1

Numer ical Study of the Subsonic Flow around NLR7301 Air foil Using a Non-Linear Turbulence Model

Pedro Simões Flores Viana1 and Luís Fernando Figueira da Silva 2 Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22453-900, BRAZIL

André Gama de Almeida3 and André Luiz Martins4 Empresa Brasileira de Aeronáutica SA – EMBRAER, São José dos Campos, SP, 12227-901, BRAZIL

The computation of lift and drag on wings at high angle of attack is of utmost impor tance for aircraft design. The exact prediction of the maximum lift coefficient, the angle of attack for which air foil stall occurs, and of the drag, is relevant ear ly in the design process. In the past, those parameters were obtained from wind tunnel tests exclusively. This paper descr ibes part of the results of an on-going effor t to compute air foil aerodynamic character istics by the use of Computational Fluid Dynamics techniques. A computational study of the flow around a classical air foil, the NLR7301, is presented. The flow conditions considered are representative of subsonic flight, and emphasis is given on the prediction of the maximum lift coefficient. Turbulence is modeled using a non-linear cubic k-εεεε model, which accounts for near wall low Reynolds number flows. A study of the sensitivity of the computed results with the size of the computational mesh is presented, as well as a compar ison between different hybr id meshes configurations. Lift and drag curves are obtained for this air foil. The computed results for pressure, lift, drag and fr iction coefficients are presented and compared to exper imental results.

I . Introduction The airfoil design process and the CFD simulations that give support to it involve different problems

related to complex flowfields. The flow around airfoils, specifically when transitional or turbulent boundary layers arise, may exhibit a large sensitivity to small variations of the freestream conditions. The accurate representation of the physical phenomena involved, such as boundary layer transition and turbulence, is crucial for a correct prediction of integral quantities, as airfoil lift and drag coeffcients. This knowledge is essential during the design phase of airplanes, since the wing design process has its base on airfoil design. The goal of airfoil design varies with the application envisaged. Generally, the wing section is required to achieve its performance goals while satifying constraints on geometry (e.g. realtive thickness), pitching moment and off-design performance. The attainement of a minimum required value of maximum lift coefficient (CLmax) is an usual constraint for transport aircraft, since it defines the minimum possible flight speeds, limited by the occurrence of wing stall. The process of wing stall involves the initiation and evolution of massive boundary layer separations from the airfoil surface, shedding vorticity into the mainstream flow. To capture that kind of phenomenon, the two-dimensional Reynolds averaged Navier-Stokes (RANS) simulation has proven to be a powerful instrument, to obtain the aerodynamic coefficients and to understand the nature of the flow. However, the requirements in terms of discretization, as well as, the prediction of the boundary layer transition, and turbulence modeling for accurate predictions of the maximum lift (CL max) and lift-to-drag ratio (L/D) still remain to be clearly established. 1,3 This paper presents a computational study of the flow around the NLR7301 airfoil. The single element airfoil (i.e. no high-lift devices) problem has been chosen not only for its own industrial importance, 1 Student, Department of Mechanical Engineering, Rua Marquês de São Vicente, 225. 2 Visiting Professor, Department of Mechanical Engineering, Rua Marquês de São Vicente, 225; Senior Member AIAA. 3 Mechanical Engineer, Computational Aerodynamics Group – Embraer, Av. Brig. Faria Lima, 2170. 4 Mechanical Engineer, Computational Aerodynamics Group – Embraer, Av. Brig. Faria Lima, 2170.

43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada

AIAA 2005-1219

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

but also because it represents a first natural step towards the solution of high-lift, multi-element airfoils. The flow conditions considered are representative of moderate Reynolds numbers in subsonic flight, and emphasis is given on the prediction of the maximum lift coefficient. At such high angles of attack, near the occurrence of stallthe important modeling issues are the presence of regions of turbulent boundary layer separation, the potential existence of a laminar separation bubble, strong adverse pressure gradients and accentuate streamline curvature. The computations which are analysed in the present work were carried on with a commercial CFD code, CFD++ from Metacomp Technologies12. The definiton of a simulation strategy is driven from a engineering point of view. In particular, an exhaustive parametric exploration involving variations of all possible variables that could affect the computed results has not been attempted. This strategy has been chosen on the basis of design experience, Metacomp suggestions and the relatively short time required to obtain those results, typical of industry environements. Additionally, turbulence is modeled using a non-linear cubic k-e model5, 6, which accounts for near wall low Reynolds number flows. A study of the sensitivity of the computed results on the size of the computational mesh is presented. Curves of lift and drag coefficients are obtained for the subject airfoil. The computed results are compared to experimental results.

I I . Mathematical formulation The flowfield around the airfoil is computed for several angles of attack, ranging from zero to values

for which the flow on the suction side is fully separated. In order to compute such a complex flow, the Reynolds Averaged Navier-Stokes (RANS) equations are solved7. The turbulent fluxes that arise on these equations are closed using a non-linear cubic k-ε model5, 6. This model provides for anisotropy of the Reynolds stresses and contains low Reynolds numbers terms which allow for computations extending to the viscous sub-layer. Following the notation where the rank two Cartesian tensor are noted in bold characters (a, b, etc.),

a ija= , ab = kjikba , abc = ljklik cba ,

a2 = kjik aa , a2 = kiikaa , I = δij ,

one may write the cubic model in canonical form as

a = µµ fC*2− S + (1a S2 – 3

1 S2 I )

+ (2a WS - SW ) (3a+ W2-3

1W2I )

+ ( 1b S2 + ( 2b W2 ) S + (3b W2S + SW2 - 3

2SW2I )

+ 4b (WS2 – S2W). (1)

In this equation, the non-dimensional forms of the strain and anisotropy tensors are

,3

2

2 ,,,

−+= ijkkijjiij UUU

kS δ

ε ),(

2 ,, ijjiij UUk

W −=ε

(2)

.3

2~~

ijji

ij k

uua δ−= (3)

Low Reynolds number damping terms are given in Table 1. In this table, the following coefficients are used8, 9

3

,9.025.1

32*

Ω++=

SCµ (4)

,2 ijij SSS = ,2 ijijWW=Ω (5)

,)/2(,1max1

1 2/1tR

RA

Re

ef

t

t

−

−

−−=

µ

µ (6)

where Rt ≡ κ2 /(νε) is the turbulent Reynolds number. This model allows for the closure of the Reynolds fluxes. One must still provide the turbulent kinetic energy and its dissipation in order to close the above equations. This is achieved by the use of a k-ε model

,)()( ρε

σµµρρ −+

∂∂

+∂∂=

∂∂+

∂∂

kjk

t

jj

j

Px

k

xkU

xt

k (7)

)()( ρερε

jjt

Ux∂∂+

∂∂

,)( 121

−+−+

∂∂

+∂∂= tk

j

t

j

TECPCxx

ρεεσµµ εε

ε

(8)

,/2* ερµ µµ kfCt = (9)

where jijik UuuP ,ρ−= is the exact turbulence production and the time scale is k/ε at large Rt, but

becomes proportional to the Kolmogorov scale, (ν/ε)1/2, for Rt<<1. The extra source term, E, is designed to increase the level of ε in non equilibrium flow regions

,0,max

∂∂

∂∂=

jj xx

k τψ (10)

,)(,max 4/12/1 υεν k= (11)

,ψερν tE TAE = (12)

Table 1: Coefficients of the non linear terms.

1a 2a 3a 1b 2b 3b 4b

31000

.3

S

f

+µ

31000

.15

S

f

+µ

31000

.19

S

f

+− µ

µµ fC 3* ).(16− µµ fC 3* ).(16− 0 µµ fC 3* ).(80

4

where τ = κ/ε, is the turbulence time scale. The model constants are Cε1=1.44, Cε2=1.92, σk=1.0, σε=1.0, Aµ=0.01 e AE=0.15. The two transport equations are subjected to the following boundary conditions at solid walls,

kw = 0, (13) εW = 2ν1k1/y1

2, (14)

where “1” denotes the centroid of the wall-adjacent cell and y1 is the corresponding normal distance. The latter boundary condition is replaced by the Neumann condition εΩ = ε1 when a wall function is invoked. This model has been validated against various flow cases across the Mach number range. Details are given in Ref. 10 for hypersonic flows and in Ref. 11 for subsonic, transonic and supersonic flows. The governing equations are discretized and solved using the techniques embedded in the CFD++ package. Steady-state solutions are sought with the aid of a pre-conditioner which increases the convergence rate.

I I I . Computational meshes used This section presents a study of the influence of different mesh configurations on the distribution of

the pressure coefficient around the NLR7301 airfoil, and on the calculated value of the lift, drag and friction coefficients. Four different values for the Alpha (angle of attack) 6.7°, 11.3°, 14.2° and 17.3° are considered, where the largest value relates to an after stall situation in the tunnel tests.

All meshes used present a hybrid configuration, i.e., they have two different cell types in its discretization: a quadrilaterals layer in the vicinity of the airfoil and triangles that span from this layer until the region of non disturbed flow, far from the airfoil (“ far field” ).

The aim of the quadrilaterals layer is to efficiently capture the viscous effects in the vicinity of the wall, i.e., at the boundary layer, and it presents an uniform height, D, along the airfoil. The height of the first cell next to the wall, d, is also constant along the airfoil and both are calculated as follows:

9.0Re893.5 −+= Cyd , 2.0Re37.0 −= CD , (15)

where Re is the Reynolds number, C is a geometric scale, in this case the chord length (C = 1m), and y+ is the classical non dimensional distance to the wall, which controls the height of the first cell and is here taken of order 1, as usual for low Reynolds numbers computations7.

The tests analyzed here present Re of 2.85.106. The values of the parameters D and d chosen are 1.89cm and 4.68 µm, respectively, which allow, according to Eq. (1), for a correct description of the boundary layer in the range 2.85 106 ≤≤ Re 6.0 106. These geometric parameters are kept constant throughout the paper, regardless the number of mesh points used.

A. Mesh A This mesh, shown in Figures 1 to

3, is made of 573210 quadrilaterals and 290483 triangles. The enlargements shown on Figures 2 and 3 denote the abrupt transition from the quadrilateral to the triangle cells, which is characterized by a change in the surface area of the computational cells of up to two orders of magnitude.

Figure 1: Mesh A around the air foil and the capsule.

5

Note that, with the purpose of achieving a better control of the mesh, an artifice dubbed “capsule” was used. That is a region around the airfoil, inside which the density of the computational cells is arbitrarily increased, with the aim of providing sufficient mesh resolution to capture the large separated, turbulent wake

expected at high angles of attack, up to post stall conditions. The effect of that technique can be observed in Figure 1. All capsules tested span from the leading-edge until 1.5 chord lengths downstream of the trailing-edge. The choice of the position of the capsule will be shown to be essential for the computations at high angle of attack. Indeed, for high values of Alpha, a correct prediction of the flow at the suction side in the vicinity of the trailing edge will be shown to be crucial for the determination of the maximum lift coefficient for this airfoil.

B. Mesh C This is the only mesh for

which the capsule was not used, as can be seen in Fig 4. The number of quadrilaterals, distributed in the boundary layer, is about 336000 (less than mesh A) and the number of triangles 36267, an order of magnitude smaller than mesh A. Without the capsule it is believed to be difficult to guarantee a good discretization and a smooth growth of the cell density when coming from the region of undisturbed flow and getting closer to the turbulent wake of the airfoil.

C. Mesh D This mesh is very similar to mesh C, with exception of the presence of a capsule, and the consequent

larger density of the cells in the upper-side and in the region of turbulent wake,. The region of boundary layer is exactly the same as for mesh C, with 336000 quadrilaterals, but the capsule used here leads to 87700 triangles. This mesh is shown in Figures 5 to 8.

A detailed exam of mesh D shows that some quality problems found in mesh A have been corrected. However, the enlargements of Figures 5 and 6 show that some mismatches of cell surface areas still exist, albeit less than those of mesh A.

Figure 3: Enlargement of Mesh A near the trailing edge.

Figure 2: Mesh A around the trailing-edge.

Figure 4: Mesh C around the air foil.

6

IV. Results and Discussion The freestream conditions for the Mach number, Reynolds number and temperature used in the present

computations are 0.201, 2.85 106 and 285K, respectively. The turbulence intensity and length scale are T =

1.0 10-2 and L = 1.0 10-3. These quantities are defined as 321 kUT −∞= and ( ) κεµ

23kCL = , with

Cµ=0.09 and =0.41. Initially it is investigated whether the numerical calculation convergence could be achieved for all

angles of attack (Alpha). The aerodynamic coefficients, particularly CL (lift coefficient) are compared to the tunnel tests results. The smallest angles of attack showed no convergence problems. However, when Alpha = 17.3°, after stall in tunnel tests, convergence could not be achieved. With the other meshes made, mesh C and mesh D, the flow is simulated in the same condition as for mesh A, yet for only one value of Alpha, the highest obtained with mesh A.

A. Computational results obtained with mesh A The computations are performed for four values of angle of attack, 6.70°, 11.32°, 14.15° and 17.29°.

For the first three values of Alpha the distribution of the pressure coefficient (CP) along the surface of the airfoil obtained with mesh A is shown in Figures 9 to 11. The corresponding values of the coefficients of lift, drag and momentum are given in Table 2, together with the measured values for these coefficients.

Figure 6: Trailing-edge of mesh D and

indication of the region with worst transition quadr ilaterals-tr iangles.

Figure 5: Mesh D around the air foil.

Figure7: Frontier quad-tr is in the the upper-side central region.

Figure 8: Detail of the worst transition region in the trailing-edge.

7

In those figures, the largest discrepancies observed between computed and measured results are found in the suction peak and near the trailing edge. Except near the trailing edge, the computations under predict CP for the upper side, while the reverse is true for the lower side. The discrepancy at the upper side is larger than that at the lower side, what, in addition to the discrepancy at the suction peak, can explain the difference on the computed CL values, which higher than the tunnel values, as shown in Table 2. Concerning CL, the highest difference, +10.15%, was obtained for Alpha equal to 14.15° and the lowest, for Alpha 6.7°, it was +8.5%.

For the drag coefficient, CD, it can be noted in Table 2 a discrepancy relative to the tunnel results considerably larger than that for CL. This discrepancy falls from 63% to 3.4%, when Alpha is raised from 6.7° to 14.15°, respectively.

The computational results obtained for CL and CD show a similar behavior to other solvers13,14 used by the Computational Aerodynamics Group at Embraer, in numerical simulations with airfoils. Particularly, the values calculated for CD at low Alpha present a worst result than those calculated at high angles.

Table 2: Comparison between the tunnel tests and the present simulations results, mesh A. test CL ∆CL CD ∆CD CM25 ∆CM25

alfa 6,70 tunnel 0.964 - 0.0108 - 0.0638 - CFD++ 1.046 8.5% 0.0176 63.0% 0.0777 21.8% alfa 11,32 tunnel 1.356 - 0.0250 - 0.0508 - CFD++ 1.489 9.8% 0.0291 16.4% 0.0675 32.9% alfa 14,15 tunnel 1.531 - 0.0385 - 0.0394 - CFD++ 1.692 10.5% 0.0398 3.4% 0.0552 40.1% alfa 17,29 tunnel 1.193 - 0.1296 - 0.0910 - CFD++ - - - - - -

-4,00

-3,00

-2,00

-1,00

0,00

1,000,00 0,20 0,40 0,60 0,80 1,00

X/C

CP

experimental

cfd++

Figure 9: Pressure coefficient along the air foil for alfa 6.70°, mesh A.

8

-8,00

-7,00

-6,00

-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,000,00 0,20 0,40 0,60 0,80X/C

CP

experimental

cfd++

Figure 11: Pressure coefficient along the air foil for alfa 14.15°, mesh A.

0,0E+00

5,0E-03

1,0E-02

1,5E-02

2,0E-02

2,5E-02

3,0E-02

3,5E-02

0,00 0,20 0,40 0,60 0,80 1,00x/c

CF

mesh A

mesh C

alfa 14.15°

Figure 12: Comparison of the fr iction coefficient along the

air foil for meshes A and C, alfa = 14.13°.

The differences observed come, at least partially, from the fact that the coefficients CL and CD are calculated from the integration of the pressure and the viscous forces along the surface of the airfoil. Consequently, for low values of Alpha, the number of cells over the airfoil in the direction perpendicular to the undisturbed flow, i.e., the direction in which the drag force acts, is relatively small, if compared to the number of cells parallel to this direction, which is the lift direction. This small spatial resolution in the transverse direction may cause a considerable error in the determination of CD. As Alpha is increased, the number of cells which contribute for the calculation of the drag also increases. Hence, this could explain the increased accuracy in the calculation of the drag when the Alpha grows. Another possible cause for the observed discrepancy in the computed values of CD is the accurate prediction of laminar to turbulent transition. Indeed, the transition is free to occur in the experiments.

The values of the moment coefficient, CM, shown in Table 2, present a large discrepancy, ranging from 22% to 40%, when compared to the tunnel tests. This coefficient also comes from the integration of the CP value along the airfoil. Therefore, the even the small discrepancies observed for CP, which are located far from the center of pressure, could influence the computed values of CM. This could contribute to the difference in CM relative to the tunnel tests.

The curve for the friction coefficient obtained for Alpha 14.1495° is given in Figure 12. In this figure it can be observed, near the trailing edge, that the curve reaches zero values, indicating that a slight separation of the flow seems to occur there.

The curve for Y+, given in Figure 13, shows a behavior similar to CF, in particular, a slight separation near the trailing edge is noted. It should also be noted that the maximal value

-7,00

-6,00

-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,00

2,00

0,00 0,20 0,40 0,60 0,80 1,00X/C

CP

experimental

cfd++

Figure 10: Pressure coefficient along the air foil for alfa 11.32°, mesh A.

9

for Y+ is approximately 3 at the suction peak. Yet, for low Reynolds number turbulence models, as used here, it is recommended that the Y+ value be below 2. The value of Y+ above 2 in the vicinity of the suction peak can explain the discrepancies observed in the determination of CP and CM.

B. Tests with different mesh configurations

Mesh A has about 850 000 cells. It was the finest mesh used in this work. Hence, the results obtained with mesh A are used as references to evaluate the results computed with meshes C and D.

In Table 3, a comparison is presented between the results obtained with meshes A, C and D for Alpha = 14.15°. It can be noted that the variation in relation to the wind tunnel value of CL is of 10.5%, for mesh A, which is the most refined one, to 12.3% for mesh C, which is the coarser one. As far as the drag coefficient is concerned, the discrepancy in relation to the tunnel value varies from 3.4% with

mesh A to 13.5% with mesh C. Figure 14 shows the

pressure distribution along the chord for the different meshes used. In this figure it can be clearly seen that only slight differences are obtained between the three meshes and that all the results overestimate the suction peak.

In Figure 15, in which are given, for the three meshes, the evolutions of CF in the vicinity of the trailing edge, it can be observed that the results obtained with meshes A and D are similar, while the values calculated for CF with mesh C are substantially higher. This discrepancy is probably due to the fact that the discretization of mesh C, downstream of the

Table 3: Compar ison of the values of the coefficients CL, CD and CM 25, obtained with the different meshes, for Alpha equal to14°.

test CL ∆CL CD ∆CD CM 25 ∆CM25 tunnel: 1,531 - 0,0385 - 0,0394 - CFD++: mesh A 1,692 10,5% 0,0398 3,4% 0,0552 40,1% mesh D 1,696 10,8% 0,0409 6,2% 0,0564 43,1% mesh C 1,720 12,3% 0,0437 13,5% 0,0615 56,1%

! !" "# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 66 67 77 78 88 89 99 9: :: :; ;< <= => >? ?@ @A AB BC CD DE EF FG GH HI IJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \\ \]^^__``aabbcdefghijklmnopqrstuvwxyz|~ ¡¢£¤¥¦§¨©ª«¬¬­­®®¯¯°°±²³´µ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÌ ÌÍ ÍÍ ÍÎ ÎÎ ÎÏ ÏÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çç çè èè èé éé éê êê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ ! !! !"#$%&'()*+,-./0123456789:;<=>?@

AABBCCDDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`aabbccddefghijkl lm mn no op pq qr rs st tu uv vw wx xy yz z | || | ~ ~~ ~ ¡¢£¤¥¦§¨©ª«¬­®®

¯¯°°±±²²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÏ ÏÐ ÐÐ ÐÑ ÑÑ ÑÒ ÒÒ ÒÓ ÓÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çè èé éê êë ëì ìí íî îî îï ïï ïð ðð ðñ ññ ñò òó óô ôõ õö ö÷ ÷ø øùúûüýþÿ

! !" "# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 11 12 22 23 33 34 44 45 55 56 66 67 77 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FGHIJKLMNOPQRSTUVWXYZ[\]^_`abcd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®® ®¯ ¯¯ ¯° °° °± ±± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁÂÃÄÅÅÆÆ

ÇÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÙÚ ÚÚ ÚÛ ÛÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çè èé éê êë ëì ìì ìí íí íî îî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûüýþÿ

! !! !" "" "# ## #$ $$ $% %& &' '( () )* *+ +, ,-./0123456789:;<=>??@@AABBCCDE EE EF FG GH HI IJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd dd de ee eff

gghhijklmnopqrstuvwxyz| ~ ~ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÌ ÌÍÍÎÎÏÏÐÐÑÑÒÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ

! !" "# #$ $% %& &' '( () )* *+ +, ,-./01234

4556677889 99 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FG GHIJKLMNOPQ QR RR RS SS ST TT TU UU UV VW WX XY YZ Z[ [\ \]^_`a ab bc cd de ef fg gh hi ij jkllmm

nnoop pp pq qr rs st tu uv vw wx xyz| ~ ~

¡¢£ £¤ ¤¥ ¥¦ ¦§ §¨©ª ª« «¬ ¬­ ­®¯° °± ±² ²² ²³ ³³ ³´ ´´ ´µµ

¶¶· ·¸ ¸¹ ¹º º»¼½ ½¾ ¾¿ ¿ÀÁ Á Âà ÃÄ ÄÅ ÅÆÇ ÇÈ ÈÉ ÉÊ ÊËËÌ ÌÌ Ì

Í ÍÍ ÍÎ ÎÎ ÎÏÏÐ ÐÑ ÑÒ ÒÓ ÓÔÕ ÕÖ Ö× ×ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åå åæ ææ æç çç çèèé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ ! !" "# ## #$ $$ $% %% %& &

& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9 9: :; ; ;< <= = => >? ?@ @A AB B BC CD DE EF FG G GH HI I I

I I IJ JJ JK KK K

L LL LM MN NO OP PQ QR RS ST TU UV V VW WX XY Y YZ Z[ [\ \ \] ]^ ^_ _ _` `a ab b bc cd de e ef fg gh h hi ij jk k kl lm mn n no op pq qr rs st t tu u uv vw wx x xy yz zz z | | || | | ~ ~~ ~ ¡ ¡¢ ¢ ¢£ £ £¤ ¤¥ ¥¦ ¦ ¦§ §¨ ¨© ©ª ª ª« «¬ ¬¬ ¬­ ­ ­­ ­ ­® ®® ®¯ ¯¯ ¯° ° °± ±² ²³ ³ ³´ ´µ µ¶ ¶ ¶· · ·¸ ¸¹ ¹ ¹º º» »¼ ¼ ¼½ ½ ½¾ ¾¿ ¿ ¿À À ÀÁ Á  Âà ÃÄ ÄÅ Å ÅÆ ÆÇ ÇÈ È ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ Ð ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ Õ ÕÕ Õ ÕÖ ÖÖ Ö× ×× ×

Ø ØØ ØÙ ÙÙ ÙÚ Ú ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá á áâ âã ãä äå åæ æç çè è èé éê êë ëì ì ìí íî î îï ïð ð ðñ ñò ò òó óô ô ôõ õö ö ö÷ ÷ø ø øù ùú ú úû ûü üý ýþ þÿ ÿ ! !" "# #$% %& &'( () )* *+ +, ,- -. ./ /0 01 12 23 345 56 66 67 77 7889 99 9: :: :; ;; ;< <

< <= == => >> >?

?@ @@ @A AA AB BB BC

CD DD DE EE EF FF FG

GH HH HI II IJ JJ JK

KL LL LM MM MN NN NO OO OP

PQ QQ QR RR RS SS ST TT TU U

U UV VV VW WW WX XX XY

YZZ[ [[ [\ \\ \] ]] ]^

^_ __ _` `` `a aa ab bb bc cc cd d

d de ee ef ff fg gg gh hh hi

ijk kl lm mn nopq qr rs st tu uvwx xy yz z | |~ ¡¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª«¬­®¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹º»¼½¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇÈÉÊËÌÍÎÏÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þßàáâãäå åæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö÷øùúûüýþÿ !"#$%&'()* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89:;<=>?@AB BC CD DE EF FG GH HI IJ JK KL LMNOP PQ QR RS ST TU UV VW WXYZ[\] ]^ ^_ _` `a abcd de e

e ef fg gg gh hh hiijj

k kk kl ll lm mm mn nn no oo opp

q qq qr rs ss stu uv vw wx xyz z | | ~ ~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥ ¥¦ ¦§ §¨ ¨ ¨© © ©ª ª« «¬ ¬­ ­® ® ®¯ ¯ ¯° ° °± ± ±² ² ²³ ³ ³´ ´ ´µ µ¶ ¶ ¶· · ·¸ ¸ ¸¹ ¹ ¹º º º» » »¼ ¼ ¼½ ½ ½¾ ¾ ¾¿ ¿À À ÀÁ Á Á  Âà ÃÄ Ä ÄÅ Å ÅÆ Æ ÆÇ Ç ÇÈ È ÈÉ É ÉÊ Ê ÊË Ë ËÌ Ì Ì ÌÍ Í ÍÎ Î ÎÏ Ï Ï

Ð Ð ÐÐ Ð Ð

Ñ Ñ ÑÑ Ñ Ñ

Ò Ò ÒÒ Ò Ò

Ó Ó Ó ÓÓ Ó Ó Ó

Ô Ô ÔÔ Ô Ô

Õ Õ ÕÕ Õ Õ

Ö Ö Ö× × ×Ø Ø ØÙ Ù ÙÚ Ú Ú ÚÛ Û ÛÜ Ü Ü ÜÝ Ý ÝÞ Þ Þß ß ßà à àá á á áâ â â âã ã ãä ä äå å åæ æ æç ç ç çè è èé é éê ê ê êë ë ëì ì ìí í í íî î îï ï ïð ð ðñ ñ ñò ò òó ó ó óô ô ôõ õ õö ö ö ö÷ ÷ ÷ø ø øù ù ùú ú úú ú ú

û û û ûû û û û

ü ü üü ü ü

ý ý ý ýý ý ý ý

þ þ þþ þ þÿ ÿ ÿ ÿ

! ! !! ! !

" " "" " "# ## #$ $% % %& & &' ' '( () )* *+ + +, ,- -. . ./ /0 01 1 12 23 34 45 56 6 67 78 89 9: :; ;< <= => >? ?@ @ @A AB BC CD DE EF FG GH HI IJ JK KL LM MN NO OP PQ Q

Q QR RR RS SS ST TT T

U UU UV VV VW WW WX XX X

Y YY YZ Z[ [\ \] ]^ ^_ _` `a ab bcd de ef fg gh hijk kl lm mn no op pqrs st tu uv vw wx xyz|~ ~ ¡¢£¤¥¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ ! !" "# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BCDEFGHIJKLMNOPQRSTUVWXYZ[\] ]^ ^_ _` `a ab bc cd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷øùúûüýþÿ ! !" "# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89:;<=>?@ABCDEFGHIJKLMNOPQRSTU UU UV VV VW WW WX XX XY YY YZ ZZ Z[ [[ [\ \\ \] ]] ]^ ^^ ^_ __ _` `` `a aa ab bb bc cc cd dd de ee ef ff fg gg gh hh hi ii ij jj jk kl lm mn no op pq qr rs stuvwxyz|~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÓ

ÔÔÕÕÖÖ××ØØÙÙÚÚÛ ÛÛ ÛÜ ÜÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ

!"#$%&'()*+,-. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FG GH HI II IJ JK KL LL LM MM MN NN NO OO OP PP PQ QR RS ST TU UV VW WXYZ[\]^_`abcdefghijklmnopqrstuuv

wwxxyyzz||~ ¡ ¡¡ ¡¢ ¢¢ ¢£ ££ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÀ ÀÁ ÁÁ Á Â Âà Ãà ÃÄ ÄÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ß

à àà àá áá áâ ââ âã ãã ãä ää äå åæ æç çè èé éê êë ëì ìí íî îïðñòóôõö÷øùúûüýþÿ !"#$%&'()*+,-.//0011223344556789:;<=>?@ABCDE EF FG GH HI IJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `` `a aa ab bb bc cc cd dd de ee ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~

-8,00

-7,00

-6,00

-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,000,00 0,20 0,40 0,60 0,80 1,00X/C

CP

experimental

mesh A

mesh D

mesh D - less turb

mesh Crefinement

Figure 14: Pressure coefficient along the air foil for different meshes,

alfa 14.15°.

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

0,0 0,2 0,4 0,6 0,8 1,0x/c

y+

mesh D - less turb

mesh A

alfa 14.15°

Figure 13: Comparison of y+ along the air foil for meshes A and

C.

10

Figure 18: Mach number contours for mesh D.

Figure 17: Mach number contours for mesh C.

trailing edge, exhibits a more abrupt variation of cell size than meshes A and D. As a consequence of the rapid growth of the cells size, there is an increased dissipation of the flow variables in the turbulent wake downstream the airfoil. This point will be discussed be in the next section.

C. Analysis of the Flow in the Vicinity of the Trailing Edge

In Figures 16 to 18 a comparison is made of the effect of the different discretizations in the vicinity of the trailing edge on the calculated Mach number contours. In Figure 17 it can be noted that the rapid growth of the mesh cells in mesh C is related to a strong dissipation of the wake downstream the trailing edge. On the contrary, the dissipation observed in Figures 16 and 18, for meshes A and D, respectively, is not so strong and the discrepancy between both is hard to be noticed.

These results confirm the importance of a “capsule” region to guarantee a sufficient density of cells in the area external to the boundary layer, but still close to the airfoil, i.e., the region near the upper side and the trailing edge.

D. Computation of L ift, Drag and Pitching Moment Curves

In this section the lift curve as a function of the Alpha (CL x ALPHA) is determined for the NLR7301 airfoil4, with the mesh that presented the best compromise between size and accuracy in the first part of the study, mesh D. Also, a calibration attempt has been made to

adjust the turbulence parameters in order to improve the agreement between wind tunnel and computational results in terms of lift.

The turbulence parameters used to obtain the initial CLxALPHA curve were turbulence intensity T= 0.01 and turbulence length L = 0.001, that leads to a turbulent to laminar viscosity ratio µt / µ = 15.782. Fourteen values of the Alpha between 0° and 20.50° are then computed. The corresponding CLxALPHA curve is shown in Figure 20. For the initial curve

Figure 16: Mach number contours for mesh A.

¡ ¡¡ ¡¢ ¢£ £ £¤ ¤ ¤¥ ¥ ¥¥ ¥ ¥¦ ¦ ¦§ § §¨ ¨© ©ª ª ªª ª ª« « «¬ ¬ ¬ ¬­ ­­ ­® ®® ®¯ ¯° °± ± ±² ² ² ²² ² ² ²³ ³ ³³ ³ ³´ ´ ´µ µ µ¶ ¶ ¶¶ ¶ ¶· · · ·· · · ·¸ ¸¹ ¹º º» »¼ ¼ ¼¼ ¼ ¼½ ½½ ½¾ ¾¿ ¿À À À ÀÁ Á Á   Â   Âà à Ãà à ÃÄ ÄÅ Å ÅÆ Æ ÆÇ Ç Ç ÇÈ È ÈÈ È ÈÉ É ÉÊ ÊË ËÌ ÌÍ Í ÍÎ ÎÏ Ï ÏÐ ÐÐ ÐÑ ÑÑ ÑÒ Ò ÒÓ ÓÔ ÔÕ ÕÖ Ö Ö× ×Ø Ø ØÙ Ù ÙÙ Ù ÙÚ ÚÚ ÚÛ ÛÛ ÛÜ Ü ÜÝ Ý ÝÞ Þ Þß ß ßà à àà à àá á áá á áâ â âã ãä ä äå åæ æç ç çè è èé éê êê êë ëì ì ìí í íî îï ïð ðñ ñò ò òó óô ôô ôõ õõ õö ö÷ ÷ø øù ù ùú ú úû ûü üý ýþ þ þÿ ÿ ÿ ! !" "# #$% %& &' '( () )* *++, ,, ,- -. . ./ /0 01 12 23 34 45 56 678 89 9: :; ;< <= => >? ? ?@ @A AB BC CD DE EF FG GG GH HH HIIJ JK KL LM M MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _`a ab bc cd de ef fg ghi ij jk kl lm mn no opq qr rs st tu uv vw wx xy yz | | ~ ~ ¡ ¡¢ ¢£ £¤ ¤¥¦§¨ ¨© ©ª ª«¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ¶· ·¸ ¸¹ ¹º º» »¼½ ½¾ ¾¿ ¿À ÀÁ Âà ÃÄ ÄÅ ÅÆÇÈÉ ÉÊ ÊË ËÌ ÌÍÎÏÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö×ØÙ ÙÚ ÚÛ ÛÜ ÜÝÞßà àá áâ âãäåæ æç çè èé éêëìíîï ïð ðñ ñò òó óô ôõöö

÷÷øøùùúû ûü üý ýþ þÿ ÿ

!"#$%&&''()*+,- -- -. ./ /0 01 11 12 23 34 45 55 56 66 67 78 89 99 9: :; ;< << <= => >> >? ?@ @@ @A AB BB BC CD DD DE EE EF FG GG GH HH HI II I

J JJ JJ J

K KK KL LL LM

0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

7,00E-04

0,80 0,84 0,88 0,92 0,96 1,00x/c

CF

mesh D - less turbN N N N N N N N N N N

mesh D

mesh C

mesh A

Figure 15: Fr iction coefficient on the suction side near the

trailing edge, alfa 14.15°.

11

obtained, until Alpha = 16.05° the maximal error for CL is 11.95%. For higher values of Alpha the tunnel test pressure distribution denoted considerable flow separation, with a corresponding decrease of the lift force (stall), while for the numerical calculation this only happens for Alpha around 17°. Hence, for Alpha above 16° the numerical results obtained have much larger discrepancies when compared to the tunnel results. The computed CLxALPHA curve is always above the tunnel results.

After the first CLxALPHA curve is obtained, one pre-stall case (in the tunnel tests, Alpha = 15.53°), has been chosen for a turbulence parameter calibration attempt. The tunnel value for the turbulence intensity was not available, but Ref. 4 states that its value is expected to be low “due to the characteristics of the tunnel” .Then, keeping the freestream turbulent intensity value constant, the value of the incoming turbulent length scale has been gradually increased from L=0.001 until 0.015, for which the CL value obtained equals the tunnel value. Using the freestream values obtained from this adjustment a corrected CLxALPHA curve is obtained for Alpha varying between 0° and 19°. The result of this procedure is shown in Figure 19.

Figure 20 shows the distribution of the pressure coefficient along the chord length when L is varied. The CP curve obtained when L = 0.015 is practically identical to the tunnel curve. An increase in the value of L causes the CL value to fall, from 1.76 (case 0) to 1.63 (case III), when L is varied from 0.001 to 0.015, respectively. The tunnel CL value for this Alpha (15.53°) is 1.59. Therefore, the result of case III is considered sufficiently close to the tunnel value (error margin of 2.5%)

After adjusting the turbulence parameter L, a second CLxALPHA curve was obtained, which presented a better agreement when compared to the tunnel tests. While the initial attempt could predict neither the CLmax. nor the ALPHAmax value, the adjusted curve, with an error margin of about 2.5%, obtained both values. Figure 19 presents the initial CLxALPHA curve, the adjusted one and the tunnel results. It should also be noted that the CLxALPHA curve, after the adjustment, presents a slope closer to the tunnel than the initial curve.

Despite this favorable result, the form of the computed curve near the stall does not reproduce the tunnel result. For high Alpha, the computed curve has a smooth fall, which is characteristic of the stall caused by a gradual propagation of the separation around the trailing edge towards the leading edge. Nevertheless, the stall observed in the tunnel tests seems to be abrupt, with a sudden fall of the CL value just after it reaches its maximal value. This behavior is typical of a leading edge stall process, caused by the onset, development and abrupt collapse of a laminar separation bubble.

O OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~~ ~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º ºº º» »» »¼ ¼¼ ¼½ ½½ ½¾ ¾¾ ¾¿ ¿¿ ¿À ÀÁ Á Âà ÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñò

òóóôôõõöö÷øùúûüýþÿ !!"# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FF FG GG GH HH HI II IJ JJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xx xyy

zz||~ ¡¢£¤¥¦¦§§¨¨©©ªª««¬­®¯°±²³´µ¶·¸¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÌ ÌÍ ÍÍ ÍÎ ÎÎ ÎÏ ÏÏ ÏÐ ÐÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æçèéêëìíîïðñòóôõö÷øùúûüýý

þþÿÿ ! !" "# #$ $% %& &' '( () )* *+ +, ,- -- -. .. ./ // /0 00 01 11 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FG GHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg gg g

h hh hi ii ij jj jk kk kl ll lm mm mn nn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞÞßßààááââããäåæçèéêëìíîïðñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ !!""

##$$%&'()*+, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FF FG GG GH HH HI II IJ JJ JK KK KL LL LMNOPQRSTUVWXYZ[\]^_`abcdefghijkl lm mn no op pq qr rs st tu uv vw ww wx xx xy yy yz zz z | || | ~ ~ ¡¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °° °± ±± ±² ²² ²³ ³³ ³´ ´´ ´µ µµ µ¶ ¶¶ ¶· ·· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á ÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àá áâ âã ãä äå åæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øùúûüýþÿ

! !" "# #$ $% %& &' '( () )* *+ +, ,- -./0123456789:;<=>?@ABCDEFGHI IJ JK KL LM MN NO OP PQ QR RS SS ST TT TU UU UV VV VW WW WX XX XY YY YZ Z[ [\ \] ]^ ^_`abcdefghijk kl lm mn no op pq qr rs st tu uv vw wx xy yz|~ ¡¢£¤¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­®

®¯¯°°±±² ²² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º»¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄÅÆÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎÏÏÐÐ

Ñ ÑÑ ÑÒ ÒÒ ÒÓ ÓÓ ÓÔ ÔÔ ÔÕ ÕÖ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞßà àá áâ âã ãäåæç çè èé éê êë ëì ìí íî îï ïð ðñ ññ ñòòó óó óô ôô ôõ õõ õö öö ö÷øù ùú úû ûüýþ þÿ ÿ

! !" "# #$ $% %& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 345 56 67 78 89 9: :; ;< <= => >? ?@ @A AA AB B BB B BC CC CD DD DE EF FG GH HI IJ J JK KL LM MN NO OP PQ Q QR RS ST TU UV VW WX X XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd de e ef fg g gh hi i ij jk k kl lm mn no o op pq qr rs ss st tt tu uu u

v vv vw ww wx xy yz z | | | ~ ~ ¡ ¡¢ ¢£ £ £¤ ¤¥ ¥¦ ¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬ ¬­ ­® ®¯ ¯ ¯° °° °± ±± ±

² ² ²² ² ²³ ³³ ³´ ´´ ´µ µ µµ µ µ¶ ¶· ·¸ ¸ ¸¹ ¹º º» » »¼ ¼½ ½¾ ¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË Ë ËÌ ÌÍ ÍÎ Î ÎÏ ÏÐ ÐÑ Ñ ÑÒ ÒÓ ÓÔ Ô ÔÕ ÕÖ Ö× × ×Ø ØÙ ÙÚ Ú ÚÛ ÛÜ ÜÝ Ý ÝÞ Þß ßà àá áâ âã ã ãä äå åæ æ æç çè èé éê êë ëì ìí íî î îï ïð ðñ ñ ñò òó óô ôõ õõ õö ö öö ö ö÷ ÷÷ ÷ø øø ø

ù ù ùù ù ùú úú úû ûû ûü üü üý ýþ þ þÿ ÿ ! !" "# #$ $% % %& &'( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= =>? ?@ @A AB BC CD DE EFG GH HI I IJ JK KL LM MNO OP PQ QR RS ST TU UV VWX XY YZ Z[ [\ \] ]^_ _` `ab bc cd de ef fg gh hi ij jk klm mn no opq qr rs stu uv vw wxy yz z | | ~ ~ ¡ ¡¢£¤ ¤¥ ¥¦ ¦§ §¨©ª« «¬ ¬­ ­® ®¯ ¯°±² ²³ ³´ ´µ µ¶ ¶· ·¸¹º» »¼ ¼½ ½¾ ¾¿ ¿ÀÁÂÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊËÌÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕÖ×ØÙÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àáâãäåæçè èé éê êë ëì ìí íî îï ïð ðñòóôõö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ !"#$%& &' '( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9:;<=>?@ABCDEFGHIJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_`abcdefghijkl lm mn no op pq qr rs st tu uv vv vw ww wx xx xyyzz

||~~ ¡ ¡¢£¤ ¤¥ ¥¦ ¦§ §¨© ©ª ª« «¬ ¬­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸¹ ¹º º»¼ ¼½ ½¾¿ ¿À ÀÁ Á ÂÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ Ü ÜÝ ÝÞ Þß ßà à àá áâ âã ãä äå åæ æ æç ç çè è èé é éê êë ëì ì ìí íî îï ï ïð ð ðñ ñ ñò ò òó ó óô ô ôõ õ õö ö÷ ÷ ÷ø ø øù ù ùú ú úû ûü ü üý ý ýþ þ þÿ ÿ ÿ ! ! !" " "# # #$ $ $% % % %& & &' ' '( ( () ) )* * *+ + +, , ,- - -. . ./ / /0 0 01 1 12 2 23 3 34 4 45 5 56 6 67 7 78 8 89 9 9: : :; ; ;< < <= = => > >? ?@ @ @A A AB B BC CD DE E EF F FG G GH H HI I IJ J JK K KL L LL L LM M MM M MN N NN N NO O OO O OP P PP P PQ Q QQ Q QR R RS ST T TU UV VW WX X XY YZ Z[ [\ \ \] ]^ ^_ _` ` `a ab bc cd de e ef fg gh hi ij j jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~

¡ ¡¡ ¡¢ ¢¢ ¢£ ££ £¤¤¥¥¦¦§§

¨ ¨¨ ¨© ©© ©ª ªª ª« «« «¬ ¬¬ ¬­­®®¯¯°°±±²²

³ ³³ ³´ ´´ ´µ µµ µ¶ ¶¶ ¶· ·· ·¸ ¸¸ ¸¹ ¹¹ ¹º ºº º» »¼ ¼½¾¿ÀÁÂÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ !"#$%&'( () )* *+ +, ,- -. ./ /0 01 12 23 34 45 56 67 78 89 9: :; ;< <= => >? ?@ @A AB BC CD DE EF FG GH HI IJ JKLMNOPQRSTUVWXYZ[\]^_`abcd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡¢£¤¥¦§¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ !"#$ $

% %% %& && &' '' '( (( () )) )* ** *+ ++ +, ,, ,- -- -. .. ./ // /0 00 01 11 12 22 23 33 34 44 45 55 56 66 67 77 78 88 89 99 9: :: :; ;; ;< << <= == => >> >? ?? ?@ @@ @A AA AB BB BC CC CD DD DE EE EF FF FG GG GH HH HIIJJKKLLMMNNOOPPQQRRSSTTUUVVWWXXYYZ[\]^_`abcdefghijkl

m mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡¢£¤¥¦§¨©ª«¬­®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõ õö ö÷ ÷ø øù ùú úû ûü üý ýþ þÿ ÿ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd de ef ff fg gg gh hh hi ii ij jj jk kk kl ll lm mm mn no op pq qr rs st tuvwxyz|~ ¡¢£¤¥¦§¨¨©©ªª««¬¬

­ ­­ ­® ®® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Âà ÃÄ ÄÅ ÅÆ ÆÇ ÇÈ ÈÉ ÉÊ ÊË ËÌ ÌÍ ÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔ ÔÕ ÕÖ Ö× ×Ø ØÙ ÙÚ ÚÛ ÛÜ ÜÝ ÝÞ Þß ßà àà àá áá áâ ââ âã ãã ãä ää äå åæ æç çè èé éê êë ëì ìí íî îï ïð ðñ ñò òó óô ôõö÷øùúûüýþÿ !"#$%&'()*+,-./0123456789:;;<<==>>??@ABCDE EF FG GH HI IJ JK KL LM MN NO OP PQ QR RS ST TU UV VW WX XY YZ Z[ [\ \] ]^ ^_ _` `a ab bc cd de ef fg gh hi ij jk kl lm mn no op pq qr rs st tu uv vw wx xy yz z | | ~ ~ ¡ ¡¢ ¢£ £¤ ¤¥ ¥¦ ¦§ §¨ ¨© ©ª ª« «¬ ¬­ ­® ®¯ ¯° °± ±² ²³ ³´ ´µ µ¶ ¶· ·¸ ¸¹ ¹º º» »¼ ¼½ ½¾ ¾¿ ¿À ÀÁ Á Â Âà Ãà ÃÄ ÄÄ ÄÅ ÅÅ ÅÆ ÆÆ ÆÇ ÇÇ ÇÈ ÈÈ ÈÉ É

-10,00

-8,00

-6,00

-4,00

-2,00

0,00

2,000,00 0,20 0,40 0,60 0,80 1,00X/C

CP

EXPERIMENTAL

CASO 0

CASO 1

CASO 2Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê

CASO 3

Figure 20: Pressure coefficient along the air foil for different values of

the turbulent length scale.

0,00

0,20

0,40

0,60

0,80

1,00

1,20

1,40

1,60

1,80

2,00

0 2 4 6 8 10 12 14 16 18 20alfa [°]

CL

CLxALFA 1 - cfd++

CLxALFA - tunnel

adjustment

CLxALFA 2 - cfd++

Figure 19: CLxALFA curve – Compar ison between the tunnel

results and the present simulation.

12

Up to Alpha 16.05°, the maximal error for CL is 6.25%, for Alpha 6.7°. Both the tunnel and the numerical result (after the adjustment) are coherently predicting a maximal value for ALPHAmax = 16.05°. However, the maximal CL value for the numerical calculation, CLmax =1.63 was 2.5% higher than the tunnel value, 1.59.

While the results of the adjusted CLxALPHA curve presented a good correspondence with the tunnel data, a different behavior is observed for the curves CDxCL and CMxALPHA, and shown in Figures 21 and 22.

In Figure 21, it can be noticed that the experimental CD values are intermediary between the computational results before and after the calibration of turbulence parameters. The simulation without calibration provides a better aproximation to experimental drag at lower CL values, while the calibrated results are a better match for CL values near stall.

In Figure 22 the pitching moment coefficient is presented as a function of alfa (CMxALFA), before and after the adjustment. It can be observed that, after the adjustment, the numerical curve is closer to the tunnel curve, and exhibits a more similar behavior, yet displaced to higher alfa values. It is also interesting to notice that the numerical results present an important discrepancy concerning the tunnel results for moderated alfa values.

V. Conclusions Initially, a mesh sensitivity study of

the flow around the NLR7301 airfoil was performed with the aim of obtaining mesh independent results for both the attached and the detached flow regimes. This required computational meshes with a large number of computational cells over the suction side of the airfoil and in the vicinity of the trailing edge. The influence of the freestream flow properties on the computed pressure distribution and on the lift and drag coefficients was assessed. In particular, the roles of the intensity of the turbulent fluctuations and of the averaged turbulence length scale were examined. The latter was shown to exert a strong influence on the aerodynamic coefficients of the airfoil, specifically in flow conditions near the aerodynamic stall.

Then, the lift coefficient curve was computed, and a good agreement was obtained between the calculated values of maximum lift coefficient and of the angle of attack where this value is observed. However, the post stall behavior of the computed solution is characteristic of a trailing edge stall, while the wind tunnel results exhibit an abrupt decrease of the lift coefficient after stall. This behavior is characteristic of the presence and subsequent coalescence of a laminar separation bubble at the suction side of the airfoil, which is probably absent from the computational results. Note, however, that for values of Reynolds number less than 10% larger than the one used in the present computations, wind tunnel results obtained for the same airfoil exhibit a completely different post-stall behavior. Indeed, there seems to exist a wind-tunnel dependence of the post-stall characteristics, indicating that the post-stall flow could not be considered as two-dimensional in the experiments.

0

200

400

600

800

1000

1200

1400

1600

0,00 0,50 1,00 1,50 2,00CL

CD CLxCD 1 - cfd++

CLxCD - tunnel

CLxCD 2 - cfd++

Figure 21: Drag polar curve.

-0,10

-0,08

-0,06

-0,04

-0,02

0,000,00 5,00 10,00 15,00 20,00 25,00

alfa [º]

CM

CMxALFA 1 - cfd++

CMxALFA - tunnel

CMxALFA 2 - cfd++

Figure 22: CMxALFA curve.

13

The absence of the laminar separation bubble in the computed results could also explain the discrepancies observed in the drag coefficient curve. This bubble would lead to a higher drag, and thus to a lesser need in terms of the adjustment of turbulence length scales. Indeed, the corrected CD curve overpredicts the drag, whereas the curve initially calculated underpredicts the value of CD.

Concerning the pitching moment coefficient curve, although the overall behavior was correctly reproduced by the computations, discrepancies were observed on the actual value of this coefficient. These discrepancies may be attributed to small differences on the computed value of the distribution of pressure coefficient close to the leading and trailing edge. These differences in CP have a large contribution to the computed value of CM, due to their large distance to the moment reference point of the airfoil.

Acknowledgments

During this work L.F. Figueira da Silva was on leave from Laboratoire de Combustion et de Détonique, Centre National de la Recherche Scientifique, with a PROFIX scholarship from CNPq-Brazil.

14

References 1Vos, J. B., Rizzi, A., Darracq, D. and Hirschel, E. H., “Navier-Stokes Solvers in European Aircraft Design,” Progress in Aerospace Sciences, Vol. 38, No. 8, 2002, pp. 601-697. 2Jameson, A., “The Present Status, Challenges and Future Developments in Computational Fluid Dynamics,” Progress and Challenges in CFD Methods and Algorithms, AGARD-CP-578, 1996. 3Hirschel, E. H., “Present and Future Aerodynamic Process Technologies at DASA Military Aircraft,” ERCOFTAC Industrial Technology Topic Meeting, October 1999. 4Han, S. O. T. H., “Two-Dimensional 16.5% Thick Supercritical Airfoil NLR 7301,” AGARD Advisory Report nº 303 ´´A Selection of Experimental Tests Cases for the Validation of CFD codes ` - vol. I – Advisory Group for Aerospace Research & Development, 1995. 5Palaniswamy, S., Goldberg, U., Peroomian, O. and Chakravarthy S., “Predictions of Axial and Transverse Injection into Supersonic Flow,” Flow, Turbulence and Combustion, Vol. 66, No. 1, 2001, pp. 37-55. 6Loyau, H., Batten, P. and Leschziner, M.A., “Modelling Shock/Boundary-Layer Interaction with Non-Linear Eddy-Viscosity Closures,” Flow, Turbulence and Combustion, Vol. 60, No. 3, 1998, pp. 257-282. 7Wilcox, D. C., Turbulence Modeling for CFD, 1993. 8Shih, T. H., Lumley, J. L. and Zhu, J., “A Realizable Reynolds Stress Algebraic Equation Model,” NASA TM-105993, 1993. 9Lien, F. S. and Leschziner, M. A., “Low Reynolds Number Eddy-Viscosity Modelling Based on Non-Linear Stress-Strain/Vorticity Relations,” Engineering Turbulence Modelling and Experiments 3, edited by Rodi, W. and Bergeles, G., Elsevier, Amsterdam, 1996, pp. 91-100. 10Goldberg, U., Batten, P., Palaniswamy, S., Chakravarthy, S. and Peroomian, O., “Hypersonic Flow Predictions Using Linear and Nonlinear Turbulence Closures,” Journal of Aircraft, Vol. 37, No. 4, 2000, pp.671-675. 11Goldberg, U., Peroomian, O., Palaniswamy, S. and Chakravarthy, S., “Anisotropic k-ε Model for Adverse Pressure Gradient Flow,” 37th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 99-0152, 1999. 12O. Peroomian and S. Chakravarthy "A 'Grid-Transparent' Methodology for CFD," AIAA Paper No. 97-0724, Reno 1997. 13Drela, M., 1989, “XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils,” in Low Reynolds Number Aerodynamics, T.J. Mueller, editor. Lecture Notes in Engineering #54, Springer-Verlag. 14Drela, M. and Giles, M. B., 1987, “ ISES: A Two Dimensional Viscous Aerodynamic Design and Analysis,” in 25th AIAA Aerospace Sciences Meeting, AIAA Paper 87-0424.