Lunds Tekniska Hogskola

Master thesisDepartment of energy

Aeroelastic flutter analysis withvarying level of accuracy

Nils Voigt Dahl

supervised byProf. Johan Revstedt

Prof. Kent PerssonAnders Karlsson

June, 2020

Examensarbete pa Civilingenjorsniva

ISRN LUTMDN/TMHP-20/5459-SE

ISSN 0282-1990

© 2020 Nils Voigt Dahl

Institutionen for Energivetenskaper

Lunds Universitet - Lunds Tekniska Hogskola

Box 118, 221 00 Lund

www.energy.lth.se

Abstract

Flutter prediction is a vital part of aircraft development which requires efficient andaccurate analysis methods. Saab AB Aeronautics is currently using linear analysisbased on the panel method and nonlinear analysis using their in-house CFD softwareprogram. These methods are in contrast to one another in terms of computing time andaccuracy. There is a need to fill the gap between these methods in terms of efficiencyand accuracy for future development of the current and next generation aircraft.

This project will evaluate the flutter analysis methods implemented in the softwareprograms ZAERO and ZEUS developed by Zona Technology, Inc. The handling ofthese software programs will be evaluated based on efficiency, accuracy and ease ofuse. JAS 39 Gripen was used for the analyses in this project. A new high fidelityaerodynamic model of the aircraft was developed to take advantage of the modellingcapabilities implemented in the software. The effect of different modelling optionssuch as airfoil thickness, fuselage fidelity and movable control surfaces was investigated.Finally, the accuracy of the predicted aeroelastic response was validated by comparisonwith previous numerical results and experimental data.

Both ZAERO and ZEUS are concluded to be overall robust in their calculationsand the results are in close correlation with previous analyses. The computationaleffectiveness is reasonable and can be further improved with some refinement of themodel and solution parameters. The evaluation shows that both software programshas the potential to complement the current methods used by Saab.

i

Preface

The subject of this thesis was composed by Saab in their search of improving theirflutter analysis methods. The current techniques implemented have proven to work andare sufficiently accurate for the problems currently analysed. But as technology rapidlyimproves and computational calculations follow the advancements both in speed andcomplexity, there is a demand for up-to-date level of complexity for the flutter analyses.This is where I come in. As a student with a love for aerodynamics and computationalaided engineering, I have set out to provide SAAB with a well-executed investigationof these new calculation methods as well as to learn about the wonderfully complexarea of flutter and aeroelasticity in the development of supersonic aircraft.

”In its simplest form, if an increase in aerodynamic load distorts a structure in such amanner that the incidence changes and inreaces the aerodynamic load further, we havean aeroelastic problem.”- A. R. Collar, 1978

ii

Acknowledgments

Firstly, I would like to express my gratitude to the colleagues at Saab. The level of ded-ication and expertise at the department is inspiring. I would like to express my specialthanks to Anders Karlsson and Par Gustafsson for their help and support throughoutworking with this thesis. I must also thank my new friends Emma Haggstrom andMarko Rosic for the enjoyable company, while they too were writing their master the-sis at the department.

It has been a real pleasure working with you all.

Nils Voigt Dahl

Linkoping, May 2020

iii

Contents

Abstract i

Preface ii

Acknowledgments iii

Nomenclature vi

1 Introduction 11.1 Project motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Methods 32.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Formulation of aeroelasticity . . . . . . . . . . . . . . . . . . . . 32.1.2 Modal approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Unsteady aerodynamics . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Aerodynamic influence coefficient (AIC) . . . . . . . . . . . . . 62.1.5 Spline theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.6 Doublet-Lattice Method . . . . . . . . . . . . . . . . . . . . . . 72.1.7 ZONA51 method . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.8 Potential equations of subsonic and supersonic flow . . . . . . . 82.1.9 Flutter calculations . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.10 Phase lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.11 Results of flutter calculation . . . . . . . . . . . . . . . . . . . . 12

2.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 MSC Nastran - modal analysis . . . . . . . . . . . . . . . . . . . 132.2.2 MSC Nastran - aeroelastic analysis . . . . . . . . . . . . . . . . 132.2.3 AEREL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 ZAERO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.5 ZEUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Ground vibration testing . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Flight test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Modelling and analysis 163.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 ZAERO bulk data . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.4 Final aerodynamic models used . . . . . . . . . . . . . . . . . . 233.1.5 Airfoil thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.6 Control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.7 Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Volume mesh for Euler equation solver . . . . . . . . . . . . . . . . . . 27

iv

3.2.1 Flow domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Parameters for solving the flutter equation . . . . . . . . . . . . 313.3.2 Euler solver parameters . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Results and discussion 344.1 Interpolation between structural and aerodynamic model . . . . . . . . 344.2 Mesh sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Comparison between fuselage models . . . . . . . . . . . . . . . . . . . 374.4 Differences when including movable control surfaces . . . . . . . . . . . 384.5 The effect of airfoil thickness . . . . . . . . . . . . . . . . . . . . . . . . 384.6 ZEUS rigid aircraft results . . . . . . . . . . . . . . . . . . . . . . . . . 394.7 Panel method vs Euler solution . . . . . . . . . . . . . . . . . . . . . . 414.8 Comparison with previous simulations . . . . . . . . . . . . . . . . . . 444.9 Problems and issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Conclusion 485.1 Achievement of the objectives . . . . . . . . . . . . . . . . . . . . . . . 485.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Future research and development . . . . . . . . . . . . . . . . . . . . . 49

A Appendix: Wagner’s function and change in lift on a 2D airfoil 52

B Appendix: Convergence rate for one reduced frequency in ZEUS 53

Nomenclature

Notationsα Angle of incidencex Modeshape matrixΦ Modal matrixG Spline matrixh Aerodynamic control pointsK Generalised stiffness matrixM Generalised mass matrixQ Generalised aerodynamic forces matrixq Generalised modal coordinatesω Frequencyωf Flutter frequencyρ Densityσ Dampinga∞ Local speed of soundb Semi-chord, b = c/2c Chord lengthCL Lift coefficientCp Pressure coefficientg Dampingk Reduced frequencyL Reference lengthLf Lifting forceM Mach numberq∞ Dynamic pressure at free streams Complex frequency variablet TimeV Free stream velocityVf Flutter velocityx Displacement

AbbreviationsAIC Aerodynamic influence coefficientAoA Angle of attackCFD Computational fluid dynamicsCFL Courant–Friedrichs–Lewy conditionDLM Doublet-lattice methodFE Finite elementGAF Generalised aerodynamic forcesGVT Ground vibration testHFG High fidelity geometry

vi

1 Introduction

The area of aeroelasticity can be described by the interaction between three main phys-ical disciplines as described by Arthur Roderick Collar[1]. Dynamics, solid mechanicsand aerodynamics all interact in the area of dynamic aeroelasticity. A deduction of oneof the physical phenomena described yields a different field of flight physics. This canbe visualized by the Collar triangle (figure 1). The triangle shows how the contributionfrom different forces can distinguish some of the important technical fields of flight.

Figure 1: Collar triangle visualizing the interaction of physical disciplines.

Static aeroelasticity covers the interaction between forces from steady flow aerodynam-ics and solid mechanics. Divergence and control reversal are some of the phenomenathat are included in static aeroelasticity. This thesis will cover flutter, which is aphenomena that comes from dynamic aeroelasticity. Flutter is regarded as one of themost important aeroelastic phenomena[2]. Both because it can lead to catastrophicstructural failure and because it is the most difficult to predict.

1.1 Project motivation

Saab has been conducting numerical flutter analyses for many years, both for develop-ment and prediction of the aircraft characteristics. The challenge lies in the massivenumber of analyses that are needed for validating every aircraft configuration at allflight conditions, while having accurate analyses with reliable results. The analysesare currently done primarily using in-house software, which is working for most casesbut is not possible to further develop. There is a state-of-the-art software systemnamed ZAERO and ZEUS developed by Zona Technology that is of interest for Saab’saeroelastic department as a compliment to the in-house capability. Flutter analysis isincluded as a part of these software systems.

Both linear and nonlinear methods are currently used by Saab, consisting of a panelmethod used for linear analysis and CFD simulations for nonlinear analysis. There isa need for complimentary methods to fill the gap in accuracy and computing time.ZAERO is a possible candidate for a more accurate linear method with higher fidelitygeometry and a new flutter equation solver. The Euler solver method implemented inZEUS could be a faster alternative to the time demanding CFD-based analyses.

1

1.2 Aims of the project

• Conduct flutter analysis on selected aircraft configurations with thesoftware ZAERO.This includes developing a new pre-processing program for creating the aerody-namic model suitable for ZAERO. Analysis should be done for different flightconditions of interest and include configurations with external stores.

• Evaluate the software ZEUS for flutter analysis.Both the ease of use and the accuracy of the results are to be evaluated byconducting multiple analyses.

• Compare both methods with each other and with previous analysesdone by Saab.Find the differences between ZAERO and ZEUS and their best application re-spectively. There are equivalent programs used by Saab for flutter analysis thatcan be used for comparison, together with experimental data.

• Produce a reference guide for ZAERO and ZEUS adapted for Saab.If Saab chooses to continue to use ZAERO and ZEUS in their research, a refer-ence guide written for the department would be desirable, in which all the mostimportant knowledge gained from this project would be included.

2

2 Methods

2.1 Theory

2.1.1 Formulation of aeroelasticity

The interaction between dynamic, elastic and aerodynamic forces is combined in theequation of motion[3],

M x(t) + Cx(t) + Kx(t) = F(t) (1)

where M is the mass matrix (part of inertial forces), C and K are the structuraldamping and stiffness matrices (part of elastic forces), and F (t) represents the aerody-namic forces. For simplicity, the structural damping matrix is excluded in the followingequations, but can easily be included when needed.

The aerodynamic forces F (t) can be split into two parts, one part induced by thestructural deformation Fa(x) and the other induced by external forces Fe(t) as follows:

F(t) = Fa(x) + Fe(t) (2)

The dependency of structural deformation in the aerodynamic force Fa(x) means thatthe system can be interpreted as a closed loop feedback system. Combining equa-tion (1) (excluding structural damping) and (2) results in a closed-loop characteristicequation:

Mx(t) + Kx(t)− Fa(x) = Fe(t) (3)

This closed-loop dynamic system can be self-excited and is the basic stability problemof flutter. The stability boundary1 can now be determined by either treating Fa(x(t)) asa nonlinear equation or by linearising the function. If the function is assumed nonlinear,the flutter equation must be solved in the time domain using time-marching iterations.This can be done using Computational Fluid Dynamics (CFD) methods, but is verycostly computationally. The other linearised function assumes amplitude linearisationof the structural deformation which states that the aerodynamic response varies linearlywith the amplitude, assuming the amplitude is sufficiently small at all times. This isa valid assumption due to the required amplitude for finding the stability boundarybeing mathematically infinitesimal for a dynamic stability problem. The linearisationmeans that the problem can be transitioned into the frequency domain and be writtenas an eigenvalue problem with an aerodynamic transfer function. The aerodynamicforces due to structural deformation Fa(x) can be written as a convolution integralto determine the output of the system function for any input. A general convolutionintegral function can be written as:

y(t) =

∫ t

0

f(λ)h(t− λ)dλ (4)

This leads to the aerodynamic force convolution integral with a transfer function H:

Fa(x) =

∫ t

0

q∞H

(V

L(t− τ)

)x(τ)dτ (5)

1The point at which the system is neutrally stable and is neither asymptotically stable or unstable.

3

where q∞ is the dynamic pressure, L is the reference length and V is the free streamvelocity. Equation (5) can be converted into a Laplace domain function as follows:

Fa(x(s)) = q∞H

(sL

V

)x(s) (6)

where H is the Laplace domain version of the transfer function H. The closed loopsystem function (3) can now be rewritten in the Laplace domain. The displacement isrewritten as a function of harmonic motion.

x(t) = xest = xeσteiωt

[s = σ + iω](7)

where σ is the damping and ω is the frequency. Aerodynamic forces due to externalload, Fe(t), can be excluded as they are independent of displacement and therefore notnecessary when finding the stability boundary. Inserting equation (7) and (6) into (3)results in:

Mx(t) + Kx(t)− Fa(x) = 0

Mxests2 + Kxest − q∞H

(sL

V

)xest = 0

(8)

This gives the Laplace domain equation that reads:[Ms2 + K− q∞H

(sL

V

)]x(s) = 0 (9)

This is the most simple form of the equation to solve for flutter. The equation will laterbe solved as an eigenvalue problem. The displacement x(s) is still expressed as thedisplacement at every point and the equation therefore contains very large mass andstiffness matrices. This can be dealt with by using the modal approach and expressingeach term by a function of modal displacement instead. The modal approach will beexplained in the next section. There are multiple methods that solve the eigenvalueproblem by making assumptions for s and making use of the frequency-domain.

2.1.2 Modal approach

As mentioned previously, flutter is the result of the unsteady interaction between in-ertial, elastic and aerodynamic forces. This can only occur when the structure movesbased on structural modes (vibration). The aeroelastic motion will be dominated bythe low-frequency structural modes[4]. This fact is used to reduce the number ofdegrees of freedom and integrating the modal matrix into equation (9). First, themodeshape matrix x from equation (7) is expressed as:

x = Φq (10)

where Φ is the modal matrix with columns containing the natural modes of the struc-ture to be analysed. q is the generalised modal coordinates which are to be determined.The values in q can be identified as the amount of the corresponding mode present inthe motion. It can however not be measured or compared as a physcial quantity as itis dependent of the mode shape.

4

The components of equation (9) are rewritten with (10) and the transpose of the modalmatrix to get generalised matrices,

M = ΦTMΦ

K = ΦTKΦ

Q

(sL

V

)= ΦTH

(sL

V

)Φ

(11)

where M, K and Q are the generalised mass, stiffness and aerodynamic force matrix,respectively. The Laplace domain closed loop equation (9) therefore reads:[

Ms2 + K− q∞Q

(sL

V

)]q = 0 (12)

This is generally referred to as the classical flutter matrix equation[3].

2.1.3 Unsteady aerodynamics

The aerodynamic forces and moment will vary in time if changes in heave or pitchare present. Furthermore, in the case of flutter calculations, it is of interest to studythe aerodynamic behaviour and motion of a single oscillation frequency, instead of thetime domain. Therefore, a transition from the time domain to the frequency domainis needed.

If an airfoil is subjected to a sudden change in heave or pitch, the new lift forcewill build up gradually[5]. This is due to the time needed for the circulation aroundthe airflow to reach a new steady flow condition and the influence from shedding ofthe starting vortex. The main factor to determine this time is the speed at which aparticle travels over the wing, which is dependent on both the chord length and thefree stream velocity. A short summary of Wagner’s function and the change in lift dueto harmonic motion can be read in appendix A describing the effects on a 2D airfoil.

It can be shown that the amplitude attenuation and the phase lag of the lift curveare a function of the frequency, velocity and chord length when assuming sinusoidaloscillation[2]. A non-dimensional parameter called reduced frequency, k, is introducedto combine these quantities:

k =ωb

V=ωc

2V(13)

where ω is the angular frequency (ω = 2πf). b is the semi-chord and V is the free-stream velocity. The reduced frequency, k, can be considered as a measure of theunsteadiness of the flow. This is more evident if the frequency is converted to periods:

k =ωb

V=

(2π

Tω

)(b

V

)= π

(2b/u)

Tω= π

TcTω

(14)

where Tω is the period of the airfoil vibration and Tc is the time it takes for a fluidparticle to travel the chord distance. The reduced frequency is the fundamental pa-rameter to calculate unsteady aerodynamics and is widely used in flutter calculations.It is said that the significance of reduced frequency in aeroelasticity can be comparedto that of Reynolds number in aerodynamics[4].

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2.1.4 Aerodynamic influence coefficient (AIC)

The generalised aerodynamic forces matrix Q from equation (11) is very complex tosolve directly in the Laplace domain. If harmonic motion is assumed, the transferfunction can be formulated in the frequency domain which reduced the complexity.The frequency dependent transfer function is called aerodynamic influence coefficient,or AIC for short. The AIC-matrix reduces the complexity further by only calculatingthe aerodynamic forces at the aerodynamic control points, instead of at the nodes usedfor the stiffness and mass matrix. The aerodynamic control points are defined by apoint in an aerodynamic box (or panel element). These boxes are used to calculatethe aerodynamic forces based on a panel method, which will be described in a latersection.

The relation between the deflection of the structural grid points x and the aerody-namic control points h can be determined by a spline matrix G.

h = Gx (15)

The spline matrix and the theory behind it will be explained in detail in a later sec-tion. The change from Laplace domain to frequency domain is done by neglecting thedamping σ (from equation (7)) for now. By combining equation (7) and (13), the termof the transfer function can be rewritten as:

sL

V=σL

V+ i

ωL

V= [σ = 0] = ik (16)

This leads to a conversion from Laplace domain with structural grid points to thefrequency domain with aerodynamic control points using the modal approach:

Q(ik) = ΦTGT [AIC(ik)]GΦ (17)

2.1.5 Spline theory

There is a need to connect (interpolate) the structural and aerodynamic model fordisplacement and force transferal. This is done by using the theory of splines[6][3]which allows for independent mesh-modeling of the structural and aerodynamic models,which often has different considerations. A spline matrix G is generated to connectthe two models as introduced earlier (15):

h = Gx

Each aerodynamic box (marked with index i) has six degrees of freedom (d.o.f), onefor each translation axis and their respective derivative with respect to the location(x/L).

hi =

hxhyhzh′xh′yh′z

i

(18)

6

The deflection matrix h is therefore a column vector of size [i x 6, 1]. The grid points ofthe structural model usually have six d.o.f. with three translational and three rotationaldisplacements. The deflection matrix x for the structural model is assembled as:

xj =

T1T2T3R1

R2

R3

j

(19)

where index j marks a grid point from the structural model used for the spline. Thespline matrix G is therefore of size [i x 6, j x 6]. The spline matrix makes it possibleto calculate the structural forces Fa from the aerodynamic forces Fh:

Fa = GTFh (20)

The relation in equation (20) can be derived from the theory of virtual work[3].

2.1.6 Doublet-Lattice Method

A three dimensional panel method can model the interaction between different liftingsurfaces and the resulting aerodynamic forces. It is a useful tool in aeroelastic analysisfor calculating the pressure distribution and the acting lifting forces due to the relativecomputational effectiveness. It can however not predict drag forces except for induceddrag which is of less importance for aeroelastic calculations. Due to the limitation ofassuming inviscid flow, the aerodynamic forces in the transonic region are not possibleto accurately calculate with the panel method without including corrections from CFDresults or wind tunnel data[2].

Figure 2: Representation of adoublet. Formed from a sourceand a sink approaching the samepoint.

A streamline in a fluid flow can be approximatedand calculated using only a few building blocks. Theseinclude uniform flow, sources, sinks, doublets and vor-tices and form the basics of potential flow calculations.A doublet can be thought of as a combination of asink and a source with the distance between them ap-proaching zero. This forms two rotational flows act-ing in opposite directions with a tangential flow vectorbetween them, as seen in figure 2. The DLM is anextension of the Vortex-Lattice method to introduceunsteady flow. The Vortex-Lattice method uses horse-shoe vortices placed along the quarter chord and theinner and outer edge of each panel element, forming a’lattice’. The quarter chord is assumed to be the loca-tion of the center of pressure for a thin airfoil. Thesepanel elements are trapezoidal boxes which form thelifting surface. Control points are placed in each panelelement’s three-quarter chord to enforce the boundarycondition of zero velocity normal to the surface. All

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vortices affect the down-wash at each panel element and the relation forms a matrixof influence coefficients.

The Vortex-Lattice method only provide solutions for steady flow. As discussedpreviously, the lifting force will vary as oscillations change the incidence and velocityof the airfoil. The unsteady aerodynamic forces are modelled using the Doublet-Latticemethod which combines doublets with an acceleration potential and the horseshoe vor-tex line at the quarter chord. The displacement of the panel can be used to determinehow the lift changes along the doublet line. The lifting pressure is assumed to beconcentrated along the quarter chord line on each panel element.

2.1.7 ZONA51 method

ZONA51 is a supersonic lifting surface theory which is similar to that of DLM, firstpublished by Liu, James, Chen, and Pototsky in 1991[7]. It is a further developmentof the harmonic gradient method presented by Chen and Liu in 1985[8]. Like theDoublet-Lattice Method, ZONA51 is an acceleration potential method which does notaccount for the characteristics of the wake. The control point is defined at the 95%chord line (compared to 75% used in the DLM) and centered span-wise. The liftingpressure for each panel element is assumed to be uniformed.

2.1.8 Potential equations of subsonic and supersonic flow

The panel method used by ZAERO to calculate the subsonic and supersonic unsteadyaerodynamics are ZONA6 and ZONA7 respectively. These methods solve the three-dimensional linearised small-disturbance potential equation. The ZONA7 method usesan identical method to calculate the lifting surface as ZONA51, but is capable to in-clude a body in the calculation. The body can be a fuselage or external stores. ZONA6adopts a higher order paneling method than the Doublet-Lattice method (DLM). Thehigher order paneling method was first implemented in Woodward’s method in the1970s, but for steady flow calculations. ZONA6 is a unsteady-flow extension of Wood-ward’s method, meaning that a linear doublet is implemented instead of a constant dou-blet. This results in a more robust code which is less sensitive to the panel resolution[3],especially at high reduced frequencies. This means that ZONA6 has less modeling re-strictions and requires less boxes to achieve a converged solution compared to DLM[9].The control point is set to 85% and 95% chord for ZONA6 and ZONA7, respectively,based on numerical experiments.

2.1.9 Flutter calculations

The flutter equation can be solved as an eigenvalue problem, as mentioned earlier.Remembering the composition and solution to a general eigenvalue problem:

(A− λI)v = 0

|A− λI| = 0(21)

where v is the eigenvector and λ is the eigenvalues. Inserting the aerodynamic forcematrix from equation (17) into (12) gives:

8

[−ω2M + K− q∞Q(ik)

]q = 0 (22)

The dynamic pressure q∞ can be expressed in terms of reduced frequency as:

q∞ =1

2ρV 2 =

1

2ρ

(ωL

k

)2

(23)

Equation (22) can now be rewritten as an eigenvalue problem by inserting (23):[M +

1

2ρ

(L

k

)2

Q(ik)− λK

]q = 0

Where λ =1

ω2

(24)

This is solved by calculating the eigenvector q and the eigenvalue λ similarly to thegeneral eigenvalue solution in (21). There is one eigenvalue for each structural modeand reduced frequency. The flutter equation (22) does not include any damping of thestructural or aerodynamic forces. This is not a physically valid assumption. Dampingneeds to be included in some form to result in an accurate solution. Three differentmethods which all include a simplified damping will be presented, all used by the in-dustry in different software.

K-methodOne way of introducing damping to the system is by adding an artificial complexstructural damping, gs. This is done to increase the stability of the system. Sincethe assumption of harmonic motion was made, and the aerodynamic damping is zero(from equations (16)(17)), the solution is only valid where the damping gs = 0. TheK-method flutter equation is written as:[

−ω2M + (1 + igs)K− q∞Q(ik)]q = 0 (25)

with the same formulation of the eigenvalue problem as equation (24), but with:

λ =(1 + igs)

ω2(26)

The eigenvalues, λ, can be solved for a given k. The components of λ can then bedecomposed as:

ωf =1√Re(λ)

gs = ω2fIm(λ)

Vf =ωfL

k=Im(λ)

Re(λ)

(27)

where ωf is the flutter frequency and Vf is the flutter velocity (air speed). Flutter veloc-ity Vf is a term used to indicate the flow velocity where the system is neutrally stable,with a small disturbance from equilibrium resulting in steady harmonic oscillations[1].

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If the flow velocity is greater than the flutter velocity, a small disturbance will initiatea growing oscillation with time. The system is then unstable.

A matched point analysis will yield a velocity corresponding (or matched) to theflight velocity.

Vf = Ma∞ (28)

The K-method uses a given pair of Mach number and air density to calculate frequencyand velocity, therefore the flutter velocity will not always equal the flight velocity andit is not a matched point analysis.

PK-methodBy estimating the frequency, eigenvalues can be found from treating the aerodynamicmatrices as real frequency dependent springs and dampers. A iteration loop is used tofind find new frequencies. The aerodynamic forces Q(ik) is treated as an oscilliatorymotion without damping (eiωt) whilst the inertial forces (generalised mass matrix) ismultiplied with the damped sinusodial motion of the eigenvalue p.

p =sL

V= γk + ik = g + ik (29)

The transient decay rate coefficient is written as γ. Inserting p into the flutter equation(22) gives: [(

V 2

L2

)Mp2 + K− q∞Q(ik)

]q = 0 (30)

This is however mathematically inconsistent since two different assumptions are imple-mented for s (see equation (7) and (16)). A modification of the PK-method was madeby Rodden[10] in 1979 by adding an aerodynamic damping matrix into the flutterequation (30). The aerodynamic forces are now split into a real and imaginary part:

Q(ik) = QR + iQI (31)

Inserted into the flutter equation together with p:[(V 2

L2

)Mp2 + K− q∞QI p

k− q∞QR

]q = 0 (32)

Rewriting the equation by inserting the definition of p gives:[(V 2

L2

)Mp2 + K− q∞QIγ − q∞Q(ik)

]q = 0 (33)

This extra term q∞QIγ is the added aerodynamic damping matrix. The solution to thiseigenvalue problem requires an iterative process which finds a reduced frequency k thatmatches with the imaginary part of p for every structural mode. This is a more timeconsuming process compared to the K-method which can solve the eigenvalues directly.The two methods produces the same prediction for flutter velocity, as the methods areidentical when there is zero damping. The PK-method can however more accuratelypredict the damping at sub-critical flight speeds (V < Vf ) which is often desired to

10

detect the aeroelastic characteristics. Which can be very useful when conducting flighttests.

The PK-method results in a matched point solution as it finds a direct solution fora given velocity and air density pair.

g-methodA relatively new method was formulated by Chen[11] in 2000 called the g-method.The damping in the aerodynamic forces is introduced in the Laplace domain using ananalytical approach. This leads to the approximation of Q(p) using the pertubationtheory for the damping:

Q(p) ≈ Q(ik) + gδQ(p)

δg

∣∣∣∣g=0

for g << 1 (34)

This thesis will not go into depth of the derivations of this analytical approach. Becausethe unsteady aerodynamics is calculated in the k-domain (neglecting damping), thesecond term in equation (34) is not available. However, the Cauchy-Riemann equationsmust still be satisfied for the partial differential equation which leads to:

δQ(p)

δg=δQ(p)

δ(ik)(35)

The relation in equation (35) can now be inserted into the second term in equation(34) and transferred to the k-domain again:

δQ(p)

δg

∣∣∣∣g=0

=δQ(p)

δ(ik)

∣∣∣∣g=0

=δQ(ik)

δ(ik)= Q′(ik) (36)

Inserted into (34) gives:Q(p) ≈ Q(ik) + gQ′(ik) (37)

This finally yields the flutter equation in the p-domain.[(V 2

L2

)Mp2 + K− q∞Q′(ik)g − q∞Q(ik)

]q = 0 (38)

The g-method is a match point analysis, similarly to the PK-method. Whilst boththe K- and PK-method only calculates number of eigenvalues equal to the number ofstructural modes, the g-method can find extra roots called lag roots. These lag rootscan have an importance when analysing aeroelastic divergence.

2.1.10 Phase lag

The flutter mode shape will not have the same motion and displacement as a struc-tural mode. This is largely due to the flutter modes being complex-valued[4]. Thecomplex part of the mode shape comes from the interaction of aerodynamic forceswhich changes which introduces phase lag in the oscillatory movement. This gives riseto the indistinguishable flutter motion, similar to that of a wave.

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2.1.11 Results of flutter calculation

There are many ways of presenting the results and solution to the flutter equation.The most important result is to find where the damping crosses zero (equals zero) andthe damping value leads to a unstable system. The visualisation of the stability of acontrol system is often represented in a Root-Locus plot for the Laplace domain or aNyquist or Bode plot for the frequency domain[2]. But for flutter evaluation, so calledV-ω and V-g plots are a useful way to visualise the frequency (ω) and damping (g) atdifferent velocities (V). The parameter along the x-axis is sometimes changed from airspeed to Mach number or dynamic pressure, based on the input parameter.

Figure 3: Example of a V-ω and V-g plot (fictional).

Figure 3 shows an example of a fictional V-g plot2 with five modes included. Fluttercan occur when two frequencies begins to converge as demonstrated here with the twohighest frequencies. The convention used in this thesis is that an unstable system hasa damping on the negative axis (g<0), which is different from the common conventionof using positive damping leading to an unstable system.

2.2 Software

This section will describe the aeroelastic software programs used to solve the equationsexplained in previous sections. These software programs are either used directly in thisproject, or have been used at Saab to produce comparable results.

2”V-g plot” are hereinafter used as a combined term describing a V-ω and V-g plot, as they areoften used in conjunction.

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2.2.1 MSC Nastran - modal analysis

The structural analysis part of MSC Nastran is used for calculating the natural fre-quencies and mode shapes which generates the modal matrix and modal coordinatestogether with the generalised mass and stiffness matrix. It can be though of as ”solv-ing the equations of motions in a vacuum”. Different materials are used together tomake a detailed structural model of the aircraft. Even the ply direction and layup ofcomposite material parts can be modeled.

2.2.2 MSC Nastran - aeroelastic analysis

MSC Nastran is a multidisciplinary structural analysis tool used for many differentengineering fields. Flutter analysis is included in the aeroelastic analysis part of thesoftware. It uses the Doublet-Lattice method (DLM) for subsonic flow and the ZONA51method for supersonic flow. The calculations are based on linearised potential flowand flat plates as lifting surfaces, which are assumed to move in harmonic motion.Non-lifting surfaces such as the fuselage or external stores can be included in thecomputation, but only as slender-body approximation[4]. The aerodynamic loads canbe solved in the frequency domain. The flutter equation is solved using the K- andP-K method.

2.2.3 AEREL

AEREL is an in-house program used by Saab, mainly for flutter and divergence cal-culation. It was originally developed by Stark[12] and has since then been improvedgradually. Similarly to MSC Nastran, the aircraft is modeled using panels parallel tothe flow. Both the resulting aerodynamic forces in subsonic and supersonic flow can becalculated, where the Advanced Doublet Element method (ADE), developed by Stark,is used for subsonic flow. A modifed version of the P-K method is used to solve theflutter equation.

2.2.4 ZAERO

ZAERO is a software system developed by Zona Technology used for aeroelastic anal-ysis and design[13]. It is a broad software aimed at the industry which covers manydisciplines for aeroelastic analysis. Such as flutter, aeroservoelasticity, flight loads, gustloads and ejection loads to name a few. ZAERO uses the K- and g-method to solvethe flutter equation.

2.2.5 ZEUS

ZEUS is an Euler Unsteady Aerodynamic Solver[14], hence the name. It is also aproduct of Zona Technology. The structure and workflow is very similar to that ofZAERO. The Euler equations are solved using a cell-centered finite volume methodwith a cartesian grid. Both inviscid and viscous effects can be included by includinga steady boundary-layer solution coupling for viscous flow. The reasoning behindit comes from the thin boundary layer that occurs at high Reynolds number flows.For wall bounded flow, the assumption is that the viscous effects are confined to the

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boundary layer. The flow domain can therefore be decomposed into a viscous andinviscid domain[15]. This method of using coupled Euler and boundary-layer equationsprovides a good alternative to the more computationally resource demanding NavierStokes based calculations such as RANS (Reynolds Averaged Navier Stokes).

One of the main parts of the software is the automated mesh generation. The carte-sian volume mesh is based on the surface mesh used by ZAERO, in which a volumemesh can grow normal from the surface. Many options for manual mesh refinementsare available together with an overset mesh scheme to improve communication betweenthe refinement boxes. The Chimera scheme[16] is used for connecting the overset grids.

Linearised Euler solverZEUS uses a non-moving grid for the linearised Euler solution. The unsteady aerody-namics are calculated using a perturbation of the shape functions to account for themoving surfaces in the normal direction from fluctuations[17]. This can be done dueto the assumption of small surface deflections needed for flutter calculations. Boththe surface deflection and airfoil thickness effect is incorporated into the boundaryconditions[15].

The resulting forces from the harmonic motions in the frequency domain is obtainedby first solving the steady mean flow problem and then adding an unsteady smallperturbation part:

q = q + q(t)

Hi = H i + Hi(t) [i = 1, 2, 3](39)

where q and q is the mean flow and perturbation flow variables, respectively. Hi

is the convective flux in the three curvilinear coordinate directions. The reason fordecomposing the unsteady flow into a mean flow and a small perturbation part is toincrease the efficiency of the calcualtions. The linearised Euler solver also facilitatesa steady flow solution instead of a full time-domain solution, which can reduce thecomputational time by an order of magnitude[15]. A pseudo-time marching scheme isused for the steady flow solver. The generalised aerodynamic forces (GAF) for eachfrequency is solved directly in the frequency domain when using the linearised Eulersolver without physical time stepping.

2.3 Experimental data

In order to validate the models, different tests are performed in the aeroelastic analysisprocess. The experimental data and methods discussed here have been used to validateprevious flutter analyses done at Saab. No experiments was done specifically to validatethe methods and results from this thesis, only previously validated models and resultsare used for comparison.

2.3.1 Ground vibration testing

The dynamic properties of the structural model is validated by conducting a groundvibration test (GVT). This is done late in the development chain using a complete air-craft. The natural frequencies of the aircraft are found from excitation of the aircraftas it is connected to shakers and mounted on air-cushions to simulate free flight. Differ-

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ent excitation types are tested until the relevant modes are found. The measurementscomes from accelerometers covering the aircraft.

2.3.2 Flight test

Flight tests are done with a special test aircraft equipped with accelerometers. Theexcitation is controlled by the control surfaces using the fly-by-wire system. During aflight test, the flight envelope3 will be opened. Flight flutter tests are considered ashigh risk test and therefore a procedure with a step-wise increase of speed at differentaltitudes is performed while monitoring the data[18]. The damping, frequency andmode shape can be estimated by processing the measurements from the accelerometers.It can then be compared to the numerical analysis. The aircraft is not taken to thepoint of theoretical flutter instability, for obvious reasons.

3The flight conditions where the aircraft is deemed safe to fly within. In this case Mach numberand altitude.

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3 Modelling and analysis

The software ZAERO and ZEUS will be evaluated by running flutter analyses of SaabJAS 39 Gripen (hereinafter referred to as Gripen). The most important objective ofthe performed analyses is to evaluate the software handling such as input, modellingaspects and output options. The accuracy of the analyses will be evaluated by compar-ison to previous flutter analyses done with the in-house programs. After a comparativeanalysis has been performed using ZAERO, new methods for flutter analysis will beinvestigated both with ZAERO and ZEUS.

3.1 Pre-processing

There are many steps involved for conducting flutter analyses. This section will gothrough the details of the pre-processing steps. The overall workflow of the pre-processing is as follows: first, a modal analysis is performed on the structural modelwhich generates the modal data. This data is later imported by the aeroelastic soft-ware program. An external program is used to generate the aerodynamic model inputdata. All input data and variables are combined into an input file. This includes theaerodynamic model, spline grid points, flight conditions, solution method and listingof output files.

ZAERO input file (.inp)

ZAERO analysis

CFD model geometry Structural model

Extract coordinates(Python script)

Modelling ofAerodynamic model

(Matlab script)

Modal analysis(MSC Nastran)

Flight conditions

Solution method

Modal data file (.f06)

Spline definition

Selectedgrid

po

ints

Output files

Figure 4: Diagram of the workflow.

3.1.1 Structural model

The structural part of the aeroelastic analysis comes from a modal analysis. An existingFE-model of Gripen was adapted and modified for the case in question. Followed bya modal analysis using Nastran to find the 30 first normal modes. The method usedwas the SOL 103 which finds the natural frequencies of the structure and outputs thevalues of displacement in form of translation and rotation for each eigenvalue in theform of a .f06-file which is then imported into the ZAERO script. The result from

16

the modal analysis was visualised in MSC Patran in order to check the validity of thesolution.

3.1.2 Aerodynamic model

Due to the panel method being used to generate the unsteady aerodynamic forces, flatpanels are used for all aerodynamic surfaces. These flat panels are required to havethe span-wise edges parallel to the x-axis, meaning zero angle of incidence. UnlikeZAERO, a ”2.5D” geometry is used in MSC Nastran and the AEREL meaning thatnon aerodynamic surfaces such as the fuselage is also modelled with flat plates followingthe criteria of zero angel of incidence in the geometry (see figure 5).

Figure 5: Example of Nastran model with ”2.5D”.

As ZAERO also uses a panel method, the aerodynamic surfaces are modelled in asimilar way with trapezoidal panels parallel to the direction of the flow. Unlike theother programs, ZAERO implements a high fidelity geometry (HFG) for the non liftingsurfaces, such as the fuselage and bodies of external stores. The panel geometry inenforced by other requirements compared to those of the flat panels (e.g. wings). Thecondition that two edges should be parallel to the flow is not required for these bodies,instead two edges must be orthogonal to the the flow direction. The convention ofthe coordinate system used in ZAERO demands that the flow direction is along thepositive x-axis. Positive z-axis is defined normal to the upper surface of the aircraftand the positive y-axis is positioned along the starboard side. The fuselage is definedby splitting the body in sections along the x-axis and defining points in the Y-Z-planemaking up the geometry of the body. There are three different methods to define abody:

• Body of revolutionA round body defined by a radius at each section along the x-axis together withthe number of divisions. Used primarily for the cylindrical bodies of missiles,rockets or external fuel tanks, but can also be used for the fuselage.

17

• EllipticalAn elliptical body defined by the semi-axis length in Y- and Z-direction. The res-olution is determined by the number of segments (along x-axis) and the numberof divisions for each ellipse.

• ArbitraryThe last option is to define all the nodes for a body at each section along thex-axis.

The nose of a body must start in a single point creating an initial section of triangularelements. A body can be split into multiple parts, to allow for different number ofdivisions in the sections. This allows for some triangular elements as well where thejoining section has a different number of divisions. This can be seen in figure 6 wherethe fuselage is split just in front of the canopy. This ZAERO model of a F-16 aircraftdemonstrates how the different methods of modelling a body can be used. The fuselageuses the arbitrary geometry modelling option4 while the external stores have cylindricalbodies using the body of revolution method.

Figure 6: ZAERO aerodynamic model of F-16 aircraft with external stores.[13]

The model in figure 6 also shows how the wings are modelled as flat plates using trape-zoidal elements with edges parallel to the flow direction. Multiple macro elements canbe used to represent the correct shape a wing, as can clearly be seen by the two macroelements making up the fin. The aerodynamic effects on a body is calculated differentlyfrom the panel method used for the wings. The volume effects of a body is calculatedusing a sheet of constant unsteady source singularity for each panel element of thebody[13]. It is prohibited to model wing-like components using the body definitionsas the source singularity method cannot satisfy the Kutta condition along the trailingedge.

4The use of the word ”arbitrary model” in this thesis refers to a body modelled with exact node-coordinates to represent the physical geometry more accurate (see the list of modelling options insection 3.1.2).

18

Modelling aspectsSince the structural grid for the unsteady aerodynamics solution need to follow strictrules, the aerodynamic model had to be assembled accordingly. A CFD model ofthe aircraft was used to construct the new panel geometry by splitting the modelinto sections and extracting the X-, Y- and Z-coordinates. The extraction of thesecoordinates was done automatically using a Python script together with the post-processing software EnSight and their Python Interpreter[19]. A Matlab script waswritten to process the coordinates to construct the panel mesh for the aerodynamicmodel. As this model should be a simplification with aerodynamic surfaces in the formof flat plates and a simplified fuselage, some modifications of the raw data modellingdata is required. First, the coordinate system used in the CFD and FE-model doesnot follow that of ZAERO. Therefore the orientation of the Y- and Z-axis is flipped.

Figure 7: Parts naming convention of JAS 39 Gripen E. Copyright Saab AB, photographerLinus Svensson.[20]

Wing-like components in forms of wings and pylons are constructed by findingthe edge locations of the leading edge, trailing edge, root edge and tip edge for eachcomponent. A macro element in form of a trapezoid is formed based on the edgelocations. Multiple trapezoidal elements are used if one is not enough to portray thecomponent accurately. A mean value is calculated for the z-location of the leadingand trailing edge to satisfy the condition of surface elements being parallel to the flowdirection. Meaning ∆z = 0 across the chord. A difference in z along the span is howeveraccepted. For the main wings, there is a small inclination along the span. However,this can be neglected which then also simplifies the modelling. The front canardsare modelled with their original inclination along the span. The fin is modelled as acontinuous surface without the fin pod (see figure 7). The leading edge on the mainwings is also modeled as a straight edge without the leading-edge extension (also calleddogtooth).

Body-like components make up the fuselage and external stores. Both the el-liptical and arbitrary body modelling option were used to model two versions of thefuselage. The raw geometric data from the CFD model was imported from multiple

19

sections along the x-axis. The y and z coordinates from each cross-section were thenconverted into polar coordinates with the origin being calculated from the mean ofthe bottom and top limits of the z-values. The fidelity of the model is based on thenumber of cross sections and the number of angular divisions ϕ. The best match fromthe converted polar coordinates to each value of ϕ is used in the final mesh when thearbitrary body modelling method was used (see figure 8). The converted raw polarcoordinates is then converted back to cartesian coordinates.

Figure 8: Using polar coordinates to model the fuselage sections.

The elliptical fuselage model used only the limiting values of the y- and z-coordinatesas the semi-radial values. The number of angular divisions is chosen together withthe number of cross sections to decide the fidelity of the model. The body of externalstores such as auxiliary fuel tanks and missiles are modelled in a similar way using anelliptical body or a body of revolution.

The fuselage modelled using the arbitrary body modelling option was renderedthrough a smoothing function to eliminate irregularities and miss placed grid points,as can be seen in figure 9 below. This was also done in order to eliminate sharpgradients in the geometry, which can cause problems when panels join at an anglelarger than the mach cone angle.

Figure 9: Smoothing of fuselage geometry.

Mesh resolutionOne of the advantages of using a panel method to calculate the aerodynamic forces isthe relatively small number of elements (panels) needed for accurate results, comparedto a CFD solution. However, the resolution of the panel mesh needs to be refinedat high reduced frequencies due to the increased number of waves the structure is

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subjected to. One of the components in the fundamental solution of the unsteadysmall disturbance equation[13] is:

e−ik(Mβ )

2( xL) (40)

where:β =

√|M2 − 1| (41)

and x is the coordinate in the flow direction. The term (Mβ

) grows rapidly when Mapproaches 1. The minimum element length ∆x can be found by normalising theperiod of (40) to be one along the full chord. This means setting:

k

(M

β

)2

= π and L =c

2

which gives e−(2πx/c)i. This results in a single period over the chord length. Theminimum resolution of panels along the chord of the wing in order to capture theoscillation is set to 12 divisions per period as can be seen in figure 10. This is used byZAERO as a criterion to get a converged solution.

Figure 10: Convergence criterion of the minimum number of boxes along the x-direction[13].

By using the minimum resolution mentioned above, a general limit can be given as:

∆x <c

12

π

k(Mβ

)2 =c

12

π

ω c2

V

(Mβ

)2 =1

12

V

f

1(Mβ

)2 (42)

For most flight conditions, this criterion is not a problem to fulfill. But for highfrequencies or Mach numbers close to 1, the number of chord divisions needed increasesdrastically.

The angle at which panels join is also monitored by the solver. It is highly recom-mended that the angle never exceed the Mach cone angle µ (eq. 43) when using theZONA7 unsteady aerodynamic method, as the linear theory fails for super inclinedboxes.

µ = sin−1(

1

M

)(43)

These kind of geometries is often present at inlets where this problem can be solved bydefining the affected panels as inlet panels and likewise with the corresponding outletpanels. Outlets and inlets was however not defined in the models used in this thesis,instead the geometry was smoothed until the inclination of the panels followed therestrictions.

21

3.1.3 ZAERO bulk data

ZAERO uses text files to define the aerodynamic model and the settings used for theanalysis. The text file (or input file) contains data in the form of bulk entry cards.A bulk data card defines a part of the model, analysis setting or output option tomention a few examples. Each card contains up to 10 columns and a finite numberof rows. The first column of the first row defines what bulk data card is used, forexample CAERO7. The following 8 columns are used for data input where every fieldhas a predefined data definition. The last column is used to mark if there are furtherrows that should be included and where to locate them. Each field has a set sizeof 8 characters, which is therefore the maximum data size for each field input. Atypical bulk data listing can be seen in the listing of the input data format below:

1 $ . . . 1 . . | . . . 2 . . . | . . . 3 . . . | . . . 4 . . . | . . . 5 . . . | . . . 6 . . . | . . . 7 . . . | . . . 8 . . . | . . . 9 . . . | . . . 1 0 . . |2 $CAERO7 WID LABEL ACOORD NSPAN NCHORD LSPAN ZTAIC PAFOIL7 CONT3 CAERO7 4001 LWING 18 19 4001 400 +CA14 $CONT XRL YRL ZRL RCH LRCHD ATTCHR ACORDR CONT5 +CA1 5000 . −2000. −4000. 6500 . 4002 1001 +CA26 $CONT XTL YTL ZTL TCH LTCHD ATTCHT ACORDT7 +CA2 7000 . −7500. −4000. 1760 . 4003

Listing 1: Example of a CAERO7 bulk data card entry

The CAERO7 card described above (listing 1) is used to model and mesh a wing.Not all data input fields are required for a simple model as can be seen by the emptyfields. It starts with declaring that this is a bulk data entry that defines a CAERO7card, followed by the identification number. LABEL is used to define a name for easieridentification when troubleshooting. NSPAN and NCHORD defines the number of divisionsalong the span and chord of the wing to define the element size and distribution. LSPAN,LRCHR and LRCHD are references to other bulk data cards with a set of fractions usedto define where the panel divisions should be located along the span, root chord andtip chord. XRL, YRL and ZRL defines the point location of the root leading edge inX,Y,Z-coordinates. The same is done for the tip leading edge point. The length unitused is defined in the beginning of the input file. In this case in [mm]. RCH and TCH

defines the chord length of the root and tip respectively. For attaching the wing to thefuselage, the field ATTCHR is used to declare that the root is attached to the part withID 1001, in this case the fuselage. Continuation markers (+CA1 and +CA2) are used todirect the compiler to the next row.

The assembly of the input file can be done manually, which is one of the reasonswhy this type of input data format has been chosen. Complex geometry quickly createsvery labor intensive work for calculating each coordinate and distance for each part ofthe model. It was decided early on that this pre-processing work should be automatedby a script. A script would also make it easier to alter the model, both geometricallyand for the panel element resolution.

The Matlab script, described in the previous sections, converts the geometry of therefined aerodynamic model to match the input format used by ZAERO.

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3.1.4 Final aerodynamic models used

A number of aerodynamic models were generated and used for the analyses. The mod-elling was done in steps, by adding parts and refining the model until the complexitywas deemed adequate and the geometry having good correlation with the physical air-craft. Only the final iterations will be presented here. It was deemed important toconduct a mesh sensitivity study to validate the resolution of the panel model. Belowis the three models used for this study together with the mesh parameters (figure 11and table 1). These models used an elliptical body as the fuselage.

(a) Iteration 1 (b) Iteration 2 (c) iteration 3

Figure 11: Aerodynamic models for clean aircraft

Model name Iteration 1 Iteration 2 Iteration 3Body type Elliptical Elliptical EllipticalNumber ofelements

2359 1811 2888

Number ofbody elements

508 820 1004

Number ofwing elements

1851 991 1884

Refined controlsurfaces

Yes Yes Yes

∆x 0.5 0.35 0.24

Table 1: Mesh parameters for aerodynamic models.

Below in figure 12 is one of the final aerodynamic models using the arbitrary bodymodelling option. This model was used to evaluate the effect of the fuselage modelaccuracy and is modelled based on the findings of the mesh sensitivity study.

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Figure 12: One of the final iterations using the arbitrary body modelling method. Config-uration with IRIS-T at the wing tips.

3.1.5 Airfoil thickness

The thickness and camber of an airfoil can be included in the analysis for supersonicspeeds when the ZONA7U method is used for calculating the generalized aerodynamicforces. In order to get the most accurate representation of the airfoil geometry, theCFD model was used to extract coordinates from a cross-section of each wing. Themain wings, canards and fin had an airfoil geometry defined. ZAERO defines the airfoilusing half thickness, camber and leading edge radius as seen below in figure 13.

Figure 13: Parameters used to define a airfoil shape for a PAFOIL7 bulk data card[13]

First, an AEFACT card is used to define x-coordinates along the wing where the camberand thickness lengths are defined. This is defined as a percentage of the chord, meaningthat it starts the first entry must be 0.0 and the last entry 100.0. Next, the camberline is defined as the distance (as a percentage of the chord) from the mean plane atthe defined points along the chord. Lastly, the thickness is given from the distance(as a percentage of the chord) from the camber line to the upper surface at the samepoints. An non constant airfoil shape along the span can be used by defining different

24

airfoil geometry at the root and tip, which is then linearly interpolated along the span.The orientation of the camber line depends on the surface normal of the wing macroelement. An example of the input data card that roughly models the airfoil in figure13 is seen below in listing 2.

1 $ . . . 1 . . | . . . 2 . . . | . . . 3 . . . | . . . 4 . . . | . . . 5 . . . | . . . 6 . . . | . . . 7 . . . | . . . 8 . . . | . . . 9 . . . | . . . 1 0 . . |2 $PAFOIL7 ID ITAX ITHR ICAMR RADR ITHT ICAMT RADT3 PAFOIL7 100 11 12 13 5 .0 12 13 5 .04 $5 AEFACT 11 0 .0 12 .5 25 .0 37 .5 50 .0 62 .5 75 .0 +AE16 +AE1 87 .5 100 .07 $8 AEFACT 12 0 .0 1 .2 2 .0 2 .2 1 .8 1 .0 0 .6 +AE19 +AE1 0 .2 0 .0

10 $11 AEFACT 13 −5.0 −2.6 0 .0 2 .4 2 .8 3 .0 3 .1 +AE112 +AE1 3 .1 3 .0

Listing 2: Example of a PAFOIL7 bulk data card entry with related AEFACT cards.

The number of points and their location used to specify the airfoil (ITAX bulk datacard) does not need to correlate to the element distribution of the wing macro element.The program automatically interpolates the points defining the airfoil to get the correctslope and thickness at each element. But it is of course most accurate to specify thepoints for the airfoil at each element intersection. This is done automatically by theMatlab script written for this thesis.

3.1.6 Control surfaces

The large and most important control surfaces of Gripen include the outer and innerelevons, outer and inner flaps, canards and the rudder (see figure 7). A control surfaceis in this case a surface that can have a discontinuous motion at the hinge linge, whichcan affect the aeroelastic response and stability. In the structural model (FE-model)the control surfaces can rotate around their respective hinge axis. Some structuralmodes clearly sets the parts in movement. Therefore, the movement of the controlsurfaces should be transferred accurately to the aerodynamic model.

Control surfaces can be defined in ZAERO from a set of elements (belonging toa macro element) and a hinge axis. The aerodynamic model was adapted to includeelement edges at the location of the hinge axis, together with a finer mesh resolutionfor each control surface. The geometry of the elevons and flaps was simplified to beequivalent to the geometry of the aerodynamic model used in Nastran and AERELwhich has been validated and proven to be accurate. This also reduces the uncertaintywhen comparing the results. The rudder was not included as a movable control surfacedue to difficulties when defining the hinge axis. This should, however, not affect theresults significantly as the rudder movement is not a critical part in the flutter analysesconducted in this project. A representation of the how the control surfaces are free tomove can be seen in figure 14 below.

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Figure 14: Inclusion of movable control surfaces.

3.1.7 Spline

As discussed previously, a transformation is needed for interpolation between the aero-dynamic and structural model. Based on a study done by Axel Baathe[21] it showsthat the number of spline points can be quite low whilst still reaching adequate res-olution of the mode shape. This is partly due to the interpolation that is occurringbetween the spline matrix and the control points of the aerodynamic boxes. All aero-dynamic boxes is given a displacement vector h with both the displacements and theirrespective slopes, as presented earlier in (15).

h = Gx

G is the spline matrix and x is the displacement vector from the spline points. Thereare four different spline methods used in ZAERO. Each with a method to project orextrapolate grid points of the FE-model to the parts of the aerodynamic model.

• Infinite Plate Spline Method (IPS)Used for flat panels, such as wings and other aerodynamic surfaces if shell el-ements are used in the FE-model. The displacement from the grid points isprojected onto the flat panel.

• Beam Spline MethodUsed if the FE-model has beam elements. Used both for body and wing elementsin the aerodynamic model.

• Thin Plate Spline Method (TPS)Used for body and wing panels if the FE-model has shell elements and the gridpoints should remain in 3-D space.

• Rigid Body AttachmentUsed if there is no particular grid point of the FE-model for the part in question,or if a single finite element with concentrated mass is used. An example of thisare the fins of a external store where the FE-model only uses a beam element forthe entire external store.

26

All four spline methods were used where applicable and automatically combined intothe spline matrix. The grid points used to define the spline can be seen in figure 15.The distribution and location of the nodes is based on previous analyses using Nastran,combined with recommendations and guidelines for ZAERO[13].

Figure 15: Grid points used for interpolating the modes from the structural model to theaerodynamic model (Spline).

3.2 Volume mesh for Euler equation solver

For running ZEUS, a volume mesh is needed to solve the Euler equations. The meshused is a structured Cartesian grid. ZEUS has developed an automated mesh generatorusing a ZAERO-to-ZEUS model which converts the aerodynamic surface mesh to asingle block of volume mesh. Refinements can be done in multiple ways using additionalbulk data cards.

3.2.1 Flow domain

The complete flow domain should have large enough dimensions to capture all flowgradients. The flow domain consists of one or more blocks. If only one body is included,one block should be sufficient[15]. Such as the case for analysing a clean airplanewithout external stores. The size of the flow domain (or global mesh block) is suggestedto be at least 8 aircraft lengths along the x-axis with at least 2 aircraft lengths in frontof the aircraft and 5 aircraft lengths aft of the aircraft[15]. The width of the globalmesh is recommended to be at least 3 times that of the full span of the aircraft, whennot using a symmetry plane. The surface mesh of all components inside each meshblock are projected onto the X-Y plane. Because all elements have matching cell faces,the mesh grid composition is constant through out the Z-direction in the block.

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3.2.2 Refinements

The structural grid mesh can be controlled and refined in multiple ways. Globalmesh parameters such as the size and growth rate in all directions can be modified.The growth rate was reduced from the default values to limit the size of the volumeelements far from the aircraft to satisfy the CFL measurement. A method called Y-Zone Technique is used for defining span wise division planes. A line-tracing methodis used to calculate how the element nodes should be connected between each Y-zoneplane. These planes are preferably located at the tip and root of each aerodynamicsurface macro element (CAERO7 entry). The grid structure close to the wings is refinedusing GAP-cards in conjunction with the Y-Zone planes. Gaps are defined similarly toCAERO7 entries. Some areas are problematic for the mesh generator to generate smoothand low skewed elements, such as between the wings and for and aft of wing edges.Here, GAP-entries can be used to enforce the distribution and alignment of the mesh.The GAP-entries used for the final mesh can be seen in figure 16 where the GAP-entriesare plotted in red together with the Y-zone lines in blue.

Figure 16: GAP definitions (red) and YZONE lines (blue). Plotted in Matlab.

Y-Zone planes where placed at every tip of a CAERO7 macro element as well as theouter edge of the fuselage to specify where the line tracing should take place. Theresulting mesh grid distribution along the X-Y-plane can be seen in figure 17 below.

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Figure 17: Top view of the final mesh used (blue). Aerodynamic model in red. Viewed inEnSight.

The effect of the line tracing method can clearly be seen as the grid stays parallel tothe sweep angle of the leading edges. Since the mesh is a fully connected structuredmesh, based on the panel distribution of the aerodynamic model, some areas are moredense in mesh elements, such as outside the middle of the pylons and the tip of thecanards. The GAP-entries also add additional grid lines that needs to be continuedthrough out the mesh. The effect of this can be seen inside the fuselage between thecanards and main wings. As the aerodynamic model is projected onto the X-Y plane,a vertical wing (such as the fin) needs a separate mesh block. Separate mesh blocksare also needed for external stores. The lower edge of the mesh block for the fin isplaced at the root of the fin and extends up, ending above then fin, as can be seen infigure 18.

Figure 18: Side view of the mesh block surrounding the fin. Aerodynamic model in red.Viewed in EnSight.

Since the aerodynamic model is projected onto the X-Y plane there are no referencegrid points in the Z-plane in which the mesh generator relates to. The default meshgenerated creates a growth rate starting from the top and bottom of the fuselage withequal distributed elements between. The height of the first cell above and below thewings is also set automatically as a small fraction of the reference chord length. Themesh distribution along the Z-axis can be controlled with GAPZ-entries which is similar

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to the GAP-entries used for the X-Y plane. These can control the number of grid linesand the distribution along the Z-axis. A GAPZ-entry was used for the wing and canardrespectively and the resulting distribution can be seen i figure 19.

(a) Front view of the mesh. (b) Side view of the mesh.

Figure 19: Mesh distribution for the clean aircraft without the mesh block for the fin.

The automated growth rate extending from the fuselages lower and upper edge can beseen in figure 19(a). The figure also shows the refined mesh distribution surroundingthe canards and main wings. The thickness of the airfoils from the aerodynamic modelis not used to define the volume mesh.

3.2.3 Mesh generation

A series of mesh iterations were used to find a suitable mesh configuration. Theparameters of the final mesh has been discussed in the sections above. Two differ-ent configurations of the aircraft were analysed, a clean Gripen without any externalstores and Gripen equipped with a IRIS-T missile at respective wing tip, the sameconfigurations analysed with ZAERO.

Figure 20: Flow domain mesh and overset mesh box for fin. 887 563 nodes, 856 963 cells.

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A set of mesh blocks are defined around the IRIS-T body and fins for this type ofaircraft configuration. A larger block is used as a flow domain for the missile and foursmaller blocks are used for the front and rear fin pairs, as can bee seen in figure 21.These four blocks are needed as the four fin pairs extends in all four directions andwould otherwise all be projected onto the same X-Y plane. A local coordinate systemis used to define a new X-Y plane for each block.

(a) Front view of the mesh. (b) Isometric view of the mesh.

Figure 21: Overset mesh block distribution for IRIS-T configuration including mesh blockfor the fin. 991 860 nodes, 948 664 cells.

3.3 Analysis

Both ZAERO and ZEUS were executed using UNIX/Linux (Windows/DOS versionsare available). The format of the input file is very similar between the both softwareprograms. It contains of three main parts:

Executive controlHere, the necessary directives for the computation are specified. This includes thefilename and format of the modal data from the FE-model and options for executionand diagnostics.

Case controlThe disciplines for the analysis is written in the case control section, such as whattype of analysis should be conducted and the order of execution (if there are multipleanalyses).

Bulk dataThe last and largest section of the input file is the data and parameters used for theanalysis. Here, the aerodynamic model is defined together with flight conditions andother variables, much of which have been presented in the previous sections.

3.3.1 Parameters for solving the flutter equation

For flutter analyses, which was the only discipline analysed in this project, there aresome parameters that can be adapted and altered to fit the case in question. The mainbulk data card can be seen below in listing 3.

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1 $ . . . 1 . . | . . . 2 . . . | . . . 3 . . . | . . . 4 . . . | . . . 5 . . . | . . . 6 . . . | . . . 7 . . . | . . . 8 . . . | . . . 9 . . . | . . . 1 0 . . |2 $ SETID SYM FIX NMODE TABDMP MLIST CONMLST NKSTEP3 FLUTTER 100 ASYML 1 30 2 3 4 25

Listing 3: Example of a FLUTTER bulk data card entry

After the reference identification (SETID) the type of model is specified (symmetric,anti-symmetric or asymmetric) together with the scheme for interpolating the gener-alised aerodynamic matrices. An asymmetric model with linear interpolation was usedfor the analyses in this thesis. The FIX entry refers to the bulk data card contain-ing the flight conditions. Depending on what type of flight condition parameters arechosen, the analysis becomes either a matched or non-matched point analysis, wherethe K-method can be applied to the non-matched analyses, otherwise the g-methodis used. NMODE determines the number of modes used for the calculations. This op-tion was used for the ZEUS analyses to reduce computing time by excluding someof the higher structural modes. If a special selection of modes is to be included, theMLIST entry refers to a list of selected modes to be used. Structural modal damp-ing can be included as a list of damping values coupled with frequency values withthe TABDMP-entry. Mass perturbation can be defined in a similar manner using theCONMLST-entry. Lastly, NKSTEP defined the number of reduced frequency steps used forthe reduced-frequency-sweep technique for calculations using the g-method.

3.3.2 Euler solver parameters

The parameters used in the Euler-Solver module in ZEUS can be changed to better fitthe analysis in question.

1 $ . . . 1 . . | . . . 2 . . . | . . . 3 . . . | . . . 4 . . . | . . . 5 . . . | . . . 6 . . . | . . . 7 . . . | . . . 8 . . . | . . . 9 . . . | . . . 1 0 . . |2 $ IDPARAM METHOD TRMSTEP FLTSTEP NEWTN NCYC LVRSMOO PRNTCOV CONT3 MKPARAM 300 3 101 50 4 2 1 +MM14 $CONT CFL GAMMA VIS2 VIS4 TVDCOEF5 +MM1 5.0

Listing 4: Example of a MKPARAM bulk data card entry

The first decision is that of which method should be used to solve the Euler equations,meaning whether or not to use the linearised Euler-solver. There is also an optionfor preconditioning used for low Mach numbers (M < 0.2) which is not consideredfor these simulations. The linearised Euler-solver is used for this thesis (METHOD =

3), which adds further parameters can be changed using the MKPARAL bulk data card.A static aeroelastic solution is first computed, where TRMSTEP defines the number oftime steps used. Next, GAF from the frequency-domain is calculated based on thenumber of time steps defined in FLTSTEP. These are the calculated forces used in theflutter analysis. For each time step, a number (NEWTN) of Newton sub-iterations areperformed. These sub-iterations are important when using multiple mesh blocks orboundary layer coupling, due to the communication between the block borders whichis done at the end of each sub-iteration. The stability of the solution can be enhanced

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using residual smoothing (LVRSMOO) which also allows for a higher CFL value. TheCFL (Courant-Friedrichs-Lewy) number is used for controlling the time-step size (seeequation 44). ZEUS allows for a maximum CFL number of 8.0. The high CFL numberis possible due to the five-stage explicit Runge-Kutta pseudo-time marching methodbeing used in combination with the residual smoothing[15][22]. A CFL number of 5.0was used in these simulations to balance robust convergence and fast computing time.

CFL =∆t∗

∆tc=

∆t∗

L/V(44)

The option to use a viscous sub-layer is specified in the VISCOUS-bulk data card. Thereare 24 parameters that can be altered for defining the boundary layer. Only twowere altered for these simulations, Reynolds number and free stream temperature,the rest was kept at their default values. The Reynolds number was calculated usingthe reference chord length defined in the AEROZ-bulk data entry and based on fluidproperties at 300 degrees Kelvin. The Reynolds number was in the order of RE =2.0 · 107.

3.4 Post-processing

The results from the simulation can be extracted as both 2D plots and plots of theaerodynamic model. ZAERO supports a number of commercial file formats for post-processing:

• Tecplot (Tecplot Inc.)

• Patran (MSC Software)

• I-DEAS (Siemens PLM Software)

• FEMAP (Siemens PLM Software)

• Ansys (ANSYS)

• Nastran (MSC Software)

The same list of formats are available in ZEUS, except for the aerodynamic modeland steady pressure results where only Tecplot is supported. The external flow of thesteady solution is only available in PLOT3D-format. In this thesis, MSC Patran hasbeen used to visualize the results from ZAERO. The resulting files from ZEUS wasvisualized using Ansys EnSight where both Tecplot and PLOT3D files can be read.The 2D plots are exported as text file tables. A Matlab script was written to plotthe data. Some result values are only accessed through the output file (.out-file), forwhich a different Matlab script was written to find and extract the values of interestautomatically.

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4 Results and discussion

The results presented in this section will follow the order in which they where produced.Starting with validation of the interpolation between the structural and aerodynamicmodel. Followed by the mesh sensitivity study and investigations of how differentparameters affect the results. The difference between ZAERO and ZEUS will then bepresented, followed by a comparison with previous analyses and experimental data.

4.1 Interpolation between structural and aerodynamic model

The accuracy of the interpolation done by the spline matrix is verified by comparingthe modes of the FE-model and the interpolated modes on the aerodynamic model.

(a) Mode 1, FE-model (b) Mode 1, aerodynamic model

(c) Mode 2, FE-model (d) Mode 2, aerodynamic model

Figure 22: Comparison of modes for the structural and aerodynamic model (iteration 1).Color gradient marks deformation.

The first 20 modes are checked for each aerodynamic model in the same way as seenabove in figure 22, both the overall shape and unique details of the deformation arecompared. Signs of nonphysical deformation could be due to an incorrect choice ofnodes, which must be addressed. The conclusion is that the spline is of high enoughresolution to capture the motion of the structural vibrations. The analysis can thereforecontinue with confidence that the displacements are accurate. The aerodynamic model

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shown in figure 22 is modelled without movable control surfaces. The effect from thiscan clearly be seen where the discontinuity from the gap between the elevons resultsin a sharp deformation gradient instead. This is most visible in figure 22(d).

4.2 Mesh sensitivity study

The effect of mesh resolution for the aerodynamic model was compared for three dif-ferent cases (see figure 11). Iteration 1 was modelled with a coarse resolution for thefuselage and fine resolution for the wings. This triggered the criterion discussed in sec-tion 3.1.2 and the minimum ∆x-value was not fulfilled for a large part of the reducedfrequency list. Iteration 2 was modelled with a finer panel resolution for the fuselageand a coarser panel resolution of the wings to minimize the number of panel elementswhilst keeping the ∆x-value low. The final model, iteration 3, was modelled as a finermesh with a lower ∆x-value. Both iteration 2 and 3 had a fuselage with elementsaligned with those of the wing roots where the wings attach. This was done in order tominimise the risk of numerical errors that can occur when an adjacent panel elementedge aligns with the aerodynamic control point.

The analysis used for the mesh sensitivity study was a matched-point flutter analy-sis at a fixed altitude and varying mach number, varying from subsonic to supersonic.A large part of the transonic mach range was excluded from these simulations as onlythe subsonic and supersonic aerodynamic solver (ZONA6 and ZONA7) was used. Alow altitude was chosen to evoke flutter. The flight conditions are outside the flightenvelope and are chosen only to evaluate the solution method of ZAERO. 19 mach num-bers and 14 reduced frequencies (for each Mach number) were used in the analysis.The computing time together with the combined file size of the stored AIC-matricescan be read in table 2.

Model name Iteration 1 Iteration 2 Iteration 3tot. Solution timeCPUh [hh:mm:ss]

07:14:12 05:08:02 12:04:36

comb. File size [GB] 25.3 13.5 34.6

Table 2: Mesh parameters for aerodynamic models.

The difference in computation time was as expected, where the model with the finestmesh taking the longest time to calculate. The computation time increases roughlyquadratically with increasing number of elements. The resulting file size come primarilyfrom saving the AIC-matrices for each mach number. Once calculated, these AIC-matrices can be reused for new calculations which saves time. It was concluded thatthe majority of the computation time (>99%) is used to calculate the AIC-matrices.The actual flutter calculations are fast in comparison. The resulting frequency andaerodynamic damping values are similar in the general sense. No large deviations werefound. It was found that the finer mesh results in a more conservative flutter velocityprediction as can be seen in plot 23 for two modes, where mode b contributes to flutter.

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Figure 23: Isolation of two modes from different panel mesh resolution.

Although the result differs between the different model iterations, the overall propertiesof the results remains the same and the difference in aeroelastic response is minimal.It was therefore decided to proceed using the mesh size of iteration 2 to minimise thecomputation time. The finest mesh of iteration 3 should arguably provide more accu-rate results, but the coarser mesh of iteration 2 is considered adequate for comparisonpurposes.

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4.3 Comparison between fuselage models

In order to fully utilize the HFG implemented in ZAERO, an aerodynamic model usingthe arbitrary body method (see figure 12) was compared to the use of an ellipticalfuselage of comparable resolution. The flight conditions and case were identical tothose of the mesh sensitivity study.

Figure 24: V-g plot on selected modes from the different fuselage modelling methods.

Only the structural modes where the fuselage had significant deformation showed adifference in the damping curves. These fuselage dominated modes are shown in figure24. The remaining modes had effectively the same aeroelastic response between thefuselage models. The reasoning for this result should come from the small influencethe fuselage has on the aeroelastic response in this case. Combining the stiffness ofthe fuselage with the relatively small contribution to aerodynamic lift means that theaccuracy of the aerodynamic forces acting on the fuselage have little influence on thestability of the system for this configuration. However, aircraft configurations withother external stores could have a larger impact on the aerolastic behaviour of thefuselage, such as auxiliary fuel tanks mounted under the fuselage. It is concludedthat an elliptical model of the fuselage is a valid option for this particular aircraftconfiguration.

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4.4 Differences when including movable control surfaces

A model with movable control surfaces (see figure 14) was analysed and compared toan identical model without movable control surfaces. The flight case was the same asfor the mesh sensitivity study.

Figure 25: Results with and without movable control surfaces.

The resulting V-g plot for 3 modes can be seen in figure 25. There is a significantdifference in the predicted damping for two of the modes and a significant deviationin the frequency for one of the modes. Different mode shapes have various effectson the control surfaces. For instance, the shape of mode a and c are dominated byanti-symmetric torsion of the main wings which actives the movement of the elevonsand leading-edge flaps. This should explain the difference in resulting aerodynamicdamping and frequency. The inclusion of movable control surfaces are implementedin the analyses done by Saab and is recommended in the ZAERO user’s manual[13].It is concluded, based on this study and the underlying theory, that the inclusion ofmovable control surfaces is needed for an accurate solution.

4.5 The effect of airfoil thickness

A comparison between ZONA7 and ZONA7U was done by running a flutter analysisat various Mach number and low altitude. The ZONA7U method is only applicablefor supersonic flows. The resulting V-g plot from some selected modes can be seenbelow in figure 26 where the blue lines is the resulting frequency and damping usingthe ZONA7U method. A large deviation in both frequency and damping can be seennear in the lower supersonic region (the first measuring points from the blue lines). Itis difficult to conclude which of the methods is most accurate in this region. ZONA7Uis presented as an unsteady aerodynamics solver for supersonic and hypersonic flow[13]and is more suitable for higher Mach numbers. The sharp gradient in the frequency

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curve could mean that this is an unreliable solution for the lower Mach numbers.However, the result of higher Mach numbers in the supersonic region gives slightlymore conservative damping values, as is previously documented with this method[9].

Figure 26: Results with and without thickness effect from airfoil (ZONA7U).

4.6 ZEUS rigid aircraft results

To validate the accuracy of the Euler solution, the results of a steady rigid aircraft(k = 0) was analysed. ZEUS gives the possibility to extract the full CFD solution forthe steady flow case and import it to a third party post-processing software (in thiscase Ansys EnSight).

Figure 27: Convergence plot for the steady flow calcu-lation. Residuals from the global mesh block solution.

It gives an insight to the solutionof the inviscid flow. The steadysolution only results in the exter-nal flow, and not the surface pro-jected values. This is due to thecartesian grid in which the aero-dynamic surfaces are still panelswith zero thickness. However,the pressure (Cp) acting on thesurface of the aircraft can be ex-ported from the unsteady aero-dynamic solution from steadyflight with k = 0. The comput-ing time for the steady flow solu-tion was approximately 20 min-utes using 4 CPU-cores. Figure27 shows the convergence for the global mesh block (which includes the fuselage and

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horizontal wings). Each time step iteration had 4 sub-iterations which solves for twoEuler cycles each. 101 time step iterations were used which results in 808 Euler cycleiterations in total, for each mesh block. The figure shows a stable and fast convergencerate.

General flowThe overall solution to the steady flow gave reasonable results. The lack of details inthe flow solution was expected due to the low mesh resolution (compared to a moredetailed CFD calculation). A comparison between an in-house CFD analysis and thesteady flow solution from ZEUS shows (see figure 28) the similarities of the overallpressure distribution. Both the main wings and canards have a good correlation to thein-house CFD results. The fuselage was modelled as an elliptical body and is not ex-pected to reach the same level of correlation. However, the magnitude of the pressureacting on the fuselage is similar to that of the in-house CFD results.

Figure 28: Pressure distribution comparison between ZEUS (top) and in-house CFD.

The flow around the wing (airfoil) did not match the in-house CFD solution in someareas for transonic flight. The top surface of the wings shows signs of a shock waveand rarefaction, similar to that of the in-house CFD solution. In contrary, the bottomsurface shows no sign of a normal shockwave which was expected in some extent.

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Viscous effectsBoth calculations with and without viscous sub-layer were conducted, but no significantdifference was noted in the results. This should be expected as the high Reynoldsnumber results in a thin boundary layer and the mesh resolution could be too courseto resolve the effects of shock-induced separation for example. The viscous effects doesnot generally affect the aeroelastic response, but there are some exceptions such as stallflutter and shock waves[23]. The viscous effects can change the position and strengthof the shock wave, which in term could result in a change in stability. The theory oftransonic flutter will not be discussed in length in this thesis, as it is a separate issue.The conclusion is that viscous effects could have an effect on flutter in special cases,and further investigation of how ZEUS calculates the viscous layer is needed. However,the inviscid Euler equation should be a valid approach for most cases of flutter analysis.

4.7 Panel method vs Euler solution

A direct comparison between ZAERO and ZEUS was made for a matched point flutteranalysis. The mach number was in the transonic region below the speed of sound, M<1.Flutter was induced by varying the altitude to well below sea level. The AoA was setto 2°. The mesh resolution for the aerodynamic model was close to the theoreticallimit of coarseness, as stated in equation (42), in order to reduce the computing time.The same flight conditions were set for both analyses with 5 reduced frequencies beingused for the solution. ZEUS calculates the resulting unsteady aerodynamic forces foreach reduced frequency (and each structural mode). Only 8 modes were used for theseanalyses for efficiency. This should not have an effect on the validity of the results.

ZEUSNbr volumeelements

887 562

CFL 5.0Iterations/k 808Nbr CPU 4Total time[hh:mm:ss]

16:07:38

GAF-file size[MB]

3.1

Table 3: ZEUS settings

ZAERONbr bodyelements

744

Nbr wingelements

1508

Nbr CPU 1Total time[hh:mm:ss]

00:11:37

AIC-file size[MB]

526.6

Table 4: ZAERO settings

The convergence rate for part of the unsteady aerodynamic calculations are presentedin appendix B. It shows a stable convergence for all 8 modes. The resulting V-g plotscan be seen in figure 33 for two aeroelastic modes. ZEUS predicts zero damping (leadingto flutter) at lower dynamic pressure. It is known that transonic effects give rise toflutter at a lower dynamic pressure (known as the transonic dip)[24]. ZEUS’s Eulersolution should be more accurate in the transonic region compared the ZONA6 methodused here by ZAERO. There are other methods implemented in ZAERO specifically fortransonic flight conditions which has not been used here. It can therefore be impliedthat ZEUS gives more accurate results for this analysis.

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Figure 29: Result from ZAERO and ZEUS flutter analysis.

It was noted that ZAERO and ZEUS makes different predictions to which structuralmode dominates the aeroelastic mode. This should not have a significant effect on theresults, other than the index of the aeroelastic modes.

However, there is currently5 no option to track the influence of the structural modeson the aeroelastic modes except for at the point of flutter. It was also noted thatthe ”mode tracking” for both ZAERO and ZEUS had some contradicting methods indetermining the index of the aeroelastic mode. This became an issue in the next studywhen comparing aeroelastic modes between different calculations.

(a) ZAERO (b) ZEUS

Figure 30: Pressure plot for symmetric wing bending mode at the point of maximumdeformation. Top view of right wing.

The difference in unsteady aerodynamic forces can be compared for each structuralmode. Figure 30 shows the pressure distribution for a symmetric wing bending mode atmaximum deformation (maximum bending). ZEUS calculates a larger and more evenlydistributed low pressure area covering the wing. The spots of low pressure locatedclose to the trailing edge in the ZAERO results (fig. 30(a)) is due to the movement

5Currently, at the time of writing this thesis. This issue may be addressed by Zona Technologiesin the future as it should be possible to implement.

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of the inner elevon. There is a significant difference in pressure acting on the fins ofthe IRIS-T missile mounted on the wing tip when comparing both results. ZAEROcalculates a more localised pressure gradient at the leading edge of all fins (front/rearand inner/outer). The Euler solution results in a more evenly distributed low pressureregion across the outer fins and a dissimilar pressure acting on the inner fins. Thiscould be an effect of the communication between the solution in the overset boxes fromthe Chimera scheme where the flow around the pylon and wing tip significantly affectsthe inner fins.

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4.8 Comparison with previous simulations

The final study is a comparison with previous analyses done by Saab. ZAERO andZEUS is compared to the linear method of AEREL, in-house CFD and experimentaldata from flight test. Both ZONA6/7 and ZONA7U are included as methods fromZAERO for calculating the aerodynamic forces. The aircraft configuration analysed isthe IRIS-T equipped Gripen, as has been described previously. The lowest order modeswere compared between the different methods and will be presented in the form of V-gplots. The calculations from ZAERO and ZEUS were done without the inclusion ofstructural damping.

Figure 31: Validation of ZAERO (ZONA6/7/7U) and ZEUS by comparing damping andfrequency to DLM, CFD and flight test results. Antisymmetric wing bending mode.

The overall agreement between the methods was quite good. Only a few distinctdifferences were noted across all modes. The first observation is the difference in theactual damping values between ZEUS and the other methods. Figure 31 shows a shiftin the damping values compared to the CFD results. The ZONA6/7 results are muchcloser to the equivalent DLM results. As both ZAERO and ZEUS uses the g-methodto solve the flutter equation to find the damping values (the DLM and CFD methoduses a modified P-K method), the difference in damping must come from a differencein the calculated aerodynamic forces. The trend of the damping curve from ZEUSis very similar to that of CFD and flight test results when neglecting the shift indamping values. The linear analysis using ZAERO shows a close relation to the linearDLM method. As observed earlier, the ZONA7U method deviates from ZONA7 inthe lower supersonic region. Another trend is deviating frequency values predicted by

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ZEUS. This occurred for multiple modes resulting in both higher and lower frequenciescompared to the other methods.

(a) Antisymmetric fin bending mode. (b) Symmetric wing bending mode.

Figure 32: Validation of ZAERO (ZONA6/7/7U) and ZEUS by comparing damping andfrequency to DLM, CFD and flight test results.

Figure 32(a) above shows how ZEUS follows the trend of the CFD based calculationsand flight test. It also shows how ZONA6/7 has a better correlation to the CFD methodthan the DLM results in the transonic region for this particular mode. However, forother modes such as the symmetric wing bending dominated mode (see figure 32(b)),a closer correlation between ZAERO and the CFD method was observed only in thehigher Mach numbers. The damping results in the lower Mach numbers has a muchcloser resemblance to the DLM method. This is also the mode where ZEUS deviatesthe most from the CFD solution with a large shift in damping values and dissimilardamping trends.

The V-g plots for the remaining aeroelastic modes has similar trends with an overallgood agreement between ZEUS and CFD, and close correlation between ZONA6/7 andDLM with some resemblance of the damping trends from the CFD method.

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4.9 Problems and issues

A collection of the problems faced is summarised and further explained in this section.

Panel resolution for transonic flightThe element size (∆x) has to be drastically decreased as the Mach number approachesM=1, as discussed in section 3.1.2. The mesh sensitivity study showed that the com-puting time increases roughly quadratically with increased number of elements. Thisis not a ZAERO-specific problem, but is derived from the unsteady small disturbanceequation which is used by ZONA6/7. In this project, the ∆x-criterion was violatedmany times and a finer panel resolution would be more appropriate to use for increas-ing the accuracy. But as ZONA6 and ZONA7 isn’t the recommended methods forsolving transonic flow in ZAERO, this can be somewhat overseen.

Fuselage modellingIt was anticipated that a large amount of work would go into modelling the fuselage.Which was correctly assumed. However, this only has to be done once. Once a modelof the fuselage is created, it can be reused. The current state of the fuselage model isquite close to the physical representation, but needs a bit more refinement. The ”2.5D”aerodynamic model used by AEREL and Nastran (See figure 5) requires significantlyless effort in modelling.

Volume mesh qualityA structured grid with matching cell faces comes with some challenges. Primarily keep-ing a high mesh quality for elements adjacent to the aircraft, where the element aspectratio is far from ideal. The volume mesh grid is based on the panel elements fromthe aerodynamic model and is therefore very coarse in the X-Y plane (compared to atraditional CFD mesh). In order to generate volume elements with a low aspect ratio,the first element layer height would surely be too large to capture the flow gradients.The Euler solver used in ZEUS seems to able to handle these high aspect ratio elements.

Memory allocation for ZEUSAlthough the calculations are very robust for ZEUS, a large stack size is needed for theGAF-calculations. The default stack size of 4096 bytes is not enough and triggers asegmentation fault if not increased. This can easily be changed temporarily for the sys-tem, but is worth to keep in mind. The maximum allocated memory allowed by ZEUSis currently 1600 MB, which has been proved to be sufficient for these calculations.However, the fraction of maximum allocated memory used for the Chimera-methodoperations needed to be increased by using the PARAM CHIMERA bulk data entry.

File storage size of AIC-matricesIt was noted early in the project that the file size of the stored AIC-matrices was verylarge, as can be seen in table 2. It was found that the file size of similar calculationsusing ZEUS resulted in much more reasonable storage usage for saving the GAF dataas can be seen in table 3 and 4. This query was forwarded to Zona Technologieswhom confirmed that it is possible to generate AIC-matrix files of that order in size.The AIC-matrix files are encoded and the content can therefore not be investigated

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in further detail. This could become a problem when hundreds of different aircraftconfigurations and flight conditions are analysed and large disk space is needed on thecomputer cluster. The option is to not save the AIC-matrices after the analysis. Thiswould however increase the time spent on computing as the whole simulation has tobe re-run for small changes, where a stored AIC-matrix could be retrieved instead.

Lack of matched point analysis options in ZEUSOne of the many benefits with ZAERO, compared to other software, is the possibilityto do a matched point flutter analysis with varying Mach numbers and a constant al-titude. The results from this analysis is then easy to compare to flight test data whichis acquired in a similar way. ZEUS does not include the same options for defining theflutter analysis flight conditions. The only matched point option for ZEUS is a con-stant Mach number and varying altitude. This is likely due to the long computing timerequired for calculating the aerodynamic forces for each Mach number. The methodused to generate the results for the comparison study (figure 31) was simply to analyseeach Mach number separately and then combine the frequency and damping values fora chosen altitude. This requires more pre- and post-processing compared to the flightcondition options in ZAERO.

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5 Conclusion

5.1 Achievement of the objectives

The main objective of this thesis was to investigate and evaluate the software ZAEROand ZEUS as flutter analysis tools. This was done primarily by conducting simulationsof Gripen and comparing both with previous analysis together with experimental data.As well as comparing ZAERO to ZEUS and finding the respective benefits and limi-tations. The ease of use and implementation was evaluated during the whole process.

As this was new software to both myself and the department at Saab, the scopeof this evaluation was quite large. This project resulted in many successful analyseswith varying levels of detail and accuracy. The success is largely due to the extensivedocumentation for both ZAERO and ZEUS together with the high level of expertiseand knowledge at the department at Saab.

Two configurations of Gripen were analysed; a clean aircraft and an aircraft con-figured with two IRIS-T missiles located at the wing tips. The aerodynamic panelmodel was created from scratch using a self developed Matlab program to adapt to therules and restrictions of the panel method. Different flight conditions were analysed.Subsonic, transonic and supersonic flight were combined with different altitudes.

Many of the modeling options and parameters were first evaluated and then used foranalyses which could be compared to the equivalent in-house methods and experimentaldata. Although there was some deviating trends in the results, compared to the resultsfrom previous analyses, the overall agreement of the results is quite good, both forZEUS and ZAERO. Many parameters can be modified or refined to increase accuracyeven further.

5.2 Concluding remarks

Both ZAERO and ZEUS show potential as software programs to fill the gap betweenthe in-house linear DLM- and nonlinear CFD-based calculations. The handling ofthe programs was aided by well-written documentation and examples, with clear errormessages and options for debugging. All computational steps seems to be robust andefficient. A few issues were found which can be worked around in most cases. Thecomputing time for these analyses is concluded to have the potential for improvement.It was shown that both ZAERO and ZEUS are robust enough to decrease the comput-ing time even more with some refinement of the aerodynamic model and the solutionmethod. Many useful results can be exported from both software and imported tomultiple post-processing programs. Since the input to the programs is written in textfile format, many processes can easily be automated and adapted to enable a highlyautomated work flow.

The final analyses in this project showed good correlation with previous analysesand experimental data, and there are many parameters and refinement options thathave not been evaluated which could be used to improve the accuracy further.

In conclusion, this evaluation finds both ZAERO and ZEUS to be performing wellfor flutter analysis and should be considered to be incorporated into Saab’s researchand development.

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5.3 Future research and development

There are many parts of the analyses that can be improved and made more effectiveif Saab chooses to continue with these software programs.

Refinement and correction of the aerodynamic modelThe aerodynamic model developed in this thesis was based on a combination of theprevious aerodynamic panel model used by Saab and by the physical geometry of theaircraft. A thorough check of the geometry is needed to find areas where the geometrycan be simplified or should be more complex to better capture the aerodynamic prop-erties. Also, the control surface of the rudder has to be defined in future analyses andthe effects of using inlet and outlet on the fuselage should be evaluated.

Develop a more user friendly program for generating the aerodynamicmodelThe Matlab script written for generating the aerodynamic model works, but is notwritten with user friendliness as a priority. The length of the script is currently ∼2700lines, much without proper documentation. As the main priority of the script was togenerate and refine the aerodynamic model, comments and documentation on how touse the script was omitted.

Model more external storesAn important part of the development of military aircraft are the flutter analyses ofdifferent aircraft configurations. This would require new aerodynamic models for theexternal stores that the aircraft is capable of carrying. Only one type of external storewas modelled and used for this project.

Transonic methods in ZAEROZAERO includes two methods for calculating the unsteady aerodynamic forces in thetransonic region and one method for sonic speed (M = 1). These methods havenot yet been evaluated. The first transonic method, ZTAIC, uses externally-providedpressure data (CFD or experimental data). The other method, ZTRAN, solves thetime-linearised transonic small disturbance equation using a local volume mesh, sim-ilar to the overset mesh block used in ZEUS with the aid of imported CFD solutionfrom a structured CFD code.

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References

[1] H. Dowell. A Modern Course in Aeroelasticity: Fifth Revised and Enlarged Edi-tion. Solid Mechanics and Its Applications. Springer International Publishing,2014. ISBN 9783319094533.

[2] J. R. Wright and J. E. Cooper. Introduction to Aircraft Aeroelasticity and Loads.John Wiley & Sons, Ltd, West sussex PO198SQ, England, 2007. ISBN 978-0470-85840-0.

[3] Zona Technology. ZAERO Theoretical Manual. Zona Technology, Inc, 9489 E.Ironwood Square Drive, Scottsdale, AZ 85258-4578, version 17.1 edition, 01 2017.

[4] D. Borglund and D. Eller. Aeroelasticity of slender wing structures in low-speedairflow. Lecture notes, KTH Aeronautical and Vehicle Engineering, 10 2010.

[5] M.F. Platzer K.D. Jones. Steady and unsteady aerodynamics. WIT Transac-tions on State of the Art in Science and Engineering, 4, 2006. doi: 10.2495/1-84564-095-0/6a.

[6] W. P. Rodden and Johnson E. H. MSC.Nastran Version 68 Aeroelastic AnalysisUser’s Guide. MSC Software Corporation, 2 MacArthur Place, Santa Ana, CA92707 USA, version 68 edition, 2004.

[7] D. Liu, P. Chen, D. James, and Anthony Pototzky. Further studies of harmonicgradient method for supersonic aeroelastic applications. Journal of Aircraft - JAIRCRAFT, 28:598–605, 09 1991. doi: 10.2514/3.46070.

[8] P.-C Chen and D. Liu. A harmonic gradient method for unsteady supersonicflow calculations. Journal of Aircraft - J AIRCRAFT, 22:371–379, 05 1985. doi:10.2514/3.45134.

[9] ZONA Technology INC. Zaero basic training - aerodynamic modeling guidelines,trim and flutter analysis. Lecture notes, ZONA Technology Inc, 9489 E. IronwoodSq. Dr. Scottsdale, AZ 85258, 08 2011.

[10] W.P. Rodden. Aeroelastic addition to NASTRAN. NASA contractor report.National Aeronautics and Space Administration, NASA, 1979. URL https:

//books.google.se/books?id=jBMxAQAAIAAJ.

[11] P. Chen. Damping perturbation method for flutter solution: The g-method. AiaaJournal - AIAA J, 38, 09 2000. doi: 10.2514/2.1171.

[12] V. J. Stark. The aerel flutter prediction system. International Council of theAeronautical Sciences Paper 90-1.2.3, Bonn, Germany, 1990.

[13] Zona Technology. ZAERO User’s Manual. Zona Technology, Inc, 9489 E. Iron-wood Square Drive, Scottsdale, AZ 85258-4578, version 9.2 edition, 01 2017.

[14] Zona Technology. About ZEUS. Zona Technology, Inc, 9489 E. Ironwood SquareDrive, Scottsdale, AZ 85258-4578.

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[15] Zona Technology. ZEUS User’s Manual. Zona Technology, Inc, 9489 E. IronwoodSquare Drive, Scottsdale, AZ 85258-4578, version 3.9 edition, 01 2017.

[16] Dougherty F. c. Steger, J. L. and J. A. Benek. A chimera grid scheme. Advancesin Grid Generation, K. N. Ghia and U. Ghia, eds., ASME FED, 5, 06 1983.

[17] D.J. Kirshman and F. Liu. Flutter prediction by an euler method on non-movingcartesian grids with gridless boundary conditions. Computers Fluids, 35(6):571 – 586, 2006. ISSN 0045-7930. doi: https://doi.org/10.1016/j.compfluid.2005.04.004. URL http://www.sciencedirect.com/science/article/pii/

S0045793005000708.

[18] M. Carlsson Leijonhudvud and A. Karlsson. Industrial application of robust aeroe-lastic analysis. Journal of Aircraft - J AIRCRAFT, 48:1176–1183, 07 2011. doi:10.2514/1.C031170.

[19] Computational Engineering International Inc. EnSight Interface Manual. Com-putational Engineering International Inc., 2166 N. Salem street, Suite 101, Apex,NC 27523 USA, 8.2-1 edition, 2006.

[20] Linus Svensson, Saab AB. Gripen e - [39-10] - maiden flight, 2019. URL https://

mediaportal.saabgroup.com/#/items/30454. [Online; accessed April 11, 2020],Modified by Nils Voigt Dahl.

[21] Axel Baathe. Transonic flutter for a generic fighter configuration. Master’sthesis, KTH, Skolan for teknikvetenskap (SCI), Farkost och flyg, Flygdynamik.,Brinellvagen 8, 114 28 Stockholm, 2018.

[22] Antony Jameson, Wolfgang Schmidt, and Eli Turkel. Numerical solution of theeuler equations by finite volume methods using runge-kutta time stepping schemes.AIAA Paper, 81, 07 1981. doi: 10.2514/6.1981-1259.

[23] Pradeepa T. Karnick and Kartik Venkatraman. Shock–boundary layer interactionand energetics in transonic flutter. Journal of Fluid Mechanics, 832:212–240, 2017.doi: 10.1017/jfm.2017.629.

[24] H. Tijdeman. Investigation of the transonic flow around oscillating airfoils. PhDthesis, 01 1977.

[25] H. Ashley R. L. Bisplinghoff and R. L. Halfman. Aeroelasticity. Dover Publica-tions, Inc, 31 East 2nd Street, Mineola, N.Y 11501, 1996. ISBN 0-486-69189-6.

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A Appendix: Wagner’s function and change in lift

on a 2D airfoil

The main factor to determine the time to reach a new steady state after a step changein pitch and/or heave is the speed at which a particle travels over the wing, which isdependent on both the chord length and the free stream velocity. A non-dimensionalparameter is introduced to combine these factors[2]:

τ =2V t

c=V t

b(45)

The value of τ is the number of semi-chords traveled at time t, where V is the freestream velocity and b = c/2 is the semi-chord. The change in lift for a simple 2D airfoilcan now be expressed by rewriting the equation of lift as:

∆Lf =1

2ρV 2ca1∆αΦ(τ) (46)

where ρ is the air density, a1 is the lift-curve slope (a1 = dCL/dα), ∆α is the changein angle of incidence (pitch) and Φ(τ) is Wagner’s function. Wagner’s function is usedto model how the effective downwash at the three-quarter (3c

4) point affects the lift

at the quarter chord ( c4) builds following a step change of angle or velocity. Wagner’s

function can be calculated exactly, but includes some terms which are not well-knownfunctions. It can however be approximated as[25]:

Φ(τ) ∼= 1− 0.165e−0.0455τ − 0.335e−0.3τ

Φ(τ) ∼=τ + 2

τ + 4, τ > 0

Φ(τ) = 0, τ ≤ 0

(47)

Equation (46) can be rewritten by introducing w = V sin(∆α) ≈ V∆α which is thechange in downwash on the airfoil.

∆Lf =1

2ρV ca1wΦ(τ) (48)

The lift of the airfoil can be determined by using the convolution approach to integratethe change in lift (48). This means modeling the excitation as very narrow impulsesof different strengths. In this case the lift is determined based on the step changes ofthe downwash w.

Lf (τ) =1

2ρV ca1

[w0 +

∫ τ

τ0=0

Φ(τ − τ0)dw

dτ0dτ0

](49)

Because the motion of the airfoil will oscillate sinusoidally based on the structuralvibration, the angle of incidence can be set as a sinusodial function.

α = α0sin(ωt) (50)

If a small enough time step (dτ0) is used, it is clear that the resulting lift will followthe same sinusodial curve as the oscillation of the airfoil, but with some phase shift(called phase lag), and with the same frequency.

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B Appendix: Convergence rate for one reduced

frequency in ZEUS

Figure 33: Convergence rate for the unsteady aerodynamic forces for one reduced frequency.Residuals from the main mesh block calculations.

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