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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
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'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
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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
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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
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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
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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.
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- 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 .
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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
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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
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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 .
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FIG. 1.2 STATIC PRESSURE VAR IATIO N NOR MAL TO FLAP SURFACE
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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
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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
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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
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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
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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 .
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FIG. 2.1 MULTI-ELEMENT AIR FOIL LOFTING PROCEDURE
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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
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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
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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 .
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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.
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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 .
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- 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
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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 ,
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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 .
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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 .
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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 .
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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
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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.
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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
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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) .
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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
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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 .
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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
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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 .
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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
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. 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 .
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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 .
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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)
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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:
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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
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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 .
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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 .
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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 .
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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
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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
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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 .
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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
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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 .
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( 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 .
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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
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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
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FIG. 5.1 COMPARISON BETWEEN NUMERICAL AND
EXACT PO TENTIAL FLOW SOLUTIONS
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FIG. 5.2 COMPARISON BETWEEN NUM ERICA L AND
EXACT POTENTIAL FLOW SOLUTIONS
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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
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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 .
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FIG. 5.5 LIFT AND MOMENT COEFFICIENTS FORNASA GA (W) -1 AIRFOIL
53
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0 EXPERIMENT RE = 6.3 X 106
LOCKHEED---- VIP
FIG. 5.6 DRAG POLAR NASA GA (W ) -1 AIRFOIL
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0 0 EXPERIMENT
LOCKHEED
. V I P----
FIG. 5.7 PRESSURE DISTRIBUTION NASA GA (W ) -1AIRFOIL (12.040 )
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FIG. 5.8 LI.FT COEF FICIENTS FOR NACA 23012 AIRFOIL
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0 NACA 23012
R E = 3.0 X lo6
M = .17
V IP
FIG. 5.9 DRAG POLAR FOR NACA 23012 AIRFOIL
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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
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. 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
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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
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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 .
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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
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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
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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 . -
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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 .
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FIG. 5.16 COMPARISON OF MEASURED AN D PREDICTED
STREAMWISE M OMEN TUM THICKNESS DEVELOPMENT
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] 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
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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
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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.
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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 .
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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
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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 .
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REFERENCES
1. Stevens, W. A., Goradia, S. H., and Braden, J. A., athe he ma tical Model for
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.
7/28/2019 Viscous-Potential Flow Interaction Analysis Method for Multi Element Infinite Swept Wings
http://slidepdf.com/reader/full/viscous-potential-flow-interaction-analysis-method-for-multi-element-infinite 79/91
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.
1 8 . C o l e s , D. E . , Measurements i n t h e Boundary Layer on a Smooth F l a t P l a t e i n
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.
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http://slidepdf.com/reader/full/viscous-potential-flow-interaction-analysis-method-for-multi-element-infinite 80/91
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
on t h e Rear of a Swept Wing", B r i t . Aer o Res. Coun. C. P. No. 107 7, 1970 .
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-
c r i t i c a l Speeds", AGARD C . P. No. 7 1, 1970 .
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..
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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 :
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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 ,
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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 .
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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
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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 ,
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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.
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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
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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
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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
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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
,
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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
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