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

When printing a copy of any digitized SAND Report, you are required to update the

markings to current standards.

Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty. express or implied or assumes any le~al liability or responsibihty for the accuracy. completeness. or usefulness of any informatlOn, apparatus, product, or process disclosed. or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product. process, or service by trade name. trademark. manufacturer. or ,otherwisebdoes not necessarily constitute or imply its endorsement. recommendation. or favoring y the United States Government. any agency thereof or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government. any agency thereof or any of their contractors or subcontractors.

Printed in the United States of America Available from

National Technical I nformation Service U. S. Department of Commerce 5285 Port Royal Road Springfield, VA 22161

NTIS price codes Printed copy: $11 .00 Microfiche copy: AOl

\

•

,

Distribution Category UC-60

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

1

1

2

4

4

5

9

10

13

15

16

20

21

21

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)

42

43

51

51

55

63

63

65

66

67

69

70

89

111

155

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

Fn

f-'i

~I

-10

++++++++++ ++ + +

++ + + + .

~ +! I' +++ + -----l

+ ! +

+

+ + + +

+ + +

+ + +

+++++

0 30 S0 90 120

+ + I

+ ! +

+

150

+ I !

Na 1 +!

TSR-5

180 210 24113 270 31130 330 3SB ROTOR ANGLE-108Bo

+

390

i I i I

+ i + !

+ ! +++!

420 450 i I

t I

. .1

~ w

•

213

------------------=-1 l \

I ! . , , i

i t,

++++++++++ ++ +

+ + + + +

+ +

10

+ + Fn 0 PF~--------------------

+ +

-r± ----------_. ! + ----·----4

-10

-213

-313

+

+

+

+

+ +

+ + +++++

+ +

+

+ +

+

+

+ + +

+

N=l

TSR-S

+

+

+

+ +

+++1

13 313 SB 913 120 1513 1813 2113 2413 2713 3130 3313 3SB 3913 4213 4513 ROTOR ANGLE-1B8Bo

-----

(S) -', In

'it + +

+ (S) + - N + 'it + + (S)

++ - 0') • (T)

+ + (S) + - to + (T) + + (S) ... · (T) + (T)

+ + (S) + · (S) + (T)

+ (S) 0

In + (S)

.... a + "" m a (k: + N (S) Z (J) + .... t- + (S) I + - 'it W + N -' + (!I

+ (S) z - .... a: + N

(k: . 0 + (S) t-+ · m 0 + .... (k: +

+ (S) + - In oj .... +

+ (S) + - N --+ +

(S) + - m +

+ +

(S) + - to + + +

+ (S) + - (T) + + + (S)

_I "'" I I I

(T) N .... (S) .... N I

~ l&..

114

f--' f--' \Jl

4

3 ~

2 ~ +

+ 1 ~

+

+

+

++ + +

+ N-l

TSR-S

+

+

+

++++++ + +

++ ++ +

+ '* + +T+' +++ Ft e I + +++++ +

t++++ -1

-2

+

+

+ +++++

+

+

+ +

e 3e se se 12e lse lee 21e 24e 27e 3ee 33e 3se 3se 42e 4se

ROTOR ANGLE-leeeo

i--'

~

20

10 I !

-------~ ++++++

+ + fn Ii! ~ + + + +++++++++ ~+++++ +

+++ + + + -----.;+%:.-- ~

-10

-20

-30

+ +

+ +

+ +

+ + + +

++++

t + + t + ! 1

++ I N ... 2

I + ; + , +++1

I

TSR-2.5

o 30 sa 90 120 150 180 210 240 270 30e 330 3SB 390 420 45e ROTOR ANGLE-108e-

+ +

+ + -I-+ + + + +

+ +

+ + -I-+ + +

+

+

1

!+ -+1

~I ~I +1 it

+ + +

+ +

+ +

+

+

+

+ +

+

117

-I-

+

N I

Z

+

+

+ + + +

+ +

• N I

0:: Ul t-

IS) N I

IS) (I') I

lSI If) ~

IS) N ~

CSJ en (I')

IS) CD (I')

IS} (I') (I')

lSI IS} (I')

IS)

"-N

IS) .". N

IS) -N

IS) CD

lSI In .....

IS) N

IS) en

IS) CD

IS) (I')

IS)

• IS) CD IS) -I W ..J (!1 Z a: 0:: 0 t-0 0::

+ + +

+ +

+

N I

IS) If) ..,. IS) N ..,. (5,) (7)

~

(5,) to (Y)

(5,) (Y) (Y)

IS) IS) (Y)

(5,) • "- (5,)

N CD (5,) ....

t'S) I ..,. W N ...J

(!I IS) Z .... a: N

0:: (S) 0

I-m 0 - 0::

t'S) If)

--(5,) N .... IS) (7)

IS) U)

IS) (Y)

IS)

f-' f-'

\D

4

3 I-

2 ...

1 ...

tt ± Ft B+ ±++

-1 I-

+

++ + ±

+ ±

+

N-2

TSR-2.5

++± T ++ ++ +++++

+++++++++++++++ ++++ ++ + ±

T++T

+

+±± ± +

-2 ~--~--~--~--~--~--~~--~--~--~--~--~--~--~~--~~ B 3B SB 9B 12B 15B 18B 21B 240 270 300 330 3S0 390 420 450

ROTOR ANGLE-lBSB·

+ +

+ +

+

+ +

CSl CSl CSl N ....

c: u... -_ .. 120

+

+

CSl ..... I

N n z

+ +

+

If)

U r.k: Ul I-

+

+

CSl N I

+

+

+ +

+ + +

+ +

+

CSl (T) I

CSl U) (T)

CSl (T) (T)

CSl CSl (T)

CSl 0

"-IS)

N CD (S) ....

CSl I 'It' W N ..J

(!) IS) Z .... a: N

0::: (S) 0

I-CD 0 - 0:::

t'Sl If')

--CSl N .... CSl en

CSl U)

CSl (T)

CSl

r····'"~· -~ .-,,-' '.'~' ''''~~.-''~' """,'- ,"-- - -'-.. '--'" _ .. r I ! !

. (S) N

• • (S) ....

i

i ! ! i I + + 1+

+t +1 +1 +f :1 + '

+ ! + i

+ I + I + I + t + I + , + I + ,

++ I .

+t+++ I ++

+ + +

• - I

(S)

C t..

121

+

+ +

+

• (S) .... I

+

+

N a z

+

+

I

+

+

+

+

Ul I ~ (f) ......

+

+

I

(S) N I

+ +

+

I

+ + + +

+

I

(S) (\')

I

(S) - N ....

(S) - en (\')

(So) - CD (Tj

(So) - (Tj (Tj

(So) - (So) (\')

(So) • ",(So) - rum (So) ....

(So) I - .... W N -I

(!)

(So) z - .... a: ru a:= (So) 0

..... - m 0 .... a:= CSl - Ul .... (So) - N .... (5) - en

(So) - CD

(So) - (\')

(So)

(S;)

+ If)

v + + (S;) + - C\J + V +

+ (S;) + · en + ('I')

ft+ (S;) + - to + ('I')

+ + (S;) + - ('I') + ('I')

+ + (S;) + · (S;)

+ ('I')

+ If) + C\J U (S;) 0 (S;) + u ~ - I'-CD + Z Ul C\J (S;) + ..... ..... + (S;) I + - V W + C\J ..J + f..!}

+ (S;) Z + · ..... CI: . C\J ~ + 0 + (S;) ..... + - CD 0 + .... ~ +

+ IS) + · If) + ....

+ + Ci) + - C\J + .....

+ + (S;) + - en + +

+ (S;) + - to + +

+ (S;) + - ('I') + +

I I -'- .J.. ++ I I I I I I

(S;)

(S;) (S;) (S;) IS) CSl (S;) C\J .... ..... C\J ('I')

C I I l&..

122

I-' r0 w

4

+++

3 + + N=2

+ + TSR-5

2 I- + + +

+ + + 1 t- +

+ + +++++

+ ++ + ++++ ++ + ++ ++++++ + Ft 0 I + + +++ + + ++ +

+ +

-1 t+++ +

-2 o 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450

ROTOR ANGLE-1080°

4

3 r

2 r +

+

+

++ + +

+

+

+

+

N=2 +

TSR=5 +

+

+ +

+ +

++++ _. __ . __ .. _. --.---------~--+_ ... _-" -" -+..±~--~+~.-.-- +

-+ ++++ +++++++ + +++ ++++

-2 I~--~--~~--~--~~--~--~----~--~----~--~----~--~----~--~--~ o 30 60 90 120 150 la0 210 240 270 300 330 360 390 420 450

ROTOR ANGLE-10a0°

~ V1

•

--l ---I !

I I 20

! I

10

Fn iii I + ++++ + + ++

+*--_._---+

"" ---~---·--··-----1

I I

I

-10

-20

-30

+ +

+ +

+ +

++++ ++

+ + + +

+ N"'2 + +

+ + TSR-7.5 + +

+ + + +

+ + + +

o 30 Eija 90 +tf20 150 le0 210 240 270 300 330 360 390 420 450 +++++ ROTOR ANGLE-10e0· +

+ ++

I

I

20

10

Fn 0

~I -10 t+ +

I- + +

-20 ~ +

-30 t + +

+

r +

-40 I +

0 30 S0

+ +

+

+ +

+ +

+ +

+++++

+

+++++ ++ +

-""'- + ++ ....

++

N-2

TSR-7.5

, --1

'T

+ ++

++++ ++

i i

+ +

--1

+

+

+

+ +

+1 -=!::I

90 120 150 180 210 240 270 300 330 3S0 390 420 450 ROTOR ANGLE-l0e0·

lSI In + V +

+ lSI + - N + V + + lSI + - CD + ('I') + • + lSI + - CD + ('I') + + lSI + - ('I') + ('I')

+ + lSI + - lSI + ('I') III t-• + lSI lSI N f\-

" · f\-(J) a I + N lSI Z r.r: + -Ul + lSI I I-

+ - v W + N ...J + <--' + lSI Z + · ... a: + N r.r: +

0 + lSI I-+ · (J) 0 + ... r.r: + lSI + + - In ..... +

+ lSI + - N + ..... +

+ lSI + + - CD + +

+ lSI + - CD oj

+ + lSI + · ('I') + + I i I i t lSI

('I') N - lSI ..... N I ..,

u... 127

4 1

3 t- N-2 ++++

TSR-7.5 + + 2 t- +

+ + +1 + I

~I 1 ~ + +1 + + + !

++ + I Ft 01 + +++ + 1 ++++

+ + I + + I +++ + + + +++ + -1 t- + I

+ + 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:

TID-4500-F66 UC-60 (283)

Aero Engineering Department (2) Wichita state University Wichita, KS 67208 Attn: M. Snyder

W. v,entz

R. E. Akins, Assistant Professor Department of Engineering Science

and Hechanics Virginia Polytechnic Institute and

State University Blacksburg, VA 24060

Alcoa Laboratories (5) Alcoa Technical Center Aluminum Company of America Alcoa Center, PA 15069 Attn: O. K. Ai

A. G. Craig J. T. Huang J. R. Jombock P. N. Vosburgh

Mr. Robert E. Allen General l'lanager Dynergy Corporation P.O. Box 428 1269 Union Avenue Laconia, NH 03246

American Wind Energy Association 1609 Connecticut Avenue NW Washington, DC 20009

E. E. Anderson South Cal;ota SCllool of Mines

and Technology Department of ~iechanical Engineering Rapid City, SO 57701

Scott /\nderson 318 r'lill is !lall University of Vermont Burlington, VT 05405

G. T. Ankrum DOE/Office of Commercialization 20 Massachusetts Avenue NW Mail Station 2221C Washillgton, DC 20585

166

•

Holt Ashley Stanford University Department of Aeronautics and

Astronautics Mechanical Engineering Stanford, CA 94305

Kevin Austin Consolidated Edison Company of

New York, Inc. 4 Irving Place New York, NY 10003

B. H. Barksdale, Jr. Hayes, Seay, Mattern, and Mattern 1315 Franklin ~oad SW Roanoke, VA 24016

Dr. P. J. Baum Institute of Geophysics and Planetary Physics

University of California Riverside, CA 92521

F. K. Bechtel Washington State University Department of Electrical Engineering College of Engineering Pullman, WA 99163

M. E. Beecher Arizona State University Solar Energy Collection University Library Tempe, A7, 85281

Leon Bjervig Civilingenior, MCIF "Osterbyhus", 6990 Dlfborg DK6990 DENMARK

K. Bergey University of Okl.ahoma Aero Fngineering Department Norman, OK 73069

Steve pJake Wind Energy Systems Route 1, Box 93-A Oskaloosa, KS 66066

Robert Brulle McDonnell-Douglas Aircraft Corporation P.O. Box 516 Departnlcnt 341, Building 32/2 St. Louis, MO 63166 167

Otto de Vries National Aerospace Laboratory Anthony Fokkerweg 2 Amsterdam 1017 THE NETHERLANDS

R. I-\'al ters West Virginia University Department of Aero Engineering 1062 Kountz Avenue Morgantown, WV 26505

E. J. I'larchol Bonneville Power Administration P.O. Box 3621 Portland, OR 97225

D. F. Warne, Manager Energy and Power Systems ERA Ltd. Cleeve Rd. Leatlwrhead Surrey KT22 7SA ENGLAND

G. R. Watson, Project Manager The Energy Center Fennine House 4 Osborne Terrace Newcastle upon Tyne NE2 INE UNI'fED KINGDOf.l

R. J. Watson Watson Bowman Associates, Inc. 1280 Niagara St. Buffalo, NY 14213

R. G. ~\atts

Tulane University Department of Mechanical Engineering New Orleans, LA 70018

W. G. Wells, P.E. Associate Professor Mechanical Engineering Department Mississippi State University Mississippi State, MS 39762

T. Wentink, Jr. University of Al.Bska Geophysical Institute Fairbanks, AK 99701

168

•

•

Texas Tech University (3) P.o. Box 4389 Lubbock, TX 79409 Attn: K. C. Mehta, CE Department

J. Strickland, ME Department J. Lawrence, ME Department

Fred Thompson Atari, Inc. 155 Moffett Park Drive Sunnyvale, CA 94086

J. M. Turner, Group Leader Terrestrial Energy Technology Program Office Energy Conversion Branch Aerospace Power Division Aero Propulsion Laboratory Department of the Air Force Air Force Wright Aeronautical Laboratories (AFSe) Wright-Patterson Air Force Base, OB 45433

United Engineers and Constructors, Inc. Advanced Engineering Department 30 South 17th Street Philadelphia, PA 19101 Attn: A. J. Karalis

University of New Mexico (2) New Mexico Engineering Research Institute Campus, P.O. Box 25 Albuquerque, N.M. 87131 Attn: G. G. Leigh

University of New Mexico (2) Albuquerque, NM 87106 Attn: K. T. Feldman

Energy Research Center V. Slog lund

ME Department

Jan Vacek Eolienne experimentale C.P. 279, Cap-aux-Meules lIes de la Madeleine, Quebec CANADA

Irwin E. Vas Solar Energy Research Institute 1617 Cole Blvd. Golden, CO 80401

Leo II. Soderholm Iowa State University Agricultural Engineering, Room 213 Ames, III 50010

Bent Sorenson Roskilde University Centery Energy JGroup, Bldg. 17.2 IMFUFA P.O. Box 260 DK-400 Hoskilde DENNARK

Southwest Research Institute (2) P.O. Drawer 28501 San Antonio, TX 78284 Attn: W. L. Donaldson, Senior Vice President

R. K. Swanson

Rick Stevenson Route 2 Box 85 Springfield, ~1O 65802

Dale T. Stjernholm, P.E. Mechanical Design Engineer Morey/Stjernholm and Associates 1050 Magnolia Street Colorado Springs, co 80907

G. w. Str icker 130 ~lerchant st. #1104 Honolulu, 81 96813

C. J. Swet Route 4 Box 358 Mt. Airy, MD

John 'l'aylor

21771

National Research Council ASBB 2101 Constitution Avenue washington, DC 20418

R. J. Templin (3) Low Speed Aerodynamics Section NRC-National Aeronautical Establishlnent Ottawa 7, Ontario O.NADA KIA OR6

170

"

•

Arnan Seginer Professor of Aerodynamics Technion-Israel Institute of

Technology Department of Aeronautical

Engineering Haifa, ISHAEL

Dr. Horst Selzer Dipl.-Phys. Wehrtechnik und Energieforschung ERNO-Raumfahrttechnik GmbH Hunefeldstr. 1-5 Postfach 10 59 09 2800 Bremen I GERHMJY

H. Sevier Rocket and Space Division Bristol Aerospace Ltd. P.O. Box 874 Winnipeg, Manitoba CANADA R3C 2S4

P. N. Shankar Aerodynamics Division National Aeronautical Laboratory Bangalore 560017 INDIA

David Sharpe Kingston Polytechnic Canbury Park Road Kingston, Surrey UNI'rED KINGDml

D. G. Shepherd Cornell University Sibley School of Mechanical and

Aerospace Engineering Ithaca, NY 14853

Dr. Fred Smith Mechanical Engineering Department Head Colorado State University Ft. Collins, CO 80521

Ken t Sm i th Instituto Technologico Costa Rica Apartado 159 Cartago COS'rA FICA

171

Wilson Prichett, III National Rural Electric Cooperative

Association 1800 Massachusetts Avenue NW washington, DC 20036

Dr. Barry Rawlings, Chief Division of Mechanical Engineering Commonwealth Scientific and Industrial

Research Organization Graham Road, Bighett Victoria, 3190 A US'J'RlILIA

Thomas W. Reddoch Associate Professor Department of Electrical Engineering The University of Tennessee Knoxville, TN 37916

Ray G. Richards Atlantic Wind Test Site P.O. Box 189 Tignish P.E.I. COB 280 CANADA

A. Robb Memorial University of Newfoundland Faculty of Engineering and Applied Sciences St. John's Newfoundland CANADA AIC 587

J. R. Rodriguez Solarwind Energy Corporation 1163 Pomona Road Unit A Corona, CA 91720

Dr. -lng. Uans Ruscheweyh Institut fur Leichbau Technische Hochschule Aachen Wullnerstrasse 7 GERHANY

Gwen Schreiner Librarian National Atomic Museum Albuquerque, NM 87185

Douglas B. Seely, P.E. U.S. Department of Energy P. O. Box 3621 102 NE llolladay Portland, OR 97208 172

•

Roger O'Eara Energy 'rimes 909 NE 43rd Suite 308 Seattle, ~A 98105

Oklahoma State University (2) Stillwater, OK 76074 Attn: W. L. Eughes

EE Department D. K. HcLaughlin

HE Department

Oregon State University (2) Corvallis, OR 97331 Attn: R. E. Wilson

ME Department R. W. Thresher

HE Department

Pat F. O'Rourke Precinct 4 County Commissioner City-County Building El Paso, TX 79901

H. H. Paalman Dow Chemical USA Research Center 2800 Mitchell Drive Walnut Creek, CA 94598

Dr. Y. H. Pao, Chairman Flow Industries, Inc. 21414 68th Ave. South Kent, \~A 98031

Ion Faraschivoiu IREQ 1800 montee Ste-Julie Varennes, Qeubec CANl\.DA JOL 2PO

R. A. Parmalee Northwestern University Department of Civil Engineering Evanston, IL 60201

flelge Petersen Riso National Laboratory DK-4000 noskilde DENHl\.HK

173

J. R. 1>1 c Con n e J 1 1umac Industries, Inc. 650 Ford St. Colorado Springs, CO 80915

James Meiggs Kaman Sciences Corporation P.O. Box 7463 Colorado Springs, CO 80933

R. N. I·leroney Colorado State University Department of Civil Engineering Fort Collins, CO 80521

G. N. Honsson Department of Economic Planning

and Development Barrett Building Cheyenne, WY 82002

Napier College of Commerce and Technology Tutor Librarian, Technology Faculty Colinton Road Edinburgh, ERIO 5DT ENGLAND

NASA Lewis Research Center (4) 21000 Brookpark Road Cleveland, 08 44135 Attn: J. Savino

R. L. Thomas W. Robbins K. Kaza

Anthony A. Nedd The Power Company, Inc. P.O. Box 221 Genesee Depot, WI 53217

v. Nelson West Texas State University Department of Fhysics P.o. Box 248 Canyon, TX 79016

Leander Nichols Natural POlver, New Roston, NH

Ronald Nousa in P.O. Cox III Rome 1132 Los Angeles, CA

Inc. 03070

90051 174

J. D. Longendyck Slltex 7 Capitol Drive Moonachie, NJ 07074

Los Alamos Scientific Laboratories P.O. Box 1663 Los Alamos, ~M 87544 Attn: J. D. Balcomb Q-DO-T

Beatrice de Saint Louvent Establissement d'Etudes et de Recherches

Meteorologigues 77, Rue de Serves 92106 Boulogne-Billancourt Cedex FRANCE

Ernel L. Luther Senior Associate PRC Energy Analysis Co. 7600 Old Springhouse Rd. McLean, VA 22101

L. fl. J. Maile 48 York Mills Rd. Willowdale, Ontario CANADA ~12 P IE 4

E. L. !1arkowski No tolola, Inc. G.E.D. Mail Drop 1429 8201 E. McDowe].l Rd. P.O. Eox 1417 Scottsdale, AZ 85252

Jacques R. Maroni Ford notor Company Environmental Research and Energy

Planning Director Environmental and Safety Engineering Staff The American Road Dearborn, MI 48121

Frank l'Jatanzo Dardalen Associates 15110 Frederick Road Woodbine, MD 21797

H. S. Matsuda, Manager Composite Materials Laboratory PiollE'erill,] H&D Laboratories Toray Industries, Inc. Sonoyama, Otsu, Shiga JAPAN 520

175

Carol Lamb 2584 East Geddes Avenue Littleton, CO 80122

Lawrence Livermore Laboratory P.O. Box 808 L-340 Livermore, CA • 94550 Attn: D. W. Dorn

M. Lechner Public Service Company of New Mexico P.O. Box 2267 Albuquerque, NM 87103

Kalman Nagy Lehoczky Cort Adelers GT. 30 Oslo 2 NORWAY

George E. Lennox Industry Director Mill Products Division Reynolds Metals Company 6601 West Broad Street Richmond, VA 23261

J. Lerner State Energy Commission Research and DevelDpment Division 1111 Bowe Avenue Sacramento, CA

L. Lilj idahl Building 303

95825

Agriculture Research Center USDA Beltsville, MD 20705

P. B. S. Lissaman Aeroenvironment, Inc. 660 South Arroyo Parkway Pasadena, CA 91105

011e Ljungstrom FFA, The Aeronautical Research Institute Eox 11021 8-16111 Bromma Sl'iEDEN

T. H. Logan U.S. Turbine Corporation 01de Courthouse Building Canfield, OH 44406

176

Indian Oil Corporation, Ltd. Marketing Division 254-C, Dr. Annie Besant Road Prabhadevi, Bombay-400025 INDIlI

JBF Scientific Corporation 2 Jewel Drive Wilmington, MA 01887 Attn: E. E. Johanson

Dr. Gary L. Johllson, P.E. Electrical Engineering Department Kansas State University Manhattan, KS 66506

B. O. Kaddy, Jr. Box 353 31 Union Street Hillsboro, NH 03244

Kaman Aerospace Corporation Old vlindsor Hoad Bloomfield, CT 06002 Attn: W. Batesol

R. L. Katzenberg 2820 Upton St. NW Washington, DC 20008

Robert E. Kelland The College of Trades and Technology P. O. Bo x 1693 Prince Philip Drive St. John's, Newfoundland CANADA AIC 5F7

S. King Natural Power, New Doston, NIl

Inc. 03070

Larry Kinnett P.O. Box 6593 Santa Barbara, CA

Samuel H. Kohler

93111

272 Old Delp Road Lancaster, FA 17602

O. Krauss Miclligan State University Division of Engineering Research East Lansing, ~II 48824 177

A. A. fJugman Kaiser Aluminum and Chemical Sales, Inc. 14200 Cottage Grove Avenue Dolton, IL 60419

Martin L. Hally, Section Manager Project Department Electricity Supply 18 St. Stephen's Green Dublin 2, IRELAND

Professor N. D. Bam Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139

C. F. Harris Wind Engineering Corporation Airport Industrial Area Box 5936 Lubbock, TX 79415

~1. L. Harris Aero/Astro Department Massachusetts Institute of Technology Cambridge, MA 02139

Terry Healy (2) Rockwell International Rocky Flats Plant P.O. Box 464 Golden, CO 80401

Helion P.O. Box 4301 Sylmar, CA 91342

Don Hinrichsen Associate Editor AMBIa KVA Fack, S-10405 stockholm SWEDEN

Sven Hugosson Box 21048 S. 100 31 Stockholm 21 SWECEN

O. Igra Department: O"f Mechanical Enqineering Ben-Gurioll University of the Negev Beer-ShevD, ISRAEL

178

Nichard G. Ferreira, Chief The Resources Agency Department of Water Resources Energy Division 1416 9th Street P.O. Box 388 Sacremento, CA 95802

D. R. Finley New England Geosystems P.O. 130 x 128 East Derry, NE 03041

James D. Pock, Jr. Department of Aerospace Engineering Sciences University of Colorado Boulder, CO 80309

Dr. Lawrence C. Frederick Public Service Company of New Hampshire 1000 Elm Street Manchester, NH 03105

H. Ger ard in Mechanical Engineering Department Faculty of Sciences and Engineering Universite Laval-Cuebec CANADA GIK 7P4

E. Gilmore Amar 1110 College Amarillo, TX 79100

Paul Gipe Wind Power Digest P.O. Box 539 Harrisburg, PA 17108

Roger T. Griffiths University College of Swansea Department of Mechanical Engineering Singleton Park Swansea SA2 8PP UNITED KINGOml

Professor G. Gregorek Ohio State University Aeronautical and Astronautical Department 2070 Neil Avenue Columbus, on 43210

I,ichard lladdad 101 Ar izona P. O. Box 530 Fl Paso, TX 79944

179

DOE Seadquarters/WESD (20) 600 E Street Nv) Washington, GC 20545 Attn: D. F. Ancona

C. E. Aspl iden L. V. Divone \'1. C. Reddick

C. W. Lodd School of Engineering Southern Illinois University Carbondale, IL 62901

D. D. Doer r Kaiser Aluminum and Chemical Sales, Inc. 6177 Sunol Blvd. P.O. Box 877 Pleasonton, CA 94566

Dominion Aluminum Fabricating Ltd. (2) 3570 Hawkestone Road Mississauga, Ontario CANP.DA L5C 2U8 Attn: L. Schienbein

C. Wood

D. P. Dougan Hamilton Standard, 1730 NASA Boulevard Room 207 Houston, TX 77058

J. B. Dragt Nederlands Energy Research Foundation (E.C.N.) Physics Department Westerduinweg 3 Patten (nh) THE NETfiEHLANDS

C. E. Elderkin Battelle-Pacific Northwest Laboratory P. O. Box 999 Richland, WA 99352

Frank R. Eldridge, Jr. The Mitre Corporation 1820 Dolley Madison Blvd. McLean, VA 22102

Electric Power Research Institute 3412 Hillview Avenue

. Palo Alto, CA 94304 Attn: E. Demeo

180

R. Camerero Faculty of Ppplied Science University of Sherbrooke Sherbrooke, Cuebec CANADA JIK 21\1

CERCFM 49 Rue du Commandant Rolland 93350 Le Bourget FRAPCE Attn: G. Darrieus

J. Delassus

Professor V. A. L. Chasteau School of Engineering University of Auckland Private Bag Auckland, NEW ZEALAND

Rovlard 'I. Clark McDonnell Aircraft Corporation P.O. Box 516 Department 337, Building 32 St. Louis, MO 63166

Dr. R. N. Clark USDA, Agricultural Research Service Southwest Great Plains Research Center Bushland, 'IX 79012

Joan D. Cohen Consumer Outreach Coordinator State of New York Executive Department State Consumer Protection Board 99 Washington Avenue Albany, NY 12210

Dr. D. E. Cromack Associate Frofessor Mechanical and Aerospace Engineering

Department University of ~assachusetts Amherst, ~1A 01003

Gale B. Curtis Tumac Industries 650 Ford Street Colorado Springs, CO

DOC/ALO (2) Albuquerque, Attn: G. P.

KM 87115 Tennyson

80915

181

West Texas State University Government repository Library Number 613 Canyon, 'I'X 79015

Wind Energy Report Box 14 102 S. Village ~ve. Rockville Centre, NY 11571 Attn: Farrell Smith Seiler

Wind Program Manager Wisconsin Division of State Energy 8th Floor 101 Soutb liebster Street Madison, WI 53702

Richard E. ~long Assistant Director Central Solar Energy Research Corp. 1200 Sixth street 328 Executive Plaza Detroit, MI 48226

1000 C. A. Fowler 17.00 L. D. Smith 2525 R. F. Clark 3141 L. J. Erikson (5) 3151 ~;. L. Garner ( 3 )

For DOEj'l'IC (Unlimited Release) 3161 J. E. Hi tcbell ( 15) 3161 P. S. Wilson 4533 J. W. Heed 4700 J. H. Scott 4710 G. E. Brandvold 4715 H. H. Braasch (200) 4715 J. D. Cyrus 4715 R. D. Grover 4715 E. G. Kadlec 4715 n C. Klimas r •

4715 11. 'f. ~la t t iSOI1

4715 R. O. Nellums 4715 \.; . N. Sullivan 4715 H. A. \\)0. tson 47 J. 5 D. F. Wolf 4715 f.l. H. Worstell 5520 '1' • B. Lane 5523 H. C. Reuter, Jr. 5523 Ir. G. Carne 5523 D. B. Clauss 5523 D. W. Lobitz 5523 P. S . Veers

182

(

5600 G. 13. Shuster 5620 M. M. Newsom 5630 R. C. Maydew 5633 H. E. Sheldahl

• 5636 J. K • Cole 5636 D. E. Berg 5636 ~! • H. Curry

)

J

183

'trU.S. GOVE RNMENT PRINTING OFFICE: 198 1-0-777-023/592