‘Suitable Mean Values’ in One-Dimensional Gas Dynamics

Abstract

The problem of the definition of mean values of flow quantities for use in the application of one-dimensional gas dynamics to non-uniform flows is considered. Previously given methods for this problem are surveyed and are shown to be considerably in error and to have some unsatisfying basis of definition. A more satisfactory basis is suggested, based upon enthalpy flux, and this is shown to be more general and to have a satisfactory thermodynamic basis, particularly in the definition of mean temperature. The significance of the form of definition of mean total pressure in relation to the energy equivalent of the irreversibilities is stressed. ‘Availability or entropy flux’, ‘mass weighted’, ‘area weighted’, ‘mass derived’ and ‘momentum mixed’ definitions of mean total pressure are considered. Data for typical velocity profiles, over a wide range of Mach number, and for all known methods of calculating mean values have been obtained and a graphical summary is presented here. At first glance, the data would seem to indicate that the percentage error in calculating total pressure by the different definitions is small, but this is not so. To assess the errors introduced in estimating the irreversibilities of a length of duct by using mass derived or other definitions of mean total pressure, then the error should be estimated as a percentage of the relative differences of mean total pressure involved. The data list which the authors hope to publish separately and the graphs in this paper enable this to be achieved.