Application of inverse mapping technique for the solution of the ideal axisymmetric impinging jet problem
Fisher, Gary D.
Master of Science
An inverse mapping technique was applied to solve an ideal axisymmetric fluid jet impinging normally against a solid plane infinite in length and width with the tube being a finite distance from the solid plane. The problem of an ideal axisymmetric jet is typical of a class of problems in fluid mechanics involving the flow of an ideal fluid with a free streamline and axisymmetric geometry. This type of problem is difficult to solve since the shape of part of the boundary of the problem is unknown a priori and conformal mapping techniques do not apply to axisymmetric geometry. Hence solution of these problems is complicated due to geometrical effects. The problem of an ideal axisymmetric jet was investigated by Duh4 with proper boundary conditions. He specified a priori the distance of the tube from the solid plane and then used a trial and error method for determining the shape of the free streamline. The solution contained significant error due to the trial and error method solution which was necessitated by the free streamline boundary condition. Davies and Aylward3 have shown by conformal mapping in the two-dimensional problem that a, an independent parameter of the problem which is defined as the ratio of the deflected velocity to the incident velocity at infinite distances from the origin, is a function of the non- dimensional tube distance. They showed that a approaches 1 when the tube is an infinite distance from the solid plane and approaches infinity as the tube exit approaches the solid plane. In this work the axisymmetric jet was transformed from the r, z plane to the Psi/^, Phi/^ plane. A non-linear form of the flow equation was derived to solve the problem in the Psi/^, Phi/^ plane. Due to the transformation, R (radial distance) was solved directly; thus avoiding a trial and error method for finding the shape of the free streamline. A value for alpha/^ was selected and the tube distance was calculated. Due to the transformation the flow equation became a second order non-linear partial differential equation and the boundary condition along the free streamline also became non-linear. A relaxation technique was applied to solve the equation. a was selected equal to 1.07 for which the tube distance was found to be 1.494. It was found that the slope of the free streamline approached infinity at the tube exit which must be the case in order for the velocity to be continuous at that point. For R z 2 the free streamline satisfies the hyperbolic equation, ZR = - ^/^ where -^/^ = .4673 in this solution.