NAVAL POSTGRADUATE SCHOOL · naval postgraduate school monterey, california

NAVAL POSTGRADUATE SCHOOLMonterey, California

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ftifttADEC 2 8 1988

ThE14SIESA COMIPUTER CODE (LSPOTF2)

FOR UN STEADY INCON IrRESSlIeBLE FLOWPAST TW~O AIRF-OILS

Chung-Khiang Pang

September 19S8

Thesis Advisor M.F. Platzer

Approved for public release; distribution is unlimited.

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I I Title (include securlv clasifleaton) A COMPLTER CODE (USPOTF2) FOR UNSTEADY INCOMPRESSIBLE FLOWPAST TWO AIRFOILS12 Personal Author(s) Chung-Khiang Pang13a Type of Report 13b Time Covered 14 Date of Report (year, month, day) 15 Page CountMaster's Thesis From To S tember 1988 169

16 Suppiementary Notation The views expressed in this thesis are those of the author and do not reflect the official policy or po-sition of the Department of Defense or the U.S. Government.17 Cosati Codcs 18 Subject Terms (continue on reverse ifnecessary and idenry" by block numberlField Group Subgrou,' Computational Fluid Dynamicsf'Panel Met)?ods,' Unsteady Aerodynamics,'Potential

WN Abstracts. continue on reverse if necessary and identfv by block number)S ,' A numenrcal code, USPOTF2, has been formulated to solve for the potential flow for two airfoils executing unsteadymotions in an inviscid incompressible flow medium. This code is an extension of an existing code U2DIIF, which does thesame calculations for the single airil case. The technique uses the weft known Panel Methods for steady flow and extendsit to unsteady flow by introducing wake model which creates a non-linear problem due to the continuous shedding ofvortices into the trailingz wake. The pesence of the second airfoil introduces a set of non-linear coupled equations for theKutta condition. Numerous case-runs be presented to illustrate the capability of the code. The case of the step change inangle of attack is compared yith Giesin 's work. All other case-runs are illustrated together with the results for the singleairfoil case.

20 Distribulion Availability of Abstract 21 Abstract Security ClassifiationM unclassified unlimited 0 sum as report 0 DTIC users Unclassified22a Name of Responsible Individual 22b Telephone (includa Area code) 22c Ofce SymbolM.F. Platzer (408) 646-2311 .67P1

DD FORM 1473,A4 MAR 83 APR edition may be used until exhausted security classification of this pageAll other editions are obsolete

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Approved for public release; distribution is unlimited.

A Computer Code (USPOTF2)for Unsteady Incompressible Flow

past Two Airfoils

by

Chung-Khiang PangCivilian, Singapore Ministry of Defence

B.Eng.(Hons), University of Adelaide, South Australia, 1981

Submitted in partial fulfillment of therequirements for the degrees of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERINGand

AERONAUTICAL ENGINEER

from the

NAVAL POSTGRADUATE SCHOOLSeptember 1988

Author:

Chung-Khiang Pang

Approved by: _____________________________

M.F. Platzer, Thesis Advisor

R.M. Howard, Second Reader

E.R. Wood, Chairman,Department of Aeronautics and Astronautics

Gordon E. Schacher,Dean of Science and Enginecring

iiJ

ABSTRACT

A numerical code, USPOTF2, has been formulated to solve for the potential flow

for two airfoils executing unsteady motions in an inviscid incompressible flow medium.This code is an extension of an existing code U2DIIF, which does the same calculations

for the single airfoil case. The technique uses the well known Panel Methods for steadyflow and extends it to unsteady flow by introducing a wake model which creates a non-

linear problem due to the continuous shedding of vortices into the trailing wake. The

presence of the second airfoil introduces a set of non-linear coupled equations for the

Kutta condition. Numerous case-runs are presented to illustrate the capability of thecode. The case of the step change in angle of attack is compared with Giesing's work.

All other case-runs are illustrated together with the results for the single airfoil case.

C-i

Accession ForrNTIS -GRkl _

OTIC TAB 0Utianou ed EJ

D i st r Co~ to~

DI tt. Lt i ''./

Availabl,iAva i F,

Dtst Speci.i

iii .....L LL i

"7"j

THESIS DISCLAIMER

The reader is cautioned that computer programs developed in this research may nothave been exercised for all cases of interest. While every effort has been made, withinthe time available, to ensure that the programs are free of computational and logic er-rors, they cannot be considered validated. Any application of these programs withoutadditional verification is at the risk of the user.

iv

TABLE OF CONTENTS

I. TRODUCTION ...........................................

A . G EN ERA L ................................................ IB. BASIC THEORY AND APPROACH ............................ 1

1. Steady Flow Problem ...................................... I2. Unsteady Flow Problem .................................... 2

C . SCO PE ................................................... 3

II. STEADY FLOW PROBLEM FORMULATION FOR TWO AIRFOILS .... 5

A . G EN ERA L ................................................ 5B. FRAMES OF REFERENCE ................................... 5

C. STEADY FLOW PANEL METHODS ............................ 5

1. Definition of nodes and panels ............................... 5

2. Distribution of Singularities ................................. 7

3. Boundary Condition ...................................... 10D. INFLUENCE COEFFICIENT ................................. 10

1. Concept of Influence Coefficient ............................. 10

2. Notation for Influence Coefficient ............................ II3. Computation of Influence Coefficient ......................... II

E. NUMERICAL SOLUTION SCHEME ........................... 14

1. Rewriting the Boundary Condition ........................... 14

2. Solving for the Strengths of Source and Vorticit" Distribution ....... 153. Computation of Velocity and Pressure Distribution .............. 154. Computation of Forces and Moments ........................ 16

Ill. UNSTEADY FLOW PROBLEM FORMULATION .................. 18

A . G EN ERA L ............................................... 18B. FRAMES OF REFERENCE .................................. 18C. UNSTEADY FLOW MODEL ................................. 19

I. Rigid Body M otion ........................................ 19

2. W ake M odel ............................................ 213. Additional Boundary Conditions ............................. 21

D. INFLUENCE COEFFICIENTS ................................ 23

1. More A's and B's Influence Coefficients ....................... 23

2. New C's Influence Coefficient ............................... 24

3. Computation of the Time-Dependent Influence Coefficient ......... 25

E. NUMERICAL SOLUTION SCHEME............................ 26

1. The Flow Tangency Condition .............................. 26

2. The Kutta Condition ..................................... 28

3. Convection of the W ake ................................... 32

4. Disturbance Potential and Pressure Distribution ................. 32

5. Computation of Forces and Moments ......................... 36

IV. DESCRIPTION OF COMPUTER CODE USPOTF2 .................. 37

A. PROGRAM STRUCTURES AND CAPABILITIES ................ 37

1. Restrictions and Limitations ................................ 37

2. Current Structures of USPOTF2 Main Program ................. 37

B. DESCRIPTION OF SUBROUTINES ............................ 43

1. Subroutine BODY ....................................... 43

2. Subroutine COEF ........................................ 43

3. Subroutine COFISH (deleted for the two airfoil case) ............. 43

4. Subroutine CORVOR ..................................... 43

5. Subroutine FANDM ..................................... 43

6. Subroutine GAUSS ...................................... 44

7. Subroutine INDATA ..................................... 44

8. Subroutine IN FL ........................................ 44

9. Subroutine KUTTA ..................................... 45

10. Subroutine NACA45 ..................................... 45

11. Subroutine NEWPOS...................................45

12. Subroutine PRESS ..................................... 46

13. Subroutine SETUP ...................................... 46

14. Subroutine TEW AK ..................................... 46

15. Subroutine VELDIS ..................................... 46

C. INPUT DATA FOR PROGRAM USPOTF2....................... 47

D. OUTPUT DATA FROM PROGRAM USPOTF2 ................... 47D. UTPT DTA ROMPRORAMUSOTF .............. i

V. RESULTS AND DISCUSSION OF CASE-RUNS .................... 53A. STEP CHANGE IN ANGLE OF ATTACK ....................... 53

1. Case Run Definition ...................................... 532. Differences between USPOTF2 and Giesing's code ............... 533. Results and Discussions ................................... 54

B. OTHER SUB-CASES ........................................ 701. Modified Ramp Change in Angle of Attack ..................... 712. Translational Harmonic Motion ............................. 713. Rotational Harmonic Motion ............................... 71

VI. CONCLUSION .............................................. 90A. GENERAL COMMENTS .................................... 90B. ENHANCING USPOTF2 PROGRAM CAPABILITY ............... 90

APPENDIX A. USPOTF2 SOURCE LISTINGS ....................... 92

APPENDIX B. EXAMPLE INPUT DATA FOR PROGRAM USPOTF2 .... 135

APPENDIX C. EXAMPLE OUTPUT DATA FROM PROGRAM USPOTF2 136

LIST OF REFERENCES .......................................... 154

INITIAL DISTRIBUTION LIST ................................... 155

vu

>"w- ..-"'' °

- . .: . ,'- .,.. . . ., . ,- - -- ------ --. ..-.-- --- -.- --. .-.--7 . :'-'';;, - 7'.i, . " : .:

LIST OF FIGURES

Figure 1. Frames of Reference for Steady Flow .......................... 6Figure 2. Panel Methods Representation for Steady Flow. .................. 8Figure 3. Influence Coefficient for Source Panel ......................... 12Figure 4. Frames of Reference for Airfoil I ............................ 20

Figure 5. Panel Methods Representation for Unsteady Flow ................ 22Figure 6. Flow Chart for USPOTF2 Computer Code ..................... 40Figure 7. List of Input Variables ................................... 48

Figure 8. List of Output Variables .................................... 51Figure 9. Giesing's Calculated Results ................................ 56Figure 10. USPOTF2 Results obtained with USPOTF2 for equal velocities at the

trailing edge . ............................................ 58Figure 11. I'SPOTF2 Results obtained with USPOTF2 for equal pressures at the

trailing edge . ............................................ 61Figure 12. Pressure Distributions with different Kutta condition .............. 64Figure 13. Pressure Distribution and Lift History as a function of the Vertical dis-

tance between the airfoils ................................. 65

Figure 14. Step Change in Angle Of Attack ............................. 67Figure 15. Modified Ramp Change in Angle Of Attack .................... 73Figure 16. 14 ,nonic Pluncing M otion ................................. 79Figure 17. Harmonic Pitching M otion ................................. 84

viii

.4

TABLE OF SYMBOLSA singularity-type indicator for uniformly distributed source

B singularity-type indicator for uniformly distributed vorticity

C singularity-type indicator for concentrated point vortex

C chord length

C, 2-dimensional drag coefficient

C, 2-dimensional lift coefficient

C. steady state value of C,

C. 2-dimensional pitching moment coefficient about leading edge

C.. steady state value of C,

C, pressure coefficient

C. x-force coefficient of local system (in the code)

C,. y-force coefficient of local system (in the code)

h. chordwise translational position (positive forward)

h, transverse translational position (positive downward)

NI Mach number

m number of core vortices

n total number of panels

i, j unit vectors along the x and y directions of respective frame of reference

n, t unit vectors normal and tangential to panel

P static pressure

P. freestream static pressure

q unit strength of uniformiy distributed panel source

SS perimeter length of airfoil

t time step

V. freestream velocity vector

V total velocity component normal to panel

V, total velocity component tangential to panel

U, V absolute velocity components in the x and y directions

x-y coordinate system fixed on the airfoil

ix

.,,77.:-'_": .- '..: :." . . .+' :, ' ---------------------- ,------,-.---.-,-....--.-----.-----,':€" -=--i.ihi-.- ,m~llimiimm iS il Hiii.....

X-Y coordinate system fixed with respect to the free stream

XM.V, mid point of panel

a, AOA angle of attack (positive clockwise from V..)

2, initial angle of attack

r dimensionless circulation strength

unit strength of uniformly distributed panel vorticity

A length of the shed vorticity panel

62 change in AOA from a, or amplitude of pitch oscillation

6h ., Jh, amplitude of chordwise and transverse oscillations

0 orientation angle of the shed vorticity panel with x-direction

0 inclination angle of the panel with the x-direction

A phase difference of the chordwise from the transverse oscillation

p incompressible density

r dimensionless rise time for ramp change in AOA

0 total velocity potential

velocity potential due to free-stream

disturbance velocity potential

velocity potential due to source distribution

velocity potential due to vortex distribution

velocity potential due to core vortices in the wake

11 pitch angular velocity of airfoil (positive counterclockwise)

0) harmonic oscillation frequency

K iteration counter

Lo

!.X

ACKNOWLEDGEMENTS

The research for this thesis was conducted at the Navil Postgraduate School Con-

puter Center facilities. Work was performed under the guidance of Professor Platzer,

thesis advisor. I would like to express my sincere appreciation to him for providing me

with the oppo,'tunity to work in the area of computational fluid dynamics and for his

patience and encouagement, both in the course of the research as well as the final

preparation of the manuscript. I would also like to thank Proiessor Howard, second

reader, who paticiitly reviewed the manusciipt and gave encouragement at a critical

stage.

M" sincere appreciation also goes to Mr. Andreas Krainer, whose total familiarity

anFd understanding of the original code made a critical difference in the coding of

L'SPOTF2: especially encouraging was the number of occasions, too numerous to recall,

where he would give expert advice as to the possible coding areas which might be in er-

ror during troubleshooting days and finally his constructive and thorough review in go-

ing through every single formula in the manuscript.

Finally. I wish. to thank my very loving wife and co-typist who bore the

tremendous extra. demands of the final months with patience, understanding and en-

couragement and who toiled painstakingly with successive drafts, cheerfully acceptingmy determination to alter every section after, but never before, it had been typed.

xi

1. INTRODUCTION

A. GENERAL

In this paper, a numerical method is formulated to solve for the flow about twotwo-dimensional airfoils which are arbitrary located and are performing an arbitrary time

dependent motion in an inviscid incompressible fluid. The original work by Teng [Ref.1] is for a single arbitrarily defined airfoil in the same potential flow condition. The ex-tension of Teng's code to two bodies is considered here. Where possible, the same con-vention and notation are adopted. As for the single airfoil case, all velocities arenon-dimensionalised with respect to the free stream and all lengths with respect to the

chord length.

A new subroutine (SUBROUTINE NEWPOS) is added to transform either of thetwo local coordinate systems to the global coordinate system. A modification is enteredinto the treatment of the Kutta condition to make it consistent with the unsteady flowKutta condition treatment. Subroutine COFISH is deleted and its role is taken over by

Subroutine COEF which normally performs the formulation for the flow tangency con-dition for the unsteady case. A more accurate method is also introduced to obtain thevelocity potential. The reader is referred to the work of Krainer [Ref. 2] for furtherimprovement of the original code.

This documentation is set up as for the original documentation in order that it willbe easier to follow and cross-referenced. While it is the intent of the author to keep thisthesis as complete as possible, the reader is well advised to review Reference I for the

work involved in the single airfoil case; no special effort will be made to reproduce ithere.

B. BASIC THEORY AND APPROACH

1. Steady Flow ProblemThe treatment for the two airfoils case in steady incompressible flow follows

closely to the single airfoil case. The governing equation, as for the single airfoil case,follows from the Conservation of Mass and the Condition of Irrotational Flow.

The continuity equation of an incompressible fluid (div V - 0) and the condi-

tion of irrotational flow (curl " - 0) leads to the well known Laplace Equation.

div( gradrk) = 0 (1.1)

t ... ........ ..... ........ .......... ,:' .. -- * -': . o'-*. :. --"'i .... .-

......

with 4) denoting the disturbance potential for the velocity. This is seen to be the classic

Neumann problem of potential theory with the usual problem of defining the boundaryconditions. The boundary conditions for the disturbance potential 4) are that its gradi-

ent normal to the surface be equal to the normal velocity of the surface and that its

gradient vanish at infinity; that is

- on S (1.2)

lim VO(P) -=0 (1.3)P-.oo

where F', is the resultant velocity of a point on the body as seen from the inertia frame

of reference, n- is the outward unit vector normal to the body, S is the body surface and

P represents a general point. Equation 1.2 holds for both airfoils i.e. on both surfaces.

The pressure coefficient is obtained through the Bernoulli's equation which de-

rives from the Momentum equation.

The approach adopted is associated with Hess and Smith [Ref 31 who devised

the popularly known PANEL method in the early sixties. In words, the boundary or

airfoil surfaces S, and S2 about which the flow is to be computed is approximated by a

large number of surface elements whose characteristic dimensions are small comparedto those of the body. Over each surface element, a uniform source distribution and a

uniform vorticity distribution is placed. The source strength (q,) varies from element to

element, while the vortex strength (y,) is the same for all elements in the same airfoil but

is different across the airfoil. The singularity strengths are determined from the flow

tangency condition on both body surfaces and the two Kutta conditions at both trailing

edges. With the determination of the singularity strengths, the relevant aerodynamic

data can then be subsequently computed.

2. Unsteady Flow Problem

The unsteady problem is similar to the steady flow problem in that they both

have the same governing equation viz. the Laplace equation, and that for both problems,

the pressure and velocity are decoupled so that the velocity and pressure calculation can

be computed separately and consecutively.

This problem differs from the steady flow in that another model is required to

simulate the continuous shedding of vorticity into the trailing wake. The existence of a

vortex sheet behind the airfoil can be explained by the Helmholtz theorem which is

basically a statement of the Conservation of Vorticity. This requires that any, change in

2

the circulation around the airfoil must be matched by an equal and opposite vortex

somewhere in the flowfield. The presence of the countervortices provides the flow with

a kind of a memor, in that the flow at a particular time is affected by the bound circu-

lation of the past. It is this non-linearity that distinguishes the numerical technique from

the simple steady flow problem of solving N linear equations in N unknowns.

The solution technique requires an iterative type solution. The present ap-

proach follows closely the original panel method of Hess and Smith as described in the

steady flow development, while with regard to the modelling of the wake, it adopts the

procedure advocated by Basu and Hancock [Ref. 41. A uniform source distribution (q,)k

and a uniform vorticity distribution [y(])J as for steady flow is placed on each panel at

time t, where j denotes the panel number and I the airfoil number. The wake consists

of a single vorticity panel attached as an additional element on each airfoil through

which the vorticities are shed into the respective wake and a series of point vortices

which are being convected downstream with the fluid. A uniform vorticity distribution

of strength (y,()), is placed on the wake panel of each airfoil. This panel is further

characterised by its length A(O, and its inclination ( ®1, ) with respect to the respective

local frame of reference. After each time step, the vorticity of the wake panel is con-

centrated into a single point vortex and convected downstream. Simultaneously a new

wake panel is formed. The downstream wake of point vortices is thus formed by the

shed vorticity of previous time steps.

C. SCOPE

Chapter II extends the original code to handle the steady flow problem for two

airfoils set at different relative distances and angles of attack.

Chapter III deals with the unsteady problem for the two airfoils system. It intro-

duces a new subroutine and a more accurate method of calculating the velocity poten-

tial. The Kutta condition for the two airfoils system is specially treated as its unique

problem of a non-linear coupled system is not seen in the single airfoil case.

Chapter IV describes the computer program, its essential capabilities and limita-

tions, its associated subroutines, its input requirements and its associated output print-

out.

Chapter V presents the results of some case-runs. Of interest is a comparison case

of a step change in angle of attack (AOA) with Giesing [Ref. 51 for the same airfoil

undergoing an impulsive start at the same AOA. Case-runs will also be run with both

airfoils at large distances apart to compare with the single airfoil case. In addition, ex-

3

_.-," .,- ---- - nl -- - - - - --II

ample cases for which no comparisons exist are given, to exhibit the capability of themethod.

Finally, Chapter VI concludes with future development efforts and the applicationpotential of this numerical method.

r"

N 11. STEADY FLOW PROBLEM FORMULATION FOR TWO AIRFOILS

A. GENERALThe modification work for the steady flow is straightforward. The revised program

allows for two arbitrarily defined airfoils placed at an arbitrary distance set at different

angles of attack. For reason of simplicity, the number of panels and nodes and the pivotlocation are set to be the same for both airfoils.

B. FRAMES OF REFERENCEThree frames of reference are involved in the two two-dimensional airfoils' case in

steady flow. These are three inertia frames of references as indicated in Figure 1.The first inertia frame of reference (also known as global frame of reference) is set

at the pivot position of the first airfoil with the X-axis pointing in the direction of the

free-stream velocity. The two other inertia frames of references, henceforth, will beknown as two frozen local frames of referencel (xyy1 and x2Y2) are fixed respectively to

each airfoil with the x-axis coinciding with the chord line originating from the respectiveleading edge. The two local frames of reference are set apart by XShift and YShift onthe global frame of reference. In steady flow, the fluid velocity and pressure depend onlyon the spatial coordinates (X,Y) and not on time.

The two airfoils are defined in the local coordinate system as input data for sim-plicity and are then transformed to the global coordinate system through a knowledge

of the relative positions of the 2 airfoils' pivots positions and the respective local angles

of attack.

C. STEADY FLOW PANEL METHODS

1. Definition of nodes and panelsEach airfoil surface is divided into n straight line segments called panels by

(n+ 1) arbitrarily chosen points called nodes. The numbering sequence begins withpanel I on the lower surface at the first airfoil trailing edge and proceeds clockwisearound the airfoil contour so that the last panel on airfoil I ends on the upper surfaceat the trailing edge. This numbering sequence then proceeds in a similar fashion for the

second airfoil ( See Figure 2). As with the single airfoil case, the numbering sequence

I For the steady case, the notation 'frozen' will be dropped

.17

P5

YY

K2

XSHIrr

Figure, 1. Frames of Reference for Steady Flow.

.7 7 M

dictates that the airfoil body always lies on the right hand side of the i* panel as one

proceeds from the iP node to the (i + 1)" Also the 1" and (n + 1)1* nodes coincide at the

first airfoil trailing edge and the (n + 2)* and (2n + 2),* coincide at the second airfoiltrailing edge. This numbering system facilitates the definitions of the unit normal vector

n, and the unit tangential vector t, for all panels with n, being directed outward from the

body into the flow and t, directed from the ,* node to the (i + 1)i node. The numbering

of the panel system is somewhat complicated by the fact that a continuous panel num-

bering sequence across the airfoil is desired. This procedure leads to the peculiar panelnumbering behaviour seen in Figure 2 where for the first airfoil, the ill panel lies betweenP and (i + l)" nodes while for the second airfoil, the il panel lies between

(i + l)" and (i + 2)* nodes2.

2. Distribution of SingularitiesOver each surface element of the two airfoils, a uniform source distribition and

a uniform vorticity distribution is placed. The source strength q varies from element to

element for each airfoil while the vorticity strength y, remains the same for all elements

in the same airfoil but is different across the airfoil. This choice of singularities follows

closely the original panel method of Hess and Smith. It automatically satisfies the

Laplace Equation (which is the governing equation for the inviscid incompressible flow)

and the boundary condition at the far field (oo). In addition, as the Laplace Equation

is a linear homogeneous second order partial differential equation, an overall compli-

cated flow field can be built up by the combination of simple flows with the condition

that the appropriate boundary condition on the airfoil be satisfied accurately.

For our case, the overall flow field (represented by the velocity potential 0) can

be built up by three simple flows. Writing this in terms of the respective local frame of

reference,

.(x ,y) - 40M(x ,y) + ,(x ,y) + 0,(x ,y) (2.1)

where 0..(x ,y) is the potential of the onset flow,

06.(x ,y) - V.xCos a(4 + ySin *()1 (2.2)

2 For the code e, has been defined out of the above convvtion with e, i 1,2 ..n as per pandnumbers for the first airfoil, but with Og.., reserved for the wake element of the first airfoil and 0,,i-n+ 2,n+ 3...2n+ I for the second airfoil with 03. reserved for the wake element of the secondairfoil for unsteady flow.

7

-3L .,." - * ..I1

I

q 2n+2

n+2

J+l j 4 1

Panel j

Source distribution =qj

Vorticity distribution = j

Note: 1.Nodal points defined by i - 1,2 ... n+l for first airfoil

and i - n+2 ... 2n+2 for second airfoil.

2. Panel number defined by j - 1,2 ... n for first airfoil

and j - n+l ... 2n for second airfoil.figure . ,Panel Methods Representation for Steady Flow.

=Z8

4), is the velocity potential of a source of distribution q(s) per unit length.

(q(s)js f .xs In r ds (2.3)

4. is the velocity potential of a vorticity distribution y (s) per unit length

O, j 27 - ds (2.4)

At this point, the disturbance potential, 4), is introduced, which is defined to be the sum

of the potential due to the source and vorticity distribution.

(2.5)

Equation (2.1) can then be read as

(2.6)

The convenience of defining the above allows for the total velocity vector to be viewed

as two components viz. the onset and the induced velocity due to the disturbance po-

tential. The total velocity is thus

-tow = V4= VO. + V4 (2.7)

It is in the introduction of the disturbance potential that leads us to the concept of in-

fluence coefficient which will be elaborated in a later section.

The pressure coefficient can be obtained from Bernouli's Equation which is

derived from the Conservation of Momentum.

{-0 V,o0 o01 'S- P ,.2 - -(2.8)CP 1 2 9

9

..... ~~~~~~~~~~~~~~~~ _7;':, ' ; i : i , i . . | I i " . . . .. ..

3. Boundary ConditionAs in the single airfoil case, the boundary conditions to be satisfied include the

flow tangency conditions and the Kutta condition. The flow tangency conditions aresatisfied at the exterior mid-points (control points). The normal velocity is taken withrespect to the respective local frame of reference (for consistency with unsteady flownotation to be introduced later):

(Vn), - 0, i- 1,2 ... n,n+ l ... 2n (2.9)

where (V), is the normal component of the total velocity 3.

The Kutta condition postulates that the pressure on the upper and lower sur-face at the trailing edge of each airfoil be equal. For steady potential flow, equal pres-sure implies equal tangential velocity in the downstream direction at the first and lastpanel of each airfoil viz the Bernoulli equation. With our definition of the tangentialvector we then have

(V')1 - - (VI)n I st airfoil

- (V) 2, 2nd airfoil (2.10)

As with the single airfoil, equations 2.9, and 2.10 lead to a linear system of(2n+ 2) simultaneous equations. With V, and V, expressed explicitly in terms of qj (j 1, 2, ... n, n+ I... 2n) and V, (l= 1, 2) we have 2n +2 unknowns in (2n+ 2) system of linearsimultaneous equations which can be easily solved.

D. INFLUENCE COEFFICIENT1. Concept of Influence Coefficient

As introduced earlier, this important concept of influence coefficient resultsfrom the presence of the disturbance potential which follows from the presence of thesingularities. Formally, an influence coefficient is defined as the velocity induced at afield point by a unit strength singularity (be it a point singularity or a distributedsingularity) placed any-where within the flow field. Recall that equations 2.9 and 2.10require the computation of the normal and tangential velocity components at all the el-

ements control points. The normal components of velocities are essential in satisfying

3 This folows Giesing's notation where the velocity with respect to the airfoil frame is denotedas total velocity. While there is no misunderstanding in the steady flow, the notation actually refersto the relative velocity with respect to the airfoil moving frame of reference for the unsteady flow.

10

--------------------~ ~ --7

the flow tangency conditions, while the tangential components of velocities are necessary

for satisfying the Kutta condition as well as computing the pressure distribution.

2. Notation for Influence Coefficient

The notation used in this documentation will be as for the notation for thesingle airfoil case with a slight modification to take into account the effects of the thesecond airfoil. As the influence coefficients are related to the geometry of the airfoil andtheir relative positions, it must, of necessity be computed with respect to the globalframe of reference for the two airfoils' case.

For steady flow, the following influence coefficients are defined.

" A,; : normal velocity component induced at the iO control point by unit strengthsource distribution on the jp panel.

* A, : tangential velocity component induced at the ill control point by unit strengthsource distribution on the ft panel.

* B,: normal velocity component induced at the i* control point by unit strengthvorticity distribution on thejA panel.

" S, :tangential velocity component induced at the P* control point by unit strengthvorticitv distribution on the JA panel.

where i and j denotes panel numbers and are defined as:i = 1 , 2 , . .n , n + I ... 2 n '

.=1, 2,..n, n+ I,...2n

3. Computation of Influence Coefficient

A single source located on thepj panel in the local frame of reference (see Figure3) induces a total velocity of

I' =- - - (2.11)

where the component perpendicular to the inducing panel is

I h 1 2 h2 (2.12)

and the component parallel to the inducing panel is

VS - - . S (2.13)+nr 2a h2 +$ 2

For a distributed source panel on the j4 panel in the local frame of reference, we have

11

*,i , ,n,'- 'e~.....................................ln

flj

r1 r r

sssi-2 jth panel S2 2

Figure 3. Influence Coefficient for Source Panel

12

f;2 V,(s) ds ~FTA N (214)

where

FTAN tan-' (xm1 - XAflm1 - yj+j) - (YM1 - vjxm, - xj+1)(I,- xj)(xm - xx,) + (myin1 -y) ym ,

xni. -112 (x,+x,+,)

yn 1 l/2 41y+ 1,) (2.15)

V' fS2~ --S)dFLOG (2.16)

where

FLOG = (In ~J+1)2 + (Mi _y,)12 (2.17)(xMIi, - xj)2 + (yin1 j)) 2

Transforming the above to global frame of reference, we have

A" = Cos(O, - Oj) - i'-Sin(6, - 6j) (2.18)

.1- I -Sin(O1 - Oj) + VV'Cos(6, - 0j) (2.19)

BU = (;'nCos(0, - Oj) - ;'Sin(O, - 0)) (2.20)

B,'=-V'Sin(0, - Oj) - V~Cos(01 - Oj) (2.21)

For the Steady Flow code, we define

A AN(JJ A BU (2.22)

B.N(lJ - A V1 BU (2.23)

13

E. NUMERICAL SOLUTION SCHEME

I. Reuriting the Boundary Condition

Though it is not critical for the steady flow problem to adopt any particular

frame of reference, we will adopt the local frame of reference in satisfying the boundary

condition for consistency with the treatment used in the unsteady case.

Even though the influence coefficients are defined in terms of the global frame

of reference, the computation of the normal and tangential velocities of each panel is

independent of the coordinate system used. Thus the normal and tangential velocities

obtained with the global coordinates system would be the same as those obtained for the

local coordinate system. Using the local coordinate system, the flow tangency condition

of equation 2.9 is defined as follows :

n 2n n2n

(V") , - .Anq + 2] A % + V(1) ++ V. Sina(O -, = 0 (2.24)

--t -

where

Si- 1,2 ... n 1 1,

n+l...2n; =2

The Kutta condition of eqn 2.10, in terms of influence coefficients, becomes for airfoil

1:

n 2n n 2n

~A'Vqj+ A'+yl '+y2 V V..Cosrx(l) - 1]=

j=1 Jln+ I j-1 -n+

2n n 2n

A + + y(2) B' + V Cosa(1) - 0(2.25)j-R+ Jm-1+

and for airfoil 2:

it 2n Rn

,Ags+ At +J + y(2) t+lJ + V., Cos(42) -O,,+,] -

J-I in+i J-I J-R+I

14

'lai

2n ( 2n (2.26)

- {ZA + A2 + y(1)Z .j + y(2) Z B2.j + v. Cos[c(2) - 02.--I fn+l J-1 mJn+l

2. Solving for the Strengths of Source and Vorticity Distribution

Equations 2.24, 2.25 and 2.26 can be written as a set of 2n + 2 linear simultane-ous equations with 2n+ 2 unknowns (q ,j 1,2 ... 2n and y,, 1- 1,2) end solved as for

the single airfoil case. However, to make the routine consistent with the unsteady flow

case, the following method is adopted.

The flow tangency condition can be rewritten explicitly for the q, in terms of

y(l), 7(2) and the free stream constant term; that is

a,,, a, 2 a13 e e a1 2 q al,2n+l [al,2n+2 b

a2,1 a2 ,2 a 2,3 e * * a 2,2,, q2 a2 ,2 ,+,i a2 2n+ 2 b2

a3,1 a3.2 a3,3 & @ * a3,2, q3 a3,2n+i a3,2,+2 b3

• * . . . .. • • = • Y(l)+ y (2)+

a2, * * "a2n.2n Rq2n J a2nn+ a2,,n+2 b2

(2.27)

Gauss Elimination is then used to solve for the q, in terms of y(l), y(2) and the

constant term. This gives

qj = bljy(1) + b21y(2) + b3j j = 1,2... 2n (2.28)

Equation 2.28 is then substituted into the Kutta condition at the two trailing edges to

form two linear simultaneous equation with two unknowns y(l) and y(2) . Gauss

Eimination is again used to solve for the Vorticity distribution y(l) and y(2) and these

are then back substituted to solve for the q.

3. Computation of Velocity and Pressure Distribution

Once the q, (j = 1,2 ... 2n) and y, (I = 1,2) are solved, the velocities at all the panel

control points can be easily obtained. The normal velocity is given by

15

_. .- ,. . *' " . ' . . .. ." r"

n 2n n 2n

- Aq, + 2 Aqj + y(l) " + (2) ]B + V*Sin[ ,,( -O1 (2.29)

J-1 Jfn+I J-l Jin+l

while the tangential velocity is given by

n 2n n 2n

(V)1- A,+ 2:Ai+Yl2Bf+v2 ~ V.. Cos[a(o - 1J (2.30)j=1 J -11+l .-l Jm-+1

with

i-,1,2 ... n ; 1 = 1, j

=n+l...2n; =2

The normal velocity will obviously satisfy equation 2.9 (used to check the code) showing

that the flow tangency condition is satisfied while the the total velocity will be given bythe tangential velocity.

Vo - (V'),, i 1,2 ... 2n (2.31)

where

n 2n n 2n

(Vt)1 = - - Bnq+ y(1)ZAf + '(2) 2 + V**Cos[ a() -O ] (2.32)

Sl in~?+l J. j- hn+l

Substituting equation 2.31 into 2.8 for C, with (V'), defined as in equation 2.32, the

pressure coefficient at the i" control point is

(C,), = I-(V')2 , i= 1,2...2n (2.33)

4. Computation of Forces and Moments

The two-dimensional aerodynamic lift (C), drag (C) and pitching moment (C.)

are calculated with respect to the global frame of reference. The moment coefficients

are computed with respect to the respective leading edges. For the code, we have

CA) =Z(Cp)Xx,+i - ,) (2.34)

16

-=:-::- dg"!Jr l' i~~li' -7 7 -77'i' 7-77- 77

Cj() - - Z(C,)A l+ - 1I) (2.35)

n

Cm(Y) Z(C,),( (X+ I - XdXm + (Y+ - Yd)Yma (2.36)

where I - 1,2 and X, X,, Y,, Y,.,, Xm,, Ym, are defined as before.

17

. . -. .. ~ - --- ---- ---

III. UNSTEADY FLOW PROBLEM FORMULATION

A. GENERALThe modification work for the unsteady flow is much more involved. It requires the

following:

1. The establishment of five frames of reference viz one fixed inertia frame of reference(global), two moving local frames of reference4 and two frozen local frames of ref-erence. 5

2. Reformulation of the two Kutta conditions which are coupled non-linearly. Thesolution requires an iterative procedure to compute for the two v(o. There are twopossible solutions to the Kutta condition due to the quadratic nature of theequations. The solution which ensures that the product of the tangential velocitiesof the first and last panels of each airfoil is negative is accepted as the solution.

3. The creation of a new subroutine (SUBROUTINE NEWPOS) which transformsall coordinates in either of the two respective local frames of reference to the globalframe of reference. This simplifies the definition of the airfoil, wake element andcore vortices relative global geometries. The code requires that the airfoil be de-fined with respect to the respective frozen local frames of reference once only.Subsequent time dependent airfoil motion, wake panel behaviour and core vorticesconvection are computer generated.

4. The introduction of a more accurate method to obtain the velocity potential byintegrating the velocity over smaller panels on the airfoil without having to storelarge arrays of influence coefficients which are not needed for satisfying the flowtangency condition.

5. Extension of the influence coefficient to include the effects of the second airfoil withits own peculiar wake. This also requires an introduction of an additional influencecoefficient, that on the wake element due to the wake element from the otherairfoil.

B. FRAMES OF REFERENCE

The inertia (global) and the two frozen local frames of reference at a specified time4 are defined as for the steady flow case. The two moving local frames of reference have

their x-y axes as for the frozen local frame of reference but this frame is moving with the

airfoil.

4 Moving local frames of reference are used to satisfy the flow tangency equation which sim-plifies to equation 2.9.

5 Frozen local frames of reference are inertia frames (used in the steady flow case) of referenceused to convert the core vortices of the previous time steps with respect to that time step frame ofreference and subsequently transformed to the current time step frozen frame of reference.

18

7- I.. -77

C. UNSTEADY FLOW MODEL

1. Rigid Body MotionThe rigid body motion for the two airfoil system is an extension to the single

airfoil system. Both airfoils are considered to have a mean velocity of - V., with a timedependent translational velocity of - IU( ) + V(f) 11 and a rotational velocity -

which can be in phase or out of phase; i and j are unit vectors in the respective local

frames of reference and 9 is positive in the clockwise direction. The flow will be deter-mined with respect to the moving local frame of reference. The flow tangency condition

takes its simplest form in the moving frame of reference. The flow tangency conditionseen from this frame of reference will satisfy equation 2.9. The unsteady stream velocity,

V,_ is made up by the vector sum of a mean velocity V., a time dependent

translational velocity + U(t) + I'()SJ and a rotational velocity £Q(O, (Figure 4).

Veavm,-V.o{ Cos[ a(ok] +Sin[ (Ok]j) + I U(Oki + V(;] +Q.1k -xj") (3.1)

The disturbance potential, of necessity, is defined with respect to the inertia frame ofreference, for it is only in this frame that the flow is irrotational. It is then transformed

to the moving frame of reference for ease in treating the flow tangency condition. The

disturbance potential is redefined to include the contributions from the wake panel and

the core vortices from both airfoils.

05s + 0, + 4. + 0 (3.2)

The total velocity is then written:

total - s'ram + VO (3.3)

The unsteady Bernoulli's equation for the pressure coefficients on the airfoil surface iswritten with respect to the moving frame of reference. Giesing showed this to be written,

in our notation as:

P - P Vr.eam 2 Vot2al _cP - I = ( (3.400 00 0

where V,,.. and V,,, are defined according to equations 3.1 and 3.3 respectively.

19

Y1Y

-0.4

0V xi

vDO1)K

ISr~ '@ Cos QC(l)7 + Sin ot(l) + (CU~ 1j (I)(y-x)

Figure 4. Frames of Reference for Airfoil I

20

m ~ ~ ~- .. . . .. I I m . .. , . . . . . .

2. Wake ModelRecall that the unsteady flow model requires an additional model to simulate

the continuous shedding of vorticity into the trailing wake. The treatment of this vortex

shedding process follows the approach of Basu and Hancock. The shed vorticity takes

place through a small straight line wake element attached as an additional panel to the

trailing edge of each airfoil with a uniform vorticity distribution [y.()oj. This shed

vorticity panel will be established if its length A(/, and inclination 19(), to the respective t -

local frames of reference satisfy the Helmholtz theorem:

A(mtlyw(oh + rk( r,_,( (3.5)

or/a (o k [ v) 1, - r,. ( - rk(I) - $ ) [ '(0-I -Y( 1 (3.6)

where SS is the perimeter of the airfoil and r, and Y,-I are respectively the total circu-lation and vorticity strength which are determined at the previous time step t*-,.

At the next time step , , the shed vorticity panel will be detached from the

trailing edge and will convect downstream as a concentrated free vortex with circulation

A(), i Y(I)], or r,_(/ - r,() at the resultant local velocity of the fluid particle. This wakeconvection process is illustrated in Figure 5 where the airfoils' subscripts are dropped

without loss of generality.

In the code, the convection of the core vortices is broken into three steps:

I. The core vortices are first convected using the resultant absolute velocity with re-spect to the frozen local frame of reference of the previous time step.

2. This is followed by a transformation to the current frozen local frame of reference.

3. Finally this is transformed to the global frame of reference by using the new sub-routine (SUBROUTINE NEWPOS).

3. Additional Boundary Conditions

The unsteady flow model has now introduced an additional boundary conditionviz. the conservation of vorticity (equation 3.5 or 3.6) through the modeling of the wake.

However, the introduction of the wake creates three additional unknowns for each

airfoil, that is, y,) 1, A(0. and Wo().. As such two additional conditions are required for

each airfoil in order to solve the system. The approach suggested by Basu and Hancock

is extended to the two airfoil case.

1. The wake panel is oriented in the direction or the local resultant velocity at thepanel midpoint.

21

'.a '

Vortev. Shedding at Time Step tk

Helmnholtz's theorem

"k (yw)k + T. - rk.1rPanelFSource Distribution (q).k

I~otictyDistribution y

rk (rk.3 - rk.2 )

(rk-4 -rk..S

(Vv,)k tak (Lk

-k (tk tk-z) ((;)k' + (Vw)k 2

Figure s. PAnel MeW todRepresntationk for Unstedy Flow

22

(V.(I)ktan O(I)t -I V()] (3.7)

where [V/(O], and [V.(/1, are the x and y velocity components at the midpoint ofthe wake panel with respect to the frozen local frame of reference6.

2. The length of the wake panel is propotional to the magnitude of the local resultantvelocity at the panel midpoint and the step size of the time step.

A~t - (,, - ,k-l)\/i(v t)olk) + ([V(o)) 2 (3.8)

We now have all the necessary equations to solve for the system.

D. INFLUENCE COEFFICIENTSAs with the boundary condition, additional influence coefficients need to be defined

as a result of the wake model. The definitions in the single airfoil case are thus extended

as follows:1. More A's and B's Influence Coefficients

Defining NP3 and NP4 as the panel number for the wake for the first and sec-

ond airfoil respectively and h as a arbitrary core vortex, the following additional influ-

ence coefficient are required.

* (B ) : normal velocity component induced at the P panel control point by unitstrength vorticity distribution on the wake panel of the first airfoil at time 1,.

* (B,,n), : tangential velocity component induced at the i& panel control point byunit strength vorticity distribution on the wake panel of the first airfoil at time t,.

* (B.,N), : normal velocity component induced at the P* panel control point by unitstrength vorticity distribution on the wake panel of the second airfoil at time t,.

* (B', tangential velocity component induced at the PA panel control point by unitstrength vorticity distribution on the wake panel of the second airfoil at time t,.

* (A - ), :x velocity component induced at the first airfoil wake panel midpointu'ith respect to the frozen local frame of reference by unit strength source distrib-ution on the j* panel at time t.

" (A-m), : y velocity component induced at the first airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength source distrib-ution on the p panel at time it.

" (Akp.), : x velocity component induced at the second airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength source distrib-ution on theP panel at time t ,

6 It is important to note that the velocities at the midpoint of the wake panels do not includethe effect due to the vorticity distributed along itself but it does include the contribution due to thewake panel of the other airfoil.

23

-- . .. . . . .. .. .. . . . . .' - -- -- t." , - . .. , = : '-;'7 : .. =" : : U ,,;,•

. , ,_

I~ -- . ". ..

(AkNJ), :y velocity component induced at the second airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength source distrib-ution on the? panel at time t,.

* (B,*p), : x velocity component induced at the first airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength vorticity dis-tribution on the jO panel at time t.

S(! Y velocity component induced at the first airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength vorticity dis-tribution on thejJ panel at time r..

*,(B PJ)k : x velocity component induced at the second airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength vorticity dis-tribution on thej4 panel at time t,.

* (B ), : y velocity component induced at the second airfoil wake panel midpointwith respect to the frozen local frame of reference by unit strength vorticity dis-tribution on the? panel at time t.

* (Axi), : x velocity component induced at the h* core vortex with respect to thefrozen local frame of reference by unit strength source distribution on the? panel at time t.

* (Aj), : y velocity component induced at the hl core vortex with respect to thefrozen local frame of reference by unit strength source distribution on thej* panel at time t.

* B,, (): x velocity component induced at the h core vortex with respect to thefrozen local frame of reference by unit strength vorticity distribution on thej' panel at time t,

* (Bl), : y velocity component induced at the h* core vortex with respect to thefrozen local frame of reference by unit strength vorticity distribution on thej" panel at time t,.

* (BL-,)k: x velocity component induced at the h core vortex by unit strengthvorticity distribution on the wake panel of the first airfoil at time t,.

S(.) :v velocity component induced at the h core vortex by unit strengthvorticity distribution on the wake panel of the first airfoil at time t,.

* (BLV,.), : x velocity component induced at the h core vortex by unit strengthvorticity distribution on the wake panel of the second airfoil at time t.

* (BL ), : y velocity component induced at the h core vortex by unit strengthvorticity distribution on the wake panel of the second airfoil at time t,.

2. New C's Influence Coefficient

The single airfoil definition is extended to include the effects of the second

airfoil.

* (C,())k :normal velocity component induced at the P* panel control point byunit strength mA core vortex at time t.

24

IF--7 .-.--.---.-- '..T;~ 7

" (CL,(), : tangential velocity component induced at the i* panel control pointby unit strength mn' core vortex in the wake of the M' at time t,

* (Ck,,(O)) : x velocity component with respect to the frozen local frame of refer-ence induced at the first airfoil wake element by unit strength mIA core vortex in thewake of the 1 at time t,.

(Cn.,()), :y velocity component with respect to the frozen local frame of refer-ence induced at the first airfoil wake element by unit strength m* core vortex in thewake of the ' at time t.

(C~,(l)), :x velocity component with respect to the frozen local frame of refer-ence induced at the second airfoil wake element by unit strength m" core vortex inthe wake of the P* at time t,.

* ,): y velocity component with respect to the frozen local frame of refer-ence induced at the second airfoil wake element by unit strength m core vortex inthe wake of the P at time r.

(CL,(O)), :x velocity component with respect to the frozen local frame of referenceinduced at the hA core vortex by unit strength m* core vortex in the wake of the Pat time t.

(Q,,()), : " velocity component with respect to the frozen local frame of referenceinduced at the h'" core vortex by unit strength mek core vortex in the wake of the illat time t.

3. Computation of the Time-Dependent Influence Coefficient

As for the single airfoil , the influence coefficients of the wake element

(),(B,.),,(Bi-) and (B,,,), are computed as for the airfoil influence coefficients

8 and B, with the subscripts NP3,NP4 replacing j.

The x and y velocity components for the respective frozen local frames of ref-

erence are obtained indirectly by first calculating for the global X and Y velocity. These

global velocities are then transformed to the frozen frames through a simple relationship

by the use of the respective angle of attack. It can be easily shown that the velocity with

respect to the frozen frame of reference is related to the global velocity by the following

relationship.

-VVi,, cos a(f)lt - (G v, sin a(Ol-(vi m), cosx(39

( ) GV- [GV sin *(Olk + [GVj' cos a((3.9)

where V denotes a generic induced velocity with respect to the frozen frame and the

precedent G denotes global velocities.

25

The global (GA 4 )),(GA.,,,(GA,I),,...(G and(GB?), are computed using equations 2.18 through 2.21 with 0, set to zero and subscripti appropriately replaced.

The global C's coefficients will be computed identically as for the single airfoilcase. The results are repeated here for the sake of completeness.

(GCLR(O)k-- Co.(Gl)k - (m),J (3.10)~2x(rj,),

(GC (O)) - - Sin[(0,)k - (0,),] (3.11)27r(r,,)k

where:

(r,.), - 1I(xm, - x.)2 + (M -Y.)

xm, = 1/2 (x, + x,.,)

ym, - I/2 (, + ,,)

x. = x coordinate of m'l core vortex at time t,

y. - y coordinate of ma core vortex at time t,

0, = tan-1( X-1-v,xt-) -, X0

= tan-'( YMx -Y. )k

xin, - x,.)

Also (GCkn^()), (GCI.,.(O)), (GC,.(O)k, (GC64,.(O)), (GC.(O), and (GCQ(O)k arecomputed with equation 3.10 with 0, set to zero and the subscript i appropriately re-

placed.

E. NUMERICAL SOLUTION SCHEME

1. The Flow Tangency Condition

The unsteady flow tangency equation for the two airfoil system is a simple ex-tension of the single airfoil case.

n 2n n2n

IA (q,)J + Z AVR(q~j~ + y( I)kZ B4i + y(2)k B~+ I( save)i*"lJmJ Jn+i J/-i Jn+i

+ yl )Jk(Bi,'P)k + [yw(2)k(Bj"Arp4)k +

26

k /-I k-I

L 1(cm)(r.-jo - + { 1(c:(o)(F,-A - rm(o)i} 2 " 0

where i - 1,2 ... 2n

(3.12)

where i - 1,2 ... 2n and . is evaluated by equation 3.1 at the i panel control point.

Rearranging the equation we can express the source strengths explicitly as a

function of the two vorticity strengths and a constant term as in equation 2.28 where:

n 2n

LI-IS. - A(qj)*1 + [Aj(qj)k ] (3.13)J-1 J.n+l

SS(l)First R.H.S. - v(l )k{( )k(BrP) - (B)k} (3.14)

2n

Second R.H.S. =(2)k (2) )k(BI.P.)k (B, (3.15)

ThirdR.H.S. AM )k(BtNP3)k

SS(2)

L-I- L(( )(rm(O - rm(O)I}tI - {KC[(O(rmF._I(t) - rm(O)I}tM2 (3.16)

where i - 1,2 ... 2n

For the code, the above is implemented in a 2N x (2N + 3) matrix in SUB-

ROUTINE COEF and is subsequently solved by GAUSS elimination to obtain an ex-

plicit expression for the source strength in terms of the vorticity distribution and a

constant term as in the steady flow case as follows:

27

, : :: ', " "~ ' 60...0684" W , •/ : . : ii . .. " : " . . " : . . '

qj blj. y(l) + b2j. y(2) + b3j , j = 1,2 ... 2n (3.17)

where

bli : part of the source strength which depends on circulation y(l)k.

b2j : part of the source strength which depends on circulation y(2),.

b3j : part of the source strength which is independent of the circulation.

Equation 3.17 is critical in the simplification of the treatment for the

non-linear coupled Kutta condition.

2. The Kutta Condition

For the unsteady flow, the Kutta condition can be written as the following:

For the first airfoil

(V) - (V,); = '(,,2- 4,2) = of-Yk( Ik1(I) (3.18)

=2SS(1) tk --tk_:

For the second airfoil

, 2 t 2 O r(2),-V2.)' - ( 2((,(0 2 n - O+) 1k )k

= 2SS(2) k(2) -k-1( 2 ) (3.19).k- 'k_-I

The tangential velocity for the first panel can be written thus:

2n n 2n- -Z (B lqj)k + y(l)kZAn+ + y(2)kZ An+

ss(I) a" ' 1,ss(2) .. ..a(I) A n 'p3[Yk,-(1) - y(l)] + S Anp.4[k_,(2) - y( 2 )l + [(uPeam)" Jkl +

k-I k-I

L[(o)(r,_p - rm(o)J 1 + r(O) }6 2 (3.20) J

Substituting equation 3.17 into 3.20, the following simplified relation for V[ is obtained.

28

.. " 7.

= Dlly(1)k +D2ly(2)k +D3 1 (3.21)

where

2mDI, l 2A SS() A

D11~~~ = Iu.i~ i ~y NP3

2n 2n

D2, I: -ZBnb2,+ Z:An, -SS() Ai ,/=, j==n+l

2n

D31 = Bnb3,+ -A(l) Ain A2'- ,.YS(2) +I(Vjtr.,),.'Il+

J-1

k-I k-I

Similar expressions can be obtained for the other trailing edge panels as follows:

F = Dly(1)k + D2 ly(2)k + D3, (3.23)

Vn+1 - Dlfl+1y(1)k + D2n+I(2)k + D3n+I (3.24)

",= Dl 2ny(l)k + D22n'( 2)k + D32n (3.25)

where the coefficients DI, D2, 3 are obtained as per equation 3.22. We require the

square of the tangential velocity for the Kutta condition

(VI) 2 _ (DII y(l)k + D21'(2)k + D31)

- DIy(l); + D2,y(2)2 + D3 (3.26)

+ 2DIID 21y()ky(2)k + 2D11D3ly(1)k + 2D2,D31y(2)k

(V -)2 (Dlny(l)k + D2ny(2 )k + D3n)2

"DI7(1)2 + D2,,y(2)+ D32 (3.27)

+ 2 DlnD 2ny(l)y( 2)k + 2DID3ny(I), + 2D2nD3ny(2)k

29

For the first airfoil equations 3.26 and 3.27 are then substituted into 3.18 togive, for the code:

AAA(I)y(1)k + BBB(I)y(2)k + CCC()y(l),k +

DDD( I)y(2)k + EEE( lW(l)ky( 2)k + FFF(1) -0 (3.28)

where

2 2BBB(1) =D21 - D2,,

CCC(l) =2[D I .D3 1 -DI,.D3,,- I _

DDD(I) =2(D2 1 .D31 - D2,, . D3.1

EEE(I) =2(D I ID2 1 -DI, D2,1

FFF(1) - D31 -D3, +2( S(yk.())(3.29)tc - Ik-1

A similar set of equation can be obtained for the second airfoil as follows:.

AAA(2)y(1)k + BBB(2)y(2)k2 + CCC(2)y(1),t +

DDD(2)y(2)k + EEE(2)y(1)ky(2)k + FFF(2) = 0 (3.30)

where

AAA(2) =D12 D-

BBB(2) D2 ,, D22n

CCC(2) -2[D1ln, D3,+, - D12m D32 n]

DDD(2) -2(D2n~, D3n.1 - D22n D32n S2-*

tk-1

EEE(2) -2[D1n~i.D2n+I D12fteD22tJ

30

FFF(2) D341- D321 + 2( -SS(2)yk-1.,2 (3.31)

The above Kutta equations are non-linear and coupled. To linearize it, we adopt

Newton's method, in which the initial iteration is linearized about the values of the

previous time step:

y() V(2): + by(2)' [,(2)c] 2 _ IY(2).... 112 + 2y(2)c-'6 y(2)" (3.32)

where:

670)'k l~i ; (l)k - (l)k-I

by(2)'4y(2)'k= ;y20-y( 2 )k-1

with K denoting the iteration counter. After dropping quadratic terms of by and col-

lecting like terms the Kutta condition for the first airfoil simplifies to the following:

{2AAA4,(l)y()... + CCC(l) + EEE(l)y(2)....) 6y(I)kt +

{2B8 B(l)y( 2)...I + DDD(l) + EEE(l)y(l)...l)by(2)kt +

AAAl~y1)~.. 1 + BBB(I)y(2)k-.. + CCC(l)y(l)k.. +

DDD()y(2)... + EEE(l)(l)...Iy(2)k... + FFF(l) - 0

(3.33)

Simnilarly the linearised Kutta condition for the second airfoil can be written:

{2A4AA(2)()jt... + CCC(2) + EEE(2)y/(2)k...,)6 y(l)kt +

{2BBB(2)y(2)j.... + DDD(2) + EEE(2)y(1)k...I)by( 2)kt +

AAA4.(2)y(l).... + BBB(2)y(2),2t-1 + CCC(2)y(l)k...l +

DDD(2)y(2).... + EEE(2 )v(Ih....,v()k.. + FFF(2) -0

(3.34)

31

The above two linear equations with two unknowns 6y(l), and 6y(2), can now be easily

solved with the same Gauss routine. The results are then back substituted into

equations 3.32. This procedure is repeated until the corrections by(l)" and by(2); are less

than a prescribed tolerance.

3. Convection of the Wake

The resultant velocities of all core vortices are calculated using the frozen localframe of reference. This is resolved into components at the h core vortex as follows:

n 2n n 2n

(U), - :[A (q)] , + [A(q)J, + y(l)Z (Bk), + y(2 ), (BiX), + I( Pj) 71t

J.-1 Jn+ JWm Jin+1

+ I.( l)]k(Bx,,,)t + [yJ2)k(Bh, - )k

k-1

+ {Zi(C m(O)(r.-(T- rm(o)Ij.., + {. (iM)(O)(r.-,(O- )

moh moh

(3.35)n 2n n 2n

(I)k - [A-(qJ)Jk + [A:(q)I + Y()kZ (Bh')k + y(2)k (O,)k + ( 67) "lk

J.-I j-n+I J-1 Jn+I

+ [y(l)1k(e.,P3)k + y,.(2)k(B %,vp4)k

k-I k-1

+ {ZI(chYm()k(rM..i( - r~(}..+ {Z:cj()ok(rM...(i) - r(~ .

m~h M~h

(3.36)

The location of the core vortices at the new time step is then computed with respect to

the current time step frozen local frame of reference and then transformed to the new

time step frozen local frame of reference. These coordinates are subsequently trans-

formed to global coordinates so as to facilitate the calculation of the influence coeffi-

cient.

4. Disturbance Potential and Pressure Distribution

The concept of disturbance potential introduced in the steady flow has played

a vital role in that it facilitates the synthesis of the flow field from simple flow field by

simple superposition of the various singularities contributions. However, there has not

been a requirement to solve directly for the disturbance potential, as our interest was

32

i T /

on the disturbance induced velocity which is the spatial derivative of the disturbance

potential. In our solutions, the concept of influence coefficient has allowed a direct

evaluation of the disturbance velocity thus nullifying the requirements of obtaining thedisturbance potential.

The treatment for the unsteady flow, though it follows the same procedure asthe steady flow case, viz the influence coefficient to obtain the disturbance velocity, still

requires the disturbance potential for the computation of the pressure coefficient as can

be seen in equation 3.4. Rewriting, we have:

(CP.)k -.- V ~G 2 - 2 (4 I), - ()-i i - 1,2 ... 2n (3.37)00

where V,,. and V,.i are defined according to equations 3.1 and 3.3 respectively.

From equation 3.37 we have written the rate of change of 0 by a backward fi-

nite difference approximation. This simplifies our iteration procedures tremendously asthe 0 from the previous time step exists at the current computation. The computation

of the disturbance potenial 0 is obtained through two steps. The difference in potential

0 from upstream at infinity to the leading edge is computed and is then combined with

the difference in the potential from the leading edge to the panel's of interest control

point.The present approach differs from the original single airfoil approach in that the

velocity potential along the airfoil surface is computed via a finer grid using Gaussian

quadrature7 . This modification improves the resultsS but the cost of doing it is a longer

computation time. The definition of infinity has also been set to 100 chord lengths in

comparison to 10 chord lengths used in the original code. Huwever, the number of

computation points and the first panel length from the leading edge upstream have not

been changed.

For completeness, the computation of the disturbance potential from the lead-

ing edge to infinity is included here for both airfoils. It is essentially the same as the

single airfoil case except that the disturbance potential is now computed relative to the

global frame of reference instead of the airfoil fixed frame of reference as used in the

7 Each panel is subdivided into 4 additional sub-panels and the tangential velocity is computedand integated over these smaller panels with a weighting function to get the disturbance potential.

S One can check the improvement by computing the y(s) obtained through the difference be-tween the trailing edges and compares them with that obtained through the Kutta condition.

33

Lill

original code. We begin by selecting a straight line extending upstream in the direction

parallel to V.. The length of the line is set at 100 chord lengths. This line is divided intoz panels ith the first element at the leading edge set equal to the single airfoil case forthe purpose of ensuring that the panel size is comparable to the airfoil panel size. Thepanel size is then subsequently increased to take advantage of the inversely decayinginduced velocities at larger distances. Using subscript f to denote these panel mid-points,

we define the folowing influence coefficients:

* (A;4), : normal velocity component induced at the f panel control point by unitstrength source distribution on thejopanel at time t,.

* (A;,), :tangential velocity component induced at theft panel control point by unitstrength source distribution on thej'panel at time t,.

*(B), : normal velocity component induced at the jt panel control point by unitstrength vorticity distribution on thepjpanel at time 'A.

" (k), : tangential velocity component at thef panel control point by unit strengthvorticity distribution on the j'panel at time t,.

* (B(.vn), : normal velocity component induced at thef panel control point by unitstrength vorticity distribution on the wake panel of the first airfoil at time t.

* (B,.,3) : tangential velocity component induced at thelf panel control point byunit strength vorticity distribution on the wake panel of the first airfoil at time t,.

" (Bjm.v), : normal velocity component induced at thefP panel control point by unitstrength vorticity distribution on the wake panel of the second airfoil at time t'.

o (B.-), tangential velocity component induced at the f panel control point by unitstrength vorticity distribution on the wake panel of the second airfoil at time t,.

o (f.,(o), : normal velocity component induced at thef panel control point of thePih by unit strength m,* core vortex at time t,.

* (C;..(O))X : tangential velocity component induced at the f panel control pointof the P" by unit strength mh core vortex at time t..

The tangential velocity at thef-panel written as for the code is then:

2n n 2,.+ 1 -- 1 r+ A )k + + Y(2)k _ I"

JulJu Ju+ .p

SS(l) ( SS(2) - k(2)) +r(l) (INVP3)kYk_-(l) (- (AN"P(i)IV(2) +

/ -1 k-1

34

;w*'- -- .- 4-, - 7 ..

valid for f- 1,2 ... z. The disturbance potential at the airfoil leading edge is the sum of

the products of the disturbance induced velocity at each panel and the panel length.2|

- - ., .,., - .,)] (3.39)

The integral over the airfoil surface, as stated before, is now done over a finer

grid. This requires the computation of the disturbance induced velocity over the smallergrids within each panels. Defining the total finer grid points in one panel by P9, we de-

fine first the refined influence coefficient

(A4(',)k

tangential velocity induced at the it, panel pl* node due to unit strengthsource distribution on theM panel at time tk

The other refined influence coefficients have the same definition. The tangential com-

ponent of the disturbance induced velocity at the jI* panel m* node is then

2n n 2n

( =- 2(B qj + y(l)kZ An + (2)k A +J-1 J.1 /,,n+n

SS(l) .SS(2)A() , -.k]l)I + A(2) I.'N.4,[ ,k-I(2) - 'k(2)I +

k-I /€-I

[LorA- rm(o)I}M + {Z(cI (O)rm.i( - r.Q))I}M2 (3.40)

which is valid for i - 1,2 ... 2n ; p - 1,2 ... P. Performing the line integration by summa-

zion, the disturbanc .otential at the P nodal point is then0P

(Owde l)k- (01e)k + [j(Vp)rjj+1 p],] for n > i i

(3.41)

S(+e)k- r+, W,, for n> i r

9 This includes the end points.

35

--.. -i -. i- .

where r,, denotes the panel length and IF, the Gaussian quadrature weighting function,

N, X X + (y+1 _ 1) 2 (3.42)

Finally the disturbance potential at the ilA panel control point is:

- 1I/2[ (4'ji)k + (4Onol+)k] , i - 1,2... 2n (3.43)

5. Computation of Forces and Moments

The C, C and C. about the leading edge of each airfoil are calculated in exactly

the same way as it is done for the steady flow problem by integrating the pressure dis-

tribution (See section D-4 of Chapter 2).

36

...................

IV. DESCRIPTION OF COMPUTER CODE USPOTF2

A. PROGRAM STRUCTURES AND CAPABILITIES

1. Restrictions and LimitationsThe restrictions and limitations listed in the single airfoil documentation [Ref.

1] still apply for the two airfoil case. Some efforts were made to optimise storage re-quirements versus computational repetitions; however as in the original code there is stillmuch room for improvement. Krainer [Ref. 6 1 has improved the original code bycombining subroutines and reducing repeated computation by introducing additionalcommon variables. Some of his improvements have been added to this program.

The computer system used is the Naval Postgraduate School IBM 3033AP. Thecurrent program fixes the maximum number of airfoil panels to 20010 and the maximumallowable time steps to 200. The computer currently has to be run with a minimumstorage requirement of 2 Mbytes. A detailed computing time study for the program isnot undertaken, for it not only changes with the number of nodal points selected butalso with the time step increment as this has an effect on the rate of convergence for thewake panel iteration. An order of magnitude is given for one case run to give an ap-

preciation of the computing time required. The system currently requires a total CPUrun time of 200 seconds using an optimising compiler for a step input with a 0.025non-dimensionalised time increment for 26 time steps,

At present, the program can run for two airfoils set at arbitrary distance and atdifferent angles of attack undergoing any of the following motions:

I. In-phase and out-of-phase Step Input

2. In-phase and out-of-phaseModified Ramp Input

3. In-phase and out-of-phase Translational Harmonic Oscillation

4. In-phase and out-of-phase Rotational Harmonic Oscillation

5. Sharp Edge Gust Field Penetration

2. Current Structures of USPOTF2 Main ProgramThe flow logic for USPOTF2 (Unsteady Potential Flow for 2 airfoils) is illus-

trated in Figure 6. It is essentially divided into three major modules, namely, the input

10 These are combined total panels for both airfoils.

37

- problem setup, Steady flow solution with its associated output and the Unsteady flow

solution ith its associated output.

The first module includes subroutines INDATA, SETUP, BODY and NACA45.

The main functions of this module are:

I. Set up the problem formulation by reading in the necessary values and flag settingfrom filecode 1.

2. If the flag for a NACA 4-digit or 5-digit of type 230XX is set, this module callssubroutines BODY and NACA45 to compute the local panel coordinates.

3. If the flag for a NACA 4-digit or 5-digit of type 230XX is not set, this module thenproceeds to read in the local panel coordinates from filecode 1.

4. The local slopes and the airfoil perimeters are then calculated in preparation for thenext module.

The second module calculates the Steady flow solution. It calls subroutines

NEWPOS, INFL, COEF, GAUSS, KUTTA, VELDIS and FANDM. The main func-

tions of this module are:

1. Transform the local airfoil coordinates into global coordinates and compute theinfluence coefficients.

2. Set up flow tangency equation as per equation 2.27 and solve for the sourcestrengths (q,) as a function of the vorticity strengths [v(0] and a constant part wherej = 1,2 ... 2n and l- 1,2.

3. Set up and solve the Kutta condition as per equations 2.25 and 2.26 for thevorticity strengths and back substitute to get the source strengths.

4. Compute the total tangential velocity in the moving frame and compute the dis-turbance potential (0,) and pressure coefficient [(),] (i - 1,2 ... 2n).

5. Finally, compute the aerodynamic coefficients of forces and moments --c,, c, c..

6. This program can terminate in this module after all the steady flow parameters areobtained without necessarily running the Unsteady flow code.

The third module calculates the Unsteady flow solution. It calls subroutines

NEWPOS, INFL, COEF, GAUSS, KUTTA, TEWAK, PRESS, FANDM and

CORVOR. The main functions of this module are:

1. For the particular unsteady motion, to compute the initial time step and new airfoilorientation with its associated local body velocities.

2. Introduce the wake panel and assume an initial length and orientation to begin it-erations.

3. Transform all local coordinates and panel slopes to global coordinates and globalpanel slopes.

38

.7, = , ..... :,::. - - . = ::.: =, -i:: . : -= 1= ? :

4. Update influence coefficient and set up flow tangency equation.

5. Solve for the source strengths in terms of the vorticity strengths and the constantpart.

6. Invoke the required Kutta condition and solve for the vorticity strengthsl1; backsubstitute to get the source strengths.

7. Update the local velocities of the wake element and compute new length and ori-entation --- check for convergence.

8. If wake element velocities have not converged, iterate with the new wake elementgeometries until convergence12.

9. Compute the total velocity, disturbance potential and the pressure coefficient.

* -10. Compute the aerodynamic lift, drag and moment coefficient.

11. Adjust time step either through an interative input' 3 or by a automatic time in-crement.

12. Compute resultant local velocities of the core vortices.

13. Convect the core vortices by the procedure described in Chapter II1 section E-3

3

'M Tere are two possible solutions; only the vorticity distribution that ensures that the prod-uct of the relative tangential velocities of the upper and lower panels of the trailing edge is negativeis accepted as solution.

12 The convergence criterion is user specified through input data for TOL.

13 This is accomplished by setting TADJ to be non-zero and is intended for use in conjunctionwith the viscous flow program.

39

!-.

READ INPUT FROM

FILECODE 1

NACA COMPUTE BODY

TYPE AIRFOIL y THICKNESS &

? CAMBER

INPUT LOCAL PANEL COMPUTE LOCAL PANEL

COORDINATE COORDINATESI I i

COMPUTE LOCAL SLOPES& PERIMETERS

INPUT MODULE

Figure 6. Flow Chart for USPOTF2 Computer Code.

40 . !.

TRANSFORM TO GLOBAL

COORDINATES & SLOPES &

COMPUTE INFL COEF

SET FLOW TANGENCY EQN AS

SYSTEM OF 2n*(2n+3) MATRIX

AS PER EQ14 2.27

SOLVE FOR SOURCE STRENGTH

IN TERMS OF VORTICITY STRENGTH

AND A CONSTANT

SET UP

KUTTA CONDITION

SOLVE KUTTA TO OBTAIN y(l) and

BACK-SUBSTITUTE FOR SOURCE STRENGTHS

COMPUTE VELOCITY (Vt)1

DISTURBANCE POTENTIAL 0

PRESSURE (Cp)i (i - 1,2 ... 2n)

COMPUTE AERODYNAMIC COEF

OF FORCES & MOMENT

c,(.t)-cd()-cl(1)(1 1. 2 )

STEADY FLOW MODULE

Figure 6 (Cont'd)

41

. .. . . . . --- 3. .

INITIALISE PARAMETERS BEFORESTARTING UNSTEADY FLOW SOLUTION

NUNSTEADY FLOW COMPUTATION ? - STOP

TIME STEP INCREMENT

T > TF ? - STOP

UPDATE INFL COEFFICIENT

SET FLOW TANGENCY EQN ASPER EQN 3.12

tSOLVE FOR SOURCE STRENGTH

IN TERMS OF VORTICITY STRENGTHAND A CONSTANT

TYPE OF KUTTA CONDITIONA

SET UP EQUAL SET UP EQUALPRESSURE KUTTA TGT VEL KUTTACONDITION CONDITION

SOLVE KUTrA TO OBTAIN y(l) andBACK-SUBSTITUTE FOR SOURCE STRENGTHS

UPDATE LOCAL WAKE PANEL VELOCITIESLENGTH & ORIENTATION

CONVERGENCE OF WAKEPANEL LOCAL VELOCITIES

fCOMPUTE VELOCITY (Vt)1 ,

DISTURBANCE POTENTIAL ojPRESSURE (Ci)i (i = 1,2 ... 2n)

fCOMPUTE AERODYNAMIC COEF

OF FORCES & MOMENT

ADJUST TIME STEP ?

WAKE ELEMENT SHED AS CORE VORTEX, COMPUTEUNSTEADY RESULTANT VELOCITIES & CONVECT CORE VORT.CES

FLOW 1MODULE UP-DATE PARAMETERS AND START NEXT TIME STEP

Figure 6 (Cont'd)

42

B. DESCRIPTION OF SUBROUTINES

1. Subroutine BODY

This subroutine is called for the purpose of obtaining the local (x,,) coordinates

of either a NACA XXXX or 230XX type airfoil. It is called by subroutine SETUP and

it in turn calls subroutine NACA45 to obtain the airfoil thickness and camber distrib-

.- utions.

2. Subroutine COEF

This subroutine was originally intended for the unsteady flow calculation for thesingle airfoil case. It is now modified to include the steady flow calculations. Its pur-

pose is to set up the flow tangency matrix as in equation 2.27 for the steady flow and

equations 3.13 through 3.16 for the unsteady flow case. These matrices are necessarily

set up in this way so that the source strengths can be solved in terms of the vorticity

strengths and a constant by subroutine GAUSS as a linear system with three right hand

sides. It is called by the MAIN program.

3. Subroutine COFISH (deleted for the two airfoil case)

This subroutine was originally set up to serve the same function as subroutine

COEF for the steady case. It is now deleted for computational efficiency.

4. Subroutine CORVORThis subroutine is called for the purpose of obtaining the convective velocities

for all the wake core vortices with respect to the frozen local frame of reference of the

current time step in accordance with equations 3.35 and 3.36 where all the influence

coefficients are now locally computed (previously done in subroutine INFL) to save

storage requirements since these are only required in this subroutine. The local influence

coefficients are obtained indirectly by first obtaining the global influence coefficients and

transforming them viz equation 3.9. This subroutine is called by the main program

nearing the end of the unsteady flow calculations before starting a new time step.

5. Subroutine FANDMThis subroutine is intended to calculate the overrall lift coefficient, drag coeffi-

cient and moment coefficient about the leading edge for the two airfoils. In order to

preserve the option of obtaining the x and y forces with respect to the respective local

frames of reference, the original method rather than equations 2.34 through 2.36 is im-

plemented. This subroutine is called by the MAIN program in both the steady and un-

steady flow computations immediately after computing the pressure coefficient.

43

6. Subroutine GAUSSThis subroutine is the standard linear system solver that employs the well-

known Gaussian elimination with partial pivoting and operates simultaneously on a userspecified number of right-hand-sides. It is called by the MAIN program in both the

steady and unsteady flow calculations. In order to use GAUSS, the coefficients of the

augmented matrix must be set up so that GAUSS will return the solutions replacing thecorresponding columns of the augmented matrix that were initially occupied by the

right-hand-sides. The coefficient set-ups are done by subroutine COEF for both steadyand unsteady flow problems.

7. Subroutine INDATAThis subroutine is intended to read in the first three to five cards of the input

data depending on whether IFLAG - 0. The first three cards contain some description

of the airfoil type, problem definition, IFLAG information as well as the number oflower and upper panels. If IFLAG - 0, it will treat the airfoils as NACA type airfoilsand will proceed to read the NACA number and to calculate the thickness parameters

that will be required by subroutine NACA45. This is the first subroutine called by the

MAIN program.8. Subroutine INFL

This subroutine generates most of the influence coefficients that are needed and

shared by the different subroutines. It has been modified to include the steady flow casefor the purpose of reducing repeated computations. It utilises the known relative ge-ometrical parameters of the singularities to carry out computation based on equations

2.18 through 2.21 for the steady flow calculations and including 3.9 through 3.11 for theunsteady flow case. The MAIN program calls this subroutine in every iteration cycle

of each time step so that the time dependent influence coefficients can be updated as

and when neccessary. Time independent coefficients are computed once in the entireflow solutionsl4.There are also time dependent influence coefficients that are independ-ent of the iterative cycle to obtain the wake panel orientationIs. Those influence coef-ficients involving the wake core vortices are also independent of the iteration cycle and

are also updated once in each time step; but the process of the update is more compli-cated. This is due to the process that the wake need first to be convected with respectto the previous time step frozen frame of reference, transformed to the present frame of

14 These are: on the airfoil panel by the panels on the same airfoil.IS These are: on the airfoil panel by the panels on the other airfoil.

44

; "" . . . . . . " ,}. . . " " : - . . . . . . . /' .- .7- .. '-' - ' . -" ' " '- - - r 7 "-. "

reference and finally transformed to the global frame of reference by subroutine

NEWPOS. Finally the influence coefficients involving the wake panels are calculated

as frequently as the number of iterations take to find a converged solution.

9. Subroutine KUTTA

This subroutine is intended to solve the Kutta equation. It has been modified

to include the steady flow solution. After the source strengths have been determined by

subroutines COEF and GAUSS in terms of the vorticity strengths and a constant, this

subroutine invokes the Kutta condition as in equations 2.25, 2.26 for the steady flow

case and 3.33, 3.34 for the unsteady flow case. The two linear equationsl 6 are then

solved again with Gaussian elimination to obtain the vorticity distributions. For the

unsteady case, we add in the additional requirement of finding the product of the

tangential velocities at the upper and lower trailing edges. These are demanded to be

negative as there are basically two possible solutions to the Kutta equations due to its

original quadratic nature.

10. Subroutine NACA45

This subroutine is intended to calculate the camber and thickness distribution

of the NACA 4-digit and the NACA 5-digit airfoils of type 230XX which share common

thickness distributions with the 4-digit airfoil having the same thickness to chord ratio.

This subroutine is called by subroutine BODY which is in turn called by subroutine

SETUP and in turn called by the MAIN program.

11. Subroutine NEWPOS

This is a new subroutine introduced as a result of setting up the five frames of

reference. Most of the coordinates computation is done with respect to the respective

local frame of reference. Its purpose then is to transform all coordinates in the respec-

tive local frames of reference to the global frame of reference17 . This is necessary for the

computation of the influence coefficients as well as in the convection of the wake core

vortices. It facilitates the simple requirements of defining the airfoil once only with re-

spect to its local frame of reference. Orientation and displacement of the airfoil in the

two-dimensional plane is henceforth calculated through this subroutine. This subroutine

16 For the unsteady case, the original two equations are non-linear and were subsequentlylinearised in the discussion in Chapter 3.

17 It has a secondary function to obtain the slopes for the airfoil panels and the wake elementpanels.

45

is called from the MAIN program in several locations immediately after the local coor-

dinates are computed.

12. Subroutine PRESS

This subroutine calculates the pressure distribution over the airfoil panels after

the iterative solution for the unsteady flow problem has successfully met the convergencecriterion. It first computes the tangential velocities at all panel control points usingP+I.equation 3.40 (with p - 2 + then performs the disturbance potential evaluation at

the current time step according to equations 3.41 through 3.43. Together with the dis-

turbance potential data obtained from the previous time step, it calculates the pressure

-* distribution using equation 3.37.

13. Subroutine SETUP

This subroutine sets up the local panel nodal coordinates for MAIN program

by reading the fourth through seventh data sets of the input fie if IFLAG - 1 is set.

It skips the data reading if IFLAG = 0 and proceeds to set up the node distribution and

calls subroutine BODY to calculate the airfoil local coordinates. The node distribution

adopts a cosine formula in order to have closely packed panels toward the leading and

trailing edges for improvements in solution accuracy. Regardless of how the nodal co-

ordinates are obtained , subroutine SETUP determines the local panel slopes and airfoil

perimeter length.

14. Subroutine TEWAK

This subroutine is intended to calculate the resultant velocity components at the

mid-point of the shed vorticity panel with respect to the current frozen frame of refer-

ence. These velocity components are necessary to ensure that the correct shed vorticity

panels' length and orientation are established. This is the governing criterion for the it-

erative solution scheme of the unsteady flow case. This subroutine is called by the

MAIN program at every iteration cycle of each time step for the unsteady flow calcu-

lation.

15. Subroutine VELDIS

This subroutine essentially performs the same function as subroutine PRESS for

the steady flow case. It is thus redundant and could be deleted with some modification

work necessary for subroutine PRESS. This work has been implemented by Krainer for

the single airfoil case. This subroutine is called by the MAIN program in the steady flow

computation.

46

...-..... . . .-- 0

C. INPUT DATA FOR PROGRAM USPOTF2

The Input data is similar to that for the single airfoil case. Program USPOTF2 reads

its input data from filecode 1. An example of an input data file is attached in Appendix

B for the case when the airfoil nodal coordinates are input by the user. Computer gen-

erated airfoil coordinates are another option that can be selected if the airfoil chosen

belongs to the family of NACA 4-digit or 3-digit airfoils of type 230XX. To do this, the

I FLAG parameter is set to zero in the first item of the 4 set of data card and replace

the 5i and 6, set of cards by two cards containing the particular airfoil NACA number

for the two airfoils using format (15). Figure 7 contains an itemised description of the

sequential input variables.

D. OUTPUT DATA FROM PROGRAM USPOTF2

The Output data is similar to that for the single airfoil case. Appendix C contains

a sample output data generated by using the input data set from Appendix B. Due to

the repetitive nature of the output as a function of time, only selective time set data are

shown. The output data file begins with writing out what the program has read from

the data file followed by the computed nodal coordinates only if they are generated by

the program; otherwise it proceeds to write the airfoil computed perimeter length. The

next set of output data are the steady flow solution parameters of distributed source

strengths, vorticity strengths, pressure-velocity distribution, force-moment coefficient,

potential at the control nodal points and the potential at the leading edge. In addition

some output is given to allow an assessment of the accuracy of the flow solution. This

consists of the normal component of the velocities at the panel control points and the

integral of the disturbance tangential velocity along the airfoil contour. If only the

steady flow solution is required, the output terminates here.

For unsteady flow, in addition to the above, the unsteady flow parameters are also

printed. This includes, at every time step, the iterative printout for the convergence of

the length and orientation of the wake element panel, the unsteady solution parameters

on the airfoil similar to that of the steady flow parameters as well as the trailing wake

core vortices data. Again the same checking mechanisms are inserted on the airfoil

panels. A detailed explanation on the output variable names is listed in Figure 8. All

output data are non-dimensional quantities.

47

C - .-. ' -. . . --- . . . ...-. ~.-. ... .

l' 7

Data Set #1 Format (15) - 1 data card.

ITITLE - Number of title cards to be used in Data Set #2.

Data Set #2 Format (20A4) - ITITLE data cards.

TITLE - Headings printed on output for case run

identification.

Data Set #3 Format (I5,2F10.6) - 1 data card.

NAIRFO - Number of airfoil, in this case m 2.

XSHIFT - Relative X distance of the 2 airfoil's pivot position

with respect to the airfoil global coordinate system.

YSHIFT - Relative Y distance of the 2 airfoil's pivot position

with respect to the airfoil global coordinate system.

Data Set #4 Format (315) - 1 data card.

IFLAG - 0 if airfoil is NACA XXXX or 230XX.

- 1 otherwise.

NLOWER - Number of panels used on both airfoil lower surface.

NUPPER - Number of panels used on both airfoil upper surface.

Data Set #5 Format (6F10.6) if IFLAG - 1 - variable data cards.

x(I),y(I) - local non-dimensionalised x-nodal followed by y-nodal

coordinates for airfoil 1. Total of Nlower+Nupper+l

nodal points divided into 6 points per data card.

Data Set #6 Format (6FI0.6) - variable data cards

x(I),y(I) - local non-dimensionalised x-nodal followed by y-nodal

coordinates for airfoil 2. Total of Nlower+Nupper+l

nodal points divided into 6 points per data card.

Figure 7. List of Input Variables

48

* - . ..--

Data Set #7 Format (6FI0.6,13) - 1 data card

ALPI(1) - Initial angle of attack (AOA) for airfoil 1 in degree.

ALPI(2) - Initial angle of attack (AOA) for airfoil 2 in degree.

DALP - Absolute change in AOA in degree for non oscillatory

motions.

- Maximum amplitude of AOA change in degree for

rotational harmomic motions.

TCON - Non-dimensional rise time (Vrtlc) of

AOA for motion involving modified-ramp change in AOA.

FREQ - Non-dimensional oscillation (wclVo)

for harmonic motions.

PIVOT - The length from the leading edge to the pivot point

for the local system.

IPHASE - Flag for in-phase and out-of-phase motion.

-0 out-of-phase motion.

- 1 in-phase motion.

Data Set #8 Format (8F10.6) - 1 data card.

UGUST - Magnitude of non-dimensional gust velocity along

global x-direction.

VGUST - Magnitude of non-dimensional gust velocity along

global y-direction.

DELHX(1) - Non-dimensional translational chordwise amplitude

for airfoil 1

DELHX(2) - Non-dimensional translational chordwise amplitude

for airfoil 2

Figure 7 (contd)

49

Data Set #8 (Cont'd)

DELUY(1) - Non-dimensional translational transverse amplitude

for airfoil 1

DELHY(2) - Non-dimensional translational transverse amplitude

for airfoil 2

PHASE(l) - Phase angle in degree between the chordwise and

transverse translational oscillation with the latter

as reference for the first airfoil.

PHASE(2) - Phase angle in degree between the chordwise and

transverse translational oscillation with the latter

as reference for the second airfoil.

Data Set #9 Format (5F10.6,I5) - 1 data card.

TF - Final non-dimensional time to terminate unsteady flow

solution.

DTS - Starting time step for non-osc. motions if TADJ - 0.0

- No. of computational steps per cycle for

harmonic motion.

- Baseline time step size for all motions if TADJ * 0.0

TOL - Tolerance criterion for checking the convergence

between successive iterations of (UW)k and (Vw)k"

TADJ - Factor by which DTS will be adjusted.

SCLA - Steady lift coefficient for the single airfoil at the

specified AOA

NGIES - Option for changing the unsteady Kutta condition to

satisfy the tangential velocity as per Gaising's case.

- 0 equal pressure at the trailing edge panels.

- 1 equal tgt velocities at the trailing edge panels.

Figure 7 (cont'd)

soo

I,50?

TK - Time step t k .

TKMl - Time step tk_ 1 .

ALPHA(L) - Angle of attack of airfoil L at time tk.

OMEGA(L) - Rotational velocity (positive counter clockwise) at time

t k for airfoil L

U(L) - Chordwise translational velocity (positive forward) at

time tk for airfoil L

V(L) - Transverse translational velocity (positive downward) at

time tk for airfoil L

NITR - Iteration number.

VXW(L) - Iterative solution of (Uw) k of airfoil L

VYW(L) - Iterative solution of (Vw) k of airfoil L

WAKE(L) - Iterative solution of shed vorticity panel length Ak

of airfoil L

THETA(L) - Iterative solution of shed vorticity panel orientation ' k

of airfoil L

GAMK(L) - Iterative solution of the strength of the current

vorticity distribution of airfoil L

J - Panel number.

XI(J) - local x-coordinate of the midpoint of jth panel.

YI(J) - local y-coordinate of the midpoint of jth panel.

X(J) - global X-coordinate of the midpoint of jth panel.

Y(J) - global Y-coordinate of the midpoint of jth panel.

Q(J) - Strength of source distribution on the jth panel.

CP(J) - Pressure coefficient at the midpoint of the jth panel.

V(J) - Total tangential velocity at the midpoint of the jth panel

with respect to the moving local system.

Figure 8. List of Output Variables.

7i

- .- z-v ' . .

VN(J) - Normal velocity at the midpoint of the jth panel

with respect to the moving local system.

PHIK(J) - Potential at the mid-point of the jth panel at

the current time step.

PHI(J) - Potential at the mid-point of the jth panel at

previous time step.

INTGAMMA - Integral of the disturbance velocity around the airfoil.

CD(L) - Drag coefficient of airfoil L.

CL(L) - Lift coefficient of airfoil L.

CM(L) - Pitching moment coefficient about leading edge

of airfoil L.

M - Trailing wake core vortex number.

XlI(M) - X-coordinate of the center of the ath core vortex

of airfoil 1 with respect to local system.

YI(M) - Y-coordinate of the center of the mth core vortex

of airfoil 1 with respect to local system.

XI(M) - X-coordinate of the center of the mth core vortex

of airfoil I with respect to global system.

Yl(M) - Y-coordinate of the center of the nth core vortex

of airfoil 1 with respect to global system.

X21(M) - X-coordinate of the center of the nth core vortex

of airfoil 2 with respect to local system.

Y21(M) - Y-coordinate of the center of the ntb core vortex

of airfoil 2 with respect to local system.

X2(M) - X-coordinate of the center of the mth core vortex

of airfoil 2 with respect to global system.

Y2(M) - Y-coordinate of the center of the mtb core vortex

of airfoil 2 with respect to global system.

CIRC(M,L)- Circulation strength of the nth core vortex of

airfoil L

Figure 8 (Contd)

52

- . ... ... ~ 7M

V. RESULTS AND DISCUSSION OF CASE-RUNS

USPOTF2 is primarily written as a follow-up to U2DIF for the single airfoil. Allthe case-runs with the exception of the gust case will be presented. The approach in thegust case is not consistent with the requirements for an irrotational flow field and willnot be treated in this report. The step change in angle of attack (AOA) will be comparedwith Giesing's for the same airfoil set at the final AOA, undergoing an impulsive startfrom rest. As there exist no comparison data for the other sub-cases, the results will thusnot be as extensive as the step input. They are documented here for the sole purposeof illustrating the capability of the code.

A. STEP CHANGE IN ANGLE OF ATTACK1. Case Run Definition

Consider a first case of two airfoils initially at zero AOA to the free stream V,which undergo an out-of-phase step change in AOA (a,,) at time to. Consider a secondcase of two airfoils at rest, set initially at an AOA of , out-of-phase with one another,gicn an impulsive start to V,.. The above two cases are equivalent within the thinairfoil approximation. Case I is computed by USPOTF2 and case 2 is obtained byGiesing's computational analysis.

2. Differences between USPOTF2 and Giesing's codeWhile both codes use an extension of the PANEL method to solve for the un-

steady potential flow solution for two airfoils, perfect correlation is not possible for se-veral reasons. The reader is referred to References 15] and 17) for a detailed description

of Giesing's approach. Some basic differences are:

0 USPOTF2 models the wake vortex sheet by a wake element and a series of pointvortices shed through the wake element. This follows the approach of Basu andHancock with the necessary assumptions on the wake element characteristics.Giesing treated the wake vortex sheet to comprise of a distributed line vortexwhich is convected at the local fluid velocity at the vortex location assuming thatthe small portion of the wake being shed does not contribute to the convection ofthe wake.

* The time step increment in LSPOTF2 is a parameter that affects the overrall resultsof the unsteady flow in that by having small time step, the point vortices being shedwill be greater in number but weaker in strength, while for bigger time step, thepoint vortices become fewer but have effectively stronger strengths. The time stepincrement in Giesing's code is important in that the assumption that the small

53

.. , . "" .. 7

portion of the wake being shed does not contribute to the convection of the wakeis only exactly true in the limit when the time step becomes zero.

9 Giesing uses the Adam's formula of varying degreeI8 to convect the wake vortexsheet while USPOTF2 uses only a predictor algorithm.

* The treatments of the circulation 1-(l) are different for both codes. USPOTF2 ex-tended the influence coefficient concept to the unsteady flow regime and solved forthe circulation using the Kutta condition of equations 3.33 and 3.34. All calcu-lations were done in the moving frame in the presence of the unsteady wake for-mations. Giesing considered the circulation to be a combination of thequasi-steady circulation and the circulation due to the vortex wakes. From Refer-ence [ 5]

r(o - r,(o + ro (5.1)

where r,() is the circulation required to satisfy the Kutta condition on body (1) asit moves through the fluid when it is assumed that the body does not shed anyvorticity and r,() is that circulation required to satisfy the Kutta condition on body(1) as it travels through the flow field generated by the vortex wakes shed by the twobodies and the flow field generated by the circulatory flow about the other body.

e Giesing's Kutta condition reqires equal tangential velocities at the upper and lowersurface panels at the trailing edges while the Kutta condition of USPOTF2 pre-scribes equal pressures. In order to have a meaningful comparison, USPOTF2 in-cludes an option for equal tangential velocities.

o USPOTF2 requires the actual computation of the total velocity potential from acombination of an onset flow potential, disturbance potential and an assumptionof a reference potential. Giesing uses the technique of the Douglas Potential Flowprogram which treats the potential as the combination of the quasi-steady potentialand the potential due to the vortex wakes where these two terms are defined asbefore. The total velocity potential. in his case, need not be computed but only thetime derivative of it which is then written in terms of previously known parameters.

* The number of panels and its distribution on the airfoils are different for bothcodes. This will be accentuated at the trailing edges and will lend itself to differ-ences in the trailing edge panel distribution.

3. Results and Discussions

Figure 9 shows Giesing's results for the Von Mises 8.4 per-cent thick, symmet-

rical airfoil undergoing an impulsive start. This can be compared with Figure 10 which

shows the result obtained with USPOTF2 for the same time step with NGIES set to

one19 . The results are not perfectly correlated but the quality and order of magnitude

agreement are excellent. Figure I I gives essentially the same result but with NGIES set

IS This is essentially a predictor-corrector algorithm.

19 This Kutta condition results in equal tangential velocities at the trailing edge panels of theairfoils.

54

to zero20. Surprisingly, the results obtained for both types of Kutta condition turn outto be quite similar. This is seen especially in the pressure coefficient plots when the twoplots are put together as seen in Figure 12. The pressure coefficient agrees over 70 per-cent of chord length at the lower surface and over 90 per-cent of chord length at theupper surface with the greatest discrepancy at the trailing edge. Further comparison forsmaller angles of attack would be of interest to see whether this similarity is generallytrue.

Figure 13 compares the aerodynamic characteristics when the vertical distanceYSHIFT is varied. Figure 14 gives the time variation for a larger time step of thenormalised lift, moment coefficient and drag coefficient for the particular case of

YSHIFT = 2.0.

20 This Kutta condition results in equal pressure coefficients at the trailing edge panels of theairfoils.

55

A- Q 7

-10

- WO AIRFOILS

0 AIRFOL~ WITHC, GROUND PLANE

-10

1 -0

00.2 0 i4 06 .6 1.0

"/C

(a) Pressure Distribution and Vortex locations.

figure 9. Giesing's Calculated Results: Impulsive Start for an 8.4 per-cent thick

Von Mises airfoil set at a - 0.8 radians for Y'SHIFT -2.0 [ reproduced

with permission from Ref. 51.

56

- 7%0AAj k

1004

oI U,

(b im io, of th-itC efiins

rb) Tre HisCotr'fte itCefiins

57

.. . ............. ......... ......... . . ..... .. ............. ... ... ."....... ........ .. .. .... .... .... ... .t.....t....

.......... .'t '....t..... ..... ' .... !.....

0 0.2 0.4 0.6 0.8 1.0

x/c

(a) Pressure Distribution at tVjc - 0.65

Figure 10. USPOTF2 Results obtained with USPOTF2 for equal velocities at the

trailing edge.: Step change in AOA for an 8.4 per-cent thick Von

Mises airfoils placed at 2 chord length vertical distance with initialAOA = 0.0 radians and final AOA a 0.8 radians pivoting at the

leading edges.

58

S- --!

C.,C

(b) Wake pattern at tl/=/c -0.65

Figure 10 (Cont'd)

59

..... . .... ......

........

02 0.I. . .

qO-IESOASDTM

(c) Tie Histry of he Lif .oeficints

Fiue1 2Cnt

I . I 6 j

III i1

0 .............. .... ....... ........ . ..........

. .. ........... .. . . . . .. "....... ;.....

0 0.2 0.4 0.6 0.8 1.0x/c

(a) Pressure Distribution at t J',c -0.65

Figure 11. USPOTF2 Results obtained with USPOTF2 for equal pressures at thetrailing edge.: Step change in AQA for an 8.4 per-cent thick VonMists airfoils placed at 2 chord length vertical distance with initialAQA = 0.0 radians and final AQA -0.8 radians pivoting at theleading edges.

61

0

K/C

(b) Wake pattern at Woo/c -0.65

Figure 1 (Cont'd)

62

... ..... .

........................... .. ......... ......... 9.- 1 1

... . ... ............

d

Fiur I I I.t

63

. ..... ..... ..........

0 0.2 0.4 0.5 0.5 1.0

x/C

NGIES = 0 : Equal Pressure Coefficient at trailing edge panels.NGIES = I : Equal Tangential Velocities at trailing edge panels

Figure 12. Pressure Distributions with different Kutta condition.

64

... . . .. .... ............ ..... . ........

ki~~H - -2GEAVI

-I- -r ISHtFT -N~JSF 1+~s~r25

- ---------

- .............

N0.2 0.4 0.6 0.8 1.0

(a) Pressure Distribution at 61.1c = 0.65x/

Figure 13. Pressure Distribution and Lift History as a function of the Vertical dis-tance betwieen the airfoils.

65

ai

C,! ... ..... ...... .... ...... ..... .. r .................. ................... ............... ..... ":..................... .. ....... .. , ....... .............. . _

-c-- UNCLE AIRFOIL

4- YSFTg6.3

o - YSHIFToh 25

.. .~. * r . ........ ... ......

............ ... .... .......... . ............. ..... ......... .......... ...... "sm ............."

S....... ....... i .... ........... ... ..... ................ . ."- - := : + .. ....... ........ ...........j t i I .:"

..... . - I

..... ........... .......................; ................... ................. i ............... .. ................... ....................... ................ ..,

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

. ....... ..... ..........o..... .,......... ... ........ ............ .. .... ......... .... ... !.. ... ............

4 ....................... ......... ........ .......... .................................. ................ ...... ........ .... :. ............. -................................

............ ..... . ... ........... .. .... .... ....................... ........ ..-............. ...........* ...... ......... ... .... .. ... o . ... ..........,.

a 0.2 0.4 0.6 D.8

NON-DIMENSIONAUSED TIME

(b) Time History of the Lift Coefficients.

Figure 13 (Cont'd)

66

... i.. :..

I I t

..... ... .... ...... ........ ......... I- ......... .... ..... ........._

d . . .. ... ... .. ... ... ... . i .. ... .. ...4 ..

. . ... .. .. .. . . .

. r... -. . .. .,! ... . .-.. ..... ... ......... .......... .......... . ...... i ....... ........ ........ ...........

12 1

S- - -.......... .......... A . -.. ....... .- .........

0 1 2 3 4 5

NON-DIMENSIONAUSED lIME

(a) Time History of the Lift Coefficients.

Figure 14. Step Change in Angle Of Attack: Step change in AOA for an 8.4per-cent thick Von Mises airfoils placed at 2 chord length vertical dis-tance with initial AOA = 0.0 radians and final AOA = 0.8 radians

pivoting at the leading edges.

67

.-. * '~. -

C , ... . ....

1 2 4

(b Tim Hitr of th2oetCofiins

Fiur 24 (Cnt

I ~ ~ 68

....... 4 ......

...... ...

0 1 2 3 4 5

NON-DIMENSIONALISED TIME

(c) Time History of the Drag Coefficients.

Figure 14 (Cont'd)

69

ai

. . . .. . . . ..

. ... .... ...... . . .. .. .... ..... ......

.......... ......... ...... .. .......... 4 ................... .........

........ ..................... ..... ... ............ .................... ..........

0 1 2 a 4 5

NON-DIMENSIONALISED TIME

(d) Time History of the Circulation

Figure 14 (Cont'd)

B. OTHER SUB-CASES

The reader is referred to the results and disussions of Reference I for the com-

parison of the single airfoil case with existing codes in the phase and magnitude re-

lationship of the aerodynamic coefficients to the forcing function as well as the wake

convection. The present results show that the same trend is maintained as in the single

airfoil case with no significant phase shift but with a general magnitude change due to

an effective ground effect caused by the presence of the second airfoil.

70

-.. . " L . . . . i . . i .. .a" '

L

I. Modified Ramp Change in Angle of Attack

As for the single airfoil, the modified ramp is defined mathematically as follows:

0 t<O

a(t) = 6a(3-2t)t 2 /T 2 0 < t <r T (5.2)6a t > T

where 6a is the magnitude of the AOA change and T is the rise time for the AOA to

reach its final value. Figure 15 treats the case of the 2 airfoils undergoing an out-of-

phase modified ramp change in the angle of attack of 0.1 radians with a time constant

of 1.5. The plots show the effects on the aerodynamic coefficients due to the presence

of the second airfoil. The corresponding aerodynamic coefficients of the single airfoil

are plotted for comparison purposes.

2. Translational Harmonic Motion

The code is capable of computing the unsteady flow solution for any general

translational motion described by a chordwise and a transverse component bearing a

given phase relationship with the restriction that the 2 airfoils move only in-phase or

out-cf-phase. The translational harmonic motion is described by

hv(t) = 6hy Sin (cot)

h,(t) = 6h. Sin (cot + i.) (5.3)

where co is the oscillation frequency, . is the phase angle between the chordwise and

transverse oscillation and 6h, and 6h, are the magnitudes of chordwise and transverse

oscillations respectively. The case-run considered in this section relates to a pure heav-

ing or plunging motion. A NACA-0015 airfoil is chosen for the case-run. The airfoils

are set at zero radian angles of attack and subsequently given an out-of- phase plunging

oscillation at an amplitude of 0.018 chord length at a non-dimensionalised frequency of

I. Figure 16 shows the aerodynamic coefficients for the two airfoils and compares them

with the single airfoil case where applicable. Note that the plots for the trailing wake

uses different scales for th,; x and y axes.

3. Rotational Harmonic Motion

The treatment of the harmonic pitching motion is similar to the modified ramp

case. As for the other sub-cases, the airfoils are restricted to in-phase and out-of phase

71

motion. The case of the out-of-phase motion will be treated here. The harmonicpitching oscillation is described by:

e(t)= 6a Sin (wt) (5.4)

where ba and co are the amplitude and frequency of the harmonic oscillation respec-

tively. Figure 17 shows the results of the 8.4% thick Von Mises symmetric airfoil os-cillating at an amplitude of 0.1 radian at a reduced frequency of Wc/V. = 20.0 about the

leading edge. Again the plots are given together with the single airfoil case undergoingthe same motion for comparison purposes.

72

..................................

L~~~~ .... . ....... i.,_l ., .. .

i

.. .. . ....... o o~ , - . ..... ........ T. .. ..... ... ......... .... . .. .. ..

i 4 " " "..-

.4.

S 0.4 0.6 1.2 1.6 2.0

NON-DIMENSIONAUSED TIME

(a) Time History of the Lift Coefficients.

Figure 15. Modified Ramp Change in Angle Of AttacLk Ramp change in AOAfor an 8.4 per-cent thick Von Mises airfoils placed at 1 chord lengthvertical distance with initial AOA = 0.0 radians and final AOA = 0.1

radians, rise time of 1.5 pivoting at the mid chord.

73

r T

.... ..... .. .. .. .......

....... . ........

c ; -----

..... a.....t.....

a 0.4 0.5 1.2 1.6 tO0NON-DIMENSIONAUSED TIME

(b) Time History of the Moment Coefficients.

Figure 15 (Cont'd)

74

.3 ......# ..... .... . .. ..... . ... T....

O L . . . .I

0 04 .5 .2 1.6 2.

SN1-D* ESONE IM

(caieHsoyo heDa ofiins

Figue 1p ....d

if75

.. .............. .........

r).., . ..... . .. ............

LO

4. ......... ... ....

a;.... .... .... ....

NON DIMENSIONALISED TIME

(d) Time History of the Circulation

Figure 15 (Cont'd)

76

. .........

...... ..... ... .. 4&

i 44

Ci/

(e) Pressure Distribution at tl-I/c -2.3

Figure 15 (Cont'd)

77

0 1 2 3 4X/C

(f) Wake pattern at tVj.c -2.3

Figure 15 (Cont'd)

78

d o *--.;i

C

............ .....~~...... ......... ... .......... ........ ..... .,........

.. ... . - - .-

.. .. ... ...... .... .. .. ... . .. .. ... ... . ... .. . .. .... ..

............ €............... ..-.. . ....-.-. -. -- .......... ........ ... ........... ..

to 4 5 12NON-DIMENSIONAUSED IME

(a) Time History of the Lift Coefficients.

Figure 16. Harmonic Plunging Motion: Translational harmonic plunging AOA

for NACA-0015 airfoils placed at 1 chord length vertical distance set

at AOA = 0.0 radians with plunging amplitude of 0.018 chord length

at a reduced frequency of 1.

79

- -.. - .. - - -

......... ..... ..... ........ ........ ........ ........ .......

~....... ......... S . ........

40 4 12NON-DIMENSIONAIJSED TIME

* (b) Time History of the Moment Coefficients.

Figure 16 (Cont'd)

80

.- 7.77*-

..............

.........

........ ... .. . .........

4 12

NOIDMNSOAUE TM

(c ieHsoyo heDa ofiinsFiue1 (otd

~~fN~J~8P

...K............. .................

ffSUAM

-I .W

0.... ..... .... n ....4. 06 0 ....6 .0

............/....c .. .......

(d). Prssr Ditrbuio at... ... .... 12.6..... ...

Figursre 6 tibto atCont'd)2.

82

0 4 B 12

x/C

Note : Different scales for x and v axes(e) Wake Pattern at f V c =12.6

Figure 16 (Cont 'd)

83

T ..... ....

. -.. . . . ......... . . .....

0 12 3NON-DIMENSIONALISED TIME

(a) Time History of the Lift Coefficients.

Figure 17. Harmonic Pitching Motion: Rotational harmonic pitching AQA forNACA-0015 airfoils placed at I chord length vertical distance set atAOA = 0.0 radians with pitching amplitude of 0. 1 radian at a reducedfrequency of 4 pivoting about the leading edges.

84

. ...........-......... ......-

..........-

2 3

NON-DIMENSIONAUSED TiME

(b) Time History of the Moment Coefficients.

Figure 17 (Cont'd)

85

...... ... 1... ..... . ...

.......

102

NON-DIMENSIONAUSED TIME

(c) time History of the Drag Coefficients.

Figure 17 (Cont'd)

86

712

-,-. - -- ---- --

..... . . .

10 12 3

NON-DIMENSIONAUSED TIME

(d) Time History of the Circulation.

Figure 17 (Cont'd)

87

... .- ... .. .

0 0. 0. 0. j. .

C.'C

(e) Pressure Coefficient at ti,. 3.0.

Figure 17 (Cont'd)

0 C -

0 1 2 3 4 5

(f) Wake pattern at tv'. -3.0.

Figure 17 (Cont'd)

89

VI. CONCLUSION

A. GENERAL COMMENTSUSPOTF2 has been developed as an intermediate stage to getting a solution for a

full cascade undergoing unsteady motion. In itself, it can be used to simulate unsteady

wing--tail, wing--canard, wing--flap, wing--aileron, horizontal tail--elevator, vertical

tail--rudder and a whole host of moving--stationary airfoil interactions. These simu-

lations require a little addition to the main program and would appear as modules

modelling the variation of the global angles of attack and relative displacements as per

all the sub-cases done. The subroutines would not be affected by the severity of the test

case except for possibly subroutine NEWPOS.Validation of USPOTF2 has been done against Giesing's code for the particular case

of a step change in angle of attack. This was shown to have good correlation. However,

it can not be said that the agreement will hold for other angles of attack and more sub-

case runs for smaller angles of attack should be made. In the same manner, the success

of the step change in AOA in no way validates the other sub-cases which, for the time

being. remain unproven.

B. ENHANCING USPOTF2 PROGRAM CAPABILITY

As noted above, USPOTF2 needs to be tried more extensively either with the exist-

ing sub-cases or with new sub-cases against existing numerical or experimental results.

The software in itself has an implicit weakness in the numerical computation of thevelocity potential. The velocity potential at the leading edge is obtained by integrating

the velocity field from the leading edge to a point 100 chord lengths upstream of the

leading edge. Two assumptions are made: First, the disturbance velocity at the point100 chord lengths of the leading edge approaches zero. Second, the contribution to thevelocity potential due to the integration from the point 100 chord lengths upstream of

the leading edge to upstream infinity must not change in time. The coefficients of pres-

sure will be accurate only, if those two assumptions hold (at least in an approximate

sense). This however does not give a very 'exact' solution as by increasing the chord

length by 900 per-cent to 1000 chord lengths, there is a change in the pressure coefficient

at the trailing edge of about 8 per-cent for the first time step. Krainer has implemented

an analytical solution to the potential problem for the single airfoil. A similar procedureto upgrade USPOTF2 should be considered.

90

-; - -- t-

Other improvements to the code would be the reduction of redundant computations.

An obvious example would be the subroutines VELDIS and PRESS which essentially

do the same work for the steady and unsteady flow respectively. Again Krainer [Ref.

6] has improved the original code for the single airfoil (U2DIIF) and though some of

his improvements were implemented in USPOTF2, there still remains a task to do the

full job completely.

Finally, the original primary objective of extending the code to solve for the un-

steady flow solution for 3,4 airfoils and leading to a full cascade still remains a big task,

not just for the programmer but also for the computer system in terms of computational

storage and time requirements.

t

91

7i-.. ~ . .

- .- . .o

APPENDIX A. USPOTF2 SOURCE LISTINGScCCCCCccccccccccCcCCCCCCcccCCCccCcccCCccccccccccCccCCccCCCCccCcc

C CC PROGRAM USPOTF2 CC VE R SIO0N 1 CC SEPTEMBER 88 CC CC UNSTEADY MOTION FOR TdO AIRFOILS IN POTENTIAL CC INCOMPRESSIBLE FLOW CC USING PANEL METHODS BASED ON THE HESS & SMITH CC AND WAKE MODEL BASED ON BASU & HANCOCK CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

COMMON /BOD/ IFLAG,NLOWER,NUPPER,NODTOT,X(202) ,Y(202),+ COSTHE(201),SINTHE(201),SS(2),NPI,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFT,NAIRFO,XI(202),YI(202),+ COSTHL(201),SINTHL(201)

COMMON /WAK/ VYW(2),VXW(2),WAKE(2),DTCOMMON /WAK2/ VYWK(2),VXWK(2)COMMON /SING/ Q(200) ,GAMMA(2) ,QK(200) ,GAMK(2)COMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TD,CVVX(2,200),

+ CVVY(2,200),XCI(2,200),YCI(2,200)COMMON /POT/ PHI(200),PHIK(200)COMMON /GUST/ UG( 200) ,VGC 200) ,XGF,UGUSTVGUSTCOMMON /EXTV/ UE(200),VN(200)COMMION /PARD/ NACAD(2),TAUD(2),EPSMAD(2),PTMAXD(2)COMMON /COF/ A(201,211),KEQNSCOMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2)IUX(2),UY(2),PIVOT,

+ XPRM,YPRMCOMMON /CPD/ CP(200),SCL,T,SCM,SGAMCOMMON /GIES/ NGIESDIMENSION XXC(2,200),YYC(2,200),TOL1(2),TOL2(2),THENP1(2),

+SINANG(2) ,COSANG(2) ,COSDA(2) ,SINDA(2) ,DHX(2) ,DHY(2) ,ALP(2),+ALPHA(2) ,ALPI(2) ,ANGLE(2) ,DELHX(2) ,DELHY(2) ,PHASE(2) ,PHA(2),+HX(2) ,HXO(2) ,HY(2) ,HYO(2)

C*C INITIALISATION

CPi 3. 1415926585M -oXPRM =0.0YPRM 0.0

CC INPUT FROM FILE CODE 5 AND SET UP LOCAL PANEL NODES AND SLOPESC

WRITE (6,1003)1003 FORMAT (ffI'DATA READ FROM FILE CODE 1',//)

CALL INDATACALL SETUPREAD (1,502) ALPI(1),ALPI(2),DALP,TCON,FREQ,PIVOT,IPHASEWRITE (6,502) ALPI(1),ALPI(2),DALP,TCONFREQ,PIVOT,IPHASE

501 FORMAT (8F10.6)

92

502 FORMAT (6F10.6,I3)READ (1,501) UGUST,VGUST,DELHX(1),DELHiXC2),DELHY(l),DELHY(2),

+ PHASE(l),PHASE(2)WRITE (6,501) UGUST,VGUST,DELHX(1),DELHX(2),DELYY(l),DELHY(2),

+ PHASE(1),PHASE(2)

READ (1,503) TF,DTS,TOL,TADJ,SCL,SCI,SGAI,NGIESWRITE (6,503) TF,DTS,TOL,TADJ,SCL,SCM,SGAM,NGIES

503 FORMAT (7F10.6,13)IF (IFLAG .EQ. 0) WRITE (6,1005)

1005 FORMAT i/I'COORDINATES OF AIRFOIL NODES',+ //,3X, X/C',6X,' Y/C',/)

IF (IFLAG .EQ. 0) WRITE (6,1010) (XI(I),YI(I)I=1,2*(NODTOT+1))1010 FORMAT (F1O.6,F1O.6)

WRITE (6,1000) NAIRFO1000 FORMAT(//,'TOTAL NO OF AIRFOILS - '.14,/)

CC STEADY FLOW CALCULATION AT ALPI( ICC INITIAL DEFINITIONC

NP1 =NODTOT + 1NP2 = 2 *NP1+ 1NP3 - NAIRFO"NODTOT+1NP4 = NP3+1NP5 - NP4+1DO 101 I = 1,2ALPHA(I) = ALPICI)IF (ALPHA(I) .GT. 90.) GO TO 200COSALF(I) - COS(ALPHA(I)*PI/180.)

101 SINALF(I) =SIN(ALPHA(I)*PI/180.)CALL NEWPOS(0)DO 1100 L - 1,NAIRFO

1100 WRITE (6,1020) L,SS(L)WRITE (6,1040) XSHIFT,YSHIFT

1020 FORMAT(//, AIRFOIL(' 12 ') PERIMETER LENGTH *',FlO.6,/)1040 FORMAT(//:: XSHIFT = I,F5.1, YSHIFT ',F5.1,/)

DO 102 I = 1,2102 WRITE (6,1030) I,ALPHA(I)1030 FORMAT (//,' STEADY FLOW SOLUTION AT ALPHA(',12,') - ',F1O.6,/)

IF (SCL.NE.O) WRITE (8,1035)1035 FORMAT(3X,'TIME',4X,'CL(1)/SCL',lX,'CL(2)/SCL',lX,'Cm(1)/SCM',

+1X,'CM(2)/SCM',3X,'CD(1)',4X,'CD(2)',4X,+'GAMK(1)/SG' ,1X,'GAMK(2)/SG' ,f/)IF (SCL.EQ.0) WRITE (8,1036)

1036 FORMAT(3X I'TIMIE',6X : CL(1)I 5x,'CL(2)',5X,'CM(l)',+5X,'CM(2) ,5X,'Cb(1)i4X, CD(2)'//)CALL INFL (0)CALL COEF (0)CALL GAUSS(3,0,0)CALL KU'TTA( NITR ,PVTAG)CALL VELDISSINi SINALF(1)SIN2 - SINALF(2)COSi - COSALF(1)COS2 aCOSALF(2)

93

CALL FANDMCSIN1,SIN2,COS1,C052)

C INITIALISATION FOR UNSTEADY FLOW CALCULATION TO BEGIN

DA = 0.0XGF = 0.0DO 100 L - 1,NAIRFOHX(L) = 0.0HYCL - 0.0HXO(L) = 0.0HYO(L) - 0.0ANGLECL) = ALPICL)*PI/180. +ATAN(VGUST/(l.+UGUST))OMEGA(L) - 0.0ALP(L) = ALPICLDHX(L) - 0.0DRY(L) = 0.0COSANG(L) = COS(ANGLE(L))SINANG(L) = SIN(ANGLE(L))UX(L) = 0.0UYCL) = 0.0COSDA(L) = 1.0SINDA(L) = 0.0KIG = (L -1)*NODTOTVXW(L) - COSALF(L)VYW(L) =SINALF(L)GAMK(L) = GAMMA(LPHA(L) = PHASE(L)*PI/180.DO 100 IG =1,NODTOTUG(IG+KIG) =0.0

100 VG(IG+KIG) =0.0T =0. 0TOLD 0.0

CC RIGID BODY MOTIONS OF AIRFOILC

IF (FREQ .NE. 0.0) GO TO 1IF (DALP .EQ. 0.0) GO TO 2IF CTCON .NE. 0.0) GO TO 3IF (IPHASE .EQ. 0) GO TO 4ALPHA(1)= ALPI(1) + DALPALPHA(2)= ALPIC2) + DALPGO TO 5

4 ALPHA(1)= ALPI~i) + DALPALPHA(2)= ALPI(2) - DALP

5 D061 =1,2COSALF(I) -COS(ALPHA(I)*PI/180.)SINALF(I) - SINCALPHA(I)*PI/180.)

6 CONTINUECALL NEWPOS( 0)

3 DT m DTSTD =DTSGO TO 60

2 IF ((UGUST .EQ. 0.0) .AND. (VGUST .EQ. 0.0)) GO TO 200DT -DTSTD = DTSGO TO 60

94

1 DT - 2.0*PI/(FREQ*DTS)TD - DT

60 T = DTWRITE (6,1051)

1051 FORMAT (IIl,' ,+ ' ** BEGIN UNSTEADY FLOW SOLUTION ****+1 . . ••. . ...A~ AA A .. A.Lo A, . A L . •. • • J

TRY = 040 M =M+ 1

IF (T .GT. TF) GO TO 200CC STORE CORE VORTEX COORDINATES FOR TIME STEP ADJUSTMENTSC

IF (M .EQ. 1) GO TO 50DO 51 L - 1,NAIRFODO 51 I - 1,M-1XXC(L,I) - XCI(L,I)

51 YYC(L,I) = YCI(L,I)50 IF (FREQ .NE. 0.0) GO TO 11

IF (DALP .EQ. 0.0) GO TO 22IF (TCON .NE. 0.0) GO TO 32

CC STEP CHANGE IN AOAC

IF (TADJ .NE. 0.0) GO TO 70C IF INCREMENTAL PROGRESSIVE TIME STEP IS REQUIREDC USE ... TD = FLOAT(M+1)*DTS ... OTHERWISE CONSTANT TIME STEP

TD =DTSGO TO 70

CC MODIFIED RAMP CHANGE IN AOAC33 IF (T .GT. TCON) GO TO 34

DAL = DALP * (3. -2.*T/TCON)*(T/TCON)**2OMEGA(1) = - (DALP*PI/180.) * (6.*T/(TCON*TCON)) * (1.-T/TCON)IF (IPHASE .EQ. 0) GO TO 41ALPHA(1)= ALPI(1) + DALALPHA(2)= ALP1(2) + DALOMEGA(2) = OMEGAM1)GO TO 42

41 ALPHA(1)= ALPI(1) + DALALPHA(2)= ALPI(2) - DALOMEGA(2) = - OMEGA(1)

42 DO 43 1 - 1,2COSALF(I) = COS(ALPHA(I)*PI/180.)SINALF(I) = SIN(ALPHA(I)*PI/180.)DA = ALPHA(I) - ALP(I)COSDA(I) = COS(DA*PI/180.)SINDA(I) - SIN(DA*PI/180.)DHX(I) = PIVOT * (1.-COSDA(I))DHY(I) = - PIVOT * SINDACI)UY(I) = PIVOT * OMEGA(I)

43 CONTINUEMTCON = MCALL NEWPOS(O)GO TO 70

95

. ...... . . .. . .. . " ' " " " ' "' ... . " : . . . "

34 DAL =0. 0IF (IPHASE -EQ. 0) GO TO 45ALPHA(1)= ALPIMi + DALPALPHA(2)= ALPI(2) + DALPGO TO 46

45 ALPHA(1 = ALPI~i) + DALPALPHA(2) = ALPI(2) - DALP

46 DO044 1=1,2COSALF(I) = COS(ALPHA(I)*PI/180.)SINALECI) = SIN(hLPHA(I)*PI/180.)DA = 0.0COSDA(I) = 1.0SINDACI = 0.0OMEGA(I) = 0.0DHXCI) = 0.0DHYCI) = 0.0UXCI) = 0.0UY(I) = 0.0

44 CONTINUECALL NEWPOS(0)IF (TADJ .NE. 0.0) GO TO 70TD = FLOAT(M+1-MTCON)*DTSGO TO 70

CC SHARP EDGE GUST (UGUST AND/OR VGUST) NOT PROVEN DUE TOC INCONSISTENT ASSUMPTIONS REQUIRING ROTATIONAL FLOWCC22 XGF =T

DO 113 L = 1,NAIRFOLIG = (L-1)*NODTOTKIG = (L-1)*NP1DO 110 IG =1,NODTOTUG(IG+LIG) =0.0

VG(IG+LIG) =0.0

XG = X(IG+KIG)XGP1 = X(IG+KIG+l)IF (IG .LT. NLOWER+1) GO TO 120IF (XGF .LE. XG) GO TO 110IF (XGF .GE. XG11) GO TO 111FAC = (X0F - XG)/(XGP1 - XG)UG(IG+LIG) = UGUST*eFACVG(IG+LIG) = VGUST*FACGO TO 110

111 UG(IG+LIG) = UGUSTVG(IG+LIG) = VGUSTGO TO 110

120 IF (XGF .LE. XGP1) GO TO 110IF (XGF .GE. XG) GO TO 121FAC = (XGF - XGP1)/(XG -XGP1)

UG(IG+LIG) = UGUST*FACVG(IG+LIG) = VGUST*FACGO TO 110

121 UG(IG+LIG) = UGUSTVG(IG+LIG) = VGUST

110 CONTINUE

96

113 CONTINUEIF (XGF .LE. COSALF(L)) MGUST MIF (TADJ .NE. 0.0) GO TO 70IF (XGF .GT. COSALF(L)) TD = FIOAT(l1+1-MGUST)*DTSGO TO 70

CC TRANSLATION HARMONIC OSCILLATIONC11 IF (DALP .NE. 0.0) GO TO 12

DO 131 I1 1,NAIRFOHX(I) -DELHX(I) * SIN(FREQ*T + PHA(I))KY(I) =DELHY(I) * SIN(FREQ*T)DHX(I) -HX(I) - HXO(I)DHY(I) -HY(I - HYOCI)UX( I) =DELHX( I)*FREQ*COsCFREQ*T+PHAc I))UY(I) =DELRY(I)*FREQ*COS(FREQ*T)

131 CONTINUEXPRM = DHX(2) - DHX(l)YPRI = DHY(2) - DHY(l)CALL NEWPOS(O)GO TO 70

CC ROTATIONAL HARMONIC OSCILLATIONC12 DAL =DALP*SIN(FREQ*T)

OMEGA~l) =- (DALP*PI/18o.) *FREQ *COS(FREQ*T)IF (IPHASE .EQ. 0) GO TO 141ALPHA(1)= ALPI(1 + DALALPHA(2)= ALPI(2) + DALOMEGA(2) =OMEGA(l)GO TO 142

141 ALPHA(1)= ALPI(1 + DALALPI{A(2)= ALPI(2) - DALOMEGA(2) - OMEGA~l)

142 DO 143 I 1,2COSALF(I) = COS(ALPHACI)*PI/18o.)SINALF(I) = SIN(ALPI{A(I)*PI1180.)DA ALPI{A(I - ALP(I)COSDA(I) =COS(DA*PI/18o.)SINDA(I SIN(DA*PI/ 180.)DI{X(I) =PIVOT * (1. -COSDA(I))DHY(I) -PIVOT * SINDAMIUY(I) =PIVOT * OMEGA(I)

143 CONTINUECALL NEWPOS(0)

CC TRANSFORM CORE VORTEX COORDINATES W. R. T. NEW AIRFOIL POSITIONC

70 IF (M.-EQ. 1) GOTO 80DO 85 L - 1,NAIRFODO 90 1Ial,M-1XCI(L,I) a XXC(L,I) + CVVX(L,I) * DTYCI(L,I) - YYC(L,I) + CYYLI) * DTXCO - XCI(LI)YCO = YCI(L,I)XCI(L,I) =XCO*COSDA(L) -YCO*SINDA(L) + DHXCL)

97

----- ' :" 7,r

90 YCI(L,I) =XCO*SINDACL) + YCO*COSDA(L) + DI{Y(L)85 CONTINUE

CALL NEWiP0S(3)80 CONTINUE

WRITE (6 1001) T,DT1001 FORMAT (;////I, TIME STEP TK = ',FlO.6,10X,'TK - TKM1 =',F1O.6,/)

WRITE (6,1004) ALPHA(1),ALPHA(2),MEGA(1),OMEGA(2),X(l)U(2),+1 UY(l),UY(2)

1004 FORMAT (/,' ALPHA(1) = ' ?F10.6,SX,' AI.PIA(2) a',FIO.6,/,+OMEGA~i) - ',F10.6,5X, OIIEGA(2) = ',F1O.6,

+/., U~i) = 1,FlO.6,SX,' U(2 ?+F1O.6,/,' V(1 = I F1O.6,5X, V(2) -,F10.6,///," 1X,,' NITR VXW(1) VYW(1) WAKE(1) THETA(1 GAMK(l)" VXW(2) VYW(2) WAKE(2) THETA(2) GAMK(2) '/

CC CALCULATE, THE TRAILING EDGE WAKE ELEMENTC

NUM = 0NITR = 0

10 DO 15 L = 1,NAIRFOKI = (L-1)*(NODTOT+l)WAKE(CL) = SQRTC VYWC L)*VYW( L)+VXW( L)*VXd( L) )*DTTHENP1(L) = ATAN2(VYW(L),VXWCL))COSTHLCNP1+KI) = COS(THENPI(L))

15 SINTHLCNP1+KI) - SINCTHENP1(L))WRITE (6,1002) NITR,VXW(1),VYW(),WAKEC),HENP1(1)GM(l),

+VXW(2) ,VYW(2) ,WAXEC2) ,THENP1(2) ,GAMK(2)1002 FORMAT C15,4F10. 6,E14. 6,4F10. 6,E14.6)

XI(NP2) = XI(NP1) + WAKEC1)*COSTHL(NPl)YI(NP2) - YI(NP1) + WAKEC1)*SINTHLCNPl)XI(NP2+1) = XICNPI) + WAKE(2)*COSTIL(2*NPI)YI(NP2+1) = YI(NP1) + WAKEC2)*SINTHLC2*NP1)CALL NEWPOS( 1)CALL INFL CNITR)CALL COEF (NITR)CALL GAUSSC3IM,NITR)

C WRITE C6,*) A(I NP)',(A(I,101),I=1,100)C WRITE (6,*) 'ACINTOT)',(ACI,102),I=l,100)C WRITE (6,*) BEFORE KlrrrA'

CALL JUTTA(NITR,PVTAG)IF (PVTAG .LT. 0.01) GOTO 13NUN - NUN + 1

C WRITE (6,*)'NUM =',NUMIF (NUM .GT. 1) GOTO 13DO 7 I - 1,2VXW(I) - 1

7 VYW(I) =0GOTO 10

13 CALL TEWAKDO 24 L = 1,NAIRFOTOLI(L) - ABS(VYW(L) - VYWK(L))fVYhlK(L)TOL2(L) = ABS(VXW(L) - VXWK(L))/VXWK(L)VYW(L) = VYWK(L)

24 VXW(L) - VXWK(L)IF ((TOLi(i) .LT. TOL) .AND. (TOL2(1) .LT. TOL)

+.AND. (TOL1(2) .LT. TOL) .AND. (TOL2(2) .LT. TOL)) GO TO 20

98

....%.....' ---- - -- -

IF (NITR .GT. 25) THENTOL - TOL*10WRITE (6,1023) TOLENDIFIF (NITR .GT. 50) STOPNITR - NITR + 1GO TO 10

20 WRITE (6,1011) NITR1011 FORMAT (//,' CONVERGED SOLUTION OBTAINED AFTER NITR - ',13)1023 FORMAT (//,' ***** TOLERANCE CRITERIA CHANGED TOL = ',F10.6)

CALL PRESSIF ((UGUST .EQ. 0.0) .AND. (VGUST .EQ. 0.0)) GO TO 300SIN1 - SINANG(1)SIN2 - SINANG(2)COS1 - COSANG(1)COS2 - COSANG(2)CALL FANDM(SIN1,SIN2,COSI ,COS2)GO TO 400

300 SINI - SINALF(1)SIN2 - SINALF(2)COS1 = COSALF(1)COS2 = COSALF(2)CALL FANDM(SIN1,SIN2,COS1,COS2)

400 CONTINUECC ADJUST TIME STEP (TADJ .NE. 0.0) IF NECESSARYC

IF (TADJ .EQ. 0.0) GO TO 95WRITE (5,2001)

2001 FORMAT (//,' DO YOU WANT TO ADJUST TIME STEP ? 0 - NO, 1 - YES')READ (5,*) IDTIF (IDT .EQ. 0) GO TO 95DT - TADJ * DTT - TOLD + DTWRITE (6,1006)

1006 FORMAT (//,' BACK-TRACK COMPUTATION AND ADJUST TIME-STEP' ,//)CC WAKE ELEMENT LEAVES TRAILING EDGE AS A CORE-VORTEXC95 DO 96 L - INAIRFO

CV(L,M) = SS(L)*(GAMMA(L)-GAMK(L))CVVX(L,M) - VXW(L)

96 CVVY(L,M) - VYW(L)XCI(1,M) - XI(NP1) + 0.5*WAKE(1)*COSTIL(NP1)YCI(1,M) - YI(NP1) + 0.5*WAKE(1)*SINTHL(NP)XCI(2,M) - XI(NP1) + 0.5*WAKE(2)*COSTHL(NPI*2)YCI(2,M) - YI(NP1) + 0.5*WAKE(2)*SINTHL(NP1*2)CALL NEWPOS(2)WRITE (6,1052)

1052 FORMAT (//,' TRAILING VORTICES DATA',//,+ 4X 'M',4X 'Xli(m)'SX, 'YII(M)' 6X 'Xl(M)' ,6X,'YI(M)',+ 5X,'CIRCI ,6X,'X21(M) 5X 1 'Y21(M)",+ SX,'X2(M) ',6X,'Y2(M) ,6X, CIRC2',/)DO 900 I - 1,M

900 WRITE (6,1050) I,XCI(1,I),YCI(1,I),XC(I,I),YC(1,I),CV(1,I),+ XCI(2,I),YCI(2,I),XC(2,I),YC(2,I),CV(2,I)

99

1050 FORMAT(15,10F11. 6)CALL CORVOR

CC RE-INITIALISE PARAMETERS FOR NEXT TIME STEP CALCULATIONC

DO 30 L 1,NAIRFOLI =(L-1)*NODTOTHXO(L) =HX(L)HYO(L) =HY(LGAMMAL)- GAMK(L)ALP(L) - ALPHA(L)DO 30 1 1,NODTOTPHI(I+LI) -PHIK(I+LI)

30 CONTINUETOLD =TDT aTDT - T+ TDGO TO 40

200 WRITE (6,1024) TOL1024 FORMAT (1X,/'***** TOLERANCE CRITERION USED: TOL ',F1O.6)

STOPEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCccCcC CC SUBROUTINE INDATA CC CC SET PARAMETERS OF BODY SHAPE CC FLOW SITUATION, AND NODE DISTRIBUTION CC CC USER MUST INPUT CC NLOWER = NUMBER OF NODES ON LOWER SURFACE CC NUPPER - NUMBER OF NODES ON UPPER SURFACE CC PLUS DATA ON BODY AND SUBROUTINE BODY CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC NACAD(I) .... NACA NUMBERS FOR THE TWO AIRFOILSC TAUD(I) .... MAX THICKNESS FOR THE IWO AIRFOILSC EPSMAD( I) ... MAX CAMBER FOR THE TWO AIRFOILSC PTMAX(D(I) .. CHORDWISE POSN FOR MAX CAMBER FOR THE TWO AIRFOILSC

SUBROUTINE INDATADIMENSION TITLE(20)COMMON /BOD/ IFLAG,NLOWER,NUPPE,tODTOT,X(202),Y(202).

+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,lqP4,+ NP5 ,XSHIFT,YSHIFT,NAIRFO,XI( 202) ,YI( 202),+ COSTHL(201),SINTHL201)

COMMON /PARD/ NACAD(2),TAUD(2),EPSHAD(2),PFhAX(2)READ (1,501) ITITLEWRITE (6,301) ITITLEDO 10 I w 1,ITITLEREAD (1,502) TITLE

10 WRITE (6,503) TITLE501 FOR1IAT(3I5)502 FORMAT(20A4)503 FORMAT(1X,20A4)

100

..... ...................

504 FORMAT(15,2F10. 6)READ (1,501) IFLAG,NLOWER,NUPPERWRITE (6,501) IFLAG,NLOWER,NU PPERREAD (1,504) NAIRFO,XSHIFT,YSHIFTWRITE (6,504) NAIRFO,XHIFT,YSHIFT

-' IF (IFLAG .NE. 0) RETURNCC COMPUTATION FOR NACA 4-DIGITS SERIESC

READ (1,501) NACAD(I)WRITE (6,501) NACAD(1)READ (1,501) NACAD(2)WRITE (6,501) NACAD(2)DO 100 I-1,NAIRFOIEPS - NACAD(I)/1000IPTMAX - NACAD(I)/100 - 1O*IEPSITAU - NACAD(I) - 1000*IEPS - 100*IP~hAEPSMAD(I) -IEPS*0.01PTMAX(I) - IPTMAX*0. 1TAUD(I) - ITAU*O.01IF CIEPS .LT. 10) GOTO 100

CC COMPUTATION FOR NACA 5-DIGITS SERIES NOTING THAT TAUD(I) ISC COMPUTED AS PER 4-DIGITS SERIES.C

PTMAXD(I) -0. 2025EPSMAD(I) 2.6595*PTMAXD(I)**3

100 CONTINUERETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE SETUP CC CC SETUP COORDINATES OF PANEL NODES AND SLOPES OF PANELS CC COORDINATES ARE READ FROM INPUT DATA FILE UNLESS CC THE AIRFOIL IS OF NACA X)0C OR NACA 230XX TYPE CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC NACAC I) .... DUMMY NACA NUMBER FOR TRANSFER TO SUBROUTINESC BODY AND NACAC TAU( I) .... DUMMY MAX THICKNESS FOR TRANSFER TO SUBROUTINESC BODY AND MACAC EPSMAX( I) .... DUMMY MAX CAMBER FOR TRANSFER TO SUBROUTINESC BODY AND NACAC PTKAX( I) .... DUMMY CHORDWISE POSH FOR TRANSFER TO SUBROUTINESC BODY AND MACAC

SUBROUTINE SETUPCOMMON /BODI IFLAG,NLOWER,NUPPER,NODTOTX(202) ,Y(202),

+ COSTHE(201),SINTHE(201),SS(2),NP,NP2,NP3,NP4,+ NP5 ,XSHIFT,YSHIFT,NAIRFO,XI(202) ,YI(202),+ COSTHL(201),SINTHL(201)COMMON /PARD/ NACAD(2) ,TAUD(2) ,EPSMAD(2) ,PTMAXD(2)COMM4ON /PAR/ NACA,TAU,EPSMAX,PThAX

101

COMMON /NUM/ PI,PI2INVPi - 3. 1415926585P121NV - .5/PI

CC SET COORDINATES OF NODES ON BODY SURFACEC

DO 210 I-1,NAIRFONODTOT - NLOWER + NUPPERKi(I-1)*(N0DTOT+1)IF (IFLAG .NE. 0) GO TO 10

CC NACA SERIES AIRFOIL CALCULATIONSC

NACA-NACAD( I)TAU-TAUD( I)EPSHAX-EPSMAD( I)PTIIAX-PTMAXD( I)NPOINTl - NLO)WERSIGN = -1.0NSTART = 0DO 110 NSURF - 1,2DO 100 N - 1,NPOINTFRACT - FLOAT( N-I) /FWAT( NPOINT)Z - .5*(1. - COS(PI*FRACT))J = NSTART +NCALL BODY(Z,SIGN,XD,YD)XI(J+K) = XDYI(J+K) = YD

100 CONTINUENPOINT - NUPPER.SIGN = 1.0NSTART = NLOWER

110 CONTINUEXI(I*(NODTOT+1)) - XI(1+K)

120 YI(I*CNODTOT+1)) - YI(1+K)GO TO 210

CC AIRFOIL DATA INPUT THAT ARE NOT NACA SERIES AIRFOIL.C

10 READ (1,501) (XI(J+K),J=1,NODTOT+1)READ (1,501) (YI(J+K),Jinl,NODTOT+1)WRITE (6,501) (XI(J+K),J-1,NODTOT+1)WRITE (6,501) (YI(J+K),J-1,NODTOT+1)

501 FORMIAT (6F10.6)210 CONTINUE

NP1 - NODTOT + 1

NP2 - NAIRFO*NP1+1C STSOE FPNL N ACLT IFI EIEEC STSOE FPNL N ACLT IFI EIEE

DO20I 1NCF

K - (I-1)*(NODTOT41)f55(I) - 0.0DO 200 J - 1,NODT7OTDX - XI(J+1+K) - XI(J+K)DY - YI(J+1+K) - YI(J+K)

102

DIST - SQRT(DX*DX +DY*DY)$S(I) - SS(I) + DISTSINTHL(J+K) = DY/DISTCOSTHL(J+K) - DX/DIST

200 CONTINUE220 CONTINUE

RETURNEND

CcCCcCCCCCcccCCCcCCCcccCCCCCCCcCCcccccCCCCcccccCcCCCCcccCCCCCCCCCCCCCcccCC CC SUBROUTINE BODY(Z,SIGN,X,Y) CC CC RETURN COORDINATES OF POINT ON THE BODY SURFACE CC CC Z - NODE-SPACING PARAMETER CC X,Y = CARTESIAN COORDINATES CC SIGN - +1. FOR UPPER SURFACE CC -1. FOR LOWER SURFACE CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

SUBROUTINE BODY( Z,SIGN,XD,YD)COMMON /PAR/ NACA,TAU,EPSMAX,PTMAXIF (SIGN .LT. 0.0) Z = 1. - ZCALL NACA45( Z,THICKCAMBER,BETA)XD = Z - SIGN*THICK*SIN(BETA)YD = CAMBER + SIGN*THICK*COS(BETA)RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE NACA45( Z,THICK,CAMBER,BETA) CC CC EVALUATE THICKNESS AND CAMBER CC FOR NACA 4- OR 5-DIGIT AIRFOIL CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

CC Z ...... ABSCISSA OF CAMBERLINE POINTC XY ...... LOCAL CARTESIAN COORDINATES OF NACA-AIRFOILC SIGN ...... SURFACE INDICATOR: SIGN - 1, UPPER SURFACEC --1, LOWER SURFACEC THICK ...... THICKNESS AT THE POSITION ZC CAMBER ...... CAMBER AT THE POSITION ZC BETA ...... ANGLE BETWEEN CAMBER LINE AND X-AXIS AT POSITION ZC TAU ...... MAXIMUM THICKNESS (INPUT)C EPSMAX ...... MAXIMUM CAMBER (INPUT)C PTMAX ...... COORDINATE POSITION OF MAXIMUM CAMBER (INPUT)C

SUBROUTINE NACA45( Z,THICKCAMBER,BETA)COMMON /PAR/ NACA,TAUEPSNAXPTHAXTHICK a 0.0IF (Z .LT. I.E-I0) GO TO 100THICK * 5.*TAU*(.2969*SQRT(Z) - Z*(. 126 + Z*(.3537

+ - Z*(.2843 - Z*. 1015))))100 IF (EPSMAX .EQ. 0.0) GO TO 130

103

:7 U, 7:31-- i I

IF (NACA .GT. 9999) GO TO 140

c CAMBERLINE OF NACA 4-DIGIT SERIESC

IF (Z .GT. PTMAX) GO TO 110CC FORWARD PART OF CAMBER LINEC

CAMBER a EPSMAX/PTMAX/PTMAX*(2. *PTnwc - Z)*ZDCAMDX - 2. *EPSMAX/PAX/JCMlW*( PTMAX - Z)GO TO 120

CC AFT PART OF CAIIBERLINEC110 CAMBER - EPSMAX/(1. -PTMAX)**2*C1. + Z - 2.*PTMAX)*C1. -Z)

DCAMDX a 2. *EPSMAX/( 1. PTMAX)**2*(PTMAX- Z)4120 BETA - ATAN(DCAIDX)

RETURN130 CAMBER - 0.0

BETA -0.0* RETURN

CC CAMBERLINE OF NACA 5-DIGIT SERIESC

140 IF (Z .GT. PTMAX) GO TO 150CC FORWARD PART OF CAMBER LINEC

W m Z/PTMAXCAMBER a EPSMAX~feW*((W - 3.)*W + 3. -PThAX)

DCAMDX - EPSMAX*3.*W*(1. - W)/PTMAXGO TO 120

CC AFT PART OF CAMBERLINEC150 CAMBER - EPSMAX*C1. - Z)

DCAIIDX - - EPSMAXGO TO 120END

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE GAUSS(NRHS,M,NITR) CC CC SOLUTION OF LINEAR ALGEBRAIC SYSTEM BY CC GAUSS ELIMINATION WITHOUT PARTIAL PIVOTING CC CC @A = COEFFICIENT MATRIX CC NEQNS - NUMBER OF EQUATIONS CC NRHS - NUMBER OF RIGHT HAND SIDES CC CC RIGHT-RAND SIDES AND SOLUTIONS STORLD IN CC COLUMNS NEQNS+1 THRU NEQNS+NRHS OF *A CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

SUBROUTINE GAUSS(NRHS ,M,INITR)COMMON /BOD/ IFLAG,NLOWER,NUPPERNODTOTX(202) ,Y(202),

104

-N-777-7 -,7,

+ COSTHE(2O1),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFT,NAIRFO,XI(202),YI(202).+ COSTHL(201),SINTHL(201)COMMON /COF/ A(201,211),KEQNS

C WRITE (9,900) M INITRC900 FORMAT (lX,M -* 13,' NITR -',13/)C WRITE (9,910)(I,A(I,NP3),A(I,NP4),A(I,NPS),I-1,2*NODTOT)910 FORMAT (lX1I5,3E14.6)

IF (M.EQ.O) KEQNS - NODTOTNEQNS - KEQNS * NAIRFONP -NEQNS+ 1NTOT - NEQNS + NRHSIF (NITR .GT. 0) GO TO 160

CC GAUSS REDUCTIONC

DO 150 I - 2,NEQNSIM mI- 1

CC ELIMINATE (I-1)TH UNKNOWN FROMC ITR THRU (NEQNS)TH EQUATIONSC

DO 150 J -I,NEQNSR = A(J,IM)/A(IM,IM)DO 150 K = I,NTOT

150 A(J,K) - A(JK) - R*A(IMK)GO TO 170

CC GAUSSIAN ELIMINATION ON ONLY THE RIGHT-HAND-SIDESC160 DO 180 1 - 2,NEQNS,

IM =I- 1DO 180 J = I,NEQNSR -A(J,IM)/A(IM,IM)DO 180 K = NP,NTOT

180 A(J,K) - A(J,K) - R*A(IM,K)170 CONTINUE

CC BACK SUBSTITUTIONC

DO 220 K =NP,NTOTA(NEQNSK) =A(NEQNS,K)/A(NEQNS,NE.QNS)DO 210 L =2,NEQNSI - NEQNS+ 1 -LIP - I+ 1DO 200 J - IPNEQNS

200 A(I,K) - A(I,K) - A(I,J)*ACJK)210 A(I,K) - A(I,K)/A(I,I)220 CONTINUE

RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcCCCCCCCCCCccC CC SUBROUTINE VELDIS(SINALF,COSALF) CC CC COMPUTE STEADY FLOW PRESSURE DISTRIBUTION AND VELOCITY C

105

~ *- -.

C POTENTIAL AT MID-POINTS OF PANELS FOR THE STEADY FLOW CASE CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC VTANG.......TANGENTIAL VELO COMP OF A FIXED POINT OF THE MOVINGC LOCAL FRAME OF REFERENCE.C PR1I) ...... DISTURBANCE VELO POTENTIAL AT THE MID POINT OF THEC I-TH PANELC PHILE(L) .... DIFFERENCE OF THE POTENTIALS OF THE LEADING EDGE TOC THE LOWER TRAILING EDGE FOR THE RESPECTIVE AIRFOILC PINCL) .... DIFFERENCE OF THE POTENTIALS AT A POINT 1000 CHORDC LENGTH UPSTREAM OF THE LE FOR THE RESPECTIVE AIRFOILC SUMC(L) .... GAMMA ASSOCIATED WITH THE INTEGRATION OF THE DISTURB-C ANCE VELOCITY AROUND THE WHOLE AIRFOILC

SUBROUTINE VELDISCOMMON /BOD/ IFLAGNLOWER,NUPPERNODTOTX(202) ,Y(202),

+ COSTHE(201),SINTHfE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFTYSHIFT,NAIRFO,XI(202) ,YI(202),+ COSTHL( 201) ,SINTHL(201)COMMON /COF/ A(201,211),KEQNSCOMMON /CPDI CP(200),SCL,TSCM,SGAMCOMMON /NUM/ PI,PI2INVCOMMON /SING/ Q(200) ,GAMMA(2) ,QK( 200) ,GAMKC2)COMMON /POT/ PHI(200),PHIK(200)COMMON /GUST/ UG( 200) ,VG( 200) ,XGFUGUST,VGUISTCOMMON /EXTV/ UE(200),VN(200)COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM,YPRMDIMENSION CKUTTA(2) ,PHITEL(2) ,PHITEU(2)

7770 DIMENSION CONTR2(2)6666 REAL *8 PIN(2),VELX

DIMENSION WGHTC5) ,PLOC(5) ,SUMC(2),+MNP1(50,50,5) ,AANP2(50,50,5) ,BBNP1(50,50,5) ,BBNP2(5O,505S),+COSTHP(102,6),SINTHP(102,6)AANP4(2),PHIL(102),UGU(100,6),+VGU(100,6) ,PHILE(2)

CC LOCATION AND WEIGHTING VALVES FOR THE GAUSSIAN QUADRATURE USEDC TO INTEGRATE FOR THE VELO POTENTIAL AROUND TH1E AIRFOILC

DATA WGHT/. 11846344,. 23931434,. 28444444,+ . 23931434,. 11846344/DATA PLOC/. 04691008,. 23076535,. 50000000,

+ . 76923466,. 95300899/CC FIND VT AND CP AT MID-POINT OF I-TH PANELC

DO 140 L- 1,NA.TRFOKI - (L-1)*NP1LI - (L-1)*NODTOTSUMC(L) - 0.0DO 130 1 - 1,NODTOTXMID - .5*(X(I+KI) +X(I+KI+1))YhID - .5*(Y(I+KI) +Y(I+KI+1))DX - X(I+KI+1)-X(I+KI)DY - Y(I+KI+1)-Y(I+KI)

106

DIST - SQRT(DX*DX+DY*DY)WTANG - COSALF(L)*COSTHL(I+KI) + SINALF(L)*SINTHLCI+KI)WNORM - SINALF(L)*COSTHL(I+KI) - COSALFCL)*SINTHL(I+KI)VTFREE - WTANGVACT - WTANG

CC ADD CONTRIBUTION OF J-TH PANELC 1. CONTRIBUTION OF J-TH PANEL TO THE VELO COMP OF THE MIDPTC OF THE I-TH PANELCC 2. CONTRIBUTION TO THE VELO POTENTIAL. THIS IS DONE BYC INTEGRATING OVER SMALLE R PANELS OF THE AIRFOIL.C

DO 155 K -1,5DX -PLOC(K)*(X(I+KI+1)-X(I+KI))

DY -PLOCCK)*(Y(I+KI+1)-Y(I+KI))XMID =X(I+KI) + DXYMID =Y(I+KI) + DYVDUM -0.0DO 150 LMI = 1,NAIRFOKJ = (LII-1)*NP1Li - (LM-1)*NODTOTDO 120 J - 1,NODTOTFLOG = 0.0FTAN - PIIF (J+KJ .EQ. I+KI) GO TO 100DXJ = XMID - X(J+KJ)DXJP = XHID - XCJ+KJ+l)DYJ - YMID - Y(J+KJ)DYJP -YMID - Y(J+KJ+l)FLOG - . 5*ALOG( (DXJP*DXJP+DYJP*DYJP) ICDXJ*DXJ+DYJ*DYJ))FTAN -ATAN2( DYJP*DXJ-DXJP*DYJDXJP*DXJ+DYJP*DYJ)

100 CTIMTJ - COSTHE(I+KI)*COSTHE(J+KJ) + SINTHIE(I+KI)*SINTHECJ+KJ)STIMTJ -SINTHE(I+KI)*COSTHE(J+KJ) - COSTHE(I+KI)*SINTHE(J+KJ)AA = P12INV*(FTAN*CTIHTJ + FLOG*STIMTJ)B - P12INV*(FLOG*CTIMTJ - FTAN*STIMTJ)VDUM - VDUM - B*Q( J+LJ) 4.AAJ4( L)*AAIF(K .EQ. 3) VACT -VACT - B*Q(J+IJ) + GAMkMA(LII)*AAIF(K .EQ. 3) VNORM - WNORM + AA*Q(J+L.) + GANMA(LM)*B

120 CONTINUE150 CONTINUE

VTANG - WTANG +VDU M *WGHT(IC)SUMC( L) - SUMC( L) +VDUM*DIST*WGIT( K)

155 CONTINUEPHI(I+LI) -(VTANG-VTFREE)*DISTCP(I+LI)in 1.0 - VACT*VACTUE(I+LI)- VACTVN(I+LI)in VNORM

130 CONTINUE140 CONTINUE

CC COMPUTE DISTURBANCE POTENTIAL AT TIE LEADING EDGE BY LINEC INTEGRAL OF THE VELOCITY FIELDC FROM UPSTREAM (AT INFINITY) TO THE LEADING EDGEC

DO 55 L - 1,NAIRFO

107

YMID = PIVOT *SINALF(L) + (L-1)*YSHIFTXLE - -PIVOT*COSALF(L)+(L-1)*XSHIFTXL = XLE

C XL -0.0NPHI 10 *NLOWERP114(L) -0.0DO 30 1 - 1,NPHIFRA CT - FLOAT(I)/FLOAT(NPHI)XLP - -100.0 * (1.0 - COS(0.5*PI*FRACT))+XLEIF (I .EQ. 1) XLP - -0. 000197+XLE

C XLP - -10.0 * (1.0 - COS(0.5*PI*FRACT))DELX - XL - XL?XlIID - 0.5*(XL+XLP)

C XMID = 0.5*CXL+XLP)*COSALF(L)C YMID - 0. 5*(XL+X(TP)*SINALj(L)

XL w XLPVELX = UGUST

CC ADD CONTRIBUTION OF J-TH PANELC

DO 40 LN = 1,NAIRPOKJ - (LN-1)*NP1Ui = (LN-1)*NODTOTDO 20 J3 1,NODTOTDXJ - XMID - X(J+KJ)DXJP - XMID - X(J+KJ+1)DYJ - YMID - Y(J+KJ)DYJP -ThID - Y(J+KJ+l)FLOG = S*A)( (DXJP*DXJP+DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))TAN - ATAN2( DYJP*DXJ-DXJP*DYJ ,DXJP*DXJ+DYJP*DYJ)CALMTJ -- COSALF(L)*COSTHE(J+KJ) - SINALF(L)*SINTHE(J+KJ)SALMTJ - -SINALF(L)*COSTHE(J+KJ) + COSALF(L)*SINTHE(J+KJ)APY = P12INV*(TAN*CALMTJ + FLOG*SALNTJ)BPY - P12INV*(FLOG*CALMTJ - FTAN*SALMTJ)VELX = VELX - DPROD(BPY,Q(J+LJ)) +DPROD(GANHA(LN),APY)

20 CONTINUE40 CONTINUE

PIN(L) -PIN(L) + VEL( DBLE(DELX)30 CONTINUE55 CONTINUECC COMPUTATION OF THE VELOCITY POTENTIAL FOR MIDPOINT OF EACH PANELC

DO 240 L - iNAIRFOLI - (L-1)*NODTOTPHP - -P114(L)

CC BEGIN WITH LOWER SURFACEC

DO 230 1 NLOWER,1,-1PHC -PHP-PHI(I+LI)

PHI(I+LI) - .5*(PHP+PHC)230 PH? PHC

PHITEL(L) -PHCC

108

C RESET FOR UPPER SURFACEC

PEP =-PIN(L)DO 250 I1 NLOWER+1,NODTOTPHC -PHP+PHI(I+LI)

PHI(I+LI) - .5*(PHP+PHC)250 PEP -PHC

PHITEU(LM PHC*240 CONTINUE

DO 334 L 1 , NAIRFOWRITE (6,1010) LWRITE (6,1000)RI - (L-1) * NP1LI - (L-1) * NODTOTSUHC(L - SUMC(L)/SS(L)DO 333 1 - 1 , NODTOTXMID - .5*(X(I+KI) + X(I+KI+I))YMID - .5*(Y(I+KI) + Y(I+KI+1))

333 WRITE (6,1050) I+LI,XMID,ThID,Q(I+LI),GAIHA(L),CP(I+LI),UE(I+LI),+ VNCI+LI),PHI(I+LI),SUMC(L)

334 WRITE (6,235) PIN(L)235 FORMAT (1X,'PHI AT LEADING EDGE U' 1F10. 6,/)1000 FORMAT(/,4X,'J',4X,'X(J)',SX I y(J? , 6X,'Q(J)',6X,'GAIMA',5X,

+ 'CP(J)',6X,'V(J)',6X,'VN(J) ,6X, PHI',4X,'INTGAIHA',/)1010 FORMAT(// AIRFOIL NUMBER' ,14,/)1050 FORIIAT(I5,9F10. 6)

RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE FANDM(SINALF,COSALF) CC CC COMPUTE AND PRINT OUT CD,CL,CM CC INTEGRATE PRESSURE DISTRIBUTION BY TRAPEZOIDAL RULE CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CP(I) ........ PRESSURE COEFFICIENT OF THE I-THl PANELC CL(L) ........ COEFFICIENT OF LIFT FOR THE L-TH AIRFOILC CD(L) ........ COEFFICIENT OF DRAG FOR THE L-TH AIRFOILC CM(L) ........ COEFFICIENT OF MOMENT FOR THE L-TH AIRFOIL WITHC RESPECT TO TH1E LEADING EDGEC CFX(L) ....... COEFFICEIENT OF TOTAL FORCE IN X-DIR OF GLOBAL SYS.C CFY(L) ....... COEFFICEIENT OF TOTAL FORCE IN Y-DIR OF GLOBAL SYS.CC

SUBROUTINE FANDM(SIN1,SIN2,COSI,COS2)COMMON /ROD/ IFLAG,NLOWER,NUPPER,NODTOT,X(202),Y(202),

+ COSTHE(201),SINTIIE(201),SS(2),NP1,NP2,NP3,NP4,+ NF5 ,XSHIFT,YSHIFT,NAIRFO,XI( 202) .YI( 202),+ COSTHL(201),SINlIL201)COMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TD,CVVX(2,200),

+ CVVY(2,200),XCI(2,200),YCI(2,200)COMMON /CPD/ CP(200),SCL,T,SCM,SGAICOMMON /SING/ Q(200) ,GAMMA(2) ,QK( 200) ,GAIIK(2)COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

109

+ XPRM,YPRMDIMENSION CM(2),CD(2),CL(2),CFX(2),CFY(2)

CC INITIALISE COEFFICIENTC

DO 110 L =1,NAIRFOCM(L) =0.0CFX(L) =0.0

110 CFY(L) -0.0

CC INTEGRATE AROUND THE AIRFOIL TO GET GLOBAL X AND Y FORCESC

DO 120 L = 1,NAIRFOKI = (L-1)*(NODTOT+1)LI =(L-1)*NODTOTDO 100 I = 1,NODTOTXMID - .5*(XI(I+KI) + XI(I+KI+1))YMID -f .5*(YI(I+KI) + YI(I+KI+1))DX = XI(I+KI+1) - XI(I+KI)DY = YICI+KI+1) - YI(I+KI)CFX(L) = CFX(L) + CP(14-LI)*DYCFY(L) = CFY(L) - CP(I+LI)*DXCM(L) -f CM(L) + CP(I+LI)*(DX*XMID + DY*YMID)

100 CONTINUE120 CONTINUE

CC DECOMPOSE INTO LIFT AND DRAG COMPONENTS W. R. T. RESP LOCAL SYSTEMC

DO 130 L =1,NAIRFOCD(L) = CFX(L)*COSALF(L) + CFY(L)*SINALF(L)CL(L) = CFY(L)*COSALF(L) - CFX(L)*SINALF(L)

130 WRITE (6,1000) L,CD(L),CL(L),CM(L)IF (M1 EQ. 0) RETURNIF (SCL .NE. 0.0) THENWRITE (8,1100) T,CL(1)/SCL,CL(2)/SCL,CM(1)/SCM,CM(2)/SCM,CD(1),

+CD( 2) ,GAMK( 1)/SGAM,GAMK( 2)/SGAMELSEWRITE (8,1100) T,CL(1),CL(2),CMC1),CM(2),CD(1),CD(2)END IF

1000 FORMAT(//,' IAIRFOIL NO 'Jo141/1' CD in',F1O.6,+ ' CL',FlO. 6, CM-' ,F1O.6)

1100 FORMAT(9F10.6)RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc CC SUBROUTINE INFL (NITR) CC CC CALCULATE INFLUENCE COEFFICIENTS CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC INFLUENCE COEFFICIENTS ON THE AIRFOIL DUE TO THE AIRFOIL:C AAN(I,J) .... NORMAL VELO AT THE MIDPOINT OF THE I-TH PANEL DUEC TO A SOURCE-DIST OF UNIT STRENGTH ON THE J-TH PANELC SUMAAN(I,L) .. NORMAL VELO AT THE MIDPOINT OF THE I-TH PANEL DUE

110

77 ... ..

C SOURCE DIST OF UNIT STRENGTH FROM THE L AIRFOILC BBN(I,J) .... NORMAL VELO AT THE MIDPOINT OF THE I-TH PANEL DUEC TO A VORTEX-DIST OF UNIT STRENGTH ON THE J-TH PANELC SUMAAN(I,L) NORMAL VELO AT H= MIDPOINT OF THE I-TH PANEL DUEC VORTEX DIST OF UNIT STRENGTH FROM THE L AIRFOILCC INFLUENCE COEFFICIENT ON THE WAKE ELEMENT:C AYNPI(LJ) ... Y-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH AIRFOIL DUE TO A SOURCE DIST OF UNIT STRE-C NGTH FROM THE J-TH PANELC AYNPI(L,NP3) . Y-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH PANEL DUE TO A SOURCE DIST OF UNIT STREN-C GTH FROM THE WAKE PANEL OF THE OTHER AIRFOIL.C (USED ONLY FOR BXNP1(L,NP3) SINCE THERE IS NO SOURCC DIST ON THE WAKE PANEL)C BYNPI(L,J) ... Y-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH AIRFOIL DUE TO A VORTEX DIST OF UNIT STRE-C NGTH FROM THE J-TH PANELC BYNPI(LNP3) . Y-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH AIRFOIL DUE TO A VORTEX DIST OF UNIT STR-C ENGTH FROM THE WAKE PANEL OF THE OTHER AIRFOIL.C CYNP1(L,N) ... Y-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH AIRFOIL DUE TO THE N-TH CORE VORTEX OFC UNIT STRENGTHC CXNPI(LN) ... X-VELO COMP AT THE MIDPOINT OF THE WAKE PANEL FROMC THE L-TH AIRFOIL DUE TO THE N-TH CORE VORTEX OFC UNIT STRENGTHCC INFLUENCE COEFFICIENTS ON THE AIRFOIL DUE TO THE WAKEC SUMCCN(I) ... NORMAL VELO AT THE MIDPOINT OF THE I-TH PANEL DUEC TO ALL POINT VORTICES OF ACTUAL STRENGTH.C SUMCCT(I) ... TANGENTIAL VELO AT THE MIDPOINT OF THE I-TH PANELC DUE TO ALL POINT VORTICES OF ACTUAL STRENGTH.CC

SUBROUTINE INFL (NITR)COMMON /BOD/ IFLAGNLOWER,NUPPERNODTOTX(202),Y(202),+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFTNAIRFO,XI(202) ,YI( 202),+ COSTHL(201),SINTHL(201)COMMON /NUM/ Pl,PI2INVCOMMON /WAK/ VYW(2),VXW(2),WAKE(2),DTCOMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),MTDCVVX(2,200),

+ CVVY(2,200),XCI(2,200),YC(2,200)COMMON /INFI/ AAN(201,201),BBN(201,201),AYNP1(2,201),BYNP1(2,201),+ SUMAAN(201,2),SUMBBN(201,2)COMMON /INF2/ SUMCCN(201),SUMCCT(201) ,CYNP1(2,200),+ CXNP1(2,200)COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM, YPRMDIMENSION GAYNP1(2,201) ,GBYNP1(2,201)IF (M .GT. 1) GO TO 510

CC INFLUENCE COEFFICIENT ON THE I-TH PANEL BY THE J-TH PANEL FROMC THE SAME AIRFOIL.C

i- III- - - - - - - - . r.

DO 200 L - lNAIRFoLI = (L- 1)*NODT0TKI - (L-1)*NP1DO 120 I = 1,NODTOTXMID - .5*(XI(I+KI) + XI(I+KI+1))YMID = .5*(YICI+KI) + YI(I+KI+1))SUMAAN(I+LI,L) =0.0SUMBBN(I+LI,L) -0.0

* DO 110 J - 1,NODTOTFLOG = 0.0FTAN a PIIF (J+LI .EQ. 1+LI) GO TO 100DXJ = XMID - XI(J+KI)DXJP - XHID - XI(J+KI+1)DYJ - YMID - YI(J+KI)DYJP - YMID - YI(J+KI+l)FLOG - . *ALOG( (DXJP*DXJTP+DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))FTAN = ATAN2( DYJP*DXJ-DXJP*DYJDXJP*DXJ+DYJP*DYJ)

100 CTIMTJ - COSTHL(I+KI)*COSTHL(J+KI) + SINTHL(I+KI)*SINTHL(J+KI)STIHTJ - SINTHL(I+KI)*COSTHL(J+KI) - COSTHL( I+KI)*SINTHL(J+KI)AANCI+LI,J+LI) -P12INV*(FTAN*CTIMTJ + FLOG*STIMTJ)BBN(I+LI,J+LI) -P12INV*(FLOG*CTIMTJ - RAN*STIMTJ)SUMMAN(I+LI,L) =SUMMANCI+LI,L) + AAN(I+LI,J+LI)SUMBBN(I+LI,L) -SUMBBN(I+LI,L) + BBNCI+LI,J+LI)

110 CONTINUE120 CONTINUE200 CONTINUE510 CONTINUE

IF (NITR .GT. 0) GO TO 271CC INFLUENCE COEFFICIENT ON THE I-TH PANEL BY THE J-TH PANEL FROMC THE OTHER AIRFOIL.C

DO 270 L - 1,NAIRFOLI = (L-1)*NODTOTKI = (L-1)*NPIDO 260 I = 1,NODTOTXMID = .5*(X(I+KI) + X(I+KI+1))YMID = .5*(Y(I+KI) + Y(I+KI+1))SUMAAN(I+LI,3-L) = 0.0SUIIBBN(I+LI,3-L) = 0.0DO 250 J = NODTOT+2 ,2*NODTOT+1DXJ = XMID - X(J-KI)DXJP - XHID - X(J-KI+1)DYJ - YNID - YCJ-KI)DYJP - YhID - Y(J-KI+1)FLOG - . 5*ALOG( (DXJP*DXJP+DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))FTAN - ATAN2( DYJP*DXJ -DXJP*DYJ, DXJP*DXJ+DYJP*DYJ)CTIKTJ - COSTHE(I+KI)*COSTHE(J-KI) + SINTHE(I+KI)*SINTH(J-KI)STIIITJ - SINTHE(I+Kl)*CGsTHE(J-KI) - C0SflfE(I+Kl)*SINTHE(J-KI)AAN( I+LI ,J-LI-1) - P12INV*(TAN*CTIMTrl + FWG*STIMTJ)BBN(I+LIJ-LI-1) - P12INV*(FLOG*CTIMTJ - FTAN*STIMTJ)SUl4MN(I+LI,3-L) - SUMMAN(I+LI3-L) + AAN(I+LIJ-LI-1)SUHBBN(I+LI,3-L) - SUMBN(I+LI,3-L) + BBN(I+LI,J-LI-1)

250 CONTINUE260 CONTINUE

112

270 CONTINUECC END OF STEADY FLOW CALhULATION FOR INFLUENCE COEFFICIENT.C

IF (M.EQ.O) RETURN271 CONTINUE

CC INFLUENCE COEFFICIENT ON THE WAKE ELEMENT FROM THE AIRFOIL.C

DO 130 L - 1,NAIRFOI - NP1*LXMID - .5*(X(I) + X(NAIRFO*NP1+L))YMID = .5*(Y(I) + Y(NAIRFO*NP1+L))DO 130 MM - iNAIRFO.i i(MM-1)*NODTOTKJ -(M.1)*NP1DO 130 J - 1, NODTOTDXJ - XIIID - X(J+KJ)DXJP -XMID - X(J+KJ+l)DYJ = YMID - Y(J+KJ)DYJTP - YMID - Y(J+KJ+1)FLOG - .5*ALOG( (DXJP*DXJP+DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))ETAN = ATAN2( DYJP*DXJ-DXJP*DYJDXJP*DXJ+DYJP*DYJ)CTIMTJ - COSTHE(I)*COSTHE(J.KJ) + SINTHE(I)*SINTHE(J.KJ)STIIITJ = SINTHE(I)*COSTHE(J+KJ) - COSTHECI)*SINTHE(J+KJ)GAYNP1(L,J+IJ) = P12INV*(FTAN*COSTHE(J+KJ) - FLOG*SINTHE(J+KJ))GBYNP1(LJ+IJ) = P12INV*(FLOG*COSTHE(J+KJ) + RAN*SINTHE(J+KJ))AYNP1(L,J+LJ) - GAYNP1(L,J+LJ)*COSALF(L)-GBYNP1(L,J+IJ)*SINALF(L)BYNP1(L,J+LJ) = GAYNP1(L,J+IJ)*SINALF(L)+GBYNPl(L,J+IJ)*COSALF(L)

1.30 CONTINUECC INFLUENCE COEFICIENT OF WAKE ELEMENT DUE TO WAKE ELEMENT FROMC THE OTHER AIRFOILC

DO 300 L - iNAIRFOI - NP1*LXMID = .5*(X(I) + X(NAIRFO*NPl+L))ThID - .5*(Y(I) + Y(NAIRFO*NPl+L))KJ = (L-1)*NP1MJ = 2*NP1NJ = MJ+3DXJ - XMID - X(MJ-KJ)DXJP - XMID - X(NJ-L)DYJ - YID - Y(MJ-KJ)DYJP -YMID - Y(NJ-L)FLOG - .5*ALOG( (DXJP*DXJP+DYJP*DYJP)/(DXJ*DXJ+DYJ*DYJ))FTAN - ATAN2( DYJP*DXJ-DXJP*DYJ ,DXJP*DXJ+DYJP*DYJ)CTIMTJ - COSTHE(I)*COSTHE(MJ-KJ) + SINTHE(I)*SINTHE(MJ-KJ)STIMTJ - SINTHE(I)*COSTHE(MJ-KJ) - COSTHE(I)*SINTHE(HJ-KJ)GAYNPi LNP3) - P1 21NV*( FTAN*COSTHE(MNJ-LI) - FLOG*SINTHE(NJ-KJ))GBYNP1( LNP3) - P1 21NV*( FLOG*COSTHE(MJ-KJ) + FTAN*SINTHE( NJ-KY))AYNP1(L,NP3) - GAYNP1(L,NP3)*COSALF(L)-GBYNPI(LNP3)*SINhLF(L)BYNP(L,NP3) - GAYNP1(L,NP3)*SINALF(L)4.GBYNP1(LNP3)*COSALF(L)

300 CONTINUEC

113

C INFLUENCE COEFFICIENT ON THE AIRFOIL BY THE WAKE ELEMENT.

DO 160 L - 1,NAIRFO

LIu(L-1)*NODTOTK1-(L- 1)*NP1DO 140 1 - 1,NODTOTXMID -. 5*(X(I+KI) + X(I+KI+1))YMID a .5*(Y(I+KI) + Y(I+KI+1))DO 145 MM -1, NhIRFOJ - NP1*MDXJ - XMID - X(J)DXJP a XMID - X(NAIRFO*NP1+MH)DYJ - YMID - Y(J)DYJP -ThID - Y(NAIRFO*NP1+MM)FLOG - 5*ALOG( (DXJP*DXJP+DYJP*DYJP) /(DXJ*DXJ+DYJ*DYJ))FTAN - ATAN2(DYJP*DXJ-DXJP*DY3 DXJP*DXJ+DYJP*DYJ)CTIMTJ - COSTHE(I+KI)*COSTHE(J) + SINTHE(I+KI)*SINTHE(J)STIHTJ - SINTHE(I+KI)*COSTHE(J) - COSTE(I+KI)*SINTHE(J)AAN(I+LI,2*NODTOT+MM) = P12INV*(FTAN*CTIMTJ + FLOG*STIMTJ)BBN(I+LI,2*NODTOT+MM) = PI2INV*(FLOG*CTIMTJ - YrAN*STIMTJ)

145 CONTINUE140 CONTINUE160 CONTINUEC

IF (M. EQ. 1) RETURNm11 aM- 1

CC INFLUENCE COEFFICIENT ON THE WAKE ELEMENT BY THE CORE VORTICES.C

DO 350 L = 1,NAIRF0

XI4ID - O.5*(X(NP1*L) + X(NP1*NAIRFO+L))YMID - 0.5*(Y(NP1*L) + Y(NP1*NAIRFG+L))DO 240 MM = 1, NAIRFOKN =(Ml1-1)*MN1DO 230 N = 1,MMIDX = XHID - XC(MM,N)DY - YMID - YC(IIM,N)DI ST2, = DX*DX+DY*DYGCYNP1 - -P12INV*DX/DIST2GCXNP1 - +P12INV'.DY/DIST2CYNPi L ,N+KN) - GCYNP1*COSALF( L)+GCXNP1*SINALF( L)CXNP 1 ( ,N+KN) = -GCYNP 1*SINALF( L)+GCXNP1*COSALF( L)

230 CONTINUE240 CONTINUE350 CONTINUE

IF (NITR. GT. 0) RETURNCC INFLUENCE COEFFICIENT ON THE AIRFOIL BY THE CORE VORTICESC

DO 400 L - 1,NAIRF0LI i(L-1)*NODTOTKI i(L-1)*N?lDO 220 I - 1,NODTOTXMID = 0.5*(X(I+KI) + X(I+KI+1))YMID - 0.5*(Y(I+KI) + Y(I+KI+1))

* 114

- *At.. ( .- C - - 0.0

SUMCCNI+LI) - 0.0DO 280 MM -1, NAIRFORN - (MM-1)*Mm1DO 210 N -1,MIDX - XGiID - XC(MM,N)DY - YMID - YC(MH,N)DIST - SQRT( DX*DX+DY*DY)COSTHN - DX/DISTSINTHN - DY/DISTCTIMTN - COSTHECI+KI)*COSTHN + SINTHE(I+KI)*SINTHNSTIMTN - SINTHE(I+KI)*COSTHN - COSTHE(I+KI)*SINTNCCN - -CTflMT/DISTCCT - -STIMTN/DISTSUMCCN(I+LI) - SUMCCN(I+LI) + CCN*CV(MMN)SUMCCT(I+LI) - SUMCCT(I+LI) + CCT*CVCMM,N)

210 CONTINUE280 CONTINUE

SUMCCN(I+LI) - P12INV*SUMCCN(I+LI)SUMCCT(I+LI) - P12INV*SUMCCT(I+LI)

220 CONTINUE400 CONTINUE

RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE COEF (SINALF,COSALF,OMEGAIJXUY,NITR) CC CC SET COEFFICIENTS OF N EQUS ARISING FROM FLOW CC TANGENCY CONDITIONS AT MID POINTS OF PANELS CC SOLVING THE N-SOURCE STRENGTHS IN TERMS OF THE CC VORTICITY STRENGTH (RESULTING IN 2 RHS) CC KUTTA CONDITION IS SATISFIED SEPARATELY TO OBTAIN CC THE VORTICITY STRENGTH CC THIS SOLUTION METHOD IS DESIRED FOR UNSTEADY FLOW CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC AC I ,J).......LHS, NORMAL VELOC ITY AT THE MIDPOINT OF THE I -THC PANEL INDUCED BY UNIT SOURCE DIST ON THE J-TH PANELC A(I,NP3) .... FIRST RHS, COMPONENT OF NORMAL VELO AT THE I-THC PANEL WHICH IS DEPENDENT ON VORTICITY STRENGTH OFC THE FIRST AIRFOILC A(INP4) .... SECOND RHS, COMPONENT OF NORMAL VELO AT THE I-THC PANEL WHICH IS DEPENDENT ON VORTIC ITY STRENGTH OFC THE SECOND AIRFOILC A(INP5) .... THIRD EMS, COMPONENT OF NORMAL VEWO AT THE I-THC PANEL WHICH IS INDEPENDENT OF TH1E CIRCULATION OFC BOTH AIRFOILSC

SUBROUTINE COEF (NITR)COMMON /BOD/ IFLAG,NLOWER,NUPPE,NODTOTX(202) ,Y( 202),

+ COSTHE(201),SINTE(201),SS(2),NPINP2,NP3,NP4,+ NP5,XSHIFT,YSHIFT,NAIRFO,XI(202) ,YI(202),

+ COSTHL(201),SINTHL(201)

COMMON /COF/ A(201,211),KEQNSCOMMON /SING/ Q(200),GAMA(2),QK(200),GAIK(2)COMMON /WAX/ VYW(2),VXW(2),WAKE(2),DTCOMMON /CORV/ CV(2,200),XC(2,200),YCC2,200),M,ThCVVX(2,200),

+ CVVY(2,200),XCI(2,200),YCI(2,200)COMMON IINF11 MN(201,201),BBN(201,201),AYNP1(2,201),BYNP1(2,201),

+ SUMMAN(201,2),SUMBBN(201.2)COMMON /INF2/ SUMCCN(201),SUMCCT(201),CYNP1(2,200),

+ CXNP1(2,200)COMM4ON /GUST/ UG( 200) ,VG( 200) ,XGFUGUST,VGUSTCOMMON /MATRIX/ AMAT(201,211)COMMON /GEOM/ SINALF(2),COSALF2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM ,YPRMNEQS NODTOT* NAIRFOIF (NITR .GT. 0) GO TO 91

CC SET/RESET LUS MATRIX A(I,J) FOR EACH NEW TIME STEPC

DO 110 I - 1,2*NODTOTDO 110 J - 1,2*NODTOT

110 A(I,J) -AANCI,J)IF (M.NE.0) GOTO 91

CC FILL IN THE RIGHT HAND SIDE FOR STEADY FLO)WC

DO 310 L=1,NAIRFOLI = (L-1)*NODTOTKI - (L-1)*NP1DO 310 I - 1,NODTOTXMID -0.5 * (XI(I+KI) + XI(I+KI+1))YMID = 0.5 * (YI(I+KI) + YI(I+KI+1))A(I+LI)NP3) = -SUMBBN(I+Ll,1)A(I+LINP4) - -SUMBBN(I+LI,2)

310 A(I+LI,NP5) - SINTHL(I+KI)*COSALF(L) - COSTHL(I+KI)*SINALF(L)RETURN

CC FILL IN THE RIGHT HAND SIDE FOR UNSTEADY FLOWC

91 DO 219 L-1,NAIRFOLI - (L-1)*NODTOTKI - (L.1)*NP1DO 210 I - 1,NODTOTXMID = 0.5 *(XI(I+KI) + XI(I+KI+1))YMID - 0.5 *(YI(I+KI) + YI(I+KI+1))A(I+LI,NP3) - -SUMBBN(I+Ll,1) + BBN(I+LI,NP3)*SSC1)/WAKE(l)A(I+LI,NP4) -- SUIIBBN(I+LI,2) + BBN(I+LI,NP4)*SS(2)/WAKE(2)A( I+LI ,NPS) a -BBN( I+LI ,NP3)*GAMA( )*SS( 1)/Wh1KE( 1)-BDN(Z+LI,NP4)

+*GAMMA(2)*SS(2)/WAKE(2) + SINTHL(I+KI)* ((1.+UG(I+LI))*COSALF(L)-+VG( I+LI)*SINALF(L)+UX(L)) - COSTHL(I+KI)* ((1. +UG( I+LI))+*SINALF( L)+VG( I+LI)*COSALF( L)+UY( L) )+OHEGA(L)*(YMID*SINTHL( I+KI)+ + XMID*COSTHL(I+I))

CC ADD CORE VORTEX CONTRIBUTIONC

IF (M .EQ. 1) GOTO 210A(I+LI,NPS) - A(I+LI,NP5) -SUMCCN(I+LI)

116

-..--.- ~-...... . .. . . . ....

210 CONTINUEDO 300 I=12*NODTOTDO 300 J=1,2*NODTOT+3

300 AMAT(I,J) - A(IJ)RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcCcCCccCCccccccCcC CC SUBROUTINE KUTTA (ALPHA, SINALF ,COSALF ,OMEGA,UX ,UY) CC CC USING KUTTA CONDITION TO DETERMINE VORTICITY CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCccCccCCCCCCcCC GAMMA ... CIRCULATION FOR THE STEADY FLO0W CASE AND ALSOC CIRCULATION FOR THE PREVIOUS TIME STEP FOR UNSTEADY CASEC GAHK ... CIRCULATION AT THE CURRENT TIME STEP FOR UNSTEADY CASEC QK ... SOURCE STRENGTH AT THE CURRENT TIME STEPC QK(I) - Bl(I)*GAIIK(1)+B2(I)*GAMK(2)+B3(I)C B1(I) .. THAT PART OF THE SOURCE STRNGTH WHICH IS INDUCED BY AC CIRCULATION OF UNIT STRENGTH FROM AIRFOIL 1C B2(I) .. THAT PART OF THE SOURCE STRNGTH WHICH IS INDUCED BY AC CIRCULATION OF UNIT STRENGTH FROM AIRFOIL 2C B3( I) ... THAT PART OF THE SOURCE STRNGTH WHICH IS INDEPENDENTC OF THE CIRCULATION FROM BOTH AIRFOILSC AA I ,J) .THAT PART OF THE TANGENTIAL VELOCITY AT THE TRAILINGC EDGE PANELS WHICH IS INDUCED BY A CIRCULATION OF UNITC STRENGTH BY AIRFOIL JC BB(I) .. THAT PART OF THE TANGENTIAL VELOCITY AT THE TRAILINGC EDGE PANELS WHICH IS INDEPENDENT OF THE CIRCULATIONC

SUBROUTINE KUrrA(NITR ,PVTAG)COMMON /BOD/ IFLAG,NLOWER,NUPPER,NODTOT,X(202) ,YC 202),

+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5 ,XSHIFT,YSHIFT,NAIRFO,XI(202) ,YI(202),+ COSTHLC2O1),SINTHL(201)COMMON /COF/ A(201,211),KEQNSCOMMON /SING/ Q(200) ,GAMMA(2),QK(200),GAMK(2)COMMON /WAK/ VYW(2) ,VXW(2) ,WAKE(2) ,DTCOMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TDCVVX(2,200),

+ CVVY(2,200),XCI(2,200),YCI(2,200)COMMON IINF11 AAN(201,201),BBN(201,201),AYNP1(2,201),BYNPI(2,201),

+ SUMAAN(201,2),SUMBBN(201,2)COMMON /INF2/ SUMCCN(201) ,SUNCCT(201) ,CYNP1(2,200),

+ CXNP1(2,200)COMMON /GUST/ UG(200) ,VGC 200) ,XGF,UGUSTVGUSTCOMMON /MATRIX/ AMAT(201,211)COMMON /BNAT/ 31,B2,3COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM ,YPRMCOMMON /GIES/ NGIESDIMENSION Bl(200),B2(200),B3(200),AA(4,2),3B(5),EE(2),FF(2),

+GG(2),RADI(2),AAA(2),BBB(2),CCV(2),DDD(2),EEE(2),FFF(2),+DG(2) ,E(2) ,GAMA(2) ,DELG(2) ,DGAMK(2,3) ,COEFL(2,3)

CC RETRIEVE SOLUTION FROM A-MATRIX FOR THE SOURCE STRENGTH IN TERMS

117

jf -. 77.

zo-r = .- 7 17

C THE TWO GANK CONTRIBUTION AND THE CONSTANT COMPONENTC

COUNT 0DO 50 1I 1,NODT0T*2B1(I) - A(I,NP3)B2(I) - A(I,NP4)

50 B3(I) - M(I,NP5)IF (M.GT.O) GOTO 400

CC STEADY KUTTA CONDITIONCC COMPUTE TANGENTIAL VELOCITIES

DO 425 L - iNAIRFOLI =CL-1)*NODTOTKI - (L-1)*NP1KK - (L-1)*2DO0430 K - 1,2IF (K .EQ. 1) 1 - 1IF (K .EQ. 2) I - NODTOTXHID = 0.5 * (XI(I+KI) + XI(I+KI+1))YMID = 0.5 * (YI(I+KI) + YI(I+KI+1))AA(K+KK,l) a SUHAAN(I+LI,1)AA(K+KK,2) - SUHAAN(I+LI,2)BB(K+KK) COSALF(L)*COSTHL(I+KI) + SINALF(L)*SINTHL(I+KI)DO 419 J =1,2*NODTOTAA(K+KK,1) -AA(K+KK1l) - BBN(I+LI,J)*B1(J)AA(K+KK,2) -AA(K+KK,2) - BBN(I+LI,J)*B2(J)BB(K+KK) -BB(K+KK) - BBNCI+LI,J)*B3(J)

419 CONTINUE430 CONTINUE425 CONTINUE

C SET UP KUTTA CONDITIONSC

DO 450 1-1,2LI a(I.I)*NAIRPO+l

DGAMK(I,l) = AA.(LI,l) + kA(LI+1,1)DGAMK(I,2) - AA(LI,2) + AA(LI+1,2)

450 DGAMK(I,3) = -(BB(LI) + BB(LI+1))CC SOLVE KUTTA CONDITION BY GAUSSC

R-DGAMK( 2, 1)/DGA1( 1,1)DO 460 K - 2,3

460 DGAMK( 2,K) - DGAIIK(2,K) -R*DGAMK( 1,K)GAIIHA(2) - DGAXIK(2,3)/DGAMK(2,2)GAIO4A(1) a (DGAMK(1,3)-DGAMK(1,2)*GANMA(2))/DGAMK(1,1)

CC CALCULATE SOURCE STRENGTHC

DO 470 L = 1,NAIRFOLI -(L-1)*NODTOTDO 470 I - 1,NODTOT

470 Q(I+LI) - GAMKA(1)*B1(I+LI) +GAIOI(2)*B2(I+LI) +B3(I+LI)RETURN

C

118

C UNSTEADY K1YITA CONDITIONCC FIND VT AT TRAILING EDGE PANELSCC INITIAL GUESS FOR GAIIK FROM PREVIOUS TIME STEP400 IF (NITR -EQ. 0) THEN

GAIIK(1 - GAIIHA(1GAMK(2) - GAIMA(2)END IFDO 125 L - 1,NAIRFOLI - (L-1)*NODTOT

KK - (L-1)*2DO 130 K - 1,2IF (K .EQ. 1) I - 1IF (K .EQ. 2) I a NODTOTXMID - 0.5 * (XI(I+KI) + XI(I+KI+1))ThID - 0.5 * (YI(I+KI) + YI(I+Kl+1))WTANG - ((1. +UG( I+LI) )*COSALF(L) -VG( I+LI)*SINALF(L)+UX(L))

+*COSTIIL(I+KI)+ ((1. +UG(I+LI))*SINALF(L)+VG(I+LI)*COSALF(L)+UY(L))+*SINTHL( I+KI) + OIIEGA( L)*( ThID*COSTHL( I+KI)+-XMID*SINTHL( I+KI))M(K+KK,1) = - AAN(I+LI,NP3)*SS(1)/WAKE(1)AA(K+KK,2) = - AAN(I+LI,NP4)*SS(2)/WAKE(2)BB(K+KK) - VTANG + AAN(I+LI,NP3)*SS(1)*GAMMA(1)/WAKEC1) +

+ AAN( I+LI ,NP4)*SS(2)*GAMHA(2)/WAKEC2)DO 119 J - 1,NODTOTM(K+KK,1) = AA(K+KK,l) + AAN(I+LIJ) - BBN(I+LI,J)*Bl(J)M(K+KK,2) - MC(K+KK,2) + AAN(I+LIJ+NODTO) - BBN(I+LI,J)*B2(J)BB(K+KK) =BB(K+KK) - BBN(I+LI,J)*B3(J)

119 CONTINUEDO 120 JJ =NODTOT+1,2*NODTOTM(K+KK,l) = MA(K+KK1l) - BBN(I+LI,JJ)*BI(JJ)M(K+KK,2) = AA(K+KK,2) - BBN(I+LI,JJ)*B2(JJ)BB(K+KK) =BBCK+KK) - BBN(I+LI,JJ)*B3(JJ)

120 CONTINUECC ADD CORE VORTEX CONTRIBUTIONC

IF (M. LE. 1) GOTO 100BB(K+KK) =BB(K+KK) + SUMCCT(I+LI)

100 CONTINUE130 CONTINUE125 CONTINUE

IF (NGIES .EQ. 1) GOTO 145CC SATISFYING KUrrA CONDITION -- SOLVE FOR VORTEX STRENGTHC

DO 135 I - 1,2LI - (I.1)*NAIRFO+1AAA(I - M(Ll,1)**2-AA(LI+1,1)**2BBD(I) = AA(LI,2)**2-AM(LI+1,2)**2ccc(I) a 2*(MA(LI1)*BB(L)-AM(LI+11)*3(L+)(2I)*SS1)/DT)DDD(I) a 2*(AA(LI,2)*BB(Ll)-AA(LI+1,2)*B(LI+1)-(I-1)*SS(2)/DT)EECI) - 2*(AA(LI,1)*AACLI,2)-AA(LI+1,1)*AA(LI+1,2))FFF(I) - BB(LI)**2-BB(LI+1)**2+2*SS(I)*GA1MA(I)/DT

119

-.- '- -- ;--~-r ....- r--- A

135 CONTINUE60 DO 140 I1 1,2

DGAMK(I,1) - 2*MA(I)*GAMK(1)+EEE(I)*GAMK(2)+CCC(I)DGAIIK(I,2) -2*BBBCI)*GAHK(2)+EEE(I)*GAMK(1)+DDD(I)

140 DGAMK(I,3) - (AA(I)*GA1K(1)**2+CCC(I)*GA1K(1)+BBB(I)*+ GAfK(2)**2+DDD(I?*GAfK(2)+EEECI)*GAM(1)*GAMK(2)+FFF(I))

C140 WRITE (6,*)'DGAM I,DGAMK(I,1),DGAMIK(I,2),DGAM(I,3)DO 144 1-1,2DO 144 KKK-1,3

144 COEFL(I,KKK) - DGAMK(IKKK)CC GAUSSIAN ELIMINATIONC

R-DGAMK(2,1)/DGAMK(l11)DO 200 K = 2,3

200 DGAMK(2,K) - DGAMKC2 ,K)-R*DGAfK( 1,K)DG(2) = DGAMK(2,3)/DGAMK(2,2)DG(1 - (DGAMK(1,3)-DGAMK(1,2)*DG(2))/DGAMK(1,1)GAMK(1) = GAMK(1)+DG(1)GAMK(2) = GAMK(2)+DG(2)IF ((ABS(DG(1)/GAIK(1)) .LT. .0001) .AND.

+ (ABS(DG(2)/GAMK(2)).LT. .0001)) GO TO 300COUNT - COUNT + 1IF (COUNT .EQ. 50) GO TO 310GO TO 60

310 WRITE(6,*)'KUrrA CONDITION NOT SATISFIED -- COUNT EXCEEDED 50'RETURN

CC SET UP KUTTA CONDITIONS FOR GIESING CONDITIONC145 DO 451 I=1,2

LI = (I-1)*NAIRFO+lDGAMK(I,1) -AA(LI,1) + AA(LI+1,1)DGAHK(I,2) - AA(LI,2) + AA(LI+1,2)

451 DGAIIK(I,3) =-(BB(LI) + BB(LI+1))CC SOLVE KUTTA CONDITION BY GAUSSC

R=DG WK(2,1)/DGAIK(1,1)DO 461 K = 2,3

461 DGAMK(2,K) =DGAflK(2,K)-R*DGAMK(1,K)GA!IK(2) -DGAMK(2,3)/DGAMK(2,2)GAMK(1) =(DGAM(1,3)-DGAMK(1,2)*GAHK(2))/DGAMK(1,l)

CC CALCULATE SOURCE STRENGTHC

300 DO 160 L - iNAIRFOLI -(L-1)*NODTOTDO 160 I - 1,NODTOT

160 QKCI+LI) - GMIK(1)*B1(I+LI) + GAMK(2)*B2(I+LI) + B3(I+LI)CC CALCULATE TANGENTIAL VELOCITY AT THE TRAILING EDGE BY BACKC SUBSTITUTION. THE PRODUCT OF THE TRAILING EDGE VELOCITYC SHOULD BE NEGATIVE . THIS IS CHECKED IN THE MAIN PROGRAM.C

VTAGI AA(1,1)*GANK(1)+AA(1,2)*GAIIK(2)+BB(l)

120

VTAG50 - AM(2,l)*GAHKC1)+M(2,2)*GAMK(2)+BB(2)VTAG5l - AA(3,1)*GAMK(l)+MA(3,2)*GAMK(2)+DB(3)VTA100 -AA(4,l)*GAMK(l)+AA(4,2)*GAMK(2)+BB(4)PVTAG - VTAG1*VTAG5O

2222 FORHAT( lX,'VTAG' ,4Fl0. 6)RETURNEND

CCcccccccccccCC~CccccCCCCccccCcCccCCCccccccccC~ccCccCCCCccCccCCcC CC SUBROUTINE TEWAK (SINALF,COSALF) CC CC COMPUTE WAKE ELEMENT AT THE TRAILING EDGE CC CccccCCcccccCcCcccCcccCcccccccccccCcccCCCccccccccccccccccCcccccCCcCC VYWK( L) .... Y-VELO COMP AT THE MID-POINT OF THE WAKE ELEMENTC WITH RESPECT TO THE LOCAL FROZEN FRAME OF REFERENCEC VXWK( L) .... X-VELO COMP AT THE MID-POINT OF THE WAKE ELEMENTC WITH RESPECT TO THE LOCAL FROZEN FRAME OF REFERENCEC

SUBROUTINE TEWAKCOMMON /BOD/ IFLAG,NLOWER,NUPPER,NODTOT,X( 202) ,Y( 202),

+ COSTHE(201),SINTH(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFTNAIRFO,XI(202) ,YI(202),+ COSTHL(201),SINTHL(201)COMMON /COF/ A(201,211),KEQNSCOMMON /SING/ Q( 200) IGAMMAC 2) ,QK( 200) ,GAIK( 2)COMMON /WAK/ VYW(2),VXW(2),WAKE(2),DTCOMMON /WAK2/ VYWK(2),VXWK(2)COMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),MTh,CVVX(2,200),

+ CVVYC2,200),XCI(2,200),YCI(2,200)COMMON /INFl/ AAN(201,201),BBN(201,201),AYNP1(2,201),BYNP1(2,201),

+ SUMAAN(201,2),SUMBBN(201,2)COMMON /INF2/ SUICCN(201),SUMCCT201),CYNP1(2,200),

+ CXNP1(2,200)COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRII,YPRMCOMMON /GUST/ UG(200) ,VG( 200) ,XGF,UGUSTVGUSTDIMENSION CONT1X( 2) ,CONTIY(2) ,CONT2X(2) ,CONT2Y(2),

+ CONT3X(2),CONT3Y(2),CONT4X(2),CONT4Y(2),CONT5X(2),CONT5Y(2)DO 20 I = 1,NAIRFONN = (I-1)*NP1XMID - 0.5 * (X(NP1+NN) + X(NP2+I-1))YMID - 0.5 * (Y(NP1+NN) + Y(NP2+I-1))UGW = 0.0VGW - 0.0XG = XMIDIF (XG .GT. XGF) GO TO 10UGW - UGUSTVGW = VGUST

10 VYWK(I) - (1.+UGW)*SINALF(I)+VGW*COSALF(I)VXWKCI) - (1.+UGW)*COSALF(I)-VGW*SINALF(I)

2221 FORMAT(1X,'SINALF :',4Fl0.5)CONTlYC I) -(1. +UGW)*SINALF( I)+VGW*COSALF( I)CONTlX( I) - (1. +UGW)*COSALFC I) -VGW*SINALF( I)CONT2Y(I) - 0

121

CONT2X(I) = 0COT3()

CONT3Y(I) = 0

CC CONTRIBUTION FROM THE SOURCE AND WAKE DIST FROM THE AIRFOILC

DO 110 L - HAIRF0KJ - (L-1)*NODTOTDO 120 J - 1,NODTOTVYWK(I) - VYWK(I) + AYNP1(IJ+KJ)*QK(J+KJ) + BYNP1(I,J+KJ)*GAMKCL)VXWK(I) - VXWK(I) - BYNP1(I,J+KJ)*QK(J+KJ) + AYNP1(IJ+KJ)*GAMK(L)CONT2Y(I) - CONT2Y(I)+ AYNP1(IJ+KJ)*QK(J+KJ)CONT2X(I) = CONT2X(I)- BYNP1(IJ+KJ)*QK(J+KJ)CONT3Y(I) - CONT3Y(I)+ BYNPI(IJ+KJ)*GAMK(L)

120 CONT3X(I) - CONT3X(I)+ AYNP1(IJ+KJ)*GAMK(L)110 CONTINUE

CC ADD WAKE ELEMENT CONTRIBUTION OF THE OTHER AIRFOILC

J = 3-IVYWK(I) = VYWK(I)+ BYNP1(INP3)*(AMlA(J)-GAMlK(J))

+ * SSCJ)/WAKE(J)* VXWK(I) = VXWK(I)+ AYNP1(I,NP3)*(GAMMA(J)-GAMK(J))

+ * SS(J)/WAKE(J)CONT4Y(I) = BYNP1(1,NP3)*CGAMHA(J)-GAMK(J))

+ * SS(J)/WAJE(J)125 CONT4X(I) = AYNP1(I,NP3)*(GAMMA(J)-GAMR(J))

+ * SS(J)/WAKE(J)CC ADD CORE VORTEX CONTRIBUTIONC

CONT5YCI) - 0CONT5X(I) = 0IF (M.EQ. 1) GO TO 140il =M -1

DO 150 L - iNAIRFOKN =CL-1)*MM1DO 130 N - 1,HH1VYWK(I) = VYWK(I) + CYNP1(IN+KN)*CVCLN)VXWK(I) = VXWK(I) + CXNP1(IN+KN)*CVCL,N)CONT5Y(I) = CYNP1(I,N+KN)*CV(L,N)

130 CONT5X(I) =CXNP1(I,N+KN)*CV(L,N)150 CONTINUE240 CONTINUE22 FORNTNUE 'XK '6F05

2223 FORMAT (1X: 'VYWK :',6Fl0.5)222 FRATRNVK 6F05

ETNDCCCCCCCCCCCCCCCCCCCCCCCCCCCCDCCCCCCCCC

cccccccccccccccccccccccccccccccccC CccC SROTN PRS(SAFCOAFO GAU Y)CC SURUIEPES(IAFCSLMGXU)CC C PUEUSAD FLWPESRDITIUINCC AOPTNSVEOCIY FLP UENT ATDIDSPINUTSIOF ANL CC C

122

,- L1

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC VTANG ..... TANGENTIAL VELO COMP OF A FIXED POINT OF THE MOVINGC LOCAL FRAME OF REFERENCE.C PHIK(I) ..... DISTURBANCE VELO POTENTIAL AT THE MID POINT OF THEC I-TH PANE AT THE CURRENT TIME STEP FOR THE UNSTEADYC FLOW CASE.C PHI(I) ..... DISTURBANCE VELO POTENTIAL AT THE MID POINT OF THEC I-TH PANEL FOR THE PREVIOUS TIME STEPC PHILE(L) .... DIFFERENCE OF THE POTENTIALS OF THE LEADING EDGE TOC THE LOWER TRAILING EDGE FOR THE RESPECTIVE AIRFOILC PINK(L) .... DIFFERENCE OF THE POTENTIALS AT A POINT 100 CHORDC LENGTH UPSTREAM OF THE LE FOR THE RESPECTIVE AIRFOILC SUMC(L) .... GAMMA ASSOCIATED WITH THE INTEGRATION OF THE DISTURB-C ANCE VELOCITY AROUND THE WHOLE AIRFOILCC THE FOLLOWING INFLUENCE COEF ARE COMPUTED FOR A FINER GRID ON THEC AIRFOIL SO AS TO OBTAIN A MORE ACCURATE VELO POTENTIAL AT THE TEC THE INFLUENCE COEF ON THE I-TH PANEL FROM THE J-TH PANEL OF THEC AIRFOIL REMAINS THE SAME FOR ALL TIME STEPC -CC AANP1(I,JK).. NORMAL VELOCITY INDUCED AT THE I-TH PANEL SUB NODE KC OF AIRFOIL 1 DUE TO UNIT STRENGTH DIST SOURCE STRENG-C TH ON THE J-TH PANEL OF AIRFOIL 1C BBNP1(I,J,K).. NORMAL VELOCITY INDUCED AT THE I-TH PANEL SUB NODE KC OF AIRFOIL 1 DUE TO UNIT STRENGTH DIST VORTICITY STR-C ENGTH ON THE J-TH PANEL OF AIRFOIL 1C AANP2(I,JK).. NORMAL VELOCITY INDUCED AT THE I-TH PANEL SUB NODE KC OF AIRFOIL 2 DUE TO UNIT STRENGTH DIST SOURCE STRENG-C TH ON THE J-TH PANEL OF AIRFOIL 2C BBNP2(I,J,K).. NORMAL VELOCITY INDUCED AT THE I-TH PANEL SUB NODE KC OF AIRFOIL 2 DUE TO UNIT STRENGTH DIST VORTICITY STR-C ENGTH ON THE J-TH PANEL OF AIRFOIL 2CCCC

SUBROUTINE PRESSCOMMON /BOD/ IFLAG,NLOWER,NUPPERNODTOT,X(202),Y(202),+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFT,NAIRFO,XI(202) ,YI(202),+ COSTHL(201),SINTHL(201)COMMON /CPD/ CP(200),SCLT,SCM,SGAMCOMMON /NUM/ PI,PI2INVCOMMON /SING/ Q(200) ,GAMMA(2) ,QK(200),GAHK(2)COMMON /WAK/ VYW(2),VXW(2),WAKE(2),DTCOMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TDCVVX(2,200),

+ CVVY(2,200) ,XCI(2,200) ,YCI(2,200)COMMON /INF1/ AAN(201,201),BBN(201,201),AYNPI(2,201),BYNP1(2,201),+ SUMAAN(201,2),SUMBBN(201,2)COMMON /INF2/ SUMCCN(201),SUMCCT(201),CYNPI(2,200),

+ CXNP1(2,200)COMMON /INFL3/ AANP(101,101,6),BBNP(101,101,6)COMMON /POT/ PHI(200),PHIK(200)COMMON /GUST/ UG(200),VG(200),XGFUGUST,VGUST

123

-. * ' , -- , .~,m ,, m r a ,,a i i m i .. . ... . . .

COMMON /EXTV/ UE(200),VN(200)COMMON /BMAT/ Bl,B2,B3COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UXC2),UY(2),PIVOT,

+ XPRM,YPRMDIMENSION PHITEL(2),PHITEU(2),WGHT(5),PLOC(5),SUMC(2),

+AANP1C5O,50,9) ,AANP2(50,50,9) ,BBNP1(50,50,9) ,BBNP2(50,50,9),+COSTHP(102,6),SINTHP(102,6),AANP4(2),PHIL(102),UGU(100,6),+VGU( 100,6) ,PDUM(2) ,DP(2) ,BBNP4(2) ,B1(200) ,B2(200) ,B3(200) ,PHILE(2)

3333 DIMENSION CON2(2),VTGT(2,50),WAKCON(2,50),VOTCON(2,50),+ DUMCON(2,50)

7733 DIMENSION CP1(200)4444 REAL *8 PINK(2),VELX

DATA WGHT/. 11846344,. 23931434,. 28444444,+ . 23931434,. 11846344/DATA PLO/. 04691008,. 23076535,. 50000000,

+ .76923466,. 95300899/WRITE (6,1000)IF (M .Gr. 1) GO TO 510

CC COMPUTE THE INFLUENCE COEFFICIENT FOR A FINER GRIDC .... ON THE AIRFOIL BY THE SAME AIRFOIL ...

CC ONLY COMPUTED ONCE)C

DO 200 L =1,NAIRFOLI = (L.1)*NODTOTKI = (L-1)*NP1DO 200 1 = 1,NODTOTDO 200 K =1,5DX = PLOC(K)*(X(I+KI+1)-X(I+KI))DY = PLOC(K)*CY(I+KI+1)-Y(I+KI))XMID = X(I+KI) + DXYMID = Y(I+KI) + DYDO 200 J = 1,NODTOTFLOG = 0.0FTAN = PIIF CJ+LI .EQ. I+LI) GO TO 100DXJ = XMID - XCJ+KI)DXJP = XMID - XCJ+KI+1)

*DYJ = YMID - Y(J+KI)DYJP - YMID - Y(J+KI+1)FLOG = . 5*ALOGC (DXJP*DXJP+DYJP*DYJP) /(CDXJ*DXJ+DYJ*DYJ))ETAN = ATAN2(DYJP*DXJ-DXJP*DYJ ,DXJP*DXJ+DYJP*DYJ)

100 CTIMTJ = COSTHECI+KI)*COSTHE(J+KI) ++ SINTHEC I+KI)*SINTHE(J+KI)STIMTJ = SINTHE(I+KI)*COSTHECJ+KI)-

+ COSTHE(I+KI)*SINTHECJ+KI)IF (L .EQ. 2) GOTO 120AANP1(I,J,K) -P12INV*(FTAN*CTIMTJ + FLOG*STIMTJ)BBNP1(I,J,K) - P12INV*(FLOG*CTIMTJ - TAN*STIMTJ)GOTO 200

120 AANP2(I,J,K) - P12INV*(FTAN*CTIMTJ + FLOG*STIMTJ)BBNP2(I,J,K) = P12INV*(FLOG*CTIMTJ - FTAN*STIMTJ)

200 CONTINUE510 CONTINUEC

124

C FIND TANGENTIAL VELOCITY AT THE MIDPOINT OF THE 1-TI PANELC

DO 600 L = 1,NAIRFOLIaC L- 1)*NOD1T

SUMC(L =0.0DO 600 I 1,NODTOTCONTR1 = 0.0

* -CONTR2 =-0.0WAKCONCLI) = 0.0VOTCON(L,I) = 0.0DUMCON(L,I) = 0.0XMID =0.5 * (XI(I+KI) + XI(I+KI+1))YMID -0.5 * (YI(I+KI) + YICI+KI+1))DX -(XICI+KI+l) - XI(I+KI))DY =(YI(I+KI+l) - YI(I+KI))DIST =SQRT(DX*DX+DY*DY)

CC ACCOUNT FOR THE FREESTREAM AND THE MOTION OF THE AIRFOILC

VSX = C1.+UGCI+LI))*COSALF(L)-VG(I+LI)*SINALF(L) + OMEGACL)*+ YMID + UXCL)VSY = Cl. +UGC I+LI) )*SINALC L)+VG( I+LI )*COSAL3( L)

+ - OMEGA(L)*XMID + UY(L)VS = VSX*VSX + VSY*rVSYVNORM = -VSX'*SINTHL( I+KI )+VSY*COSTHL( I+KI)VTANG m (((1. +UG(I+LI))*COSALF(L)-VG(I+LI)*SINALF(L)+UX(L))

+ *COSTHL(I+KI)+ ((1. +UG(I+LI))*SINALF(L)+VG(I+LI)*COSALFCL)+UY(L))+~ *SIfL( I+KI)+ OIEGA(L)*(YMID*COSTHLCI+KI)+ - XMID*SINTHL(I+KI)))

VTFREE = VTANGVACT = VTANG

CC INTRODUCE SMALLER GRIDS FOR THE PURPOSE OF THE VELO POTENTIAL.C VELO ONLY CALCULATED AT THE MIDPT OF THE PANEL WHERE K =3C

DO 260 K = 1,5DX = PLOC(K)*(X(I+KI+l)-X(I+KI))DY -= PLOC(K)*(Y(I+KI+1)-Y(I+KI))XINT - X(I+KI) + DXYINT = Y(I+KI) + DYVDUM = 0.0

CC INFLUENCE COEF ON I-TH PANEL DUE TO TilE WAKE ELEMENTSC

DO 245 MM = 1, NAIRFOJ = NP1*MMDXJ = XINT - X(J)DXJP - XINT - X(NAIRPO*NP1+MM)DYJ - YINT - Y(J)DYJP = YINT - Y(NAIRFO*NP1+MM)FLOG - .5*AWOG( (DXJP*DXJP+DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))YIAN - ATAN2( DYJP*DXJ -DXJP*DYJDXJP*DXJ+DYJP*DYJ)CTIMTJ = COSTHEI+KI)*COSTHE(J) +

+ SINTHEC I+KI )*SINTHE( J)STIMTJ = SINTHE(I+KI)*COSTHE(J)-

125

+ COSTHEC I+KI)*SINTHE(J)AANP4(MH) = P12INV*(FTAN*CTIMTJ + FLOG*STIMTJ)

245 BBNP4(MH) = P12INV*(FILOG*CTIMTJ -FTA*STIlTJ)CONTRi = SSC 1)*(GAMMA( 1)-GAMK( 1))*AANP4( 1)

+/WAKE( 1)+SS(2)*(GAMMA(2)-GAMK(2))*MANP4C2)/WAKE(2)CC CONTRIBUTION TO VELO COMPONENT BY WAKE ELEMENT.C

IF (K .EQ. 3) THENVACT - VACT + SS(1)*(GAMHA(1)-GA11K(l))*AAN(I+LI,

+NP3)/WAKE(1)+SS(2)*(GAM1NA(2)-GAMKl(2))*MANCI+LI,NP4)/WAKE(2)WNORM = WNORM + SS(1)*(GAMMA(l)-GAMK(1))*BBNP4(l)

+/WAKE( 1)+SSC 2)*(GAMMA( 2) -GAIK( 2) )*BBNP4( 2)/WAKEC 2)END IF

CC EFFECTS ON AIRFOIL BY THE SAME AIRFOILC - - - - - - - - - - - - - - - - - - - -CC INTEGRATION AROUND FIRST AIRFOILC CONTRIBUTION TO VELO COMP BY AIRFOIL 1 WHEN K = 3C

DO 300 J = 1,NODTOTIF (L .EQ. 2) GOTO 270VDUM = VDUM-BBNP1CI,J,K)*QKCJ)+AANP1CI,J,K)*GAMK(l)IF (K .EQ. 3) THENVACT = VACT-BBN(I+LI,J)*QK(J)+

+AAN( I+LI ,J)*GAMK( 1)VNORM = VNORM+(AANP1(I,J,K)*QK(J))+

+CBBNP1(I,J,K)*GAIKC1))END IFGOTO 300

CC INTEGRATION AROUND SECOND AIRFOILC CONTRIBUTION TO VELO COMP BY AIRFOIL 2 WHEN K =3C

270 VDUM =VDUM-BBNP2(I,J,K)*QK(J+NODTOT)+AANP2CI,J,K)*GAMKC2)

IF (K .EQ. 3) THENVACT = VACT-BBN( I+LI ,J+NODTOT)*QK(J+NODTOT)+" AANC I+LI )J+NCDTOT)*GAMK(2)

VNORM' = VNORM+(AANP2( I,J,K)*QK(J+NODTOT) )+" (BBNP2(I,J,K)*GAMK(2))ENDIF

300 CONTINUECC EFFECTS ON AIRFOIL BY THE OTHER AIRFOILC-- - - - - - - - - - - - - - - - - - - -CC INTEGRATION AROUND BOTH AIRFOILC CONTRIBUTION TO VELO COMP BY BOTH AIRFOIL WHEN K -3C

DO 350 J =NODTOT+2,2*NODTOT+1DXJ - XINT - X(J-KI)DXJTP - XINT - X(J-KI+l)DYJ = YINT - Y(J-KI)DYJP = YINT - Y(J-KI+l)FLOG = .5*ALOG( (DXJP*DXJP+DYJP*DYJP) /(CDXJ*DXJ+DYJ*DYJ))

126

...-.

FTAN = ATAN2(DYJP*DXJ-DXJP*DYJDXJP*DXJ+DYJP*DYJ)CTlMTJ - COSTHE(I+KI)*COSTHE(J-KI) ++ SINTHE(I+KI)*SINTHE(J-KI)STIMTJ = SlNTHE(I+KI)*COSTHE(J-KI)+ COSTHE(I+KI)*SINTHE(J-KI)AANP3 = PI2INV*(FTAN*CTIMTJ + FLOG*STIMTJ)BBNP3 - PI2INV*(FLOG*CTIMTJ - FTAN*STIMTJ)IF (K-EQ.3) THENVACT - VACT-BBN(I+LIJ-LI-1)*QK(J-LI-1)++ AAN(I+LIJ-LI-1)*GAMK(3-L)WORM - VNOR14+(AANP3*QK(J-LI-1))++ (BBNP3*GAMK(3-L))ENDIF

350 VDUM VDUM - BBNP3*QK(J-LI-1) + AANP3*GAMK(3-L)CC CONTRIBUTION BY CORE VORTICESC 1. TO VELO POTENTIALC 2. TO VELO COMPONENT ONLY WHEN K 3C

IF (M. EQ. 1) GOTO 150mmi = M - 1SUMCN = 0.0SUMCT = 0.0DO 400 MM = 1, NAIRFOKN = (MM-1)*MmlDO 400 N = 1,MM1DXC = XINT - XC(MMN)DYC = YINT - YC(MMN)DISTC = SQRT(DXC*DXC+DYC*DYC)COSTHN = DXC/DISTCSINTHN = DYC/DISTCCTIMTN = COSTHE(I+KI)*COSTHN + SINTHE(I+Kl)*SINTHNSTIMTN = SINTHE(I+KI)*COSTHN COSTHE(I+KI)*SINTHNCCN = -CTIMIN/DISTCCCT = -STIMTN/DISTCSUMCN - SUMCN + CCN"tCV(MMN)

400 SUMCT - SUMCT + CCT*CV(MMN)IF (K EQ. 3) THENVACT ac VACT+SUMCT*PI2INVWORM - VNORM+(PI2INV*SUMCN)ENDIFCONTR2 = PI2INV*SUMCT

150 CONTINUEWANG - WANG + DBLE((CON7Rl+CON7R2+VDUM)*WGHT(K))SUMC(L) - SUMC(L)+VDUM*DIST*VGNT(K)WAKCON(LI) - WAKCON(LI)+CONTR1VOTCON(LI) - VOTCON(LI)+CONTR2DUIMCON(LI) - DUMCON(LI)+VDUM

260 CONTINUEPHIK(I+LI) - ('(VTANG)-VTFREE)*DrSTCP(I+LI) - VS - (VACT*VACT)

7755 CP1(I+LI) - CP(I+LI)UE(I+LI) a VACTVN(I+LI) - VNORM

600 CONTINUE6688 FORMAT(2I5,7El5.7)

127

3335 FORMAT (215,7Fl0.6)CC COMPUTE DISTURBANCE POTENTIAL AT THE LEADING EDGE BY LINEC INTEGRAL OF THE VELOCITY FIELDC FROH UPSTREAM (AT INFINITY) TO THE LEADING EDGEC

TRY = 1PINK(1) - 0.0PINK(2) -m.

130 DO 56 L - 1,NAIRFOYMID - PIVOT * SINALF(L + (L-1)*YSHIF-rXLE = - PIVOT*COSALF(L)+(L1)*XSHIFTXL w XLE

C XL =0.0NPHI = 10 * NLOWERPDUM(L) - 0.0DO 30 1I 1,NPHIFRACT =FWAT(I)/FLOAT(NPHI)

XLP - 100.0 *TRY * (1.0 - COS(0.S*PI*FRACT))+XLEIF (I .EQ.-1) XLP - -0. 000197+XLE

C XLP = -10.0 * (1.0 - COS(0.5*PI*FRACT))DELX = XL -XLPXMID = 0.5*(XL+XLP)

C XMID = 0.5*(XL+XLP)*COSALF(L)C YMID = 0.5*CXL+XLP)*SINALF(L)

XL =XLPVELX = UGUST

CC ADD CONTRIBUTION OF J-TH PANELC

DO 40 MM = 1,NAIRFOLi =(MM-1)*NODTOTKJ in(MM-1)*NP1DO 20 J = 1,NP1IF (J .EQ. NP1) GO TO 24DXJ - XMID - X(J+KJ)DXJP = XMID - X(J+KJ+l)DYJ = YMID - Y(J+KJ)DYJP = YMID - Y(J+KJ+1)FLOG = . *ALOG( (DXJP*DXJP+DYJP*DYJP) /(DXJ*DXJ+DYJ*DYJ))FTAN = ATAN2 (DYJP*DXJ-DXJP*DYJ, DXJP*DXJ+DYJP*DYJ)

C CALMTJ = -COSTHE(J+KJ)C SALMTJ - SINTHE(J+KJ)

CALMTJ -- COSALF( L)*COSTHE( J+KJ) - SINALF( L)*SINTHE( J+KJ)SALMTJ - -SINALF(L)*COSTHE(J+KJ) + COSALF(L)*SINTHE(J+KJ)APY = P12INV*(TAN*CALMTJ + FWOG*SAI&ITJ)BPY - P12INV*(FLOG*CALMTJ - TAN*SALMTJ)VELX - VELX -DPROD(BPY,QK(J+LJ)) + DPROD(GANK(MM),APY)GO TO 20

24 DXJ - XMID -X(J+KJ)

DXJP - XMID -XC2*NPI+MH)

DYJ - YMID -Y(J+KJ)

DYJP - YMID -Y(2*NP1+MM)

FLOG - . 5*ALOG( (DXJP*DXJP4DYJP*DYJP) /( DXJ*DXJ+DYJ*DYJ))FTAN = ATAN2 (DYJP*DXJ-DXJP*DYJ ,DXJP*DXJ+DYJP*DYJ)CALMTJ - -COSALF(L)*COSTHE(J+KJ) - SINALF(L)*SINTHE(J+KJ)

128

7- - - - - - -- 'Zti .r

*~. 7k.-

SALMTJ - -SINALF(L)*COSTHE(J+KJ) + COSALF(L)*SINTHE(J+KJ)APY = P12INV*(FTAN*CALMTJ + FLOG*SALMTJ)DUMMY - SS(MM)*(GAlHACHN)-GAfK(HM))/WAKE(MMl)VELX - VELX + DPROD(APYDUMMY)

20 CONTINUE40 CONTINUE

1CC ADD CORE VORTEX CONTRIBUTIONC

IF (m .EQ. 1) GO TO 50

DO 70 11 - iNAIRFO

DO 60 N - iMMIDX -XHID - XC(IIN)DY - YMID - YC(IIN)DIST - SQRT(DX*DX+DY*DY)COSTHN - DX/DISTSIN7TIN = DY/DISTSALMTN = -SINALF(L)*COSTHN + COSALFCL)*SINTHNCPT - Pl21NV*SALNTN/DIST

60 VELX - VELX + DPROD(CPT,CV(IIN))70 CONTINUE50 CONTINUE

PDUM(L) - PDUM(L) + VELX * DBLECDELX)7771 FORMAT (215,6E14.6,2F10.6)30 CONTINUE

DP(L) = PDUM(L)-PINK(L)56 CONTINUE

PINK(l) - PDUM(1)PINK(2) = PDUM(2)

C COMPUTATION OF THE VELOCITY POTENTIAL FOR MIDPOINT OF EACH PANELC

DO 240 L - 1,NAIRFOLI -(L-1)*NODTOTPH? - -PINK(L)

CC BEGIN-WITH LOWER SURFACEC

DO 230 1 NLOWER,1,-1PHC =PHP-PHIK(I+LI)

PHIK(I+LI) U0.5*(PHP+PHC)

230 PH? PHCPHITEL(L) =PHC

CC RESET FOR UPPER SURFACEC

PH? -PINK(L)DO 250 I1 NLOWER+1,NODTOTPHO PHP+PHIK(I+Ll)PHIK(I+LI) -0.5*(PHP+PHC)

250 PH? PHCPHITEU(L) -PHC

240 CONTINUE

129

CC COMPUTE CP AT MID POINT OF I-TH PANELC

DO 295 L - INAIRFOLI = (L-1)*NODTOTKI = (L-1)*NPISUMC(L) - SUMC(L)/SS(L)DO 290 I - 1,NODTOTXIMID - .5*(XI(I+KI) + XI(I+KI+I))XMID - .5*(X(I+KI) + X(I+KI+1))YMID - .5*(Y(I+KI) + Y(I+KI+1))CP(I+LI)- CP(I+LI) - 2. *(PHIK( I+LI)-PHI( I+LI))/DTWRITE (6,1050) I+LI,XIMID,XMID,YMID,QK(I+LI),GAMK(L),CP(I+LI),+ UE(I+LI),VN(I+LI),PHIK(I+LI),PHI(I+LI),SUMC(L)

290 CONTINUEWRITE (6,235) PINK(L)

295 CONTINUE235 FORMAT (1X,'PHI AT LEADING EDGE -',F1O.6,/)1000 FORMAT(/,4X 'J' ,4X, XIJ) ',5X,'X(J) ',6X, Y(J)',6X,'Q(J)',6X,

+'GAMMA' ,4X, CP(J) ';X, V(J)',6X,'VN(J)',5X,'"PHIK',6X,+'PHI', 6X, INTGAMK ,/)

1050 FORMAT(I5,12F10. 5)1200 FORMAT(lX,'LENGTH OF LEADING EDGE INTEGRATION IN CHORD -',FIO.6/)

RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE CORVOR (SINALF,COSALF) CC CC COMPUTE THE LOCAL VELOCITIES OF CORE VORTICES CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCVX(I,N) .... X-VELO OF THE I-TH AIRFOIL, N-TH CORE VORTEX WITHC RESPECT TO THE CURRENT FROZEN FRAME OF REFERENCEC CCVY(I,N) .... Y-VELO OF THE I-TH AIRFOIL, N-TH CORE VORTEX WITHC RESPECT TO THE CURRENT FROZEN FRAME OF REFERENCEC XC(I,N) .... X-COORD OF THE LOCATION OF THE I-TH AIRFOIL, N-THC CORE VORTEX W.R.T. THE GLOBAL FRAME OF REFERENCE.C YC(1,N) .... Y-COORD OF THE LOCATION OF THE I-TH AIRFOIL, N-THC CORE VORTEX W.R.T. THE GLOBAL FRAME OF REFERENCE.C GAMY(I) .... GLOBAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A SOURCE DIST OF UNIT STRENGTHC ON ONE PANELC GBMY(I) .... GLOBAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A VORTEX DIST OF UNIT STRENGTHC ON ONE PANELC AMY(I) .... LOCAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A SOURCE DIST OF UNIT STRENGTHC ON ONE PANELC BMY(I) .... LOCAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A VORTEX DIST OF UNIT STRENGTHC ON ONE PANELC SUMAMY(I) .... LOCAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A SOURCE DIST OF UNIT STRENGTHC ON ALL PANELS

130

C SUMBMY(I) .... LOCAL Y-VELO AT A LOCATION OF THE I-TH AIRFOILC CORE VORTEX DUE TO A VORTEX DIST OF UNIT STRENGTHC ON ALL PANELSC GCMX .... GLOBAL X-VELO AT A LOCATION OF A CORE VORTEX DUEC TO ANOTHER POINT VORTEX OF UNIT STRENGTHC GCMY .... GLOBAL Y-VELO AT A LOCATION OF A CORE VORTEX DUEC TO ANOTHER POINT VORTEX OF UNIT STRENGTHC CMX .... LOCAL X-VELO AT A LOCATION OF A CORE VORTEX DUEC TO ANOTHER POINT VORTEX OF UNIT STRENGTHC CMY .... LOCAL Y-VELO AT A LOCATION OF A CORE VORTEX DUEC TO ANOTHER POINT VORTEX OF UNIT STRENGTHCC

SUBROUTINE CORVORCOMMON /BOD/ IFLAG,NLOWER,NUPPERNODTOTX(202) ,Y(202),

+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5,XSHIFT,YSHIFTNAIRFO,XI(202) ,YI( 202),+ COSTHL(201),SINTHL(201)COMMON /SING/ Q(200) ,GAMMA(2) ,QK(200),GAMK(2)COMMON /WAK/ VYW(2),VXW(2),WAKE(2),DTCOMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TD,CCVX(2,200),+ CCVY(2,200),XCI(2,200),YCI(2,200)COMMON /POT/ PHI(200),PHIK(200)COMMON /GUST/ UG(200) ,VG(200) ,XGF,UGUST,VGUSTCOMMON /NUM/ PI,PI2INVCOMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM,YPRMDIMENSION AMY(2),BMfY(2),SUMAMY(2),SUMBMY(2),

+VX(2) ,VY(2),GAMY(2),GBMY(2)IF (M.EQ.1) GOTO 40

CC ACCOUNT FOR THE FREE STREAM INCLUDING GUST EFFECTS VC

UGC = 0.0VGC = 0.0DO 15 I = 1,NAIRFOKN = (I-1)*Hm1DO 10' N = 1,MM1XG XC(I,N)*COSALF(I) + YC(I,N)*SINALF(I)IF (XG .GT. XGF) GO TO 5UGC - UGUSTVGC = VGUST

5 CONTINUEVY(I) - (1.+UGC)*SINALF(I)+VGC*COSALF(I)VX(I) - (1. +UGC)*COSALF( I)-VGC*SINALF(I)XMID - XC(I,N)YMID - YC(I,N)

CC CALCULATE THE INFLUENCE COEFFICIENT DUE TO THE AIRFOILSC

DO 25 L - INAIRFOLJ =(L"1)*NODTOTKJ -(L-1)*NP1SUMAMY(I) = 0.0SUMBMY(I) = 0.0

131

DO 20 J - ,PDXJ - XMID - X(J+LJ)DYJ - YHID - Y(J+KJ)IF (J .EQ. NP1) GO TO 11DXJP = XHID - X(J+KJ+l)DYJP = YMID - Y(J+KJ+1)GO TO 12

11 DXJP = Q4ID - X(2*NP1+L)DYJP = YMID - Y(2*NP1+L)

12 FLOG = .5*ALOG( (DXJP*DXJP+DYJP*DYJP) /(DXJ*DXJ+DYJ*Dyj))YUAN = ATAN2(DYJP*DXJ-DXJP*DYJ ,DXJP*DXJ+DYJP*DYJ)GAIIY(I) - P12INV*(FTAN*COSTHE(J+KJ) - FLOG*SINTHE(J+KJ))GEMY(I) = P12INV*(FLOG*COSTHE(J+KJ) + FTAN*SINTHE(J+KJ))AMY(I) - GAMYCI)*COSALF( 1)-GBMYC I)*SINALFC I)EMY(I) = GAMYC I)*SINALF(I)+GBMYCI)*COSALF( I)IF (J.EQ.NP1) GOTO 20SLIMAMYI) = SUHAMY(I) + AKY(I)SUMBMY(I) - SUMDmY(I) + BMYCI)VY(I) - VY(I) + AMY(I)*QK(J+LJ)VX(I) = VX(I) - BMY(I)*QK(J+IJ)

20 CONTINUECC ADD THE CONTRIBUTION DUE TO THE WAKE ELEMENTSC

VY(I) = VY(I) + SUIIBIY(I)*GAIIK(L) + SS(L)*BMYCI)*+ (GAIIMA(L)-GAMK(L))/WAKE(L)

VX(I) = VX(I) + SUMAMY(I)*GAMK(L) + SS(L)*AMY(I)*+ (GAMMA(L)-GAMK(L))/WAKE(L)

25 CONTINUECC CALCULATE INFLUENCE COEFFICIENT DUE TO THE WAKEC

DO 35 L = 1,NAIRFOK01C =(L-1)*M1DO 30 MC = 1,MM1IF ((L.EQ.I) .AND. (MC.EQ.N)) GOTO 30DX - XMID - XC(L,MC)DY - ThID - YC(L,MC)DIST2 = DX*DX+DY*DYGCMY - -P12INV*DX/DIST2GCMX = +P12INV*DY/DIST2CHY - GCMY*COSALF( I )4.~CM*SINAF( I)CMX = -GCMY*SINALF(I)4GCMX*COSALF(I)

CC ADD CORE VORTEX CONTRZIBUTIONC

VY(I) - VY(I) + CMY*CV(L,MC)VX(I) - VX(I) + CHX*CV(LMC)

30 CONTINUE35 CONTINUE

CC CONVECTION VELOCITY OF CORE VORTICES AT NX TIME STEPC

CCVX(I,N) - VX(I)CCVY(I,N) - VY(I)

10 CONTINUE

132

............-~ -7-=

* . ~ .. ....1

15 CONTINUE40 CONTINUE

RETURNEND

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CC SUBROUTINE NEWPOSN CC CC TRANSFORM THE RESPECTIVE FROZEN LOCAL COORDINATES AND PANEL CC ANGLES TO THE GLOBAL FRAMES OF REFERENCE CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCccCCCC XSHIFT .... GLOBAL X LOCATION OF THE 2-ND AIRFOIL PIVOT POSITIONC YSHIFT .... GLOBAL Y LOCATION OF THE 2-ND AIRFOIL PIVOT POSITIONC ITYPE .... FLAG FOR DEALING WITH THE VARIOUS COMPONENTSC - 0 : FOR TRANSFORMING AIRFOIL COORDINATESC - 1 : FOR TRANSFORMING WAKE ELEMENTSC - 2 : FOR TRANSFORMING MOST RECENT CORE VORTEX SHEDC = 3 : FOR TRANSFORMING ALL PREVIOUS CORE VORTICESC XPRM .... X GLOBAL RELATIVE MOVEMENT OF THE PIVOTS POSITIONC YPRM .... Y GLOBAL RELATIVE MOVEMENT OF THE PIVOTS POSITIONCC

SUBROUTINE NEWPOS(ITYPE)COMMON /BOD/ IFLAG,NLOWER,NUPPERNODTOT,X(202) ,Y(202),+ COSTHE(201),SINTHE(201),SS(2),NP1,NP2,NP3,NP4,+ NP5 ,XSHIFT,YSHIFTNAIRFOXI(202) ,YI(202),+ COSTHL(201),SINTHL(201)COMMON /CORV/ CV(2,200),XC(2,200),YC(2,200),M,TD,CVVX(2,200),

+ CVVY(2,200),XCI(2,200),YCI(2,200)COMMON /GEOM/ SINALF(2),COSALF(2),OMEGA(2),UX(2),UY(2),PIVOT,

+ XPRM, YPRMCC CHECK FLAG TO ASCERTAIN TYPE OF TRANSFORMATIONC

IF ( ITYPE .EQ. 1 ) GOTO 30IF ( ITYPE .EQ. 2 ) GOTO 40IF ( XTYPE .EQ. 3 ) GOTO 45

CC AIRFOIL COORDINATES AND LOCAL ANGLES TRANSFORMATIONC

DO 10 1 = 1,NP1X(I) - (XI(I)-PIVOT)*COSALF(1)+YI(I)*SINALF(I)Y(I) - (XI(I)-PIVOT)*(-SINALF(l))+YI(I)*COSALF(1)

10 CONTINUEDO 20 1 - NP1+1,2*NP1X(I) - (XI(I)-PIVOT)*COSALF(2)+YI(I)*SINALF(2)+

+ XSHIFT + XPRMY(I) - (XI(I)-PIVOT)*(-SINALF(2))+YI(I)*COSALF(2)+

+ YSHIFT + YPRM20 CONTINUE

CC TRANSFORM LOCAL ANGLESC

DO 222 L = 1,2

133

,, ,. .:-, :, -. - ...- -- " ...... . :'- . ,' <.-r - - --!: , i . ': i~ ; i .; "! i _,: -,'..----.C . - :' : -!; i

K -(L-1)*(NODTOT+l)DO 222 J - 1,NODTOTDX a X(J+l+K) - X(J+K)DY - Y(J+1+K) - Y(J+K)DIST - SQRT(DX*DX +DY*DY)SINTHE(J+K) - DY/DIST

222 COSTHE(J+K) - DX/DISTGOTO 50

CC WAKE ELEMENT TRANSFORMATIONC

30 I NP2X(I) - (XI(I)-PIVOT)*COSALF(1)+YI(I)*SINALFC1)Y(I) - (XI(I)-PIVOT)*C-SINALF(1))+YI(I)*COSALFC1)I - NP2+ 1XCI) - CXI(I)-?IVOT)*COSALFC2)+YI(I)*SINALF(2)+

+ XSHIFT + XPRMY(I) -(XI(I)-PIVOT)*(-SINALFC2))+YI(I)*COSALFC2)+

+ YSHIFT + YPRMCC WAKE ELEMENT ANGLES TRANSFORMATIONC

DO 223 L = 1,2K - (L-1)*CNODTOT+l)J NP1DX - X(NP2+L-1) - X(NP1*L)DY - Y(NP2+L-1) - Y(NP1*L)DIST - SQRT(DX*DX +DY*DY)SINTHE(J+K) = DY/DIST

223 COSTHE(J+K) - DX/DISTGOTO 50

CC MOST RECENT CORE VORTEX SHED TRANSFORMATIONC40 XCC1,M) - (XCI(1,M)-PIVOT)*COSALF(1)+YCIC1,t1)*SINALF(l)

YC(1,M) - CXCI(1,M)-PIVOT)*(-SINALF(l))+YCI(1,1)*COSALF(l)XC(2,M) = (XCI(2,M)-PIVOT)*COSALF(2)+YCIC2,11)*SINALF(2)+

+ XSHIFT + XPRIYC(2,M) - (XCI(2,M)-PIVOT)*(-SINALFC2))+YCI(2,M)*COSALF(2)+

+ YSI{IFT + YPRIIGOTO 50

CC ALL PREVIOUS CORE VORTICES TRANFORMATIONC

45 DO 46 1- 1,M-1XC(1,I) = CXCI(l,I)-PIVOT)*COSALF(1)+YCIC1,1)*SINALFC1)YC(1,I) - (XCIC1,l)-PIVOT)*(-SINALF(l))+YCI(1,I)*COSALF(l)XC(2,I) - (XCI(2,I)-PIVOT)*COSALF(2)+YCI(2I1)*SINALF(2)+

+ XSHIFT + XPRMYC(2,I) = (XCI(2,1)-PIVOT)*( -SINALF(2))+YCI(2,I)*COSALF(2)+

+ YSHIFT + YPRM46 CONTINUE50 CONTINUE

RETURNEND

134

APPENDIX B. EXAMPLE INPUT DATA FOR PROGRAM USPOTF25

AIRFOIL TYPE : 8.4% THICKNESS SYMMETRICAL MISES AIRFOILAIRFOIL COORDINATES ARE USER INPUT DATA (IFLAG - 1)NLOWER = 25 , NUPPER - 25

01 25 252 0.0 2.0

1.000000 0.994858 0.980866 0.958884 0.929536 0.8934550.851308 0.803815 0.751753 0.695948 0.637271 0.5766200.514918 0.453098 0.392084 0.332794 0.276105 0.2228650.173861 0.129819 0.091393 0.059146 0.033560 0.0150100.003767 0.000000 0.003767 0.015008 0.033560 0.0591460.091393 0.129819 0.173861 0.222865 0.276105 0.3327910.392082 0.453095 0.514915 0.576617 0.637266 0.6959460.751750 0.803815 0.851308 0.893455 0.929536 0.9588840.980866 0.994858 1.0000000.000000 -0.000782 -0.002784 -0.005721 -0.009351 -0.013459-0.017837 -0.022285 -0.026618 -0.030671 -0.034289 -0.037341-0.039712 -0.041314 -0.042083 -0.041979 -0.040979 -0.039096-0.036360 -0.032820 -0.028555 -0.023651 -0.018220 -0.012379-0.006259 0.000000 0.006259 0.012379 0.018220 0.0236510.028555 0.032820 0.036360 0.039096 0.040979 0.0419790.042083 0.041314 0.039712 0.037341 0.034289 0.0306710.026618 0.022285 0.017837 0.013459 0.009351 0.0057210.002784 0.000782 0.0000001.000000 0.994858 0.980866 0.958884 0.929536 0.8934550.851308 0.803815 0.751753 0.695948 0.637271 0.5766200.514918 0.453098 0.392084 0.332794 0.276105 0.2228650.173861 0.129819 0.091393 0.059146 0.033560 0.0150100.003767 0.000000 0.003767 0.015008 0.033560 0.0591460.091393 0.129819 0.173861 0.222865 0.276105 0.3327910.392082 0.453095 0.514915 0.576617 0.637266 0.6959460.751750 0.803815 0.851308 0.893455 0.929536 0.9588840.980866 0.994858 1.0000000.000000 -0.000782 -0.002784 -0.005721 -0.009351 -0.013459-0.017837 -0.022285 -0.026618 -0.030671 -0.034289 -0.037341-0.039712 -0.041314 -0.042083 -0.041979 -0.040979 -0.039096-0.036360 -0.032820 -0.028555 -0.023651 -0.018220 -0.012379-0.006259 0.000000 0.006259 0.012379 0.018220 0.0236510.028555 0.032820 0.036360 0.039096 0.040979 0.0419790.042083 0.041314 0.039712 0.037341 0.034289 0.0306710.026618 0.022285 0.017837 0.013459 0.009351 0.0057210.002784 0.000782 0.0000000.0 0.0 -45.80000 0.00 0.0 .0000 00.0 0.00 0.0 0.00 0.0 .000 0.0 0.05.0000 0.065 0.001 0.00 4.855698 -.921370 1.216290 0

135

APPENDIX C. EXAMPLE OUTPUT DATA FROM PROGRAM

USPOTF2DATA READ FROM FILE CODE 1

5

AIRFOIL TYPE : 8.47 THICKNESS SYMMETRICAL PUSES AIRFOILAIRFOIL COORDINATES ARE USER IPUT DATA ('rFLAG a 11NLOMER a 25 , MPPER a ZS

1 ?.5 ?.5

2 0.0 2.01.000000 0.99486 0.96086 0.95884 0.9t9536 0.8934550.651306 0.803815 0.751753 0.695948 0.637271 0.576620O.514916 0.4S3098 0.392084 0.332794 0.276105 0.222"650.173841 0.129819 0.091393 O.059146 0.033560 0.015010

* 0.003767 0.000000 0.003767 0.015008 0.033560 0.0591460.091393 0.129619 0.173841 0.22265 0.27610S 0.3327910.392082 0.45309S 0.514915 0.576617 0.637266 0.6959460.751750 0.80385 0.851308 0.893455 0.929536 0.9588840. 90866 0.994858 1.0000000.000000 -0.000782 -0.002784 -0.005721 -0.009351 -0.013459-0.017637 -0.022285 -0.026618 -0.030671 -0.034289 -0.037341-0.039712 -0.041314 -0.042083 -0.041979 -0.040979 -0.03994-0.036360 -0.032820 -0.0285S -0.023651 -0.018220 -0.012379-0.006 59 0.000000 0.006259 0.012379 0.018220 0.0236520.028555 0.032620 0.036360 0.039094 0.040979 0.0419790.042083 0.041314 0.039712 0.037341 O.034289 0.0306710.026618 0.022285 0.017837 0.013459 0.0093S1 O.005721

0.002784 0.000782 0.0000001.000000 0.99465W 0.960866 0.956664 0.929536 0.8934550.851308 0.803815 0.751753 0.695948 0.637271 0.5766200.514918 0.453096 0.392084 0.332794 0.276105 0.222S650.173861 0.29819 0.091393 0.059146 0.033560 0.0150100.003767 0.000000 0.003767 0.01500 0.033560 0.0591460.091393 0.129819 0.173841 0.2265 0.276105 0.3327910.392082 0.453095 0.514915 0.576617 0.637266 0.69594.0.7517S0 0.803815 0.651308 0.893455 0.929536 0.956664

0.96066 0.994856 1.0000000.000000 -0.000782 -0.002784 -0.005721 -0.009351 -0.013459

-0.017837 -0.022265 -0.024418 -0.030671 -0.034289 -0.037341-0.039712 -0.041314 -0.04063 -0 .041979 -0.060979 -0.03909-0.036360 -0.032820 -0.026555 -0.03651 -0.018220 -0.012379-0.006I59 0.000000 0.006?.9 0.012379 0.018220 0.023651

0.028555 0.03220 0.036360 0.039096 0.040979 0.0419790.042083 0.041314 0.039712 0.037341 0.034209 0.0306710.024418 0.022285 0.017837 0.013469 0.009351 0.0087210.002784 0.000762 0.0000000.000000 0.000000-4S.800003 0.000000 0.000000 0.000000 0.000000 00.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000O.60000 0.025000 0.001000 0.000000 4.65596 -0.921370 1.21629 0

136

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LIST OF REFERENCES

1. Teng, N.H., The Development of a Computer Code (U2DIIFJ for the Numerical

Solution of Unsteady, Inviscid and Incompressible Flow over an Airfoil, Master's

Thesis, Naval Postgraduate School, Monterey, California, Jun 1987, pp. 1-135.

2. Krainer, A., A Numerical Method For Computing Unsteady 2-D Boundary Layer

Flows, Naval Postgraduate School, Monterey, California, Feb 1988, pp. 5-11.

3. Hess, J.L. and Smith, A.M.O., Calculation of Potential Flow about Arbitrary Bodies,

Progress in Aeronautical Sciences, Vol. 8, Pergamon Press, Oxford, 1966, pp. 1-138.

4. Basu, B.C. and Hancock, G.J., The Unsteady Motion of a Two-Dimensional Aerofoil

in Incompressible Inviscid Flow, Journal of Fluid Mechanics, Vol. 87, Jul 1978, pp.

159-168.

5. Giesing, J.P., Unsteady Two-Dimensional Potential Flow for Two Bodies with Lift.

Technical Report Contract No. 4308(00), Report No. DAC-33552, Douglas Air-

craft Division, Long Beach, California, Mar 1967, pp. 1-60.

6. Krainer, A., Naval Postgraduate School, Monterey, California. Private Communi-

cation, Mar-Sept 1988.

7. Giesing, J.P., Unsteady Two-Dimensional Potential Flow with Lift, Technical Report,

Report No. LB-32144, Douglas Aircraft Division, Long Beach, California, Mar

1965, pp. 1-67.

154

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INITIAL DISTRIBUTION LIST

No. Copies

Defense Technical Information Center 2Cameron StationAlexandria, VA 22304-6145

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, CA 93943-5002

3. Chairman, Dept. of Aeronautics, Code 67 1Naval Postgraduate SchoolMonterey, CA 93943-5000

4. Professor M.F. Platzer 10Naval Postgraduate SchoolMonterey, CA 93943-5000

5. Professor R.M. Howard INaval Postgraduate SchoolMonterey, CA 93943-5000

6. Mr. Andreas Krainer IViktor Adlerstr.52A-2345 Brunn, Austria

7 Mr. Pang Chung Khiang 5Air System Program OfficeDefence Material OrganisationPaya Lebar AirportSingapore 1953Republic Of Singapore

8 Professor T. CebeciStaff DirectorResearch & TechnologyMcDonnell Douglas Corporation3855 Lakewood BoulevardLong Beach, CA 90846

9 Mr. Joseph GiesingSenior Staff EngineerAirframe Technolog"McDonnell Douglas Corporation3855 Lakewood BoulevardLong Beach, CA 90846

155

" " : "''"T ' "- .. .'

; l: ":'i ' : l' ' : | .l '-,' '" . : " t? : : , L, . ,- -_ -:. i : ! .2 '- }: *. . .. -:.: -

10 Mr. James O'BrienResearch ScientistHeat Transfer BranchMail Stop 5-11NASA-Lewis Research Center21000 Brookpark RoadCleveland, OH 44135

11 Mr. Campbell HendersonHead, Aircraft Performance Analysis BranchCode 6051, Naval Air Development CenterWarminster, PA 18974-5000

12 Ms. Lisa J. CowlesCode 6051, Naval Air Development CenterWarminster, PA 18974-5000

13 Mr. H.A. BermanCode 931M. Naval Air System CommandWashington, DC 20361

14 Mr. G. DerderianCode 931 E, Naval Air System CommandW\ashington, DC 20361

'

156