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Design of a Supersonic Natural Laminar Flow Wing–Body
E. Iuliano,∗D. Quagliarella,†and R. S. Donelli‡
Italian Aerospace Research Center, 81041 Capua, Italy
I. Salah El Din§
ONERA, 92190 Paris, France
and
D. Arnal¶
ONERA, 31055 Toulouse, France
DOI: 10.2514/1.C031039
The present investigation has been carried out within the SUPERTRAC project funded by the European
Community in the 6th Framework Programme and aimed at investigating various techniques within laminar flow
technology applied to supersonic drag reduction. In particular, this work deals with natural laminar flow shape
design of a supersonic high-sweep wing–body configuration. The reference geometry has been provided by Dassault
Aviation, one of the two industrial partners together with Airbus. Two different design are presented, produced by
the Italian Aerospace Research Center and ONERA, respectively, using numerical optimization procedures based
on evolutionary computing techniques and robust aerodynamic analysis tools. Results show how shape optimization
can be effective in changing the boundary-layer characteristics of a supersonic high-swept wing and enhancing
natural laminar flow on the wing surface.
Nomenclature
A= wave amplitude function
CD= wing drag coefficient
CL= wing lift coefficient
CM= wing pitching moment coefficient
Cp= pressure coefficient
c= airfoil chord
fi= shape-modification functions
lam = laminar
ler = leading-edge radius
M= Mach number
N= amplification factor
R= attachment-line Reynolds number, We=
~
Ue
q
Rex= Reynolds number based on xabscissa
R= compressible attachment-line Reynolds number,
We=
~
Ue
q
sep = separation
T= temperature
Te= boundary-layer edge temperature
Tr= recovery temperature
Tw= surface wall temperature
T= reference temperature,
Te0:1TwTe0:6TrTe
t= time
tea = trailing-edge angle
tr = transition
Ue= chordwise velocity component
~
Ue= chordwise velocity gradient at stagnation, dUe=dx
We= spanwise velocity component
wi= shape-modification function weights (design variables)
x= abscissa along the airfoil chord
y= abscissa along the wingspan
z= abscissa normal to airfoil chord
x= abscissa normal to leading edge
y= abscissa normal to wall
z= abscissa parallel to leading edge
= wave number along
x
= wave number along z
= sweep angle
= air kinematic viscosity
= air kinematic viscosity evaluated at T
= imaginary part of the disturbance growth rate
!= wave frequency
I. Introduction
THE supersonic civil transport aircraft is still a challenge for the
aeronautical scientific community. At supersonic speeds, the
high amount of drag and noise levels requires a careful design
process that is capable of providing innovative aircraft configurations
and aerodynamic shapes. Since the Concorde and Tu-144 aircraft,
several design ideas have been proposed, but noneof them have been
put into practice. This has been mainly due to the loss in aerodynamic
efficiency and the economic implications that are embedded in the
supersonic flight. In recent years, many government/industrial-
funded projects have been devoted to find new engineering solutions
to these well-known problems. In this context, one of the main
objectives of the last 10 years has been to maximize the laminar flow
portion on supersonic wings. As friction drag can account for about
20–40% of the total drag at supersonic speeds, a delay of the laminar-
to-turbulent transition could allow large weight and fuel savings. It is
well known that boundary-layer transition on swept wings is caused
by small perturbations that grow as they propagate downstream. A
standard method to predict transition, often used in industrial
applications, is the eNmethod [1]. The eNmethod has been applied in
this design process, since it has already been used in the supersonic
regime [2,3].
Presented as Paper 2009-1279 at the 47th AIAA Aerospace Sciences
Meeting, Orlando, FL, 5–8 January 2009; received 25 March 2010; revision
received 8 November 2010; accepted for publication 9 November 2010.
Copyright © 2010 by the American Institute of Aeronautics and Astronautics,
Inc. All rights reserved. Copies of this paper may be made for personal or
internal use, on condition that the copier pay the $10.00 per-copy fee to the
Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923;
include the code 0021-8669/11 and $10.00 in correspondence with the CCC.
∗Aerospace Research Engineer, Applied Aerodynamics and Aeroacoustics
Laboratory, Via Maiorise; e.iuliano@cira.it.
†Aerospace Research Engineer, Applied Aerodynamics and Aeroacoustics
Laboratory, Via Maiorise; d.quagliarella@cira.it.
‡Aerospace Research Engineer, Aerodynamics and Aeroacoustics
Methods Laboratory, Via Maiorise; r.donelli@cira.it.
§Aeronautical Research Engineer, Applied Aerodynamics Department, 8
Rue des Vertugadins; itham.Salah_el_din@onera.fr.
¶Aeronautical Research Engineer, Aerodynamics and Energetics
Modeling Department, 2 Avenue Edouard Belin; Daniel.Arnal@
onecert.fr.
JOURNAL OF AIRCRAFT
Vol. 48, No. 4, July–August 2011
1147
A design task aiming only at delaying transition could be inef-
fective, or even negative, when looking at global wing performance;
other features should also be addressed as key elements, such as
pressure drag reduction, noise reduction, structural design, fuel
volume, and subsonic performance. For instance, the leading-edge
sweep angle should be carefully evaluated, because it has a strong
effect on low-speed aerodynamics, drag characteristics, and
structural design. Indeed, the possibility of maintaining extensive
laminar flow over a low-sweep wing is confirmed by experimental
data [4–6] and theory [7,8]. However, as the wing would have high
inviscid drag, the corresponding airfoil section should be designed to
be thin with a sharp leading edge. As a consequence, the structural
stiffness and the subsonic stall behavior would be negatively
affected. On the other hand, a high-sweep wing design with subsonic
leading edge does not present these negative features, but obtaining
large regions of laminar flow is a very difficult challenge, due to high
Reynolds numbers, adverse pressure gradients, and highly-three-
dimensional flow structures. In other words, a supersonic wing
design requires a multidisciplinary and multipoint approach.
Recently, the NEXST project by the Japan Aerospace Exploration
Agency [9,10] proved the possibility to achieve extended natural
laminar flow both numerically and experimentally, even for highly
swept wings. In their work, the wing airfoil shapes and twist
distribution were designed using an inverse method and boundary-
layer stability analysis tools [3,11]. The numerical and experimental
campaigns were performed on a subscaled airplane model and hence
at a very low Reynolds number (about 11% of the full-scale concept).
The present work is mainly focused on the aerodynamic shape
optimization of a high-sweep wing for a supersonic business-jet
configuration in order to achieve natural laminar flow (NLF). A fully
3-D transition prediction procedure is presented here as the core
analysis tool in an optimization framework based on evolutionary
algorithms. Indeed, current computational fluid dynamics (CFD)
solvers, mostly based on Reynolds-averaged Navier–Stokes (RANS)
equations, and viscous–inviscid interaction tools are not able to
predict the onset of transition, because they lack the information
about the growth of boundary-layer disturbances. Hence, in order to
accurately predict the transition location, three chained computations
are generally required: an aerodynamic flow solution to obtain the
external mean flow, a boundary-layer computation, and a stability
analysis of boundary-layer velocity profiles.
Several tools may be used to perform such computations with
different levels of fidelity in flow physics modeling. The design chain
and tools used in the present research will be detailed in the following
sections. The design task is carried out within the SUPERTRAC
(Supersonic Transition Control) project, funded by the European
Community within the 6th Framework Programme (2005–2008).
The objective of the aerodynamic shape design is to numerically
demonstrate that the laminar portion of the wing can be increased
even in challenging design conditions by means of suitable param-
eterizations and accurate flow solvers. Two partners have been
involved in the present work: namely, CIRA and ONERA; even
though they share nearly the same tools for aerodynamic evaluation,
the adopted optimization strategy and the wing geometry param-
eterizations are different. Both approaches and results are presented
and discussed.
II. SUPERTRAC Project Framework
The study of laminar–turbulent transition is of great importance in
many fluid-dynamic applications, since the magnitude of viscosity
and heat transfer increases considerably as the flow becomes turbu-
lent. The accurate prediction of the transition location (for instance,
on a wing surface) is then a prerequisite for accurately predicting the
drag. In the last 20 years the interest in laminar flow technology has
grown rapidly because of the large benefits envisaged in its
application on modern aircraft, due to energy savings and envi-
ronmental advantages [12]. For commercial transport aircraft, the
achievement of laminarization would significantly reduce the drag of
wings, tails, or nacelles with respect to conventional design, which is
usually fully turbulent. The feasibility and potential benefits of
laminarization for subsonic/transonic transport configurations have
been evaluated and demonstrated in many wind-tunnel and flight
experiments [13–15]. The problem has been and is still widely
studied for subsonic and transonic flows, but little is known about the
feasibility of laminar flow control techniques at supersonic speeds.
The extension of the low-speed laminar flow techniques to
supersonic configurations is not obvious and requires specific inves-
tigations. The objective of the SUPERTRAC project is to investigate
the possibilities to apply the laminar flow technology to supersonic
aircraft wings, such as micron-sized roughness, anticontamination
devices, laminar flow control by suction, and natural laminar flow
shape optimization. Indeed, future air traffic vision aims at severe
requirements on emission and noise that are directly related to
weight, size, fuel burn and drag reduction. The SUPERTRAC
project, coordinated by ONERA, involved nine partners, including
two European aircraft manufacturers. All the partners have already
participated in previous European research projects devoted to
laminar transonic aircraft development [16,17] (ELFIN, HYLDA,
HYLTEC, EUROTRANS, and ALTTA).
III. Transition Mechanisms: Theory Background
A. Phenomenological Description
In this section the phenomena causing a laminar boundary layer to
become turbulent on a swept wing are discussed. There are at least
three main mechanisms that govern transition phenomena and that
have to be considered [18] (see Fig. 1).
First, the attachment-line contamination that occurs when
turbulence convected along the fuselage propagates along the swept
leading edge and then contaminates the wing surface. This kind of
instability is strongly affected by the leading-edge wing radius, as
observed by Pfenninger [19] during the laminar flow control flight
tests with the X21 aircraft. He formulated a criterion, later perfected
by Poll [20,21], based on the attachment-line Reynolds number
R<245. For compressible flows, a modified Reynolds number
Ris
used [22]. The criterion attachment-line contamination to be avoided
if an appropriate sweep and curvature of the leading edge are
selected.
The other two mechanisms have a common origin. Close to the
leading edge, in a region known as the receptivity region, according
to Morkovin [23,24], the external disturbances, such as freestream
turbulence, engine noise, acoustic waves, enter the boundary layer
and generate unstable waves. This is obtained through the effect of
the boundary layer growing in the neighborhood of the leading edge,
wall roughness, wall waviness, and blowing/suction. The above-
introduced receptivity mechanism can trigger the amplification of
Tollmien–Schlichting (TS) and crossflow (CF) waves. TS waves are
the result of the instability of the streamwise mean velocity profile,
i.e., the component of the mean velocity profile in the external
streamline direction. These waves are unstable in regions of zero or
positive pressure gradients. Their evolution is fairly well described
by the linear stability theory [25,26]. CF waves are the result of the
laminar turbulent
Flow
Leading edge sweep
3D disturbance amplification
Higher−order instabilities
Tollmien−Schlichting instability
Cross−flow instability
Attachment−line instability
Fully turbulent flow
Fig. 1 Transition phenomena on a swept wing.
1148 IULIANO ET AL.
instability of the crossflow mean velocity profile. These waves are
unstable in regions of negative pressure gradient, typically in the
vicinity of the leading edge where the flow is strongly accelerated.
When the wave amplitude becomes finite, nonlinear interactions
occur and lead rapidly to turbulence.
B. Transition-Delaying Strategy
Delaying the onset of transition requires to modify the mean
velocity field and/or the instability mechanisms in such a way that the
growth rate of TS and CF waves is reduced. This objective may be
achieved by using three strategies:
1) Natural laminar flow control consists of optimizing the wing
shape in order to maximize the laminar flow portion.
2) Laminar flow control modifies the boundary-layer velocity
profile by applying a small amount of suction at the wall.
3) Hybrid laminar flow control combines the previous two
approaches.
The present work deals with the natural laminar flow control
strategy. To design a shape with extended laminar flow, the
hypothesis of a linear evolution of the boundary-layer disturbances is
sufficient. In this context, the most common transition prediction
method is the eNmethod [1,2,27]. This method is based on the
relative amplification of the discrete frequency disturbance, which
first reaches a preset transition level of eN. The eNmethod involves
the stability analysis of the laminar boundary layer by solving
the linear equations of the nonstationary disturbances superposed to
the basic motion. Under the assumption of quasi-parallel flow, the
linearized Navier–Stokes equations admit a normal mode solution in
the form of a harmonic wave:
’
x;
y; z; tA
yei
xz!t(1)
where ’is a velocity, pressure, or density fluctuations. Introducing
the previous expression in the linearized Navier–Stokes equations, a
system of ordinary differential equations for compressible flow is
obtained: these equations and the related boundary conditions are
homogenous and represent an eigenvalue problem that admits a
nontrivial solution only when the dispersion relation !!; is
satisfied [25]. Usually, the problem is solved by following two
different theories: a spatial theory, where and are assumed to be
complex and !real, and a temporal theory, where and are
assumed to be real and !complex.
The solution of the linear system of equations allows the growth
rate of the disturbances to be determined. The Nfactor represents the
amplitude ratio for each frequency obtained by integrating the spatial
amplification rate as follows:
NlogA
A0Zx
x0
xdx(2)
where is the imaginary part of the disturbance growth rate, and A0is
the wave amplitude at xx0.
IV. Design Target
The objective of the present work is the design of a supersonic
wing with enhanced natural laminar flow characteristics, as well as
the understanding of the aerodynamic and geometric features that
can drive the optimization toward NLF enhancement. Indeed, such a
kind of optimization has not been attempted in previous works;
therefore, a solid expertise and scientific background on the results
cannot be obtained. This makes it of paramount importance to give a
physical interpretation to the results coming from the optimization
process.
The reference configuration is the wing–body shape optimized
within the framework of Supersonic Business Jet project of Dassault
Aviation and made available within SUPERTRAC project. A sketch
of the geometry is given in Fig. 2. The inboard wing has a 65
leading-edge sweep angle, and the outboard-wing sweep is 56. The
wing semispan is 9.35 m and the aspect ratio is 3.5. The cruise flight
Mach number is 1.6. Previous investigations [28] showed that the
baseline airfoil presented a very short extent of laminar flow, and
transition was triggered by both CF and TS waves. A redesign of the
wing airfoil at a selected station was agreed upon among the
SUPERTRAC partners in order to build a new reference shape for
the wing design phase, which is described here. The target is to
optimize the wing section airfoils and twist angle in order to
maximize the laminar flow region while monitoring and controlling
the pressure (vortex and wave) drag.
To this aim, given the aerodynamic design problem, a numerical
optimization procedure is used. Generally speaking, the problem of
searching a solution to a minimization problem strongly depends on
the shape of the objective functions and on the set of constraints. As
neither the objective function nor the imposed constraints can be
a priori cast in closed form, the exploration of the design space can
get very computationally intensive, especially if the target function
has a highly multimodal profile. Indeed, the presence of several local
minima can trap a gradient-based algorithm in a suboptimal solution
or negatively impact the convergence of an evolutionary-based
algorithm, even if the naturally explorative characteristics of the
latter have shown to be potentially able to approximate the true
optimum or the set of optima. Moreover, the specific problem under
analysis usually requires a proper parameterization and many design
variables to control not only the profile shape, but also its distribution
along the spanwise direction, possibly the twist angle and the
thickness ratio distribution. Therefore, the designer has to deal with
the multimodal and high-dimensional character of the objective
function, which greatly increases the complexity of the search
approach. The problem gets even more complicated considering the
partial lack of literature in the peculiar field of supersonic laminar
flow wing design, which does not provide sufficient guidelines
toward the optimal solution. Hence, a proper, highly explorative,
design strategy must be defined and assessed. The two partners,
CIRA and ONERA, have split the design space exploration into two
different optimization strategies, named global and local. It must be
emphasized once again that the two approaches do not refer to the
search method in itself, but to the investigation of the relationship
between physical parameters and design variables, i.e., the adopted
parameterization. Indeed, the global search aims at finding the
optimal solution by modifying the wing profiles shape along the full
chord and span. The advantage of this approach is to let the whole
shape vary according to the variation of the aerodynamic conditions
along the wing, e.g., local Mach number, Reynolds number, and
downwash. This feature allows more flexibility especially in
satisfying aerodynamic constraints on total lift, drag, and pitching
moment coefficients while optimizing for natural laminar flow. On
the other hand, the local approach is focused on local shape
modification around the wing leading edge, a region in which it is
more likely, according to previous design experience, that a shape
change can introduce a different flow behavior in terms of laminarity.
The local search aims at analyzing in detail the sensitivity of natural
laminar flow improvement to local shape change, where this
sensitivity is as high as possible. In other words, this is like a trust-
region approach, where the region to trust is not sought iteratively,
but is supposed to be already known from past knowledge. Even
though sharing a similar evolutionary-based optimization method,
the two approaches are supplementary, because they look at the
Fig. 2 Baseline geometry.
IULIANO ET AL. 1149
design space from two different perspectives (global and local) while
aiming at the same goal. In principle, this choice is made possible just
by the peculiar problem under study: indeed, previous works [3,11]
already showed that the design of a supersonic swept wing for natural
laminar flow under aerodynamic and geometrical constraints has to
properly and carefully balance leading-edge design, overall wing
profile shape and twist distribution. As reported in later sections, this
design hints are also the core experience and knowledge drawn by the
authors during the presented design research.
V. Design Methodology
Before getting into the specification of the problem formulations,
the design methodology adopted by both approaches, which are quite
close in the principle, is given. The method used to perform the wing
design is based on numerical optimization. The whole process can be
split into two main components, the optimizer, controlling the scan of
the design space and the analyzer, evaluating the performance of a
configuration on the optimizer request. A stochastic evolutionary-
based approach has been chosen to solve this problem considering
the numerous tools involved in the evaluation of the performance
process of a single aircraft design and the design space wideness.
Apart from accuracy considerations, the main drawback of a classical
gradient approach (via finite differences) is the numerous compu-
tations needed for one optimizer iteration considering the great
number of design variables involved in the parameterization. More
up-to-date approaches for gradient computations such asadjoint state
resolution [29] would allow this problem to be overcome. Unfor-
tunately, even though effective for the CFD solver, such techniques
must still be made compatible with the three-dimensional transition
prediction tools. Objective and constraint functions have been
defined in order to ensure the geometrical and aerodynamic fitness to
the requirements. The fitness is evaluated through an automatic pro-
cedure that performs sequentially a 3-D Euler aerodynamic analysis,
a 3-D boundary-layer computation and a boundary-layer stability
analysis. Hence, global aerodynamic coefficients, boundary-layer
parameters and laminar-to-turbulent transition predictions are
simultaneously available to build up the cost function to minimize.
VI. Design Chains and Tools
A. Chains Overview
This section is dedicated to the detailed description of the various
stages and tools of the two optimization processes that are sketched in
Fig. 3. Basically, the loop is made up of an optimizer, a configuration/
mesh generator and an evaluation module, each exchanging data and
information with only one of each other: the optimizer and the
configuration generator communicate through design variables; the
configuration geometry is submitted to the evaluation module, which
returns the values of constraints and objectives to the optimizer.
The fitness function can take into account several components of
the aerodynamic performance, such as inviscid drag, transition and
laminar separation location, and lift and pitching moment coef-
ficients. These terms can be computed for one or more design points
and can be freely combined into one or more objective functions and
constraints. Furthermore, each performance term can be either
directly summed (through a weight) into a given objective or con-
straint either filtered through some predefined penalty functions.
Constraints may also be of geometrical nature, such as maximum
thickness or wing box layout and volume. Some constraints, like lift
force, may also be directly satisfied through the variation of a free
parameter, such as the incidence angle.
B. Optimization Algorithms and Process Management
Genetic algorithms (GAs) are based on evolutionary concepts
implying operators such as several types of mutation and crossover
operators. Even though both approaches use a GA, they do not share
the same tools. The global approach calls an in-house CIRA
evolutionary optimization tool, GAW [30–33], which can handle, in
addition to the GA management, multipoint and multiobjective
constrained problems and is also able to use hybrid operators
adopting classical gradient-based algorithms. When facing a
multiobjective problem, the GA can directly use the defined domi-
nance criteria to select elements appointed to reproduction and to
hence drive the evolution of the population toward the Pareto front.
The selection operator used here is random walk. The locally
nondominated population elements met in the walk are selected for
reproduction. If more nondominated solutions are met, then the first
one encountered is selected. At the end of every new generation, the
set of Pareto optimal solutions is updated and stored. A kind of
extension of the elitism strategy to multiobjective optimization is
obtained by randomly selecting an assigned percentage of parents
from the current set of nondominated solutions.
Concerning the second approach, called local, the GADO [34]
library, based on a genetic algorithm and integrated in the DAKOTA
[35] open source optimization suite was chosen. The flexibility of
DAKOTA, which integrates various capabilities from the manage-
ment of typical gradient-based approaches to complex surrogate
models, has proven useful and efficient in previous studies [36].
C. Parameterization and Automatic Configuration: Mesh Generation
The key feature of both design systems is the capability of
performing the automatic remeshing of new configurations, for fixed
topological connections and grid tuning parameters. The surfaces of
new configurations are generated on the base of geometrical param-
eters, which are linked to some of the design variables controlled by
the optimizer. When dealing with an optimization problem the
parameterization refers to the link between the design variables and
the global shape of the configuration under study: in an aeronautic
frame like the present, the parameterization acts on the real geometry,
which has to be modeled according to the modification of the design
variables. Once chosen a reference configuration (here convention-
ally corresponding to the configuration obtained by setting all design
variables to zero), links between geometric parameters and design
variables are set in the problem definition phase and arranged in a
database file. In principle, it would be possible to link each config-
uration modification to a design variable, but it can be useful to
exploit general rules or functions to modify a set of parameters with
one design variable. The capability of modifying the reference
geometry in different ways, depending on the specific problem, can
Data manager
(GAW / DAKOTA)
Optimizer
(GAW / GADO)*
Modeler
(GEORUN / In house generator)
Domain and mesh generator
(ENDOMO, ENGRID / Dedicated in house generator)
Flow solver (ZEN / ElsA)
BL analysis (BL3d / 3C3D)
BL stability (PARAB_3d)
Design variables
Design variables
Wing geometry
Wing-body intersection
Wing geometry
Wing-body intersection
GRID
GRID
Objective / constraints
Objective / constraints
*: (global / local)
Fig. 3 Wing design chain.
1150 IULIANO ET AL.
improve the efficiency of the optimization process. As an example,
the database format for aerodynamic shape of wings has been
designed in such a way that parameters are familiar to the aero-
dynamicist. Taper ratio, sweep angle, twist angle, maximum
thickness distribution along the wingspan are typical parameters.
Each of these geometrical items can be activated and linked to a
specific design variable. The airfoil shapes throughout the wing are
controlled by using a list of points. They can be modified either by
adding predefined shape functions to the reference geometry and by
moving some control points through a bispline approach [37].
To generate the new mesh, specific set of input data are required in
both cases: the topology of the domain (i.e., how blocks are
connected to each other), the geometry (i.e., the shape of each block),
the grid dimensions (how many cells have to be generated in each
block) and the grid parameters (how the grid cells have to be sized
and clustered). To generate a family of meshes that share the same
characteristics during the design process, the topology data, grid
sizes and parameters are kept fixed. This is useful within the auto-
matic procedure and also for better comparisons of flow solutions.
Indeed, drag computation is very sensitive to grid density, partic-
ularly when meshes are not very fine. The procedure helps to
maintain a standard grid quality during the remeshing of new
configurations in the optimization loop.
For the global approach, in-house software is used to build
computational volume grids. The GEORUN [30] geometry gen-
erator, the ENDOMO [38] domain modeler, and the ENGRID
[39,40] multiblock structured-grid generator have been chained to
generate topologically similar grids on geometrically different
aerodynamic shapes. For the second approach, the surface meshing
was generated using ICEM HEXA mesh generator. Then an ana-
lytical deformation tool is coupled with an automated mesh generator
dedicated to supersonic configurations to build the volume mesh.
D. CFD Flow Solvers
The aerodynamic analysis system is based on a structured
multiblock approach [41–44]. The in-house-developed ZEN (CIRA)
and elsA (ONERA) flow solvers are multizonal Euler–RANS
solvers, with several turbulence models implemented. Spatial
discretization is performed by finite volume, central schemes, or
decentral schemes, with second- and fourth-order artificial dissi-
pation. Convergence toward steady solution is obtained using
explicit Runge–Kutta formulas, with implicit residual smoothing and
multigrid acceleration techniques. The flow solvers are fully
vectorized. These features represent a compromise choice between
efficiency and accuracy, since an optimization procedure requires the
analysis of hundreds of configurations.
E. Three-Dimensional Boundary-Layer and Stability Computation
To estimate the laminar-to-turbulent transition onto a wing
surface, several methods can be used, but the designer usually aims at
obtaining an acceptable compromise between accuracy and speed,
because the number of computations to perform in a design process
may be very high. The present approach consist of a 3-D Euler
inviscid solution coupled to a fully-three-dimensional finite differ-
ence boundary-layer analysis. This choice allows the basic features
of both the external aerodynamic flowfield (like shock waves,
pressure gradient, and wing loading) and the laminar/turbulent
boundary-layer flow (both streamwise and crossflow velocity
profiles, temperature profiles, and integral quantities) to be caught.
Once known the spatial evolution of the laminar boundary layer, a
method based on the linear stability theory coupled to the eNmethod
is applied to check for laminar-to-turbulent transition. A general view
of the transition prediction cascade is given in Fig. 4.
1. Three-Dimensional Boundary-Layer Computation Method
The three-dimensional boundary-layer method is based on the
finite difference scheme developed by Cebeci [45] for the global
approach and on the 3C3D [46] solver for the local approach. Both
codes are able to solve compressible laminar/turbulent flows on
finite wings with and without suction through either the full three-
dimensional or the quasi three-dimensional boundary-layer
equations [47].
The solution of the three-dimensional equations, where
streamwise velocity uand spanwise velocity wcomponents are
positive, can be achieved easily with the finite difference methods of
TRANSITION
LOCATION
Pressure Distribution
Numerical Computation
Database Method
“Parabole” Code
Local Stability Analysis
“COSAL” code
Non Local Stability
Analysis
Boundary Layer Calculation
eNmethod – Correlation with experimental data
Flight Data Wind Tunnel Data
Fig. 4 Flowchart of transition prediction on wings.
IULIANO ET AL. 1151
Crank-Nicolson and Keller , as described in [48]. When the spanwise
velocity component contains regions of flow reversal, however, the
solution of three-dimensional boundary-layer equations is not so
straightforward and requires special procedures to avoid the
numerical instabilties that can result from reversal in w.For
simplicity and robustness, the zigzag box scheme [48] is used in this
case.
The boundary-layer method is formulated in the standard mode,
i.e., it requires that the external velocity distribution is given in
addition to the freestream conditions and wing geometry. It assumes
that there is no flow separation on the wing; if one is found, then the
calculations are terminated at the location where the chordwise wall
shear force vanishes.
2. Boundary-Layer Stability Analysis
Many industrial tools based on the local linear stability theory,
coupled with eNmethod have been developed in these years. These
tools are very fast, but the huge quantity of calculations necessary to
predict the transition location is often time consuming. Furthermore,
most of these methods require a convergence procedure, which
cannot be easily executed automatically. For these reasons ONERA-
CERT developed the so-called database method [49,50]. This
method is based on the idea to analytically represent the growth rate
of unstable disturbances as a function of some relevant boundary-
layer parameters [51–53]. The database method is able to compute
the growth rates of Tollmien–Schlichting and crossflow waves. The
main notations are given in Fig. 5. At a given point on the wing, is
the local angle between the external streamline and the xaxis and is
the angle between a chosen direction and the external streamline. The
coordinate system is referenced as x; z; y, where yis the normal to
the wall. The basic flow, resulting from a laminar boundary-layer
computation, provides the local velocity profiles
Uyand
Wy.
These profiles are projected in a given direction (corresponding to the
wave angle direction ) leading to a 2-D projected velocity profile
U2Dy. The latter is inflexional for a projection angle sufficiently
large. The main idea of the database is to guess that the growth rate
may be approximated with only a prescribed frequency, the projected
Reynolds number R2D(i.e., calculated with the external projected
velocity
Ue;2D, and the displacement thickness 1;2Dcalculated with
the projected velocity profile
U2Dy) and two parameters:
Ui
U2DyiPyidU2D
dyyyi
where yiis the distance of the inflexion point from the wall. It is
implicitly assumed that all the quantities in the previous expressions
are dimensionless according to
Ue;2Dand 1;2D. Then for a given
value of these two key parameters Ui;P, that means for a given
physical mean boundary-layer abscissa
Uy;
Wy, it is guessed
that for a given frequency, the growth rate 2Dof the projected profile
has a universal dependence with respect to the Reynolds number R2D
in the form of two half-parabolas (see Fig. 5). These two half-
parabolas may be characterized by R0,Rm,R1, and mfor,
respectively, the critical Reynolds number, the Reynolds number
corresponding to the maximum of amplification, the second critical
Reynolds number (branch II of the neutral curve), and the value of the
maximum of amplification. All of these quantities refer to the
projected mean velocity profile. These four coefficients are simple
analytical functions of the frequency with coefficients that only
depend on Uiand P. It has been proven that the final expressions
obtained work rather efficiently, with a maximum of 10% of
difference in comparison with exact stability results (and with a ratio
of computation effort in the range of 105!). The above expressions do
not work for the zero frequency, but the approach has been extended
by ONERA in cooperation with CIRA [51]. In fact, the two param-
eters have been computed with the assumption that, for practical
cases, the projected Reynolds number and the corresponding growth
rate are more or less limited in a region close to R0(as given in Fig. 5).
A new simple analytical relationship between the growth rate and the
projected Reynolds number for given values of Uiand Phas been
found [54]. Figure 5 shows the comparison of the estimated and exact
spatial amplification rate, when the asymptotic value of the projected
growth rate 1is approximated with an analytical relationship.
3. Choice of Critical NFactor
It is well known to the aeronautical scientific community that the
use of the eNmethod is subject to the choice of the critical Nfactor at
transition. Several investigations have been performed to find cali-
brations and exploit correlations able to accurately predict transition
through eNmethod in transonic flows [13–17]. However, to the
authors’knowledge, no similar data are available for supersonic
flows. Hence, as the main objective of the present research is to
investigate the potential to enhance the laminar flow on a supersonic
leading-edge wing through shape optimization rather than to design a
new wing, the Nfactor at transition is expected and assumed to be a
plausible and approximate (but tested) value, rather than a precise
threshold. As a consequence, the expertise and knowledge acquired
in past successful research projects have been exploited to this aim. In
X
αι
0 0.1 0.2 0.3 0.4 0.5 0.6
0
1
2
3
4
5
6
7
8
9
10
11
12
database
exact results
a) Coordinate system b) Two half parabolas express the Reynolds
dependance of the local growth rate, for
the nonzero frequencies
c) Comparison of the exact and the estimated
growth rate 3D for fixed dimensional
spanwise wave numbers = 1000 m-1
σ
β
Fig. 5 Database method.
1152 IULIANO ET AL.
particular, during the ELFIN II European project, numerical
computations using the database envelope method were compared
with Fokker 100 flight-test data; one of the conclusions of such an
investigation was that [55],
The scatter of Nfactors is quite reasonable, despite whether the
flow has been treated as compressible or incompressible, with or
without the presence of curvature effects. The calibrated Nfactors,
in most cases, are in the range of 15 or above for compressible
without curvature....Compressibility has a moderate effect on
stabilising the disturbances.
Since compressibility has a damping, and hence stabilizing, effect
on boundary-layer disturbances, a transition Nfactor of 15 is
assumed to be a suitable value for supersonic cases and is used here to
detect the transition location on a supersonic wing throughout the
design process.
VII. Parameterization and Optimization
Problem Formulations
The problem definition phase includes the identification of design
variables, objectives and constraints to be handled in the opti-
mization process. As mentioned above, CIRA and ONERA applied
two different approaches in the problem definition phase, while
sharing the same design conditions and constraints. The global
approach (CIRA) consisted of investigating a wide design space
made of about 200 variables to allow for a wide range of possible
shape modifications, along both the wing chord and the wingspan.
After a preliminary sensitivity analysis, the local approach (ONERA)
consisted of performing local airfoil shape modifications applied in
the vicinity of the leading edge, combined with global spanwise
geometrical deformations.
A. Global Approach (CIRA)
1. Parameterization
The global search approach followed by CIRA aims at
investigating and successively defining the wing shape character-
istics that could promote the laminar flow in supersonic conditions.
Much is known [12], even in the supersonic regime [3], about how
the airfoil pressure distributions should be to reach wide regions of
laminarity, but it is not straightforward to find the corresponding
shape, which also depends on the flow conditions, wing planform,
and actual design constraints. Moreover, it is expected that the
fuselage presence effects and the varying aerodynamic conditions
along the wingspan (e.g., the Reynolds number varies from about
80 106at the wing root to 16 106at the wing tip) would radically
modify the resulting shape of the wing in the spanwise direction.
Hence, a proper parameterization is introduced either to catch this
feature and to ensure the geometry smoothness throughout the
wingspan. Wing section shape modification is carried out by
following the modification functions approach: 68 predefined
functions, acting both on airfoil mean line and thickness distribution,
are added to the baseline shape by using weight coefficients through
the following formula:
zxz0xX
n
i1
wifix
where zxis the modified shape and z0xis the baseline shape.
Using this approach, the shape design variables are represented by
the weights of the modification functions and the corresponding
ranges are chosen following designer experience or a trial and error
process. The fiare chosen among the classical airfoil design modi-
fication functions, like the Hicks–Henne, Wagner, and Legendre
functions [37], but they include also wave, sinusoidal, and rear
loading functions to locally modify the airfoil shape. To correlate the
shape modifications along the wingspan, design variables involving
section geometry modifications (shape and twist angle) are
recombined into new design variables by using a linear trans-
formation matrix. This allows the uniformity and regularity of shape-
modification distribution throughout the wing and the reduction of
the total number of design variables handled by the optimizer. The
general law for design-variable transformation is given by the
following matrix form:
v1
v2
.
.
.
vm
2
6
6
6
4
3
7
7
7
5
W11 W12 W1p
W21 W22 W2p
.
.
..
.
..
.
..
.
.
Wm1Wm2 Wmp
2
6
6
6
4
3
7
7
7
5
V1
V2
.
.
.
Vp
2
6
6
6
4
3
7
7
7
5
where Vi(i1;...;p) are the old design variables, vi(i1;...;m)
are the new design variables, and Wik (i1;...;m and
k1;...;p) are the transformation weights, with m<p. Among
the various functionalities that this approach allows, for instance, is to
obtain any twist angle distribution law (e.g., linear, quadratic, and
piecewise linear) along the span length or to connect the shape
modification between different portions of the wing. Twenty-five
wing sections have been selected for parameterization and wing
geometry definition, but only three of them (namely, the root, kink,
and tip sections) are effectively involved and controlled in the global
shape modification. Indeed, the adoption of the transformation
matrix allows the spread of each design variable throughout the
wingspan, e.g., by using piecewise linear functions, as shown
schematically in Fig. 6. The left-hand figure depicts the piecewise
weight functions for shape and twist design variables versus the span
length. For instance, the modification of the inboard section shape
(dotted line, labeled as inboard-wing shape) does not have any effect
on the outboard wing, because in that region the weight of each
a) Global approach - design variables spread along
the wing span
b) Local approach
Fig. 6 Parameterization details.
IULIANO ET AL. 1153
root-section shape design variable is zero. The total number of design
variables is 208, resulting from 68 airfoil shape variables for each
control wing section (204 total variables), three variables to control
the twist angle, and one additional variable to modify the wing-
fuselage setting angle.
2. Optimization Problem
A synoptic view of design condition, constraints and objective as
set up by CIRA is given in Table 1. The objective function
GCD;C
L;C
M;ler;tea;xlamincludes some penalty functions to
take into account for constraints, and it is defined as
GkPCD~
CDnP~
CLCLmP~
CMCM
qPxlam‘P~
ler lertP~
tea tea
where k,n,m,q,‘, and tare constant values that define the relative
importance of the corresponding aerodynamic/geometric perform-
ance component. The quadratic penalty function is activated in a
quadratic mode only if its argument is positive. Hence, Phas the
following expression:
Pxx2if x>0
0ifx0
The function xlam is introduced to estimate the transition and
laminar separation position on the whole wing, and it is defined as
xlam X
n
i1
xlu xll xsu xsl
where
xlu max0;X
i
tr Xi
trupper
xll max0;X
i
tr Xi
trlower
xsu max0;X
i
sep Xi
sepupper
xsl max0;X
i
sep Xi
seplower
Here, Xi
tr and Xi
sep are the computed values of transition and
separation point at span section i,Xi
tr and Xi
sep are the desired values
of transition and separation point at span section i, and nis the
number of streamwise stations defined along the wingspan.
Separation point is the chordwise abscissa at which the laminar
boundary-layer calculation stops for each section: this can occur
either because of a laminar separation (e.g., caused by a separation
bubble) or because the boundary-layer solution does not converge at
that point for some reason. Separation is taken into account in the
xlam objective function, because when it occurs upstream of the
transition location, transition is automatically switched, even if the N
factor has not yet reached the critical value. This approach is used in
order to delay the laminar separation point as much as possible.
The constraint value on inviscid drag and the set of desired
transition locations have been assigned following preliminary
studies and past experiences in laminar wing design. In particular, the
drag penalty is activated with a huge weight when the inviscid drag
exceeds the baseline value (195 drag counts). On the other hand, the
transition specifications that have been imposed in the design
problem represent a sort of utopia point, i.e., the actual threshold
above which laminar flow would become really beneficial on aircraft
performances and emissions.
B. Local Approach (ONERA)
1. Parameterization
The chosen parameterization is based on 14 design variables,
among which one is the angle of attack, five are the twist angles in
five control sections chosen along the span, four are the relative
leading-edge camber deformation extents, and the remaining four
correspond to the maximum deformation amplitudes (Fig. 6, right).
The parameterization of the leading-edge camber is the result of a
preliminary study that has shown that the Cppeak is very sensitive to
such leading-edge region modification. The drop in the Cppeak
amplitude is significant, and so is the following recompression. Even
though the outboard NLF extent is considered, the whole wing
design is parameterized in order to maximize this extent.
2. Optimization Problem
In the local approach, the drag reduction is not an active constraint,
but a filtering constraint. This means that the final population
generated by the optimizer that satisfies the constraints is filtered in
order to satisfy the filtering constraints. The objective function is
Table 1 CIRA problem definition
Parameters Values
Design variables
Wing twist 3to 3
Wing section shape 0:25 to 0:25a
Design point
M11.6
Re151:8106
Reference chord 6.27 m
Altitude 44,000 ft
Lift coefficient CL0:182
Constraints
Lift coefficient ~
CL0:180
Pitching moment ~
CM0:05
Trailing-edge angle ~
tea 0:050 rad
Leading-edge radius ~
ler 0:002 m
Laminar extent usbIWcXtr 0:35
Laminar extent LS IW Xtr 0:35
Laminar extent US OW Xtr 0:45
Laminar extent LS OW Xtr 0:35
laminar separation Xsep 0:60
Drag coefficientd~
CD0:0195
Objective
GCD;C
L;C
M;ler;tea;x
lamTo be minimized
aThis is the range of variation of the weights withat multiply the
shape-modification functions, as mentioned in Sec. VII.A.1.
bUS stands for upper side; LS stands for lower side.
cIW stands for inboard wing; OW stands for outboard wing
dVortex and wave drag.
Table 2 ONERA problem definition
Parameters Values
Design variables
Wing twist (five control sections) nui3to 3
LEadeformation extent
(four control sections) xi
20% to 50% of local chord
LE deformation amplitude
(four control sections) zi
0 to 0.1 m
Angle of attack 0to5
Design point
M11.6
Re151:8106
Reference chord 6.27 m
Altitude 44,000 ft
Lift coefficient CL0:182
Constraints
Lift coefficient ~
CL0:180
Pitching moment ~
CM0:02
Filtering constraints
Nonviscous drag coefficientb~
CDCD(baseline, CL0:182)
Trailing-edge angle (always satisfied) ~
tea 0:050 rad
Leading-edge radius ~
ler 0:002 m
Objective
G: laminar extent on upper outboard To be maximized
aLeading edge.
bVortex and wave drag.
1154 IULIANO ET AL.
directly expressed as a laminar-extent surface ratio. The optimization
problem formulation is summarized in Table 2.
VIII. Results
A. Baseline Wing–Body Analysis
Before getting into the design process, the reference configuration
has been analyzed in order to have a comparison to quantify the
achievable performance gain during the optimization task. More-
over, a preliminary mesh tuning and sensitivity analysis has been
carried out in order to find the best parameters and size to get a good
compromise between the level of accuracy and the computational
speed. Both structured meshes are made of nearly 106nodes and are
depicted in Figs. 7. The flow conditions are shown in Table 3.
The aerodynamic analysis of the baseline reference configuration
showed that a very limited amount of laminar flow can be achieved on
such a configuration. Figure 8 shows the pressure coefficient curves
at the wingspan stations reported in the same figure. A strong
expansion region is observed around the kink section, and the
discontinuity in the leading-edge sweep angle gives rise to an oblique
shock wave that propagates toward the wing tip with an angle equal
to the inboard-wing sweep (65). At the same time, the presence of an
expansion peak on the suction side downstream of the attachment
line is evident throughout all the wing stations, and its intensity is
very important in the inboard region. This phenomenon, together
with the oblique shock wave, completely inhibits the possibility to
find a long run of laminar flow on the wing surface. Indeed, the
laminar boundary-layer computation stops quite early, due to laminar
separation problems, except near the reference station (at
y5612 mm), because the wing airfoil was designed to match the
conditions in this section. Furthermore, the Nfactor is already very
high in the computed region, triggering the transition very early
along the whole wingspan. The N-factor distributions, with a cutoff
at NNcrit, on the upper and lower sides of the baseline wing are
presented in Fig. 9.
As can be seen in Fig. 9, the laminar region (N<N
crit) is mainly
confined near the leading edge. The NLF extent in this region
represents less than 4% of chord length of the outboard upper surface.
The laminar extent is detected using a critical Nfactor of 10. This
value is very conservative, as it is typically used for 2-D transonic
configurations. As mentioned in Sec. VI.E, a more realistic value of
the Nfactor for a 3-D transonic configuration would typically be
around 15. On the other hand, the slope of the computed N-factor
curves on the baseline shape (and hence the spatial growth of
boundary-layer disturbances) is so steep that even a variation on the
choice of the critical Nfactor would not bring significant differences
in transition prediction. For the above-described reasons, the refer-
ence configuration represents a critical starting point for the design
optimization, and strong airfoil shape modifications are expected
near the wing leading edge in order to prevent unfavorable
aerodynamic features.
B. Optimization Behavior
1. Global Optimization
The optimized wing–body configuration is the result of a complex
and articulated design process. It has involved several genetic
Fig. 7 Sketch of structured meshes.
Table 3 Flow conditions
Parameters Values
Mach 1.6
Reynolds 51:8106
Lref , m 6.27
Wing area, m250
AOA, 3.65
CL0.183
Fig. 8 Baseline pressure distribution at selected spanwise stations.
IULIANO ET AL. 1155
optimization cycles, both single objective and multiobjective, each
with different settings and strategies. A first optimization stage has
been performed considering the wing alone (without the fuselage) in
order to investigate the effect of shape and twist modification on
supersonic laminar flow in a simpler and rather ideal case. Indeed, it
was expected that a homogeneous incoming flow in terms of Mach
and Reynolds numbers would have enhanced the possibility to get an
extensively laminar boundary layer on the wing surface. The
performance-degradation effect of the fuselage was considered in the
final design step. Here, the starting point was the already-optimized
isolated wing coupled to the Dassault Aviation fuselage. The
convergence history of this last genetic algorithm run is depicted in
Fig. 10; it can be observed that the ratio between the initial value of
the objective function and the final one is about 4. This figure shows a
further strong improvement in the fitness, even in the presence of the
fuselage. In fact, the optimal design (wing and fuselage) achieves the
same level of performance as the initial configuration of the last run
(optimized isolated wing), which was designed without taking into
account the fuselage effect. This last result has been obtained by
changing only the wing shape, and hence it shows how proper shape
modifications can be effective to balance the laminar performance
loss due to the fuselage.
2. Local Optimization
A family of enhanced shapes was found after nearly 500
evaluations (Figs. 11 and 12), with each evaluation implying flow
and transition analysis that required up to 20 min on a NEC SX8
supercomputer. The final optimal shape was then obtained by
filtering the design that underwent the minimum deformation extent
in terms of twist (the most relevant parameter in this case), compared
with the reference case, with a maximum laminar extent. As
mentioned previously, even though the drag was not taken into
consideration during the optimization process, it has been checked
that the inviscid pressure drag variation, compared with the baseline,
was negligible.
The optimizer reached 7.1% of laminar surface instead of 4.7% for
the initial shape for a critical Nfactor set to 10. The laminar extent
corresponding to a higher value of this Nfactor would, of course, be
more important. For instance, for Ncrit 15 the laminar extent is
multiplied by 2. Both lift and pitching moment constraints are
satisfied.
C. Enhanced Configurations Analysis
1. Global Optimization
The impact of the optimization process on wing pressure dis-
tribution is shown in Fig. 13; the contour lines and the Cpcurves
highlight the continuity and the low level of flow expansion on the
suction side. As a consequence, the recompression phenomena are
present only around the trailing edge and in the wake region where
Fig. 9 N-factor maps on baseline wing.
400
600
800
1000
1200
1400
1600
1800
2000
2200
0 1000 2000 3000 4000 5000 6000 7000
Objective Function
Individuals
Fig. 10 GAW final run convergence history.
CL
Laminar extent on external wing (%)
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-2
0
2
4
6
8
10
Fig. 11 CLvs laminar extent.
CM
Laminar extent on e xternal wing (% )
-0.04 -0.02 0 0.02
0
1
2
3
4
5
6
7
8
9
Fig. 12 CMvs laminar extent.
1156 IULIANO ET AL.
the contact between the upper and lower flows occurs. The
comparison with the pressure solution on the baseline gives evidence
of a completely new redesign of the whole wing upper side.
Surface pressure distribution strongly influences the mechanisms,
which leads to laminar-to-turbulent transition onto a wing. In fact, a
careful design of the pressure gradient can delay the amplification of
boundary-layer disturbances and passively control attachment line,
crossflow,and Tollmien–Schlichting instabilities. In the present case,
the evolutionary optimization algorithm has been used to naturally
enhance the pressure coefficient characteristics by shape design
toward the maximization of natural laminar flow in real supersonic
flight conditions. Past studies [56] and present optimization results
show that two most important design features can be identified: the
chordwise pressure gradient near the attachment line and aft the
leading edge. Only the combined optimization of both can result in
an effective performance improvement. Indeed, the attachment-line
Reynolds number (
RWe=
~
Ue
q) is directly proportional to the
spanwise velocity component Weand inversely proportional to the
chordwise velocity gradient ~
Ueevaluated on the attachment line.
Obviously, the latter depends on the shape of the airfoil around the
leading edge and, in particular, on the leading-edge radius. Hence,
through the design of a strong acceleration profile with small leading-
edge radius, it is possible to prevent the attachment-line instability.
However, a tradeoff must be found with the imposed constraint on
minimum leading-edge radius value and with the need for avoiding
strong leading-edge pressure peaks, which would feed the
streamwise disturbances. Such a tradeoff has been found in the
present research and highlighted in Fig. 14, where the attachment-
line velocity component We, the chordwise velocity gradient ~
Ue, and
the resulting
Rare plotted against the wingspanwise coordinate. Both
the compressible and incompressible formulation of
Rare reported,
showing negligible differences. The value of
Rat each wing section
has been drawn from the CFD solution. The CFD Cartesian velocity
components are transformed by the boundary-layer code and its local
reference system into the local spanwise and chordwise velocity
components; hence, it is straightforward to draw Weand ~
Ue. The
latter is obtained through a simple derivation operation, which
provides more accurate results with a smaller spatial step size x=c.
x/C
Cp
18.06.04.02.00
Sec=02
Sec=04
Sec=06
x/C
Cp
18.06.04.02.00
Sec=09
Sec=12
Sec=17
Sec=20
a) Cut sections along span on Cp contour lines
b) Sectional pressure coefficient, inner wing
c) Sectional pressure coefficient, outer wing
Fig. 13 Global optimum aerodynamic solution.
IULIANO ET AL. 1157
Weand ~
Uehave been computed at various chordwise abscissa steps
(x=c) near the attachment-line location. In Figs. 14a and 14b, the
curves obtained for three values of x=c (0.001, 0.0005, and
0.00035) are reported:
Rfalls well below Poll’s threshold of 245 as
the accuracy of the calculation increases. The chordwise velocity
gradient in Fig. 14b shows the effect of the proper shape design
carried out by the evolutionary algorithm; the gradient levels are very
high at each wing section and the trend to increase along the span
direction balances the growth of the spanwise velocity due to the
high-sweep angle effect, thus causing the observed decrease of
R.
Just a little farther downstream, pressure rapidly falls in order to
minimize the chordwise extent of maximum crossflow growth in the
leading-edge region. A slight peak can appear at the end of this
limited region to minimize the crossflow growth, as discussed in [57].
Thereafter, the pressure gradient is very small and favorable
(negative pressure gradient), to damp the Tollmien–Schlichting wave
amplification. Here, another tradeoff has been found, as this
continuous pressure decrease must not be strong enough to cause
excessive shock losses farther downstream and even flow separation
at the design point. The pressure drag constraint was added just to
control this feature and effectively acted to avoid big drag rises at the
design point.
Another issue toward NLF enhancement is related to the wing
planform. The double-leading-edge sweep angle and the strong
taper-ratio distribution contribute to set significantly different aero-
dynamic conditions at each wing section. Indeed, not only is the
crossflow pattern very sensitive to sweep angle change and taper
ratio, which alter the spanwise pressure gradient, but the Reynolds
number also varies from about 75 106at the root section to
16 106at the tip. Hence, as a result of the attempt to passively
control the NLF amplification mechanisms under modified external
conditions through shape optimization, the wing surface is shaped
from sections having completely different geometric characteristics.
An example is the tendency to nose-droop that is not observed in the
inner sections, but is present in the kink midwing region: here,
indeed, an interesting and not-expected tradeoff solution has been
found by the optimizer. On one hand, the optimizer reduced the
leading-edge radius as much as possible to avoid crossflow and
attachment-line contamination in a critical zone where a dis-
continuity in attachment-line flow has been detected; on the other
hand, moderate pressure peaks arose and the optimizer tried to damp
them through nose-drooping. More details about the optimized wing
airfoil’s shape can be found in SUPERTRAC project technical
reports [28,58].
The main result of global search optimization is the general
observable decrease of N-factor levels along the whole wing. This is
a key feature because, even if the choice of the critical value of N(and
hence of the transition location) can be discussed, the advantages of
having lower disturbance growth rates are clear and make the
application of additional (active) boundary-layer flow control
techniques much more effective. The envelope curves as computed
by the database method for several wing stations are reported in
Figs. 15–18. Both on the upper and lower wing surfaces, transition is
triggered mainly by combined TS-CF wave amplification, except
near the wing-fuselage intersection, where the leading-edge shape
dulls the CF waves. Adopting 15 as the critical Nfactor, the transition
lines onto the optimal wing surface are extracted and plotted in
Fig. 19 as the chord percentage of laminar flow extent along the
wingspan. As expected, the transition location is placed more down-
stream for airfoil sections located near the wing tip, due to the
Reynolds number decreasing effect. The surface of laminar flow has
been increased from 4.7 to 21% on the suction side and from 10 to
33% on the pressure side.
Y [mm]
R
Rinc
1000 2000 3000 4000 5000 6000 7000 8000 9000
60
80
100
120
140
160
180
200
220
240
260
280
300
60
80
100
120
140
160
180
200
220
240
260
280
300
R , Dx/c at LE = 0.001
R , Dx/c at LE = 0.0005
R , Dx/c at LE = 0.00035
Rinc
, Dx/c at LE = 0.001
Rinc
, Dx/c at LE = 0.0005
Rinc
, Dx/c at LE = 0.00035
_
_
_
_
_
_
_
_
Y [mm]
We
Ue / 1.0E+05
1000 2000 3000 4000 5000 6000 7000 8000 9000
260
280
300
320
340
360
380
400
420
0.0
1.0
2.0
3.0
4.0
5.0
6.0
We
, Dx/c at LE = 0.001
We
, Dx/c at LE = 0.0005
We
, Dx/c at LE = 0.00035
Ue
, Dx/c at LE = 0.001
Ue
, Dx/c at LE = 0.0005
Ue
, Dx/c at LE = 0.00035
~
~
~
~
a) Compressible and incompressible R
b) We velocity component and Ue velocity gradient
−
∼
Fig. 14 Global optimum, Poll criterion parameters computation for
attachment-line contamination vs spanwise axis.
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N factor
x/c
Envelope curve
TS curve
CF curve 0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N factor
x/c
Envelope curve
TS curve
CF curve
Fig. 15 Stability curves for optimal wing section shape SEZ02: upper (left) and lower (right) sides.
1158 IULIANO ET AL.
Figure 20 shows the N-factor evolution map on the upper wing
surface for the global optimum wing, as computed by ONERA using
the 3C3D code and the database method; results are very close to
CIRA predictions.
2. Local Optimization
The impact of the optimization process on the Cpdistribution
(Fig. 21) is very instructive. The Cppeak on the upper side has been
very consequently reduced. The distribution has even been flattened
for the inner wing section, and is very similar to the 2-D distribution.
The counterpart of this flattening at the leading edge is a progressive
expansion until 30%, with strong spanwise variation in the slope. It
can also be observed that the shocks have been smoothed on the
upper side. On the lower side, note stronger shocks at the trailing-
edge level and hence increasing the drag penalty.
Regarding the performance in terms of laminar extent, the
resulting optimal shape performance gain, as can be seen on the
N-factor distribution on the upper side (Fig. 22), is not as important,
as expected. A narrow low-N-factor area around the kink (blue
region) that extends throughout the whole wing chord can be noted;
here, the boundary-layer stability computation failed due to a flow
separation induced by the pressure peak that can be observed in
Fig. 21. To check the validity of the laminar-extent evaluation for
such a configuration, exact linear stability computations have been
performed at two sections, y2:25 and 6.57 m. The comparison
between exact and simplified calculations demonstrates that the
database method provides a good estimation of the Nfactor (Fig. 23).
The streamwise coordinates used are the curvilinear abscissa
measured from the attachment line; however, the difference between
sand xstill remains small. Extra configurations have also been
studied to understand how a modification of the parameterization
could enhance the research of an optimal configuration. The
conclusions that have been drawn are that a splinelike twist law
instead of linear-by-part law would induce additional drag penalty,
but would also increase the laminar flow robustness to the flow
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N factor
x/c
Envelope curve
TS curve
CF curve 0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
N factor
x/c
Envelope curve
TS curve
CF curve
Fig. 16 Stability curves for optimal wing section shape SEZ06: upper (left) and lower (right) sides.
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N factor
x/c
Envelope curve
TS curve
CF curve 0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
N factor
x/c
Envelope curve
TS curve
CF curve
Fig. 17 Stability curves for optimal wing section shape SEZ12: upper (left) and lower (right) sides.
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N factor
x/c
Envelope curve
TS curve
CF curve 0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
N factor
x/c
Envelope curve
TS curve
CF curve
Fig. 18 Stability curves for optimal wing section shape SEZ20: upper (left) and lower (right) sides.
IULIANO ET AL. 1159
condition variations. The sweep angle, which was not considered in
both optimizations, is also a parameter that has to be taken into
account in future design processes.
IX. Cross-Comparison
In both local and global optimization approaches, it is a fact that
the wing laminar extent remains limited in the perspective of an
2y/b
x/c at transition
00.20.40.60.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Baseline wing - S uction side
Baseline wing - P ressure side
Global optimalwing - Suction side
Globaloptimal wing - Pressure side
Fig. 19 Transition lines: database method, Nfactor at transition 15.
Fig. 20 Global optimization: N-factor contour map.
a) Pressure distribution, inner wing
b) Pressure distribution, outer wing
Fig. 21 Local-approach enhanced pressure distribution in five span
locations.
Fig. 22 Local optimization: N-factor distribution of optimum.
s/c
Nfactor
00.05 0.1 0.15 0.2
0
5
10
15
20
10.00kHz
9.00kHz
8.00kHz
7.00kHz
6.00kHz
5.00kHz
4.00kHz
3.00kHz
2.00kHz
1.00kHz
data base
ONERA air foil
Boundary layer ONERA
s/c
Nfactor
00.1 0.2 0.3 0.4 0.5
0
5
10
15
20
25
30
35
10.00kHz
9.00kHz
8.00kHz
7.00kHz
6.00kHz
5.00kHz
4.00kHz
3.00kHz
2.00kHz
1.00kHz
0.00kHz
data base
ONERA air foil
Boundary layer ONERA
a) Section y=2.25m
b) Section y=6.57m
Fig. 23 Local optimization: N-factor distributions.
1160 IULIANO ET AL.
appreciable friction-drag reduction. On the upper side, the global-
approach wing gives better results on the inboard and the outboard
near the tip, whereas the local-approach wing provides better results
on the outboard center part. On the lower side, neither the global nor
the local approach designs show a significant laminar extent on the
inboard wing, whereas the global approach performs better on the
outboard region. The inboard has a very limited laminar extent in
both cases. However, global-approach N-factor levels are lower,
which is more convenient for a hybrid transition control application.
As a matter of fact, a detailed study of both wings performance at
different flight points with and without suction has been achieved
within the same project [59] and has shown that wings designed
toward NLF enhancement are more suitable than nonoptimized
configurations for active flow control.
X. Conclusions
The objective of the presented study was to verify the possibility to
perform shape optimization for natural laminar flow maximization in
the supersonic regime. One important result is that for such a
configuration (high Mach number, high-sweep angle, and low aspect
ratio) the three-dimensional effects are dominant and strongly affect
the inviscid pressure distributions and the boundary-layer velocity
profiles. Hence, having a full set of 3-D tools during the optimization
process is mandatory.
The chosen strategies and computational tools have proved to be
sufficiently robust to successfully perform both optimizations.
Indeed, the use of fast stability analysis using analytical approx-
imations and database techniques was very efficient and robust,
considering the three-dimensional complexity of the problem. The
comparison of both wings using a single method (envelope Nfactor)
and common tools has shown that achieving substantial NLF-extent
improvement with shape optimization remains a challenge for such
configurations. However, both 3-D optimizations have led to
enhanced configurations that show superior performances with
respect to the reference in terms of hybrid laminar flow control.
From the design-solution point of view,the common conclusion of
the presented optimizations is that sensitivity to NLF of such a
configuration is clearly driven (for a given planform, twist angle, and
thickness law) by the leading-edge parameters such as the leading-
edge radius and camber. Future investigations should look in the
direction of a deeper knowledge of the correlations between these
and other parameters that have not been varied in the present study,
e.g., the sweep angle, and their combined effects on supersonic
laminar flow control.
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