a vortex model of the darrieus turbine by thong van …
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A VORTEX MODEL OF THE DARRIEUS TURBINE
by
THONG VAN NGUYEN, B.S. in M.E,
A THESIS
IN
MECHANICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Approved
Accepted
December, 1978
/}c^
* ' , » . . • • • •
7
ACKNOWLEDGEMENTS
I am deeply indebted to Dr. James H. Strickland for his
direction of this thesis and to other members of my committee.
Dr. Clarence A. Bell and Dr. Allen L. Goldman, for their help
ful criticism. My thanks also go to Ms. Kathryn Carney for her
help in typing this thesis.
n
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
ABSTRACT v
LIST OF TABLES vi
LIST OF FIGURES .- vii
NOMENCLATURE x
I. INTRODUCTION 1
1.1 Purpose of Research 1
1.2 Previous Work 3
1.3 Relationship of Project to the Present State of
the Art 4
1.4 Research Objectives 5
II. AERODYNAMIC MODEL 6
2.1 Vortex Model 6
2.1.1 Lattice point notation 15
2.1.2 Rotor geometry 15
2.1.3 Induced velocities at lattice points ... 17
2.1.4 Blade element bound vorticity 19 2.1.5 Vortex shedding and convection of wake
lattice points 21
2.1.6 Blade loading and rotor performance . . . . 23
2.2 Numerical Analysis 25
2.2.1 Computational procedure 25 • • •
Page
2.2.2 VDART2 code 29
III. METHODS FOR REDUCING CPU TIME 33
3.1 Frozen Lattice Point Velocities 33
3.2 Fixed Wake Grid Points 34
3.3 Continuity Considerations 36
3.4 Vortex Proximity 37
IV. COMPARISONS OF ANALYTICAL AND EXPERIMENTAL RESULTS . . 38
4.1 Rotor Performance 44
4.2 Blade Forces 46
4.3 Wake Structure 56
V. CONCLUSIONS 67
5.1 Summary of Results 67
5.2 Recommendations for Future Work 68
BIBLIOGRAPHY 70
APPENDIX A 72
A.l Polynomial Interpolation of Wake Velocities 73
A.2 Linear Interpolation 76
APPENDIX B 79
B.l VDART2 80
B.2 Listing of VDART2 with Time-Saving Feature (polynomial interpolation of wake velocities) 87
B.3 Listing of VDART2 with Time-Saving Feature (linear
interpolation of wake velocities) 98
B.4 Listing of the Program for Extrapolation of C Value . 106
APPENDIX C 108 iv
ABSTRACT
An aerodynamic performance prediction model for the Darrieus
turbine has been constructed. The primary purpose was to allow
reasonable prediction of aerodynamic blade forces and moments to
be made. Previous aerodynamic models based on simple momentum
principles are inadequate for predicting blade loading. In
addition, accurate overall performance predictions for large tip
to wind speed ratios cannot be made with the simple momentum
models. Detailed prediction of the near wake structure is also
within the capability of the present model.
Results were obtained from the present vortex model for the
one-, two- and three-bladed rotors operating at various tip to
wind speed ratios to study the effect of number of blades and
tip to wind speed ratios on the rotor performance. Power coef
ficients predicted by the present vortex model were compared to
the vortex model due to Fanucci and the simple momentum or
"strip theory". In addition, experimental results were used to
compare with predictions obtained from the analytical model in
an attempt to validate the analytical model.
LIST OF TABLES
Page
Table 1. Input Data for VDART2 31
Table 2. Output Data for VDART2 32
Table 3. Lift and Drag Coefficients for NACA 0012 (Re=40,000) . 45
VI
LIST OF FIGURES
Page
Figure 1. Two Dimensional Vortex System 7
Figure iZ, Velocity Induced at a Point by a Vortex Filament . . 10
Figure 3. Velocity Profile of Vortex with Viscous Core . . . . 12
Figure 4. Equivalence of Vortex Sheet and Discrete Shed
Vortices 13
Figure 5. Two Dimensional Rotor Geometry 16
Figure 6. Velocity Induced at a Point by a Vortex 18
Figure 7. Blade Coordinate System . . ; 20
Figure 8. Vortex Shedding Diagram 22
Figure 9. Computer Flow Diagram 27
Figure 9A. Computer Flow Diagram 28
Figure 10. Arrangement of Grid Points in the Wake 35
Figure 11. Schematic of General Test Setup 39
Figure 12. Sketch of Carriage and Rail Cross-section 40
Figure 13. Dye Injection System 42
Figure 14. Motion of Markers in a Fixed Frame of Reference . . 43
Figure 15. Comparison of Calculated C Values for a One-Bladed Rotor (C/R = 0.1) . . . . P 47
Figure 16. Comparison of Calculated Cn Values for a One-Bladed Rotor (C/R = 0.15, Re = 40,000) 48
Figure 17. Comparison of Calculated Cn Values for a Two-Bladed Rotor (C/R = 0.15, Re = 40,000) 49
v n
Page
Figure 18. Comparison of Calculated C values for a Three-Bladed Rotor (C/R = 0.15, ^Re = 40,000) 50
Figure 19. Effect of Tip to Wind Speed Ratio on Normal Force (C/R = 0.05, Re = 40,000, NB = 2) 52
Figure 20. Effect of Tip to Wind Speed Ratio on Tangential Force (C/R = 0.15, Re = 40,000, NB = 2) 53
Figure 21. Effect of Number of Blades on Normal Force (U^/U^ = 5.0, C/R = 0.15, Re = 40,000) 54
Figure 22. Effect of Number of Blades on Tangential Force (U^/U^ = 5.0, C/R = 0.15, Re = 40,000) 55
Figure 23. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.15, U^/U^ = 2.0, Re = 40,000). 57
Figure 24. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.15, U.p/U = 6.0, Re = 40,000) . 58
Figure 25. Effect of Tip to Wind Speed Ratio on Streak Line (C/R = 0.15, Re = 40,000, NB = 2) 60
Figure 26. Effect of Number of Blades on Streak Line (Uy/U^=5.0, C/R = 0.15, Re = 40,000) 61
Figure 27. Calculated Streamwise Perturbation Velocity Profiles at One Rotor Diameter Downstream of a One-Bladed Rotor (C/R = 0.15, U^/U^ = 6.0, Re = 40,000) . . . . 62
Figure 28. Calculated Streamwise Perturbation Velocity Profiles at One Rotor Diameter Downstream of a One-Bladed Rotor (C/R = 0.15, U.p/U = 2.0, Re = 40,000) . . . . 63
Figure 29. Solid Particle Marker Motion (UT/U = 5.0, C/R = 0.15. Re = 40,000, NB = 2) . . . 65
I 00
Figure 30. Linear Interpolation of Wake Velocities 77
Figure 31. Calculated Blade Forces on a One-Bladed Rotor (C/R = 0.150, Uy/U^ = 2.5, NR = 4, Re = 40,000) . . 109
Figure 32. Calculated Blade Forces on a One-Bladed Rotor (C/R = 0.150, Uy/U^ = 5.0, NR = 4, Re = 40,000) . . 110
• • vm
Page
Figure 33. Calculated Blade Forces on a One-Bladed Rotor (C/R = 0.150, U.j./U = 7.5, NR = 4, Re = 40,000) . . Ill
Figure 34. Calculated Blade Forces on a Two-Bladed Rotor CC/R = 0.150, UT/U = 2.5, NR = 4, Re = 40,000) . . 112
I 00
Figure 35. Calculated Blade Forces on a Two-Bladed Rotor (C/R = 0.150, UT/U = 5.0, NR = 4, Re = 40,000) . . 113
I 00
Figure 36. Calculated Blade Forces on a Two-Bladed Rotor (C/R = 0.150, UT/U = 7.5, NR = 4, Re = 40,000) . . 114
I 00
Figure 37. Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, UT/U = 2.5, NR = 4, Re = 40,000) . . 115
I oo
Figure 38. Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, UT/U = 5.0, NR = 4, Re = 40,000) . . 116
I oo
Figure 39. Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, UT/U = 7.5, NR = 4, Re = 40,000) . . 117
I 00
Figure 40. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, UT/U = 2.5, Re = 40,000) . . . 118
I 00
Figure 41. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, UT/U = 5.0, Re = 40,000) . . . 119
i oo
Figure 42. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, U^/U^ = 7.5, Re = 40,000) 120
Figure 43. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R = 0.150, U^/U^ = 2.5, Re = 40,000) 121
Figure 44. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R = 0.150), Uj/U^ = 5.0, Re = 40,000) 122
Figure 45. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R = 0.150, U^/U^ = 7.5, Re = 40,000) 123
Figure 46. Calculated "Streak Line" Development for a Three-Bladed Rotor (C/R = 0.150, U^/U^ = 2.5, Re = 40,000) 124
Figure 47. Calculated "Streak Line" Development for a Three-Bladed Rotor (C/R = 0.150, Uj/U^ = 5.0, Re = 40,000) 125
Figure 48. Calculated "Streak Line" Development for a Three-Bladed Rotor (C/R = 0.150, U^U^ = 7.5, Re = 40,000) 126
ix
NOMENCLATURE
Tg - Non-dimensional bound vorticity
r^ - Non-dimensional spanwise shed vortex strength
V - Induced velocity at point p
h - Distance from the vortex filament to a point at which induced velocity is to be obtained.
h^ - Vortex core c
R - Rotor radius
V - Maximum induced velocity at the vortex core c -
9D - Angular position of the blade
p - Fluid density
Up - Local relative velocity fluid velocity in the plane
of the airfoil section
L - Lift force per unit span
C - Airfoi 1 chord length
1 section lift coefficient
1 section drag coefficient
1 section tangential force coefficient
1 section normal force coefficient
NB - Number of blades
BN - Blade number
x, z - Coordinate axis
C - Airfoi
e. - Airfoi d
C - Airfoi
C - Airfoi n
NT - Time step at which the vortex originated.
1 - Unit vector in the x direction
k - Unit vector in the z direction
c - Unit vector in the chordwise direction
n - Unit vector in the normal direction
U - Undisturbed freestream velocity 00 "'
Uj - Tangential speed of the blade
U - Perturbation velocity in the x direction
W - Perturbation velocity in the z direction
a - Airfoil angle of attack
Ax, Az - Distance traveled by a given lattice point in one time step
At - Time step
FI - Tangential force per unit blade length
F' - Normal force per unit blade length
F. - Non-dimensionalized tangential force per unit blade ^ length
F - Non-dimensionalized normal force per unit blade length n
T"** - Non-dimensional torque produced by a single blade e
C - Power coefficient contribution of a single blade pe
C - Average power coefficient for the entire rotor during P a single revolution
NT I - Number of time increments per revolution of the rotor
AS - Angle through which the blade moves between vortex sheddings
xi
CHAPTER I
INTRODUCTION
Wind power is one of the unlimited, non-polluting sources of
energy and it is available in many regions throughout the world.
One of the more economically viable wind machines that converts
such power into a usable form is the Darrieus turbine which was
invented in 1920 by G. Darrieus, a Frenchman. This machine was
subsequently re-invented in 1970 by scientists with the National
Research Council of Canada.
In recent years, several aerodynamic performance prediction
models have been formulated for the Darrieus turbine. The models
of Tempi in [1], Wilson and Lissaman [2], Strickland [3], and
Shankar [4] have all been used to predict the performance of three-
dimensional Darrieus rotors. Each of these models (the latter three
being virtually identical) are based upon equating the forces on the
rotor blades to the change in streamwise momentum through the rotor.
The overall performance can be predicted reasonably well with these
models under conditions where the rotor blades are lightly loaded
and the tip to wind speed ratios are not high.
1.1 Purpose of Research
While these models are moderately successful at predicting over
all performance trends they are totally inadequate from several stand
points. Major deficiencies associated with these simple momentum
models can be summarized as follows: 1
2
*Accurate performance predictions for large tip to wind
speed ratios cannot be made because the momentum equations
used in these models become invalid. This situation
deteriorates with increasing rotor solidity.
•Predicted blade loads are most probably ^ery inaccurate
since these models (1) assume a quasi-steady flow
through the rotor, (2) cannot distinguish between
1, 2 or 3 bladed rotors of constant solidity and
blade Reynolds' number, (3) assume a constant stream-
wise velocity as a function of streamwise position in
the vicinity of the rotor, and (4) assume that the
flow velocities normal to the freestream direction
are zero.
*It is doubtful that meaningful information concerning
the near wake structure of the rotor can be obtained
from the present models. This information may be
important with regard to the placement of rotors
in close proximity to each other and in making
assessments of the environmental impact of large
scale rotors on downstream areas.
It goes without saying that these deficiencies hamper the
design process associated with the Darrieus turbine. It is also
apparent that accurate predictions concerning the aerodynamic
performance, the structural and vibratory characteristics, and the
environmental impact of the Darriues turbine becomes increasingly
important as the scale of the turbine is increased to large sizes.
Alteration of the simple momentum models to alleviate the
listed objections presently appears to be hopeless. On the other
hand, the development of a model based on a vortex lattice analysis
can potentially eliminate all of the list objections of the present
prediction models.
1.2 Previous Work
Several vortex models for vertical axis wind machines have
been developed in the past. Models which typify previously
developed vertical axis vortex models are those due to Fanucci [5],
Larsen [6], and Holmes [7].
In reference [5] Fanucci presents a two-dimensional vortex
model which is applicable to straight bladed rotors of large
height to diameter ratios with unstalled blades. A transient
analysis is formulated which allows one to simulate unsteady free-
stream conditions and to observe the development of the rotor wake.
A distribution of vortices along the blade camber line is used
to model the potential flow near the airfoil. This in turn allows
one to determine the pressure distribution along the airfoil sur
face as well as the lift forces acting on the blade. This repres
entation of the airfoil is only valid for unstalled conditions.
Larsen's cyclogiro, unlike the Darrieus turbine, has blades
which articulate and which in fact "flip" twice per revolution.
The blades are articulated such that strong vortices are shed from
the airfoils only when they are flipped (the Darrieus turbine blade
sheds vortices continuously). Larsen performs a steady state
analysis by assuming a fully developed wake. This wake structure
is subsequently modified by an iterative process which ensures that
blade-wake interactions are self consistent. Airfoil lift and
drag forces are calculated based on the local airfoil angle of
attack.
The vortex model presented by Holmes in reference [7] is
applicable to a vertical axis wind turbine with straight blades
and a large height to diameter ratio. The analysis is strictly
valid for rotors which have a large number of unstalled blades
operating at large tip to wind speed ratios. The assumption of
a large number of blades allows the unsteady flow problem to be
replaced by a steady flow problem.
1.3 Relationship of the Project to the Present State of the Art
While none of the previously developed vortex models are valid
for stalled conditions, they do provide valuable insight into
concepts which can be used in this two-dimensional vortex model
for the Darrieus turbine. Thus, the present work is simply a
logical extension of previous work which requires that aerodynamic
stall be considered in the model. Both Fanucci [5] and Holmes [7]
assumed that the rotor blades were always at angles of attack
sufficiently small that aerodynamic stall was not encountered.
Larsen's cyclogiro [6] has articulating blades which operate at
angles of attack which are less than the stall threshold levels.
1.4 Research Objectives
The major research objective was to formulate an analytical
model for the straight-bladed Darrieus turbine of large height
to diameter ratio using a vortex lattice method.
The analytical model was to be formulated such that it
possesses the following features or capabilities:
*The overall rotor performance for a wide range of rotor
solidities and for all normal tip to wind speed ratios
was to be accurately predicted.
*The aerodynamic blade forces and moments were to be
accurately predicted as a function of rotor position.
*The rotor wake was to be described in some detail.
Careful attention was to be given in formulating a numerical
solution scheme which will require a minimum amount of computer
time. Results from this model will be compared in some detail
with results from the simple momentum models. In addition, the
experimental results will be used to compare with predictions
obtained from the analytical model in an attempt to validate the
analytical model.
CHAPTER II
AERODYNAMIC MODEL
Based on the aforementioned research objectives, the analyti
cal model is to be constructed so as to require only moderate
computational time while adequately describing forces on the
airfoil blade sections. The analytical model which will be des
cribed herein is expected to meet these requirements.
The production and convection of vortex systems springing from
the individual blades will be modeled and used to predict the
"induced velocity" or "perturbation velocity" at various points in
the flow field. The induced or perturbation velocity is simply
the velocity which is superimposed on the undisturbed wind stream
by the wind machine. Having obtained the induced velocities, the
lift and drag of the blade can be obtained using airfoil section
data.
2.1 Vortex Model
A simple representation of the vortex system associated with
a blade element is shown in Figure 1. The airfoil blade element
is replaced by a "bound" vortex filament sometimes called a "sub
stitution" vortex filament [8] or a "lifting line" [9]. The use
Fiaure 1. Two Dimensional Vortex System
8
of a single line vortex to represent an airfoil segment is a simpli
fication over the two dimensional vortex model of Fanucci [5] which
uses three to eight bound vortices positioned along the camber line.
The use of a single bound vortex represents the flow field ade
quately at distances greater than about one chord length from the
airfoil [8]. It can be argued on several bases that this is probably
an adequate representation.
As indicated in this figure, the strengths of the shed vortices
have changed on several occasions. On each of these occasions a
spanwise vortex is shed whose strength is equal to the change in
the bound vortex strength as dictated by Kelvin's theorem [10]. In
terms of the circulation around a closed contour, Kelvin's theorem
can be written as:
DI = 0 ( ) Dt
Thus if the contour encloses both the airfoil and its wake it
is seen that any change in the bound circulation must be accompanied
by an equal and opposite change in circulation in the wake. The
center of each shed vortex is convected in the fluid at the local
fluid velocity.
The fluid velocity at any point in the flow field is the sum
of the undisturbed wind stream velocity and the velocity induced
by all of the vortex filaments in the flow field. The velocity
induced at a point in the flow field by a single vortex filament
can be obtained from the Biot-Savart law which relates the induced
velocity to the filament strength. For an arbitrarily shaped
filament of strength r and length Z, the induced velocity V ^t P
a point p not on the filament is given by [9]
w _ r / r x d l , ,
Here r is the position vector of points on the filament with res
pect to the point p and r is the magnitude of the position vector.
Upon referring to the case shown in Figure 2, equation (2) can be
written as:
^p " ^ 4 W ^^°^®1 •" °^^2^ (2)
where the unit vector e is in the direction of r x dt. For an
infinitely long vortex filament, the angles 6, and e^ approach
zero. The velocity at a point p is then given by
\ - U (4)
It should be noted that if the point p should happen to lie on
a vortex filament that equation (4) yields indeterminate results
since e cannot be defined and the magnitude of V is infinite. The P
velocity induced by a straight vortex filament on itself is in fact
equal to zero [11] and equation (4) is valid only for points that
lie outside of the "vortex core". The velocity induced by a
straight vortex filament on a point within the vortex core is
10
vortex filament
Figure 2. Velocity Induced at a Point by a Vortex Filament
11
proportional to the distance from that point to the vortex filament
and reaches a maximum value at the edge of the vortex core.
Although experiments performed by Ciffone and Orloff [12] show
that the velocity profile tends to be like that given in Figure 3,
it does not appear that there have been other analytical or exper
imental works that quantitatively give good prediction of the max
imum induced velocity. If one assumes that the maximum core velocity
V is the velocity associated with the vortex sheet springing from
the airfoil an estimate of V can be made. By referring to Figure
4 and the definition of circulation [9], the following expression
can be obtained
3r = 2V 3x (5) c
or
V = 1 — (6) c 2 ax ^ '
where V represents the maximum induced velocity in the vortex, c
For spanwise vortices shed from any given blade element
it ~ ^ (7) 3x RAe ^ '
where r is the spanwise shed vortex strength and RAe is the dis-s
tance through which the blade element moves between vortex shed-
dings. Combining equations (6) and (7), the maximum velocity can
be given as:
12
Figure 3. Velocity Profile of Vortex with Viscous Core
13
A
Y
vortex sheet
• ' ^
l i n e of shed vortex centers
v o r t e x cores
Figure 4. Equivalence of Vortex Sheet and Discrete Shed Vortices
14
Referring again to Figure 3, the vortex core can be obtained
by equating V in (4) to the maximum velocity V given in (8): p c
h, = ^ (9)
Thus the velocity induced by a straight vortex filament on a
point within the vortex core is given by:
\ - MK-^ no)
In order to allow closure of the proposed vortex model, a
relationship between the bound vortex strength and the velocity
induced at a blade must be obtained. A relationship between the
lift L per unit span on a blade and the bound vortex strength r
is given by the Kutta-Joukowski law [10] as
L = pU^r (11)
where p is the fluid density, and Uj is the local relative fluid
velocity in the plane of the airfoil section. The l i f t can also
be formulated in terms of the airfoil section l i f t coefficient C,
as:
L = | P C , C U / (12)
where C is the airfoil chord length. Combining equations (11) and
(12) the bound vortex strength can be given as
r = Ic.cu, (13)
15
Equation (13) provides the required relationship between the bound
vortex strength and the induced velocity at a particular blade
since the lift coefficient as well as the velocity U^ are functions
of the induced velocity. In other words, if the induced velocity
(produced by all of the vortex filaments in the flow) at an air
foil is known, the local velocity vector can be obtained. The air
foil angle of attack can then be computed using the local velocity
vector, the blade velocity, and the blade orientation. The lift
coefficient can then be obtained from airfoil data. Equation (13)
can then be used to calculate the bound vortex strength for the
blade in question. It should be noted that the effects of aero
dynamic stall are automatically included using the method.
2.1.1 Lattice point notation
Vortex centers are represented by "lattice points" in this
model. Therefore, a lattice point numbering convention is required.
All variables associated with a particular lattice point or vortex
center such as lattice point coordinates and velocities as well
as vortex strengths bear double subscripts. The first subscript
denotes the blade from which the lattice point originated. The
second subscript denotes the time step at which the vortex originated.
2.1.2 Rotor geometry
The rotor geometry is very simple. By referring to Figure 5
the following relationships for points on the blade quarter chord
line can be obtained.
16
Figure 5. Two Dimensional Rotor Geometry
17
I = - sin [^BN-D + 9] (14)
I = - cos [^BN-D + e] (15)
Here NB is the number of blades making up the rotor while BN is the
blade number in question.
2.1.3 Induced velocities at lattice points
The velocity induced at a point within the vortex core by an
infinitely long straight vortex filament has been given previously
in equation (10) as
Vp = ^2SZe (10)
For points that are outside of the vortex core, the induced
velocity has been given in equation (4) as
- > • r \ - ^m ^'^ Using the notation of Figure 6, the unit vector e in the
->•
direction of V can be replaced with the following results: P
" ^ P points inside vortex core 2(RA0)'^
" ^ A points outside vortex core 2TTh
18
Figure 6. Velocity Induced at a Point By a Vortex
19
The total perturbation velocity at a lattice point V(i,j)
is obtained by summing the induced velocities from all other vor
tices in the flow. Using lattice point notation this can be
written as:
NB NT
v(i,j) = J2 £ \^^^^^ (1 ) k=l il=l
2.1.4 Blade element bound vorticity
From equation (13) it is seen that the bound vorticity is a
function of the airfoil section lift coefficient and the local
relative velocity. The lift coefficient is in turn a function of
the local airfoil angle of attack. As a step toward obtaining the
relative velocity vector and airfoil angle of attack, a blade
coordinate system is set up as shown in Figure 7. The unit vectors
c and n represent the positive chordwise and normal directions
respectively. These vectors can be formulated by
c = cos0n t - sinOg t (18)
"n = -sineg T - cose„ k
Here the unit vectors in the x and z directions are denoted by i
and t respectively.
The relative velocity of the blade element with respect to the
oncoming wind stream is given by
UR = (UOO + U + U.pCoseg)t + (W-U^sineg)l< (19)
20
Figure 7. Blade Coordinate System
21
where U^ is the undisturbed freestream velocity in the x direction and
U and W are the perturbation velocities in the xandzdirections res
pectively. The tangential speed of the blade element is given by Uj.
The magnitude of Un is given by
UR = IK + U + U.^cos9g)2 + (W - U^sineg)2]^/2 (2QJ
The angle of attack can be obtained by noting that
- Up-n Tana = ^
3 -c ^R ^ (21)
(U+Ujsin9g + Wcos0g
^ "" " (U+U^)cos9g - Wsin9g + U^
Finally the non-dimensional bound vorticity can be calculated
from
m- - I'i (f) (r) (22) oo 00
2.1.5 Vortex shedding and convection of wake lattice points
Vortices are shed from each blade element in such a way so as
to satisfy equation (1). Vortices which are shed during any given
time period can be related to the change in bound vorticity with
respect to time and position along the blade. Referring to the
diagram of Figure 8, the spanwise shed vortex strengths can be
written as
r^(i, NT-1) = rg(i, NT-1) - rg(i, NT) (23)
22
bound vortex at present time
( ^ = -
rg(I,NT)
spanwise shed vortex
O r^d.NT-l)
Figure 8. Vortex Shedding Diagram
23
The initial velocity of a lattice point which is being shed
from a blade element is assumed to be equal to the sum of the per
turbation velocity at the blade and the freestream velocity. During
the first time increment, after leaving the "lifting line", the
lattice point travels a distance given by:
Ax(i,j) = [U(i,j) + UjAt (24)
Az(i,j) = [W(i,j)]At
During successive time increments, an "open" or explicit inte
gration formula is used in calculating the distance traveled by a
given lattice point.
Ax(i,j) = [|u(i,j) - ij(i,j) + UjAt t=NT ^ r t=NT-l "•
Az(.i,j) = [|w(i,j) - 4l(i,j) ]At
(25)
]At t-NT-1
2.1.6 Blade loading and rotor performance
Two aerodynamic forces are considered in the present analysis.
The first is the tangential force per unit blade length F' acting
along the chord line of a blade element in the direction of motion.
The second is the normal force per unit blade length F' acting in
the direction of the unit normal vector shown in Figure 7. A com
plete set of two-dimensional aerodynamic forces would also include
a pitching moment about the spanwise axis. In general, this moment
is small and is thus neglected. The forces F^ and F^ can be
expressed in terms of the fluid density p, the airfoil chord length
24
C, and the relative velocity U of the fluid moving onto the air
f o i l .
F; = 1VCU,2 (26)
1; = 2'n^<
In non-dimensional form, these forces can be written as
F = I = c {—) ^ l / 2 p C u J ^^00 J 27)
00
" l / 2 p C u 2 n'LI» 00
The coefficients C. and C„ are related to the more common air-t n
foil lift and drag coefficients C and C. by
C. = C„sina - C .cosa t X, a
C = - C.cosa - CjSina
(28)
The torque produced by a single blade element can be written in non-
dimensional form by
t - -^ - UiK (29) e pM 2 2 R t
00
The contribution of a single blade to the instantaneous rotor
power coefficient is given by
" 00
where Uj is the tangential blade velocity. The average power coef
f i c ien t for the entire rotor during a single revolution is given by
NTI NB^
Cp = NTT $ ' ? S e ^''^
where NTI is the number of time increments per revolution of the
rotor.
2.2 Numerical Analysis
The numerical analysis closely follows the concepts presented
in the previous section. In all cases, variables are non-dimen-
sionalized to provide economy in utilization of the resulting com
puter codes. Velocities are normalized with U^, distances with R, 2
area with R , time with R/U and circulation with RU_. Force, ' 00 CO
torque and power are non-dimensionalized as indicated in section
2.1.
2.2.1 Computational procedure
The general procedure requires that calculations be made at
small time increments until a periodic solution is obtained.
Initially there is no wake structure and it is only as the wake
develops sufficiently that a periodic solution is obtained.
Based on Fanucci's experience with the two-dimensional vortex
model [5], it appears that the wake must propagage 3 or 4 rotor
26
diameters downstream for a periodic solution to be achieved. An out
line of the computational procedure is given in Figure 9.
The computation is initialized by setting the bound vor
ticity in each blade element to zero. The perturbation velocities
at each blade element are then calculated based on all of the vortex
filaments in the flow using equation (16). These velocities will
initially be equal to zero since no wake structure exists and since
the bound vorticity has been set equal to zero. The bound vorticity
is then calculated for each blade element using equation (22)
and the last calculated value of the induced velocity. This pro
cess is repeated in order to correct the predicted values of the
induced velocities and bound vorticities. The next step is the
calculation of blade element performance (torque and power output).
These values are output at this point along with the induced
velocities at each blade element. To assist in comparing the ana
lysis with the experiment, calculation of velocities and positions
of solid particle markers placed in the flow are made and output
at this point. The next major step is to recalculate the posi
tion of all the wake vortex filaments using equations (24) and (25).
Prior to this step, however, velocities at each wake lattice point
must be calculated. Time is incremented and new shed vortices
are created using equation (23). At the end of a preselected number
of time increments, velocities at selected fixed points in the wake
are calculated and output along with the location and velocities
of wake lattice points. If a complete revolution has been completed
27
Calculate Blade Element Positions
Predict Induced Velocities at Blade Elements Using Old r„ Values **
->-
Predict Bound ^ Vortex Strengths Using Predicted Induced Velocities
•>'
Correct Induced ^ Velocities at Blade Elements Using Predicted Fp Values
Calculate Average Rotor Power Coeff ic ient
Create Shed Vortices
Increment Time
E Correct Bound Vortex Strengths Using Corrected Induced Velocities
Calculate ^ Instantaneous Blade Forces and Rotor Performance
T Details are Shown in Figure 9A.
.'
Convect Wake Lattice Points New Positions
Figure 9. Computer Flow Diagram
28 Calculate Instan- ,
I taneous Blade Forces and Rotor Performance
I
#
Calculate Velocities at Wake Lattice Points
Calculate Velocities at Markers
I Convect Markers to New Positions
Calculate Velocities at Selected Fixed Points
Print out the Velocities and Posi tion of Wake Lattice Points
No
Convect Wake Lattice Points to New Positions
Figure 9A. Computer Flow Diagram
29
the rotor performance for the revolution is output. The process
is repeated for the desired number of revolutions of the rotor.
2.2.2 VDART2 code
The VDART2 computer code was written using the V^ortex method
of solution for the DARrieus Turbine in 2, dimensions as outlined
previously. The program consists of a main program and twelve
subroutines including two subroutines that are used to calculate
velocities at selected fixed points in the wake as well as velo
cities and locations of solid particle markers to assist in com
paring the analysis with the experiment. While no attempt is made
to discuss the program in detail, the general features of the
input-output characteristics will be given along with a listing
of the program.
The turbine configuration is input in terms of the number of
blades NB, the chord to radius ratio C/R, and the rotor height
to radius ratio H/R. Airfoil section data at a selected Reynolds
number is input in tabular form in terms of the lift and drag
coefficients, C» and C^ for various values of the angle of attack
a. Location of fixed points and the original locations of markers
are input in terms of XF, ZF and XM, ZM respectively. Finally, the
tip to wind speed ratio U-^/U^ is selected thus forming a complete
set of input data.
At each time increment, forces and aerodynamic parameters are
output for each blade element. The angular location of each blade
30
element along with the calculated angle of attack is given. The
local perturbation velocities are also given along with the forces
F and F. and the torque T . In addition, the instantaneous rotor n t e
torque coefficient and power coefficient are given. The location
and velocities of markers are also output at each time increment.
At the end of a preselected number of time increments the location
and velocities of wake lattice points are given. Input-output
variable code names are given in Tables 1 and 2 and a listing is
given in Appendix B.l.
Table 1. Input Data for VDART2
31
INPUT
function (
rotor geometry
airfoil data
speed selection
fixed point selection
initial marker position selection
:ode name
NB
CR
RE
TA
TCL
TCD
NTBL
UT
XF
ZMAX
NDEL
XMAR
ZMAR
MDEL
quantity
number of rotor blades
C/R
Reynolds' number x 10~
tabular value of a
tabular value of C.
tabular value of C.
number of tabular values
UT/U T <»
Fixed point position
Half of the distance over which fixed points are equally spaced.
Number of spacings between fixed points (Number of fixed points less one)
Initial position of markers
Half of the distance over which markers are equally spaced
Number of spacings between markers (Number of markers less one)
Table 2. Output Data for VDART2
32
OUTPUT
function
blade
identification
aerodynamic
parameters at
each blade
blade
forces
lattice
point
behavior
fixed point
behavior
marker
behavior
code name
BLADE
THETA
ALPHA
U, W
FN
FT
T
NT
X, z
U, W
POINT
UUF, WWF
MARKER
XM, ZM
UM, WM
quantity
blade number
angular position
blade angle of attack a
U W 7j—, M—; perturbation velo-" " cities
n
e
time step origin of lattice points
^, p- lattice point notation
u w , . 77— , 7j—, lattice point per-« oo turbation veloci
ties
fixed point number
rj—, rr- fixed point perturba-cjo oo tion velocities
marker number
~ , -^ marker positions
7p, Tj— marker perturbation oo oo velocities
CHAPTER III
METHODS FOR REDUCING CPU TIME
Computer processing time for the VDART2 model is presently
moderately long. The major portion of the CPU time is required
for calculating the velocities of wake lattice points. Therefore,
reduction of the number of computations required to calculate
velocities at wake lattice points appears to be most fruitful.
The subroutine FIVEL which calculates induced velocities has
already been written in an efficient format and thus reduction
in computational time will be obtained by reducing the number
of times which FIVEL might be called. For example, consider a
two-bladed rotor. If twenty time increments per revolution are
used and the rotor rotates through seven revolutions, then FIVEL
will be called 3.66 x 10 times. Several methods for reducing
CPU time, both tried and untried are presented in the sections
below.
3.1 Frozen Lattice Point Velocities
One approach was used to update lattice point velocities on
a less periodic basis. It was assumed the lattice point moves
with a velocity on the order of the freestream velocity and that
the perturbation velocity should be updated when the lattice point
travels a distance equal to the distance traveled by the rotor
blade in one time step. Using this criteria, thewake velocities
33
34
were updated after approximately ewery U-/U^ time step. Obviously
a certain amount of danger is present with this method with regard
to both numerical accuracy and stability. Several cases were run
initially with and without this time-saving feature and the power
coefficients were in good agreement for moderate tip to wind speed
ratios. One case was run at a very high tip to wind speed ratio
(Uy/U^ = 20) and numerical instabilities were seen to result.
3.2 Fixed Wake Grid Points
A method which was used successfully on the VDART2 program
to reduce CPU time utilized a number of grid points arranged as
shown in Figure 10. Perturbation velocities were calculated at
each of these grid points instead of at the vortex lattice
points in the wake. The velocities at the vortex lattice points
were then obtained by either linear or polynomial interpolation
of the velocities at the 50 grid points. Potentially this method
can reduce the CPU time by a factor of NT/NG where NG is the number
of fixed grid points. In reality, the interpolation procedure
reduces the reduction factor to approximately NT/2NG. For cases
where a vortex lattice point happens to fall outside the grid
pattern, its velocity is calculated in the usual way. Power coef
ficients were calculated for a 1, 2 and 3 bladed rotor at tip to
wind speed ratios of 2.5, 5.0, and 7.5. Agreement between this
method and the conventional method was quite good except for the
three-bladed rotor operating at a tip to wind speed ratio of 7.5.
35
35 34 33 32 31
40 39 38 37 36
45 44 43 42 41
I 50 49 48 47 46
Figure 10. Arrangement of Grid Points in the Wake
36
The resulting numerical instability was reminiscent of the one
encountered using the method of 3.1. The calculated power
coefficient values first dropped with each revolution in good
agreement with the more exact techniques and then at a certain
point in time began to increase to some positive value. In any
event, this technique holds especially good promise since it
reduces the dependence on the number of time steps from a cubic
function to a square function. Detailed discussion of this method
is given in Appendix A.
3.3 Continuity Considerations
This technique could be used in conjunction with the method
of 3.2. Basically, this method would take advantage of the
continuity equation given by:
M + 9 V 8w (32) 3x 9y 9Z "
to allow calculation of one of the velocity components in terms of
the others. For example, consider the two-dimensional case where
the lateral velocity w could be obtained from a difference equation
of the form
AW = . (33) AZ Ax
The values of U would be calculated as usual at the fixed
wake grid points based on the cumulative perturbation velocities
from vortices in the wake. The values of w along the wake centerline
37
would also be calculated in the same fashion. All other values
of W would be calculated efficiently using equation (33). In
this case, the CPU time would be approximately one-half of that
for the method of 3.2.
3.4 Vortex Proximity
A technique of combining vortices whose centers pass in
close proximity to each other could be useful. Logic to "skip"
the absorbed vortex when calculating perturbation velocities
should be carefully developed to avoid loss of time due to exces
sive use of logic "if" statements.
Conversely, vortices whose centers are far away from the
point at which perturbation velocities are being calculated could
be neglected. Some criteria based on a combination of range and
vortex strength could be used.
CHAPTER IV
COMPARISONS OF ANALYTICAL AND EXPERIMENTAL RESULTS
The major purpose of this section is to present analytical
and experimental results from the present work. Analytical results
from the present two-dimensional vortex lattice model VDART2
can be compared to the vortex model due to Fanucci [5] and the
simple momentum or "strip theory" models [2, 3, 4]. Aerodynamic
forces predicted by the VDART2 model are compared with experi
mentally measured forces. Finally, dye streak lines and marker
patterns predicted by VDART2 are compared with experimental results.
No attempt to discuss the experimental work in detail will
be made since it is reported elsewhere [14]. It is worthwhile,
however, to describe briefly how the experimental data were
obtained. A simple rotor configuration was built and operated
in a water tow tank. The use of water as a working fluid greatly
facilitates the ability to visualize the flow structure while
working at appropriate blade Reynolds numbers. In addition, blade
forces can more easily be measured since blade internal forces are
small. As depicted in Figure 11, the rotor is mounted on a carriage
which slides along a fixed rail. A cross section of the rail and
carriage assembly is shown in Figure 12. The carriage is provided
38
39
Roller Chains ^
Moving Carriage
Fixed Rail
/
Rotor
/
[y
Figure 11. Schematic of General Test Setup
Figure 12. Sketch of Carriage and Rail Cross Section
41
with instrumentation slip rings which transmit signals from the
rotor to the carriage housing. Instrumentation signals from the
moving carriage housing are transmitted to a fixed frame of refer
ence via cables suspended from the laboratory ceiling with surgical
tubing. For flow visualization, a stationary frame of reference
was used to observe the motion of dye and surface markers. A 35
mm camera was mounted over the center of the tow tank. The camera
was fitted with an autowind which was triggered by an inter-
velometer at a rate of 1/3, 2/3, or 1 Hz depending upon the rotor
tip to wind speed ratio. In order to observe a "streak line"
consisting of particles which flow over the trailing edge of the
airfoil, dye was injected through the trailing edge of one of the
airfoils, the dye injection system is shown in Figure 13. The
use of buoyant solid particles to view the flow structure was
also undertaken. An example of the motion of these solid markers
is given in Figure 14.
Validation of the transient blade loading prediction capa
bility of the analytical model was perhaps the most important
goal of the experimental work. It was intended that the normal
and tangential forces acting on the airfoil blade as well as the
moment about the quarter chord position be measured as a function
of time and blade position. Force measurements were made by
using a Whetstone bridge with four active strain gage elements.
The signals produced by the bridge circuits were first amplified.
The amplified signals were then passed through slip rings and
42
pressure relief valve
freon 12 y container
0.57 mm hypodermic needle
Figure 13. Dye Injection System
43
Figure 14. Motion of Markers in a Fixed Frame of Reference
44
monitored on a strip chart recorder and a dual trace storage
oscilloscope.
It should be noted that all of the results used in this section,
except for the case shown in Figure 15, were obtained for the
airfoil NACA 0012. Data for the NACA 0012 airfoil lift and drag
coefficients are given in Table 3.
4.1 Rotor performance
Comparisons between the present vortex model VDART2 and the
vortex model due to Fanucci and the strip theory (multiple stream-
tube model) are shown in Figure 15. The coefficient of drag was
chosen to be the same for each model. The lift coefficient for
the strip theory and the present vortex model was selected to
be equal to that for a thin airfoil at a low angle of attack.
The vortex model of Fanucci generates its own lift coefficient
by utilizing a series of vortices along the chord line and by
requiring that the vortex strengths be such that the fluid velocity
is tangential to the airfoil camber line. This method yields a lift
coefficient equal to that input into the present vortex model and
strip theory for thin airfoils in a uniform wind stream at low
angles of attack. The agreement between the strip theory and the
present vortex model is remarkably good. The rather high values
of Cp predicted by Fanucci's vortex model are somewhat mysterious.
It is possible that there is a numerical error in Fanucci's computer
code which results in low values for the induced velocities.
45
Table
AIRFOIL DATA
(RE = 0.04 MILLION)
ALPHA
0.0 2.0 5.0 8.0
10.0 11.0 15.0 18.0 21.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
100.0 110.0 120.0 130.0 140.0 150.0 154.0 160.0 164.0 168.0 170.0 172.0 175.0 180.0
! 3. L i f t and Dr
' t .
0.0 0.2500 0.5175 0.7300 0.7800 0.7650 0.7175 0.7000 0.6975 0.9546 1.1200 1.1000 0.9700 0.7100 0.4100 0.0900
- 0.2300 - 0.5300 - 0.8000 - 0.9800 - 1.0500 - 0.9400 - 0.8400 - 0.7000 - 0.6800 - 0.7100 - 0.7400 - 0.8400 - 0.5000
0.0
•ag Coefficients f
• d
0.0180 0.0188 0.0236 0.0355 0.0880 0.1080 0.1905 0.2580 0.2855 0.6666 1.0100 1.3700 1.7000 1.9300 2.0500 2.0700 2.0400 1.8900 1.6900 1.4100 1.0900 0.7200 0.5600 0.3700 0.2700 0.2100 0.1800 0.1500 0.0800 0.0300
or NACA0012 (Re = 40,000)
46
Analytical results for the one, two. and three bladed rotors
which correspond to the experimental setup are shown in Figures
16, 17 and 18. These results were obtained using the Texas Tech
IBM 370 computer. Run times on the order of an hour were required
to obtain five to ten revolutions of the rotor (depending upon the
number of blades). Since the relationship between the rotor power
coefficient C and the inverse of the number of rotor revolutions P
(1/NR) appears to be linear, some estimate of the long-term average
C value can be made. A computer program for extrapolating the
long-term average C value was written using the least-square NR
method with respect to the weighting function e , this program can
be easily converted into a subroutine to be used in VDART2. The
listing of the program is given in Appendix B.4,
From examination of Figures 15 and 16, it can be seen that
the agreement between strip theory and the VDART2 model is quite
good. For the higher solidity cases shown in Figures 17 and 18,
the agreement between strip theory and the VDART2 model is
reasonably good for low to moderate tip to wind speed ratios but
quite poor at high tip to wind speed ratios. The deviant behavior
of the strip theory at high tip to wind speed ratios and high
solidities is to be expected due to the complete breakdown of the
simple momentum principles used in that theory.
4.2 Blade Forces
A test matrix consisting of three rotor configurations (1, 2
and 3 blades) and three tip to wind speed ratios (2.5, 5.0, and
47
^ Fanucci Vortex Model [5]
•Strip Theory y^OLJiy II
J •""" Present Work C, = 27rsina CQ = 0.25 + .026 Cj_ 2^0
U^/U,
Figure 15. Comparison of Calculated C Values for a
One-Bladed 2-D Rotor (C/R =0.1)
48
0.3
0.2
0.1
U.U
0.1
0.2
1 1—
Present work 1 (VDART2) -Jp
1 • 1
^^^^^^"'"^^v * M
-
t 1
/o^ 1 1
1 • 1
\ ^strip theory Y [2, 3. 4]
1 « J
, ,
8 10
Uj/U^
Figure 16. Comparison of Calculated C Values for a
One-Bladed Rotor (C/R = 0.15, R^ = 40,000)
49
0.4
0.2
0.0
-0.2
-0.4
-0.6
G) present work
strip theory
o 0 8 10
Figure 17. Comparison of Calculated C Values for a
Two-Bladed Rotor (C/R = 0.15, R^ = 40.000)
50
0.4
0.2
0.0
-0.2
-0.4
-0.6
present work
0
strip theory
o^ 8 10
Uj/Uc
Figure 18. Comparison of Calculated C Values for a
Three-Bladed Rotor (C/R = 0.15, R^ = 40,000)
51
7.5 was used. Blade forces are shown in Figure 19 through 22 for
five of the nine test cases which were run. In each case, the
fourth revolution of the rotor was chosen to compare experimental
and analytical results. In each case the basic features of the
periodic waveforms were reasonably well developed by the fourth
revolution.
From Figures 19 and 20 it can be seen that at moderate to
large tip to wind speed ratios the downstream (e = 180° to 360°)
blade forces are reduced significantly from those upstream. It
can also be noted that the minimum value of the non-dimensional
tangential force F. and the zero value of F occur at values of
e > 0° instead of 9 = 0° as might be expected. This occurs due
to a significant lateral flow velocity W near 9 = 0 ° . A minimum
value of F. also occurs at 9 < 180° due to lateral flow in the
opposite direction. The effect of aerodynamic stall is clearly
seen at the lowest tip to wind speed ratio, especially with
regard to F,. Predicted stall regions for the upstream and down
stream area extend from 9 = 45° to 165° and 9 = 195° to 330°
respectively. Experimental data show a delay in the onset of stall
indicating that the dynamic stall phenomenon should be included
in the analytical model.
From Figures 21 and 22, the effect of rotor solidity can be
seen. The major effect is a progressive retardation of the flow
in the downstream area. Retardation in the upstream area is a much
weaker function of the number of blades.
n
n
n
52
180 270 360 450
10
0
10
20
\o
0
- c
1
\*^r
fcT
- J _ . .
(6 - 1080)" 1 1
Jo Co
Q
Of
^ \^J/\}^ = 5.0 1 oo
• i
1
\o
»
I •
-
-
-
ssX^ ^
_ J — 0 90 180 270 360 450
(0 - 1080)°
20 -
0
-20 -
(6 - 1080)'
Figure T5. Effect of Tip to Wind Speed Ratio on Normal Force (C/R = 0.15, Re = 40,000, Ng = 2)
53
180 270 360
(e - 1080)°
450
1.0 -
0.0
-1.0
180 270 360 450
(6 - 1080)°
2.0 C
1.0 -
0.0 "
-1.0 -
-2.0 -
0 90 180 270 360 450
(e - 1080)°
Figure 20. Effect of Tip to Wind Speed Ratio on Tangential Force (C/R = 0.15, Re = 40,000, NB=2)
n
n
n
20
10
0
10
20
-
- P
I
A L 0 90 180 270
(6 - 1080)° T
-20 -
n
-J i— 360 450
0" 50 TBO 270 360 450"
(e = 1080)°
10
0
10
20
1
f V
[
• 1
b
Ng = 3
1 1
1
0=\_
1
I
-
lo
\p ^ -
f
0 90 180 270 360 450
(e - 1080)°
54
Figure 21. Effect of Number of Blades on Normal Forces (U^/U^ = 5.0, C/R = 0.15, Re = 40,000)
55
1.0 -
0.0
-1 .0 -
1.0
0.0 -
-1 .0
180 270 360
(e - 1080)°
450
-
V F
-O
tb
cP\ ®
/ ® \
\g<^
NB = 2
o
®v/o
,
o
1
90 180 270 360 450
(6 - 1080)'
90 180 270
(6 - 1080)°
360 450
Figure 22. Effect of Number of Blades on Tangential Force (Uj/U^= 5.0. C/R = 0.15, Re = 40,000)
56
Experimental data for the normal force F are seen to be in n
reasonably good agreement with the analytical model except as noted
in the stall region. Experimental data for the tangential force
F^, however, is in poor agreement with the analytical model. This
disagreement is believed to result from problems encountered in the
experiment.
4.3 Wake Structure
Several aspects of the wake structure were examined briefly
using experimental and/or analytical data. "Streak lines" produced
by particles flowing over the trailing edge were obtained both
experimentally and analytically. Velocity profiles were obtained
in the near wake of the rotor using the VDART2 computer code.
Positions of solid particle markers placed in the flow ahead of
the rotor were also obtained using VDART2 for comparison with
experimental results.
"Streak lines" produced by particles flowing over the trailing
edge of a one-bladed rotor are shown in Figures 23 and 24. These
streak lines were produced using the VDART2 model. The streak lines
shown in Figure 23 depict the developing wake of a lightly loaded
rotor (i.e., low tip to wind speed ratio). As can be seen from
this figure, the streak line signature near the rotor is little
changed as the rotor completes 1, 2, 3, and 4 revolutions. This
indicates that a periodic analytical solution, with regard to blade
loading, is reached after only one or two revolutions. The streak
lines shown in Figure 24, on the other hand, depict the developing
57
V
"T 1 r
y
Figure 23. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.15, U^/U^ =2.0, Re = 40,000)
1 1 r T T 1 1 r
58
Figure 24. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.15, Uj/U = 6.0, Re = 40,000)
59
wake of a highly loaded rotor (i.e., large tip to wind speed ratio).
The streak line structure near the rotor is seen to be relatively
stationary only after about five revolutions although some change
can be noted between the seventh and ninth revolution. Therefore,
a periodic near wake structure is obtained only after a relatively
large number of revolutions at the higher tip to wind speed ratios.
Streak lines obtained from the experiment are given in Figures
25 and 26 along with their analytical counterparts. The photographs
are actually negative prints produced from color slides which gives
one the impression of smoke issuing from the blade as opposed to
red dye. Comparison between analytical and experimental results
show good agreement in regions where dye patterns have not become
too diffuse. Streak lines for each of the five cases depicted
were also recorded using a movie camera. Examination of these
films revealed several aspects of the flow which are not apparent
in the still pictures. Notable among these observations was the
presence of large well-organized vortices at the edges of the wake
structure especially at the higher tip to wind speed ratios and
higher solidities. The celerity or velocity of the vortex centers
appeared in most cases to be quite small while the center portion
of the wake moved at a nearly constant velocity. At low tip to
wind speed ratios distinct starting and stopping vortices could
be noted as the blade went into and out of aerodynamic stall.
Non-dimensional perturbation velocities in the streamwise
direction are shown in Figures 27 and 28 for the two cases given
e = 840° e = 849°
UT/U =2.5
6 = 1560' e = 1569'
U^/U = 7.5 e = 1929° e = 1920O
Figure 25. Effect of Tip to Wind Speed Ratio on Streak Line
e = 1605 e = 1628'
0 = 1560° 0 = 1570°
0 = 1560O e = 1628°
Figure 26. Effect of Number of Blades on Streak Line
62
-.5
0
.5
V"-
.5
0
.5
-.5
T — I — I r T 1 1 r T — I 1 r T 1 1 » r
I \
X 3
/ / I I \ \
V /
I \
/
• • - — - ' ^
V —
•• \
\
/ t \ \ \
\
/ '9
•.MKJ^^^^ -.5 JL—J L__J I 1 1 1 1 «- t « ' « I t 1 1 i L.
Figure 27. Calculated Streamwise Perturbation Velocity Profiles at One Rotor Diameter Downstream of a One-Bladed Rotor (C/R = 0.15, U^/U^ = 6.0, Re = 40.000)
T I I I 1 1 1 1 1 1 1 T
y ^^ / \
/ \ I
'^9
T r 1
r" .-A
\ I
y
T 1 r
rN •
63
1
0 .iim
u/u_ V I ^
- .1
0 U
.1
_/mTiv ,-.<frm\
f I
^ - ^ _ -
nm .JL_JI <-
Figure 28. Calculated Streamwise Perturbation Velocity Profiles at One Rotor Diameter Downstream of a One-Bladed Rotor (C/R = 0.15, Uj/U^ =2.0, Re = 40,000)
64
in Figures 24 and 23 respectively. These velocities are given at
a location which is one rotor diameter downstream from the rotor
center. In both cases, the wake has been allowed to develop through
ten revolutions of the rotor.
For the case shown in Figure 27 (U-^/U^ = 6.0) the streamwise
perturbation velocity distribution U/U^ is relatively invariant with
time except near the edge of the wake. The lateral perturbation
velocity distribution W/U^ is, on the other hand, quite variable
with respect to time. The largest variations appear near the edges
of the wake and possess peak magnitudes on the order of ± 0.10 which
is about 20% of the maximum streamwise perturbation velocity.
For the case shown in Figure 28 (Uj/U^ =2.0) the maximum
streamwise perturbation velocities are relatively small (U/U^ =0.1)
but are more variable with respect to time than the more highly
loaded case of Figure 27. The lateral perturbation velocities are
highly variable with respect to time and are also on the order of
± 0.10. Therefore, in this case, the maximum lateral perturbation
velocity is about 100% of the maximum streamwise perturbation
velocity.
Experimental verification of predicted wake perturbation
velocities was obtained. An example of solid particle marker motion
is shown in Figure 29. As can be seen from this figure, comparison
between analytical and experimental results is good if one takes
into account the experimental shortcomings. The agreement between
analytical and experimental results for the marker is not particularly
/ /
4 \ \
-f
\ \
V /
/ •
65
0 = 1260° 0 = 1267°
. • • • \
\
X
\
0 = 1755° 0 = 1770°
0 = 1950° 0 = 1959°
Fiaure 29. Solid Particle Marker Motion ^ ru /U = 5.0, C/R = 0.15, Re = 40,000)
66
good due to the problem encountered in the experiment with regard
to collisions between the particles and the rotor blades. In some
cases, the markers became impaled on the leading edge of the airfoil.
At higher tip to wind speed ratios and solidities the problem
becomes more severe due to the fact that the probability of collision
increases linearly with both tip to wind speed ratio and the number
of rotor blades. An additional complication was that approximately
30 cm of the center portion of the flow field was not visible due to
the towing mechanism support structure. The net result was that a
large percentage of the markers could not be used to display the
fluid motion.
CHAPTER V
CONCLUSIONS
The results from the present model are doubtlessly superior as
compared to the vortex model due to Fanucci [5] and the simple momen
tum or "strip theory" models [2, 3, 4]. Comparisons between analytical
and experimental results show reasonable agreement in most cases
except for the tangential force components. Comparisons between
the present model (VDART2) with experimental results and previous
models are summarized in the section that follows. Some suggestions
for future work are also given in section 5.2
5.1 Summary of Results
Several statements can be made with regard to the results from
the VDART2 mode:
*The rotor power coefficient predicted using the multiple
stream tube model is in good agreement with the vortex
models except at high tip to wind speed ratios for high
solidity rotors.
*The blade loading distribution with respect to rotor
position is significantly different for the two models
at moderately high tip to wind speed ratios. The vortex
model shows significant retardation of the flow in the
downstream area of the rotor.
67
68
*The streamwise velocity defect appears to be reasonably
stationary with respect to time while the lateral velocity
components are quite variable with respect to time. The
lateral components are normally less than 10% of the free-
stream velocity values.
•Several techniques appear promising with regard to signi
ficant reductions in the CPU time associated with running
the VDART2 code.
In addition, several statements can also be made with regard to
the comparisons of experimental and analytical results:
•Measured normal force components are in good agreement with
analytical predictions.
•Agreement between measured tangential force components
and analytical predictions are quite poor due to problems
encountered in the experiment.
•Streak lines produced by dye injection are in good agree
ment with streak lines predicted analytically in regions
of flows where diffusion of the dye is not too severe.
•Velocities in the wake as indicated by the motion of the
solid particle markers are in reasonable agreement with
analysis considering the shortcomings of the experiment.
5.2 Recommendations for Future Work
Several suggestions can be made with regard to the extension of
the present analytical work.
69
•The three CPU time-reducing techniques discussed previously
in section 3.0 (frozen lattice point velocities, continuity
considerations and vortex proximity) should be tried to
determine which technique yields the greatest reduction of
CPU time.
•Some experiments should be re-run after some modification of
the test rig. Some additional data not previously taken
should be obtained (i.e., velocity profiles in the wake).
•Experimental results indicate that the dynamic stall phen
omenon, which is not predicted by the model, occurs at low
tip to wind speed ratios. Therefore, it is suggested that
the dynamic stall phenomenon be taken into consideration in
any future models.
BIBLIOGRAPHY
1. Templin, R. J., "Aerodynamic Performance Theory for the NRC Vertical Axis Wind Turbine," National Research Council of Canada Report LTR-LA-160, June (1974).
2. Wilson, R. E., Lissaman, P. B. S., Applied Aerodynamics of Wind Power Machines, Oregon State University, May (1974).
3. Strickland, J. H., "The Darrieus Turbine, A Performance Prediction Model Using Multiple Streamtubes," Sandia Laboratory Report SAND 75-0431, October (1975).
4. Shankar, P. N., "On the Aerodynamic Performance of a Class of Vertical Shaft Windmills," Proceedings Royal Society of London, A.349, pp. 35-51, (1976).
5. Fanucci, J. B. and Walters, R. E., "Innovative Wind Machines: The Theoretical Performances of a Vertical Axis Wind Turbine," Proceedings of the Vertical-Axis Wind Turbine Technology Workshop, Sandia Laboratory Report SAND 76-5586, pp III-61-93, May (1976).
6. Larsen, H. C , "Summary of a Vortex Theory for the Cyclogiro," Proceedings of the Second U.S. National Conferences on Wind Engineering Research, Colorado State University, pp. V-8-1-3, June (1975).
7. Holmes, 0., "A Contribution to the Aerodynamic Theory of the Vertical-Axis Wind Turbine," Proceedings of the International Symposium on Wind Energy Systems. St. John's College, Cambridge, England, pp. C4-55-72, September (1976).
8. Milne-Thomson. L. M., Theoretical Aerodynamics, Second Edition, Macmillan and Co., (1952).
9. Karamcheti, K., Principles of Ideal Fluid Aerodynamics, John Wiley and Sons, (1966).
10. Currie, I. G., Fundamental Mechanics of Fluids, McGraw-Hill, (1974).
11. Tietjens, 0. G., Fundamentals of Hydro- and Aeromechanics, Dover Publications, (1957).
70
12. Ciffone, D. L., Orloff, K. L., "Far-Field Wake-Vortex Charac- , \ 464-470^(1975')' *"^ '" ^°"^"^1 Aircraft. Vol. 12, No. 5, pp. --^^^
13. Barr, A. J . , Goodnight, J. H., Sail, J. P., Helwig, J. T., A User's Guide to SAS '76, Sparks Press of Raleigh, North Carolina (1976).
14. Webster, B. T. , "An Experimental Study of an Airfoil Undergoing Cycloidal Motion," M.S. Thesis, Texas Tech University (1978).
71
5b)
APPENDIX A
FIXED WAKE GRID POINTS
As a matter of terminology, it should be understood that the term
"exact velocities" used in this appendix refers to velocities that are
obtained from equation (16) in section 2.1.3.
Calculation of wake velocities is the most time-consuming step
of all (Subroutine WIVEL in Appendix B.l). Therefore, it is necessary
to develop a numerical scheme which can give the approximation of
wake velocities without having to apply equation (16) to all lattice
points in the wake. Two methods, the polynomial interpolation and
the linear interpolation of wake velocities, are suggested. Both
require a number of grid points to be set up in the wake. The
arrangement of these grid points should cover the wake as much as
possible but not be too sparse in order to yield fairly accurate
results. In the early period of wake development, there are more
vortices near the rotor. Since vortices near the rotor have a strong
effect on the blade forces, there should be more grid points placed
near the rotor than in the region far away from the rotor. Figure 10
shows the arrangement of 50 grid points with 5 rows parallel to the
X-axis and 10 rows parallel to the z-axis. With such an arrangement
of grid points, the polynomial interpolation and the linear interpo
lation methods take approximately 37 and 31 minutes, respectively
72
73
compared to 60 minutes CPU time required to give "exact velocities"
for the case of a one-bladed rotor with tip to wind speed ratio of
8.
It should be noted that any reduction of CPU time made by the
two previously described methods acutally takes place only when
the calculation of more than 50 lattice points in the wake is
required. In other words, both methods are applied to approximate
wake velocities only after there were 50 lattice points in the wake.
Listings for the computer codes using the polynomial interpola
tion and the linear interpolation of wake velocities are given in
Appendices A.2.2 and A.2.3 respectively.
A.l Polynomial Interpolation of Wake Velocities
The following discussion shows how the U component of wake
velocities are interpolated. Interpolations of the W-components
of wake velocities are made in a similar fashion.
The arrangement of 50 grid points is as shown in Figure 10.
The perturbation velocities at these grid points are calculated from
equation (16). Wake velocity components are assumed to be a poly
nomial of the form:
U = c + C2Z + C3Z^ +...+ CgZ^ + CgX + c^x +...+c^^x +...
4 9 ^ C50Z X
The velocities at the 50 grid points can thus be expressed as:
74
U. "=1 * V l ^ =3 1 *• +C5Z; + cgx^ + C7x2+-.-..+c,4X?*...+c5oZ;x^
U2 = <:i+C2Z2+C3Z2+...+C5Z*+CgX2+C7x2+...+c^4x5+...+c5QZ^x^
Ucn = Ci+CoZm+C^Z 4 .9 50 = ^r°2^50^°3^56---^V50"'V50^V50^---^^14W---^^50^50^ 50
r», u,
. 50
In matrix form:
"1 z z2 z"* X x2
^ h 4 z* X x2
9 4 9 A "I • • • An
9 4 9 Xrt...^QXA
1 z z^ z'' X x2 x^ z^x^ 50 ^50"' 50"" 5O"'/50-' 5 0 " 50 50
=1
•
•
•
. ' 5 0
[A]
•50
Here A represents the indicated coefficient matrix
Then c^, c^, .... C^Q can be given by:
75
U.
[A] -1
U.
'50 U 50
The coefficient matrix [A] may be inverted using the subprogram
MATRIX PROCEDURE by SAS (Statistical Analysis System) [13]. It is
interesting to note that the coefficient matrix needs to be inverted
only once and the results can be used repeatedly to evaluate the 50
constants in the polynomial at different points in time.
Having evaluated the 50 constants in the polynomial, perturbation
velocities of lattice points whose positions are within the range
of the 50 grid points can be simply computed by substituting their
coordinates into the polynomial. Perturbation velocities
of lattice points whose positions are otherwise outside the range
covered by grid points are calculated directly from equation (16).
Polynomial interpolation of wake velocities does reduce CPU
time by a factor of approximately 50 percent for the case of one
bladed rotor and tip to wind speed ratio of 8. CPU time may be
longer for a given number of revolutions for lower tip to wind speed
ratios because at low tip to wind speed ratios, lattice points are
quickly convected outside the range covered by grid points. As
discussed previously, these perturbation velocities will be calculated
from equation (16) which is s^ry time consuming. Fortunately, fewer
76
revolutions are required for periodicity at the lower tip to wind
speed ratios. Polynomial interpolation of wake velocities is
quite good at points within a distance of 2.5 rotor diameters down
stream. Further away from the rotor, x becomes larger, truncation
errors resulting from the evaluation of the 50 polynomial constants
are magnified in each succeeding term of the polynomial which results
in large errors in the interpolated wake velocities.
A.2 Linear Interpolation
Similar to the polynomial interpolation, 50 grid points are
arranged as shown in Figure 10. The perturbation velocities at
these grid points are calculated using equation (16). Referring to
Figure 30, let
RX = (Xp - x^)/(x,^+5 - x^)
RZ = (Zp - Z^)/(ZN+I - ^N^
RZRX = (RZ)x(RX)
The U component of the perturbation velocity at point p whose position
is within the range covered by the 50 grid points can be given by
U = (RZ-RZRX)U^^i + (1-RZ-RX+RZRX)U^ +(RZRX)U^^g + (RX-RZRX)U^^5
Again, the W component of the perturbation velocity can be interpo
lated in a similar fashion. In case the lattice points are outside
the range covered by the 50 grid points the perturbation velocities
will be calculated directly from equation (16).
77
N +
N + 6 N + 5
Figure 30. Linear Interpolation of Wake Velocities
78
Like polynomial interpolation, linear interpolation of wake
velocities reduces computer time to approximately 50 percent for
the case of a one-bladed rotor with a tip to wind speed ratio of
8. The polynomial interpolated velocities and the linear interpo
lated velocities were compared to the "exact velocities". The
comparison indicates that the linear interpolation of wake velocities
in general yields more accurate results than the polynomial inter
polation of wake velocities due to the round-off errors in the
polynomial scheme.
APPENDIX B
COMPUTER CODE LISTING
79
80
B.1 VDART2
81
I
2
3
m
?s
•in
15
7 70 60
21
70
6 ^0
Cf):-'Mn«^/Lnc/x ( 3. <.'^n), z (T . 4nr ) WIT , ' . o n ) r.()'<Mnn/vrL/ij( i . / .ori i t W C {
r.n ( I T C L ( 3 0 ) 7F J X( ?r!)
COMMf i . i /VKO/uni^ .ACO) r.n"r*.ijrj/r,Arvn.S( A . ^ C f ) Cnf'".nfj/CLT.*.?./T ^ ( 3 0 ) , coHMOfj/f I x / x r i x c ^ n i r .n"vr ; i - j / "AR/yf i ?s ) ,7rM ps i t<EAn(S. 1 ) NC. r .P .UT r n : r ' A T { i ) , 2 F i o . ' t ) K F a D ( 5 . 7 ) N T ; I L . R E F n K M A T ( I 2 , F i n . 3 ) no 10 1=1,M7UL R F A f ) ( 5 , l ) TA( T ) , T C L ( I ) , T C D ( 1 ) F O R M A T ( 3 F 1 0 . 4 ) NSWl=2 x«aR=-s .n Z r A R = 2 . 0 MDEL=2A r O E L l = M n ^ L * l n E L M = 2 . 0 * Z M A q / M 0 E L 00 ? t J = 1 . M 0 E L 1 XMIJ)=XMAR ZM ( J )=-ZMAR>K J - 1 ) •DEL ' ' COMTINUP I\'C = 3 ZMAX=1.5 N0EL=9 X F = 2 . 0 MCELl=NOEL+l DFLZ = ?*ZM/',X/NDPL NT 1*24 D E L T = 6 . 2 « i 3 2 / N T I '1T=1 N R = l l on 25 I = l , N ! ) E L l X F l r ( 1 ) = X F ? F I X ( I ) = - Z M A X + ( I - 1 ) * D « ^ L Z CnriTlNijF CO 50 1 = 1,M"^ GS< l . l ) = 0 . 0 C»GH( I ) = 0 . C CONTINUE
'.OO)
T C n i 3 0 ) . N T R L
WRITECftt-V) N n , U T , C R , P F Fn '< ' - ' iT (30X , 'RCTHR r i^Ta '
* ' T I P TO WI'-ID SP«EEU « , F 4 . 3 , / / / / / / 3 0 » : , ' A l R F f : i
/ / /?ox; R A T i n = ' - F A . I ,
CL L D«TA« ' C O ' )
I ) I
T C L ( 1 ) 7 F 1 0 . ' r cn I)
*27X, 'ALPHA • ,5X. CO 15 I = l.MT'iL WRIT=r6f5) ^A( FlJR"^T(;>Gx,FlO, COMTINU? L=l DO 4 0 X=1,NR CP!:U"=C.O on 20 1=1,NTI J = L*I- IC CALL HGFQM{NT,NR,D'=LT)
DIVEL (NT.NB) RVORT(MT,N»UCR.UT) aiVEL(NT,rjn) PERF{NT,NB,CR,tJT,.MTI
CPSur-.=CPSU.M+CPL CALL U'lVFL (NTtNR.UT.MSWl ) CALL MARKER!MDEL1,NK,NT.DE IF (riT.MH.J) GO TO 21 CALL FPIVFL (MDcLUNh
FORM AT? 5 !• ,l5X.'BLi0F' ,fl>J, 'NT' ,12X,
'MIJMRE'^ OF 5'LAr5S= ', I2,/20y. /20X,'CHnRC TO RADIUS DAT 10='
' ,F 5.2 , •V IL LI ON) • ,/ / / /?/X,•(RE=
CALL CALL CALL CALL r?L )
T , U T )
NT)
, 1 5 X . DO 60 M=1,MB UO 70 N = l , J * . . , . . K W I T E ( 6 , 7 ) M , M . X ( M , N ) , Z ( . 1 FO'?*'AT( ! 3 X , 1,1 ,fl.><, I A , P X , F B CONTINUE CONTINUE L = L*1 CALL C O f i L P ( N T , N « , D r L T , l J T ) f;T=NT*l CALL S H : : O V R ( N T , N I J )
CUNT I niF CP = CPSU'VNTI W R I T : J 6 , 6 ) C P . K „^T..n Fa.J>r-AT(10X,'AVFRAG«= nCTPR CCNTircuF FNP
IIX l l ) t , ' U ' .<^X, ' K ' / )
N) , U ( M , N ) . W J r . N ) 4 , 4 X , F 7 . 3 3 X , F 7 . 3 )
r . P = ' , F ? . 4 , ' FOk RFVOLL'TICN ^MJ ' F R * , I 2 )
82
SUBRCUTINE RGCCM(NT,Ne , CELT) C O M y C N / L Q C / X ( 2 . ^ C C ) , Z ( 5 . ^ C C ) T H E T = « N T - 1 ) * C E L T C T E = f t . 2 3 3 2 / N B CO IC I ' l t N B T H F . T A = T H E T * ( I - 1 ) » C T B X( I . N T ) = - S I N { l h E T A ) 2( l . N T ) = - C r S ( T H E T A )
IC CCNTINUE RETURN END
SUBROUTINE RIVEL(NT.NB) Cn^,'J0N/L0r./X(3.400).Z(3.40O) C 0 W 0 N / y E L / U I 3 . 4 0 0 ) , W ( 3 . 4 0 ) ) Cn?^MCIN/r,U1/f;S(3,4C0),GP(14),0Gr,(l4)
J = .WT usu^-^o.o wsuM=o.n DO 10 K=1,N3 on 10 L=1,MT
usuSi^Ssu-ti.V'''^'''''^'-'''^'''''-''-''''^'*^^''^'^'*^^-''^^' WSUM*WSUM*UW
10 CONTINUE UC I, J)=USU«« w( I, j)=wsu;<
11 CONTINUE RETURN END
SURHOUTINF BVO»T(NT,Nn.CR.iT) CO^.*"0-J/LOC/XI3,40r » ,Z(^.AOn) C0MM0N/VEL/U(3,40 0),W(3.40C) C0''?1UN/GAf/GS( 3,4C0),G'^(l4l,CG°M4) C0MM0N/CLTAR/TA(30),TCL(30 ),TCO(30),NT3L 00 10 I=l,NP URON=-(U( r,NT) + 1.0)*X(I.NT)-W{T.NT)«Z(I.NT) UROC = -(U( I ,NT)-»-1.0)*Z( I .NT. •WC I ,.'JT)*X( I ,NT ) + UT UR=SCRT('JKDN««2-»-URDC**2) ALPHA=ATAN(URnN/URDC) CALL ACLIALPHA.CL) GB( I )=CL«CR-UR/2.C GS(I,NT)=GR(1)
10 CONTINUE RETURN END
83
SUMRnuTINE P F R F I N T , N r > , r R , U T , N T I ,CPL ) C l ) M M n . N / L 0 C / X ( 3 . 4 0 0 ) , Z 3 , 4 0 0 ' • * - " - ' COMMON / V E L / U ( 3 . 4 C C ) , W ( 3 . 4 0 0 ^2!it^'^^''C'^" /GS l i . 4 0 0 ) , CH ( 14 J , OG'i ( 1 4 ) C n M M n N / C L T A R / T A ( 3 0 ) , T C L I 3 0 ) , T C D ( 3 0 ) . N T R L
^ *!Qx'^*Tn''^i3^*'^"''"''*'»?'^» 'RLADE' .2X. 'ALPHA- ,3X. 'FN' .IIX, *'FT',llX,'T',lir.,tu',9<.'W') TR=0.0 CPL=0.0 00 10 I=1,N3 TH=(NT-l)-360.0/WTl*( I-l )« 160.0/NR URDN=-(U(I,NT)*1.0J*X(I,NT)-W(I,WT)»Z(I.NT) URDC=-(U« I ,NT) + 1 .0)*Z( 1 ,NT )•••«( I ,NT1*X( I .NT )*UT U^.=SORTlURON*«2+URnC«*2 ) ALPHAa4TAN(URnN/URnC) AL=57.206*ALPHA CALL ACL(ALPHA,CL) CALL ACNCTC4LPHA,CN,CT) G« ( 1 ) = CL*CR*UR/2.0 GS( I,NT) = GRl I) FN=CN«UR**2 FT = CT*ur<««;> TE=FT*CR/2.0 W.';iTE(6,2) T H . I . AL,FN,FT,TF,IJ( 1 .NT) .W( I .NT)
2 F0RMAT(F<1. l.I(S,F7.1,3X.F10.1.3r,F10.3,3X,Fl0-3,'^X,F7.3,3X,F7.3) T'5 = TR*TE CPL=CPL+TE*WT
10 CONTINUE WRITe(^,3) TR.CPL
3 FORMAT{//10X,'ROTOR TORQUE COEFFICIENT= • ,E10.3./, lOx, »'kOTO« POWEK COEFFICIENT^' .H10.3 ) RETURN END
SUP^.OUTINF MARKF:I(*^OELI .. IH . W T . O P L T . U T ) Ci)M^1O.>J/L0C/X(3.400),Z(3l4Cr) ' C0^".0N/GAM/GS{3,4C'?),Gn( 14 J.OGRI 14) COH>MO.N/MAR/XM(25) , 7 M ( 2 *5 ) DIMENSION LM(25J ,W»'(25) .Uf'r.l25) .WVO(25) WKITE(6,1)
1 FORMAT!//, 13X.'MARKER',lOX.' XM',14<,' ZM',14X.' LM«.15X,' WM
00 11 I=l,.yDELl USUM=0,0 WSUM=0.0 IF (NT.LF.I) GO TO 12 UM0(I)=UMII) w*<0( I )=w:u I)
12 DO 10 K=l,NB 00 10 L=1,NT CALL FIVEL (X ( K. L ) .X" ( I J . Z (Ji. L ) . Z". I I ) , GS I K. L ) ,UU .V, V. ) USUM = UU+USU'^ 'jsu*<=ww*wsur*.
10 CONTINUE UM{I)=USUM K>«( I ) = WSUM VRITE{6,2) I .XM(I l,ZM( I ),u.'»'( n .WVC I )
2 FORMAT{15X,I2,9X.F7.3tl0X,F7.3.1CX,F6.4,10X,F«.4) IFINT.LE.l1 GO TO 13 XM( I ) = XM(I )*(3.0*U'M I )-U"0(I)+?.0)*OT/2.0 Z»'(I l = ZM(I )-»-(3.0*WM( I )-WM0{l ) )*DT/2.0 GO TO 11
13 XM( r ) = XM(I )*(UM( 1 )*1.0)*DT Z'M I ) = ZM( I )*HM( I J*OT
11 CONTINUE l t TuRN END
84
SURRDUTINF FPIVFL (NOFLl.Nl' NTI J n y M n N / i n C / x ( 3 . 4 0 r i ; z n : 4 C 5 ^ COMMnN/GAM/GSl3.4C0).GP(14),0CD(l4) CnMMON/ri^X/<F IX(20) .7FlX(2(n
* ^g"Y{^{^(^^^^J''Pn'NT'.lOX.'UUF'.14X.'WUF'.//) USU". = 0.0 WSUMsO.O 00 10 K=l,NR 00 10 L=l.NT
USUMXSUIIUSUM''^*^''^''''^'''''''^''-''"''''^'^''^^''^''-'*'-^'^"' WSUM=WW*WSUM
10 CONTINUE W R I T E { 6 , 7 ) I ,USU?4,WSUM
7, ^ORWAT(15X,I2,9X,F7.3.10X,F7.3) 11 CONTINUE
RETURN END
^HSSQyTJKS WIVEL (NT,NP.UT.NSWn COMMON/L0C/X(3.400).7(3.400) COMMON/VEL/Ut3.400),W(3.40C) coMMON/vca/uo(3,4co),wn(3,4oo) COy?'.ONyGAM/GS(3.4C0).GP( 14 ),QGB( 14) TF (NT.LF.1) GC TO 12 NTl=NT-l 00 11 I=1.NR DO 11 J=1.NT1 U0(I.J)=U( I.J) W0(I.J)=U( 1.J) IF (NSWl.EC.O.OR.NT.F'.NSWl) GO TO 30 GO TO U
30 USU.V = 0.0 WSU"=0.0 IFCJSvVl.FO.O) GO TO 9 00 10 <=l.!^n 00 10 L=1,NT CALL FIV!;L (X(K.L).X'(I,J),Z(K.L).Z{l.J).GS{K.L).UU.kW) USUM=USUM+UU wsuM=viSu;i*Kw
n CONTINUE 9 U(I,J)=USUM
W( I.J)=WSUM 11 CONTINUE
IF (NT.EO.NSWl) NSW1=NT+1 12 RETURN
END
?! ;URROUTINE CONLP(NT,^:R,D£LT.UT) :0^MON/LOC/X(3.4C0).7(3,40r) CO''MON/VEL/UC3.40 0),W(3.40r) COMMIJN/VEO/UO(3.400) .V.0(3. 00) DT=0ELT/UT NT1=NT-1 DO 20 1=1,N9 IF (NT.LF. 1J GO TO 11 00 10 J=l,NTl X( I, J) = X( I .J)-»-(3.0*U( I..j)-lin( I , J)42.C)*0T/2 Z( If J)=Z{ I .J)*(3.0«V,( I . J)-un( I ,J) )*DT/2.0
10 CONTINUE 11 X( l.NT) = X( I.NT)-'-(L;( I.NT )+1.0)«=DT
Z( I.NT)=Z( I,NT)+H( I.NT)»OT 20 CONTINUE
HE TURN END
85
10
Uni jTINF SHFOVR (NT .NP ) y(>N/ Ai',/r,<; ( 3.4rn) , G " (i 4 ) ,nGH( i4 ) 10 I=1,NR I .NT )=G".( I ) I ,NT-l)=OGl'( I )-GP( I ) (1)=GR(I) TINUE URN
<;u":inijT INF C ( ) ' ^ ' "••
no GSI 35( OGw CON RET HMD
10
5
SURROUTINE FIVFL ( X 1. X? . 7 1 .22 .GA>i.yA.UU, WW) NT I=24 DFLT=6.2832/NTI R L l M = 2 . 0 / r i T I DX=X1-X2 Dl=Zl-Z2 SD=DX«*2+0Z*«2 SRSD=SORT{SD) IF (CRSD.LE.RLIV) GC TO 10 UU=-OZ*GAMMA/(SD«6.2832) WW=OX*GAVMA/ ( SC -b .2332 ) GO TO 5 VELTAN=(3. 1416*GAMMA)/(2.0=»DELT**2) UU=-OZ=»VELTAN WW=DX«VELTAN RETU'lN END
10 20
?H328HJrLlA§5VmRVff8[-.l30,.TCn(30),r.,TEL NTRL1=NTPL-1 An=57.29^*ALOH^ IF(AD.LE.O.O) aO=APi-36r'.0 IF(AD.GE.0.0) AL=AO IF(A0.GE.180.0) AL=36C.0-A; IFIAO.GE.360.0) AL=AG-360. no 10 I=l,NTBLl
iF(AL.GE.TAlIl.ANC.AL.LE.T-II+l)) GO TO 20 CONTl.NUR XA=(AL-TA(J))/(TA(J*1)-TA(J)) CL = TCL(J) + XA*(Tr.L( JM)-TCLrJ) ) IF(AD.GT.180.0.ANC.AD.LT.3'.0.0) CL = -CL RGTURN END
86
10 2 0
<;ur'M»UTINE AfrjCT ( AL PHi , CN , f.T ) Cr»M»Mvj/CLT/>ri/Ti( 3 0 ) , ir .L I iO) , TCn( 10 ) . N T r a NT»IL1 = NTHL-1 A 0 = 5 7 . ' 0 6 « A L H H A I F ( A O . L E . 0 . 0 ) AD=AD+36r .O I F M O . G F . O . O ) AL = ao I F C A U . G E . 1 8 0 . 0 ) A L = 3 6 0 . 0 - A G I F ( A O . G E . 3 6 0 . 0 ) AL = A n - 3 6 0 . ) DO 1 0 l = l . N T I ) L l J = I I F I A L . G E . T M I ) . A N D . A L . L e . T a { 1 * 1 ) ) GO TO 20 CONTINUE XA=( AL-TA( J l ) / ( T A ( J * n - T A ( i ) ) C L = T C L ( J ) + < A * ( T C L ( J + 1 ) - T C L ( J ) ) C D = T C D ( J ) + X A * ( T C O ( J * l ) - T C D J ) ) l F ( A O . G T . 1 8 O . O . A N n . A C . L T . 3 f c 0 . 0 ) CL=-CL C N = - C L * C O S ( A L P H A ) - C O « S I N ( A l P H A ) CT = CL'»SIN( ALPHA)-CP*COS(ALPHA) RETURN END
'•' l - 1 o n " ^ o r w I K : ; ? ? t I i r - ^ - " ^ ^ - ^ " - ^-^-°™1a, .nta.po-''
88
XI ( 50) .M (50) 01 "C'.";i DM xr ( 10) «'=AO(',.ii) xs
11 FORf.AT ( inPB.3) 7 I MA r= I .s neLTA=2.0"7lMAX/4.0 on JO ! = 1 , S ZS=-ZIMAX*( l-n»0ELTA 00 30 j=i,m K= I* ( J-l )*S ZI(K)=ZS
30 CONTINUE "=0 00 eo 1=1.10 DO -so J=1.5 K = M+J XI (K ) = XG(I)
TO CONTINUE ^=•••5
30 CONTINUE 0 0 1 0 1 = 1 , 5 0 C l = 1 . 0 Z l = Z I ( I ) Z 2 = Z I ( I ) * * ? Z 3 = / I ( I ) * * 3 Z 4 = 2 I ( I ) * * 4 X I = X I ( I ) X 2 = X I ( I ) * * 2 X 3 = X 1 ( I ) * « 3 X4 = XI ( I ) '»*4 X 5 * X 1 ( I ) « « 5 X6-=XI ( I ) » » 6 X 7 = X 1 ( I ) » « 7 <P=XI ( I ) « * R X9 = XI ( r ) " ' T ? X 1 = Z 1 « X 1 Z i r 2 = Z l * x 2 Z X 3 = Z 1 * X 3 7X4^=Z1*X4 Z X 5 = 2 1 = X 5 Z < 6 = 7 1 * X 6 Z X 7 = Z 1 * X 7 ZXRsZl^X-^ Z X 9 = Z l » x q Z 2 X I = Z 2 * X 1 ^ 2 X 2 = Z 2 « X 2 Z 2 X 3 = Z 2 * X 3 Z 2 X 4 = Z 2 * X 4 Z 2 X 5 = Z 2 * X 5 Z 2 X 6 = Z 2 * X 6 Z 2 X 7 = Z 2 * X 7 Z 2 X e = Z 2 * x e Z 2 X ^ = Z 2 * X 9 Z 3 X 1 = Z 3 * X 1 Z 3 X 2 = Z 3 * X 2 Z3X3 = Z3'»X3 Z 3 X 4 = Z 3 * X 4 7 3 X 5 = Z 3 * X 5 Z 3 X 6 = i : 3 * X 6 Z3X7=Z3'»X7 Z 3 X a = Z 3 * X 8 Z 3 X 9 = Z 3 * X 9 Z 4 X 1 = Z 4 - X 1 Z 4 X 2 = Z 4 * X 2 Z 4 X 3 = Z 4 « X 3 2 4 X 4 = Z 4 * X 4 Z 4 X 5 = Z 4 * X 5 Z 4 X 6 = Z 4 « X 6 Z 4 X 7 = Z 4 * X 7 Z 4 X 8 = Z 4 * X R Z 4 X 9 = Z 4 « X 9 W R I T E ( 8 . 1 ) C l . Z l . Z 2 , Z 3 , Z 4 , x i , X 2 . X 3 , X 4 , X 5 . X 6 . X 7 . X 8 , X 9 .
* Z X 1 . 2 X 2 , Z X 3 , Z X 4 , Z X 5 . Z X ' . . Z X 7 , Z X i l , 7 X 9 , Z 2 X l , Z 2 X 2 , Z 2 X 3 , Z 2 X 4 , Z 2 x 5 . Z 2 X 6 , * Z 7 X 7 , Z 2 X 8 . Z 2 X 9 . Z 3 X 1 . 7 3 X 2 , 7 " ? X 3 . Z 3 X 4 . Z 3 X 5 . Z 3 X 6 . Z 3 X 7 , Z 3 X a . Z 3 X 9 , *l^Kl, ' 4 X 7 , ' 4 X 3 . ! 4 X 4 . 3 4 X 5 , !< .X6 ,7 4 X 7 . 7 < , X 6 . ?4X9
1 FnR^^AT•( 1 2 ( 4 F 2 0 . 5 / ) 10 CONTINUE
STOP END
89
INFILt I N ; Z4X1-Z4XV)
CM A n ^ X 7^Xl -Z3X9 ( 2 0 . ) : F>-r;c M l k i X JRINJ ; FCTCH MATP.IX nAT.• = ^^T,<X; MAT=lNV(M\Ty I X) ; C'lTPlH I'«MT 0 n = l . iAi LUT ; K U N ; O'.TA -JUL L_; SET l .JAluuT; F I LU °UT (CCL1-C0L50) i z O . i o )
l w P J T ( l . i Li-L-f A I - X 9 ZX1-2X9 IZXi-lZX^
b j l
90
C n M M 0 N / V E 0 / U 0 ( 2 . 1 C C C ) , K 0 ( 2 . 1 0 0 0 ) C n M H O N / G A M / G S ( 2 , l C 0 0 ) , G B ( 5 0 ) . 0 G B ( 5 0 ) C O M M O N / C L T A B / T A ( 3 0 ) . T C L ( 3 0 ) . T C D ( 3 0 ) . N T B L C n M M n N / L 0 C I / X I ( 5 0 ) , Z I ( 5 0 ) , R M A T ( 5 0 . 5 0 ) C 0 M M 0 N / M A T R X / A ( 5 0 ) , B ( 5 0 ) DI>«ENSION X S d O J R E A 0 ( 5 . l ) N 8 , C R . U T
1 FORMATd l t 2 F l 0 . 4 ) R E A n ( 5 , 2 ) NTBL.RE
2 F O R f A T ( I 2 , F 1 0 . 3 ) CO 10 1=1,NTBL R E A D ( 5 , 3 ) T A d J . T C L d ) , T C D ( I I
3 F O R M A T ( 3 F 1 0 . 4 ) 10 CONTINUE
INC=1 NT 1 = 24 DeLT=6.2832/NTI NT=1 NR = 3 DO 50 I=1,NB GS( I,1) = 0.0 OGB(1)=0.0
50 CONTINUE WRITE(6,4J NB,UT,CR,RE
4 F0kMAT(30X.'ROTOR OAT A'.///20X,'NUMBER OF BLADES='.I 2,/20X. *'TIP TO WIND SPEED RAT 10='.F4.I./20X. 'CHORD TO RADIUS RATIO' *,F4.3.//////30X.'AIRFOIL DATA',/27X.'(RE='.F5.2.'MILLION)'./, »27X. •ALPMA'.5X, ' CL',8X. 'CD' ) CO 15 1=1,NTBL WRrTE(6,5) TA(IJ.TCL( I ) .TCDtI)
5 FORMAT(20X,F10.1,2F10.4) 15 CONTINUE
REA0I5.ll) XS 11 FORMAT!10F8.3)
ZIMAX=1.5 PELTA=2.0*ZI MAX/4.0 DO 30 1=1,5 ZS=-ZIMAX*(I-1I*CELTA 00 30 J=l, 10 K=I*(J-l)*5 i n K ) = ZS
30 CONTINUE M=0 00 80 1 = 1, 10 00 90 J=l,5 -K = M + J XI(K)=XS(I)
90 CONTINUE M^M+5
80 CONTINUE READ(9,47) ( (RMAT(I.J ) ,J=1 , 501,1 = 1,50)
47 FOR.'1AT(50F20.16) L=l DO 40 K=l,NR CPSU."=0.0 00 20 1=1.NTI
91
i -L*INC ALL BUE0M(NT,NB,CELT ) CALL BIVEL (NT,N8)
CALL PVORT(NT.NR.CR.UT) CALL RIVEL(NT,NB) CALL PERF(NT,NB,CR,UT,NTI,CPL) CPSUM«CPSUM-t.CPL CALL WIVEL(NT,NR,UT) IF (NT.NE.J) GO TO 22 WRITE(6,9) no 60O H«1,NB 00 700 N=1,J WRITE<6,7) M,N,X(M,N),Z(M,N),U(M.N),W(M,N)
700 CONTINUE 600 CONTINUE
L = L*1 22 CALL CONLP (NT.NB.DELT,UT)
NT=NT*1 CALL SHEOVR<NT,NBI
20 CONTINUE CP=CPSUM/NTI WSIT0(6,6) CP,K
6 FORMATdOX,* AVERAGE ROTOR CP«'.F7.4,' FOR REVOLUTION NUMBER',12) 40 CONTINUE
NR2=11 DO 400 K=4,NR2 CPSUH»0.0 DO 200 1=1,NTI J=L«INC CALL BGEOM(NT,NB,0ELT) CALL BIV6L (NT,N8> CALL BVnRTlNT,N8,CR,UT) CALL BIVEL(NT,NB) CALL PERF(NT,NB,Ca,UT,NTI,CPL) CPSUM=CPSUM-t.CPL CALL C02FF(NT,N3) CALL SWIVEL(NP,NT) IF (NT.NE.J) GO TO 21 WRITE,6.9, ... . _ UX.'U..9«..W/.
F7.3)
9
7 70 60
21
200
400
FORMAT! • 1' ,15X,'BLAD!! • . 8X . 'NT ' , 12X, 'X • , 1IX , • Z ' , DO 60 M=l,NB no 70 N=l,J WRITE(6,7) M,N,XIM,N).Z(M,N),U(M,N).W(M,N) FORMAT(lax, I 1,8X,I4,BX,F8.4,4X.F8.4,4X,F7.3,3X, CONTINUE CONTINUE L = L*l CALL CONLP(NT,Nn,DELT,UT) NT«NT*1 CALL SHEDVR'NT.NB) CONTINUE CP=CPSUri/NTI WRIT6(6,6) CP,K CONTINUE STOP END
THET-INT-l )*OELT nTB=6.2812/NB DO 10 1=1.NR THeTA=THET-K I-l)*OTe XII,NT|«-SIN(THETA) 7.{ I,NT)=-COS(THETAJ
10 CONTINUE RETURN END
92
r^S^^I^^!l^I^li£R^''»^«'50),OGB(50) C0MMON/veL/U(2.10CO),W(2,1000) COMMON/LOC/X(2.lOCO),Z(2:1000) DO 11 1=1,NO J = NT USUM=0.0 WSUM=0.0 00 10 K=l,NB no 10 L=l,NT USUM=USuSii3**'-^'-''*''-'''^"^-^»-^'''J>-^SIK.L).UU,WW) WSUM = WSUM+V.W
10 CONTINUE U( I.J)«USUM W( I.J) = WSUM
11 CONTINUE RETURN END
COMMON/VEL/U(2,10CO),W(2,1000) C0V?^0N/GAM/GS(2,1CCC) ,GB(50) .0GB (50) COMMON/CLTAB/TA(30),TCL(30),TCD(30),NTBL DO 10 1=1,NB URON=-(U( I,NT) + 1.0>*X d ,NT)-W(I,NT)«Zd.NT) -UROC=-(U( I .NT) + 1.0)-»Z d,NT)+W( I .NT)*Xd ,NT)+UT UR=SQRT(URON**2+UROC**2) ALPHA=ATAN(URDN/URDC) CALL ACLIALPHA.CL ) GR( I ) = CL*CR*UR/2.0 GS(I.NT)=GR(I )
10 CONTINUE RETURN END
93
SURROUTINF P E R F ( N T . N B . C R . U T , N T I . C P L ) c n M M O N / L n c / x ( 2 . i o c o ) . i ( 2 , i o 6 o ) CUMMON/V£L/U(2.10CO), W(2,1000) C0MM0N/GAM/GSI2,1CC0),GB(50),0GB(50) C0MM0N/CLTAB/TAI30),TCL(30),TCDI 30),NTBL W^ITE(6,1}
1 FORMAT(///,3X,'THETA' ,2X, 'BLADE',2X,'ALPHA',8X *'FT',11X,'T'.11X,'U',9X,'W«) TR=0.0 CPL=0,0 00 10 1=1,NB TH=tNT-l)*360.0/NTl4.( I-l ) •360. 0/NR URDN=-(U( I ,NT)*l.O)*X( T.NT)-Wd,NT)*Z(I,NT) URDC = -<Ud ,NT) + 1.0I«Z I I.NT)+Wd,NT)«X{ I,NT)+UT UW=SORTCURDN**2*UROC**2) ALPHA=ATAN(URON/UROC) AL=57.296*ALPHA CALL ACL(ALPHA,CL) CALL ACNCT(ALPHA,CN,CT) G R ( I ) = C L * C R * U R / 2 . 0 3 S ( I , N T ) = G B ( I ) F N = C N * U R * * 2 F T « C T * U R * * 2 T E = F T * C R / 2 . 0 H R I T E ( 6 , 2 ) T H , I . A L , F N , F T , T E , U d , N T ) . W ( I . N T )
2 F O R M A T ( F 8 . 1 . 1 6 . F 7 . l . 3 X . E 1 0 . 3 , 3 X . E 1 0 . 3 , 3 X , E 1 0 . 3 . 3 X . F 7 . 3 . 3 X . F 7 . 3 ) TRaTR-^TE -CPL=CPL*TE»UT
10 CONTINUE W R I T E ( 6 . 3 ) TR.CPL . ^ , ,
3 F O R M A T ( / / I O X , ' R O T O R TOROUE C O E F F I C I E N T = ' , E 1 0 . 3 , / , l O X , *»ROTOR POWER COEFFICI E N T = • , E l O . 3 )
RETURN END
?8SXRfl?/S!l!!^i4?53^!j^Tiooo, C O M M O N / V E L / U I 2 . 1 0 C 0 ) . W ( 2 , 1 0 0 0 ) C 0 M M 0 N / V E 0 / U 0 ( 2 , ICCC) , W O ( 2 , 1 0 0 0 ) C0Mr '0N /GAK/GS(2 .1CCC) , GB( 5 0 ) . 0 G 8 ( 5 0 ) I F ( N T . L E . l ) GO TO 12 N T 1 = N T - 1 no 11 I = l . N 8 DO 11 J = l t N T l U O d t J ) = U ( I , J W O d . J ) = W( I , J ) USUf^=0.0 WSUM=0.0 DO 10 K = 1 , N B ? 2 L L ° F I V E L ' ( X ( K . L ) . X ( I , J ) . Z ( K . L ) . Z d , J ) . G S ( K . L ) , L U , W W ) USUM=USUM-t-UU KSUM=WSUM4.V,W
10 CONTINUE U ( I . J ) = U S U M W( I , J ) = WSUM
11 CONTINUE 12 RETURN
END
94
!HPR8HJi8E/IVir!big9:?Ii.iooo. COMMON/VEL/U(2tl000),W(2,1000) C0MM0N/VE0/U0(2.ICCC),W0(2,1000) COMMON/GAM/GS(2,1 COO ,GB(50),0GB I 50) C0MMON/MATRX/A(5O).R(50) NT1«NT-1 00 10 1=1,NB DO 10 J=1,NT1 UOd,J)=U( I,J) W0(I,J)=W(I,J) ZAR=ABS(Z(I,J)1 IF (ZAB.GT.1.5) GO TO 21 Cl=1.0 Zl-Z(I,J) Z2=Z(I.J)**2 Z3=Z(I.J)**3 Z4=Z(1,J)**4 Xl«Xd,J) X2=X(I,J)**2 X3=X(I,J)**3 X4=X(I,J)**4 X5«X(I,J)«*5 X6=X(I,J)*«6 X7=X(1,J)«*7 X8=X(I.J)*«8 X9=X{I,J)**9 ZXl=Zl*Xl ZX2=Zl*X2 ZX3=Z1*X3 ZX4=Z1«X4 ZX5=Z1*X5 ZX6=Z1*X6 2X7=21*X7 2X8«Z1*X8 7X9=Z1*X9 Z2Xl=Z2*Xl Z2X2=Z2«X2 Z2X3«Z2*X3 Z2X4=Z2*X4 Z2X5=Z2*X5 22X6=Z2*X6 Z2X7=Z2*X7 Z2X8=Z2*X8 Z2X9=Z2*X9 Z3X1=Z3*X1 Z3X2=Z3*X2 Z3X3=Z3»X3 Z3X4=Z3*X4 Z3X5 = Z3-'X5 Z3X6«Z3*X6 Z3X7=Z3*X7 Z3X8=Z3*X8 Z3X9=Z3«X9 Z4X1^Z4*X1 Z4X2=Z4«X2 Z4X3=Z4*X3 24X4=Z4«X4
95
7 4 X 5 = 7 4 * X 5 2 4 X 6 = Z 4 * X 6 Z 4 X 7 « Z 4 * X 7 Z 4 X R = Z 4 * X 8 Z 4 X 9 = 7 4 * X 9 U ( 1 . J ) = A ( 1 ) * C 1 + A { 2 ) * Z 1 * A ( 3 ) * Z 2 * A ( 4 ) * Z 3 + A ( 5 ) * Z 4 + A ( 6 ) * X 1 * A ( 7 ) * X 2 * A ( 8
* ) « X 3 - i - A ( 9 ) * X 4 > A ( 10 ) * X 5 + A( l l ) * X 6 - » - A ( 1 2 ) « X 7 * A ( 1 3 ) * X 8 * A ( 1 4 ) * X 9 * A ( 1 5 ) * Z X • 1 » A ( 1 6 ) * Z X 2 + A ( 1 7 ) * Z X 3 * A { 1 8 ) * Z X 4 + A ( 1 9 ) * Z X 5 * A ( 2 0 ) • Z X 6 * A ( 2 1 ) * Z X 7 * A ( 22 • ) * Z X 8 * A ( 2 3 ) » Z X 9 * A ( 2 4 ) « Z 2 X 1 * A ( 2 5 > • Z 2 X 2 * A ( 2 6 ) • Z 2 X 3 + A ( 2 7 ) • Z 2 X 4 * A ( 2 8 ) • 1 Z 2 X S * A 1 2 9 ) * Z 2 X 6 * A ( 3 0 ) « Z 2 X 7 + A I 3 1 ) * Z 2 X 8 + A ( 3 2 ) * Z 2 X 9 + A ( 3 3 ) * Z 3 X 1 * A ( 3 4 ) « 2 Z 3 X 2 * A ( 3 5 ) * Z 3 X 3 + A < 3 6 ) * Z 3 X 4 - t - A ( 3 7 » * Z 3 X 5 * A ( 3 8 ) * Z 3 X 6 * A ( 3 9 ) * Z 3 X 7 * A ( 4 0 ) * 3 Z 3 X 8 * A ( 4 l ) * Z 3 X 9 * A ( 4 2 ) * Z 4 X 1 * A ( 4 3 ) * 2 4 X 2 * A ( 4 4 ) « Z 4 X 3 * A ( 4 5 ) * Z 4 X 4 - » A ( 4 6 ) * 4 Z 4 X 5 + A ( 4 7 ) •Z4X6-»-A(48) * Z4X7+A ( 4 9 ) • Z 4 X 8 * A ( 50 > • Z4X 9
W ( I , J ) = n d ) * C l * R ( 2 ) * Z l * B { 3 ) * Z 2 * f l ( 4 ) * Z 3 ' ^ 8 ( 5 ) * Z 4 * B ( 6 ) * X l * B ( 7 ) * X 2 * B ( 8 • ) * X 3 - ' - f l { 9 ) * X 4 * B d 0 ) * X 5 - ^ P ( l l ) * X 6 - » - R ( 1 2 ) * X 7 * e ( 1 3 ) * X 8 * B ( 1 4 ) * X 9 * B ( 1 5 ) * Z X * l * R ( 1 6 ) * Z X 2 * B d 7 ) « 2 X 3 * R ( 1 8 ) * Z X 4 * B ( 1 9 ) * Z X 5 * B t 2 0 ) * Z X 6 * B ( 2 1 ) * Z X 7 * 8 ( 2 2 • ) * Z X B - t - B ( 2 3 ) * Z X 9 + B 1 2 4 I » Z2X 1*8 ( 25 ) • Z 2 X 2 * B ( 26 ) * Z2X3 + B ( 27 ) * Z 2 X 4 * B ( 28 ) * 1 Z 2 X 5 * R ( 2 9 ) * Z 2 X 6 + B « 3 0 ) * Z 2 X 7 * B < 3 1 ) * Z 2 X 8 * R ( 3 2 ) * Z 7 X 9 * B ( 3 3 ) * Z 3 X 1 * 8 ( 3 4 ) • 2 Z 3 X 2 * B < 3 5 ) » Z 3 X 3 - ' - B ( 3 6 ) * Z 3 X 4 * 0 ( 3 7 ) * Z 3 X 5 + B ( 3 8 ) * Z 3 X 6 * e ( 39 ) * Z 3 X 7 + B ( 4 0 ) * 3 Z 3 X 8 + B ( 4 l ) * Z 3 X 9 + 3 ( 4 2 ) • Z 4 X 1 + B ( 4 3 ) * Z 4 X 2 * 8 ( 4 4 ) * Z 4 X 3 * 0 ( 4 5 ) * Z 4 X 4 * B ( 4 6 ) * 4 Z 4 X 5 * B ( 4 7 ) * Z 4 X 6 * 3 ( 4 8 » < ' Z 4 X 7 * R ( 4 9 ) * Z 4 X 8 * B ( 5 0 ) * Z 4 X 9
GO TO 10 21 U S U M « 0 . 0
WSUM=0.0 0 0 3 0 K=l ,NB 0 0 3 0 L=1,NT CALL F I V E L ( X ( K . L ) . X ( I , J ) , Z ( K , L ) , Z ( I . J ) . G S ( K , L ) , L U , W W ) USUM=USUM*UU -WSUM = WSUM-mW
30 CONTINUE •U( 1 , J )=USUM W( 1 .J )=WSUM
10 CONTINUE RETURN END
2«?S8aJfg!/5?3V?S?5):'SiS?i56Hr COMMON/VEL/UI2,1000),w(2,1000) CnM«'ON/VEO/UO 12,1 COG) ,W0(2, 1000) PT=0ELT/UT rjTl=NT-l DO 20 1=1,NB IF (NT.LE. 1) GO TO 11 X?lIj)=Xd^J) + C3.0*UC 1,J)-UO(I,J)*2.0)*OT/2.0 2( I,J)»7<I•J)*<3.0*WI I,J)-WO(I,J) )*0T/2.0
10 CONTINUE . „^ 11 X( 1,NT) = X( I,NT)*(UC I,NT)-H.O)*OT
Zl I,NT) = Z( I,NT) + Wd,NT>*OT 20 CONTINUE
RFTURN FNO
96
SUBROUTINE COEFF(NT,NR» ciDMMnN/LOC/xY2,idco), Z I2 ,1000) C0MM0N/L0CI/XI(50)*ZI(50),RMAT(50,50) C0MM0N/MATRX/A(50),D( 50) COMMON/GAM/GSI2.1CC0),GB(50),0GB(501 DIMENSION Ul(50),WI(50) WRITE<6,3)
3 FORMAT!'l'.9X,«GklC POINTS',1IX,'Ul»,15X,•Wl',/) DO 20 1=1,50 USUM=0.0 WSUM=0.0 00 10 K=1,NB 00 10 L«l,NT CALL FIVEL(X(K,L).XI ( I ) . Z(K,L).ZId),GS(K,L),UU,kW) USUM=USUMt-UU WSUM=WSUM*WW
10 CONTINUE UI{I)=USUM WI (I )>WSUH WRITE(6.2) l,UI(n.WId)
2 FORMAT!12X,I2.12X.F8.4.9X,F8.4) 20 CONTINUE
00 40 1=1.50 ASUM=0.0 8SUM=0.0 DO 30 J=l,50 ASUM=ASUM-t-RMAT!I, J)*UI (J) RSUM=flSUM"t-RMAT! I , J)*WI (J)
30 CONTINUE A! I ) = ASUM H( I )=RSUM
40 CONTINUE RETURN END
^.|\8HJiXS/gSf5i:?485]'?gi!50).OGB!50) DO 10 I»1»NB,, GS! I.NT)=GBtI) GS!I,NT-l)=OGBd)-GB( I) 0G5II)=GB(I)
10 CONTINUE RFTURN END
97
SyjRQUTlNE FIVEL 1X1.X2.Zl.Z2.GAMMA,UU,WW)
bELT»6.2832/NTl RLIM=2.0/NT! DX=X1-X2 07=Z1-Z2 S0*DX**2+07**2 SRS0=S0RT(S0) IF !SRSD.LE.RLIM) GO TO 10 UU=-D2»GAMMA/ISn*6.28 32) WW=DX»GAMMA/(SD»6.2832) GO TO 5
10 VELTAN«!3. 1416*GAKMA)/I 2.0*DELT**2) UU=-0Z*VELTAN WW«OX*veLTAN
5 RFTURN END
AD=57.296*AL«>HA iciAR'J:i*2'5* An=A0*360.0 IF!AD.GE.0.0) AL = AD TcJ.R'^i-i^S'O' AL»360.0-AD IF!AD.GE.360.0) AL=AD-36C.O no 10 I=l,NTBLl
10 ^g.^$y;^g|-^A'»>-ANO.AL.LE.TAd + l)) GO TO 20
20 XA=(AL-TA!J))/ITA!J+l)-TA(J)) CL = TCL( J)-»-XA*(TCL(J + l )-TCL( J) ) J^l JD^GT. 180.0. AND. AD. LT. 360. C) CL—CL END
SUBROUTINE ACNCT(ALPHA.CN.CTi COMMON/dLTAB7T4l30T,TCLT30T,TCDI30),NTBL NTBL1«NTBL-1 An=57.296*ALPHA IFIAD.LE.0.0) A0=AD+360.0 IF!AO.GE.O.O) AL=AC IFIAD.GE.180.0) AL=360.O-A0 IFIAD.GE.360.0) AL=AD-360.0 00 10 I=1.NT8L1 J=I IF(AL.GE.TA{ I ).AN0.AL .LE.TAd + 1) ) GO TO 20
10 CONTINUE 20 XA=IAL-TA(J))/ITAlJ*l)-TA(J))
CL=TCLIJ)*XA«ITCLIJ*1)-TCL!J)) CD=TCD(J)+XA*!TCDIJ*1>-TCD(J)) IF!AD.GT.180.0.AND.AD.LT.360.C) CL=-CL CN»-CL»COS!ALPHA)-CD*SIN!ALPHA) CT=CLaSIN(ALPHA)-CD*COS<ALPHA) RETURN END
98
B.3 Listing of VDART2 with Time-Saving Feature (linear interpolation of wake velocities)
99
><EAD!5. 1 ) • NR.CR.UT 1 F O R M A T ! ! 1 , ? F 1 0 . 4 )
TTEAD(5.2) NTOL.RE 2 F 0 R ^ A T d 2 , F 1 0 . 3 )
DO 10 I = l , tJT3L . P ^ * n i 5 , 3 ) T A d ) , T C L ! I ) . T C D ( l )
3 F 0 R . M A T ( 3 F 1 0 . 4 ) 10 CONTINUE
NT 1 = 24 n E L T = 6 . 2 r . 3 7 / N T I NT = 1 IF I N B . C C . I ) NR=3 IF I N R . E 0 . 2 ) Nn=2 I F ( N B . E 0 . 3 ) NR=l INC=3 XMA.'>=-5.0 7MAR=2 .0 MDEL=24 M0£L1=;^DI!L + 1 DELM=2.0«ZMAR/MDEL DO 76 J = 1 , M 0 E L 1 X M ( J ) = X M A R Z M ( J ) = - 7 M A R + ( J - 1 ) * D E L M
26 CONTINUE 0 0 50 I=1,NH G S d . 1 ) = 0 . 0 O G H I I ) = 0 . 0
l i r CONTINUE WRITE! 4 , 4 ) NfJ,UT.C;N.P.F
4 F0nrAT(30X.'-^OTO^ OAT A ' .///20X , • NU 'f E OF PL AOE S= ' , I 2 ,/?0X . *'TIP TO Wivn SPEED RATin=' F4. 1,/20X,'CHORD TO RACIUS o.ATIC' *,F4.3.//////30X.'AMFCIL C A T A ' . / ? 7 X , ' (RE = ' .F5.?,'»'ILLI ON ) ' . / / / *27X. 'ALPHA '.5X. ' C L ' . S X , 'CO') 00 15 I=1,NTRL WRITE(6,5) T A d ) , T C L ( n .TCL( I )
5 FnR'1AT(20X.F10.l,2FlO.'i) 15 CONTINUE
REAOIS.U) XS 11 FORMAT(10F3.3)
ZIM&x=1.5 CELTA = 2.0*71 MAX/4.0 on 30 1=1,5 Z?=-ZI MAX*(I-l)*nELTA CO 30 J=l, 10 K=I*(J-1)*5 7 I (K) = Z«
30 CONTINUE •••=0 no 30 1 = 1. 10 on 90 J=l.5 X I ( K ) = X S I I )
90 CONTINUE
JJO CONTINUE L=l CO AO K=1,NR CPSU**=0.0 DO 20 1 = 1 . N T I J = L « I N C CALL 3 G ' n f M N T , M R , D H L T ) C^LL H IVFL ( W T . N H ) CALL P.VnriT ( N T . N R . C R . U T ) T I L L ttIV=L(NT,NB) ^ . , ^ , ^ „ . , C \ L L PPf^F ( NT , NP., CR . UT , NT I , CPL ) CP<;u"=C01l.'f1+CPL CALL '..-IVFL ( N T . N ^ . U T ) r. ^LL ;iARi<EK!*':JELl . N ^ . N T . O ^ I T . u d
100
II- (NT.NE.J) GO TC 22 W^Irh(6.^) m 600 Mr: 1 ,N(3 no 700 N=l ,J WRITF(6.7) M.N.X!K,N),Z(M,.0 ,UIM,N) ,W(r,N)
700 CONTINUF 600 CONTINUE
L = L*1 22 CALL CDNLP(NT.ND,DELT,UT)
NT=NT*l CALL SHFOVR!NT.NQ)
20 CONTINUF CP=CPSUM/NTI V;RITe!6,6) CP,K
6 FORMAT! lOX. • AVERAGE IIOTC^ ro=',F7.4,' FOR REVOLUTION NUM8FR'.I2) 40 CONTINUF
NRl=NR*l NR2=11 no 400 K=NRl,NR2 CPSUM=0.0 00 200 1=1,NTI J=L*INC CALL HGFOM(NT,NR,DELT) CALL RIVEL (NT.NP.) CALL BVOKTINT.NB.CR.UT) CALL KIVEL(NT.NH) CALL PE'\F(NT,N3.CR.UT.NTI.rP(.) CPSUM=CPSU^' + CPL CALL SWIVFL!NT.N3) CALL MAKKER(MOELl .."415, NT .DE I T.UT) IF (NT.NE.J) GO TO 21
51 FORMATd 1' .9X. 'GRID PP I NTS ' . 11 X , 'U I ' . 15X , ' W I • , / ) DO 5? IK = l , ' 3 0 •/.•« I TE ( 6 , 5 3 ) I K . UI ( I K ) , W I ( 1 K )
S < FUR: 'AT ( 15>'. . I 2 . 1 7 t . F H . 4 , Q r . , F H . 4 ) 5? CONTINUE
^ F O R ' / A T d l ' , 1 5 X . ' 3 L A r e . , p x . 'NT ' . 1 2X , ' r ' , 1 IX , ' Z • . 1 I > . ' U ' . ^X , ' W • / ) nil *,o n= 1 , Nf
urilTC(6'^M''M.N.<(K.N) ,"7(M,.0 .U(M.N) ..J(^,N) 7 FOR." AT ( lSX.Il,8X,l4,ex,FG.4.4X.F8.4,4X.F7.3.3X.F7.3) 70 CONTINUE 60 CONTINUE
L-=L*1 21 CALL CONLP (NT.NP.,OELT.l'T)
NT=NT*1 CVLL SMEPVRINT.NU)
200 C0.4TINUI: C'>=CPSUM/NTI W " I T E ( 6 , 6 ) CP.K
4 0 0 CONTlNUli STOP ENO
101
*;u»i7nuTiN6 • GFnM( NT.Nn,nE(.T) CnvMnrj /LOC/X ( 3 . 4 O O ) . Z ! 3 . 4 0 C ) T H = T = ! N T - l )->OELT n T R = 6 . 7 r t 3 ? / N P nrj 10 1 = 1. NT THFTA=TH5T+!l-l)*DTR X! I,NT)=-5TN(THETA) 2( I,NT)=-CnS(THETA)
10 CONTINUE RETURN ENO
C( jURROUTINF RTV«=L(NT,N8) . O M " 0 N / G A M / G S ( 3 , 4 C 0 ) ,G ' ' ( 3 0 ) , 0 G n ( 3 0 )
C O M M n N / V E L / U ( 3 . 4 0 0 ) , W ( 3 . 4 0 r j CUM.^O.J/LnC/X( 3 , 4 0 0 ) , Z ( 3 . 4 0 . ) no 11 I = 1 , N 3 J=NT U S U " = 0 . 0 •*SU'«' = 0 . 0 DO 10 Ksl .Nf^ 0 0 10 L = l , N T CALL F I V E L ( X ( K . L ) . X ( I , J ) , Z (K, L ) , Z ( I . J ) . GS( K , L ) . UU. v»W ) USU^=USUM*L:'J VSUV = WSU''+WW
10 CONTINUE U! I . J ) = M S U f r ! I . J)=WSU^*
11 CONTINUE •'ETURN END
SUB7.0UT 1 NF RVORT ! NT . NR , C:i, I T ) CnMMOU/LOC/X!3,400).7(3,4UC) COMMON/VEL/U!3.4CC),W(3.4Cr) rnMMON/GA.'VGS(3.<»C0).GP(30: ,OGR( 30) C0MK0N/CLTAH/TA(30).TCL(30),TCO(30),NTPL
u2DN2-(rjli"riT)*l.C)*XI I ,NT)-W! I.MT)*Z!1 .NT) U^nC = -!U!I.'!T)*1.0)*7( I. NT )*W( I , NT ) f d , NT ) •UT UR = SORT(L'P.DN*s«2-»URnC*-7 ) ',LPHA=ATAN !I)RCN/URDC ) CALL ACLIALPHA.CL) GK(I )=CL*Cn*uR/2.0 GS(I,NT)=G3!I)
10 CONTINUE RETURN END
102
5»!'^nUTl JF I ' F R r i N T . N H . C . U T . N T I ,CPL) ( n.'vu) i / L n r / x ( 3 .40U) .7 1 3 , 4 o r I COMMON/VEL/U(3.400).W(3.400) C'JMM(jN/0 AM /GS ( 3. 4 CO ) , GM ( 30 ), OGR ( 30 ) CUVyON/CLTA(3/TA(30).TCL(30),TCD(30)..NTQL
1 FORMAT(///.3X.'THFTA',2X,'PL AUG',2X.'ALPHA',8X. 'FN • ,11X, »'FT',11X,'T»,11X,'U',9X,'W') TP=0.0 CPL=0.0 DO 10 I=1,NR TH=(NT-1)*360.0/NTT*( I-l )* ^60.0/NR URDN«-IU! I ,NT)*1.C)*XI I ,.NT )-W( I,NT)*Zd .NT) UaDC=-(U!I,NT)*l.0)«7(I.NT)+W(I.NT)«X(1,NT)+UT UR=SORT(URON**2*Uanr**7) ALPHA=ArAN!U«nN/U'<OC) AL=57.296*ALPHA CALL ACLIALPHA.CL)
.. CALL ACr)rT!ALPHA,CN,CT) G.3!I ) = CL*CR*U'</2.C GS!I,NT)=GB!I) FW = CN*U?.**7 FT=CT*L'R**2 TE=FT*CR/2.0 WRITE!6,2) TH,I,AL,FN,FT,TF,U!I.NT),Wd.NT)
2 FORMAT(F9.l.I6.F7.1,3X,ei0.3,3X,E10.3,3X,El0.3.3X,F7.3,3X.F7.3) TH=TR*Tf: CPL = CPL*-T!E*UT
10 CONTINUF WR1TF(6.3) TR.CPL
3 FnKMAT{//10X. 'ROTOW TCHOUE COEFFI CIENT=' .£10 .3,/, lOX. *"'xnrci DUWE.^ COEFFICIENTS'.E 10.3 ) RETURN FNn
SUBROUTINE WIVELI NT,N!\.UT) COMMON/Lnc/x(3.4on).z(3,4nr) COMMON/VEL/U!3,400).K(3.40C) Cn"M(lN/VEU/UO(3.4C0) .^0(3. 00) CO VMO.N / G AM /GS ( 3 . 4 CO ) . GR ( 30 ), OGB { 3 0 ) IF (NT.LF.1) GC TO 12 NT1=NT-1 DO 11 1 = 1, N'l no 11 J=l.NTl UOII,J)=U!I.J) WO!I.J)=W!I,J) USUM=0.0 KSUM=0.O on 10 K=l,NB no 10 L=l,NT CALL FIVEL!X!K,L).X!I,J).2(<.L).2d.J).GSIK.L).LU.WW) USUM=USUM*UU WSU.M = WSU".+ WW
10 CONTINUE U! I,J)=USUM W( I,J) = WSU.'
11 COfJTiNUK 12 RFTURN
END
103
SUMRnUTINF S W I V E L I N T . N H ) COMMON/l O C / X ( 3 . 4 0 0 ) . Z ( 3 .40f») C O M M O N / V E L / U I 3 , 4 0 0 ) . W ! 3 . 4 0 r ) C O M M r v > i / v y E ( ) / i j O ! 3 . 4 C 0 ) , H r ! 3 , 4 0 0 ) CO[^Mr)N/LOC I / X M 5 0 ) . Z I I 5 0 ) . i ; i ! 50 ) , u I ( 50 ) C r ) M ' * q . N / ( - . A M / r . S ( 3 . 4 C O ) . G ' ' ( 3 0 ) . O G R ( 3 0 ) N i i = r j T - i no 60 1=1 ,50 usu/'=o.o WSUM=0.0 DO 10 K = 1 . N 0 DO 10 L = 1 . N T F.rKii f . l y . . ^ . ' - ' ^ " ^ ' L ) . X l ( I ) . Z ( K . L ) , 2 I d ) , G S ( K . L ) .UU .WV) U.»U"=wS' I I I * (J l I WSUM=WSUM+KW
10 CONTINUF. UI d )=USUM '.VI ! I ) = WSUH
to . • CONTINUF DO 100 I=1.NR DO 100 J = l ,NT1 U P ! I, i)=U( I.J) WO!I.J)=WII.J) ZAR = All'; ( 7( I.J)) IF ! ZA2.GE.7I 15) ) GC TO 21 P'3 4 0 K.= 2.4 IF !Z! I , J).LE.7I !K) ) GC TO 11
40 CONTItJU'E K = K • 1
11 » =K DO 50 L=6,46.5 I ? (X ( I , J ) . L E . X I ! L ) ) GO TO 22
5 ^ CONTINUE GO TO 21
2? U = L * ' ' - ' ' B 7 = ( 7 { I , j ) - 7 I C N ) ) / ( 7 ! I N * I ) - Z I (N ) ) R X = ( X ( I , J ) - X I ( N ! ) / ( X | ( N * S ) - X I ( ' J n RZ.'lX = rxZ«?.X U! I . J ) = { R Z - R Z n X ) * U l ( N * I ) * ( I . O - R Z - R X * R Z R X ) « U I ( N ) * R /RX*U1 ( N * 6 ) * ( R X - R
1 2 R . r ) * U I ( N * 5 ) W( I , J ) = ( K Z - R Z R X ) * W 1 ( W * ! ) * ( 1 . 0 - ' > Z - R X * R Z P X ) » W I (N ) •RZRX^WI ( N * 6 ) * ( R X - R
1 Z I I X ) ^ W I ( N * 5 ) GO TO 100
21 usu:i=o.o WSUM. = 0 . 0 no 30 K = 1 . N H no 30 L = 1 . N T CALL F I V F L ( X ( K . L ) . X ( I . J ) ,Z «.<.L) .Z ( I . J ) . G S ( K . L ) .UU.WK) USUM=USUM*UU WSU'* = WSMM*WW
30 COWTlNUf^ U( I . J)= 'J ' ' .U" W! I . J ) = WSU'^
100 CONTINUE RETURN ENO
104
^UHROUTINE MARKFR C'OFLI ,MR.NT.DELT.UT ) COMMON/Lor./x( 3 . 4 0 0 1 , 7 ( ' . 4 c r ) Cnr*M0N/GAM/GS(3.4CC),GB( 14 ),OGB( 14) COMMON/MAR/XM(75) .TV(75) DIMENSION UM( 25) , 125) ,U.-'. (25),WMO(25) WKl T e(6. 1 )
I FORMAT(//, 13X.'MARKER' , lOK.' XM',14X,' ZM',14X,' LM',15X,' WM',//) nT = ()ELT/UT 00 11 1=1,M0EL1 USUM=0.0 WSUM=0.0 IF (NT.LE. 1) GO TO 12 UMO!1)=UM(I) WMO!I)=WM(1)
12 DO 10 K=1,NR no 10 L=1,NT CALL FIVEL (X(K.L) ,XMd ) ,2Ii',L),2Md ) ,GS(K,L) ,UU,WW) USUM=UU*USUM WSUV=WW*WSUM
10 CONTINUE UM!n=USUM. WM!I)=WSUM WRITE!6,2) I ,X.".! I ),ZM! I ),U.^d) ,WMI )
? FOKMAT!15X,I?,9X,F7.3,10X,F7.3,lOX,Fa.4.10X.F8.4) I F I N T . L E . l ) GO TO 13 X M I I ) = XM( I ) * ! 3 . 0 * U . ^ ! I ) -UMOd ) *? . 0 ) * n T / 7 . 0 7 M d ) « 2 M I I ) * ! 3 . 0 a w M ( l ) - W M O ( I ) ) » D T / 2 . 0 GO TIJ I I
13 X M d ) = XM( 1 ) * ( U M ( I ) * 1 . 0 ) * 0 T Z.-M I ) = ZH! f )*WM( I ) *0T
I I CONTINUF PETUKfJ END
SU«:^OUTINF SKEDVR(NT,Nn) COMMON/GAM/GS(3,4CO),GP(30),0Gfl(3O) no 10 1=1,NR GS! I ,NT1=GP! I ). GS!l.NT-l)=nGR!I)-GE(I) OGH(I)=G3( I)
10 CONTINUE RETU=^N ENO
SURRPUTINE CnNLO(NT,NR.n«:LT,UT) CO'*MOIJ/(.nC/X! 3.400),Z(3.40r) C0"M0W/VEL/U!3,40C).W(3.4Cr) cn" :MiN /vFn /uo ( j , 4 C P ) t W C i 3 . oo) DT=DELT/UT NTl= .NT- l DO 20 1=1.NR I F ( N T . L E . 1 ) CO TO 11 x ¥ l ! S ) = X ( i " j ) * ! 3 . O » U d , J ) - « 0 ! I , J ) * 7 . 0 ) * D T / 2 . 0 7 ! l . J ) = Z l l , J ) * ! 3 . C * W ! I , J ) - w O d , J ) ) ^ 0 T / 2 . 0
11 X ? ' J ' I N T V = X ! I , N T ) * I U ( I . N ' ' ) * 1 . 0 ) * O T 7A I . N T ) = Z! I . N n * W d . N T ) * C T
20 CONTI.NUF •»ETURN FNO
105
SyjRQ^TINE FIVEL !X1.X2,21,22,GAMMA,UU,WW) 05LT«6.2832/NTI RLIM=2.0/NTI D X = X 1 - X 2 0 7 = 2 1 - 2 2 S 0 - 0 X * * 2 * O Z * * 2 SRSD=SORT!SD) I F I S R S D . L E . R L I M ) GO TO 10 U U » - 0 Z * G A M M A / ! S n * 6 . 2 8 3 2 ) WW«OX*GAMMA/(SD*6.283 2 ) GO TO 5
10 V E L T A N = ! 3 . 1 4 1 6 * G A M M A ) / ! 2 . 0 * D E L T * * 2 ) U U = - 0 Z * V E L T A N WW«OX*VELTAN
5 RETURN END
20
SJ^?H5^^'''S>-f?t'30..TCO.,0,.NTBU AD=57 .296*ALOHA I F ! A D . L E . 0 . 0 ) A D = A D * 3 6 0 . 0 I F ! A C . G E . 0 . 0 ) AL=AD r i i ^ R * & i * i 8 ° * 0 ' A L = 3 6 0 . 0 - A D
AS*t8-?!i?g?BL°{ *^='^°-36c.o 10 ^g..!,?^;^g|-TA<I>-ANO.AL.LE.TAd*l), GO TO 20
r, 4^.'-7l?'*^P''''''A!J*l )-TA(J)) CL=TCL!J)*XA«!TCL!J*1)-TCL!J ) J^|j5D-GT.ie0.0.ANC.A0.LT.360.C) CL—CL END
COMMON/CLtAB/TAl30T,TCLT36y,tcDI30),NT3L NT3Ll=NT8L-l A0='i7. 296*ALPHA IFIAD.LE.O.OJ AD=AO*360.0 IFIAD.GE.O.OI AL=AO IFIAD.GE.180.0) AL=360.O-AD IFIAD.GE.360.0) AL=AD-360.0 00 10 I=l.NTBLl J=I IFIAL.GE.TA!Il.AND.AL.LE.TAII+l)) GO TO 20
10 CONTINUE 20 XA=IAL-TA! J) )/!TA(J*l )-TA!J))
CL=TCL!J)+XA«ITCL!J+1)-TCL!J)) C0=TC0IJ)+XA*!TCD{J*1)-TCD!J)) !F!An.GT.18O.O.AND.A0.LT.360.C) CL=-CL CN«-CL*COS!ALPHA)-CD*SIN!ALPHA) CT=CL*SINIALPHA)-CO*COS(ALPHA) RETURN END
106
B.4 Listing of the Program for Extrapolation of C Value
107
DIMENSION Y!9) DO 20 M=L.18 REA0!5,l) N.Y
1 F0RMAT!I8.9F8.5) SUMX=0.0 SUMY=0.0 SUMXY-0.0 SUMX2=0.0 SUME=0.0 0 0 10 1 = 1 . N X = 1 . 0 / I R I = F L O A T d ) E = E X P ! R I ) SUMX=SUMX*X*E S U M Y » S U M Y * Y ( I ) * E SUMXY=SUMXY*X*>! I ) * E SUMX2=SUMX2+E*X* *2 SUME=SUME+E
10 CONTINUE A=SUMX2 B=SUMX C=SUMXY D=SUME F=SUMY DELTA=SUMX2*SUME-SUMX**2 AA=!SUMXY*SUME-SUMX*SUMY)/DELTA BR=!SUMX2*SUMY-SUMX*SUMXY)/DELTA W R I T E I 6 , 2 ) AA,BB ^^ , ^ . „ „ , _ , . _ ,
2 F O R M A T ! / / / , 4 X , ' A A = ' , F 1 0 . 5 , / / , 4 X , • B B = ' , F 1 0 . 5 ) 20 CONTINUE
STOP ENO
APPENDIX C
ADDITIONAL ANALYTICAL RESULTS OF VDART2
108
0.5 -
0 -
-0.5 -
n
10.0
5.0
0 -
-5.0
-10.0
• 1 .
• 1 ; • ;
, '
1 '
1 •
(
1 1 t
1
. . •
1 ' ' ' 1
r ' 1 1 I I
\ 1 1 • •
1 J. .,
' \ 1 \ 1 1 .\ 1
-"TY" 1 l \ ' 1 \ 1
- - h \ -• • ! 1 \
' • N I , ; i ' I 1
1 i • I T :•!
• 1 ^ !
T-— '—^i-r--t++-!-• • . I I I i M i
. • • . ' i i 1 • • ' 1 '
• • ' 1 1 : 1 1 M
1 1 1 i i i l l ' i 1 ' ' ! .
. : 1 ! 1 I I ' ' ' 1 ' 1 ' • . • . i . 1 1 ' i • • 1 ' ; 1
' • '
; . 1 1 ' ' i 1 . . 1 1
• ' '
1 • • ' • 1 1 1 .
' ' . : . i 1 1 ! . ; 1 ' i ' * ' .i M i l l /
1 1 ^ I ' M ! V i 1 i 1 ' i , ' ' 1 ' i l l t V
' ' / • ' /
. . . y 1 . ! • > i I ' • • '
I r 1 ' / 1 / i t ! r 1
/ 1 1 1 / i 1 ' .
/ • 1 1 J - .. ' 1 j, --4 - j X - -4-} 1 1
1 . 1 1
TH i i
i _» . ' • 1 i 1 1 1
± ,.. 1 ' i i I I -H \—M—^-M-^
, 1 1 ^ - r i - - i — r — _ l _ l . . . 1 ; . 1 . : • _l_i L i_ i 1 ' ' ,-
t t 1 • ! ; 1 • I 1 ' ' I I ' M 1 , ' ' • V 1 ' 1 i 1 ' . / r I ' M 1 i X ' ; 1 1 1 y ' 1 1 1' y ••
1 , . ' . • i
1 • i 1 ^ • ' ' 1 >* M • •
I I ^' I X I . . . 1 • .
1 1 1 . - ' 1 1 1 ' • . 1 J M 1 1 1 • ' X I 1 ' 1 '
/ ' I ' l l . r ! ' ' , ' •
/ 1 1 1 1 i / ' • '
' _ i _4 ' I C
' 1 1
1 1 ' 1 ' '
' : ' , ' '
I t - I » 1 1 1 • ! ' •
1 1 : . 1 1 1 ! 1 • • I I I I I I I . :
' ' •
} 1 i 1 1
1 < i '
1 1 1 1 • • t 1 ' •
1 I I I ' ' 1 1 1 . • 1 > 1 ' i ' i 1 1 1 '
•'i : ' 1! 4+Vr
• , 1 1 r + • X ! ' 1 ' 1
K Y ! 1 1 1 1 j \ 1 • ' 1 1 1
1 • \ > ' 1 t 1 • ' \ 1 1 ' 1 i '
- i l 1 , M • '"• 1 1 1 !
• \ ' ' 1 1 ; ! • \ •• I I I ,
1 \ - I I I ' ' 1 1 1 I i \ 1 1 ( 1 i \ - " '
1 1 \ i_
: 1 i ' 1 1
•
! ' ! , \ i I 1 ' \ • t 1 1
: ' j i l l Vi • ' i \ i
1 1 1 '
' ' ' ' A L \ 1 ' [ • \ 1/7 I 1 \ /^ • 1 1 ' \ / I 1 . V . ^ 1 : 1 1 ' ' ' I • • 1
"7 " " 7 " ' •-
- ^ - -M- I - f - - r - ! -M-
0 90 180 270
e (degree)
360 450
Figure 31. Calculated Blade Forces on a One-Bladed Rotor (C/R - 0.150, UT/U = 2.5, NR = 4, Re = 40,000)
no
2.0
1.0 -
0
-1.0 J
n
20.0
10.0
0
-10.0
-20.0
0 90 180
6 (degree)
270 360
t 1 M 1 • I I I
M i l ; • 1 ' M 1 •
) 1
1 1 . t ( '
' 1 ' > 1 I
1 1
1 1 1 i 1 1
1 1 ! ! 1 I I
! 1 . 1 ' \ i ' 1 .\' ! I l ' \ I ' I I '
1 1 1
1 \ 1 \ \ ' ' I I I
' \ ' 1 ' l\i 1
i \ ' ; i \ ;
— •\ \ 1 \ •
_ — — —TT-
r r " r • ' 1
--M-!-
• • 1 1 " 1 I 1 1 1 1 1 1 1 • 1 . , 1 1 • :
j 1 . I I 1 1 : ^ 1 , 1 • — , ' i : 1 1 I - l . . 1 M 1 i 1 ] ; 1 ' ! 1 1 ' ' • ' \ j • ' ' ' ' ' '
1 i 1 1 1 I I I . 1 I I I ; • , . i ' M ' ' ! • ' • 1 1 1 1 1 1 ^ , , i 1 ! • ;
1 ^ ^ ' T T N > ^ \ M i M i l I > ' \ \ 1 , 1 . . 1 \ \ •- y\ • -r y : i i • ' ' 1 1 / 1 I X l 1 ; 1 . M ! M l / | 1 1 N 1 ; 1 1 1 I I I / \ ! ' ' 1 ' 1 ^ ' \ 1.1 1 1 ' / ' 11 t i t 1
/ 1 M 1 1 \ I 1 1 1 I 1
! I / 1 1 ' i I r M l 1 M '1 1 '1 1 / 1 1 ' 1 ' !
I I /• ; 1 i i M l ! t 1 ' i / i 1 i 1 ' 1 \ ' '
I ' l l ' 1 t M i l l M ' I I I ' 1 i' ! M 1 1 1 1 I ! '• • ' \ \ 1 1 V
M l 1 1* 1 1 \ \ \ [ M l 1 i 1 J i \ 1 1 ! \i \ 1 1 1
I i / \ ' 1 / ' l\ ' I M l / , 1 l \ ' , 1
*' • / 1 '' \ r 1 1 1 1 1 1 M : ^ 1 i • ' ' ' I \ 1
' - ' / ' 1 ' l\l 1 I , 1 ! / 1 1 1 1 ly • 1 I ' ' ' ' ' / M 1 \ • ' f 1 \ ' • " > 1 K T i n— . . T i i rr± 4: Xi T M ! I """> ! \ 1 • 1 / - - r 1 \ j I I I "7 . __ q i 4 : 4 1 - 1 - 4v I t
V • y - ^ -•— - - " T 1 11 1 K 1 '> 1
-T-! - r i — 1 1—1——rr n—n—^—^ - H — " - H - -r- - - H - ^ 1 i 1 - ! - H - 1 1 1 1—^-^rf-\—H-H-
450
Figure 32. Calculated Blade Forces on a One-Bladed Rotor (C/R = 0.150, U^/U^ = 5.0, NR = 4, Re = 40,000)
Ill
2.0
1.0 -
0
-1.0
20.0
0
n
-20.0
-40.0
0
i ' 1 1 • ' ' 1 1 1 1 1 1
1 • , 1 1 V
1 ' . 1 / M
• / ! 1 i l l / ' 1 ! \' : 1 M / i [ ! \ . 1 1 \ ,, _,. \ \ \ ,, In J , , , 1
1 1 ' y ' • \ 1 1 ' . 1 1 / . A • 1 ••• ~ r / i - • •
••!"\ — . ( { • \ M — : • T 1 1 / • \ / l ' > t L
\ 1 _J
—H4::i..ilLL—-4EEE
' 1 1 i 1 ; • ! 1 1 O ' ' '
--^ "~~A' i ! ""*" i i >" 1 \ ' ' ' ' ' 1 ! ' ' i\' 1 ' ' I 1 i \ 1 ; !
'L . - \ ! 1 t T T |*-P^>) 1 * 1 ' 1 1 V ' ["'
'T ' ' ' 1 M 1 \ ' ! 1 1 1 1 1 1 —1\
I , . , I , i ,M
T I ' l l 1 \
1 i ' T _ 1 1 A
• \ 1 i} \ 1 1 \ 1
__( h-h 1 1 1 1 1 VH-h - -- - —^—^—n—rrrr
I ' M 1 1 \ 1 , 1 M M M \ 1
: 4 : i i i i [ : ±q±n : - :± . j<± —H—H4--K^-;-H--L+H-— H H-i 1—*—r--; 1- 1 1 1 1
± 90 180
6 (degree)
270 360 450
Figure 33. Calculated Blade Forces on a One-Bladed Rotor C/R = 0.150, Uy/U„ = 7.5, NR = 4, Re = 40,000)
112
0.5 -
0.0 _
-0.5 -
1 1 I -
! ! 1 , . .
1
IX''- - iH J|.;":|V , - ; ^ ^ | | - : . | , . J 4 • -.r
4- _ . . . _ . _ _ \- -[ - | - - M - h- T -Lj
1 1 1
1 1
•' M L I 1 ' nV •
1 'L/li-f VIj I'M % 1' ^j{ j - . . j : : j j^ . : i , -\it/ " - T - t - i - - h i r - T ^ - 1 \--
1 1 1 1 T
. 1 _L L. 1 _ _ __ LJ _
. ! 1 I 1
n
10.0
5.0
0
-5.0
-10.0
0 90 360
H ' ' ' ' ' • ' f M i M 1 . I ' ' 1 1 ' 1 1 M 1 1 1 M , . . 4 . 1 ' 1 ! ' M ! M M i 1 j 1 1 1 1 1 M l . ' 1 1 i 1 M i i ' ' i 1 1 M 1
' ' 111 i ' ' ' 1 ' ; ' ' M M 1 I I I 1 t t M : M . 1 1 , i 1 . ' I 1 M ' 1 ' ' ' ' ' ' 1 ! • 1 1 1 1 i 1 1 ' 1 ' M 1 1 >"
i l l 1 ' T" > ' . I l l I X ' ! 1 1 1 / . 1 ;
I 1 M 1 1 ! / 1 i \ ' ! 1 1 1 \ \ f . : ,
' 1 1 / ' • I 1 ' ' f i \ i j^"^ 1 1 1 - . ^ ' 1 \ l 1 1 1 / ' I I \ ' 1 ^ ' M V • • [ • • • - • / , (
\ \ ' ' ' ••• \ 1 . • ^ 1 ' M : . - • !\ 1 X ' 1 1
\ y ~ • 1 ' 1 A xi rt 1 \ i 1 / M
- \ / j i 111 _ ' i:
r "T
> 1 ! > ! I 1 1 i i 1 ' M I I i l l i l l 1 : i 1 ! { I I ' ! • 1 ' I 4 M : 1 ! , ' 1 ' ! | 1^ ' ^ . ^ 1 1 1 1 '
: 1 \\y' \\ \ \ i l 1 1 y \ 1 1 \ ' M l M j >*"M 1 M \ 1 i l
1 > M 1 1 1 1 \ 1 I I 1 L^ • ' 1 1 ' I I I '
j ^ i • \ ' i -^i ' 1 • i l l
y \ 1 1 ' 1 1 - M l ' : I , 1 \ 1 t I , 1 : ' : M 1 ' t : 1 M M , 1 1 M l W
1 r ' ' ' i \' 1 ' ! M 1 \ M 1 , 1 1 , 1 ^, , .
1 ' ' ' \ 1 1 M l \ 1
! ! I 1 \ i l l \ i
1 I' ! M l" I ^
1 ' '\
' 1 1 ' \ ' L-I I 1 1 \' 1 X 1 1 • 1 \/\ ' 1 . x -1 V / '
\ ' r^ 1
4:_ Jl _ 4 _ I t 1 1 • 1
450
Figure 34.
180 270 0 (degree)
Calculated Blade Forces on a Two-Bladed Rotor (C/R = 0.150, U /U = 2.5, NR = 4, Re = 40,000)
2.0
1.0
0.0
-1.0 -
113
20.0
10.0
n 0
-10.0
-20.0
0
Figure 35
90 180
0 (degree)
270
- L i - I T ... 1 I M 1 M i 1 •• - r " T T ^ j H — r
t r q i _ _L - 1 ! i !i 1 1 u ' J4Z-,+^-4: _ L _ ! _ 4 _ _ u — L _ L j . _ | . 1 1 , ! ' 1 I y^ • , j • 1 -r-H 1—1—r-t-M—^^T-J—[^t-^—U-l M—U-/-I-; ^ 4- j
1 1—H—' t—i—'—1 ' ' ^ n — 1 \\ 'y\—rr~H—n— H-r • —H r . 1 1 —+7^ - - i - ^ i i j L U - _ ^ ^ ^
\ i 1 . , , * . , . . - . , ' • / ' W 1 1 1 1 1 i
\-- - ; 1 / i 1 i 1 ' ' 1 I- ' ' / 1 1 ' ' \ 1 1 1 1 1 ' ' M 1 ' • ' -• '•• • t ' \ M ; i 1
1 / 4 i l ' i 1 / i i 1 '
-J- 4 - I i—j- , "i^M ' . 1 1 / 1 1 l \ ' ! I 1 i \ ' ! ! / ' ' ' ' V / 1 \ \ t 1 f 1 1 1 \
i / ' 1 1 ! \ • ' ' i ' / ' ' \ ' 1
\ /• 1 1 iV ' ' ! \ l \ V - • - • • • 1 4 ' 1 \ 1 i
L / i i M 1 \ M l / - M ! \ 1 \ , , 1 ,/, 4 ' 1 i V 1 \ 11 y i l 1 1 \ i
. ,, , 1 , 1 1 , / ( V V 1 v' M I | \ \ 1 / 1 ' 4- 111
["/ 1 • ; 1 1 1 M M V ' ' / ! M 1 ' 1 V' • • \ 1 / 1 1 1 \
\l...4:±4±:4.ii--:+:.::: +:ii.s±:ii V r-' ' ' ' ' M r——
:;;^i+:::___^-^44—-n ^-+h-THiiiiii:+iiiiixt:iiiiiiiiiiiiii^^==±i:iKi
360 450
Calculated Blade Forces on a Two-Bladed Rotor (C/R = 0.150, Uj/U^ = 5.0, NR = 4, Re = 40,000)
114
1.0
0
-1.0 -
20.0
0
n
-20.0
-40.0-
: 1 1
- ^ I -1 1
I •
' 1 1 1 1
V • 1
\ ' ' \ ' '
\ " V 1
\ rt 1 \ 1 \
[ I A l \
I 1 • 1 N
U-. 1 1 1 1 1 ;
1 ' 1 1 I 1
' ! f- ' T r
. \ ' ' •>
'• • • i l l ' ' - M 1 M t 1 : 1 1 1 1 1
M : . j 1 1 • 1 r 1 i , 1 1 1 M 1 , ; • ! i 1
1 - 1 ; 1 • • M M M ' • i
, = ' 1 ' • ' 1 I • 1 I I I I
M M I I
! / 1 ' /
M . 1 / 1 ! /
1 M r ; T • / 1 1 /
i' > - I . . . . , / . \ ' * J
rt ' / 1 V 1 1 / I • ^ ^ ^ - ' ^ l • ' X \ . 1 M l 1 ' ' M J 1 t ; ' : 1 M
- i - j - r ' 111 1
! M 1 • ' M l M M
•' \ • M ! 1 M l 1 t I I I 1 1 '
I I ' t. i . . 1 . - -
/ I N L X xJ>=^ i f ' M 1 1
1 / M ' /
! /i 1 1 1 V^ M i 1 1 j \ t i l l 1 4
/ I I I M l ' I I I 1 1
1 1 >
1 1 I 1 1
i I '
f ' 1 1 1
: It IT: I T -i
1 ' ' 1 i . I I 1 1 . I I 1
' ' ' 1 M i ; M l
1 I 1 1 1 i 1 i
1 1 1 1 1 1 1
,
' ^ J M M > "^" *' T;
M M
1 1 1 ' 1
1 ' i
1
; 1 ! 1
•• • M I 1
1 ;
1 I
!
I I , : I ' M I . I ,
1 • ' 1 1 1 i • • . I ,
, , 1
• 1 1 1 - t 1
i ^ «. '
/ r \ • r M I \
M I I I i M -Ml '
' \ ^r\ • \ / ^
i /• 1 1 - / ' 1 . 1 1
• 1 1 1 • \
• 1 '
4 I : i 1
1
1 ' ' 1 ' ' M M ' ' M 1 i | 1
' ' M • • \ \
M M \ M l ' ' 1 M l , : , ' 1 I I
1 1 i i
' 1 1 1 1 1 1 1 - • • f t r -
( 1
I 1 ' 1 1 ! ! 1 '
\ M l \ M \ 1 1
11 7 1
1
liii i f t ittiti :TI . •A ^ •JX -
4I *
1
1 \ r
1 1
1
1 T 1 1
0
Figure 36
90 180 270 360 450
0 (degree)
Calculated Blade Forces on a Two-Bladed Rotor (C/R = 0.150, U^/U^ = 7.5, NR = 4, Re = 40,000)
115
0.5 -
0.0 -
-0.5 -
10.0 -
5.0 -
n 0
-5.0
-10.0 -
1
• ^
1 1 , 1
' ' 1 1 1 1 .
; 1 ,
• 1 1
1 • • •
1 1 i . M . .
i • 1 1 1 4 1 1 1 :
r M 1 1
I ' l l 1 MM I M 1 I ' i ' 1 1 1
M ' 1 \ l 1
1 t 1
\ '
\ " •
\ ^ I ,- \-
* 1
*
•
— \
4—h-.-
' M • ' M 1 1 M 1 • i 1 ;
• : 1 1 1 : : M . : . 1 •
M . • 1 1 ' •
: . 1 ! ' •• 1
M M M l '
1 i 1 1 I j
• . 1 ;
i i 1 •
< M ' 1 • ! . M l . . : 1
i 4 1 • \ \ •
M M 1 M 1
• ' • ' .
1 ' X
I . y • X I
\ / ' '.
^ ' 1 • •
\ •- I 1 1 1
• M j l 1 ' • • ' i ' '
' 1 ' • ' !
1 i 1 ' • - i ! 1 M 1
• • ' ' ' ' '
M ' • • • ' '
'.:. ; ., y • M • i 1 y ' ; ' ; ' y • 1 4 [y\
• M ' -^, i . ; ' j y M t
1 ' > 1 1 1 / 1 ' ' ' / M l , / 1 M M ! ' '
M ' ' J\ 1 i M M / I t . M l / 1 M • >- 1 i l l . . ' "^r^ 1 , M ! • ' ! / 1 ' . i X ] _ i X I 1
X, 1 1 1 1 • • • • i i l l ! • . M 1 1
M l 1 M I. i . . 1 ' ' • f t • M, I' M M ' . • t ' ,_, . (
1 '.—r- ^ —
.JM—J r-----i 4 • 1 • 1 M 1 • M l"| 1 J J i _ l
, 1 1_
1 M 1 i i '
I 1 1 1 1 1 M 1 I • M 1 1 ! 1 1 1 1 M l M x > 1 1 M 1 1 i > - ^ 4 ^ \ i 1 ixT 1 n ,\ 1 i
• 1 , 4 i\ M ' i l l 11 P M l 1 i 1 1 1 1 i ' 1 1 1 \ 1 I I I 1 I ? t M I I
1 i M 1 1 ^^4^ 4^ 4^4^ 1 1 ' j 1 M i l 1 i M i l ' ' I I M \ 1 M IV 1 i 1 M T 1 1 ' 1 J
M ' i 1 \ t i l l \
' 1 i i t l1 1 1 \ ! 1
' ' \ r ' 1 ; ; \ y 1 W / I 1 \ ' ^\ ' \ / . • M Y > M M r 1 1 ^ + 4 ^ ' IJi lLiJ.l. l i lJjUl -
• 1
0 90 180 270
0 (degree)
360 450
Figure 37. Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, U-p/U = 2.5, NR = 4, Re = 40,000)
116
1.0
0
-1.0
l i J I I j l L L 1 *ii.-|J-') ! 1
_ 1 _ _ / _ "J , . . . , , . 1 , , J 4 ' ••f"'- ~ —^ r^
7 M V ' 1 1 ' i / 1
4T--^--J/ -M-- l - - - \ i H H + i - i - - ^ H t-- / ^-r-:t •-T- -^ - hl^ 'T' 1 ' / •' 4- F-1- - \ 4 )- ^ J 1_ /|_
-4^1 - -i- ^1 /1^J^\ 'J\ ^rr \ y \\ i j' ; ' ' _ V / n" Ky n i ^ i\ ' / — _U _i_ V--'^! 1 ' 1 NL J \ M / I . . . . . 1 1 M 11 M 1 i i M 1 1 i \ I A ! V ' ^ \ 1 } 1 ; ! ' ! ^ M 1 ' ! '^'
V ' / J ." .... M 1 i M • ; 1 1 "^-jy 1 1 r^ r 1 M M 1 1 ; t y
A '. 1 ! 1 M 1 ' 1 1 1 1 • - I i 1 1 M ' ' 1 i 1 1
1 1 i 1 ! 1 1 M ! 1 1 1
n
10.0
-10.0-
-20.0-
Figure 38.
0 (degree)
Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, Uy/U^ = 5.0, NR = 4, Re = 40,000)
X
4.0
2.0
0
-2.0
117
J
n
40.0
0
-40.0
0 90 180 270
0 (degree)
360 450
Figure 39. Calculated Blade Forces on a Three-Bladed Rotor (C/R = 0.150, U^/U_ = 7.5, NR = 4, Re = 40,000)
oo
118
Figure 40. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, U^/U^ = 2.5, Re = 40,000)
Figure 41. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, U^/U^ = 5.0, Re = 40,000)
120
^trtlt ! ! I ' ' I ' I 1111 i i 11 n i i • I
TTTTM,,, -H- I I '
I Ii n . . i i M i ; ' ; i i | i i M -^^ rttttTTTTtf-TO I i! I!!:! 11 i: I c r c
-,
1 y
X /
I
'
^ ' ^ •
' • 1
1 _ \
• • , r
—u
1 • ' • I
r —
T 1
.
v!^— ^ x
• \
1 - >
r
y
• ' 1 .
h+T+j-
' i I
• , "
i
I ;
• » •
1 • 1 1
H-H-H I I ' ' I
1 ' 1
' 1 1 1
l i t .
' . ' • 1 . 1
- T — TT
- T T -
-H T-r
4 f—
4
T -r
h i j i l l • '
r t '
J 1 1 1 1 '
^ ; , | , , ; !
1 ; M ! 1 1 ' t '
' I ' l l ' ' ' T [ 1 • 1 1 ( 1 t - 1 i 1 '
tew
44|+
-4
ri-r
hrttt
' ' I I I
T t ^ 1
4_* *
T
4-
t
TT
4,1
t t t t t t M M I ' I I ' •
—»-T±
t '
p+
\ i -
T t t
t t :
4 "
t t t t i 1
1
T--(-f-j 1
n
frrtt ' M l 1 1 ' m 1
44^ "Ht" 1 t j !
4
t^ t t f f i 1: r t
^
m\^-' ' "
r i i i i M ! 1 1 ! 1 I I
1 ' '
44 44 11 1 1
r4 -4
tttt "t 7" r T
t T t
1 X T 1 1 I '
nf
+tt+ 1 1 1 1 I 1 ' 1 T 1 t * T
1 4-4-4-4-T T
4 .-X
) n 11 J J t I t I T 1 t 1 T M T T
.X - I l -•4- -r4
T I 1 I ' '
ml
It" -4-' • i t : : V"
-.) H (j j l t 1 T I 1
t t t t T
: : :X ; : x : : : 4 I
— X -J "i j "
• - 4 r '
4W
1 +j--11
11 1 1
r
' i i i i j i i ' i
! ; M f 11 j *
I I 111,
4-Ui-1 -44 4 j ^ ± L ^
r T
'
H-l-M
m
111 I I I
I . I ,
: m i
----X-
"'tT'" 4 t T
Figure 42. Calculated "Streak Line" Development for a One-Bladed Rotor (C/R = 0.150, U^/U^ = 7.5, Re = 40,000)
121
t-
Figure 43. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R = 0.150. U.p/U„ = 2.5, Re = 40,000)
p —
[ 1
i
1 ^ —
1 1
-1 'r-
. 1 •
1 • ' 1
- T - f -
:;44-—i-H-
H4 1 1 . . 1
M M 1-r-t-
H+rf I ' l l
1 • M 1 I ' l l
- IT-
bt4 ^ ^
--4-
k M M
• + r - -
4= Li
.-14-
M 1 -1 M 1 m
1 U l
4: T-l
X - . -- H X -M 1 ' M M
M i l l
-44-
-liH-g:-
m m h Ii II11 H
I if hil-Htt
II 1 lijl 41 1 1 1 I m]
"T '^TrT M
TT-X M i l -H4 ~rr~r i ' 11 -n^r ~H--h 1 M t -|--f
m
m\\\ i||iiii|Wf;
Figure 44. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R = 0.150, U^/U^ = 5.0, Re = 40,000)
123
1
Figure 45. Calculated "Streak Line" Development for a Two-Bladed Rotor (C/R =0.150, U^/U^ = 7.5, Re = 40,000)
124
i_x:
i ^j44;r^tig^4^!.:''i: : i! '4-!!p::x 4 ita
Figure 46. Calculated "Streak Line" Development for a Three-Bladed Rotor (C/R = 0.150, U^/U^ = 2.5, Re = 40,000)
125
i.—
++
4-
i-;-
Tt -tj-t-
-U-L
r r r ±4i
m M i l
-rr M M
Figure 47. Calculated "Streak Line" Development for a Three-Bladed Rotor (C/R = 0.150, U^/U^ = 5.0, Re = 40,000)
126
-i-H
*++4
zrr
Fiaure 48 Calculated "Streak Line" Development for a Three-Bladed • Rotor (C/R = 0.150. U ^ U ^ = 7.5. Re = 40.000)
