An Explicit Primitive Conservative Solver for the Euler Equations with Arbitrary Equation of State

TL;DR

This work introduces an explicit-in-time primitive-conservative solver for the Euler equations that updates a generic thermodynamic variable $\varphi$ (such as $T$, $P$, $e$, $h$, or $s$) instead of the total energy $E^{t}$, and is agnostic to the equation of state. Conservation is enforced by a secant-root search that selects an optimal thermodynamic state $\overline{\varphi}$ to match the exact total-energy change, yielding energy conservation at machine precision while remaining first-order in time. The approach markedly accelerates simulations with complex EoS (notably Span-Wagner via CoolProp) by reducing the cost of thermodynamic evaluations, with temperature-based updates delivering the largest speed-ups; results are validated across CO$_2$ Span-Wagner and N$_2$ Van der Waals tests, including NICFD regimes. The method is generalizable to higher dimensions and provides a practical path toward efficient, accurate NICFD simulations without sacrificing conservation or requiring bespoke EoS solvers.

Abstract

This work presents a procedure to solve the Euler equations by explicitly updating, in a conservative manner, a generic thermodynamic variable such as temperature, pressure or entropy instead of the total energy. The presented procedure is valid for any equation of state and spatial discretization. When using complex equations of state such as Span-Wagner, choosing the temperature as the generic thermodynamic variable yields great reductions in the computational costs associated to thermodynamic evaluations. Results computed with a state of the art thermodynamic model are presented, and computational times are analyzed. Particular attention is dedicated to the conservation of total energy, the propagation speed of shock waves and jump conditions. The procedure is thoroughly tested using the Span-Wagner equation of state through the CoolProp thermodynamic library and the Van der Waals equation of state, both in the ideal and non-ideal compressible fluid-dynamics regimes, by comparing it to the standard total energy update and analytical solutions where available.

Figures (19)

• Figure 1: Variable arrangement and MUSCL reconstruction sketch for a generic variable $q$ on two neighbouring control volumes $C_i$ and $C_j$.
• Figure 2: Example plots of $F({\overline{\varphi}})$ for various $\varphi$ choices at the initial time step discontinuity of a Nitrogen shock tube using the Span-Wagner EoS Span_2000.
• Figure 3: Carbon-dioxide shock tube test with $\Delta x = 0.025 \;\mathrm{m}$ using the Span-Wagner EoS Span_1996 through the CoolProp thermodynamic library CoolProp. Compressibility factor profile and numerical solution on the on the pressure-density thermodynamic plane at $t=0.001\;\mathrm{s}$, with isentropes in yellow and the vapour saturation curve in black.
• Figure 4: Carbon-dioxide shock tube test results using MUSCL with $\Delta x = 0.025\;\mathrm{m}$ using the Span-Wagner EoS Span_1996 through the CoolProp thermodynamic library CoolProp. Density, pressure, and velocity profiles at $t=0.001\;\mathrm{s}$ and absolute value of the total energy imbalance in time. Comparison between the $\varphi$ update scheme for temperature, pressure, specific energy, specific enthalpy, specific entropy, and the standard total energy update (both using the analytical EoS and a look-up table).
• Figure 5: Carbon-dioxide shock tube test 1st order results without using MUSCL with $\Delta x = 0.025\;\mathrm{m}$ using the Span-Wagner EoS Span_1996 through the CoolProp thermodynamic library CoolProp. Density and pressure profiles (first row) and their dimensionless variations with respect to the standard total energy update (second row) at $t=0.001\;\mathrm{s}$. Comparison between the $\varphi$ update scheme for temperature, pressure, specific energy, specific enthalpy, specific entropy, and the standard total energy update (both using the analytical EoS and a look-up table).