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
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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
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
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^
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)
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
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)
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)
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
^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
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
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-''
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
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
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 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
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
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)
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'^
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)
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)