Vortex pannel numerical method.pdf

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O R I G I N A L A R T I C L E

Engineering the race car wing: application of the vortex panelnumerical method

Pascual Marques-Bruna

Published online: 1 April 2011

International Sports Engineering Association 2011

Abstract This study examined aerodynamic properties

and boundary layer stability in five cambered airfoilsoperating at the low Reynolds numbers encountered in

motor racing. Numerical modelling was carried out in the

flow regime characterised by Reynolds numbers 0.82

1.29 9 106. The design Reynolds number of 3 9 106 was

used as a reference. Aerodynamics variables were com-

puted using AeroFoil 2.2 software, which uses the vortex

panel method and integral boundary layer equations.

Validation of AeroFoil 2.2 software showed very good

agreement between calculated aerodynamic coefficients

and wind tunnel experimental data. Drag polars, lift/drag

ratio, pitching moment coefficient, chordwise distributions

(surface velocity ratio, pressure coefficient and boundary

layer thickness), stagnation point, and boundary layer

transition and separation were obtained at angles of attack

from -4 to 12. The NASA NLF(1)-0414F airfoil offers

versatility for motor racing with a wide low-drag bucket,

low minimum profile drag, high lift/drag ratio, laminar

flow up to 0.7 chord, rapid concave pressure recovery, high

resultant pressure coefficient and stall resistance at low

Reynolds numbers. The findings have implications for the

design of race car wings.

Keywords Airfoil

Boundary layer

Race car

Reynolds number Validation Vortex panel method

Abbreviations

c Chord (m)

cd Profile drag coefficient

cl Lift coefficient

cl=cdmax Maximum lift/drag ratiocm a/c Pitching moment coefficient about the

aerodynamic centre

cp Pressure coefficient

i, j Control point/panel designation

n Panel number

ni Unit vector

PR Resultant pressure coefficient

v/V Surface velocity ratio

V?

Free-stream velocity (m/s), car velocity (km/h)

Re Chord-related Reynolds number

Rex Boundary-layer Reynolds number

Rexcr Critical Reynolds number

T Air temperature (Kelvin)

Sj Panel length (m)

x, y Cartesian coordinates/distance from origin

xcr Boundary-layer transition (critical) point (x/c)

xsep Boundary-layer separation point (x/c)

xstag Boundary-layer stagnation point (x/c)

a Geometric angle of attack ()

a0 Zero-lift angle of attack ()

bi Angle between V? and ni ()

cj Vortex strength at panel j (m2/s)

d Boundary layer thickness (mm)

hij Angle between panels i and j ()

l?

Air viscosity [kg/(m s)]

q?

Air density (kg/m3)

1 Introduction

Motor racing requires the application of principles of aero-

nautical engineering for the design of downforce-generating

P. Marques-Bruna (&)

Faculty of Arts and Sciences, Edge Hill University,

St Helens Road, Ormskirk, Lancashire L39 4QP, UK

e-mail: [email protected]

Sports Eng (2011) 13:195204

DOI 10.1007/s12283-011-0064-5


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wings. A wing is a three-dimensional lifting surface of finite

span (Fig. 1) composed of one or more two-dimensional

airfoil sections of theoretical infinite span [1]. The Reynolds

number (Re) characterises the fluid flow regime over objects

of geometric similarity [2] and the lowest Re used in airfoil

design for aviation has typically been 3 9 106 [35]. Airfoil

data based on very low Re are also available in the open

literature. Data obtained at Re as low as 0.17 9 106 arereported by Katz [6] and the lift (cl) and profile drag (cd)

coefficients and boundary layer development are different to

those observed in full-size aircraft. Simons [7], in his rare

work on model aircraft aerodynamics and unmanned aerial

vehicles (UAVs), also provides aerodynamic data for air-

foils of small chord (c) functioning at very low Re

(0.02 B Re B 0.25 9 106). However, race car wings oper-

ate in the low-Re flow regime ([0.5 9 106). Airfoils

belongingto different periods in thehistory of aerodynamics,

ranging from early-design to natural laminar flow (NLF)

airfoils, feature in contemporary race cars [6]. However,

there is limited information regarding aerodynamic proper-ties and, particularly, boundary layer stability of small-chord

airfoils originally designed for aviation but operating at the

low Re encountered in motor racing.

It is well known that the performance of airfoils

designed for Re[ 0.5 9 106 deteriorates considerably at

very low Re due to the formation of a separation bubble

immediately aft of the minimum pressure point [7, 8]. At

high a, the adverse pressure gradient is severe, the bubble

bursts and the airfoil stalls. Laminar separation bubbles are

likely to cause significant departures of cl and cd from

theoretical predictions [7, 8]. In large aircraft, laminar flow

rarely persists far behind the leading edge because the Re is

high and transition occurs without separation. Similarly,

race car wings operate at Re[ 0.5 9 106 and the literature

[68] suggests that the boundary layer makes a natural

unforced transition, therefore aerodynamic coefficients

may be obtained by calculation. Also, Simons [7] has

explained that in thicker laminar flow airfoils favourable

flow conditions are preserved over a greater range of cl. In

fact, the minimum drag is higher; however, the low drag

bucket is wider in thicker airfoils. A similar effect is caused

when lowering the Re, whereby the minimum drag isslightly higher but the bucket is wider. This is because the

relative viscosity of air at low Re is greater compared with

density, velocity and chord factors and the boundary layer

is laminar for a greater distance, which widens the bucket.

Thus, the low drag bucket of laminar flow airfoils is

expected to vary according to profile thickness and Re.

The vortex panel method assumes an ideal flow (invis-

cid, incompressible [4]) and eliminates the restrictions of

thin airfoil theory, limited to airfoils of thickness B12% at

geometric angles of attack (a) below the stall [9, 10]. The

vortex panel method permits the computation of lift, since

vortices have circulation [1]. However, the Kutta conditionmust be satisfied [2], whereby flow from the upper and

lower airfoil surfaces joins smoothly at the trailing edge

producing vortex strength of zero. Boundary layer stability

may be examined using aerodynamics theory [1, 1113].

Boundary layer thickness (d) can be approximated using

equations for low-speed incompressible laminar flow and

the corresponding equations for turbulent boundary layers

[4]. The d grows parabolically with distance from the

leading edge, whereby turbulent layers grow at a faster rate

than laminar layers [2]. Laminar-turbulent transition occurs

at a point downstream the leading edge where the laminar

boundary layer becomes unstable and microscopic bursts of

turbulence begin to form [5]. Thus, the vortex panel

method and boundary layer equations may be used in the

theoretical analysis of airfoil aerodynamics.

The vast majority of experimental airfoil data have been

obtained using Re C 5 9 106 intended for aircraft [3, 5,

1418]. Several studies have revealed the complexity of

boundary layer stability in model aircraft and UAVs that

function at very low Re [7, 8]. However, reports of airfoil

aerodynamics in the low-Re flow regime characteristic of

race cars are sparse [6]. Also, it remains to be shown

whether laminar flow airfoils are superior to earlier types

when operating at low Re. Thus, this study aimed to

examine aerodynamic properties and boundary layer

stability in five airfoils originally designed for aviation but

operating at low Re. It was hypothesised that: H1the

aerodynamic properties of airfoils deteriorate and the

boundary layer tends to destabilise when operating at

off-design Re and H2laminar flow airfoils possess

superior aerodynamic properties than earlier types in the

flow regime that applies to race car wings. The findings

have implications for the design of high-downforce wingsFig. 1 A rear wing assembly with variable incidence settings, a

Gurney tab and side fins mounted on a Grand Touring sports car

196 P. Marques-Bruna


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that allow the race car to take the corners at high speed and

accelerate and brake effectively, thus enhancing car

performance and safety.

2 Method

2.1 Description of the airfoils

Five airfoils representative of different periods in the history

of aerodynamics [3, 14] ofc = 0.3 m were examined. These

included four National Advisory Committee for Aeronautics

(NACA) airfoils of the 4, 5 and 6-series (designations NACA

2412, 23012, 64206 and 651-412) and one National Aero-

nautics and Space Administration (NASA) airfoil with

extensive NLF (designation NLF(1)-0414F). Airfoil ordi-

nates [3, 15, 18, 19] were reconstructed using Delcam Pow-

erSHAPE-e 8080 computer-aided-design (CAD) software

(Fig. 2). The airfoils zero-lift a (a0) is also shown in Fig. 2.

The NACA 2412and 23012 are classical early-design airfoils,the NACA 64206 and 651-412 have a small-radius leading

edge [3, 18] and the NACA 651-412 and NLF(1)-0414F

possess advanced laminar flow characteristics [7, 18, 20].

2.2 Numerical analysis

Aerodynamics variables were obtained using AeroFoil 2.2

software [21], which uses the vortex panel method [4, 10]

and integral boundary layer equations [1, 11, 13, 22]. The

vortex panel method is based on the philosophy of covering

the airfoil surface with a vortex sheet of such strength that

the airfoil surface becomes a streamline of the flow (Fig. 3;based upon Anderson [4]). The airfoil geometry is recon-

structed using a series of vortex panels.

The vortex panel method is governed by the equation

[4, 10]

V1Cosbi Xnj1

cj

2P

Zj

ohij

onidsj 0;

where V?

is the freestream velocity, bi the angle between

V?

and ni, i the control point at which the vortex strength is

being calculated, n the panel number, j the panel which is

inducing some vortex at i, cj the vortex strength at j, ni the

unit vector normal to the ith panel, sj is the length of panel j,

and hij is given by

hij tan1 yi yjxi xj

where x and y are the coordinates of the control points at

panels i and j, respectively.

Boundary layer thickness (d) for low-speed incom-

pressible laminar flow was approximated using the soft-

ware [21] which utilises the equation [4, 22]

d 5:0xffiffiffiffiffiffiffi

Rexp ;

where x is distance from the airfoil leading edge and

Rex is the boundary-layer Reynolds number. For turbulent

boundary layers, the corresponding equation is [4, 22]

d 0:37xffiffiffiffiffiffiffiffiffiffiRe1:5x

p :Location of the transition point (xcr), in effect a finite

transition region, was calculated using the software [21]

NACA 2412

Camber: 2.00%

0: - 2

NACA 23012

Camber: 1.83%

0: - 1

NACA 64206

Camber: 1.12%

0: - 3

NACA 651-412

Camber: 2.14%

0: - 3

NASA NLF(1)-0414F

Camber: 2.70%

0: - 4

x/c

y/c

chord line mean camber line

Fig. 2 The five airfoils with their corresponding maximum camber,

a0, and chord and mean camber lines

(a)

(b)

Fig. 3 Distribution ofa a vortex sheet and b a series of vortex panels

over the surface of an airfoil

Race car wing engineering 197


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based upon air viscosity (l?

), air density (q?

) and the

experimentally determined critical Reynolds number

Rexcr at which transition occurs [1, 13].

xcr l1Rexcrq1V1

:

2.3 Validation of the numerical method

Validation tests of AeroFoil 2.2 software were carried out

using drag polars, pressure coefficient (cp) and pitching

moment coefficient about the aerodynamic centre (cm a/c) at

Re of 2, 3 and 3.1 9 106 and using the five airfoils.

Agreement between the numerical method and wind tunnel

experimental data [3, 5, 18, 20] was determined using root

mean square error (RMSE) values. The mean RMSE values

for cl, cd and cm a/c for -4 B a B 12, a t 1 intervals,

were calculated. The mean cp RMSE values of the

NLF(1)-0414F upper and lower surfaces were calculated at

a0 ? 2 and a0 ? 8.

2.4 Aerodynamics variables

It was determined that race car wings operate typically in

the flow regime of 0.82 B Re B 1.29 9 106. For example,

International Standard Atmosphere (ISA) thermodynamic

conditions ofq?

= 1.225 kg/m3 and l?

= 17.89 9 10-6,

car V?

= 200 km/h (55.6 m/s) and c = 0.3 m yield

Re = 1.14 9 106; obtained using [2]

Re q1

V1

c

l1 ;

where l?

was calculated using standard air temperature

(T) in Kelvin, thus 288.15 K, as follows:

l1 1:458 106T1:5

T 110:4

:

The following variables were calculated for each air-

foil (at 0.82 B Re B 1.29 9 106 and -4 B a B 12 at

1 intervals): cl, cd, maximum lift-to-drag ratio cl=cdmax;a for cl=cdmax ; cm a/c, surface velocity ratio (v/V) dis-

tribution, cp distribution, d distribution, flow stagnation

point (xstag), xcr, and flow separation point (xsep) [2, 4, 6, 11,

15, 17, 20]. The v/V is the surface flow velocity relative to

V?

. Drag polars were constructed, including drag polars

for a simulated design Re of 3 9 106 [3]. The v/V, cp and

d distributions, xcr and xsep were obtained for both surfaces.

The v/V, cp and d distributions at a0 ? 2 and a0 ? 8 were

selected to display airfoil aerodynamics at low and high

a and plotted against c. The cp was integrated over each

airfoil surface and the resultant cp for the complete airfoil

(PR) calculated at a0 ? 2 and a0 ? 8.

3 Results


Calculated and published experimental data were in very

good agreement (Fig. 4). RMSEs for cl and cd (Table 1)

were smaller than the changes in the magnitude of these

coefficients associated with sampling at a = 1 intervals.The cp RMSEs for the NLF(1)-0414 were slightly greater

in upper surface calculations.

3.2 Drag polars, cl=cdmax and cm a/c

Theoretical drag polars are presented in Fig. 5. The partial

stall of the NACA 23012 at low Re and of the NACA

651-412 at the design Re is shown (i.e., a sudden decline in

cl accompanied by increasing cd, at high cl settings). All

airfoils show decreased aerodynamic efficiency at low Re,

typified by lower peak cl and generally greater cd for a

given cl. The thin-profile NACA 64206 produces very lowdrag at cl = 0, but shows a narrow range of effective cl at

Fig. 4 Experimental (Exp) cl and cd data and calculated values

obtained using AeroFoil 2.2 software (AF) for two selected airfoils

Table 1 RMSE for aerodynamic coefficients

cl cd cm a/c cp

NACA 2412 3.1 9 106 0.04 0.0004 0.009

NACA 23012 3 9 106 0.05 0.0004 0.005

NACA 64206 3 9 106 0.04 0.0009 0.005

NACA 651-412 3 9 106 0.04 0.0016 0.018

NLF(1)-0414F 3 9 106 0.03 0.0008 0.011

NLF(1)-0414F 2 9 106 0.05 0.0013 0.007

NLF(1)-0414F 3 9 106

a0 ? 2 Upper 0.12

a0 ? 2 Lower 0.08

a0 ? 8 Upper 0.11

a0 ? 8 Lower 0.10



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low Re as it stalls abruptly at cl & 0.40.5. The NACA

651-412 stalls abruptly in the vicinity of cl = 0.9 at

Re = 0.82 9 106; however, it reaches cl = 1.3 before

stalling abruptly at Re = 1.29 9 106. In the NACA

651-412 and NLF(1)-0414F the low-drag bucket becomes

narrower at low Re. The NLF(1)-0414F attains very low cd(minimum cd = 0.0036 at cl = 0.425, design Re) and

shows stall resistance. Outside its low-drag bucket there is

a steep linear increase in cd with cl at Re = 0.82 9 106.

The cl=cdmax declines with decreasing Re in all airfoils(Table 2). The highest cl=cdmax is attained by the NLF(1)-

0414F in any flow regime. However, a for cl=cdmax increases

in some airfoils but decreases in others with changes in Re.

The NACA 651-412 yields the highest cm a/c, with a sudden

increase in cm a/c at the stall (a & 6; Fig. 6). A similar

pattern is observed for the other airfoil with a sharp leading

edge, NACA 64206, where a more gradual increase in cm a/cat the stall (a & 3) occurs.

3.3 v/V, cp and d distributions

Figure 7 shows v/V, cp and d distributions at Re = 1.29 9

106 only, since data at Re = 0.82 9 106 were similar. At

a0 ? 2, peak upper surface v/V occurs at 0.2c in the

Fig. 5 Drag polars at low Re (0.82 and 1.29 9 106) and the design Re of 3 9 106

Table 2 cl=cdmax and the a atwhich it occurs

Re = 0.82 9 106

Re = 1.29 9 106 Design Re = 3 9 10

6

cl/cd max a () cl/cd max a () cl/cd max a ()

NACA 2412 90 5 98 6 98 4

NACA 23012 83 7 93 7 110 8

NACA 64206 46 2 60 3 83 4

NACA 651-412 86 2 95 3 125 3

NASA NLF(1)-0414F 119 6 125 5 156 3



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NACA 2412, nearer the leading edge (0.1c) in the NACA

23012, late in the 6-series airfoils (0.5c in the NACA

64206 and 0.6c in the NACA 651-412) and as late as 0.7c

in the NLF(1)-0414F. Over most of the upper surface, theNACA 2412 and 23012 show an adverse pressure gradient

and the NACA 64206 a flat gradient. However, the NACA

651-412 and NLF(1)-0414F are capable of maintaining a

favourable cp gradient up to 0.6c and 0.7c, respectively.

The lower surface cp distribution contains regions of

negative pressure in all airfoils. At a0 ? 8, there is rapid

acceleration of the upper surface flow around the leading

edge in all airfoils, and more prominently in the NACA

64206 (v/V = 0.74). In the NACA 64206 and 651-412, xcroccurs immediately aft of their sharp leading edges. The

NACA 651-412 shows increased pressure recovery from

&0.8c and the NLF(1)-0414F shows the rapid concave

pressure recovery from 0.7c characteristic of NLF airfoils.

The d distribution displays a logarithmic development of

the boundary layer (Fig. 7). A thick turbulent boundary

layer develops over the upper surface of the NACA 23012,

reaching 2.3 mm (a0 ? 2) and 2.8 mm (a0 ? 8) at xsep.

Interestingly, in the NACA 651-412, d is greater on the

lower surface than on the upper surface (at a0 ? 2), and in

the NLF(1)-0414F there is a sharp increase in d from 0.7c.

Table 3 shows integrated cp and the PR. The NACA

651-412 attains a high PR at a0 ? 8, whereas the NLF(1)-

0414F yields high PR at any a.

3.4 xstag, xcr and xsep

The xstag migrates aft of the leading edge as a deviates from

0 (Fig. 8). For a[ 0, xstag is observed to migrate farther

downstream in the airfoils with a small-radius leading edge

(NACA 64206 and 651-412).

Figure 9 shows xcr and xsep as a function of a and Re.

For xcr, only data at Re = 0.82 9 106 are shown, since data

at Re = 1.29 9 106 were similar. In contrast, xsep was

affected by Re. With increasing a, xcr shifts forward on the

upper surface and aft on the lower surface. In the NACA

64206, xcr migrates rapidly on both surfaces as a increases

from -1 to 4 and the airfoil stalls. The NACA 651-412 is

capable of restraining the upper surface xcr migration (up to

a = 2) and the lower surface migration (from a = 0).

Similarly, in the NLF(1)-0414F, xcr remains near 0.7c for

a B 5 (upper surface) and for a C -1 (lower surface).Increased Re (1.29 9 106) had the effect of delaying and

even preventing separation, thus higher a was required for

separation to occur (Fig. 9). When separation did occur, it

took place nearer the trailing edge. The NACA 64206

experiences a rapid shift in upper surface xsep from a = 0

at Re = 0.82 9 106 and from a = 4 at Re = 1.29 9 106.

This is the only airfoil that shows lower surface separation

near the leading edge, which occurs at -4 B a B -1.

The other airfoil with a small-radius leading edge (NACA

651-412) also experiences a rapid shift in upper surface xsep(from a & 6 at Re = 0.82 9 106 and from a & 9 at

Re = 1.29 9 106). In the NLF(1)-0414F, the upper surfacexsep shows some to-and-fro migration at a beyond the upper

boundary of the low-drag bucket, but remains at&0.8c. At

Re = 0.82 9 106, the lower surface xsep migrates towards

the trailing edge at a outside the low-drag bucket.

However, at Re = 1.29 9 106 there is late (C0.8c) lower

surface separation for -3 B a B 1.

4 Discussion


The calculated coefficients showed very good agreement

with experimental data [3, 5, 18, 20] This suggests that the

accuracy of the numerical method [21] is acceptable for the

analysis of airfoil aerodynamics (Fig. 4). Accuracy was

lowest in the calculations for the NACA 651-412 and

the upper surface cp distribution for the NLF(1)-0414

(Table 1); due perhaps to the complex geometry and subtle

viscous effects in these two airfoils [8, 19, 20].

4.2 Drag polars, cl=cdmax and cm a/c

When operating at cl within its low-drag bucket, the

NLF(1)-0414F is the most efficient airfoil (Fig. 5). This is

due primarily to its rearward position of the minimum

pressure that decreases cd [4, 5]. All airfoils are less effi-

cient at low Re. This is expected, as Abbott et al. [5] and

Bertin [2] reported that cf and cd decline with increasing Re

up to Re & 20 9 106. The airfoils with a small-radius

leading edge, NACA 64206 and 651-412, have a narrow

range of operational cl below the stall and may be used as

stabilisers [2, 14]. In off-design conditions, the two laminar

Fig. 6 cm a/c at low Re for the five airfoils



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flow airfoils show higher minimum drag, in agreement with

previous observations [7, 8]. However, the low-drag bucket

shrinks with decreasing Re, which differs from previous

experimental data obtained at very low Re (\0.5 9 106

[7, 8]) and very high Re (3 B Re B 9 9 106 [5]). In the

NLF(1)-0414F, the rapid increase in cd beyond the upper

Fig. 7 v/V, cp and d distributions over the upper and lowersurfaces of the airfoil (Re = 1.29 9 106). xcr (white circles) and xsep (black circles)



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boundary of the low-drag bucket at low Re (Fig. 5) can be

attributed to its thick profile (14% [3]), moderate camber

(2.70%; Fig. 2) and steep aft pressure recovery at high cl

[5, 18]. Nonetheless, the NLF(1)-0414F is stall resistant athigh a due to a thicker leading edge than typical for NLF

airfoils [20]. Both laminar flow airfoils retain acceptable

aerodynamic properties at low Re, provided that they

operate at cl within their low-drag bucket.

The cl=cdmax drops with decreasing Re (Table 2).

However, the findings suggest that the NLF(1)-0414F is

specially suited for low-Re operation. At low Re, the early-

design NACA 2412 shows a higher cl=cdmax than the 5 and

6-series airfoils, which is not the case at high Re [5]. In

particular, the NACA 64206 attains a very low cl=cdmax at

Re = 0.82 9 106 despite its thin profile. The NACA

651-412 shows sizeable pitching tendency (up to cm a/c =

-0.10 at the stall at a & 6; Fig. 6), mainly due to its

considerable maximum camber (2.14%) [5]. Large cm a/ctends to cause geometric twist, which decreases a and can

reduce downforce [6, 7]. Thus, a wing with NACA 651-412

sections should be constructed with sufficient torsional

rigidity (Fig. 1) to prevent geometric twist. The analysis

unveils the greater versatility (wide low-drag bucket, low

minimum cd, high cl=cdmax and stall resistance) of theNLF(1)-0414F for motor racing.

4.3 v/V, cp and d distributions

The classical NACA 23012 has its maximum camber far

forward on the airfoil (Fig. 2). This explains the pressure

peak near the leading edge (Fig. 7) and the extensive

region of adverse pressure gradient [5]. The small leading

edge radius of the NACA 64206 helps achieve low drag

and suppress leading edge negative cp peaks at low a.

However, at high a the sharp leading edge causes a large cppeak due to centripetal forces turning the air molecules

around the leading edge [5, 18, 20]. The large cp peak

generates a steep pressure gradient just aft of the leading

edge, immediate transition [22] and early flow separation

(at 0.2c). This escalates form drag [2] and leads to an

abrupt stall. The thick turbulent boundary layer over the

upper surface of the NACA 23012 increases the airfoils

effective camber, which generates more lift at the expense

of greater profile drag [7]. The effects of the greater

thickness of the NLF(1)-0414F (14%) are evident at

a0 ? 2 (Fig. 7), including high maximum v/V and a long

favourable cp gradient, based upon Abbott et al. [5].However, its thin rear end (see CAD-generated profile;

Fig. 2) produces an inflection point in the v/V curve

at&0.7c and subsequent rapid adverse cp gradient. This is

suggestive of high dynamic instability [13, 19] and

explains the sudden thickening of the boundary layer in this

region. However, the concave-type pressure recovery used

in the NLF(1)-0414F helps lessen the severity of turbulent

separation [18]. The high PR of the NACA 651-412 and

NLF(1)-0414F at large a (Table 3) indicates the greater

capacity of the laminar flow airfoils to generate downforce.

4.4 xstag, xcr and xsep

Migration ofxstag from the leading edge (Fig. 8) adversely

affects pressure gradients and boundary layer stability [19,

20]. To control xstag migration, airfoils with a sharp leading

edge may be fitted with a small-chord (0.100.15c) trailing

edge flap of the same airfoil geometry as the main element.

A flap helps trade lift due to a for lift due to flap deflection

by loading the aft section of the main airfoil [15, 17,

19, 20]. Thus, high cl can be achieved while still keeping

Table 3 Integrated cp and PR(Re = 1.29 9 106)

a0 ? 2 a0 ? 8

cp Upper cp Lower PR cp Upper cp Lower PR

NACA 2412 -0.29 -0.07 -0.22 -0.68 0.24 -0.92

NACA 23012 -0.29 -0.06 -0.23 -0.67 0.21 -0.88

NACA 64206 -0.24 0.00 -0.24 Stall Stall Stall

NACA 651-412 -0.33 -0.12 -0.21 -0.86 0.24 -1.11

NASA NLF(1)-0414F -0.38 -0.04 -0.34 -0.80 0.24 -1.04

Fig. 8 Migration of xstag with a



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xstag near the leading edge. This maintains favourable cpgradients on both surfaces and delays both transition and

separation. Particularly, the NACA 651-412 shows early

transition at a0 ? 8 (Fig. 7) which can be delayed using

the flap. Given the thick leading edge of the NLF(1)-

0414F, the use of a small flap should inhibit xstag migration

and widen the low-drag bucket, based on Vicken et al.

[18, 19].

In the NACA 2412, 23012 and 651-412, xcr moves

upstream with increasing a (Fig. 7). However, the copious

interchange of momentum within the turbulent boundary

layer allows the layer to remain attached despite increased

Rex and d [5]. Increased Re also helps delay and even

prevent separation (Fig. 9), since high Re is favourable to

the development of turbulence which energises and adds

stability to the boundary layer [5, 11, 13]. Interestingly, in

the NACA 651-412, xcr occurs earlier in the lower surface

than in the upper surface at a0 ? 2 (Fig. 7), which is

uncommon at high Re [14]. According to Murri et al. [20],

the onset of upper-surface trailing-edge separation for the

NLF(1)-0414F is a = 4 (Re = 2 9 106). At lower Re,

flow separation is observed at any geometric a (Fig. 9). In

agreement, Murri et al. [20] predicted turbulent flow sep-

aration in the pressure recovery region to occur at off-

design conditions for the NLF(1)-0414F, unless a boundary

layer energiser is used. Installation of a spanwise row of

vortex generators at 0.6c may improve lift and reduce drag

[20]. However, the effect of vortex generators on boundary

xcr

xcr

xsep

xsep

xsep

xsep

Fig. 9 xcr (top) and xsep (middle and bottom) as a function ofa and Re



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Documents

Vortex pannel numerical method.pdf