The study of flow separation from the surface of a solid body, and the determination of global changes in the flow field that develop as a result of the separation, are among the most fundamental and difficult problems of fluid dynamics. It is well known that most liquid and gas flows observed in nature and encountered in engineering applications involve separation. This is because many of the "common" gases and liquids, such as air and water, have extremely small viscosity and, therefore, most practical flows are characterised by very large values of the Reynolds number; both theory and experiment show that increasing Reynolds number almost invariably results in separation. In fact, to achieve an unseparated form of the flow past a rigid body, rather severe restrictions must be imposed on the shape of the body.
The difference between a separated flow and its theoretical unseparated counterpart (constructed solely on the basis of inviscid flow analysis) concerns not only the form of trajectories of fluid particles, but also the magnitudes of aerodynamic forces acting on the body. For example, for bluff bodies in an incompressible flow, it is known from experimental observations that the drag force is never zero; furthermore, it does not approach zero as the Reynolds number becomes large. On the other hand, one of the most famous results of the inviscid flow theory is d'Alembert's paradox which states that a rigid body does not experience any drag in incompressible flow. It is well known that this contradiction is associated with the assumption of a fully attached form of the flow; this situation almost never happens in practice.
Separation imposes a considerable limitation on the operating characteristics of aircraft wings, helicopter blades, turbines, etc., leading to a significant degradation of their performance. It is well known that the separation is normally accompanied by a loss of the lift force, sharp increase of the drag, increase of the heat transfer at the reattachment region, pulsations of pressure and, as a result, flutter and buffet onset.
It is hardly surprising that the problem of flow separation has attracted considerable interest amongst researchers. The traditional approach of studying the separation phenomenon is based on seeking possible simplifications that may be introduced in the governing Navier-Stokes equations when the Reynolds number is large. The first attempts at describing separated flow past blunt bodies are due to Helmholtz (1868) and Kirchhoff (1869) in the framework of the classical theory of inviscid fluid flows, but there was no adequate explanation as to why separation occurs. Prandtl (1904) was the first to recognise the physical cause of separation at high Reynolds numbers as being associated with the separation of boundary layers that must form on all solid surfaces.
In accordance with the Prandtl's theory, a high Reynolds number flow past a rigid body has to be subdivided into two characteristic regions. The main part of the flow field may be treated as inviscid. However, for all Reynolds numbers, no matter how large, there always exists a thin region near the wall where the flow is predominantly viscous. Prandtl termed this region the boundary layer, and suggested that it is because of the specific behaviour of this layer that flow separation takes place. Flow development in the boundary layer depends on the pressure distribution along the wall. If the pressure gradient is favourable, i.e. the pressure decreases downstream, then the boundary layer remains well attached to the wall. However with adverse pressure gradient, when the pressure starts to rise in the direction of the flow, the boundary layer tends to separate from the body surface. The reason for separation was explained by Prandtl in the following way. Since the velocity in the boundary layer drops towards the wall, the kinetic energy of fluid particles inside the boundary layer appears to be less than that at the outer edge of the boundary layer, in fact the closer a fluid particle is to the wall the smaller appears to be its kinetic energy. This means that while the pressure rise in the outer flow may be quite significant, the fluid particles inside the boundary layer may not be able to get over it. Even a small increase of pressure may cause the fluid particles near the wall to stop and then turn back to form a recirculating flow region characteristic of separated flows.
It might seem surprising that the clear understanding of the physical processes leading to the separation, could not be converted into a rational mathematical theory for more than half a century. The fact is that the classical boundary-layer theory, which was intended by Prandtl for predicting flow separation, was based on the so called hierarchical approach when the outer inviscid flow should be calculated first ignoring the existence of the boundary layer, and only after that one can turn to the boundary layer analysis. By the late forties it became obvious that such a strategy leads to a mathematical contradiction associated with so called Goldstein's singularity at the point of separation. The form of this singularity was first described by Landau & Lifshitz (1944) who demonstrated that the shear stress in the body surface upstream of separation drops as the square root of the distance from the separation, and the velocity component normal to the surface tends to infinity being inversely proportional to the shear stress. This result was later confirmed based on more rigorous mathematical terms by Goldstein (1948). Goldstein also proved -- and this result appeared to be of paramount importance for further development of the boundary-layer separation theory -- that the singularity at separation precludes the solution to be continued beyond the separation point into the region of reverse flow.
The Goldstein's theoretical discovery came at the time when an important development was taking place in experimental investigation of separated flows. The most disputable was the effect of upstream influence through the boundary layer in supersonic flow prior to separation. It might be observed, for example, when a shock wave impinges the boundary layer on a rigid body surface. Starting with Ferri (1939) a number of researches demonstrated that instead of simple reflection of the shock wave from the body surface, as it would happen in a fully inviscid flow, more complicated shock structure, called the "lambda-structure", develops. It consists of the primary shock on impinging the boundary layer at some point on the body surface, and the secondary shock which forms some distance upstream of this point. The secondary shock is provoked by the thickening of the boundary layer which, in its turn, is caused by propagation of disturbances through the boundary layer from the region of higher pressure downstream of the main shock. This process, obviously, can not be explained in the framework of classical boundary-layer theory. Indeed, following the Prandtl hierarchical strategy, one has to consider the external inviscid flow region first and then the boundary layer. In the case of supersonic flow the external region is governed by the hyperbolic equations. The boundary-layer equations, when solved with prescribed pressure gradient, as it is required by classical Prandtl theory, are known to be of the parabolic type. Therefore neither the external flow nor the boundary layer allow upstream propagation of disturbances.
Although the boundary-layer theory in its classical form was found to be insufficient for describing the separation phenomenon, Prandtl's insight into physical processes leading to separation and, even more so, the mathematical approach suggested by Prandtl for analysing high Reynolds number flows, laid a foundation for all subsequent studies in the asymptotic theory of separation. In a broader sense Prandtl's idea of subdividing the entire flow field into a number of regions with distinctively different flow properties, proved to be a beginning of one of the most powerful tools in modern asymptotic analysis, the method of matched asymptotic expansion.
Significant progress in theoretical study of separated flows has been achieved in the last thirty years, and a comprehensive description of the underlying ideas and the main results of the theory may be found in a monograph by Sychev et al (1998). A key element of the separation process, which was not fully appreciated in the classical Prandtl's (1904) description, is a mutual interaction between the boundary layer and the external inviscid flow. Because of this interaction, a sharp pressure rise may develop "spontaneously" at a location on the body surface where in accordance with the Prandtl's theory the boundary layer would be well attached. This pressure rise leads to a rapid deceleration of fluid particles near the wall and formation of the reverse flow downstream of the separation. The interaction precludes development of the Goldstein singularity.
The asymptotic theory of viscous-inviscid interaction, known now as the triple-deck theory, was formulated simultaneously by Neiland (1969) and Stewartson & Williams (1969) for the self-induced separation in supersonic flow and by Messiter (1970) for incompressible fluid flow near a railing edge of a flat plate. Based on the asymptotic analysis of the Navier-Stokes equations they demonstrated the region of interaction is $O(Re^{-3/8})$ long and has a three-tiered structure being composed of the viscous near-wall sublayer (region 1), main part of the boundary layer (region 2) and inviscid potential flow region 3 situated outside the boundary layer.
Characteristic thickness of the viscous sublayer is estimated an $O(Re^{-5/8})$ quantity, i.e. it occupies $O(Re^{-1/8})$ portion of the boundary layer being comprised of the stream filaments immediately adjacent to the wall. The flow velocity in this region is $O(Re^{-1/8})$ relative to the free-stream velocity, and due to the slow motion of gas here the flow exhibits high sensitivity to pressure variations. Even small pressure rise along the wall may cause significant deceleration of fluid particles there. This leads to thickening of flow filaments, and the streamlines change their shape being displaced from the wall.
The main part of the boundary layer, the middle tier of the interactive structure, reprsents a continuation of the conventional boundary layer into the region of interaction. Its thickness is estimated as $O(Re^{-1/2})$ and the velocity is an order one quantity. The flow in this tier is significantly less sensitive to the pressure variations. It does not produce any noticeable contribution to the displacement effect of the boundary layer, which means that all the stream lines in the middle tier are parallel to each other and carry the deformation produced by the displacement effect of the viscous sublayer.
Finally, the upper tier is situated in the potential flow region outside the boundary layer. It serves to "convert" the perturbations in the form of the stream lines into perturbations of pressure. These are then transmitted through the main part of the boundary layer back to the sublayer.
Later it became clear that the triple-deck interaction region, while being small, plays a key role in many fluid flows. It, for instance, governs upstream influence in the supersonic boundary layer, development of different modes of instabilities, bifurcation of the solution and possible hysteresis in separated flows. As far as separation phenomena are concerned, the theory has been extended to describe boundary-layer separation from a smooth body surface in incompressible fluid flow, supersonic flow separation provoked by a shock wave impinging upon the boundary layer, incipient and large scale separations at angular points of the body contour both in subsonic and supersonic flows, separation at the trailing edge of a thin aerofoil appearing as a result of increase of the angle of attack or the aerofoil thickness, leading-edge separation, separation of the boundary layer in hypersonic flow on a hot or cold wall, separation provoked by a wall roughness, etc. (see Sychev et al. 1998 and references in this book.)
However, despite obvious progress in this field, many aspects of the boundary-layer separation theory remain unresolved. Most notably, the theory remains predominantly restricted to incipient or small scale separations when the entire recirculating region together with the separation and reattachment points `fits' into the $O(Re^{-3/8})$ long region of interaction. Even in the studies specifically amed at desribing developed separations (see Neiland 1969; Stewartson & Williams 1969; Sychev 1972; Ruban 1974), the analysis is confined to the "local" flow behaviour near the separation point.
Very little is still known about developed separations, separation in transonic flow, three-dimensional boundary-layer separation, unsteady separation, etc. Future research in these areas should be exciting.