The disturbance energy and Rayleigh criterion in a non-ideal compressible fluid

Abstract

Small perturbation theory neglects non-linear terms and allows for analytical predictions and identifications of physical trends. It can then be used for the understanding and modelling of more complicated nonlinear phenomena, e.g. turbulence and fluid instabilities. In compressible fluids these theories generally rely on the ideal gas assumption. This article theoretically examines the fluctuations in a non-ideal compressible fluid, utilizing the Navier–Stokes–Fourier model, with an arbitrary equation of state. The linearized governing system is deduced, thereafter elucidating the amplification or attenuation of fluctuations through an analysis of energy disturbance, extending the framework established by Chu (Chu, 1965) to encompass arbitrary non-ideal compressible fluids. It is demonstrated that the proposed disturbance energy embodies the same advantageous properties as Chu’s ideal-gas one. The study of the time evolution of the disturbance energy then allows to derive a simple stability criteria generalizing the Rayleigh criterion (Rayleigh, 1878) to steady and non-uniform flows of non-ideal compressible fluids. Similar to the Rayleigh criterion, this criterion offers novel physical insights into the amplification or attenuation of arbitrary fluctuations within non-ideal compressible fluids. Moreover, the present work represents an advancement attempting to extend classical findings derived under the ideal gas assumption to integrate alternative equations of state. The analysis and equations derived are expected to allow improvements in both the understanding and modelling of disturbances in non-ideal compressible fluids.