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AD881981
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TOApproved for public release, distributionunlimited
FROMDistribution: Further dissemination onlyas directed by Eustis Directorate, U.S.Army Air Mobility R & D Laboratory, FortEustis, Virginia 23604; Dec 1970 or higherDoD authority.
AUTHORITY
USAAMRDL ltr, 11 Jun 1971
THIS PAGE IS UNCLASSIFIED
GOIM AD
an USAAVLABS YECHNICAL REPORT 69-66
SAN ACTUATOR-DISC ANALYSIS OF HELICOPTER WAKEGEOMETRY AND THE CORRESPONDING BLADE RESPONSE
I . By
S"N. lsiekarR. Loewy
Li.
S..December 1970C=: L
EUSTIS DIRECTORATEU. S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORY
FORT EUSTIS, VIRGINIACONTRACT DAAJ02-68-C-0047 J £" w
UNIVERSITY OF ROCHESTER
ROCHESTER, NEW YORK
export ,ontrol, and vah t ran1cnttal
t' f- vn ol ~,~rflfl 'fl or I� 1n
ii! C~~r'. t Dr, it, S
L.
a3~
DEPARTMENT OF THE ARMYEUSTIS DIRECTORATE
U.S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORYFORT EUSTIS, VIRGINIA 23604
This report has been reviewed by the Eustis Directorate,U. S. Army Air Mobility Research and DevelopmentLaboratory, and is considered to be technically sound.The report is published for the dissemination of in-formation and the stimulation of thought.
ABSTRACT
Assuming that a helicopter rotor in forward flight can be repre-sented by a flat circular disc with an appropriate steady pressurediscontinuity across it, expressions were developed for relatingthe pressure field (1) to the tctal aerodynamic thrust and thetotal steady pitching and rolling moments experienced by the rotoror (2) to the time-dependent aerodynamic flapping moments at theroots of the blades as they rotate. The selected pressure field(1) satisfies the requirement of a loss of lift at the center andperiphery of the disc, (2) accounts for first and second azimuthalharmonic variations in the lift density, and (3) satisfies Laplace'sequation, which is the governing equation for the linearized steadyflow of an incompressible fluid.
A digital computinc program was written for calculating the velocityfield from the selected pressure field and for calculating thepositions of streamlines emanating from points above the disc. Twosample cases were chosen, corresponding to the UH-l rotor atadvance ratios,A , of 0.26 and 0.08 respectively. The computedstreamlines for both cases show the tip-vortex phenomenon, indicat-ing Prandtl-Lanchester trailing vortices emanating from the tipareas of the circular wing.
The geometry of the vortex wake that is shed and trailed from theblades was predicted from the above-menticned streamline calcula-tions. The computed deviations from a helical wake were greaterfor the/i = 0.08 case than for the/I = 0.26 case.
In order to examine the sensitivity of blade-load calculations towake geometry, an existing computer program for calculating bladeairloads was modified to accept the computed wake geometry, ascontrasted to the rigid helix usually assumed. The blade-loadspredictions of this program using the computed wake geometry inputwere compared with the predictions resulting from a helical wake-geometry input and with experimentally measured airloads for thetwo cases mentioned above. The comparison indicates that for theUH-I two-bladed teetering rotor system, the computed airloads aremore sensitive to the wake geometry for the slower flight • = 0.08)than for the faster one (/S = 0.26). This weaker dependence athigher forward speeds may be due to (1) the wake being blown fartherbehind the rotor and/or (2) the relative predominance of cyclicpitch input, tangential velocity variation, and blade flapping motionin the total time-varying environment of the blades at a higherspeed.
l iii
FOREWORD
The subject investigation was performed under the direction ofMr. William Nettles of U. S. Army Aviation Materiel Laboratories,Fort Eustis, Virginia. The work was authorized by DA Task1F162204A13904.
In preparing parts of the report, material was drawn freely fromthe work of Mr. R. A. Piziali, listed as Reference 2 iii thisreport. The authors are grateful for the cooperation of CornellAeronautical Laboratory, and particularly to Mr. R. A. Pizialifor making available the source decks for their blade-loadsprogram BLP2.
÷V
TABLE 02 CONTENTS
Page
ABSTRACT ................... .......................... .. iii
FOREWOLD ............... ............................ v
LIST OF ILLUSTRATIONS ............ ................... ... ix
LIST OF TABLES ............... ....................... ... xii
LIST OF SYMBOLS ........................................ .. xiii
CHAPTER 1 INTRODUCTION ............... .................. 1
1.1 A First View of the Helicopter Wake-GeometryProblem .................. ..................... 1
1.2 Some Relevant Physical Phenomena ..... ......... 5
1.21 Blade Dynamics ........... ............... 51.22 Blade Aerodynamics ................... 61.23 Wake-Induced Velocity Relations .. ...... . 111.24 Wake-Transport Relations .... .......... .. 12
1.3 Review of Various Approaches to Handling Rotary-Wing Aerodynamics ........ ................ ... 13
1.31 Analytical ......... ................. ... 131.32 Experimental ......... ................ ... 141.33 Computational ........ ............... ... 141.34 Some Complicating Factors Encountered in
the Approaches Cited ......... ............ 151.341 Physical Aspects ........... .............. 151.342 Theoretical Difficulties . .......... 151.343 Experimental and Computational Difficulties 16
1.4 Motivation for the Present Work ... ......... .. 16
CHAPTER 2 THEORETICAL MATERIAL ....... .............. ... 19
2.1 Theoretical Background ........... .............. 19
2.2 Capsule Statement of the Work Done ........... ... 19
2.3 Theoretical Scheme ......... ................ ... 23
2.31 Equations of Fluid Motion .. ......... 232.32 Coordinate Transformations and Solutions
of Laplace's Equation .... ........... .. 26vii
2.33 Choice of the Pressure Field, Based onTotal Thrust and Net Moments ... ......... ... 30
2.34 Choice of the Pressure Field, Based onBlade Flapping Moments ........... .......... 36
2.35 Induced Velocity Field ..... ........... ... 44
CHAPTER 3 THE COMPUTER PROGRAM ......... .............. ... 48
3.1 General Description .......... ................ ... 48
3.2 Representation of Various Physical Phenomenain the CAL Program ......... ................. ... 53
3.21 Rotor Dynamics ........... ............... ... 533.22 Blade Aerodynamics ....... .............. 553.23 Wake-Induced Effects ....... ......... .... 583.24 Wake Transport ........... ........... .... 59
3.3 Algorithm for Computing the UR Wake Geometry . . . . 60
3.31 Subroutine RWAKE ........... ............... 613.32 Subroutine STMLN(KEND) ......... ............ 643.33 Subroutine FRWRD ....... ............... ... 66
3.4 Integration With the CAL Program .......... 66
3.5 Peripheral Program .......... ............... .. 69
CHAPTER 4 COMPUTED RESULTS AND INTERPRETATION ....... ... 70
: 4.1 UH-l at,� = 0.26. Injection on Flat Disc. ....... 70
4.2 UH-l atA = 0.26. Injection at SteadyDeflected Positions of Blades ..... .......... . . 78
4.3 UdH-! at, = 0.08. Injection at SteadyDeflected Positions of Blades ..... ............ ... 82
CHAPTER 5 CONCLUSIONS AND POSSIBILITIES FOR FURTHER WORK . 93
LITERATURE CITED ............... ....................... ... 97
APPENDIXES
I. Associated Functions of Legendre and Some ofTheir Properties ....... ................ ..... 99
II. FORTRAN IV Source Listings ........ ............ ... 106
III. Induced Velocity at Selected Positions on the Disc 194Due to Selected Segments of the Nearest Tip Trail .
DISTRIBUTION .................... ........................... ... 197
viii
LIST OF ILLUSTRATTONS
Figure Page
1 Schematic Representation of the HelicopterRotor Aerodynamics Problem .................... 3
2 Physical and Mathematical Blade ElementCharacteristics ............... ................. 7
3 Induced Velocity Over Disc for AzimuthallySymmetric Lift Distribution Disc Incidence = 00 20
4 Induced Velocity Over Disc for AzimuthallySymmetric Lift Distribution. DiscIncidence = 150 ..................... ..... ..... 21
5 Induced Velocity Distribution According toMangler and Squire. Disc Incidence = 150 ........ 22
6 Cartesian Coordinate Systems XZ("Windsystem"?) and X 'Z'("Disc system") InDimensional Fovrm ........... .................. .24
7 Ellipsoidal Coordinate System, Viewed intheX ' Plane ............. ................... .. 28
8 P~rx))Plotted as a Function of Radius ron the Disc ............. .................... .. 34
9 Typical Steady and First Two HarmonicComponents of Experimental Blade Loads ......... .. 37
10 Time Histories of the Lift Density at a
Point (,k) of the Disc ..... ............. .. 40
11 Flow Diagram for Part I of BLP2 ...... ...... .. 49
12 Major Steps in the Iterative Procedure forSolving the Equations in Part 2 of BLP2 .......... 5o
13 Flow Diagram for Part 2 of BLP2 .... .......... .. 51
ix
Fiqure Page
14 Schematic of the Modified BLP2 .... ......... ... 52
15 K-Subscript iotation, Wake Configurationand Wake Vortex Strengths .... ............... 56
16 Flow Diagram for Subroutine RWAXE ............ ... 62
17 Flow Diagram for SubrDutine STMLN(YEND) ........ .. 65
18 Flow Diagram for Subroutine FRWRD ... ....... ... 67
19 Flow Diagram for Subroutine SUB14 as Modifiedat UR .............. ...................... .. 68
20 Z -Component of the Induced Velocity on aLifting Disc, Using the UR Program ........... .. 71
21 Typical Instantaneous Geometry of WakeTrails (UH-l,# = 0.26,NA =2, One of theBlades at) k= 750 ) ........ .............. .. 73
22 Geometry of Streamlines in the SteadyFlow Field (UH-1,,# = 0.26) .... ......... ..... 74
23 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, One RotorRevolution After Release (UH-ilA = 0.26). ..... 75
24 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, Two RotorRevolutions After Release (UH-I,,, = 0.26) .... 76
25 Time Histories of the Unsteady Parts ofMeasured and Computed Airloads (UH-1,,& = 0.26).. 77
26 Harmonic Analyses of Measured and ComputedAirloads (UH-l,& = 0.26) ..... ............. ... 79
27 Time Histories of the Unsteady Parts ofMeasured and Computed Blade-Flap'-iseBending Moments (UH-l,,,& 0.2b6 ............ ... 80
28 Harmonic Analyses of Measured and ComputedBlade-Flapwise Bending Moments (UH-l,/e = 0.26).. 81
X
Figure Page
29 Typical Instantaneous Geometry of aWake Trail (UH-l%# = 0.08,AN = 1,Blade at fie = 75 ) ...... ................ ... 84
30 Typical Instantaneous Geometry of aWake Trail (UH-1,o = 0.08,Ah = 1,Blade at = 2550) ..... ............... .... 85
31 Geometry of Streamlines in the SteadyFlow Field (UH-I,#a = 0.08) .............. .... 86
32 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, One RotorRevolution After Release (UH-l, = 0.08). 87
33 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, Two RotorRevolutions After Release (UH-I,# = 0.08) . 88
34 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, Three RotorRevolutions After Release (UH-I,A = 0.08). 89
35 Three Views of a Fluid Contour, ReleasedFrom 90% Radius on the Disc, Four Rotor 90Revolutions After Release (UH-l,/ = 0.08). ...
36 Time Histories of the Unsteady Parts ofMeasured and Computed Airloads (UH-1,A = 0.08). . 91
37 Time Histories of the Unsteady Parts ofMeasured and Computed Blade-FlapwiseBending Moments (UH-1,/W = 0.08) ............ ... 92
38 Positions of Selected Blade-Stations and SelectedWake-Trail Segments (UH-IA = 0.26) ......... ... 195
xi
LIST OF TABLES
T3 ble Pageý
I Rotor Blade Vibration Natural Frequenciesand Comparison With Rotor Speeds, forUH-1 and H-34 . . . . . . . . . . . . . . . . . . 4
II The Terms Retained in the Expression (62)for the Disturbance Pressure P . . . . . . . . . 38
III Various Options for Choosing the SteadyPressure Field P Through the Coefficients 63
C and P &W . . . . . . . . . . . . . . . . .
IV Comparison of Contributions of the Quasi-Steady and Wake-Induced Quantities to theBound Vorticity Strength Components at the85% Radial Station, UH-1,/M = 0.26 and
,,A = 0.08 . . . . . . . . . . . . . . . . . . . 94
V Induced Velocity at Selected Blade-Staticnsdue to Selected Tip-Trail Segments (UH-1,
0.26). See Figure 38 . . . . . . . . . .196
xii
LIST OF SYMBOLS
cosine coefficients of Fourier expansion
Ok ' nk of Gk
cosine coefficients of Fourier expansion,Ok of qk
A 0k Al coefficients in trigonometric
expansion for Yk
b blade semi-chord
B vector magnetic induction field
B LP2 blade airload digital computer programdescribed in Reference 2
sine coefficients of nth harmonic inBnk Fourier Expansion of Gk
sine coefficients of nth harmonic in6rik Fourier Expansion of lk
M "arbitrary" constant in the solution ofC % Legendre's equation, multiplying the
cosine term
CT thrust coefficient T TCT ir,. R 441
CMr coefficient of rolling moment
CMp coefficient of pitching moment =-RT'R
Ct Mangler thrust coefficient =
xiii
L
D m "arbitrary" constant in the solutionn of Legendre's eqjuation, multiplying
the sine term
d A elemental area of rotor disc
ft-. measured lift curve slope
RrO) blade aerodynamic lift per unit span
structural damping coefficient in the9k kth natural mode
generalized force in the kth naturalGk mode
imaginary number a V-,
circulation associated with the part
IK of the bound vortex due to all partsof the airfoil angle of attack exceptthat resulting from wake vorticity
J vector current distribution
"ka azimuthal position of blade whenshedding vorticity
implies the total of line filaments
L of vortex strength
I length along vortex filament
generalized mass in the kth naturalmode
"constant of separation" in solutionof a partial differential equation
Mr rolling moment (about the X' axis)
MIP pitching moment (about the y axis)
xiv
M (y) moment applied by rotor blade to
shaft at angle V/
M 9Mo, ,M5, harmonic components of M(;(r)
total number of normal modes chosenN h9 to represent total blade response
total number of harmonics retained inNh Fourier expansion
flI unit vector, normal to a surface
"constant separation" in solution ofa partial differential equation
NI total number of blades in a rotor
total number of azimuthal stations onrotor disc
total number of radial stations onrotor disc
blade aerodynamic pitching moment perP(r~t) unit span
P disturbance pressure
PO static pressure
associated Legendre function of thefirst kind
nondimensionalized disturbanceP pressure a P
Q(U,V,W) nondimensional induced velocity CI(.v,),S
- associated Legendre functions of thesecond kind
q metric tensor
xv
generalized displacement (a function
k of time) in the kth normal mode
induced velocity vector
component of fluid velocity in theI-axis direction
Y, radial distance from the rotor centerto a point on a blade
r - vector distance between source pointand field point
R radius of actuator disc
point in the three-dimensional wake
Shift. induced velocity coefficients in the
jtrip expansion forWK
1 (independent) time coordinate (sec.)
T total thrust
induced velocity coefficients in trigT MXK1 expansion for W.
U disturbance velocity in x-direction
U nondimensionalized disturbance veloci-
U ty in the x - direction = UV 1"irpressed" normal flow due to all
effects except wake vorticity
VOk Vn coefficients in trig expansion for W
"tangential" relative airflow velocityV, component in X-direction
total relative fluid velocity normalV, to the blade leading edge
V volume of fluid
xvi
1A
V nondimensionalized disturbance ve-locity in the y-direction = V
VF.SVi induced velocity
VFs free-stream velocity
V disturbance velocity in the y-direction
nondimensionalized disturbance ye-W locity in the z-direction = W
VFsnormal relative velocity component
(X) (positive in the plus z-direction)
w disturbance velocity in the z-direction
"impressed" normal flow component dueWk to wake vorticity
Sw (assumed) uniform induced velocity-- over rotor disc
x X/Rxw "wind coordinates"
blade chordwise coordinate, positiveaft
Cartesian coordinate in horizontal,Ix longitudinal direction, positive for-
ward
longitudinal Cartesian coordinate inplane of rotor disc, positive forward
j unit vector in direction of positivex-axis
y displacement of blade normal to thechord plane
YI Cartesian coordinate in lateraldirection, positive starboard
xvii
LJ
YOC built-in coning angle plus steadyblade bending deflection (plus, up)
Y/R
ZZ z/R, geIR
Cartesian coordinate in verticalZ direction, positive down
Cartesian coordinate normal to rotordisc, positive down
angle of attack corresponding toWA airfoil stall
geometric angle of attack of bladeA'S (airfoil section (positive, nose up))
incidence angle of rotor disc,
0 •(positive, nose down)
maximum value of total bound, circula---i ~k tion, corresponding to blade stall
vector distribution of vorticity
total circulation around the blader section
maximum bound vorticity on a blade atazimuth station i
strength of vortex sheet per unitY length
scalar vorticity per unit length inI, the wake
fraction of time interval which wakeS _is advanced from blade position at
which it was shed, trailed
difference in ( ) between upper anda• ( lower surface, or increment in ( )
circulation associated with boundvorticity resultinq from wake vortic-ity trailed and shed when blade wasat azimuthi
xviii
% torsional displacement of blade
£9Glauert coordinate
W k k natural frequency (rad/sec)
in ellipsoidal coordinates
"dummy" variable of integration inchord or X- axis direction
disc coordinates
velocity potential '
pressure difference at r Y k whena blade is at that point on the disc
S10 air mass density
electromagnetic induction coefficient/o defined by Vx' B 'ok
dT elemental volume
elemental surface area (of a vortexSZ tube)
vector operator = - + + -
ax T
curvilinear coordinates (see Figure 7),p F'the last corresponds to angle of
or azimuth on the rotor disc
A' tip speed ratioVFs/CIR
Subscripts
identifies quantity evaluated aroundC a closed contour
k identifies k* natural mode
identifies a finite number (say ofm normal modes)
i identifies induced
xix
I
U)L identifies upper, lower airfoilsurface,, respectively
T identifies trailed vortex
BU Ty bound vortex
S shed vortex
Q field point
K It particular point on the rotordisc
indicates partial derivative with respectto variable(s) represented by subscript(s)which follow(s)
Superscripts
Wk identifies O*natural mode
It vector
perturbation quantity
unit vector
steady component
xx
CHAPTER 1
INTRODUCTION
1.1 A First View of the Helicopter Wake-Geometry Problem
The desirability of an accurate determination of the wake geometryfor a helicopter rotor stems primarily from the necessity of pre-dicting the airloads experienced by the blades. These airloads, inaddition to providing the lift and propulsive force, are a majorsource of vibratory excitation and drive the structurally importantoscillatory blade bending moments. Since aerodynamic forces aredetermined by the flow field, the relative flow field must be knownif one is to satisfactorily predict the airloads on a prescribedairfoil executing a prescribed motion.
It has been demonstrated from fixed-wing experience (see, for ex-ample, the reduced polar diagrams and reduced lift/angle of attackcurves in Reference 1), that for a finite lifting body, the velocityfield perturbation resulting from the distribution of the "free"vorticity in the wake must be taken into account in order to predictthe aerodynamic forces (aerodynamic force is sufficiently definitive)on the body. It may be argued that this wake-induced velocity iseven more important for rotary wings since (1) rotation causes thewake vorticity to remain in the vicinity of the (succeeding) bladesfor a much longer time and (even for the " ,hest known advanceratios) is not blown behind the wing so quickly as with fixed-wingcraft, and (2) as a result of a lower forward speed in the case ofa pure helicopter, the change in angle of attack for a given down-wash velocity will be larger over most of the disc than in thefixed-wing case.
In addition to the determination of the rotor airloads, the wake-induced velocity field is likely to be influential in the determina-tion of (1) interference effects between rotors, (2) positioning andeffectiveness of auxiliary surfaces, (3) interference with the fuse-lage and other nonlifting bodies, (4) debris entrainment, (5) pro-jectile trajectories, etc; hence, the continuing interest in thestudy of helicopter wake geometry and in the resultant wake-inducedvelocity field.
One of the difficulties in an accurate determination of the wake-induced velocity should be immediately apparent: the interdepend-ence of the wake geometry and the induced velocity field. The wake-
1
induced velocity field can be expressed only as an integral (overthe wake) of a fun ction of the wake geometry and the vorticitystrengths, while the wake geometry is obtained by time integrationsof the flow field along particle paths. As can be seen from subse-quent sections, this one complication alone is sufficient to makethe general wake-geometry determination problem difficult. Thisinterdependence between wake geometry and induced velocity fielddoes not arise in the customary treatment of the fixed-wing problembecause one assumes there that the induced velocity is quite smallas compared to the forward speed and hence the trailed vorticity isassumed to lie on a "rigid, horizontal" surface behind the wing.
Another complicating factor is the flexibility of rotor blades.Most helicopter blades currently have their first flapwise naturalfrequencies within a few percent of rotor speed. Table I summarizesthe flapping natural frequencies for the UH-l and H-34 rotor systems.It has been demonstrated that significant variations in downwashexist up to the 4th azimuthal harmonic. Under such circumstances,the azimuthal variations in aerodynamic loads are almost certain toexcite substantial oscillatory flapping displacements as well astorsional displacements, and the resulting "plunging" velocitieswill affect the aerodynamic loads on the blades to the first order,while the torsional displacements (i.e., angle of attack changes dueto torsion) will affect them to the zeroth order. Thus, the situa-tion is not one of a "prescribed blade executing a prescribed motion".As a final note, it may be mentioned that the rotor control settingsand attitude are themselves affected by th periodic variations inthe hub loads.
Figure 1 indicates the interrelations of these various aspects ofthe problem. The figure can be looked upon as an information flowdiagram*, the rectangular blocks representing various phenomena ascustomarily treated in isolation. Thus, for example, the airloadsand wake vorticity strengths resulting from the phenomenon of bladeaerodynamics can be calculated if the (1) blade structural displace-ments and motions, (2) control settings, (3) vehicle attitude andmotion, and (4)/ wake-induced velocity (inflow) are all known as"inputs" to the blade aerodynamic equations. These equations are,for unstalled subsonic motion, simple algebraic relations. (Theassumption that makes this possible is the customary strip theorypostulation that each spanwise section of the airfoil has a two-dimensional pattern of aerodynamic behavior.) We are not so fortu-nate in regard to some of the other phenomena; e.g., the wake-
*By "information", we mean here "knowledge about the values (steadyor otherwise) of specific physical quantities of interest".
2
05 E -4 ROTOR_ _ _
blade structurO STRUCTURALdisplacement DNICSand motions
airloads(0) control setting -•-- BLADE wake vortex strengthposition
AEROYNAMICS due to unp'erturbed flow
r compressibility, stall,-------- I and reverse flow
woke-induced WAKE-INDUCED IL effects may enter here
velocity field VELOCITY SRELATIONS
I WAKE-TRANSPORT wake vorticilyiRELATIONS positions (geometry)
4interferenQe ,wake interaction,"and viscosity effects mayenter here
vehicle attitude VEHICLEand motion DYNAMICS
Quantities of Interest,
I. Control settings 5. Blade displacements
2. Airloads and motions
3. Vortex strength, position 6. Vehicle attitude and
due to unperturbed flow motion4. Induced velocity field 7. Woke vorticity position
(wake geometry)
Figure 1. Schematic Representation of the HelicopterRotor Aerodynamics Problem.
3
41- tT, C) W U ) r- *H 'U *HC) w7 10 z 0
b) -rI -r 0) t3 Z (Ql 0 41' K U0r_ ) ,q (3)*Hj 4-) a) U tn~ (j)% r- -r1 I ,~
r- 0 ) r U) () 9 *HU--i() 14 I 04 I 00 P4 rH -H, A 4-) A 04 (D to 04 n.H r '0 4j~ >.i a : ro H 0) H-
4.. H 'Z 04 (f)I:! 0)04 r-i H -4-1 H 14.4L4.-4 Co 4-1 (1)O 4) 0 Uo r ) i 4-4 '4-4 q0OH A0 H 'o q 1 '-4 10
H a .1-;1 4J3- CLI S 1 4-4 10 41) z 10 4-)a)Hr *rl C/) " U) 0 04 0 r-I U) 0 P- ~"o00 b ý4 ~ra p 00 ) 00 r- b -i C.) rj40 U) *HI *HH *HH a)H (1- *ri r-I U) I, 0
0 4-44-1 q 4- U) 4-1 U)V- p4 4-A U) 4- 44
0
00~ I. C C%4 Cl) Q) 0D 0') 0 LA) Ho- 0)HO' j C'J C) N 0 C)0 LA
U. U) Co o C Hr Cl 11; L.4 A c'. L4 cO i'1
ciO p 'U) LA) LA t ) ~co QO Itt 00 w t0 It 00 H- H-
rNI Ni (D H- f iC4tTI JCOr- H H~
0'0
0
.-W 10 H cv NIL- AH lN
00
[-, CA, C))
CO 04
11 0 4 0o0 ) 4
PCI f) ~ 4-- LA rNi C0 C'
H j- )H C
t3))
H- Er 4-)Il - C) uH a)) 0) -H H
4r)cf 4-) tn V-i HO' A
a) 4.)d 4-4 P_____q __ _ _ _ _ _ __ _ _ _ _ _ _ CA I '-
1L ~C . . ~ ' C l _ _ _ __4
induced velocity relations (i.e., Biot-Savart law integrated overthe wake) and the wake-transport relations. These are integralexpressions, as mentioned above.
1.2 Some Relevant Physical Phenomena
Equations representing some of the phenomena shown in Figure 1 willbe stated in this section. More specific description of these phe-nomena as represented in the computer program appears in Chapter 3.
1.21 Blade Dynemics
The oscillatory response of a rotor blade to variations in airloadscan usually be satisfactorily represented in terms of its responsein each of a judiciously chosen finite number, N, , of its normalmodes of displacement (rigid as well as elastic). If ¶kCt) is thegeneralized displacement in the kV normal mode, the flapwise andtorsional displacements at a point radius r can then be written as*
kA1
(kk)
where Y(v), 9(k) (r) is the mode shape in the k normal mode.
*We are here assuming the blade to be infinitely rigid in the chord-
wise direction. The computations reported herein used the additionalstructural simplifications of Reference 2, ignoring the dependenceof torsional displacements on airloads, thus assuming the torsionalresponse of the blades to be completely known beforehand.
5
The differential equation for determining the generalized dis-placements %(t') is of the form
- .W ( 3 )Mk jk + ' k9 k k k 1k + Mkk 1k GkG(t) (3)
k.-'-,N,
MkK,•.j and Gk are the generalized mass, structural damping,natural frequency, and generalized forcing function respectivelyin the k0 mode. The first three of these are dynamic "proper-ties" of the rotating blade, as is the mode shape, and can be pre-determined for each mode. The forcing function depends on theairloads experienced by the blade:
re
Gk([) = [F(r-•, (k)(\ + (<_r " t ) (4)
and "P(-,i) represent the time-varying aerodynamiclift and pitching moment per unit span at radius r .
1.22 Blade Aerodynamics
Consider a blade segment with its leading edge at x •- - andtrailing edge at •- -+-b in a blade-attached coordinatesystem; see Figure 2(a). Let the relative airflow have a"tangential" component V1 in the XI direction and an"impressed" normal flow component W(X) in the z direction.W(x) is the combined effect of (1) the component of V,
normal to the blade chord as a result of the geometrical angle ofattack ae and from the local camber of the airfoil section,(2) the component of the forward speed normal to the blade chordas a result of shaft tilt, spanwise blade slope, etc., (3) in-stantaneous flapping velocity of the blade section, (4) instan-taneous torsional velocity of the blade section (due to bladeflexibility as well as cyclic control inputs), and (5) inducedvelocity due to the wake vorticity. We wish to estimate the aero-dynamic forces experienced by this blade segment.
Since no through-flow can exist on the impermeable airfoil, thepresence of the airfoil must give rise to such a "disturbance"that its effect will exactly cancel the "impressed" normalvelocity V(x 1in the region -bdX 1 i4o . It can be shown(see, for example, Reference 3) that for a thin airfoil, thisdisturbance can be represented by a vortex sheet over the chordlength, as shown in Figure 2(b). The distribution of the
6
strength r(Xi) of this vortex sheet must be such that
c ,O(5)
--6
z1
(a) blade element and (b) thin airfoil repre-"impressed" normal sentation with boundvelocity distribution vorticity distribution
xo-" h-g, Xo- 4b
(c) Glauert coordinate •
x,. -b CoSO
Figure 2. Physical and Mathematical Blade Element Characteristics.
7
In order to uniquely specify the circulation around the airfoilsection, it is customary to further require that the flow besmooth over the sharp trailing edge (Kutta hypothesis)*. This canoccur only if
Y(+b) = 0 (6)Subject to the condition (6), equation (5) has now to be solved.
The solution, by Sohngen's formula is
+by(X,) = X . ;;(0 .A (7)
-b
After thus determining the bound vorticity strength Y'(,) from theimpressed normal velocity W (x,) , it now remains to deduce thepressure difference between the two faces of the airfoil. Assumingthat (1) the disturbance velocities are small compared to V/ 1(2) the flow is irrotationel outside of the vortex sheet, and (3)the flow is incompressible, the momentum equation for unsteady flowcan be reduced to
V, X, a-(8)
where p is the disturbance pressure and 6 is a velocity
*This is tantamount to requiring a finite velocity at the trailingedge. It has been corroborated by experiment, in that the pressuredistributions on the surfaces of the airfoil as predicted fromsuch a choice of the bound vorticity distribution agree with experi-mental measurements (see, for example, Reference 4).
potential, such that the disturbance velocity U equals -Changing the order of differentiation in the first term and aiCintegrating from x, .-- to a point on the upper face of thevortex sheet,
I' + (9)
and similarly at a point on the lower face,
-~~ 0)L 10
I b
But it can be shown (by considering an elemental rectangular con-tour around a point x, ) that
L!
_'!- ==UU = c
and, by integrating,
X1
9L-U. - P Cie, (12)
-b
Using equations (9) through (12), one gets the aerodynamic pressuredifference across the vortex sheet:
which is the unsteady form of Zhukovskii t s theorem.
9
The total circulation r around the blade section, the aerodynamiclift F per unit span, and the aerodynamic pitching moment aboutmid-chord 1• per unit span are obtained by integration over thechord:
Y (X,c dI~, (14)
P \X, ax,~dC (15)
+6
PC X\X (16)
The other "output" from the blade aerodynamics block in Figure 1is the wake vortex strength. The strength of free vorticestrailed and shed from a blade can be deduced by logical extrapola-tion of the fixed-wing "lifting line" theory. Thus, when theinstantaneous bound vorticity varies along the blade span, trailedvortex strengths are given by
Xt Jr., -. (17);_v,
where YT(Y ) is the strength of the trailed vortex sheet andn Tv(0) is the strength of the bound vortex at the spanwise loca-tion 1 . Similarly, for a blade with continuously changing(tempo•al change) bound vorticity, the shed vortex strength is givenby
7'3 (rt) • (18)
VT Ait
10
where 26t',r) is the strength of the shed vortex sheet as a
function of spanwise position r and time t , and V is thetotal relative fluid velocity normal to the blade leading edge.
When the strength of the bound vortex varies both along the spanand with time, the above expressions for YT and Y$ willrepresent the components of the strength of the vortex sheet inthe chordwise and spanwise directions respectively.
1.23 Wake-Induced Velocity Relations
These relations are used for determining the "induced" velocitySa at a field point Q due to a distribution of vorticityr in a volume V . This is directly analogous to the problemof determining the magBptic induction field §Q at Q due to acurrent distribution - in V . The latter problem has thesolution given by the Biot-Savart field law:
"ga- - .Y
where Y is the vector distance from the source point to the fieldpoint. See, for example, Reference 5.
Analogously, for our induced velocities,
(The /^o is a characteristic of the units used for the electro-magnetic quantities such that 17'B x &.-T
If the source distribution of vorticity is over line segments asin our wake model, our formula is modified to
11 1 x(0
e L
where -r is the vector from c..to Q , and T is the scalarvorticity per unit length along JA , an elemental length of vortexfilament in the wake. The integration is over L. , which is theaggregate of all the wake filaments.
1.24 Wake-Transport Relations
Kelvin's theorem of conservation of circulation (see Reference 6)states that for an idea) fluid, the substantial derivative of thecirculation I'e fj.Z" around a closed contour C is zero.
S_ (21)
Ot
Now, if C lies on time surface Z of a vortex tube,
, -f. ff (v x .VaCI . (22)
using Stoke's theorem and the fact that V x c is at right anglesto the unit normal 'A on the surface E . Therefore, usingKelvin's theorem, one concludes that a contour on a vortex-tubesurface will always remain on a vortex-tube surface. Using twosuch intersecting surfaces, it is easy to see that vortex linesalways travel with the fluid (Helmholtz theorem; see Reftrence 7).
We may consider all vorticity in the wake model to have originatedfrom the blades. Thus, if we can calculate the positions of fluidparticles released into the floy field from discrete locations onthe disc*, at all subsequent (for our purposes, discrete) instants,we will be able to predict the wake geometry (i.e., time-dependent
*The "disc" here means the surface described by the blades as theyrotate. Due to the preconing built into the blades, the steadydeflection under load, and the flapping displacements, this surfacewill not, in general, be a plane circular surface.
12
positions of the wake-filament end points in Figure 15(b) ).
1.3 Review of Various Approaches to Handling Rotary-WingAerodynamics
1.31 Analytical
Among existing theoretical approaches are the actuator theories,the blade element theories, and the wake theories.
a. The early actuator disc theory for a propeller assumed auniform pressure distribution and induced velocity over thedisc, and explained their interrelationship through a simplemomentum transfer approach, leading to a relation like
2irR v V1V +,v" (23)
where T, vi , and VP, are the total thrust, the inducedvelocity at the disc, and the forward velocity respectively.Glauert modified this assumption of the uniform pressuredistribution to one increasing linearly from the front to therear of the rotor disc (see Reference 8). A modification dueto Kinner (Reference 9) and Mangler (Reference 10) assumes anaziuiuthally symmetric lift distribution satisfying therequirement of a loss of lift both at the center and at theperiphery of the disc.
All the actuator disc theories implicitly assume an infinitenumber of blades. The inherently unsteady nature of theaerodynamics associated with a finite number of blades is,therefore, ignored, as are all effects concerned with theshape of airfoil sections.
b. The blade element theories, on the other hand, considerthe forces experienced by individual blades in their motionthrough the fluid; they are thus intimately concerned withairfoil section characteristics. The blade is divided into anumber of radial elements, and airfoil properties and momentumbalance are used for each element to yield the thrust andinduced velocity (see Reference 11). An interesting combina-tion of momentum and blade-element theories for forward flight
13
i • dm "A
has been suggested by Wood and Hermes (Reference 12). Theypostulate that the instantaneous induced velocity at a pointon the disc can be obtained by superposition of transientmomentum theory induced velocity distributions correspondingto the number of blade passages over that point in space.
c. In contrast to the above, the early vortex wake theoriesrepresented the wake by a series of circular vortices. Thegeometry, however, was based on an assumed induced velocityfield and on an infinite number of blades. The wake vortextheories then attempted to obtain the induced velocities withthe aid of the Biot-Savart formnula. Recent modificationsof the vortex wake theories assume helical vortices trailedfrom near the blade tips and a "?warped-.pie"? type of mesh-vortex system that is shed from the blade. These vortex waketheories may be looked upon as a modification of Prandtl'stheory for a finite fixed wing.
1.32 Experimental
Considerable experimentation has resulted in a good deal of testdata both for structural loads and aerodynamics of rotary wings.Excellent smoke-trail photographs for a model rotor in a windtunnel were obtained by Tararine (Reference 13). Very extensiveblade-load and blade-motion data were presented by Scheimann(Reference 14) and by Burpo and others (Reference 15). Much ofthe experimentation has been carried out primarily to gain insight,and rigid model rotors have frequently been used (e.g., Tararine).A recent water-tunnel study of rotary-wing flow visualization wascarried out by Lehman (Reference 16). An interesting electromag-netic analog was reported by Gray (Reference 17).
1.33 Computational
There appear to be very few numerical solutions of the simultaneousequations of wake motion resulting from blade loading, vortexstrength, and vortex-induced velocity as described by the Biot-Savartlaw. One such calculation was accomplished by Crimi (Reference 18)with an extensive iterative calculation for the time-varying flowand the vortex geometry.
Reference 2 (and its subsequent extension, Reference 19) is anexample of the elaborate blade-loads programs that have beendeveloped in recent years. Reference 19 numerically simulatesmost of the phenomena shown in Figure 1. (A notable exclusion is
14
the wake-geometry computation.) We will Cescribe some details ofthe Reference 2 program in Chapter 3 of this report. Anotherexample is the program reported in Reference 20, which comparedthe blade-load predictions using the rigid wake-induced velocitywith similar predictions using a uniform inflow velocity at thedisc for a four-bladed, fully articulated rotor.
1.34 Some Complicating Factors* Encountered in the ApproachesCited
1.341 Physical Aspects
As a result of the inherently unsteady aerodynamics of the rotorin forward flight, the "shed" vortices are likely to be signifi-cant, and this requires that considerable detail be retained inthe wake model (see Figure 15(b) ). Also, due to rotation thewake geometry is more complex than for a fixed wing with a similarlyvarying bound vorticity.The flexibility of the blades, as mentioned earlier, makes unpre-
scribed blade motions inevitable.
" ... For example, a typical blade of a flapping, three-bladed, 25 foot radius rotor has a first bendingfrequency of about 2.5 times the rotational speed foroperating tip speeds; by comparison, a rotating stringor chain has a corresponding frequency of about 2.4 timesits rotational speed. Certainly, for airloads withfrequencies of third harmonic and over, the blade isquite flexible and will respond elastically. Thisinteraction between dynamic structural deformations andairloads is the second sdrious physical complicationin the blade airloads problem." (Reference 21)
1.342 Theoretical Difficulties
Even if the equations were not coupled as regards blade motion and
*Material for this section has been drawn from Reference 21.
15
aerodynamics, the problem is difficult to solve because of theinterdependence of the wake geometry and the induced velocity.To avoid this complication, drastic assumptions have been maderegarding vortex geometry. Momentum theories, which bypass wakeeffects completely, have been subject to other limitations. TheKinner-Mangler theory, for example, which is one of the moresophisticated actuator disc thecries, ignores the nonuniformity oflift with azimuthal position.
Another disturbing factor is the questionable validity of smallperturbation assumptions in linearizing the basic flow equations.For such assumptions to be valid, the downward velocity inducedby the lifting surface, which from the momentum point of view isthe basic source of lift, must be very small compared with forwardspeed. This is the case for a fixed wing and makes possible theassumption that the trailed vorticity lies on a rigid cylindricalsurface through the trailing edge with generators parallel to thedirection of flow. It is much less obvious that this assumptionwill give good results when using an "actuator disc" to representa lifting rotor at any forward speed. Although the downward flowcomponent near the blade tip may be small compared to the totalforward velocity of the tip, there are likely to be points on therotor where this is not the case. In any event, the "actuator disc"'ar.;iot recojnize blade rotation, but only free-stream velocities,-,o that "perturbations" may not be small, and linearized flow
'iuaitlorin could lead to large er'ors.
1. 14 ' Ii;rrimental and Corrputational Difficulties
., iiriiI i I ,tidy of blade airloads has to be very elaborate,ii, 'If (t i it ,; ftifficult to design a blade which is both aero-
,,i irl- (t dyrnamically similar to the full scale item;i* ,,,. !. pia',irenments have to be made to define streamlines,
, i, lijt onr'illatory blade pressures, drag, and power;ili,,,, lri I 'I tunnel wall effect is especially difficult where
-i, ai I kit ion over the rotor disc and unsteady effects
A w, ta-F,,,•v ty ' •ompijt.tion such as that done in Reference 18,.Ii, ii, f ?,,, -r11,If, . 1': lenqthy even or a high-speed computer becauseII IV n i, i(-rt frJi i equations (involving unprescribed velocities)
, , .4 , ,I .
.4 Mut i.'o-t ion for tme Present Work
Tite 'muoe-trail photographs taken by Tararine (Reference 13) for aSi 1 mo,!el r'otý) in a wind tunnel showed significant departures
16
from the traditionally assumed helical tip trail, especially overthe rear of the disc. Numerical results of blade-loads computations(at the Cornell Aeronautical Laboratory*) reported by Piziali inReference 2 suggested that a more accurate description of the wakegeometry might lead to a better correlation between the computedand measured blade loads. A further modification of the Reference 2work was performed by Chang at CAL (Reference 19), who incorporated(1) the adjustments in the pitch control settingsand shaft tiltfor the given flight condition (thus "representing" the pilot inthe steady-state computational algorithm) and (2) torsional degreesof flexibility of the blades into the computer program. BothReference 2 and Reference 19 used a rigid helical representationfor the wake, however. The results of Reference 19
"...are not significantly closer to the measured resultsthan those obtained in Reference i.** Therefore, it app(that, at least for these configurations, further improv,_.. cin the correlations will probably require the establishmentof a better wake model and an even more complete representa-tion of the blade motions...." (Reference 19)
Further evidence of the possible importance of wake geometry isreported in Reference 20. This comparison of the results of twoblade-load computations, one using a uniform inflow and the otherusing a variable inflow resulting from a CAL type rigid wake, showedmarked differences in the two blade-load predictions.
It was concluded that a blade-load computation using a more realis-tic wake-geometry input for calculating induced velocities would beworthwhile. Since an accurate wake-geometry computation (such asReference 18) from a time-varying velocity field resulting fromwake interaction, fuselage effects, and blade bound circulationwould be quite extensive, a modified actuator theory approach wasadopted. The approximate method reported here uses a Kinner-Manglertype actuator disc with an improved lift distribution to get anapproximation to the time-averaged induced velocity field. The liftdistribution is chosen so as to realistically represent the radialvariation typical of helicopter rotors and to accommodate up to the
*Cornell Aeronautical Laboratory, Inc., Buffalo, New York,
hereafter abbreviated as CAL.
"**Meaning Piziali's work, Reference 2, according to our list ofReferences.
17
second harmonic components of the azimuthal variation. The methodthen predicts the spatial positioning of individual vortices releasedfrom a finite number of blades, without any a priori assumptionsregarding the wake geometry.
18
CHAPTER 2
THEORETICAL MATERIAL
2.1 Theoretical Background
Kinner (Reference 9) noticed that Laplace's equation, which isthe governing equation for the pressure field in case of linearized,steady, incompressible fluid flow, is separable in the ellipsoidalcoordinates and that it is possible to choose the coordinate systemand the harmonics in such a manner that the solution is discontinuousacross a finite, flat, circular disc. Thus, he related the pressurefield to the total lift developed by a circular wing. He also gaveapproximate values of the induced downwash on the disc. His analy-sis did not allow any disc-incidence angle, however, nor did hecalculate the sideways induced velocities. In addition, thediscontinuities Kinner dealt with yield a variation over the discfor certain combinations of these solutions that appears quite similarto the lift distribution on a typical rotor.
Mangler (Reference 10) modified Kinner's work to include the disc-incidence angle, but he restricted his solutions to azimuthallysymmetric lift distributions. For such a restricted flight condition,he obtained numerical values of the induced downwash on the disc.Figures 3 and 4 show the distribution of the induced velocity overthe disc for a disc incidence of 00 and 150 respectively. Figure 5shows the same results as Figure 4, but they are displayed throughthe variation of the induced velocity along three longitudinal lineson the disc.
2.2 Capsule Statement of the Work Done
In order to evaluate the importance of the details of the wakegeometry, it was decided to compare the blade-load predictions froman existing blade-load program, using two different wake-geometryinputs. The program used is that* reported in Reference 2. The two
*Referred to as BLP2 of CAL (Blade-Loads Program 2 of Cornell
Aeronautical Laboratory). Some details of this program appear inSections 3.1 and 3.2.
19
X Y/R
DIRECTION
00.33
Total Thrust
Ct 0
W 4 induced velocitynormal to disc
Figure 3. induced Velocity Over Disc for Azimuthally SynsZricLift Distribution, Expressed as Contours for VCt "
For Disc Incidence = 00. (From Mangler,
Reference 10)20
SY/R
VFS DIRECTION .
S o-. .2 000.1+0.2 +0.2+0.1 01
Total ThrustCt = "VR
induced velocitynormal to disc
Figure 4. Induced Velocity Over Disc For Azimuthally SymmetricLift Distribution, Expressed as Contours fors.--WFor Disc Incidence = 150. (From Mangler, VriCtReference 10)
21
14US LATERAL STATION~ FROM WUB
0062 AOI5
O-S J# /
tI
jb ELer
0no"
Figure 5. Induced Velocity Distribution According toMangler and Squire. Disc Incidence = 150.(From Reference 8)
22
wake-geometry inputs were (1) a rigid, helical wake geometry result-ing from the assumption of a uniform induced velocity. (This isthe type of wake geometry used in the CAL computations in Reference2 and will be termed the CAL wake geometry.) (2) A distorted wakegeometry resulting from the steady velocity field of an actuatordisc with a lift distribution that includes up to the second azimuth-al harmonic variations. (This wake geometry will be termed the URwake geometry.) The steady velocity field used here includes inducedvelocities in all directions. Specifically, it does not ignore thesideways velocities. This is considered to be a definite advantage,since even small sideways distortions of the wake may be of impor-tance, especially where the trailed vortex from one blade passesnear the succeeding blade(s).
No changes were made in BLP2 other than to modify it to suit the IBMSystem 360/65 at the Computing Center of the University of Rochester*and to modify it to interface with the UR wake computing program.The representation of rotor dynamics, blade aerodynamics, and wake-induced velocity relations was retained in its entirety.
Section 2.3 describes the theoretical scheme leading to a determina-tion of the steady induced velocity field (used for calculating theUR wake geometry) for a given flight condition.
2.3 Theoretical Scheme
2.31 Equations of Fluid Motion
Consider a stationary, flat, circular disc, radius R , in aninfinite expanse of an incompressible fluid. Establish a Cartesiancoordinate system x yz such that the velocity of the fluid at aninfinite distance upstream of the disc is -- Vf at a disc-incidence angle a (see Figure 6). The equations for incompressi-ble fluid flow are
momentum: =i 1 (24)
continuity: (25)Ti o ...i,j i ,,
*Hereafter referred to as UR.
23
x ' J •Port Side View
z Z
WindDirection
X -- •Plan
Figure 6. Cartesian Coordinate Systems xy' ("Wind System") andx'yk' ("Disc System") in Dimensional Form.
24
where •i are the components of the fluid velocity, /o is theconstant density of the fluid, and P, is the pressure. For steadyflow, the momentum equations become
¶ ,Po,' "' 1- ,2,3 (26)
Noting that the components of A are (-VS÷u,v,w) , where (.,W)is the velocity perturbation, and assuming a lightly loaded rotor,i.e., assuming
(u,v,w) << (27)
allows linearizing equation (26) to
Differentiating (28) with respect to
x /0
But the left-hand side of this equation vanishes as a consequenceof the continuity condition (25). Hence,
P0, 0 (29)
i ~i.e., Vp : 0
which is the governing equation for the pressure field in a steadyincompressible fluid flow. Hence, the disturbance pressure field
p must satisfy Laplace's equation.
25
2.32 Coordinate Transformations and Solutions of Laplace's Equation
Since we are interested in obtaining a pressure field that satisfiesthe boundary conditions of discontinuity across the disc, we attemptto transform Laplace's equation in 3-space to a coordinate systemsuitable for that purpose.
Define a Cartesian coordinate system xy ' z , which is obtainedby rotating the xy'z -system through the angle a about the y'-axis(see Figure 6). For convenience, let us nondimensionalize ourCartesian systems on the disc radius R , i.e., define new systemsSXYZ and X'Y'Z' , such that
(x,Y',Z) - xY Z ,, )',(R (30)
and I I I
The transformations between the wind system XYZ and the disc system)('y'ZI are:x dos °l 0 -I
[ Cos a]0 VN(31)
Z = 1 (32)
Define a curvilinear coordinate system such that
26
y" -- VI (33)Z- VP
It may be notec that this aiO:VO1 coordinate system will cover theentire 3-space once, if we restrict v' , and 3 (seeFigure 6) to tl.e ranges
o _ I- : 5,V--< +0 •~ CC)ý :L0 (34)
Figure 7 shows the O coordinate system viewed in the X)7-Zplane (pitch plane of the helicopter). The V a constant surfacesare hyperboloids of one sheet and the r a constant surfaces areellipsoids, both families of surfaces being azimuthally symmetricabout the Z' axis. 3t6 is the azimuthal angle measured fromthe negative )(' axis, counterclockwise looking along the plus z'axis. V- 0 represents the two faces of the disc. The coordinateV, changes sign as one crosses the disc.
The inverse of the transformation (33) is
r ZI (35)
-XI
The matrix of first partials of the Cartesian coordinates X ,Y' Z'(represented by ti ) with respect to the curvilinear coordinatesV, 7S (represented by ) is
-1/ (36)
-v 0
27
II
Figure 7. Ellipsoidal Coordinate System, Viewed in thexz'z Plane.
28
The metric tensor for the WVJ system is
I -I a
0 W ,,4.2 0- 1+ ?I (37)
0 0 +
which shows that the a,47" coordinates form an orthcgonal system.This makes writing out Laplace t s equation especially easy, sincein orthogonal coordinates (see Reference 22),
F ( 1e \ at (eie. * .L3j5
where *k * (no summation).
For our ellipsoidal coordinate system, then, Laplace's equationVZ• = 0 takes the form
Letting
,,) +,v). . +,(3•) (39)
it is found that (38) separates into the three equations
S + m- 1 = o )o+ rn (40)
~. [~0(n.] '1 (41)
29
S+ 42)
where V" and Y% arc the constants of separation. The single-valued solutions of (40) are
L=OSml/f, /,r n[ ... Yn= iteger (43)
Equations (41) and (42) are recognized as forms of Legendre'sassociated differential equation (see Reference 23), and theirsolutions are
P n n (V Owl(44)
12 3) P '7) Q(')]" (45)
Thus, we find that the mos÷ -eneral single-valued solution ofLaplace's equation in our e.lipsoidal coordinate system is
SV [P~ Ccosmi (46)
where mn is an integer.
2.33 Choice of the Pressure Field, Based on Total Thrust and NetMoments
We shall deal with the disturbance pressure in its nondimensional-ized form P , normalized on twice the free-stream dynamic pressure,i.e.,
P (47)
3 A
The general solution of Laplace's equation is given by equation (46).Appendix I, whizh was prepared using Reference 24, shows some of theassociated functions of Legendre, P ' and 0- , and their proper-ties. Before concluding that equation (46) is the general solutionfor the pressure field, we must examine it for any contradictionswith the physical restrictions on the disturbance pressure P
Thus, we notice from Appendix I that the Q'k'(^) would be infiniteat W 2±1 , since they contain terms with denominators thatvanish at V.-± . This leads to an infinite disturbance pressureon the Z@ a,.is. Hence, we must discard the Ql'(1i) type ofcomponent from equation (44), thus reducing it to
(48)
This form of eliminates disturbances of infinite magnitudealong the rotor shaft axis.
Similarly, it is noted from Appendix I that P()--e as p-o.Since this would mean an infinite disturbance at infinity, thiscondition must also be ruled out on physical grounds. This reducesequation (45) to
= (49)
Appendix I shows that (49) tends to zero as Y tends to o
Equations (39), (43), (48), and (49) are combined to obtain thegeneral form of the disturbance pressure:
Z P''W)CM(-) IC n
whereCP and DO are arbitrary censtants. From among the infinityof terms in this general form, we will retain only those terms thatare necessary to simulate a reasonable lifting rotor or disc.
31
Since over the disc area the lift density should correspond to thedifference in the pressures p just below (9ao, vcO) and above(y=O.v>O) the disc, we drop all combinations of wi,b in (50) thatresult in an even (vw.n). This is because P",w) is an even func-tion if (Cn ) is even.
Let us set the practical requirements that (1) an integration of thelift density over the disc area should correspond to the total(steady) thrust developed by the rotor, (2) the lift density shouldvanish at the center and at the periphery of the disc, (3) theexpression for the pressure distribution should be able to representnet pitching and rolling moments supplied by the lifting disc tothe craft, (4) the pressure distribution should reasonably repre-sent the first harmonic components of lift density as experiencedby lightly loaded rotor blades, and (5) there should be some pro-vision for simulating the second harmonic components of the bladebending moments.
Total Thrust
Using equation (50), the total thrust T is
T sf JA -- 4dA
co w 'b r" sin M 'O I LO:("W ter]
rVA": X c: O: (o P: (.,) rdr
From eouations (33), we find rt.RS(¶-u) for a point on the disc.Hence, while integrating on the disc,
rdr[
32
+1.'T - zi "" E-;S*o: -(-,o) Pn (W.) S)dip
Noting that W) P*(i) and using the orthogonality relations inAppendix I, only the (noi) term will have a nonvanishing integral,and we get
T=: uIr Fe 122 (5.1)
Defining a coefficient of thrust Cr,
CT T (52)
we get
0
_ (53)
1 is the rotor speed in radians per second, and,4,& is calledthe tip-speed ratio
SWR (54)
Thus, the combination(hi=O, Ii4) is necessary for getting a steadythrust from the pressure field.
From Figure 8, it is seen that P,0(Z))does not vanish at the centerof the disc, i.e., at V)=t I . In order to have a nonvanishingthrust, and still to satisfy condition (2) above, we must includethe combination (M=O) t i along with any other combination(s)having (m= O) . Their coefficients will have to be so relatedthat the composite lift density vanishes both at r=O and atr=f .Let us try a combination of(mn-o)tj1)and (rn Ojhi-S). Sincel).Oatthe periphery and -L)a t I at the center, we require that
C P. (tl1) (i0) 4-
P;• cp(a).- U, 0) + c;.; o). 00 U=O) =
CO.c C 9 CTX (55)
33
a. .lo
0,6,
Figure 8. ,(;C)Plotted as a Function of Radius r on the
Disc.
34
Thus, the combinations (t11% 0) r i) and (mO, i -- ) can
together satisfy conditions (1) and (2) on page 32 ifC - C1.
Rolling and Picching Moments
Using equation (50), the rolling moment about the X axis is
mt% rd o (cow IT--uPMeM
Using orthogonality of the circular function over 0 - V
(LLower.-•ter)
while integrating on the disc, we have
•., Mr p~r Oo
Noting that I (JOSE-3.VTbh and using the orthogonality
relations in Appendix I only the (n.2) term will survive inte-
gration, and we ge4
r r 3 1 V5 (56)
Defining a coefficient of rolling momentMr (57)
CMr r"
35
we get
.5 CMr -CrD' 6 ,,p (58)
A similar calculation for the pitching moment MP leads to
MP Aij-rRýpV, s-Ce jY axis (59)
5 Vre -- CZ j• axis
S(60)2 Ax
c 4P MP (61)
Thus, the combination (raIj t1=Z)will satisfy condition (3) onpage 32.
Figure 9 shows typical plots of the steady and first two harmoniccomponents of experimentally measured blade loads. Since the radialvariation of Pnj) in Figure 8 is quite similar to the variationof the first harmonic components in Figure 9, it was decided toretain the combination(MM-If 4 )in equation (50). Also retainedwas the combination(m.2,n,3)--)
Table II shows the combinations ofm and n to which the steadydisturbance pressure field P was restricted.
40 %Q'"ef),c"c'so vsL4"srn, rAV (62)
2.34 Choice of the Pressure Field, Based on Blade Flapping Moments
n a j aIn Section 2.e 3 the coefficients Clt , C; , rpe , and thwerecalculated fromct o, h, andC .afelb The latter represent the "total",or integrated, effect o the disc as felt by the craft, in the formof the thrust, rolling moment,and pitching moment respectively.
While this sort of a determination of the pressure field may seem
36
LBS/IN.
X"s ste
4.
STEADY COMPONENT
1 ST COS
4 ST SIN
2 WN COS
2 N SIN
Figure 9. Typical Steady and First Two Harmonic Components ofExperimental Biade Loads. (From Scheimann, Reference 14.H-34, Flight 20, V = 122 Knots.)
37
TABLE II. THE TERMS RETAINED IN THE EXPRESSION
(62) FOR THE DISTUJRBANCE PRESSURE P
m nR
0 1 This is the only term that gives rise to anon-zero total lift (but does not have zerolift density at the center of the disc).
0 3 Together with the above term, will give zerolift density at the center of the disc if
1 2 This is the only term that will give rise tonet pitching and rolling moments.
1 4 Although the net moments over the disc arezero for this term, the radial distribution ofpressure due to this term is quite similarto experimentally observed pressure distri-butions.
2 3 To account for second harmonic variations inlift density and blade-flapping moments.
38
quite obvious, .:t certainly is not the only way to relate thepressure field to physical quantities. For a real helicopter witha finite number of blades, barring the case for which experimentalmeasurements of blade differential pressures are available (as incase of Referenie 14 and 15), it may be quite difficult to accu-rately determine the steady components of total thrust, rolling, andpitching moments, because of uncertainties in fuselage drag, vehicletrim, etc. it would probably be advisable, then, to be able tobase the determination of the steady pressure field on some moreeasily (and conFidently) estimable quantity. The flapwise momenton the blade could be one such quantity. Even for an experimentalcase, it is probably as easy, if not easier, to get measurementsof the blade-flapping moments as it is to measure the differentialpressures. In any event, since lift variations higher than thefirst harmonic do not result in any "integrated" effect at all, itis clear that expressing the C.,174min terms of the instantaneousflapping moments on the blade would be helpful in simulating higherharmonic variations in lift in any actuator disc approach. Thissection deals with this approach.
There are two possible ways in which the coefficients C and D,can be expressed in terms of the flapping momentsAI(o*) experiencedby the blades at their roots as they rotate:
(1) Assuming that the value (•J"A)-(P(,Pe)7of the theoretical,..a- up$*.steady lift density at a point fr,$) on the disc is the timeaverage of the lift supplied to the rotor when there is and isnot a blade at that point, or
(2) Assuming that theM(*)is simply the flapping moment ona blade, when it is at the azimuthal position 00. , and subjectedto the steady lift dens ityrP, i)-• )at .
It may be noted that with assumption (1), the total momentum trans-ferred from the fluid to the disc will equal the total momentumtransferred from the fluid to the Nh blades. Assumption (2) doesnot satisfy this condition.
Making assumption (1), let us further assume that the time historyof the blade lift density at a point(.r, )of the disc is a seriesof pulses as s'hown in Figure 10. ./7 r,;p)represents the pressuredifference experienced by each of the blades at Oý when atp Crj-)represents the assumed steady lift density at (r, i .
/oweP " lower
39
N6 = number of bladesA• = rotor speed, radians/sec6 = blade semichord
IT --
+4b LI.TTassumed pulsating lift density for afinite number of blades
- --------- assumed steady lift density for anactuator disc
Fiure IQ. Time Histories of the Lif t Density at a Point (P,#)c f the Disc.
4 j
Assumption (1) requires that the areas under the t-wo time-hiftorycurves be equal.
bAN6 /ower
and M ý. 2h(r -). 2 .dr0
oo
-4
.f,, r kd .(64)
Using expression (62) fore, we get
Evaluating the integrals for the five combinations of (M, .*)shownin Table IT tbis red,-•es to
N(*)4 - CM[•.)~ ~ -f-7- Cz -A j- 5 1-0{ •*2/,
IAYY/6% s a , ?
Or, remembering that C , we nave
" 16N 1
(66)
I SZ W6(67)
D =�. *Al. (68)
I VA . a (69)
2 - A/o A (70)
where . 0 MM, 1 ,,Aeanc Ae are the harmonic components of
M{) M +M~ Cs +M5Sl'ý7 *I 2 MC2 eo i,+M~i T7l)
Equations (65) through (70) relate the steady pressure field to theflapping moments resulting from the airloads on the blades, and wereso used in the UR wake computations reported herein. These rela-tions w.re obtained under the assumption (1) of equal average
Smomentum transfer on page 39
if we choose the simpler assumption (2) on page 39 , thus ignoringthe average momentum transfer requirement, we get
N (Y) J', s 9.*' dr(72)
R oSince on the disc r RV we ha3.e
.. =(, - 00
R2
4 i
Using (C32) for R , we get
M~ 4 hb Rpk L5 /0C YTs S/
Evaluating the five integrals,
jC*4#?7~)O *
.22
Noting that both C Iand C contribute to the first harmonic cosineflapping moment, Ieneed fn additional. "constraint" to determinethem. Let us choose the total pitching momentM,.b from equation (59)for the purpose. Similarly, equation (56) will be used to determine.27. and. 3
. The results are:
2R
iiS
Co - - (74)
;2 ?7~p -R5
227EFS
i4
mA
G (77)
1r. b 1sa (80)
Equations (73), (74) and (77) through (80) do not satisfy theaverage n'orentum transfer condition.
2.35 Induced Velocity Field
We shall deal with the induced velocity in its nondimensionalizedform •(uL/,I,) , normalized on the free-stream velocity&/h-6 , i.e. ,
Using (81) and (47), equation (28) becomes
2?, i.. ,~ (79)
which is the complete~ly dimensionless torm of the linearized momentumequations. Integrating (82) and noting that all disturbancesvanish at an infinite distance upstream of the disc, we get the
in25ue n velocity field
44
We~~~~~~~~~ ~ ~ ~ ~ ~ ~ shl elwt h nue eoct nisnniesoai
Q.(xyýY, Z) =f _
The subscript i represents the Cartesian "wind coordinates"X.,Yand X In order to evaluate (83), then, for, a field point dPo
(X~ ~) we must first obtain the partials 6"a and --of the pressure field P , and then carry out thie integratio~n from
16 CWo to x . Using the chain rule for first partials,
-,., =k ,2,3 (84)
where Mrepresent the "wind coordinates"? (X,YZ )
f "'represent the "disc coordinates" (X;YZ' )
r~represent the "ellipsoidal coordinates" ~)
The are available from equation (3?2 d.4 h are ob-cained
by inverting the matrix of first partials/ 7 in equation (36):
[i =/ V"'1 0 W 7) & 0 V Z("' (85)
~~/n~C 0.fiz P~~[zl~)1f 0
The ae obtained by using the expression (62) for P and theformulads for derivatives of Legendre's associated functions inAppendix I:
2ýC Q"Cas a, ;kV-SyA -W L' (86)
45
mA
( 71) 7 Z7COS c /0, In (88)
We now have all the necessary formulas for numerically integratingequations (83). Some simplification, however, is possible in case
of the X-component of equation (83):
x(x I Y, •Z,) -- d - (' (89)
So long as the path of integration does not pass through pointswhere P is discontinuous, evaluation of the integral on the right-hand side yields
4./(x, Y,Z) = P(x,Yz) (90)
Since the only place where the horizontal path of integration
encounters a discontinuity in P is the disc surface, (90) holds forall points upstream of the disc. For a point (X,YZ ) on a stream-line which crosses the disc, we use the physical condition that thefluid velocity at a point on the lower face of the disc is the sameas that at its image on the upper face (i.e., the actuator ispermeable)*, an- cv3luate the integral for U in two parts:
4U(x,Y,z) =J- ,f.rY~Z)d0 #J/(WfZd
he-. U r (x , Y, z) P(x , Y, Z) V re CA,•0ý ýt)'- q,,o !,,,ijake lmr(91)
Avee lace
*or, "no finite -hanqes in velocity can occur in the infinitesimaltime it takes a particle to cross the disc."
i46
or, since for our disturbance pressure P of equation ('02), P--P Iwe get a~iee lower
U (x, Y,z) = P(xY,-Yz) -2P otewei1, itlWak~e /owor eace
The term (intersection point) indicates the point at which thehorizontal path of integration up to the field point (XhYZ )crosses the disc.
The induced velocity components Y and W are evaluated numerically.
The wake geometry resulting from this steady induced velocity field
is also calculated numerically, as described in Chapter 3.
47
CHAPTER 3
THE COMPUTER PROGRAM
3.1 General Description
This section briefly describes the blade-loads program, BLP2 ofCAL, as modified at UR. This program used to calculate blade loadsbased on the two different wake geometries.
The BLP2 is in two parts. Part 1 first generates the blade segmentcoordinates and the wake element coordinates for the given flightcondition. Using these coordinates, Part 1 then calculates (usingthe Biot-Savart field law) the induced velocity coefficients.,Smand T7 appearing in equations (105) and (106),
'V" (MO, j; &I -. , N,, ; J isIpg d',, Na). TheseS and7 coefficients are then written in binary form on TAPE 2. Figure 11
shows the logical flow diagram for Part 1 of the unmodified CAL pro-grai.t.
Part 2 uses the TAPE 2 data set generated in Part 1 as input. Inaddition, all the dynamic properties of the blades and controlinputs to the craft are supplied in Part 2. It then solves theequations of rotor dynamics, blade aerodynamics, and wake-inducedeffects by iteration. (The essential elements of the representationof these three phenomena in the numerical program are summarizedin Sections 3.21, 3.22, and 3.23 respectively.) After achievingconvergence, the blade loads and other quantities of interest (aswell as -heir harmonic analyses) are written on TAPE 7. Figure 12shows the major steps in the iterative procedures for solving theequations. Figure 13 shows the flow diagram for Part 2 of BLP2. Itwill be noticed from Figure 12 or 13 that the Part 2 procedure in-volves two iterations, one "inside" the other. The iterative solu-tion of the simultaneous equations (109) in the bound circulations1Z takes place "inside" the iterative solution of the blade dynami-caI equations (96).
Figure 14 schematically shows the BLP2 as modified at UR. Bycomparison with Figure 2, it may be noted that the control settings,vehicle attitude, and motion have ceased to be unknowns. (Thismeans that BLP2 ignores the "Pilot" and "Vehicle Dynamics" blocksin Figure 1) Similarly, the uniform induced velocity is needed ifthe CAL rigid wake geometry is used; and some information on theblade loads is needed if the UR wake geometry is to be computed.Since the modification to the CAL procedure is essentially the useof an alternative wake geometry, the necessary changes and/oradditions to the CAL program occur in Part 1 of BLP2, as can be
48
| mL-L
READ C6010,11 I'llYI
~I LIS ~MIDOIT It. IAAILOO,014" IOug I tmA. tm l A W on HE imi,
VAXE [L8 11
Figure~~~~ 11. Flo diagra for Part 1 fBL (rmReeene2
49 111116 tKT
JNW #~ A L SL DE PU ETI ES Fl pJ . T IPRIM100 TIE WU INITESRTI OO1911 TI; iTN&
WNPTE'tsyAPPROXIA IO
CW ON~PI SITANT, 1k, OFTVQ11 IATIONS CrEEPt NII1
TN ~Iujj ~l~A IPII 01918T
I SLY INULTAIIEUI 81T OF
r-EIUATIMII]
I VL=AT Tot ".ARET CEPPICEN
L M SACH AZINUT POSITIO1H
MMTUS GENERALIZED AIJULOA& IN EA=
1163PH U AZIWM POSITION
IEAATE THE RAIDISH IC CUONHNENTS OF
KNUIUAIALOAD IN ELACA WKH
T MIITEI A110116 A$1111
MITEE RESPLTSE NFGD N
Figure 12. Major Steps in the Iterac:ive Procedure for Solvingthe Equations in Part 2 of BLP2. (From Reference 2)
SUBROUTINE SUB21:
COMPUTES CNSITANTS AND 1VARIABLES USED REPEATEDLYTN THE PROCRAT I
SUBROUT NE'SUB2
DIAGONAL ELEMENTS OF W
"I" ELEMENTS ort r QATION, INITTAL TRIAL r's ANDGENERALIZED COORDIdJATES
1TM =1INDEX FOR KE NU1MBEROF ITERATIONS PERMITTED ON
THE MOINS. NIAT OIO'MAX,•' =., NITM . INIAEFAILED TO OVRE-
.....2----- HoE P• CONVERC I--~ PRESRIBE• •NO. OF ITERATIONS
I ~ ~~~(1) ITERATIVELY SOLVES THE I• EQUATIONS •SEFgr 4
[ OI•:•D:TN•~ vT~! I (2) cOMPUTES LIFT, MOMENT, GEN'ERA•LIZED
RECOPUTETHE I''s (3) COMPUTES FOURIER COEFFICIENTS OF THE GENERALIZEDOF HE ' EUATONSCOORDINATES AND THEIR TIME NIf-.1'ORIES.
B ONE(4) PRINT RESULTS.OF TE GE. CORDS. (5) HARMONICALLY ANAL'YZES RESULTS.
Figure 13. Flow Diagram for Part 2 of ?LP2. (From Reference 2)
51
®CA I
STRUCTURAL
IýA E -Y NFAMI I
AE-ODNAM1.Voricty I I
WANKL-INDVCEDI
40 BND VORTLIcTyO a
LL
r Wk-nue eoiyefcso on iclto@-U ifrm ndued eloityforpostioingtheCALwak
s-Information~~8ITCKAI obld oad orcosigprsue ilfor calculating R Owk(See Figure 1 forI oTher quanFWtite fitrs
Figur 14.Scheaticof te Moifie BPET
A52offc;t
noticed from the lower right-hand portion of Figure 14. Appendix IIshows the FORTRAN IV source listings for subroutine SUB14 of Part 2,as modified at IR, and for subroutines RWAKE, COEFS, STMLN, VLCTS,CHAMP, FRWRD, ADVNC, and CRSNG, which comprise the UR written wake-geometry program.
Essentially no changes were made in Part 2 of BLP2.
3.2 Representation of Various Physical Phenomena in the CALProgram
This section briefly describes the representation of the phenomenaof rotor dynamics, blade aerodynamics, wake-induced effects, andwake transport in the original BLP2. An alternative to the CALwake transport was added at UR, while the other three representationswere left unchanged.
3.21 Rotor Dynamics
For a numerical procedure in which all pertinent quantities arecomputed at a finite number Na of discrete instants around theazimuth for a finite number Ale of blade radial stations, it isconvenient to harmonically analyze the generalized forcesmentioned in Section 1.21:
0,,W)= A (Z(A ,Cos PJ2t7A 5aiý )7 (92)
whreA is the number of harmonics rets. ed. (The maximum numberwould be- . The components A,,( NAkNmkl'1,NA) are definedby equatioffs (4) and (92), except that hereafter, we shall neglectthe torsional displacements @Ot), so that the aerodynamic pitchingmoments will not do any "work" on the blade modes, which now involveonly flapping displacements. Thus,
- A":4 j -r (93)
N.-
53
whepre:iio isPe length of the j th spanwise segment about radialposition 'is the flapping displacement atr j in the * thnormal mode; and P 1 1j and Ali are the #? th azimuthal cosine and sinecomponents of the lift density P at the j th spanwise station,such that
pc., = j+ (jc.n t•• A,, Zt (94)n-Z ... j =J, Ne
The equations of motion (3) for the NM,, modes -an now be solvedseparately for each of the harmonic components of the steady-stateresponse, resulting from the corresponding component of the forcingfunction, yielding:
(2d 2
4,.. _*L 2 P 4
These define the generalized flapping displacements
Nh
k o1 . A, A
The flapping displacement and corresponding "plunging" velocity of
a blade :,adial station can now be computed according to equation (1):
1 60 (Nh 7 (97)
k', ( Go cos-i z'*+b,,, si
54
ý(.joL Z 4a, cost7J2to l 1.2 t
) () Z, -, ( 72t (98)
k -"z ,,
The plunging velocity affects the quasi-steady component 1 K of thebound circulation, as can be seen from Figure 14.
3.22 Blade Aerodynamics
The blade representation of Reference 2 used trigonometric seriesexpressions for the distributions r(x,)and (*,)me-tioned in Section1.22:
X.(69) =fA co ZA, (99)
K 1 , ,Na
cp cs i (100)
where 9 is the blade-chord based Glauert coordinate (see Figure
2(c)) defined by
X#4 =-be 09 (101)
The subscript IC refers to the particular discrete disc point towhich the C-,,,AeAnW 8),P 2 above belong. K ranges from1 toA/ra , where Aeo.ANp.rN. See Figure 15(a). (Note that thevorticity distribution in equation (99) identically satisfies theKutta-condition of equation (6).) In the numerical computations,the infinite series in equations (99) and (100) were truncated afterthe third term, i.e.,/0 ranged from 1 to 3. (It turns out that thehigher harmonic components in Lg(*)do not affect the theoretical7" , P , andP5/1 as evaluated from equations (14), (15) and (16).)
Substituting equations (99) and (100) into equatiorn_(5), thecoefficientsAve can be related to the components P. of the im-pressed normal velocity:
55
RELATIVL
f f$
y I0/K
0 2
(a) Example of the K-
¥.o" subscript notation for
airload points in rotor
disc when0
0,A b
(b) Wake
odel for
to-
bladed
rotor. Both
tip
and root
concentratedp,
vortices shown.
' (c) Relation between mesh-r. r r.- ra vortex strengths and
the bound-vorteystrengths when Hr.9,
Figure 15. K-Subscript Notation, Wake Configuration and Wake
Vortex Strengths. (From Reference 2)
56
I,,
ii i
A14 z. (102)
A-
The total bound circulation is now obtained by evaluating the"integral in equation (14):
-' ) 6 (i4#* -A (103)
weheren is the measured lift curve slope for the airfoil
The numerical program also limits the total bound circulationat disc point K to a maximum value row, , whirh is determined fromthe previously known stall angle of attackOC, 0 1 for the particularblade section:
r"A Id OC A (104)
When the 71( as computed from equation (103) is found to exceedthis l7 p, , it is recognized that the blade is stalled at thatposition and the computed rle is replaced by e .. The circulatorylift, as computed using equation (15), is augmented at the stalledsegment by a cross-flow drag force.
The other "output" from the blade aerodynamics block in Figure 1is the wake vorticity strengths. The wake configuration used inthe CAL program is shown in Figure 15(b). The wake consists of amesh wake behind each blade for A'SV azimuthal segments, as wellas concentrated tip (and root) vortices behind these meshes. Theactual locations in space of the various points in the wake model
57
are determined in the computations of the "wake vorticity positions";see Figure 1. The strengths of the mesh-vortex segments are relatedto the bound vorticities ac the/Mr disc points as shown inFigure 15(c). The strength of a concentrated tip (or root) vortexsegment is taken to be simply the maximum value of the bound vortic-icy on the corresponding blade when at the appropriate azimuthalposition.
3.23 Wake-Induced Effects
For the purposes of the numerical program, the vorticity strengthson all the filaments in the mesh wake can be expressed in terms oftheNrpg bound vorticity strengths as shown above. Further, thewake-induced normal velocities 40P( were needed only at the N,, pointson the disc in the form of components similar to the in equa-tion (100). For a given wake geometry, therefore, it was possibleto evaluate coefficients SOW,. such that the induced normal velocityat the •th disc point due tY the mesh wake is given by
Similarly, for the concentrated trailed vortices part of the wake,coefficients r.. are computed such that the corresponding inducednormal velocity due to the trailed wake is given by
Wr (e) i O7,Cno )? (106)
is the maximum bound vorticity on a blade when at azimuthalposition S . The induced velocity coefficientsS. and 7
b,*.. areevaluated in Subroutine SUB14 of Part 1 of the CAL program.
The "impressed" normal flow component (e)is broken up into twoparts: one, z(G), resulting from all the contributing quantitieson page 6 except the wake vorticity; and the other, 10(,), resultingexclusively from the wake vorticity. Thus,
&=, (107)I jo 4ý/;
CJ, N'
58
and (6 (1G8)
Combining equations (102), (103), (107), and (108), it is possibleto write
2,, * .O .. , .- ,a J J
which is a set of Yra equations for the Nra unknowns . I isthe bound circulation resulting from the normal flow o K andrepresents the bound circulation at disc point AC resu rom
the induced velocity due to the wake vorticity segments whosestrengths depend on the bound circulation at disc point j . Itmay be noted that the XK depend on the blade response, since theA• include the blade-flapping velocities, torsional displacements,
and torsional velocities. Equations (109) are solved by iterationin subroutine GAMMA of Part 1 of the CAL program.
3.24 Wake Transport
Representation of the wake transport in the CAL program consists ofdetermining the positions of the points defining the mesh andtrailed vortices in the wake model shown in Figure 15. The rigid-wake calculations reported in Reference 2 used the following formu-las for the x',I and Z' coordinates of the point Ri j, in thewake model,(i, )
51,o= k" - ) 4- " 1 C,?-,',), (2 jr'
"i" ~~Z( z,.• ( 5,, Q(•-_) 7 ,- -ýoc, >2 ..,•!re~in
i5
where,1 4 is the azimuthal increment,•'k-•. ha is the azimuthalposition of the blade shedding the wake.#j is the semichord atthe blade segment end point; t (P•') .is the built-in coning plussteady deflection of the blade segment end points, positive upwards._& is the assumed uniform induced velocity (quantity 4" in Figure14) over the disc. 6' is the fraction of a time interval throughwhich the wake is advanced with respect to the true blade position, .• "d•a' ,which is quasi-empirical, was determined in such a waythat for a two-dimensional oscillating airfoil, this discrete
representation of the shed wake gave results that were in "reasona-ble agreement" in the "reduced frequency range of interest" withthe classical solution of the problem. (See Reference 2 for detailsof this determination of 6".) For i = 1, the coordinates (X4,Z')i•kare simply the points on the trailing edge of the blade at azimuthalposition kaIt will be noticed that this wake transport ignores the displace-
ments in the sideways direction. Also neglected is the variation,over the disc, in the induced velocity in the Z" direction.
3.3 Algorithm for Computing the OR Wake Geometry
It can be seen from equations (110) that the point on the disc*(•t-', Pz')9 where the fluid particle now at the point Rijk&
'lij*4 in the wake used co be ('-7o'J) time intervals beforethe blades assumed the azimuthal position *a, has the coordinates
x 0 =- r. CO.sfrka, if') A V,7- b Shl, s/(_k4-*
=~ ~ i 6), s"''k "A' V7 ~cf~~~6ji7
0/Z. . =
"/Sh'"isc'" is not a flat surface, since blade steady deflection
an(! built-in coning are accounted for, through
,6, 6D
The difference between the expressions (110) and (;.ll) Jefinesthe movement of this fluid particle from RQl. (x, F •Z) to
under the influence of and LU, in (s--)time intervals.
In the UR wake-geometry computations, the points of "injection",1"k4 , can be retained* as in (111), but the movement under the
influence of both Vlxs and the induced velocity was computed differ-ently, through an actual calculation of the streamlines through theinjection points. The steady pressure field used for calculatingthe streamlines always had the pressure discontinuity on the flatdisc.
3.31 Subroutine RWAKE
Subroutine RWAKE, which is the entry point for calculating the URwake geometry, first chooses the steady pressure field throughformulas such as those developed in Sections 2.33 and 2.34. Figure16 shows the flow diagram for subroutine RWAKE. Table III showsthe various options for the choice of the steady pressure fieldbased on various physical quantities.
It may be noticed from Appendix I that the values of are
purely imaginary quantities when n is even. The correspondi. C,-and' & also turn out to be purely imaginary quantities. (P (.)are a¶ways real.) In such cases, the quantities actually handledunder QMN MJ) were e" . To offset this factor " , the corre-spondirg uantities CMN(J) and DMN(J) in core were set equal to-C'"and-z'V., respectively. Thus, all the computations were done in"reals" , and the products CMN(J)*QMN(J), DMN(J)*QMN(J) always agreedwith their theoretical counterparts Con'(_vm B" ,_ 7i,) for all
~J; i.e., for all combinations of w an'ln. 07 0
After choosing the steady pressure field, RWAKE sequentially calcu-lates, using equations (111), the disc-injection points 'Jk, forthe mesh and trailed wake grid positions (see Figure 15(b) ); itthen calls subrcutine STMLN, which calculates the streamline througha given disc point. At the end of 1MAKE, all the wake coordinates,disc inj-ction point coordinates, total normal velocity values atdisc points, and induced velocity values at the disc points arewritten on TAPE8.
*An option in the UR wake computation program also enables theparticles to be injected on the flat disc, thus ignoring
61
RIAD AC UftACY
W AftlkMSTUS POOL
WA X& CA TAP "AITON A14JT1A LL POS 1T1O
OF TIP-1miLC31OIJ P7.
COLFS
7T
& L ft R aA D A L. P O S T O -ftu
v AMA WAKL D&4ve JAZI-t
POIMNT.A JI.tA -JA Zlt* TJAI~ aa WO .00TAP6S.6 T"t
TiIIA .WA 4SIT'...
1JARI!AR. I , A
SIot 4~iOA SyatmaA
WMIT5.440 TAPS 4, 1NSSTMTREAMLIN 0000111 FOR
# a " a ¶TII~s Pon T111IMns 111M-NA4mALLAT
STREMLIN calculateR
(See a.bl onIII)8.T
Figure 16. Flw ns Digame r urutn RA
TABLE III. VARIOUS OPTIONS FOR CHOOSING THE STEADY PRESSUREFIELD P THROUGH THE COEFFICIENTS C' AND D"
P ,;' a Cos.
KKSW Remarks
I Subroutine COEFS bypassed. C'and •, remainunchanged.
2 .C1 and D. to be read in as input to sub-routine COEFS.
3 " anc-Dm calculated from CT , Cf andA (See Sec~ion 2.33.)
4 C: and.P calculated from TrX1,Me),R,Prs Vand A n(See Section 2.33)
5 CZ'• and.Vm calculated from input in the formof harmonic components of the flapping momentexperienced by the blade as it rotates. (SeeSection 2.34.) Average momentum transfer con-dition satisfied.
6 CZ' and.D calculated from input in the formof harmonic components of the blade flappingmoments, plus Rpf, V'.,b,M,. Averagemomentum transfer condition not satisfied.
7 C0 and.V' calculated from R.f, Vo'., N/6b,plus the harmonic components of the blade flap-ping moments which are computed from the har-monic components of the experimentally measuredlift density on instrumented blades. Averagemomentum transfer condiCion satisfied.
8 C and." calculated from the same inputs as7 above, plus Mdp and Af . Average momentumtransfer condition not satisfied.
63
3.32 Subroutine STMLN(KEND)
Subroutine STMLN(KEND) follows the fluid injected from a given discinjection point for a number (KEND-l) of time intervals, saving thecoordinates of the position after each time interval. The field isassumed steady, and the particle paths and streamlines are identical.A time interval is equivalent to the rotation of a blade from oneazimuthal position to the next. Thus,
A2.: 5' secoloO' (112)
To nondimensionalize time, we must use a normalization that is com-patible with the normalizations (30) and (81) for distance and veloc-ity respectively. This calls for normalizing t on the time takenby V,- to travel a distance R ; i.e.,
-4f= • V...±L = . (113)
whereA1*is the angle (radians) between two azimuthal positions andA is, again, the tip-speed ratio. For accurate streamline calcu-lations, the program has the provision for further reducing the timeincrement through a factor NINC. Thus, each time increment TINC is(I/NINC) times the time interval, and
T/INC = 43 (114)
Equation (114) represents the "usual" time increment value. ALLexception is the very first interval of a streamline. (See Figure 13)For a fraotional wake advance6d, the TINC for the first NINC incre-rents of each streamline will be (1-d) times that in equation (114).NINC is one of the accuracy parameters read in at the beginning ofRWAKE.
Subroutine VLCTS numerically calculates the induced-velocity com-ponents V and W at a given point (X, Y, Z) by the procedure indica-ted in Section 2.35. XUP (the location of "infinity" upstream ofthe disc) and XINC (the value of the incremental length used for thenumerical integration of (83) ) are the accuracy parameters used inVLCTS. Both XUP and XINC are also read in at the beginning of RWAKE.
64
#JJRCTIDWPITIIU
).NINC.
COMPUTL NORMALCOT~APOgT Yva6PF SAVE coo..Or'
TOTAL VtLOCT PONTme a
JPT Im r-=
PO~SMssp NO. ONSTARAhL 'NIL
C14AM K.I.NCHAMPQ~g
we~ *ogo a
vP-P imu Uo poes SeIIWT101
infiitel upstramTO jNOtam.c=e.
VLCpTS calculates, V n, at nd
at a given point.
Figure 17. Flow Diagram for Subroutine STMLN(XEND).
65
The quantities KFWD and KRVRS appearing in Figure 17 are "counters"for the number of flat-disc crossings made by -he particle beingfollowed, in the forward (i.e., normal, or downward) and reverse(i.e., upward) directions respectively.ap 1)PSubroutine CHAMP computes P , -•-at a given point( ,
.Vand (ata ivn oitusing the formulas from Section 2.35. It he noticed that themultiplier appearin- in equation (87) ftr U will make the quantityreal since, as explained in Section 3.31, qntherCt 1,i or iQ7 ,but not both, will be imaginary for each value of n. The quanrtityhandled in QMNMI(J) will always be real, being equal toi Q1,'"'/) if12 is odd, and equal toog ' (07)if n is even. his leads to agree-ment between the products CM'Ni(J)*QMNMi(J), DMN(J)*qMiM(J) and theirtheoretical counterparts
3.33 Subroutine FRWRD
Subroutine FRWRD, which is called in subroutine STMLN, computes thenext position on the streamline by invoking the (call ADVNC - callCHAMP) cycle NINC times. Figure 18 shows the flow chart for sub-routine FRWRD. Subroutine ADVNC saves the current values ofcoordinates, velocities, pressure, etc., and calculates the coordi-nates of the particle after a time increment TINC, using the currentvalues of the velocity. If a flat-disc crossing is recognized onADVNC-ing, FRWRD compensates for it by calling CRSNG. SubroutineCRSNG assures that the values of the velocity are continuous acrossthe flat disc, although the pressure has a discontinuity at theflat-disc surface. This is achieved in CRSNG by first finding outthe exact point of crossing, and then modifying the L/ component ofthe velocity through an operation similar to equation (91).
3.4 Integration With the CAL Program
It is seen from Figure 11 that the CAL wake coordinates are generatedin subroutine SUB14 of Part 1 of the original BLP2. (SUB14 callsanother subroutine COORD for actually computing these coordinates.)Since our modifications to the BLP2 essentially consist of generat-ing and using an alternate wake geometry, the interfacing betweenthe UR-written wake-computing part of the program and the BLP2occurs in subroutine SUP14. The FORTRAN IV source listing forSUB14 appears in Appendix II. Figure 19 shows the flow diagram
66
.JINC. I
INCfOLLI. NT NM -
No0 .JPTTOTAL - NINC
SU111MOUTTIEE
V *. C 7
no
C~~u.JIMA .4W COIiJ EA Ci
EP S
*ADVNflC saves the values of various quantities and calculatescoordinates of new position (after a time incremeint of TIEt) usingcurrent velocity values.**CRSNG compensates for crossing of the discs so that velocityvalues on either side of the disc are continuous, although pressureis not.
Figure 18. Flow Diagram for Subroutine FRWRD.
67
r r
1111.10
r 9-4u
I~ yuA
WAD ALL PUS1V
Figure~~~~--.- 19. JlwDarmfrSuruieSB4aModifie Mt UF.1468
for subroutine SUB14 as modified at UR. The changes and addi-tions to the original program are shown in dotted lines. Thechanges were made so as to make it possible to compute and useeither of the two wake geometries. The UR wake geometry is gener-ated (and written on TAPE8) in subroutine RWAKE. Provision wasalso made to make it possible to generate just the mR wake and thenterminate the job. Conversely, a UR wake generated in a previousrun can be directly read off TAPE8 and used for computing theinduced velocity coefficients in Fart 1. All these options areaccomplished through an input integer variable MMSW. (See thesource listing for subroutine RWAKE in Appendix II for a descriptionof these options.)
3.5 Peripheral Program
Two peripheral programs had to be written in addition to the URwake computation described in Section 3.3. These were namedRWAKEPIC and RSLTSPLT. Both make extensive use of the plottersubroutines package e ailable on the University of Rochester disclibrary.
RWAKEPIC reads the UR-Lco.nputed wake geometry from the TAPE8 dataset generated in subroutine RWAKE of Part 1 and, using the plottingroutines, pictorially displays the wake geometry. Three optionsare available as to the type of picture produced: pictures ofstreamlines, instantaneous pictures of the wake generated by a fi-nite number of blades, and progressive distortion of stream-tubesoriginating (from points equispaced around the azimuth) on the disc.RWAKEPIC, which is a separate main program, was found to be usefulin interpreting the UR wake geometry.
RSLTSPLT, which is also a separate main program, reads the con-verged airloads, blade-bending moments, bound vorticity strengths,etc., from the TAPE3 data set generated toward the end of Part 2(see page 48). Using the plotter routines, RSLTSPLT then createscurves showing the time histories as well as harmonic analyses ofthese converged quantities which are the results of the BLP2.RSLTSPLT was found to be helpful in comparing the blade-loadspredictions resulting from a UR wake geometry with those from theCAL wake geometry and with experimentally measured blade loads.
69
CHAPTER 4
COMPUTED RESULTS AND INTERPRETATION
The UR-written wake-geometry program described in Chapter 3 wasfirst used for calculating the induced velocity at a small numberof points on a flat, lifting disc at an angle of attack of 2.50Figure 20 shows the computed Z -component of this velocity. Theinduced velocity is upward on a crescent-shap ci area near theleading edge of the disc and on an area immediately behind thecenter. These results agree well with the results of Mangler'scomputations shown in Figiures 3 and 4.
The wake-geometry program was then used to predict three wakegeometries for two flight conditions of the UH-l rotor:
(1).A 0.26. (Wake geometry computed from streamlinesemanating from the flat disc, thus ignoring the steadydeflections j(PC) See footnote, page 61.)
(2,U= 0.26. (Streamlines emanating from the steadydeflected positions of the blades.)
(3)A, = 0.08. (Streamlines emanating from the steadydeflected positions of the blades.)
The CAL-computed values for the airloads and blade flapping momentsare available (Reference 2) for these flight conditions. (Thesewere computed using a wake geometry inpu~t calculated from a uniformdownwash.) Also available are experimental measurements for theairloads and flapping moments.
The modified BLP2 was then used to obtain the blade-loads predic-tions resulting from the UR-computed wake geometries. Except forthe wake-geometry inputs, all other inputs for these computationswere identical to those in the corresponding CAL runs. Even forthe wake-geometry inputs, the wake models used were the same.
4.1 UH-l atA =0.26 (Condition 67 of Reference 15), Injectionon Flat Disc
j~rs =188 feet per second (forward speed)
0( - 5.8 negrees (disc angle of attack)
!a
o= 2-5
, CtZ1t.21!-o• -' .'. •C-r 0 o'°055"
II , @o+ 0 -4 - 49I -o € •1.0 -• • I' h-.,
\\ -*o.4o •.;;
PLAN
Contours of zero downwash are shown by dotted lines. Numbers shownin the figure are Z -direction induced velocity values, positivedownward, normalized on the forward velocity V 1..
Comparison with Mangler's induced velocityvalues from Figure 3
location Mangler's UR
number W-Yet l .
0.2 0.1940.24 0.1820.06 0.040.34 0.24
Figure 20. . -Component of the Induced Velocity on a Lifting
Disc, Using the UR Program.
71
III
/V• = 2 (number of revolutions of wake)
0.7 (wake advance)
The induced velocity on the disc, as computed by RWAKE, showed anupwash area near the forward edge and just behind the center of thedisc. Figure 21 shows a typical instantaneous "picture" of waketrails. It is clear that the trails deviate appreciably from askewed circular helix. Figure 22 shows streamlines starting from90% radius on the flat disc. All the streamlines leave the discin the downward direction, but those in the region /Y/> 0.8soon come under the influence of a tip vortex effect, the onesfarther out doing so sooner. Streamlines in the /Y>< 0.8region continue downward. The result is a separation of thestreamlines into two wing-tip type systems emanating from the tipareas, and a body of streamlines emanating from the inboard areaof the circular wing. The "wing-tip" grouping of streamlinesexhibited a tendency to go higher (and move laterally) relativeto the latter group, which moves downward comparatively undis-turbed by the presence of the disc. Similar separation wasobserved in the water-tunnel experiments reported in Reference 16.The rolling up of the tip vortices is clearly visible in Figures 23and 24, which show the progressive distortion of an initiallycircular fluid contour, released from 90% radius on the flat disc,one and two rotor revolutions after release, respectively.
Figure 25 shows a comparison of the unsteady parts of the airloads(time histories), computed from this UR wake geometry, with theCAL-computed airloads and with experimentally observed values. Itcan be seen that the airloads predictions from the two differentwake geometries are only slightly different. The UR wake airloadsdo not seem to suggest a decidedly better (or worse) agreement withexperiment.
In Figure 21, a tip-trailed vortex is seen passing c;ose to aboutthe 75% radial station when the blade is at 3 = 75 . The inducedvelocity at that station due to the nearest tip-vortex trail seg-ment was -10.13 fps(CAL)/-3.93 fps(UR). (The negative sign indicatesan upward velocity, since our convention has positive Z downward.See Figure 6) The induced velocity at this station due to the entirewake was +9.48 fps(CAL)/+5.745 fps(UR). As compared to these, thenormal component Y' OC• due to the control input was 92,0 fps atthat station. The to~al normal component* was +43.74 fps(CAL)/+48.60fps(UR).
jnormal to the blade chord. See Figure 2(a).
72
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-. - - -CO#4PUIEW, ULt' WAKL 0"Jectio.,~ lo~fit d
Figure 25. Time Histories of the Unsteady Parts of Measuredand Computed Airloads (UH-1, IA.= 0.26).
77
Conversely, at 21.2% radius and 2550 azimuth, where significantdifferences between the two computed airloads are apparent inFigure 25, the induced velocity due to the three nearest segmentsof the tip-trail from the preceding blade is found to be -4.85fps(CAL)/-3.41 fps(UR). By comparison, the induced velocity dueto the entire wake is -25.1 fps(CAL)/-9.918 fps(UR). This differ-ence of about 15 fps is probably due to the closeness of themesh wake, which curls around close to the retreating blade atsmall radii, combined with the fact that this UR wake computationused streamlines emanating from the flat disc, thus placing thewake lower than the CAL computation which accounted forgoe .
The slight differences between the airload predictions from thetwo wake geometries are more observable at smaller radii. This isprobably attributable to the smaller flapping (plunging) velocitiesand smaller tangential velocities (and hence smaller ".9*Og contri-bution) at small radii, which would make differences in t?e wake-induced velocity more noticeable as seen at the blade aerodynamicsection. Contributions to induced velocity by specific vortexfilament segments are given in Appendix III.
Figure 26 shows a comparison of harmonic breakdowns of the air-loads. Figures 27 and 28 show the flapwise bending moments(time histories of unsteady parts and harmonic components, respec-tively) for the two computations and the experimental results.
4.2 UH-1 at 4 = 0.26 (Condition 67 of Reference 15), Injectionat Steady Deflected Positions of Blades
J/. = 188 feet per second (forward speed)
S= 5.8 degrees (disc angle of attack)
NW = 2 (number of revolutions of wake)
C = 0.7 (wake advance)
The computed wake geometry for this case showed all the qualitativefeatures of Figures 21 through 24. The tip-vortex effect, however,was delayed (i.e., became discernible farther downstream) sincethe streamlines, which emanated from the "coned" blade surface thistime, had to travel some distance before they entered the regimeinfluenced by the pressure discontinuity on the flat disc which isthe source of the tip vortices.
SWhen the computed geometry was used as input to BLP2, the blademotions did not converge. The iterative procedure in Part 2exhibited a sustained-oscillations type of behavior. These itera-tion-to-iteration oscillations in the generalized blade displacements
78
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Figure 26. Harmonic Analyses ef Measured and Computed
Airloads (UH-l, /iL= 0.26).
79
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Figure 27. Time Histories of the Unisteady Parts of Measuredand Computed Blade-Flapwise Bending Moments(UH- 1, .L- 0. 26).
80
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Figur?, 28. Harmonic Analyses of Measured and Computed Blade-Flapwise Bending Moments (UH-l,A." = 0.26).
81
were less than 1.5% for all structural modes except the high est(2nd antisymmetric), for which they were about 8.60%. The noncon-verged loads, however, were 0found to be quite close to the CAL-computed loads. At the 255 azimuth of the 21.2% radial station,the airload prediction agreed better with the CAL-computed valuethan did the prediction of Section 4.1 (see Figure 25).
4.3 UH-1- at& = 0.08 (Condition 65 of Reference 15), Injectionat Steady Deflected Positions of Blades
X =55.1 feet per second (forward speed)
=2.5 degrees (disc angle of attack)
NW = 6 (number of revolutions of wake)
6'ý = 0.7 (wake advance)
Realizing that the diffarences in the com~puted airloads for the,,"= 0.26 flight condition from the tw,.o wake-geometry' inputs wereonly slight, it was decided to carry out a similar comparison fora slower Q,0U 0.08) flight condition. For this ca.se, the inducedvelocity-to-forward speed ratio should be higher than for thefaster flight case (with comparab~e disc loading). Airload pre-dictions for this slower flight condition, therefore, would beexpected to show more marked differences for one wake geometry ascompared to the other.
During the wake-geometry computation, the coownwash-to-forwardspeed ratio was found to be as high as 1.11 in certain regionstoward the rear of the disc. T1his makes the ent-re calculationof the induced velocity field (and the resultant streamlines andwake geometry) suspect, since the theory was baseci on a lineariza-tion of the momentum equations. By way of comparison, the highestvalue of the downwash-to-forward speed ratio was about 0.12 for the,A= 0.26 flight condition. lhus-, the degree of confidence in the
computed wake geometry for theA,&= 0.08 fflight condition must belower than for.,,,= 0.26.
As expected, the upwash area of the disc was about the same asbefore, viz., a thin crescent near the leading edge and a smallarea behind the disc center. Many of the streamlines emanatingfrom these regions started in the upward direction. (This did nothappen in the,,a =0.26 case because the induced upwash, being trulya small perturbation, to the fluid velocity, was more than compen-sated for by V-'j -5/9'hO' , which is the niormnal (downward) componentof the forward speed. For the/ 0.26 case, not only was ~hi~gher,
82
but O was also somewhat larger, 5.80, than the 2.50 for the..,&0.08 flight.)
Figures 29 and 30 show typical wake trails for the A = 0.08 flightcondition. The deviations from a helical wake are more prominentin this case because of the higher ratios of all the induced veloc-ity components to the forward speed. Figure 31 shows the stream-lines starting from 90% radius. The wing-tip phenomenon is againevident. (The upward-starting streamlines mentioned above are notnoticeable in this picture because the leading edge crescent withlarge induced upwash was located outboard of about 92% radius,and the upwash region behind the center of the disc extended onlyup to about 50% radius.) This rolling-up-of-the-tip-vortex effectis more clearly seen in Figures 32 through 35, which show theprogressive distortion of an originally circular fluid contourreleased from 90% radius, one through four rotor revolutions afterrelease, respectively.
Figures 36 and 37 show plots of the unsteady parts of the airloadsand bending moments, respectively, for A = 0.08. It can be seenthat the airload predictions from the two different wake geometriesdiffer sigrificantly. The airload predictions from the UR wakeseem to be in better agreement with experiment in the region around900 azimuth. While the rest of the azimuthal positions show theairload histories to be qualitatively similar, the increase inhigher harmonic content with the UR wake seems clear. This isemphasized in the blade response in Figure 37.
The wake-induced velocity at the 75% radius, when the blade is at) = 750, is found to be 14.17 fps, and the total normal velocity
at that point is +55.81 fps. Thus, for this slower flight condi-tion, the wake-induced velocity is, as expected, a more significantfraction of the normal velocity experienced at the blade section.
83
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92SUE
CHAPTER 5
CONCLUSIONS AND POSSIBILITIES FOR FURTHER WORK
The wake-geometry computations reported in Chapter 4 show that theflow field behind a lifting actuator disc features tip vorticessimilar to those from a lifting wing.
The computed airloads for the& = 0.26 flight condition of theUH-l teetering rotor seem to be less sensitive to wake geometrythan are those for the,/t = 0.08 flight condition. This is indicat-ed by the slight differences between the airload predictions result-ing from two different wake geometries (CAL rigid helical wake andUR distorted wake) for the, = 0.26 condition and by the signifi-cant differences between similar predictions for theA = 0.08condition. The accuracy of the UR-computed wake geometry forthe slower (,,& = 0.08) flight condition is questionable becauseof applying linearized equations to an inapplicable amplituderegime. Still, the airload predictions of the blade-loads program,even if this wake geometry input is considered "arbitrary", areuseful in providing some insight as to the differences which willresult from wake-geometry differences.
This stronger sensitivity to wake geometry in the case of aslower flight condition calls for some quantitative discussion. Assuggested near the end of Chapter 4, the wake-induced velocity inthe case of theA= 0.08 condition forms a more significant fractionof the normal velocity experienced by the blades. This suggestsan examination of the variation of contributions due to wake effectsand those due to all other effects.
The faster flight condition certainly has larger contributions fromthe geometric pitch input and of the blade flapping to the totalnormal velocity at the blades. Thus, for example, at the 85% radialstation, the collective pitch was 0.2011 rad at = 0.26, ascompared to 0.1209 rad at/ = 0.08. The amplitude of the cyclicpitch input was 0.1195 rad at,. = 0.26, as compared to 0.04 radat,, = 0.09. Similarly, the amplitude of the first azimuthalharmonic component of the displacement in the rigid flapping of theteetering blade (normalized on tip deflection) was 0.6096 atA = 0.26, as compared to 0.1830 at/ = 0.08. At the rotor speedof 33 rad/sec, this flapping component results in a maximum flappingvelocity of 20.1 fps at/Z = 0.26, as compared to 6.04 fps at ,a0.08.
93
The corresponding quasi-steady bound vorticity strength IK compari-sons (see Figure 14 and equation (109) ) for the 85% radial stationare (sq ft/Sec):
steady component 331.3 (.A = 0.26) as compared to 249.9 (,M = 0.08)first harmonic 158.3 (,4 = 0.26) as compared to 75.34 (A = 0.08)second harmonic 33.2 (A = 0.26) as compared to 8.9 (& = 0.08)third harmonic 28.91 = 0.026) as compared to 13.75 (U = 0.08)
These values show that in the case of the faster flight condition,the variations in the quasi-steady bound vorticity are larger. Thissame point is illustrated in another way in T..ble IV, which com-pares the contributions of the two terms on the right-hand side ofequation (109). The ratio of the influences of the quasi-steadyand wake-induced quantities on the bound vorticity is quite highfor the first two harmonic variations ata = 0.26, but not so inthe case ofd = 0.08. For the 3rd, 4th,and 5th harmonics, theratio is q-nller at either value of,& , because there are no cyclicvariations in 01.0 or Vf above the 1st harmonic, and the bladeflapping displacements are smaller in harmonic components above the2nd.
TABLE IV. COMPARISON OF CONTRIBUTIONS OF THE QUASI-STEADYAND WAKE-INDUCED QUANTITTES TO THE BOUNDVORTICUTY STRENGTH COMPONENTS AT THE 85% RADIALSTATION, UH-l,/A = 0.26 AND/ = 0.08
The Ratio 1 4*- ;. 1 (See Equation (109))Pg c--
Harmonic Component = 0.26 A = 0.08
Steady 1.17: -0.17 1.16: -0.161st harmonic 1.34: -0.34 2.7: -1.72nd harmonic 1.25: -0.25 0.497: 0.5033rd harmonic 1.76: -0.76 2.94: -1.944th harmonic 0.684: 0.316 0.157: 0.8435th harmonic 0.204: 0.796 3.59: -2.59
As regards the amount of wake vorticity remaining close to the blades,Figures 21, 29, and 30 show how much more rapidly the distancebetween a given blade and the wake increases for the higher speedcase. The strength of the wake vorticity, of course, also entersthese considerations, but this is a more complex and less influ-ential variable as regards the effects of forward speed.
ýJ4
It can therefore be concluded that the relative insensitivity ofthe blade loads to wake geometry in the case of the faster flightcondition (,X = 0.26) is due mainly to (1) the wake being blownaway relatively quickly and (2) a predominance of the cyclic pitchinput, tangential velocity variation, and blade flapping displace-ments in the total time-varying aerodynamic environment at theblade sections.
It may be note- that all the computations reported here pertainedto the UH-1 two-bladed teetering rotor. It would probably beworthwhile to carry out similar comparative airload computationsfor a fully articulated rotor such as the four-bladed Sikorsky H-34.This might facilitate comparisons of the importance of bladedisplacements, cyclic pitch and tangential velocity variationsand wake-induced effects. Since the H-34 has four blades, therewill be more wake vorticity near the disc, and hence wake geometrycould be more influential than it seemed in theA = 0.25 casefor the UH-l. If this does seem worthwhile, the wake computationcan be repeated using the converged blade loads to establish thetime-average pressure field which is the basic input to the URwake geometry determination.
If the UR wake-computing program is to be extensively used withthe CAL BLP2, considerable saving in running time could beachieved by writing the UR wake geometry onto a temporary direct-access data set at the beginning of subroutine SUB14 (see Figure 19)and reading it from this data set instead of repeatedly readingit sequentially from TAPE8.
As alluded to earlier, a fundamental difficulty exists in tl.e workreported here. Linearizing the equations of motion on the basis ofdisturbance velocities being small compared to forward speed meansthat the wake computations will be more reliable, for a given tipspeed, at the higher advance ratios. On the other hand, since weare using an assumed s pressure field for calculating the pathsof particles which are rea ly unsteady, sufficient time must elapseto allow an averaging of the unsteady effects to take place. Thus,the streamline positions--and the wake geometry obtained therefrom--will be most realistic (in the case of a finite number of blades)only when each vorticity element leaving the blades is subjected tothe effect of a large number of blade passages. For a given numberof blades and a given tip speed, this calls for a small advanceratio,,&. Put another way, the method explored here will be mostaccurate for a lightly loaded rotor (i.e., low thrust per discarea) with many blades at high forward speed and high tip speed.Like all other rotor vortex aerodynamic theories, the results willreflect changes in vortex strength and position only when the numberof blades is sufficiently high to leave vortices close to succeeding
95
blades and yet low enough to keep the strength of the vorticesfrom becoming too low. The scope of this research has notallowed definition of these limits, if in fact they exist. Ithas suggested, however, that wake details are important for tipspeed ratios of 0.1 and below, and are unimportant at around
= 0.25 and above.
96
LITERATURE CITED
1. Prandtl, L., and Tietjens, 0. G., APPLIED HYDRO-AND-AERO-MECHANICS, Dover Publications, 1957, p. 210.
2. Piziali, R. A., A METHOD FOR PREDICTING THE AERODYNAMICLOADS AND DYNAMIC RESPONSE OF ROTOR BLADES, Cornell Aeronau-tical Laboratory, Inc.; USAAVLABS Technical Report 65-74,U.S. Army Aviation Materiel Laboratories, Fort Eustis, Vir-ginia, January 1966, AD 628583.
3. Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. AERO-ELASTICITY, Addison-Wesley Publishing Co., Inc., Reading,Massachusetts, 1957, pp. 213-217.
4. Karman, Theodore von, AERODYNAMICS, McGraw-Hill Book Company,Inc., New York, 1957, p. 45.
5. Panofsky, W. K. H., and Phillips, Melba, CLASSICAL ELECTRICITYAND MAGNETISM, Addison-Wesley Publishing Company, Inc., Read-ing, Massachusetts, 1962, p. 125.
6. Landau, L. D,, and Lifshitz, E. M., FLUID MECHANICS, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1959,p. 8.
7. Lamb, Horace, dYDRODYNAMICS, Dover Publications, New York,1945, pp. 203-204.
8. Payne, P. R., HELICOPTER DYNAMICS AND AERODYNAMICS, Pitmanand Sons Ltd., London 1959, Chapter 2.
9. Kinner, W. DIE KREISFORMIGE TRAGFLACHE AUF POTENTIALTHEORE-TISCHER GRUNDLAGE, Ingenieur-Archiv, VIII Band, 1937.
10. Mangler, K. W., CALCULATION OF THE INDUCED VELOCITY FIELD OF AROTOR, Royal Aircraft Establishment Report, No. Aero. 2247 1948.
11. Glauert, H., AIRPLANE PROPELLERS, Aerodynamic Theory, edited byW. F. Durand, Dover Publications, New York, 1935, Chapter XDivision L, Vol. IV.
12. Wood, E. R., and Hermes, M. E., ROTOR INDUCED VELOCITIES INFORWARD FLIGHT BY MOMENTUM THEORY, presented at AIAA/AHS VTOLResearch, Design, and Operations Meeting, Atlanta, Georgia,February 1969.
97
13. Tararine, S., EXPERIMENTAL AND THEORETICAL STUDY OF LOCAL INDUCEDVELOCITIES OVER A ROTOR DISC FOR ANALYTICAL EVALUATION OF THEPRIMARY LOADS ACTING ON HELICOPTER ROTOR BLADES, European Re-search Office, U. S. Army Contract, October 1960.
14. Scheimann, James, A TAP11TAIT.ON OF HELICOPTER ROTOR-BLADE DIFFER-ENTIAL PRESSURES, STRESSE6, AND MOTIONS AS MEASURED IN FLIGHT,NASA TMX-952, March 1964.
15. Burpo, F. B., et al, MEASUREMENT OF DYNAMIC AIRLOADS ON A FULL-SCALE SEMIRIGID ROTOR, TCREC Technical Report 62-42, December1962.
16. Lenman, August F., MODEL STUDIES OF HELICOPTER ROTOR FLa•' PATTERNS,USAAVLABS Technical Report 68-17, April 1968.
17. Gray, Robin B., EXPERIMENTAL SMOKE AND ELECTROMAGNETIC ANALOGSTUDY OF INDUCED FLOW FIELD ABOUT A MODEL ROTOR IN STEADY PLIGHTWITHIN GROUND EFFECT, NASA TND-458, August 1960.
18. Crirni, Peter, PREDICTION OF ROTOR WAKE FLOWS, presented at CAL/!USAAVLABS Symposium, Buffalo, New York, June 1966.
19. Chang, T. T., A METHOD FOR PREDICTING THE TRIM CONSTANTS AND THEROTOR-BLADE LOADINGS AND RESPONSES OF A SINGLE-ROTOR HELICOPTER,Cornell Aeronautical Laboratory, Inc.; USAAVLABS Technical Report67-71, U.S. Army Aviation Materiel Laboratories, Fort Eustis,Virginia, November 1967, AD 666802.
20. Rabbott, John P., and Paglino, Vincent M., AERODYNAMIC LOADINGOF HIGH SPEED ROTORS, presented at CAL/USAAVLABS Symposium,Buffalo, New York, June 1966.
21. Loewy, Robert G., Styers, D. J., and Gabel, R., A REVIEW OFTHEORETICAL AND EXPERIMENTAL INVESTIGATIONS OF THE ROTOR AERO-DYNAMIC PROBLEM, Vertol Division, Boeing Airplane CompanyReport No. R-217, October 1960.
22. Sokolnikoff, I. S., TENSOR ANALYSIS, Second Edition, John Wiley& Sons, Inc., New York, 1964, p. 268.
23. Whittaker, E. T., and Watson, G. N., A COURSE OF MODERN ANALYSIS,Fourth Edition, Cambridge University Press, 1965, p. 324.
24. Robin, Louis, FONCTIONS SPHERIQUES DE LEGENDRE ET FONCTIONSSPHEROIDALES, Gauthier-Villars, Paris, 1957.
98
APPENDIX I
ASSOCIATED FUNCTIONS OF LEGENDRE AND SOME OF THEIR PROPERTIES*
I-1 Legendre Functions of the First Kind, R (z)
P '(j) - (-3p
zz
Particular values:
( -IY• __-,_ ... n even
) ... n odd
Asymptotic values for lzl- " Do:
Reference 24.**(2n..l)!A 1*3*5o....(2n-1), with i1. (Schuster's notation).
99
1-2 Legendre Functions of the Second Kind, 4,,(Z)
For t outside the segment (-l, +-):
7 e(Z)- -3Z 1 I
~ j-7
For -1 < 4 +1, replace/ 7 -A in the above formulas by.1.
Particular values for 6-eQ (6>0):
Asymptotic values for/Z/-. 00
(Note: from the above formulas, Q,,(,~take the forms:
of)= -t'"' -7
.00
and the values on the disc ( 0-O) are:
i•, (I'V) = -,?
1-3 Associated Functions of Legendre of the First Kind,Por(Z)
For -:•4 :
'0"• d"'Pdz) (Hobson's notation)
"di '
N'z - 1 iT7
pz, 'Cz)= -(1r-Z)
i (Z) - -5Z(7-,-
ForZ outside the segment (-1* '1), replace (-1) (1-1 Wk
in the above formulas by (a-1)"VI
Par'ticular values:
4010
S.. -,odd
101
For Lo)
A symptot ic va lue s f or /c/ooa
1-4 Associated Functions of Legendre of the Second Kind,
For Z outside the segment (-1, +1):
A"I A Zý '
'A
-~ ~ L VIZ- 7 XYY__
7 Z1,-
-__ -,
102
ZL
For -1 < Z ( +1, replace the rnultip~ic tive factor (Z2-1)o/2
in the above formulas by (-i)S (1-Z )m/2; and replace log X24by log U'A. 7-7I-zParticular values fore-PO, (6 O,9P)O :
Asymptotic values for/Z/-- do:
Note: From the above formulas, (4"(i,) take the forms:
i(, Jeo vrr,-.7 0 tow,-'Q,; (*7) - v',? ,• top. '-' '- -f"--- ,,•,
j~ ~ ~~V oe• V - 17p SAS) -,, o • ,,-7R &o, -? • -5
(.e.
Q ,;C~~ 7 - • (,÷ 'Vta, " -, -' /57'i"7O- -•--10
i~03
and the values on the disc (7 0) are:
(iiv
Z (eo) = -1 z
P,7(• - ) ( 1)
J 0" 7~~j~~ d7 10'rIof
Normalization
I4 0
ae
104
iI
Differentiation
:=. _ F - p ."'
dz - *
For Z outside the segment (-1, +1),
dz- ,
* da,,eL. 7 (,')'10
105
APPENDIX II
FORTRAN IV SOURCE LISTINGS
Page
Subroutine SUB14 (as modified at UR) .. .. ....................107
Subroutine RWAKE. .. ........................................ 118
Subroutine COEFS. .. ........................................ 127
Subroutine SIMLN. .. .. ......................................137
Subroutine VLCTS. .. .. ........................... ...........145
Subroutine CHA14P. .. .. ......................................iso
Subroutine FRWRD. .. .. .................................. . .155
Subroutine ADVNC. .. .. ......................................159
Subroutine CRSN.G. .. ........................................162
Subroutine VARFMT .. .. ......................................166
Mainprog RWAKEPIC .. .. ......................................167
Mainprog RSLTSPLT .............................................. 180
106
iii
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APPENDIX III
INDUCED VELOCITY AT SELECTED POSITIONS ON THE DISC DUE TO
SELECTED SEGMENTS OF THE NEAREST TIP TRAIL
Table V shows the induced velocity due to nearby filaments of the
trailed vorticity. Five instantaneous positions were chosen for
points on the blades (see Figure 38), and the induced velocities
resulting at these points due to selected short segments of the
tip vortices (as positioned by the two geometries, CAL and UR)
were computed for the /4: 0.26 flight condition of UH-1.
194
Blade at•i- 750
101
3-
-
9--6'1e6
/ / /
/ y / / N
/
/\
8 1 y
//
J O l
II I
7
4
Blade atS&= ZL5 °
Coordinates (x', y, z') of blade stations 1, 2, 3:
1. (+1.514, -4.434, -0.-44)2. (-3.649, +12.40, -0.679)
3. (-4.574, +15.86, -0.854)
See Table V for coordinates of wake segments as representedin the two wake geometries.
Figure 38. Positions of Selected Blade Stations and SelectedWake Trail Segments (UH-1, AL = 0.26).
195
'4 0 LC 'o Ln m
4 N N 1- '0) N 00 a'
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NJ--^aflCV
UNCLASSIFIEDSecurity Classification
DOCUMENT CONTROL DATA • R & D(S ec u rity c la s si t h a tio n o f ti tle , bo d y o f a b . f 'a , I a nd In d .e In g a -n o l~ tio -m u - (I b e *e te re d . oo ihe. o T ra. ! rIIIo. I- .asa l "ie dj
I ONI GINA TING AC TIVI TV (Co/• alp e author) 2,. REPORT SICURITVY CLASSIFICATION
University of Rochester Unclas sified
Rochester, New York OUP
3 REPORT TITLE
AN ACTUATOR-DISC ANALYSIS OF HELICOPTER WAKE GEOMETRY
AND THE CORRESPONDING BLADE RESPONSE
4. DESCRIPTIVE NOTES (T•ype of report and inclusive date&)
Final ReportS. AUTHORIS) (Flrmi name, middle initial, l.it name)
M. JoglekarR. Loewy
6, NFOORT DATE 7.. TOTAL NO. OF PAGES b7. NO. OF REFS
December 1970 215 24SS* CONTRACT OR GRANT NO. 9. ORIGINATOR'S REPORT NUM'SERISI
DAAJ0Z-68-C-0047 J ,I6. PROJECT NO. USAAVLABS Technical Report 69-66
C. Task IF16LZ04A13904 9) OTHER REPORT NOIS) (AnY Dtntnat. Slat 6D hald
d.
10. DISTRIBUTION STATE!MENT
This document is subject to special export controls, and each transmittal to foreign
governments or foreign nationals may be made only with prior approval of Eustis
Directorate, U.S. Army Air Mobility R&D Laboratory, Fort Eustis, Virginia 23604.Il. SUPPLEMENTARY NOTES |12. SPONSORING MILITARY ACTIVITY
Eustis Directorate
U. S. Army Air Mobility R&D Laboratory
_ Fort Eustis, Virginia13. ABlSITRNAC€T
In this analytical research, it is assumed that a helicopter rotor in forward flight can
be represented by a flat plate. Expressions are developed relating the pressure field
to the steady aerodynamic thrust and moment and the time-dependent flapping moment
This pressure field satisfies the zero pressure condition at the center and edge of thedisc, the first and second harmonic variation in lift, and Laplace's equation.
A computer program was written to calculate the velocity field and streamlines. Twosample cases corresponding to high- and low-speed conditions for the UH-I rotorwere chosen for computation. Both cases showed the tip-vortex phenomenon.
In order to examine the sensitivity of blade load calculations to wake geometry, anexisting airloads program was modified to accept the computed wake geometry. A
comparison of results of this modified program with experimental data indicated thatthe effect of distorted wake is greater at low advance ratios than at high advanceratios. The reduced sensitivity at high advance ratios may be due to the wake beingblown farther behind the rotor and/or the predominance of other aeroelastic param-eters.
DDD 1'.*1.,147 73_ - 1.. ..-. ,C._,,DD.ElS7 FORI ANNY wSE. UNCLASSIFIED
Ueu rtry ClasessicaUce
UNC LASSIFIEDSecurity Classification
4. LINK A LINK 6 LINK CKFY WOROS
ROLE WT ROLE WT ROLE WY
Helicopter Rotor WakeWake GeometryDistorted WakeHelicopter Blade LoadsHelicopter Aerodynamics
UN( I..A'.SI'I IED
Security Classification
