Experimental Observations

Boundary-layer separation at the leading edge of a thin airfoil is the principal factor that limits the lift force acting on an airfoil in a fluid stream. Jones (1934) was the first to describe this kind of separation. Since that time, many researchers have turned to the experimental study of flow around the leading edge of an airfoil. In addition to a great number of original studies, several surveys have been devoted to this theme. The reviews by Tani (1964) and Ward (1963) can be regarded as the most complete.

Experiments show that as the angle of attack increases the picture of the flow around the airfoil changes in the following way. When the angle of attack is small, the flow over the profile is attached. Then the pressure has its maximum at the stagnation point O of the flow, where the zero streamline divides into two – one branch lies along the lower surface of the airfoil, and the second one bends around the leading edge of the airfoil and then lies along its upper surface. As we move from the stagnation point along the upper branch, the pressure first falls rapidly, reaching a minimum at some point M (Figure 4.1a), and then starts to increase, so that the boundary layer downstream from point M finds itself under the influence of an adverse pressure gradient. Its magnitude increases with growth of the angle of attack, finally resulting in boundary-layer separation, which occurs earlier for smaller relative airfoil thickness.

When the boundary layer separates, one can observe the appearance of a closed region of recirculating flow on the upper surface of the airfoil (Figure 4.1a).

The Background of the Problem

This chapter will be devoted to the analysis of one of the fundamental problems of hydrodynamics: ascertaining the limiting state for the steady flow behind a body of finite size as the Reynolds number Re → ∞ in cases where unseparated flow over the body is impossible, for example, behind a blunt body such as a circular cylinder or a plate placed normal to the oncoming flow. Although in reality such flows already become unsteady at Reynolds numbers of the order of 101–102, and undergo transition to a turbulent state with further increase in Reynolds number, the solution of this problem is of great interest in principle. Moreover, one might anticipate that such a solution would allow the study of fluid flows at moderate Reynolds numbers when a steady flow regime is still maintained but the methods of the theory of slow motion (Re < 1) are no longer applicable.

There are several points of view about possible ways of solving this problem. According to the first of these, already stated by Prandtl (1931) and then developed by Squire (1934) and Imai (1953, 1957b), the limiting flow configuration as Re → ∞ is the classical Kirchhoff (1869) flow with free streamlines and a stagnation zone that extends to infinity and expands asymptotically according to a parabolic law (Figure 6.1). This picture has been severely criticized both because of poor quantitative agreement with experimental results (for measurements of body drag) and also for some fundamental reasons.

If a solid-body contour (Figure 2.1) has a sharp corner forming a convex angle γ < π, then an incompressible fluid flow around it cannot be unseparated. It is well known that for unseparated flow over a convex corner an unrealistic situation arises, when the speed of fluid elements increases without limit as the corner is approached. According to potential-flow theory, the speed is proportional to r−α, where α = (π − γ)/(2π − γ), and r is the distance to the corner O. The pressure decreases as the point O is approached and, according to Bernoulli's law, must become negative in some vicinity of the point.

The physical reason for flow separation from a corner is the viscosity of the medium. If the flow around the corner remained unseparated, then the fluid acceleration ahead of the point O would be accompanied by a deceleration downstream of that point. The boundary layer next to the wall immediately behind the corner would then be subjected to an infinitely large adverse pressure gradient, which would lead to flow separation. There can be an exception in the case of slight surface bending, when the adverse pressure gradient proves to be insufficient for boundary-layer separation. This case will be considered in Section 3, devoted to determining the conditions for the onset of separation. As will be shown, the unseparated state of the flow near the corner is maintained up to angles π − γ = O(Re−1/4). A further increase of the surface bending angle π − γ leads to boundary-layer separation.

To determine the sound produced by turbulence near an elastic boundary, it is necessary to know the response of the boundary to the turbulence stresses. These stresses not only generate sound but also excite structural vibrations that can store a significant amount of flow energy. The vibrations are ultimately dissipated by frictional forces, but they can contribute substantially to the radiated noise because elastic waves are “scattered” at structural discontinuities, and some of their energy is transformed into sound. Thus, flow-generated sound reaches the far field via two paths: directly from the turbulence sources and indirectly from possibly remote locations where the scattering occurs. The result is that the effective acoustic efficiency of the flow can be very much larger than for a geometrically similar rigid surface, even when only a small fraction of the structural energy is scattered into sound. Typical examples include the cabin noise produced by turbulent flow over an aircraft fuselage and the noise radiated from ship and submarine hulls, from duct flows, piping systems, and turbomachines [26]. Interactions of this kind are discussed in this chapter.

Sources Near an Elastic Plate

The simplest flexible boundary is the homogeneous, nominally flat, thin elastic plate, which supports structural modes in the form of bending waves. The effects of fluid loading are usually important in liquids, where the Mach number M is small, and in this section, it will be assumed that M ≪ 1 and, therefore, that mean flow has a negligible effect on the propagation of sound and plate vibrations.

Aerodynamic Sound

The sound generated by vorticity in an unbounded fluid is called aerodynamic sound [60, 61]. Most unsteady flows of technological interest are of high Reynolds number and turbulent, and the acoustic radiation is a very small by-product of the motion. The turbulence is usually produced by fluid motion relative to solid boundaries or by the instability of free shear layers separating a high-speed flow (such as a jet) from a stationary environment. In this chapter the influence of boundaries on the production of sound as opposed to the production of vorticity will be ignored. The aerodynamic sound problem then reduces to the study of mechanisms that convert kinetic energy of rotational motions into acoustic waves involving longitudinal vibrations of fluid particles. There are two principal source types in free vortical flows: a quadrupole, whose strength is determined by the unsteady Reynolds stress, and a dipole, which is important when mean mass density variations occur within the source region.

Lighthill's Acoustic Analogy

The theory of aerodynamic sound was developed by Lighthill [60], who reformulated the Navier–Stokes equation into an exact, inhomogeneous wave equation whose source terms are important only within the turbulent (vortical) region. Sound is expected to be such a very small component of the whole motion that, once generated, its back-reaction on the main flow is usually negligible. In a first approximation the motion in the source region may then be determined by neglecting the production and propagation of the sound.

Influence of Rigid Boundaries on the Generation of Aerodynamic Sound

The Ffowcs Williams–Hawkings equation (2.2.3) enables aerodynamic sound to be represented as the sum of the sound produced by the aerodynamic sources in unbounded flow together with contributions from monopole and dipole sources distributed on boundaries. For turbulent flow near a fixed rigid surface, the direct sound from the quadrupoles Tij is augmented by radiation from surface dipoles whose strength is the force per unit surface area exerted on the fluid. If the surface is in accelerated motion, there are additional dipoles and quadrupoles, and neighboring surfaces in relative motion also experience “potential flow” interactions that generate sound. At low Mach numbers, M, the acoustic efficiency of the surface dipoles exceeds the efficiency of the volume quadrupoles by a large factor ∼O(1/M2) (Sections 1.8 and 2.1). Thus, the presence of solid surfaces within low Mach number turbulence can lead to substantial increases in aerodynamic sound levels. Many of these interactions are amenable to precise analytical modeling and will occupy much of the discussion in this chapter.

Acoustically Compact Bodies [70]

Consider the production of sound by turbulence near a compact, stationary rigid body. Let the fluid have uniform mean density p0 and sound speed c0, and assume the Mach number is sufficiently small that convection of the sound by the flow may be neglected. This particular situation arises frequently in applications. In particular, M rarely exceeds about 0.01 in water, and sound generation by turbulence is usually negligible except where the flow interacts with a solid boundary [111].

Jets and shear layers are frequently responsible for the generation of intense acoustic tones. Instability of the mean flow over of a wall cavity excites “self-sustained” resonant cavity modes or periodic “hydrodynamic” oscillations, which are maintained by the steady extraction of energy from the flow. Whistles and musical instruments such as the flute and organ pipe are driven by unstable air jets, and shear layer instabilities are responsible for tonal resonances excited in wind tunnels, branched ducting systems, and in exposed openings on ships and aircraft and other high-speed vehicles. These mechanisms are examined in this chapter, starting with very high Reynolds number flows, where a shear layer can be approximated by a vortex sheet. We shall also discuss resonances where thermal processes play a fundamental role, such as in the Rijke tube and pulsed combustor. The problems to be investigated are generally too complicated to be treated analytically with full generality, but much insight can be gained from exact treatments of linearized models and by approximate nonlinear analyses based on simplified, yet plausible representations of the flow.

Linear Theory of Wall Aperture and Cavity Resonances

Stability of Flow Over a Circular Wall Aperture

The sound produced by nominally steady, high Reynolds number flow over an opening in a thin wall is the simplest possible system to treat analytically. Our approach is applicable to all linearly excited systems involving an unstable shear layer, and it is an extension of the method used in Section 5.3.6 to determine the conductivity of a circular aperture in a mean grazing flow.

Fluid motion in the immediate vicinity of a solid surface is usually controlled by viscous stresses that cause an adjustment in the velocity to comply with the no-slip condition and by thermal gradients that similarly bring the temperatures of the solid and fluid to equality at the surface. At high Reynolds numbers, these adjustments occur across boundary layers whose thicknesses are much smaller than the other governing length scales of the motion. We have seen how the forced production of vorticity in boundary layers during convection of a “gust” past the edge of an airfoil can be modeled by application of the Kutta condition (Section 3.3). In this chapter, similar problems are discussed involving the generation of vorticity by sound impinging on both smooth surfaces and surfaces with sharp edges in the presence of flow. The aerodynamic sound generated by this vorticity augments the sound diffracted in the usual way by the surface. However, the near field kinetic energy of the vorticity is frequently derived wholly from the incident sound, so that unless the subsequent vortex motion is unstably coupled to the mean flow (acquiring additional kinetic energy from the mean stream as it evolves) there will usually be an overall decrease in the acoustic energy: The sound will be damped. General problems of this kind, including the influence of surface vibrations, are the subject of this chapter. We start with the simplest case of sound impinging on a plane wall.

Damping of Sound at a Smooth Wall

Dissipation in the Absence of Flow

The thermo-viscous attenuation of sound is greatly increased in the neighborhood of a solid boundary where temperature and velocity gradients are large.

This book deals with that branch of fluid mechanics concerned with the production and absorption of sound occuring when unsteady flow interacts with solid bodies. Problems of this kind are commonly known under the heading of aerodynamic sound but often include more conventional areas of acoustics and structural vibration. Acoustics is here regarded as a branch of fluid mechanics, and an attempt has therefore been made in Chapter 1 to provide the necessary background material in this subject. Elementary concepts of classical acoustics and structural vibrations are also reviewed in this chapter. Constraints of space and time have required the omission or the curtailed discussion of several important subareas of the acoustics of fluid-structure interactions, including in particular many problems involving supersonic flow. The book should be of value in one or more of the following ways: (i) as a reference for analytical methods for modeling acoustic problems; (ii) as a repository of known results and methods in the theory of aerodynamic sound and vibration, which have tended to become scattered throughout many journal and review articles over the past forty or so years; and (iii) as a graduate level textbook. Chapter 1 and selected topics from Chapters 2 and 3 have been used for several years in teaching an advanced graduate level course on the theory of acoustics and aerodynamic sound.

Theoretical concepts are illustrated and sometimes extended by numerous examples, many of which include complete worked solutions. Every effort has been made to ensure the accuracy of formulae, both in the main text and in the examples. The author would welcome notification of errors detected by the reader and more general suggestions for improvements.

Scientific instruments carried aboard spacecraft often have to be equipped with mechanisms to operate shutters, protective covers, filter wheels, aperture changers and devices that focus, scan and calibrate, just as spacecraft themselves may have to be equipped with mechanisms such as deployable booms, reaction wheels and gas valves for manœuvrability, and driven shafts to steer antennae and solar arrays. This chapter focusses on the design principles of the former, treating them as a special branch of mechanical engineering, although somewhat paradoxically the system designer's first duty is to avoid the use of mechanisms wherever possible to reduce complexity and the risk of end–of–life failure.

We define a mechanism as a ‘system of mutually adapted parts working together’. It may provide useful relative movement, as for a focussing device in a photographic instrument. In the absence of a human operator, it may include a drive motor to overcome friction, or perhaps to perform controlled amounts of useful work. For example, the instrument might be a rock sample drill on a planetary lander. High powered machines such as rocket–engine turbopumps are not considered. The development of robotics (Yoshikawa, [1990]) for manufacturing industry has been widespread in an era in which the remotely directed manipulator arm has been useful in space.

The electronics are usually digital, and have been largely dealt with in Chapter 4 already. (In a functioning space mechanism they can be so well integrated that the combined system is described by some authors as ‘mechatronics’.)