Interpolation in transient polytropic flow

Abstract

An accurate method of space-line interpolation between known parameter values at discretely spaced points in a numerical grid is introduced. The method–designated ‘quasi-steady’ or ‘physical’ interpolation—is conceptually simple, being based on a physical interpretation of flow. Its greatest value is in spatially varied regions of flow. Potential applications include free-surface flows, two-phase flows and general 3D gas flows, but this paper is restricted to the analytically simple case of polytropic gas flows. In sufficiently simple steady flows, the proposed method is potentially exact. It is used herein to assess the accuracy of (a) linear interpolation and (b) some cubic interpolation schemes related to those discussed by Murray in Ref [1]. The latter are found to be potentially more accurate than the former, but less “safe”—special care must be taken to avoid undesirable behaviour at high Mach numbers. No method of interpolating between discrete points can be universally exact. In unsteady flows, for example, an infinity of alternatives could be physically possible, particularly when discontinuities exist between adjacent grid points. Two proposed implementations of the quasi-steady method lead to plausible approximations to unsteady flow distributions. It is shown that they will typically be more accurate than linear or cubic methods. Despite its greater accuracy, the quasi-steady method should not always be used in preference to other methods. Alternative criteria as simplicity can also be relevant.