THESIS.dviaan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 15 december 2003 om 13.00 uur door
Willem Hendrik Jan Joseph VAN STAVEREN ingenieur luchtvaart en ruimtevaart
geboren te Sittard
Prof.dr.ir. J.A. Mulder.
Prof.dr.ir. J.A. Mulder, Technische Universiteit Delft, promotor
Prof.dr.ir. P.G. Bakker, Technische Universiteit Delft
Prof.dr.ir. P.M.J. van den Hof, Technische Universiteit Delft
Prof.dr.ir. Th. van Holten, Technische Universiteit Delft
Prof.dr.ir. J.H. de Leeuw, University of Toronto, Ontario, Canada
Prof.dr.-Ing. G. Schanzer, Technische Universitat Braunschweig, Duitsland
Dr.ir. J.C. van der Vaart, Technische Universiteit Delft
Prof.dr.ir. M.J.L. van Tooren, Technische Universiteit Delft, reservelid
Dr.ir. J.C. van der Vaart heeft als begeleider in belangrijke mate aan de totstandkoming
van het proefschrift bijgedragen.
Delft University Press
P.O. Box 98
2600 MG Delft
Email: info@library.tudelft.nl
ISBN 90-407-2453-9
windshear / Computational Aerodynamics / CFD / elastic aircraft / fixed wing aircraft /
flight test / flight dynamics / loads / panel method / parameter identification / potential
flow / simulation / system identification / unsteady aerodynamics
Copyright c©2003 by W.H.J.J. van Staveren
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system, whithout
written permission from the publisher: Delft University Press
Printed in The Netherlands
Summary
The response of aircraft to stochastic atmospheric turbulence plays an important role
in aircraft-design (load calculations), Flight Control System (FCS) design and flight-
simulation (handling qualities research and pilot training). In order to simulate these
aircraft responses, an accurate mathematical model is required. Two classical models will
be discussed in this thesis, that is the Delft University of Technology (DUT) model and
the Four Point Aircraft (FPA) model. Although they are well estabilished, their fidelity
remains obscure. The cause lies in one of the requirements for system identification; it
has always been necessary to relate inputs to outputs to determine, or identify, system
dynamic characteristics. From experiments, using both the measured input and the mea-
sured output, a mathematical model of any system can be obtained. When considering an
input-output system such as an aircraft subjected to stochastic atmospheric turbulence, a
major problem emerges. During flighttests, no practical difficulty arises measuring the air-
craft motion (the output), such as the angle-of-attack, the pitch-angle, the roll-angle, etc..
However, a huge problem arises when the input to the aircraft-system is considered; this
input is stochastic atmospheric turbulence in this thesis. Currently, during flighttests it
still remains extremely difficult to identify the entire flowfield around an aircraft geometry
subjected to a turbulent field of flow; an infinite amount of sensors would be required to
identify the atmospheric turbulence velocity component’s distribution (the input) over the
vehicle geometry.
In an attempt to shed some more light on solving the problem of the response of aircraft to
atmospheric turbulence, the subject of this thesis, it depends on the formulation of two dis-
tinct models: one of the atmospheric turbulence itself (the atmospheric turbulence model),
and the other of the aircraft response to it (the mathematical aircraft model). As concerns
atmospheric turbulence, stochastic, stationary, homogeneous, isotropic atmospheric tur-
bulence is considered in this thesis as input to the aircraft model. Models of atmospheric
turbulence are well established. As for mathematical aircraft models, many of them have
been proposed before. However, verifying these models has always been extremely difficult
due to the identification problem indicated above. As part of the mathematical aircraft
model, (parametric) aerodynamic models often make use of (quasi-) steady aerodynamic
results, that is all steady aerodynamic parameters are estimated using either results ob-
tained from windtunnel experiments, handbook methods, Computational Aerodynamics
(CA) which comprises Linearized Potential Flow (LPF) methods, or Computational Fluid
Dynamics (CFD) which comprises Full-Potential, Euler and Navier-Stokes methods.
ii Summary
In this thesis the simplest form of fluid-flow modeling is used to calculate the time-
dependent aerodynamic forces and moments acting on a vehicle: that is unsteady Li-
nearized Potential Flow (LPF). The fluid-flow model will result in a so called “unsteady
panel-method” which will be used as a virtual windtunnel (or virtual flighttest facility)
for the example discretized aircraft geometry, also referred to as the “aircraft grid”. The
application of the method ultimately results in the vehicle’s steady and unsteady stability
derivatives using harmonic analysis. Similarly, both the steady and unsteady gust deriva-
tives for isolated atmospheric turbulence fields will be calculated. The gust fields will be
limited to one-dimensional (1D) longitudinal, lateral and vertical gust fields, as well as
two-dimensional (2D) longitudinal and vertical gust fields. The harmonic analysis results
in frequency-dependent stability- and gust derivatives which will later be used to obtain
an aerodynamic model in terms of constant stability- and gust derivatives. This newly
introduced model, the Parametric Computational Aerodynamics (PCA) model, will be
compared to the two classical models mentioned earlier, that is the Delft University of
Technology (DUT) model and the Four-Point-Aircraft (FPA) model. These three para-
metric aircraft models are used to calculate both the time- and frequency-domain aerody-
namic model and aircraft motion responses to the atmospheric turbulence fields indicated
earlier. Also, using the unsteady panel-method the aircraft grid will be flown through
spatial-domain 2D stochastic gust fields, resulting in Linearized Potential Flow solutions.
Results will be compared to the ones obtained for the parametric models, i.e. the PCA-,
DUT- and FPA-model.
From the results presented, it is concluded that the introduced PCA-model is the most
accurate for all considered gust fields. Compared to the Linearized Potential Flow solution
(which is assumed to be the benchmark, or the model that approximates reality closest)
the new parametric model shows increased accuracy over the classical parametric models
(the DUT- and FPA-model), especially for the aircraft responses to 2D gust fields. Fur-
thermore, it shows more accuracy in the aircraft responses to 1D longitudinal gust fields.
Although results will be presented for a Cessna Ce550 Citation II aircraft only, the the-
ory and methods are applicable to a wide variety of fixed-wing aircraft, that is from the
smallest UAV to the largest aircraft (such as the Boeing B747 and the Airbus A380).
As an overview of this thesis, after the introduction given in chapter 1, a short summary
of the applied atmospheric turbulence model is given in chapter 2. Next, the theory
of steady incompressible Linearized Potential Flow is given in chapter 3. Chapter 4
continues with a similar treatment as in chapter 3, discussing unsteady incompressible
Linearized Potential Flow. Both analytical frequency-response functions (or aerodynamic
transfer functions) and numerical frequency-response functions for isolated wings will also
be discussed in this chapter. In chapter 5 the definition of specific aircraft motion per-
turbations and atmospheric turbulence inputs will be given. Chapter 6 discusses the
aircraft grid for the example aircraft. This grid will be used for both steady and unsteady
Linearized Potential Flow simulations. For aerodynamic model identification purposes, the
aircraft grid defined in chapter 6 is used in chapter 7 where the numerical symmetrical
Summary iii
aerodynamic frequency-response functions are given for the PCA-model. They are deter-
mined with respect to aircraft motions in surge and heave, and to both longitudinal and
vertical gusts. All perturbations in aircraft motion and gusts are of harmonic nature. Re-
sults of the analytical continuation of frequency-response data for time-domain models will
also be given (aerodynamic fits). Next, in this chapter the concept of frequency-dependent
stability derivatives and frequency-dependent gust derivatives for complete aircraft config-
urations is discussed. Furthermore, the steady symmetrical aerodynamic model is defined
in this chapter. Chapter 8 treats, along the same lines as in chapter 7, the numeri-
cal asymmetrical frequency-response functions and unsteady asymmetrical aerodynamic
model for the PCA-model. The (harmonic) degrees of freedom considered are now with re-
spect to swaying aircraft motions and antisymmetrical longitudinal-, asymmetrical lateral-
and anti-symmetrical vertical gusts. In chapter 9 the aircraft grid defined in chapter 6
is flown through 2D spatial-domain gust fields. First, the aerodynamic force and moment
coefficients acting on the aircraft geometry are calculated assuming a recti-linear flight-
path (no aircraft motions will be considered). Next, additional theory is given for the
so-called “coupled-solution”, that is the aircraft equations of motion are now coupled with
the potential flow solution. Chapters 10, 11 and 12 discuss the equations of motion of
aircraft subjected to both 1D longitudinal, lateral and vertical gusts and 2D longitudinal
and vertical gusts. In chapter 10 the mathematical aircraft model for the “Parametric
Computational Aerodynamics model” (or “PCA-model”) is introduced, and it includes the
equations of motion using both aerodynamic frequency-response functions (or frequency-
dependent stability- and gust derivatives) and an aerodynamic model in terms of constant
stability- and gust derivatives. Chapters 11 and 12 will discuss the equations of motion
for parametric aerodynamic models in terms of constant stability- and gust derivatives.
The aircraft models are based on the Delft University of Technology gust-response theory,
the “DUT-model” (chapter 11), and Etkin’s “Four-Point-Aircraft model” (or “FPA model”,
chapter 12). In these chapters, the constant stability derivatives obtained in chapter 10
will be used for simulations. A comparison of results of the PCA-, the DUT- and the
FPA-model is given in chapter 13. In this chapter both time- and frequency-domain
results, given in terms of aerodynamic coefficients, will be compared to the ones obtained
from a time-domain Linearized Potential Flow simulation (the LPF-solution). In this
case no aircraft motions are taken into account (the aircraft (-grid) is traveling along a
prescribed recti-linear flightpath), thus the aerodynamic response is limited to gust fields
only. Also, time-domain aircraft motion results will be compared to results obtained for the
LPF-solution. First, the PCA-, the DUT- and the FPA-model aircraft motion simulations
will be compared to the ones obtained for the LPF-solution. These simulations make use
of the gust-induced aerodynamic coefficients obtained for a recti-linear flightpath (exclud-
ing aircraft motions). Next, the PCA-, DUT- and FPA-model aircraft motion simulations
are compared to results obtained from a Linearized Potential Flow simulation which is
coupled to the equations of motion (the so-called “coupled-solution”, designated as the
LPF-EOM-model). This simulation, in which the aerodynamic grid will be flown through
stochastic 2D longitudinal, lateral and vertical gust fields, will be the ultimate test for
the parametric models presented in chapters 10, 11 and 12. Chapter 13 is followed by
iv Summary
conclusions and recommendations in chapter 14.
Since the research conducted for this thesis involved multiple disciplines, some of them are
explained in detail for their educational value. For example, the developed panel-methods
are described as a one to one mapping of the applied software codes. Furthermore the
recipe for determining the novel PCA-model equations of motion, including its parameters,
is outlined in detail.
I Atmospheric Turbulence Modeling 7
2 The atmospheric turbulence model 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Fundamental atmospheric turbulence correlation functions . . . . . . 14
2.3 The atmospheric turbulence covariance function matrix . . . . . . . . . . . 15
2.3.1 The general covariance function matrix . . . . . . . . . . . . . . . . 15
2.3.2 A 2D spatial separation example . . . . . . . . . . . . . . . . . . . . 18
2.4 The atmospheric turbulence PSD function matrix . . . . . . . . . . . . . . . 22
2.4.1 The general PSD function matrix . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Reduced spatial frequency dimension examples . . . . . . . . . . . . 26
2.5 Atmospheric turbulence model parameters . . . . . . . . . . . . . . . . . . . 31
2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Steady linearized potential flow simulations 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Short summary of steady linearized potential flow theory . . . . . . . . . . 37
3.2.1 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.4 A general LPF solution . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Numerical steady linearized potential flow simulations . . . . . . . . . . . . 42
3.3.1 Body surface discretization . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Quadri-lateral panels . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.4 Wake separation and the numerical Kutta condition . . . . . . . . . 50
3.3.5 General numerical source- and doublet-solutions . . . . . . . . . . . 51
3.3.6 Velocity perturbation calculations . . . . . . . . . . . . . . . . . . . 58
3.3.7 Aerodynamic pressure calculations . . . . . . . . . . . . . . . . . . . 61
3.3.8 Aerodynamic loads and aerodynamic coefficients . . . . . . . . . . . 61
3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Numerical unsteady linearized potential flow simulations . . . . . . . . . . . 76
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.4 Unsteady wake-separation and the numerical Kutta condition . . . . 79
4.3.5 General numerical source- and doublet-solutions . . . . . . . . . . . 81
4.3.6 Velocity perturbation calculations . . . . . . . . . . . . . . . . . . . 89
4.3.7 Aerodynamic pressure calculations . . . . . . . . . . . . . . . . . . . 89
4.3.8 Aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Aircraft motion perturbations and the atmospheric turbulence inputs 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Atmospheric turbulence input definitions . . . . . . . . . . . . . . . . . . . . 111
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Aerodynamic frequency-response data . . . . . . . . . . . . . . . . . . . . . 126
5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 PCA-model steady-state aerodynamic results . . . . . . . . . . . . . . . . . 135
6.4.1 A PCA-model steady-state solution . . . . . . . . . . . . . . . . . . 135
6.4.2 (Quasi-) Steady stability derivatives . . . . . . . . . . . . . . . . . . 136
6.4.3 Stability derivatives obtained from flight tests . . . . . . . . . . . . . 139
6.5 Unsteady wake geometry definition . . . . . . . . . . . . . . . . . . . . . . . 141
6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2.1 Initial condition definitions . . . . . . . . . . . . . . . . . . . . . . . 154
7.2.2 Time-domain simulations . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2.3 Effect of the discretization time on frequency-response data . . . . .…