A Free Streamline Theory for Two-Dimensional Fully Cavitated Hydrofoils

The problem of cavity flows received attention early in the development of hydrodynamics because of its occurrence in high speed motion of solid bodies in water. Many previous works in this field were mainly concerned with the calculation of drag in a cavitating flow. The lifting problem with a cavity (or wake) arose later in the applications of water pumps, marine propellers, stalling airfoils, and hydrofoil crafts. Although several formulations of the problem of lift in cavity flows have been pointed out before, these theories have not yet been developed to yield general results in explicit form so that a unified discussion can be made. The problems of cavitating flow with finite cavity demand an extension of the classical Helmholtz free boundary theory for which the cavity is infinite in extent. For this purpose, several self-consistent models have been introduced, all aiming to account for the cavity base pressure which is in general always less than the free stream pressure. In the Helmholtz-Kirchhoff flow these two pressures are assumed equal. Of all these existing models, three significant ones may be mentioned here. The first representation of a finite cavity was proposed by Riabouchinsky in in which the finite cavity is obtained by introducing an "image" obstacle downstream of the real body. A different representation in which a reentrant jet is postulated was suggested by Prandtl, Wagner, and was later considered by Kreisel and was further extended by Gilbarg and Serrin. Another representation of a free streamline flow with the base pressure different from the free stream pressure, was proposed recently by Roshko. In this model the base pressure in the wake (or cavity) near the body can take any assigned value. From a certain point in the wake, which can be determined from the theory, the flow downstream is supposed to be dissipated in such a way that the pressure increases gradually from the assigned value to that of the free stream in a strip parallel to the free stream. Apparently this model was also considered independently by Eppler in some generality. Other alternatives to these models have also been proposed, but they do not differ so basically from the above three models that they need to be mentioned here specifically. The mathematical solutions to the problem of flow past a flat plate set normal to the stream have been carried out for these three models. All the theories are found to give essentially the same results over the practical range of the wake underpressure. That such agreement is to be expected can be indicated, without the detailed solutions for the various models, from consideration of their underlying physical significance, as will be discussed in the next section. In the present work the free streamline theory is extended and applied to the lifting problem for two-dimensional hydrofoils with a fully cavitating wake. The analysis is carried out by using the Roshko model to approximate the wake far downstream. The reason for using this model is mainly because of its mathematical simplicity as compared with the Riabouchinsky model, or the reentrant jet model. In fact, it can be verified that these different models all yield practically the same result, as in the pure drag case; the deviation from the results of one model to another is not appreciable up to second order small quantities. The mathematical considerations here, as in the classical theory, depend on the conformal mapping of the complex velocity plane into the plane of complex potential. By using a generalization of Levi-Civita's method for curved barriers in cavity flows, the flow problem for curved hydrofoils is finally reduced to a nonlinear boundary value problem for an analytic function defined in the upper half of a unit circle to which the Schwarz's principle of reflection can be applied. The problem is then solved by using the expansion of this analytic function inside the unit circle together with the boundary conditions in the physical plane. In order to avoid the difficulty in determining the separation point of the free streamline from a hydrofoil with blunt nose, the hydrofoils investigated here are those with sharp leading and trailing edges which are assumed to be the separation points. Except for this limitation, the present nonlinear theory is applicable to hydrofoils of any geometric profile, operating at any cavitation number, and for almost all angles of attack as long as the wake has a fully cavitating configuration. As two typical examples, the problem is solved in explicit form for the circular arc and the flat plate for which the various flow quantities are expressed by simple formulas. From the final result the various effects, such as that of cavitation number, camber of the profile and the attack angle, are discussed in detail. It is also shown that the present theory is in good agreement with the experiment.