Numerical Optimization and WindTunnel Testing

of Low ReynoldsNumber

y

Th Lutz W Wurz and S Wagner

University of Stuttgart D Stuttgart Germany

Abstract

A numerical optimization to ol has been applied to the design of low Reynolds

number airfoils  Re  The aero dynamic mo del is based on the

Eppler co de with ma jor extensions A new robust mo del for the calculation of short

n

transitional separation bubbles was implemented along with an e transition criterion

and Drelas turbulent b oundarylayer pro cedure with a mo died shap efactor relation

The metho d was coupled with a commercial hybrid optimizer and applied to p erform

unconstrained high degree of freedom optimizations with ob jective to minimize the drag

for a sp ecied range One resulting was tested in the Mo del WindTunnel

MWT of the institute and compared to the classical RG airfoil

In order to allow a realistic recalculation of exp eriments conducted in the MWT the

limiting nfactor was evaluated for this facility A sp ecial airfoil featuring an extensive

instability zone was designed for this purp ose The investigation was supplemented by

detailed measurements of the level

Toget more insight in design guidelines for very low Reynolds numb er airfoils the

inuence of variations of the leadingedge geometry on the aero dynamic characteristics

was studied exp erimentally at Re

Intro duction



For the Reynoldsnumber regime of wingsections Re   sophisticated di

 

rect and inverse metho ds for airfoil analysis and design are available In the hands of

exp erienced users these metho ds allow a carefully directed airfoil design Usually sp ecic

airfoils are b eing develop ed for any new aircraft in order to maximize the p erformance for



the intended range of application Windtunnel tests mainly serve for verication of the

predicted aero dynamic characteristics and are necessary means to determine the b ehav

ior and to optimize turbulators or the ap settings Doubtless the total number of design

lo ops was signicantly reduced due to the availability of reliable prediction metho ds

This do es not hold for the airfoil design at low Reynolds numb ers Re  In

this regime transition usually takes place in a separated b oundarylayer if the clean airfoil is

considered To predict the inuence of laminar separation of unsteady transitional separa

tion bubbles and of the viscousinviscid interaction is still a challenge This is esp ecially true

in the range of the critical where strong nonlinearities o ccur In general

the uncertainty in the airfoil analysis increases with decreasing Re For this reason it is

necessary to include the lessons learned from windtunnel tests in the design metho dology of

low Reynolds number airfoils Extensive exp eriments physical interpretation of the results

Research Assistant Institute for Aero dynamics und Gas Dynamics IAG Pfaenwaldring

y

Professor Head of Institute Institute for Aero dynamics und Gas Dynamics IAG Pfaenwaldring

and the transfer into successively improved airfoil designs as p erformed e g by Selig et

 

al is therefore the most promising approach

Being aware of the ab ove mentioned problems an attempt was made to p erform direct nu

merical shap e optimizations of low Reynolds numb er airfoils Re This approach

is considered to be instructive for two reasons First the optimizer may nd unconven

tional solutions which could inspire new ideas for an improved manual design Second weak

points of the aero dynamic mo del can be identied if the optimized airfoils are examined in

a windtunnel test This in turn may lead to improvements of the prediction metho ds

The intention of the present optimizations was to p erform an unconstrained design of

completely new airfoils rather than to mo dify existing ones This necessitates considering

a large number of design variables in order to enable a detailed representation of the airfoil

shap e and the pressure distribution Furthermore sto chastic optimization algorithms should

b e applied to reduce the p ossibility of getting trapp ed in a lo cal optimum A high degree of

freedom problem in combination with a sto chastic optimization strategy however requires

a very large number of airfoils to be generated and analyzed until the optimization pro cess

converges If the optimizations are to be p erformed with acceptable computation time a

most ecient aero dynamic mo del is needed For the present investigations a p otentialow

metho d coupled with an integral b oundarylayer pro cedure a simplied bubble mo del and

n

an e databasemetho d for transition prediction was applied Viscousinviscid interaction

was neglected in order to minimize the computational eort which actually is roughly two

orders of magnitude b elow an XFOIL analysis The complete airfoil optimization to ol has

  

b een proven successful for the design of high Reynolds number Re   NLF airfoils

and is routinely applied for various design tasks

In the following sections the ingredients of the metho d will briey be describ ed There

after optimization examples of low Reynolds numb er airfoils including windtunnel verica

tion will b e discussed

Supplementary to this theoretical approach to airfoil design fundamental windtunnel

tests for very low Reynoldsnumber airfoils Re  will b e presented First detailed

exp erimental investigations of airfoil characteristics for this Reynoldsnumber regime were



published by Schmitz He proved that thin airfoils featuring a sharp leading edge have

a lower critical Reynolds number than conventional airfoil designs In the exp eriments of

Schmitz the cambered plate Go a shows the best p erformance up to Re 

The problem of how to provoke b oundarylayer transition for Re in order



to avoid extensive laminar separation was examined by Hamma Among other designs

a thick symmetrical airfoil with a sharp leading edge was tested in a water tunnel

and showed promising results However very complex ow phenomena were observed For

example laminar separation with laminar reattachment partly followed by a transitional

separation bubble o ccurred for a sp ecic and Reregime Furthermore the separation

angle abruptly increased for a distinct angle of attack Such complex ow phenomena cannot

be predicted reliably with available theoretical mo dels The present authors therefore

prefer an exp erimentbased approach with the intention to establish guidelines for the design

of very low Reynoldsnumber airfoils In a recent test campaign the inuence of relevant

geometric parameters like the leadingedge radius on the aero dynamic p erformance has

been examined First results have been obtained and will b e discussed

Aero dynamic Mo del

For the present investigations the determination of the airfoil characteristics during the



optimization pro cess is based on the airfoil design and analysis metho d of Eppler Ma jor

extensions with resp ect to b oundarylayer computation were added to this to ol

In the direct mo de the inviscid outerow is evaluated by means of a higherorder panel

metho d whereas an inverse conformal mapping pro cedure is used in the design mo de Based

on the inviscid velo city distribution the b oundarylayer development is calculated with a



rstorder integral pro cedure The metho d is based on the numerical integration of von

Karmans integral momentum equation and Wieghardts energy equation with the momen

tum thickness and the energy thickness as dep endent variables For attached laminar



ow algebraic closure relations are used which were derived from FalknerSkan selfsimilar

proles in regions with adverse pressure gradient whereas solutions for the at plate with

suction serve as a basis in accelerated ow regimes



For the calculation of turbulent b oundarylayers the metho d prop osed by Drela was

implemented in the Eppler co de with a new shap efactor relation H H Re With

 



this approach Greens lagequation is intro duced as a third governing equation in order to

account for history eects in case of nonequilibrium ows To close the problem Drela pro

p osed an algebraic shap efactor relation whichwas derived from an evaluation of Swaords

twoparameter b oundarylayer proles For very low lo cal Reynolds numbers Re 



and large values of H this relation yields H This b ehavior however should only o c

 

cur with tangential blowing which is not considered in the present investigations Therefore

  

an alternative empirical relation was derived which ensures H for any Re and





furthermore satises the asymptotic condition H H  for a rectangular velo city

 

prole

Reliable transition prediction is of essential imp ortance for successful design and opti

mization of natural laminar ow airfoils For the present investigations the semiempirical

n  

e metho d according to van Ingen and Smith Gamb eroni was implemented This

approach takes advantage of the fact that the transition pro cess in subsonic Dows is typi

cally asso ciated with the growth and breakdown of TollmienSchlichting waves TSwaves

As long as the disturbance amplitude A is small enough the amplication can b e predicted

n

by means of linear stability theory The basic idea of the e metho d is that transition

may be assumed when the most amplied frequency reaches a certain critical amplication

factor n ln A A The value of n dep ends on freestream conditions the

cr it cr it initial cr it

receptivity mechanism and on the denition of the transition p oint Therefore n has

cr it

to be correlated individually for each windtunnel or for freeight conditions taking the

numerical mo del used for the airfoil analysis into account compare Sec With the present

n

e implementation spatial disturbance growth is considered Since stability analysis based

on a direct solution of the OrrSommerfeld equation requires to o much computational eort

for the purp ose of numerical airfoil optimization a database metho d was implemented This

database contains the amplication rates for shap e factors incl separated ows at

dierent Reynolds numb ers and dierent frequencies

To account for the inuence of transitional separation bubbles which are short according

    

to the denition of Tani a new simplied bubble mo del was develop ed The ob jec

tive was not to enable a detailed calculation of the ow prop erties inside the bubble but

to eciently determine realistic initial values for the turbulent b oundarylayer calculation

downstream of the bubble Thereby a minimum numb er of sup erimp osed empirical relations



was aspired Following the classical approach prop osed by Horton a constant outerow

velo city U is assumed in the laminar part of the bubble For this region a new family of

e

b oundarylayer proles has been intro duced which is based on a FalknerSkan separation

prole b eing successively shifted in wall normal direction The wall distance of the dividing

streamline and accordingly the magnitude of and H are assumed to increase linearly

 

with the arc length s dep ending on the separation angle which is determined by means of



an empirical correlation Downstream of transition a strong abstraction of the real physics

is intro duced The reattachment p oint is assumed to coincide with the transition p ointwhich

in turn is connected to an abrupt drop of the outerow velo city to the inviscid value If

furthermore const and c is supp osed the integral momentum equation can be

 f

solved analytically in the reattachment region which yields the increase over the whole



bubble compare Fig The main advantage of this approach is that the use of uncertain

dissipation laws in the unsteady recirculation region is avoided and a direct noniterative

calculation is p ossible

The drag co ecient of the airfoil is nally determined from the b oundarylayer prop er

ties at the trailing edge by means of the SquireYoung formula If turbulent separation is

predicted the inviscid lift co ecient is corrected in a semiempirical way as prop osed by Ep



pler In principle the airfoil design metho d enables an iterative viscousinviscid coupling

For the present investigations however this option is not used in order to minimize the

computational eort during the optimization and to reduce the roughness of the ob jective

function considered see Sec A more detailed description of the aero dynamic mo del along

 

with validation examples can be found in other publications of the authors

Exp erimental SetUp

 

The exp eriments were carried out in the Mo del WindTunnel MWT of the IAG The

MWT Fig is an op en return tunnel with a closed test section It is placed inside a large

test hall which ensures stable op eration conditions

Figure The Mo del WindTunnel MWT of the IAG



The test section measures  m and is m long The D airfoil mo dels span

the short distance of the test section A thyristor regulated kW engine and an adjustable

b elt drivegives control over the rpm of the bladed Due to the high contraction ratio

of the turbulence level Fig is between Tu    dep ending on

the tunnel sp eed ms U ms Typical chord lengths of the mo dels are in the

range of m  m which results in p ossible Reynolds numb ers from Re to

Re The airfoil mo dels are built of reinforced

0.10

b erglass in NCmilled molds to ensure the

Tu [%]

necessary surface accuracy A sp ecial milling

frequency range: 10 - 5000 Hz

hnique is used to reduce the remaining

tec 0.08

surface roughness of the mold to mm

Therefore only a small amount of work is

to smo oth and p olish the sur

necessary 0.06

face of the mo dels The airfoil mo dels are

mounted horizontally between two circular

plates which are ush with the tunnel

end 0.04

side walls A sp ecial construction of the gap

geometry of the end plates reduces leakage

y dierent static pressure in

eects caused b 0.02

side and outside the test section The end plates are connected to a balance with a high

0.00

load cell The lift is determined

precision 0 10 20 u [m/s] 30

directly by this force measurement whereas

the drag is measured indirectly by means of

Figure Turbulence level of the Mo del Wind

 

an integrating rake The wakerake

Tunnel

is p ositioned of the chord length down

stream of the trailing edge and is traversed in spanwise direction to account for drag varia

tions due to longitudinal inherent structures in the boundary layer Four Furness Controls

Ltd micromanometers are used to measure simultaneously the dynamic head the mean

pressure loss in the wake the static pressure dierence and the maximum pressure loss in

the center of the wake The complete measurement system is highly automated and con

trolled by a PC equipp ed with a bit AD converter Standard windtunnel corrections are

applied and the results are monitored online

Correlation of the Critical Amplication Factor for

the Mo del WindTunnel

n

The application of the e metho d describ ed in Sec requires the determination of a li

miting nfactor for the test facility or free ight conditions in order to enable a realistic

recalculation of exp eriments Dierent pro cedures to determine n are p ossible One di

cr it

rect approach is to use the measured turbulence level of the wind tunnel considered together

 

with available correlations for n The main disadvantage with this approach is that

cr it

the integral value of the turbulence level is used and the sp ectral distribution of the velo c

ity uctuations is neglected Typically the unstable TS waves are connected with a small

frequency range in comparison to the whole range used for the evaluation of the turbulence

level The direct measurement of the velo city uctuations in this critical frequency range

is almost imp ossible b ecause of its small amplitude for low turbulence facilities which is

in the order of the electronic noise of the hotwire equipment In addition it is imp ortant

to distinguish between vorticity and pressure uctuations which can have the same contri



bution to the measured turbulence level but due to dierent receptivity mechanisms the

generated initial TS amplitudes in the boundary layer can dier signicantly

To avoid these problems an indirect calibration is commonly preferred The predicted

transition p ositions for several test cases are compared to the exp erimentally determined

lo cation and the nfactor is adjusted until a go o d match is achieved This pro cedure also

takes small dierences in the numerical evaluation of the nfactor into account b ecause the

numerical mo del is directly involved A priori it is not clear if a similar metho d

1.4 4.5

be adopted to the low Reynolds num can Experiment MWT u/u H

8

regime To establish a well dened

ber Theory 12

case we designed a sp ecial symmetri

test 4.0

airfoil which provides transition with

cal 1.0

out a separation bubble at zero angle of at

3.5

tack Duetothelow test Reynolds numb ers

H12

Re and Re this is

hallenging problem The advantage quite a c 0.6

3.0

of a symmetrical mo del is that the zero angle

of attack can b e adjusted very accurately by

the static pressure dierence be measuring 2.5

Contour

ween the lower and upp er surface near the

t 0.2

leading edge In addition the pressure dis

more precisely the circulation of tribution 2.0

0.0 0.5 s/s 1.0

mo del is not aected by p ossible inu

the max

ences from the windtunnel b oundary layer

Figure Velo city distribution and develop

The airfoil was designed to achieve an almost

ment of the shap e factor airfoil GW

constant shap e factor of H between



Re

xc and xc Toavoid distur

bances resulting from pressure orices the

pressure distribution was measured with the help of a miniaturized static pressure prob e p o

sitioned at the b oundarylayer edge The agreement with the calculated inviscid distribution

is very go o d see Fig Based on the theoretical distribution a b oundarylayer calculation

was p erformed and used as input for the spatial stability analysis The resulting amplitude

development is depicted in Fig

The exp erimental determination of the transition p oint is a serious problem b ecause

transition is a more or less continuous pro cess Therefore the evaluated p osition dep ends

strongly on the metho d used for transition detection This again has a nonnegligible inu



ence on the correlated value of n For the present investigations a classical pro cedure

cr it

was applied which is based on the change of the wall shearstress A miniature pitot tub e

with mm op ening is traversed on the surface of the mo del and the pressure dierence

to a reference pressure is recorded Fig shows the measured data together with a curve

t For the higher Reynolds number of Re the wall shearstress increases at

ss and reaches its maximum at ss The corresp onding nfactors

max max

resulting from Fig are n and n The upstream p osition is similar to the

station where rst sound can be heard with a stethoscop e The transition p osition was

uniform in spanwise direction

15 0.5

upitot n Re=200000 end of 0.4 Re=250000 transition

10 0.3

’onset’ of transition 0.2 5 ’onset’ of end of transition transition 0.1

0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.7 s/s 0.9

s/smax max

Figure Developmentofthewall shearstress

Figure Amplitude development for dif

in the transition region airfoil GW

ferent TSfrequencies airfoil GW

arbitrary scale

Re

The comparisons for Re yields similar nfactors For the numerical mo del

a rounded value of n is used to predict the lo cation for the switch from laminar to

turbulent closure relations According to the considerations ab ove this p osition indicates

the onset of transition and therefore the rst station where the mean velo city prole is

inuenced by the transition pro cess It must be noted that basically an inviscid pressure

distribution was used for the comparison with the exp erimental distribution First of all

b ecause the agreement is very good see Fig and second to be consistent with the nu

merical mo del applied for the present airfoil optimizations Sec Tocheck the results an

analogous deduction was p erformed on the basis of a calculated viscous pressure distribution

The inuence of the b oundarylayer displacement is very small b ecause of the zero angle of

attack and therefore the evaluated nfactors are only slightly lower n andn

Numerical Optimization of Low ReynoldsNumber

Airfoils

Optimization Algorithm

With direct numerical optimization an automated search for an optimal solution with resp ect

to a usersp ecied ob jective function e g minimization of the drag co ecient is p erformed

This is done by means of an iterative variation of the design variables chosen For airfoil

optimizations it is usually necessary to intro duce geometric or aero dynamic constraints like

a limitation of the thickness or of the moment co ecient

Regarding the optimization algorithm deterministic metho ds like gradientbased ap

proaches and sto chastic metho ds likeevolution strategy or genetic algorithm can b e distin

guished In general gradient metho ds converge fast for a simple top ology of the ob jective

function but may get trapp ed in a lo cal optimum As a result the optimized airfoil can

lo ok very similar to the initial shap e Actually this is true for most of the numerical airfoil

optimizations published so far Sto chastic algorithms oer a greater chance to nd a bet

ter optimum and furthermore can cop e with complex top ologies of the ob jective function

However usually much more iterations and therefore airfoil analysis are required Since

the ob jective function considered with the presentinvestigations shows multimo dal b ehavior

 

and strong nonlinearities ahybrid optimizer was applied The POINTER to ol chosen en

ables constrained optimization and consists of a combination of genetic algorithm downhill

simplex and gradient metho d

Airfoil Parameterization

To p erform numerical optimizations prop er design variables have to be chosen i e the

airfoil has to b e parameterized in an adequate way Typically geometric shap e functions like

Legendre p olynomials Wagner functions or b ezier splines are applied with the resp ective

co ecients used as design variables The shap e functions may directly dene the airfoil

contour or represent p erturbations of a given basic shap e

For the presentinvestigations another approachwas preferred Instead of optimizing the

contour in a direct way the inverse conformal mapping pro cedure according to Eppler has

been applied to generate the airfoil shap e The input parameters of this approach were

i

values represent the angle of used as design variables for the optimization pro cess The

i

attack relative to the zerolift line for which the outerowvelo city in a certain section i is

intended to b e constant This approach has the advantage that the values directly control

i

the lo cal pressure gradient and nally the b oundarylayer characteristics Furthermore a

spline representation of the sensitive leadingedge region is avoided

Optimization Examples

The ob jective of the present investigations was to optimize airfoils with minimized average

drag co ecient for a set of given angles of attack relativetothe inviscid zerolift line

X

c Re Min

d desig n desig n

rel to zerolift line

desig n

 can be estimated From the sp ecied regime a design lift range of  c

l

desig n

Prescribing xed values for was preferred to a direct sp ecication of c in order to

l

desig n

avoid an iteration or an interp olation of the predicted drag p olar

Altogether three dierent Reynolds numb ers were considered namely Re  

and  Since an exp erimental verication in the mo del windtunnel of the institute was

planned a value of n b eing representative for this facility compare Sec was

cr it

sp ecied for transition prediction

A rst optimization run for Re desig n

I TL 132-O (6.34 %)

 was p erformed starting from a re

designed NACA  section as the ini



airfoil No geometric or aero dynamic

tial TL 133-O (7.43 %)

constraints were intro duced at all Thus the

optimization pro cess was driven solely by

ob jective to minimize the aero dynamic

the TL 134-O (7.77 %)

drag In order to enable a detailed represen

tation of the airfoil shap e a large number of

Figure Shap es of the optimized airfoils

values were chosen as design variables

for Re   and 

i

desig n

During the optimization run up to

from top to b ottom

airfoils were generated and analyzed which

required almost h CPU time on a desktop workstation Further optimizations were per

formed for Re  and Re  The resulting airfoil contours are

desig n desig n

II III

depicted in Fig It is obvious that the shap es completely dier from the thickNACA

airfoil As exp ected the airfoil thickness reduces with decreasing design Reynolds numb er

The absolute value of tc is furthermore determined by the design lift range which is identical

for all present optimization examples

TL O airfoil b eing opti

The 2

mized for Re  will now desig n

II TL 133-O (7.43%) In Fig

b e discussed in more detail TL 133-O-smooth (7.56%)

the inviscid velo city distributions for

1.5

dierent angles of attack dashed

lines are given On the lower side

U / Uoo

the airfoil shows no signicant pres

sure recovery and for Re the desig n

II 1

b oundarylayer is predicted to be

laminar up to the trailing edge On

other hand a steep almost lin the ο ο ο ο

α = 0 , 3 , 6 , 9 (rel. to zero-lift line)

ow deceleration was intro duced

ear 0.5

on the suction side Upstream of the

pressure recovery a remarkable valley

can b e observed in the inviscid velo c

0

y distribution The upstream ank it 0 0.25 0.5 0.75 1

x / c

of the valley causes laminar separa

tion and reattachment is predicted to

Figure Shap es and inviscid velo city distributions

o ccur at the end of the valley just

of the original and the mo died TL O airfoil

upstream of the main pressure recov

ery By these means transition is xed to the ideal p osition The turbulent b oundary layer

is then abletoovercome the pressure rise without separation up to the trailing edge

For the sp ecied design conditions obviously transition initiated by a provoked separation

bubble is preferred by the optimizer instead of a continuous destabilization of the attached

b oundary layer However this result may be attributed to the limited design range or to

the simplied bubble mo del applied which determines the increase over the bubble only



from local ow prop erties at the separation and reattachment point resp ectively compare

Sec Therefore with the present aero dynamic mo del the velo city distribution inside of

the bubble do es not aect the predicted drag directly

It was decided to examine the physical characteristics of such a sp ecial valleyshap ed ve

lo city distribution in future exp erimental studies For the presentinvestigation we preferred

to lo cally smo oth the optimized velo city distribution and to apply an articial turbulator

upstream of the main pressure recovery The corresp onding airfoil shap e was calculated by

means of a mixedinverse metho d see Fig

Exp erimental Verication and Discussion

The lo cally mo died shap e was tested in the MWT with a bump tap e turbulator Nopp en

band on the suction side at chord length A comparison of the measured drag p olar

and the theoretical result assuming forced transition at the turbulator p osition is given in

Fig The measurement is repro duced quite reasonably by the present aero dynamic mo del

The drag polar shows a distinct laminar bucket which coincides with the design lift region

At rst view the optimization seems to be successful with resp ect to the ob jective function

considered To further examine the quality of the airfoil comparative windtunnel tests were

1 1

TL 133-O-smooth, turbulator at 77% US Theory, ncrit. = 12, xtra, US / c = 0.77 Experiment IAG, turbulator at 77% US RG 15, clean 0.8 0.8 Re = 300,000 Re = 300,000

0.6 0.6

cl cl

0.4 0.4

0.2 0.2

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025

cd cd

Figure Comparison of the measured drag

Figure Drag p olars for the optimized air

p olars for the optimized airfoil TL O

foil TL Osmo oth exp eriment MWT

smo oth and the RG section exp eriment

Re 

MWT Re 

p erformed for a manually designed airfoil namely the RG section Fig This airfoil

has proven very successful in mo del glider applications but was not esp ecially designed for

the design conditions considered with the present optimization Therefore it should be ex

p ected that it is inferior at least in the design lift region according to Eq However the

comparison of the measured drag p olars as depicted in Fig shows that the average drag

c for the RG and co ecient in this region is almost identical for both airfoils

d

c for the TL Osmo oth The RG oers a lower minimum drag co ecient

d

whereas the optimized airfoil has some gain near the upp er edge of the laminar bucket

This result in fact is somewhat disapp ointing The reason for the mo derate p erfor

mance of the present optimization result may rst of all be attributed to weak p oints of

the aero dynamic mo del applied for the optimization Furthermore the mo dication of the

optimized shap e may have a negative impact on the drag characteristics which has to be

veried by supplementary windtunnel tests Actually the mo dication causes an increase

of the predicted minimum drag level compared to the original TL O

further examination of the RG shows

A 2

that long but shallow separation bubbles o c

cur on both airfoil sides in the minimum

drag region Whereas on the suction side

1.5

transition is observed at x c 

tr a

b oundary layer remains laminar up to

the U / Uoo

the trailing edge on the pressure side As

1

discussed in the last section this approach

seems to be advantageous compared to a

with distinct transition and a subse design ο ο ο ο α = 0 , 3 , 6 , 9 (rel. to zero-lift line)

0.5

quent steep pressure recovery This problem



was discussed for example by Pfenninger

RG 15 (8.93%)

who preferred a distinct concave main pres

very in combination with transition sure reco 0

0 0.25 0.5 0.75 1

trol by means of a suitable turbulator

con x / c

To investigate this problem on a theoreti

Figure Shap e and inviscid velo city distri

cal basis it is necessary to account for vis

butions of the RG airfoil

cousinviscid interaction eects Further

more the application of the SquireYoung formula to determine the drag is questionable

in case of separation bubbles which extend almost into the wake Direct numerical opti

mizations considering these problems e g by using the XFOILco de mayb e for a reduced

number of design variables would therefore be valuable to obtain more insight into the

problem of optimum shap e design for low Reynoldsnumber airfoils

Exp erimental Investigations on Very Low Reynolds

Number Airfoils



At Reynolds numb ers ab ove Re  the design of the airfoil leadingedge is an imp ortant

task b ecause of the fundamental inuence on the width of the low drag bucket and the stall



behavior If the leading edge is to o sharp then a pronounced suction p eak o ccurs directly

at the leading edge for angles of attack slightly ab ove the laminar bucket This suction p eak

will cause an almost rapid jump of the transition p osition towards the leading edge which

in turn pro duces a fast forward movement of the turbulent b oundarylayer separation The

stall characteristics of such an airfoil design are p o or i e an abrupt breakdown in lift has

to be exp ected On the other hand if the nose is to o blunt overvelo cities are induced

downstream of the leading edge and the transition p osition moves upstream for lower angles

of attack than with an ideal nose geometry This reduces the width of the low drag bucket

However the stall b ehavior will b e more gentle compared to an airfoil with small nose radius

In any case at higher Reynolds numbers a laminar separation bubble near the leading edge



should be prevented b ecause of p ossible bubble bursting which will cause a sudden stall

Atlow Reynolds numb ers the situation is

10

somewhat dierent First of all the inuence H

12 T

of laminar separation bubbles is much more

20

pronounced Second the stability charac

8 18

of the laminar b oundary layer are teristics 16 δ δ

2 / 2 14

favorable therefore it is much easier more R S

12

hieve a long laminar run For Reynolds to ac 6 10

8

numbers b elow Re  the problem

6

arises how to obtain a turbulent b oundary

4

layer prior to an extended laminar separa

tion which will cause signicant additional

The common way to solve this prob

drag 2

lem is the application of an articial turbu

1.00 1.10 U /U 1.20 1.30

on the suction side In general the

lator S R

lower surface is designed to b e fully laminar

Figure Relative increase over a transi



A xed turbulator on the upp er surface has

tional separation bubble in dep endence of the

one main disadvantage if it has to b e eec

maximum shap e factor H and the outer



T

tive at higher lift co ecients then the turbu

ow velo city ratio U U according to the

S R

lator has to be placed far upstream So if

present bubble mo del

a multip oint design is considered at lower

values of c the turbulator will force earlier transition than achievable with a clean airfoil

l

A p ossible approachtoavoid this disadvantage may b e the controlled use of a transitional

separation bubble near the leading edge The bubble can b e forced to o ccur at higher angles

of attack by a steep pressure rise resulting from a sharp leading edge For lower angles of

attack the suction p eak is not present and a longer laminar run results A leadingedge bubble

must not increase the drag signicantly if no premature turbulent separation is caused This

can b e seen in Fig where the relative increase over a transitional separation bubble is



depicted as it results from the simplied bubble mo del describ ed in Sec The growth of

which nally determines the drag increase is prop ortional to the momentum thickness

 

S

at the laminar separation p oint S which is small near the leading edge

Three airfoils were designed to investigate the eect of such a forced leadingedge bubble

Because of the low design Reynolds number of Re itwas preferred to examine the

inuence of systematic geometric mo dications of a known airfoil rather than to design a

completely new shap e using theoretical metho ds To reduce the numb er of parameters to b e

varied a symmetrical airfoil NACA was chosen as initial shap e -2.0 -2.0 α α c NACA 0009 original cp 0.00 original p -0.35 -1.5 3.00 original -1.5 0.65 0.00 medium 1.65 3.00 medium 2.65 -1.0 0.00 sharp -1.0 3.65 3.00 sharp 4.65 5.65 -0.5 -0.5 6.65 7.65

0.0 0.0

0.5 0.5

contour 1.0 1.0 0.00 0.10 0.20 0.30

x/c 0.0 0.2 0.4 0.6 0.8 x/c 1.0

Figure Measured pressure distribu

Figure Inviscid pressure distributions of

tions NACA suction side exp eriment

the NACA and mo dications

MWT Re

-2.0 -2.0 α α c NACA 0009 medium c NACA 0009 sharp p -0.35 p -0.30 -1.5 0.65 -1.5 0.70 1.65 1.70 2.65 2.70 -1.0 3.65 -1.0 3.70 4.65 4.70 5.65 5.70 -0.5 6.65 -0.5 6.70 7.65 7.70

0.0 0.0

0.5 0.5

1.0 1.0

0.0 0.2 0.4 0.6 0.8 x/c 1.0 0.0 0.2 0.4 0.6 0.8 x/c 1.0

Figure Measured pressure distributions Figure Measured pressure distributions

NACA medium suction side exp eri NACA sharp suction side exp eri

ment MWT Re ment MWT Re

With the present study the inuence of the leadingedge radius r was investigated Two

le

mutations of the NACA r c were generated by sp ecifying a reduced value

le

for r but retaining the original contour for xc  The co ordinates inb etween

le

were obtained from a p olynomial approximation spline which ensures a smo oth curvature

distribution near the junction The resulting airfoils show nose radii of r c

le

designated sharp and r c designated medium resp ectively In Fig the

le

shap es and the inviscid pressure distributions for the airfoils examined are presented The

mo died airfoil sections exhibit a small overvelo city near the junction and also the desired

suction peak near the leading edge

The windtunnel mo dels were built with pressure orices on one side of the airfoil in

order to measure the pressure distribution A sp ecial NCmilled template was used to drill

the holes with a diameter of d mm exactly p erp endicular to the surface The pressure

orices were connected to a scanivalve system and a micromanometer The pressures were

measured according to c  qq An integration time of seconds was sucient to

p

obtain a reliable mean value No windtunnel corrections were applied to the data Figs

show the measured distributions for dierent angles of attack Due to a small oset in

the calibration the distributions were not measured at the same angles of attack

For the distributions show the same characteristics as the inviscid calculations

despite the fact that there is a laminar ow separation without reattachment in the rear part

The transition p osition was checked with a stethoscop e With increasing angle of attackthe

laminar separation moves upstream and forms a closed separation bubble This forward

movement starts earlier for the original NACA and the medium contour For

the separation bubble b egins just after the leading edge As exp ected the total length of

the bubble is shortest for the sharp leading edge However due to the high overvelo city at

the separation p oint a higher pressure recovery is observed in the reattachment region This

increases the momentum thickness It can also be seen that the gradient of the pressure

recovery has b ecome steep er for the mo died leading edges The measured distributions for

 show no clear picture for all cases examined From the general behavior it lo oks as

if an unsteady separation bubble exists between xc  This was checked for the

original airfoil with the stethoscop e If the stethoscop e was placed at xc and at the

b oundarylayer edge a apping can b e heard which indicates a rapid upstreamdownstream

movement of the transition p osition Due to the small diameter of the pressure orices and

the necessary tub e length unsteady eects can not b e resolved and the c distribution shows

p

mean values

Fig shows the lift co ecient plotted versus angle of attack As a result of the laminar

separation without reattachment sub critical state at angles of attackbelow the gradient

of the lift curve is much lower than the theoretical inviscid lift slop e dc d for a at

l

plate Ab ove the turbulent reattachment pro cess started i e a separation bubble is

formed whichmoves upstream with increasing angle of attack A steep c increase results for

l

this regime For the lift slop e again is clearly b elow which may be caused by

the turbulent b oundarylayer displacement eect The c is highest for the airfoils with

l

max

the mo died leading edge The stall b ehavior is mo derate for all congurations

Fig shows the measured drag characteristics Despite of the strong mo dications in the

leading edge region the drag p olars are quite similar The sub critical range of  c

l

shows no increase in drag due to separation eects A small advantage of the medium airfoil

can be observed for c 

l

The measurements show that the intended b ehavior could not b e achieved for the Reynolds

numb er considered with the variation of only one parameter namely the leadingedge radius 1.0 1.0 original sharp cl cl medium

0.5 0.5

0.0 0.0 -10 -5 0 5 α 10 0.01 0.02 0.03 0.04 0.05 cd

α π -0.5 dcl/d =2 -0.5 original sharp medium

-1.0 -1.0

Figure Measured lift co ecients vs Figure Measured drag p olars NACA

angle of attack NACA and mo di and mo dications exp eriment MWT

cations exp eriment MWT Re

Re

The main problem is that signicant mo dications of the inviscid pressure distribution can

b e egalized due to the dominating viscous eects at suchlow Re More combined theoretical

and exp erimental investigations are necessary in future to establish a straightforward design

methology for very low Reynolds number airfoils

Conclusion and Outlo ok

A numerical optimization to ol was applied to the design of low Reynolds number airfoils

with minimized drag for a sp ecied lift region The airfoil optimized for Re

desig n

was tested in the Mo del WindTunnel of the institute The predicted drag characteristics is

in go o d agreement to the measured polar and the resulting laminar bucket coincides with

the design lift region The comparison to a classical manual design RG however shows

similar integral drag values for the design lift range considered An improvement could not

be achieved with the present optimization which may be attributed to diculties to mo del

exactly the inuence of transitional separation bubbles extending into the wake

In order to deduce guidelines for the design of very low Reynoldsnumber airfoils wind

tunnel tests have b een p erformed at Re Thereby the inuence of relevant geomet

ric parameters was investigated In a rst campaign the NACA plus two mo dications

showing smaller leadingedge radii were examined The measurements show that strong vis

cousinviscid interaction eects o ccur along with unsteady ow phenomena resulting from

large laminar separation bubbles To get more insightinto the very complex owphysics the

leadingedge separation phenomena will b e further studied applying dierentow visualiza

tion techniques A careful interpretation and a comparison of measurement and theoretical

prediction will be necessary to nally establish prop er airfoil design criteria for this sp ecial

Reynoldsnumber regime

References



Drela M Xfoil An Analysis and Design System for Low Reynolds Number Airfoils

Low Reynolds Number Conference Proceedings edited by T J Mueller

University of Notre Dame



Eppler R Airfoil Design and Data Springer Verlag BerlinHeidelb ergNew York

ISBN X

Althaus D Niedriggeschwindigkeitsprole Friedrich Vieweg Sohn Verlagsge

sellschaft mbH BraunschweigWiesbaden ISBN

Bo ermans L M M and van Garrel A Design and Windtunnel Test Results of a

Flapp ed Laminar Flow Airfoil for HighPerformance Sailplane Applications Technical

SoaringVol No pp

Drela M LowReynoldsNumb er Airfoil Design for the MIT Daedalus Prototyp e A

Case Study Journal of Aircraft Vol No



Pfenninger W and Vemuru C S Design of Low Reynolds Number Airfoils I

Journal of Aircraft Vol No March



Selig M S Donovan J F and Fraser D B Airfoils at LowSpeeds Soartech No

H A Stokeley North Horsesho e Circle Virginia Beach Virginia USA



Selig M S et al Summary of LowSp eed Airfoil Data Volume Departmentof

Aeronautical and Astronautial Engineering University of Illinois at UrbanaChampaign



Lutz Th Berechnung und Optimierung subsonisch umstromter Prole und Rota

tionskorp er Dissertation Institut fur Aero dynamik und Gasdynamik Universitat

Stuttgart VDI FortschrittBerichte Reihe Stromungstechnik Nr ISBN



Lutz Th and Wagner S Numerical Shap e Optimization of Subsonic Airfoil Sections

In ECCOMAS European Congress on Computational Methods in Applied Sciences

and Engineering September Barcelona Spain Tobe published



Schmitz F W Aero dynamik des Flugmo dells Luftfahrt und SchuleReihe IVBand

C J E Volckmann Nachf E Wette BerlinCharlottenburg



Hamma W Neu entwickelte Tragugelprole fur Re mit den zugehorigen Polaren

und Ergebnissen aus Grenzschichtb eobachtungen an diesen Prolen Diploma thesis Teil

II Institut fur Aero dynamik und Gasdynamik Technische Ho chschule Stuttgart



Weigold W Untersuchungen zur halb empirischen Ermittlung des Umschlags in lam

inaren Abloseblasen und Programmierung eines vereinfachten Blasenmo dells Diploma

thesis Institut fur Aero dynamik und Gasdynamik Universitat Stuttgart



van Ingen J L A Suggested SemiEempirical Metho d for the Calculation of the Bound

ary Layer Transition Region Rep ort VTH VTH Department of Aeronautical

Engineering Delft University of Technology



Smith A M O and Gamb eroni N Transition Pressure Gradient and Stability The

ory Rep ort ES Douglas Aircraft Company



Tani I LowSp eed Flows Involving Bubble Separations Progress in Aeronautical

SciencesVol pp Pergamon Press New York



Horton H P A SemiEmpirical Theory for the Growth and Bursting of Laminar Sep

aration Bubbles Current Pap er CP No Ministry of Technology Aeronautical

Research Council Queen Mary College University of London



Althaus D Prolp olaren fur den Mo dellug I Neckar Verlag VillingenSchwenningen



Althaus D Prolp olaren fur den Mo dellug I I Neckar Verlag Villingen

Schwenningen



Abb ott I H von Do enho A E and Stivers L S Summary of Airfoil Data NACA

Rept NACA



Mack L M A Numerical Metho d for the Prediction of HighSp eed BoundaryLayer

Transition Using Linear Theory NASA SP NASA



van Ingen J L and Bo ermans L M M Research on Laminar Separation Bubbles at

Delft UniversityofTechnology in Relation to Low Reynolds Numb er Airfoil Aero dynam

ics Proceedings Conference on Low Reynolds Number Airfoil Aerodynamics UNDAS

CPBpp Indiana USA



Saric W S LaminarTurbulentTransition Fundamentals AGARD Report

Special Course on Skin Friction Drag Reduction AGARD



Schubauer G B and Skramstad H K Laminar Boundary Layer Oscillation and

Transition on a Flat Plate NACA Rept NACA



Synaps Inc Pointer Optimization Software Release Users Guide Clairmont

Road Suite Atlanta GA USA



Althaus D Eects on the Polar due to Changes or Disturbances to the Contour of the

Wing Prole Technical SoaringVol No



Gaster M The Structure and Behaviour of Laminar Separation Bubbles AGARD

CP Flow Separation Part IIpp NATO AGARD