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Theoretical framework and jump relations for supersonic supercritical-fluid flows through shock waves

Kevin Foley-Valledor and Lluís Jofre*

  • *Contact author: lluis.jofre@upc.edu
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Phys. Rev. Fluids 11, 113401 – Published 5 November, 2026

DOI: https://doi.org/10.1103/cksm-mzc4

Abstract

Shock waves have historically been modeled assuming ideal-gas behavior, yielding universal jump relations and a viscous structure that depends almost exclusively on the Mach number. However, many fluids of physical interest operate in dense or supercritical regimes where ideal-gas thermodynamics no longer applies. Therefore, this work investigates the structure of shock waves at supercritical-fluid conditions without invoking ideal-gas assumptions. Starting from the Navier-Stokes equations coupled with a real-fluid thermophysical model, the governing relations are derived for the spatial evolution of thermodynamic variables across the shock. Using supercritical carbon dioxide as a representative fluid, it is shown that nonideal thermodynamics strongly modifies the pressure-density jump relations and internal profiles of shocks, producing substantial deviations from classical Rankine-Hugoniot predictions. Interestingly, despite these pronounced thermodynamic effects, the physical thickness of the shock remains primarily controlled by the upstream Mach number. Taking both the structural and thermodynamic observations together, these results ultimately reveal how nonideal fluid behavior fundamentally alters compressive discontinuities, thereby establishing the strict limits of classical shock theory in dense and supercritical fluids.

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Phys. Rev. Fluids 11, 113401– Published 5 November, 2026
  • Received 7 May 2026
  • Accepted 14 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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