I

AIAA 89-0266 Evaluation of an Analysis Methodfor Low-Speed Airfoils by Comparison with Wind Tunnel Results

R. Evangelista and C. S. Vemuru, Analytical Services & Materials Inc., Hampton, Virginia.

27th Aerospace Sciences Meeting January 9-1 2,1989/Reno1 Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics (1 370 L'Enfant Promenade, S. W.. Washington. D.C. 20024

Evaluation of an Analysis "cethod for Low-Speed Airfoils by Comparison with Wind Tunnel Results

Raquel ~vangelista' and Chandra S. Vemuru Analytical Services and Materials, Inc.

Hampton, VA 23666

ABSTRACT

The airfoil analysis method ISES developed by Drela and Giles at MIT has been evaluated for low Reynolds number applications through a comparison of its predictions to wind tunnel test results for several low-speed airfoils. Airfoils analyzed were the Eppler 387, NASA NLF(1)-1015 and NASA NLF(1)-0416. The ISES analysis method was verified as a reliable method in predicting the aerodynamic characteristics of low-speed airfoils in the chord Reynolds number range from 200,000 to 3 million. It was found that a high value for the transition criterion (the critical disturbance amplification ratio), ncr, from 13 to 15, was necessary to analyze a case with laminar separation problems accurately. Otherwise, cases without laminar separation bubbles were analyzed more accurately using low values for ncr, from 7 to 11. As indi- cated by Drela and Giles, the method is limited to handle cases with thin separated regions. It was found that the method was not reliable for cases with large separated regions such as at Cll, ma* conditions, or at Reynolds numbers below 200,000.

LIST OF SYMBOLS

c airfoil chord

Cd airfoil section drag coefficient

Cx airfoil section lift coefficient

Cm airfoil section pitching moment coefficient

Cp pressure coefficient

ncr critical disturbance amplification ratio R Reynolds number based on chord

x airfoil abscissa

a angle-of-attack relative to chord line

(degrees)

INTRODUCTION

The design of airfoils at low Reynolds numbers requires an analysis method that can predict lami- nar separation, and more importantly, its effects on the aerodynamic characteristics of an airfoil. At low Reynolds numbers, if laminar separation occurs, the flow may reattach if transition occurs downstream of the separation point thereby forming a laminar (transitional) separation bubble. The greatest diffi~ulty~in low Reynolds number airfoil analysis is how to incorporate the effect of a lam- inar separation bubble in the calculation of air- foil drag.

Drela and Giles at MIT have recently developed an airfoil analysis method incorporated in an air- foil design/analysis program named ISES [ I 1 , [21 , which has the potential for handling the low Reynolds number problems of laoinar separation. The significance of ISES as a design tool lies in its potential to calculate separation bubble ef- fects on drag. One low-speed airfoil design and analysis code which was developed earlier by Eppler and Somers [ 3 1 can only predict the severity of laminar separation, but it cannot predict the drag increment due to the flow separation.

In the interest of verifying the application of ISES to low-speed, low Reynolds number airfoils, the ISES analysis has been applied to three air- foils, all of which have been tested in the Low Turbulence Pressure Tunnel (LTPT) at NASA Langley Research Center. The results obtained from the ISES analyses have theh been compared to experimen- tal data.

BRIEF PROGRAM DESCRIPTION

The airfoil design/analysis program ISES is described in detail in [I] and [2]. It solves for the flow field around an airfoil by a simultaneous solution of the Piscretized Euler equations which describe the inviscid flow, and the discretized two-parameter integral boundary-layer equations which describe the viscous effects on the flow. Because the flow field is solved as a fully coupled system, this method is expected to handle flows with strong viscous-inviscid interactions such as low Reynolds number and transonic flows.

A streamline grid is used to discretize the Euler equations which are derived from the inviscid Euler formulation of the steady-state mass, momen- tum, and energy conservation laws in integral form. The boundary-layer problem is also expressed in integral form, as this requires less computation than a finite difference approach. The integral boundary-layer equations are then discretized using two-point central differencing and a form of back- ward Euler differencing.

Transition is predicted by a spatial amplifi- cation theory based on the Orr-Sommerfeld equation of linear stability theory. The critical distur- bance amplification ratio for transition. ncr, is specified by the user. The user also has the option to specify transition locations on the airfoil.

A criterion based on the local kinematic shape parameter determines separation. Laminar separated regions are computed using an inverse method until the shape parameter no longer satisfies the separa- tion criterion; at this point, direct boundary- layer calculations proceed downstream.

+ Research Engineer, Member AIAA This paper is declared a work of the U.S. Government and is not subject to copyr~ght protection in the Un~ted Stares.

AIRFOILS AND WIND TUNNEL DESCRIPTION

The a i r f o i l s , Eppler 3 8 7 , NASA NLF(1)-1015, and NASA NLF(1)-0416, were chosen f o r t h i s s tudy because they have a l l been t e s t e d ex tens ive ly a t the LTPT [ 4 ] , [ 5 ] , [ 6 1 , and were a l l found t o have laminar separa t ion bubbles a t low Reynolds numbers.

The Eppler 387 is a low Reynolds number a i r - f o i l design. The NLF(1)-1015 is a n a t u r a l laminar flow a i r f o i l designed f o r h igh-a l t i tude , long- endurance appl ica t ions . The NLF(1)-0416 is a natu- r a l laminar flow a i r f o i l f o r general a v i a t i o n appl ica t ions .

The LTPT is a closed-throat , s ing le - re turn tunnel which can be operated a t s tagna t ion pres- s u r e s from 0.10 t o 10 atmospheres. The t e s t sec- t i o n i s 7 . 5 f e e t high, 7.5 f e e t long, and 3 f e e t wide. The cont rac t ion r a t i o is 17.6 t o 1 and nine ant i - turbulence screens a r e i n s t a l l e d i n t h e s e t - t l i n g chamber. The low turbulence l e v e l s i n the tunnel and i ts l a r g e s i z e toge ther with the prec i - s i o n pressure ins t rumenta t ion make i t an i d e a l a i r f o i l t e s t i n g f a c i l i t y . For the t e s t s of these a i r f o i l s , su r face pressure measurements were used t o determine s e c t i o n l i f t and pitching-moment char- a c t e r i s t i c s . Wake rake measurements were used t o determine s e c t i o n drag. The wake measurements were taken a t spanwise l o c a t i o n s away from t h e row of o r i f i c e s on the model, i n o rder t o avoid any pos- s i b l e o r i f i c e in f luence on t h e drag.

DISCUSSION OF RESULTS

ISES is run according t o s e v e r a l input param- e t e r s t h a t def ine t h e computational space and t h e na ture of the problem. Drela and G i l e s have recom- mended values t o be used f o r these parameters i n [ 7 ] . As p a r t of t h e p resen t i n v e s t i g a t i o n , a pre- l iminary s tudy was conducted t o determine t h e e f - f e c t s of changing those parameters which did not concern t h e physics of the problem. It was found t h a t the c a l c u l a t i o n s were not very s e n s i t i v e t o changes i n t h e parameters r e l a t e d t o the computa- t i o n a l space. Therefore, a l l a i r f o i l ana lyses presented i n t h i s paper were run using the paraa- e t e r s recommended by Drela and Giles . A l l t h e input parameters used f o r t h i s s tudy a r e gqven i n Appendix A.

One input parameter which a f f e c t s t h e physics of the problem is t h e n a t u r a l t r a n s i t i o n c r i t e r i o n , t h e c r i t i c a l d i s tu rbance a r p l i f i c a t i o n r a t i o , n,,. Values between 7 and 15 f o r ncr a r e t y p i c a l l y used i n codes with a t r a n s i t i o n model based on dis- turbance a m p l i f i c a t i o n theory. A high va lue f o r ncr r e s u l t s i n de lay ing t r a n s i t i o n ; conversely, a low value f o r ncr r e s u l t s i n e a r l y t r a n s i t i o n . P a r t of t h e p resen t code v e r i f i c a t i o n e f f o r t was t o determine t h e range of va lues f o r ncr t h a t can c o r r e c t l y model the d i f f e r e n t t r a n s i t i o n phenomena which occur a t low Reynolds numbers - with and without laminar s e p a r a t i o n - thereby p r e d i c t i n g a i r f o i l drag c o r r e c t l y .

The f i r s t s e t of r e s u l t s t o be discussed here corresponds t o cases with laminar s e p a r a t i o n bubbles. Experimental and t h e o r e t i c a l p ressure d i s t r i b u t i o n s f o r the Eppler 387 a i r f o i l a t

-.

R = 200,000 and a = 4" a r e shown i n Figure 1." I n Figures l a and b, the t h e o r e t i c a l pressure d i s - t r i b u t i o n s were obtained using ncr = 9 and ncr = 14 respec t ive ly . The presence of a laminar s e p a r a t i o n bubble i s ind ica ted by the so-called pressure p la teau on the upper s u r f a c e , which i n t h i s case s t a r t s a t about x/c 0.4. The agree- ment between the experimental and t h e o r e t i c a l pressure d i s t r i b u t i o n s i s good i n both cases , except f o r the separa t ion bubble region on t h e upper surface. When ncr = 14, the ex ten t of the t h e o r e t i c a l pressure p la teau agrees very well with the experiment, but when ncr = 9 t r a n s i t i o n i s p red ic ted s l i g h t l y e a r l i e r , r e s u l t i n g i n a s h o r t e r separa t ion bubble. Furthermore, t h e drag predic- t i o n agrees b e t t e r with experiment when ncr = 14 then when ncr = 9, so t h a t i n t h i s case, t h e h igher ncr appears t o model t h e laminar separa- t i o n bubble cor rec t ly .

The t h e o r e t i c a l l i f t and p i t c h i n g moment coef- f i c i e n t s f o r both t h e o r e t i c a l cases a r e i n f a i r agreement with each other . Unlike drag, l i f t and p i tch ing moment a r e not very s e n s i t i v e t o s l i g h t changes i n s e p a r a t i o n bubble length. The reason why the t h e o r e t i c a l l i f t c o e f f i c i e n t s a r e s l i g h t l y higher than t h e experiment is t h a t t h e r e i s a s l i g h t mismatch hetween t h e angle-of-attack d e f i n i - t i o n s between theory and experiment. This exp la ins the s l i g h t d i f f e r e n c e s i n t h e p ressure d i s t r i b u - t i o n s between theory and experiment. Wowever, t h e au thors cannot exp la in t h e pressure drop across t h e t h e o r e t i c a l p ressure p la teau which does not show i n t h e experimental data .

Another case with laminar s e p a r a t i o n bubbles is shown i n Figure 2, where t h e a i r f o i l NASA NLF(1)-1015 is a t R = 500,000 and a = 2 O . In Figures 2a and b, the t h e o r e t i c a l p ressure d i s t r i - bu t ions were obtained using ncr = 9 and ncr = 14 respec t ive ly . Again, t h e p ressure d i s t r i b u t i o n from t h e higher ncr matches experiment b e t t e r than t h a t from t h e lower ncr; t h e main d i f f e r e n c e between t h e 2 t h e o r e t i c a l p ressure d i s t r i b u t i o n s is again i n t h e l eng ths of t h e pressure p la teaus cor- responding t o t h e s e p a r a t i o n bubbles. The drag p r e d i c t i o n a t t h e higher ncr a l s o agrees b e t t e r wi th experiment.

Figures 3 and 4 a r e a l s o c a s e s with laminar s e p a r a t i o n bubbles. I n Figure 3, the Eppler 3 8 7 a i r f o i l i s a t a = 4 O and R = 300,000. In Fig- u r e 4, the NLF(1)-1015 a i r f o i l is a t a = 2O and R = 1 mil l ion. As with t h e previous cases with s e p a r a t i o n bubbles, t h e t h e o r e t i c a l drag predic- t i o n s us ing t h e h igher ncr agree b e t t e r with experiment. However, t h e p r e s s u r e d i s t r i b u t i o n s obtained using the h igher ncr do not agree with experiment a s wel l a s when t h e lower ncr is used. This c o n t r a d i c t i o n may be a t t r i b u t e d t o poss ib le o r i f i c e in f luences on bubble l eng th a t t h e s e Reynolds numbers. It has experimental ly been found [61 t h a t d i s tu rbances from pressure o r i f i c e s can reduce laminar s e p a r a t i o n bubble l e n g t h s by t r i g g e r i n g e a r l i e r t r a n s i t i o n . The agreement of t h e experimental p ressure d i s t r i b u t i o n t o t h e p ressure d i s t r i b u t i o n a t a lower ncr may i n d i c a t e

*e have chosen not t o match t h e o r e t i c a l l i f t c o e f f i c i e n t s t o experimental values because of convergence d i f f i c u l t i e s : t h e same Ca can be obtained f o r one case wi th a bubble a s f o r ano ther case without.

t h a t a lower ncr can model the e f f e c t s of d i s t u r - bances from various sources such a s the freestream or pressure o r i f i c e s . As f o r the drag calcula- t i o n s , i t i s c o r r e c t t h a t the t h e o r e t i c a l drag values match experiment a t the higher ncr, s ince t h e experimental drag was obtained ou ts ide o r i f i c e inf luences.

For a case without laminar separa t ion bubbles, Figure 5 shows the NLF(1)-1015 a i r f o i l a t a = 6' and R = 1 mi l l ion . Here, t h e drag p r e d i c t i o n from the lower ncr a n a l y s i s i s c l o s e r t o experiment. S imi la r ly , the t h e o r e t i c a l pressure d i s t r i b u t i o n a t ncr = 10 agrees b e t t e r with experiment than with n,, = 14. Unlike the cases with laminar separa t ion bubbles, i t i s the lower ncr value t h a t p r e d i c t s t r a n s i t i o n and drag more c o r r e c t l y i n t h i s case

' without a separa t ion bubble.

Pfenninger and Vemuru have a l ready observed t h a t a higher ncr is required f o r cases with laminar separa t ion bubbles than f o r cases without bubbles [81. They p o s i t t h a t bubble cases requ i re higher values f o r ncr because ncr grows rap id ly a t t h e end of a laminar separa t ion bubble, whereas ncr grows more gradua l ly otherwise.

I n t h i s fol lowing d i scuss ion of r e s u l t s , p l o t s of experimental l i f t , drag, and pitching-moment c h a r a c t e r i s t i c s f o r a wide range of ang les of a t t a c k a r e compared t o ISES ana lyses r e s u l t s using d i f f e r e n t values f o r ncr.

Figures 6 t o 9 a r e low Reynolds number cases where laminar s e p a r a t i o n occurs over most of the angle-of-attack range shown. The l i f t and pitching-noment curves agree q u i t e well wi th experiment, regard less of the ncr used, s o d i s - cuss ion w i l l be focused on t h e drag polars .

The case of the Eppler 387 a t R = 200,000 i s shown i n Figures 6a and b. Between ncr = 9 and ncr = 14, t h e t h e o r e t i c a l drag po la r a t t h e higher ncr agrees much b e t t e r with experiment.

The Eppler a i r f o i l a t R = 300,DOO is shown i n Figures 7a and b. Again, it is the h igher ncr t h a t p r e d i c t s a drag po la r c l o s e t o t h e experimen- t a l drag polar.

The r e s u l t s shown i n Figure 8 a r e t h e NLF(1)-1015 a t R = 500,000. Figures 8a, b, c , and d correspond t o a n a l y s i s cases with ncr = 9, 11, 13, and 15 respec t ive ly . Here, t h e ncr = 9 and ncr = 11 ana lyses underpred ic t drag, whereas ncr = 13 and ncr = 15 p r e d i c t d rag q u i t e accu- r a t e l y f o r t h e e n t i r e low drag CA range.

The r e s u l t s shown i n Figure 9 a r e f o r NLF(1)-1015 a t R = 1 mil l ion. A t t h i s Reynolds number, laminar separa t ion bubbles s t i l l e x i s t over most of the l o r d r a g CA range. Therefore, i t is the higher ncr drag po la r t h a t agrees c l o s e l y with experiment.

Figures 10 and 11 a r e cases where t h e r e is l i t t l e evidence o f ' l a m i n a r separa t ion . The NASA NLF(1)-0416 a i r f o i l has been analyzed a t R - 2 mi l l ion using ncr = 7, 9, and 11. (See Fig. 10.) The agreement between theory and experi- ment is q u i t e good f o r a l l th ree values of ncr, a l though the case with t h e lowest ncr agrees bes t with experiment f o r the widest range of l i f t coef- f i c i e n t s , i n p a r t i c u l a r f o r Cl < 0.4 and

CL > 0.8. A t these l i f t c o e f f i c i e n t s , drag predic- t i o n s a t the higher ncr values a r e no t iceab ly lower than a t ncr 7 . The higher ncr values have delayed t r a n s i t i o n which has r e s u l t e d i n lower drag where there a r e no separa t ion bubbles. For 0.4 <Cl 0.8, t h e r e is some l a n i n a r separa t ion , so i n t h i s C1-range drag does not decrease with an increase i n ncr.

The r e s u l t s shown i n Figure 11 a r e a l s o f o r the NLF(1)-0416, but a t R = 3 mil l ion. Fig- u r e s l l a , b, and c correspond t o a n a l y s i s cases with ncr = 7 , 9 , and 11 respec t ive ly . A s i n t h e previous case, t h e r e i s a no t iceab le decrease i n drag with an increase i n ncr f o r most of the Cl range shown, and ncr = 7 g ives r e s u l t s t h a t b e s t match t h e experiment. A t t h i s higher Reynolds number, the re i s l i t t l e evidence of laminar separat ion.

It can be concluded from t h e above comparisons t h a t ISES can accura te ly p r e d i c t low Reynolds number a i r f o i l c h a r a c t e r i s t i c s . A high value f o r t h e c r i t i c a l ampl i f ica t ion r a t i o , ncr (13 t o 15) . works well f o r cases with laminar separa t ion bub- b l e s , whereas a lower value f o r ncr ( 7 t o 11) i s more adequate f o r cases t h a t do no t develop separa- t i o n bubbles.

This a n a l y s i s method is not without l imi ta - t i o n s , however. Drela and Gi les po in t out t h a t t h e method i s l imi ted t o t h i n l y separa ted regions. From our own experiences, we have found t h a t t h e code indeed encounters d i f f i c u l t i e s where separated. regions become too t h i c k , such a s near Cl,max, and t h e low Cl range o u t s i d e t h e low drag bucket. This can be seen i n a l l t h e p l o t s f o r a i r f o i l sec- t i o n c h a r a c t e r i s t i c s i n Figures 6 t o 11. A s an a d d i t i o n a l t e s t of t h e code's l i m i t a t i o n s , ISES was run f o r t h e Eppler 387 a t R = 100,000. A t t h i s low Reynolds number, t h e laminar separa t ion bubbles a r e q u i t e t h i c k an& extensive. Figure 12 shows pressure d i s t r i b u t i o n s f o r t h e Eppler a i r f o i l a t a = 0" and R = 100,000. Unlike t h e o ther cases where ncr 13 is good enough t o model the laminar s e p a r a t i o n bubble l eng th , t h e d rag i n t h i s case is very much underpredicted. Even when ncr = 20, t h e t h e o r e t i c a l p ressure p l a t e a u is s t i l l s h o r t e r than t h a t of t h e experiment, a l though t h e agreement i n d rag i s good. Apparently, t h e code cannot handle t h e severe laminar s e p a r a t i o n problema i n t h i s low Reynolds number case.

The next i tem t o be d i scussed i n t h i s s e c t i o n is the e f f e c t of g r i d s i z e on the ca lcu la t ions . Drela and G i l e s have used 132 x 32 g r i d s t o analyze low Reynolds number a i r f o i l s i n [ I ] . However, s i n c e a 132 x 32 g r i d case runs about f i v e times a s long a s f o r a 96 x 16 g r i d on a VAX 8300, cases have been analyzed using both g r i d s izes . A typi- c a l CPU run time f o r a 9 6 . x 16 g r i d case f o r 4 i t e r a t i o n s is 2 112 mlnutes; f o r a 132 x 32 g r i d case f o r 4 i t e r a t i o n s , 11 minutes. I n Figures 13 t o 15, r e s u l t a n t p ressure d i s t r i b u t i o n s from t h e ISES analyses on 96 x 16 and 132 x 32 g r i d s a r e superimposed on experimental p ressure d i s t r i b u - t ions . The f i r s t two cases have laminar s e p a r a t i o n bubbles whereas t h e t h i r d case has no bubbles.

Figure 13 shows t h e pressure d i s t r i b u t i o n comparisons f o r t h e Eppler 387 a t a = 4' and R = 200,000. The t h e o r e t i c a l p ressure d i s t r i b u - t i o n s show the same e x t e n t of t h e experimental

pressure p la teau f o r both g r i d s izes . A comparison of the a i r f o i l s e c t i o n c h a r a c t e r i s t i c s shows t h e expected d i sc repanc ies i n Cl and C, due t o the s l i g h t pressure d i s t r i b u t i o n d i f fe rences . On the o ther hand, the t h e o r e t i c a l Cdts have an excel- l e n t match with experiment. The t h e o r e t i c a l r e s u l t s from the 96 x 16 and 132 x 32 g r i d s agree a t l e a s t within 2% of each other .

Figure 14 shows the pressure d i s t r i b u t i o n comparisons f o r the NLF(1)-1015 a i r f o i l a t R = 500,000 and a = 2". The o v e r a l l agreement i n the pressure d i s t r i b u t i o n s i s very good f o r both t h e o r e t i c a l g r i d s i z e cases. However, a r e s u l t of using the coarse r g r i d is t h a t the 96 x 16 r e s u l t s show kinks a t the end of the pressure p la teaus corresponding t o the laminar separa t ion regions on both the upper and lower sur faces of the a i r f o i l . I t appears t h a t these kinks have caused the drag for t h e 96 x 16 case t o increase by 2% over t h e 132 x 32 case.

F igure 15 shows t h e pressure d i s t r i b u t i o n c o w parisons f o r t h e NLF(1)-1015 a i r f o i l a t R 1 m i l - l i o n and a = 6". Again, the o v e r a l l agreement i n t h e p ressure d i s t r i b u t i o n s is very good f o r both t h e o r e t i c a l g r i d s i z e s , and t h e 2 s e t s of theore t i - c a l r e s u l t s agree a t l e a s t within 2% of each o ther .

From t h e above d i scuss ion , one can see t h a t by using t h e 96 x 16 g r i d , t h e q u a l i t y of the a n a l y s i s r e s u l t s is not compromised by the speed of t h e ana lys i s .

DIFFICULTIES I N RUNNING THE CODE

The au thors ' experiences i n running many anal- y s i s cases were t h a t ob ta in ing a converged s o l u t i o n was not always s t ra igh t forward , p a r t i c u l a r l y i f a case involved laminar separa t ion . In o rder t o ob- t a i n converged so lu t iona f o r t h e s e problem cases , i t was necessary t o f i r s t o b t a i n a converged solu- t i o n with t r a n s i t i o n forced e a r l i e r than t h e expected l o c a t i o n of t h e s e p a r a t i o n bubble. This s o l u t i o n was then used a s input t o run a case w i t h t r a n s i t i o n f u r t h e r downstream. This procedure of moving t h e forced t r a n s i t i o n l o c a t i o n s rearward had t o be repeated s e v e r a l times before a s u c c e s s f u l l y converged s o l u t i o n f o r a n a t u r a l t r a n s i t i o n c a s e with s e p a r a t i o n bubbles could be obtained.

Also, when an a n a l y s i s case a t a p a r t i c u l a r angle-of-attack presented convergence problems be- cause of laminar s e p a r a t i o n , i t was necessary t o f i r s t t r y so lv ing f o r o t h e r angle-of-attack cases where separa t ion problems were not a s severe. Then these converged s o l u t i o n a were used a s input t o ob- t a i n t h e s o l u t i o n f o r t h e a c t u a l case of i n t e r e s t .

A s an average, converged s o l u t i o n s f o r cases with s e p a r a t i o n bubbles requ i red 20 i t e r a t i o n s , a s opposed t o cases with no bubbles which only took 10 to 12 i t e r a t i o n s . D i f f i c u l t cases required as much a s 48 i t e r a t i o n s f o r convergence.

CONCLUSIONS

The ISES a n a l y s i s method has been v e r i f i e d a s a r e l i a b l e method i n p r e d i c t i n g the aerodynamic c h a r a c t e r i s t i c s of low-speed a i r f o i l s i n t h e chord Reynolds number range between 200,000 and

3 mil l ion. The nethod can handle t h i n separated regions very wel l , but i t may not be r e l i a b l e f o r cases with l a r g e separa ted regions such as a t

Cl,,,, condi t ions , o r a t Reynolds numbers below

200,000.

I t was found t h a t a high value f o r the t r a n s i - t i o n c r i t e r i o n ( the c r i t i c a l ampl i f ica t ion r a t i o ) , nc r , from 13 t o 15, was necessary t o analyze a case with laminar separa t ion problems accura te ly . Otherwise, cases without laminar separa t ion bubbles were analyzed more accura te ly using low values f o r ncr, from 7 t o 11. Low values fo r the t r a n s i t i o n c r i t e r i o n nay be used t o nodel the e f f e c t s of d i s - turbances on low Reynolds number flows.

ACKNOWLEDGMENTS

Authors a r e g r a t e f u l t o Drs. Mark Drela and Michael Gi les f o r the ISES a i r f o i l des ign /ana lys i s code, and to Mr. Dan Somers and M r . Robert J. McGhee f o r providing t h e experimental data. This work was performed under c o n t r a c t s NAS1-18235 and NAS1-18599 a t NASA Langley Research Center.

REFERENCES

1. Drela, M. and Gi les , M. B., "Viscous-Inviscid Analysis of Transonic and Low Reynolds Number A i r f o i l s , " A I A A Journa l , Vol. 25, NO. 10, pp. 1347-1355.

2. Drela , M. , "Two-Dimensional Aerodynamic Design and Analysis Using t h e Euler Equations," Massachusetts I n s t i t u t e of Technology, Gas Turbine Laboratory Rept. 187, Feb. 1986.

3. Eppler , R. and Somers, D. M. : "A Computer Program f o r t h e Design and Analysis of Lowspeed ~ i r f o i l s , " NASA TM 80210, 1980.

4. McGhee, R. J . , Walker, B. S., and Mi l la rd , B. F., "Experimental Resu l t s f o r the Eppler 387 A i r f o i l a t Low Reynolds Numbers i n t h e Langley Low-Turbulence Pressure Tunnel," NASA TM 4062, October 1988.

5. Maughmer, M. and Somers, D. H., "An A i r f o i l Designed f o r a High-Altitude, Long-Endurance, Remotely-Piloted Vehicle," AIAA paper 87-2554CP.

6. Somers, D. M., "Design and Experimental Resu l t s f o r a Natural Laminar Flow A i r f o i l f o r General Aviation Applicat ions," NASA Technical Paper 1861, June 1981.

7. Gi les , M. and Drela , M., "A User 's Guide t o ISES," Jan. 31, 1986.

8. Pfenninger , U. and Vemuru, C. S., "Design of Low , Reynolds Number A i r f o i l s - I," AIM-88-2572CP.

APPENDIX A - VALUES FOR THE INPUT VARIABLES TO RUN ISES

A d e s c r i p t i o n of t h e requ i red input v a r i a b l e s t o ISES is given i n "A User 's Guide t o ISES" [TI. The fol lowing is a l is t of t h e va lues f o r the input v a r i a b l e s used i n t h i s study.

To i n i t i a l i z e g r i d s , t h e input v a r i a b l e s be-

4

sides the airfoil coordinates used are:

SINL = SOUT = 0. (as required for isolated airfoil cases)

CHINL = CHOUT = 2.5 CHWID = 6.

The grid-generation program was always run in the "geometry-driven" mode.

For 96 x 16 grids,

NINL = 10 NBLD - 78 NOUT = 10 JJ = 16 JJJ = 9.

For 132 x 32 grids,

NINL = 10 NBLD = 114 NOUT = 10 JJ = 32 JJJ = 16.

The approximate streamvise grid spacings for 96 x 16 grid cases were LE, TE ds/c = .005, whereas for 132 x 32 grid cases, LE, TE ds/c = .010.

To run the analysis, the following were used:

GVAR(~), GVAR(4), GVAR(5) GCON(3), CCON(4), GCON(5) MACH = 0.10 CL = 0.0 (dummy variable) SINL = SOUT - 0.0 (dummy variables) ALFA = angle-of-attack ISMOM = 2 PCWT = 0.05 IFFBC = 2 REYNIN = chord Reynolds number TRANS1 = usually 1 for natural transition,

unless transition was being forced on the upper surface

TRANS2 = usually 1 for natural transition, unless transition was being forced on the lower surface

MCRIT = 0.95 MUCON = 1 ACRIT = between 7.0 and 20.0 for the critical

amplification ratio.

EPPLER 387 AIRFOIL M = 0.10, Rc = 200,000, a = 4"

0.2

- ----- Theory (11-14) Exvenmenr

1'00-0 XIC XIC

Figure 1. Comparisons of pressure distributions

between experiment and ISES theory where (a) Theory ncr - 9 (b) Theory ncr - 14.

NASA NLF (1)-1015 AIRFOIL M = 0.10, Rc = 500,000, a = 2 O

Figure 2. Comparisons of pressure distributions between experiment and ISES theory where (a) theory nCr - 9 (b) theory ncr - 14.

EPPLER 387 AIRFOIL M = 0.10, Rc = 300,000, a = 4 O

Figure 3. Comparisons of pressure distributions between experiment and ISES theory where (a) theory ncr - 9 (b) theory ncr - 14.

NASA NLF (1)-1015 AIRFOIL M = 0.10, Rc = 1.0 x lo6, a = 2'

NASA NLF (1)-1015 AIRFOIL M = 0.10, Rc = 1.0 x lo6, a = 6 O

2 5 % Theory (n.101 1.323 0087 - 1786 Theory (n=14) 1.375 0083 . 1897 Exwr~ment 1.349 0094 ..I830

Theory ,972 ,0073 -.I% - 1 A r Theory(n.13) ,983 .W77 -.I930

L Expanmen1 ,968 .W76 ..I910

- ----- Theory (n.10) Exper~ment

c p !b .0.6

0.2

0.6

1 0 0 0.2 0.4 0.6 0.8 1.0

r C

ta)

Theory 1nn14) Exper~ment

?PC.+

0 0.2 0.4 0.6 0.8 1 0 x;c

(b) Figure 5. Comparisons of pressure distributions

between experiment and ISES theory where (a) theory ncr - 10 (b) theory ncr - 14.

- . -- Figure 4. Comparisons of pressure distributions

between experiment and ISES theory where (a) theory ncr - 9 (b) theory ncr - 13.

EPPLER 387 AIRFOIL Rc = 200,000 EPPLER 387 AIRFOIL

- R c = 200,000 Theory (n = 9)

2 Expanmen1 1 . 4 r I

EPPLER 387 AIRFOIL Rc = 300,000

- Thwry (n : 91 3 Expanmow

1 . 4 r I

EPPLER 387 AIRFOIL Re = 300,000

- Theory (n = 14) 3 Exparimmt

Cb) Figure 7. Comparisons of airfoil section characteristics between experiment and ISES theory where (a) theory ncr - 9 (b) theory ncr - 14.

NASA NLF (1) - 1015 AIRFOIL Rc = 500,000

NASA NLF (1) - 1015 AIRFOIL Rc = 500,000

- Thwry (n : 9) 1.61- : E x ~ f l m ~ n l

NASA NLF (1)-1015 AlRFOlL

R c = 500,060

Figure 8. Comparisons of airfoil section characteristics between experiment and ISES theory whre (a) theory ncr - 9 (b) theory ncr - 11 (c) theory ncr - 13 (dl theory ncr - 15.

NASA NLF (1)-1015 AIRFOIL Rc = 1 . 0 ~ l o 6 NASA NLF (1)-1015 AIRFOIL

Rc. l .Ox 106

? X I x I

1 ( I

1 \ >

Figure 9. Comparisons of airfoil section characteristics between experiment and ISES theory whre (a) theory ncr - 9 (b) theory ncr - 11

NASA NLF (1)-1015 AIRFOIL R c = 1.0 x 106

.0.4 , 1 a A 1 . I 2 -8 4 0 4 8 12 16 20 004 W 8 0 1 2 . 0 1 6 0 2 0 0 2 4 0 2 8 - 2 - 1 0

a. dog

Cc > Cd c m - Fig. 9 (Condaded)

(c) theory ncr - 13 (d) theory ncr - 15

NASA NLF (1)4416 AIRFOIL

1 . 8 7 R, = 2.0 x lo6 - Thew (n = 7) !

1 . 6 t 3 Exwnnum

1 O C f P

NASA NLF (1)-1015 AIRFOIL R c = l.Ox 106

1 6 r - Theory (n = IS) 2 Exponrnent

1.4C

1 . 2 i

1.Ob

0.8 +

NASA NLF (1)-0416 AIRFOIL 1 . 8 r Rc = 2.0 x lo6

NASA NLF (1)-0416 AIRFOIL 1 . 8 r A,= 2.0 x l o 6

Figure 10. Comparisons of airfoil section characteristics between experiment and ISES theory where

(a) theory ncr - 7 (b) theory ncr - 9 (c) theory ncr - 11.

NASA NLF (1)-0416 AIRFOIL R,= 3.0 x 106

1.8-

NASA NLF (1)-0416 AlRFOlL

i a- R, = 3.0 x l o6

.0.4 .12 4 -4 0 1 8 12 16 20 .W4.008.012.016.020.024028 .2=

a. deg (b)

6 Cm

NASA NLF (1)-0416 AIRFOIL

1.8- R, = 3.0 x l o6

Figure 11. Comparisons of airfoil section characteristics between experiment and ISES theory where (a) theory n,, - 7 (b) theory ncr - 9 (c) theory ncr - 11.

EPPLER 387 AIRFOIL M = 0.10, Rc = 100,000, a = 0"

3 cd c m ~ h . o r y (nzl3) .a .0G5 -.@ Theory (13.20) ,410 ,0171 ..OW

. l T

Exp.nrnmt ,390 ,0170 - .wn [

Figure 12. Comparison of pressure distributions between experiment and ISES theory where (a) theory ncr - 13

EPPLER 387 AIRFOIL M = 0.10, Rc = 200,000, a =

Figure 13. Comparison of pressure distributions between experiment and ISES theory where (a) theory 96 x 16 grid

(b) theory 132 x 32 grid.

NASA NLF(1)-1015 AIRFOIL M = 0.1 0, Re = 500,000, a = 2'

Figure 14. Comparison of pressure distributions between experiment theory where (a) theory 96 x 16 grid

(b) theory 132 x 32 grid.

and ISES

NASA NLF (1>1015 AIRFOIL M = 0.10, Rc = 1.0 x lo6, a= 6 O

C# C C-

Figure 15. Comparison of pressure distributions between experiment and ISES theory where (a) theory 96 x 16 grid

(b) theory 132 x 32 grid.