ARL-FL IGHT-MECH-T'Y'-408 AR-005-586

(0)

00N DEPARTMENT OF DEFENCEI DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORY

MELBOURNE, VICTORIA

Flight Mechai{cs Techrico( Memorndum 4139

INCORPORATION OF VORTEX LINE AND VORTEX RING HOVERWAKE MODELS INTO A COMPREHENSIVE ROTORCRAFT

ANALYSIS CODE (U)

DTIC byS- .ECTE R P . Tottolueto, N.E. Gilbert,MAY 2 4 19 59 5. Hill, K.R. Reddy

.D

Approved for Public Rekase

(C) COMMONWEALTH OF AUSTRALIA 1989JANUARY 1989

80 -,-

/TCNICAL 1'T NATOAI s AVTm<)IS INFORM4ATIONA I

REF IUC )SE LL r~ N SL OvAT

This work is copyright. Apart from any fair dealing for the purpose of study,research, criticism or review, as permitted under the Copyright Act, no partmay be reproduced by any process without written permission. Copyright is theresponsibility of the Director Publishing Rnd Marketing, AGPS. Lquiries shouldbe directed to the Manager, AGPS Press, Australian Government PublishingService, GPO Box 84, Canberra, ACT 2601.

I. ]

hI

AR-005-586

DEPARTMENT OF DEFENCEDEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORY

Flight Mechanics Technical Memorandum 408

INCORPORATION OF VORTEX LINE AND VORTEX RING HOVER

WAKE MODELS INTO A COMPREHENSIVE ROTORCRAFT

ANALYSIS CODE

by

R. TOFFOLETrO, N. E. GILBERT, S. HILL, and K. R. REDDY

SUMMARYThe incorporation of simplified hover wake models into the comprehensive rotorcraft

analysis code CAMRAD is described and examples are given on their use. Theaxisymmetric models, in which vortices are represented by either straight lines or rings, area more generalized form of the free wake models of R. T. Miller at MIT, with the wakegeometry also able to be prescribed. Incorporation has allowed access to the tabularrepresentation in CAMRAD of airfoil section characteristics as functions of angle of attackand Mach number, and has broadened the range of rotor wake models in the code to includea free wake hover model that does not have the convergence problems of the existing freewake model when used for hover.

DSTO4MELBOURNE

© COMMONWEALTH OF AUSTRALIA 1989

POSTAL ADDRESS: Director, Aeronautical Research Laboratory,P.O. Box 4331, Melbourne, Victoria 3001, Australia

CONTENTS

NOTATION

1. INTRODUCTION 1

2. WAKE MODELS 2

3. PROGRAM MODIFICATIONS 4

4. TEST CASES 5

5. CONCLUDING REMARKS 7

REFERENCES 8

APPENDICES

Appendix A - Computational Procedure 9

Appendix B - Velocity Components Induced by Wake VortexElements 16

Appendix C Block Diagram for Free Wake Method 17

Appendix D - Block Diagram for Prescribed Wake Method 18

Appendix E - Modified CAMRAD Subprograms 19

Appendix F - Added CAMRAD Subprograms 25

Appendix G - Input Description 39

Appendix H - Test Case Command File 40

Appendix J - Test Case Output File 41

DISTRIBUTION

DOCUMENT CONTROL DATA

B~y

__ __ L .

NOTATION*

c blade chord **

Ct blade lift coefficient

Cr/o ratio of thrust coefficient to rotor soliditydb, core burst test parameterf factor introducing lag in solution

fr empirical scale factor for concentrated vortices inboard of tip vortexK1 , K2 axial settling rates of tip vortex before and after passage of following bladeK3, K4 radial contraction parameters for tip vortexM number of aerodynamic segments

M blade tip Mach numberN number of bladesr, z radial and axial displacement coordinates (origin at rotor hub centre; z positive

down)rA[i] r at mid-points of aerodynamic segments, for i = 1.... MrA[i] r at edges of aerodynamic segments (from root to tip), for i = 1 .... M+lrb burst vortex core radiusT. vortex core radiusr., vortex core radius limited to a minimum of 0.005

r. unburst vortex core radiusS number of concentrated vortices along the bladeT number of vortex line or ring levels in intermediate wakeu, w net radial and axial induced velocity components tuF, WF radial and axial induced velocity components due to far wake ±

u1, w1 radial and axial induced velocity components due to intermediate wake :Wb downwash at blade due to trailing near wakew,.tf self-induced downwash at bladea blade angle of attack **r blade bound circulation

A incremental change in appropriate quantityE tolerance for induced velocity convergence

0 blade pitch angle **X, X., 7. longitudinal, lateral, and vertical induced velocity components (funct's of r, W)

* blade inflow angle **blade azimuth angle

All quantities are dimensionless (based on density, rotor rotational speed, and rotor radius). Quantitiesused only locally to simplify expressions are not included hereFunction of r

* Function of r, z

Subscriptsm,n as for subscipt (s,t) but at a source of induced velocity, i.e. when calculating

induced velocity at (r,.,, z,), contributions are summed over (mn)max maximum valuenew value at current iterationold value at previous iterationst value at concentrated vortex number s (from tip), where 1 _< s < S, and at vortex

line or ring level number t at which induced velocity is to be calculated (i.e.object). Note: I - s 5 S and 1 < t _< T in intermediate wake; t = 0 at blade, inwhich case subscript is dropped (e.g. r,,0 - r,); t = T + 1 for far wake

1. INTRODUCTION

In response to requests from the Australian Services to evaluate performance character-istics, especially for hover, of helicopters presently operated, as well as those beingconsidered for procurement, Aeronautical Research Laboratory (ARL) has developed ananalysis capability in the area of hovering rotor aerodynamics which includes both inhouseand acquired codes.

In 1987, Reddy and Gilbert compared predicted hover performance with flight data forfour helicopters.' Comparisons were also made of main rotor blade loading distribution forone of the helicopters, a Sikorsky S-58 (equivalent to Westland Wessex). Predictions wereobtained using three nonuniform inflow rotor wake models and a uniform inflow modelbased on momentum theory. The nonuniform wake models used were a

" helical vortex lattice prescribed wake model,

" vortex line prescribed wake model, and

" vortex ring free wake model.

The first of these models is incorporated in CAMRAD (Comprehensive AnalyticalModel of Rotorcraft Aerodynamics and Dynamics), a code developea by Johnson at AmesResearch Center. 2,3 ,4 ,5 The code, which was acquired in 1984 as part of a cooperativeprogram with the US Army, uses straight-line vortex elements joined in the form of ahelical vortex lattice to represent the trailed and shed vorticity. CAMRAD was also used toprovide uniform inflow predictions (based on momentum theory), a preliminary process indetermining the trimmed prescribed wake solution.

The second model, which uses infinite and semi-infinite straight-line vortices, wasdeveloped by Reddy 6,7 at ARL independently of very similar work by Miller 8.9 atMassachusetts Institute of Technology (MIT) and Beddoes (unpublished) at WestlandHelicopters Limited (WHL). An acquired free wake hover code that was implemented atARL by Hill and Reddy(unpublished) was used as the third model. The method, whichgenerally represents infinite line vortices by rings, is based on one of the variations inMiller's method.

To investigate the consistency of these methods, the same parameters and empiricalcorrections were used within each model in applying to each helicopter, and wherepossible, consistency of appropriate quantities between models was also maintained. Themajor identified inconsistency between the wake models was the different representation ofairfoil characteristics. In CAMRAD, the two-dimensional airfoil section characteristics arerepresented in tabular form as functions of angle of attack and Mach number, compressibil-ity effects arc therefore effectively incorporated. In the more simplified vortex line andvortex ring models, the characteristics are represented by a constant lift curve slope and aquadratic drag polar without corrections for compressibility. It was planned therefore toincorporate the vortex line and vortex ring models into CAMRAD, principally to allow the

two-dimensional airfoil data to be available to these simpler models.

Since Miller's models are formulated in a way that allows a free wake geometry for

both the vortex line and vortex ring models, it was decided to incorporate his models, but ina more generalized form, allowing the geometry to also be prescribed using the options inCAMRAD, as well as some additional features. The main purpose of this report is todocument the model formulation used and to provide the information necessary to run themodels.

I ' I[ I I II I ! it aNaI J, . .

2. WAKE MODELS

Comprehensive codes such as CAMRAD allow the analysis of a complete rotorcraft,usually allowing for two separate rotors. However, it is generally assumed sufficient toconsider only the main rotor in the case of hover for a conventional helicopter with a singlemain rotor and anti-torque tail rotor. For performance predictions, standard estimates arethen made for the power requirements of the tail rotor, accessories and transmission, andaerodynamic interference. By assuming an axisymmetric wake, the harmonics of blademotion and the shed wake can be neglected, and only collective control needs to be adjustedto trim to a specified thrust.

The helical vortex lattice model in CAMRAD follows the common approach of closelytracing the three-dimensional helical shape of both the strong tip vortex and inboard vortexsheet. Unfortunately, this apparently straight-forward approach results in a computationprocess that is complex and computationally demanding. This is especially so for the hovercase where there is no large uniform relative wind due to the translational velocity of thehelicopter. This means that the wake is not swept away from the rotor and hence more ofthe wake must be considered, It also means that wake induced velocities are the onlyvelocities present, resulting in a greater sensitivity of the wake geometry to changes ininduced velocity. This increased sensitivity leads to instabilities and slow wakeconvergence if free wake models are used.' 0

The simplified axisymmetric methods described here are an attempt to overcome theabove problems for the hover case. The basis of these methods iz that the continuouslydescending helix is represented by vertically separated horizontal vortex lines (for ahorizontal rotor disc), which are either straight or circular. The wake is divided into threeregions, which are defined as near, intermediate, and far.

Root vorices Tip vortices

Rotor blade~{Rotor blade

Near wake Near wake

a) plan view a) plan view Rotor bladeRotor blade , ,, ,oo ld

- InlermediateIntermediate wake

wake wk

74' I I Far wake

b) side view Far wake b) side view

Fig. 1 Vortex Line Wake Model Fig. 2 Vortex Ring Wake Modelwith Concentrated Far Wake with Distributed Far Wake

2

The three wake regions are illustrated in Figs 1 and 2 for the vortex line and vortex ringmodels respectively. These are first described in relton to a prescribed wake geometryusing either of the options in CAMRAD, i.e. (a) Landgrebet l or (b) Kocurek andTangler.12 Both are based on model rotor flow visualization data.

For both vortex line and vortex ring models, the near wake is represented by semi-infinite straight lines attached to, and in the plane of, the blade, with a greater concentrationtowards the tip. Based on the observations in Kocurek and Tangier's experiments of fourwell defined tip vortices below the blade, the intermediate wake is represented at each of thecorresponding four axial levels by either two straight infinite vortex lines (Fig. 1) or twovortex rings (Fig. 2). However, the generalized manner in which these methods areimplemented here allows the number of these levels to be varied (up to 36). One of these.concentrated' rolled-up vortices is located at the outside boundary of the prescribedcontracted wake and represents the strong tip vortex, and the other is located directlybeneath the root cutout and represents the inboard vorticity.

Below the intermediate wake region, it is believed that the wake is unstable; the tipvortices undergo viscous dissipation, resulting in wake expansion. To account for thisregion, which is still close enough to the rotor disc to induce significant inflow, Kocurekand Tangler proposed a vortex ring with radius equal to the rotor radius, axial location atthe same level as the fourth tip vortex beneath the rotor, and strength of four times that ofthe tip vortex. This concept is adopted as an option for the far wake (referred to as'concentrated far wake') in the form of either a ring for the vortex ring model, or an infinitestraight li,-.e replacing the ring for the vortex line model, as shown in fig. 1. The otheroption for the far wake included is the one given by Miller (referred to as 'distributed','sheet', or 'distributed sheet') using two seini-infinite vortex planes (for vortex line) orcylinders (for vortex ring - as in Fig. 2) with strength determined by the geometry of theintermediate wake, and positioned one wake spacing below each of the last inner and outerrings of the intermediate region. This latter option is the only one used for the free wakemethod. For the prescribed wake method, the far wake may be neglected.

When the wake geometry is allowed to be free in Miller's simplified models, thedifficulties of convergence typical of vortex lattice models are not generally experienced.The radius and axial spacing of each concentrated vortex in the intermediate wake, with itsconsequent effect on the distributed far wake, is determined by the requirements forequilibrium of the velocities, this being the essence of the free wake method. In Miller'smethod, up to three concentrated vortices are allowed in the intermediate wake though onlytwo are used in the prescribed intermediate wake here and in Ref. 1 by Reddy and Gilbertwhen using Miller's vortex ring free wake model. The generalized formulation here allowsthis number to be increased up to ten.

The computational procedure is outlined in Appendix A, with the prescribed wakemethod incorporated as part of the complete method, and expressions for the velocitycomponents induced by wake vortex elements are given in Appendix B (see Ref. 8 forderivations). Block diagrams showing the separate structures of the free and prescribedwake methods are given in Appendices C and D respectively.

3t

3. PROGRAM MODIFICATIONS

Modifications to the standard VAX 780 version of CAMRAD are given. Because thenew axisymmetric models are intended to be used for a single rotor configuration,modifications made to Rotor 1 subprograms are not similarly made to Rotor 2subprograms.

The following subprograms (each stored as a separate Fortran file, e.g.GEOMRI.FOR) in CAMRAD (see Ref. 5, Part 11) have been modified (see Appendix E):

GEOMRI Calculate wake geometry distortion

INPTW 1 Read wake namelist

PRNTWI Print wake input data

RAMF Calculate rotor/airframe periodic motion and forces

TRIM Trim

TIMER Program timer

Changes to the VAX VMS operating system since 1984 have resulted in the output ofnull component CPU times at the end of the CAMRAD output file. Modifications toTIMER, while not necessary for implementation of the models, are therefore included inAppendix E.

The following added subprograms (all included in the file WAKERI.FOR) form thebasis of the new models (see Appendix F):

WAKERI Determine induced velocity at rotor using vortex line or ringmodel

VTXIF Calculate induced velocity in intermediate wake due tointermediate and far wake

IVTERP Calculate induced velocity along blade at concentratedvortices

ILINE Evaluate expressions for velocity induced by vortex line inintermediate wake

IRING Evaluate elliptic integral expressions for velocity induced byvortex ring in intermediate wake

ELLIPCON Calculate constants used in elliptic integral expressions

FRING Evaluate elliptic integral expressions for velocity induced bysemi-infinite vortex cylinder in far wake

FLINE Evaluate elliptic integral expressions for velocity induced bysemi-infinite vortex sheet in far wake

PRESWG Determine prescribed wake geometry

The above modified files and added file are first compiled. After then obtaining themain program object file CAMRAD.OBJ and library object file CAMRAD.OLB containingall original compiled subprograms, the new executable file CAMRAD.EXE is given ontyping

$LINK CAMRAD, GEOMR1, INPTW1, PRNTW1 RAMF, TRIM, TIMER, WAKER1, CAMRAD/LIB

4

Nine new input variables, all in namelist NLWAKE, have been added (see Appendix Gfor description and default values). Also included in Appendix G are some comments onexisting CAMRAD variables.

4. TEST CASES

To demonstrate the various model options and illustrate the effect of includingcompressibility, comparisons are made of blade loading distribution for the S-58 usingScheiman's test data' 3 as in Ref. I. Main rotor performance and blade loads are obtainedfrom CAMRAD using the basic H-34 (i.e. S-58) data deck and NACA 0012 airfoil tables.

Fig. 3 shows the effect of compressibility using the vortex ring, free wake model.Since the tip Mach number is relatively low (= 0.56), the effect is only minimal in thisexample. For the 'compressible' case in Fig. 3, the command file and resulting output file(the latter in abbreviated form) are given in Appendices H and J respectively . The vortexring, free wake model is selected by setting OPMODL = 2 and LEVEL(l) = 2. Tworolled-up vortices (NIBV = 2) and four vortex levels in the intermediate wake (NIVL = 4)are specified. The free wake model only allows the distributed sheet far wake mode!. Theempirical factor scaliog the rolled-up concentrated vortices inboard of the one at the tip is setto the default value of 0.6. By setting inputs DEBUG(14) and DEBUG(24) to 1, additionalinformation on the induced velocity and free wake geometry is printed. Because thesolution is independent of azimuth, the number of azimuth steps per revolution (MPSI undalso MPSIR) is set to the minimum value of 4, the number of b1ades.

400"

* Co'prmbI.

300 ._,rcomp++[,es

Blade load - ScIan Tes Data

100

0 F02 04 06 08 10

Blade radial station

Fig. 3 Effect of Compressibility on Blade Load Distribution for S-58 usingVortex Ring, Free Wake Model (CT/" = 0.0817)

Operating conditions and main rotor data, which are common to all the test casespresented, are included in the output file listing (Appendix J). Values of the new inputvariables for the new models are included in the 'Main Rotor' subsection of the 'Input Data'section of this file.

5

400

0~~-- Free Wake

---a PreScnbed Wake (wtr coecerwrated far wake)

300 *a Prescnbed Wake (wah disrbuted far wz~e)

P~t'de load I. Scarman Tst Data(lb/It) I (wth 1256&67.5 percentles)

200-

100,

0 102 04 06 08 1 0

Blade radial slation

Fig. 4 Effect of Wake Model Variations on Blade Load Distribution forS-58 using Vortex Line Model (CT/a = 0.0817)

Figs 4 and 5 show the effect of the same wake model variations applied to the vortexline and vortex ring models respectively, each with cormpressibility included. In Table I,the maximum blade !oading (at a radial station of 0.95) is tabulated for these variations, butboth with and without compressibility includedi. Each model gives reasonably similardistributions, but the maximum loading given by the vortex line, prescribed wake modelwith a concentrated far wake is about 10% less than that given by the other models.

400

-V- Free Wake

30.9- Prescnbe4 Wake )wltt corteruraled I ar wake)

300 .a Prownrbed Wake (artS drsinbuted far wake)

Blade load Sctreima Test Data(lb/ft) j (arth 1256617.5 percentfles)

200

100

002 04 06 08 1 0

Blade radial station

Fig. 5 Effect of Wake Model Variations on Blade Load Distribution forS-58 using Vortex Ring Model (CT/a= 0.0317)

6

TABLE IEffect of Wake Model Options on Maximum Blade Load for S.58

Compressible IncompressibleWake Model Vortex Line Vortex Ring Vortex Line Vortex Ring

Free 388 375 365 353Prescribed (concentrated far wake) 349 380 316 338Prescribed (distributed far wake) 388 385 346 346

5. CONCLUDING REMARKS

By incorporating the simplified hover wake models described here into CAMRAD, themodels themselves have been enhanced by allowing access to compressibility effectsincluded in the two-dimensional airfoil tables used by CAMRAD. In addition, the range ofrotor wake models in CAMRAD has been broadened and now includes a free wake hovermodel (either vortex line or vortex ring) that does not have the convergence problems of theexisting free wake model in CAMRAD when used for hover.

In deciding which of the simplified models to use, consideration should be given tomaintaining a balance between the degrees of approximation used within the wake modelitself. Specifically, when representing vortices by just straight lines in all wake regions, theadded complexity of a free wake solution may not be warranted. The more physicallyaccurate representation by rings would therefore seem to be more consistent with a freewake.

REFERENCES

1. Reddy, K. R. and Gilbert, N. E., "Comparison with Flight Data of HoverPerformance Using Various Rotor Wake Models," Proceedings of the ThirteenthEuropean Rotorcraft Forum, Arles, France, September 1987, Paper No. 2-12.

2. Johnson, W. J., "Development of a Comprehensive Analysis for Rotorcraft - I. RotorModel and Wake Analysis," Vertica, Vol. 5, 1981, pp. 99-129.

3. Johnson, W. J., "Development of a Comprehensive Analysis for Rotorcraft - 11.Aircraft Model Solution Procedure and Applications," Vertica, Vol. 5, 1981, pp. 185-216.

4. Johnson, W. J., "Assessment of Aerodynamic and Dynamic Models in a Compre-hensive Analysis for Rotorcraft," Computers & Mathematics with Applications, Vol.12A, No. 1, 1986, pp. 11-28.

5. Johnson, W. J., "A Comprehensive Analytical Modl of Rotorcraft Aerodynamicsand Dynamics," Parts 1, 11, and I, NASA TM-81182, 3, and 4, June/July 1980.

6. Reddy, K. R., "The Vortex Flow Field Generated by a Hovering Helicopter,"Proceedings of the 7th Australasian Conference on Hydraulics and Fluid Mechanics,Brisbane, Australia, Aug. 1980, pp. 553-556.

7. Reddy, K. R., "The Effect of Rotor Wake Geometry Variation on Hover InducedPower Estimation for a UH-1H Iroquois Helicopter," ARL-AERO-TM-384, October1986.

8. Miller, R. H., "A Simplified Approach to the Free Wake Analysis of a HoveringRotor," Proceedings of the Seventh European Rotorcraft and Powered Lift AircraftForum, Garmisch-Partenkirchen, W. Germany, September 1981, Paper No. 2; also,Vertica, Vol. 6, 1982, pp. 89-95.

9. Miller, R. H., "Simplified Free Wake Analysis for Rotors," FFA TN 1982-07,Aeronautical Research Institute of Sweden, 1982; also, MIT ASRL TR 194-3.

10. Bliss, D. B., Wachspress, D. A., and Quackenbush, T. R., "A New Approach tothe Free Wake Problem for Hovering Rotors," Proceedings of the 4 1st Annual Forumof the American Helicopter Society, Ft. Worth, Texas, May 1985, pp. 463-477.

11. Landgrebe, A. J. "The Wake Geometry of a Hovering Helicopter Rotor and itsInfluence on Rotor Performance," Journal of the American Helicopter Society,Vol. 17, No. 4, October 1972, pp. 3-15.

12. Kocurek, J. D. and Tangier, J. L., "A Prescribed Wake Lifting Surface Hover Per-formance Analysis," Proceedings of the 32nd Annual National VISTOL Forum of theAmerican Helicopter Society, Washington, D.C., May 1976, Preprint No. 1001.

13. Scheiman, J. "A Tabulation of Helicopter Rotor-Blade Differential Pressures,Stresses, and Motions as Measured in Flight," NASA TM X-952, March 1964.

8

APPENDIX AComputational Procedure

Initial Step Free Wake only

For r = r[i] at i = 1,...M, set

w(r) = X.(r)

u(r) = 0

wherer = radial displacement coordinate along the blade

rA[il = r at mid-points of aerodynamic segments

M = number of aerodynamic segments

u, w = net radial and axial induced velocity components

X, = vertical component of induced velocity, given by uniform inflowmethod based on momentum theory (from CAMRAD)

Iteration Loop

STEP 1 - Free Wake only

Calculate bound circulation distribution r(r) along the blade for r = rA~i] at i = 1,...M:

r(r) = r c(r) ct (r, a, r Mip)

wherec = blade chord

cl = blade lift coefficient

The latter is interpolated from 2-D airfoil tables as a function of angle of attack a andMach number r Mip, with a given by

a(r) = O(r) - O(r)

where0(r) = blade pitch angle (specified)

O(r) = blade inflow angle ( = w(r)/r)

9

STEP 2Representing the near wake by semi-infinite line vortices at r = rA[i] for i = 1,...M+I,

calculate, using the Biot-Savart law, the induced velocity wb(r) along the blade at r = rA[ilfor i = 1,...M:

w(r) =- [r(rA[i]) 1 +r4n- rAE[11 + r)

where

AF(rAE[il) = r(rA[i]) - r(rA[i-1]) for i = 2,...M

Ar(rAE[I]) = r(rAIll)

Ar(rAE[M+lD)= - r(rA[MI)

The above bound circulation distribution is given in Step 1 for the free wake, and by theuniform inflow method based on momentum theory (from CAMRAD) for the prescribedwake.

STEP 3Determine the circulation strength r. and location r, of each rolled-up, concentrated

vortex along the blade for s = I ...S (tip to root), where S is the total number of thesevortices. The boundaries for each region to be represented by concentrated vortices arefirst defined as follows.

The outboard boundary of the tip vortex region is at r.A[Mo], where Mo = M +1, and theinboard boundary is at rAE[MI], where r(rAIMI]) is the maximum of the circulationstrengths (r..,) calculated in Step I along the blade at the aerodynamic segment mid-points.

The boundaries for the inboard regions are defined to be at the closest aerodynamicsegment boundary inboard of the values defined by the user. The array indices are definedas M2,...Ms. 1, corresponding to boundaries rAE[M2,...rAIE[Ms.-1. The most inboardboundary is at the blade root, where the index is Ms, and the boundary rA[Ms].

a) Free Wake

The circulation strength r, and location r, (centroid of circulation distribution over theregion rAEJMSI to rAEFM,_1 ] ) are given by

M,1I

r, = - Ar(rAE[i])

.z= j X rAE[il Ar(rAE[i)

For s = 2,...S, r, is then scaled by an empirical factor fr (default value of 0.6).

10

b) Prescribed Wake

Here S = 2, and the concentrated vortices are assumed to be at the boundary extremities(tip and root), with the magnitude of the circulation strength of each equal to r_,., i.e.

t = rF,, r2 = - r,-.,

r= rA[M+l] r2 = r,,[Il]

STEP 4 - Free Wake only

By interpolating induced velocity components along the blade calculated at the previousiteration in Step 8 (zero initially), determine values at each concentrated vortex position, fors = I,...S:

{ulJ, = ul at r=r,

= u(rAji-l) + (uJ(rA[i]) - u(rA[i-1])) [i for rAti-l] < r, < rail

and similarly for (w),, IUF),, and {wp},.

STEP 5 - Free Wake only

Set intermediate wake geometry, defining radial and axial positions of the concentratedvortices at each vortex line or ring level, i.e. r,., and z,, for s = I,...S and t = 1,where T is the number of levels:

r, = r.,t,- + Ar,

44 = z.t.1 + Az.,,

where the incremental displacements Ar,., and Az,,, are given from the previous iteration(Step 13), but are approximated initially by

Ar,, = 0

AZ,. = L (rA[M])N

STEP 6 - Free Wake only

Calculate radial and axial components of the induced velocity at each vortex position inthe intermediate wake, i.e. for s - I,...S and t = 1,...T, (a) due to the intermediate waketo give (ul), and (wi),.,, and (b) due to the far wake to give (uF},, and (w),,. Only theradial components are shown below; the axial components are given by substituting w for uin all expressions:

~11

S T

[Ulls~t--E E uj(r. r., z... - z. r.m) .. .. ..

at -I (.. ) n -1 (,, )

S

uF., = A uF(r.,, r,,r.1, Z.,TI - z.,, r.)rn-I C,. )

Expressions for the above right-hand side velocity components, together withequivalent axial components, are given in Appendix B for both vortex line and vortex ringmodels.

STEP 7 - Free Wake only

Using the Biot-Savart law as in Step 2, calculate the induced velocity {Wb), atconcentrated vortices on the blade due to the trailing near wake for s = 1,...S:

(Wb}.= wj(r) atr=r,

S S

- X'(r . + X L r rrn- (i.' ) rn-I (,. )

STEP 8 - Free Wake only

Determine net radial and axial components of the induced velocity at concentrated vortexpositions on the blade (i.e. at t = 0 ) and at each vortex position in the intermediate wake:

Us, = {UI + UF), for t=0= (uI + uF},., for t = I,...T

wIA= {WI+Wi:+Wb), + ( w}, for t=0

= {wl + wF },. + (w,,}), for t =

The self induced velocity {w,}, of a vortex ring of radius r, is given by8 1 ]{w'.}, = L. kr ') -

where

r" = vortex core radius r. limited to a minimum of 0.005, i.e.max(0.005, r.)

rc = rb. (burst vortex core radius) if dbV_ 0 or z,,, <db,

= ru, (unburst vortex core radius) otherwise

db. = core burst test parameter (< 0 to suppress bursting) - from CAMRAD

12

STEP 9

a) Free WakeUsing net induced velocity components at the blade and in the intermediate wake,

determine new wake geometry for s = 1,...S and t = 1,...T:

r= r.,_ + 6Ar,,

4A= 7,,j + AZ., t

where the incremental displacements 6r,, and Az,, are given by

,r.., =,(u,., + u.")

N1(W.3-1 + Wsa)

b) Prescribed WakeUsing prescribed wake geometry based on either of the options in CAMRAD, i.e.

(a) Landgrebe or (b) Kocurek and Tangier, set -ip and root vortex positions (noting S = 2)for t = 1,...T:

r,., = KY4 + (1 - K) e -2nK3t/N

r,,, = rA[l]

= = L [ K, + K2(t- )

whereKI, K2 =axial settling rates of tip vortex before and after passage of following blade

K3, K 4 = radial contraction parameters for tip vortex

These parameters are given by CAMRAD on specifying the appropriate option, i.e.value of OPRWG in Namelist NLWAKE.

STEP 10Calculate radial and axial components of the induced velocity along the blade (a) due to

the intermediate wake to give u,(r) and w,(r), and (b) due to t le far wake to give uF(r) andwF(r) for r = rA[il at i = 1,...M. As in Step 6, only the radial components are shownbelow, with axial components obtained by substituting w for u:

S T

u,(r)= u(rr r... z ... , r.) d2

M -1 1

13

-- ''-- -- mmm m m~ m llllI m m.tl

S

uF(r) = K UF(r, r,,,T.l,, Z.T+,l, r) fordistributed far wake. -1

= u1(r, ri, ZT, 4r,) + ul(r, r2, ZT, r 2) forconcentrated far wake(prescribed only)

where

dI = (r=., - r)2 + (z.. - z)2

Expressions for the above right-hand side velocity components, together with theequivalent axial components, given in Appendix B are again used.

STEP 11

Determine net radial and axial components of the induced velocity along the blade togive u(r) and w(r) for r = rAli] at i = I,...M:

u(r) = ul(r) + uF(r)

w(r) = w,(r) + wF(r) + wb(r)

For prescribed wake, go to Step 14.

STEP 12 - Free Wake only

Test for convergence of induced velocity; if the maximum number of iterations has beenreached or

M

I,_, [{Iw(rA[il)}..,. { w(rA~iD)}old]' < IL W... r]d

where

w.., = max Iw(rAli])I

= tolerance for induced velocity convergence

and then go to Step 14.

STEP 13 - Free Wake only

To help prevent numerical instability in the iterative procedure, lag new (non-convergent) solution for (a) induced velocity u(r) and w(r) along the blade for r = rA[i] at i= 1,...M, and (b) wake geometry incremental displacements Ar,., and Az,. for s =,..S

and t =,...T:

u(r) = f {u(r))n1- + (1 - f) (u(r) 1,d

w(r) = f {w(r)),.. + (1 - f) {w(r))}Id

14

Ar,= f (A.) + (1 - f) (Ar.A 1 ,

A7,, f (Ar.j. + (I -f) (Az.J).

where the factor f used to introduce lag into the solution is typically 0. 1.

Having completed an iterative cycle for the free wake, go back to Step 1.

STEP 14Transform induced velocity components u(r) and w(r) along the blade to longitudinal,

lateral, and vertical components X,(r, 'p), X,(r, 'V), and 4(r, ip) used by CAMRAD:[X,] u v csi]

15

APPENDIX BVelocity Components Induced by Wake Vortex Elements

Vortex Line

w(r,P, h, 0)= n (+02 + ( 7 -2+ T]

u(r, p, h,r) =r p+r +

UF(r, p, h,r) = - [~h + ac( -- T + r)2,\-.-) frp

~- aartan + r)] for p = r

- -- rctn~L' + rtn(.- for p > r

Vortex Ringrhfk 2 rE(2 -k 2)-2K

u(r, p, h, r) = 4Ic~'p 1 k 2 K

wi(r, p,h, r) = \J K - E(I k2(l + pr))]4m (1 kl2)

uF(r, ph, r) = r ( 1 - 2E]

It

7cJ '+ p2 -2p rcos 0 r+p2+ h2- 2prcosowFrp h ) j~2P( rcs) ~~ 2

+ 0]j

where the latter is integrated numerically, and

k2 = 4pr(p + r)

2 + h2

E = I + !-(F-7)(l-k 2) +~( )( c)

K = F + L(F- I) (I -k2) + -L(F-Z) (I k2)2

16

APPENDIX C

Block Diagram for Free Wake Method

ga.~itj _ _ _ _ _ _ __ _ _ _ _ _ _

ii~iE

allAC

LC4 E

171~

APPENDIX DBlock Diagram for Prescribed Wake Method

W19

8t

18

APPENDIX EModified CAMRAD Subprograms

SUBROUTINE GEOMR1 (LEVEL)

COMMON /KTIP/KT MOD

CC CALCULATE WAKE GEOMETRY DISTORTIONC

CC FOR KOCUREX AND TANGLER AND LANDGREBE MODELS, FWGT(1) AND FWGT(2) MODC ARE USED AS FACTORS FOR KT(1) AND KT(2) MODCC KOCUREK AND TANGLER

FB-. 000729*7WFC.2.3-~.206*TWFM-1.-~.25*ExP( .04*Tw)FN.. 5-.0172*1WKT(1)=U('B+FC*(ABS(CTG),/LOATNBLADE)**FN)**FM)*FVWGTU.) MODKT(2)-SQRT(ABS(CTOG-FLOAT(NBLADE)**FN*(ABS(-FB/FC) )**(1./FM)))1 *FWGT (2) MODKT(3)-4.*CTHKT(4)-.78GO TO 17

C LANDGREBE16 KT(l)-.25*(CTOS+.001*IW)*FwGT(l) MOD

KT(2).(1.+.01*'IW)*CTH*FWGT(2) MODKT( 3)..145+27.*CTGKT(4)-.78

SUBROUTINE INPTWICOMMON /WltATA/FACNWN ,OPVXVY, KNW, KRW, KFW, KID,RRU, FRU PRU,FNW, DVS D

ILS ,CORE( S) ,OPCORE( 2) ,WKMODL( 13) ,OPNWS) 2) ,LHW,OPH-W,OPRTS ,VELB, DPHIB2,DBV,QDEBUG,RG,N'G(30),MRL,NL(30),OPWKBP(3),KRWG,OPRWG,FWGT(2),FWG3SI(2) ,F~vSO(2) ,KWGT(4) ,KWSI)4) ,KWGSO) 4)INTEGER OPVXVY,OPCORE ,WKMOD)L,OPNWS ,OPHW ,OPRTS ,OPWKBP ,OPRWG

1 ,NIVL,NIBV,WFMODL,OPMODL, ITERV MODREAL KWGT,KWGS1 ,KWSO1 ,RIBB(8),FGAMMA MOD

COMMON /G1DATA/KFWG,OPFWG, ITERWG, FAC'IW,WGMODL( 2) ,RIW( 2) ,COREW(4

1),MRV,LDMG,N!MG36),IWGDB2) ,QGDB,D~QG(2)INTEGER OPFW. ,WGMODLCOMMON /ThIDATA/M=OC( 182)INTEGER DEBUGEQUIVALEN4CE (TTVO(41) ,DEBUG)COMPEt /UNITNO/NFDAT, NFAF1 ,NFAF2 ,NFRS ,NFEIG ,NFSCR ,NUDB, !'UOT,NUPP,INULIN,NJINCOMMON /RING/ NIBV,RIBB,NIVL,FACTIV,EPVL,WFMODL,OPMODL,FAMiMA, MOD1 ITERV MOD

19

CC READ WAKE NAMELISTC

NAMELIST ,4NLWAKE/FAC , OPVXVY, KNW, KRW,KFW, KIW, RRU, FRU, PRU, ENW, DVS1 ,DLS,CORE,OPCORE,WI'MODL,OPNWS,LHW,OPHW,OPRTS,VELB,DPHIB,DBV,QDEBUG2 ,MRG, NG,MRL,NL ,OPWKBP, KRWG,OPRWG, FWGT, FWGSI ,FWGSO, KWT,KWSI , KWSO3, KFW ,OPFWG, ITERWG, FACIW ,WGMODL, RIWG, COREWG, MRVBWG, LDMWG, NDM~r, IP4WGDB , WGDB, D~wG5 ,NIVL,RIBB,NIBV,FACTIV,EPIVEL,WFMO0DL,OPMODL,FGAMMA, ITERV MOD

C MODC----------------- DEFAULT VALUES FOR VORTEX LINE/RING-------------------- MODC MOD

NIBV=2 MODDO I=1,8 MOD

RIBB(I)=0.0 MODEND DO MODNIVL=-4 MODFACTIV=0 .1 MODEPIVEL-0 .05 MODWFMODL=2 MODOPMODL=-0 MODFGAMMA= 0.6 MODITERV=200 MOD

C------------------- EN----- D ---------------------------------- -MOD999 FORMAT (1X,33HREADING NAMELIST NLWAKE (ROTOR 1))

WRITE (NUOUT,999)READ (NUIN,NLWAKE)IF (DEBUG .GE. 2) WRITE (NUDB,NLWAKE)RETURNEND

SUBROUTINE PRN'1W1

COMMON /R1DATA/R)O(932) MODEQUIVALENCE (RflXX(81) ,RROOT) MODEQUIVALENCE (TMXXO(77),MPSI),(TMXO((157),LEVEL)COMMON /UNITNO/N'FDAT, NFAF1 ,NFAF2 ,NFRS ,NFEIG,NFSCR,NUDB ,NUOUT,NUPP,

iNULIN ,NUINREAL RIBB(8),FGAMMA MODINTEGER NIVL,WFMODL,OPMODL,NIBV,ITERV MO0DCOMMON /RING/ NIBV,RIBB,NIVL, FACTIV,EPIVEL,WFMODL,OPMODL, FGAMMA, MOD

1 TTERV MODCC PRINT WAKE INPUT DATAC

CC--------------------- VORTEX RING/LINE ---------------------------C991 FORMAT(//1X, 'VORTE LINE AND VORTEX RING MODELS (PRESCRIBED AND FR

IEE)'/'5X,'NUMBER OF INTERMEDIATE VORTEX LEVELS, NivL =,,13/5x,'FAR2 WAKE MODEL (0 TO OMIT, 1 FOR CONCENTRATED, 2 FOR SHEET), WFMODL3=',T3//5X,'FOR FREE WAKE ONLY'//5X,'FACTOR INTRODUCING LAG IN INDU4CED VELOCITY, FACTIV =' ,F10.4/5X, 'TOLERANCE FOR INDUCED VELOCITY,5 EPIVEL =',F10.4,'SX,'ROLLED-UP VORTEX WEIGHTING FACTOR (EXCLUDING6TIP), FGAMMA =',F9.4/5x,'MAxium NUMBL.2 (IF INDUCED VELOCITY ITERAT710NS, iTERV =,,14,/5x,'NUMBER OF ROLLED--UP VORTICIES, NIBV =,,13)

20

986 F'OFWAT(/5X, 'INBOARD EDGE OF ROLLED-UP VORTICIES, EXCLUDING ROOT AN1D TIP (NIBV-2 VALUES),'//15X,'RIBB =',8(F9.4))

CC-------------------------------- END--------------------------------------C

999 FORMAT (//1X,23HNONUNIFORM INFLOW MODEL! MOD*5X,'VORE LINE MODEL IF 1, VORTEX RING MODEL IF 2, OPMODL =,,13A4oD

* 5X,27HEXTENr OF NEAR WAKE,MOD

1 KNW =,IS/5X,33HEXTENT OF ROLLING UP WAKE, KRW =,15/5X,26HEXTENT2 OF FAR WAKE, KBW =,15/5X,30HEXTENT OF DISTANT WAKE, K2w =,iS/5x3,37HROLLUP INITIAL RADIAL STATION, RRU =,F10.4/5X,42HROLLUP INITI4AL TIP VORTEX FRACTION, FRU =,FlO.4/5X,27HROLLUP EXTENT (DEG), P4R0 =,FlO.2/5X,37HNEAR WAKE TIP VORTEX FRACTION, FNW =,FIO.4/5X,5OSHNUMBER OF SPIRALS IN AXISYMMETRIC FAR WAKE, LHW -,I5/5X,4OHAXISY6MMETRIC WAKE GEOMETRY IF 0, OPH-W =,13)

WRITE (NUOUT,990) KRWG,OPRWG, (FWGT( I) ,FWGSI( I) ,FWGSO( I), I 1,2) ,(K

WRITE (NUOUT,991) NIVL,WFMODL,FACTIV,EPIVEL,FGAMA, ITERV,NIBV MODIF (NIBV .GT. 2) THEN MOD

IF (RIBB(l) .LT. RROOT) THEN MODDRI= (0.9-BRR)T) /FLOAT(NIBV-1) MODDO I=1,NIBV-2 MOD

RIBB( I)=RROOT+I*DRI MODEND DO MOD

END IF MODWRITE(NUOUT,986) (RIBB(I),I1,NIBV-2) MOD

END IF MODIF (LEVEL .LE. 1) GO TO 1

SUBROUTINE RAMF (LEVEL1 ,LEVEL2 ,OPLMDA)

COMMON /RING/RIXX( 16) MODINTEGER OPRTR2,DEBUG,MHARM(2),MHARMF(2)

1 ,OPMODL MODEQUIVALENCE (VOM(77),MPSI),(TMWO(179),MARI(l) ), (TII1XX(91),ITEM),(T'flO(92),EPMOTN),(TVO(93),ITERC),(TMXX(94),EPCIRC)2,(TMXDX(49),DEBUG),(TMX(90),MREV),(T1MVO(89),MPSIR),(TRIMXX(11),OPR

3TR2),(RTRlX(4),CMEAN),(RTRXX(6),NBM1),(RTRXX(7),NT 1),(RTRlXOX(48),NGM1),(RTR2XOX(4),CMEAN2),(RTR2Xx(6),NBM2),(RTR2xX(7),NTM2),(RTR52XX(8),NGM2),(BODYXCX(254),NAM),(ENGNXX(11),NDM),(TMXX(181),MHARM(61) ),(R1)OX(24),SIGMAI) ,(R2XX(24),SIGMA2)7 , (RI)OF(14),OPMODL) MODCOMMON /LJNI'110NFDAT,NFAF1 ,NFAF2 ,NFRS,NFEIG,NFSCR,,NUDB,NTUOUT,NUPP,

1NULIN,NUINCC CALCULATE ROTOR/AIRFRAME PERIODIC MOTION AND FORCES

C

C CALCULATE INDUCED VELOCITYC

CALL WAKEU1

21

C ----- VORTEX RING/LINE --- MODMOD

IF(OPMODL.GT.0.AND.LEVEL1.NE.0) THEN MODCALL WAKERI (LEVELl) MODGOTO 14 MOD

ENDIF MODC - END - MOD

CALL WAKEN1(LEVEL1)

C END MOTION ITERATIONC TEST CIRCULATION CONVERGENCE

OUT=0IF (LEVELl .EQ. 0) GO TO 53IF ((LEVEL1 .EQ. 2) .AND. (OPMODL .GT. 0)) GO TO 205 MODGIMS=0.

SUBROUTINE TRIMCOMMON /TMDATA/MXX( 182)COMMON /TRIMCM/TRIMXX( 1604)COMMON /CASECM/CASEXX( 9)COMMON /RING/RIXX( 16) MODINTEGER RESTRT, RSWRT,OPRTR2

1 ,OPMODL MODEQUIVALENCE (TMXX(157),LEVEL1), (TMXX(158),LEVEL2), (TMXX(159),ITERU

1 ), (TMXX( 160 ), ITERR), ( TX( 161 ), ITERF ), ( TM(162 ), NPRNTT ), (TMXX( 1632),NPRNTP),(TMXX(164),NPRNTL),(CASEXX(i),RESTRT),(CASEXX(5),RSWRT),3(TRIMXX(11),OPRTR2)4 ,(RIXX(14),OPMODL) MODCOMMON /UNITNO/NFDAT,NFAF1,NFAF2,NFRS,NFEIG,NFSCR,NUDB, NUOUT,NUP,INULIN,NUIN

CC TRIMC

C NONUNIFORM INFLOW AND PRESCRIBED WAKE2 IF (LEVEL .EQ. 1) ITERR=MAX0(ITERR,1)

IF (ITERR .LE. 0) GO TO 3DO 20 IT=l,ITERRIF (LEVELl .EQ. 0) GO TO 22LEV1=1

C ------------------- VORTEX LINE/RING ----------------------------------- MODIF(OPMODL.GT.0) THEN MOD

CALL GEOMR1(LEV1) MODGOTO 22 MOD

END IF MODC---------------------------END --------------------------------------- MODC

CALL WAKEC1(LEV1)

22

C NONUNIFORM INFLOW AND FREE WAKE3 ITERF=!MAXO (ITERF, 1)

DO 30 IT=1,ITERFIF (LEVELl .EQ. 0) GO TO 32LEVI-LEVELI

CIF(OPMODL.GT.0) GO TO 32 MOD

CCALL WAKEC1(LEV1)

SUBROUTINE TIMER(N, I,T)COMM ,"IMECM/rSTART(14),TSUM(14),NCALLS(14),IDBSAV(23),ICNTCOMMON /TMDATA/TVC( 182)EQUIVALENCE (TMXXO(64),DEBUIG),(TYVOX(41),IDB(l) ),(TDVO((40),ITDB)INTEGER DEBUG, IDB( 23)INTEGER*4 ITIME,HANDLE ADR,CODE MODCOMMON /UNI 1NO/NFDAT, NFAF1 ,NFAF2 ,NFRS ,NFEIG,NFSCR, NUDB,NuouT,NUPP,1NULIN,NUIN

CC PROGRAM TIMERC

REAL TFRACT(14),TCALL(14)INTEGER ID(2..14)DATA MT/14/DATA ID/4HCASE,4H ,4HTRIM,4H ,4HFLUJT,4H ,4HSTAB,4H ,4HITRAN,4H ,4HSTAB,4HL ,4HFLU;T,4HL ,4HWAKE,4H-C ,4HGEOM,4HR2 ,4HRAMF,4H1 4HM4ODE,4H ,4HMOTN,4HR ,4HPERF,4H ,4HLOAD,43H/

999 FORMAT (1H1,17HCOMPUTATION TIMES//50X,8HCPU TIME,4X,7HPERCENT,8X,61HNUMBER,3X,9HTIME/CALL/52X,5H(SEC),19Xf8HOF CALLS,4X,5H(SEC)/)

998 FORMAT (1X,6HTIME =,F12.3,4H SEC)997 FORMAT (1X,6HTIME =,Fl2.3,4H SEC,5X,7H(START ,2A4,lH))996 FORMAT (lX,6HTIME =,Fl2.3,4H SEC,5X,5H(END ,2A4,lH),5X,12H(CALL NU

1MBER,13,18H, TIME INCREMENT =,F12.3,5H SEC))901 FORMAT (10X,4HCASE,31X,2F12.3,I12,F12.3)902 FORMAT (1OX,11HTRIM (TRIM),24X,2F12.3,Il2,Fl2.3)903 FORMAT (10X,14HFLUTrER (FLUTr),21X,2Fl2.3,I12,F12.3)904 FORMAT (1OX,22HFLIGHT DYNAMICS (STAB),13X,2F12.3,Il2,Fl2.3)905 FORMAT (10X,16HTRANSIENT (TRAN),19X,2F12.3,Il2,Fl2.3)906 FORMAT (1OX,23HLINEAR ANALYSIS (STABL),12X,2Fl2.3,I12,F12.3)907 FORMAT (10X,23HLINEAR ANALYSIS (FLUTL),12X,2F12.3,I12,F12.3)908 FORMAT (1OX,25HNONUIFORM INFLOW (WAKEC),10X,2F12-.3,I12,F12.3)909 FORMAT (1OX,21HWAKE GEOMETRY (GEOMR),14X,2Fl2.3,I12,Fl2.3)910 FORMAT (1OX,25HVIBRATORY SOLUTION (RAMF),1OX,2F12.3,112,F1?.3)911 FORMAT (1OX,18HROTOR MODES (MODE),17X,2Fl2.3,Il2,Fl2.3)912 FORMAT (1OX,23HROTOR EQUATIONS (MOTNR),12X,2Fl2.3,Il2,F12.3)913 FORMAT (10X,18HPERFORMANCE (PERF),17X,2Fl2.3,Il2,r12.3)914 FORMAT (IOX,12HLOADS (LOAD),23X,2F12.3,I12,F12.3)

CODE-2IF (N .EQ. 0) GO TO 10IF (N .EQ. 1) GO TO 11IF (N .EQ. 2) GO To 12IF (N .EQ. 3) GO TO 13

C RETURN PRESENT TIMEIERROR=LIB$STAT TIMER (CODE,ITIME,HANDLE ADR) VAXT=.01*FLOAT( ITiE) VAXIF (DEBUG .GE. 1) WRITE (NUOUT,998) TRETURN

C INITIALIZE10 CONTINUE

IERROR=LIB$INITTIMER (HANDLEADR) VAX

23

DO 100 JT=1l,MTTSUM(JT)=O.NCALLS(JT)=0

100 TSTART(JT)=0.ICNT-0RETUlRN

C START TIMER11 CONTINUE

IERROR=LIB$STAT TIMER( CODE, ITIME, HANDLEADR) VAXT=.01*FLPLT( ITIAE) VAXTSTART( I)=TIF (DEBUG .GE. 1) WRITE (NUOUT,997) T,ID(1,I),ID(2,I)IF (I .LE. 1) GO TO 113IF (ICNT .EQ. 1) GO TO 111DO 112 II=1,23IDBSAV( II )=IDB( II)

112 IDB(II)=0ICNT=1

111 IF (ITIME .LT. ITDB*1000) GO TO 113DO 114 II=1,23

114 IDB(II)-IDBSAV(TI)113 CONTINUE

RETURNC STOP TIMER

12 CONTINUEIERROR=LIB$STAT TIMER(CODE,ITIME,HANDLEADR) VAXT*.01*FLOAT( ITIME) VAX

DT'=T-TSTART( I)TSUM( I) =TSUM( I )+DTNCALLS( I)=NCALLS( I)+1IF (DEBUG .GE. 1) WRITE (NIJOUT,996) T,ID(1,I),ID(2,I),NCALLS(I),DTRETUJRN

C PRINT TIMES13 CONTINUE

TCASE=TSUM( 1)IF (TCASE .NE. 0.) TCASE=100.7ICASEDO 130 JT=1,MTTFRAC-T( JT)=TSUM( JT) *TAETCALL(JT)=0.IF (NCA.LLS(JT) .NE. 0) TCALL(JT)=TSUM(JT)/FLOAT(NCALLS(JT))

130 CONTINUEWRITE (NUOUT,999)WRITE (NUOUT,901) TSUM(1),TFRACT(1),NCALLS(1),TCALL(1)WRITE (NUOUT,902) TSUM(2) ,TFRACT(2) ,NCALLS(2) ,TCALL(2 )WRITE (NUO(JT,903) TSUM1(3),TFRACT(3),NCALLS(3),TCALL(3)WRITE (NUO)UT,904) TSUM(4) ,TFRACT(4) ,NCALLS(4) ,TCALL(4)WRITE (NUOUT,9fG:) TSUM(5),TFRACT(5),NCALLS(5),TCALL(S)WRITE (NUJOUT,906) TSUM(6),TFRACT(6),NCALLS(6),TCALL(6)WRITE (NUOUT,907) TSUM(7),TFRACT(7),NCALLS(7),TCALL(7)WRITF (NUOUT,908) TSUM(8),TFRACT(8),NCALLS(8),TCALL(8)WRITE (NUOUT,909) TSUIM(9),TFRACT(9),NCALLS(9),TC-ALL(9)WRITE (NUOUT,910) TSUM(10),TFRACT(10),NCALLS(1)),TCALL(10)WRITE (NUOUT,911) TSUIM(11) ,TFRACT(11) ,NCALLS(11) ,TCALL(11)WRITE (NUOUT,912) TSUM(12),TFRACT(12),NC-ALLS(12),TCALL(12)WRITE (NUOUT,913) TSUM(13) ,TFRACT(13) ,NCALLS(13) ,TCM.LL(13)WRITE (NUOUT,914) TSUM(14),TFRACT(14),NCALLS(14),TCALL(14)

24

APPENDIX FAdded CAMRAD Subprograms

SUBROUTINE WAKERi (LEVEL)C -----------------------------------------------------C VlE-'ERMINEF THE INDUACED) VELO)CITY AT THE ROTrOR USING A VORTEXC LINE OR RING MODELCC THE WAKE MAY BE EITHER PRESCRIBED OR FREECC THE FAR WAKE MAY BE EITHER NEGLECTED OR REPRESENTED AS AC DISTRIBUTED OR CONCENTRATED VORTEX

INTEGER DEBUGG,DEBUJGV,I,IR,IT,J,JPSI,JR,JT,LEVEL,M,MPSI,MRA,N,1 NBLADE,NIBV,NTM,NTM1,OPCOMP,OPMODL,S,SMAX,T,NIVL,WFIODL,2 MS(0:10),ICOUNT,ITERVREAL ALPHA,AREA,BETAC,BETAS,CL,COREB,CVERT,D5,DA,DBV,DELW,DGA4,1 DGAMR,DU,D,E,EPIVEL,EPR,ERR,FACTIV,FACOR,FOLD,GAMR,H,2 LAMBDA,MACH,MTIP,PI,Q1,RM',RRO(OT,T75,URES,WMAX,ZST,ANGL(30),3 DR(10,36) ,DZ(10,36),GA4A(30) ,GS(10) ,R(10,36) ,DROLD(10,36),4 RS(10),RUB(0:10),U(30),UIF(10,36),UJNW(10,36),UOLD(30),US(10),5 W(30),WB(30),WIF(10,36),WIFR(30),WNW(10,36),WOLO(30),WRU(10),6 WS(10),WSELF(10) ,Z(10,36) ,DZOLD(10,36) ,RIBB(8) ,FGAMHA

CHARACTER CHAR*2CC----------------CANRAD COMMON BLOCK -------

COMMUON /RlDATA/RlX( 932)COMMON /RTR1CM/RTR1XX( 1070)COMM /CONTCM/CONTXX( 32)COMMON /WKV1CM/WKVIC( 8165)COMMON /TMDATA/T'vO(182)COMMON /'IRIMCM/l'RIMX(1604)COMMON /W1DATA//WlXX( 126)COMMON //QR1CM/QRlX( 1139)COMMON /IM1CM/M1XX( 6773)COMMON /AEMNCM/AEMNXO(78)

COMMON /RING/ NIBV,RIBB,NIVL,FACTIV,EPIVEL,WFM.ODL,OPMODL, FGAMKA,1 ITERV

REAL RA(30) ,'WIST(30),CHORD(30),VIND(3,30,36),GAMOLD(30,36),1 SINPSI(36),COSPSI(36),THiETZL(36),DRA(30) ,RAE(31),GAI4(30,36),2 ZETA(5,30) ,P1(5) ,CRCOLD(36) ,CRC(36) ,CORE(2)

CEQUIVALENCE (R1XX(150),MRA),(RTRXC(20),RA(1)),(CONTXOX(1),T75),1 (R1,OX(272),IWIST(1)),(R1,O(302),THETZL(1)),(W1XX(l),FACTOR),2 (R1O((182),CHORD(1)),(QRIXX(24),GAM(1,1) ),(R1)OX(81) ,RROOT),3 (WKVflOC(1120),VIND(1,1,1)),(R1)Oc(26),NBLADE),(TvO((77) ,MPSI),4 (TRIrVOX(57) ,SINPSI(1) ) ,(R1)oc(80) ,OPCOMP) ,(RTR1XX(7) ,NTM),5 (TRIXXO(21),COSPSI(l)),(WIXOC(13),CORE(1) ),(W1)OC(40),DBV),6 (RTR1XX(2),MTIP),(WKV1,0((4360),LA~MDA),(QR1XX(1104),CRC(l)),7 (QR1XX(22),BETAC), (QR1XX(23),BETAS), (RTRIXO((50),v)RA(l)),8 (R1XX(151),RAE(1)),(WKV1XX(4),GAMOLD(1,1) ),(TM2O((53),DEBUGV),9 (TMXX(63),DEBUGG),(MD1XX(6148) ,ZETA(l,1)),(AEEOO(31),P1(1) ),A (WKV1)C( 1084) ,CRCOLD(1))

CCOMMON /UNITNO/NFDAT,NFAF1 ,NFAF2 ,NFRS ,NFEIG,NFSCR,NUYDB ,NUOUT,NUPP,iNULIN ,NUIN

CC ------------------- END CANRAD COMMON BLOCK ---------

25

C COMMION /AlELICOM/Ql,W,U,GAMA,RS,GS,WS,US,DZ,R,DRZwIFRSMM

899 FORI4AT(/1X,'RING/LINE LEVEL ',10(19))900 FORMAT(/1x,'voRTEX NO.',14,' R=',10(F9.5))

901 FORmAT(18X,'Z=',1O(F9. 5 ))

902 FORMAT(/X,'STRENGTH OF ROLLED UP VORTEX' ,F1O.6)

903 FORM.AT(5X,10(F9. 5))904 FuRmAT(ix,A2,2x ,10(F9-

5 ))

CC INITIALIZE VARIABLESC

DATA Pi/3.14159265/

IF (LEVEL EQ2. 1) THENSMAX=2ELSESMAX=NIBV

END IFMS(0)=MRA+lRUB(0)=RAE(MRA+1)MS( SMAX)-1RUB( SMAX )=RROIF (SMAX .LE. 2) GO TO 20

C TEST DATA FOR FATAL ERRORSDO I=1,NIBv-2

IF (I .EQ. 1) THENIF (RIBB(I) .LT. RR(XYE) GO TO 10ELSEIF (RIBB(I) .LT. RIBB(I-1)) THEN

WRITE(NUDB,*) 'ERROR IN DATA :LOCATION OF INBOARD VORTEX

IBOUNDARIES'GO TO 10

END IFEND IF

END DODO I=1,NIBV-2

S=SMAX-IRUB(S)=RIBB( I)

END DOGo To 20

10 DRI=(0.9-RROOT)/FLOAT(NIBV-l)DO I=0,NIBv-2

S=SMAX-IRUB( S-I )=RUB( S)+DRI

END DO20 NTM1=MAXO(1.NTM)

ICOUNT=lCVERT=180 ./PIQ1'-2 0*pI/FLOAT( NBLADE)IT=0o

CC GEOMETRIC PITCH AND INDUCED VELOCITY FROM CAMRAD

C (FREE WAKE ONLY)C

IF (LEVEL .EQ. 2) THENDO IR=1,MRA

PNGL( IR)-~T75+(!TWIST( IR)+THETZL( IR) )/CVERT

W(IR)=-viND(3,IR,l)DO JT'=1,NTM1

ANGL( IR)='ANGL(IR)+ZETA(JT, IR) *P1(JT)

END DO

26

END DOEND I F

CC POSITIONS OF ROLL-UP BOUNDARIESC

IF (SMAX .GT. 2) THENDO I=2,SMAX-1

5=IDO JR=1,MPA

J=JRIF (RAE(J) .GE. (RUB(S)-1.E-4)) Go TO 101

CC RUB(S) IS INBOARD LIMIT OF THE VORTEX ROLL-UP BOUNDARYC

END DO101 MS(S)=J

END DOEND IFMS(0)=MRA+1RUB( 0)=RAE(MRA+1)MS(SMAX)-1RUB (SMAX )=RROOT

CC BEGINNING OF LOOP FOR NEXT ITERATIONCC COMPUTE BLADE BOUND CIRCULATIONC70 IF(LEVEL .EQ. 1) THENCC FROM CANRADC

FOLD-i 0-FACTORDO JPSI=1,MPSI

CRCOLD( JPSI )=FOLD*CRCOLD( JPSI )+FACTOR*CRC( JPSI)DO IR=1,MRA

GAM'OLD(IR,JPSI )=FOLD*GAMOLD( IR,JPSI )+FACTOR1 *G24(IR,JPSI)

GAMA( IR).-GAM'OLD( IR,JPSI)END DO

END DOELSE

CC FROM INDUCED VELOCITY

Do 80 JR-i ,MRACC DETERMINE ALPHA,URES,MACH,COSL

ALPHA-(AN4GL(JR)-W(JR)/RA(JR) )*CVERTURES=W(JR )*W( JR )+RA (JR )*RA( JRIF(URES.NE.0.0) URES-SQRT(ABS(URES))MACH-URES*MTIPIF(OPCOMP.EQ.0) MACH0.0

C LIFT COEFFICIENT

CALL AEROT1(ALPHA,MACH,RA(JR) ,l,CL,CD,CM)

C CIRCULATION

GAIIA( JR)=0.5*CL*tIRES*CHORD( JR)

C

27

s0 CONTINUEEN~D IF

CC LOOP TO FIND STATION OF MAX. CIRCULATIONC

DO 90 JR=MRA,2,-1J=JRIF(GAMA(J) .GT. GANA(J-1)) GO TO 100

90 CONTINUE100 MS(1)=J KM1 IS LOCATION OF MAX. CIRCULATION ON BLADE

RUB( 1) RAE (J)

C COMPUTE INDUCED VELOCITY AT~ BLADE DUE TO NEAR TRAILING WAKEC (DUE TO SHEET)C

105 Do 111 IR=1,MRAWB(IR)=0.0DO 110 JR=1,MRA+1

IF (JR .EQ. 1) THENDELW=-1/(4.*PI)*(GAMA(JR) )*

1 (1.0/(RAE(JR)-RA(IR))+1.0/(RAE(JR)+RA(IR)))ELSE IF (JR .EQ. (DMA.1)) THENDELW=1/(4.*Pl)*(GA4A(JR-1) )*

I (1.0/(RAE(JR)-RA(IR))+1.0/(RAE(JR)+RA(IR)))ELSEDELW=-1/(4.*PI)*(GAMA(JR)-GAMA(JR-1) )*

1 (1.0/(RAE(JR)-RA(IR))+1.0/(RAE(JR)+RA(IR)))END IFWB( IR)=WB( IR)+DELW

110 CONTINUE111 CONTINUE

CC PRESCRIBED WAKE GEOMETRYC

IF(LEVEL .EQ. 1) THEN

CALL PRESWG(MS(1))GO TO 306

END IFCC FIND CENTROID OF VORTEX, RS(S)C

DO S=1,SMAXGS(S)=0.0GAMR=0.0DO 120 JR-MS(S)+1,MS(S-1)

IF (JR .EQ. 2) THENi DGAt4=GAMA( JR)ELSE IF (JR .EQ. (MRA+1)) THENDGAII=-GA4A( JR-i)ELSEDGAI4=GAMA( JR)I-GAMA( JR-I)

END IFGS (S )=GS (SI -DGAM

C GS(S) IS THE CIRCULATION OF THE VORTEX, SC

DGM jDA*pAE( JR)GAMR='GAMR+DGAMRi120 CONTINUE

IF (ABS(GS(S)) .LT. i.E-10) THENRS(S)=(RUB(S)+RUB(S-1) 1/2.0ELSE

28

RS(S)=ABS(GAMR/GS(S)) ILOCATION OF VORTEX, SIF (S .NE. 1) GS(S)-~GS(S)*FGA4MA

END IFEND DO

CC FIND VELOCITIES AT RS(S), WS(S), AND US(S)C

DO I=1,SMAX5=1DO J=MS(S),MS(S-1)

JR=JIF (RS(S) .LT. RAE(JR)) Go To 160

END DOCC INDUCED VELOCITYC160 CALL IVTERP(S,JR)C

IF (IT .EQ. 0) THENWS(S)=0.0US (S 1=0.

END IFEND DOIF (IT .EQ. 2) THEN

DO S=1,SMAXDO T=1l,TMAX

IF (T .EQ. 1) THIENZ(S,T)=DZ(S,T)RS,T) =RS (S I+DR IS,T)

ELSEZ(S ,T) =Z (S. T-I)+DZ (5, T)R(S,T)=R(S,T-1 )+DR(S,TI

END IFEND DO

END DOELSE

CC ESTABLISH INITIAL WAKE GEOMETRY FROM MOMENTUM THEORYC

DO 221 S=1,SMAXDo 220 T=1l,NIVL

IR=MS( 1)219 D5=W(IR)*Ql

C CHECK TO SEE IF INFLOW AT STATION IR IS .NE. TO ZERO.C IF INFLOW AT IR .EQ. ZERO THEN MOVE IN TO NEXT RADIAL STATION.C THIS PREVENTS PROGRAM FROM CRASHING & HELPS CONVERGENCE.

IF (D5 .EQ. 0.0) THENIR=IR+1Go TO 219

ENDI FDZ(S,T)=D5R(S,T)=RS(S) INITIALISE RADIAL VORTEX POSITONSDR(S,T)=0.0IF (T .EQ. 1) THEN

Z(S,T)=DZ(S,T) INITIALISE VERTICAL VORTEX POSITONSELSEZ(S,T)=Z(S,T-1 I+DZ(S,TI

END IF220 CONTINUE221 CONTINUE

29

END I FCC COMPUTE VELOCITIES IN WAKE DUE TO INTERMEDIATE AND FAR WAKESC

230 DO 271 S=1,SMAXDO 270 T=1I,NIVL

DZOLD(5, T) =DZ (5,T)DROLD(S,T)=DR(S,T)

CC CALCULATE VELOCITES AT R(S,T), Z(S,T) DUE TO INTERMEDIATE AND FAR WAXESC

CALL VTXIF(R(S,T),Z(S,T),0,WIF(S,T),UIF(S,T))CCC VELOCITY IN WAKE DUE TO ROLLED-UP NEAR WAKEC

H=-Z(S,T)IF (OPMODL .EQ. 1) GO To 260WNW(S,T)=0.0UNW(S,T)=0.0DO M=1,SMAX

RMN=RS(M)CALL IRING(R(S,T),RMN,H,GS(M),DU,Dq)WNW(S,T)=WNW(S,T)+ONUNW(S,T)=UNW(S,T)+DU

END DOGO To 666

260 WNW(S,T)=O.0UNW( S ,T) =0.0DO M=1,SMAX

CALL ILINE(R(S,T),RS(M),H,GS(M),DU,DW)WNW(S,T)=WNW(S,T)+DqUNW(S,T)=UNW(S,T)+DU

END DO666 COREB=CORE(l)

IF (DBV .GE. 0.0 APND. Z(1,1) .LT. DBV) COREB=CORE(2)IF (COREB .LT. 0.005) COREB=0.005

CC SELF INDUCED VELOCITYC

WSELF(S)=1/(4.*PI)*GS(S)*(LOG(8.0*RS(S)/COREB)-0.25)WIF(S,T)=WIF(S,T)+WNW(S,T)*0.5+WSELF(S)UIF(S,T)=UIF(S,T)+UNW(S,T)*0.5

270 CONTINUE271 CONTINUE

CC COMPUTE VELOCITY AT A ROLLED-UP NEAR WAKE DUE TO OTHER ROLLED-UPC NEAR WAKESC

DO 305 S=1,SMAXWJRU(S)=0.0DO M=1,SMAX

IF (RS(S) .GT. (RS(M)-1.E-4) .AND. RS(S) .LT. (RS(M)-I-.E-4)1) THEN

DM=GS(M)/(2.0*RS(M))ELSE

END IFWRU(S)=WRU(S)+DW

END DOCC COMPUTE NEW WAKE GEOMETRY

30

CZ(S,1)=((WS(S)+WIF(S,1)+.WRt(S)*/(4.*PI))*0.5+WSELF(S)*0.5)*Q1DR(S,1)-(US(S)+UIF(S,1) )*0.5*O1DZ (S,1 ) -Z(S,1)R(S,1)=D)R(S,1)+RS(S)

290 DO 300 T-2,NIVLDZ(S,T)=Q*(WIF(S,T)sWIF(S,T-1) )*0.5Z(S,T)=Z(S,T-1)+DZ(S,T)DR(S,T)=Q1*(UIF(S,T)+UIF(S,T-1) )*O.5R(S,T)=R(S,T-1 )+DR(S,T)

300 CONTINUE305 CONTINUE

IF (DEBUGG .GE. 2) THENWRITE(NUDB,*) 'WRITE(NUDB,*)' VORTEX LINE/RING GEOMETRY'WRITE(NUDB,*)' ITERATION NUMBER' ,ICOUNTWRITE(NUDB,899) (T,T=0O,NIVL)DO S-3.,SMAX

zS=0.0WRITE(NUDB,900) S,RS(S),(R(S,T),T=1INIVL)WRITE(NUDB,901) ZS,(Z(S,T),T=1l,NIVL)WRITE(NUDB,902) GS(S)WRITE(NUDB,*)

END DOEND IF

CC COMPUTE NEW INDUCED VELOCITIES AT ROTORC

WMAX=0.0ERR=O .0

306 DO 330 IR=1,MRAZST=0 .0UOLD( IR)=U( IR)IJOLD( IR)=-W( IR)IF (IT .EQ. 0) UOLD(IR)=0.0

CC VELOCITIES AT RA(IR) DUE TO0 INTERMEDIATE AND FAR WAKESC

CALL VrXIF(RA( IR) ,ZST,LEVEL,WIFR( IR) ,U( IR))C

W( IR)=WIFR( IR)+WB( IR)IF ((W(IR)*W(IR)) .GT. WMAX) WMAX=(W(IR)*W(IR))ERR=ERR+(W(IR)-WOLD(IR) )**2

330 CONTINUECC NO LOOP IF PESCRIBED WAKEC

IF (LEVEL .EQ. 1) THENIT-3GO To 380

END IFCC COUNT ITERATIONS AND CHECK CONVERGENCEC

ICOUNT-ICOUNT+lER-ERR/RA/?4RAEPR-EPIVEL*EPIVEL*WMAX/4 .0IF (ERR .GT. EPR) GO TO0 350IT-3GO TO 380

31

CC WEIGHT NEW VELOCITIES AND DISPLACEMENTS FOR NEXT ITERATIONC

350 DO 360 IR=1,MRAFOLD=1.0-FACTIVW( IR)=FOLD*WOLD( IR)+FACTIV*W( ER)U( IR)=FOLD*UOLD( IR)+FACTIV*U( IR)

360 CONTINUEI T12DO 375 S=1,SMAX

DO 370 T=0,NIVLDZ(S,T)=F'OLD*DZOLD(S,T)+FACTIV*DZ(S,T)DR( S,T)-FOLD*DROLD( S,T)+FACTIV*DR( S,T)

370 CONTINUE375 CONTINUE

CC IT=0O :FIRST ITERATION, IT='2 :PROCEEDING ITERATIONS, IT-3 :CONVERGEDC380 IF (IT .EQ. 3) Go To0 395

IF (ICOUNT .GT. ITERV) GO TO 396GO TO 70

C395 CONTINUE

IF (DEBUGV .GE. 1) THENWRITE(NUDB, *) 'INDUCED VELOCITIES FOR VORTEX LINE/RING MODEL'WRITE(NUDB,*) 'RADIAL STATIONS'WRITE(NUDB,903) (RA(IR),IR=1,MRA)WRITE(NUDB,*) 'BLADE AXES'WRITE(NUDB,*) 'AXIAL VELOCITY'WRITE(NUDB,903) (W(IR),IR=1,MBA)WRITE(NUDB,*) ' RADIAL VELOCITY'WRITE(NUDB,903) (U(IR),IR=1,MRA)WRITE(NUDB,*)

END IFDO JPSI=1,MPSI

DO IR=1,t4RA

VIND(3,IR,JPSI)--U(IR)SNS(PIVIND(1,IR,JPSI)-U( IR)*CSPSI(JPSI)

END DOIF (DEBUGV .GE. 2) THEN

WRITE(NUDB,*) ' SHAFT AXES'PSI'=FLOAT(JPSI )*360.0/FLtn1T(MPSI)WRITE(NUDB,*) 'PSI =',PSICHAR= 'LX'WRITE(NUDB,904) CHAR,(VIND(1,IR,JPSI),IR=l,MRA)CHAR=' LY'WRITE(NTJDB,904) CHAR,(VIND(2,IR,JPSI),IR=l,MRA)CHAR= 'LZ'WRITE(NUDB,904) CHAR,(VIND(3,IR,JPSI),IR=l,MRA)WRITE(NUDB,*)

END IFEND DOIF (DEBUGG .EQ. 1) THEN

WRITE(NUDB,*) 'WRITE( NUDB, *)' VORTEX LINE/RING WAKE GEOMETRY'WRITE(NUDB, *)' NUMBER OF ITERATIONS' ,ICOUNTWRITE(NUDB,899) (T,T=0ONIVL)DO S=l,SMAX

zS=0.0WRITE(NUDB,900) SRS(S),(R(S,T),T--l,NIVL)

32

WRITE(NUDB,901) ZS,(Z(S,T),T=1,NIVL)WRITE(NUDB,902) GS(S)WRITE(NUDB,*)

END DOEND IF

C CALCULATE MEAN INDUCED VELOCITYLAMBDA-O.0AREA=O.ODO IR=1,MRA

DA=RA( IR) *DA( IR)AREA=AREA+DADO JPSI=1,MPSI

LAMBDA=LAMBDA-(viND(3,IR,JPSI)-B5ETAC*VIND(1,IR,JPSI)1 -BETAS*VIND(2,IR,JPSI) )*DA

END DOEND DOLAl4BDA=LAMB)A/( FLOAT(MPSI ) *AE)

396 WRITE(NUDB,*) '****** SOLUTION NOT CONVERGING (INDUCED VELOCIT

GO TO 395

ENDC ---------------------------------------------------------------------

SUBROUTINE VTXIF(RST,ZST,LEVEL,WIE,UIF)CC SUBROUTINE TO CALCULATE THE VELOCITY AT (RST, ZST) DUE TOC THE INTERMEDIATE AND FAR WAKESC

REAL UIF,WIF,UI,WI,RM,N,DW,DUCORE(2),KT,1 COREB,DBV,RST,DSFACT,WA-,UF,RIBB(8)

INTEGER M,SMAX,N,NIVL,OPMODL,Wt:MODL,LEVEL

COMMON ,'W1DATAA41X( (126)

EQUIVALENCE (W1)DX(13),CORE(l) ),(W1)O((40),DBV)

COMMON /RING/ NIBV,RIBB,NIVL,FACTIV,EPIVEL,WFMODL,OPMODL, FGANY4A,1 ITERVCOMMON /XTIP/KT(4)

REAL Q1.W(30),GAMA(30),RS(10),GS(10) ,WS(10),US(10),DZ(10,36),

1 R(10,36) ,DR(10,36) ,z(10,36) ,WIFR( 30) ,U(30)

COMMON /4iELICOM/Q1 ,W,U,GAMA,RS,GS,WS,US,DZ,R,DR,Z,WIE'P,SMAX

UIF=0.0WIF'=0.0DO M=1,SMAX

WI=0 .0UI=0 .0

CC INTERMEDIATE WAKEC

DO N=1,NIVLRMN=R(M,N)H=Z(t',N)-ZSTIF (OPMODL .EQ. 1) THEN

CALL ILINE(RST,RMN,H,GS(M,DU,DW) VORTEX LINEELSE

33

CALL IRIG(RST,RMN,HGS(M) ,D,) ! VORTEX RING

END IFCC EFFECT OF VISCOUS CORE USING APPROXIMATION BY SCULLY

CIF (LEVEL .GE. 1) THEN

COREB=CORE( I)IF (DBV .GT. 0.0 .AND. DBV .GT. Z(1,1)) COREB=CORE(2)DS=( RMN-RST) **2+H*H

FACT=DS/( DS+COREB*COREB)ELSE

FACT=I.0END IFWI=WI+DW*FACTUI=UI+DU*FACT

END DOCC FAR WAKEC

H=Z(M,NIVL)-ZST+DZ(M,NIVL)RMN=R(M, NIVL) +DR(M, NIVL)IF (OPMODL .EQ. 1) THEN

CALL FLINE(RST,RMN,H,GS(M),UF,WF) ! VORTEX LINEELSECALL FRING(RST,RMN,H,GS(M),UF,WF) VORTEX RING

END IFIF (LEVEL .EQ. 1) THEN : PRESCRIBED WAKE

IF (WFMODL .EQ. 1) THEN ! CONCENTRATED FAR WAKE

IF (M .EQ. 1) THEN ! TIP VORTEXGAMM=4.0*GS(M)IF (OPMODL .EQ. 1) THEN

CALL ILINE(RST,RMN,H,GAMM,UF,WF) VORTEX jTNE

ELSECALL IRING(RST,RMIW,H,GAMM,UF,WF) I VORTEX RING

END IFELSE ROOT VORTEX

IF (OnMODL .EQ. 1) THENCALL ILINE(RST,RMN,H,GS(M),UF,WF) ! VORTEX LINEELSECALL IRING(RST,RMN,H,GS(M),UF,WF) I VORTEX RING

END IFEND IF

END IFIF (WFMODL .EQ. 0) THEN NO FAR WAKE

WIF=WIF+WI

UIF-UIF+UIELSE IF (WFMODL .EQ. 1) THEN ! CONCENTRATED FAR WAKEWIF=WIF+(WI+WF)

UI F-UI F+ ( UI+UF)ELSE

WIF=WIF+(WI+WF/DZ(M,NIVL)) [ DISTRIBUTED FAR WAKEUIF-UIF+(tUI+UF/DZ(M,NIVL))

END IFELSEIF (ABS(DZ(M,NIVL)) .LT. 0.0001) DZ(M,NIVL)=0.1WIF=WIF+(WI+WF/DZ(M,NIVL)) ! FREE WAKEUIF=UIF+(UI+UF/DZ(M,NIVL))

END IFEND DORETURNEND

C ---------------------------------------------------------------------

34

SUBROUTINE IVTERP( 5,1)CC SUBROUTINE TO CALCULATE THE INDUCED VELOCITY AT RS (5) BYC INTERPOLATING THE VELOCITIES AT RA(I) (I=1,MRA)C

REAL RA(30),GRAD

INTEGER S,I

COMMON /RTR1CM/RTR)O(1070)

EQUIVALENCE (RTH1)c( 20) ,RA) 1))

REAL Ql,W(30),GA4A(30),RS(10),GS(10),wS(10),US(10) ,DZ(I0,36),1 R(10,36) ,DR(10,36),z(10,36),wIFR(30),U(30)

COMMON /4ELICOM/01,W,U,GANA,RS,GS,WS,US,DZ,R,DR,Z,WIFR,SMAX

GBAD=(RS(S)-RA( I-i) )/(RA(I)-RA( I-1))C VERTICAL VELOCITY

C RADIAL VELOCITYUS(S)-U(I-1).(U(I)-U(I-1) )*GRADRETURNEND

C-----------------------------------------------------------------------SUBROUTINE ILINE(R,P,H,GANMA,UILINE,WILINE)

CC FUNCTION FOR RADIAL AND AXIAL COMPONENT OF VELOCITY INDUCEDC BY VORTEX LINEC

REAL R,P,H,GANMA,UILINE,WILINEDATA PI/3.14159265/IF (ABS(H) .LT. 1.E4 .AND. ABS(P-H) .LT. 1.E-4) THEN

UILINE=0.0WILINErO .0RETURN

END IFC

WIIE-AM/2*I*(-)(PR)*+*)(+)'(+)*+*)UILINE=-GAI1MA/(2.*PI)*(H/)(P-R)**2+H*H)-~H/U(P+R)**2+H*H))

CRETURNEND

C-----------------------------------------------------------------------SUBROUTINE IRING(R,P,H,GANNA,UIRING,WIRING)

CC FUNCTION FOR RADIAL AND AXIAL COMPONENT OF VELOCITYC INDUCED BY VORTEX RINGC

REAL R,P,H,K2,E,K,TEMP2

DATA P1/3.14159265/IF (ABS(H) .LT. 1.E-4 .AND. ABS(P-R) .LT. 1.E-4) THEN

WIRING-0.0UIRING=0 .0RETURN

END IFCALL ELLIPCON(R,P,H,K2,E,K)

35

CTEMP2=K2/( R*P)IF (TEMP2 .LE. 0.0 ) THEN

wRiTE(6,*) ' wA SIGN. MODULE IRINC'STIOP

END IFC AXIAL VELOCITY

WIRING=SQRT(TEMP2)*(K-E*(1.0-.5*K2*(..0+P/R))/(1.0-K2))WIRING=GANMA/( P1*4. ) *WIRIN

C RADIAL VELOCITYUIRING-H/(2.0*R)*SQRT(TEMP2)*(E*(2.0-K2)/(1.0-K2)-2.0*K)UIRING=-GA.1IA/(4.*PI)*UIRING

CRETUJRNEND

C ----------------

SUBROUTINE ELLIPCON(R,P,H,K2,E,K)

C EVALUATE CONSTANTS USED IN INDUCED VELOCITY COMPONENTSC DEFINED BY ELLIPTIC INTEGRALSC

REAL R,P,H,K2,E,K,TEMP2,F

K2-4 .0*R*P/( (R+P) **2+H*H)IF (K2 .GE. (1.0-2.E-8)) GO TO 100

CTEMP2=1 .0-1(2F=LOG(4.0/SQRT(TEMP2))

K=F+.25*(F-1.0)*(1.0-K2)+9./64.*(F-7./6.)*(1.0-K2)**2C

REUNC

100 E--1.0K=10 .0K2123..0-3. OE-8

C

ENDC -----------------------------------------------------------

SUBROUTINE FRING(R,PH,GANMMA,UFRING,WFRING)CC RADIAL AND AXIAL COMPONENT OF VELOCITY INDUCED BY SEMI-C INFINITE VORTEX CYLINDERC

REAL R,P,H,PI,P3,P4,X3,14,K2,E,K,TEmp2

DATA PI/3.14159265/IF(ABS(P) .LT. 1.OE-6) P=1.OE-6

CC AXIAL VELOCITYC

WFRING=0O.0P3=0. 2*PIDO 20 P4=P3*0.5,PI.P3*0.5,P3X3=P*P+R*RIH*H-2 .0*R*P*COS( P4)14=P3*P*(P-R*C.OS(P4) )/( P*P+R*R-20*R*P*COS(P4))

* *(1.041/SQRT(X3))C

WFRING=WFRING+I 420 CONTINUE

36

WFRING=GAMM/( 2. *PI ) *WRNCC RADIAL VELOCITYC

CkLL ELLIPCON(R,P,H,K2,E,K)TEM.P2=P/R/K2IF (TEMP2 .LE. 0.0) THEN

WRITE(6,*) P,R,K2WRITE(6,*) 'BAD SIGN. MODULE FRING'STOP

END IFUFRING=-GAMMA/(2.*PI)*SQRT(TMP2)*(K*(2.0-K2)-~2.0*E)RETIURNEND

C -- ----------------------------------------------------------REAL FUNCTION FLINE(R,P,H,GAMMA,UFLINE,WFLINE)

CC RADIAL AND AXIAL VELOCITY DUE TO VORTEX SHEET

REAL R,P,H,PI,E,GAMMA,UFLINE,WFLINE

DATA PI/3.1415927/C AXIAL VELOCITY

WFLINE=-PI/2.-ATAN(H/(R+P))IF(P .GT. (R+1.OE-8)) WFLINE-PI-ATAN(H/(P-R))-ATAN(H/(P+R))IF(P .LT. (R-1.OE-8)) WFLINE-ATAN(H/(R-P))-ATAN(H/(R+P))WFLINE=GAMMA/( 2. *PI ) *WLINE

C RADIAL VELOCITYUFhLINE=GAMMA/(4.*Pl)*LOG((H*H+(R+P)**2)/(H*H+(P-~R)**2i)

RETURNEND

C --------------------------------------------------------------SUBROUTINE PRESWG( I)

CC DETERMINES PESCRIBED WAKE GEOMETRY FOR VORiiiX RING AND LINE METHODSC

REAL HV,RT,RROOT,KT(4),RIBB(B)INTEGER IR,NIVL,WFMODL

C----------------- CAMRAD COMMON BLOCKS------------------------------COMMON /R1DATA/R,X( 932)COMMON /KTIP/KTEQUIVALENCE (RlXX(81) ,RRO'r)

C ----------------------- END -------------------------------------COMMON /RING/ NIBV,RIBB,NIVL, FACTIV,EUTVEL,WFMODL,OPMODL,FGAMMA,

1 ITERVREAL Ql,W(30),GAMA(30),RS(10),GS(10),WS(10),US(I0),DZ(10,36),1 R(10.36) ,DRI(10,36) ,Z(10,36) ,WIFR(30) ,U( 30)

COMMON /HELICOM/Q1,W,U,GAMA,RS,GS,WS,US,DZ,R,DR,Z,WIFR,SMAX

RS(1)=1.0GS(1)=GAMA(I)RS (2) =RROOTGS (2) =-GS (1)DO IR=1,NIVL+1

HV=KT( 1)*Q1+KT( 2)*(Ql*IR-~Q1)RT=KT(4)+(1l-KT(4))*(1.0/EXP(Q1*IR*KT(3)))R(1,IR)=RTR(2,IR)=RROOTZ(1,IR)=HV

37

Z(2,IR)-HVEND DO

DZ!2,NIVL)-DZ(1 .NIVL)DR( 1,NIVL)=R( 1,NIVL+1)-R(1,NIVL)IF (WFMODL .EQ. 1) THEN'

DR( 1,NIVL)=1.O-R(1,NIVL)DR(2,NIVL)=O.ODZ(1,NIVL)=O.ODZ(2,NIVL)=O.O

END TRETU RN

38

APPENDIX GInput Description

New Variables in Namelist NLWAKE

Variable Default Description

OPMODL 0 Inflow model0 vortex lattice (existing CAMRAD model)I vortex line2 vortex ring

NIVL 4 Number of vortex levels in intermediate wake; maximum 36WFMODL 2 Far wake model in prescribed wake vortex line and ring models

0 no far wake1 concentrated2 distributed sheet

FACTIV 0.1 Factor introducing lag in induced velocity iterationEPIVEL 0.05 Tolerance for induced velocityFGAMMA 0.6 Roll-up weighting factor

ITERV 200 Maximum number of induced velocity iterationsNIBV 2 Number of rolled-up vortices; minimum 2, maximum 10RIBB(NIBV-2) Inboard edge of rolled-up vortices from root to tip, excluding

root and tip; must be between root and 0.9 (default: evenlydistributed)

Comments on Important Existing CAMRAD Variables

Namelist Variable(s) CommentNLTRIM MPSI, MPSIR Because solutions are independent of azimuth, can be

as low as number of bladesNLTRIM DEBUG(14),(24) Additional debug information for new models

Prescribed wake Free wakeNLTRIM LEVEL(I) 1 2NLTRIM ITERU I 0 or INLTRIM ITERR 1 0 or 1NLTRIM ITERF N/A 1NLWAKE OPRWG Defines prescribed wake geometry, as with existing

vortex lattice model in CAMRAD

NLWAKE CORE(I),(2), DBV As with existing vortex lattice model in CAMRAD

NLWAKE FWGT(l),(2) For Kocurek & Tangier (8 < OPRWG I 11) andLandgrebe (8 < OPRWG : 11) models, factors forvortex settling rates KT(1) and KT(2); equal I forunchanged experimental rates

39

APPENDIX HTest Case Command File

$ASSIGN (USERNAME.CAMRAD.AIRFOIL]NACA0012 .TAB AFTABLE).$ASSIGN [USERNAZE.CAMRAD.INPUTJH34 .DAT INPUTFILE$DEFINE/USER MODE SYS$OUTPUT [USERN.CAMRAD]H34.OUTr$RUN [USERNARE. CAMRAD ICAI4RAD

&NLCASE NCASES=1 ,BLKDAT=0 ,NFRS=-1 ,NFEIG-1 ,NFAF2=41 ,&END&NLTRIM MPSI=4,MPSIR=4,DEBUG(24)=,DEBJG(14)=1,VKTS=0.0,FACTOR.=O.5,ITERC=40,ITERM=40,FACTM-0.5,MTRIM=80,LE'vEL(1)=1,ITERR=1,ITERF=0,DELTA=0.5,ITERU~l,OPDENS=3 ,TEMP=56 .0 ,DENSE=0 .00232,OPTRIM=11,EPTRIM=0.001,CTRIM=0.817,MHARM-0,NROTOR=1,COLL-9 .1,DOF=54*0,DOFT=8*0,NIPRNTP=0 ,NPRNTL-0 ,NPRNTT=1 ,NPRNTI=1,OPREAD=1,1, 2*0, 1,1,&END&NLRTR VTIPN=621.6,BTIP=1l.00,KHLMDA=1.15,KFLKDA=1.O,FXLMDA=1.0,FMLNDA=0. ,EACIWU=0.5,OPCOKP=I,&END&NLWAKE KFW=120 ,OPHW=0 ,OPRWG=8, FAcTN=0 .1,WKMODL(2)=0,WKMODL(4)=,WKMODL(6)=0,WK4ODL(8)=0,WKMODL( 12)'=0,CORE(1)=0.05,CORE(2)=0.05,OBV=-1.,FWGT( 1) =1.0, FWGT(2 )=1. 0,EPIVEL-0 .02 ,WFMO0DL-2 ,OPMODL=1 ,NIBV=2 ,FGAMrMA=0 .7,FACTIV=0.1,ITERV=200,&END&NLBODY CONFIG=O,NSHAFT(l1)=0,&END&NLLOAD MALOAD=1 , NWKG4P=-4*0 ,MWKGMP=0,NPwI'-75*0,&END

40

APPENDIX J

Test Case Output File

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AL 149 DEPARTMENT OF DEFENCE PAGE CLASSIFI NTIONUNCLASSIFIED

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4. TITLE 5. SECURITY CLASSIFICATION 6. NO. PAGESIncorporation of Vortex Line and (PLACE APPROPIATE CSSxIFICATION 62Vortex Ring Hover Wake Models into IN Bam(s) IE. SEE IS), C.(Ca Comprehensive Rotorcraft Analysis R TR (R), UNCLASSIFIE (U) .

Code

DICUMENT TITLE ABSTRACT

8. AuIfKO(S) 9. DaINGRADING/DELIITING INSTRUCTIONS

R. Toffoletto, N.E. Gilbert, Not applicable.S. Hill, K.R. Reddy

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14. DESC(RIPIU 15. DRDA SUBJECTCATEGORIE

Helicopter wakes CAMRAD <mathematical model) 0051AHoveringVorticesRotor Aerodynamics

16. ABSRCT The incorporation of simplified hover wake models into thecomprehensive rotorcraft analysis code CAMRAD is described and examplesare given on their use. The axisymmetric models, in which vortices arerepresented by either straight lines or rings, are a more generalized form ofthe free wake models of R.T. Miller at MIT, with the wake geometry also ableto be prescribed. Incorporation has allowed access to the tabularrepresentation in C AM R AD of airfoil section characteristics as functions ofangle of attack and Mach number, and has broadened the range of rotor wakemodels in the code to include a free wake hover model that does not have theconvergence problems of the existing free wake model when used for hover.

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Flight Mechanics 51 1130Techrical Memorandum 408

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