Two-Dimensional Problems in Hydrodynamics and Aerodynamics

Abstract

REVIEWED BY W. R. DEAN 1 THIS BOOK gives an introduction to the hydrodynamics of inviscid fluids and is, after the preliminary chapters, mainly concentrated on two-dimensional motions. There is a full treatment of the general two-dimensional motion of an immersed cylinder with particular reference to the flow past a thin airfoil and past an infinite array of airfoils, and a full treatment of discontinuous motions and free surface flows. The analysis that is usually given of the stability of two-dimensional vortex systems is here significantly extended by some work with equations accurate to the second order of the arbitrary displacements, and interesting conclusions are drawn from this extension; this important section is based on work attributed to A. M. Liapunov.

On the Geometry of the Domain of the Solution of Nonlinear Cauchy Problem

“…In the case of equation (3), we can admit in the role of characteristics the families of characteristic curves along which relations (19) and (20) are fulfilled. Characteristic families can be therefore given a priori in an arbitrary manner.…”

“…Analogously, we denote by D 2 the domain of the plane (x, y) which is completely covered by the characteristic lines given by equation (22). Also, we introduce the notation (3) and its derivative with respect to the normal direction on the interval I of the straight line y = 0 if the families of plane curves given by equations (21) are the characteristics corresponding to invariants (19), while (22) is the family of characteristics corresponding to invariants (20). Proof.…”

“…if the family of plane curves (52) represents the characteristic curves corresponding to invariants (19), while family (53) corresponds to the characteristic invariants (20).…”

“…It is one of the most important problems of mathematical physics to study the properties of solutions of equations of mixed type (cf. [2], [21], [19], [22]). The first fundamental results in this direction were obtained by the Italian mathematicians Francesco Tricomi in the twenties of the 19th century.…”

Origin of ejecta in the water impact problem

“…First, it is worth mentioning the studies by Joukowsky on the impact between two spheres, one of which is half submerged in liquid, 3 and of sea plane landing in 1898 also by Joukowsky 4 as well as by von Karman. 5 Later, Wagner 6 developed irrotational inviscid flow models for the impact loads and dynamics, and Sedov 7,8 laid out the impact theory in its classical form where motions with discontinuous changes in the velocity field are considered. Motivation for the problem also came from its relevance to other applications such as slamming of ships 9, 10 to characterize structural and dynamic responses and impact of water waves on coastal structures.…”