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CANADIANAERONAUTICS ANDSPACE JOURNAL

JOURNALAÉRONAUTIQUE ET

SPATIAL DU CANADA

Published by • Publié parthe Canadian Aeronautics and Space Institute

l’Institut aéronautique et spatial du Canada105 – 1750, croissant Courtwood Crescent

Ottawa, Ontario, Canada K2C 2B5Tel./Tél. : 613-234-0191, Fax/Téléc. : 613-234-9039

[email protected] www.casi.ca

December 2005 décembre Volume 51, No. 4

ISSN 0008-2821

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http://pubs.casi.ca/action/showImage?doi=10.5589/casj5104fi&iName=master.img-001.jpg&w=369&h=215

ScopeThe Canadian Aeronautics and Space Journal, theofficial publication of the Canadian Aeronauticsand Space Institute, is devoted to timely reportingon recent archival research and development ofaerospace sciences to the international community.Contents include a wide spectrum of technicalmaterials, ranging from advanced theoreticaltreatises to dissemination of recent experimentalresults, as well as book reviews and abstracts ofrecent reports and publications of Canadianaerospace research organizations.

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Canadian Aeronautics and Space Journal Journal aéronautique et spatial du Canada

EDITORIAL BOARD / CONSEIL ÉDITORIAL

Editor / Directeur scientifique

Mr. Stewart BaillieDirector Flight Research Laboratory

IAR/NRCDirecteur, laboratoire de recherches en vol

IRA/CNRC

Associate Editors / Directeursscientifiques associés

Dr. Susan SkoneAssociate Professor

Department of Geomatics EngineeringUniversity of Calgary

Dr. David ZinggCanada Research Chair in

Computational AerodynamicsInstitute of Aerospace Studies

University of Toronto

Dr. Nashed YoussefChair, CASI Propulsion Section

Pratt & Whitney, Canada

Dr. Alex JablonskiChair, CASI Astronautics Section

Canadian Space Agency / Agence spatialecanadienne

Mr. Graeme EastaughResearch Officer, Structure and Materials

Performance Laboratory IAR/NRC /Laboratoire des structures, des matériaux et

de la propulsion IRA/CNRCSMPL/IAR/NRC LSMP/IRA/CNRC

Assistant to the Editor / Adjoint audirecteur scientifique

Angelo FatoricFRL/IAR/NRC LRV/IRA/CNRC

[email protected]

PRODUCTIONPublishing Services, NRC Research Press

National Research Council of CanadaServices d’édition

Presses scientifiques du CNRCConseil national de recherche du Canada

PUBLISHED / PUBLIÉ PARCanadian Aeronautics and Space InstituteInstitut aéronautique et spatiale du Canada

1750 Courtwood Cr., Suite 105Ottawa, ON K2C 2B5

(613) [email protected]

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CANADIAN AERONAUTICS AND SPACE JOURNAL

JOURNAL AÉRONAUTIQUE ET SPATIAL DU CANADA

Turbulence Model Studies to Investigate the Aerodynamic Performance of a NASADual Control Missile at Supersonic Mach NumbersM. Khalid, A. Dujardin, P. Hennig, L. Leavitt, F. Leopold, M. Mendenhall, S. Prince . 153

Aerodynamic Forces Approximations using the Chebyshev Method for Closed-LoopAero-servoelasticity StudiesAlin Dorian Dinu, Ruxandra Mihaela Botez, Iulian Cotoi . . . . . . . . . . . . . . . . . . . . 167

Applying Fast Fourier Transform Analysis and Data Window in Software GlobalPositioning System Receivers to Mitigate Continuous Wave Interference underDynamic ConditionsZ. Jiang, G. Lachapelle, C. Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Adjoint-Based Sonic Boom Reduction for Wing-Body Configurations in SupersonicFlowSiva K. Nadarajah, Antony Jameson, Juan Alonso. . . . . . . . . . . . . . . . . . . . . . . . . . 187

i

Published by ⋅ Publié parCanadian Aeronautics and Space Institute • Institut aéronautique et spatial du Canada

150 - 1750, croissant Courtwood Crescent, Ottawa, ON, Canada K2C 2B5Tel./Tél. : 613-234-0191, Fax/Téléc. : 613-234-9039, E-mail/Courriel : [email protected]

Publication Mail Registration No. 231452 ISSN 0008-2821

Cover Image: The ability to manage the shock waves and the resulting sonic boom from a supersonic business jet will be adetermining factor in the overall acceptance of this aircraft type by the general public and its ability to over-fly populated areas.

Nadarajah et al. (pp. 187–199) presents a numerical method for determining how the aircraft geometry alters these far-field pressuredistributions from a supersonic aircraft, as shown above.

� � �

Sur la couverture : La maîtrise des ondes de choc et du bang supersonique produits par un réacté d’affaires constitue un critèrecrucial pour l’acceptation générale de ce type d’avion par le grand public et l’obtention de la permission de survol des zones

habitées. Nadarajah et coll. (pages 187–199) présentent une méthode numérique permettant d’établir de quelle façon la géométried’un avion supersonique modifie les distributions de pression dans le champ lointain, tel que montré ci-dessus.

This journal is indexed or abstracted in / Ce journal est signalé ou résumé dans AIAA Aerospace and High Technology database,Cambridge Scientific Abstracts Mechanical Engineering and Transportation Abstracts database, Cambridge Scientific Abstracts Civil

Engineering Abstracts, the Engineering Information Inc. Compendex, the National Research Council of Canada — CISTI Source.

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Turbulence Model Studies to Investigate theAerodynamic Performance of a NASA Dual

Control Missile at Supersonic Mach NumbersM. Khalid * A. Dujardin ** P. Hennig *** L. Leavitt **** F. Leopold *****

M. Mendenhall ****** S. Prince *******

* Institute for Aerospace ResearchNational Research Council Canada

Ottawa, ON K1A 0R6, Canada.E-mail: [email protected]

** Deutsches Zentrum für Luft- und Raumfahrt(DLR)

Cologne, Germany.

*** EADSLenkflugkörpersysteme GmbH (LFK)

Germany.

**** NASALangley, Virginia, USA.

***** ISLFrance.

****** Nielsen EngineeringUSA.

******* QinetiQUK.

Received 12 December 2005.

NOMENCLATURE

Cl, CL lift coefficient (L/qS)

Cm pitching moment M/qSl

CN normal force coefficient

Cd drag coeffiecient

CA(p) axial force coefficient based on pressure

CA(f) axial force coefficient based on frictionforces

D missile diameter

L lift

l moment reference length (2.6 in) (1 in =2.54 cm).

q dynamic pressure(½ρU2)

X axial distance

u /U, vx /V, u /V ratio of horizontal component of velocitywith respect to free stream

v/U, vz /V, v/V ratio of normal velocity with respect to thefree stream

S reference area (πD2/4 = 5.31 in2)

INTRODUCTION

At supersonic Mach numbers, the classic problemsassociated with drag prediction and such characteristics asshock shape and shock boundary layer interactions togetherwith transition and heat transfer effects continue to take thecentral stage. At higher angles of attack, the flow separationproblems become unavoidable and the long slender missilegeometry gives rise to the presence of unsteady vortex regions,which evolve into Karman vortex sheets pervading over the aftregions. Here, the wing CLMAX is often compromised as theseparated flow interacts with the developing vortices. At evenhigher Mach numbers M > 5, other non-equilibrium and realgas effects become important. Higher heat transfer rates lead tosurface ablation and other associated unsteady flowcomplexities. Leading-edge regions on wings and (or) controlsurfaces bear the brunt of higher temperature effects, whichoften lead to loss of control.

From a design viewpoint the resulting optimizedconfiguration, in many ways would be a blend of excellencefrom many disciplines. The best design would have conformed

© 2005 CASI 153

Vol. 51, No. 4, December 2005 Vol. 51, no 4, décembre 2005

AbstractThis paper contains an investigation of the ability of someof the current turbulence models to predict viscous effectsin supersonic Mach number regimes. In most cases, theseturbulence models have been derived and validated intraditional subsonic to transonic Mach number regimes andtheir application to supersonic and hypersonic regimes isassumed valid without a proper recourse to understandingthe wider implications of the viscous flow field at highMach numbers. In such flow regimes, such characteristicsas the strong shock interactions with control surfaces andevolving vortices present in the flow, or other large entropygradient effects brought about by the forebody or other finsurfaces, produce non-trivial challenges for the turbulencemodels. This study was aimed at interrogating the strengthof the turbulence models to model such physics. TheApplied Vehicle Technology Panel Group 082 of the

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to the weight/structural and aero elastic issues. Its structuralcomponents would be well suited for higher heat-transfer rates.Its intake would be well embedded into the geometry andplume and other vectored jet emissions controlled for minimumradar cross section. Owing to advances in modeling techniquesand increased computer power, there is a strong compulsion inmissile design to rely more and more on numerical simulationrather than opt for more exhaustive and expensive wind tunneltesting. Knowledge gained per design cycle and faster iterationtimes make strong arguments for using mathematical-basedmodels.

It is equally well recognized that the quality of turbulencemodels in Navier–Stokes codes is pivotal to efficient andaccurate design of missiles. Towards this end, The AppliedVehicle Technology (AVT) Panel of the Research TechnologyOrganisation (RTO) tasked Technical Group 082 to investigatethe applicability of various turbulence models to address thecomplex viscous problems associated with the design ofmodern missile configurations. The main objective of thepresent research was to provide an assessment of the relativecost and accuracy of the turbulence models used in modelingviscous terms in Navier–Stokes solutions when predictingmissile aerodynamics at high subsonic and supersonic speeds.The suggested Mach number range 0.8 < M < 4 made the scopeof the activity somewhat unique as it covered the challengingtransonic regimes where the flow is not so easy to resolve andextended to upper supersonic regimes, where thecompressibility effects, shock wave/turbulent boundary layerinteractions and separation effects become important. It wasalso recommended that during the course of the group activity,a number of test-cases be defined and that various turbulencemodels be applied to the selected set of test-cases to gauge theaccuracy and the relevant costs associated with differentclosure schemes. The group was to study the predicted balance,pressure, and other flow field data in light of the measuredquantities and arrive at some definitive conclusions. In the firstphase the group surveyed the literature for the previousresearch in this area and researched into previous experimentaland modeling exercises carried out to study some of theaforesaid problems. The team members made documentedpresentations to the group, which are contained in the finalAVT/RTO Report (2004) of Task Group 082.

A more defined purpose of this particular study was toinvestigate the capability of the turbulence models to accuratelyperform CFD simulations of the flow past the NASA dualcontrol missile at two angles of attack, α = 6° and 24°. Theturbulence modeling studied during the course of thisinvestigation included the one equation Spalart–Allmaras, thetwo-equation standard k–ε , and the k–ω turbulence models aswell as the Baldwin–Lomax turbulence model enhanced toaccount for the cross-flow separation due to Degani–Schiff(DS) or other curvature effects from Qin–Jayatunga. Some purelaminar (lam) computations were also carried out to isolate theviscous effects from complete computations. The group usedboth structured and unstructured type grids to identify anystrengths or advantages of a particular method. The evolution

154 © 2005 CASI


suite de la page 153

Research Technology Organisation selected a dual controlNASA missile to investigate the capabilities ofComputational Fluid Dynamics (CFD) to predict the flowfield and performance characteristics of complex-shapedprojectiles at high Mach numbers.There are two types of difficulties that are encounteredwhen computing such problems. The first type aregeometry based, and are related to the shape of the noseand forebody, number and types of strakes or forwardcontrol mechanisms, fin geometry deployment, and shapeand design of the cowls and intakes. The second type ofdifficulty deals with flow complexities such as the heattransfer (M > 4) implications; the vortices shed from theforebody and other control surfaces; the interaction ofthese vortices with the body structures, the free stream, andwith each other; the base flow interaction with the freestream; the boundary-layer development; and, in the caseof supersonic flows, the shock boundary-layer interactions.Air-breathing missiles with intakes or missiles with jet-controlled guidance systems add to the flow complexities.Various turbulence models employed in current CFD codesare tested for their ability to address these viscous issues.

RésuméCet article présente notre recherche sur la capacité decertains modèles actuels de turbulence à prédire les effetsvisqueux, aux nombres de Mach correspondant auxrégimes supersoniques. La plupart de ces modèles deturbulence ont été dérivés et validés pour une gamme derégimes allant du subsonique au transsonique — donc pourles nombres de Mach peu élevés. On a présumé qu’ilss’appliquaient aux régimes supersoniques ethypersoniques, mais sans tenir compte des conséquencesplus larges pour le champ d’écoulement fluide, aux grandsnombres de Mach. Certaines caractéristiques de telsrégimes d’écoulement, notamment les interactionsintenses entre le choc et les gouvernes et l’évolution destourbillons présents dans l’écoulement, ou les effetsconsidérables sur le gradient d’entropie causés par lefuselage avant ou les autres surfaces de gouverneconstituent des difficultés dont la résolution est ardue pourles modèles de turbulence. Cette étude visait à examiner lacapacité des modèles de turbulence à reproduire cesphénomènes physiques. Le Groupe 082 sur la technologieappliquée aux véhicules, de l’Organisation pour larecherche et la technologie, a choisi un missile à doublecommande de la NASA pour déterminer si la dynamiquenumérique des fluides pouvait prédire le champd’écoulement et les caractéristiques comportementales desprojectiles aux formes complexes, se déplaçant à desnombres de Mach élevés.Le calcul de tels problèmes comporte deux types dedifficultés. Le premier relève de la géométrie et découle de

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of the vortex formed by the front fins was of particular interestas the vortices will undoubtedly interact very strongly with theforward strakes at higher angles of attack and affect thecontrollability of the projectile. Also, the flow deflected by theforward strakes will subsequently impinge upon the aft controlsurfaces, which could possibly lead to control failures if themissile was not designed appropriately.

In the final analysis, based on the extent of wind tunnel dataavailable the group decided to use the NASA Tandem ControlMissile shown in Figure 1 as the main model for the groupstudy. Most balance and pressure data were to come fromNASA Langley, US, whereas other flow field data weregenerated by ISL, France. During the test campaign at NASA,two Mach numbers, M = 1.75 and M = 2.5 were investigated forangles of attack varying from –4° to 28°. The Reynolds numberfor both velocities was set at 6.56 × 106/m (2 × 106/ft). For thenumerical study the first Mach number M = 1.75 was utilizedfor most computations with two angles of attack settings: α = 6°and 24°. To provide greater confidence, at least one participantcompleted a full drag and lift polar using the less time-consuming coarser grid and appropriate turbulence model. Interms of grid type studies, members agreed to work with bothstructured and unstructured type grids. The originalunstructured grid was supplied by NASA, whereas QinetiQ inUK, NLR from the Netherlands, and IAR from the NationalResearch Council of Canada supplied the structured grids thatwere individually further adapted by various other members ofthe team. While a more detailed analysis of the results arereserved for a AVT/RTO report (2004), a synopsis of theimportant findings are included in this paper. Except for thefirst-order methods (DATCOM, MISL3), which make use ofthe complete missile geometry and were only included toprovide a rough estimate of the global forces, most detailed

computations in the absence of the yaw angle had adopted areflection plane model for Navier–Stokes computations. Inbroader terms, the paper includes a comparison of variousviscous flow-field characteristics as computed using differentturbulence models. In some cases, specific velocity profiles atsome critical junctions were also compared against measureddata. Most computations recovered the force and momentvalues quite successfully and these were also compared againstEuler and other empirical-based algorithms.

To measure the full impact of the turbulence modeling onsimulation of flows at high Mach number, the group decided tolook at results from lower order methods and quantify theinfluence of turbulence modeling in higher order Navier–Stokes solutions. Towards this end, some members opted forapplying empirical-based and Euler codes to obtain the baseline results. Parabolized Navier–Stokes (PNS) codes andReynolds Averaged Navier–Stokes (RANS) codes withdifferent turbulence models were expected to provide mostaccurate flow field and force and moment predictions.

The group mandate had asked for turbulence modelingassessment studies in the Mach number range 0.8 < M < 4.Owing to the limited resources available for such groupactivities, it was decided to conduct CFD-based computationsat a free stream Mach number of M = 1.75, with unit Reynoldsnumber of Re/ft = 2 × 106 and two settings of angle of attack,α = 6° and 24°. At the lower angle of attack α = 6°, the flowshould largely be attached on majority portions of the missileand most CFD codes should have little problem in reaching asatisfactory solution. At the higher angle of attack, there willundoubtedly be regions of separated flows with well-developedvortex structures invoking various control problems as theyinteract with missile fins. Prediction of such flows will test theaccuracy, the economy, and the reliability of most turbulencemodels. Viscous flows not withstanding, where possible,results from simpler empirical based methods from DATCOMor MISL3 would be introduced when comparing ordinary forceand moment predictions, as these tools provide a quick andready method of evaluating design performance under mostcruise conditions.

Finally, it was concluded that for the flow cases studied, theSpalart–Allmaras turbulence model was able to resolve thephysics as well as the two-equation turbulence model. Awayfrom the wall, most turbulence models captured such viscousfeatures as the velocity profiles quite well, however, suchviscous characteristics near the wall region were not asaccurately predicted.

SOLVERS

WIND CodeThe computations at IAR were carried out using the WIND

CFD code version 2.0, supplied by the National Project forApplications-oriented Research in CFD (NPARC) Alliance(Bush et al., 1998). This code evaluates second-order accuratefinite differences of the governing Navier–Stokes partial

© 2005 CASI 155



la forme de la pointe et de l’avant du fuselage, du nombreet des types de listons ou de dispositifs de gouverne avant,de la géométrie de l’empennage, et de la forme et laconception des capots et des admissions. Le second typeprovient des complexités de l’écoulement, telles cellesdécoulant du transfert de chaleur (pour M > 4); lestourbillons détachés du fuselage avant et des autres plansde gouverne; l’interaction de ces tourbillons avec lesstructures du fuselage et l’écoulement libre, et entre eux;l’interaction de l’écoulement de culot avec l’écoulementlibre; la formation d’une couche limite; et, en présenced’écoulements supersoniques, les interactions entre lacouche limite et le choc. Les admissions d’air des missilesaérobies ou les commandes d’orientation par jets ajoutent àla complexité de l’écoulement. Nous avons testé lacapacité des différents modèles de turbulences utilizés parles logiciels actuels de dynamique numérique des fluides àtraiter ces problèmes de viscosité.

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differential equations in conservative form. The explicit non-viscous terms were calculated using Roe’s second-order flux-difference splitting algorithm; smoothing was used to dampennumerical instabilities. Certain types of explicit operators — forexample, the central difference operator for structured grids —may require the addition of numerical smoothing to dampeninstabilities that are a natural part of the scheme. Smoothingmust also be added explicitly when utilizing Wind’sconvergence acceleration capability. Values for varioussmoothing parameters are specified in the input data file. Thedefault smoothing values for Jameson-type Euler solutions are1/4, 1/256, and 0. For viscous solutions these values are asmuch as 1, 1/64, and 2, respectively. The implicit viscous termswere evaluated using either a full block implicit operator or aparabolized Navier–Stokes (PNS) operator, depending on thesimulation. The Spalart–Allmaras (1992) model was used tocalculate the turbulent viscous terms. For economy, the gridwas constrained to remain within y+ ~ 5 near the wall regionsand was thus thought to provide adequate sublayer resolutionfor such viscous characteriztics as shear stress, skin friction,and velocity profiles u/U and v/U. For a more accurate solutionbased on the Spalart–Allmaras model the wall distance y+ isrecommended to be less than 1.

CFD-FASTRAN CodeLFK (Lenkflugkörpersysteme GmbH) made use of

commercial code CFD-FASTRAN as well as the DLR FLOWercode. CFD-FASTRAN is a density based Navier–Stokes solverfor compressible flows using Van Leer’s FVS scheme (VanLeer, 1982). Among several turbulence models provided asoptions in CFD-FASTRAN the standard k–ε turbulence modelfor high Reynolds number flows (Launder and Spalding, 1974)was selected. Therefore, logarithmic wall functions were usedto avoid the necessity of resolving the laminar sublayer thuspermitting considerably fewer grid points in the near-wall

region. Strictly speaking, the assumptions underlying thelogarithmic law of the wall are compromised whereverboundary layer separation occurs. Therefore, the results mightprovide useful hints regarding a possible loss in overallaccuracy for the missile configuration considered here causedby the much more economic wall-function approach whencompared to turbulence models requiring the resolution of thelaminar sublayer.

TAU CodeThe TAU code as used by DLR is a finite volume

Euler/Navier–Stokes solver that can use structured,unstructured, and hybrid meshes and has already been appliedto the study of various configurations as reported in Hannemanand Mack (2002) and Dujardin et al. (2002). The Reynolds-averaged Navier–Stokes equations are discretised by a finitevolume technique using tetrahedral, pyramids, prisms, andhexahedral elements. Prismatic elements are used for theboundary-layer region while the tetrahedral ones are used ininviscid flow regions. The internal data storage is based on anefficient edge-data structure of Galle (1999), using theNETCDF format. Having a fine mesh y+ < 1, the DLR wereable to resolve the flow accurately in the viscous sublayer.

Beside the central scheme, different upwind solvers such asVan Leer, AUSM/Van Leer, and AUSMDV are implemented.Some acceleration procedures such as multi-grid or residualsmoothing are available and have been modified for hypersonicflows. Also, modifications on the limiter have been done toachieve a higher accuracy and better convergence. Shock andleeside treatments especially for Euler computations have beenadded. A three-stage Runge–Kutta scheme is used for timediscretisation including local time-stepping. Due to theimplemented explicit residual smoothing, CFL number up to 4can be used. To reduce the computational time for two-dimensional flows, a special two-dimensional model is

156 © 2005 CASI


Figure 1. Dual control missile (all dimensions are in mm).

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activated that suppresses the flux calculation in the span wisedirection. Beside the common boundary conditions such asconstant wall temperature and adiabatic wall, radiativeequilibrium is available. Laminar flow as well as turbulent flowcan be considered, as various turbulence models have beenimplemented. The DLR had used both the k–ω and the Spalart–Allmaras turbulence models during their studies. Concerningthe turbulence parameters used for the computation one shouldbe careful in the choice of the µt /µ l coefficient as well as theturbulence intensity. Indeed the k–ω is relatively freestreamdependant (Menter, 1994), and the influence of these factorscan be important. For this study both factors have been chosenaround 1%. Higher values somehow exaggerate the resultingviscosity and the turbulent behaviour of the flow (thickerboundary layer and shorter separation length, high turbulentenergy), but two low values lead usually to numerical problemsas poor convergence and shock resolution.

UMS3DNS CodeNASA Langley had used USM3Dns for their computations

on an unstructured grid. USM3Dns is a tetrahedral cell-centered, finite-volume Euler and Navier–Stokes solver (Frink,1998). The inviscid flux quantities are computed across the cellfaces using Roe’s flux-difference splitting scheme and thespatial discretization is accomplished by a novel analyticalreconstruction process. The solution is advanced in time tosteady state using an implicit backward-Euler time-steppingscheme. Flow turbulence is modeled by the Spalart–Allmarasone-equation model, which is optionally coupled with a wallfunction to reduce solution stiffness and the number of cells inthe sublayer of the boundary layer. All computations presentedin this paper are performed using the wall-function feature ofUSM3Dns in a fully turbulent mode. Using a fine mesh withwall function may not be justified, and thus viscous solutionsfrom NASA Langley were not included in meticulouscomparisons in the wall regions.

USM3Dns runs on massively parallel computers andclusters of personal computers (PCs). A single processorversion is available for vector processors such as the Craysuper-computers (with multi-tasking) and single processorworkstations. The parallel version (Bhat and Parikh, 1999;Parikh, 2001) is the version of choice because of its rapidturnaround time for large problems. The code requires175 eight-bit words of memory per tetrahedron. It runs with aspeed of 34 µs/cell/cycle/processor on a Cray C90 and230 µs/cell/cycle/processor on the SGI Origin 2000 parallelcomputer.

USM3Dns supports standard boundary conditions such asthe flow tangency, no-slip solid surface, characterizticinflow/outflow (for subsonic flows), and freestream–inflow/extrapolation–outflow (for supersonic flows). Inaddition, some special boundary conditions including wallfunctions, wake flow, jet engine, and propeller are available inthe code, and are reviewed in Frink et al. (2000).

IMPNS CodeQinetiQ in the UK had used an iterative PNS solver

“IMPNS” for its computations (Birch et al., 2000). Thegoverning equations, solved by IMPNS, are the steady-statecompressible Navier–Stokes equations. The perfect gas lawand Sutherland’s law for laminar viscosity are employed toclose the system. When necessary, turbulence is handled usingthe Favre mass-averaged form of the Navier–Stokes equationswith either a Baldwin–Lomax (1978) algebraic turbulencemodel, enhanced to account for crossflow separation (bymodifications due to either Degani–Schiff (1986) or thecurvature method of Qin and Jayatunga (1998)), or the one-equation model of Spalart and Allmaras (1992). For caseswhere viscous effects are unimportant, the viscosity may beswitched off and the corresponding Euler equations may besolved with appropriate boundary conditions.

The governing equations are discretised using a cell-centredfinite-volume scheme in a generalized co-ordinate system foruse with structured, body-conforming grids. To enable a stablewell-defined space-marching solution, the discretised flowequations are parabolized by neglecting viscous terms in thestreamwise direction. In addition, for single-sweep calculationsthe flow outside the boundary layer must be supersonic in thestream wise direction and a portion of the stream wise pressuregradient within subsonic regions is assumed negligible. Severalinterpretations of the stabilizing approximation by Vigneron etal. (1978), in which a portion of the stream wise pressuregradient is neglected, have been investigated and implementedin the solver.

FLUENT 5 CodeThe QintiQ team had also made use of the FLUENT 5

commercial flow solver to compute the test cases to comparethe accuracy and relative efficiency of the PNS space-marchingtechnique. The FLUENT flow solver is based on the finite-volume method and can utilize structured, unstructured, orhybrid grids. The coupled solver, which simultaneouslycomputes both the continuity and momentum equations, wasemployed in the present study. FLUENT uses an upwind, flux-difference splitting and can operate using either implicit orexplicit time-marching schemes. For the present investigationthe spatial accuracy was set to second order, and the Spalart–Allmaras turbulence model was used for comparison with theSpalart–Allmaras results from IMPNS.

STRUCTURED AND UNSTRUCTURED MESHES

The computational domain for the structured grid is shownin Figure 2. The inlet was located 0.1 in downstream from thetip of the missile to avoid the grid quality problems associatedwith modeling flow in this region. The fact that the actual tipwas not included in the computational domain should not havea large effect on the results. The outlet was located 2.0 indownstream from the end of the actual missile shown inFigure 1; the body of the missile model used in the

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computations was extended downstream from the rear controlfins to the outlet. The azimuth free stream boundary waslocated approximately 4.75D from the centreline at the front ofthe missile, and approximately 22D from the centreline at therear of the missile.

Figure 3 shows the side view of a typical structured meshused in the computations. The grid lines were continuousthrough all the blocks of the computational domain. The mesharound the front fins was similar to the mesh around the rearfins. The density of the grid lines was high near the missilesurfaces where the viscous resolution of the flow had a directbearing on the accuracy of the computations. The grid lineswere farther apart in outer regions of the domain where the flowwas not affected as much by viscosity. The average y+ value atthe first grid point above the missile surface was approximately5. A total of 4.0 million control volumes were used for eachsimulation.

A specimen of (coarse and fine) unstructured grids wassupplied by NASA Langley. They are completely unstructured,even if an anisotropic distribution resulting from the splitting ofprismatic cells into tetrahedral cells is used in the boundarylayer. These shapes are, however, more adapted to viscouscomputation and allow better control of the near-wall normaldistribution and particularly the normal first spacing. For thisreason y+ ≤ 1 has been kept almost everywhere, a y+ peakoccurring inevitably at some edges, and for both coarse and fine

grids. The coarse grid is composed of about 700 000 nodeswhile the fine grid get about 1 500 000 points. A view of thefine grid is shown in Figure 4.

RESULTS

As a first test, the aforementioned codes were tested for theirability to compute ordinary lift force and moment coefficientsfor free stream Mach number of M = 1.75, with unit Reynoldsnumber of Re/ft = 2 × 106 and two settings of angle of attack,α = 6° and 24°. Within various degrees of accuracy almost allthe codes were able to resolve these coefficients to asatisfactory value. In some cases, see Figure 5, the participants

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Figure 2. The computational domain.

Figure 3. Side view of the computational structured grid near themissile body.

Figure 4. General views of the fine unstructured grid.

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were able to produce a complete polar ranging from α = 0° to24°. Table 1 provides a detailed study of the exact force andmoment coefficients as computed by various participants.While it was satisfying to note that at α ≈ 6°, the normal forceseems to be well predicted by all computations the differencesin axial drag results varied from as low as 0.4296, as predictedby Price and Moule (QINETIQ) to as much as 0.731 by Hennig(LFK) when compared to an experimental value of 0.5594. It isplainly obvious that the turbulence models are unable to copewith complex vortex-induced separated flow that evolves alongthe length of the missile. At higher angle of attack α ≈ 23.98°,the computations seem to match the normal force reasonablewell, but the predictions for axial drag are no better. It was

equally interesting to note from computed results fromDATCOM and MISL3 as published by Lesieutre et al. (2002)that for lower angles of attack even empirical-based codes arevery economic design tools.

The predicted Mach number contours as obtained from theWIND Code computations using the Spalart–Allmarasturbulence model by McIlwain and Khalid (2004), on thevertical (a) and horizontal (b) planes bisecting the missilecentreline, are shown in Figure 6 for α = 24°. These planes liemidway between the fins. The bow shock wave is clearlyvisible in both plots. The flow between the top two front finswas accelerated, creating a region of higher Mach number flowtrailing out into the free stream. There were regions of slower

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Figure 5. Evolution of the body force and moment coefficients versus the angle of attack for M = 1.75 as obtained using the TAU code.

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Tab

le1.

Com

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dfo

rce

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coef

fici

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flow underneath the missile, especially beneath the tip andbetween the two sets of fins. Between the front and rear finsthere was also a region of low-speed flow near the body on thetop and bottom of the missile, and a region of high-speed flownear the body along the sides of the missile. This flow remainedclose to the body and did not trail into the free stream like theshock waves extending from the tip and the front fins. Note thatthe same flow computed by Pirzadeh and Leavitt (2004) usingthe Spalart–Allmaras turbulence model on an unstructured gridshows very similar flow patterns in Figure 7.

When comparing the computed results against the velocityfield as depicted in the measured shadowgraph (Figure 8)obtained by Leopold et al. (2003) the flow patterns are in goodagreement with experiment. It is useful to note that theshadowgraph has picked up the complex shock interactionfeatures near the forward control surface and the distinct bowline, which runs from the oblique shock at the forebody

indicates the interaction between the shock and the outer edgeof the vortex. These features have been well resolved by mostcomputations.

It is apparent that the flow on top surface and on the sidesevolves into some very distinct vortices. Figures 9 and 10 takenfrom Dujardin (2004) and Nöding and Hennig (2004),respectively, show the total pressure contours at a number ofstation cuts along the axial length from different computationsusing three different turbulence models. While the flowfeatures are discussed in greater detail in the AVT/RTO Report(2004), it is apparent from the reflection plane flow field asdepicted in Figure 11 that the vortices start evolving from theregion of the missile. They seem to be energized andstrengthened somewhat from the presence of the forwardcontrol surfaces. Although the computations were performed torepresent a steady-state case, there appears to be a numericalperiodicity in solution showing a vortex pattern on the topsurface, which appears to stretch outwards and give birth to

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Figure 6. Mach number contours for theα = 24° simulation; (a) z = 0 (vertical) plane and (b) y = 0 (horizontal) plane (McIlwain and Khalid, 2004).

Figure 7. Mach contours for tandem controls missile at α = 24°, M =1.75 (from Pirzadeh and Leavitt, 2004).

Figure 8. Shadowgraphs at 100 ms (Leopold et al., 2004).

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further vortices closer to the surface, mimicking the classicvon-Karman vortex behaviour. It must also be recognized thateven though, time-marching iterative schemes may beemployed in some of the CFD codes, the solutions remainsteady in nature and can only be an approximate reflection ofthe true vortex flows at high angle of attack. The vortexgrowing on the side of the missile becomes increasinglydistinct towards the middle length of the missile and standsproud from the system developing on the centre upper surface.The vortices coalesce together near the trailing edge. Thecomplete vortex pattern appears to maintain its main featuresfor some distance past the aft trailing surfaces and merges intothe mainstream flow. These distinct vortex patterns willdecidedly impact upon the aerodynamic performance of themissile at higher angles of attack.

Another aspect of this study addresses the question ofviscous resolution of solvers and the included turbulence

models to predict velocity profiles near the missile surfaces.Profiles of the x and y components of velocity for the α = 24°case as obtained from the WIND code are compared toexperimental data in Figures 12 and 13. These profiles arelocated at x/D = 5.8 (slightly downstream from the front set offins), 10.3 (between the two sets of fins), and 13.8 (in front ofthe rear set of fins). There is considerable disagreementbetween the experimental and numerical velocity profiles nextto the missile surface. The relatively coarse numerical grid thatwas used next to the missile surface (y+ ≈ 5) and the simple one-equation turbulence model (Spalart–Allmaras) that was used inthe simulations were likely contributing factors towards thisdiscrepancy. There was better agreement between the two setsof results farther away from the surface of the missile, at y/D >1.75, especially for those stations located far downstream fromthe front set of fins.

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Figure 9. Total pressure contours at α = 24° and M = 1.75 obtained with the k–ω model (left) and the Spalart–Allmaras model (right).

Figure 10. Total pressure contours at M = 1.75 and α = 24° obtained using the k–ε turbulence model. Left: x/D = 6; right: x/D = 10.

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http://pubs.casi.ca/action/showImage?doi=10.5589/casj5104fi&iName=master.img-020.png&w=386&h=137



Figure 14 shows another set of comparisons, which hadused the DLR TAU code (Dujardin, 2004) on an unstructuredgrid using two different turbulence models to compute the axialand normal components of the velocity profiles. From thevelocity profiles shown in Figure 14 and indeed those studiesearlier in Figures 12 and 13, it is obvious that U components ofvelocity are better resolved away from the surface where theviscous effects tend to be weaker. For the V component ofvelocity it can be observed that the Spalart–Allmaras model

does a far better job as employed in WIND code to predict thevelocity profiles in the near regions at stations x/D = 10.3 andx/D = 13.8. The corresponding predictions from the TAU codeusing k–ω or the Spalart–Allmaras model did not do as well atboth stations. The station at x/D = 5.8 is immediatelydownstream of the control surfaces and suffers from possiblevortex-induced flow separations and complex interactions withthe shock. The V component of velocity at this station is,therefore, not as accurately resolved. The fact that

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Figure 11. Distribution of the flow field Mach number (left) and corresponding total pressure(right) (Dujardin, 2004) on several cross plane.Visualisation of the vortices for M = 1.75, α = 24°, k–ω model.

Figure 12. Comparison of numerical and experimental (Leopold et al.,2003) U-velocity profiles for α = 24° at x/D = 5.8 (red), 10.3 (green), and13.8 (blue).

Figure 13. Comparison of numerical and experimental (Leopold et al.,2003) V-velocity profiles for α = 24° at x/D = 5.8 (red), 10.3 (green), and13.8 (blue).

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measurements were recorded at M = 2.0 rather than thecomputed free-stream input of M = 1.75 must also contribute tothe discrepancy between the predicted and experimentalvalues.

The authors from QinetiQ, UK (Prince and Birch, 2004)further investigated the question of velocity profilecomparisons at different Mach numbers. To resolve the issue ofhow the ISL experimental results, obtained for Mach 2.0, and0.43 million per D Reynolds number flow, compares with theNASA results from a Mach 1.75, Re = 0.63 million per D, flow,two additional PNS calculations were performed. Solutionswere obtained for α = 6° and 24° using the Spalart–Allmarasturbulence model on the finest PNS grid such that the results

could be compared with the ISL velocity measurements at theequivalent conditions. These calculations also allowed for acomparison of the PNS results at the two conditions to assessthe differences in the flowfields for the two cases.

As expected, the solutions performed by QinetiQ (seeIMPNS solutions Figures 15 and 16) at the ISL wind-tunneltest conditions best matched the experimental data, and thedifferences between the computed profiles for the two differentconditions are seen to be significant. The implications of thisare that the ISL data, while useful for qualitative analysis of theflow physics, are not good for comparison with CFD forvalidation of the various turbulence models employed in thisNATO–RTO study.

Figure 15 presents a cross plot of earlier axial and normalvelocity component profiles when compared against the resultsfrom the UK. The axial velocity profiles at stations x/D = 5.8,10.3, and 13.8, as obtained from the DLR TAU code and theWIND code are compared against the UK IMPNS code and themeasurement in Figure 15. In most cases the comparisons werecarried with the unified use of the Spalart–Allmaras model as itseemed to perform better than the two-equation model.Individual attempts (AVT/RTO Report, 2004) at using othertwo-equation models had shown that the Spalart–Allmarasmodel had performed better, and thus the cross plots in thispaper compare results from this turbulence model. For the axialvelocity comparisons it is noted that DLR and QinetiQ havedone very well in being able to resolve the flow close to thewall. The results from the WIND code near the wall regionespecially at the aft stations are not as accurate. The normalcomponents of the velocity ratio v/U, from various codes at thesame axial stations of x/D = 5.8, 10.3, and 13.8 are shown inFigure 16. The computed and experimental velocity profilescapture the highly turbulent and vortical nature of the flow. Thenormal components of velocity show a large range in v/Uvalues extending from v/U = –0.3 to 0.4 with at least twoinflexion regions in between.

It is thus convenient to deduce that IMPNS computationswith Spalart–Allmaras model and TAU based computationsfrom DLR with fine grids (y+ ~ 1), provide the best resolutionof viscous flows at high supersonic Mach numbers. The WINDsolutions from IAR/NRC had used a relatively coarser grid y+ =5, and did not resolve the axial viscous components close to thewall as accurately. The IMPNS-based solutions are the morewelcome as the parabolised version of the Navier–Stokesequations tend to be less stiff and provide faster convergence.

CONCLUSION

Most CFD computations were able to capture the globallarge-scale features of the flow; the turbulence models as yet,are not able to accurately resolve the complexity of the viscousflow in regions of high shock–shock, shock–vortex, and othershock–boundary-layer interactions; or other regions of large-scale separation at high angle of attack. While the economy ofvarious computational methods has not been a subject of this

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Figure 14. (a) Velocity profiles for the vz component along the z axis attwo positions of the symmetry plane. (b) Velocity profiles for the vxcomponent along the z axis at two positions of the symmetry plane.

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paper, it was found during various deliberations of this groupthat the PNS-based Navier–Stokes computations are far moreefficient in producing satisfactory results than those based onthe full Navier–Stokes equations.

There is evidence from the literature at large (Lesieutre etal., 2002) that for most design purposes it would suffice toconduct a majority of preliminary design exercises using theDATCOM-type algorithms and resorting to higher ordercomputations where the flow becomes increasingly challengingfor the lower order methods.

Of the various turbulence models investigated, it appearsthat despite the empiricism of the single equation approach, theSpalart–Allmaras model appears to resolve the physics as well

as the two-equation k–ε model at higher Mach numbers.Almost all the turbulence models fared well when computingvelocity profiles away from the region of strong flowinteractions from the control surfaces or the principal shockstriggered from the forebody and the control surfaces. On thesame token the axial component of velocity seemed to be bettermodelled than the normal component.

For this particular missile, the study found the existence ofcomplex vortex systems that would impact upon the missileperformance at higher angles of attack.

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Figure 15. Comparison of measured and predicted profiles of axialvelocity ratio, u/V, at x/D = 5.8, 10.3, and 13.8.

Figure 16. Comparison of measured and predicted profiles of normalvelocity ratio, v/V, at x/D = 5.8, 10.3, and 13.8.

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ACKNOWLEDGEMENT

The authors are grateful to Peter Tonn and Sandra Cheyne ofNATO’s AVT/RTO panel for facilitating meetings anddiscussion sessions, which led to the completion of this work.

REFERENCES

AVT/RTO Report. (2004). “Assessment of Turbulence Modeling for HighSpeed Air Vehicles”.

Baldwin, B., and Lomax, H. (1978). “Thin-Layer Approximation andAlgebraic Model for Separated Turbulent Flows”. Proceedings of the AIAA16th Aerospace Sciences Meeting, Huntsville, Alabama, 16–18 January 1978.

Bhat, M.K., and Parikh, P.C. (1999). “Parallel Implementation of anUnstructured Grid-Based Navier-Stokes Solver”. AIAA Paper 99-0663.

Birch, T.J., Ludlow, D.K., and Qin, N. (2000). “Towards an Efficient,Robust and Accurate Solver for Supersonic Viscous Flows”. Proceedings ofthe ICAS 2000 Congress, Harrogate, United Kingdom, 27 August –1 September 2000. CD-ROM. International Council of the AeronauticalSciences, Reston, Virginia. pp. 242.

Bush, R.H., Power, G.D., and Towne, C.E. (1998). “WIND – TheProduction Flow Solver of the NPARC Alliance”. AIAA Paper 98-0935.

Degani, D., and Schiff, L.B. (1986). “Computation of Turbulent SupersonicFlows Around Pointed Bodies Having Crossflow Separation”. J. Comput.Phys. Vol. 66, pp. 173–196.

Dujardin, A. (2004). “Flow Computation on a Dual Control Missile Usingthe DLR Unstructured TAU Solver”. In Assessment of Turbulence Modelingfor High Speed Air Vehicles. AVT/RTO Rep.

Dujardin, A., Gülhan, A., Longo, J.M.A., and Mack, A. (2002).“Numerical/Experimental Investigation of a Wedge-Compression Corner withGap-Flow under Cold-Hypersonic Gas Condition”. Proceedings of the 4thEuropean Symposium on Aerothermodynamics for Space Applications, Capua,Italy, pp. 717–723.

Frink, N.T. (1998). “Tetrahedral Unstructured Navier-Stokes Method forTurbulent Flows”. AIAA J. Vol. 36, No. 11, pp. 1975–1982.

Frink, N.T., Pirzadeh, S.Z., Parikh, P.C., and Pandya, M.J. (2000). “TheNASA Tetrahedral Unstructured Software System (TetrUSS)”. Aeronaut.J. Vol. 104, No. 1040, pp. 491–499.

Galle, M. (1999). “Ein Verfahren zur Numerischen SimulationKompressibler, Reibungsbehafteter Strömungen auf Hybriden Netzen”.Deutsches Zentrum für Luft- und Raumfahrt, Cologne, Germany. ReportDLR-FB-1999-04.

Hanneman, V., and Mack, A. (2002). “Validation of the Unstructured DLRTAU-Code for Hypersonic Flows”. AIAA Paper 2002–3111.

Launder, B.E., and Spalding, D.B. (1974). “The Numerical Computation ofTurbulent Flows”. Comput. Methods Appl. Mech. Eng. Vol. 3, pp. 269–289.

Lesieutre, D., Love, J., Dillenius, M., and Blair, A.B., Jr. (2002). “RecentApplications and Improvements to the Engineering Level AerodynamicPrediction Software MISL3”. Proceedings of the 40th AIAA AerospaceSciences Meeting and Exhibit, Reno, Nevada, 14–17 January 2002. AmericanInstitute of Aeronautics and Astronautics, Reston, Virginia. AIAA Paper2002-0275.

Leopold, F., Demeautis, C., and Faderl, N. (2003). “ExperimentalInvestigations of the RTO Missile Configuration at High Angles of Attack”.ISL PU 657/2003.

McIlwain, S., and Khalid, M. (2004). “WIND CFD-Based PerformanceStudy of a Dual Control Missile”. In Assessment of Turbulence Modeling forHigh Speed Air Vehicles. AVT/RTO Report.

Menter, F.R. (1994). Two-Equation Eddy-Viscosity Turbulence Models forEngineering Applications”. AIAA J. Vol. 32, No. 8, pp. 1598–1605.

Nöding, P., and Hennig, P. (2004). “LFK Results for the Dual ControlMissile”. In Assessment of Turbulence Modeling for High Speed Air Vehicles.AVT/RTO Report.

Parikh, P.C. (2001). “Application of a Scalable, Parallel, Unstructured-Grid-Based Navier-Stokes Solver”. AIAA Paper 2001-2584.

Pirzadeh, S., and Leavitt, L. (2004). “Performance Studies on a DualControl Missile Using Unstructured Grids”. In Assessment of TurbulenceModeling for High Speed Air Vehicles. AVT/RTO Report.

Prince, S., and Birch, T. (2004). “Application and Validation of theParabolized Navier-Stokes Method for the Investigation of the NASA DualControl Missile Configuration in Supersonic Flow”. In Assessment ofTurbulence Modeling for High Speed Air Vehicles. AVT/RTO Report.

Qin, N., and Jayatunga, C. (1998). “Algebraic Turbulence Modeling forVortical Flows Around Slender Bodies”. In Missile Aerodynamics. NATORTO-MP-5, Paper 20.

Spalart, P.R., and Allmaras, S.R. (1992). “A One-Equation TurbulenceModel for Aerodynamic Flows”. AIAA Paper 92-0439.

Van Leer, B. (1982). “Flux Vector Splitting for the Euler Equations”. In 8thInternational Conference on Numerical Methods in Fluid Dynamics. Edited byE. Krause. Lect. Notes Phys. Vol. 264, pp. 667–683.

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Aerodynamic Forces Approximations usingthe Chebyshev Method for Closed-Loop Aero-

servoelasticity StudiesAlin Dorian Dinu * Ruxandra Mihaela Botez * and Iulian Cotoi *

* Department of Automated Production EngineeringÉcole de technologie supérieure

1100 Notre Dame WestMontreal, QC H3C 1K3, Canada.

E-mail: [email protected]

Received 6 May 2005.

NOMENCLATURE

C modal damping matrix

c wing chord length

K modal elastic stiffness matrix

k reduced frequency

M modal inertia or mass matrix

M Mach number

T Chebyshev polynomial

Q modal generalized aerodynamic forces matrix

QI imaginary part of the modal generalizedaerodynamic forces matrix

QR real part of the modal generalized aerodynamicforces matrix

q non-dimensional generalized coordinates (withrespect to time t)

qdyn dynamic pressure

V true airspeed

VE equivalent airspeed

Vp matrix of eigenvectors

V0 reference true airspeed

η generalized coordinates

λ vector of eigenvalues

ν airspeed ratio

ρ true air density

ρ0 reference air density

σ air density ratio

� modal transformation matrix

ω natural frequency

INTRODUCTION

The aero-servoelasticity represents the combination ofseveral theories regarding different aspects of aircraftdynamics. Studies of aero-servoelastic interactions on anaircraft are very complex problems to solve, but essential foraircraft certification. The instabilities issued by the adverseinteractions among the flexible structure, the aerodynamicforces, and the control laws acting on it could appear at anytime inside the flight envelope, so we can state that the aero-

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AbstractThe approximation of unsteady generalized aerodynamicforces from the frequency domain into the Laplace domainacting on a Fly-By-Wire aircraft presents an importantchallenge in the aero-servoelasticity area. The aerodynamicforces in the reduced-frequency domain are approximatedin the Laplace domain, to be able to study the effects of thecontrol laws on the flexible aircraft structure. In this paper,we present a new method for the approximation of thegeneralized aerodynamic forces by use of Chebyshevpolynomials and their orthogonality properties. Acomparison of this new method with the Padé method usedto calculate an approximation of the generalizedaerodynamic forces from the frequency domain into theLaplace domain is presented. This comparison shows thatthis new method gives excellent results with respect to thePadé method and is applied on the Aircraft Test Modelfrom NASA Dryden Flight Research Center.

RésuméL’approximation de forces aérodynamiques généraliséesnon-stationnaires du domaine de la fréquence dans ledomaine du Laplace, forces qui actionnent sur un avion acommandes électriques, représente une importantechallenge pour le domaine de l’aéroservoélasticité. Lesforces aérodynamiques du domaine de la fréquence réduitesont approximées dans le domaine du Laplace pour étudierles effets des lois de contrôle sur la structure flexible del’avion. Dans cet article nous présentons une nouvelleméthode pour l’approximation de forces aérodynamiques

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servoelastic interactions concern mainly the research fieldlocated at the intersection of the following three disciplines:aerodynamics, aeroelasticity, and servo-controls. One mainaspect of the aero-servoelasticity is the conversion of theunsteady generalized aerodynamic forces Q(k, Mach) from thefrequency domain into the Laplace domain Q(s), where krepresents the reduced frequency, Mach is the Mach number,and s is the Laplace variable. There are mainly three classicalmethods for approximating the unsteady generalized forces byrational functions from the frequency domain to the Laplacedomain (Tiffany and Adams, 1984, 1988; Edwards, 1977;Roger, 1977; Karpel, 1982): Least Square (LS), Matrix Padé(MP), and Minimum State (MS). All three methods use rationalfunctions under Padé form.

Several aero-servoelastic analysis software codes aredeveloped for the aerospace industry. One of the computerprograms used for aero-servoelasticity analyses is the Analogand Digital Aeroservoelasticity Method (ADAM) that wasdeveloped at The Flight Dynamics Laboratory (Noll et al.,1986). ISAC (The Interaction of Structures, Aerodynamics, andControls) was developed at NASA Langley Research Center(Adamsand Hoadley, 1993). At the Boeing Company, the aero-servoelastic software FAMUSS was used (Pitt, 1992). Anaeroelastic code, ZAERO, has been developed at ZonaTechnology, which has been used for aero-servoelastic studies(Chen and Sulaeman, 2003). The STARS code was developedat NASA Dryden Flight Research Center (Gupta, 1997).

Among all these computer codes, we have chosen to workwith the STARS code. The STARS program is an efficient toolfor aero-servoelastic interactions studies and has an interfacewith NASTRAN (Newsom et al., 1984; Rodden et al., 1979) acomputer program frequently used in the aeronautical industry.In this paper, the lateral dynamics of a half aircraft test modelATM is modeled in STARS. Following finite element modelingand the doublet lattice method application on the ATM inSTARS, the unsteady aerodynamic forces are calculated asfunctions of reduced frequencies k and Mach number M. Wehave provided here a bibliographical research on the softwareused in aero-servoelasticity. All these codes exploit two mainclassical methods for aerodynamic-force approximations from

the frequency domain (aeroelasticity) into the Laplace domain(aeroservoelasticity): Least Squares (LS) and Minimum State(MS).

We will now present in detail bibliographical research onother existing methods in the literature.

The approximation of the unsteady generalizedaerodynamic forces is a must for the control analysis of oursystem. Due to the fact that Q(k, Mach) can only be tabulatedfor a finite set of reduced frequencies, at a fixed Mach numberM, it must be interpolated in the s domain to obtain Q(s). In thispaper, we describe such an interpolation method that uses theChebyshev polynomials and its results. In the subsonic regime,the unsteady generalized aerodynamic forces Q(k, Mach) arecalculated, using finite elements computer programs such asSTARS or NASTRAN, by the Doublet Lattice Method (DLM).We further need to convert these forces into the Laplace domainwhere they will be denoted as Q(s). The aerodynamic forcesdependence of s may be written as an irrational function evenfor simple cases such as two-dimensional potentialincompressible flows on an airplane wing profile. Theodorsen(1933) proved that Q(s) could be expressed by use of Hankel’sfunctions. A few years later, Wagner found the first rationalapproximation (Dowell, 1995) for Q(s). Another approach usedthe approximations of unsteady aerodynamic forces by Padépolynomials. This approach was based on a fractionalapproximation of the form P(s)/R(s), where P and R are twopolynomials in s, for every term of the unsteady force matrix. Inthis way, every pole of R(s) showed a new state called theaugmented state, in the final linear invariant aero-servoelasticsystem. Thus, if the initial square matrix had N dimensions, andif a Padé approximation of M order is used, then there will beintroduced N(N + M) augmented states. The number ofaugmented states was reduced by Roger (1977). In hisformulation, only N × M modes were introduced, where N is thenumber of initial modes. Roger’s method is based on the factthat the aerodynamic lag terms remain the same for eachelement of the unsteady aerodynamic force matrix. Thisiscalled the LS method and is used in computer aero-servoelasticcodes such as STARS and ADAM.

Another method was derived from the LS method and wasproposed by Vepa (1977). This method used the samedenominators for every column of the aerodynamic matrix Q,and was called MP method.

Various improvements were made to the two methodspresented above. One such type of improvement is thatdifferent conditions (restrictions) were imposed onapproximations to pass through certain points. Generally, theapproximations were restricted to be exact approximations atzero and at two other chosen points. Generally, the first pointwas chosen to represent the estimated flutter frequency and thesecond point to represent the gust frequency. Then, theimproved methods were renamed: ELS method (ExtendedLeast Squares) (Tiffany and Adams, 1984) and EMMP method(Extended Modified Matrix Padé) (Dunn, 1980). Later, Karpel(1998) proposed a completely different approach to solve theabove approximations. He knew from the beginning that the

168 © 2005 CASI



généralisées en utilizant les polynômes de Chebyshev etleurs propriétés d’orthogonalité. Une comparaison de cettenouvelle méthode avec la méthode de Padé qui esthabituellement utilizée pour calculer l’approximation deforces aérodynamiques généralisées du domaine de lafréquence dans le domaine du Laplace est aussi présentée.Cette comparaison nous montre que cette nouvelleméthode donne des résultats excellents par rapport à laméthode Padé en l’appliquant sur l’Aircraft Test Model(l’ATM) de NASA DFRC (Dryden Flight ResearchCenter).

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goal was to find a linear invariant system in the time domainand he decided to integrate this information directly in theequation giving the unsteady aerodynamic forces values, byadding a term resembling a transfer function of a linear system.Further, as he wanted to find a linear system of reasonabledimensions, he wrote the approximation under the MS form.The advantage of this method with respect to Roger’s methodresided in the fact that it allowed an excellent approximation tobe obtained, but with a smaller number of augmented states. Allof the methods described above allow for the approximation ofunsteady aerodynamic forces for one Mach number at a time.To obtain approximations for several Mach numbers, weshould perform the approximation approach given by the MSmethod for each Mach number, which might be expensive interms of computing time. A valid approximation for a range ofMach numbers could be useful for military Fly-By-Wireaircraft, where the Mach number varies rapidly during high-speed manoeuvres, and where aero-servoelastic interactions areextremely important. Poirion (1995, 1996) constructed anapproximation allowing for the calculation of unsteadyaerodynamic forces for Mach number values contained in aspecific interval and for a frequency domain. He used the MSmethod and considered a regular dependence with the Machnumber. He used several MS approximations, obtained forseveral fixed Mach numbers, and a spline interpolation methodfor Mach number dependence. Thus, he obtained a formula thatallows for the computing of unsteady aerodynamic forces forany couple (k, Mach), where k is the reduced frequency, theequations remaining valid for a range of (k, Mach) ∈ [kmin,kmax]*[Mmin, Mmax].

The approximation methods should satisfy two opposedcriteria simultaneously: an excellent (exact) approximation,which could be obtained by increasing the number of lag termsand a linear invariant system in the time domain of a very smalldimension (with the smallest possible number of lag terms).There was no method satisfying both criteria until now. In tworecent papers, Cotoi and Botez (2001, 2002) have proposed anew approach based again on a precise Padé approximation.The two authors used order-reduction methods for the last termof the approximation, which could be seen as a transferfunction of a linear system. The approximation error for thisnew method is 12–40 times lower than for the MS method forthe same number of augmented states and depends of thechoice made for the model reduction method. However, thismethod remains very expensive in terms of computing time.Dinu et al. (2005) presented open-loop flutter analysis resultsusing a new method based on Chebyshev polynomials theoriesof the ATM (Aircraft Test Model). Control laws were notconsidered in their previous paper (where open-loop flutteranalysis was performed), while in the present paper, this newmethod is applied to the ATM— on which ailerons and rudderact — so that closed-loop flutter analysis results aredetermined.

Aircraft Equations of MotionThe flexible aircraft equations of motion, where no external

forces are included, may be written in the time-domain asfollows:

~��

~�

~( , )M C K Qη η η η+ + + =q k Machdyn 0 (1)

where ρ is the air density, V is the true airspeed, and qdyn =0.5ρV2 is the dynamic pressure; η is the generalized coordinatesvariable defined as q = �η where q is the displacement vectorand � is the matrix containing the eigenvectors of the followingfree-vibration problem:

M K��q q+ = 0 (2)

The following transformations were used in Equation (1):

~,

~,

~M M C C K K= = =� � � � � �T T T

Q( , ) ( )k Mach A k= � �Te (3)

Here, M, K, and C are the generalized mass, stiffness, anddamping matrices; k, the reduced frequency is written as k =ωb/V where ω is the natural frequency and b is the wing semi-chord length. Ae(k) is the aerodynamic influence coefficientmatrix for a given fixed Mach number M and a set of reducedfrequencies values k. The Laplace transformation is furtherapplied to Equation (1), and we obtain:

[~ ~ ~

] ( ) ( ) ( )M C K Qs s s q s s2dyn 0+ + + =η η (4)

Q(s) are the unsteady aerodynamic force approximations ofQ(k, Mach) in the Laplace domain. In this paper, we describe anew approximation method that uses Chebyshev polynomialsand its results.

Chebyshev Polynomial TheoryThese polynomials (Rivlin, 1990; Weisstein, 1999–2005)

are a set of orthogonal polynomials defined as the solutions tothe Chebyshev differential Equation (10) and are denoted asTn(x). They are used as an approximation to a least squares fit,and are closely connected to trigonometric multiple-angleequations. The Chebyshev polynomials of the first kinddenoted by Tn(x) are implemented in Mathematica asChebyshevT [n, x], and are normalized so that Tn(1) = 1.

Any continuous function may be expressed by use ofChebyshev polynomials using the following equation:

f x c c T xjj

j( ) ( )= +=

∞

∑12

01

(5)

where Chebyshev polynomials used in Equation (5) areexpressed under the following form:

© 2005 CASI 169


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T x j xj( ) cos( ( ))= arccos (6)

and the coefficients cj used in Equation (5) are expressed asfollows:

cf x T x

xx jj

j=−

=−∫2

1d and 1, 2,

2π( ) ( )

1

1

� (7)

The Chebyshev polynomials have orthogonality properties thatallow us to keep the approximation’s error within apredetermined bandwidth.

The following recurrence relationships are used in the newapproximation method:

T x

T x x

T x xT x T xr r r

0

1

1

2

( )

( )

( ) ( ) ( )

=== −

⎧

⎨⎪

⎩⎪

+ −1 1

(8)

and the following condition in the aim to find the Chebyshevpolynomials solution is imposed:

T xr( ) = 0 (9)

where r specifies the rank of the Chebyshev polynomial.Equation (9) gives the following solution:

xj

r= +

cos(2 1)

2π

(10)

Tr(x) is a function defined by cosines, which lets us concludethat between two solutions of this function we find an extremeof |1| amplitude in the middle of the interval, specifically at:

xjr

j r= =cos , ...,π

0,1, (11)

Methodology for the ChebyshevApproximation Method

To develop our approximation method, we used thepredefined functions for the Chebyshev polynomials expressedin Equation (6) that have been implemented in the Maplekernel, in MATLAB.

These functions (chebpade and chebyshev) allow theconstruction of a polynomial interpolation for the unsteadygeneralized aerodynamic forces, acting on the ATM for 14values of reduced frequencies k and 9 values of Mach number.The elements forming the matrices of the unsteady generalizedaerodynamic forces calculated by the Doublet Lattice MethodDLM in STARS are denoted by Q(i, j) with i = 1…8 and j =1…8 for the first eight elastic modes.

The approximation by means of this method is obtained byuse of a similar path to the one used for the Padé method. For

each element of the unsteady aerodynamic force matrix, wefound a power series development under the following form, byuse of Maple’s “chebyshev” function:

Qijij

nij

nij

n

s c c T s( ) ( )( ) ( ) ( )= +=

∞

∑12 0

1

(12)

where

cs T s

ss nn

ij ij nij

( )( )( ) ( )

( )=

−=

−∫2

1d for 0,1,

2πQ

1

1

�

We found an approximation by rational fractions by use of the“chebpade” function:

� ( )

( )

( )

( ) ( )

( ) ( )

Qij

nij

n

M

nij

nij

n

P

nij

s

a T s

b T s

=+

=

=

∑

∑0

1

1

(13)

where M = P + 2.This new form integrates the orthogonality properties of

Chebyshev polynomials and allows the variation of the degreeof the numerator M and the denominator P, to obtain a verygood approximation.

In Equation (13), an approximation order [M, P] = [16, 14]gives M = 16 and P = 14 where M is the maximum rank ofChebyshev polynomials at the numerator and P is themaximum rank of Chebyshev polynomials at the denominator.

We compared the results found by means of our Chebyshevapproximation method with the results given by the Padémethod. These results are expressed in terms of a totalnormalized approximation error.

The Padé method uses a parameter identification solution todetermine a polynomial fractional form that identifies anorthogonal polynomial interpolation. This fractional form is thekey aspect of this method, due to the fact that it allows the orderreduction system.

The Padé polynomials are used in the LS method ofimplementation, which is considered the most classical andmost used method now for aero-servoelastic interactionsstudies. The LS method was implemented in most aero-servoelasticity software ISAC, ADAM, and STARS.

For various aircraft types (such as CL-604 or F/A-18),classical aero-servoelasticity studies by use of the LS or MSmethods were performed. Following an analysis of the resultsobtained and the algorithms used, we found that the LS methodwas easier to implement and execution time was faster than thatof the MS method. Our computer programs were written inMATLAB. For this reason, in this paper, a comparison isperformed between flutter results obtained with the Padémethod and flutter results obtained with our method based onChebyshev polynomials properties.

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Figures 1 and 2 show the real and imaginary parts ofaerodynamic forces elements approximated by Padé andChebyshev methods versus their initial values calculated inSTARS in the frequency domain. Figures 3 and 4 show that ournew approximation method gives the best approximation erroron an interval chosen in the proximity of each approximationpoint. The effect of the Chebyshev polynomials properties isseen in these types of results. Due to these properties, we areable to impose a bandwidth for the error convergence for eachelement of the unsteady generalized aerodynamic forcematrices.

Both figures show the overall normalized approximationerror by Chebyshev and Padé methods. The differencesbetween Figures 3 and 4 are based on the model order:Figure 3 shows the results for the [16, 14] polynomial modelorder while Figure 4 shows the results obtained for the [15, 13]polynomial model order. In these examples, the results wereobtained using the ATM data generated in the STARS code atMach number M = 0.5 and for 14 reduced frequencies k =[0.0100 0.1000 0.2000 0.3030 0.4000 0.5000 0.5882 0.62500.6667 0.7143 0.7692 0.8333 0.9091 1.0000].

The Padé method gives a small error near the middle of theapproximation interval and an increased error towards each endof it. The Chebyshev approximation method demonstrates analmost constant value of the error all along the approximationinterval. The total normalized approximation errors differencesat both ends of the approximation interval are noticed in bothfigures. Those differences are higher for k14 = 1 than for k1 =0.01.

The total normalized approximation errors were calculatedfor different values of the polynomial approximation orderusing the Padé and the Chebyshev polynomial fraction methods(polynomial model order should be equivalent for bothmethods) for all the Mach numbers and the same differenceswere observed. The total normalized approximation errorobtained with the Chebyshev method was found to be much

© 2005 CASI 171


Figure 1. The real part of the aerodynamic forces calculated by Padéand Chebyshev methods versus their initial values in the frequencydomain.

Figure 2. The imaginary part of the aerodynamic forces calculated byPadé and Chebyshev methods versus their initial values in thefrequency domain.

Figure 3. The total normalized approximation errors for the [16, 14]model order.

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smaller with respect to the total normalized approximationerror given by the Padé method.

In Table 1, a few examples regarding the values of the totalnormalized approximation errors provided by the Chebyshevmethod and by the Padé method are given. In Table 1, threedifferent orders of approximation were used: [16, 14], [15, 13],[10, 8] for the same fight condition and the results are presentedfor the real aerodynamic forces part errors denoted by JQ_REALand for the imaginary aerodynamic forces part errors denotedby JQ_IMAG. It can be clearly seen that no matter the order of thepolynomial approximation, the total normalized error for theChebyshev method is lower than the total normalized errorgiven by Padé method. These errors were calculated using thefollowing formula:

Jij ij

ijj

N

iQ

R RQ Q

Q_ REAL

new old

2

modes

=−

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟==

∑11

N

k

modes

∑∑⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

×=1

14

100%(14)

Jij ij

ijj

N

iQ

I IQ Q

Q_ IMAG

new old

2

modes

=−

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟==

∑11

N

k

modes

∑∑⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

×=1

14

100%

where QRold and QIold are the real and the imaginary parts of theunsteady aerodynamic forces given by STARS for the ATMmodel and QRnew and QInew are the real and the imaginary partsof the unsteady aerodynamic forces approximated by

Chebyshev or by Padé theories. Nmodes is the total number ofmodes (also the dimension of Q), k is the index of the reducedfrequency, and J is the total normalized error.

Open-Loop Flutter Results Obtainedusing the Chebyshev Method

To validate our method, we used the STARS ATM developedby the NASA Dryden Flight Research Center. This lateralmodel (only anti-symmetric modes are provided) includesaero-structural elements (flexible aircraft) and control surfaces(ailerons and elevator). First, a free vibration analysis wasperformed in the absence of aerodynamics to obtain the freemodes of vibration. We obtained the same frequencies andmodes of vibration using our MATLAB method as withSTARS.

Then to calculate aerodynamic forces in the frequencydomain by the DLM, the same simulation parameters wereconsidered as the ones considered in the STARS computerprogram: reference semi-chord length b = 38.89 in (1 in =2.54 cm), reference air density at sea level ρ0 = 1.225 kg/m3,altitude at sea level Z = 0 ft (1 ft = 0.3048 m), reference soundairspeed at sea level a0 = 340.294 m/s.

In Table 2, the speeds and frequencies at which flutteroccurs are calculated by two methods, Padé and Chebyshev, forthree different types of approximation orders. The executionspeed is three times smaller for the Chebyshev polynomialsmethod than the execution speed used for the Padé method.


172 © 2005 CASI

Figure 4. The total normalized approximation errors for the [15, 13]model order.

Order Method JQ_REAL JQ_IMAG

[16, 14] Chebyshev 0.065065 0.037611Padé 0.065271 0.287657

[15, 13] Chebyshev 0.055391 0.040053Padé 0.260147 0.775989

[10, 8] Chebyshev 0.061466 0.035109Padé 0.136338 0.055638

Table 1. Total normalized approximation errors by Chebyshev andPadé methods.

Flutter results (fuselage first bending mode)

MethodSpeed(knots)

Frequency(rad/s)

Computationtime (s)

Pk-Padé [8, 6] 445.5 77.5 122Pk-Padé [9, 7] 445.5 77.5 134Pk-Padé [10, 8] 445.8 77.5 144Pk-Chebyshev [8, 6] 446.5 77.5 40Pk-Chebyshev [9, 7] 446.6 77.5 47Pk-Chebyshev [10, 8] 446.6 77.5 53

Table 2. Flutter results comparison for ATM in open loop.

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Closed-Loop Flutter Results Obtainedusing the Chebyshev Method

We applied our new approximation method for the closed-loop aero-servoelastic analysis, using the transfer functioninformation provided by the aircraft’s control laws (aileronsand elevator are used for lateral aircraft control). Theconversion from the frequency domain into the Laplace domain(as closed-loop calculations should be realized in the Laplacedomain) was done this time by use of the LS method for theapproximations obtained with Chebyshev polynomials. Toachieve this type of conversion, we used the following variablechange:

s k ks

s= ⇒ = = −jj

j (15)

where “j ” is the complex number j = −1.We rewrote the approximation of the unsteady generalized

force matrix in the Laplace domain as follows:

Q( ) ( ) ( )s A A s A s As

s b= + − + − + −

− +0 1 22

31

j jj

j

+ −− +

+As

s b4

2

jj

� (16)

and we took into account that Q(s) has a real part QR(s) and animaginary part QI(s), such as:

Q Q Q( ) jR Is = +( ) ( )s s (17)

We obtained then the following expressions for theseaerodynamic forces:

Q

Q

R 02

2

2

2 2

I 1 2

( )

( )

s A s As

s bA

s A ssb

s b

n

= + ++

= − −+

=

∞

+∑n

n

n

n

12

2n

A=

∞

+∑

⎧

⎨⎪⎪

⎩⎪⎪

12n

(18)

where A0, A1,…, An+2 are the LS decomposition matrices and bnare the lag terms introduced by the LS method.

Figures 5, 6, and 7 show the flutter closed-loop results. InFigure 5, the frequency is plotted versus the damping, inFigure 6, the frequency is plotted versus the EquivalentAirspeed and in Figure 7, and damping is plotted versus theEquivalent Airspeed. From these three figures, the flutterfrequency, damping, and equivalent airspeed are calculated(flutter occurs where damping is zero). Their numerical valuesare given in Table 3.

In Table 3, a comparison of the results obtained with ournew method (a combination of the P flutter method with theChebyshev approximations) with the results obtained by a

combination of the P flutter method with the Padéapproximations is provided for 2 to 6 lag terms.

Our new approximation method provides the same valuesfor the flutter speeds and frequencies as the classical methods(which takes Padé approximations into account, because thesemethods are actually based on Padé) no matter the number oflag terms considered. These good results were expected tooccur, following the properties of the Chebyshev polynomials.In addition, the execution time required by the P-Chebyshevmethod is three times smaller than the execution time requiredby the P-Padé method.

It is well known that a larger number of lags imply on onehand an increased number of calculations and on the other handit introduces new poles, thus modifying the initial system and

© 2005 CASI 173


Figure 5. Frequency versus damping calculated by the P-Chebyshevmethod for the closed loop ATM analysis.

Figure 6. Frequency versus equivalent airspeed calculated using the P-Chebyshev method for the closed loop ATM analysis.

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generating approximation errors. For this reason, the advantageof using this new method with respect to classical methods isthat using a small number of lags (2 lag terms) the same resultsare obtained as if using a large number of lags (6 lag terms).

CONCLUSION

The Chebyshev approximation method provides excellentflutter results for a small number of lag terms. However, due tothe fact that the Chebyshev polynomials were generated by useof the ATM data, there are quite large differences between thevalues of the elements contained in the unsteady generalizedaerodynamic force matrices (1e +10). Restraints regarding thethreshold of the approximation error had to be imposed, i.e., forsmaller elements we imposed an error value of 1e –4 and forlarger elements an error value of 1e –2. Without theserestraints, the Chebyshev polynomials cannot be generated.

We could see that by use of the Chebyshev method in openloop, we were able to find very good values for the flutterspeeds and the frequencies at which flutter occurs. One of themost important achievements of our new method, if not the

most important, is the fact that the computation time for theopen-loop case is up to 3 times smaller than in the Pk–Padémethod and up to 30 times smaller than in our Pk–LS case, evenfor an increased approximation order.

As for the closed-loop case, we observed that no matter theinitial Chebyshev approximation order, it would be enough tomake use of only 2 lags when converting to the Laplace domainto obtain excellent approximation results.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. Gupta from NASADryden Flight Research Center for allowing us to use the ATMon STARS. We would also like to thank the other members ofthe STARS engineering group: Mr. Tim Doyle, Mr. MartyBrenner, and Dr. Shun Lung for their precious assistance andcollaboration.

This work was made possible due to funds received from theNatural Sciences and Engineering Research Council of Canada(NSERC), and from the Ministère du Développementéconomique, de l’innovation et de l’exportation (MDEIE).

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Dowell, E.H. (1995). “A Modern Course in Aeroelasticity”, KluwerAcademic, Dordrecht, The Netherlands.

Dunn, H.J. (1980). “An Analytical Technique for Approximating UnsteadyAerodynamics in the Time Domain”. NASA Tech. Pap. TP-1738.

Edwards, J.W. (1977). “Unsteady Aerodynamic Modeling and ActiveAeroelastic Control”. Department of Aeronautics and Astronautics, StanfordUniversity, Stanford, California. SUDAAR Rep. 504, February.

Gupta, K.K. (1997). “STARS – An Integrated, Multidisciplinary, Finite-Element, Structural, Fluids, Aeroelastic, and Aeroservoelastic AnalysisComputer Program”. NASA Tech. Memo. TM-4795, pp. 1–285.

Karpel, M. (1982). “Design for Flutter Suppression and Gust AlleviationUsing State-Space Modeling”. J. Aircr. Vol. 19, No. 3, pp. 221–227.

Karpel, M. (1998). “Reduced-Order Models for Integrated AeroservoelasticOptimization”. American Institute of Aeronautics and Astronautics, Inc.

174 © 2005 CASI


Figure 7. Damping versus equivalent airspeed calculated using the P-Chebyshev method for the closed loop ATM analysis.

Flutter results (control mode 2)

MethodSpeed(knots)

Frequency(rad/s)

Computationtime (s)

P-Padé [4, 2] 287.7 51.15 354P-Padé [5, 3] 287.6 51.13 361P-Padé [8, 6] 287.6 51.10 392P-Chebyshev [4, 2] 287.6 50.92 97P-Chebyshev [5, 3] 287.6 50.92 102P-Chebyshev [8, 6] 286.7 50.72 126

Table 3. Flutter results comparison for ATM in closed loop.

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Newsom, J.R., Adams, W.M., Jr., Mukhopadhyay, V., Tiffany, S.H., andAbel, I. (1984). “Active Controls: A Look at Analytical Methods andAssociated Tools”. In Proceedings of the 14th ICAS Congress, ICAS 84-4.2.3,Toulouse, France, 9–14 September 1984. Edited by B. Laschka and R.Staufenbiel. International Council of the Aeronautical Sciences and AmericanInstitute of Aeronautics and Astronautics, New York, N.Y. pp. 230–242.

Noll, T., Blair, M., and Cerra, J. (1986). “ADAM, An AeroservoelasticAnalysis Method for Analog or Digital Systems”. J. Aircr. Vol. 23, No. 11,pp. 852–858.

Pitt, D.M. (1992). “FAMUSS: A New Aeroservoelastic Modeling Tool”.AIAA Pap. 92-2395-CP.

Poirion, F. (1995). “Modélisation Temporelle des SystèmesAéroservoélastiques. Application à l’Étude des Effets des Retards”. LaRecherche Aérospatiale, No. 2, pp. 103–114.

Poirion, F. (1996). “Multi-Mach Rational Approximation to GeneralizedAerodynamic Forces”. J. Aircr. Vol. 33, No. 6, pp. 1199–1201.

Rivlin, Th.J. (1990). “Chebyshev Polynomials: From ApproximationTheory to Algebra and Number Theory”. 2nd ed. Wiley, New York, N.Y.

Rodden, W.P., Harder, R.L., and Bellinger, E.D. (1979). “AeroelasticAddition to NASTRAN”. NASA Rep. CR-3094.

Roger, K.L. (1977). “Airplane Math Modeling Methods for Active ControlDesign”. In Conference Proceedings on Structural Aspects of Active Controls.AGARD, Neuilly-sur-Seine, France. CP-228, pp. 4.1–4.11.

Theodorsen, T. (1933). “Interference on an Airplane of Finite Span in anOpen Rectangular Wind Tunnel”. NASA Tech. Rep. TR 461.

Tiffany, S.H., and Adams, W.M., Jr. (1984). “Fitting Aerodynamic Forces inthe Laplace Domain: An Application of a Nonlinear Nongradient Technique toMultilevel Constrained Optimization”. NASA Tech. Memo. TM 86317.

Tiffany, S.H., and Adams, W.M. (1988). “Nonlinear ProgrammingExtensions to Rational Function Approximation of Unsteady Aerodynamics”.NASA Tech. Pap. TP-2776, July.

Vepa, R. (1977). “Finite State Modeling of Aeroelastic System”. NASARep. CR-2779, February.

Weisstein, E.W. (1999–2005). “Chebyshev Polynomial of the First Kind”.MathWorld – A Wolfram Web Resource [On-line]. Available from: [retrievedMarch 2004].

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Applying Fast Fourier Transform Analysis andData Window in Software Global Positioning

System Receivers to Mitigate ContinuousWave Interference under Dynamic Conditions

Z. Jiang * ** G. Lachapelle * and C. Ma *

* Position, Location and Navigation (PLAN) GroupDepartment of Geomatics Engineering

The University of CalgaryCalgary, AB T2N 1N4, Canada.

** Present address: NovAtel Inc.1120 68th Avenue NE,

Calgary, AB T2E 8S5, Canada.E-mail: [email protected]

Received 9 March 2005.

INTRODUCTION

The Global Positioning System (GPS) uses a direct sequencespread spectrum (DS-SS) signal that incorporates some degreeof anti-jamming capability in the signal structure itself.However, GPS signals, normally in the range of –160 to –156 dBW for the C/A-code, are weak and well below thebackground RF noise level and thus make it easy for aninterference signal to overcome the inherent anti-jammingcapability of the DS-SS signal owing to the fact that theprocessing gain is not high enough to compensate theinterference.

For narrow-band interference, which usually arises fromspurious signals generated by inadequately shielded electricalequipment or from narrow-band radio links adjacent to GPSfrequencies, a so-called frequency-domain interferenceexcision algorithm based on Fast Fourier Transform (FFT)analysis has been proved to be a good approach to mitigate suchsusceptibility. Several techniques based on this concept havebeen proposed, including adaptive transversal filters (ATF)(Przyjemski et al., 1993), FFTs, and filter banks (FB) (Rifkinand Vaccaro, 2000).

Generally speaking, the objective of this FFT-based narrow-band interference mitigation algorithm is to reduce the level ofthe interference at the expense of introducing some degree ofdistortion on the desired signal before signal correlation isconducted. The estimation and the suppression of interferenceare in fact performed in the frequency domain. So, threeoperations are obviously necessary herein. The first oneconsists of transforming the time domain signal into thefrequency domain by fixed-length Discrete FourierTransform/Fast Fourier Transform (DFT/FFT), the second oneis the actual interference identification and suppression, and thethird one consists of transforming signal from the frequencydomain back to time domain for conventional GPS signalprocessing.

The power distribution of CW interference suggests thatonly a small number of frequency domain cells contain nearlyall of the interference power within the band (Dipietro, 1989).Based on this conclusion, one possible strategy could be to setthe weights on all cells containing large interference to zero,

© 2005 CASI 177


AbstractTraditionally, most Radio Frequency Interferencemitigation methods have been implemented and testedusing conventional hardware receivers. With the rapiddevelopment of computer technologies, the signal-processing computational load is becoming less of aconcern, and thus it becomes feasible to develop and testnew interference mitigation methods based on softwarereceivers together with modern digital signal-processingtechniques. This paper describes an investigation into thedevelopment of a narrow-band Continuous Wave (CW)interference mitigation algorithm and performance testingusing a software Global Positioning System receiver basedon Fast Fourier Transform analysis and data windowing.

In this paper, some strategies were proposed to improvethe anti-jamming performance. First, for high-levelinterference, a fixed detection threshold suggested inprevious literature is not optimal. An adaptive detectionthreshold that is of better performance and is a function ofthe standard deviation of the normalized spectrum and thecorrelator power output was used in lieu. Second, a prioriinformation-based soft thresholding techniques for both bitsynchronization and enhanced signal acquisition wereapplied under high dynamic conditions and high-levelinterference environments. The factors that are crucial forweak-signal detection and tracking, namely, coherent

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while leaving the others at unity. The computational challengein this case lies in determining which cells constitute largeinterferences. One straightforward way is establishing athreshold, and then comparing it to signal components in allcells; when the magnitude of a signal component exceeds thethreshold, interference is assumed and the component isremoved by setting its corresponding weighting function tozero. The threshold can be established on the basis ofknowledge of the interference distribution or on the basis ofheuristic experience, such as setting the threshold to excise afixed percentage of cells or total interference power. Anotherstrategy for the weighting function derivation could be to setany cell value exceeding the threshold to the background noiselevel, and thus whiten the interference spectrum. This approachmay yield improved performance, because the cell valuecontaining interference will be reduced only to the backgroundlevel, hence retaining most of the signal power. The drawbackof this approach, however, is the requirement of thebackground noise level estimation, thus further increasing thecomputational burden.

Applying FFT analysis and data windowing to narrow-bandCW interference mitigation is actually not a brand newtechnique. Since the early 1980s, quite a lot of research hasbeen conducted, e.g., Li and Milstein (1982), Dipietro (1989),Young and Lehnert (1994), Wang and Amin (1998). However,these previous investigations have focused mainly on spreadspectrum communications systems, and only in recent yearshas such a frequency domain analysis-based interferencemitigation method been applied to GPS by some researchers,namely, Peterson et al. (1996), Badke and Spanias (2002), andCutright et al. (2003). The studies are far from completed andmany implementation issues, such as the determination of thedetection threshold for different kinds of interference anddifferent interference levels, the impact of narrow-bandinterference, the effectiveness of the mitigation algorithm in asoftware receiver for signal acquisition, tracking and positionfixing, have not been fully addressed. Thus, to enhance theapplication of the frequency excision algorithm in GPS, furtherresearch into this algorithm, aided by the flexibility that asoftware GPS receiver provides, becomes necessary.

THEORETICAL BACKGROUND

The received baseband data stream has three components,namely, the signal samples sk, the broadband noise samples ηk ,and the narrow-band interference sample θk . Each componentis assumed to be uncorrelated with the other two andcharacterized by a zero mean. The individual componentcorrelations are (Dipietro, 1989)

[ ]E s sk m

S k mk m

* =≠=

⎧⎨⎩

0(1)


178 © 2005 CASI

suite de la page 177s

integration time, tracking loop bandwidth, and integrationtime in the loop filter were evaluated to assess theeffectiveness of this algorithm. Some interferencesuppression strategies for spread spectrum systems,namely, windowing and overlap processing, were alsoinvestigated. Test results proved that the proposedalgorithm is effective to mitigate narrow-band CWinterference under certain power level. Windowing andoverlapped processing are good strategies to furtherimprove the anti-interfernce capability by 2 dB.

RésuméTraditionnellement, la plupart des méthodes de réductiondu brouillage RF ont été mises en œuvre et testées aumoyen de récepteurs matériels conventionnels. Grâce àl’évolution rapide des technologies de l’informatique, lacharge de traitement du signal devient de moins en moinsproblématique, et par conséquent, il est devenu réalisablede mettre au point et de tester de nouvelles méthodes deréduction du brouillage basées sur des récepteurs logiciels,alliés à des techniques modernes de traitement du signalnumérique. Cet article décrit une étude qui vise la mise aupoint d’un algorithme de réduction du brouillage associé àun signal continu à bande étroite ainsi que de tests deperformances au moyen d’un récepteur GPS logiciel basésur l’analyse FFT et le fenêtrage des données.Dans cet article, nous proposons un certain nombre destratégies qui visent à améliorer le rendement desméthodes antibrouillage. Tout d’abord, en ce qui concernele brouillage de haut niveau, le seuil de détection fixesuggéré dans la littérature antérieure n’est pas optimal.Nous avons utilisé à la place un seuil de détection adaptatifqui offre un meilleur rendement et est fonction de l’écarttype du spectre normalisé et de la puissance de sortie ducorrélateur. Ensuite, nous avons appliqué, dans desconditions hautement dynamiques à fort niveau debrouillage, des techniques de seuillage logiciel basées surdes informations a priori pour la synchronisation des bits etl’acquisition d’un signal amélioré. Afin d’évaluerl’efficacité de cet algorithme, nous avons évalué lesfacteurs qui sont essentiels dans la détection et le suivid’un signal faible, à savoir un temps d’intégrationcohérent, la bande passante de la boucle de poursuite et letemps d’intégration dans la boucle de filtrage. Nous avonségalement étudié un certain nombre de stratégies desuppression du brouillage dans les systèmes à étalementdu spectre, à savoir le fenêtrage et le traitement simultané.Les résultats des essais effectués démontrent quel’algorithme proposé se révèle efficace pour réduire lebrouillage associé à un signal continu à bande étroite, pourcertains niveaux de puissance. Le fenêtrage et le traitementsimultané constituent de bonnes stratégies pour améliorerde 2 dB supplémentaires les capacités antibrouillage.

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[ ]Ek m

k mk mη η

σ* =

≠=

⎧⎨⎩

02

(2)

[ ]E rk m m kθ θ* − (3)

where E[] is the expectation operator and * denotes thecomplex conjugate. Obviously, the signal and noise samples areeach uncorrelated, while the narrow-band interferences arecorrelated. Based on this characteristic, the spectral peaks ofthe interference can,therefore, be discriminated and suppressedfrom the GPS signal and the thermal noise (Gaussiandistribution) through an adaptive threshold power level and anadaptive notch filter.

The first stage of the FFT-based interference mitigationalgorithm is interference detection The correlation power,which indicates the average post-correlation signal-to-noiseratio (SNR), is used in this paper to detect the existence ofinterference. It is computed as (Ndili and Enge, 1997):

SNRExpected Noise Floor

pc

2 2= +I Q

(4)

where I and Q are the prompt 1 ms in-phase and quadraturecorrelation signals. The mean and the variance of the SNRpc arefunctions of noise and interference in the signal, and, therefore,are candidates for interference detection.

The change in the SNR before and after interferencemitigation is often used to show the performance improvementsince the objective of the algorithm is to identify and mitigatenarrow band interference based on its statistical properties. TheSNR of the output of a correlator is given by (Dipietro, 1989)

SNRVar

2= ( [ � ])

( �)E s X

s XkH

kH

(5)

where Var () is the variance operator and H is the conjugatetranspose. Replacing the corresponding components inEquation (6) with the assumed statistical properties of thesignal components and the characteristics of the DFTtransformation matrix, the SNR of the correlator outputbecomes

SNR1 2 3

2

=⎛⎝⎜

⎞⎠⎟

+ +=∑ αkk

NS

d d d

1(6)

with

d SN

kk

N kk

N

1 = 2

2

αα

=

=∑∑

−⎛⎝⎜

⎞⎠⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟1

1(7)

d kk

N2 2 2=

=∑σ α1

(8)

dN

kk

Nk3

1 2==∑ ( )Θ

1α (9)

where d1 is a “self noise” term that is analogous to that seen inlinear prediction suppression processors (Ketchum andProakis, 1982), a term that vanishes in the case of no filtering,d2 is the residual broadband noise, d3 is the narrow-bandinterference power, and Θk is the kth FFT bin component of thenarrow-band interference signal.

If the value of αk is set to unity, the pre-suppression SNR is

SNR12 2

= ⋅

+=∑

N S

N k

Nkσ θ( )

1

(10)

where N is the FFT length for suppression processing and S isequal to E s s[ ]*

k k .As mentioned before, the ratio of Equations (6) to (10)

yields the improvement factor used to assess interferencemitigation performance.

The influence of CW interference in the frequency domain isnot a single line because interference spreads out through theentire spectrum due to the finite FFT analysis period. Thisphenomenon is called spectral leakage. Thus, simply removingone frequency component with the largest power spectrum lineis not sufficient, and a suitable detection method is needed toidentify which frequency component contains stronginterference and should be removed. The judging criterion isbased on the statistical analysis of the input signal.Traditionally, the standard deviation of the resultingnormalized spectrum is multiplied by a fixed value to set adetection threshold for determining the presence of RadioFrequency Interference (RFI) (Cutright et al., 2003), and theestimate of this fixed value is determined empirically.

The traditional FFT-based interference mitigation algorithmcan remove most of the interference energy in the frequencydomain. But it has the drawback that it can suppress energyonly in the main-lobe spectral bins, which is not adequate dueto the residual interference distributed throughout the spectrumby these side-lobes, especially when the interference powerlevel is high.

FFT operations assume the signal processed to be a periodicextension of a finite sequence. For each block of Np samples ofthe input sequence that is used to do the FFT analysis, adiscontinuity will occur at the FFT block boundary if the Npsamples are not periodic in the FFT window of observation.Therefore, the FFT of a signal often exhibits high spectral side-lobes due to the finite processing interval. The behaviour ofthese side-lobes is very similar to a broadband interferencesource in terms of contributing energy to all FFT spectral bins.As a result, no threshold can be selected that will allow theremoval of the interference energy without also removing

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excessive signal spectrum. This phenomenon also constitutesspectral leakage.

Spectral leakage that survives the frequency domain filteringoperation may severely degrade the performance of themitigation algorithm. To mitigate the effect of spectral leakage,a data window is often applied prior to FFT computation.Windowing smoothes the discontinuities at the block boundaryand therefore lessens the effect of spectral leakage. Thewindowed FFT can be expressed as:

X k w n x nn

N j kn

Np

p( ) ( ) ( )==

− −

∑0

1 2

e

π

(11)

k Np= −0 1 1, , ...,

The window function w(n) determines the amount of spectralleakage in the DFT output. If w(n) is set to identity, namely,

w n n Np( ) , , , ...,= = −1 0 1 1 (12)

then Equation (12) reverts to non-windowed processing, whichis the same as the traditional FFT output. In this situation, it isequivalent to using a rectangular window. The Fouriertransform of the rectangular window is a Sinc function with thefirst side-lobe reduced by 13 dB relative to the main lobe.

When the frequency of a signal is not exactly one of the DFTfrequencies, the signal energy will be spread across thespectrum proportional to the width of the main lobe and theheight of the side-lobes of the window (Capozza et al., 2000).In the traditional FFT, which is equivalent to a rectanglewindowed FFT, the first side-lobe attenuation is only –13 dB.The interference frequency is much easier to mix with thesignal frequency, causing degradation of the mitigationperformance. So selecting a window with lower side-lobes willreduce the amount of spectral leakage. However, a lower side-lobe usually results in a wider main lobe, causing reducedspectral resolution.

The objective of using a data window in an interferencemitigation algorithm is to minimize the spectrum spreading ofthe interference frequency component to minimize the numberof frequency bins that need to be excised. At the same time,when no interference is presented, the windowing operationshould minimize the SNR loss of the GPS signal. Thus, windowselection requires a trade-off between the reduction in SNR dueto the signal attenuation caused by introducing windowoperation and the effectiveness of the spectral containment forinterference.

Overlapped processing techniques can be used to minimizethe degradation of the SNR due to data windowing whilemaintaining frequency content. The overlapped processingallows the tails of the window to be eliminated, therebysubstantially reducing the output SNR loss. With 50%overlapped processing, the SNR loss can be reduced to as lowas 0.6 dB (Capozza et al., 1999).

PROPOSED METHOD

The fixed interference detection threshold as proposed in theprevious literature is not optimal in some case, especially whenthe interference is strong. In this paper, an adaptive interferencedetection threshold determination method that is a function ofthe standard deviation of the normalized spectrum and the post-correlation SNR is used since the latter is a good indicator ofthe interference level. All of the results reported herein arederived from this adaptive detection threshold. After thedetection threshold is determined, the normalized spectrum isthen compared against the threshold and bins exceeding thedetection level are identified. The bins containing RFI, alongwith a variable number of surrounding bins, are then set to zeroin the original frequency domain spectrum. The effect ofremoving the frequency bins is equal to applying band-passfilters in the time domain. The Inverse Fast Fourier Transform(IFFT) of this spectrum is finally taken to yield a new timedomain signal without RFI or with mitigated RFI. To obtain theoptimal anti-jamming performance, three parameters have to becarefully chosen, namely:

• the average interval to remove the bias

• the detection threshold, and

• the number of samples to be removed near the bin containingthe RFI.In GPS applications, the interference levels are normally

well above the –13 dB side-lobes of the rectangular window.The spectrum herein exhibits significant spectral leakage. Thewindowing operation reduces the amount of the spectrum thatmust be excised, thus preserving more of the desired signalspectrum. In this paper, a four-term Blackman–Harris windowwith a –92 dB side-lobe was carefully chosen. The equation forcomputing the coefficients of this window is as follows (Harris,1978):

w kk

n[ cos+ = −

−⎛⎝⎜

⎞⎠⎟

1] 21

1 2α α π

+−

⎛⎝⎜

⎞⎠⎟

−−

⎛⎝⎜

⎞⎠⎟

α π α π3 441

61

cosk

nk

n(13)

where 0 ( – 1)≤ ≤k n , α1 = 0.35875, α2 = 0.48829, and α4 =0.01168The performance of frequency containment for the interferencecomponent is excellent with this window because of the verylow side-lobe that essentially restricts the spectral leakage.However, high signal attenuation on the FFT block transitionswill be introduced as the same time that may result in a 3 dBSNR loss. But this relatively high SNR loss can becompensated for by an overlapped processing technique.

To reduce the bit synchronization error when using a FFT-based interference mitigation algorithm under high-dynamicconditions and a high-level interference environment, somechanges to the tracking loop have been made, namely, soft

180 © 2005 CASI


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thresholding in bit synchronization and improved acquisitionbased on earlier information (details can be found in the resultsand analysis section).

TEST SET-UP

To obtain repeatable and controllable GPS signals withnarrow-band CW interference, a hardware GPS simulator and ahardware arbitrary signal generator were used to generate theGPS signals and specified interference. These two signals werethen combined in an interference combiner unit. The outputwas fed to a GPS front-end, namely, a Signal Tap, which down-converted the RF signals to intermediate frequency (IF) signalsand sampled them; the resulting data were used in a softwarereceiver where a mitigation algorithm was used to assess theperformance.

Two synchronous 12-channel L1-only hardware signalsimulation units (Spirent GSS STR 6560) were used. They canreproduce the signal propagation environment of a receiverinstalled on a dynamic platform, simulating the effects of high-dynamic host vehicle motion, navigation satellite motion, andionospheric and tropospheric effects. An Agilent ESG E4431Bsignal generator generated the narrow-band CW interferencesimulated in the test. The GPS and interference signals werecombined in a GSS 4766 interference combiner unit thatfacilitates the use of commercial-off-the-shelf (COTS) signalgenerators as fully integrated interference sources withhardware simulators, such as the Spirent GSS 6560. The COTSsignal generators were controlled through an IEEE-488-compliant (GPIB) bus, via Spirent SimGEN for Windows,hosted on the control PC. The interference signal is definedalong with all the other scenario parameters from withinSimGEN’s normal user GUI environment. The RF outputs arecombined with the satellite signal generators (SSG) in the GSS4766 interference combiner unit (ICU). The system hardwareconfiguration is shown in Figure 1.

A hardware front-end, GPS Signal Tap, made by AccordSoftware & Systems Private Limited, was used to down-convert, quantize, and log GPS signals for base-band signalprocessing with the software GPS receiver. The softwarereceiver used herein was that described by Ma et al. (2004).Owing to the limited capacity of the on-board RAM of theSignal Tap, only 80 s of data could be collected per test, whichis nevertheless sufficient to test the above procedures becausein the worst case, only 60 s of data are needed to obtain theephemeris.

From a receiver design point of view, to maximize toleranceto dynamic stress, the pre-detection integration time should beshort and the carrier-loop filter bandwidth should be wide;however, to receive a weak signal or to combat interference, thepre-detection integration time should be long and the carrier-loop filter-noise bandwidth should be narrow. To evaluate theperformance of this algorithm under extreme conditions, thetest was designed to apply interference and dynamic stress tothe receiver at the same time.

Two-dimensional constant speed circular trajectories weresimulated. The circle radius was set to 1000 m and a constantvelocity of 80 m/s was used to simulate high dynamics. Theinterference power was changed from a J/S ratio of 18 dB to28 dB and performance was compared. To ensure that theresults were stochastically repeatable, all the tests wereconducted six times.

RESULTS AND ANALYSIS

1. Traditional FFT-Based Mitigation AlgorithmTo investigate the impact of integration time on the loop

filters in dynamics and under interference conditions, aconstant vehicle velocity of 80 m/s with a J/S of 21 dB wereused. Integration times of 1, 2, 4, and 6 ms were tested, and theresults are shown in Table 1.

As shown in Table 1, the use of a 6 ms integration timecould successfully mitigate the influence of interference andyielded a reasonable solution. But only four satellites could betracked, constituting a low availability and poor reliability. Thisconfirmed that longer integration time results in lower satelliteavailability in high-dynamic conditions due to the longerresponse time in the tracking loop. When the integration timewas reduced to 2 ms or 4 ms, the dynamic capability wasimproved and six satellites could be tracked. But a largeposition error resulted in this case due to the impact ofinterference. If the integration time further reduces to 1 ms, thesatellite availability goes back to 4 (under interferenceconditions, the signal is weak and longer integration time isactually needed). So for GPS receiver design, the compromisemust be made in choosing the integration time to get a balance

© 2005 CASI 181


Figure 1. Test set up.

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between dynamic stress and interference mitigation ability. Inthe situation of 2 and 4 ms, if the problem of the large positionerror can be solved, both high accuracy and an acceptable levelof availability can be achieved. The tracking of one or moresatellites that might have sizeable range errors caused the largeposition error in the test. Figure 2 shows the pseudorangeerrors of different satellites using a 2 ms integration time.

In the test, an integration time of 2 ms allowed the trackingof six satellites. However, the coordinates could not converge totheir true values after a LS adjustment. One large pseudorangeerror existed and thus caused divergence of the estimates. Thiserror could however be removed using a fault detection andexclusion algorithm as sufficient measurement redundancy wasavailable in this case.

From Figure 2, it is obvious that PRN 20 was abnormal. Thepseudorange measurement is off from its true value by about600 km, which represents a 2 ms transmission time error. Thiserror is an integer millisecond error caused by the bitsynchronization error. A bit synchronization error will notaffect signal tracking, especially when the integration time is1 ms. That is to say, this error will continue to persist if thesignal is not re-acquired again. To eliminate this error, furtherinvestigation into the bit synchronization process is needed.

The histogram method was used in the software receiver forbit synchronization determination. In Figure 3, the histogramof PRN20 is shown. It can be seen that only cell 17 exceededthreshold 2, and no other cell reached threshold 1. Thus, the bitsynchronization process was declared successful and the data

bit transition point was assumed to be at cell 17 which was 2bins away from the true value of cell 15.

To improve bit synchronization performance, the followingtwo methods can be used:

• Use a soft threshold for threshold 2. This threshold isassociated with the noise floor. If the noise level is high, thethreshold will be lower. There will be a high possibility oftwo cells exceeding threshold 2 before one cell exceedsthreshold 1. The bit synchronization will restart again insteadof a bit being synchronized at an incorrect point.

• If 2 cells exceed threshold 2, go back to acquisition instead ofstarting bit synchronization again, because the requiredacquisition C/N0 is about 6 dB higher than tracking. The bitsynchronization error is most probably caused by aninaccurate acquisition result. If acquisition is re-started, theacquisition detection threshold can be adjusted through thenewly calculated noise floor. The acquisition result will bemore precise and there is a high possibility that the bitsynchronization result will be true.Figure 4 shows the bit synchronization result with

combination of the above two methods. The synchronized bittransition point is at the correct place. Figure 5 shows thepseudorange measurement errors when using the improved bitsynchronization procedure. After least-squares adjustment, theerrors of all the satellites converge to their zero mean.

Two scenarios, warm start and cold start, were defined andused herein. The definition of warm start is to start acquisitionin a “clean” environment, and interference is applied after the

182 © 2005 CASI


Integration time (ms) 1 2 4 6

Horizontal RMS error (m) 1 066 097 216 309 278 293 15Vertical RMS error (m) 222 133 50 234 888 585 6Tracked PRN 1, 14, 16, 25 1, 14, 16, 20, 25, 30 11, 14, 20, 25, 30 14, 16, 25, 30

Table 1. Impact of integration time on loop filter.

Figure 2. Pseudorange errors compared with true value.

Figure 3. Bit synchronization result for PRN 20.

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convergence of signal tracking. In contrast, cold start meansacquisition is started after the interference is applied. Thecomparison of the position errors between warm-start and cold-start scenarios in kinematic mode with FFT-based interferencemitigation and improved bit synchronization are shown inFigure 6.

It can be seen that, when the mitigation algorithm is applied,the RMS position error increases. This is to be expectedbecause the mitigation algorithm also removes part of thesignal energy when the interference frequency component isremoved. It in turn causes a distortion of the correlation peaksand larger pseudorange errors on all satellites and finallyincreases the position error.

In a warm start environment, no improvement in the positiondomain was observed when the mitigation algorithm wasapplied. On the contrary, the RMS horizontal error worsened.

This is because when the interference frequency componentwas excised, some of the true signal components were removedas well, since true signals always mix with interference to acertain degree. If the interference power level is not high, theinherent anti-jamming property of the spread spectrum systemcan mitigate the interference without any extra processing. Butwith mitigation, the resulting position solution may be slightlydegraded due to the signal loss upon frequency excision.

Under cold-start conditions, acquisition was performed in aninterference environment. Acquisition was more difficult thantracking since the signal strength required in acquisition is 6 dBhigher than that required in tracking. Thus, under cold-startconditions, the use of the mitigation algorithm is essential. Itcan be seen in Figure 6 that the maximum tolerance in coldstart was 18 dB without interference mitigation while itincreased to 22 dB with interference mitigation, a 4 dBimprovement in this case.

To investigate the stability of the algorithm, six independenttests were carried out under the same conditions. Figure 7shows the rate of success of these tests in obtaining positionsolutions under different interference levels when themitigation algorithm was applied. If no position could becalculated, or if the position error exceeded 50 m (meaning thatthe error is well above the normal value and the result is notreliable), the algorithm declared a failure.

Under cold-start conditions, when the J/S was no greaterthan 21 dB, the success rate in obtaining a position solution was100%. If the J/S increased to 22 dB or 23 dB, the success ratewas reduced to 33.3% or 16.7%, respectively. If the J/Scontinued to increase, no solution could be obtained. Themitigation algorithm is statistically stable when the J/S is nogreater than 21 dB in a cold start condition.

Under warm-start conditions, when the J/S was no greaterthan 26 dB, a 100% success rate in obtaining a solution couldbe achieved. If the J/S increased to 27 dB, the success rate wasreduced to 33.3%. If the J/S continued to increase, it was notpossible to obtain a solution. The mitigation algorithm wasstatistically stable when the J/S was no greater than 26 dB inthis case.

2. Implementation of Data Windowing Before FFTFigure 8 presents a zoomed section of the spectra obtained

with the 4 ms FFT processing of the non-windowed andwindowed data, respectively. It is clear that the windowingoperation increased the concentration of the interferencefrequency and reduced the number of frequency bins that needto be excised. Windowing reduced the amount of spectrum thatmust be excised and thus preserved more of the desired signalspectrum. As a result, less signal energy loss and less distortionin the correlation peak improved the accuracy of the positionestimation and increased the maximum tolerance of theinterference mitigation algorithm. The plots in Figure 9 showthe effect of windowing on position estimation.

From the position domain results shown in Figure 9, onecan say that the RMS position error decreases slightly whenusing the windowing operation. The RMS vertical error

© 2005 CASI 183


Figure 5. Pseudorange errors using the improved bit synchronizationprocedure.

Figure 4. Histogram of bit synchronization using the improvedprocedure.

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decreased from 4.9 to 3.7 m and the RMS horizontal errordecreases slightly from 5.6 to 5.4 m. The number of satellitesthat can be tracked increases from six to seven, which meansthat employing a windowing operation allowed all of thesimulated satellites to be tracked.

To check the performance stability, six independent testswere conducted under each J/S ratio condition, and the resultsare shown in Figure 10. When the J/S ratio was below 23 dB,the position error was within 10 m in all of the six tests. Whenthe J/S ratio increased to 23 dB, the position error alsoincreased. The position error varied between different tests dueto the random noise. In some tests, the RMS error reached20 m; however, the success rate in terms of obtaining

reasonable position solutions was still 100% with the windowoperation. Thus, the interference mitigation algorithm of thedata window, combined with frequency excision, wasstatistically stable when the J/S ratio was no greater than 23 dB.

184 © 2005 CASI


Figure 6. Comparison of mitigation results with CW interference.

Figure 7. Rate of successful position fixing.

Figure 8. Enlarged spectra of windowed and non-windowed data.

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When the J/S ratio increased to 24 and 25 dB, even with awindowing operation, the success rate reduced to 16.7%, whichmeans that only one out of six tests could achieve a solution.Under this circumstance, the mitigation result was notstochastically repeatable. The maximum tolerance of thismitigation algorithm with a data window is 23 dB under cold-start conditions. If the J/S ratio is greater than 25 dB, a positioncannot be obtained.

From the above analysis, the advantage of using a datawindow before FFT transform is apparent. Although theimprovement in position accuracy is slight, the receiversensitivity is improved, since more satellites can be effectivelytracked. Using of a data window together with the frequencyexcision algorithm can successfully mitigate interference of J/S

ratios values no greater than 23 dB under cold-start conditions.The maximum tolerance increases by 2 dB, as compared to atraditional FFT based mitigation algorithm, thus improving thesensitivity.

CONCLUSIONS

The FFT-based algorithm is effective to mitigate narrow-band CW interference with a certain power level. An adaptivedetection threshold that is a function of the standard deviationof the normalized spectrum and the correlator power output hasa better interference mitigation performance than that of thefixed detection threshold, as was suggested previously in theliterature.

© 2005 CASI 185


Figure 9. Effect of windowing on GPS position estimation.

Figure 10. Stochastic repeatability test results with data windowing.

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In high-dynamic mode (velocity of 80 m/s), by using a softthreshold for bit synchronization, an enhanced acquisitionmethod, and a careful selection of the integration time in theloop filter, the maximum mitigation capability for positionfixing can reach 21 dB in a cold-start environment. Windowingand overlapped processing before FFT transform have beenshown to be a good strategy to improve the performance of theFFT-based interference mitigation algorithm. Although theimprovement in position accuracy is slight, more satellites canbe effectively tracked. Data windowing along with FFTanalysis can successfully mitigate interference with a J/S ratioup to 23 dB under cold-start conditions, a 2 dB improvementcompared with traditional FFT based algorithms.

To achieve optimal interference mitigation performance,some key receiver parameters should be adjusted as well, suchas the adaptive selection of the coherent integration time, thetracking loop bandwidth and the integration time in the loopfilter for different interference levels and receiver dynamics.These parameters were chosen manually in this paper and theadaptive selection of these parameters according to differentdynamic stress and interference power level is recommendedfor future works.

REFERENCES

Badke, B., and Spanias, A.S. (2002). “Partial Band Interference Excisionfor GPS Using Frequency-Domain Exponents”. Proceedings of the 2002 IEEEInternational Conference on Acoustics, Speech, and Signal Processing,(ICASSP ‘02), Orlando, Florida, 13–17 May 2002. Vol. 4. Institute ofElectrical and Electronics Engineers, Piscataway, New Jersey. pp. 3936–3939.

Capozza, P.T., Holland, B.J., Li, C., Moulin, D., Pacheco, P., and Rifkin, R.(1999). “Measured Effects of a Narrowband Interference Suppressor on GPSReceivers”. Proceedings of the 55th Annual Meeting of the Institute ofNavigation on Navigational Technology for the 21st Century, 28–30 June1999, Cambridge, Massachusetts. CD-ROM. The Institute of Navigation,Alexandria, Virginia. pp. 645–651.

Capozza, P.T., Holland, B.J., Hopkinson, T.M., and Landrau, R.L. (2000).“A Single-Chip Narrow-Band Frequency-Domain Excisor for a GlobalPositioning System (GPS) Receiver”. IEEE J. Solid-State Circuits, Vol. 35,No. 3, March, pp. 401–411.

Cutright, C., Burns, J.R., and Braasch, M. (2003). “Characterization ofNarrow-Band Interference Mitigation Performance versus Quantization Errorin Software Radios”. Proceedings of the ION 59th Annual Meeting, 23–25June 2003, Albuquerque, New Mexico. CD-ROM. The Institute of Navigation,Fairfax, Virginia.

Dipietro, R.C. (1989). “An FFT Based Technique for Suppressing Narrow-Band Interference in PN Spread Spectrum Communications Systems”.Proceedings of the 1989 IEEE International Conference on Acoustics, Speech,and Signal Processing (ICASSP-89), Glasgow, Scotland, 23–26 May 1989.Vol. 2. Institute of Electrical and Electronics Engineers, New York, New York.pp. 1360–1363.

Harris, F.J. (1978). “On the Use of Windows for Harmonic Analysis with theDiscrete Fourier Transform”. Proc. IEEE, Vol. 66, No. 1, January, pp. 51–83.

Ketchum, J.W., and Proakis, J.G. (1982). “Adaptive Algorithms forEstimating and Suppressing Narrow-Band Interference in PN Spread-Spectrum Systems”. IEEE Trans. Commun. Vol. COM-30, No. 5, May,pp. 913–924.

Li, L., and Milstein, L. (1982). “Rejection of Narrow-Band Interference inPN Spread-Spectrum Systems Using Transversal Filters”. IEEE Trans.Commun. Vol. COM-30, No. 5, May, pp. 925–928.

Ma, C., Lachapelle, G., and Cannon, M.E. (2004). “Implementation of aSoftware GPS Receiver”. Proceedings of GNSS 2004, Session A3, LongBeach, California, 21–24 September 2004. The Institute of Navigation,Fairfax, Virginia.

Ndili, X. and Enge, P. (1997). “Receiver Autonomous InterferenceDetection”. In Proceedings of 53rd Annual Meeting of the Institute ofNavigation on the Future of Navigation: Facing the Challenges, 30 June – 2July 1997, Albuquerque, New Mexico. The Institute of Navigation,Alexandria, Virginia.

Peterson, B., Hartnett, J., Fiedler, R., and Nebrich, A. (1996). “FrequencyDomain Techniques for Fast GPS Acquisition and InterferenceDetection/Rejection”. J. Inst. Navig. Vol. 43, No. 3.

Proakis, J.G. (1996). “Interference Suppression in Spread SpectrumSystems”. Proceedings of the IEEE 4th International Symposium on SpreadSpectrum Techniques and Applications, Mainz, Germany, 22–25 September1996. Vol. 1. Institute of Electrical and Electronics Engineers (IEEE),Piscataway, New Jersey. pp. 259–266

Przyjemski, J., Balboni, E., and Dowdle, J. (1993). “GPS Anti-JamEnhancement Techniques”. Proceedings of the 49th ION Annual Meeting onFuture Global Navigation and Guidance, Cambridge, Massachusetts, 21–23June 1993. Institute of Navigation, Alexandria, Virginia. pp. 41–50.

Rash, G.D. (1997). “GPS Jamming in A Laboratory Environment”.Proceedings of the 10th International Technical Meeting of the SatelliteDivision of the Institute of Navigation, ION GPS-97, 16–19 September 1997,Kansas City, Missouri. The Institute of Navigation, Alexandria, Virginia.pp. 389–398.

Rifkin, R., and Vaccaro, J.J. (2000). “Comparison of Narrowband AdaptiveFilter Technologies for GPS”. IEEE 2000 Position Location and NavigationSymposium, San Diego, California, 13–16 March 2000. Institute of Electricaland Electronics Engineers (IEEE), Piscataway, New Jersey. pp. 125–131.

Wang, C., and Amin, M.G. (1998). “Performance Analysis of InstantaneousFrequency-Based Interference Excision Techniques in Spread SpectrumCommunications”. IEEE Trans. Signal Process. Vol. 46, No. 1, January,pp. 70–82.

Young, J.A., and Lehnert, J.S. (1994). “Sensitivity Loss of Real-Time DFT-Based Frequency Excision with Direct Sequence Spread-SpectrumCommunication”. Proceedings of the 1994 Tactical CommunicationsConference on Digital Technology for the Tactical Communicator, 10–12 May1994, Fort Wayne, Indiana. Vol. 1. Institute of Electrical and ElectronicsEngineers (IEEE), Piscataway, New Jersey. pp. 409–420.

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Adjoint-Based Sonic Boom Reduction forWing-Body Configurations in Supersonic

FlowSiva K. Nadarajah * Antony Jameson ** Juan Alonso **

* CFD LaboratoryDepartment of Mechanical Engineering

McGill University688 Sherbrooke Street West, Room 711

Montreal, QC H3A 2S6, Canada.

** Department of Aeronautics and AstronauticsStanford University

Durand Building, 496 Lomita MallStanford, CA 94305, USA


Received 12 April 2005

NOMENCLATURE

A flux Jacobian matrix

B boundary

C non-dimensional coefficient

c chord length

D domain

D discrete adjoint artificial dissipation flux

d artificial dissipation flux

E internal energy

F design variable

F Euler numerical flux vector

FF far field

f Euler flux vector

G gradient

G smoothed gradient

h numerical flux across cell interface

I cost function

n outward normal

p pressure

q flux velocity

R residual

S shape function

S face areas of computational cell

s arc length

T temperature

t time

U scaled contravariant velocity component

u velocity (physical domain)

w state vector

x coordinates (physical domain)

z altitude

α angle of attack

� adjustable constant for artificial dissipation scheme

Λ numerical spectral radius of the flux Jacobian matrix

λ spectral radius of the flux Jacobian matrix

ξ coordinates (computational domain)

© 2005 CASI 187


AbstractThis paper presents an adjoint method for the calculation ofremote sensitivities in supersonic flow. The goal is todevelop a set of adjoint equations and their correspondingboundary conditions to quantify the influence of geometrymodifications on the pressure distribution at an arbitrarylocation within the domain of interest, away from thesurface of the aircraft. First, this paper presents theformulation and discretization of the adjoint equations. Thespecial treatment of the adjoint boundary condition toobtain remote sensitivities is also discussed. Second, wepresent results that demonstrate the application of thetheory to a three-dimensional remote inverse designproblem using a supersonic business jet wing-bodyconfiguration.

RésuméDans cet article, nous présentons une méthode adjointe decalcul des sensibilités à distance dans le flot supersonique.L’objectif est la mise au point d’une série d’équationsadjointes et de leurs conditions aux limitescorrespondantes, afin de quantifier l’influence desmodifications de la géométrie sur la distribution de lapression à une position arbitraire dans le domaine d’intérêt,

continued on page 188

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ρ density

ψ Lagrange multiplier

ϖ weighting coefficients

SubscriptsI contributions associated with variation of the state

vector

II contributions associated with variation of the shapefunction

D drag

i, j, k cell indices

I contributions associated with variation of the statevector

II contributions associated with variation of the shapefunctions

max maximum index

NF near field

T target

W wall

+, – cells across the cell face

INTRODUCTION

A 2001 US National Research Council study (Committee onBreakthrough Technology for Commercial Supersonic Aircraft2001) concluded that, “… the sonic boom is the major barrier tothe development of the supersonic business jet and a major, butnot the only, barrier to the development of supersonic transportswith overland capability”. The Committee also determined thatthere was a potential market for at least 200 supersonic businessjets over a 10-year period. The 8–15 passenger jets willprobably fly at approximately Mach 1.8 with a range of 3000–4000 nautical miles (1 nautical mile = 1.852 ×103 m).

Concorde, a commercial supersonic aircraft built by Franceand Britain was in operation for more than 30 years. It cruisedat Mach 2.0 with a total range of 4090 nautical miles at60 000 ft (1 ft = 3.048 × 10�1 m) and consumed 6200 gallons offuel per hour (1 gallon = 4.54609 × 10�3 m3). The aircraftretired from commercial service on the 24 October 2003 due to

both commercial and technical factors. The new breed ofsupersonic transports must possess superior performancecharacteristics apart from low sonic boom capability, comparedto its predecessors to compete in the commercial jet industry.These include improvements in structures, aerodynamics, andpropulsion. In particular, the experience of NASA’s HSRprogram suggests that it should be possible to improve the lift-to-drag ratio from the value of 7.5 attained by the Concorde toaround 9 (Cliff et al., 2001). Designs of supersonic transportsof the future will benefit from multidisciplinary optimizationtechniques that were not available during the design andconstruction of aircraft like the Concorde.

Before the sonic boom reduction problem can be attempted,it is important to have the capability of calculating the sonicboom or ground pressure signature accurately. For typicalcruise altitudes required for aircraft efficiency, the distancefrom the source of the acoustic disturbance to the ground istypically greater than 50 000 ft. A reasonably accuratepropagation of the pressure signature can only be obtained withsmall computational mesh spacings that would render theanalysis of the problem intractable for even the largest parallelcomputers. An approach that has been used successfully in thepast is the use of near-to-far field extrapolation of pressuresignatures based on principles of geometrical acoustics andnon-linear wave propagation. These methods are based on thesolutions of simple ordinary differential equations for thepropagation of the near-field pressure signature to the ground.The Whitham F-function (Whitham, 1952) and Thomas’equivalent wave-form parameter method (Thomas, 1972) arecommon methods of choice.

Figure 1 is a schematic of the propagation of the aircraftpressure signature. “CFD Far Field” indicates the far-fieldboundary of the mesh. At a pre-specified distance below theaircraft and still within the CFD mesh, the location of a near-field plane can be seen. This plane is the effective interfacebetween the CFD solution and the wave propagation program.At the near-field plane, the flow solution wo is representedusing a number of parameters, M, which can be taken, forexample, as the number of mesh points on which the pressurewave-form has a value significantly different from the freestream. The lower portion of the domain between the CFD nearfield and the ground plane is where the pressure signaturepropagation method will be active. Given the conditions, wo,the propagation altitude, and the altitude-dependentatmospheric properties ρ(z), p(z), T(z), the propagation methodproduces a pressure signature at the ground plane we areinterested in, which can be used to determine any of a variety ofmeasures of sonic boom impact such as overpressures, risetime, impulse, etc.

Traditional methods to reduce the sonic boom signaturetarget aircraft weight reduction, increases in lift-to-drag ratio,better specific fuel consumption, etc. Seebass and Argrow(1998) revisited sonic boom minimization and provided adetailed study of sonic boom theory and figures of merit for thelevel of sonic booms. Diverse methods have been employed inthe design of low-boom aircraft configurations. The following

188 © 2005 CASI



à distance de la surface de l’avion. Cet article présente,premièrement, la formulation des équations adjointes etleur dicrétisation. Nous y discutons également dutraitement spécial de la condition aux limites adjointepermettant de dériver les sensibilités à distance. En secondlieu, nous montrons l’application de la théorie à unproblème inverse tridimensionnel de conception tenantcompte des effets à distance, visant la configuration dufuselage et des ailes d’un réacté d’affaires supersonique.

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are a selected number of papers on this topic. Orr et al. (2002)proposed a rather exotic concept to reduce the sonic boom.Their goal was to increase the apparent length of the aircraft byoff-axis volume addition. A swept forward keel placed normalto the Mach plane increased the apparent length and proved tobe effective in reducing the sonic boom strength. Komadina etal. (2002) evaluated twelve different configurations. Theground sonic boom signature, aircraft aerodynamics, massproperties, and flight performance were evaluated for all twelveconfigurations using empirical methods. The two mostpromising concepts were then chosen and higher fidelitymethods were used to compute the vehicle performance andcharacteristics. Argrow et al. (2002) argued that most sonicboom minimization techniques shape the aircraft equivalentbody of revolution in the vertical plane and not the truegeometry to reduce the far-field pressure signatures. Theauthors used a combination of linearized and non-linearaerodynamic theories, computational fluid dynamics based onthe Euler equations, and a gradient-based method to optimizethe shape of the aircraft. The design variables were nose tiltangle and canard and wing dihedral, sweep, and twist angles.

Through the support of the DARPA QSP Program, advancedalgorithms for the design and optimization of quiet supersonicplatforms have been developed during the last several years.DARPA’s vision for the project was to develop conceptualaircraft designs that produced initial overpressures of 0.3 psf(1 psf = 4.88243 kg/m2), while cruising at Mach 2.5 with arange of 6000 nautical miles and with a weight around 100 000lbs (1 lb = 4.53592 × 10�1 kg). This is an ambitious reduction in

the initial overpressure compared to the Concorde’s 1.5–2.0 psf. Our experience has indicated that large reductions inthe ground peak pressure cannot be achieved with minor shapemodifications of the baseline configuration. Alternative designmethods such as genetic algorithms have been used in a multi-level design environment to reach the neighborhood of theoptimum design before switching over to a gradient-basedmethod to refine the design. Promising results have beenachieved by using genetic algorithms in a linear methodenvironment. Non-linear methods are needed to meet severalgoals: first, to verify if not improve the results of the linear-based method; second, to improve the design by using thetechniques of optimal control; lastly, to allow the introductionof more objective functions to improve the final design.

The concept presented in this work proposes the idea that theground pressure signature could be adjusted by modifying theaircraft surface geometry to control the near-field pressuredistribution. It is not at all clear what type of changes to thesurface geometry would produce near-field pressuredistributions whose propagation to the ground would generatesonic booms with lower peaks. It appears, however, that theproblem might be separated into two parts: first, theidentification of near-field pressure distributions that are bothfeasible and lead to acceptable ground signatures; and second,the design of the surface geometry such that it will produce thedesired near-field pressure distribution.

Traditional adjoint implementations have been aimed atreducing a cost function computed from the pressuredistribution on the surface that is being modified. In this case,

© 2005 CASI 189


Figure 1. Schematic of the propagation of the aircraft pressure signature.

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however, we would like to obtain the sensitivity of pressuredistributions at points located at a distance from the surfacewhere the geometry is being modified. This type of sensitivitycalculation has not been attempted before in aerodynamicshape design, but is closely related to inverse-scatteringproblems in acoustics and electromagnetics. In such anapproach, a target near-field pressure distribution must bespecified. The cost function may then be chosen as the integralof the square of the difference between the current and targetnear-field pressure distribution. The gradient of the costfunction with respect to the design variables such as theposition of the surface mesh points is calculated, and adirection of improvement is obtained from an optimizationalgorithm. The procedure is repeated until a new aircraftsurface geometry is produced that provides a near-fieldpressure signature that approaches the specified target nearfield pressure distribution, provided that it is actuallyrealizable. The design procedure should also include otherobjective functions and constraints to maintain or improveother aircraft performance parameters such as lift-to-drag ratioand the total amount of lift. The possibility that the adjointmethod could be adapted to solve the remote inverse problemwas demonstrated by Nadarajah et al. (2001) for a two-dimensional internal flow problem. The method was thenextended for three-dimensional wing and wing–bodyconfigurations in supersonic flow by Nadarajah et al. (2002a,2002b, 2002c).

The issue of choosing a near-field signature to produce adesired ground signature was addressed by Alonso et al.(2002). The work accomplished in this research focuses oncontrolling the near field signature, and not the groundsignature. A future extension of the method would be to includethe wave propagation program in the design procedure, suchthat the ground pressure signature is considered as the targetpressure distribution instead of the near-field pressuredistribution.

THE REMOTE INVERSE DESIGN PROBLEM

USING CONTROL THERAPY

The aerodynamic properties that define the cost function arefunctions of the flow field variables, w, and the physicallocation of the boundary, which may be represented by thefunction S.

Suppose that the performance is measured by a cost function

I w w= +∫ ∫ϖ ϖξ ξ1 2M S B N S BB B

( , ) ( , )W NF

d d (1)

containing both wall boundary (BW) and near field boundary(BNF) contributions, where dBξ includes the surface and near-field elements in the computational domain, while ϖ1 and ϖ 2are the weighting coefficients. The coordinates ξi that describethe fixed computational domain are chosen so that eachboundary conforms to a constant value of one of these

coordinates. In general, Mand N will depend on both the flowvariables w and the metrics S defining the computational space.

The design problem is now treated as a control problemwhere the boundary shape represents the control function,which is chosen to minimize I subject to the constraints definedby the flow equations. A shape change produces a variation inthe flow solution, δw, and the metrics, δS, which in turnproduce a variation in the cost function

δ ϖ δ ϖ δξ ξI w w= +∫ ∫1 2M S B N S BB B

( , ) ( , )W NF

d d (2)

with δM = [Mw]I δw + δMII , δN = [Nw]I δw + δNII ,where we use the subscripts I and II to distinguish betweenthe contributions associated with the variation of the flowsolution δw and those associated with the metric variationsδS. Thus, [Mw]I and [N w]I represent ∂ ∂M / w and ∂ ∂N w/with the metrics fixed, while δMII and δNII represent thecontribution of the metric variationsδS toδMandδN with theflow solution fixed. The weak form of the Euler equations forsteady flow is

∂∂

=∫ ∫ψξ

δ ψ δT

Td di

i i iF n FD B

D B

where the test vector ψ is an arbitrary differentiable function,and ni is the outward normal at the boundary. If a differentiablesolution, w, for this equation is obtained, then it can beintegrated by parts to give

ψξδT d

∂∂

=∫i

iFD

D 0 (3)

Since this is true for any ψ, the differential form can berecovered. Here, δFi can be split into contributions associatedwith δw and δS using a similar notation

δ δ δF F w F F Sf

wi iw i iw ij

j= + =∂∂

[ ] [ ]I II Iwhere

The domain can then be split into two parts as shown inFigure 2. First, the near-field domain (D1) whose boundariesare the wing surface and the near-field boundary plane. Second,the far-field domain (D2), which borders the near-field domainalong the near-field boundary plane and the far-field boundary.Thus equation (3) can be written as

ψξδ ψ

ξδξ ξ

T T

1 2

d d∂∂

+ ∂∂

=∫ ∫i

ii

iF FD D

D D 0

190 © 2005 CASI


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This may be integrated by parts to give

n F Fi ii

iψ δ ψξ

δξ ξT

T

W 1

d dB D

D D∫ ∫− ∂∂

+ − − ∂∂

=∫ ∫n F Fi ii

i( )–ψ ψ δ ψξ

δξ ξ+ T

T

NF 2

d dB D

D D 0 (4)

where ψ+ and ψ– are the values of the Lagrange Multiplier, ψ,above and below the boundary. Since the left-hand expressionequals zero, it may be subtracted from the variation in the costfunction (2) to give

δ ϖ δ − ψ δ

ϖ δ − ψ ψ ) δ

ξI n F

n F

i i

i i

=

+

∫∫

[ d

[ d

1T

2+ –

W

NF

B

B

M B

N B

]

( – ]Τξ

+ ∂∂

+ ∂∂∫ ∫∫ψ

ξδ ψ

ξδξ ξ

T T

1 12

d di

ii

iF FD DD

D D (5)

Sinceψ is an arbitrary differentiable function, it may be chosenin such a way that δI no longer depends explicitly on thevariation of the state vector δw. The gradient of the costfunction can then be evaluated directly from the metricvariations without having to re-compute the variation δwresulting from the perturbation of each design variable.

Comparing equations (2) and (4), the variation δw may beeliminated from Equation (5) by equating all field terms withsubscript I to produce a differential adjoint system governingψ

∂∂

⎡

⎣⎢

⎤

⎦⎥ =

+

ψξ

T0 in

iiwF[ ]I

D DD

1 2

(6)

The corresponding wall and near-field adjoint boundaryconditions are produced by equating the subscript I boundaryterms in equation (5) to produce

n Fi iw wψ ϖTWon[ ]I M B= 1 (7)

n Fi iw w( ) [ ]ψ ψ ϖ+ – TNFon− =I N B2 (8)

The remaining terms from equation (5) then yield a simplifiedexpression for the variation of the cost function, which definesthe gradient

δ ϖ δ ψ δ ξI n Fi i= −∫ { dW

TB II IIM B1 [ ] }

+ − −∫ +{ d2NF

Tϖ δ ψ ψ δ ξB II IIN Bn Fi i( ) [ ] }–

+ ∂∂

⎧⎨⎩

⎫⎬⎭+∫ ψ

ξδ ξ

T

1

di

iF[ ]IID DD

2

(9)

The details of the formula for the gradient depend on the way inwhich the boundary shape is parameterized as a function of thedesign variables and the way in which the mesh is deformed asthe boundary is modified. The boundary conditions satisfied bythe flow equations restrict the form of the left-hand side of theadjoint boundary conditions (7) and (8). Consequently, theboundary contribution to the cost functionsMand N cannot bespecified arbitrarily. Instead it must be chosen from the class offunctions that allow cancellation of all terms containing δw inthe boundary integral of equation (5). In this research, the costfunction is the weighted sum of the drag coefficient and theSobolev norm of the difference between the current and targetremote pressure distributions. From equation (1),Mand N canbe defined as

M S( , ) =1c

pw Cy x∂∂

− ∂∂

⎛⎝⎜

⎞⎠⎟

ξα

ξαcos sin

and

N S(w, ) =12

T( )p p− 2

The cost function can then be written as

Ic

y x= ∂∂

− ∂∂

⎛⎝⎜

⎞⎠⎟∫ϖ

ξα

ξα ξ1

1Cp

W

dB

Bcos sin

+ −∫ϖ ξ221

2( )p pT

NF

dB

B

and further simplified to

© 2005 CASI 191


Figure 2. Near-field and far-field domains.

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192 © 2005 CASI


I C p p= + −∫ϖ ϖ ξ1 221

2D T

NF

d( )B

B (10)

The values of the weighting coefficients are selected basedon the relative magnitude of the gradients of the dragminimization and the remote inverse cost functions. The remoteinverse gradient is typically an order of magnitude smaller thanthe gradient due to drag minimization. Therefore, the weightsare chosen to increase the magnitude of the gradient from theremote inverse cost function. In practice, larger weights areused for the remote inverse gradient, since the primary designobjective is to reduce the near-field pressure signature. Thedisadvantage of this approach is that the weights must bechosen at the beginning of the design process and if the userdoes not have prior knowledge of the magnitude of thegradients, then generally an initial guess is taken. The weightsare altered for subsequent runs.

An alternative method for problems with more than oneobjective function is to develop separate adjoint equations, onefor each objective function. Both gradients are then calculatedseparately, multiplied by weights, and summed. A direction ofimprovement is then based on the new gradient. This methodhas the advantage that the user is better equipped withknowledge regarding the difference in magnitude between thetwo gradients. Appropriate weights are chosen to achieve thedesired compromise. A disadvantage is the need to calculate aseparate adjoint solution for each objective function.

In this work, we prefer to use a composite cost function,since we had a priori knowledge regarding the magnitude of thegradient contribution from the remote inverse and the dragminimization cost functions.

REMOTE INVERSE DESIGN VIA THE DISCRETE

ADJOINT METHOD

The remote inverse adjoint method can be formulated eitherby the continuous or discrete adjoint method. The previous

section offers an overview of the continuous adjoint approach.The continuous remote adjoint boundary condition wouldrequire an update of the adjoint variables at the near-field cells.Since the near-field cells, do not lie in a specific row or columnof cells in the vicinity of the near field, but rather as a group ofcells scattered along the near-field plane, the implementation ofthe continuous adjoint approach would prove to be far tocomplex. However, the discrete adjoint approach proved to beless complicated, and offered a manageable approach to theimplementation of the remote adjoint boundary conditionproblem.

The discrete adjoint equation is obtained by applying controltheory directly to the set of discrete field equations. Toformulate the discrete adjoint equation, we first define the costfunction I as such,

I C p p s= + −∑ϖ ϖ1 221

2D T

NF

( ) ∆

where CD is total wing drag coefficient, p is the current nearfield pressure, and pT is the target near-field pressure. Next, wetake a variation of the residual term, which can be written as

δ δ δ δR w h d hi j ki j k i j k i j k

( ) , ,, , , , , ,

= − ++ − +1

2

1

2

1

2

− + −− + −

δ δ δh h hi j k i j k i j k, , , , , ,

1

2

1

2

1

2

(11)

with δ hi+1/2,j,k = δ fi+1/2,j,k – δ di+1/2,j,k, where f is the convectiveflux and d is the artificial dissipation flux. We then pre-multiplythe variation of the discrete residual by the Lagrange Multiplierand sum the product over the computational domain to producethe following:

ψ δi j k, ,T

maxmaxmax

k

k

j

j

i

i

i j kR w===∑∑∑

222

( ) , ,

The variation of the cost function, δI, can then be augmented by the product of the variation of the discrete governing equationand the Lagrange Multiplier ψ , ,

Τi j k.

δ ϖ δ ϖ δ ψI C p p p sk

k

j

j

i

= + − +∑ ∑∑===

1 222

D TNF

, ,T

maxmax

( ) ∆ i j k2

i

i j kR wmax

∑ δ ( ) , , (12)

To eliminate δw from equation (12), terms multiplied by the variation δwi,j,k of the discrete flow variables are collected andequated to zero. The following is the resulting discrete adjoint equation:

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Vt

S A S Ai j

i j k i j k i j k i j k=∂∂

= +− −

ψ ,

, , , , , , , ,

12 1

2

1

2

11 1T

12 2T

13 3T+

⎛⎝⎜

⎞⎠⎟ −

⎡

⎣⎢⎢ −

−S Ai j k i j k

i j k i j k1

2

1, , , ,

( ), , , ,ψ ψ

+ + ++ + +

S A S A Si j k i j k i j k i j k i j

11 1T

12 2T

131

2

1

2

1

2, , , , , , , , , , , ,

( ), . , ,k i j k

A i j k i j k3T⎛

⎝⎜

⎞⎠⎟ −+ψ ψ1

+ + ++ + +

S A S A Si j k i j k i j k i j k i j

21 1T

22 2T

23, , , , . , , , .

1

2

1

2

1

2, , ,

( ), , , ,k i j k

A i j k i j k3T⎛

⎝⎜

⎞⎠⎟ −+ψ ψ1

+ + +S A S A Si j k i j k i j k i j k i j

21 1T

22 2T

23, – , , , . – , , , , –

1

2

1

2

1

2, , ,

( ), , , – ,k i j k

A i j k i j k3T⎛

⎝⎜

⎞⎠⎟ −ψ ψ 1

+ + ++ + +

S A S A Si j k i j k i j k i j k i j k

31 1T

32 2T

33, , , , , , , , , ,

1

2

1

2

1

2

1Ai j k

i j k i j k3T

, ,( ), , , ,

⎛⎝⎜

⎞⎠⎟ −+ψ ψ

+ + +S A S A Si j k i j k i j k i j k i j k

31 1T

32 2T

33, , – , , , , – , , , , –

1

2

1

2

1

2

1Ai j k

i j k i j k3T

, ,( ), , , , –

⎛⎝⎜

⎞⎠⎟ −ψ ψ

+ − −D D D D Di i i,j i,j i,j,k+

1

2–

1

2+

1

2–

1

2+

1

2

+ +, , , , , ,j k j k k k

− Di,j,k–

1

2

(13)

Here, V is the cell area and

Di+ i+

1

2

1

2

T T

, , , , ,, , , ,( – )

j k i j k j ki j k i j k

iv v= −+

+1

2

21Λ ψ ψ

++ +3

2

42 1

j k j ki j k i j k

,

( )

, ,, , , ,( – )Λ

i+3

2

T Tψ ψ

+ −+

+2 1

2

41 1

2

v vi j k j k

i j k i j ki j k,

( )

, ,, , , ,

– ,( – )Λ

i+1

2

T Tψ ψ ( )

, ,, , – , ,( – )4

1Λi–

1

2

T T

j ki j k i j kψ ψ

is the discrete adjoint artificial dissipation term thatcorresponds to the discretization of the inviscid flow equationsby the Jameson–Schmidt–Turkel (JST) scheme (1981). Thedissipation coefficients ν(2) and ν( )4 are functions of the flowvariables, but, to reduce complexity, they are treated asconstants. The effect of this partial discretization has beenexplored by Nadarajah and Jameson (2000, 2001).

Discrete Adjoint Boundary ConditionTo develop the discrete adjoint boundary condition for the

calculation of remote sensitivities for supersonic flow, theδwNFterm from the discrete cost function is added to thecorresponding term from equation (12). The discrete boundarycondition appears as a source term in the adjoint fluxes. At theNF cell, the source term ΦNF for inverse design is added toequation (13) and can be written as,

Φ ∆NF T NF NF= − −ϖ δ1( )p p s p

The wall boundary condition appears as source terms in theadjoint fluxes along the cells above the wall. The derivation ofthis boundary condition was explored by Nadarajah andJameson (2000). For a first-order dissipation scheme, the

discrete adjoint equations are completely independent of theco-state variables in the cells below the wall. However, if weuse the blended first-and-third-order scheme, these flowvariable values are required. A simple zeroth-orderextrapolation across the wall has produced good results, asshown by Nadarajah and Jameson (2000).

OPTIMIZATION PROCEDURE

In this paper, the inverse design boundary condition isapplied to the near field, while sensitivity derivatives or thegradient are calculated on the airfoil surface. The gradient forthe discrete adjoint is obtained by perturbing each point on thelower wall. Once the gradient G has been determined, it can beused to drive a variety of gradient-based search procedures. Thesearch procedure used in this work is a descent method inwhich small steps are taken in the negative gradient direction.Let F represent the design variable, andG the gradient. Then animprovement can be made with a shape change

δ λF G= −

© 2005 CASI 193


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However, it is better to replace the gradient G by a smoothedvalue G in the descent process. This acts as a preconditionerthat allows the use of much larger steps and ensures that eachnew shape in the optimization sequence remains smooth. Toapply smoothing in the ξ1 direction, the smoothed gradient Gmay be calculated from a discrete approximation to

G G G G– , = 0 at end points∂∂

∂∂

=ξ

θξ1 1

where θ is the smoothing parameter. If the modification isapplied on the surface ξ2 = constant, then the first-order changein the cost function is

δ δ ξ

λξ

θξ

ξ

λ θξ

I = −

= − − ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟

= − + ∂∂

∫∫

∫∫

G F

G G G

G G

d

d

1

1 11

2

11d

⎛⎝⎜

⎞⎠⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

<

∫∫2

0

ξ

again guaranteeing an improvement unless G G= = 0 andassuring an improvement if λ is sufficiently small and positive.

In some problems, it turns out that the Hessian can berepresented as a second-order differential operator, so that witha proper choice of the smoothing parameter, the methodbecomes the Newton method. Search methods were intensivelyevaluated in a recent study by Jameson and Vassberg (1999),and it was verified that these sample problems (which may havea high linear content) could be solved with a number of searchsteps independent of the number of design variables.

RESULTS

This section presents the results of remote inverse and dragminimization for wing–body configurations in supersonic flow.The objective is to reduce the peak pressure at the near-fieldplane and thus reduce the ground signature peak. Viscouseffects are likely to be very small in these examples, so it issufficient to use the Euler equations. The calculations wereperformed with the new SYN88-MBC multiblock code thattakes advantage of the FORTRAN 90/95 derived data typearchitecture. The flow solver is augmented with an adjointsolver and shape modification routines to allow for automaticshape optimization.

Wing–Body Configuration: Sonic Boom Reduction,Without Lift Constraint, Wing Redesign Only

The wing–body supersonic business jet configuration wassized to accommodate between 6 to 8 passengers with a grosstake-off weight of 100 000 lbs and a fuselage length of 100 ft.The supersonic flight condition at which all designs were

calculated is Mach 1.5. Figure 3 shows the wing–bodyconfiguration. The fuselage is cylindrical and the maximumdiameter occurs at 31% of the fuselage length measured fromthe nose. The wing is a biconvex wing with a 7.125° leadingedge sweep, an aspect ratio of 3.0, and a taper ratio of 0.218.The root airfoil is a 3% thick biconvex airfoil and the tip is1.5% thick. The biconvex profile in the center sections wasobtained by interpolating between the root and tip. The airfoilswere constructed to accommodate deep spars at the 10% and80% chord locations. The baseline wing does not havegeometric twist. The computational mesh has eight blocks with193 × 49 × 33 nodes on a C–H grid. The fuselage has 25 pointsin the cross-streamwise-direction and 144 points in thestreamwise-direction. The wing contains 97 points in thestreamwise-direction and 17 sectional cuts in the spanwise-

194 © 2005 CASI


Figure 3. Business jet wing–body configuration: biconvex wing,193×49×33 C–H grid, 8 blocks.

Figure 4. Convergence history of the flow solver.

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direction. The far-field boundary is located at approximately 5root chord lengths. In Figure 4, convergence history of theflow solver is illustrated for the Mach 1.5 test case.

To obtain an accurate representation of the far-field pressuresignature through a wave propagation method, a precise near-field pressure distribution must be acquired. The location of thenear-field boundary is largely influenced by the number ofmesh points. A large number of mesh points would result insmaller mesh spacings in the region of the near field andproduce a pressure distribution of greater accuracy, however,the computational cost would increase dramatically. If locatedtoo close to the aircraft, a proper far-field representation of theaircraft pressure signature may not be feasible. In this work, thenear-field boundary is located at approximately one root chordlength below the aircraft on the 193 × 49 × 33 C–H mesh.

To illustrate the possibility of sonic boom reduction, a targetpressure distribution was obtained by re-scaling the initial near-field pressure distribution. Ultimately, this step will be replacedby a method that produces a target near-field pressure basedupon the desired ground pressure signature. The target pressurewas obtained by taking the result of SYN88-MBC at a flightcondition of M∞ = 1.5 and scaling the resulting pressuredistribution to 40% of its original value.

The objective function is the integral of the differencebetween the current and target near-field pressures. Theminimum permissible thickness constraint is imposed at eachchordwise cut between the 10% and 80% chord locations at theend of each design cycle. Points from the leading edge upto the10% chord location and from the 80% chord location to thetrailing edge are not constrained and free to move in anydirection. The lift coefficient in this case is not constrained. Inthis design, the design variables are the locations of all of thepoints on the surface of the wing. Therefore, only the secondpeak in the near-field pressure profile will be expected tochange. The flow is calculated at Mach 1.5 at a fixed angle ofattack of 1.62°.

Figure 5 illustrates the initial and final root airfoil profiles.The lower surface of the final airfoil contains a slightly largerexpansion region when compared with the original biconvexairfoil. It is this modification that allows the near-field wingpeak pressure (second peak) to be reduced. The largerexpansion region weakens the strength of the leading edgeattached shock in the near-field region. Figure 6 shows theinitial near-field pressure in blue (�) and the target pressure inblack (+). After 50 design cycles, the final near-field pressuredistribution is obtained and illustrated as the red (*) line. Thewing peak pressure has been reduced by 40%. The largemodifications on the wing upper surface are a result of thethickness constraint. Since the lift coefficient was notconstrained, CL reduced from 0.1 to 0.073. The baseline wingdrag coefficient is 0.00568 and the final wing drag increasedslightly to 0.00582. Even if drag due to lift has decreased due to

© 2005 CASI 195


Figure 6. Target, initial, and final near-field pressure distribution after50 design cycles.

Figure 7. Convergence of the objective function.

Figure 5. Initial and final root airfoils at M∞ = 1.5.

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the decrease in the lift coefficient, the stronger attached leadingedge shock has increased the wing wave drag.

In Figure 7, the convergence of the objective function isshown. Here, the objective function as defined in equation (10)is computed at each design cycle. For this test case, ϖ1 is set tozero and thus the objective function only represents the norm ofthe difference between the current and target near-fieldpressure. The near-field pressure quickly reaches the vicinity ofthe final result within 20 design cycles. A total of 50 designcycles are computed to ensure that the objective function hasreached its minimum value.

The complete shape optimization procedure for sonic boomreduction requires the determination of desirable ground boomsignatures. In Figure 8, we show the initial and final groundsignature profiles. The PC Boom software for far-fieldpropagation developed by Wyle Associates was used tocalculate the ground signatures. The results clearly indicate that

arbitrarily scaled near-field signatures may not result in moredesirable behavior at the ground.

Wing–Body Configuration: Sonic Boom Reduction, LiftConstraint

We now repeat the same design case but with the followingthree changes: first, the lift coefficient is constrained at 0.1.Second, gradients are calculated for points on the surface of thefuselage and thus allowed to be modified. Third, the objectivefunction is the weighted sum of the drag coefficient and integralof the difference between the current and target near-fieldpressures, where ϖ1 is the weight on the drag coefficient andϖ 2 is the weight on the remote inverse cost function. In thiscase, the drag coefficient weight is, ϖ1 = 0.005 and the remoteinverse cost function weight is set to ϖ 2 = 1.

The value of the lift coefficient is maintained by adjustingthe angle of attack to attain the desired lift coefficient of 0.1.The wing thickness constraint is imposed in the same manneras the previous case.

Figure 9 illustrates the baseline and optimized airfoil.Figure 10 shows the target, initial, and final near-field pressuredistributions. The desired target pressure distribution is notachieved in contrast with the unconstrained case illustrated inFigure 6. In this case, there is a struggle between the near-fieldpeak pressure reduction versus maintenance of constant lift.Each design cycle, produces a shape modification that shifts thenear-field pressure distribution towards the target pressure.Unfortunately, this also causes a reduction in the liftcoefficient. This must be compensated by an increase in theangle of attack to maintain the total lift coefficient, which inturn leads to an increase in the near-field peak pressure. After50 design cycles, the solution converges to the (*) line inFigure 10. The final fuselage peak pressure has been reducedto almost 18% its original value and the wing peak pressure isreduced by 22%.

To maintain the lift coefficient, the angle of attack wasincreased from 1.62° to 2.39°. The wing drag increased slightly

196 © 2005 CASI


Figure 8. Initial and final ground signatures after 50 design cycles.

Figure 9. Initial and final root airfoils at M∞ = 1.5. (a) Wing root section and (b) mid-span section.

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from 0.00568 to 0.00574. In an alternate test case, where thedrag coefficient weight was set to zero, the near-field peakpressure for both the fuselage and wing were reduced by 18%and 25%, however, the wing drag coefficient increased to0.00610. Table 1 contains a comparison of the two designcases. Table 1 clearly shows that a cost function that does notinclude the drag coefficient will result in larger reductions inthe wing near-field peak pressure. However, in a multi-

disciplinary design environment, it is critical that otherimportant parameters are kept within acceptable amounts and atrade-off between the various design goals are met. Table 1clearly shows that a composite cost function that includes thedrag coefficient is unable to reduce both the near-field peakpressure and drag coefficient but it is able to reduce the peakpressure while maintaining the wing drag coefficient. A moredetailed study of the effect of the weights on the cost functionsis presented by Nadarajah et al. (2002).

In Figure 12, both the initial and final pressure contours areplotted. The majority of the changes in the shape are localizedaround the lower surface wing–fuselage intersection. Thelarger expansion regions on the lower surface of the wing areillustrated in these plots by the shorter red region (compression)and the longer green-orange region. This has the effect ofweakening the strength of the shock, and thus reducing the peakof the near-field pressure. Figure 13 illustrates the initial andfinal fuselage mesh. The larger expansion region on theunderside of the fuselage around the wing–fuselageintersection is clearly due to the increase in the fuselagecurvature.

CONCLUSIONS

The results presented in this paper demonstrate thefeasibility of remote inverse calculations using the adjointmethod. An application to the sonic boom minimizationresulted in a 40% reduction in the near-field peak pressure forthe unconstrained biconvex wing. In the constrained problem,the fuselage peak pressure was reduced by 18% and the wingpeak by 22%. It proved highly beneficial to use a compositecost function consisting of the sum of the weighted remoteinverse and drag minimization cost functions, resulting in finaldesigns that had a reduction in the peak pressure whilemaintaining constant inviscid drag. Cases with no dragcoefficient added to the integral of the near-field pressure

© 2005 CASI 197


Figure 10. Target, initial, and final near-field pressure distribution after 50 design cycles. (a) Wing root section and (b) mid-span section.

Figure 11. Sonic boom reduction: initial and final ground signaturesafter 50 design cycles. M∞ = 1.5, α = 2.39°, CL = 0.1.

CaseFuselage peakreduction

Wing peakreduction

Wing,CD

Baseline 0.00568Remote inverse 18% 25% 0.00610Drag and remote inverse 18% 22% 0.00574

Table 1. Near field peak pressure reduction and wing drag coefficientfor various design cases.

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difference in the objective function saw an increase in the dragcoefficient.

AcknowledgmentsThis research has benefited greatly from the generous

support of the AFOSR under grant number AF F49620-98-1-022 and DARPA QSP Program under grant number MDA972-01-2-0003.

ReferencesAlonso, J.J., Jameson, A., and Kroo, I. (2002). “Advanced Algorithms for

Design and Optimization of Quiet Supersonic Platforms”. Proceeedings of the40th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 14–17January 2002. American Institute of Aeronautics and Astronautics, Reston,Virginia. AIAA Pap. 02-0144.

Argrow, B., Farhat, C., Maute, K., and Nikbay, M. (2002). “A ShapeOptimization Methodology for Reducing the Sonic Boom Initial PressureRise”. Proceedings of the 40th AIAA Aerospace Sciences Meeting and Exhibit,Reno, Nevada, 14–17 January 2002. American Institute of Aeronautics andAstronautics, Reston, Virginia. AIAA Pap. 02-0145.

Cliff, S.E., Reuther, J.J., Saunders, D.A., and Hicks, R.M. (2001). “Single-point and Multipoint Aerodynamic Shape Optimization of High-Speed CivilTransport”. J. Aircr. Vol. 38, No. 6, November-December, pp. 997–1005.

Committee on Breakthrough Technology for Commercial SupersonicAircraft. (2001). “Commercial Supersonic Technology: The Way Ahead”.National Research Council (US), National Academy Press, Washington, D.C.Tech. Rep., March.

Jameson, A., and Vassberg, J.C. (1999). “Studies of Alternative NumericalOptimization Methods Applied to the Brachistochrone Problem”. OptiCON99.

Jameson, A., Schmidt, W., and Turkel, E. (1981). “Numerical Solutions ofthe Euler Equations by Finite Volume Methods with Runge-Kutta TimeStepping Schemes”. AIAA Pap. 81-1259, January.

198 © 2005 CASI


Figure 12. Pressure contour at M∞ = 1.5. (a) Initial and (b) final.

Figure 13. Fuselage mesh. (a) Initial and (b) final.

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http://pubs.casi.ca/action/showImage?doi=10.5589/casj5104fi&iName=master.img-077.png&w=340&h=172

Komadina, S., Drake, A., and Bruner, S. (2002). “Development of a QuietSupersonic Aircraft with Technology Applications to Military and CivilAircraft”. Proceedings of the 40th AIAA Aerospace Sciences Meeting andExhibit, Reno, Nevada, 14–17 January 2002. American Institute ofAeronautics and Astronautics, Reston, Virginia. AIAA Pap. 02-0519.

Nadarajah, S., and Jameson, A. (2000). “A Comparison of the Continuousand Discrete Adjoint Approach to Automatic Aerodynamic Optimization”.Proceedings of the 38th AIAA Aerospace Sciences Meeting and Exhibit, Reno,Nevada, 10–13 January 2000. American Institute of Aeronautics andAstronautics, Washington, D.C. AIAA Pap. 2000-0667.

Nadarajah, S., and Jameson, A. (2001). “Studies of the Continuous andDiscrete Adjoint Approaches to Viscous Automatic Aerodynamic ShapeOptimization”. A Collection of Technical Papers: 15th AIAA ComputationalFluid Dynamics Conference and Exhibit, Anaheim, California, 11–14 June2001. American Institute of Aeronautics and Astronautics, Reston, Virginia.AIAA Pap. 2001-2530.

Nadarajah, S., Jameson, A., and Alonso, J.J. (2001). “An Adjoint Methodfor the Calculation of Non-collocated Sensitivities in Supersonic Flow”.Proceedings of the 1st MIT Conference on Computational Fluid and SolidMechanics, Cambridge, Massachusetts, 12–15 June 2001. Vol. 2. Edited byK.J. Bathe. Elsevier, New York, New York. pp. 921–925.

Nadarajah, S., Jameson, A., and Alonso, J.J. (2002a). “An Adjoint Methodfor the Calculation of Remote Sensitivities in Supersonic Flow”. Proceedingsof the 40th AIAA Aerospace Sciences Meething and Exhibit, Reno, Nevada,14–17 January 2002. American Institute of Aeronautics and Astronautics,Reston, Virginia. AIAA Pap. 2002-0261.

Nadarajah, S., Jameson, A., and Alonso, J.J. (2002b). “Sonic BoomReduction using an Adjoint Method for Wing-Body Configurations inSupersonic Flow”. Proceedings of the 9th AIAA/ISSMO Symposium onMultidisciplinary Analysis and Optimization Conference, Atlanta, Georgia, 4–6 September 2002. Vol. 2. American Institute of Aeronautics and Astronautics,Reston, Virginia. AIAA Pap. 2002-5547.

Nadarajah, S.K., Kim, S., Jameson, A., and Alonso, J.J. (2002c). “SonicBoom Reduction using an Adjoint Method for Supersonic Transport AircraftConfiguration”. Proceedings of the IUTAM Symposium Transsonicum IV,Göttingen, Germany, 2–6 September 2002. Edited by H. Sobieczky. KluwerAcademic Publishers, Dordrecht, The Netherlands. p. 355.

Orr, M., Mozingo, J., Marconi, F., Bowersox, R., and Schetz, J. (2002).“Sonic Boom Alleviation using Keel Configurations”. Proceedings of the 40thAIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 14–17 January2002. American Institute of Aeronautics and Astronautics, Reston, Virginia.AIAA Pap. 02-0149.

Seebass, R., and Argrow, B. (1998). “Sonic Boom Minimization Revisited”.Proceedings of the 2nd AIAA Theoretical Fluid Mechanics Meeting,Albuquerque, New Mexico, June 1998. AIAA Pap. 98-2956.

Thomas, C. (1971). “Extrapolation of Sonic Boom Pressure Signatures bythe Waveform Parameter Method”. NASA TN D-6832.

Whitham, G.B. (1952). “The Flow Pattern of a Supersonic Projectile”.Commun. Pure Appl. Math. Vol. 3, pp. 301–348.

© 2005 CASI 199


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AAdjoint-Based Sonic Boom Reduction for Wing-Body

Configurations in Supersonic Flow (No. 4, December/décembre2005): S.K. Nadarajah, A. Jameson, J. Alonso . . . . . . . . . . . . . . 187

Aerodynamic Forces Approximations using the Chebyshev Methodfor Closed-Loop Aero-servoelasticity Studies (No. 4,December/décembre 2005): A.D. Dinu, R.M. Botez, J. Cotoi . . . . 167

Alonso, J., S.K. Nadarajah, A. Jameson: Adjoint-Based SonicBoom Reduction for Wing-Body Configurations in SupersonicFlow (No. 4, December/décembre 2005). . . . . . . . . . . . . . . . . . . 187

Application of Three Combustion Models to a Model Combustor(No. 1, March/mars 2005): L.-Y. Jiang, I. Campbell . . . . . . . . . . . . 1

Applying Fast Fourier Transform Analysis and Data Window inSoftware Global Positioning System Receivers to MitigateContinuous Wave Interference under Dynamic Conditions (No. 4,December/décembre 2005): Z. Jiang, G. Lachapelle, C. Ma . . . . . 177

Assessing the Global Availability and Reliability of the MarsNetwork, a Proposed Global Navigation Satellite System forMars (No. 1, March/mars 2005): K. O’Keefe, G. Lachapelle,S. Skone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

BBackman, D., N. Bellinger, G. Shi, G. Li: Numerical Modeling of

a Single Aluminum Sheet Containing an Interference FitFastener (No. 3, September/septembre 2005). . . . . . . . . . . . . . . . 107

Bellinger, N., G. Shi, G. Li, D. Backman: Numerical Modeling ofa Single Aluminum Sheet Containing an Interference FitFastener (No. 3, September/septembre 2005). . . . . . . . . . . . . . . . 107

Benay, R., A. Bourgoing: Investigation of an AsymmetricalShock-Wave Boundary-Layer Interaction in a Supersonic PlanarNozzle (No. 2, June/juin 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bisson, S., L. Del Ciotto, C. Legare, S. Rutherford, O. Underhill,C. Mathias, D. Lee, L. Smyth, V.D. Nguyen, B. Tanguay,Y. Mébarki, M. Deslauriers, A. Rebaine, Y. Cronier: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Book Review: Orbital Mechanics for Engineering Students writtenby Howard D. Curtis (No. 2, June/juin 2005): A.M. Jablonski . . . . 87

Book Review: Stress, Strain, and Structural Dynamics — AnInteractive Handbook of Formulas, Solutions, and MATLABToolboxes written by Bíngen Lang (No. 3,September/septembre 2005): V.K. Wickramasinghe . . . . . . . . . . . 145

Botez, R.M., I. Cotoi, A.D. Dinu: Aerodynamic ForcesApproximations using the Chebyshev Method for Closed-LoopAero-servoelasticity Studies (No. 4, December/décembre 2005) . . . 167

Bourgoing, A., R. Benay: Investigation of an AsymmetricalShock-Wave Boundary-Layer Interaction in a Supersonic PlanarNozzle (No. 2, June/juin 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CCampbell, I., L.-Y. Jiang: Application of Three Combustion Models

to a Model Combustor (No. 1, March/mars 2005) . . . . . . . . . . . . . . 1Chesser, H., Y. Soucy: Force Limited Vibration Testing Applied to

the MOST Spacecraft (No. 2, June/juin 2005) . . . . . . . . . . . . . . . 47Contantinescu, I., I. Ilie, R. Jr. Landry: Simulation of GPS and

Galileo Architectures for Anti-jamming and Multipath Analysis(No. 1, March/mars 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Cotoi, I., A.D. Dinu, R.M. Botez: Aerodynamic ForcesApproximations using the Chebyshev Method for Closed-LoopAero-servoelasticity Studies (No. 4, December/décembre 2005) . . . 167

Cronier, Y., S. Bisson, L. Del Ciotto, C. Legare, S. Rutherford,O. Underhill, C. Mathias, D. Lee, L. Smyth, V.D. Nguyen,B. Tanguay, Y. Mébarki, M. Deslauriers, A. Rebaine: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

DDujardin, A., P. Hennig, L. Leavitt, F. Leopold, M. Mendenhall,

S. Prince, M. Khalid: Turbulence Model Studies to Investigatethe Aerodynamic Performance of a NASA Dual Control Missileat Supersonic Mach Numbers (No. 4, December/décembre 2005) . . 153

Del Ciotto, L., C. Legare, S. Rutherford, O. Underhill, C. Mathias,D. Lee, L. Smyth, V.D. Nguyen, B. Tanguay, Y. Mébarki,M. Deslauriers, A. Rebaine, Y. Cronier, S. Bisson: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Deslauriers, M., A. Rebaine, Y. Cronier, S. Bisson, L. Del Ciotto,C. Legare, S. Rutherford, O. Underhill, C. Mathias, D. Lee,L. Smyth, V.D. Nguyen, B. Tanguay, Y. Mébarki: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Dinu, A.D., R.M. Botez, J. Cotoi: Aerodynamic ForcesApproximations using the Chebyshev Method for Closed-LoopAero-servoelasticity Studies (No. 4, December/décembre 2005) . . . 167

FFloryan, J.M., P. Luchini, M. Quadrio: Modification of Turbulent

Flow Using Distributed Transpiration (No. 2, June/juin 2005) . . . . . 61Fluid Flow and Thermodynamic Analysis of a Wing Anti-Icing

System (No. 1, March/mars 2005): J. Hua, H.H.T. Liu. . . . . . . . . . 35Force Limited Vibration Testing Applied to the MOST Spacecraft

(No. 2, June/juin 2005): Y. Soucy, H. Chesser . . . . . . . . . . . . . . . 47

GGiroux, R., R. Gourdeau, R. Jr. Landry: Inertial Navigation

System/Global Positioning System Fusion Algorithm Design ina Fast Prototyping Environment: Towards a Real-TimeImplementation (No. 3, September/septembre 2005) . . . . . . . . . . . 133

Vol. 51, No. 4, december 2004 vol. 51, no 4, décembre 2004

207

Canadian Aeronautics and Space JournalJournal aéronautique et spatial du Canada

Published by / Publié parCanadian Aeronautics and Space Institute / Institut aéronautique et spatial du Canada

150 - 1750, croissant Courtwood CrescentOttawa, ON, Canada K2C 2B5

Tel./Tél. : 613-234-0191, Fax/Téléc. : 613-234-9039E-mail/Courriel : [email protected], Website: 247.casi.ca

Index to Volume 51, 2005Number 1 - March/mars pp. 1–45 Number 3 - September/septembre pp. 95–151Number 2 - June/juin pp. 47–93 Number 4 - Decembre/décembre pp. 153–209

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Gourdeau, R., R. Jr. Landry, R. Giroux: Inertial NavigationSystem/Global Positioning System Fusion Algorithm Design ina Fast Prototyping Environment: Towards a Real-TimeImplementation (No. 3, September/septembre 2005) . . . . . . . . . . . 133

Greatrix, D.R., J. Karpynczyk: Rocket Vehicle Design forSmall-Payload Delivery to Orbit (No. 3, September/septembre2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

HHennig, P., L. Leavitt, F. Leopold, M. Mendenhall, S. Prince,

M. Khalid, A. Dujardin: Turbulence Model Studies toInvestigate the Aerodynamic Performance of a NASA DualControl Missile at Supersonic Mach Numbers (No. 4,December/décembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Hua, J., H.H.T. Liu: Fluid Flow and Thermodynamic Analysis of aWing Anti-Icing System (No. 1, March/mars 2005) . . . . . . . . . . . . 35

IIlie, I., R. Jr. Landry, A. Constantinescu: Simulation of GPS and

Galileo Architectures for Anti-jamming and Multipath Analysis(No. 1, March/mars 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Inertial Navigation System/Global Positioning System FusionAlgorithm Design in a Fast Prototyping Environment: Towardsa Real-Time Implementation (No. 3, September/septembre2005): R. Giroux, R. Gourdeau, R. Jr. Landry. . . . . . . . . . . . . . . 133

Investigation of an Asymmetrical Shock-Wave Boundary-LayerInteraction in a Supersonic Planar Nozzle (No. 2, June/juin2005): R. Benay, A. Bourgoing . . . . . . . . . . . . . . . . . . . . . . . . . 71

JJablonski, A.M.: Book Review: Orbital Mechanics for Engineering

Students written by Howard D. Curtis (No. 2, June/juin 2005) . . . . 87Jameson, A., J. Alonso, S.K. Nadarajah: Adjoint-Based Sonic

Boom Reduction for Wing-Body Configurations in SupersonicFlow (No. 4, December/décembre 2005). . . . . . . . . . . . . . . . . . . 187

Jiang, L.-Y., I. Campbell: Application of Three Combustion Modelsto a Model Combustor (No. 1, March/mars 2005) . . . . . . . . . . . . . . 1

Jiang, Z., G. Lachapelle, C. Ma: Applying Fast Fourier TransformAnalysis and Data Window in Software Global PositioningSystem Receivers to Mitigate Continuous Wave Interferenceunder Dynamic Conditions (No. 4, December/décembre 2005). . . . 177

KKhalid, M., A. Dujardin, P. Hennig, L. Leavitt, F. Leopold,

M. Mendenhall, S. Prince: Turbulence Model Studies toInvestigate the Aerodynamic Performance of a NASA DualControl Missile at Supersonic Mach Numbers (No. 4,December/décembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Karpynczyk, J., D.R. Greatrix: Rocket Vehicle Design forSmall-Payload Delivery to Orbit (No. 3, September/septembre2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

LLachapelle, G., C. Ma, Z. Jiang: Applying Fast Fourier Transform

Analysis and Data Window in Software Global PositioningSystem Receivers to Mitigate Continuous Wave Interferenceunder Dynamic Conditions (No. 4, December/décembre 2005). . . . 177

Lachapelle, G., S. Skone, K. O’Keefe: Assessing the GlobalAvailability and Reliability of the Mars Network, a ProposedGlobal Navigation Satellite System for Mars (No. 1,March/mars 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Landry, R. Jr., A. Constantinescu, I. Ilie: Simulation of GPS andGalileo Architectures for Anti-jamming and Multipath Analysis(No. 1, March/mars 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Landry, R. Jr., R. Giroux, R. Gourdeau: Inertial NavigationSystem/Global Positioning System Fusion Algorithm Design ina Fast Prototyping Environment: Towards a Real-TimeImplementation (No. 3, September/septembre 2005) . . . . . . . . . . . 133

Leavitt, L., F. Leopold, M. Mendenhall, S. Prince,M. Khalid, A. Dujardin, P. Hennig: Turbulence Model Studiesto Investigate the Aerodynamic Performance of a NASA Dual

Control Missile at Supersonic Mach Numbers (No. 4,December/décembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Leopold, F., M. Mendenhall, S. Prince, M. Khalid, A. Dujardin,P. Hennig, L. Leavitt: Turbulence Model Studies to Investigatethe Aerodynamic Performance of a NASA Dual Control Missileat Supersonic Mach Numbers (No. 4, December/décembre2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Lee, D., L. Smyth, V.D. Nguyen, B. Tanguay, Y. Mébarki, M.Deslauriers, A. Rebaine, Y. Cronier, S. Bisson, L. Del Ciotto,C. Legare, S. Rutherford, O. Underhill, C. Mathias: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Legare, C., S. Rutherford, O. Underhill, C. Mathias, D. Lee,L. Smyth, V.D. Nguyen, B. Tanguay, Y. Mébarki,M. Deslauriers, A. Rebaine, Y. Cronier, S. Bisson,L. Del Ciotto: Recent Improvements to the NRC 9 m × 9 mWind Tunnel (No. 3, September/septembre 2005) . . . . . . . . . . . . . 95

Li, G., D. Backman, N. Bellinger, G. Shi: Numerical Modeling ofa Single Aluminum Sheet Containing an Interference FitFastener (No. 3, September/septembre 2005). . . . . . . . . . . . . . . . 107

Liu, H.H.T., J. Hua: Fluid Flow and Thermodynamic Analysis of aWing Anti-Icing System (No. 1, March/mars 2005) . . . . . . . . . . . . 35

Luchini, P., M. Quadrio, J.M. Floryan: Modification of TurbulentFlow Using Distributed Transpiration (No. 2, June/juin 2005) . . . . . 61

MMa, C., Z. Jiang, G. Lachapelle: Applying Fast Fourier Transform

Analysis and Data Window in Software Global PositioningSystem Receivers to Mitigate Continuous Wave Interferenceunder Dynamic Conditions (No. 4, December/décembre 2005). . . . 177

Mathias, C., D. Lee, L. Smyth, V.D. Nguyen, B. Tanguay, Y.Mébarki, M. Deslauriers, A. Rebaine, Y. Cronier, S. Bisson,L. Del Ciotto, C. Legare, S. Rutherford, O. Underhill: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Mébarki, Y., M. Deslauriers, A. Rebaine, Y. Cronier, S. Bisson,L. Del Ciotto, C. Legare, S. Rutherford, O. Underhill,C. Mathias, D. Lee, L. Smyth, V.D. Nguyen, B. Tanguay:Recent Improvements to the NRC 9 m × 9 m Wind Tunnel(No. 3, September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . 95

Mendenhall, M., S. Prince, M. Khalid, A. Dujardin, P. Hennig,L. Leavitt, F. Leopold: Turbulence Model Studies to Investigatethe Aerodynamic Performance of a NASA Dual Control Missileat Supersonic Mach Numbers (No. 4, December/décembre2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Modification of Turbulent Flow Using Distributed Transpiration(No. 2, June/juin 2005): M. Quadrio, J.M. Floryan, P. Luchini . . . . 61

NNadarajah, S.K., A. Jameson, J. Alonso: Adjoint-Based Sonic

Boom Reduction for Wing-Body Configurations in SupersonicFlow (No. 4, December/décembre 2005). . . . . . . . . . . . . . . . . . . 187

Nguyen, V.D., B. Tanguay, Y. Mébarki, M. Deslauriers,A. Rebaine, Y. Cronier, S. Bisson, L. Del Ciotto, C. Legare,S. Rutherford, O. Underhill, C. Mathias, D. Lee, L. Smyth:Recent Improvements to the NRC 9 m × 9 m Wind Tunnel(No. 3, September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . 95

Numerical Modeling of a Single Aluminum Sheet Containing anInterference Fit Fastener (No. 3, September/septembre 2005):G. Li, D. Backman, N. Bellinger, G. Shi . . . . . . . . . . . . . . . . . . 107

OO’Keefe, K., G. Lachapelle, S. Skone: Assessing the Global

Availability and Reliability of the Mars Network, a ProposedGlobal Navigation Satellite System for Mars (No. 1,March/mars 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

PPrince, S., M. Khalid, A. Dujardin, P. Hennig, L. Leavitt,

F. Leopold, M. Mendenhall: Turbulence Model Studies toInvestigate the Aerodynamic Performance of a NASA Dual

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Control Missile at Supersonic Mach Numbers (No. 4,December/décembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

QQuadrio, M., J.M. Floryan, P. Luchini: Modification of Turbulent

Flow Using Distributed Transpiration (No. 2, June/juin 2005) . . . . . 61

RRebaine, A., Y. Cronier, S. Bisson, L. Del Ciotto, C. Legare,

S. Rutherford, O. Underhill, C. Mathias, D. Lee, L. Smyth,V.D. Nguyen, B. Tanguay, Y. Mébarki, M. Deslauriers: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Recent Improvements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005): V.D. Nguyen, B. Tanguay,Y. Mébarki, M. Deslauriers, A. Rebaine, Y. Cronier, S. Bisson,L. Del Ciotto, C. Legare, S. Rutherford, O. Underhill,C. Mathias, D. Lee, L. Smyth . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Rocket Vehicle Design for Small-Payload Delivery to Orbit (No. 3,September/septembre 2005): D.R. Greatrix, J. Karpynczyk . . . . . . 123

Rutherford, S., O. Underhill, C. Mathias, D. Lee, L. Smyth,V.D. Nguyen, B. Tanguay, Y. Mébarki, M. Deslauriers,A. Rebaine, Y. Cronier, S. Bisson, L. Del Ciotto, C. Legare:Recent Improvements to the NRC 9 m × 9 m Wind Tunnel(No. 3, September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . 95

SShi, G., G. Li, D. Backman, N. Bellinger: Numerical Modeling of

a Single Aluminum Sheet Containing an Interference FitFastener (No. 3, September/septembre 2005). . . . . . . . . . . . . . . . 107

Simulation of GPS and Galileo Architectures for Anti-jamming andMultipath Analysis (No. 1, March/mars 2005): I. Ilie,R. Jr. Landry, A. Constantinescu. . . . . . . . . . . . . . . . . . . . . . . . . 13

Skone, S., K. O’Keefe, G. Lachapelle: Assessing the GlobalAvailability and Reliability of the Mars Network, a ProposedGlobal Navigation Satellite System for Mars (No. 1,March/mars 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Smyth, L., V.D. Nguyen, B. Tanguay, Y. Mébarki, M. Deslauriers,A. Rebaine, Y. Cronier, S. Bisson, L. Del Ciotto, C. Legare,S. Rutherford, O. Underhill, C. Mathias, D. Lee: RecentImprovements to the NRC 9 m × 9 m Wind Tunnel (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Soucy, Y., H. Chesser: Force Limited Vibration Testing Applied tothe MOST Spacecraft (No. 2, June/juin 2005) . . . . . . . . . . . . . . . 47

TTanguay, B., Y. Mébarki, M. Deslauriers, A. Rebaine, Y. Cronier,

S. Bisson, L. Del Ciotto, C. Legare, S. Rutherford,O. Underhill, C. Mathias, D. Lee, L. Smyth, V.D. Nguyen:Recent Improvements to the NRC 9 m × 9 m Wind Tunnel(No. 3, September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . 95

Turbulence Model Studies to Investigate the AerodynamicPerformance of a NASA Dual Control Missile at SupersonicMach Numbers (No. 4, December/décembre 2005): M. Khalid,A. Dujardin, P. Hennig, L. Leavitt, F. Leopold, M. Mendenhall,S. Prince . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

UUnderhill, O., C. Mathias, D. Lee, L. Smyth, V.D. Nguyen,

B. Tanguay, Y. Mébarki, M. Deslauriers, A. Rebaine,Y. Cronier, S. Bisson, L. Del Ciotto, C. Legare, S. Rutherford:Recent Improvements to the NRC 9 m × 9 m Wind Tunnel(No. 3, September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . 95

WWickramasinghe, V.K.: Book Review: Stress, Strain, and Structural

Dynamics — An Interactive Handbook of Formulas, Solutions,and MATLAB Toolboxes written by Bíngen Lang (No. 3,September/septembre 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . 145


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JOURNAL AÉRONAUTIQUE ET SPATIAL DU CANADA

Turbulence Model Studies to Investigate the Aerodynamic Performance of a NASADual Control Missile at Supersonic Mach NumbersM. Khalid, A. Dujardin, P. Hennig, L. Leavitt, F. Leopold, M. Mendenhall, S. Prince . 153

Aerodynamic Forces Approximations using the Chebyshev Method for Closed-LoopAero-servoelasticity StudiesAlin Dorian Dinu, Ruxandra Mihaela Botez, Iulian Cotoi . . . . . . . . . . . . . . . . . . . . 167

Applying Fast Fourier Transform Analysis and Data Window in Software GlobalPositioning System Receivers to Mitigate Continuous Wave Interference underDynamic ConditionsZ. Jiang, G. Lachapelle, C. Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Adjoint-Based Sonic Boom Reduction for Wing-Body Configurations in SupersonicFlowSiva K. Nadarajah, Antony Jameson, Juan Alonso. . . . . . . . . . . . . . . . . . . . . . . . . . 187

Postage paid at Ottawa Port payé à OttawaPublications mail Poste-publicationRegistration No. 40069141 Enregistrement no 40069141

Published by ⋅ Publié parCanadian Aeronautics and Space Institute • Institut aéronautique et spatial du Canada

150 - 1750, croissant Courtwood Crescent, Ottawa, ON, Canada K2C 2B5Tel./Tél. : 613-234-0191, Fax/Téléc. : 613-234-9039, E-mail/Courriel : [email protected]

Publication Mail Registration No. 231452 ISSN 0008-2821

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