ABSTRACT
The preceding potential flow problems have concerned free flows or flows forced by sources, sinks, and vortices, in all space or in regions bounded by rigid boundaries, plus the analogue problems, for example, electro (magnetostatic) fields due to electric charge (current) distributions. Another type of problem (Diagram 38.1) is the field, for example, electric, due to a given surface potential, for example, on a disk or on a plane (Section 38.1), where again the boundaries are fixed. Yet another class of problems are those of free jets, for that the boundary is not known a priori and must be determined as part of the solution to the problem. In the potential flows considered before the boundaries were rigid impermeable walls placed at positions known “a priori” and where the velocity is tangential. The same boundary condition, namely tangential velocity, applies to free jets for which the surface of separation from the surrounding fluid at rest is not determined “a priori,” that is, the shape of the free surface is part of the solution of the problem. Since a free surface separates a fluid at rest from a flow, it must be an isobar, that is, a line of constant pressure; by the Bernoulli law the velocity is constant in modulus along an isobar, although the direction generally varies for a curved free surface. If the jet issues from a hole in a reservoir (Sections 38.2-38.4) or impinges on a rigid obstacle (Sections 38.5-38.8) the direction of the velocity is tangential, hence known, although its modulus generally varies because the pressure on the obstacle is usually nonuniform. Thus a free jet with confining walls, or impinging upon obstacles or with given direction (Section 38.9) can be represented using lines of constant modulus or angle of the velocity that are the coordinates in the hodograph plane. Mapping the region occupied by the free jet in the hodograph plane conformally onto the physical plane leads to a relation between the complex velocity and complex potential. The latter would have to be integrated to obtain explicity the complex potential in the physical plane; even if the explicit complex potential is not available it is possible to obtain by integration some results, such as the shape of the free surface or the pressure distribution on obstacles. The problems with free jets considered include: (i/ii) the jet issuing from a slit in a wall, without or with (Section 38.3) a reentrant tube, including the calculation of the contraction ratio (Section 38.2); (iii/v) the jet perpendicular (Section 38.4) [oblique (Section 38.5)] to a flat plate or incident symmetrically on an arrow (Section 38.6) leading to a calculation of the drag; (vi/vii) the attachment (deflection) of a jet to (from) a wall, due the Coanda effect (Section 38.7) [a fluidic source (Section 38.8)]; (viii) the collision or merging of several free jets (Section 38.9). Some of the problems have multiple interpretations, for example, (iii) the deflection of a flow by a plate perpendicular to wall (Section 38.5); (v) the thrust on a surfboard riding on a stream; (viii) a jet incident normal to a wall. The hodograph method specifies both the shape of free jets and the forces and moments they exert on obstacles.
