1
Introduction

Unsteady aerodynamics refers to flow of air over bodies whose velocity field is changing in time. The causes of unsteadiness can be

• Translational and rotational acceleration of the body relative to the fluid. The vast majority of this book is devoted to this type of unsteadiness.
• Upstream or free stream unsteadiness. In the atmosphere, this phenomenon is referred to as atmospheric turbulence and is caused by a variety of meteorological and geographical phenomena. In the laboratory, we encounter wind tunnel turbulence. There will be no discussion of this type of unsteadiness in this book.
• Turbulence in boundary layers, which is a ubiquitous source of unsteadiness in most practical flows. A short discussion of laminar and turbulent boundary layers is included in Chapter 7.
• Flow that separates from the surface of the body, either instantaneously or permanently. This type of flow is inherently unsteady, even when the motion of the body is steady. We will discuss this type of unsteadiness in Chapter 7.

Other sources of unsteadiness, such as acoustics, jet impingement, wake interaction, and thermal effects, lie beyond the scope of this book.

The vast majority of practical airflows will feature some degree of turbulence upstream in the boundary layer and in the wake of the body. As a consequence, nearly all aerodynamics is unsteady. However, aircraft, rotors, wind turbines and other engineering structures are generally designed to operate under attached flow conditions, so that the turbulence is confined to a thin layer of fluid in contact with the surface. Under such conditions, the effect of turbulence is averaged and therefore the flow can be treated as steady. Then, the major source of unsteadiness becomes the motion of the body itself. Conversely, civil engineering structures are mostly aerodynamically bluff bodies; even though they seldom move, they are subjected to significant unsteadiness due to separated flow and upstream turbulence.

This book deals mostly with wings and therefore the source of unsteadiness it will address most of the time is body motion. Our ancestral prototype of flight is bird flight, which involves flapping wings. Yet the first man-made flying objects were kites which in their simplest form do not flap or deform in any way. The first gliders and aircraft also had fixed wings, and flapping blades were introduced in helicopter rotors much later. From a practical point of view, it is clearly easier to work with steady aerodynamics. This is also the case from a mathematical point of view; the flow equations are simpler and easier to solve. From the experimental point of view too, setting up and measuring a steady flow is more straightforward.

Even fixed-wing aircraft undergo unsteady motion, both rigid and flexible. Rigid aircraft motion is the field of study of flight dynamics; aircraft have both oscillatory and non-oscillatory rigid body eigenmodes that cannot be predicted adequately using purely steady aerodynamic analysis. We will give an example of the calculation of aerodynamic stability derivatives in Chapter 5. Furthermore, aircraft structures are flexible and are becoming increasingly so. The study of vibrating structures in an airflow is the subject area of aeroelasticity. Again, a steady or quasi-steady aerodynamic analysis is insufficient to predict aeroelastic phenomena. Chapter 3 includes one example of a direct application of unsteady aerodynamics to flutter prediction. Nevertheless, all of the methods presented in this book can be used for flight dynamic, aeroelastic or combined aeroservoelastic analysis.

1.1 Why Potential and Vortex Methods?

The equations of fluid flow are notorious for being unsolvable. The Millennium Prize (Clay Mathematics Institute, [2000]) for proving the existence and smoothness of solutions to the 3D Navier-Stokes equations was still unclaimed at the time of writing of this book and the original US$1 million prize money had already depreciated to US$575,000 due to inflation. Numerical solutions of these equations are possible, but turbulence renders them impractical. In order to capture all the spatial scales of turbulence at a Reynolds number encountered in aeronautical practice, the computational requirements of a direct numerical simulation of the Navier-Stokes equations exceed the capabilities of even the fastest and biggest modern computers. Therefore, in order to model practical problems, we resort to solving easier equations. These can be averaged or filtered versions of the original Navier-Stokes relations or simpler equations that are developed after making assumptions about the physics of the flow.

The fastest solutions are obtained for potential flow equations, whereby the flow is assumed to be inviscid, irrotational and isentropic, if not incompressible. Even though a significant amount of the physics of fluid flow is discarded in order to obtain such solutions, their range of validity can include many aeronautical applications under nominal operating conditions. For example, potential flow methods are the industrial standard for aircraft aeroelastic calculations. As long as the flow remains attached to the surface, its Reynolds number is high and there are no strong shock waves, potential methods can provide fast and reliable solutions to practical engineering problems. Their main advantage is that they do not require the calculation of the solution in the entire flowfield; calculations on the surface of the body and in its wake are sufficient, and the computational cost of such solutions is very low. Even separated flows can be approximated in this manner, by shedding vortices from the separated flow region of the body into the wake.

Potential flow approaches for steady aerodynamics are presented in detail in many textbooks, notably Katz and Plotkin ([2001]). Gülçat ([2016]) discusses many potential flow methods for unsteady aerodynamics for various flow conditions, from incompressible to hypersonic. Potential flow techniques for unsteady transonic flows are also presented in Landahl ([1961]) and Nixon ([1989]). The present book focuses on application; each method is presented in detail and applied to practical, usually experimental test cases. Furthermore, the book is accompanied by computer codes in the Matlab programming environment that can be used to solve these test cases. The text and computer codes should be studied in parallel. It is hoped that this application-based approach will help the reader to develop a deeper understanding of the various methodologies.

The focus on potential and vortex methods means that the reader will not find any information in this book on what is commonly referred to as Computational Fluid Dynamics (CFD). The latter requires the numerical solution of the flow in a very wide region around the body, usually by means of finite volume, finite element or finite difference discretization. Even though most of the methods discussed in this book are numerical, they only require the discretization of the surface of the body; the wake is treated in a Lagrangian manner so that its vorticity propagates at the local flow velocity. Readers interested in unsteady CFD can consult alternative texts, such as Tucker ([2014]).

1.2 Outline of This Book

Chapter 2 constitutes an introduction to the mathematics of unsteady flow. The full flow equations are presented, and their simplifications using specific flow assumptions are derived. Both compressible and incompressible flow equations are presented, and their boundary conditions are discussed. Solutions to these equations are developed, and their implementation by means of Green's theorem is described in detail. The chapter finishes with a discussion of vorticity and the viscous flow equations.

Chapter 3 introduces the classical unsteady aerodynamic theories for 2D incompressible inviscid flow. The modelling of a flat plate airfoil oscillating in a flow is presented, and analytical equations for the resulting aerodynamic loads are derived. The example of impulsive airfoil motion is used in order to introduce the Wagner function while oscillating motion is used for the definition of Theodorsen's function. It is shown how general small amplitude motion can be represented using these theories and the generation of thrust or drag due to unsteady phenomena is explained. Finally, finite state theory is derived in detail.

Chapter 4 introduces numerical methods that can be used to model 2D inviscid unsteady flow with higher fidelity, for example by modelling more accurately the wake behind an oscillating airfoil or by representing the geometry of airfoils with non-negligible thickness. Three such methods are presented, all with their advantages and disadvantages. They are used in order to demonstrate the physics of the wake behind oscillating airfoils and the mechanisms of thrust generation. Comparison of the predicted aerodynamic loads to experimental results demonstrates how the higher fidelity of numerical methods can represent more of the physics of the real phenomenon. Furthermore, it is shown that all numerical panel methods can be linearized and transformed to the frequency domain in order to obtain faster aerodynamic load predictions for harmonically oscillating wings.

Chapter 5 presents unsteady aerodynamic theories for 3D finite wings. It starts with a description of finite wing geometry. Then, analytical solutions are developed,...