The potential flow of an axisymmetric jet impinging normally against a flat plate

Abstract

A mathematical treatment of an ideal axisymmetric fluid jet flowing out of a tube and impinging normally against a flat plate was made in this study. The problem of an axisymmetric impinging jet was solved by Schach using the integral equation method of Trefftz. However Schach assumed implicitly that the velocity ratio was equal to I, where V infinity and Uoo are the velocities of the incident stream and deflected stream, respectively, at infinity from the origin. This assumption considerably simplifies the problem since it requires that the magnitude of the velocity along the tube wall and the magnitude of the velocity along the free streamline be equal to the magnitude of V infinity. The validity of this assumption has not been investigated prior to the present study for the axisymmetric geometry. In the two-dimensional problem, it has been shown by conformal mapping that a is a function of the ratio h/R, where h is the distance of the tube exit from the plate and R is the tube radius, a approaches 1 when the tube exit is infinitely far from the plate and becomes infinite as the tube exit approaches the plate. In this work, the effects of a on the velocity distribution were taken into consideration. The value of ok was determined according to the ratio h/R. A simple procedure was developed to check the assumed value of infinity and to justify the validity of the assumed shape of the free streamline. A reasonable form of the velocity distribution along the tube wall was also proposed to satisfy the boundary conditions of the velocity at the tube exit and at infinity. The flow field of an axisymmetric jet of an ideal fluid impinging normally against a flat plate with h/R = 0.7072 was solved as an example to apply the method developed in this work. The results show that the value of c4 is equal to 1 and that the free streamline can be suitably represented by the hyperbola rz = constant for r > 3, with R = 1 and h = 0.7072, where r and z are radial and axial coordinates respectively.