SAND81-7017 Unlimited Release UC-60
A Vortex Model of the Darrieus Turbme: An Analytical and Experimental Study
J. H. Strickland. T. Smith. K. Sun Texas Tech University Lubbock. TX 79409
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SAND8l-70l7 Unlimited Release Printed June 1981
FINAL REPORT A VORTEX MODEL OF THE DARRIEUS TURBINE:
AN ANALYTICAL AND EXPERIMENTAL STUDY
J. H. Strickland T. Smith
K. Sun
Texas Tech University Lubbock, Texas 79409
ABSTRACT
Improvements in a vortex/liftin9, line-based Darrieus wind turbine, aerodynamic performance/loads model are described. These improvements include consideration of dynamic stall, pitching circulation, and added mass. Validation of these calculations was done through water tow tank experiments. Certain computer run time reduction schemes for the code are discussed.
Prepared for Sandia National Laboratories under Contract No. 13-5602
iii/iv
,
1.0
2.0
TABLE OF CONTENTS
INTRODUCTION . . . . . 1.1 Purpose of Research
1.2 Summary of Previous Work
1.3 Research Objectives
AERODYNAMIC MODEL
2.1 Summary of the Vortex Model
2.2 Dynamic Effects •
2.2.1 Added mass effects
2.2.2 Circulatory effects
2.2.3 Dynamic stall
2.2.4 Computation of unsteady force, moment, and torque coefficients
2.3 Methods for Reducing CPU Time
2.3.1 Frozen lattice point velocities
2.3.2 Fixed wake grid points •.
2.3.3 Continuity considerations
2.3.4 Vortex proximity
3.0 TWO-DIMENSIONAL EXPERIMENT •
3.1 Wake Velocity Profiles
3.1.1 Velocity transducer
3.1.2 Calibration of hot-film probe
3.1.3 Data acquisition and test matrix
3.2 Improved Blade Force Measurements •
3.2.1 Modified procedure to obtain forces
3.2.2 Signal processing
v
Page
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26
28
29
30
32
34
36
39
40
42
4.0 COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS
4.1 Blade Forces
4.2 Wake Structure
4.2.1 Two-dimensional rotor
4.2.2 Three-dimensional rotor
5.0 SUMMARY OF RESULTS
5.1 Summary ..
5.2 Conclusions
5.3 Recommendations
6.0 BIBLIOGRAPHY
7.0 APPENDIX . 7.1 VDART2 Computer
7.2 VDART3 Computer
7.3 Blade Force Data
Code
Code
7.4 Wake Velocity Data
Listing (FWG
Listing (FWG
vi
Version)
Version)
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1.0 INTRODUCTION
A number of aerodynamic performance prediction models have been formu-
lated for the Darrieus turbine in the past. In general, these models are
based either upon simple momentum principles [1,2,3,4] or upon some form
of vortex theory [5,6,7,8,9]. The major disadvantages of the simple momen-
• tum models are associated with their inability to adequately predict
blade loadings and near-wake structure. They do, however, provide
reasonable predictions of the overall rotor performance for situations
where the rotor is not heavily loaded. The vortex model [9], on the other
hand, appears to be capable of adequately predicting blade loading and
wake structure, but previously required a large amount of computing time.
Additional experimental work was needed to supplement that obtained from
the study given by reference [9] to fully validate the vortex model.
1.1 Purpose of Research
This work is an extension of work performed under Sandia contract
1106-4178, "A Three-Dimensional Aerodynamic Vortex Model for the Darrieus
Turbine" [9]. Based on that study, it was ascertained that additional
work should be performed with regard to both analysis and experiment.
The major need in the analysis area was found to be with regard to
the sometimes excessive computer processing times required. For instance,
it was found that computer processing times for the two- and three-
dimensional vortex models were much longer than for their simple momentum
counterparts. In some cases, the computer processing times for the three-
dimensional vortex model were so long that a periodic solution could not
be obtained in a reasonable time (less than one hour on the CDC 6600). #
The excessive computer processing time greatly limited the utility of
thes e computer codes.
1
The major need with regard to experimental work was to obtain quanti
tative data which describes the wake region, in addition to the qualitative
data already obtained from flow visualization studies. This new data is in
the form of velocity profiles taken in the wake region at several streamwise
locations behind the rotor. Another need with regard to experimental work
was to obtain reliable blade force measurements. While normal blade forces
predicted by the vortex model were found to be in good agreement with the
original experimental data, this was not the case for tangential blade
forces. Tangential blade forces predicted by the vortex model were found
to agree rather poorly with the original experimental data. The tangential
force is, in general, an order of magnitude smaller than the normal force.
An uncertainty analysis revealed that with the previous experimental
arrangement very large errors in the experimental data would likely exist.
Therefore, the purpose of this research was to enhance the utility
and credibility of the present vortex model. The successful completion of
this work provides a valuable analytical tool for Darrieus wind turbine
designers.
1.2 Summary of Previous Work
Several aerodynamic performance prediction models have been formulated
for the Darrieus turbine. The models of Templin [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 wel~ with
these models under conditions where the rotor blades are lightly loaded
and the rotor tip to wind speed ratios are not high.
2
..
t
While these models are moderately successful at predicting overall
performance trends, they are inadequate from several standpoints. Accu
rate 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 inaccurate since these models assume a quasi-steady flow
through the rotor, a constant streamwise velocity as a function of stream
wise position in the vicinity of the rotor, and that the flow velocities
normal to the freestream direction are zero. There has been some doubt
in the past as to whether or not meaningful information concerning the
near-wake structure of the rotor could be obtained from these 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 environ
mental impact of large-scale rotors on downstream areas. Alteration of
the simple momentum models to alleviate all of the listed objections
presently appears to be unlikely.
Another class of prediction schemes are the "vortex models" which do
not have the disadvantages associated with the simple momentum models.
Several of these vortex models for vertical axis machines have been
developed in the past. Models which typify previously developed vertical
axis vortex models are those due to Fanucci [5], Larsen [6], Wilson [7],
Holmes [8], and Strickland [10]. The vortex models of Fanucci [5], Wilson
[7], and Holmes [8] are strictly two-dimensional models. The vortex model
of Larsen [6] is not strictly two-dimensional if the vortices trailing
from the rotor blade tips are considered. Both Fanucci [5] and Holmes [8]
assumed that rotor blades were always at angles of attack sufficiently
small such that aerodynamic stall was not encountered. The giromill [6,7]
3
has articulating blades which operate at angles of attack that are less
than the stall threshold levels. The vortex model of Strickland [10] is
formulated for three-dimensional rotors and considers aerodynamic stall.
Experimental work with regard to overall rotor performance has been
conducted by several groups. Notable among these works are those con
ducted by Sandia Laboratories on small rotors placed in a wind tunnel [II]
and large rotors operating in the natural wind environment [12,13]. The
experimental work reported in [9] and summarized in [10] produced results
with regard to details of the flow structure near the rotor, as well as
instantaneous blade force measurements. Streak lines produced by fluid
particles leaving the trailing edge of two-dimensional rotor blades were
successfully visualized and instantaneous normal blade forces were success
fully measured.
1.3 Research Objectives
The major research objectives for this work were as follows:
* The computer processing times required by the original vortex
model were to be reduced by an order of magnitude.
* Spanwis e velocity profiles \,ere to be taken in the wake of the
two-dimensional experimental rotor.
;, Satisfactory measurement of the tangential blade forces on the
two-dimensinal experimental rotor was to be made.
* A user's guide for the two- and three-dimensional vortex model
computer codes was to be written.
2.0 AERODYNAMI C liODEL
The basic vortex model developed prior to this work due to Strickland
[9,10] will be described only briefly. The reader is referred to the
4
•
•
indicated references for more details. The original work was based upon
quasi-steady aerodynamics which gives rise to some error in predicting
blade forces and performance. Dynamic effects and their inclusion into
the original model are discussed in section 2.2. Methods for reducing CPU
time are discussed in section 2.3.
2.1 Summary of the Vortex Model
The general analytical approach requires that the rotor blades be
divided into a number of segments along their span. The production, con
vection, and interaction of vortex systems springing from the individual
blade elements are modeled and used to predict the "induced velocity" or
"perturbation velocity" at various points in the flow field. The induced
or perturbation velocity at a point is simply the velocity which is super
imposed on the undisturbed wind stream by the wind machine. Having
obtained the induced velocities, the lift and drag of the blade segment
can be obtained using airfoil section data.
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 "substitution" vortex filament
[14] or a "lifting line" [15]. The use of a single line vortex to repre
sent an airfoil segment is a simplication 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 adequately at distances greater than about one chord length
from the airfoil [14]. The strengths of the bound vortex and each trailing
tip vortex are equal as a consequence of the Helmholtz theorems of vorti
city [16]. As indicated in Figure 1, the strengths of the shed vortex
5
Figure 1. Vortex System for a Single Blade Element
6
•
,
•
systems 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 [16].
Each portion of a vortex filament making up the shed vortex system
is convected in the flow at the local fluid velocity. Therefore, the
vortex filament will be distorted in a number of ways as it moves through
the fluid. As a first approximation, it can be assumed that the vortex
filament may stretch, translate, and rotate as a function of time.
The fluid ve10ci ty 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.
Referring to the case shown in Figure 2, for a straight vortex filament
of strength r and length ~ the induced velocity V at a point p not on p
the filament is given by [15]
-+ v
p
-+ -+ -+
(1)
where the unit vector e is in the direction of r x~. It should be noted
that if point p should happen to lie on the vortex filament, equation (1)
• -+ yields indeterminate results, Slnce e cannot be defined and the magni-
-+ tude of v is infinite. The velocity induced by a straight vortex filament
p
on itself is, in fact, equal to zero [17].
In order to allow closure of the vortex model, a relationship between
the bound vortex strength and the velocity induced at a blade segment must
be obtained. A relationship between the lift, L, per unit span on a blade
7
vortex filament
Fi gure 2. d at a Point by 1 . ty Induce Ve oc~ Vortex Filament
•
8
•
•
segment and the bound vortex strength, fB
, is given by the Kutta-Joukowski
law [16]. The lift also can be formulated in terms of the airfoil section
lift coefficient, C~. Equating these two expressions for lift yields the
required relationship between the bound vortex strength and the induced
velocity at a particular blade segment •
( 2)
Here the blade chord is denoted by C, and DR is the local relative fluid
velocity in the plane of the airfoil section. It should be noted that the
effects of aerodynamic stall are automatically introduced into equation
(2) through the section lift coefficient.
2.2 Dynamic Effects
Unsteady aerodynamic loading of an airfoil can arise due to several
effects. In the absence of aerodynamic stall, these effects can be loosely
categorized as those due to fluid inertia "added mass" and unsteady wake
circulation "circulatory lift and moment." For conditions where boundary
layer separation can occur, a combination of viscous, inertial, and un-
steady wake effects give rise to "dynamic stall."
Added mass effects are normally small for the C/R values commonly
us ed on full-scale Darrieus turbines (i. e., C/R '" 0.07). For the present
experiment where C/R = 0.15, added mass effects are important at the higher
tip to wind speed ratios. These effects were unaccounted for in the
original vortex model and should be added to more accurately predict
aerodynamic loads.
The circulatory lift produced by the unsteady wake has been included
in the vortex model in an approximate way. The effect of downwash and
9
upwash produced by discrete vortices models this effect. The pitching of
the airfoil does require an adjustment in the bound vorticity in order to
satisfy the Kutta condition. For small values of C/R, this adjustment is
negligible. At larger values of C/R, this effect must be considered.
While the original vortex model included the effect of steady stall, it
did not include any dynamic stall model. Klimas [18] has included an
approximate dynamic stall model in the Sandia version of VDART. It appears
that dynamic stall may be an important phenomenon even in turbines with
small values of C/R. Neglect of the dynamic stall effect makes small
differences in the magnitude of the rotor power coefficient, but large
differences in plots of power output versus windspeed.
2.2.1 Added mass effects
The added mass effect will be obtained by utilizing an analytical
technique found in Milne-Thomson [19]. For simplicity, it is assumed that
the airfoil can be approximated as a flat plate with a coordinate system
as shown in Figure 3. The x-y axes are fixed in the plate. The origin
of this coordinate system (body coordinates) moves with a relative velocity,
UR' with respect to an inertial reference frame fixed in the fluid at
"infinity." The body reference frame rotates with an angular velocity of
w with respect to the inertial reference frame.
The complex potential for the flat plate undergoing the prescribed
motion can be written in the following form with respect to the inertial
reference frame:
if a l F(Z) = -- log Z + Z + 2'TT
10
(3)
•
•
y
b b
c
Figure 3 Flat Plate Airfoil Motion
11
In particular from article 9.65 of reference [19], the complex potential
for a flat plate rotating and translating with respect to an inertial
reference frame is given by
'r C -1 C2
_2 F(s) + ~TI log s + (i 2 UR sin a') s + i 16 Ws
where
C 1 Z = I; (s + -;:) .
For large values of Z, the complex potential can be written as
F(Z)
Following the procedure in article 9.53 of reference [19], the complex
force on the flat plate can be \vri tten as
x + iY da
l iprcU + iV) - 2TIp(iwal +~).
(4)
(5)
( 6)
(7)
Here X and Yare the forces along the body coordinate axes, and U and V
are the velocities of the origin of the body coordinate system with respect
to the inertial system. The velocities, U and V, represent instantaneous
velocities along the x and y axis, respectively. By comparing equations
3 and 4, it is seen that the value of al
required in equation 7 is given
by
(8)
Inserting this expression into equation 7 yields
X + iY ipr (U + iV) 4 ( C) 2 ( . dV) TIp 7; wV - l dt • (9)
12
•
The added mass effects are seen to be those represented by the "non-circu-
latory" part of the solution. The forces aris ing due to the added mass
effects can be given as
X am
y am
C2 dV - TIP 4 d t .
(10)
The pitching moment about the mid-chord can be obtained in a similar
fashion by using the development in article 9.52 of reference [19]. The
results are given as
M (ll)
where the positive sense is counterclockwise.
The moment about the mid-chord does not depend upon the circulation
strength and, thus, might be thought of as an added mass effect. Tradi-
tional1y, the moment coefficient is based on the moment at the quarter-
chord with the lift force assumed to act through that point. Adopting
this tradition yields the following formulation for the moment at the
quarter-chord:
C2
f y u V 4 - TIp 4" • (12)
Since, for steady flow over a flat plate airfoil, the right-hand side of
equation 12 is zero, this term in the unsteady case can be viewed as an
added mass effect.
2.2.2 Circulatory effects
The pitching of the airfoil gives rise to changes in the bound
13
circulation strength as a result of required adjustments to satisfy the
Kutta condition and as a result of the shed vortex sheet. As mentioned
previously, the shed vortex sheet is modeled in an approximate way in the
VDART model with a series of discrete vortices. It remains then to
satisfy the Kutta condition for the pitching airfoil.
The Kutta condition will be satisfied if a stagnation point is forced
to coincide with the sharp trailing edge of the airfoil. This can most
easily be accomplished by considering the flow in the circle plane as given
by equation 4. The complex velocity in the circle plane can be obtained
by taking the derivative of the complex potential.
dF (L;) di';
if -1 C 2 C2 _3 2n i'; - i 2" DR sin a.' r,- - i 8' Wi';
From equation 5, it can be seen that for the trailing edge condition,
(Z = C/2), <;; = 1. It should be noted that this choice rather than
Z = -C/2 for the trailing edge condition limits the range of aT to
(13)
n/2 < aT < 3n/2. Since the stagnation condition requires that the complex
velocity be equal to zero, then the bound circulation strength must be
equal to
f (14)
The last term represents the additional circulation due to the pitching
motion. It should be noted that this same result is obtained if one
calculates the circulation based on the motion of the three-quarter chord
point.
From equation 9, the forces due to circulation are given by
14
2.2.3 Dynamic stall
2 C2
- TIpCV - TIp -- wV 4
C2 TIpCUV + TIp ---;;- wU.
(15)
Dynamic stall is a highly complex problem and will be handled in an
approximate manner, using a modified Boeing-Vertol method [20]. The ob-
served hysteresis in lift and moment coefficients is obtained in this
method by utilizing two-dimensional static wind tunnel data, along with
an empirically derived stall delay representation.
The method utilizes a modified blade angle of attack for use in
entering two-dimensional force coefficient data. The modified angle of
at tack a is given by m
a m
(16)
where ab
is the effective blade angle of attack, y and Kl are empirical
constants, and S. is the sign of d. This modified angle of attack is used CI.
to calculate force coefficients in the following manner:
CL (a a b
) CL (am) - abO m
CM CM(am) (17)
Cd cd(am) .
Here abO is the effective blade angle of attack for zero lift and CL, CM'
and Cd are the coefficients of lift, moment, and drag, respectively.
For low Mach numbers and for airfoil thickness to chord ratios greater
than 0.1, the value of y for lift stall (YL) and moment stall (YM
) are
15
given by
1.4 - 6(.06 - tic) (18)
YM = 1.0 - 2.5(.06 - tic)
where t is the maximum airfoil thickness. Therefore, the value of a used m
to calculate the lift coefficient is obtained using YL
, whereas am for the
moment and drag coefficients is based on Ym
• The empirical constant, Kl
, can
be obtained from
(19)
Therefore, Kl is equal to 1.0 for a positive and 0.5 for a negative.
This formulation is applied when the angle of attack, aB
, is greater
than the static stall angle or when the angle of attack is decreasing after
having been above the stall angle. The stall model is turned off when the
angle of attack is below the stall angle and increasing. Dynamic effects
are accounted for, as outlined in the previous two sections, when the stall
model is turned off.
2.2.4 Computation of unsteady force, moment, and torque coefficients
Use of the material developed in the preceding sections allows a
rational method for computing force and moment coefficients, using airfoil
section data corrected for dynamic effects. The idealized potential flow
models can be used to suggest how the data should be corrected for dynamic
effects.
The conventional nomenclature and positive sense for forces, angles,
velocities, etc., are shown in Figure 4 for an airfoil blade on a Darrieus
turbine. It should be noted that the attachment point, ~, is extremely
16
t
F' t
c/4
+
c
Figure 4 Blade Forces and Mounting Geometry
17
important ~,hen considering dynamic effects, since it introduces what some
have referred to as "virtual angle of incidence" [21]. The forces and
moment per unit length of blade can be defined in terms of thrust force,
normal force, and moment coefficients as follows:
F' t
F' n
(20)
For the idealized flat plate case neglecting terms containing dV/dt,
the three coefficients can be ,vritten as
C 2'IT U(V + .~ w) (2'IT sin a3 / 4 ) cos a3 / 4 (21)
n U2 4 R
C 'IT U~ W
m U2 8 R
where the subscripts on (J, indicate the fraction of the chord at which the
angles are taken.
From equation 21, it can be deduced that if one calculates the thrust
coefficient, Ct
, using the angle of attack at mid-chord, dynamic effects
are included. The normal coefficient, C , should be calculated based on n
the three-quarter chord angle of attack, since the velocities indicated in
equation 21 occur at the three-quarter chord point. In order to clarify
this last statement, it should be noted that the velocity, V + Cw/4, is
the relative velocity normal to the airfoil at the three-quarter chord
point and that the velocity, U, is the relative velocity tangent to the
airfoil at all chord points, as well as the three-quarter chord point.
18
t
An order of magnitude analysis shows that the term containing dV/dt in
equation 10 contributes little to the total normal force and, thus, can
be justifiably neglected in constructing equation 21. It also should be
noted that the moment coefficient given in equation 21 is on the order of
lIC/8R and is, therefore, reasonably small for normal C/R values.
In conclusion, the following rules were followed in the calculation
of the force and moment coefficients:
1) The tangential force coefficient was calculated using the mid-
chord angle of attack to obtain airfoil section data.
2) The normal force coefficient was calculated using the three-
quarter chord angle of attack to obtain airfoil section data.
3) The moment coefficient about the quarter chord was assumed
to be negligible.
The steady state coefficients are to be evaluated at either the mid-
chord or three-quarter chord angle of attack, which are given to first
order in terms of the angle of attack at the attachment point, a~, as
follows:
+ (10 -0 UJ a l / 2 at;, 2 U
R
at;, + (l - f,) i.ll
a3 / 4 4 UR .
For cases where the airfoil is stalled, the method of section 2.2.3
is used to calculate CL
, CM' and Cd from which
C n
19
sin
(22)
(23)
The effective blade angle of attack, ((b' used in the method of section
2.2.3 is equal to ((3/4 for calculation of Cn
in equation 23 and a l / 2 for
the calculation of C and C . t m
The torque produced by a unit length of blade can be given in terms
of a torque coefficient, C , which in turn can be defined in terms of the T
force and moment coefficients as follows:
T'
C T
C + (1 t 4
2.3 METHODS FOR REDUCING CPU TIME
f) C r n
+ .f C r m
Original computer processing times for the Vortex DARrieus Turbine
(VDART) models were moderately long. The CPU time in minutes for the
original model on the CDC 7600 computer were given approximately by
t NT 3
(100) , VDART3
t
(24)
(25)
where NE is the number of blade element ends, NB is the number of blades,
and NT is the total number of time steps. The VDART2 and 3 denote two-
and three-dimensional models. A major portion of the CPU time was required
for calculating the velocities of wake vortex lattice points. The sub-
routine FIVEL, \vhich calculates induced velocities, had already been
written in an efficient format and, thus, reduction in computational time
was obtained by reducing the number of times the subroutine FIVEL is called.
As an example of the number of times which FIVEL might be called, consider
a two-bladed rotor with ten elements. If 20 time increments per revolution
20
are used and if the rotor rotates through seven revolutions, then FIVEL
will be called 666 x 106
times. For a two-bladed rotor in two dimen-
6 sions, this figure will drop to 3.66 x 10. Several methods for reducing
CPU time are presented in the sections below. Some of these methods have
been tried and found to be successful while others have not.
2.3.1 Frozen lattice point velocities
One approach which has been used in the VDART2 and VDART3 programs
is 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 free-
stream 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, the wake velocities
should be updated after approximately every UT/Uoo
time step. Therefore,
the CPU time given in equation 25 can be reduced by a factor of UT/UOO
•
Obviously, a certain amount of risk 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 low to moderate tip to wind speed ratios. One case
was run at a very high tip to wind speed ratio (UT/Uoo
= 20), and numerical
instabilities were seen to result. This feature should be used with
caution.
2.3.2 Fixed wake grid points
A method which has been used on the VDART2 and VDART3 programs to
reduce CPU time utilizes a number of grid points arranged in the flow field
as shown in Figure 5. Any number of grid points can be placed anywhere
in the flow; however, the arrangement shown in Figure 5 has been used most
21
5 4 3 2 1
10 I v: 8~ ~7 6
I \ 15 14 13 12 1 1
20 \ 19 18 17 I 1 6
~ V
25 2~ 22 2 1
30 29 28 27 2 6
35 34 ~3 32 3 1
40 39 38 37 3 6
45 44 43 42 4 1 .
50 49 ~8 47 4 6
Figure 5. Arrangement of Grid Points in the Wake
22
extensively. Perturbation velocities are calculated at each of these grid
points instead of at the vortex lattice points in the wake. The velocities
at the vortex lattice points are then obtained by linear interpolation of
the velocities at tile 50 grid points (250 for VDART3). Potentially, this
method can reduce the CPU time by a factor of (NE)NT/NG where NG is the
number of fixed grid points. In reality, the interpolation procedure reduces
this factor to approximately (NE)NT/2NG. For cases where a vortex lattice
point happens to fall outside the grid pattern, its velocity is calculated
in the usual way. It also should be pointed out that the lattice point
velocities are calculated in the usual way until 50 (250 for VDART3) vortex
lattice paints are shed into the wake.
A typical comparison of blade forces obtained from VDART2 by the
standard vortex model method (SVM) and the fixed wake grid method (FWG)
is shown in Figure 6. A similar plot using VDART3 is shown in Figure 7.
+ + The non-dimensional tangential and normal forces F and F are defined
t n
in terms of the tangential and normal forces per unit blade length, F~ and
F'; the fluid density, p; the airfoil chord length, C; and the freestream n
velocity, Uoo
' by
F+ F'
t t 1/2pCU:
(26)
F+ F'
n n
1/2PCU2
00
As can be seen from these figures, the agreement between the two methods
is reasonably good. The largest variations appear in the downstream
portion of the blade traverse. For these particular examples, the number
of time increments per revolution of the rotor NTI was chosen to be 24
for the VDART2 case and 20 for the VDART3 case. For 2040° of rotation,
23
1.0
F+ 0.0 t
-1.0
10
o
-10
-20
.0 90 180 270
(8 - 1080) degrees
o 90 180 270
(8 - 1080) degrees
360
f d !I ;.
450
Figure 6. Comparison of Calculated Blade Loads (VDART2) ( - SVt~ model, --- FWG model, Re = 40,000, NB = 2, C/R = 0.15, UT/Uoo = 5.0, NTI = 24)
24
3.0
2.0
1.0
0.0
o
20
10
o
-10
-20
-30
o
90 180 270 360 450
(8-720) degrees
90 180 270 360 450
(8-720) degrees
Figure 7 Comparison of Calculated Blade Loads (VDART3, mid plane, -- SVM model, --- FWG model, Re = 0.3 X 10 6
, NB = 2, C/R = 0.135, UT/Uoo
= 6.0, NT! = 20) 25
the SVM method was found to require approximately three times more CPU
time than the FWG method.
Power coefficients were calculated for a two-dimensional one-, two-,
and three-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 reasonably
good except for the three-bladed rotor operating at a tip to wind speed
ratio of 7.5. The results are depicted in Figure 8. The solid and
dashed lines are visual aids only with the symbols denoting the actual
information obtained from the computer runs. In each case, the FWG
method yields slightly lower values of C. The difference between the p
two methods is accentuated at higher tip to wind speed ratios and higher
solidities •
In conclusion, this technique holds especially good promise, since
it reduces the time dependence on the number of time steps from a cubic
function to a square function. A variation of this method would allow
the fixed grid point system to expand into the downstream region as the
wake expands into the downstream area. It is planned to continue to in-
vestigate the advantages and disadvantages of the FWG method by running
additional cases.
2.3.3 Continuity considerations
This technique was intended to be used in conjunction with the FWG
method given in the previous section. Basically, this method was intended
to take advantage of the continuity equation given by
dU + dV + dW dX dy d Z °
to allow calculation of one of the velocity components in terms of the
26
(27)
.3
.2
C P
.1
0
-.1
.3
C .2 P
. 1
0
-.1
.3
.2
C . 1 P
0
-.1
-.2
NB
UT/Uoo
UT/Uoo
NB = 3
UT/Uoo
Figure 8. Comparison of Calculated Power Coefficients (VDART2)
(0 SVrvl model, A FWG model, sol id 1 ine for visual aid, Re = 40,000, C/R = 0.15)
27
others. For example, in the two-dimensional case the lateral velocity, w,
can be obtained from a difference equation of the form
/':;W
6z 6u 6x
(28)
The values of u were to be calculated as usual at the fixed wake grid points,
based on the cumulative perturbation velocities from vortices in the wake.
The values of >'1 along the wake centerline were also calculated in the same
fashion. All other values of w l.Jere to be calculated efficiently using
equation 28.
It was originally expected that the CPU time would be reduced by one-
half for the two-dimensional case and by one-third for the three-dimensional
case. The expected time savings did not materialize and, in fact, this
feature appeared to be a time consumer. The reason is apparent when one
examines the calculation procedure in the subroutine FIVEL, which calculates
the induced velocities arising from an individual vortex filament. In this
procedure, several parameters are calculated which are common to the calcu-
lation of each of the components of velocity. Each component of velocity
is then calculated using a single multiplication of parameters common to
the calculation of the other velocities. In other words, the calculation
of the velocity component, which was to be obtained from continuity con-
siderations, requires only a single multiplication in the original scheme.
Therefore, this technique was considered to be unworthy of additional
consideration.
2.3.4 Vortex proximity
A technique of combining vortices whose centers pass in close proximity
to each other could be quite useful in the two-dimensional case. The
28
•
spatial coincidence of vortex filaments in the three-dimensional case is,
perhaps, too rare an occurrence to consider. The occasional coincidence
of vortex filaments ends might, on the other hand, justify their combina
tion. Logic to "skip" the absorbed vortex when calculating perturbation
velocities should be carefully developed to avoid loss of time due to
excessi ve use of logic "if" statements.
Conversely, vortices whos e centers are far mvay 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.
Implementation of this time-saving feature did not occur as part of
this work, but has been used in one form by D.E. Berg [25] of Sandia
Laboratories. In his use of the method, the vortices downstream of a
certain streamwise position were simply neglected, since these vortices
had very little effect in the vicinity of the rotor. This scheme was used
in conjunction with the fixed wake grid technique and was easily applied
by neglecting vortices downstream of the last set of grid points. The
reported results were good.
3.0 TWO-DIMENSIONAL EXPERIMENT
In order to check the validity of the analytical model with regard
to instantaneous blade loading and wake structure, an experiment was set
up as described in reference [9]. Only a brief description of the experi
mental arrangement will be given herein, and the reader is referred to
the indicated report for more details.
In general, a straight~bladed rotor with one, two, or three blades
was built and operated in a water tow tank with a depth of 1.25 meters,
29
a ,~idth of 5 meters, and a length of 10 meters. The rotor blades extended
to within 15 centimeters of the tank bottom. This simple rotor appears to
be adequate for validating the major features of the analytical model. The
use of water as a working fluid greatly facilitates the ability to visual
ize the flow structure while working at appropriate blade Reynolds numbers.
In addition, blade forces are more easily measured. In order to use avail
able section data for the NACA 0012 airfoil, the rotor blades are required
to operate at a Reynolds number of 40,000 or greater. An airfoil chord
length of 9.14 em and a rotor tip speed of 45.7 em/sec were chosen to yield
a blade Reynolds number of 40,000. Three towing speeds of 18.3 em/sec,
9.1 em/sec, and 6.1 em/sec were chosen to yield tip to wind speed ratios
of 2.5, 5.0, and 7.5, respectively. The rotor diameter was chosen to be
1.22 meters, thus giving solidity values (NC/R) of 0.15, 0.30, and 0.45
for one-, two-, and three-bladed rotors, respectively.
Two types of measurements were to be made to obtain quantitative data
useful in further evaluation of the analytical model. The first of these
was the measurement of velocity profiles in the rotor wake using a hot
wire anemometer system. The second measurement involved a second attempt
to accurately measure the tangential blade forces. Methodology associated
with making these measurements is given in the following sections.
3.1 Wake Velocity Profiles
Velocity profiles were obtained in the rotor wake by towing a velocity
transducer along behind the rotor as depicted in Figure 9. This was re
peated a number of times with the probe being placed at various spanwise
locations. The distance behind the rotor at which measurements were made
is fully adjustable. Probe cables were suspended from the laboratory
30
•
I I I
I
\
/ /
".. /'
(rotor
" , \
\ \
wake rake f carriage
\ I
/ / \
" ,/ "..
o
CvelOCity transducer
Figure 9. Schematic of General Arrangement for Obtaining Wake Velocity Profiles
31
ceiling using surgical tubing and, thus, allowed the anemometer instrumen-
tation to be placed in a fixed laboratory reference frame.
3.1.1 Velocity transducer
The total velocity was obtained by using a TSI model l239W Quartz
coated hot film probe which has a hemispherical sensing element mounted
on the end of the probe support. When the probe support axis is vertical,
the sensor measures the total velocity in the two-dimensional plane of
interest. The probe was used with a DISA 55M hot-wire anemometer system.
This probe was selected due to its rugged nature and its applicability for
use in slightly contaminated water flows.
The direction of the flow was measured using a small vane mounted to
a precision ultra low torque potentiometer, as shown in Figure 10. This
potentiometer is a BOw}~R model PS 091-105 with a maximum starting and
running torque of 0.005 oz. in. In order to ascertain the sensitivity of
the direction indicator, it is appropriate to calculate the angle of attack
of the vane with respect to the oncoming field which will produce the
starting torque of the potentiometer. In terms of the given parameters,
the torque can be expressed by
T ~ DCtCL
(a - 3/4 C) u~ (29)
If the torque is maximized with respect to C, it is found that C 2a/3.
Assuming that the lift coefficient is given by CL
0.10:, then
-a (30)
By choosing C equal to 1 inch and ~ equal to 5 inches in equation 30, it
can be shown that the flow direction should be measureable to within about
32
u 00
__ "'T" bar
ultra low torque 2 K ~ potentiometer
~~D~
counter
.....-vane
a
Figure 10 Vane Direction Indicator
33
± 0.3 degrees at a towing speed of 18.3 em/sec and about ± 3.0 degrees
at a towing speed of 6.1 em/sec.
This method of measuring flow velocities appears to yield much more
reliable results than the originally proposed method using a cross-wire,
hot-wire anemometer probe. According to both prominent manufactures of
hot-wire equipment (TSI and DISA) , the calibration drift problems assoc
iated with small hot-wires in water make their use impractical. A
calibration run would have been required over the complete range of
velocities and angles normally encountered by the probes prior to each
run lasting more than a few minutes had the hot-wire been used. Using
the large cylindrical hot-film probe and direction indicator required
that the hot-film probe and direction indicator be calibrated over the
appropriate velocity range only occasionally. The calibration was checked
during the course of each run by noting the zero velocity value of the
anemometer voltage output and the towing speed velocity voltage which
occurred prior to the probe's passage into the wake. While there is not
enough information in these two measurements to construct a calibration
curve, correlation of these two data points with previously obtained
calibration curves provides a useful check.
3.1.2 Calibration of hot-film probe
The hot-film probe was calibrated in the probe calibrator shown in
Figure 11. This calibrator was designed and constructed as part of this
project and utilizes water from the towing tank. The water enters the
calibrator through a valve which is used to control the inlet flow rate.
Before flowing over the probe, the water passes through a short section
of honeycomb with screens on the upstream and downstream sides. The
34
calibrator calibration chamber
manometer
orifice plate
Figure 11 Probe Calibrator
35
honeycomb and screens inlet
valve
hydrostatic pressure in the calibrator is measured with a simple glass tube
manometer mounted into the top wall. An orifice plate is used to constrict
the flow so that maximum readings on the manometer are on the order of 25
cm for the particular flow range of interest.
The calibrator itself is calibrated by closing the outlet valve and
noting the fill rate of the calibrator calibration chamber as a function
of the calibrator manometer reading. Curves for two different orifice
plates are shown in Figure 12.
The calibration move for the TS1 l239H probe obtained from the cali-
brator is shown in Figure 13. In addition, data obtained by simply towing
the probe in the tow tank at various speeds is shown in Figure 13. The
agreement between these two sets of data is very good. As can be seen from
this curve, the data can be represented by
(31)
where E is the probe output voltage, V is the measured fluid velocity, and
E is the probe output voltage for no flow. By using a least-squares curve o
fit through the data, the constants were found to be A = 1.888 x 10-6 ftl
2n 2 (sec volts ), n = 2.214, and E
o 2
177 volts .
3.1.3 Data acquisition and test matrix
Data were acquired using the Mechanical Engineering Department HP9835A
desktop computer coupled to a four-channel HP593l3A analog to digital con-
verter and a HP7225A plotter. The system is capable of acquiring analog
signals from an experiment at rates of up to 200 Hz, which was quite
adequate in light of the rotor rotational speed of 0.12 Hz. A real time
clock, coupled with the AID pace rate, provided adequate time monitoring.
36
.8 .4
0- 0.93" orifi ce
'1- 0.54" orifice
.6 .3
V (ft/sec)
.4 .2
.2 .1
o o o 2
Figure 12 Calibrator Calibration Data
37
E2_E 2 o
(volts)2
400
300
200
o calibrator data
CI - tow tank da ta
0.1
VCft/sec)
Figure 13. Calibration Curve for TSI l239W Hot Film Probe (overheat ratio = 1.1)
1.0
The synchronization of the rotor position for various runs was extremely
important. The rotor has a transducer (Waters Mfg. Analyzer APT 55)
mounted on the main shaft, which allowed the rotor angular position to
be monitored and recorded along with whatever other parameter was being
measured. Calibration and input data for each run was stored on magnetic
tape cartridges which are compatible with the HP9835A. Each cartridge
is capable of storing 256 K Bytes of information or about 128 K data
points on 42 files.
The test matrix selected for the wake velocity measurements was a
compromise between several factors, such as spatial resolution, time re
quired to perform the experiment, data manipulation and storage, and
compatibility with previous computer code output format.
More than 260 wake velocity runs consisting of five blade number tip
to wind speed ratio combinations, two streamwise probe locations behind
the rotor, and 13 spanwis e probe locations \vere made. Forty-eigh t data
points per revolution were acquired. The experiment was run for an average
of ten revolutions. Three measurements were made each time data were
acquired (rotor position, velocity, and flow angle). "ith this test matrix,
more than 249,600 pieces of data were collected and placed on eight 256
K Byte data cartridges.
3.2 Improved Blade Force Measurements
The original experimental arrangement has been modified to allow more
accurate measurements of the tangential blade forces to be made. In addi
tion, more accurate measurements of the normal force have been made, although
they were measured in a reasonably successfully manner at an earlier date.
39
3.2.1 Modified procedure to obtain tangential forces
Tangential forces are, in general, on the order of 0.01 lbf
maximum
and are, thus, quite small. Experimental uncertainties for the original
experiment were estimated to be on the order of ± 70 percent. Several
experimental errors ~vere responsible for this large uncertainty. Mechan
ical noise caused by the meshing of gear and sprocket teeth produces
dynamic forces which are the same order of magnitude as the tangential
force. Fortunately, this noise was successfully filtered out of the
signal. The original placement of the strain gage bridges required that
signals from two sets of bridges be summed in order to obtain the tangential
force. This was previously done by hand, which introduced errors asso
ciated with the digitizing process and phase misalignment of the two
signals. Signal levels from the strain gage circuits were on the order
of 50 vV max. These signals were amplified by a factor of 1,000 before
being passed through a set of slip rings. Thus, the signal level through
the slip rings ~vas only about 50 mV, which is felt to be relatively low
considering that the slip rings were not of especially high quality. Some
shifting and drifting of output signals was noted, which was thought to be
caused by variations of the resistance in the slip rings.
Several modifications were made to address the above difficulties. A
strain gage bridge >vas placed on a modified vertical support arm shown in
Figure 14. This allowed the tangential forces to be obtained using a single
bridge and, thus, eliminated the need to sum signals from two bridges.
The strain gages used >vere 350 ohm gages, >vhich allowed the bridge voltage
to be increased from 5 volts to 15 volts with a corresponding factor of
three increase in sensitivity. In addition, the cross section thickness
at the point of measurement was decreased by a factor of two so that the
40
CD Ft strain bridge F gage n
CD F strain gage bridge n
CD M strain gage bridge
Figure 14. Blade Force Measurement
41
bridge output was increased by a factor of four. Therefore, the signal
level was increased by a factor of 12. A new set of silver slip rings was
obtained, which eliminated the oxide buildup problems associated with the
old copper slip rings. The experiment was modified to accept the new slip
ring assembly. The strain gage bridge amplifiers were also upgraded.
3.2.2 Signal processing
Force measurements at each of the three indicated positions were made
using a Whetstone bridge with four active strain gage elements. The signals
produced by the bridge circuits were amplified using CAL EX model 176L ampli
fiers mounted on the rotor arm. A gain of approximately 1000 was used to
amplify the millivolt signals into the 0.1 to 10 volt range. The amplified
signals were passed through slip rings and monitored with the HP9835A data
acquisition system mentioned in section 3.1.3. A KRONITE filter (model 335)
was used to eliminate mechanical noise at approximately 2 Hz from the Ft
signal whose frequency content is approximately 0.1 to 0.2 Hz. The low-pass
filter was set on 0.6 Hz.
Ten blade force runs consisting of five blade number tip to wind speed
ratio combinations and two force measurements (normal and tangential) were
made. Forty-eight data points per revolution were acquired for an average
of ten revolutions per run. Three measurements were made each time data
were acquired (time, force, rotor position). With this test matrix, 14,400
pieces of data were collected which required approximately 28,800 Bytes of
memory on a data cartridge.
4.0 COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS
The major purpose of this section is to present analytical and experi
mental results from the present work. Where applicable, these results will
42
be presented in light of previous analytical models and experimental data.
A comparison of analytical and experimental results pertaining to blade
forces on the two-dimensional rotor will be presented in the first section.
Comparisons for both the two-dimensional rotor and a three-dimensional rotor
will be made with respect to wake velocity profiles in the second section.
4.1 Blade Forces
Typical blade force measurements taken with the modified experimental
setup are shown in Figure 15 and 16. The quality of these data appear to
be much better than those data presented in reference [9]. Repeatability
was found to be very good as evidenced by the very close agreement between
data taken during two different runs. A complete set of data for the blade
force measurements can be found in the appendix (section 7.3).
From preliminary comparisons of these data with VDART2 output, it
became apparent that the dynamic effects discussed in section 2.2 were
significant. At the lowest tip to wind speed ratio (2.5), dynamic stall
was found to be important. At the highest tip to wind speed ratio (7.5),
added mass effects and pitching circulation were found to be important, while
at the moderate tip to wind speed ratio (5.0), both effects played a role.
It was also ascertained that the normal blade force data should be
corrected by subtracting out the centrifugal forces induced in the experi-
ment. This correction can be given approximately by
tbt
i b ;
VT
2 (-) V
00
(32)
where PB/Pf
is the blade density to fluid density ratio, tic is the thick
ness to chord ratio, and -;/~f is the total blade length to the blade
length immersed in the fluid ratio. The numerical coefficient is equal
43
28 r
r 18 r
r +
f'n 8
r -18 1"'+
+ r
-28 r r
-30
-40
8
+ +
+ +
30
+ + + +
+ + + +
+++++
+
,
+
+ +
....
++++ + +
++ +
TSR-7.S
68 ge 128 IS8 180 2113 240 270 300 ROTOR ANGLE-1880'
+ ++
+++ ++
+ +
+ +
,
338 368 398
Figure 15 Normal Force Data From Two-Dimensional
+ +
+ +
++
428 4S8
Experiment (UT/Uoo
7.5, C/R = 0.15, Re = 40,000)
44
4r---------------------------------------------------,
3 t-
TSR-7.5
2 I-+ +
+ +
+ + +
+ + + ++ +
rt 0~----~--------------~---------+-+--~+~T~.+~--------------~+~-; ++ +
-1 r
+ +
-2 rr+
+ +
o 30
+
S0
+ +
..J.. ..J..
+++ +++ +
,
+ + +
+ ++++++ + +++
-'- -'- -'-
+
90 120 150 180 210 240 270 300 330 3S0 390 420 450
ROTOR ANGLE-10B0·
Figure 16 Tangential Force Data From Two-Dimensional Experiment (UT/Uoo = 7.5, C/R = 0.15, Re = 40,000)
45
to twice the airfoil cross sectional area divided by the thickness chord
product. This correction is insignificant at the lower tip to wind speed
+ ratios producing a downward shift in the F curve of only 0.48 at a tip to n
wind speed of 2.5. At a tip to wind speed ratio of 7.5, the shift is equal
to about 4.29.
Results at a tip to wind speed ratio of 7.5 are shown in Figure 17.
At this tip to wind speed ratio, only the dynamic effects discussed in
sections 2.2.1 and 2.2.2 are present; dynamic stall does not occur. As
can be noted from this figure, these dynamic effects produce a significant
downward shift in the F+ curve and an amplification in the F+ curve. It n t
is apparent that these effects should be included in the analytical model.
The agreement between the VDART2 model and this experiment is reason-
ably good in light of the uncertainties. The hump seen in the experimental
curve near 1080° + 270° may be partially due to misalignment errors in the
blade mounting. Errors on the order of 1° in the blade angle of attack
could cause this level of deviation from the analysis. A slight phase
shift is also apparent between analysis and experiment. The exact cause
of this shift is unknown, but may be partially due to the time step size
used in the analytical model. Since calculations are smeared over a
particular time step which represents about 15° of rotor rotation, the
shift due to this cause could potentially be up to 15°.
Results at a tip to wind speed ratio of 2.5 are shown in Figure 18.
At this tip to wind speed ratio, the dominant dynamic effects are those
due to dynamic stall which was discussed in section 2.2.3. It is apparent
from Figure 18 that some sort of correction to the quasi-steady analysis
is required to adequately predict the experimental results. Strict appli
+ cation of the method suggested in section 2.2.3 yielded values of Ft
which were
46
20
10
0 , , \ \
-10 \
F+ \ \
n \
-20 \
-30 0
-40
-50
0 90
4
3
2
1 F+
t
0
-1
-2 0 90
Figure 17.
180
,-, r, I \
I J \
" ... " I \ I \ I" I, ,"\ I \ \'" ..... oJ 'J \
\ I \ \
I \ \ \ \ \
270
e - 1080° 360
\ \ \ I ,-,'
450
G ,_, , , 0,' \
",I \ ~ \ I
\ /J I
. ''''0 ~
180 270 360 450
Blade (Re = data,
8 - 1080°
Force Data For a Two-Dimensional Rotor 40,000, NB = 2, UT/Uoo = 7.5,0 tow tank
quasi-steady model. --- dynamic model)
47
10
5
0 F+
n -5
-10
-15
-20 0 90 180 270 360 450
8 - 1080°
3 ~------~------~-------P------~------~~---.
2
1
'fo <:) I \ ~<i)\
o \ Q \ ... '--~.
-1 ~ ______ ~ ______ ~ ______ -L ______ ~ ______ ~~ __ ~
o 90
Figure 18.
180 270 360 450
Blade (Re = data,
8 - 1080°
Force Data for a Two-Dimensional Rotor 40,000, NB = 2, UT/U", = 2.5, @ tow tank ___ quasi-steady model, - dynamic model)
48
on the order of 3.5. Several modifications were tried in an attempt to
bring the results into closer agreement. These modifications were primarily
aimed at adapting the time delay coefficients given in equation 18. Uni-
form adjustment of these coefficients, where the ratio, YL/YM
, was held
constant, could be used to produce a F+ curve which was satisfactory, while n
F+ . f . t was unsatls actory or Vlse versa. The most satisfactory adjustment of
the time delay coefficients is shown in Figure 18. In this case, YL
was
used as given in equation 18, whereas YM
(used to calculate delay in drag
coefficient) was set equal to zero. The reason that this adjustment yields
better results is unclear. Even with this adjustment, the peak value of
F: is overpredicted by 66 percent. Therefore, while the modified Boeing
Vertol dynamic stall model does appear to yield improvement in prediction
of normal and tangential forces, the results are not totally satisfying.
Figure 19 represents data taken at a moderate tip to wind speed ratio
(5.0), where each of the dynamic effects (added mass, pitching circulation,
and dynamic stall) are important. In this case, the values of YL
and YM
used in the analysis are as given in equation 18. If YM
is set equal to
zero as in the previous case, stall occurs prematurely and "chops off" the
F: curve at about 2.0. The undershoot in the predicted F: curve at around
1080 0 + 150 0 is a result of the hysteresus loop in the Boeing-Vertol dynamic
s tall model.
It goes almost without saying that several anomalies exist with regard
to the dynamic stall model presently being employed. A review of this and
other dynamic stall models given by McCrosky [26] indicates that the inc on-
sistencies in such models are rather commonplace. It is apparent that
additional work needs to be done in this area. It should be noted that
the effects of dynamic stall are strongly related to the chord to radius
49
20
10
0 F+
n
-10
-20
-30
-40
4
3
2 F+
t
1
0
-1
-2
\ 0\ 0 d
\ Q.
0\ .-\.. '" 0-
o 90
0 90
Figure 19. Blade eRe = data,
- ."..;""'" ---' - , ~,- - , ".,- ,
I I I
180 270 360 450
e - 1080°
180 270 360 450
e - 1080°
Force Data for a Two-Dimensional Rotor 40,000, N = 2, UT!U = 5.0, ® tow tank ___ quasi~steady mod~l, - dynamic model)
50
ratio, C/R, as are other dynamic effects. In general, the effects are
strongest for large C/R values. The two-dimensional experiment conducted
in the present work represents a rather large C/R value equal to 0.15, as
opposed to about 0.05 for most full-scale rotors. Therefore, the configur-
ation studied in the present work represents a rather severe test with
regard to dynamic effects.
4.2 Wake Structure
Results from the present two-dimensional tow tank experiment, as well
as results from the wake measurements behind a three-dimensional Darrieus
turbine made by Vermeulen [22], will be compared with analytical results in
this section. The test conditions are vastly different for the two sets
of experiments with the tow tank experiment representing a two-dimensional
low turbulence level flow, while the measurements made by Vermeulen repre-
sent a three-dimensional high turbulence level atmospheric flow.
4.2.1 Two-dimensional rotor
Velocity profiles were taken at one and two rotor diameters downstream
of the rotors used in the tow tank test series. These experimental data
can be compared not only with the VDART analytical model, but also with the l
simple momentum model [3]. The simple momentum model can be used to estimate
the fully developed wake by multiplying the velocity defect computed for the
"actuator" disk by a factor of two. As will be shown in section 4.2.2, the
wake behind a Darrieus turbine reaches a fully developed condition within
about one rotor diameter downstream of its vertical axis.
Figures 20, 21, and 22 illustrate that the level of agreement beuveen
both analytical models and the experimental data is reasonably good so long
as the perturbation velocities are small. However, the momentum model is
51
l-u/u co
, I I
2.0 _ o experimental data -- VDART profile
-- simple momentum
1.0 ... -
o ---o
-1.0~ ______ ~1 _____________ L-' ____________ ~1~ ____ ~
1.0
Figure 20. Experimental
(NC/R =
o -1.0
Z/R
Comparison Between DART, VDART2 and Data at One Rotor Diameter Downstream 0.30, uT/Uoo = 2.5, Re = 40,000)
52
2.0 l-
l-U/U 00
1.0 :-
0 I~
-1.0
• UT/Uoo 5
NB = 1
0 experimental data
- VDART profile
-- simple momentum
... .----" t;:L .... '-0_ ~
Q 0
I
1.0
I
----------Q Q
I
o Z/R
Q
I
---o ..... "Q .... ,
~ "-
I
-1.0
Figure 21. Comparison Beuleen DART, VDART2 and Experimental Data at One Rotor Diameter Downstream
(NC/R = 0.15, UT/ll"" = 5, Re = 40,000)
53
•
•
u Iu = 5 T 00
o experimental data
2.0 - 'JDART profile
l-u/u 00
l.0
o
-l.0
-- simple momentum
1.0 o -1.0
Z/R
Figure 22. Comparison Between DAH.T, VDA:.~T2 and Experimental Data at One Rotor Diameter DOl'mstream
(NC/R = 0.30, UT/Uoo = 5, Re = 40,000)
unable to predict a reasonable wake velocity profile for cases where the
perturbation velocity approaches 1.0. It is well known that the momentum
model breaks down for these cases. The perturbation velocity profile shown
for the case in Figure 22 is completely unrealistic, since it indicates
that the fluid in the wake is moving upstream into the rotor. Since the
level of the perturbation velocity is a strong function of tip to ~vind
speed ratio and rotor solidity (NC!R), it can be said, in general, that the
simple momentum model will not predict a reasonable wake at high tip to
wind speed ratios or for large rotor solidities.
The vortex model, on the other hand, continues to predict reasonable
results for the average streamwise velocity perturbations at the higher tip
to wind speed ratios and for larger rotor solidities as shown in Figure 22.
The vortex model is also capable of predicting both instantaneous streamwise
and lateral perturbation velocities as illustrated in Figure 23 and 24.
4.2.2 Three-dimensional rotor
Wake measurements on three-dimensional Darrieus turbines have been made
by Blackwell [11] on a 2-meter turbine operating in a wind tunnel and by
Vermeulen [22] on a 5.3-meter turbine operating in a natural wind environ
ment. The results of Blackwell's measurements have been reported by Giles
[23] and more recently by Sheldahl [24] and will not be presented herein.
The full-scale wake measurements obtained by Vermeulen [22] behind a
5.3-meter Darrieus turbine in a natural wind environment were chosen for
the purposes of this report as a test case for the near-wake prediction
capability of the VDART model. The work of Vermeulen allows the VDART
model to be checked on a three-dimensional rotor of moderate size operating
55
VI 0'\
1-u/u 00
rotor angle 1110° 1155 0 1245 0
1290° 1380 0 1425°
_0.51L-__ ~L-__ ~L-____ ~ ______ ~ ____ -L ____ ~ ______ ~~ __ ~~ __ ~~ ________ ~
1 o -1 1 o Z/R
-1 1 o
Figure 23. Streamwise Perturbation Velocities (Re = 40,000, NB = 3, UT!U
oo = 5.0, EJ tow tank data, - VDART2)
-1
\J1 ~
1-w/u 00
1.0~1--------------------------------------------------------------------------~
rotor angle 1110 0 1155 0 1245°
0.5
0.01 00}~~~~S: 0 ~'iliA(!U!f0~1----
1290° 1380 0 1425°
o.o~g (£) @f X>~-~~ ~ ~~0 1- ,_ ...,~- .......... 1.") 0~ , ~e~"""",00>----
-0.5 1 0 -1 1 0 -1 1 0 -1
z/R
Figure 24. Lateral Perturbation Velocities (Re = 40,000, NB = 3, Ur/Uoo = 5.0, ® tow tank data, --- VDART2)
in a turbulent natural wind environment. The measurements of Vermeulen
consist of wake velocity profiles taken at 1.1, 3.0, 5.0, and 8.0 rotor
diameters downstream at a few selected tip to wind speed ratios. All of
these profiles were taken at "hub" height behind the rotor.
Calculations using the VDART computer code were performed for a two
bladed rotor operating at a tip to wind speed ratio, uT/Uoo
' of 4.0. The
cilord to radius ratio, CIR, was selected to be equal to 0.10. Data for a
NACA 0012 airfoil operating at a Reynolds number of 0.3 x 106
was used.
These conditions match those reported by Vermeulen [22] reasonably well.
Predicted perturbation velocity profiles at six streamwise locations
in the near wake are shown in Figure 25. As can be seen from this figure,
the development of almost self similar velocity profiles occurs fairly
quickly downstream of the rotor. There appears to be some broadening of
the shear layers at the edges of the wake as one goes downstream. Based
on the small number of data points used to define the profiles, this
broadening may be somewhat the result of artistic license. The lack of
symmetry is apparent with a greater velocity deficit occurring on the
advancing side of the rotor.
The centerline velocity reaches a nearly constant value for points
downstream of X/D = 1.0 as shown in Figure 26. The constant centerline
velocity is, of course, characteristic of the near wake region of most
bluff bodies. The centerline perturbation velocity increases linearly as
a function of the streamwise distance X/D in the immediate vicinity of
the rotor. The reason for this latter occurrence is not apparent.
It is interesting, at this point, to note the velocity defect profile
predicted by the simple momentum model [3]. This profile agrees most
closely with the vortex model profile at X/D = 0 as shown in Figure 27.
58
o
- 0.5
o o
1 - U/Uco
o o 0.5 0 0.5 I
I ~
§§
o 0.5 1 XI D 2
Figure 25 Calculated Velocity Defect Profiles for a Three-Dimensional Two-Bladed Rotor 6
(C/R = 0.10, UT/Uco = 4.0, Re = 0.3xlO )
59
3
.4
1-U/U 00
.2
o L-~~ ____ ~~~ ____ ~~~~~~~~
-1 0 2 3 4 5 X/D
Figure 26 Ca1cu1ated Center Line Ve10city in the Near Wake of a ThreeDimensiona1 Two-81aded Rotor (same case as Figure 25)
60
.6
·4
l-U/Uoo
.2
.4 l-U/U
00
.2
2x simple momentum profile '-.-------, ,-' , , \
:;:~~~.~ \
X/O = 1,0 2,'1 3,A
simple momentum profile
. \ \
J --------- ..... . - " -.- , // \
\ \ \ \ \
O~~~~~~~~~~~~~~~--~ o
I/O
Figure 27 Comparison between VOART and Simple Momentum Profiles (same case as Figure 25)
61
-1
It is also interesting to note that if one doubles the magnitude of the
velocity defect at X/D = 0, the velocity defect profiles downstream of
X/D = 1.0 are approximated. This is not too surprising, since it is well
known that one-half of the velocity deficit occurs as the flow "passes
through" the rotor and the other half further downs tream.
The VDART code pres ently does not have any turbulence model built into
it and, therefore, the resulting velocity profiles must be corrected for
those effects. The uncorrected profiles would be consistent with measure-
ments made in a low turbulence wind tunnel. As Vermeulen [22] points out,
the diffusion or broadening of the velocity defect in the wake will, in
general, be a function of the wind direction fluctuations in the approach
flow. The root mean square, aS, of the direction fluctuations were about
eight degrees for the cases under study.
A relatively simple correction can be applied if one assumes that the
direction fluctuations are Gaussian in nature and that the broadening can
be calculated based on the average of a number of quasi-steady wakes pro-
duced by different upstream wind directions. The probability that the
streamwise wake axis will deviate by an angle, Sc' from the mean angle and
lie between S c
013 12 and S + 013 12 is given by c c c
ycSS c
2 2 exp(-S 12 (J )cSS
c c
The corrected velocity defect profile as a function of the angle 13
(33)
(13 is measured from the streamwise ~vake axis) can, therefore, be calculated
from
U(S)corr r yU(S - J3 )d J3 c c (34)
62
Here U(S S ) is the uncorrected velocity defect profile evaluated at an c
angle, S - S. The angle, S - S , is also measured from the streamwise c c
wake axis.
Corrected velocity profiles at X/D = 1.1 and 3.0 are shown in Figure
28, along with the experimental results due to Vermeulen [22J. It should
be noted that the profile at X/D = 1.1 is modified very little by this
correction, while there is a significant modification of the profile at
X/D = 3.0. Figure 28 indicates that the corrected velocity profiles pre-
dicted by VDART are in some reasonable agreement with the experimental data.
Dynamic effects are not included in the analysis and may have a modi-
fying effect. It is anticipated that these effects may be minimal at
locations away from the immediate vicinity of the rotor.
5.0 SUMMARY OF RESULTS
In this section, a summary of the work which has been accomplished
will be given, along with a list of conclusions that have been reached.
In addition, recommendations for future work will be presented.
5.1 Summary
Experimental data have been taken in the following areas:
* Transient normal and tangential blade forces have been measured
on a two-dimensional rotor for five different tip to wind speed
ratio and rotor configuration combinations.
* Total velocity data were obtained at one and two rotor diameters
downstream of a two-diamensional rotor. Data were taken at 13
spanwise positions across the wake. Five cases which represented
three different rotor geometries (number of blades) and three
different tip to wind speed ratios were run.
63
.4
l-U/Uco
.2 X/D = 1.1
0
So
.4 l-U/U co
.2
0
-20 0 SO
20
Figure 28 Comparison Between VDART and Experimental Data of Vermeulen [22J ( - VDART data corrected for wind direction fluctuations, same case as Fi gure 25 )
* Flow direction measurements were obtained to coincide with each
of the total velocity measurements made. The total velocity was
then decomposed into its streamwise and lateral components.
Analytical work has been conducted along the following lines:
* Time saving methods in the form of FLP (frozen lattice point
velocities) and FWG (fixed wake grid points) were incorporated
into the two- and three-dimensional VDART codes.
* The effects of unsteady aerodynamics have been formulated and
incorporated into the VDART codes.
* The VDART codes have been exercised in order to assess their
agreement with experimental results.
5.2 Conclusions
Several statements can be made with regard to the analytical model,
the experimental work, and comparisons between the analytical and exper
imental results.
* Use of the time-saving feature FWG yields good results in te~s of
maintaining accuracy, while reducing computer time by a factor of
three for six revolutions and by a factor of six for 12 revolutions.
* Use of the time-saving feature FLP yields good results in general,
but must be used with caution, especially at high tip to wind speed
ratios and/or high rotor solidi ties.
* Use of continuity considerations as a time-saving feature is not
justified as no savings can be realized.
* Dynamic effects are important at C/R values on the order of 0.1 or
greater and are manifest in the form of stall delay, added mass
effects, and circulation due to pitching.
65
." Good results are obtained in predicting the effects of added mass
and pitching circulation on blade forces.
* Poor results were obtained in predicting the effect of dynamic
stall on blade forces in that constants in the dynamic stall model
(Boeing-Vertol) had to be adjusted in several cases to yield agree
ment between analysis and experiment.
* Reasonable results were obtained with regard to predicting the near
~vake velocity profiles behind a two-dimensional rotor operating in
a uniform low-turbulence-Ievel stream.
* Reasonable results were obtained with regard to predicting the near
wake velocity profiles behind a three-dimensional rotor operating
in a natural wind environment after simple wind direction fluctuation
corrections were made.
5.3 Recommendations
* A better approach to the dynamic stall formulation should be under
taken.
* Additional blade force (or instantaneous rotor torque) data should
be obtained on full-scale rotors.
,~ Additional wake veloci ty profiles behind full-scale rotors should
be obtained.
* The role of freestream turbulence with respect to performance and
near wake structure should be studied •
." The effects of the ground plane should be inves tignted.
66
•
6.0 BIBLIOGRAPHY
1. Templin, R.J., "Aerodynamic Performance Theory for the NRC VerticalAxis Wind Turbine," National Research Council of Canada Report LTRLA-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., 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. 111-61-93, May (1976).
6. Larsen, H.C., "Summary of a Vortex Theory for the Cyclogiro," Proceedings of the Second U.S. National Conferences on ,Hnd Engineering Research, Colorado State University, pp. V-8-l-3, June (1975).
7. Wilson, R.E., "Vortex Sheet Analysis of the Giromill," J. Fluids Engineering, Vol. 100, No.3, pp. 340-342, September (1978).
8. Holmes, 0., "A Contribution to the Aerodynamic Theory of the VerticalAxis Wind Turbine," Proceedings of the International Symposium on ,Hnd . Energy Sys terns, St. John's College, Camb ridge, England, pp. C4-55-72, September (1976).
9. Strickland, J.H., "A Vortex Model of the Darrieus Turbine: An Analytical and Experimental Study," Final Report submitted to Sandia Laboratories on Contract #06-4178, January (1979), Sandia Report SAND 79-7058, February (1980).
10. Strickland, J.H., Darrieus Turbine: Engineering, Vol •
Webs ter, B. T., Nguyen, T., "A Vortex Model of the An Analytical and Experimental Study," J. Fluids
~-'--'-==-101, No.4, pp. 500-505, December (1979).
11. Blackwell, B.F., Sheldahl, R.E., Feltz, L.V., "Wind Tunnel Performance Data for the Darrieus Wing Turbine wi th NACA 0012 Blades," Sandia Laboratory Report SAND 76-0130, May (1976).
12. Sheldahl, R.E., Blackwell, B.F., "Free-Air Performance Tests of a 5-Metre-Diameter Darrieus Turbine," Sandia Laboratory Report SAND 77-1063, December (1977).
13. Wors tell, M. H., "Aerodynamic Performance of the l7-Metre-Diameter Darrieus Wind Turbine," Sandia Laboratory Report SAND 78-1737, January (1979).
67
14. Milne-Thomson, L.M., Theoretical Aerodynamics, 2nd Ed., Macmillan and Co., (1952).
15. Karamcheti, K., Principles of Ideal-Fluid Aerodynamics, John Wiley and Sons, (1966).
16. Currie, I.G., Flllldamental Mechanics of Fluids, McGraw-Hill, (1974).
17. Lamb, H., Hydrodynamics, 6th Ed., Dover, (1945).
18. Klimas, P .C., "Possible Aerodynamic Improvements for Future VAWT Sys tems," Proceedings - VAWT Design Technology Seminar for Indus try, Sandia Lab oratories, April (1980).
19. Hilne-Thoms on, L.M., Theoretical Hydrodynamics, 4th Ed., Macmillan and Co., (1960).
20. Garmont, R.E., "A Hathematical Hodel of Unsteady Aerodynamics and Radial Flow for Application to Helicopter Rotors," U.S. Army Air Mobility R&D Lab. Report on Boeing-Vertol Contract DAAJ02-7l-C-0045, May (1973).
21. Migliore, P.G., Wolfe, W.P., "Some Effects of Flow Curvature on the Aerodynamics of Darrieus Wind Turbines," Prepared for DOE under contract EY-76-C-05-5l35 by West Virginia University, April (1979).
22. Vermeulen, P., Builtjes, P., Dekker, J., Bueren, G., "An Experimental Study of the Wake Behind a Full-Scale Vertical-Axis Wind Turbine," TNO Report 79-06118, June (1979).
23. Giles, R., "Wind Tunnel Test Data Reduction for Darrieus VerticalAxis Wind Turbines," Civil Engineering Res earch, Uni versi ty of New Mexico, (1977).
24. Sheldahl, R.E., Proceedings - VAWT Design Technology Seminar for Industry, Sandia Laboratories, April (1980).
25. Berg, D.E., Private Communication, November (1980).
26. McCroskey, ~v.J., "Recent Developments in Dynamic Stall," Presented at the Symposium on Unsteady Aerodynamics, Tuscon, 18-20 Harch (1975).
68
•
7.0 APPENDIX
Listings of the VDART2 and 3 computer codes, along with blade force
data are given in this appendix.
69
7.1 VDART2 Computer Code Listing
This appendix contains the VDART2 computer code listing with the FWG
option. The Fortran code is suitable for execution on the Texas Tech
University ITEL AS6 computer.
70
COMHON/XWAKE/XFW( 10) ,ZFH( 5) , IFW ,KFH COMMON/UWAKE/UFW( 10,5) , WFWC 10,5) COMMON/LOC/XC3,400),ZC3,400) COMMON/VEL/U(3,400),WC3,400) tOMMON/VEO/UO(3,400),WO(3,400) COMMON/GAM/GS(3,400),GBC14),OGBC14),OALPC14) COMMON/CLTAB/TA(30) ,TCL(30),TCD(30) ,NTBL,ASTAL,TCR DIMENSION NPT(5),NPP(5),SPP(5),XO(5),ZO(5) NR=12 NTI=24 NSW1=2 NSW2=2 READ(5,1) NB,CR,UT,XIP,TCR FORHATCI1,4F10.4) READC5,2) NTBL,RE,ASTAL
2 FORMATCI2,2F10.3) DO 1 0 I= 1 , NTBL READC5,3) TA(I),TCLCI),TCD(I)
3 FORMATC3F10.4) 10 CONTINUE
INTEGER PSW1,PSW2,PSW3 READ(5,16) PSW1,PSW2,PSW3
16 FORMAT(3I1 ) PRINT 60,PSW1,PSW2,PSW3
60 FORMATC3I2) READC5,55) NPR
55 FORHAT(I2) IFCNPR.EQ.O) GO TO 23 DO 12 I=1,NPR READC5,13) NPTCI),NPPCI),SPPCI),XOCI),ZOCI)
13 FORMATC2I3,3F7.3) 12 CONTINUE 23 CONTINUE
DELT=6.2832/NTI NT=1 DO 50 I=1,NB GS<r,1)=0.0 OGB(I)=O.O OALPC I) =0.0
50 CONTINUE PRINT 4,NB,UT,CR,XIP,RE,TCR,ASTAL
4 FORMATC30X,'ROTOR DATA' ,11127X,'NUMBER OF BLADES=',I2,/27X, $'TIP TO WIND SPEED RATIO=' ,F4.1,/27X,'2-DIMENSIONAL ROTOR' $,/27X,'CHORD TO RADIUS RATIO=' ,F4.3,/27X, 'BLADE MOUNTING' $,' LOCATION=',F6.3,' CHORDS' ,11111130X,'AIRFOIL DA~A',11127X, $'RE=',F5.2,'HILLION',/27X,'THICKNESS TO CHORD RATIO=',F5.2, $/27X,'STATIC STALL ANGLE=',F5.2,' DEG',11127X,'ALPHA',5X, $' CL' ,8X, 'CD')
DO 15 I= 1 , NTBL PRI~~ 5,TACI) ,TCLCI) ,TCDCI)
5 FORMATC20X,F10.1,2F10.4) 15 CONTINUE
DO 40 K=1,NR 71
CPSUt1=0.0 DO 20 1= 1 , NTI CALL BGEot'l( NT, NB, CR,DELT, XIP) CALL BIVEL( NT, NB, NTI) CALL BVORT(NT,NB,CR,UT,NTI,XIP) CALL BIVEL(~~,NB,NTI) CALL PERF(NT,NB,CR,UT,NTI,CPL,XIP,PSW2) CPSUM=CPSUM+CPL NPW=NT*NB NFPW=IFW*KFW IF(NPW.LE.NFPlrll GO TO 42
41 CALL SWIVEL(NT,NB,UT,NSW1 ,NTI,PSW3) GO TO 43
42 CALL WIVEL(NT,NB,UT,NSW1,NTI) 43 CONTINUE
IF(NSW2.EQ.NT) GO TO 9 GO TO 11
9 CONTINUE IF(NPR.EQ.O) GO TO 24 DO 14 J=1,NPR CALL PROFIL(NT,NB,NPT(J),NPP(J),SPP(J),XO(J),ZO(J),NTI)
14 CONTINUE 24 CONTINUE
NSW2=NT+UT 11 CONTINUE
CALL CONLP(NT,NB,DELT,UT) NT=NT+1 CALL SHEDVR(NT,NB)
20 CONTINUE CP=CPSUN/NTI PRINT 6,CP,K
6 FORl"IAT(10X,'AVERAGE ROTOR CP=',F5.4,' FOR REVOLUTION NUMBER',I2) IF(PSW1.NE.1) GO TO 22 PRINT 7
7 FORMAT(12X, 'NB' ,3X, 'NT' ,4X, 'X' ,6X, 'Z' ,6X, 'U' ,6X, 'W') NRI=NTI*K DO 45 M=1,NB DO 45 N=1,NRI,5 PRINT 8,M,N,X(M,N),Z(M,N),U(N,N),W(H,N)
8 FORHAT( lOX,2I5,4F7 .3) 45 CONTINUE 22 CONTINUE 40 CONTINUE
END
72
BLOCK DATA COMMON/XWAKE/XFW(10),ZFW(5),IFW,KFW DATA IFW/101,KFW/51 DATA XFW/-1.0,-O.5,O.O,O.5,1.0,2.0,3.0,4.0,5.0,7.01 DATA ZFW/-1.5,-O.75,O.O,O.75,1.51 END
73
SUBROUTINE BGEOM(NT,NB,CR,DELT,XIP) COMMON/LOC/X(3,400),Z(3,400) THET=(NT-1)*DELT DTB=6.2832/NB DO 10 I=1,NB THETA=THET+(I-1)*DTB X(I,NT)=-SIN(THETA)-(XIP-.25)*COS(THETA)*CR Z(I,NT)=-COS(THETA)+(XIP-.25)*SIN(THETA)*CR
10 CONTINUE RETURN END
74
SUBROUTINE BIVEL(NT,NB,NTI) COMMON/LOC/X(3,400),Z(3,400) COMMON/VEL/U(3 ,400), W(3 ,400) COMMON/GAI'IIGS( 3,400) , GB( 14) , OGB( 14) , OALP( 14) DO 11 I=1,NB J=NT USUM=O .0 WSUM=O.O DO 10 K=1,NB DO 10 L=1,NT CALL FIVEL(X(K, L) , X( I, J) , Z (K, L) ,zer, J) , GS(K, L) , UU, WloJ, NTI) USUM=USUM+UU WSUM=WSUM+\-J1,~
10 CONTINUE U( I, J) =USUM wer, J) =WSUM
11 CONTINUE RETURN END
75
SUBROUTINE BVORT(NT,NB,CR,UT,NTI,XIP) COH~10N/LOC/X( 3,400) ,Z<3, 400) COHNON/VELlU( 3,400) ,W<3, 400) COI1HON/GllJ1/GS( 3,400) ,GB( 14) ,OGB( 14) ,OALP( 14) C0l1HON/CLTABITA(30), TCL(30), TCD(30) ,NTBL,ASTAL, TCR REAL K DO 10 I=1,NB URDN=-(U(I,NT)+1.0)*X(I,NT)-H(I,NT)*Z(I,NT) URDN=URDN-(XIP-.25)*CR*UT URDC=-( U( I, NT) + 1 .0) *Z( I, NT)+IrJ( I ,NT) *X(I, NT)+UT UR=SQRT(URDN**2+URDC**2) ALPHA=ATAN2(URDN,URDC) K=CR*UT/(2.0*UR) CADOT=K*NTI*(ABS(ALPHA)-ABS(OALP(I»)/(2.0*3.1416) URDNN=URDN+O.5*CR*UT ALPN=ATAN2(URDNN,URDC) CALL ALDNT(ALPN,CADOT,CL,CD,CN,CT) GB(I)=CL*CR*UR/2.0 GS(I,NT)=GB(I)
10 CONTINUE RETURN END
76
SUBROUTINE PERF(NT,NB,CR,UT,NTI,CPL,XIP,PSW2) COMMON/LOC/X(3,400),Z(3,400) COMMON/VEL/U(3,400),W(3,400) COMMON/GAM/GS(3,400),GB(14),OGB(14),OALP(14) COMMON/CLTAB/TA(30),TCL(30),TCD(30),NTBL,ASTAL,TCR REAL K INTEGER PSW2 IF(PSW2.NE.1) GO TO 4 PRINT 1 FORMAT(!II ,3X, 'THETA' ,2X, 'BLADE' ,2X, 'ALPHA' ,SX, 'FN', 11X,
$'FT', 11X, 'T', 11X, lUI ,9X, 'WI) 4 TR=O.O
CPL=O.O DO 10 I=1,NB TH=(NT-1)*360.0/NTI+(I-1)*360.0/NB URDN=-(U(I,NT)+1.0)*X(I,NT)-lV(I,NT)*Z(I,NT) URDN=URDN-(XIP-.25)*CR*UT URDC=-(U(I,NT)+1.0)*Z(I,NT)+lV(I,NT)*X(I,NT)+UT UR=SQRT(URDN**2+URDC**2) ALPHA=ATAN2(URDN,URDC) AL=57.296*ALPHA K=CR*UT/( 2.0*UR) CADOT=K*NTI*(ABS(ALPHA)-ABS(OALP(I»)/(2.0*3.1416) OALP( 1) =ALPHA URDNN=URDN+0.5*CR*UT URDNT=URDN+.25*CR*UT ALPN=ATAN2(URDNN,URDC) ALPT=ATAN2(URDNT,URDC) CALL ALDNT(ALPN, CADOT, CL,DUMCD, CN , DUMCT) CALL ALDNT(ALPT, CADOT ,DUl1CL, DUMCD, DUl1CN, CT) GB(I)=CL*CR*UR/2.0 GS(I, NT):GB(I) FN=CN*UR**2 FT=CT*UR**2 TE=FT*CR/2.0-FN*CR*(XIP-.25)*CRl2.0 IF( PSVJ2. NE. 1) GO TO 5 PRINT 2,TH,I,AL,FN,FT,TE,U(I,NT),lV(I,NT)
2 FORMAT(FS.1,I6,F7.1,3X,E10.3,3X,E10.3,3X,E10.3,3X,F7.3,3X,F7.3) 5 TR=TR+TE
CPL=CPL+ TE*UT 10 CONTINUE
IF(PSlV2.NE.1) GO TO 6 PRINT 3,TR,CPL
3 FORMAT(1110X,'ROTOR TORQUE COEFFICIENT=',E10.3,1,10X, $'ROTOR POlVER COEFFICIENT=',E10.3)
6 RETURN END
SUBROUTINE WIVEL(NT,NB,UT,NSW1,NTI) COMMON/LOC/X(3,400),Z(3,400) COHMON/VEL/U(3 ,400), WC3 ,400) COMHON/VEO/UO(3, 400) ,WO(3 ,400) COHtvlON/GAH/GS( 3,400) ,GB( 14) ,OGB( 14) ,OALP( 14) IF(NT.LE.1) GO TO 12 NT1=NT-1 DO 11 1=1, NB DO 11 J=1,NT1 UOCI,J)=U(I,J) WOCI, J) =1:JO, J) IF(NT.EQ.NSW1)GO TO 30 GO TO 11
30 CONTINUE CALL PIVEL(NT,NB,X(I,J),Z(I,J),U(I,J),W(I,J),NTI)
11 CONTINUE IF(NT.EQ.NSW1) NSW1=NT+UT
12 CONTINUE RETURN END
78
SUBROUTINE SWIVEL ( NT, NB, liT, NSW1 , NTI, PSVJ3) COMNON/XWAKE/XFW( 10) , ZFlrJ( 5) , IFIrJ, KFW COMNON/UWAKE/UFW( 10,5) , WFlrJ( 10,5) COMNON/LOC/X(3,400),Z(3,400) COMNON/VEL/U(3,400),W(3,400) COMNON/VEO/UO(3,400),WO(3,400) COMMON/GAWGS(3 ,400) , GB( 14) ,OGB( 14) , OALP( 14) INTEGER PSW3 THETA=(NT-1)*360.0/NTI XMIN=XFW(1) ZMIN=ZFW( 1) XMAX=XFW( Inn ZMAX=ZFW(KFW) IF(NT.EQ.NSW1) GO TO 5 GO TO 13
5 CONTINUE IF(PSW3.NE.1) GO TO 2 PRINT 1, THETA
1 FORMATC/II,5X,'VELOCITIES AT FIXED WAKE POINTS',/SX,'(ROTOR AN' $, 'GLE=' , F6. 1 , 'DEG. ) , , I I, 3X, 'X' , 6X, , Z' , 6X, 'U' , 6X, 'W' )
2 DO 10 IW=1,IFW DO 10 KVJ= 1 , KFlrJ CALL PIVEL(NT,NB,XF'vV(I'vV) ,ZFW(K\V) ,UFVJ(IW,K\V),
$'vVFW(IW,K'vV),NTI) IF(PSVJ3.NE.1) GO TO 16 PRINT 15,XFWCI'vV) ,ZFW(lI.'W) , UFWCIW,KlV) ,
$WFW( IW, K'vV) 15 FORMAT(4F7.3) 16 CONTINUE 10 CONTINUE 13 CONTINUE
NT1=NT-1 DO 11 1=1, NB DO 11 J=1,NT1 UO(I,J)=U(I,J) WO(I,J)=H(I,J) IF(NT.EQ.NSH1) GO TO 30 GO TO 11
30 CONTINUE IF(X(I,J).LT.XMIN.OR.X(I,J).GT.XMAX) GO TO 40 IF(Z(I,J).LT.ZMIN.OR.Z(I,J).GT.ZMAX) GO TO 40 GO TO 41
40 CALL PIVEL(NT,NB,X(I,J),Z(I,J),U(I,J),H(I,J),NTI) GO TO 11
41 CONTINUE CALL INTERP(X(I,J),Z(I,J),U(I,J),H(I,J))
11 CONTINUE IF(NT.EQ.NSW1) NS'vV1=NT+UT
12 CONTINUE RETURN END
79
SUBROUTINE PROFIL(NT,NB,NPT,NPP,SPP,XO,ZO,NTI) COMt'lON/LOC/X(3,400) ,Z(3,400) COMHON/GAWGS(3 ,400) ,GB( 14) ,OGB( 14) ,OALP( 14) THETA=( NT-1) *360 .01 NTI PRINT 1, THETA FOR~lliT(III,10X,IWAKE VELOCITY PROFILE',/11X,'(ROTOR ANGLE='
$,F6.1, 'DEG.)' ,II ,3X, 'X' ,6X, 'Z' ,6X, 'U' ,6X, 'W') DO 10 I=1,NPP XP=XO+SPP*(I-1 ) *( 2-NPT) *( 3-NPT)/2.0 zP=ZO+SPPlf( 1-1) *( NPT -1) *( NPT -2)/2.0 CALL PIVEL(NT,NB,XP,ZP,UP,WP,NTI) PRINT 2,XP,ZP,UP,WP
2 FORHAT(4F7.3) 10 CONTINUE
RETURN END
80
SUBROUTINE PIVELCNT,NB,XP,ZP,UP,WP,NTI) COMMON/LOC/XC3,400),ZC3,400) COMMON/GAH/GS(3,400) ,GBC 14) ,OGBC 14) ,OALPC 14) USUM=O.O WSUM=O.O DO 20 K=1,NB DO 20 L=1,NT CALL FIVELCXCK,L),XP,ZCK,L),ZP,GSCK,L),UU,WW,NTI) USUM=USUM+UU WSUM=WSUM+l'M
20 CONTINUE UP=USUM WP=WSUM RETURN END
81
SUBROUTINE INTERP(XP,ZP,UIN,WIN) COMMJN/XWAKE/XFH( 10) , ZFVJ( 5) , IFH, KFW COMMON/UWAKE/UFH( 10,5), WFH( 10,5) DIfvIENSION UX( 2) , WX( 2) , UQ( 2,2) , WO( 2,2) IW=1 KW=1
2 IF(XP.LE.XFH(IW)) GO TO 6 IW=IW+ 1 GO TO 2
6 IF(ZP.LE.ZFH(KW)) GO TO 8 KH=!0d+ 1 GO TO 6
8 CONTINUE IW=1H-1 KH=KH-1 HX=XFH( IW+ 1 )-XFVJ( IW) HZ=ZFH(KH+1)-ZFH(KH) DO 50 1Q=1,2 DO 50 KQ=1,2 1S=IW+IQ-1 KS=KH+KQ-1 UQ(1Q,KQ)=UFH(1S,KS) v.JQ(1Q,KQ)=I.JFH(1S,KS)
50 CONTINUE DHX=(XP-XFW(IW))/HX DHZ=(ZP-ZFH(KW))/HZ DO 1 0 J=1,2 UX(J)=UQ(1,J)*(1.0-DHX)+UQ(2,J)*DHX WX( JhHQ( 1 , J) *( 1.0-DHX)+\I/Q( 2 ,J) *DHX
10 CONTINUE U1N=UX( 1)*( 1.0-DHZ)+UX(2)j('DHZ vJIN=ltJX( 1) *( 1 .O-DHZ) +WX( 2) *DHZ RETURN END
82
•
SUBROUTINE CONLP(NT,NB,DELT,UT) C~~IDN/LOC/X(3'400)'Z(3,400) C MMON/VEL/U(3,400),W(3,400) C ~!ON/VEO/U0(3,400) ,WO(3 ,400) DT=DELT/UT NT1=NT-1 DO 20 I=1,NB IF(NT.LE.1)GO TO 11 DO 10 J=1,NT1 X(I,J)=X(I,J)+(3.0*U(I,J)-UO(I,J)+2.0)*DT/2.0 Z(I,J)=Z(I,J)+(3.0*W(I,J)-HO(I,J))*DT/2.0
10 CONTINUE 11 CONTINUE
X(I,NT)=X(I,NT)+(U(I,NT)+1.0)*DT Z( I, NT) =Z( I, NT) +l.J( I, NT) *DT
20 CONTINUE RETURN END
SUBROUTINE SHEDVR( NT, NB) COMt-DN/GAWGS(3,400) ,GB( 14) ,OGB( 14) ,OALP( 14) NT1=NT-1 DO 10 I=1,NB GS( I, NT) =GB( 1) GS(I,NT1)=OGB(I)-GB(I) OGB(I)=GBCI)
10 CONTINUE RETURN END
84
•
SUBROUTINE FIVEL(X1,X3,Z1,Z3,GAMMA,UU,WW,NTI) DELT=6.2832/NTI RLIM=2. 01 NTI CX=X1-X3 CZ=Z1-Z3 CCAV=CX*CX+CZ*CZ SRLIM=RLIM*RLIM IF(CCAV.LT.SRLIM) GO TO 10 VF=GAMMA/(6.283185*CCAV) GO TO 11
10 VF=(3. 1416*GAHMA)1 (2 .O*DELT*DELT) 11 UU=-CZ*VF
WW=CX*VF RETURN END
85
SUBROUTINE ALDNT(ALPHA,CADOT,CL,CD,CN,CT) COMMON/CLTAB/TA(30) ,TCL(30) ,TCD(30) ,NTBL,ASTAL,TCR REAL K PI=ARCOS(-1.0) ASTR=ASTAL*PI/180.0 IF(ABS(ALPHA).GE.ASTR) STALL=1.0 IF(CADOT.GE.O.O.AND.ABS(ALpr~).LE.ASTR) STALL=O.O IF(STALL.EQ.1.0) GO TO 1 GO TO 2 CONTINUE SALPH=SIGN(1.0,ALPHA) SADOT=SIGN(1.0,CADOT) K=0.25*SADOT*SALPH*(3+SADOT) CHEATL=1 .0 CHEATD=1.0 GL=CHEATL*(1.4-6.0*(0.06-TCR)) GD=CHEATD*(1.0-2.5*(O.06-TCR)) ADL=(ALPHA-K*GL*SQRT(ABS(CADOT)))*180.0/PI ADD=(ALPHA-K*GD*SQRT(ABS(CADOT)))*180.0/PI CALL ATAB(ADL,CL,DUM) CALL ATAB(ASTAL,CLS,DUM) SL=(CL-TCL(1))/(ADL-TA(1)) SLM=(CLS-TCL(1))/(ASTAL-TA(1)) IF(SL.GT.SLM) SL=SLM CL=TCL(1)+SL*(ALPHA*180.0/PI-TA(1)) CALL ATAB(ADD,DUM,CD) GO TO 3
2 AD=ALPHA*180 .01 PI CALL ATAB(AD,CL,CD)
3 CONTINUE CN=-CL*COS(ALPHA)-CD*SIN(ALPHA) CT= CL*SIN(ALPrill)-CD*COS(ALPHA) RETURN END
86
SUBROUTINE ATAB(ALD,CL,CD) COMMON/CLTAB/TA(30),TCL(30),TCD(30),NTBL,ASTAL,TCR AD=ALD NTBL1=NTBL-1 IF(AD.LE.O.O) AD=AD+360.0 IF(AD.GE.O.O) AL=AD IF(AD.GE.180.0) AL=360.0-AD IF(AD.GE.360.0) AL=AD-360.0 DO 1 0 1= 1 , NTBL 1 J=I IF(AL.GE. TA( I) .AND.AL.LE. TA(I+ 1)) GO TO 20
10 CONTINUE 20 XA=(AL-TA(J))/(TA(J+1)-TA(J))
CL=TCL(J)+XA*(TCL(J+1)-TCL(J)) CD=TCD(J)+XA*(TCD(J+1)-TCD(J)) IF(AD.GT.180.0.AND.AD.LT.360.0) CL=-CL RETURN END
87
2 0.15 2.5 0.25 0.12 30 0.04 8.0
0.0 0.0000 0.0180 2.0 0.2500 0.0188 5.0 0.5175 0.0236 8.0 0.7300 0.0355 10.0 0.7800 0.0880 11.0 0.7650 0.1080 15.0 0.7175 0.1905 18.0 0.7000 0.2580 21.0 0.6975 0.2855 30.0 0.9546 0.6666 40.0 1.1200 1.0100 50.0 1.1000 1.3700 60.0 0.9700 1.7000 70.0 0.7100 1.9300 80.0 0.4100 2.0500 90.0 0.0900 2.0700
100.0 -0.2300 2.0400 110.0 -0.5300 1.8900 120.0 -0.8000 1.6900 130.0 -0.9800 1.4100 140.0 -1.0500 1.0900 150.0 -0.9400 0.7200 154.0 -0.8400 0.5600 160.0 -0.7000 0.3700 164.0 -0.6800 0.2700 168.0 -0.7100 0.2100 170.0 -0.7400 0.1800 172.0 -0.8400 0.1500 175.0 -0.5000 0.0800 180.0 0.0000 0.0300
111 2 3 10 .3333 2.0 -1.5 3 10 .3333 4.0 -1.5
88
7.2 VDART3 Computer Code Listing
This appendix contains the VDART3 computer code listing with the FWG
option. The Fortran code is suitable for execution on the Texas Tech
University rTEL AS6 computer.
89
COMMON/XWAKE/XFWC 10) , YFWC 5) , ZFWC 5) , IFW, JFV1, KFW COMMON/UWAKE/UFW(10,5,5),VFWC10,5,5),WFWC10,5,5) COM~DN/LOC/X(11,200),YC11,200),ZC11,200) COMMON/VELlU(11,200),V(11,200),W(11,200) COHMON/VEO/UO( 11 ,200) , VO( 11 ,200) , woe 11 ,200) COHt~N/GAM/GT(11,200),GS(11,200),GB(14),OGBC14),OALPC14) COMMON/CLTAB/TA(30) ,TCL(30),TCD(30) ,NTBL,ASTAL,TCR DD'IENSION NPT(5) ,NPP(5) ,SPP(5) ,XO(5) ,YO(5) ,ZO(5) NR=12 NBE=5 NTI=16 NSW1=2 NSW2=2 READ(5,1) NB,CR,HR,UT,XIP,TCR
1 FORMAT(I1,5F10.4) READC5,2) NTBL,RE,ASTAL
2 FORMATCI2,2F10.3) DO 10 1= 1 , NTBL READC5,3) TA(I),TCL(I),TCD(I)
3 FORMAT(3F10.4) 10 CONTINUE
INTEGER PSW1,PSW2,PSW3 READC5,16) PSW1,PSW2,PSW3
16 FORMAT(3I1) PRINT 60,PSW1 ,PSW2,PSW3
60 FORMAT(312) READ(5,55) NPR
55 FORMAT(I2) IF(NPR.EQ.O) GO TO 23 DO 12 I=1,NPR READC5,13) NPT(I),NPP(I),SPP(I),XO(I),YO(I),ZOCI)
13 FORHAT(213,4F7.3) 12 CONTINUE 23 CONTINUE
DELT=6.2832/NTI NE=NBE*NB+1 NE1 =NE-1 NT=1 DO 50 I=1,NE1 GS(I,1)=0.0 OGB(I)=O.O OALP(I)=O.O
50 CONTINUE CALL AREA C NE, NB, HR ,A.T) PRINT 4,NB,UT,HR,CR,XIP,RE,TCR,ASTAL
4 FORMAT(30X,' ROTOR DATA' ,I I 127X, 'Nm1BER OF BLADES=' ,12 ,1Z7X, $'TIP TO WIND SPEED RATIO=',F4.1 ,1Z7X, 'HEIGHT TO RADIUS RATIO=' $,F4.1 ,1Z7X, 'CHORD TO RADIUS RATIO=' ,F4.3,IZ7X, 'BLADE MOUNTING' $,' LOCATION=' ,F6.3, 'CHORDS' ,11111130X, 'AIRFOIL DATA' ,11127X, $'RE=',F5.2,'HILLION',IZ7X,'THICKNESS TO CHORD RATIO=',F5.2, $1Z7X, 'STATIC STALL ANGLE=', F5.2, 'DEG' ,I I 127X, 'ALPHA' ,5X, $ , CL' ,8X, , CD I )
DO 15 1= 1 , NTBL 90
PRINT 5,TA(I),TCL(I),TCD(I) 5 FORMAT(20X,F10.1,2F10.4)
15 CONTINUE DO 40 K=1,NR CPSUM=O.O DO 20 I=1,NTI CALL BGEot1( NT, NE, NB, CR, HR,DELT ,XIP) CALL BIVEL(NT,NE,NTI) CALL BVORT(NT,NE,NB,CR,UT,NTI,XIP) CALL BIVEL(NT,NE,NTI) CALL PERF(NT,NE,NB,CR,UT,AT,NTI,CPL,XIP,PSW2) CPSUM=CPSUM+CPL NPW=NT*NE NFPW=IFW*JFW*KFW IF(NPW.LE.NFPW) GO TO 42
41 CALL SWIVEL ( NT, NE, UT, NSW1 ,NTI, PSW3) GO TO 43
42 CALL WIVEL(NT,NE,UT,NSW1,NTI) 43 CONTINUE
CALL CONLP(NT,NE,DELT,UT) IF(NSW2.EQ.NT) GO TO 9 GO TO 11
9 CONTINUE IF(NPR.EQ.O) GO TO 24 DO 14 J=l,NPR CALL PROFIL(NT,NE,NPT(J),NPP(J),SPP(J),XO(J),YO(J),ZO(J),NTI)
14 CONTINUE 24 CONTINUE
NSW2=NT+UT 11 CONTINUE
NT=NT+1 CALL SHEDVR(NT,NE)
20 CONTINUE CP=CPSUM/NTI PRINT 6,CP,K
6 FORMAT(10X,IAVERAGE ROTOR CP=',F5.4,' FOR REVOLUTION NUMBERI,I2) IF(PSW1.NE.1) GO TO 22 PRINT 7
7 FORMAT(12X, 'NEI ,3X, 'NTI ,4X, IXI ,6X, 'Y' ,6X, 'ZI ,6X, 'U' ,6X, $'V' ,6X, 'WI)
NRI=NTI*K DO 45 M=4,B,2 DO 45 N=1,NRI,5 PRINT B,M,N,X(M,N),Y(M,N),Z(M,N),U(M,N),V(M,N),W(M,N)
B FORMAT(10X,215,6F7.3) 45 CONTINUE 22 CONTINUE 40 CONTINUE
END
91
BLOCK DATA Cm1MON/Xv,IAKE/XFlJ( 10) , YF1v( 5) , ZFW( 5), IFW, JF1J, KFW DATA IFVJ/101, JFW51 ,KFlv/51 DATA XFW/-1.0,-O.5,O.O,O.5,1.0,2.0,4.0,6.0,10.0,16.01 DATA YFW/-O.5,O.25, 1.0,1.75,2.51 DATA ZF\v/-1.5,-O.75,O.O,O.75, 1.5/ END
92
SUBROUTINE AREA(NE,NB,HR,AT) NBE= (NE-1) I NB DELY=HR/NBE RRSUM=O.O H=O.O NBE1=NBE-1 DO 10 1=1,NBE1 H=H+DELY RR=1-4*(H/HR-0.5)**2 RRSUM=RRSDt1+RR
10 CONTINUE AT=2.0*RRSUM*DELY RETURN END
93
SUBROUTINE BGEOM(NT,NE,NB,CR,HR,DELT,X1P) COM~10N/LOC/X( 11,200) , y( 11,200) , Z( 11 ,200) THET=(NT-1)*DELT NBE=( NE-1 )/NB DELY =HRI NBE
. DTB=6.2832INB DO 10 1=l,NB THETA=THET+(1-1)*DTB X(1,NT)=-(X1P-.25)*CR*COS(THETA) Y(l,NT)=O. Z(1,NT)=(X1P-.25)*CR*SIN(THETA) NEI=1+(1-1)*NBE DO 10 J=l,NBE NEJ=NEI+J NEJ1=NEJ-1 Y(NEJ,NT)=Y(NEJ1,NT)-DELY*(-1.0)**1 RR=1-4*(Y(NEJ,NT)/HR-0.5)**2 X( NEJ, NT) =-RR*S1N( THETA) -(X1P-.25) *CR*COS( THETA) Z(NEJ,NT)=-RR*COS(THETA)+(X1P-.25)*CR*SIN(THETA)
10 CONTINUE RETURN END
SUBROUTINE BIVELCNT,NE,NTI) COMMJN/LOC/XC 11,200) ,YC 11,200) ,ZC 11,200) COMHON/VELlUC 11 ,200) , VC 11 ,200) , WC 11 ,200) COMMJN/GAH/GTC11,200),GSC11,200),GBC14),OGBC14),OALPC14) NT1=NT-1 DO 11 I=1,NE J=NT VSUM=O.O USUM=O.O WSUM=O.O IFCNT.LE.1) GO TO 21 DO 20 K=1,NE DO 20 L=1,NT1 L1=L+1 CALLFIVELCXC K, L) , XCK, L1) , xer, J) , YCK, L) ,YCK, L 1) , YC I, J) ,
$ZCK,L) ,ZCK,L 1) ,Zer,J) ,GTCK,L) ,UU, VV ,WW,NTI) USUM=USUM+UU VSUM=VSUM+VV WSUM=WSUM+WW
20 CONTINUE 21 CONTINUE
NE1=NE-1 DO 10 K=1,NE1 DO 10 L=1,NT K1=K+ 1 CALL FIVELCXCK,L),XCK1,L),XCI,J),YCK,L),YCK1,L),YCI,J),
$ZCK,L) ,ZCK1 ,L) ,ZCI,J) ,GSCK,L) ,UU,VV,WW, NTl) USUM=USUM+UU VSUM=VSUM+VV WSUM=WSUM+WW
10 CONTINUE 9 CONTINUE
UCI,J)=USUM VCI,Jl=VSUM WCI,J)=WSUM
11 CONTINUE RETURN END
95
SUBROUTINE BVORT(NT,NE,NB,CR,UT,NTI,XIP) COMMON!LOC!X(11,200),Y(11,200),Z(11 ,200) CmmJN!VEL!U( 11,200), V( 11,200), vJ( 11 ,200) COM}10N!GAI'I!GT( 11 ,200) ,GS( 11 ,200) ,GB( 14) ,OGB( 14) ,OALP( 14) C0I1~10N!CLTABITA(30), TCL(30), TCD(30) ,NTBL,ASTAL, TCR REAL K NBE=( NE-1 )!NB DO 10 I=1,NB NEI=1+(I-1)*NBE DO 10 J=1,NBE NEJ=NEI+J NEJ1=NEJ-1 RR1=SQRT(X(NEJ1,NT)**2+Z(NEJ1,NT)**2) RR2=SQRT(X(NEJ,NT)**2+Z(NEJ,NT)**2) DX=X(NEJ,NT)-X(NEJ1,NT) DY=Y(NEJ ,NT)-Y(NEJ1 ,NT) DZ=Z(NEJ,NT)-Z(NEJ1,NT) EL=SQRT«RR1-RR2)**2+DY**2) SINT=-(X(NEJ,NT)+X(NEJ1,NT))!(RR1+RR2) COST=-(Z(NEJ,NT)+Z(NEJ1,NT))!(RR1+RR2) UTAVE=UT*(RR1+RR2)!2.0 UAVE=(U(NEJ,~~)+U(NEJ1,NT))!2.0 VAVE=(V(NEJ,NT)+V(NEJ1,NT))/2.0 ~vAVE=(W( NEJ, ~~)+H( NEJ1 ,NT) )12.0 URDN=«1.0+UAVE)*DY*SINT-VAVE*(DX*SINT+DZ*COST)+WAVE*DY*COST)!EL URDN=URDN-(XIP-.25)*CR*UT*DY/EL URDC=(1.0+UAVE)*COST-WAVE*SINT+UTAVE UR=SQRT(URDN**2+URDC**2) ALPHA=ATAN2(URDN,URDC) K=CR*UT/(2.0*UR) CADOT=K*NTI*(ABS(ALPHA)-ABS(OALP(NEJ1)))/(2.0*3.1416) URDNN=URDN+0.5*CR*UT*DY/EL ALPN=ATAN2(URDNN,URDC) CALL ALDNT(ALPN,CADOT,CL,CD,CN,CT) GB(NEJ1)=CL*CR*UR!2.0 GS(NEJ1 ,NT)=GB(NEJ1)
10 CONTINUE RETURN END
SUBROUTINE PERF(NT,NE,NB,CR,UT,AT,NTI,CPL,XIP,PSW2) COMMON/LOC/X(11,200),Y(11,200),Z(11,200) COMMON/VELlU(11,200),VC11,200),WC11,200) COMMON/GAMlGT(11,200),GSC11,200),GBC14),OGBC14),OALPC14) COMMON/CLTAB/TA(30), TCL(30), TCD(30) ,NTBL,ASTAL, TCR REAL K INTEGER PSW2 IF(PSW2.NE.1) GO TO 4 PRINT 1
1 FORMATC/ I I, 1X, 'THETA' , 1X, 'ELEMENT' , 1X, 'ALPHA' ,5X, 'FN' ,8X, $'FT' ,8X, 'T' ,8X, 'U' ,6X, 'V' ,6X, 'W')
4 NBE=(NE-1)/NB TR=O.O CPL=O.O DO 10 I=1,NB TH=(NT-1)*360.0INTI+(I-1)*360.0/NB NEI=1+CI-1 )*NBE DO 10 J=1,NBE NEJ=NEI+J NEJ1=NEJ-1 RR1=SQRT(X(NEJ1,NT)**2+ZCNEJ1,NT)**2) RR2=SQRT(X(NEJ,NT)**2+ZCNEJ,~IT)**2) DX=X(NEJ,NT)-X(NEJ1,NT) DY=Y(NEJ,NT)-Y(NEJ1,NT) DZ=Z(NEJ,NT)-Z(NEJ1,NT) EL=SQRT«RR1-RR2)**2+DY**2) SINT=-(X(NEJ,NT)+X(NEJ1,NT»/CRR1+RR2) COST=-(Z(NEJ,NT)+ZCNEJ1,NT»/CRR1+RR2) UTAVE=UT*CRR1+RR2)/2.0 UAVE=CUCNEJ,NT)+UCNEJ1,NT»/2.0 VAVE=CV(NEJ,}IT)+V(NEJ1,NT»)/2.0 WAVE=(W(NEJ,NT)+W(NEJ1,NT»)/2.0 URDN=«1.0+UAVE)*DY*SINT-VAVE*(DX*SINT+DZ*COST)+WAVE*DY*COST)/EL URDN=URDN-(XIP-.25)*CR*UT*DY/EL URDC=C1.0+UAVE)*COST-WAVE*SINT+UTAVE UR=SQRT(URDN**2+URDC**2) ALPHA=ATAN2(URDN,URDC) AL=57.296*ALPHA K=CR*UT/(2.0*UR) CADOT=K*NTI*(ABS(ALPHA)-ABS(OALP(NEJ1»))/(2.0*3.1416) OALP(NEJ1)=ALPHA URDNN=URDN+O.5*CR*UT*DY/EL URDNT=URDN+.25*CR*UT*DY/EL ALPN=ATAN2CURDNN,URDC) ALPT=ATAN2(URDNT,URDC) CALL ALDNTCALPN,CADOT,CL,DUMCD,CN,DUMCT) CALL ALDNT(ALPT,CADOT,DUMCL,DUMCD,DUMCN,CT) GBCNEJ1)=CL*CR*UR/2.0 GS(NEJ1,NT)=GB(NEJ1) FN=CN*UR**2 FT=CT*UR**2 TE=FT*CR*EL*CRR1+RR2)/C2.0*AT)-FN*CR*DY*CXIP-.25)*CR/AT IF(PSW2.NE.1) GO TO 5
97
PRINT 2,TH,NEJ1,AL,FN,FT,TE,UAVE,VAVE,WAVE 2 FORMAT(F7.1,I7,F7.1,3E10.3,3F7.3) 5 TR=TR+TE
CPL=CPL+TE*UT 10 CONTINUE
IF(PSW2.NE.1) GO TO 6 PRINT 3,TR,CPL
3 FOR~ffiT(1110X,'ROTOR TORQUE COEFFICIENT=',E10.3,1,10X, $'ROTOR POWER COEFFICIENT=',E10.3)
6 RETURN END
SUBROUTINE WIVEL(NT,NE,UT,NSW1,NTI) COMMON!LOC!X(11,200),Y(11,200),Z(11,200) COI1MJN!VEL!U( 11 ,200), V( 11,200), W( 11,200) COMI1JN!VEO!UO( 11,200), VO( 11,200), woe 11,200) COMMJN!GAM!GT(11,200),GS(11,200),GB(14),OGB(14),OALP(14) IF(NT.LE.1) GO TO 12 NT1=NT-1 DO 11 I=1,NE DO 11 J=1,NT1 UO(I,J)=UCI,J) VOCI,J)=V(I,J) WOCI, Jh\VCI, J) IF(NT.EQ.NSW1)GO TO 30 GO TO 11
30 CONTINUE CALL PIVEL(NT,NE,X(I,J),Y(I,J),Z(I,J),U(I,J),V(I,J),W(I,J),NTI)
11 CONTINUE IF(NT.EQ.NSW1) NSW1=NT+UT
12 CONTINUE RETURN END
99
SUBROUTINE SWIVEL( NT , NE, UT , NSvl1 , NTI, PSW3) COMI10NIX101AKEIXFW( 10) , yn-I( 5) , ZFlrI( 5) , IFW, JFW, KFlrJ COMt-ION/UWAKE/UFlrJ( 10,5,5) , VFW( 10,5,5) , WFlrJ( 10,5,5) COHI10N/LOC/X( 11,200), Y( 11,200) ,Z( 11,200) COM}lON/VELlU( 11 ,200), V( 11,200), W( 11,200) COMMON/VEO/UO(11,200),VO(11,200),WO(11,200) CQtJjt·DN/GAM/GT( 11,200) ,GS( 11,200) ,GB( 14) ,OGB( 14) ,OALP( 14) INTEGER PSW3 THETA=(NT-1)*360.0/NTI XMIN=XFW( 1) YI1IN= YFW( 1) ZMIN=ZFlrJ( 1) XMAX=XFW( IFIV) YHAX=YFW(JFW) ZMAX=ZFW(KFH) IF(NT.EQ.NSW1) GO TO 5 GO TO 13
5 CONTINUE IF(PSW3.NE.1) GO TO 2 PRINT 1, THETA FORMAT(/ I I ,5X, 'VELOCITIES AT FIXED WAKE POINTS' ,/8X,' (ROTOR AN'
$, 'GLE=' ,F6.1, 'DEG.)' ,II ,3X, 'X' ,6X, 'Y' ,6X, 'Z' ,6X, 'U' ,6X, 'V' ,6X, 'W') 2 DO 10 IW=1,IFW
DO 10 Ji'i=1,JFW DO 10 K\v=1 ,KFiv CALL PIVEL(NT,NE,XFW(IW), YFW(JW) ,ZFW(KW) ,UFW(IW,JVJ,KW),
$VFW( IW, JW, KW) , WFlrJ( IW, Ji>J, KW) , NTI) IF(PSW3.NE.1) GO TO 16 PRuiT 15 ,XFW( IW) , YFW( JW) ,ZFW(KW) , UFW( IW, JW ,KW) , VFW(IW, JW, KW) ,
$WFW(IW,JW,KW) 15 FORMAT(6F7.3) 16 CONTINUE 10 CONTINUE 13 CONTINUE
NT1=NT-1 DO 11 1= 1, NE DO 11 J= 1, NT1 UO(I,J)=U(I,J) VO(I,J)=V(I,J) WO(I, J) =iv( I, J) IF( NT. EQ. NSW1) GO TO 30 GO TO 11
30 CONTINUE IF(X(I,J).LT.XMIN.OR.X(I,J).GT.XMAX) GO TO 40 IF(Y(I,J).LT.YMIN.OR.Y(I,J).GT.YMAX) GO TO 40 IF(Z(I,J).LT.ZMIN.OR.Z(I,J).GT.ZHAX) GO TO 40 GO TO 41
110 CALL PIVEL( NT, NE,X( I, J) , Y(I, J) ,Z(I, J), U( I, J), V(I, J) , W(I, J) ,NTI) GO TO 11
41 CONTINUE CALL INTERP(X(I,J),Y(I,J),Z(I,J),U(I,J),V(I,J),W(I,J))
11 CONTINUE IF( NT. EQ. NSW1) NSW1 =NT+UT
100
12 CONTINUE RETURN END
101
SUBROUTINE PROFIL(NT,NE,NPT,NPP,SPP,XO,YO,ZO,NTI) COMMJN/LOC/X(11,200),Y(11,200),ZC11,200) COMIvION/GAMlGT( 11 ,200) ,GSC 11 ,200) ,GBC 14) ,OGB( 14) ,OALP( 14) THETA=(NT-1)*360.0/NTI PRINT 1, THETA FORHAT(!II,10X,'WAKE VELOCITY PROFILE',/11X,'(ROTOR ANGLE='
$,F6.1, 'DEG.)' ,II ,3X, 'X' ,6X, 'Y' ,6X, 'Z' ,6X, 'U' ,6X, 'V' ,6X, 'WI) DO 10 I=1,NPP XP=XO+SPP*(I-1)*(2-NPT)*(3-NPT)/2.0 yp=yo+spp*( 1-1) *( NPT -1) *C3-NPT) ZP=ZO+SPP*(I-1)*(NPT-1)*CNPT-2)/2.0 CALL PIVELCNT,NE,XP,YP,ZP,UP,VP,WP,NTI) PRINT 2,XP,YP,ZP,UP,vP,WP
2 FORHAT(6F7.3) 10 CONTINUE
RETURN END
102
SUBROUTINE PIVEL(NT,NE,XP,YP,ZP,UP,VP,WP,NTI) COMMON/LOC/X(11,200),Y(11,200),Z(11,200) COMMON/GAHlGT(11,200),GS(11,200),GB(14),OGB(14),OALP(14) NT1=NT-1 USUM=O.O VSUM=O.O IVSUM=O .0 DO 10 K=1, NE DO 10 L=1,NT1 L1=L+1 CALL FIVEL(X(K,L),X(K,L1),XP,Y(K,L),Y(K,L1),YP,
$Z(K,L),Z(K,L1),ZP,GT(K,L),UU,VV,WW,NTI) USUM=USUM+UU VSUM=VSUM+VV WSUM=\~SUM+WW
10 CONTINUE NE1=NE-1 DO 20 K=1,NE1 DO 20 L=1,NT K1=K+1 CALL FIVEL(X(K,L),X(K1,L),XP,Y(K,L),Y(K1,L),YP,
$Z(K,L),Z(K1,L),ZP,GS(K,L),UU,VV,WW,NTI) USUM=USUM+UU VSUM= VSUM+ VV WSUM=WSUM+~I
20 CONTINUE UP=USUM VP=VSUM WP=WSUM RETURN END
103
SUBROUTINE INTERP(XP,YP,ZP,UIN,VIN,WIN) COM~{)N/XWAKE/XFW( 10) , YFH( 5) , ZFW( 5) , IFW, JFltI, KFH COMNON/UHAKE/UFW(10,5,5),VFW(10,5,5),WFW(10,5,5) DIMENSION UX(2,2),VX(2,2),HX(2,2),UQ(2,2,2),VQ(2,2,2),HQ(2,2,2) DIMENSION UY(2),VY(2),WY(2) IW=1 JW=1 Kl>l= 1
2 IF(XP.LE.XFH(IH» GO TO 4 IW=IW+ 1 GO TO 2
4 IF(YP.LE.YFIoI(JtV)) GO TO 6 JW=JW+1 GO TO 4 .
6 IF(ZP.LE.ZF1v(KH» GO TO 8 KH=KtI+ 1 GO TO 6
8 CONTINUE IW=IW-1 JW=JW-1 KlrJ=Ktv-1 HX=XFW(IW+1)-XFW(IW) HY=YFW(JW+1)-YFH(JW) HZ=ZFW(KH+1)-ZFW(KH) DO 50 IQ=1,2 DO 50 JQ=1,2 DO 50 KQ=1,2 IS=IW+IQ-1 JS=JW+JQ-1 KS=KlrJ+KQ-1 UQ(IQ,JQ,KQ)=UFW(IS,JS,KS) VQ(IQ,JQ,KQ)=VFW(IS,JS,KS) WQ(IQ,JQ,KQ)=WFW(IS,JS,KS)
50 CONTINUE DHX= (XP-XFltJ( IW) ) IHX DHY=(YP-YFW(JW»/HY DHZ=(ZP-ZFW(KH»/HZ DO 10 J=1,2 DO 20 1=1,2 UX(I,J)=UQ(1,I,J)*(1.0-DHX)+UQ(2,I,J)*DHX VX(I,J)=VQ(1,I,J)*(1.0-DHX)+VQ(2,I,J)*DHX HX(I,J)=WQ(1,I,J)*(1.0-DHX)+WQ(2,I,J)*DHX
20 CONTINUE UY(J)=UX(1,J)*(1.0-DHY)+UX(2,J)*DHY VY(J)=VX(1,J)*(1.0-DHY)+VX(2,J)*DHY h~(J)=WX(1,J)*(1.0-DHY)+WX(2,J)*DHY
10 CONTINUE UIN=UY(1)*(1.0-DHZ)+UY(2)*DHZ VIN=VY(1)*(1.0-DHZ)+VY(2)*DHZ WIN=WY(1)*(1.0-DHZ)+WY(2)*DHZ RETURN END
104
SUBROUTINE CONLP(NT,NE,DELT,UT) COMMON/LOC/X(11,200),Y(11,200),Z(11,200) COMMON/VEL/U(11,200) ,V(11,200),W(11,200) COMMON/VEO/UO(11,200),VO(11,200),WO(11,200) DT=DELT/UT NT1 =NT-1 DO 20 I=l,NE IF(NT.LE.l)GO TO 11 DO 10 J=l,NTl X(I,J)=X(I,J)+(3.0*U(I,J)-UO(I,J)+2.0)*DT/2.0 Y(I,J)=Y(I,J)+(3.0*V(I,J)-VO(I,J))*DT/2.0 Z(I, J) =Z( I, J)+<3 .O*W( I, J)-1t10(I, J) *DT/2.0
10 CONTINUE 11 CONTINUE
X(I,NT)=X(I,NT)+(U(I,NT)+1.0)*DT Y(I,NT)=Y(I,NT)+V(I,NT)*DT Z( I, NT) =Z( I, NT)+I.J( I, NT) *DT
20 CONTINUE RETURN END
105
SUBROUTINE SHEDVR(NT,NE) COMMON/GAM/GT(11,200),GS(11,200),GB(14),OGB(14),OALP(14) NT1=NT-1 NE1=NE-1 DO 10 1= 1 , NE 1 GS(I,NT)=GB(I) GS(I,NT1)=OGB(I)-GB(I) OGB( I) =GBCI)
10 CONTINUE GT(1,NT1)=GB(1) GT(NE,NT1)=-GB(NE1) DO 20 1=2, NE1 11=1-1 GT(I,NT1)=GB(I)-GB(I1)
20 CONTINUE RETURN END
lOb
SUBROUTINE FIVEL(X1,X2,X3,Y1,Y2,Y3,Z1,Z2,Z3,GAMMA,UU,VV,WW,NTI) AX=X2-X1 AY=Y2-Y1 AZ=Z2-Z1 BX=X2-X3 BY=Y2-Y3 BZ=Z2-Z3 CX=X1-X3 CY=Y1-Y3 CZ=Z1-Z3 CCAX=CY*AZ-AY*CZ CCAY=AX*CZ-CX*AZ CCAZ=CX*AY-AX*CY CCAV=CCAX*CCAX+CCAY*CCAY+CCAZ*CCAZ IF(CCAV.LT.0.0001) GO TO 10 B=SQRT(BX*BX+BY*BY+BZ*BZ) C=SQRT(CX*CX+CY*CY+CZ*CZ) ADB=AX*BX+AY*BY+AZ*BZ ADC=AX*CX+AY*CY+AZ*CZ VF=(ADB/B-ADC/C) *GAHMA/( 12.56637*CCAV) UU=CCAX*VF W=CCAY*VF WW=CCAZ*VF
9 GO TO 11 10 UU=O.O
W=O.O WW=O.O
11 CONTINUE RETURN END
107
SUBROUTINE ALDNT(ALPHA,CADOT,CL,CD,CN,CT) COMMJN/CLTAB!TA(30), TCL( 30) ,TCD( 30) ,NTBL, ASTAL, TCR REAL K PI=ARCOS(-1.0) ASTR=ASTAL*PI/180.0 IF(ABSCALPHA).GE.ASTR) STALL=1.0 IF(CADOT.GE.O.O.AND.ABSCALprill).LE.ASTR) STALL=O.O IF(STALL.EQ.1.0) GO TO 1 GO TO 2 CONTINUE SALPH=SIGNC1.0,ALPHA) SADOT=SIGNC1.0,CADOT) K=0.25*SADOT*SALPH*(3+SADOT) CHEATL=1.0 CHEATD=1.0 GL=CHEATL*C1.4-6.0*(O.06-TCR)) GD=CHEATD*(1.0-2.5*(O.06-TCR)) ADL=(ALPHA-K*GL*SQRTCABS(CADOT)))*180.0/PI ADD=(ALPHA-K*GD*SQRTCABS(CADOT)))*180.0/PI CALL ATAB(ADL,CL,DUM) CALL ATAB(ASTAL,CLS,DUM) SL=(CL-TCLC1))/(ADL-TA(1)) SLl1=( CLS-TCL( 1) )/CASTAL-TA( 1) ) IF( SL. GT. SU1) SL=SLM CL=TCL(1)+SL*(ALPHA*180.0/PI-TAC1)) CALL ATABCADD,DUM,CD) GO TO 3
2 AD=ALPHA*180.0/PI . CALL ATAB(AD,CL,CD)
3 CONTINUE CN=-CL ;!COS( ALPHA) -CD*SIN( ALPHA) CT= CL*SIN(ALPHA)-CD*COS(ALPHA) RETURN END
lOd
SUBROUTINE ATAB(ALD,CL,CD) COHHON/CLTAB/TA( 30) , TCL(30) ,TCD(30) ,NTBL, ASTAL, TCR AD=ALD NTBL1=NTBL-1 IF(AD.LE.O.O) AD=AD+360.0 IF(AD.GE.O.O) AL=AD IF(AD.GE.180.0) AL=360.0-AD IF(AD.GE.360.0) AL=AD-360.0 DO 10 1= 1 , NTBL 1 J=I IF(AL.GE.TA(I).AND.AL.LE.TA(I+1» GO TO 20
10 CONTINUE 20 XA=(AL-TA(J))/(TA(J+1)-TA(J))
CL=TCL(J)+XA*(TCL(J+1)-TCL(J)) CD=TCD(J)+XA*(TCD(J+1)-TCD(J)) IF(AD.GT.180.0.AND.AD.LT.360.0) CL=-CL RETURN END
109
2 0.1000 2.0 4.0 .25 .12 30 0.30 10.9
0.0 0.0000 0.0085 2.0 0.2000 0.0094 5.0 0.5000 0.0125 7.5 0.7700 0.0177
10.9 0.8500 0.0465 11.0 0.8600 0.0935 15.0 0.8200 0.1705 17 .5 0.7900 0.2335 21.0 0.7500 0.3285 30.0 0.9800 0.6300 40.0 1.1200 1.01 00 50.0 1.1000 1.3700 60.0 0.9700 1.7000 70.0 0.7100 1.9300 80.0 0.4100 2.0500 90.0 0.0900 2.0700
100.0 -0.2300 2.0400 110.0 -0.5300 1.8900 120.0 -0.8000 1.6900 130.0 -0.9800 1.4100 140.0 -1.0500 1.0900 150.0 -0.9400 0.7200 154.0 -0.8400 0.5600 160.0 -0.7000 0.3700 164.0 -0.6800 0.2700 168.0 -0.7100 0.2100 170.0 -0.7400 0.1800 172.0 -0.8400 0.1500 175.0 -0.5000 0.0800 180.0 0.0000 0.0300
111 5 3 6 0.75 0.0 1.0 -1.875 3 6 0.75 2.0 1.0 -1.875 3 6 0.75 6.0 1.0 -1.875 3 6 0.75 10.0 1.0 -1.875 3 6 0.75 16.0 1.0 -1.875
110
7.3 Blade Force Data
Blade force data from the two-dimensional experiment is presented in
this appendix. A minimum of two runs for five different combinations of
tip to wind speed ratio and blade geometry cases were made. The first set
of data displays are non-dimensional normal and tangential forces taken
during the fourth revolution. The second set of data displays are non
dimensional normal and tangential forces for each entire run. The
resolution of these plots is, of course, low in comparison with the plots
for a single revolution.
111
20
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+ + I + + +++++ + !
-2 t*-+ + I
+++ + I I , I
0 30 60 90 120 150 IB0 210 240 270 300 330 360 390 420 450
ROTOR ANGLE-leBe-
----
lSI r-----------.--.. -"--.-.----.~--~------------- ~----- If) , + '\I"
+ lSI + + - N
+ '\I"
+ lSI + + - m + ('I')
+ lSI + + - to
+ ('I')
+ lSI + + - ('I')
+ ('I')
+ lSI + - lSI + If) +
('I')
N "- + lSI • • I + - "- lSI Z ~ + N CD
Ul + lSI ..... + ....
+ lSI I + - '\I" W
+ N .J + ~
+ lSI Z - .... a: + N + ~
+ lSI 0
+ t-+ - CD 0 .... ~ + + lSI + - If) + --+
+ lSI + + - N
+ --+ lSI +
+ - m + + lSI +
-I - to
+ + lSI + - ('I') + + I I I I t lSI
('I') N .... lSI .... N I
¥ La..
129
I I I
(S) (S) N .....
++ +
+
+ +
+ + + + :
.f+ ++
+ + + t + + + + + + + +.
++ +
+ I ++'
(S)
C I.&..
130
+ +
+ +
I
(S) .....
+
('I) D Z
+
+
I
+
+
+
+
t +
+
+
+
, (S) N J
+ + + + + +
+ +
I I
(S) ('I) J
(S) It) v
(S)
- N V
(S) - en ('I)
(S)
· CD ('I)
(S) - ('I) ('I)
(S)
- (S) ('I)
(S) 0
· "- (S)
N 00 (S) -'
(S) J - v W N ...J
t!1 (S) Z
· ..... a: N~ (S)O
- 005 .... ~ (S)
- It) ..... (S) - N ......
(S) - en
(S) - CD
(S)
· ('I)
(S)
20
10
Fn 0
~I r++ +
-10 ~ + +
-20
-30
0 30
+ +
+
L
+ + +++++
i
+ +
+
I
+ +
~~1-
+ +
+++++++++++ + + + ++ • -of
+ ++++ + ++
+ + + +
N=3
TSR-5
~L --'- _L I --'- i L --'-
+ +
I
- --,
+ +
I
+ + ..
60 90 120 150 le0 210 240 270 300 330 360 390 420 450 ROTOR ANGLE-10e0·
(SJ __ .. ___ • ___ - ......-.._~ ___ _...."'M.'<O
It') + v + + + (SJ
- N • V + + (SJ + - 0) + (T) + + (SJ + - CD + (T)
+ + (SJ + - (T) + (T) +
+ (SJ + - (SJ
+-1 (T)
III + (SJ 0 (T) D + . f'.. (SJ D Q: + N m Z U) + (S) ~ ... + (S) I + - V W - N -l + CJ + (S) Z + - ... a: + N
Q: + + (S) 0 + ~
+ -t - m 0 ..... Q:
+ (S) + - III + ..... +
+ (S) + - N + .... ++
m + - 0) + + + (S) + - CD + +
+ (SJ + - (T) + +
I I I t. (SJ
(T) N -- (S) .... N I
~ u..
132
--l 4~i -------------- I
I 3 r +++
+ + 2 ... + +
N=3
TSR-5
+ + +
~I 1 l + +
+ +
+
+
+ + + + f"t e I +
+ +
t+++ -1
+++++++ +
+ + +' + + +++ + + +++++++ + ++++++
-2 e 3e se se 12e lse lee 21e 24e 27e 3ee 33e 3se 3se 42e 4se
ROTOR ANGLE-leeeo
.... Ln U U zf}j
I-
•
-----.---.-----------r--------------~ I() f ~
--~
135
-I() D a zll:::
(J) I-
(J) z o H I~ ...J o > W Il:::
L.. o Il::: W ~ 1: ~ Z
---~-·---l ".--~ -~---.. -----~----.. ---.~-,----
.---- - It) I
·-~··---I~--~-·---------- I
I I
tn Z
Ul
0
- I • 0: Z en t-
t-I t-::l ...l 0 > W 0:
La.. 0
t:t: W III 1: ::J Z
•
~--------~~:=====~m~==------~--------~7 _____ _ ~--------
m -
----"---.---~--------,
~
I i
In I - I I I Q:
~ Z U)
r-- 0 t-t I-::J -l 0 > LaJ Q:
U. 0
Q: l.aJ Pl %: :l Z
CS) cg - -I cg .., h..
_~_. ___ ,_' ___ ___ .. ___ 4_=-______________ _ 137
~----------------------~----------------------~~ -
N ~ • • 0:: Z U) U)
~ Z 0 H ~ :J
S > 1.&.1 0::
u.. 0
ei ~ ::J Z
CSt CSt IS) IS) CSt CSt an c an IS) - I -L.. I
.- .. _--_.''''''-_. 138
r------------------------r----------------------~~ ...
N ~ n n ~ z 00 m r z
0 ~
r ~ ~ 0 > W ~
~ 0
~ w m L ~ Z
~-------------------------------------------.------------------------~ 139
N o z Ul • ct: Ul I-
140
Ul
.... Ul Z o 1-1 I:J ..J o > w ct:
u.. o ct: W tIl 1:
~
IS) .....
N D Z
Ul
~ Ul t-
IS)
~ La..
141
IS) ..... 1
Ul Z o H t::J -1 o £:j DC
La.. o DC W ~ ~ :J Z
~--------------------~----------------------,~ ~
~ • N ~ • • Z ~ m
m z ~ 0
~ ~ ~ ~ 0 > W ~
~ 0
~ W ~ ~ ~ Z
142
r------------------------r-----------------------,~ ..
~ • N ~ I I Z ~ ~ ~ Z ~ 0
~ ~ ~ ~ 0
~ ~ 0
~ ~ M ~ ~ Z
143
--------
~---r-----
II)
m
•
z
N r0-
o • •
H
Z D!; m I-
I-:l ..J 0 > I.&J D!;
I.&. 0
D!;
Itt 1:
~
~----:========-------~m~--------~--------~7 ~
m -144
111
~
•
0
N I'-
H
I I Z 0: en t-
I-
:3 o > W 0:
u.. 0
0:
III 1:
2
... 1
145
U')
(J) z
•
0
N ""
.... • m Z
t-t-:::::I ..J O > l&J ~
U. 0
~ l&J = 4 :::::I Z
146
~----------------~----~----------------------~~ ~
~
N •
• N Z • 00 ~
00 Z ~ 0
~ ~ ~ ~ 0 > ~ ~
~ 0
ffi ~ ~ ~ z
147
N I Z
-
Ul •
N I tt: U) t-
(S) an
(S)
c: u.. 148
______________ . __ -.Ul -
gJ -I
U) Z o H t-:J ..J 0
~ u.. 0
tt: 1aJ til ~ :J Z
III
~
•
0
~ C\I
H
~ Z en ....
.... :J ..J g ~ La. 0
0:
Jd :E
~
r-
an • N N • • Z 0:
(J) t-
... ~ >
~ -)
~ .-->
~ -~
<p ~
r-
- I
lSJ lSJ -150
--· -· · · · · · · · · · · · ·
· · · · · · · ·
· · · ·
I J lSJ -I
II) -'II" -p) -N ---lSJ -en
CD
"" to
II)
'II"
N
-
(J) Z o H
~ ..J o ~ 0:
U. o 0: W
~ Z
~-----------------------r------------------------'~ ..
~ ~ I I ffi z m r z
0 ~
~ ~ 0
~ ~
~ 0
~ ~ ~ ~ ~ z
151
r----------r---,-.-----'--!'! --l
C') a z
It) I
D:': (I) t-
CSl 111
152
CSl 111 I
CSl CSl .... I
!
(I) Z 0 H t-::J ..J 0 > W D:':
U. 0
D:': W I:Q l: ::J Z
------
CS) ....
.----~------
It') (') I I 0: Z U)
r U) z 0 H r :J .J 0 > ~ lL. 0
0: W ~ 1:
~
cg cg .... I ~
lL.
153
, I I (I) I
Z
It)
0
C') I U ~ Z (I) .... H ~ :::l ...J 0
I > w , ~
I I..&-0
,
~
I
w I ~ :::l Z I
I
7.4 Wake Velocity Data
Wake velocity data from the two-dimensional experiment are presented
in this appendix. All of the data presented herein were taken at one
rotor diameter downstream of the axis of rotation. Data for five different
combinations of tip to wind speed ratio and blade geometries (number of
blades) are presented. The first set of data displays the streamwise
perturbation velocities measured during the fourth revolution of the rotor.
The second set of data displays the lateral perturbation velocities meas
ured during the fourth revolution of the rotor. Each set of data also
contains a plot of the velocities predicted by VDART2.
155
I-' \Jl (J\
-1.0
-.5
o ~ I
.5
1.0
U/U'"
-La 4 i
- 5 I
TSR-2.~ ] NB=2
0
.5 I 1.0 ~
ROTOR ANGLE=1110° 1155° 1245°
* * * * * * * * * * * ~ / 1 I~ ~*II~ f"'ijfl :-+ ff *+==+ rwt ~
* * * * * * *
12913° 13813° 1425°
* * * * * * * * * * * ~ J*~ 11\ ~ r~~
1 R* ~ + )(IP
* * * * * * *
+ --ANALYTICAL DATA, * --EXPERIMENTRL DATA
I ~I --.J
1
-loB
-.5
13 "l
.5
loB
W/UID
-loB
-.5
TSR-2.5
NB"2
.: ~ 1.0
ROTOR ANGLE=11113° 1155°
* ~* )( *~ * "W=P IMI.I *~¥I*I)( *'* .... * ;;
12913° 1380°
1245°
* * ~ * ,If ¥ "I *~ 1 .!+..'7
1425°
* *** ** * ~I*I*I*I I;ALI ~¥* ,'tK,*P.f. ~VI*I*I-~
+ --ANALYTICAL DATA, * --EXPERIMENTAL DATA
i--' \Jl cc
-1.13 f·-- ROTOR ANGLE-=11-1-13 0 1155 0 1245 0
-0: 7* .... \ * ~ * ~. ;::;. ~ ~ * ~ ,0fF~¥C+
I .5 -t
I
1. 13 .
U/U", 1 129130 138[:)0 1425 0
-1.13
?,*,*~ o *
* * \*-+
? .. ,,~ ;;:::-- 1-
-.5
TSR=5
NB=1 0
.5
1. 0 -i I
+ --ANA~yTICP~ DATA, * --EXPERIMENTRL DRTA .__ I
------ I ROTOR ANGLE-1110" 1155° 1245° -1.0 -,
-.5
i I * * * * * * I 1 * * * ~ ~ j!",. H. *.JL* * * * * I 0 1 "'~' I~ + ~ r ~ +=f"!l;lji;: t "'III iii! I~:::l-
I .5 J I I
1-'1 \.Jl1
\01 I
I i
I I I
La 1 w/u", I
1290°
-l.0 ]
-.5 1
TSR"'5 I * ... if 7<,. NB
c
1 a l ...... ;;I
I
.5
1380° 1425°
* * ~~~* I ~ ~ * ***~*~ II"'" 'I-
ROTOR ANGLE~1110· 1155· 1245° -1.0
-.5
o
if if
.S
1.0 -
U/U .. - 1
gl -1.0 1 1290· 1380·
if If If If If if if if
1425°
if If If If If
.>l- I 1-
-.S
TSR"'S NB .... 2
0
if If If If
.5
+ --VDART2 DATA, If --EXPERIMENTAL DATA 1.0
•
• • In In -r N N -r ...
a: I-a: ~
.J a: I-z W :E H 0:: W 0.
• • X In (Sl W In CD I .... (T) I .... ....
* a: I-a: ~
N I-0:: a: ~ • (Sl > I .... ,
I + W • .J (Sl l:) en Z N a: ... ~ 0 I-0 0::
CSl In CSl In CSl CSl In CSl In CSl
.... 8 In N ... ::J I a " ~ In 3: Ul .z I-
161
i-' IJ\ I\l
-1.0
-.S
o
.S
1.0
-1.0
-.S
TSR-S
NB-3 o
.S
1.0
ROTOR ANGLE-l 110- 11SS- 1245-
1290- 1380- 1425-
+ --VDART2 DATA, H --EXPERIMENTAL DATA
~ UJ
-1.0 -
-.5 -
o -
.5 -
1.0 -
Iol/U ...
-1.0 -
-.5 -
TSR-5 NB-3 o -
.5 -
1.0 -
ROTOR ANGLE~1110° 1155-
C K*K ~ *** ~~ -~'.:A=3V 1+ 1+ 1+ 1+
1290-
It 1+
~It ~ ~
1380-
Qt! ~* .~
1245-
1+ 1+ ¥ "~~H\L:
1425-
~I* ~ "t 1+ ~
+ --VDART2 nATA, * --EXPERIMENTAL DATA
J--~ -ROTOR ANGLE"111l1l0 1155° 1245° -1.0 I
* -.5
o ~ r~ .. + "* +-itf* ~ +J'·r ~~
I * * * * *
.5 ~ I
f-' 1.0 J 0'\
+-U/U
ID
I 1290° 1380° 1425° -1.0 l
I I
- 5 1 TSR"7.5 I
NB"'2 01 >
.5 J * * * I I
I
I + --ANALYTICAL DATA, * --EXPERIMENTAL DATA 1. 13 -\
\. --1 . ~ ROTOR RNGLE-1110° 1155 0 1245 0
-1.0 I
~i 0\1 Vl'
i !
-.5 ~
I~ ~ ~ i "'v ~~K ,K ",J" 13 I~ * ~~ * * * * I r
.5
loB
\'VU m
-loB
-.5
TSR~7.5
NB=2 o
I
! I
1 ~ !
.5 ~ loB
I
I , I I
129130 138130 1425 0
~ M.~ *.* * * **~~. *
* '*"
+ --ANALYTICAL DATA, * --EXPERIMENTAL DATA
DISTRIEUTION:
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Arnan Seginer Professor of Aerodynamics Technion-Israel Institute of
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