Approximate Analysis of a Laminar Elliptical Jet

Abstract

Theme TPHE past several years have witnessed an upsurge JL of interest in thrust augmenting ejectors for V/STOL aircraft and ejector pumps for lasers. In each of these devices, as well as many others, energy is transferred from a high velocity primary flow to a lower velocity secondary flow in such a compact dimension that the near field of the jet is critical. A parameter of basic importance in determining the success or failure of this transfer of energy is the momentum distribution due to the shape and location of the primary nozzles. The momentum distribution may be characterized by a (defined as the lateral position of the mean velocity), since a majority of the jet's momentum is contained within the half widths. In the case of turbulent nonaxisymmetric jet shapes, Trentacoste and Sforza1 have shown experimentally that the major axis half width initially decreases while on the minor axis, the half width grows monotonically. Such a difference in growth rate has not been predicted by either a laminar or turbulent treatment of the problem. Pai and Hsieh2 have treated the linearized case of the laminar rectangular jet by applying the boundary-layer assumptions in both the major and minor axis directions. The result is a rnonotonic half width growth on both axes. In the present analysis, the Navier-Stokes equations are derived in elliptical coordinates. It is then assumed that at any position within the jet, one of the orthogonal coordinate surfaces coincides with the elliptical isovel passing through that position. At this point it is possible to make some rational assumptions concerning the functional form and order of magnitude of each of the variables employed. These assumptions reduce the equations to boundary-layer (i.e., parabolic) type. They also reduce the 4 equations and 4 unknowns to 2 equations and 2 unknowns. These 2 equations are of twodimensional form with several additional terms, and may be described as quasi-two-dim ensional. This form does not require that the equations be solved throughout the threedimensional field but rather that they be solved on the major and minor axes of the jet cross section at each streamwise position. In this way the required computations are greatly reduced. Although the present analysis is laminar, references are made to turbulent experiments to point out qualitative agreement between the two.