ABSTRACT
The basics of plane steady incompressible irrotational flow (Chapters I.12 and I.14) have been applied to a number of problems: (1) flows past bodies (Chapter I.28); (2) sources, sinks, and vortices in corners, channels, slits, and wells (Chapter I.36); (3) the aerodynamics of airfoils and wings (Chapter I.34); and (4) jets with free boundaries (Chapter I.38). Some consideration was given to compressible (Sections I.12.4, I.12.5, 1.1, and 1.2) and rotational (Sections 1.3 through 1.9) flows, and confined and unsteady flows are addressed next (Chapter 8). A flow is confined if it is limited to a compact region, such as a cavity, and free if it extends to infinity in all directions, for example, an airfoil in a free stream; an intermediate case is a duct, where a flow is partially confined by walls. The flow due to a rigid body moving uniformly in a fluid is steady in a reference frame moving with the body and unsteady in any other reference frame relative to which the body is moving; for example, a pulsating body will cause an unsteady flow. The examples of confined and/ or unsteady flows start with a cylinder moving in a large cylindrical cavity (Sections 8.1); the flow is confined by the cavity, and the cylinder moves relative to the cavity. The case of two cylinders not contained in one another and in relative motion leads to a flow that includes the particular cases where the cylinders move along (across) the line of centers [Section 8.2 (Example 10.10)]. The case of cylinders with the same radius and velocity is equivalent to a cylinder moving perpendicular (parallel) to a wall at an equal distance from the two cylinders, since each cylinder is the image of the other on the wall. The flow past two cylinders is specified exactly by an infinite number of images and approximately by the first few images if the distance between the centers is much larger than the radii. A third case (Section 8.3) is two cylinders at rest with a noncoincident parallel axis. This problem can be solved using coaxial coordinates that also apply to cylindrical log standing on a wall. If the cylinder recesses into the wall, it becomes a circular mound, and if it is then removed, it creates a ditch; by symmetry this is equivalent to a biconvex or biconcave cylinder. Another type of obstacle on a wall is a backward (forward) ramp or step (Section 8.4); the symmetry with regard to a line parallel to the wall leads to a parallel-sided duct with a ramp or step-like expansion (contraction). A parallel-sided channel may contain a thick semi-infinite plate (Section 8.5) with a sharp or blunt front end; the latter can also exist in a free stream, and in the limit of zero thickness becomes the edge of a flat plate. The flat plate is the simplest airfoil (Section I.34.1) and serves to study two aspects: (1) the increase in lift of an airfoil due to the use of a flap (Section 8.6) and (2) the lift loss due to flow separation (Section 8.7). The flap may have a slot separating it from the wing, which helps delay flow separation until a larger angle-of-attack. A set of parallel instead of tandem airfoils forms a cascade that can change the speed and direction of an incident stream (Section 8.8). The separated flow, either from the trailing edge of an airfoil or from farther upstream, forms a wake; its simplest representation, as a vortex sheet, is unstable to small amplitude sinusoidal disturbances (Section 8.9). The vortex sheet may separate two jets with different velocities and/or mass densities.
