Viscous-Potential Flow Interaction Analysis Method for Multi Element Infinite Swept Wings

7/28/2019 Viscous-Potential Flow Interaction Analysis Method for Multi Element Infinite Swept Wings

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N A S A C R - 2 4 7 6

R E P O R T

VISCOUS/POTENTIAL FLOW

SWEPT WINGS

I

F. A. Dvorak and F. A. Woodward

by - -

RESEARCH, INC.

Wash. 98031

A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D. C. ' N O V E M B E R 1974



'For sale by the National Technical Inform atio n Service, Springfield, Virginia 22 15 1

I

3. Recipient's Catalog No.

5. Report Date

November 1974

6. Performing Organization Code

8. Performing Organization Report No .

10. Work Unit No.

11. Contract or Grant No.

NAS 2-7048

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

1. Report No.

NASA CR - 2476- 2. Government Accession No.

4. Title and Subtitle

A VISCOUS/POTENTIAL FLOW INTERACTION ANALYSIS METHOD

FOR MULTI-ELEMENT INFINITE SWEPT WINGS, VOLUME I

7. Author(s)

F. A . Dvorak and F . A . Woodward

9. Performing Organization Name and Address

Flow Research, In c.

Kent, Washington 98031

12. Sponsoring Agency Name and Address

National Aeronaut ics and Space Administrat ion

Washington, D.C . 20546

15. Supplementary Notes

16. Abstract

A n analysis method and computer pro gra m have been developed for the calculation of the visco sity dependent

aerodynamic characterist ics of mult i-element inf in ite swept win gs i n incompressible f low. The wing conf ig-

urat i on consist ing at the most of a slat , a main element and double slot ted f lap i s represented i n the method

by a la rge number o f pane ls. The i nv isc id pressure d is t r ibu t ion about a g iven conf igura t ion in the normal

cho rd direct ion is determined using a two dimensional potent ia l f low pr ogram employing a vortex lat t ice tech-

nique. The bound ary l aye r development over each ind ivi dual element of th.e hi gh l i f t conf igurat ion is deter-

mined us ing e i ther in tegra l or f in i te d i f fe rence boundary layer techniques. A source d is t r ibut i on is then

determined as a funct ion o f the calculated boundar y la yer displacement th ickness and 'pressure dis tr ibut ions.

Th is source d is t r ibu t ion i s included in the second calculat ion of the potential f low about the confi gurati on.

Once the solution has conver ged (usua lly after 2-5 itera tions between the potential f low and bounda ry lay er

calculat ions) l i f t , drag, and pitch ing moments can be determined as funct ions of Reynolds number.

17. Key Words (Suggested by A uth orb ))

Aerodynamics, Air fo i ls, Boundary Layers,

Flaps, H ig h Lif t, Potential Flow, Slats, Wings,

Swept Wings, Viscous Interaction, Viscous Flows

18. Distribution Statement

UNCLASSIFIED-UNLIMITED

CAT.01

22. Price'

4.00

21. NO. of Pages,

86

19. Security Classif. (of this repo rt)

Unclassif ied-Unlimited

20. Security Classif. (of th is page)

Unclassif ied-Unlimited



TABLE OF CONTENTS

TABLE O F CONTENTS

LIST O F F I G U R E S

SUMMARY

INTRODUCTION

L I ST OF SYM BOL S

POTENTIAL now METHOD

c o n £ i g u r a t on D e f i n i t i o n

Inv isc id M e t h o d .

~ i s c o u s / P o t e n t i a l l o w In te rac t ion

BOUNDARY LAYER CALCULATION METHODS

Stagnat ion L i n e I n i t i a l C o n d i t i o n s

I n t eg r a l B o u n d a r y L a y e r M e t h o d sB o u n d a r y L a y e r T r a n s i t i o n

F i n i t e D i f f e r e n c e B o u n d a r y L a y e r M e t h o d

CALCULATION PROCEDURE

CALCULATIONS AND DI SCUSSI ON OF RE SUL T S

CON CLU SIO NS AND RECOMMENDAT IO NS

REFERENCES

APPENDIX I POTENTIAL now THEORY

APPENDIX I1 SOLUTION OF BOUNDARY CONDITION EQUATIONS

APPENDIX 111 PROGRAM MACRO now CHARTS

Page

i



LIST OF FIGURES

T i t l ei gu r e

1.1 Comparison Between TheoretPcal and Experimental PressureD i s t r i b u t i o n s

S t a t i c - P r e s s u r e V a r i a t i on Normal t o t he S u r fa c e o f a

S l o t t e d F l ap

FloG About -a ~ u l t i - ~ l e m e n t i r f o i l

Mul t i E lement Ai r f of I Lof t i ng Procedure

comparison of Exact and Numerical Cal cu la t io ns of Surf ace

Radius of Curva ture

A ir fo i l Geometry Using Planar Panel s

Aerodynamic Inf luence Coeff ic ients

V or te x D i s t r i b u t i o n o n A i r f o i l

Kut ta Condi t ion - Modif ied by Source Di s t r i bu t i on

Flow Chart f o r Boundary Layer Cal cu la t io ns

I n i t i a l V el o ci ty D i s t r i b u t i o n f o r a S l o t t e d A i r f o i l

Conf igura t i on

P os s i b l e Eddy V i s c o s i t y P r o f i l e s on a S l o t t e d F l a p

Comparison of Methods f o r C a l c u la t i n g P r e s s u r e C o e f f i c i e n t s

Computation Pro ced ure f o r Aerodynamic Forc es

Over l ay S t ruc ture

Comparison of Numerical and Exact Po te n t i al Flow Solu t io ns

Comparison of Numerical and Exact Po te n t i al Flow Solu t io ns

Comparison of Calculated and Measured Turbulent Boundary

Layer Developments on a Ci rcu l a r Cyl inde r .

Comparison of Cal cul ated and Measured Ve loc i ty P r o f i l e

Developments Downstream of a Blowing Slot.

L i f t and Moment Co ef fi ci en ts f o r NASA GA(w)-1 A i r f o i l



LIST OF FIGURES ( c o n t ' d )

Drag Polar for NASA GA(w) -1 A i r f o i l

Comparison o f Pre ss ur e D is tr ib ut io ns f o r NASA GA(w)-1

A i r f o i l

L i f t C o e f f i c ie n ts f o r NACA 23012 Air fo i l

Drag Polar for NACA 23012 Ai r fo i l

Comparison of Measured and Predi c te d Pres sur e Di st r ib ut ion s

f o r NACA 23012 A i rf o i l wit h 25%C. S l o t t e d F l ap

Comparison of Measured and Pred ic te d P res sur e Di st r ib ut ion s

f o r NACA 23012 Ai r f o i l wi th L.E . Sl ot and 25%C. Sl o t te d Fla p

Comparison of P red ic t ed P r essu re D is t r ibu t ion s f o r NACA 64A010

A i r f o i l w i t h L.E. Sl o t and Double S lo t te d Flap

Comparison of Measured and Predic ted Pres sur e Dis t r i bu t io ns f o r

Fo s t e r ' s A i r f o i l Flap Combina tion

Comparison of Measured and Pr ed ic te d Velo cit y P ro f i l e s on Fla p

Upper Surf ac e

Comparison of Measured and Pred ic t ed P res sur e Dist r ib ut ion s f o r

an Inf ini te Swept Wing

Comparison of Measured and Predicted Streamwise Momentum

Thickness Developments f o r an I n f i n i t e Swept Wing

Comparison of Measured and Pre di ct ed Shape Fa ct or and Angle BDevelopments fo r an Inf i n i t e Swept Wing

P r e d ic t e d P r e s s u re D i s t r i b u t i o n f o r F o s t e r ' s A i r f oi l - F la p

Configuration Swept 25 Degrees

Pre dic t ed Streamwise and Cross-Flow Veloc i ty Pr o f i le s a tF l a p T r a i l i n g Edge f o r F o s t e r ' s A i r f o i l F la p C o n fi g ur a ti o n

Swept 25 Degrees



BYF. A. Dvorak and F. A. Woodward

Flow Research, Inc.

SUMMARY

An an a l y si s method and computer program have been developed fo r t h e

ca l cu l a t i o n of t h e v i s c o s i t y d ep en dent a er od yn am ic ch a r ac t e r i s t i c s o f

mu1t i-elemen t i n f i n i t e swept wings i n incompres s ib le f low.

The wing co n f i g u r a t i o n co n s i s t i n g a t most o f a s l a t , . a main element

a n d d o u b l e s l o t t e d f l a p i s r eprese n ted i n t h e method by a l a rg e number o f

pane l s . The i n v i s c i d p r e s s u r e d i s t r i b u t i o n a bo u t a g i v en c o n fi g u r at i o n i n

t h e no rm al ch o rd d i r e c t i o n i s determined us ing a two d imens iona l po ten t i a lf low program employing a vo r t ex l a t t i c e technique. The boundary la ye r

d ev elo pm en t o v e r - e ach i n d i v i d u a l e lem en t o f t h e h i g h l i f t co n f i g u r a t i o n

i s d et er mi ne d u si n g e i t h e r i n t e g r a l o r f i n i t e d i f f e r e n c e boun dary l a y e r

techn iques .

0n ce . th e boundary l ay er development i s known, a s o ur c e d i s t r i b u t i o n i s

de te rmined a s a - fun c t io n o f th e c a lc u la ted boundary l a ye r d i sp lacement .

t h i c k ne s s a nd p r e s s u r e d i s t r i b u t i o n s . T h i s s o u r ce d i s t r i b u t i o n i s inc luded

i n t h e s ec on d c a l c u l a t i o n o f t h e p o t e n t i a l ' f l o w a b ou t t h e c o n f ig u r a t i o n ,

and r e p r e s e n t s t h e e f f e c t o f t h e bo un da ry l a y e r i n t h e m o di f ic a ti o n o f t h e

p o t e n t i a l fl ow . Once t h e s o l u t i o n h a s co nv er ged ( u s u a l l y a f t e r 2-5 i t e r a t i o n s

b etw een t h e p o t e n t i a l f l o w and b ou nd ar y l ay e r c a l cu l a t i o n s ) l i f t , d r ag , andpi tc hi ng moments can be determined a s fu nc ti on s of Reynolds number.

The new method ha s a number of fe a t u re s and ca p a b i l i t i e s which make i t

a unique method a t t h i s t ime . Some of the se f e a t u r es inc lude :

-The in c l us ion o f methods capab le o f c a l cu la t i ng th e boundary l a ye r

development over i n f i n i t e yawed wings .

-The i n c l u s i o n o f no rm al p r e s s u r e g r ad i en t and l o n g i t u d i n a l cu r v a t u r e

t er ms i n t h e f i n i t e d i f f e r e n c e program . T h i s h a s l e d t o much

improved p r ed ic t io ns of t he performance o f mul ti - el ement a i r f o i l s in

two d imens ions a s compared t o t he p re d ic t io ns o f o t he r methods ,e s p e c i a l l y i n t h e . c a l c u l a t i o n of p r o f i l e d ra g.

-The u s e of s o u r ce d i s t r i b u t i o n s r a t h e r t h an t h e di s pl acem ent t h i ck n es s

d i r e c t l y t o r e p re s e n t t h e e f f e c t o f t h e bo undary l a y e r on t h e p o t e n t i a l

f low. This approach i s much more e f f i c i e n t t h an t h e a l t e r n a t e

p ro ce du re s i n c e t h e i n f l u e n c e c o e f f i c i e n t m a t r i x r e p r e s e n t i n g t h e

geometry o f th e con f igura t i on need be inv er te d on ly once. Computer

t i m e ex p en d i t u r e s are consequently much less . wi th t h e new method.



- In t h e f u t u r e , t h e e f f e c t s of t a n g e n t i al i n j e c t i o n a

boundary layer control on aerodynamic performance may be calculated,

a s w e l l a s t h e e f f e c t s of f u l l y t h r e e d ime ns io na l f l ow .

The computer program i s w r i t t e n in Fortran I V f o r t h e CDC 6600 and

7600 fa mi ly of computers. The program occu pies 100,000 ( o c t a l ) words of.

s torage and opera tes in the over lay mode. The program ha s been s tr uc tu re di n such a way th at extension o r replacement of indi v idu al c a l c u l a t i o n

procedures i s s t ra igh t fo rward .



INTRODUCTION

Background

The mult i-elem ent wing i s an e s se n t i a l component o f t h e h i gh - l i f t sys tems

of e x i s t i n g co mm ercial and m i l i t a r y a i r c r a f t . H i s t o r i c a l l y t h e d e s ig n of t h e s esys tems has been dependent upon expe r imenta l v e r i f i c a t i o n of p red i c t ed ae ro-

dynamic performance. This approach has been and con t in ues to be a v e r y c o s t l y

and time consuming ve nt ur e. The adv ent of h igh speed computers and of advanced

numerical methods i s however g ra dua l ly reduc ing t h e re l i an ce on t he expe r imenta l

method. Ca lcu la t io n methods now e x i s t which perm it t h e so lu t i o n of many prac-

t i c a l en g i n e e r in g p ro bl em s. The p r e d i c t i o n of t h e a er od yn am ic c h a r a c t e r i s t i c s

o f t w o- di me ns io na l l i f t i n g m u lt i- el em e nt a i r f o i l s i n c l u d i n g t h e e f f e c t of

v i s c o s i t y i s an impor t an t example o f t h i s ca pa b i l i t y .

The a v a i l a b i l i t y o f a three-dimensional ve rs io n of such a method would be

o f c o n s i d e r a b l e ~ v a l u e o t h e d e s i g n e r s of m odern a i r c r a f t h i g h - l i f t s ys te ms ,

pa r t i c u l a r ly wi th re s pec t t o STOL a i r c r a f t , where t he des ig n probl ems appearto be t h e mos t fo rmidab le . Trade-off s t u d ie s could be made fo r the de sign and

ana lys i s o f i nd iv i du a l components such a s t he l e ad in g edge dev i ces o r t h e

s l o t t e d f l ap s . A mul t i -element wing a na ly s i s would have oth er important

a p p l i c a t i o n s , a n d i t i s because of th e usefu lness of such a method th a t th e pro-

c e d ur e d e s c r ib e d i n t h i s r e p o r t was d e ve lo pe d.

The method i s c u r r e n t l y v a l i d f o r t h e i n f i n i t e yawed w in g ca s e , b u t h a s b e en

s t r u c t u r e d i n su ch a way t h a t a t a l a t e r d a t e , i t c an b e ex te nd ed t o t h e f u l l y

t h r e e d i m en s io n a l c a s e . The a d d i t i o n o f v i s c o u s e f f e c t s i s accomplished using

di s t r i bu t ed sources de t ermined from the boundary l ay e r ca l c u l a t i on s . The need

t o a dd v i s c o u s e f f e c t s i s c l e a r l y d e mo n st ra te d by t h e r e s u l t s shown i n F i g u r e 1.1

(Ref . 1 . ) . Obvious ly , t h e i n v i sc id so l .u t i on gr os s l y ove r e s t i ma t e s t he pe rformanceof. t h e a i r f o i l s e c t i on .

I t was P ran dt l who f i r s t sugges t ed add ing t he boundary l ay e r d i sp l acement

t h i c k n e s s t o t h e o r i g i n a l g e om et ry t o a cc o un t f o r t h e d is p l ac e m en t o f t h e i n v i s c i d

f low s t reaml i nes by t he boundary l ay e r . This approach has s in ce been used suc-

ce ss fu l l y by many resea rch ers . A p r a c t i c a l c om pu ta ti on al d i f f i c u l t y a r i s e s w i th

t h i s approach however , and th a t i s t h e need i n t h e p o t e n t i a l f lo w c a l c u l a t i o n t o

r e - i n v e r t a t e ac h p a s s a l a r g e m a t r i x r e s u l t i n g i n l a r g e c om pu ter t i me e x p en d i t u r e s.

I n o r d e r t o o b t a i n s mooth p r e s s u re d i s t r i b u t i o n s i t i s a l s o u s u a l l y n e c es s ar y t o

smooth t he new geometry be fo re each po t e n t i a l f l ow ca l cu l a t i o n re su l t i ng i n

fu r t he r computer t ime expe ndi tu res . An a l te r n a t iv e procedure stemming from an ide a

f i r s t s ug g e st e d by P r e s t o n , R e f . -, has been s ucc ess fu l l y adopted i n t he computer

p ro gram d e s c ri b e d i n t h i s r e p o r t . B r i e f l y , a s o u r c e o r s i n k ( n e g a t iv e s o u r c e )

d i s t r i b u t i o n i s determined a s a fun c t io n of th e known displacement thic knes s ,

e n tr a in m en t r a t e , and v e l o c i t y d i s t r i b u t i o n s ( q = d/ds (peue6*). W ith t h e i n t r o -

d u c t i o n of a s o u r ce d i s t r i b u t i o n a new v o r t e x d i s t r i b u t i o n i s determined given



h11I II II I

0 EXPERIMENT

- - - - - INVISCID CALCULATION

VISCOUS / POTENTIA L FLOW

1NTE RAC Tl ON

FIG. 1.1 COMPARISON BETWEEN THEORE TICAL A NDEXPERIMENTAL PRESSURE DISTRIBUTION



th e o r ig in a l geometry, consequen t ly , the re i s no need t o in ve r t the mat r ix a

s econ d ti m e. I f s e v e r a l i t e r a t i o n s b et ween i n v i s c i d and v i s co u s fl ow a r e

r eq u i r ed , t h e p o t e n t i a l co mp ut er time sav ing i s s u b s t a n t i a l .

Problem Definit ion

The ca lc u la t io n o f . the po te n t i a l f low about a mul t i- e lement conf ig ura t ion

r e p r e s e n t s t h e f i r s t t a s k o f a ny a n a l y s i s method. Because t he ana ly s i s i s

l im i te d t o i n f i n i t e swept wings a two-d imensional po te n t i a l f low method i s

adequa te . Fu ture exp ans i on- to th e fu l l y th ree-d imensiona l case sugges ted ,

however , t h a t any prograni-be w r i t t e n i n modular form i n order t h a t t he two-

dimens ional method could be re a d il y replac ed by a thre e-dime nsional method.

I n th e two-dimensional ca se, a mathematical model i s r e q u ir e d f o r t h e fl ow f i e l d

about a series of a r b i t r a r i l y s haped b o d ie s i n an i nco m p r ess i b l e, i n v i s c i d f lo w.

The model m ust a s a f u r t h e r r equ i rem ent b e ab l e t o p r ed i c t t h e p r e s s u r e d i s t r i -

b u t i o n a t s e l ec t e d o ff -b od y p o i n t s abo ve t h e f l a p segm en ts . , T h i s i s neces sary

b ecau se d ownstream o f t h e wing t r a i l i n g ed ge t h e s t a t i c ' p r e s s u r e normal t o t h e,

f l a p s u rf a ce i s g r e a t l y i n f l uen ced b o th by t h e pr o xi m it y of t h e f l a p t o t h e

wing t r a i l i n g edg e and by t h e . l a r g e s u r f ace cu r v a t u r e i n t h e f l a p le ad i n g ed ge

r e gi o n. T hi s s t a t i c p r e s s u r e v a r i a t i o n ( s e e F i gu r e 1 . 2) h a s a conside.rable

in fl ue nc e on th e development of the combined wing wake-flap boundary la ye r

downstream of th e wing t r a i l i n g edge.

With t h e p o t en t i a l f l o w f i e l d s p e c i f i e d , i t i s n e c es sa r y t o p r e d i c t t h e

boundary la ye r development over th e mul t i -e lement conf igu rat i on. Cal cul at io ns

must i n c l u d e s t a g n a t i o n l i n e i n i t i a l c o n d i t i on s , l a mi n ar , t r a n s i t i o n and t u rb u-

le n t boundary l aye r developments and l aminar o r tu r bu le n t s epa ra t i on p re d ic t ion s

fo r each e lement of t he i n f i n i t e swept wing h igh l i f t system. The ca lcu la t ion s must

inc lud e accu ra te p r ed i c t i ons o f boundary l a yer development i n th e r eg ions where

wing o r f i r s t f l ap upper su r fa ce and cove boundary l a ye r s merge wi th th e down-

s t r eam f l ap upper su r f ace boundary l ay er . Th i s r equ ir ement i s a n a b s o l u t en e c e s s i t y i f a c c u r a t e d ra g p r e d i c t i o n s a r e t o . b e made. Both longi tudinal curva-

tu re and normal p res sure g ra d ie n t terms must be included i n t he governing

boundary layer equat ions as e ac h e f f e c t h a s a s i g n i f i c a n t i n f l u e n c e on t h e

boundary la ye r development and subsequent ly on th e se ct io n drag co ef f i c i en t . These

e f f e c t s a r e p a r t i c u l a r l y i m p or ta nt i n t h e wing t r a i l i n g e dg e- fl ap l e a d in g edge

re gi on . Once th e boundary lay er development i s known, i t s e f f e c t on t h e e x t e r n a l

flow must be determined.

A comple te an a l ys i s p rogram f o r th e ae rodynamic cha ra c t e r i s t i cs o f mul t i -

e lem en t i n f i n i t e s w ep t. wi ng s i s developed by combining th e s e par a te p o t en t i a l

f low and boundary l a ye r ca lc u l a t io n p rocedures . . I t e r a t i o n b etw een t h e s ep a r a t e

procedures r es u l t s i n th e p re d ic t ion o f v i s co s i ty dependen t ae rodynamic fo rces .

The d i f f e r en t par t s o f the f low about a mult i -e l ement i n f i n i t e swept

wing h igh l i f t system cons i s t in g o f a l ead i ng edge s l a t , t he main wing and

d o ub l e s l o t t ed f l a p s a r e shown i n F i g u re 1 . 3 . The d i f f e r e n t c a l c u l a t i o nschemes th at fcrm th e elements o r modules of th e in te gr at ed computer program

ar e p r es en ted . i n t h e f o l l ow i n g s ec t i o n s .



FIG. 1.2 STATIC PRESSURE VAR IATIO N NOR MAL TO FLAP SURFACE



SLOTTED A l RFOlLTURBULENT BOUNDARY

LAYER CALCULATED B YIINITE DIFFERENCE

PROGRAM

4

METHOD OF SlNGULARlTlES

FIG. 1.3 FLOW ABOUT A MULTI-ELEMENT INFINITE SWEPT WING



LIST' F SYMBOLS

Aerodynamic influence coefficient

Function in Van Driest damping factor

Normal velocity due to external source

Eddy Reynolds number U U/Vd t

Reynolds number at stagnation line

Profile drag coefficient =D

2+PU,

TLLift coefficient = 2

+PU, c

Moment coefficient

Airfoil normal chord

Local skin friction coefficient

Streamwise skin friction coefficient

Resultant skin friction coefficient .

Cross flow skin friction coefficient

Pressure coefficient

Drag force/unit span

Universal functions in Curles laminar method

Correction term to Thwaites laminar method

Shape factor, ratio of displacement to momentum thicknesses (a*/8)

Shape factor (6 - 6*)/8

Non-dimensional pressure gradient parameter

Von Karman's mixing length coefficient

Mixing length 1 = ky inches

Lift forcelper unit span

Local Mach number

Free stream Mach number

~otal ormal velocity

8



P S t a t i c p r e s s u r e , p ou nd s p e r s q u a r e i n c h a b s o l u t e

q S ource s t r eng th

r Radius of curvatu re , inch es

R Loca l r ad i us of c u rva tu re

Re i n s

Re

t r a n s

Chord Reyno lds number U,c/v

Momentum thickness Reynolds number Ug/v

Streamwise momentum th i ckn ess Reynolds number a t i n s t a b i l i t y p oin t

Streamwise momentum th ic kn es s Reynolds number a t t r a n s i t i o n

R e s u lt a n t v e l o c i t y

Loca l s t reamwise ve lo c i ty

F r e e s t r ea m v e l o c i t y

T a ng en ti al v e lo c i t y a t a i r f o i l s u r fa c e

Components of v el oc it y i n %, y and z d i r e c t i o n s

F r i c t i o n v e l o c i t y (T /p )J f2

W

Components of len gth i n th e chord , normal and spanwise d i re c t i o ns

D i s t a n c e a lo n g a s t r e a m l i n e

Three-d imens ional boundary la ye r th i ck ne ss parameters def in ed i n

Equat ion 3 . 2 2 .

6 Boundary la ye r th i ck ne ss

P D e n s it y of a i r

T S h e a r s t r e s s

T L oc al s u r f a c e s h e a r s t r e s sW



Y i V o r t e x s t r e n g t h

a Angle between s t r eam l ine a t ou te r edge o f .boundary l a ye r and wing

normal chord

6 ' A ng le be tw ee n s u r f a c e s t r e a m l i n e and e x t e r n a l s t r e a m l i n e d i r e c t i o n s

v K i n e m a t i c v i s c o s i t y

v Eddy v i s cos i tyt

Y ( Y > I n t e r m i t t e n c y f u n c t i o n

0 S t a nd a r d d e v i a t i o n of i n t e r r n i t t e n c y f u n c t i o n

S u b s c r i p t s

e Value a t edge of boundary l ay e r

i ith a l u e

i n I n co m pr e ss ib l e

i n s I n s t a b i l i t y

j j th v a l u e

L L o c a l v a l u e

1 lower

t r a n s T r a n s i t i on

Streamline component

T u r b u l e n t

Upper



POTENTIAL now METHOD

Conf igura t ion Def in i t ion

The m u lt i- el em ent a i r f o i l co n f i g u r a t i o n i s r epresen ted by pa i r s of

sur fac e coord ina te po in t s . Each e lement wi t h th e excep t ion o f th e main

wing, may be sp ec i f ie d i n i t s own o r a r e fe rence coord ina te sys tem. The

main e lement mus t be g iven i n t he r e f e re nce coord ina te system. I n d i v i d u a l

coord ina te sys tems are r e l a t e d t o t h e r e f e r e n c e c o o rd i n at e s ys te m by p i v o t

p o i n t s . The p i v o t p o i n t s are p r e s c r i b ed i n b o t h t h e e lement an d r e f e r en ce

c o o rd i n at e sy st em s. I n or d e r t o l o f t t h e c o n fi g u r a t io n , e le me nt r o t a t i o n

an g l e s must a l s o b e p r e s c r i b ed . G iven t h e p i v o t p o i n t s and r o t a t i o n an g l e s

a ny e le me nt may b e t r a n s l a t e d and r o t a t e d t o t h e d e s i r e d l o c a t i o n r e l a t i v e

t o t h e m ain e l em en t.

Piv ot po in ts may be determined based on such requirements a s a

sp ec i f i e d s l o t gap and wing-f lap over l ap . Leading edge coord ina tes usu a l l y

provide a conven ient e l ement p ivo t p o in t a l though i n some cases t he h inge

p o i n t o f a f l a p on i t s mechanical t r ac k o r l ink age mechanism gi ves a ready

r e f e r en c e p o i n t . F igure 2 .1 shows a fo ur e l ement c onf igu ra t ion in bo th

i n p ut and l o f t e d p o s i t i o n s .

I f t h e c o n f i g u ra t i o n i s made up of a main elem ent and one o r more

s l o t t e d f l a p s , t h en a d d i ti o n a l a na l y s i s i s r eq u i r ed t o d e te rm i ne f l ap u p p er

s u rf a c e lo n g i t u d in a l r ad i u s of c u rv a tu re f o r l a t e r u s e i n t h e f i n i t e

d i f f e r enc e boundary l a ye r ca l cu l a t i on methods . Accura te ca lc u l a t io ns o f

curva tu re r e qu i re t he use of very smooth inpu t d a ta . Because o f th i s , i t

was fo un d n e c e s s a ry t o u se s p l i n e f u n c t i o n s t o r e p r e s e n t t h e s u r f a c e b e in g

analyzed. A s p l i n e u n de r t en si o n* i s f i r s t p a ss ed t hr ou gh t h e c o o r d i n at e

p o i n t s r e p r e s e n t i n g t h e f l a p u pp er s u r f ac e . F i r s t d e r i v a t i v e s dy /d x aret h en d e te rm i ned fro m t h e s p l i n e d cu r v e u s i n g a n a l y t i c ex p r e s si o n s . A

second sp l in e under t en s io n i s now used t o rep re se nt a curve through th e

c a l cu l a te d f i r s t d e r iv a ti v e s. T hi s s p l i n e i s l i ke w i se d i f f e r e n t i a t e d u s i n g

an a l y t i c ex p r e s s i o n s . Once b o t h f i r s t an d s econ d d e r i v a t i v e s o f t h e

s u r f ace a r e known t h e r ad i u s o f cu r v a t u r e c an b e r e ad i l y c a l cu l a t ed . F i g u r e

2.2 i n d i c a t e s t h e s u c c e ss of t h i s t e c h n iq u e i n r e l a t i o n t o known v a l ue s o f

r ad i u s o f cu r v a t u r e f o r t h e NACA 4412 a i r f o i l u pp er s u r f ac e co nt o u r.

*, "Sp lin es' Under Tension" - a technique developed by D r . A. C l i n e o f t h e

Nat io nal Center f o r Atmospher ic Research, Boulder , Colorado, fo r ob ta in in g

smooth continuous curves from sets of imput coord ina te po i n t s .



FIG. 2.1 MULTI-ELEMENT AIR FOIL LOFTING PROCEDURE



NACA 4412 UPPER SURFACE

EXACT SOLUTl ON

0. NUMERICAL

I I I I I I

20 40 60 80 100 120

CHORD WISE DISTANCE X (%)

FIG. 2.2 COMPARISON OF EXAC T 81 NUMERICAL CALCULATION

OF SURFACE RAD IUS OF C U R V A T U R E

1



Inviscid Method

The a i r f o i l a nd a s so c i a t ed f l a p sy stem i n t h e i r l o f t e d c o n fi g u ra t io n

i s approximated by a l a rg e number of p l a na r segments , o r pane l s , wi t h corne r

p o i n t s l o c a t e d on t h e a c t u a l a i r f o i l o r f l a p s u r f a c es . The g eo metry of

a typica l two e lement sys tem i s i l l u s t r a t e d below:

Fig. 2 . 3 Airfoi l Geometry Using Planar Panels

A t r i a n g u l a r d i s t r i b u t i o n of v o r t i c i t y i s l oc a t e d on each ad j acen t

p a i r o f p a n e l s, as shown above . The vo r t ex di s t r ib u t i o n i s i d e n t i f i e d b y

th e . in d e x of th e common edge, and . is g i ve n un i t m a gn it ude a t t ha t po i n t .

The no rm al c omponent o f v e l o c i t y i nduc ed by t he j t h vo r t e x d i s t r i b u t i o n a t

t h e c e n t e r o f p a n e l i i s de s i gna t e d t h e a er odynam ic i n f l ue nc e c oe f f i c i e n t

a and i s c a l c u l a t e d a s f o l l o ws :i j ,

'panel (j-1)

Fig. 2 . 4 Aerodynamic Inf luenc e Co ef f i c ie nt s



F i r s t , t he ho r i z on t a l and v e r t i c a l components of ve l oc i t y u and wi i ja r e ob ta ined by summing the in f luenc es of a l i n e a r l y vary ing Jo r t ex

d i s t r i b u t i o n o n pa n el ( j- 1) h av in g u n i t v a l u e a t t h e t r a i l i n g ed ge , a nd a

l i n e a r l y v a ry i n g vo r t e x d i s t r i b u t i o n on p an e l j h a vi ng u n i t v a lu e a t t h e

lea din g edge. Formulas f o r th e u and w components induced by these

vo r t e x d i s t r i b u t i o ns i n t erms of t he primed coo rd i na t e sys tem as soc i a t ed

wi th the in f lue nc ing pane l a r e g iven i n Appendix I .

u = u w c0s6 - w l ' s n6 -l + U i j cos6 - w' si n6 (2.1)i j - i , j - 1 j-1 i , j - 1 j i j j

The normal v el o ci ty a i s theni

A series of o v er la p pi ng t r i a n g u l a r v o r t e x d i s t r i b u t i o n s a r e pl ac ed o n t h e

upp er a nd lo we r s u r f a c e s o f t h e a i r f o i l , a s i n d i c a t e d :

,.- V or t ex L a t t i c e

4

Fi g . 2.5 V ort ex D i s t r i bu t i o n on A i r f o i l

I t shou ld be noted t h a t t he number of pa nel s on th e upper and lower

s u r f a c e s a r e n o t n e c e s s a r i l y e qu al . A t t he l ead i ng edge , t he vo r t ex

s t r en g t hs o f t he uppe r and lower vo r t i c e s a r e se t equa l , t o i n s u re a sm ooth

flow around the leading edge. A t t h e t r a i l i n g edge t h e "K ut ta " cond i t ion

sp ec i f i e s t h a t t h e m agni tudes o f t h e su r fac e ve l oc i t y on t he upper and lower

surfaces have a common l i m i t . T h is i m p l ie s t h a t t h e v o r t e x s t r e n g t h s on t h e

upper and lower s ur fa ce s must be equal and oppo si te .



I n t h e ab ov e exam ple, t h e a i r f o i l h a s 8 panels on th e upper s ur f ac e and

7 on th e lower , f o r a t o t a l o f 1 5 p a n el s . I f t h e l e a d i ng e dg e v o r t i c e s 1 and

1' a r e s e t e qu al (y = y;) and the t r a i l i n g edge vo r t i c es 9 and 16 a r e s e t

e q u a l a n d o p p o s i t e l y g = - y16) a t o t a l of 15 unknown vo rt ex st re ng th s

remain . The unknown vo r te x s t re ng th s a r e determined by sp ec i fy ing th a t the

sum of t h e induced ve lo ci ty and th e normal component of th e f r e e s t ream ve lo ci ty

go t o z e ro a t e ac h p a ne l c o n t r o l p o i n t . For a n a i r f o i l h av in g N p a n e l s , t h e

t o t a l norm al v e l o c i t y a t p an el i may be w r i t t en

The f i r s t t e rm rep r ese n t s th e normal component o f a u n i t f r ee s t r eam

v e l o c i t y a t t h e c o n t r o l po i nt o f p a ne l i , and th e second i s th e sum of t h e

p r od u c ts o f t h e i n f l u e n c e c o e f f i c i e n t s and t h e N unknown vortex strengths.

Wr i t ing t h i s boundary co nd i t ion equa t ion fo r each of th e N p a n e l s r e s u l t s

i n a l i n e a r sy s t e m o f N e q ua t io n s i n t h e N unknown vortex strengths.

I n m a t r i x f or m,

I.;Ii n

Th i s m a t r i x e q u a t i o n ca n t he n b e so l v e d f o r t h e v o r t e x s t r e n g t h s .D i r e c t

i n v e r s i o n i s employed f o r s i n g l e e le m en t a i r f o i l s , a nd e i t h e r a d i r e c t

method or an i t e r a t i v e procedure (descr ib ed i n Appendix 1I ) ca n be employed

f o r m ul ti -e le me nt a i r f o i l s .

A i r f o i l s w i t h b l u n t t r a i l i n g e d g es c an b e a na ly ze d s u c c e s s f u l l y u si ng

t h e K u t ta c o nd i ti o n t h a t t h e t r a i l i n g edge v o rt e x s t r e n g t h s a r e e q ua l and- y i n F ig ur e 2 .5 ). I f t h e t r a i l i n g e dg e c l o s e s t op p o s i t e ( i . e . yg - -a p o i n t , t h e s tr e n g t h s bt t h e t r a i l i n g edge v o r t i c e s must g o t o z e ro , s i n c e

t h e t r a i l i n g e dg e w i l l b e a s t a g n a ti o n po i nt i n t h e f lo w. Al though th is

r e s u l t i s g iven au tomat i ca l ly by th e so l u t io n o f t h e above sys tem of equa t ions ,

it h a s b ee n f ou nd t h a t a n a l t e r n a t e f o r m u la t io n of t h e se e q u a t i o n s i s d e s i r a b l e

f o r a i r f o i l s w it h t r a i l i n g e dge c l o su re . I n t h i s c a se t h e c o e f f i c i e n t s i n

t h e l a s t column of th e matr ix (eqn. 2.5) become very s m a l l r e s u l t i n g i n a

poo r ly cond i t ioned system of equat ion s .

I n t h e a l t e r n a t e f o rm u la t io n , t h e K u t t a . c on d it i on i s s p e c i f i e d b y

s e t t i n g t h e s t r e n g t h s of t h e v o r t i c e s a s so c ia t ed wi th t h e t r a i l i n g e ge pa n el s

e q ua l t o z e r o ( i . e . , y = y = 0 i n F igure 2 .5) . However, t h i s procedure

e l i m i n a t e s t h e l a s t co?umn !it inf l uen ce co ef f i c i en ts i n Eqn. (2:5) lea ving an

inde te rm ina te sys tem of N e q u a t i o n s i n N-1 unknowns.



An addit ional unknown i s prov ided by add ing th e in f lu ence o f a cons tan t

s t r e n g th so u rc e d i s t r i b u t i o n j u s t i n s i d e t h e a i r f o i l s u rf a ce . The so ur ce

i s d i s t r i b u t e d on t h e i n n e r s i d e of e ac h p a ne l us ed t o r e p r e s e n t t h e a i r f o i l .

The ve l oc i t i e s induced by a c o ns t a nt s t r e n g t h s o u r c e d i s t r i b u t i o n a r e g iv en

i n Appendix I , an d a r e u sed t o c a l c u l a t e new v a l u e s f o r t h e l a s t c olumn o f

i n f l u e n c e c o e f f i c i e n t s i n t h e bo un da ry c o n d i t i o n e q u a t i o n s . The unknown

so u r c e s t r e n g t h i s added t o th e r emain ing N - 1 unknown vor t ex s t ren gt hs t og i v e a w e l l cond i t ioned set of equa t ions . I t shou ld be no ted t ha t th e

unknown source strength i s alwa ys v e ry c l o s e t o z e r o f o r a i r f o i l s w i th

t r a i l i n g edge c l o su r e.

Th e p r e s su r e c o e f f i c i e n t a t th e mid point of p anel i i s c a l c u l a t e d as

fo l lows :

where

and uwi are g iven by Eqns (2 -1) and (2.2) . The l i f t and p i t c h i n g

momen$'aoet$icients a r e obta in ed by in te gr at in g t h e pre ssu res around th e

a i r f o i l c o n f i g u r a t i o n .

Viscous /Po ten t i a l F low In te r ac t ion

The i n v i s c i d f lo w ar ou nd a n a i r f o i l c a n ' b e m o di fi ed t o a c c ou nt f o r v i s c o u s

e f f ec t s th rough the add i t io n of th e boundary l a ye r d i sp lacement th i ckness 6*

t o t h e o r i g i n a l a i r f o i l geometry. The p o t e n t i a l . f l o w method d e sc r ib e d i n t h e p r e v i -

ous sec t ion can then be used t o ca lc u l a t e t he f low f i e i d abou t t he new geomet ry .

A m o di fi ed p r e s su r e f i e l d r e s u l t s , w hich i n t u r n c a u se s a c ha ng e i n t h e c a lc u -

la ted boundary layer development . A ft er s e v e r a l i t e r a t i o n s b o th t h e p r e s s u re

f i e l d and boundary la ye r developments should become conve rgent , This

procedure i s used i n bo th t h e Lockheed and McDonnel-Douglas programs (Re fs1and A ) , and while i t w ould seem a t f i r s t g l a n c e t o b e a s t r a i g h t f o r w a r da pp ro ac h , s e v e r a l r e q ui r em e n ts a r e n e c es sa r y t o e n su r e a s a t i s f a c t o r y s o l u t i o n .

These include:

- The n e c e s s i t y t o c a l c u l a t e and i n v e r t a new i n f l u e n c e c o e f f i c i e n t

m a t r ix a t e a ch i t e r a t i o n d ue t o t h e c hang e i n r e s u l t a n t geometry.

- The ne ce ss i ty t o smooth th e geometry each t ime t he d isplacement

t h i c k n e s s i s added , t o ensure a smooth p ress u re d i s t r i bu t io n .



- The ne c e s s i t y du r i ng ea ch i t e r a t i o n t o modify t o a c ons i der a b l e

ex t en t t h e ca l cu l a t e d pre ssure and bounda ry l ay e r deve lopments i n t h e

t r a i l i n g e dg e r e g i o n i n o r d e r t o e n s ur e a c on ve rg en t c a l c u l a t i o n

procedure .

An a l t e r n a t e p ro ce du re i s a v a i l a b l e which u s e s t h e same i n f l ue nc e c o e f f i c i e n t

m a t r i x t h r oughou t t he c a l c u l a t i o n , and i t i s t h i s method which i s u sed i n t h e

present program.

The e f f e c t o f t he bounda ry l a ye r on t he po t e n t i a l f l ow i s repre sen t ed by

a d i s t r i b u t i o n o f s o u rc e s on t h e p an e l s c on n ec ti ng p o i n t s o f t h e o r i g i n a l a i r -

f o i l s u r f a ce ( F ig u re 2 . 3 ) . The s t r e n g t h q j of t h e s o u r c e d i s t r i b u t i o n i s made

t o t h e r a t e of e n tr a inm e n t of m ass i n t o t he bounda ry l a ye r ( i . e .

6* ) )I Thus t he c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n and boundar y

l a ye r d i sp l acement t h i ckn ess developments can be used t o de t e rmine t h e source

s t r e n g t h s f o r t h e n e x t i t e r a t i o n of t h e a n a l y s i s . , The s ou r ce d i s t r i b u t i o n

ha s t h e e f f e c t o f m od if yi ng t he boundar y c ond i t i ons t o t he o r i g i na l pr ob le m

b y a l t e r i n g t h e r i g h t h and s i d e of Eqn 2 . 5 . The i n f l ue nc e c oe f f i c i e n t s a i j -

(o r u i j , w i . ) r ema in unchanged a s does t he geometry of t h e conf igu ra t i on

being analy?zed.

The e f f e c t o f t h e s ou r c e d i s t r i bu t i o n on t h e boundar y c ond i t i ons i s

determined i n t h e fol lowing manner. Consider a pa ne l r e p r e s e n t i ng a po r t i on

o f t h e a i r f o i l g eom etry; t h e s o u r c e s t r e n g t h i s known as i s t he norma l ve loc i ty

. induce d by t he s ou r c e d i s t r i b u t i o n a t t he bounda ry po i n t on t h e pa nel . T h is

normal vel .oci ty i s t h e new boundar y c ond i t i on t o be s a t i s f i e d by a l l s ou r c e s

and v o r t ic es rep re se nt in g th e geometry and the boundary lay er e f f ec ts . However,

t h e s ou r c e d i s t r i b u t i o n o f t he same pa ne l a l r e a dy s a t i s i f i e s t h i s new boundar y

c ond i t i on , t he r e f o r e , t he re m ai ni ng s ou r c e s and vo r t i c e s must s a t i s f y t he

boundar y c ond i t i o n o f t a n ge n t i a l f l ow t o t he s u r f a c e .

S ou rc e i n £ u e n ce c o e f f i c i e n t s u s i j and wsij a r e de f i ned a s i nduc ed

v e l o c i t i e s p e r u n i t s o ur ce s t r e n g t h q j a t a c or n er p o i n t where t h e

s o u rc e d i s t r i b u t i o n on a. panel i s repre sen t ed by two ove r l apping t r i a ng le s .

T h i s d e f i n i t i o n i s c omple te ly a na logous t o t he vo r t e x i n f l ue nc e c oe f f i c i e n t s

U i j and ~ij

The t o t a l i nduc ed ve l o c i t i e s a t t h e i - t h boundar y po i n t c an be de s c r ibe d by

?With t h i s t echniqu e the norma l ve lo c i t y component a t t h e sur fac e n and theis o u rc e s t r e n g t h q , a r e e q ua l.i



Because o f t he i n t rodu c t ion o f a s o u rc e d i s t r i b u t i o n , t h e K u tt a c o n d it i o n a t

t h e t r a i l i n g edge of t h e a i r f o i l t a ke s a f o r m d i f f e r en t f rom those used for

b l u n t and c l o se d t r a i l i n g e dg es ' i n t h e i n v i s c i d f lo w c a l c u l a t i o n . With t h e

t r a i l i n g edge b e in g a s t ag n a t i o n p o i n t t h e sum o f t h e v o r t ex and so u r ce

v e l o c i t i e s i s ze r o . With th e t r a i l i n g edge sources known th e vor t ex s t r e ng t hs

yU and y1 can be determined by t h e co ndi t io n t h a t t h e component of ve lo ci ty

n orm al t o t h e t r a i l i n g e dg e pa ne l i s z e ro a t t h e t r a i l i n g e dg e on t h e u pp erand lower su r faces .

Fig. 2.6 Ku tta Condi t ion - Modif ied by Source Dist r ibut ion

From th e preceeding f i gu re ,

Y u = (-ql + qu cos8) /s in 8 (2.9)

I t should be no ted t ha t t he above equa t ions imply equa l p ressure

c o e f f i c i e n t s on t h e up pe r and l ow er s u r f a c e s a t t h e t r a i l i n g e dge. Taking

t h e d i f f e r en c e o f t h e sq u a r e s of e ach eq u a t i on ,



The a d d i t i o n o f t h e s o u rc e d i s t r i b u t i o n e x t e r n a l t o t h e a i r f o i l m o d if i es

t h e norm al v e l o c i t y a t t h e c o n t r o l p o i n t of p an e l i. Refer r ing to Eqn. (2 .4) ,

where

i s t h e normal ve lo c i ty induced by the e x te rna l source on pane l j .

Since t h e q a r e known, t he r i g h t hand s i d e of Eqn. (2.5) becomesj

S i n c e , t h e v o r t e x s t r e n g t h s a t t h e t r a i l i n g e dg e o f t h e u pper and lower

s u r f a c e s a r e a l s o s p e c i f i e d i n t erm s o f t h e e x t e r n a l s o u r ce s t r e n g t h by E qns.

(2.9 and (2 .10) ', an ad d i t io na l unknown cons tan t source d i s t r ib u t io n in s i de

ea ch a i r f o i l s u r f a ce i s r e q u i r e d t o so l v e Eqn ( 2 .5 ) , a s de s c ri b ed i n t h e pr ev io u s

s e c t i o n .

The advanta ge i n computer time of t h i s p rocedure r e su l t s from hav ing t o

c a l c u l a t e t h e i n f l ue n c e c o e f f i c i e n t s o nl y on ce . A t e a c h s u c c e s s i v e i t e r a t i o n

.on ly m a t r i x m u l t i p l i c a t i o n i s r equ i r e d to de te rmine th e new vor t ex s t r e ng t hs .

A s i n the d i sp lacemen t method th e e f f e c t of th e boundary l ayer co r r ec t io ns t ends

t o c a us e a n o ve rs ho o t o r c o r r e c t i on i n t h e p r e ss u r e f i e l d s o l u t i o n a t e ac h

i t e r a t i o n . Th i s u n d e s i r a b l e f e a t u r e i s avoided and rapid convergence assured

i f t h e bo un da ry l a y e r d e ve lo pm en t and r e su l t a n t so u r c e d i s t r i b u t i o n i s determined

from a p r e s s u r e f i e l d w ei gh te d u s i n g f i f t y p e r c e n t of t h e c u r r e n t s o l u t i o n and

f i f t y p e r c e n t ' o f t h e pr ev io u s s o lu t i o n.

Pil though t he precedure does no t re qu i r e ex te ns iv e smoothing as ir, t h e

displacement methods some l i m i t a t io n on the sourc e s t re ng th i s r e qu i re d i n t h e

t r a i l i n g e dg e r e g i o n i f r a p i d c on ve rg en ce i s t o be ach ieved . Rapid growth

o f t h e bounda ry l a y e r a pp ro ac hi ng s e p a r a t i o n i n s t r o n g a d v e r s e - ~ r e s s u r e r a d i e n t s

( t y p i c a l of c o n f i g u r a t i o n s a t h ig h an g le s -o f -a t ta c k) c a u se a bn o rm a ll y f a s t gr ow th

of t he di sp lacemen t th i ckness , and in tu r n the source s t r e ng t h . Numer ical exper -

i me nt s i n d i c a t e t h a t i f a l i m i t i s , p l a c e d on th e maximum sourc e s t re ng th conver-

g en ce c a n o c cu r b etw ee n two a nd f i v e i t e r a t i o n s . The c a l c u l a t i o n s a l so i n d i c a t e

t h a t t h i s l i m i t i s d i f f e r e n t f o r s l o t t e d a i r f o i l c a se s , w here t h e boundary l a y e r

g rowth on the f l a p i s very much g re a te r than th a t t y p i ca l of s i ng le el emen t cases .

More w i l l b e s a i d of t h i s l i m i t i n a f o ll o wi n g s e c t i o n .



Because t h e p r e s su r e - co e f f i c i e n t s a r e d et er mi ned from t h e v e l o c i t i e s o n t h e

boundary po in t s o f a pane l r a th er t h an f rom t h e v or t e x s t r e n g t h s a t t h e c o r ne r

p o i n t s , t h e t r a i l i n g edg e p r e s s u r e s a r e n o t known a p r i o r i . T h er e fo r e, t h ey a r e

ca l cu l a t ed b y si m p l e l i n e a r ex t r ap o l a t i o n o f t h e p r e s s u r e s f ro m t h e l a s t two

boundary po i n t s on the upper and lower su r f ace s r es pec t ive ly .

A s p e c i a l s i t u a t i o n a r i s e s i f any e l em ent of t h e g eom etry d o es n o t h av e a

c l o s e d t r a i l i n g ed ge . I n t h i s c a s e t h e s o l u t i o n f o r t h e i n v i s c i d f lo w a bo ut t h e-a r t i c u l a r e l e m e n t i s de te rmined us ing th e Kut ta c ond i t io n yu - - yl. No

i n t e r n a l d i s t r i b u t e d s ou rc e i s r equ i re d to complete the prob lem def in i t io n .

Consequent ly, when bounc!ary l a ye r e f f e c t s a r e inc luded i n the f i r s t i t e r a t i o n

(wi th th e Kut ta co nd i t ion dete rmined f rom Eqns. (2 .9 ) and (2 . l o ) ) , an in te rn a l

s o u r ce i s r equ i re d t o complete th e prob lem def in i t io n and i t is neces sary to

r e c a l c u l a t e t h e i n fl u en c e c o e f f i c i e n t t o i n c l ud e t h e e f f e c t of t h e d i s t r ib u t e d

s o ur ce. S ub sequ en t i t e r a t i o n s r eq u i r e o n l y m a t r ix m u l t i p l i c a t i o n t o o b t a i n

t h e v o r t e x s t r e n g t h s .



BOUNDARY LAYER CALCULATION METHODS

The b ou nd ar y l ay e r d eve lo pm en t i s - c a l c u l a t e d f ro m th e s t ag n a t i o n l i n e o f

each e l em ent . F or t h e i n f i n i t e s wept w in g ca s e two s ep a r a t e c a l c u l a t i o n p r o-

c e d u re s a r e u s ed , e a ch f o r a p a r t i c u l a r r e g i o n of t h e f lo w. On the upper

su r f ac es of th e l ead i ng edge s l a t and main wing and fo r t he lower su r f ace ofevery e lement of th e conf i gura t ion an in te gr a l method i s used. This method

i s ab o u t 1 0 0 t im es f a s t e r t h an t h e co r re s po n din g f i n i t e d i f f e r en ce method i n

two di me ns in ns . Economy of computer time i s e s s e n t i a l i n a n i t e r a t i v e m ethod

p a r t i c u l a r l y when a m u lt i -e l em ent co n f ig u r a t i o n i s cons idered , i f the method i s

t o be of p r a c t i c a l u se t o t h e d e si g ne r . I n a l l c a s e s w her e p a s s iv e b lo wing

(s lo t t ed f l ap s ) o r powered b lowing i s c o ns i de r ed , t h e f i n i t e d i f f e r e n c e m ethod

i s used . Th is method cou ld be used f o r th e complete boundary lay er an a l ys i s

i f d e s i r e d by t h e u s e r .

D es c r ip t i o n s of t h e i n d iv id u a l bo un da ry l ay e r and t r a n s i t i o n an a ly s e s arep r e se n t e d i n t h e f o l l o wi n g s e c t i o n s .

S t agn a t io n ~ i n e n i t i a l C on di ti on s

. I t ha s been p red ic ted t h e or e t i ca l ly by Cumpsty and Head 4 and Bradshaw 5amongst o t h e r s t h a t f l o w a l on g t h e s t a n g a t i o n l i n e of a i n f i n i t e yawed wing

ap p r ao ch es an a s y m p to t i c co n d i t i o n . T h i s co n d i t i o n i s one where the r a t e of

g ro wt h of t h e b ou nd ar y l a y e r du e t o f r i c t i o n a l f o r c e s i s balanced by the

d ivergen ce of t h e f low f rom th e spanwise t o th e s t r eamwise d i rec t io n . Cumpsty

and Head l a t e r demons t rated i n an exper imenta l s tudy (Ref 6) h e i r e a r l i e r

t h e o r e t i c a l p r e d ic t i on . They were a b l e ' t o show th a t whether th e f low i s laminar

o r t u r b u l e n t i t s i n t e g r a l p r o p e r t i e s c an b e d et er mi ne d d i r e c t l y a s a f u n c t i o n of

a s in g le non-d imensiona l parameter C*. The parameter C* ( 2 v 2 / v d ~ / d x ) ,where

V ' i s t h e s pa nw is e v e l o c i t y , V t h e k in em at ic v i s c o s i t y and d ~ / d x he chordwise

v e l o c i t y g r a d ie n t a t t h e s t a gn a t i on l i n e ) i s a form of Reynolds number which

c o r r e l a t e s w e l l w i t h t h e s t re am w is e sh ap e f a c t o r H , momentum thickness 8 and

s k i n f r i c t i o n c o e f f i c i e n t C f/2 . The c o r r e l a t i o n s f o r H and 8 a r e p r es e nt e d i n

t a b u l a r f or m i n T a b le 1. I n i t i a l i n t e g r a l bo un da ry l a y e r p a r a me t er s a r e de te rm in ed

from t h e t a b l e f o r t h e c a lc u l a t e d C*. I f C* < 1 .35 x l o 5 t h e f lo w i s l aminar

o t h e r w i s e i t i s t u r b u l e n t . The ap p r o p r i a t e c a l c u l a t i o n method i s then used t o

d e t e r m in e t h e downstream boundary l ay er growth (See Fig ure 3.1).

Integral Boundary Layer Methods

Laminar Method

A v a r i e t y o f me th od s e x i s t f o r t h e c a l c u l a t i o n o f l a mi n ar bou nd ary l a y e r

d eve lop m en t s, t h e mo st g en e r a l o f t h e s e b e in g b a s ed on f i n i t e d i f f e r en c e m etho ds .

I n t h e c a s e o f t h e i n f i n i t e swept wing s u b s t a n t i a l r e g i o n s of l a mi n ar f l ow (1 0%

c ho rd o r mo re ), a r e l i k e l y on l y a t t h e low er Reynolds numbers and sweep an gl es

a nd i n c a s e s where l a r g e f l a p d e f l e c t i o n s r e s u l t i n c o n s id e r a bl e l am in ar f lo w

on t h e l ow er s u r f a c e s of t h e f l a p s . I n t h e s e i n s t a n c e s t h e e f f e c t o f t h e l am in ar

f l o w on t h e e x t e r n a l f l ow i s n e g l i g i b l e a n d t h e d r a g c o n t r i b u t i o n v e r y s m al l .



o r Rell along a

I n s t a b i l i t y

aI T r a n s i t i o n I

l i tu r b u l e n t C a l c '

Fig. 3.1 Flow Chart f o r Boundary Layer Cal c u l a t on s



Tab le 1. S t a g n a t i o n L i n e C o r r e l a t i o n of C*

w i t h H and Rel1.



B ec au se o f t h i s , a l l l a m in a r b ou nd ar y l a y e r c a l c u l a t i o n s a r e made u s in g a two-

d imens io na l in t e g r a l - app ro ach a lo n g ex te r n a l s t r e aml in es . The f i n i t e d i f f e r e n c e

m ethod t o b e d e s c r i b e d i n a f o l l o w i n g s e c t i o n c an a l s o b e us ed t o d e t er m in e t h e

lamin a r b o un d ary l ay e r d ev elop ment b u t f o r th e i n f i n i t e swept win g ca se a s w i l l

b e se en l a t e r i t a p p ea r s t o be u n ne ce ss ar y f o r p r a c t i c a l c a l c u l a t i o n s .

The two d im ens ion al method i s an ad a p ta t io n by C u r le 2 of a method

developed by Thwaites 4. In T hwai te ' s method t he momentum i n t eg ra l e quat ion

i s w r i t t e n i n t h e f orm

d / d x ( ~ / u ) = L/U

where

T h wa i te s u s ed e x a c t s o l u t i o n s t o a v a r i e t y o f l a mi n ar f l o w s t o d e t e rm i n e

t h e r e l a t i o n s h i p betw een L and K ,

C u r le h as p o in ted o u t th a t Eqn. 3 . 4 i s n o t ad eq u a te i n f lo ws ap pro ach in g

s e p a r a t i o n , and h e h as su g g es ted an ex ten s io n o r co r r ec t io n t o Eqn.3 . 4

g i v i n g :

The parameter p i s a f u n c t i o n of b o t h t h e p r e s s u r e g r a d i e n t and t h e c u r v a t u r e

o r s ec on d d e r i v a t i v e of v e l o c i t y .

Curle rewrote Eqn. 3.5 i n t h e f o r m

L. = Fo (K) - ) Go(K) 3.7

where Fo and Go a r e u n i v e r s a l f u n c t i o n s d e t e rm i ne d from a s e r i e s of e x a c t

so lu t i o n s t o l amin a r f lo ws i n th e same way a s d id Th wai te s f o r Eqn. 3 . 4 . .-

A f t e r s u b s t i t u t i o n o f Eqn. 3 - 5 i n t o Eqn. 3 . 2 a n d w i t h s u b s e q u e n t , i n t e g r a t i o n ,

t h e r e s u l t c a n b e r ea r ra n ge d i n t h e f orm



T h i s e q u a t i o n i s c o n ~ re n i e nt l y o l v e d by i t e r a t i o n , g i n i t i a l l y e qu al t o

zero . With va lue s o f Y and u d et er mi ne d i n t h e f i r s t i t e r a t i o n . a s econd

i t e r a t i o n i s ca r r i e d o u t u s in q Eqn. 3 . 7 . A t e a c h , s t e p i n t h e c a l c u l a t i o n

t h e l o c a l s k i n f r i c t i o n c o e f f ic i e n t , C 12 and t he shape fa c t o r H can be

ca l cu l a t ed u s in g E q n 3 .3 . The l o c a l s i i n f r i c t i o n c o e f f i c i e n t h a s been

d e f i n e d a s

Cf = (lJIp0U)R 3 . 9

where R i n Eqn. 3 . 3 i s de te rmined i n a s im i l a r manner t o L fro m a s e r i e s of

known s o l u t i o n s t o g iv e

The fun c t i on s Fo, F l y Go and G1 a r e t a bu la ted i n t he computer p rogram.

C a l c u l a t i o n s b eg i n a t t h e s t a g n a t i o n l i n e i n t h e s wept wing c a s e , w i t h t h e

i n i t i a l momentum th i ck n es s 0 g iven a s a f u n c t i o n of C*. I f t h e f lo w i s two

dimens iona l K t a ke s a i n i t i a l v al ue KO = 0 .0 855 a t t h e s t a g n a t i o n p o in t f ro m

which th e i n i t i a l momentum th ickn ess 0, i s

The c a l c u l a t i o n p r oc ee ds e i t h e r t o l a mi na r s e p a r a t i o n o r t o t h e end of

t h e a i r f o i l whichever o cc ur s f i r s t . The calculated boundary layer development

i s t h e n i n t e r r o g a t e d t o d et er mi ne i f t r a n s i t i o n , l am in ar s e p a r a t i o n o r f o rc e d

t ra n s i t io n (boundary la ye r t r i pp ing ) has t aken p lace . I f any of t he se phenomena

have occu rred th e downstream f low i s assumed t o be t u rbu len t .

Turbulent Method

Se ve ra l methods have been developed f o r t h e c a l c u l a t i o n of i n f i n i t e sweptwing th r e e dime nsio nal boundary la ye rs . Among t h e more u se fu l of t h e methods

a r e t ho se by Cumpsty and Head 2, Nash 10, nd Bradshaw 2. Na shl s method (a

f i n i t e d i f f e r e n c e p r oc ed ur e) i s a l s o a p p l i c a b l e t o f u l l y t h r e e d im en si on al

b ou nd ar y l a y e r s b u t i n t h e w ord s of t h e o r i g in a to r i s cumbersome and inflexible

when ap pl i ed t o complex geometr ies . I n p ra c t ic a l ca ses th e methods of Cumpsty

and Head and of Bradshaw appear t o g i ve s im il ar r e s u l t s , wi th Cumpsty and Heads

method having a co ns i der ab le advantage both i n speed and convenience. Because

of t h i s , t h e i r method was ch os en f o r u s e i n t h e v i s c o u s / p o t e n t i a l f lo w i n t e r a c t i o n

program.

In deve loping t h e i r method, Cumpsty and Head chose an or th ogo nal cu rv i l in ea r-

s y st em o f c o o r d in a t e s b as ed o n t h e p r o j ec t i o n s o f e x t e r n a l s t r e am l in e s o n t h es u r f a c e . I n t h i s sy st em s tr ea mw is e t u r b u l e n t bou nd ary l a y e r v e l o c i t y p r o f i l e s

r e s em b le v e r y c l o s e l y two -d im en sion al p r o f i l e s . When th e s t r eamwise p r o f i l e s a reknown th e cr o s s- f l ow v e l o c i t y p r o f i l e s c an b e ca l cu l a t ed a s f u n c t i o n s of t h e

s t r eam w ise p r o f i l e s and t h e an g l e b etween t h e s u r f ace s t r ea m l in e and t h e p r o j ec t i o n

of t h e e x t e r n a l s t r e a m l i n e on t h e s u r f a c e ( a n g le 6) .



Cumpsty and Head wrote t he s t reamwise and cro ss f low equa t ions i n i n t e g ra l

equa t ion form a s fo l lows:

Streamwise Momentum Equation

Cross Flow Momentum Equation

where k= tan a .

Equat ions (3 .12) and (3.13) con tai n se ve ra l unknowns and bef ore tu rb ul en t

boundary l ay er p red ic t i ons can be made fu r th er r e l a t i on sh ip s are r equ i r ed

between th e streamw ise and cr os s f low momentum thic kn es se s, th e streamwise shape

f a c t o r H and t h e s t re amw ise sk i n f r i c t i o n co e f f i c i e n t Cf '

Entrainment Equations

The f a c t t h a t t h e s tr ea mw is e v e l o c i t y p r o f i l e s are similar t o two

dimens iona l ve lo c i ty p r o f i l es l e d Cumpsty and Head t o assume th a t th e r a t e of

entra inment a long a s t r eam l i n e i n t h e i n f i n i t e sw ept wing ca se co ul d b e d e t e r -

mined usin g re la t i on s hi ps developed fo r two-dimensional f low. Thi s i s ac r e d i b l e a s sum p ti on s i n ce t h e en tr a in m ent p r o ce s s i s a f u n c t i o n of t h e v e l o c i t y

d e f ec t i n t h e o u t e r p a r t o f t h e b ou nd ar y l ay e r , a r eg i o n w here t h e s t r eam wi se

and two dimensional p r o f i le s are expected t o ag re e most cl os el y. Cumpsty and

Heads ent ra inment equat ion takes the form

where H1 = (6-6;)/O11.

The f un ct io n F(H ) i s taken i n t he form present ed by Head 11.1



Lik ew ise t h e e x p r e s s io n r e l a t i n g H t o t h e more u su a l sh a p e f a c t o r H i s a l s o1

given by Head.

The fun ct i on s F and G i n Eqns 3 .15 and 3 .16 can be an al y t ic a l ly def ined

a s f o l l o w s :

F(H1) = exp [-3.512 - 0.617 l n (HI-3) 1. (3.17)

f o r H 5 1 . 6 o r

G(H) = . 3.3 + exp[0.4383 - 3.0 64' 1n (H-0.6798) (3.19)

f o r H > 1.6.

-

Streamwise Ve loc ity P r o f i l e s

Cumpsty and Head demonstrated that the streamwise tur bul en t boundary lay er

v e l o c i t y p r o f i l e s c ou ld b e r e p r e se n t e d q u i t e a c c u r a t e l y by t h e two d im e ns io na l

ve lo ci ty p r o f i l e fami ly der ived by Thompson 12. The law of t h e w a l l - law of

th e wake ve l oc i t y p r o f i l e f ami ly of Coles 1 T g i v e s r e s u l t s which a r e i n good

agreement with Thompsons profi les and could e a s i l y b e us ed a s a n a l t e r n a t e

approach.

Cross Flow Pr o f i l e s

The c r o s s f l o w p r o f i l e s h a ve b ee n sp e c i f i e d by t h e s i m p le r e l a t i v n sh i p

between streamwise and c ro ss f l ow v e lo c i t i e s sugge sted by Mager 14,

2V / U = (1 - 5/61 tan6 (3.20)

where i s t h e a n g l e between t h e su r f a c e s t r e a m l in e ( r e su l t i n g sk i n f r i c t i o n

d i r e c t i o n ) a nd t h e p r o j e c t i o n of t h e e x t e r n a l s t r e a m l in e o n t h e su r f a c e ,5 i s

t h e d i r ec t i on normal t o t he s u r f a ce , and u and v a r e t he s t r eamwise and c ro ss

f l ow v e l o c i t i e s .



Cross Flow Thicknesses

The cross flow thicknesses have been defined using a power law velocity

profile instead of one of the more complicated two parameter profile relation-ships, in the streamwise direction. This approach greatly simpli~ied he

definition of the cross flow thicknesses without any great loss in accuracy.The use or the power law relationship

in Equation (3.20) gives the cross flow thicknesses as defined by:

Skin Friction Coefficient

The streamwise skin friction coefficient is determined using Thompson'stwo parameters skin friction law although here,again ole's skin friction law

could also have been used. The relationship is of the rorm

C = f (H, Re) and is given as C = exp (AH + B)f(3.23)

fl

where 2A = .01952 - .38682 + ,028342 - .00072

B = .I9151 - ,83492 + ,062592~ .001953z3

The cross flow skin friction coefficient Cf2 is then determined from Cf as

Cf2= Cfl tanB and the resultant skin friction coefficient Cf as C =Ic /COPB.

fl



Calcu la t i on Procedure

W ith i n i t i a l v a l u e s of 8 and H known ei ther from the laminar boundary11l a y e r c a l c u l a t i o n o r t h e s ta g n a ti o n l i n e i n i t i a l c ond i t i ons Equat ions 3 .12, 3 .13

and 3 . 14 a re i n t e g r a t e d u si ng s ta nda r d i n t e g r a t i on p r oc edu r es . The pa r am e t er s

8 H and 6 i n c o nj un c ti o n w i th t h e s k i n f r i c t i o n c o e f f i c i e n t , and t h e c r o s s

f low th i ckne sses a r e det e rmined a long s t ream l ines a s fun c t ion s of t h e known

p r e ss u r e d i s t r i b u t i o n .

The s t reamwise f re e s t ream ve lo c i ty U and ang le .a ar e determined a s showni n t he s ke t c h , a l s o s hown i s t h e a n g l e 6 , %he angle be tween t h e pr o j e c t i o n of

t h e e x t e r n a l s t r e am l i ne on t h e s u r f a c e , a nd t h e r e s u l t a n t s k i n f r i c t i o n d i r e c ti o n .

s n a

O >us/ua.

i n a0

The sum of th e two ang les a ando@ i s cont inuous ly moni tored dur ing the

c a lc u la t io n . I f t h i s sum reaches 90 t he f l ow i s completely spanwise and

by d e f i n i t i o n t u r b u l e n t s e p a r a t io n h a s o cc ur re d. . T he c a l c u l a t i on i s s topped

a t t h i s p o i n t.

Boundary Layer Transi t ion and Laminar Separat ion

Boundary layer t r a n s i t i o n i s a ve ry complex phenomenon and t o d a t e no

r e l i a b l e t h e o r e t i c a l m ethod ha s be en de ve lope d f o r i t s pred ic t i on . Reynolds

number i s a c o n t r o l l i ng par am e te r , bu t i t has been shown t h a t th e Reynolds

number a t t r a n s i t i o n c a n b e i n c re a s ed a cons ide rab l e amount by ca re fu l

e l i m i n a t i on of d i s t u r ba nc e s . A t ve ry low Reynolds numbers, lam ina r boundary

l a y e r s a r e s t a b l e t o s m a l l d i s t u r ba nc e s , how ever , a t h i ghe r R eyno lds num bers

t h e b ou nd ar y l a y e r i s uns t ab l e , and sma l l d i s tu rban ces can be ampl i f i ed .



Ampl i f i ca t ion o f these d i s tu rbance s cause th e f low t o become tu rb u le n t . The

p o i n t a t which f low bre ak down occ urs depends on t h e s tr en gt h and dominant

f re qu e nc y of t h e i n i t i a l di s t u r ba n c e . Disturbances may b e due t o f r ees t rea m

t u r b u le n c e , su r f a c e r ou gh n es s , n o i s e o r v i b r a t i o n o f t h e su r f a c e . A s t h e r e

i s n o d e t a i l e d a n a l y s i s of t h e t r a n s i t i o n p ro c es s , t r a n s i t i o n p r e d i ct i o n i s

accomplished by means of empir i ca l co r re la t io ns . Gra nvi l le 15 has developeda p ro c ed ur e b as e d on t h e d e t e rm i n at i on o f t h e n e u t r a l s t a b l i E t y p o i n t a nd

t h e t r a n s i t i o n p o in t . The n e u t r a l s t a b i l i t y p o in t i s d e f i n e d as t h a t p o in t

downstream of which s mal l d i s tu rba nce s a r e ampl i f ied wi t h in t h e boundary

l a y e r . I t i s t h i s a m p l i f i c a t i o n o f s m a l l d i s t u rb a n c e s t h a t u l t i m a t e l y l e a d s

t o t r a n s i t i o n . The n e u t r a l s t a b i l i t y p oi nt i s reache d when t h e Reynolds

number based on th e l o c a l momentum th i ckn ess and t h e l o c a l f low prop er t ie s

a t t a i n s some c r i t i c a l v a lu e , ROgn2 . .Schl i cht ing and Ulr ich (Ref . 16) have

shown2that Rg c an b e c o r r e l a e w i th t h e l o c a l p re s s u r e g r a d i e n t p a r a m e t e r

K = €3 / u ( d ~ / d & ~ ? or re l a t ion s by Smi th 17 and o th er s have been reduced

t o a n a l y t i c a l form a s f o l l ow s :

I n s t a b i l i t y Curves

< 650o r 0 < Reins -and K = 0.69412 - 0.23992 I n Re

2+ 0.0205 I n Re

for 650 < RBins ( 0,000.

I f f o r a g iv en Re t h e p r e s su r e g r a d i e n t p a ra m et er K a s c a l c u la t e d by

Eqn. 3.24 o r 3.25 i s gr ea te r than t ha t de te rmined by th e boundary l ay er

development th e f low ha s passed from a s t a b l e t o a n u n s t a b l e r eg i on . Once

t h e f l ow p a s s e s i n t o t h e u n s t a b l e r e gi o n , t h e t r a n s i t i o n p r oc e s s b e g i n s ,

a nd G r a n v i l l e h a s be e n a b l e t o show t h a t a c o r r e l a t i o n s imilar t o the ' i n s t a b i l i t y

p r o c e s s c an b e u se d t o d et e rm i n e t h e t r a n s i t i o n p o i n t .

~ r a n v i l l e ormed a n a v er a ge p r e s su r e g r a d i e n t p ar am e te r d e f i n e d a s

f t r ansJ . K d s- --

- si n s

K =s - S

t r a n s i n s



. which co r r e l a t ed r easonab ly w e l l wi th t h e momentum th ic kn es s Reynolds number

a t t r a n s i t i o n R T h i s c o r r e l a t i o n i s p r es e n te d i n a n a l y t i c a l f orm a s

fo l lows: ' t r ans

Tr a n s i t i o n C u r v e s

5k = -0.0925 + 7.0 x 10- Rg

< 750 ,or0 < % t r a n s -

- 2and K = 1.59381 - 0.45543 In Re+ 0.032534 In Re (3.29)

f o r 1100 c Ret rans - 3000.

When t h e k ca lc ula te d by one of th e above expres sions f o r a g iven Rg i s

gr ea te r th an th e val ue determined f rom the boundary l ay er development,

t r a n s i t i o n i s p r e d i c t e d .

With t r a ns i t io n p red ic t ed , i n i t i a l va lues .of t he momentum th ickne ss 9

and t h e sh a p e f a c t o r H a r e re q u i re d t o s t a r t t h e t u r b u l e n t b oundary l a y e r

c a l c u l a t i o n . Because th e boundary l ay er growth i s co nt inu ou s t h e momentum

t hi ck ne ss a t t r a n s i t i o n i s used a s th e i n i t i a l t u rb u le n t momentum th ickness .

S i n c e t h e s h ap e f a c t o r v a r i e s from v a l u e s g r e a t e r t h a n 2 . 0 t o l e s s t h a n 1. 5

t hr ou gh t h e t r a n s i t i o n r e g i on a n e m p i r i c a l e x pr e ss i on i s used t o d e t e r m i n e t h e .

i n i t i a l t u r bu l en t s hape f a c t o r . The emp i r i ca l r e l a t io n between H and Rgt r a n s

w a s determined from da ta obtai ned by Coles 18:

, - 1.4754H t - + 0.9698

.. L O g l ~ R erans

I n many cases the p r essu re g rad ien t i s of s u f f i c i e n t s t r e n gt h t o

s e p a r a t e t h e l a m in ar b oundary l a y e r p r i o r t o t r a n s i t i o n . Except Zn extreme

c a s e s t h e b ou nd ar y l a y e r w i l l t h en r e a t t a c h ; u s u a l l y as a tu rb ule nt boundary

la ye r . Only r ec en t l y have r es ea r che r s been ab l e to ana lyze th i s phenomenon

(Ref . 19 ) and as y e t t he p rocedure i s extremely complicated and cumbersome,

c o n s e q z n t l y e m pi r ic a l r e l a t i on s h i p s , a r e s t i l l r e q u i r e d . From t h e measurements

of Gaster (Ref . g),nd o t h e r s a c o r r e l a t i o n i s formed which i s capable of

p r e d i c t i n g b o t h t h e o c c u rr e nc e of ' s e p a r a t i o n and l a t e r t h e re a tt a ch m e nt a s a

t u r b u l e n t bo un da ry l a y e r o r t h e c a t o s t r o p h i c s e p a r a t i o n .



T h e co r r e l a t i o n i s of t h e form:

f o r Re 7 125

and

f o r R e < 125. (3.32)

I f K becomes l e s s than -0.09 sep ara t io n occu rs , and i f R e i s l e s s

than 125 th e boundary la ye r i s no t ab le t o r e -a t tach . However, i f i s

g r e a t e r t h a n 12 5 t h e v a l u e of K determined by th e boundary la y e r development

m us t b e l e s s t h an t h a t c a l cu l a t ed b y Eqn. 3 .3 1 b e f o r e s ep a r a t i o n w ith o u t

rea t tachment i s p r e d i c t ed . ' I f r ea t ta c hm e nt i s p r e d i c t ed , t h e t u r b u l e n t

b ou nd ar y l ay e r c a l cu l a t i o n i s i n i t i a t e d us ing t he momentum th ickness c a lc u la ted

a t t h e s e p a r a t io n p o i n t .



Finite Difference Boundary Layer Method

On el em en ts of a h ig h l i f t c o n f i g ur a t i on where t h e f l ow f i e l d i s

pa r t i c u l a r ly complex , such as i n th e r eg ion where t l i e .wing wake mixes o r

i n t e r a c t s w i t h t h e f l a p u ppe r s u r f a c e , i n t e g r a l bou nd ar y l a y e r methods a r e

no t capab le o f complete ly ana lyz ing th e f low. A more s a t i s f a ~ t o r ~ m e t h o dan be

d ev el op ed u s i n g f i n i t e d i f f e r e n c e m eth od s. Such a method i s d es cr ib ed i n t h e

fo l lowi ng pa rag raphs.

Governing Equations

The gov er n in ge qu at ion s of mean mot ion for th ree-d imensional incomp ressib lef l o w i n a g e n e r a l s ys te m of c u r v i l i n e a r o r t h og o n a l c o o r di n a t e s a r e :

Con t inu i ty Equa t ion

a a a- h3pu) + q hlh3pv) + (hlpw) = 0a

(3.36)



The s h ea r s t r e s s t e rm s i n Eq s. (3.33) and (3.35) a r e :

I f a p r ac t i c a l t hr ee -d im ens io na l h i g h - l i f t s ys tem i s considered where i t can

b e assumed t h a t c u r v a t ur e e f f e c t s i n t h e s pa nw is e d i r e c t i o n a r e n e g l i g i b l e

compared t o t he normal chord di re ct io n, a su r fa ce coor dinate sys tem can be

employed, where

x = a h = l + k y1

k = f (x)

where k i s t h e l o n g i t u d i n a l s u r f a c e c u r v a t u r e.

The equa t io ns can then be wr i t t en i n t he fo l lowing form:



C on t i nu i t y

Equa t ions (3 .39) t o (3.42) rep re s en t t h e l amina r boundary l ay e r equa t io ns i n

incompress ib l e f l ow. The c o rr e s pond i ng e qua t i ons i n t u r bu l e n t f l ow a r e :

- -he co n t i nu i t y eq u at io n - i s unchanged. The te rms u 'v ' and v 'w'

r e p r e s e n t t h e Re ynolds s t r e s s e s i n t h e no rm al c ho rd and s pa nw is e d i r e c t i o ns .

The s h e a r s t r e s s t er m u'v' i s repre sen t ed by th e express ion

- - = v (au/ay - uk/l+ky)u ' v ' tx

wherev tx

i s t h e eddy v i s c o s i t y , w hich i n t h i s c a se i s determined us ing a two

d i m e ns i ona l m odel. Then, i f t he s he a r s t r e s s ve c t o r i s c ons i de r ed t o be a l igne d

w i t h t h e r a t e of s t r a i n v e c t o r (Nash an d P a t e l (Zl)), t h e eddy v i s c o s i t y i n t h ellz I 1

di re c t i o n may be de t e rmined f rom th e express ion



o r t h e eq u i v a l en t fo rm

Eddy V is c os it y Model

The eddy v i sc os i t y model used i n t he ca lc u l a t io n p rocedure i s a m o d i f i c a t i o n

of one developed fo r two-dimens ional tu rb ul en t bounda ry ' lay ers and w a l l je t s

over curved sur f ac es (Ref. 22) . The ba s ic two laye r model co ns is ts of inn er . ando u t e r r eg i o n s . The i n n e r r eg i o n p r o f i l e i s ca l cu l a t e d u s i n g t h e m od if ied

Van Driest r e l a t i o n

where

The o u t e r r eg i o n eddy v i s c o s i t y p r o f i l e i s determined from the Eddy

Reynolds number (ud a/vt) and t h e i n t e r m i t t en c y f u n c t i o n ( y (y) ) . fo rmula t ions

d es c ri b ed i n R e fe r en ce-2 . These fu nc t i on s a r e combined t o giv e:

w here t h e v e l o c i t y s c a l e ud and t h e l e n g th s c a l e a ( st a n da r d d e v i a t i o n o f '

t h e i n t e r m i t t e n c y f u n c t i o n y ) a r e known fu nc t ion s o f th e shape f ac to r H and

the d i splacement th icknes s 6* f o r conven t iona l boundary l ay er s .



The ext ens ion of th e eddy v is co s i t y model t o th e mixed f low cas e

a s s o c i at e d wi t h s l o t t e d f l a p s i s based on obse rv at io ns made of th e develop-

m en t o f w a l l s j e t s e xh i b i t i n g bo t h ve l o c i t y maxima and minima i n t he p r o f i l e

( s e e s k e t c h ) .

I n t h e r e g i on below t h e ve l o c i t y maxima

(Region A) t h e c onve n t iona l eddy v i s c os i t y

model i s us ed t o de s c r i be t he f low . The

ou t e r r e g i on o f t h e p r o f i l e .( Reg ion C)

re pr e s en t s t he remnant of t h e upst ream

boundar y l a y e r ha v i ng a l a r g e va l ue o f

t h e s h a pe f a c t o r H. Measurements of the

tt a n d a r d d e v i a t i o n o f t h e i n t e r m i t t e n c y 0

i n d i c a t e t h a t a n a s ym p to ti c v a l u e i s

a ppr oac hed a t h i gh va l ue s of H. S i m i l a r Ybe ha v i o r i s o bs er ve d f o r t h e v e l o c i t y

d e f e c t Ud. By employing t h e as ym pt ot ic

v a l u e s o f a a nd Ud i n t h e Eddy R eyno lds

number re l a t i on u do /v t, a t h i r d l a y e r i s

es t a b l i s he d which when jo ined t o t h e conven-

t i o n a l two l a y e r eddy v i s c o s i t y p r o f i l e

p r o v i d e s a comple ted eddy v is co s i ty model .

0u-n

Wall j e t s ex h i b i t i ng on ly a maximum

i n ve lo c i ty have been s tud i ed by many

re se ar ch er s . Measurements of th e s tand-

a r d d e v i a t i o n a of t he i n t e r m i t t e nc y

f u n c t i o n y f o r t h e s e f l o ws a ll o w

pr e d i c t i on s t o be made of t h e eddy

v i s c o s i t y i n t h e o u t e r re gi on o f t h e Yp r o f i l e ( R e g i o n B of sk e tch ) . The

v e l o c i t y d e f e c t Ud i n t h i s c a s e i s

s imply U - U . Region A i s aga indesc r ibed u s ing th e convent iona l boundary

0

l a y e r m o del f o r t h e eddy v i s c o s i t y .



The s l o t t e d a i r f o i l ( p as s i ve b.lowing) c a s e r e s u l t s i n v e l o c i t y p r o f i l e s

which a r e v e r y s i m i l a r - o : t h os e f o r . ai ge -n ti al i n j e c t i o n ; p r o f i l e s w i t h

v e l o c i t y maxima i n t h e c a se o f t h e : le a d i n g e dg e s l o t t e d ' s l a t , a nd p r o f i l e s w i t h

ve lo c i ty maxima and minima f o r ' s lo t t e d f l a ps . A s a consequence it was assumed

t h a t t h e same ed dy v i s c o s i t y a pp ro ac h a s d ev e lo p ed f o r t a n g e n t i a l i n j e c t i o n

co uld b e a p p li e d t o t h e s l o t t e d ' a i r f o i l c as e. The i n i t i a l v e l o c i t y p r o f i l e f o rt h e s l o t t e d c a s e i s shown i n F igure 3 .2 . Two f a c to rs make th e problem s l i g h t l y

d i f f e r e n t f ro m t h e t a n g e n t i a l i n j e c t i o n c a s e . T he se ar e :

( i ) With s l o t t e d c o n f i gu r a t io n s , t h e p e r s i s t e n c e of t h e p o t e n t i a l

co re , and

( i i ) on. t h e f l a p s u r f a c e t h e p o s s i b i l i t y of c o n si d er a bl e l a m in a r f lo w

a t l e a s t t o t h e s u c t io n peak.

Consequently t h e f low may co ns is t of a laminar boundary layer , above which i s

a p o t e n t i a l c o r e . Above t h e p o t e n t i a l c o r e . i s t h e remnant of t he cove and

upper su r f a ce tu r bu l en t boundary l ay er s o f th e wing o r p receding f l a p segment .To accoun t f o r th e p resence of th e l aminar boundary l ay er and th e po t en t i a l

c o r e , t h e e ddy v i s c o s i t y i s s e t t o z er o i n t h e s e r eg io ns , I n t h e p o t e n t i a l

co re r eg ion t he r emain ing v i scous t e rms a r e neg l ig ib l e i n compari son wi th th e

i n e r t i a t er ms , r e s u l t i n g i n - a form of B e r n o u l l i s ' e q u at i on . T hr ee d i f f e r e n t

ed dy v i s c o s i t y d i s t r i b u t i o n s a r e p o s s i b l e d e pe nd in g on t h e f l ow re gi me

(Figure 3 .3) .

The inc lus io n o f cu rva tu re t erms i s e s s e n t i a l t o t h e s uc ce ss d f t h e c a l -

cu la t i on method a s i s t h e ne c e s s i ty o f i n cl u di n g t h e s t a t i c p r e s s u re v a r i a t i o n

i n t h e d i r e c t i o n norm al t o t h e a i r f o i l s u r f a c e . I n r e gi o n s away from t h e wing

t r a i l i n g e dge - f l a p lea din g edge Eq. (3.44) i s adequate ; however , i f th e afo re-

mentioned region i s o f i n t e r e s t , t h e n a p r e s s u r e f i e l d P (x ,y ) i n t h e two-d imens iona l and i n f i n i t e swept wing cas e o r P (x ,y , z ) i n t h e f u l l t hr ee -d im en si on al

cas e mus t be p res c r ibed . The p r e s s u r e f i e l d a bo ve t h e i n d i v i d u a l f l a p s u r f a c e s

i s de te rmined d i r e c t ly f rom the known induced ve loc i ty f i e l d .

Once t h e ed dy v i s c o s i t y d i s t r i b u t i o n , t h e su r f a c e c u r v a t u r e and t h e p r e s su r e

f i e l d P(x,y) a r e known, Eqns. 3.43, 3 .,44 and 3.45 can be sol ved i n con jun ctio n

w i th t h e - c o n t i n u i t y e q u at i on . I n t h e p r e s e n t c a l c u l a t i o n a l l s pa nw is e g r a d i e n t s

have been neg lec ted . The r e su l t in g equations a r e so l v e d us i ng a m o d i f i c a t i o n of

the Crank - Nichol son p rocedure (23) f i r s t desc r ibed i n Ref. 24.-The i n i t i a l v e l o c i ty p r o f i l e i s det erm ine d by combining known o r assumed

v e l o c i t y d i s t r i b u t i o n s i n t h e wing t r a i l i n g edge - p o t e n t i a l c o r e r e g i o n .

The in te gr a l method CIBL) i s used t o c a l cu la t e th e boundary l a ye r development .t o t r a n s i t i o n o r t o some p o i n t on t h e f l a p su r f a c e downstream of t h e wing t r a i l i n g

edge i f t r a n s i t i o n t a ke s p l ac e i n t h e s l o t r eg io n . Th e c a l c u l a t e d i n t e g r a l

p ar am e te rs a t t h e s l o t e x i t a r e t h en u se d t o d e te rm in e t h e l am in ar o r t u r b u l e n t

b ou nd ary l a y e r v e l o c i t y d i s t r i b u t i o n a nd t h i c k n e s s . I f t h e f lo w i s l a m i n a r , t h e

l a mi n a r b ou nd ar y l a y e r p r o f i l e on t h e f l a p u p pe r su r f a c e i s represented by a

Pohlhausen polynomial . I f t h e f l ow i s t u r b u l e n t , ~ h o m p s o n ' sv e l o c it y p r o f i l e

fami ly i s u se d t o c a l c u l a t e t h e v e l o c i t y d i s t r i b u t i o n .



Y

WING OR FIRST FLAP

UPPER SURFA CE B. L.

. WING OR FIRST FLAP---- LOWER SURFACE B.L.

B POTENTIAL CORE

A FLAP LA M INA R B.L.

FIG. 3.2 INI TIA L VELOCITY D ISTRIBUTION FOR A SLOTTED

AIRFOIL CONFIGURATION



FLAP LAM INAR B.L. FLAP TUR BUL ENT B.L.

POTENTIAL CORE POTENTIAL CORE

WING OR TURBU LENT SLOTTED

FLAP B.L.

FIG. 3.3 POSSIBLE EDD Y V ISCOSITY PROFILES O N A

SLOTTED FLAP



The p o t e n t i a l c o re v e l o c i t y d i s t r i b u t i o n i s determined froxn th e ca lc ul a t ed

of f-body pre ssu re s (P (x ,y) ) , The wing lower su r f ac e boundary l a ye r p ro f i l e

i s r e p r e s e n t e d by a power l aw p r o f i l e , w h i l e t h e w ing upper s u r f a c e t r a i l i n g

e dg e v e l o c i t y p r o f i l e i s determined us ing Thompson's ve lo c i ty p r o f i le f ami ly

(12) and th e va lues o f H , R and C a t t h e wing t r a i l i n g e dg e.8 f

Aerodynamic Forces

The a er od yn am ic l i f t c o e f f i c i e n t f o r a g i ve n c o n f i g u r at i o n c a n b e

de te rm i ne d i n s e v e r a l ways. F or c l o s e d t r a i l i n g e dge a i r f o i l s , t he m ost

a c c u r a t e p r oce du re i nvo l ve s summing t h e i nd i v i du a l vo r t e x s he e t s t r e ng t h s .

from which

where

r = c i r c u l a t i o n ab ou t a i r f o i l

yi , = v o r t e x s t r e n g t h o f ith i n g u l a r i t y

U = f r e e s t r e a m v e l o c i t ym

C = re fe rence chord .

When t h e a i r f o i l t r a i l i n g edge r e ma ins open Eqn. 3 . 9d oe s no t ne c e s s a r i l y

g i v e t h e c o r r e c t c i r c u l a t i o n ev en th ou gh t h e v o r t e x d i s t r i b u t i o n i s a v a l i d

s o l u t i o n f o r t h e g i v en bo un dary c o n d i t i o n s. The p r e ss u r e d i s t r i b u t i o n d e t e r -

m ined f rom t h e vo r t e x d i s t r i bu t i o n i s i n very poor agreement wi th exper iment

( s e e F i g u r e 3 . 4 ) . Consequently i t ha s be en f ound t h a t more s a t i s f a c t o r y

p r e s s u r e d i s t r i bu t i o ns c an be de t er m ine d from t he e xp r e s s i on

The l i f t i s t h e n d et er mi ne d b y' i n t e g r a t i o n o f t h e p r e s s u r e c o e f f i c i e n t s

ab out t h e a i r f o i l .



0 N A C A 65 210 (REF 26)

= 2 .50 M = .4

FIG. 3.4 COMPARISON OF ME THODS FORCALCU LATING PRESSURE COEF FICIENTS



The f lo w model u se d t o r e p r e s e n t open t r a i l i n g e dg e a i r f o i l s i s be ing

reviewed. T t i s p o s s i b l e t h a t w i t h t h e K u t ta c o n d i t i o n y = - Y 1 a smallU

amount of c i r c u l a t i o n i n s i d e t h e a i r f o i l g i v e s r i s e t o n on -zero t a n g e n t i a l

v e l o c i t i e s i n s i d e t h e a i r f o i l and c on se qu en tly v o r t i c i t y s t r e n g t h which i s

g r e a t e r t ha n i f t h e f low i n s i d e t h e a i r f o i l w a s s t agnant .

P i t c h i ng moment c ha r a c t e r i s t i c s a r e det er mi ne d a s f o l l ow s : I t i s

presum ed t h a t i nc r e m e n t a l r e s u l t a n t p r e s s u r e f o r c e s a c t a t t h e c e n t e r of

pre ssu re of a sma l l pane l o f a p re s c r ibe d leng th . Consequently , t he pre s sure

fo rc e t imes t h e moment arm t o some re f e re nce po in t g iv es t h e i nc rement i n

p i t c h i ng m oment f o r t ha t po i n t . The sum of th e in cre men tal p i t ch in g moments

f o r e a c h c a l c u l a t e d p r e s s u r e g i ve s t h e p i t c h i ng moment f o r t he s y st em .

The p r o f i l e d r a g i s de te rmined for a s t reamwise sec t i on of t he i n f i n i t e '

span wing by us in g t h e Sq ui re and Young drag formula ( 2 5 ) .

The st re am wi se momentum t h ic k n es s o s , v e l o c i t y Us and shape fa c t or

a r e u se d i n Eqn. 3 . 5 4 . The s k i n f r i c t i o n d ra g c o e f f i c i e n t i s determinedHs

by t h e s ummation of t h e l o c a l s k i n f r i c t i o n f o r c e s i n t h e d r ag d i r e c t i o n .

P r e s s u r e d r a g i s de te rm i ne d by t a k i ng t h e d i f f e r e nc e bet we en p r o f i l e d r a g

and f r i c t i o n d r ag .

CALCULATION PROCEDURES

A l l of t h e c a l c u l a t i on me thods, po t e n t i a l f l ow a nd boundar y l a ye r , a r e

i n c o r p o r a t e d i n t o a s i n g l e computer program. The ca l c u l a t i o n sequence i s

ou t l i n e d beiow:

( i ) The p o t e n t i a l f lo w p r e s s u r e f i e l d i s computed fo r a mul t i -

e l em e n t i n f i n i t e s w e pt wing c on f i gu r a t i on ( c on s i s t i n g of up t o f ou r e l em e n ts ,

a l e a d i n g ed ge s l a t , t h e main a i r f o i l , a nd a d o ub l e- sl o tt ed f l a p ) .( i i ) The boundary l a ye r p ro pe r t i e s a r e t hen computed fo r each e l ement

of t h e c o n f i g u r a ti o n a s a f u n c t i o n of t h e p o t e n t i a l f lo w p r e s s u r e d i s t r i b u -

t i o n . I n cl u de d i n t h e s e c a l c u l a t i o n s a r e t h e l o c a t io n s o f t r a n s i t i o n o r

l a m in a r s e p a r a t i o n an d t u r b u l e n t s e p a r a t i o n , i f p r e s e n t .

( i i i ) S o u r c e d i s t r i b u t i o n s are de te rmined to re pr e se n t t h e d isp l acement

e f f e c t s o f t h e bounda ry l ay e r on each e lement and of the wing wake-flap

b ou nd ar y l a y e r i n t e r a c t i o n .



( i v ) A new po te n t i a l f low so lu t i on i s then computed tak ing in to account

t h e so u r c e d i s t r i b u t To n computed i n s t e p ( i i i ) a bo ve .

S t e p s ( i i ) t hr ou gh ( i v ) a r e - r e p e a t e d u n t i l c on ve rg en ce ( ba sed on t h e p r e s su r e

d i s t r i b u t i o n and l i f t c o e ff i ci e nt ) i s a ch ie v ed , o r u n t i l t h e c a s e i s abandoned

fo r r easons such a s l a rg e separ a t io n zones. L i f t , d r ag and p i t ch ing moment

c o e f f i c i e n t s a r e t h e n c a l c u l a t e d f o r t h e g i ve n c o n f i g u r a t i o n . The a pp ro ac h

i s i l l u s t r a t e d i n F ig ur e 4.1.

The a c t u a l p rogra m o v e r l a y s t r u c t u r e i s ' g i v e n i n F ig u r e 4.2. The main

supe rv i so r p rogram has been ca l l ed VIP ( fo r v i s cou s /p o te n t i a l f low in te r a c t io n ) .

Th i s progra m d i r e c t s t h e o v e r a l l f lo w o f t h e c a l c u l a t i o n . The other programs

i n c l u d e POTFLOW (p o te n t ia l flo w) , IBL ( i n t e g r a l boundary la y e r method,

INSPAN ( in f i n i t e span f i n i t e d i f f e r en ce method, and ~ E L D P T f i e l d p o i n t

c a l c u l a t i o n f o r o ff-b od y p r e s s u r e s ) .

CALCULATIONS AND DISCUSSION OF RESULTS

The u l t ima te t e s t of any an a l ys i s method i s i n how w e l l does i t p r e d i c t

actual aerodynamic performance. Th i s c an b e d et er mi ne d i n t h e c a se of p o t e n t i a l

f low methods by comp&ison wit h ex ac t so lu t i on s; f o r boundary la ye r methods,

the usual recourse however , i s comparison wit h experiment. Evalu ation of th e

o v e r a l l v i s c o u s / p o t e n t i a l f lo w i n t e r a c t i o n a n a l y s i s c a n a l s o b e made o n l y

through comparison wit h experiment. The comparisons t h a t fol low re pr es en t a

c r o s s - s e c t i o n o f t h e p o s s i b l e c o n f i g u r a t i o n s t h a t c a n b e t r e a t e d by t h e a n a l y s is

met hod.

The po t e n ti a l f low method developed as p a r t o f t h e c o n t r a c t e f f o r t h as

been compared wi th se ve ra l exact . po te nt ia l f low analy ses . Of con sidera ble

i n t e r e s t i s th e comparison fo r t he h ighly cambered Karman-Tref ftz a i r f o i l

shown i n F igure 5 .1 . Hess (27) has used th i s case to demonst ra te th e degre e

of agreement between h i s new method and oth er c l a s s i c a l methods. I t i s t he re -

f o r e e nc o ur ag in g t o n o t e t h a t o ur a n a l y s i s i s i n almost t o t a l agreement wi th th e

exa ct cas e i n compar ison wi t h o th er methods. A second comparison wit h a n

e x a c t s o l u t i o n i s f o r t h e two el em en t s l o t t e d f l a p a i r f o i l co n f i g ur a t i o n o f

Will iams ( 2 8 ) . Here ag ai n agreement between th e numerical approach and th e

e x a c t s o l u t i o n i s e x c e l l e n t ( F i g u re 5.2 ).

Two cal cu la t i on s have been included t o demonst ra te some of the ca pa bi l i ty

of th e f i n i t e d i f fe re nc e boundary Layer method i n two-dimensions (se e Ref . 22) .The e f f e c t o f lon g i tu d in a l su r f ac e cu rva tu re on the boundary l a ye r deve lopment

i s shown i n Fi gur e 5.3. I t w i l l b e s e e n t h a t when c u r v a t u r e e f f e c t s a r e i g n o re d

t h e c a l c u l a t i o n i s i n poor agreement wi th th e dat a . The ca lc ul a t io ns shown i n

F igure 5 .4 demons tr a t e th a t ve l oc i ty p r o f i l e s ty p i ca l o f thoge found on th e

u pp er su r f a c e s of s l o t t e d o r blown f l a p s c a n b e p r e d i c t e d q u i t e a c c u r a t e l y .



INPUT WING GEOMETRY

ANGLE-OF-ATTACK

SWEEP ANGLE AND

REYNOLDS NUMBER

LOFT INPUT GEOMETRYCOMPUTE POTEN TIAL CALCU LATE FLAP

FLOW SOLUTION SURFACE CURVATURE

CALCULATE BOUNDARY

LAYER PROPERTIESINCLUDING TRANSITION

AND SEPARATION

DETERMINE SOURCE

DISTRIBUTION REPRESENTING

DISPLACEMENT EFFECTS

OF BOUN DARY LAYER STOP

NO

COMPUTE POTENTIALh

SO LU TlONFLOW ABOUT CONFIGURATION

CONVERGEDINCLUDING VISCOUS EFFECTS

A YESCALCULATE LIFT, DRAG

AND PITCHING MOMENT SOLUTl ON

STAB LECOEFFICIENTS

FIG. 4.1 COMPU TATION PROCEDURE FOR A ERO DYN AMIC FORCES



FIG. 4.2 VISCOUS/POTENTIAL FLOW PROGRAM OVE RLAY STRUCTURE

Overlay (0,O)

PROGRAM

VIP

<

Overlay (4,O)verlay (1,O) Overlay (2,O) Overlay (3.0)

C4

i

PROGRAM

I B L

PROGRAM

POTF LOW

Overlay (3,l)

PROGRAM

lNSPAN

/

Overlay (3,2)

PROGRAM

FELDPT

.

PROGRAM

BOUND

-PROGRAM

DEVELOP

h



FIG. 5.1 COMPARISON BETWEEN NUMERICAL AND

EXACT PO TENTIAL FLOW SOLUTIONS



FIG. 5.2 COMPARISON BETWEEN NUM ERICA L AND

EXACT POTENTIAL FLOW SOLUTIONS





0 A 0 0 EXPERIMENTAL DATA(McGAHAN)

THEORETICAL CALCULATIONS

1.!5*

1.0.

Y-

lNCHES

.5

01

-SLOT START

.

---- START AT STATION 2

0

-STATION 2

FIG. 5.4 COMPARISON OF CALCULATED AN D MEASURED

VELOCITY PROFILE DEVELOPMENTS

DOWNSTREAM OF A BLOWING SLOT



Calcu la t ions were made f o r a se r i e s o f ang les -of - a tt ack f o r t he new

NASA GA(w)-1 a i r f o i l . Comparisons were made w it h t h e NASA measurements and

wi th c a lc u l a ti o n s made by Morgan (29) us ing t h e Lockheed program. Ther e s u l t s a r e shown on Figures 5 .5, 5 .6 and 5 .7 . Both methods g ive ex ce l l en t

agreement wi th exper iment i n comparison wi th measured l i f t and p i t ch ing moment

coef . f i c i en t s (F igure 5 .5 ) . A t t h e h igher angles-of -a t tack th e source method

:. a p p e a rs t o b e i n b e t t e r a gr ee me nt w i t h e xp er im en t. I n a l l c a se s however t h e

Lockheed program i s i n s l i g h t l y b e t t e r a gre em en t w i t h e x pe ri me nt s i n t h e

t r a i l i n g e d ge r e g i on i n t h e p r e d i c t i o n of p r e s su r e c o e f f i c i e n t s . I n t h e

p r e se n t pr og ra m (VIP) t h e p r e s su r e c o e f f i c i e n t s a r e c a l c u l a t e d o n t h e o r i g i n a l

a i r f o i l s u rf a ce , a nd i t i s b e l ie v e d t h a t i f t h e p r e s su r e s a r e d et er mi ne d

a t off-body po in ts def ined by t he d isp lacement th ickness th a t improved agreement

w i t h e x p e ri m e n t al p r e s su r e s w i l l r e s u l t . T hi s p ro ce du re w i l l b e t r i e d a t a

l a t e r da te . Measured and ca l cu la t ed d rag po la r s a r e shown i n F igure .5 .6 .

No a ll ow a nc e h a s be en made f o r t r i p d r a g o r s e p a r a t i o n e f f e c t s i n t h e c a lc u-l a t i o n s a l th o ug h t h e s e a r e p r e s en t i n t h e m easurements. Calculated and measured

p r e s s u r e d i s t r i b u t i o n s a r e compared i n F i gu re 5 . 7 . C u r r e nt l y n e i t h e r t he o-

r e t i c a l a p p r o a c h i s c a pa b le o f p r e d i c t i n g t h e e f f e c t s of s e p a r a t i o n p r e s e n t i n

the measurements. A f u r t h e r s e t o f c a l c u l a t i o n s we re made f o r t h e NACA 23012

a i r f o i l . C omparisons betw een t h e o r y and ex pe ri me nt f o r l i f t c o e f f i c i e n t v e r su s

ang le -o f - a t t ack (F igure 5 .8 ) and l i f t ve r sus d rag (F igure 5 .9 ) a r e i n good

agreement.

The mul t i-e lement pre dic t io n c ap ab i l i t y of th e program i s demonst ra ted i n

F igures 5 .10 through 5 .14. T he . f i r s t cas e cons ide red i s t h a t of t h e NACA

23012 a i r f o i l w i th a 25 pe rcen t chord s l o t t e d f l a p (Ref . 30 ) . A s shown i n

Figu re 5 .10 th e presen t method i s i n be t t e r ag reemen t wi th exper iment thani s t h e Lockheed program. I t i s b e l i ev e d t h a t t h e u s e of a f i n i t e d i ff e re n ce

boundary l a ye r method inc lud in g th e e f f ec t s of cu rva tu re and normal p re ssu re

g r a d i e n t s r e su l t s i n an improved p h y s i c a l r e p r e se n t a t i o n of t h e fl o w i n t h e

wing t r a i l i n g e dg e- fl ap u p p e r . s u r f a c e r e g i on . T h is i n t u r n r e s u l t s i n a n

improved p red ic t io n o f th e c i r cu la t i on abou t the complete conf igura t ion when

v i scou s e f f e c t s a r e inc luded . S imi la r conc lus ions can be drawn f rom the r e s u l t s

of..F i g u r e 5 .11 f o r t h e NACA 23012 con f igu rat i on having a leadin g edge s l o t

a nd a s l o t t e d f l a p (R ef. 3 1 ).

The NACA 64A010 a i r f o r 1 wi th l ead ing edge s l o t and a d ou bl e s l o t t e d f l a p

(Ref. 32) i s considered i n F igure 5 .12 , Also shown f o r compar ison ar e t h e

r e s u l t s f o r th e same con f igu rat i on f rom th e Lockheed program. The comparisoni s s i m i l a r t o t h a t o f F ig ur e 5 . 11 i n t h a t the g r e a t e s t d i f f e r e n c e b etw ee n

t h e t wo r e s u l t s i s i n t h e p r e di c t io n o f t h e p re s s ur e d i s t r i b u t i o n f o r t h e

main element. I t i s b e l i ev e d t h a t t h e d i f f e r e n c e i s a r e su l t o f t h e way i n

which t h e two programs t r e a t t h e mixing between t h e main element and th e

d ou bl e s l o t t e d f l a p s .



FIG. 5.5 LIFT AND MOMENT COEFFICIENTS FORNASA GA (W) -1 AIRFOIL

53



0 EXPERIMENT RE = 6.3 X 106

LOCKHEED---- VIP

FIG. 5.6 DRAG POLAR NASA GA (W ) -1 AIRFOIL



0 0 EXPERIMENT

LOCKHEED

. V I P----

FIG. 5.7 PRESSURE DISTRIBUTION NASA GA (W ) -1AIRFOIL (12.040 )



FIG. 5.8 LI.FT COEF FICIENTS FOR NACA 23012 AIRFOIL

56



0 NACA 23012

R E = 3.0 X lo6

M = .17

V IP

FIG. 5.9 DRAG POLAR FOR NACA 23012 AIRFOIL

57



0 NACA 23012 WlTH .25C SLOTTED FLAP

CY = 8' &F LA P = 20' Re = 2.2 X l o 6

CL = 2.01

LOCKHEED PROGRAM CL = 1.94

,,,, I P CL = 2.024

FIG. 5.10 COMPARISON OF MEASURED AND PREDICTED

PRESSURE DISTRIBUTIONS FOR NACA 23012

AIRFO IL WITH 25% SLOTTED FLAP



. 0 NACA 23012 WlTH L.E. 'SLAT

AND SLOTTED FLAP CL = 2.08

-5

CY = 8O 6 SLAT = o0

6 FLAP = 20° Re = 2.2 X lo6

-4

LOCKHEED PROGRAM CL = 1.76-- V I PCL = 2.11

-3

P

-2

-0

10 .1 .2 .2 .3 .4 .5 .6 .7 .8 .8 .9 1 O

FIG. 5.11 COMPARISON OF MEASURED A ND PREDICTED

PRESSURE DISTRIBUTIONS FOR NACA 23012

AlRFOl L WITH L.E. SLAT AN D 25%SLOTTED FLAP



CONFIGURATION

64A010 + SLAT + DOUBLE SLOTTED FLAP

SSLAT = -3.3'. SFLAP= 20°, a = 40

LOCKHEED PROGRAM CR = 1.34---- VIP CR = 2.033

. .

FIG. 5.12 COMPARISON OF PR EDICTE D PRESSUREDISTRIBUTIONS FOR NACA 64A010 AIR FO ILWITH L.E. SLAT AND DOUBLE SLOTTED FLAP



A s ec on d and c ~ n s i d e r a b l ymore cha l l en g ing case has been inv es t iga ted

u s i n g t h e same ba s i c fou r e l emen t geomet ry , In th i s example the s l o t and

d ou bl e s l o t t e d f l a p s a r e p os i ti o ne d i n a l a n d i n g c o n f i g u r a t i o n -- 26.1Q, & = 52.7". Two problems ha ve been e nco unt ere d a t

' slat

t h i s t ime 6AaPthey w i l l r e q u i r e f u r t h e r s t u dy . The f i r s t p ro bl em c e n t e r s on

t h e i t e r a t i v e p ro ce du re f o r s o l u t i o n o f t h e i n fl u e n c e c o e f f i c i e n t m a t ri x .

Each element i s h i t i a l l y an al yz ed I n i s o l a t i o n w ith r e s pe c t t o i t s neigh-mponents . Tn te r f e r ence e f f e c t s a r e then de te rmined and th e ana lys i s

con t inued un t i l convergence i s achieved. The problem a r i s e s when in te r fe ren ce

e f f e c t s are added t o t h e s l o t l o a di n g ( i n i t i a l l y s t r o n g l y n e g a t i v e l y lo ad ed

ecause o f the nega t ive ang le -o f - a t t ack ) , and change th e load ing t o s t rong ly

p os i t iv e l i f t . The r esu l t i s a d i v e r g e n t so l u t i o n . A nother p o s s i b l e d i f f i c u l t y

i th t h e conf igu ra t ion a na ly s i s may be wi th t he method o f so l u t i on . L i n e a r l y

vary ing v o r t i c i t y methods do no t have a s t ro ng d iagona l ly dependen t ma t r ix as

i n th e ca se o f cons tan t source pane l methods . I t e r a t i v e me thods o f so l u t i o n

r e l y t o a c o n si d er a bl e e x t e n t o n t h e s t r o n g d i ag o n al f o r t h e i r s u cc e s s. More

ork i s d e f i n i t e l y needed i n t h e a r e a o f f a s t r e l i a b l e s o l u t i o n t ec hn iq ue s.

When the p rob lem wi th th e i t e r a t i v e s o lu t ion w a s e n co u n te r ed , t h e d i r e c tt echn ique w a s u se d t o o b t a in t h e i n v i s c i d p r e ss u r e d i s t r i b u t i o n . A s a r e s u l t

of a v e r y hi g h s u c t i o n peak i n t h e t r a i l i n g e dg e r e gi o n o f t h e u p pe r s u r f a c e

of th e main el emen t (p robably due t o th e ve ry h igh camber e f f e c t r e s u l t i ng

from t h e h ig h ly d e f le c te d f l a p s) t h e s t a r t i n g v e l o c it y p r o f i l e t o t h e f i n i t e

di f f er en ce program had unacceptably h igh ve lo ci t i es . More work i s needed t o

r e so l v e t h i s p ro ble m.

The RAE 2815 co nfi gu ra t io n te s t e d by Fo st er (33) co ns is t i ng of a main

e lemen t and a s in g l e s lo t t ed f l ap has been cons idered i n F igures 5 .13 and 5 .14.

he comparisons i n F igure 5 .13 inc lud e measurements and ca lc ul a t io ns fo r two

c o n f i g u r a t i o n s . The spi ke i n press ure measured by Fo ste r i s n o t d u p l i c a t e d

y NASA Ames, n o r i s i t r ep ro du ce d i n t h e c a l c u l a t e d p r e ss u r e d i s t r i b u t i o n s .Th i s h a s a m arked e f f e c t on t h e v e l o c i t y p r o f i l e s shown i n F i g u r e 5.14 .

A lthough t h e i n i t i a l v e l o c i t y p r o f i l e s a r e i n r ea s o na b le ag re em en t, t h e

d i f f e r e n t p r e s su r e g r a d i e n t c o n d i t i o n s e x p er i en c e d b y t h e me asure d and

ca l cu l a t e d boundary l ay er deve lopment s r e su l t s i n qu i t e d i f f e r en t downs tr eam

p r o f i l e s . I t i s i n t e r e s t in g t o no te , however, t h a t t he ae rodynamic load

comparisons shown i n Table 2 a r e i n gen er al ly good agreement. Comparisons a r e

a ls o made f o r t h e RAE 2815 con f igu rat i on having a drooped leadi ng edge, and as l o t t e d f l a p d e f l e c t e d 30 de gr ee s. A l l comparisons were a t 9' angle-o f-at tac k

and a t a Reynolds number of 3.8 mi l li o n .



0 FOST ER 2.5% GAP, 3.1% O.L.

' 0 NASA AMES 1.5% GAP, 1.0% O.L.

VI P 2.5% GAP, 3.1% O.L.

---- VI P 1.5% GAP, 1.0% O.L.

- FIRST MEASURED1 VELOCITY PROFILE

FIG. 5.13 COMPARISON OF MEASURED AN D PREDICTEDPRESSURE DIST RIB UTIO N FOR FOSTER'SAIRFOIL FLAP COMBINATION



FOSTER (lo0 FLAP DEFLECTION)

2.5% GAP, 3.1% 0 . L .

a = 90

0 EXP'T XIC = .894

EXP'T XIC = 1.240 (FLAP T.E.)

VIP XIC = .883

- - - - VIP XIC = 1.240

U/U I N V

FIG. 5.14 COMPARISON OF MEASURED A ND PREDICTEDVELOCITY PROF1LES ON FLAP UPPER SURFACE



Table 2 - RAE 2815

Fla p Conf ig ur a i o nC~

CC~

C

n e a s . L~ l c. meas . D ~ a l c .

1 0 " f l a p

.025C Gap

1 0 " f l a p

.015C Gap

30" f l ap

.020C Gap

The i n f i n i t e swept wing c ap ab i l i t y o f th e program i s c o n si d e re d i n F i gu r es

5.15 through 5.19. Ca lc ul at ion s have been made fo r comparison wi th Cumpsty and

Heads measurements (Ref. 34) on a 61.1" swept wing. Unf ortu nat ely th e con-

f i g u r a t i o n t e s t e d h a d a l a rg e r eg ion o f separ a ted f low on bo th th e upper and

lower su r f aces . A s shown i n F igure 5 .15 th e measured and ca lc ul a t ed pr es su re

d i s t r i b u t i o n s a r e i n re as o na b le a gree me nt i n t h e forwa rd s e c t i o n o f t h e a i r f o i l

a l th o u gh t h e s e p a r a te d f l ow g r e a t l y m o d if i es t h e p r e s s u re s i n t h e t r a i l i n g ed ge

reg ion . The pre dic ted s t reamwise momentum th i ckn ess va r i a t io n i s i n good agree-

ment wi th exper iment away f rom th e se pa ra t io n regi on a s shown i n F igur e 5 .16 .

The comparison between pre di ct ed and measured val ue s of th e ang le B ' i s good

o n l y i n t h e r e g i o n f a r removed fro m se p a r a t i o n , w h i l e t h e p r e d i c t e d sh ap e

f a c t o r H i s i n poor agreement wi t h exper iment (F igure 5 .17) . The behavior

o f H i s a r e s u l t o f much l a r g e r c a l c u l a t e d p r e s s u r e g r a d i e n t s t h a n e x i s t i n

t h e . exper imen ta l case .

F i g u r es 5. 18 a nd 5. 19 r e p r e se n t t h e r e s u l t s o f c a l c u l a t i o n s f o r t h e RAE

2815 a i r f o i l f l a p c o n f i g u r a t i o n swe pt 25 d e g r e e s. The p r e s su r e d i s t r i b u t i o n

r e s u l t i n g f r o m 5 i t e r a t io ns o f p rogram V IP i s shown i n Fig ur e 5.18. Also

i n c l u de d a r e t h e c a l c u l a t e d a ero dy na mi c l i f t d r a g and moment c o e f f i c i e n t s f o r

ko th t he 25 degree and ze ro degree cases . I n th i s compar ison th e l i f t and moment

c o e f f i c i e n t s a r e re du ce d s l i g h t l y , w h il e t h e d ra g a s a r e s u l t o f t h e i n c re a s ed

s t r e a m w i se d i s t a n c e i s cons ide rab ly g re a t e r fo r th e swept wing. F la p t r a i l i n g

edge s t r eamwise and c ros s f low ve lo c i ty p r o f i l e s a r e shown i n F igure 5 .19 . For

t h i s p a r t i c u l a r c a s e t h e c ro ss -f lo w p r o f i l e s o n th e f l a p upper s u r f a c e a r e v e ry

sm a l l , a r e s u l t both of th e moderate sweep ang le , and the moderate loading on thef l a p . -



o CUMPSTY AND HEAD

(AIRFOIL UPPER SURFACE)

Re = 1.377 X lo6

VIP ( INVISCID CALCULATION)---- VIP ( ITERATION NO. 6)

FIG. 5.15 COMPARISON OF MEASURED AN D PREDICTED

PRESSURE DISTRIBUTIONS FOR AN INF INIT E

SWEPT WING .



FIG. 5.16 COMPARISON OF MEASURED AN D PREDICTED

STREAMWISE M OMEN TUM THICKNESS DEVELOPMENT



] CUMPSTY AND HEADA

VIP ( ITERATION NO. 6)

FIG. 5.17 COMPARISON OF M EASURED AN D PREDICTEDSHAPE FACTOR AN D ANGLE P DEVELOPMENTSFOR AN IN FIN ITE SWEPT WING



CALCULATIONS (VIP)

RAE 2815 (lo0 FZAP DlZFLEXTION)

2.5% GAP, 3.1% O.L.

CU = go, Cro = 2

CALCULATED AERO DYNAMIC COEFFICIENTS

SWEEP ANG LE 25O-

FIG. 5.18 PREDICT ED PRESSURE DISTRIBUTION S FORFOSTER'S A1 RFO I L-FLAP CON FIGU RATIO NSWEPT 25 DEGREES



FIG. 5-19 PRED ICTED STREAMWISE AN D CROSS-FLOW

VELOCITY PROF1LES AT FLAP TRAILING

EDGE FOR FOSTER'S AIRFO IL FLAP

CONFIGUR ATION SWEPT 25 DEGREES.



PROGRAM LIMITAT IONS

Although i t i s bel iev ed t h a t t h e ca l cu la t i on procedure and computer program

i s c a pa b le of a n al y zi n g a wide v a r i e t y of a i r f o i l c o n f i g u r a t i o n s , l i m i t a t i o n s

b o t h i n t h e o r e t i c a l methods and du e t o program s t r u c t u r e r e s t r i c t t h e r a ng e

o f a p p l i c a t i o n . These l i m i t a t i o n s in c l u d e r e s t r i c t i o n t o :

- In f in i t e swep t wings con s i s t ing o f a t mos t fou r e l ement s two of

w hi ch c an b e s l o t t e d f l a p s .

- I n co m p re s s ib l e fl ow ; a l t h ou g h t h e p r e s su r e d i s t r i b u t i o n s a r e

c or re ct ed f o r ~ a c humber e f f e c t s us ing Gother t s ru l e .

- Smal l regions of ' sep ara t ion . Although the so urce method lend s

i t s e l f r ea d i l y t o the deve lopment o f a separa ted f low model, t he

cu r r e n t model does no t have t h i s c ap ab i l i t y ( as apparen t from

t h e r e s u l t s o f F i gu r e 5 .1 5) . I f s e p a r a t i o n i s p r e d ic t e d t h e e x i s t -

in g approach i s t o s im pl y e x t r a p o l a t e t h e s o ur ce s t r e n g t h t o t h e

t r a i l i n g edge of t h e a i r f o i l .

O s c i l l a t i o n s i n l i f t o cc ur re d i n e a r l y c a l c u l a t io n s h a vi ng p r e d ic t ed

r e g i o n s of s e p a r a t i o n . I t was f in a l l y de termined tha t t h i s was due to un- .

a c c ep t ab l y l a r g e s ou rc e s t r e n g t h s a t t h e t r a i l i n g edge of t h e a i r f o i l .

A numer ical exper iment demonst ra ted t h a t monotomical ly convergent so lu t i on s

c an b e o b t a i n e d i f t h e maximum v a l u e o f t h e so u r c e s t r e n g t h a t t h e t r a i l i n g

e dg e o f s i n g l e el em en t a i r f o i l s b e l i m i t e d u s in g t h e c r i t e r i o n

'%ax= 0.115 - 0 O l (ITR-1)

where 1- ITR ITRMAX

For con£ gu rat io ns having s lo t t e d f l ap s t he maximum va lue of q . c u r r e n t l y

a l lowed . in the p rogram i s 0.15. In a l l c a se s a n al yz ed t o d a t e t h e b eh a vi o r

of the l i f t coe f f i c i e n t has been a monoton ic dec rease wi th inc reas ing number

o f i t e r a t i o n s .



CONCLUSIONS AND RECOMMENDATIONS

The v i s c o u s / p o t e n t i a l f lo w i n t e r a c t i o n program d e s cr i b ed - i n t h i s r e p o r t

e mpl oy s p o t e n t i a l f l o w and b ou nd ar y l a y e r p r o c e d ur e s w h ic h a r e c u r r e n t l y

un ique t o t h i s me thod and i t i s b e l ie v e d t h a t b ec au se of t h i s t h e a n a l y s i s

r ep res en t s a cons ide r ab le advance on o t he r me thods o f i t s ty pe . The program

i n i t s pres en t fo rm i s a p p l i c a b l e t o a wi de v a r i e t y o f p ro bl em s, i n p a r t i c u l a r ,

t o t h e i n f i n i t e s w ep t wing c a s e . There i s however c ons id e rab le s cope fo r

e x t e n s i o n t o a g e n e r a l t h re e -d i me n si o na l w ing c a l c u l a t i o n pr o c ed u r e.

Sp ec if ic conclu s ions r ega rd i ng each of t he major components of th e program

a r e g i v e n i n t h e f o l l o w i n g p ar a g ra p h s.

' - The v o r t e x l a t t i c e p o t e n t i a l f l ow m ethod d e ve lo pe d f o r t h i s p ro gra m

i s an ac cu ra te num er ic a l approach , adapted fo r two-dimens ions, f rom a

g e n e r a l t h r e e d i m e n s i o n a l l i f t i n g p o t e n t i a l f l o w m ethod d e ve lo pe d by o n e o f

t h e a u t h o r s . The e x t e n s i o n t o t h e t h r e e di m e n si o n al c a s e i s t h e re f o re r e l a t i v e l y

s t r a i g h t f o r w a r d .

- The i n t e g r a l bo un da ry l a y e r method us ed f o r s i n g l e el em e nt a i r f o i l s

and c u r r e n t l y o n a l l b u t t h e f l a p u pp er s u r f a c e s i n t h e m ul ti -e le me nt mode,

i s q u i t e a c c u r a t e , a s w it n es se d by t h e good. d r a g p r e d i c t i o n c a p a b i l i t y . A t

th e same t ime t he method us es on ly a f r ac t i on o f a second of computer tim e

per boundary la ye r development .

The i n c l u s i o n . o f c u r v a t u r e a nd no rm al p r e s s u r e g r a d i e n t e f f e c t s i n t h e

f i n i t e d i f f e r e n c e b ou nd ar y l a y e r m ethod e n a b l e s c o m p li c a te d bo un da ry l a y e r

f l ow s t o b e r e p r e s e n t e d m ore a c c u r a t e l y t h a n i s p o s s i b l e ' b y o t h e r p ro c ed u re s

now av ai la bl e. I t i s b e l i e v e d t h a t t h e i mp rov ed p h y s i c a l r e p r e s e n t a t i o n o f

t h e f lo w o v er s l o t t e d f l a p s i s r e s p o n s i b l e f o r t h e g r e a t e r d e g r e e o f a gr ee me nt

with exper iment than i s achieved by o ther methods .

-. The us e o f s o u r c e s t o r e p r e s e n t t h e d i s pl a c em e n t e f f e c t s o f t h e bo un da ry

l a y e r o n t h e p o t e n t i a l f l ow , w h i l e n o t n e c e s s a r i l y more a c c u r a t e a procedure

t h a n t h e d i r e c t em ploym ent of t h e d i s pl a c em e n t t h i c k n e s s , h a s two d i s t i n c t

advan tages . The f i r s t a d va n ta g e i s i n t h e c o m p u t a ti o n al s u p e r i o r i t y of s uc h a

procedure . The i n f l u e n c e c o e f f i c i e n t m a t r i x ne e d by i n v e r t e d o n l y o nc e, w i t h

s uc ce ed in g' i t e r a t i o n s r e q u i r i n g o n ly m a t r i x m u l t i p l i c a t i o n . I f o n e i s e v e r

t o c o n t em p la t e a v i s c o u s / p o t e n t i a l f lo w i n t e r a c t i o n p rogram a p p l i e d t o g e n e r a l

t h r e e d im e n s io n a l a i r p l a n e c o n f i g u r a t i o n s s u ch a p ro c e du r e w i l l b e a l m o s t

mandatory. The second advan tage i s r e l a t e d t o t h e m o de l li n g o f s e p a r a t e d f lo w

r e g io n s i n a p o t e n t i a l f lo w a n a l y s i s by d i s t r i b u t e d s o u rc e s . I t h a s b e en

d em on st ra te d i n t h e l i t e r a t u r e t h a t s uc h an a pp ro ac h i s p o s s i b l e , a n d i t i s

recommended t h a t one o f th e f i r s t ex tens ions o f th e p re s en t p rogram s hou ld

b e t o i n c l u d e s e p a r a t i o n e f f e c t s . The program would th en be c apa ble of

p r e d i c t i n g C R a s a fu nc ti on of Reynolds number.

max



Another u se fu l exten s ion t o t he program would be th e inc lu s io n of

com pre s s ib i l i t y e f f e c t s i n the boundary l ay er deve lopment. Even a t low f r e e

stre am Mach numbers th e hi gh su ct io n peaks experienced by a s l o t o r m ain e lement

o f a h i g h l i f t s ys tem can l e ad t o co m p r es s i b i l i ty p ro bl em s.

With some .mo dif i cat ion th e f i n i t e d i f fer en ce boundary la ye r program

module can be used to p red ic t the e f f ec t o f t an ge n t i a l blowing o r boundary

l a y e r s u c t i o n i n t h e o v e r a l l c o n t ex t o f a v i s c o u s / p o t e n ti a l f lo w i n t e r a c t i o n

method f o r h ig h l i f t s ys tems .



REFERENCES

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Two-Dimensional Multi-Component Airfoils in Viscous Flow", NASA CR-1843,

July 1971,

2. Preston, J. H., "The Effect of the Boundary Layer and Wake on the Flow Past

a Symmetrical Airfoil at Zero Incidence", A. R. C. R. and M 2107, 1947.

3. Callaghan, J. G., and Beatty, T. D., "A Theoretical Method for the Analysis

and Design of Multielement Airfoils:, J. Aircraft, Vol. 9, No. 12, December

1972.

4. Cumpsty, N. A., and Head, M. R., "The Calculation of Three-Dimensional

Turbulent Boundary Layers, Part 11: Attachment - Line Flow on an Infinite

Swept Wing", Aero Quart ., Vol. XVIII, May 1967.

s

5. Bradshaw, P., "Calculation of Three-Dimensional Turbulent Boundary ~ayers",J. Fluid Mech., Vol. 46, 1971.

6. Cumpsty, N. A. and Head, M. R., "The Calculation of the Three-Dimensional

Turbulent Boundary Layer, Part 111: Comparison of Attachment - Line Cal-

culations with Experiment", Aero Quart . , Vol. XX, May 1969.

7. Curle, H., "A Two Parameter Method for Calculating the Two-Dimensional

Incompressible Laminar Boundary Layer", J. R. Aero Soc. , Vol. 71, 1967.

I8. Thwaites, B., Approximate,Calculation f the Laminar Boundary Layer",

Aero Quart., Vol. I, 1949.*

9. Cumpsty, N. A. and Head, M. R., "The Calculation of Three-Dimensional

Turbulent Boundary Layers, Part I: Flow Over the Rear of an Infinite

Swept Wing", Aero Quar t . , Vol. XVIII, February 1967.

10. Nash, J. F., "The Calculation of Three-Dimensional Turbulent Boundary

Layers in Incompressible Flow", J. Fluid Mechanics, 37, 1969.

11. Head, M. R., "Entrainment in the Turbulent Boundary ~ayer", &M 3152,

Aero Research Council, Great Britain, 1958.

12. Thompson, B. G. J., "A New Two Parameter Family of Mean Velocity Profiles

for Incompressible Turbulent Boundary Layers on Smooth Walls", RCM3463, Aero Research Council Great Britain, 1965.

13. Coles, D. E., "The Law of the Wake in the Turbulent Boundary Layer",

J. Fluid Mech., Vol. 1, 1956.

14. Mager, H., "Generalization of Boundary Layer Momentum Integral Equations to

Three-Dimensional Flows Including Those of Rotating ~ ~ s t e m s " ,ACA TR 1067,

1952.



15. Gr an vi l l e , P . S . , "The Cal cu la t ion of th e Viscous Drag of Bodies of Rev-

o l u t i o n " , D a v i d W. Ta yl or Model Basin Report 849, 1953.

16 . S c h l i c h t i n g , J . and Ul r i ch , A. , "Zur Berechnung Des Un se ll ag es Laminar-

Turbu l en t en" (On t h e C a l cu l a t i on o f Lam inar -Turbu len t T r an s i t i on ) , J ah rbuch

1942 Der Deutschen Luf t fahr t -Fo rschung .

17 . S m i th , A. M. O . , " T r a n s i t i o n , P r e s s u r e G r a d i e n t a nd S t a b i l i t y T he or y" ,P roc . 9 t h In t e rn a t . C ongress o f App l. Mech., B ru s se l s , Vo l. 7 , 1957.

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Supersonic Flow", J e t Pr op ul si on Lab Rep ort No. 20-69, 1953.

1 9 . B r i l e y , W. R., "An An aly sis of Laminar Se pa ra t i on - Bubble Flow using the

Navier-S tokes Equat ions" , P roceedings - Flui d: Dynamics of Unstead y, Three-

Dimens ional and Separa te d F lows , Georg ia Tech . , June 1971.

20. G a s t e r , M . , "The S t ru ct ur e and Behavior of Laminar Sep ar a t ion Bubbles" ,

ARC 28-226, 196 7.

21. Nash, J . F. and P a t e l , V. C . , "A G e n e r a l i z e d Method f o r t h e C a l c u l a t i o n o f

Three-Dimensional Tur bule n t Boundary Layers1 ' , P roc eed ings - Workshop on

Fl ui d Dynamics of Unsteady, Three-Dimensional and Se pa ra te d Flows, G eorgia

T e c h . , A t l a n t a , J u n e 1 97 1.

22. Dvorak, F . A., "Ca lcu l a t io n of Turbu len t Boundary Layer s and Wall J e t s

ov er Curved Surfac es" , AIAA Jo ur na l , Vol . 11, No. 4, A p r i l 1973.

23. Crank, J . and Nicholson , P . , "A P r a c t i c a l Method f o r Numer ica l Eva lua t i on o f

S o l u t i o n s o f P a r t i a l D i f f e r e n t i a l E q u a ti o n s o f t h e H ea t Co nd uc ti on Ty pe ",

Pro c . Cambridge Ph i l . Soc . , 43 , 1947.

24. Dvorak , F. A . , and Head, M. R ., " He at T r a n s f e r i n t h e C o ns t a n t P r o p e r t y

Tur bule n t Boundary Layer", I n t . J . Heat Mass Tr an sf er , Vol . 10 , 1967 .

25 . S qu i re , H. B. and Young, A. D . , he C a l c u l a t i o n of t h e P r o f i l e Drag of

~ i r f o i l s " , B r i t . Aero Res. Coun. R and M 1838, 1937.

26. McLellan, C . H . , and C ange los i , J . I . , " E f f e c t s o f N a c e l l e P o s i t i o n o n

Wing Na ce l l e In te rf er en ce ", NACA TN 1593, June 1948.

27 . H e s s , J . L. , "C a l cu l a t i on o f P o t e n t i a l Flow About Ar b i t r a r y Three -

Dim ens iona l L i f t i n g Bodies", Repo rt No. MDC J 5 6 7 9 - 0 1 , D o u g l a s A i r c r a f t

Company, Oct. 1972.

28. W i l l i a m s , B. L . , "An Exact T es t Case fo r t h e P lan e Po t e n t i a l F low About

Two A d j a c en t L i f t i n g A i r f o i l s " , RAE Tech. Report 71197, S ept . 1971.

29. Morgan, H . , NASA Langley Res ea rc h Cen te r Pr i v a t e Communication.

30. Wenzinger, C . J . and Delano, J . B . , " P r e s s u r e D i s t r i b u t i o n O ver a n NACA

23012 A i r f o i l w i t h S l o t t e d a nd P l a i n F la p , NACA TR 633.



31. Harr is , T. A . , and Lowry, J . G . , "Pre ssur e D i s t r ib u t i o n Over an NACA 23012

A i r f o i l w i th a F i x ed S l o t a nd a S l o t t e d Fl a p" , NACA TR 732, 194 2.

32 . Kel ly , J . A . and Hayter , N . B ., "L if t and Pi t ch in g Moment a t Low Speeds of

t h e NACA 64A010 Ai r fo i l Se ct io n Equipped wi t h Var ious Combinations of a L e d i n g

Edge S l o t , Leading-Edge F lap , S p l i t F lap and Double-S lo t ted la^", NACA TN

3007, 1953.

33 . F os t e r , D . N . , I r w i n , H. P. , and Wi l l i ams , B. R . , "The Two Di me ns io na l Flow

Around a S l o t t e d Flap ", RAE Tech. R epo rt 70164, Sep t. 1970.

34. Cumpsty , N . A. and Head, M . R., "The Ca lc ul at io n of Three-Dimensional Turbu-

l e n t Bou nd ar y L a y e r s , P a r t I V : Compari son of Measurements wi th C al cu la t i on s

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35. La bru jer e , Th . E . , Loeve, W. and S loo f , J . W . , "An Approxima te Method f o r t h e

C a l cu l a t i o n of t h e P res su r e Di s t r i b u t i o n on Wing-Body C ombina ti ons a t Sub-

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36 . Dvorak, F. A. and Woodward, F. A., "A ~ i s c o u s / P o t e n t i a l low I n t e ra c t i onAn aly s i s Method f o r Mul t i -Element I n f i n i t e Swept Wings" , Volume I.1,

NASA CR 137550, Apri l 1974..



APPENDIX I

POTENTIAL FLOW THEORY

The po te n t i a l f l ow theory i s used to de r ive t he i n f luen ce of cons t an t

a nd l i ne a r d i s t r i bu t i o ns o f s ou r c es and vo r t i c i t y on p l a na r two d im ensi onal

s u r f a c e s . C ons ider a n el e me n ta r y l i n e s ou r c e o r l i n e vo r t e x l oc a t e d a t a po i n t

5 on the x a x i s a nd pe r pend i c u la r t o t he x , z plane . In i ncompress ib l e f l ow,

th e magnitude of t h e v e l o c i t y i n du ce d by e i t h e r s i n g u l a r i t y a t a n a r b i t r a r y

p o i n t P(%,z) i s given by:

where

The geometry i s i l l u s t r a t e d by t h e f o ll o wi n g s k e tc h :

For an e lementary source , t he ve lo c i t y V i s i n t h e d i r e c t i o n o f t h e l i n es

j o i n i n g 6 and P , whi l e f o r an e l ementa ry vor t ex , t he ve lo c i t y V i spe r pe nd i c u l a r t o t h i s l i n e . The ho r i z o n t a l and v e r t i c a l components xf v e l o c i t y

c or re sp o nd in g t o t h e l i n e s o u r c e o r v o r t e x a r e g i ve n by :



u = - w = vcose = x-S

s v 2nd 2

The c o n t r i b u t i o n of a c o n s t a n t d i s t r i b u t i o n of s o u r c e s o r v o r t i c e s a l o ng t h e

x a x i s i s ob t a i ne d by i n t e g r a t i ng e qua t i ons ( 2 ) a nd . ( 3 ) from 0 t o c .

1 [ - 1 2= -2n t a n -

x Ctan-' 2 ]

1= - - l o g

The e f f ec t s o f com pr ess ib i l i t y may be ob t a ined by apply ing Got he r t ' s Rule ,

w i t h B =

and

w = u = --1 Bz

s v ; [ t a n - tan-' 5

F or a s o u r c e o r v o r t e x d i s t r i b u t i o n v a ry i ng l i n e a r l y w i t h x, w i t h z er o s t r e n g t h

a t t h e o r i g i n and u n i t s t r e n g t h a t x = c ,



1 z J ( x K 2 - - [ t a n-1;- tan-' 1 (91m- X-C

-1 z Xl + l [ t a n I - - t a n ] + - o g

4x-Cl2 + zC

C o m p res s ib i l it y e f f e c t s a r e o b t a in ed a s b e f o r e , i . e . , by m u l ti p l y in g z and

wv by B , and di vi di ng us by B.

A s o ur c e o r v o r t e x d i s t r i b u t i o n ha vin g u n i t s t r e n g t h a t t h e o r i g i n , and

z er o s t r e n g t h a t x = c , can be ob ta ined by sub t r a c t in g the prev ious ly der ived

l i n e a r l y v a ry i n g d i s t r i b u t i o n s from t h e c o ns t a nt d i s t r i b u t i o n s .



A P P E N D I X TI

SOLUTION OF BOUNDARY CONDITION EQUATIONS

For s ing le o r mul ti - el emen t a i r f o i l s , t h e boundary cond i t ion equa t ions ' can

b e so l ve d b y d i r e c t i n v e r s i o n . F or m ul ti -e le me nt a i r f o i l s , u se ca n a l s o b e

made of a r ap id ly convergen t i t e r a t io n scheme r epor t e d i n Refe rence (35) . I n

t h i s me thod th e mat r ix i s s u bd iv id ed i n t o s m a l le r p a r t i t i o n s , o r b lo c k s, w i th

each b lock repres ent ing th e inf lu enc e of one e lement of t he mult i -e lement a i r -

f o i l . The d iagona l b lock s r ep res en t th e in f luence o f th e e l emen ts on themse lves ,

t h e o ff - d ia g o n a l b l o c k s r e p r e se n t t h e i n t e r f e r e n c e o f o ne e le me nt on t h e o t h e r s .

The ord er of any blo ck i s r e s t r i c t e d t o 6 0 , t h e maximum number of pa ne ls on t h e

upper and lower s ur fac e of th e e lement .

The i n i t i a l i t e r a t i o n c a l c u l a t e s t h e s ou rc e and vo r t e x s t re n g t h s

c o rr e sp o nd i ng t o e a ch b l o c k i n i so l a t i o n . For t h i s s t e p , o n ly t h e d ia g o na l

b locks a r e p rese n t i n th e aerodynamic mat r ix . Once the i n i t i a l approx imation

t o t h e s o ur c e an d v o r t e x s t r e n g t h s i s d et er mi ne d , t h e i n t e r f e r e n c e e f f e c t of e a c h

b lo c k on a l l t h e o t h e r s i s c a l c u l a t e d b y m a t r ix m u l t i p l i c a t i o n . The incremental

n or ma l v e l o c i t i e s o b t a in e d a r e su b t r a c t e d f ro m t h e n or ma l v e l o c i t i e s sp e c i f i e d

by th e boundary cond i t ions . Th i s p rocess i s r e p e a te d 1 5 t o 2 0 t im e s, o r u n t i l

t h e r e s i d u a l i n t e rf e r e nc e v e l o c i t i e s are smal l enough t o ensure th at convergence

has occur r ed .

The procedure i s i l l u s t r a t e d be lo w f o r a n a ero dy na mi c m a t r i x c o n s i s t i n g

of n ine b locks . The unknown s i ng u l a r i t y s t r e ng th s a r e des igna ted y t h e

s p e c i f i e d n o r m a l v e l o c i t i e s Cj

i 'To s o l v e

where



P u t

Therefore [ D + E ] Eyl = {C )

-1o r t y ) = - D [ C - E E y l l

F i r s t ap pr ox im a ti on :

{v)-' = D - l {c )

C al cu l a t e AC' = E { Y )1

Second approximation:

Si mi la r ly , kth approximation: '

{ Y } ~ = D - ~ C - A C k-1)

Note that ,



APPENDIX 111

PROGRAM MACRO FLOW CHARTS

The genera l program ov er la y s t ru ct ur e of each .over lay i s desc r ibed

b r i e f l y i n t h e f o l l o w in g p ar a gr a ph s a nd f l o w c h a r t s . *

OVERLAY (0 ,O) Program VIP

Program VIP co nt ro ls t h e e n t i r e computer program (F igu re Al) . A l l

pr imary over l ays a r e ca l l e d from VIP. Over l ay (0,O) a l s o con t a ins o the r

sub rout ines which ar e commonly used i n t he o th er ove r lay s .

OVERLAY (1,O) Program POTFLOW

Program POTFLOW c ~ n t r o l s he l o f t in g of th e con f igu rat i on , th e ca lcu la t ion

o f f l a p s u r f a c e c u r v at u r e, t h e c a l c u l a t i o n of t h e p o t e n t i a l f l o w p r e s s ur e s , a s

w e l l a s t h e c a l c u l a t i o n o f t h e l i f t a nd moment c o e f f i c i e n t s .

OVERLAY (2,O) Program IBL

Program IBL con tr o l s th e i n te g r a l boundary la ye r ana l .ys is f rom the

c a l c u l a t i o n of i n i t i a l co n d i ti o n s a lo ng a s t a g na t i o n l i n e t o t h e t u r b ul e n t

boundary la ye r an al ys is . The program lo gi c f low i s shown i n F igu re A3.

OVERLAY (3,O) Program INSPAN

Program INSPAN co nt ro ls th e i n f i n i t e swept wing f i n i t e d i f fe re nc e boundary

la ye r ana ly si s. The ov er la y c a l l s two secondary ov er la ys , OVERLAY ( 3 , 1 ) Program

BOUNDRY (F igure A4 ) and OVERLAY ( 3 , 2 ) , Program DEVELOP (Figure A5). Program

BOUNDRY i n i t i a l i z e s the g r i d network normal t o the f l a p su r f a ce upon which t he

f i n i t e d i f f e r e n c e m ethod i s a p p l i e d . Normal chord and spanwise ve lo ci ty p ro f i l e s

a r e i n i t i a l i z e d i n p r ep a ra t io n f o r t h e a n al y si s .

Program DEVELOP con tr o l s th e a c tu a l c a l cu la t i on procedure used i n determinin g

t h e downstream development of t h e boundary la y e r.

OVERLAY (4,O) Program FELDPT

Program FELDPT cal cu la te s th e of f-body pres sure d i s t r ib u t i o n s P(x ,y) f o r

i n p u t t o INSPAN.

* Program inpur /outp t i t d i sc r i p t io n and a complete p.rogram l i s t i n g i s g i ve n i n '

Reference 36.



INPUT

'4POTFLOW GEOMETRY LOFTING

CALCULATIONCURVATURE CALCULATIONS

MATRIX

INVERSION

SOURCE

CALCULATES

CALCULATED

DEVELOPMENTS

LIFT, DRAGFELDPT IBL

A N D CALCULATES

MOMENTOFF-BODYPRESSURES

SUMMARY OVER FLAPUPPER SURFACE LAYER ANALYS IS

INSPANFINITE

DIFFERENCEBOUNDARY

LAYER ANALYS IS

F IG . A 1 O V E R L A Y ( 0 , O ) P R O G R A M V I P



SOLVE

CALCULATES PRESSURE

COEFFICIENTS AND LIFT

AND MOMENT COEFFICIENTS

I GEOMETRY INPUT

-

LIFT FILLSORTS PRESSURE

CL & CM' DISTRIBUTION FOR RETURN

BOUNDARY LAYER

CALCULATION.

ROTATE

LOFTS GEOMETRY AND

CALCULATES FLAP SURFACE

CURVATUREA

FIG. A - 2 OVERLAY (1 ,O) PROGRAM POTFLOW

CALCULATION OF

AERODYNAMIC INFLUENCE

COEFFICIENTS

4



BOUND

DIRECTS INPUT

F IG . A 3 O V E R L A Y ( 2 . 0 ) P R O G R A M I B L

N O YES

I N TSTAGNATION'.

L IN€CALCULATION Hie;-

LAMINAR

CALCULATESLAMINAR N O

BOUNDARY 4LAYER

DEVELOPMENT

YES

TRCALC

TRANSITIONCALCULATION

TRANS IT INSTAB

f, RANS ITIO N f, NSTABILITYSEARCH POINT

i

TU RB

TURBULENT 4 L

BOUNDARYLAYER

CALCULATION

DRAGCALCULATES

SKINFRICTIONPRESSURE

AND PROF,ILEDRAG

PRINTER

PRINT OUT

OF BOUNDARY

LAYERPARAMETERS

r

RETURN



CALCULATES

I NI T I AL NO RM AL

CHORD AND

SPANWISE VELOCITY

PROFI LES-FR OM

INITIAL STREAMWISE

PROF1LE

H j e i C f .I

A ND U i

FROM VIP

RETURNiELIN

INITIA L SLOT

PROFILE COUPLED

TO WING OR FLAP

UPPER SURFACE

T.E. PROFILE

FIG. A - 4 OVERLAY (3.1) PROGRAM BOUNDARY

4READ LAST

PROFILE OF

1ST FLAP -PCA LC

CALCU LATES

SURFACE PRESSURE

G RADIENT AN D

VELCAL

CALCULATE

VELOCITY

PROFILE ON

WlNG T.E. AND

FLAP UPPER

SURFACE IF B.L.

IS TURBULE NT

,



INPUT DATA AND

'PARAMETER

Vl NPUT

INITIAL NORMAL CHORD

AND SPAN WISE

VELOCITY PROFILES

( c t =Om KIN FRICTIONCALCULATION

I E X I T IT H I C K

LOCATES EDGE OF

BOUNDARY LAYER

D E R l VCALCULATES

VELOCITY GRADIENTS

SHAPE

CALCULATES H, 8,6

FOR REGION

BELOW UMAX

P F l E L DCALCULATES

OPTION

ADJUSTS Ax

ON BASIS OF

Documents

Viscous-Potential Flow Interaction Analysis Method for Multi Element Infinite Swept Wings