Design of maximum thrust plug nozzles with variable inlet geometry

Abstract

An optimization analysis is presented for axisymmetric plug nozzles with varible inlet geometry. The analysis is based on the governing gas dynamic relations for a rotational flow of a frozen or equilibrium gas mixture. The problem is formulated to maximize the axial thrust produced by the plug nozzle for a general isoperimetric constraint, such as constant nozzle length or constant nozzle surface area. The effects of base pressure and ambient pressure are included in the thrust expression to be maximized. The governing gas dynamic equations and the differential and integral constraints that the solution must satisfy are incorporated into the formulation by means of Lagrange multiples. The formalism of the calculus of variations is applied to the resulting functional to be maximized. The results of the optimization analysis are a set of partial differential equations for determining the Lagrange multipliers in the region of interest and a set of equations for determining the necessary boundary conditions for the solution. The complete set of equations for the gas dynamic properties and the Lagrange multipliers are system of first order, quasi-linear, non-homogeneous partial differential equations of the hyperbolic type, which can be treated by the method of charac- teristics. The characteristic and compatibility equations for the system are presented. A numerical solution procedure is presented to determine wether or not a given plug nozzle geometry is an optimal solution. An iteration technique is developed which systematically adjusts the plug nozzle geometry until the optimal solution is obtained. Selected parametric studies are presented. These studies illustrate the effect of the specific heat ratio, the design pressure ratio and the base pressure model on the thrust peformance and nozzle geometry of optimal, fixed length, plug nozzles.