S&thangt, Vol. 18, Parts 3 & 4, August 1993, pp. 553-574. © Printed in India. Three-dimensional flow separation V C PATEL Department of Mechanical Engineering & Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa 52242-1585, USA Abstract. Separation of three-dimensional flow, although much more common than its two-dimensional counterpart, has defied precise description and definition in spite of numerous attempts. Here, we briefly review the grammar that is used to describe various facets of the phenomenon, and use some recent numerical and experimental results to illustrate the outstanding difficulties of the subject. Keywords. Three-dimensional flow separation; flow visualization; kinematic and dynamic complications. 1. Introduction Practically all fluid flows are three-dimensional and, among these, few are free from separation. In fact, no flow can properly be described without considering the possibilities of separation and attendant unsteadiness. While the topic of three- dimensional flow separation is of interest in a vast array of fields and many aspects of the problem are common to them all, here we shall concentrate on certain basic aspects of flow structure. For this purpose, we need not distinguish between laminar and turbulent flows even though the most common ones are turbulent. Similarly, distinction between internal and external flows is also of little consequence. Separation from sharp edges and separated flows, per se, including the dead horse, the backward-facing step, are also excluded. In two-dimensional flow, the criterion or definition of separation is the vanishing of skin friction or wall shear. Unfortunately, such a precise and simple definition is not possible in a three-dimensional flow, because one additional space dimension leads to flow states which are inconceivable in two dimensions. Three-dimensional flow separation has attracted a great deal of attention but the kinematic and dynamic complications of three-dimensional flows are so formidable that the phenomenon is still beyond the reach of definitive theoretical analyses or numerical calculations. The situation in experimental studies is not much better. Detailed measurements in three- dimensional separated flOWS are rare because of limitations of instrumentation, on the one hand, and the immense amount of information that is required for proper documentation and resolution of such flows, on the other. Laser-Doppler velocimetry (LDV), particle-image velocimetry (PIV), and holography have yet to develop the necessary resolution capabilities. Up to the present time, flow visualization in the 553 554 V C Patel laboratory has been the most popular method in the study of separation in three- dimensional flows. Indeed, flow visualization, although qualitative and sometimes ambiguous, has contributed the most to our knowledge of the subject. A number of criteria or definitions have been proposed for three-dimensional flow separation, each of which emphasizes one particular symptom or another to characterize the whole phenomenon. Often, different terms are used to describe the same flow feature, and there is as yet no unified, clear definition. However, it is generally recognized that separation.is that phenomenon by which the vorticity generated in the boundary layer near the surface is ejected into the mainstream. 2. Concepts from topology In three dimensions, flow phenomena such as flow reversal, shear layers, and vortices can often be very difficult to identify and describe, especially if the flow is unsteady. Understanding complex kinematics and devising a proper means to describe it clearly is as important as developing physical models reflecting the dynamics of flow separation. The kinematic aspects of many complex flows, observed in experiments or calculations, are best described by specifying the flow topology using the concepts of singular or critical points, usually on the basis of the velocity or the wall-shear-stress (or skin-friction) vector field. Singular points are the salient features of continuous vector fields. A vector field is mathematically defined as a map which assigns to each point in the field a vector (e.g., wall shear stress, velocity, vorticity). A continuous vector field has the property that at any regular (nonsingular) point in the field there is one, and only one, field line (e.g., friction line, streamline, vortex line) which passes through that point. A singular point is one at which the magnitude of the vector is zero, and its direction accordingly is indeterminate. A finite number of singular points may occur within a vector field. Singular points are classified mathematically according to the behaviour of the field lines in their vicinity, and have been extensively explored in the theory of ordinary differential equations in connection with phase-plane studies. The ideas of a continuous vector field are applicable to the limiting streamlines or skin-friction lines on a body, and also to the flow above the body surface. Singular points are classified into two main types, nodal points (N) and sadd;e points (S). Nodal points may be further subdivided into nodes and loci (spiral nodes). In the flow over a surface, an infinite number of friction lines are directed either into or away from a node. If the friction lines spiral into or out of a node, it is called a focus. Nodes or loci with friction lines directed into them correspond to points of flow detachment or separation (Ns), and those with friction lines directed outward are points of flow attachment (Na). A saddle point has two common frictior lines, called separatrices, which intersect at the point, one of which is directed to~, ard and the other away from the singular point. All other neighbouring friction lines are asymptotic to these separatrices. Singular points in friction lines are observed in surface-flow visualizations, and are among the most prominent features looked for in numerical solutions. The questions that follow naturally are: what types of singular points occur in a given flow, where they are located, and how they are connected to one another. The study of flow topology in terms of these singular points provides a language that enables complex three-dimensional flow patterns to be described and interpreted in a systematic Three-dimensional flow separation 555 manner, whether they are obtained from experiments or computations. Furthermore, theories of topology of continuous vector fields provide the rules which apply to flows that are kinematically possible. Tobak & Peake (1982) summarize the following two rules for skin-friction lines on bodies: EN - YS = 2, (1) for skin-friction lines on an isolated closed body; and EN - Es = 0 (2) for skin-friction lines on a closed three-dimensinal body connected simply to a plane wall, where N and S denote nodal (node or focus) and saddle points, respectively. It is also customary to examine three-dimensional flow fields by plotting (projections of) streamlines in certain cross-sections of the flow or in symmetry planes. Although these provide limited (and sometimes distorted) views of the total three-dimensional flow, the streamline fields bear a certain topological resemblance to the skin-friction field, and therefore, topological laws are frequently applied to them also. Thus, for flow streamlines in a crossflow plane or in a symmetry plane cutting a three-dimensional body, Tobak & Peake (1982) give the following topological law: + EN' - Es where N' and S' denote half-nodes and half-saddles, which arise at intersections of the planes with the body. It should be noted here that a point that appears as a singularity (N or S) in such cross-sectional views is not a singular point of the velocity-vector field because the velocity in the direction normal to the plane of view is not necessarily zero. The topological concepts and their relation with the flow separation are summarized in a recent article by Chapman & Yates (1991). On the basis of topology, they identify three basic types of separation. (i) Type I separation, illustrated in figure 1. Such a separation originates at a saddle point (Ss) of the skin-friction lines and the fluid surface of separation rolls up to form a vortex. The vortex centre (N) is a spiral node (focus) in the streamlines viewed in the plane of symmetry. The separation line (the friction line on the body originating from the saddle point Ss) divides the body surface into separate regions, and therefore, a node of attachment (Na) is required to allow fluid into the separated region. (ii) Type II separation, illustrated in figure 2. Such a separation also originates at a saddle point (Ss) of the skin-friction lines but there is no singular point in the flow. Instead, such a separation leads to vortices with origins at the spiral nodes (N,) of the skin-friction lines. The separation line (the friction line on the body originating from the saddle point S,) does not divide the body surface into separate regions. (iii) Crossflow separation, illustrated in figure 3. This separation does not originate from a singular point. It is similar to the open separation of Wang (1974) and the local separation of Tobak & Peake (1982), and not even separation in the terminology of Lighthill (1963, p. 72). Of practical interest in the study of three-dimensional flow separation is how separation patterns originate and how they succeed each other as some flow parameter, such as the Reyndds number, angle of attack, or body geometry, is varied. The structural changes taling place in the pattern of flow separation as a flow 556 V C Patel S$ _.~ot,, No u A-~r../ S' $' plane of cross-flow Figure 1. Type I separation (Chapman & symmetry plane, A-A Yates 1991). cross-flowS Figure 2. Type II (horn) separation (Chapman plane, A-AA & Yates 1991). U No~ cross-flow Scross-flow Figure 3. Cross-flow (open) separation (Chapman plane, A-A plane, B-B & Yates 1991).