[american institute of aeronautics and astronautics 45th aiaa aerospace sciences meeting and exhibit...

American Institute of Aeronautics and Astronautics

1

A Study of Transport Aircraft High-Lift Design Approaches

Alexandre P. Antunes1 and Ricardo Galdino da Silva2

Universidade de São Paulo, São Paulo, SP, 05508-900, Brazil

João Luiz F. Azevedo3

CTA/IAE/ASA-L, São José dos Campos, SP, 12228-903, Brazil

The high-lift design process is focused in obtaining the best aerodynamic shape and the optimized

position for the high-lift devices. The main requirement for the design process is strongly connected

with the need to achieve a maximum target lift coefficient for the landing and take-off maneuvers. In

this context, the present work has the objective of presenting two numerical methodologies to predict

the maximum lift coefficient, namely, the quasi-3D and the 3D approaches. The NLR7301 airfoil

configuration has been chosen for the study, since it is a non-proprietary geometry. The numerical

simulations are performed with the CFD++ and VSAERO commercial codes. The meshes are created with the ICEM mesh generator. The work also presents a sensitivity analysis of the aerodynamic

parameters, such as the global 3-D lift coefficient or the section lift coefficient, with regard to changes

in the numerical setup. In the present study, the results obtained by the two methodologies showed a certain level of discrepancy. A dispersion of about 6% in the prediction of the maximum CL was

obtained by the quasi-3D analysis, while the 3D methodology presented a premature stall.

I. Introduction

he high-lift design process is focused in obtaining the best aerodynamic shape and the optimized position for the

high-lift devices. The main requirement for the design process is strongly connected with the need to achieve a

maximum target lift coefficient (CL) for the landing and take-off maneuvers. The definition of the target

landing/take-off maximum CL depends on the mission to be carried out by the airplane. The design process must be

effective in achieving the target CL due to the penalization that a bad design can cause on some macro variables

related with the performance or the operation of the airplane. The subject of high-lift devices has always been an

area of special interest to airplane designers. The accurate prediction of pressure distributions, boundary layer

confluences, and detached flow regions over the multi-element high-lift wings plays a fundamental role in the design

of high-lift devices. The complex physical phenomena involved in such flowfield are responsible for the difficulty

associated with high-lift design. It is important to observe that currently most of the aerodynamic design is

performed with the aid of CFD.

Lately, the advance of CFD has produced an important change in the form in which aerodynamic analysis and

design are performed in the aeronautical industry. CFD has reached such a level of importance that one cannot

imagine an aerodynamic department without a CFD group. The constant development of CFD, as a tool capable of

producing complex analyses, has been responsible for the spread of its use all over the world. Until a few years ago,

aerodynamic design was driven by simplified analytical methods and empirical formulations, together with massive

wind tunnel campaigns to validate the aerodynamic design.

Today the fierce competition between the aeronautical manufacturing companies does not allow such a costly

design procedure anymore. The current tendency encourages the use of wind tunnel tests just as a way to corroborate

the aerodynamic design, previously performed with CFD.1-2

The world aeronautical manufacturing companies are

eager to reduce the inherently high cost associated with wind tunnel tests. In order to give an idea of the amount of

money required for a wind tunnel campaign, the milled model costs might reach the order of hundreds of thousands

of dollars. This cost intrinsically depends on whether the model is a full or a half model, and its geometric

complexity. The cost of the test might start at a few hundreds of thousands of dollars and reach the mark of millions

1 Graduate Student, [email protected].

2 Graduate Student, [email protected] 3 Currently, Director for Space Transportation and Licensing, Brazilian Space Agency, [email protected].

Associate Fellow AIAA.

T

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-38

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

of dollars, depending on the desired range of Reynolds number, and the size of the wind tunnel testing cross section.

The increase of test price is associated with the increase of Reynolds number for a given fixed number of test runs.

On the other hand, as the wind tunnel Reynolds number approaches the real flight Reynolds number, the more

representative the experimental results become of the real flight aerodynamic coefficients. This is the reason for the

existence of some cryogenic wind tunnels such as the Nation Transonic Facility (NTF) at NASA Langley Research

Center, and the European Transonic Wind Tunnel (ETW) near Cologne, Germany. However, the price required to

use these highly specialized cryogenic wind tunnels is prohibitive, and they do not meet the productivity required for

a development program.3 Despite the desire to reduce the number of wind tunnel campaigns, it is important to have

in mind that wind tunnels will not be retired at least in the next decades. Hence, one can appreciate the importance

of CFD in the cost minimization for the development of new products.

During the aerodynamic design phase of airplane components, high-lift devices represent one of the most

laborious items in the process due to the stringent geometrical constraints and the flowfield complexity.4 Figure 1

shows the physical phenomena present in a flowfield over high-lift configurations. One can notice the presence of

shock waves, laminar bubbles, transition from laminar to turbulent flow, boundary layer confluence, and flow

separation. These various flowfield physical phenomena and their interactions are responsible for the attributed

complexity in high-lift device analyses.

Figure 1. Characteristic physical phenomena present in the flowfield over high-lift devices.

The high-lift design process is usually coupled with the cruise wing design and, frequently, the cruise wing

design specifies most of the important design parameters. The typical supercritical wing airfoils are characterized by

a relative thin rear-end shape, which creates difficulties to the flap design due to the small flap thickness. It is worth

mentioning that, usually, the upper and the lower rear geometrical shapes of the flaps and the slat geometry are

already determined from the aerodynamic cruise design. Figure 2 provides some idea of the geometrical constraints

in the flap design task. One can notice the limitation on the chord length of high-lift devices caused by the location

of the front-spar and the rear-spar. The spar positioning is dictated by the fuel volume requirement.

Figure 2. Typical geometrical constraints during the flap design.

Whenever there is an antagonism, the cruise design directives surpass the high-lift design directives. This is

completely acceptable since cruise is the most important phase of flight for an airliner.5 Furthermore, the high-lift

devices are not only affected by the constraints from cruise design, but they are also affected by constraints coming

from others areas such as systems, structures, loads, and weight distributions, among others.


3

Figure 3. Gap/overlap definition.

In practical terms, much of the high-lift design effort is concentrated in finding the best gap/overlap positions for

the slats/flaps and the aerodynamic shape for the forward part of the flaps. The gap is defined as the radius of the

circumference centered in the trailing edge of the main element and tangent to the flap profile at a certain position.

The overlap is defined as the projection in the streamline direction of the distance from the leading edge of the flap

to the main element trailing edge. Figure 3 explains the geometrical definition of the gap/overlap. During the

preliminary design phase, changes on these parameters are performed to obtain the best aerodynamic performance.

A small modification on the gap/overlap is sufficient to generate considerable changes in the aerodynamic

coefficients. In Figure 4 one can observe the changes in the 2-D pressure coefficient, Cp, due to modifications on the

gap/overlap.

Figure 4. Change in the Cp distribution caused by changes in the gap/overlap of the high-lift devices.

The choice of the wing area is mainly based on performance requirements, and it is determined to allow the

achievement of the best ML/D at a specific speed and altitude during the cruise flight phase. As consequence of this

optimization, higher CL values are required at the landing and the take-off flight maneuvers. In order to achieve

these target higher CL values, the aerodynamic designer can use a combination of several sets of leading edge and

trailing edge high-lift devices. For the leading edge, one can choose which device is more appropriate among the

following possibilities: hinged leading edge, variable camber leading edge, fixed slot, simple Krueger flap, folding,

bull-nose Krueger flap, two-position slat, three-position slat. On the other hand, for the trailing edge, the available

possibilities include the use of: split flap, plain flap, simple slotted flap, fixed vane/main double slotted flap,

main/aft double-slotted flap, triple-slotted flap. Figure 5 shows a few devices that can be implemented in the

airplane in order to allow the realization of the target performance. Another possibility would be the increase of the

wing area to achieve the required lift at the landing and take-off maneuvers. However, this would implicate in a non-

optimized ML/D during the cruise flight phase, causing an increase in fuel consumption.

From the 50’s to the 70’s, the increase of the complexity of high-lift devices was a tendency. However, in the

late 70’s, the beginning of CFD use allowed a way to decrease the high-lift device complexity by means of shape

and gap/overlap optimization. Such advances have led to a reduction in the high-lift devices weight, an increase in

its useful load, an improvement in take-off L/D, and a reduction in airframe noise, among others. Figure 6 shows,

for several airplanes, a relation between the maximum CL and the complexity of the high-lift system. One can notice


4

that there is almost no difference, in terms of maximum achievable CL, for the double-slot flap and the triple-slot

flap. The evolution in the ability to achieve the same maximum CL with simplest high-lift configuration is a result of

the geometrical and gap/overlap optimization process.

Kruger Flap

Figure 5. Examples of high-lift devices that can be implemented in the airplane to guarantee the target maximum CL.

Figure 6. Relation between the maximum CL and the complexity of the high-lift devices.

This work has the objective of presenting an overview of two high-lift methodologies, the quasi-3D approach

and the 3D approach. Moreover, the work also presents an analysis of the sensitivity of the aerodynamic parameters

with regard to changes in the numerical setup. For this sensitivity study, two turbulence models, two sets of

freestream conditions for the turbulence parameters, one mesh for the two-dimensional simulations and two meshes

for the three-dimensional simulations are used. No gap/overlap variation is studied in these simulations.

In order to evaluate these design methodologies, one fictitious airplane in a high-lift configuration is the object

of study in this work. In this study case, the airplane loft created does not intend to represent any of the actual flying

commercial airplanes. The construction of the fairing geometry and the fuselage geometry are based on scaled views

of a few airplanes. The wing is generated by the extrusion of the NLR7301 airfoil6 in the spanwise direction, which

creates a non-conventional configuration.

YAK-40

VFW-614

G.II

BAC-111

Caravelle

Cit. II

CJ1

F.28

DC-9-10

TU 134

F-100 CBA-123

ERJ-145

AVRO RJ85

B.707

VC 10

C 5

Falcon 20

A320

DC-9-30

Trident

B.777

A 321

A 300B

L 1011

MD-11

A 340

B.737-200

B.727TU 154

B.747

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 5 10 15 20 25 30 35

Complexity Of The High-Lift System

CL

_m

ax

Single Slot Flap

Double Slot Flap

Single Slot Flap + Slat

Double Slot Flap + Slat

Triple Slot Flap +Slat


5

II. High-Lift Methodologies

In the aeronautical manufacturing industry, the prediction of the maximum CL generated by high-lift devices is

obtained mainly by the quasi-3D and the 3D methodologies.7 The first one is very useful and consists in a simplified

methodology for quick and effective analyses whereas the other is considerably more complex and time consuming.

In the very early stages of the high-lift design process, there are considerably large numbers of proposals to be

analyzed and only some of them will generate a feasible component to be implemented in the airplane. The quasi-3D

methodology is more applicable on the early design stages due to the fast capability to indicate which proposals are

more likely to achieve the design requirements. At this stage, the quasi-3D methodology allows design optimization

of the geometric shape and the gap/overlap positioning without a prohibitive computational cost.

As the design process evolves, the urge for a refined analysis arises. At the advanced stages of the design

process, the 3D methodology is applicable to perform verifications at very specific design aspects, which are

neglected by the simplified quasi-3D methodology. Issues such as the three-dimensionality of the flowfield and the

aerodynamic interaction among the high-lift devices are considered on the 3D analyses. Usually, when the design

process reaches the advanced stage, the 3D methodology is only used to analyze a few potential proposals and,

hence, despite the computational costs, the amount of helpful information generated by this methodology is worth

the effort.

A. The Quasi-3D Methodology

In the quasi-3D methodology, two-dimensional viscous simulations are coupled to a three-dimensional panel

method solver. The 2D simulations are used to determine the maximum Cl at a certain number of wing spanwise

stations, while the three-dimensional simulation is used to determine the spanwise wing loading at a given CL. With

these two sets of information, one can determine the distribution of maximum achievable Cl along the wing span

and, whenever the wing loading curve reaches any of these local maximum Cl, it is assumed that, at this point, the

wing achieved the stall condition. One can observe in Fig. 7 the matching between the wing loading curve and the

maximum local Cl values, as previously mentioned.

Figure 7. Matching between the wing loading curve and the maximum local Cl values for determination of the

maximum CL.

One can notice that a slight misprediction of Cl brings the wing loading curve to a wrong matching and,

consequently, CL is altered. The obtained results for the maximum CL are as good as the accuracy in the 2-D

prediction of the Cl, because all the error associated with the 2-D simulation is transferred to the maximum CL

prediction. Theoretically, the quasi-3D procedure is very simple and clear but, to obtain good results with the

methodology, a calibration process is usually required. Furthermore, there are the errors associated with the use of a

panel method for the determination of the wing loading distribution.

Absolute certainty is never achievable, so designers have to work with some margins to cover the risk. The

calibration process intends to reduce the margin of wrongly predicting the maximum local Cl values. Basically, it

consists in searching the best numerical setup by means of comparisons between available two-dimensional

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

y

Cl

Wing Spanwise Load

Maximum Section Lift Coefficient


6

experimental data and parametric 2-D simulations. These comparisons are carried out for a series of geometrical

configurations and, based on such results, one can stipulate the numerical setup that better represents the physical

phenomena being simulated. Once this calibration procedure is finished, the 2-D simulations for the geometry of

interest should be performed using the numerical setup determined from the calibration.

In the present work, the maximum section lift coefficients are obtained with the CFD++ code8, which uses a

finite volume approach and has the capability to work with various cell shapes, i.e., hexahedra, tetrahedra, and

pyramids, among others. The discretization is based on a multi-dimensional, total variation diminishing (TVD)

scheme, and Riemann solvers are used to define the interface fluxes. A variety of turbulence models are available

ranging from one to three-equation transport models, including both linear and nonlinear closures.

The three-dimensional analyses are obtained with the VSAERO9 surface panel method. VSAERO uses

piecewise constant source and doublet singularities on quadrilateral panels representing the surface of the

configuration. Source strengths are solved directly from the external Neumann boundary condition using the normal

component of velocity of the external flow. A set of linear equations is obtained with doublet strengths as the

unknowns by imposing the internal Dirichlet boundary condition of zero perturbation potential inside the

configuration. The set of linear equations is solved either by a direct method or by a block Gauss-Seidel iterative

procedure, depending upon the number of unknowns in the equations. The gradient of the doublet potential

distribution is used to obtain the surface perturbation velocities. The wakes downstream of the trailing edges of the

multi-element system are modeled as thin wake panels. The wake shapes change at the end of each iteration in order

to satisfy the force-free condition on the wake panels. A converged wake shape is obtained when the wake shapes

cease to change with the iterations and, only after this is achieved, the surface pressure distributions are calculated.

The integrated aerodynamic forces can be obtained by appropriate surface integration of the pressure distributions.

B. The 3D Methodology

The use of CFD during these preliminary stages of the airplane design is not a privilege of a few airplane

manufacturing companies anymore. CFD has become a well-known and widely used tool to design and to support

daily aerodynamic issues. In this methodology, CL is obtained directly from a 3-D CFD simulation. The quality of

the results is highly dependable on the mesh, the turbulence model, and the code capability to accurately represent

the Navier-Stokes equations, without the introduction of too much dissipation.

The main difficulty with this methodology is associated with mesh generation, especially when complex high-lift

configurations are the object of study. The need to accurately capture the boundary layers and the merging among

the wakes of the several elements leads to the use of hexahedral meshes, due to a better control of mesh quality in

such complex flowfield regions. On the other hand, the complexity of the geometry does not allow the generation of

such type of meshes. For these cases, the practical decision might be the use of tetrahedral meshes with prism layers

to better represent the boundary layers, or the generation of overset multiblock meshes.10 In either case, there are

difficulties intrinsic to the mesh generation process.

In the application of this methodology, one can clearly see that the obtained CL is quite sensitive to the mesh

element type. Another important point in the 3D methodology consists in the capability of the turbulence model to

capture relevant physical phenomena such as transition. The most important aspect to achieve a reliable CL

prediction consists in the correct calculation of transition over the slat upper surface.11

For the present work, it is

assumed that the flow is fully turbulent. The assessment of the effects of having the transition from laminar to

turbulent flow is beyond the scope of the present paper.

III. Numerical Formulation

A. Governing Equations

The flow of interest in the present context are modeled by Reynolds-averaged Navier-Stokes (RANS)

equations, written in dimensionless form and assuming a perfect gas, as

( ) 0=−⋅∇+∂

∂ve PP

t

Q , [ ]T

ewvuQ ρρρρ= , (1)


7

where Q is the dimensionless vector of conserved variables, ρ is the fluid density, U={u,v,w} is the Cartesian

velocity vector and e is the fluid total energy per unit of volume. The inviscid flux vector, Pe, and the viscous flux

vector, Pv, are given as

( )

+

+

+

+

=

Upe

ipwU

ipvU

ipuU

U

P

z

y

x

e

ˆ

ˆ

ˆ

ρ

ρ

ρ

ρ

,

( )( )( )

+

+

+

=

ii

itzi

lzi

ityi

lyi

itxi

lxi

v

i

i

i

i

P

ˆ

ˆ

ˆ

ˆ

0

β

ττ

ττ

ττ

, (2)

where i = x, y, z are the indices used within the Einstein indexing notation; and { }zyx iiii ˆ,ˆ,ˆˆ = is the Cartesian

coordinate unit vector. The other relations can be given as

( )

∂

∂

+−=−+=

∂

∂−

∂

∂+

∂

∂=+=

j

i

t

tljij

tij

lijiij

m

m

i

j

j

il

lij

tij

lijij

x

eqqu

x

u

x

u

x

u

PrPr,,

3

2,

µµγττβδµττττ (3)

where iu and ix are the Cartesian velocity and coordinate components, respectively, and ijδ is the Kronecker delta.

In the previous definitions, lµ is the molecular dynamic viscosity coefficient, computed by the Shutherland law.

The pressure, p, can be calculated from the perfect gas equation of state. Furthermore, ei is the internal energy and γ

is the ratio of specific heats. Details on the theoretical and numerical formulations can be found in Refs. 12 and 13.

B. Turbulence Modeling

The turbulence effects are included into the RANS equations through the Reynolds-stress tensor, defined as

_____

''uutij ρτ −= . (4)

Eddy viscosity models compute the Reynolds stresses through the Boussinesq hypothesis, which states that the

turbulence stresses are a linear function of the mean flow straining rate times a modifying constant such as

∂

∂−

∂

∂+

∂

∂= ij

m

m

i

j

j

it

tij

x

u

x

u

x

uδµτ

3

2 , (5)

where tµ is the eddy viscosity coefficient, computed by the chosen turbulent model.

1. One-Equation SA Model

The Spalart-Allmaras (SA) 14 model has proven to be a numerically robust approach, and generally good results

have been demonstrated for a wide variety of flows. This model requires one single transport equation for the

modified eddy viscosity, given as

( ) ( )[ ] ( ){ } 21221

2221

~~1~~~1~~1

~Uf

df

cccSfc

Dt

Dtt

bwbtb ∆+

−−∇+∇+⋅∇+−=

ν

κνννν

σν

ν , (6)

where the eddy viscosity is formulated as

1~

ννν ft = , 31

3

3

1

νν

χ

χ

cf

+= ,

ν

νχ

~= . (7)


8

The production term is given by

( )

22

~~ν

κ

νf

dSS += ,

12

11

vff

χ

χν

+−= . (8)

The dissipation term involves

6

1

63

6

631

+

+=

w

ww

cg

cgf , ( )[ ]11

52 −+= rcrg w

, ( )2~

~

dkSr

ν= , 10≤r . (9)

In order to allow laminar regions in the flow, the dissipation term also includes

( )24

32χtc

tt ecf−= . (10)

Here, tν is the eddy viscosity, S is the mean strain and d is the distance to nearest wall boundary. The model

constants are given as

1355.01 =bc , 32=σ , 622.02 =bc , 41.0=κ ,

( )σκ

2

2

11

1 bbw

ccc

++= , 3.02 =wc , 23 =wc , 1.71 =νc , 1.13 =tc e 24 =tc .

Furthermore, the CFD++ code uses 01 =tf .

2. Two-Equation SST Model

The “shear-stress-transport” model, SST,15

is a hybrid model that uses a blending function to combine the best

aspects of both the k-ω16 and the k-ε17 turbulence models. Near solid walls, a k-ω model formulation is used

allowing integration to the wall without any special damping or wall functions. Near the outer edge of the boundary

layer and in shear layer, the model blends into a transformed version of the k-ε formulation, thus, providing good

prediction for the free shear flows. In SST model, the Reynolds stresses are given by Eq. (5), and the eddy viscosity

is given as

{ }21

1

,max SFwa

kat =ν . (11)

The SST model uses a transport equation for the kinetic energy and a transport equation for the time-scale

given, respectively, as

( ) ( )[ ]kwkPDt

kDtkk ∇+⋅∇+−= µσµρβ

ρ *~ , (12)

( ) ( )[ ] ( ) ω

ωρσωµσµωρβ

ν

γωρωω ∇∇−+∇+∇+−= ..

ˆkFP

Dt

Dtk

t

112 21

2 . (13)

The production term is given as

j

iijij

n

n

i

j

j

itk

x

Uk

x

U

x

U

x

UP

∂

∂

−

∂

∂+

∂

∂+

∂

∂= δρδµ

3

2

3

2. (14)

For the turbulent kinetic energy transport equation, the production term is limited by the following form


9

( )ωρβ kPP kk*

,min~

10= . (15)

The model constants are given as ( ) 2111 1 θθθ FF −+= , where 1θ represents the constants for the k-ω model and 2θ

represents the constants for the k-ε model. The F1 variable is a blending function that turns on the k-ω closure near

walls and the standard k-ε model outside boundary layers, and it is defined as

=

2

2

21

4500

dCDdd

KF w

ω

ρσ

ω

ν

ωβ,,maxmintanh

*,

∇⋅∇= −10

2 101

2 ,max ωω

ρσ ωω kCDk. (16)

The F2 variable is another blending function that turns on the eddy viscosity definition inside boundary layers

=

2

22

5002

ω

ν

ωβ dd

KF ,maxtanh

* . (17)

The models constants are given by

85.01

=kσ , 501

.=ωσ , 075.01 =β , **

β

κσ

β

βγ ω

211

1 −=t, 0.1

2=kσ , 8550

2.=ωσ , 0828.02 =β ,

**β

κσ

β

βγ ω

222

2 −=t, 09.0

* =β , 41.0=κ .

IV. Mesh Generation

The arrival and the evolution of the commercial mesh generators introduced the possibility to use a large variety

of elements types to mesh complex realistic configuration. Nowadays, these realistic geometries, such as airplanes,

helicopters, cars, among others, can be meshed without the unacceptably high time consumption. The key drivers in

stimulating the development of reliable and efficient mesh generators are computational fluid dynamics and

computational aerodynamics18

.

In the present work, several meshes are generated using the ICEM-CFD tool19: a surface quadrilateral mesh, a

2-D quadrilateral mesh, a hybrid volumetric tetrahedral-prism mesh, and a hybrid volumetric hexahedral-tetrahedral-

prism mesh. The following described 2-D and surface meshes are used with the quasi-3D methodology, while the

described 3D meshes are associated with the 3D analyses. Different concepts of grid generation are applied for each

type of mesh, but the main objective is to produce the best suitable grid for each predicting methodology.

The 2-D mesh can be seen in Fig. 8. For this mesh, the farfield boundary is positioned at 100 chords from the

airfoil. This is the minimum distance to avoid reflexive numerical problems associated with the imposition of the

boundary condition. The total number of points and the distribution of such points over the elements, in order to

generate the mesh, are determined by the experience of the authors,19

using well-known best practices such as the

fact that one should concentrate more points in regions of higher gradients. One can also observe a mesh

concentration behind the trailing edge of each element. This concentration is an attempt to better capture the wake

confluence of adjacent elements. The bunching in the region near the wall is suitable to allow at least one point in

the viscous sub-layer of the boundary layer. The stretching applied in the direction normal to the wall is smooth

enough to avoid an undesirably fast growth of the mesh.


10

Figure 8. Two-dimensional mesh for the wing profile.

In Fig. 9, one can observe the details of the mesh generated for the VSAERO9 code. The concept consists in

dividing the airplane into regions, which are known as patches. The patches correspond to a group of panels and, for

each patch, the bunching distribution is prescribed. It is advisable, in regions of higher pressure gradients, to avoid

neighbor mesh elements with different sizes and presenting hanging nodes. This lofting is a non-conventional

configuration but, as the authors are also aiming to demonstrate a different approach to mesh complex high-lift

geometries, this configuration is adequate for the purpose.

Figure 9. Surface mesh generated for the VSAERO panel method solver.

In Fig. 10, one can observe the surface details of the initial hybrid volumetric mesh for Case 1. A tetrahedral

mesh is initially generated and, afterwards, prism layers are grown to better represent the near wall region. Figure 11

shows a cut at a certain spanwise station along the wing. Details of the grown prism layers over the main and the

flap elements can be observed in Fig. 11. One can further observe a fast growth in the size of the tetrahedral

elements just a few chords away from the prism layers. This fast growth is caused by the absence of constrains in the

volumetric growth ratio at the ICEM-CFD mesh toolbox. Such large tetrahedral elements are not recommended to

capture with accuracy the various wake interactions present in high-lift devices. On the other hand, the refinement of

these elements would considerably increase the total mesh elements and the computational cost.


11

Figure 10. Surface discretization of the initial 3-D hybrid volumetric mesh.

Figure 11. Spanwise station cut showing the details of the hybrid tetrahedral/prismatic 3-D mesh.

An alternate approach to adequately represent the flowfield is to use an hexahedral block mesh to discretize the

high-lift devices and the region around the body. In Fig. 12, one can observe how the configuration in Case 1 is


12

divided into regions in order to allow the generation of the hybrid tetrahedral-hexahedral-prismatic mesh. In order to

have a better control of the volume mesh, support curves are created to accommodate the hexahedral block mesh.

These support curves are generated with the aid of CAD tools. Figure 13(a) shows the constructions curves used to

support the block of the hexahedral meshes, and Fig. 13(b) show the resulting hexahedral volumetric mesh over the

wing.

Figure 12. Approach to mesh the configuration with a tetrahedral-hexahedral-prismatic mesh.

Figure 13(a) Construction lines. Figure 13(b) Hexahedral mesh for the wing.

Figure 13. Details of the construction lines to support the volumetric hexahedral mesh.

From this hexahedral block mesh, the tetrahedral mesh, and prismatic layers for the fuselage and a small part of

the wing, which is responsible to make the transition from the high-lift wing to the fairing surface, are grown. In Fig.

14, one can observe the above-mentioned portion of the wing where the transition among different mesh elements

occurs. Figure 15 represents a cut along a spanwise station showing the overall characteristics of this resulting

volumetric mesh. One of the major advantages of the hybrid hexahedral-tetrahedral-prismatic mesh is the reduction

in the computational costs due to the possibility of working with a coarser mesh when compared to a refined

tetrahedral-prismatic grid. The hybrid tetrahedral-prismatic mesh has approximately 5.0 million elements and the

hybrid hexahedral-tetrahedral-prismatic grid has around 2.3 million elements, thus demonstrating the gain in

computational costs while still providing a better discretization of the region away from the body. The decrease in

the total number of elements in the hybrid hexahedral-tetrahedral-prismatic volumetric mesh occurs because the

surface discretization of the wing and the flap requires much less elements with an hexahedral mesh than with a

tetrahedral grid. In the present work, much of the hybrid mesh study is based on the trade-off verification18

presented in Fig. 16. This trade-off relates the mesh accuracy and the ease of use. One must have in mind that the

feasibility of the 3D methodology is consolidated by the grid generation process without the excessive time usage.

For this reason, the authors explore different mesh generation procedures.

Tetrahedra + Prisms Hexahedra


13

Figure 14. Surface discretization of the 3-D tetrahedral/prismatic hybrid volumetric mesh.

Figure 15. Spanwise station cut showing the details of the hybrid hexahedral/tetrahedral/prismatic three-

dimensional mesh.


14

Figure 16. Comparison for different mesh types for RANS computations.

V. Results

The 2-D simulations are performed using a viscous formulation and tests are accomplished using both the SA

and SST turbulence models. Moreover, two different freestream conditions for the turbulent parameters, namely

turbulence intensity (I) and turbulence length scale (L), are also tested. These variations in the turbulent parameters,

I and L, are presented as the ratio of turbulent eddy viscosity by laminar eddy viscosity. Figure 17 presents the

comparison between experimental and numerical results for the NLR7301 airfoil configuration with a gap of 2.6%.

This comparison is performed for a flight condition with freestream Mach number 0.185 and Reynolds number of

2.51 million. The best agreement with experimental data is obtained by the SA model and the variations in the

turbulent parameters, I and L, do not cause any significant difference in results.

Figure 17. Comparison between the maximum experimental Cl and the maximum Cl obtained with the simulations

with the SA and the SST models.

Because the wing has no sweep, no taper ratio and is constructed with the same profile from the root to the tip,

without any incidence change, the estimated achievable Cl is the same for the entire span. The wing loading curve is

obtained by integration of the Cp distribution provided by the VSAERO9 panel method for a three-dimensional

airplane model. In Fig. 18 one can observe the Cp distribution over the surface of the airplane.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00

AOA

Cl

Experimental

SA - mit/mi = 10

SA - mit/mi = 50

SST - mit/mi = 10

SST - mit/mi = 50


15

Figure 18. Cp distribution from the VSAERO panel method.

Figure 19 shows the matching between the wing loading curve and the maximum local Cl values obtained over

the span from the SA and SST numerical simulations. Using such information, the quasi-3D methodology predicts a

maximum CL value about 2.66 for the SA simulations and about 2.50 for the SST simulation. Considering that the

SA simulations show a better agreement with the experimental data in terms of the section pressure coefficient

distributions, one would have a tendency to consider that the right value for the maximum CL is 2.66. Here, one can

observe how the dispersion on the two-dimensional maximum local Cl values changes the predicted wing CL. For

this reason, the calibration of the methodology with the wind tunnel results is essential. As previously mentioned,

the calibration tries to minimize the dispersion found in the two-dimensional simulations. For this study case, an

empirical-numerical method, called the pressure difference rule, PDR,21

is used for a verification of the lift

coefficient predicted by the quasi-3D methodology. The PDR method tries to connect low fidelity CFD runs with

physical criteria observed during wind tunnel testing. This method considers the existence of a maximum pressure

difference between the suction peak and the trailing edge at the maximum lift condition. The maximum CL obtained

with the PDR method is 2.35, which indicates a certain level of discrepancy with the quasi-3D methodology results.

Actually, the differences are about 11% for the SA results and 6% for the SST results.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 1 2 3 4 5 6 7 8 9

Y

CL

SST

SA

Wing_loading SST

Wing_loading SA

Figure 19. Wing loading curve matching to determine the maximum CL.

Results are included in the present paper, for the 3D methodology, for both meshes previously described for this

configuration. Both simulations considered the SA turbulence model. In Fig. 20, one can observe the stall angle of

attack in the CL coefficient versus alpha curve for the three methodologies here discusses, namely, the 3D

methodology, the quasi-3D methodology and the PDR method. The maximum predicted CL using the 3D

methodology for the tetrahedral-prismatic mesh is 2.03, which is an unexpectedly low value. The results for the

hexahedral-tetrahedral-prismatic mesh produced a maximum CL value of 2.33. Such predicted maximum CL is

closer to those obtained with the quasi-3D methodology analyses.


16

The comparison between the three different methodologies is performed without intention to assume that one

approach is better than the others. The focus is on verifying the level of agreement obtained by different numerical

methodologies. This agreement is a very important issue once it provides confidence about the achievement of the

target aerodynamic performance. Depending on the magnitude of the dispersion presented in the numerical

simulations, one can have a situation in which the quasi-3D methodology might predict the accomplishment of the

target performance whereas the 3D methodology might not. In such situation, one must decide between alterations

in the design, which can obviously impact the schedule of the project, or pushing forward with the current proposed

design, with all the inherent risks of leading to a worse situation in the future. Hence, the consistency of the results

obtained by different numerical methodologies, in predicting approximately the same aerodynamic coefficient, is

important to minimize possible surprises revealed in future wind tunnel tests.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.0 5.0 10.0 15.0 20.0 25.0

AOA ( 3D )

CL

3D Methodology Hexahedral + Prism + Tetrahedral

3D Methodology Tetrahedral + Prism

Vsaero

PDRSST

SA

Figure 20. CL versus angle of attack curve.

VI. Conclusions

The present results show a certain level of discrepancy among the predictive methodologies. In the case of the

quasi-3D methodology, one should observe that the results obtained with the SA turbulence model are in better

agreement with the available data and, in particular, they better reproduce the maximum experimental Cl. The results

with the SST model are an example of how disperse the maximum CL prediction can be if the two-dimensional

simulation does not work adequately. In the present case, the dispersion presented by the quasi-3D methodology was

about 6%. This is the main reason why most of the quasi-3D success relies on a good calibration process. The

authors’ experience, about high-lift two-dimensional simulation, is in favor of the SA model due to the lower

computational time costs and the ability to yield good results, at least, most times.

The results with the PDR method indicated a better agreement with those obtained with the 3D methodology. In

the authors’ opinion, this method must be further investigated in order to reach more conclusive guidelines about its

use and behavior. At least for the current geometry of study, hence, this method is based on physical verification

during wind tunnel tests for conventional configurations, which is not the case of the present aircraft.

The 3D methodology has begun quite recently to be more widely used by industry for high-lift device design.

This methodology seems to be able to predict many complex multi-element flowfields at angles of attack below the

stall. However, the prediction near the wing stall strongly depends on the configuration being studied, the mesh

quality and the numerical setup. Unfortunately, considering the current computational results, it is not possible to

precise the causes for the errors observed in the maximum lift coefficient and the airplane stall angle of attack. For

this reason, the authors are beginning to conduct systematic mesh refinement studies in order to try to develop

guidelines for adequate grid refinement to produce reasonable maximum CL predictions. Such studies are, however,

beyond the scope of the present paper.

The computation of the 3D analysis is considerably more expensive in comparison with the quasi-3D

methodology. One of the greatest bottlenecks of the 3D process is the process of cleaning up imperfections of the

CAD description. The computation costs depend on the number of cases to be analyzed and the computational

power available. Sometimes, if only a few analyses are being simulated, the computation costs might be of the same


17

order of the cost of the CAD work for a complex geometry. Usually, although the computational power presently

increases by a factor of 10 every five years, the amount analysis and the attention to small geometric details also

increase to minimize the uncertainties of the design. For this reason, the authors believe that the 3D analysis will

always be costly regardless of the focus of the investigation.

Acknowledgments

The authors acknowledge the partial support of Conselho Nacional de Desenvolvimento Científico e

Tecnológico (CNPq) through the Integrated Project Research Grant No. 501200/2003-7. Support for the present

research was also provided by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) through Grant

No. 2000/13768-4.

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