The major successes achieved since the late 1960s in the development of the theory of separated flows of liquids and gases at high Reynolds numbers are in many respects associated with the use of asymptotic methods of investigation. The most fruitful of these has proved to be the method of matched asymptotic expansions, which has also become widely used in many other fields of mechanics and mathematical physics (Van Dyke, 1964, 1975; Cole, 1968; Lagerstrom and Casten, 1972). By means of this method, important problems have been solved concerning boundary-layer interaction and separation in subsonic and supersonic flows, the nature of these phenomena has been made clear, and basic laws and controlling parameters have been ascertained. The number of original papers devoted to different problems in the asymptotic theory of separated flows now amounts to many dozens. We can confidently speak of the appearance of a new and very productive direction in the development of theoretical hydrodynamics. However, it has not yet received an adequate systematic account, and the objective of this book is to make an attempt to partially fill this gap.

Of necessity, the authors have restricted themselves to consideration of a range of problems most familiar to them, associated with two-dimensional separated flows of an incompressible fluid, assumed to be laminar. Thus, the book excludes the results of many investigations of separated gas flows at transonic, supersonic, and hypersonic speeds, although chronologically the problem of boundary-layer separation in supersonic flow was solved earlier than the others (see Chapter 1).

Here we leave completely untouched the study of turbulent boundary-layer interaction and separation, as well as the large class of problems of internal flows in channels and tubes.

The problem of studying flow separation from the surface of a solid body, and the flow that is developed as a result of this separation, is among the fundamental and most difficult problems of theoretical hydrodynamics. The first attempts at describing separated flow past blunt bodies are known to have been made by Helmholtz (1868) and Kirchhoff (1869) as early as the middle of the nineteenth century, within the framework of the classical theory of ideal fluid flows. Although in the work of Helmholtz one can find some indication of the viscous nature of mixing layers, which are represented in the limit as tangential discontinuities, nonetheless before Prandtl (1904) developed the boundary-layer theory at the beginning of the twentieth century there were no adequate methods for investigating such flows. Prandtl was the first to explain the physical nature of flow separation at high Reynolds number as separation of the boundary layer. In essence, Prandtl's boundary-layer theory proved to be the foundation for all further studies of the asymptotic behavior of either liquid or gas flows with extremely low viscosity, i.e., flows of media similar to air or water, with which one is most commonly concerned in nature and engineering. Therefore we begin with a brief description of the main ideas of this theory.

Boundary-Layer Theory

The Navier–Stokes equations with corresponding boundary and initial conditions are usually used as the basic system of relations describing a viscous fluid flow. These equations reflect sufficiently well the behavior of real liquids and gases.

The Analogy between Unsteady Separation and Separation from a Moving Surface

For the steady fluid motions considered in the preceding chapters, flow separation leads to a change in the flow structure as a whole. The limiting state when the Reynolds number tends to infinity is defined by the Helmholtz–Kirchhoff theory for ideal fluid flows with free streamlines, and the location of the separation point, in accordance with the criterion of Prandtl, coincides with the point where the shear stress at the surface of the body vanishes (see Chapter 1).

The situation is somewhat different if the flow is unsteady. To illustrate this fact we consider the example of flow past a circular cylinder that is set into motion instantaneously from rest. The starting motion of a body in a viscous fluid can be likened to the introduction of the no-slip condition at the surface of a body as it moves through a fluid that has no internal friction. Therefore, at the first instant the flow is potential and is described by the well-known solution for unseparated steady flow past a cylinder with zero circulation. At the body surface, where the no-slip condition is imposed, there arises the process of vorticity diffusion and convection, which leads to the formation of a boundary layer.

Blasius (1908) formulated this problem and gave an approximate solution. It turns out that at a certain instant the point of zero skin friction starts moving upstream along the body surface from the rear stagnation point. At the same time a region of reverse flow appears within the boundary layer (Figure 5.1).

The authors are honored by the publication of the English edition by Cambridge University Press, edited by two colleagues who are well known for their work in fluid mechanics – Professor A. F. Messiter of the University of Michigan and Professor M. D. Van Dyke of Stanford University. The authors are deeply indebted to them for their interest in this book, the initiative for the translation into English, and for their considerable effort in the editing, which has no doubt improved the original text. Our work rests heavily on the present state of development of the asymptotic analysis of problems in fluid mechanics, which in many respects is due to Professor Van Dyke and Professor Messiter.

Since the publication of the Russian edition of this book in 1987, there have appeared numerous papers devoted to the asymptotic theory of separated flows. Therefore, in the preparation of the English edition the authors have attempted to bring the book more up to date. In particular, Chapters 2 and 6 have been expanded by the addition of two new sections, (Section 4 and Section 6, respectively) and Chapter 7 has been completely revised. Elsewhere we have confined ourselves to improvements and short additions in the text and to expanding the list of references.

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.