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.
Highlights
Dense vapours are single-phase gases featuring large heat capacities relative to their molecular weights
Part of the expansion occurs close to the thermodynamic critical point (TCP), where the sound speed substantially decreases, making the expander flow highly supersonic and causing shockwaves to form in the expander
In the dense-vapour regime, close to the liquid–vapour equilibrium, the classical ideal-gas assumption has to be reconsidered to take into account the thermodynamic critical point and the so-called associated non-ideal-gas effects
Summary
Dense vapours are single-phase gases featuring large heat capacities relative to their molecular weights. Refraction properties of compression shocks in non-ideal gases gas dynamics theory to BZT fluids Most of these studies have focused on the mixed-nonlinearity effects on wave formation in idealised inviscid flow conditions (without any boundaries), yet have revealed significant departure from classical theory. The non-admissible branches a–b and b–c from figure 1(iii) are part of a loop in M2(M1), which is consistent with the expression of the Liu–Oleinik maximum wave speed criterion (for each point on the b–c branch there exists an intermediate point on the H-line with higher wave speed) It is the consequence of a change of variation of the R-line slope (i.e. the mass flow rate j) when the post-shock state point is moving from sonic condition on the non-convex part of the H-line in the (p, θ)-diagram (red segment in figure 1(iii)).
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