One-dimensional refraction properties of compression shocks in non-ideal gases

Abstract

Non-ideal gases refer to deformable substances in which the speed of sound can decrease following an isentropic compression. This may occur near a phase transition such as the liquid–vapour critical point due to long-range molecular interactions. Isentropes can then become locally concave in the pressure/specific-volume phase diagram (e.g. Bethe–Zel’dovich–Thompson (BZT) gases). Following the pioneering work of Bethe (Tech. Rep. 545, Office of Scientific Research and Development, 1942) on shocks in non-ideal gases, this paper explores the refraction properties of stable compression shocks in non-reacting but arbitrary substances featuring a positive isobaric volume expansivity. A small-perturbation analysis is carried out to obtain analytical expressions for the thermo-acoustic properties of the refracted field normal to the shock front. Three new regimes are discovered: (i) an extensive but selective (in upstream Mach numbers) amplification of the entropy mode (hundreds of times larger than those of a corresponding ideal gas); (ii) discontinuous (in upstream Mach numbers) refraction properties following the appearance of non-admissible portions of the shock adiabats; (iii) the emergence of a phase shift for the generated acoustic modes and therefore the existence of conditions for which the perturbed shock does not produce any acoustic field (i.e. ‘quiet’ shocks, to contrast with the spontaneous D’yakov–Kontorovich acoustic emission expected in 2D or 3D). In the context of multidimensional flows, and compressible turbulence in particular, these results demonstrate a variety of pathways by which a supplied amount of energy (in the form of an entropy, vortical or acoustic mode) can be redistributed in the form of other entropy, acoustic and vortical modes in a manner that is simply not achievable in ideal gases. These findings are relevant for turbines and compressors operating close to the liquid–vapour critical point (e.g. organic Rankine cycle expanders, supercritical $\text{CO}_{2}$ compressors), astrophysical flows modelled as continuum media with exotic equations of state (e.g. the early Universe) or Bose–Einstein condensates with small but finite temperature effects.