'8'3W2ICI'll NOIAI1A/OD NOIAI1A/OD 81`LI'91'S111711EliZIll'A0ThldN'MT 81`LI'91'S111711EliZIll'A0ThldN'MT TIE TIE A/V93140*121=1X A/V93140*121=1X ZIE ZIE LIE LIE 31-1=1-111 31-1=1-111 0171T-C 0171T-C OT OT 171E 171E OC1 OC1 V=V1AS V=V1AS STE STE dV=V1-130 dV=V1-130 91E 91E ((lX*(AV-1--T))/(IS)SOD*(V-'T))NVIV=IHd ((lX*(AV-1--T))/(IS)SOD*(V-'T))NVIV=IHd LIE LIE (IHd)S9V=VVIHd (IHd)S9V=VVIHd 81E 81E (VVIHd)NISAVVIHd)SOD=1XX (VVIHd)NISAVVIHd)SOD=1XX 6TE 6TE Th/11*1XX=01XX Th/11*1XX=01XX IHA=21IHd IHA=21IHd TZE TZE d13H1-131-11-IHd=VHd1V d13H1-131-11-IHd=VHd1V ZZE ZZE TO(T0+VHd1V=CIVHd1V TO(T0+VHd1V=CIVHd1V EZE EZE 17ZE 17ZE . . SIN3I3IAA303 SIN3I3IAA303 IAI1 IAI1 1VNOI1D3S 1VNOI1D3S NOI1V111D1VD NOI1V111D1VD AO AO SZE SZE C1NV C1NV D D DV11C1 DV11C1 9ZE 9ZE 01 01 LZE LZE (8T171:7'b3.A01:1dN)AI OD 0017 0017 00 00 01 (ZTOO'bTA01:1dN)AI (ZTOO'bTA01:1dN)AI 009 009 8ZE 8ZE 01 01 (66662:19.A01:1dN)AI (66662:19.A01:1dN)AI OSS OSS 6ZE 6ZE OD OD (0091T9)31I2IM (0091T9)31I2IM ION ION 31IAM:Id 31IAM:Id NI NI BM_ BM_ CIBIAIDBdS CIBIAIDBdS VDVN )39011d )39011d V V C131;101S C131;101S noA noA 009 009 AAVH AAVH TEE TEE )1VIts1110A )1VIts1110A -ram -ram 3sn 3sn VDVN VDVN (.8.I1717 VIVI:1901d VIVI:1901d ZEE ZEE 31-11 31-11 (C1D1D'VHd1V)1717VDVN (C1D1D'VHd1V)1717VDVN 11VD 11VD LEE LEE 0017 0017 (CICIDIC1101CIVHd1V)1717V3VN (CICIDIC1101CIVHd1V)1717V3VN 1'EE 1'EE 11VD 11VD 01 01 008 008 OD OD SEE SEE (03IDIVHd1V)00VDVN (03IDIVHd1V)00VDVN 009 009 11V3 11V3 9EE 9EE (CICI3'O13'CIVHd1V)00VDVN (CICI3'O13'CIVHd1V)00VDVN 11V3 11V3 LEE LEE 01 01 8£E 8£E 008 OD OD (CI3i13IVHd1V'IS'XI:11111)XXVDVN (CI3i13IVHd1V'IS'XI:11111)XXVDVN 6£E 6£E OSS OSS 11VD 11VD (OCI3'O13IOVHd1VIIS'XIIIII)XXV3VN (OCI3'O13IOVHd1VIIS'XIIIII)XXV3VN 11VD 11VD 017E 017E 0.0=13 0.0=13 (CZT/Id*7).3D'(VHd1V)SEIV)AI (CZT/Id*7).3D'(VHd1V)SEIV)AI 008 008 1i7E 1i7E Zi7E Zi7E dIl dIl enH enH NOI1V111D1V3 NOI1V111D1V3 S3SSO1 S3SSO1 AO AO C1NV C1NV E17E E17E 3 3 1717E 1717E (1-11:11IHd'Thili'Id1H10948/A101XXI1XX)S01dI1 (1-11:11IHd'Thili'Id1H10948/A101XXI1XX)S01dI1 11VD 11VD Si7E Si7E 917E 917E (I1-1d)SO3*CID-(IHd)NIS*13=X3 (I1-1d)SO3*CID-(IHd)NIS*13=X3 Li7E Li7E (IS)SO3*(IHd)NIS'C13+(IS)SO3*(IHd)S03*13 (IS)SO3*(IHd)NIS'C13+(IS)SO3*(IHd)S03*13 =AD =AD 8.17E 8.17E (121*Id)/(3*3)=DIS (121*Id)/(3*3)=DIS 617E 617E SLS SLS 01 01 (O'bTODWV)AI (O'bTODWV)AI OD OD OSE OSE (Z**(IHd)NIS)/(A3*DIS*SZT.0)=213A (Z**(IHd)NIS)/(A3*DIS*SZT.0)=213A ISE ISE C(IHd)SO3*(IHd)NIS*A)/(XD*DIS*SZT.0)=IiVA C(IHd)SO3*(IHd)NIS*A)/(XD*DIS*SZT.0)=IiVA ZSL ZSL (A--T)*A*V3A*17+A*A=NVD (A--T)*A*V3A*17+A*A=NVD ESE ESE 0'0=NV3 0'0=NV3 (0'0'11'NV3)1I (0'0'11'NV3)1I {7SE {7SE ((A*A-1-219A)*.Z)/((NVD)111OS-A.21E1A*.Z)=V ((A*A-1-219A)*.Z)/((NVD)111OS-A.21E1A*.Z)=V SSE SSE (11VA'T)/2/VA=dV (11VA'T)/2/VA=dV 9SE 9SE 01 01 089 089 LSE LSE OD A3*DIS*5ZT'0=):18A A3*DIS*5ZT'0=):18A 85E 85E SLS SLS X3*DIS*SZT'0=1:1VA X3*DIS*SZT'0=1:1VA 6SE 6SE ('elAA+Z**(IHd)NIS*A)/1113A=V ('elAA+Z**(IHd)NIS*A)/1113A=V 09E 09E (IIVA-(II-IcI)SOD*(IHd)NIS*A)/AVA=dV (IIVA-(II-IcI)SOD*(IHd)NIS*A)/AVA=dV 19E 19E 66 66 ZI'E ZI'E 31V3 31V3 N0IS2GA N0IS2GA 17L/L1/S0 17L/L1/S0 Nt0:11):10A Nt0:11):10A 61701 61701 ESO ESO ((IS)SOD*X1:1)/111=2:1Dd ((IS)SOD*X1:1)/111=2:1Dd 08S 08S Z9E Z9E 2101DVA 2101DVA £9£ £9£ 3 3 NINJdVJV NINJdVJV 1VIXV IIV1119NV IIV1119NV 331\13113A1131NI 331\13113A1131NI AO AO CINV CINV 'SN0I1V2131I 'SN0I1V2131I 179£ 179£ 59£ 59£ 06'017'0£ 06'017'0£ (17-C)AI (17-C)AI 99E 99E 19£ 19£ (0T-OdI OTT/017'0E OTT/017'0E (ST-0AI (ST-0AI 89E 89E 0£'01710£ 0£'01710£ OTT OTT S(V1BEI+V)=V S(V1BEI+V)=V 69£ 69£ OV OV S(V11313+c1V)=AV S(V11313+c1V)=AV OLE OLE TIE TIE 09 09 01 01 (1.39.NV)AI (1.39.NV)AI OL OL 0£ 0£ ZLE ZLE (09/19)31I2:1M (09/19)31I2:1M 'V 'V AV AV ELE ELE 09 09 (8"51AZ)1V141:10A (8"51AZ)1V141:10A VLE VLE SLE SLE 1S31 1S31 3DN3911gANOD 3DN3911gANOD 110A 110A 91E 91E D D ILE ILE 09 09 01 01 (I000-31.(dVAN/11213-dVDSEIV)AI (I000-31.(dVAN/11213-dVDSEIV)AI OS OS 81£ 81£ OL OL 61E 61E 08£ 08£ 311NI1NOD 311NI1NOD OT OT 18E 18E NOdfl NOdfl 1VIXV CINV CINV Z8£ Z8£ NOI1V11131VD NOI1V11131VD SNOLLONfld SNOLLONfld 11\1gC1N3d30 D D AO 51101DVA 51101DVA £8£ £8£ 3DNa213A1:131NDIV1119NV 3DNa213A1:131NDIV1119NV D D .178£ .178£ ((IHA)NIS)/((IS)SOD*A*(V-. ((IHA)NIS)/((IS)SOD*A*(V-. T))=M T))=M OS OS 58£ 58£ SIA/3*M*OH21.--91:1 SIA/3*M*OH21.--91:1 98E 98E (D*(Z**NO*OH11*S.0)=1SNIOD (D*(Z**NO*OH11*S.0)=1SNIOD L8£ L8£ XD*1SNOD=AXA XD*1SNOD=AXA 88£ 88£ 68E 68E AD*1SNOD=AAA AD*1SNOD=AAA (E1H-111)*AXA=AXAINX (E1H-111)*AXA=AXAINX 06£ 06£ (8H-1?:1)*AAA'''dAdlnlX (8H-1?:1)*AAA'''dAdlnlX 16E 16E (M*M)*(D*El*OHI:1*S*0)=T1D (M*M)*(D*El*OHI:1*S*0)=T1D Z6£ Z6£ XD*Td*T.LD=AO XD*Td*T.LD=AO E6E E6E AD*11D=d1 AD*11D=d1 176£ 176£ 56£ 56£ 00:1*.Z)h1C1=1:1DACI 00:1*.Z)h1C1=1:1DACI 96£ 96£ X21/0=1:ID X21/0=1:ID 0/(1D-C110)=V1D 0/(1D-C110)=V1D TOO' TOO' L6£ L6£ 86£ 86£ 1OO*0/(0D-OOD)=-VCID 1OO*0/(0D-OOD)=-VCID AD-1-(IHA)SODANCID-(1-1c1)NIS*V1D=c1X0 AD-1-(IHA)SODANCID-(1-1c1)NIS*V1D=c1X0 66£ 66£ XD-(I1-1d)NIS*VCID-1-(IHA)SOD*V13=dAD XD-(I1-1d)NIS*VCID-1-(IHA)SOD*V13=dAD 0017 0017 ?:IDA*AID*(V-"T)*((IHcOANV1VicIXD+XD*7)=1A ?:IDA*AID*(V-"T)*((IHcOANV1VicIXD+XD*7)=1A 1017 1017 0c1*21D*(V-.T)*((iHd)ANV1V/dAD+AD*"Z)=NA 0c1*21D*(V-.T)*((iHd)ANV1V/dAD+AD*"Z)=NA Z017 Z017 (669/19)B1n1M (669/19)B1n1M NdilA NdilA £017 £017 (L'OZAZ`X017)1V141:10d (L'OZAZ`X017)1V141:10d 669 669 17017 17017 Ti= Ti= T1 T1 1:1DACI*1A+ 1:1DACI*1A+ 6017 6017 T)*1A+Z1=Z1 T)*1A+Z1=Z1 :13c1C1(V-. :13c1C1(V-. 9017 9017 IIDAO*Dd*NA+£1=E-1_ IIDAO*Dd*NA+£1=E-1_ 1017 1017 1DACI*1:1DA*(V-. 1DACI*1:1DA*(V-. T)*NA+171=171 T)*NA+171=171 8017 8017 liDdOIDA*1A+S1=S1 liDdOIDA*1A+S1=S1 6017 6017 21DACI*):1Dc:1*(V-I1)*1A+91=91 21DACI*):1Dc:1*(V-I1)*1A+91=91 0117 0117 +CID*E**(IHA)SOD*.Z-(IHA)NIS*1D*(Z**(IHd)SOD-7)))+LI=L1 +CID*E**(IHA)SOD*.Z-(IHA)NIS*1D*(Z**(IHd)SOD-7)))+LI=L1 TTfr TTfr 7/1:10*(D*E**11:1*Z**(V-T)*(V10*(IHA)SOD'ZT 7/1:10*(D*E**11:1*Z**(V-T)*(V10*(IHA)SOD'ZT ZI-17 ZI-17 ((IHA)NIS*V1D+(II-Id)NIS*OD-1D*(Z**(IHA)NIS+ ((IHA)NIS*V1D+(II-Id)NIS*OD-1D*(Z**(IHA)NIS+ )))+ )))+ 81= 81= I I 81 81 £117 £117 "Z/1:113*(D*17*111*(V--I)*(dV+'T)*T "Z/1:113*(D*17*111*(V--I)*(dV+'T)*T 1711' 1711' ST17 ST17 1\rdniAll 1\rdniAll 9117 9117 aNa aNa 1117 1117 ZI*£ ZI*£ ES° ES° N0IS213A N0IS213A 17L/LT/S0 NV211210d NV211210d Si7O1 Si7O1 8T17 8T17 ANuncmans ANuncmans 61t7 61t7 (1-121qHd112112:11Id111-110DVA1011`11)SO1dI1 (1-121qHd112112:11Id111-110DVA1011`11)SO1dI1 OZ17 OZ17 - - In' In' sassoi sassoi SOldIl SOldIl dI1 dI1 3H1 3H1 S3NI1421319C1 S3NI1421319C1 EinH EinH C1NV C1NV 3 3 S#NI31S0109 S#NI31S0109 NOdfl NOdfl 1A2103H1 1A2103H1 'A2109141 210 210 S#11CINV21d CI9SVO CI9SVO ZZ.17 ZZ.17 "sassoi "sassoi 210d 210d 210 210 ON ON BSVD BSVD AO AO 31-11 31-11 £Z17 £Z17 17Z.17 17Z.17 wo=nAtris wo=nAtris SZ17 SZ17 cro=vms cro=vms 9n7 9n7 =NV =NV LZ17 LZ17 T T 8Zi7 8Zi7 "T=VILAIV "T=VILAIV 0.0=1A1V 0.0=1A1V 6Z17 6Z17 00 00 01 01 996 996 (C/Z*..1.D.0)I (C/Z*..1.D.0)I 0E17 0E17 (olybaoo)di (olybaoo)di op op 01 00? 00? T£17 T£17 01 01 (0.Tb3.0D)dI (0.Tb3.0D)dI OD OD OOT OOT ZE.17 ZE.17 (0.Z.b3'10D)dI (0.Z.b3'10D)dI 01 01 OD OD ££17 ££17 1717t7 1717t7 01 01 (0.Z.b3.0D)dI (0.Z.b3.0D)dI HIV HIV OD OD 996 996 17£17 17£17 00? 00? 311NI1NO3 311NI1NO3 S£17 S£17 Z**(IHd)NIS)121bS*121*.Z)/((121-'d)*b)-)dX3)ASODV*(IdtZ)=A Z**(IHd)NIS)121bS*121*.Z)/((121-'d)*b)-)dX3)ASODV*(IdtZ)=A (a(T000-+ (a(T000-+ 9£.17 9£.17 L£17 L£17 01 01 OD OD SOT SOT =A =A 8£17 8£17 T T 171717 171717 01 01 OD OD SOT SOT 6E17 6E17 01 01 00? 00? (T000-11*(((IHd)NIS)S9VDdI (T000-11*(((IHd)NIS)S9VDdI OD OD OUT OUT Ot7-17 Ot7-17 T1717 T1717 aoHlaw aoHlaw 0 0 SNI31SC11OD SNI31SC11OD Z1717 Z1717 EH' EH' EIT=LAI EIT=LAI OT OT OC1 OC1 171717 171717 (T-Fwv*z)=A (T-Fwv*z)=A Si717 Si717 9t+ 9t+ n*on=oz n*on=oz A*A=ZA A*A=ZA L-1717 L-1717 Z*Z=ZZ Z*Z=ZZ 6t717 6t717 (IViA1Z)13SS2E1 (IViA1Z)13SS2E1 11VD 11VD OS-17 OS-17 11VD 11VD (OIVIA/OZ)13SS3E1 TS17 TS17 (SD2)di (SD2)di 01 01 OD OD 00£ 00£ ZS-17 ZS-17 £S17 £S17 '17,07=8 '17,07=8 17S17 17S17 "9*9=0 "9*9=0 SS17 SS17 9S17 9S17 *(ZA-9)*(ZA-V))/(E**ZZ)+((ZA-8)*(ZA-V))/(ZZ*ZZ)+(ZA-V)/ZZ=ZAII *(ZA-9)*(ZA-V))/(E**ZZ)+((ZA-8)*(ZA-V))/(ZZ*ZZ)+(ZA-V)/ZZ=ZAII 1S17 1S17 UZA-C1)*(ZA-D)*(ZA-43)*(ZA-V))/(17**ZZ)+((ZA-D)T UZA-C1)*(ZA-D)*(ZA-43)*(ZA-V))/(17**ZZ)+((ZA-D)T 8S-17 8S-17 aId*A*S.)NIS*.Z)/(IV*Id*A)=ZAT1D aId*A*S.)NIS*.Z)/(IV*Id*A)=ZAT1D ZAT1- ZAT1- 6917 6917 OD OD 0.1. 0.1. 0017 0917 0917 (n*n-FT)/(n*n)-al (n*n-FT)/(n*n)-al 00£ 00£ T917 T917 v**(n*n-F-T))*n>kn,o7-zi v**(n*n-F-T))*n>kn,o7-zi Z917 Z917 (c**(n*n+-T))/(9**n*'i7 (c**(n*n+-T))/(9**n*'i7 v**n*-Tz+n*n*17-r--r)*n*n*'9I=i71 v**n*-Tz+n*n*17-r--r)*n*n*'9I=i71 - - £917 £917 (OT**11*'9£-8**11*.0917-4-9**11*"590T-17**11*.£09+1411**SG--T)*fl*114:179=9-1. (OT**11*'9£-8**11*.0917-4-9**11*"590T-17**11*.£09+1411**SG--T)*fl*114:179=9-1. 17917 17917 (OT**(11*11+'T))/T (OT**(11*11+'T))/T S9.17 S9.17 (£**ZA)/91+(Z**ZA)41+ZA/Z1+01=ZAT12 (£**ZA)/91+(Z**ZA)41+ZA/Z1+01=ZAT12 9917 9917 (n*n±-r)/(n*n)=nAd (n*n±-r)/(n*n)=nAd ZAT1D ZAT1D - - oov oov t9-14. t9-14. zninm+vms=vms zninm+vms=vms 8917 8917 (otyaNlAw)i (otyaNlAw)i 01 01 OD OD 69-17 69-17 T T (899.**011)/860'0-=3 (899.**011)/860'0-=3 0L17 0L17 01 01 (01.31\l'InIV)AI (01.31\l'InIV)AI OD OD Z Z TL17 TL17 T T (S8Z.T**OCI)/T£0.0=3 (S8Z.T**OCI)/T£0.0=3 ZL17 ZL17 00=3 00=3 (0.-1.19*LAIV)AI (0.-1.19*LAIV)AI £L.17 £L.17 Z (onv/n)*(2- (onv/n)*(2- (on*on-vT)/(mv*on*on))+ziAins=ztAins (on*on-vT)/(mv*on*on))+ziAins=ztAins vti7 vti7 SZt. SZt. 'T+1A1V=IAIV 'T+1A1V=IAIV (14V*.Z)/(NV*(T-1,A1V*'Z))=NV (14V*.Z)/(NV*(T-1,A1V*'Z))=NV 9L17 9L17 TOT TOT SOldIl SOldIl VaLT/S0 VaLT/S0 SVOT SVOT NOIS'tlgA NOIS'tlgA Ntql1?:10d Ntql1?:10d ZT*£ ESO ESO (-r+kivz)hiv=kmv (-r+kivz)hiv=kmv OT OT Z.L17 Z.L17 vms*((id*Id)/13)-(n*n+t)/(n*n)=9 vms*((id*Id)/13)-(n*n+t)/(n*n)=9 8Li7 8Li7 zinins*(idtz)-D=DIED zinins*(idtz)-D=DIED 6LV 6LV Di13,K((no)/(n*n+-0)=d Di13,K((no)/(n*n+-0)=d ost. ost. T817 T817 D ssmanH ssmanH SNOIlVinDivD SNOIlVinDivD 3 3 z8i7 z8i7 E8.17 E8.17 (0'Tb3-1H)dI (0'Tb3-1H)dI 01 01 005 005 OD OD SOT SOT 17817 17817 0-1=U 0-1=U S817 S817 01 01 006 006 OD OD 9817 9817 +Z**(IHd)NIS)121bS*H1:1*.Z)MHI:1-120*b).-)dX3)dSODV*(IdtZ)=Id +Z**(IHd)NIS)121bS*H1:1*.Z)MHI:1-120*b).-)dX3)dSODV*(IdtZ)=Id 005 005 L817 L817 ((T000'T ((T000'T 88-17. 88-17. Id*d=d Id*d=d 006 006 6817 6817 osv osv Nntru..] Nntru..] CINB CINB T617 T617 05/17/74 05/17/74 3.12 3.12 1045 1045 053 053 VERSION VERSION FORTRAN FORTRAN 492 492 BESSEL(Z,V,AI) BESSEL(Z,V,AI) SUBROUTINE SUBROUTINE 493 493 C C FUNCTIONS FUNCTIONS GOLDSTEIN GOLDSTEIN CALCULATES CALCULATES BESSEL BESSEL 494 494 THE THE BESSEL BESSEL FOR FOR C C 495 495 MODEL MODEL TIP TIP LOSS LOSS 496 496 C C S=0.0 S=0.0 497 497 AK=0.0 AK=0.0 498 498 499 499 C= C= 1. 1. K=1,10 K=1,10 500 500 30 30 DO DO B=(.25*Z*Z)**AK B=(.25*Z*Z)**AK 501 501 D=V+AK D=V+AK 502 502 503 503 P=1. P=1. TK=D-1. TK=D-1. 504 504 5 5 IF(TK.LE.0.0) IF(TK.LE.0.0) 40 40 TO TO 505 505 GO GO P=D*TK*P P=D*TK*P 506 506 D=D-2. D=D-2. 507 507 TO TO 508 508 GO GO 5 5 509 509 40 40 E=P E=P S=B/(C*E) S=B/(C*E) 510 510 S S + + 511 511 AK=AK+1. AK=AK+1. C=AK*C C=AK*C 512 512 513 513 CONTINUE 30 30 AI=((.5*Z)**V)*S AI=((.5*Z)**V)*S 514 514 515 515 RETURN RETURN 516 516 END END 05/17/74 05/17/74 1045 1045 VERSION VERSION 3.12 3.12 0S3 0S3 FORTRAN FORTRAN NACA00(ALPHA,CL,CD) NACA00(ALPHA,CL,CD) SU8ROUTINE SU8ROUTINE 517 517 518 518 C C LIFT LIFT AND AND COEFFICIENTS COEFFICIENTS OF OF DETERMINES DETERMINES THE THE DRAG DRAG NACA NACA 519 519 C C - - 0012 0012 ALPHA; ALPHA; AIRFOIL. NACA NACA FOR FOR ATTACK, ATTACK, GIVEN GIVEN ANGLE ANGLE A A 520 520 A A AT AT OF OF C C ORTHOGNAL ORTHOGNAL POLYNOMIAL POLYNOMIAL OBTAINED OBTAINED A A BY BY EQUATIONS EQUATIONS WERE WERE 521 521 THE THE C C REPORT REPORT IN IN 529. 669,PAGE 669,PAGE PUBLISHED PUBLISHED NACA NACA NO. DATA DATA 522 522 CURVEFIT CURVEFIT NACA NACA OF OF C C 523 523 C C A0=5.73 A0=5.73 524 524 A2=7.*A0 A2=7.*A0 525 525 SDO=0.0058 SDO=0.0058 526 526 SD1-0.0006 SD1-0.0006 527 527 SD2=.130 SD2=.130 528 528 SD3=0.0168 SD3=0.0168 529 529 SD4=0.0006 SD4=0.0006 530 530 SD5=12570. SD5=12570. 531 531 AMAX=0.218 AMAX=0.218 532 532 A=ALPHA A=ALPHA 533 533 IF(A.GT.AMAX) IF(A.GT.AMAX) TO TO 24 24 GO GO 534 534 535 535 CL=A0*A CL=A0*A CD=SD0+(SD1+A)+(SD2*A*A) CD=SD0+(SD1+A)+(SD2*A*A) 536 536 537 537 25 25 TO TO GO GO CL=(AO*A)-(A2*(A-AMAX)**2) CL=(AO*A)-(A2*(A-AMAX)**2) 538 538 IF(CL.GE.0.0) IF(CL.GE.0.0) 61 61 TO TO 539 539 GO GO CL=0.0 CL=0.0 540 540 +SD5*(A-AMAX)**4 +SD5*(A-AMAX)**4 CD=SD3+SD4*(A-AMAX)**2 CD=SD3+SD4*(A-AMAX)**2 541 541 IF(CD.LE.1.0) IF(CD.LE.1.0) TO TO 25 GO GO 542 542 CD=1.0 CD=1.0 543 543 544 544 RETURN RETURN 25 25 545 545 END END 05/17/74 05/17/74 1045 1045 3.12 3.12 053 053 VERSION VERSION FORTRAN FORTRAN (ALPHA,CL,CD) (ALPHA,CL,CD) SUBROUTINE SUBROUTINE NACA44 NACA44 546 546 547 547 C C COEFFICIENTS COEFFICIENTS LIFT OF OF DETERMINES DETERMINES THE AND AND DRAG DRAG 548 548 NACA NACA C C - - 4418 4418 AIRFOIL. AIRFOIL. FOR FOR ATTACK, ATTACK, ALFHA; NACA NACA OF OF ANGLE ANGLE A A GIVEN GIVEN 549 549 AT AT C C A A ORTHOGNAL ORTHOGNAL POLYNOMIAL POLYNOMIAL A A OBTAINED OBTAINED WERE WERE BY BY EQUATIONS EQUATIONS 550 550 THE THE C C 824, 824, 401. REPORT REPORT IN IN NO. PAGE PAGE PUBLISHED PUBLISHED DATA DATA NACA NACA OF OF 551 551 NACA NACA CURVEFIT CURVEFIT C C 552 552 C C ALP=ALPHA*180./3.141593 ALP=ALPHA*180./3.141593 553 553 IF(ALP.GE.8.0) IF(ALP.GE.8.0) 20 20 TO TO GO GO 554 554 0.3975 0.3975 CL=0.099375*ALP CL=0.099375*ALP 555 555 + + 10 10 IF(ALF.6G.+2.0) IF(ALF.6G.+2.0) 556 556 TO TO GO GO 0.000000704*ALP**3 0.000000704*ALP**3 0.000028188623*ALP**2 0.000028188623*ALP**2 CD=0.00001644&ALP CD=0.00001644&ALP 557 557 + + - - 1+0.00661 1+0.00661 558 558 100 100 GO GO TO TO 559 559 0.0000023229*ALP**3 0.0000023229*ALP**3 0.00002732*ALP**2 0.00002732*ALP**2 CD=0.0001695356*ALP CD=0.0001695356*ALP 660 660 10 10 + + + + 0.00629752 0.00629752 1+ 1+ 661 661 100 100 TO TO GO GO 562 562 30 30 IF(ALP.GE.12.0) IF(ALP.GE.12.0) TO TO GO GO 20 20 563 563 0.6078 0.6078 0.0731*ALP 0.0731*ALP 564 564 CL= CL= + + TO TO 10 10 565 565 GO GO 50 50 GE.15.5) GE.15.5) TO TO GO GO 30 30 566 566 IF(ALP. IF(ALP. 0.0248 0.0248 0.00738*ALP**2 0.00738*ALP**2 0.214377*ALP 0.214377*ALP 567 567 CL= CL= - - - - TO TO 568 568 10 10 GO GO IF(ALP.GE,16.0) IF(ALP.GE,16.0) 60 60 TO TO GO GO 50 50 569 569 3.23 3.23 -0.11*ALP -0.11*ALP CL= CL= 570 570 + + 10 10 TO TO GO GO 571 571 +1.934 +1.934 -0.029*ALP -0.029*ALP 60 60 CL= CL= 572 572 0.1851985 0.1851985 CD=+0.0131686686494*ALF CD=+0.0131686686494*ALF 573 573 - - RETURN RETURN 100 100 574 574 575 575 END END 05/17/74 05/17/74 1045 1045 3.12 3.12 VERSION VERSION FORTRAN FORTRAN 0S3 0S3 NACAXX(ALPHA,CL,CD) NACAXX(ALPHA,CL,CD) SUBROUTINE SUBROUTINE 576 576 577 577 C C PROFILE PROFILE FOR FOR FOR FOR USE USE SUBROUTINE SUBROUTINE A A EMPTY EMPTY IS IS AN AN NACAXX NACAXX 578 578 C C EQUATICNS EQUATICNS FIT FIT CURVE CURVE INSERT INSERT MUST MUST ONE ONE STORED. STORED. PREVIOUSLY PREVIOUSLY NOT NOT 579 579 C C OF OF FUNCTION FUNCTION AS AS A COEFFICIENTS COEFFICIENTS DRAG DRAG AND AND LIFT LIFT SCCTIONAL SCCTIONAL FOR FOR 580 580 C C DEGREES. DEGREES. IN IN ATTACK ATTACK OF OF ANGLE ANGLE 581 581 C C 582 582 C C ALP=ALPHA*188./3.141593 ALP=ALPHA*188./3.141593 583 583 CD CD AND AND CL CL FOR FOR FIT FIT PROGRAM PROGRAM CURVE CURVE ADD ADD 584 584 C C RETURN RETURN 585 585 END END 597 597 NACAXX NACAXX FOR FOR ERRORS ERRORS NO NO 00026 00026 SUBPROGRAM. SUBPROGRAM. OF OF LENGTH LENGTH xiaNadav xiaNadav II II THE THE "F" "F" USE USE FACTORS THE THE OF OF circulation circulation vorticity vorticity the in conserved, conserved, the is is Since Since wake wake equal equal the the be be circulation circulation must must to to "generated" "generated" by the the blades blades Hence Hence ds ds 'total 'total = = path path Using Using circular circular a a 2n 2n {2a1rf2}rd0 {2a1rf2}rd0 "total "total = = wake wake tangent tangent velocity velocity i.e., i.e., is is (0), (0), of of a' a' infinite infinite number number blades a' a' a' a' constant. constant. For For # # an an a'r0 a'r0 47r 47r FtotalB FtotalB 1 1 in in decreasing decreasing between. between. blades blades and and increasing increasing the the of of function function 0, 0, is is blades blades finite finite a' a' For For near near a a of of circulation circulation calculated calculated above the the function function circulation circulation the the is is F F Hence, Hence, a a a'rS2 a'rS2 4nrF 4nrF blade blade F F per per = = total total Kutta-Joukowski Kutta-Joukowski from from this this time, At At same same W W CL CL Frft Frft = = 2 2 107 107 is is of of FL FL ratio ratio rtotal rtotal the the circulation circulation to to generated generated included included the the drag drag rtotal rtotal NOW NOW rT rT sin(I) sin(I) CL CL (ixd) (ixd) d-C)x d-C)x Since Since = = pW2 pW2 pcVxfdr pcVxfdr cCxdr cCxdr dFx dFx = = = = , , Combining Combining ID ID airS2 airS2 (I) (I) 47trF 47trF cos cos Cx Cx 1+a' 1+a' W W CLW CLW Bc Bc sin sin CL CL dp dp Bc Bc since since a a aCx aCx 1 1 a' a' sirul) sirul) (I) (I) 8F 8F 1+ 1+ cos cos by by determined determined is is thrust thrust Now Now dTblade dTblade dTmomentum dTmomentum element element -B -B 2 2 (1 (1 dr dr a)2a\T,27urdr a)2a\T,27urdr C C pW pW c c pVco pVco = = y y then then also, also, disk disk the the localized localized rotor rotor at at be be "a" "a" Consider Consider to to aCY aCY a)2 a)2 (1 (1 2 2 (1. (1. aF)aF aF)aF S S - - a.' a.' = = 8sin2 8sin2 (1) (1) VF2 VF2 F) F) (1 (1 4SF 4SF F F 2S 2S + + = = a= a= 2(S+F2) 2(S+F2) 109 109