One-dimensional treatment of non-uniform flow

Abstract

For one-dimensional flow of a perfect gas, conditions at a station of a duct are defined by any four independent properties. Standard methods exist for the calculation of any other desired property from the given four independent properties. The object of this paper is to illustrate the errors likely to arise when the simple one-dimensional flow methods are applied to a circular section duct in which a boundary layer exists. Graphical results are presented, for the case of a one-seventh power law boundary-layer velocity profile, showing the ratio of the true mean values calculated with allowance for the boundary-layer, to quantities derived from the simple one-dimensional calculation. Various boundary layer thicknesses and a range of Mach numbers are dealt with. Specifically three examples are worked out in detail, with different selections of the four independent variables, the selections being chosen to cover problems of common interest. The results of the first two examples might be applied, for instance, to the problem of the performance or design of a duct discharging adiathermally to atmosphere, from a reservoir with known stagnation conditions. The errors are usually small. Thus calculations by simple one-dimensional theory differ by less than about 21 per cent up to a 'one-dimensional' Mach number M = 1, and 5 per cent up to M = 2, from the values obtained by assuming a boundary-layer thickness at exit of 10 per cent of the duct radius. For other boundary-layer thicknesses the errors are roughly in proportion. The results of the third example indicate the errors likely to arise in the analysis of other quantities at a station, from measurements of mass flow, area, total temperature and static pressure. Here the accuracy of the one-dimensional method is within 2 per cent up to a freestream Mach number M' = 2 for any boundary-layer thickness. Total pressure is an exception, the error in this case approaching 10 per cent at M' = 2. General equations are presented for use in cases not covered by these examples. They are analogous to the onedimensional equations, and give ratios of mean flow quantities to their sonic values, as functions of Mach number and correction factors, graphically presented, which depend on the velocity distribution. As a further illustration of possible application of the theory, the correction factors may be used for the calculation of momentum fltix or kinetic energy flux from the mean velocity and mean density.