# Low Speed Unsteady Airfoil Flows

he standard textbooks on aerodynamics usually omit any discussion of unsteady aerodynamics or, at most, consider it only in a single chapter, based on two justifications. The first is that unsteady aerodynamics should be regarded as a specialized subject required “only” in connection with understanding and analyzing aeroelastic phenomena such as flutter and gust response, and therefore should be dealt with in related specialist books.

The second reason appears to be reluctance to discuss aerodynamics with the inclusion of the time-dependent terms in the conservation equations and the boundary conditions for fear that added complications may discourage the reader. We take the opposite view in this book and argue that a full understanding of the physics of lift generation is possible only by considering the unsteady aerodynamics of the starting vortex generation process.

Furthermore, certain “steady” flows are inherently unsteady in the presence of flow separation, as for example the unsteady flow caused by the Karman vortex shedding downstream of a cylinder and “static” airfoil stall which is an inherently unsteady flow phenomenon. Therefore, it stands to reason that a unified treatment of aerodynamics that yields steady-state aerodynamics as a special case offers advantages.

This reasoning is strengthened by the developments in computational fluid dynamics over the past forty years, which showed that accurate steady-state solutions can be obtained efficiently by solving the unsteady flow equations. We have, however, chosen to concentrate on unsteady low-speed flows over airfoils in order to present a reasonably comprehensive coverage while limiting the size of the book. This implies that the content is restricted to the discussion of two-dimensional incompressible flows and, as a consequence, the book is structured as described in the following paragraphs.

The introductory first chapter describes the physics of unsteady flows by explaining the unsteady flow mechanisms underlying the generation of lift on two-dimensional airfoils and finite-span wings and the generation of thrust on flapping airfoils and wings. This is followed by a demonstration that airfoils VI Preface capable of pitch and plunge oscillations can extract energy from the air stream instead of generating thrust, provided the amplitudes of oscillation and the phasing between the pitch and plunge oscillations reach certain critical values.

This phenomenon can lead to the destruction of an aircraft wing within seconds due to explosive flutter. The possibility of airfoil flutter due to pitch oscillations only is then considered and, in this case, the unsteady aerodynamic effects caused by the vortex shedding from the airfoil are shown to be an essential component in the explanation of this phenomenon. An understanding and incorporation of unsteady flow concepts is also required for the determination of the loads caused by wind gusts. The chapter ends by drawing attention to the dynamic airfoil stall and stall flutter phenomena, as caused by flow separation effects.

It is well recognized that the “rational” analysis of separated flows, i. e. , an analysis other than empirical or semi-empirical, needs to be based on the solution of the viscous flow equations stemming from the Navier-Stokes equations. In addition, most flows of practical importance are partly or fully turbulent and therefore require the use of Reynolds averaging in order to evolve a practically useful computational tool. For these reasons, the second chapter begins with a presentation of the Navier-Stokes equations and their Reynolds-averaged form.

Furthermore, since many flows can be analyzed efficiently by the use of reduced forms of the Navier-Stokes equations, the thin-layer Navier-Stokes, boundary layer and inviscid flow equations are also included in this chapter. Since inviscid, boundary layer, and Navier-Stokes methods are now widely used, separate chapters are devoted to describe the three methods for the computation of steady and unsteady airfoil flows. The computation of inviscid airfoil flows benefited enormously, both conceptually and computationally, by the introduction of the so-called panel method, pioneered at the Douglas Aircraft Company in the 1960s.

Thus, a panel method for the calculation of the flow over an airfoil executing a general time-dependent motion is described in chapter three. It is known that the viscous flow effects can be included with the pressure distribution obtained from an inviscid flow solution as input into the boundary layer equations. This concept can be further refined by interaction between the inviscid and boundary layer computations, thus making it possible to analyze mildly separated flows as described in chapter five. The ourth, sixth and seventh chapters describe applications of the inviscid, boundary layer and viscous-inviscid interaction codes, respectively, to provide the reader with an appreciation for the usefulness and range of validity of each method by comparing the computations with available experimental results. The eighth and ninth chapters consider the analysis of strongly viscous and separated flows by means of the Reynolds averaged Navier-Stokes equations by describing first the various solution methods for both incompressible and compressible flows and t