Generalized Tadmor Conditions and Structure-Preserving Numerical Fluxes for the Compressible Flow of Real Gases

Abstract

We generalize Tadmor's algebraic numerical flux condition for entropy-conservative discretizations of conservation laws to a broader class of secondary structures, i.e. possibly non-convex secondary quantities whose evolution can consist of both conservative and non-conservative contributions. The resulting generalized Tadmor condition yields a discrete local balance law for secondary structures alongside the discrete conservation law that is solved. In contrast to the convex entropy setting, non-convex secondary quantities can have singular Hessians and non-injective gradients; this introduces an additional necessary structural requirement, which we term (discrete) null-consistency. Null-consistency constrains admissible numerical work terms and is required for the existence and well-posedness of fluxes satisfying the generalized Tadmor condition. To construct such fluxes in practice, we show how discrete gradient operators provide systematic construction methods even when some of the functions entering the secondary structure are arbitrary, as in compressible flow closed by an arbitrary equation of state. As an application, we derive an entropy-conserving and kinetic-energy-consistent numerical flux for the Euler equations with an arbitrary (non-ideal) equation of state. We demonstrate the performance of the resulting scheme on a set of supercritical/transcritical compressible-flow test cases using several non-ideal equations of state, including a fully turbulent transcritical flow with a state-of-the-art equation of state and models for viscosity and heat conductivity. Computations are performed with our new open-source, flexible, JAX-based, multi-GPU compressible flow solver for Helmholtz-based equations of state available at github.com/rbklein/HelmEOS2.

Figures (12)

• Figure 1: Illustration of the null-consistency property at a point $\bm{u}_s \in \mathcal{U}$ where $\mathcal{H}_q(\bm{u}_s) := \frac{\partial^2 {q}}{\partial {\bm{u}}^2}(\bm{u}_s)$ is singular for $n=2$ and $d=1$. Note how $f_1(\bm{u}_s) \nabla_u \xi_1(\bm{u}_s) + f_2(\bm{u}_s) \nabla_u \xi_2(\bm{u}_s) = \nabla_u \psi_q(\bm{u}_s) - c_q(\bm{u}_s)\nabla_u \mathcal{G}_q(\bm{u}_s)$ to satisfy the dual compatibility condition.
• Figure 2: Comparison of the speed of sound $c$$[\text{m} \cdot \text{s}^{-1}]$, reduced density $\rho_r = \rho / \rho_c$ and isobaric heat capacity $c_p$$[\text{J} \cdot \text{kg}^{-1} \cdot \text{K}^{-1}]$ as a function of reduced temperature $T_r = T / T_c$, using different EoS (ideal gas (IG), Van der Waals (VdW), Peng-Robinson (PR), Kunz-Wagner (KW)) over an isobar $p=1.1 \cdot p_c$$[\text{Pa}]$.
• Figure 3: Results of the convergence analysis, the relative errors in mass density $\varepsilon_{\rho}$, momentum density $\varepsilon_m$, and total-energy density $\varepsilon_E$ are shown and compared against a reference. All errors are seen to be of order $\mathcal{O}(\Delta x^2)$.
• Figure 4: Results of the conservation analysis, the relative conservation error of total entropy $\varepsilon_{\mathcal{S}}$, and total kinetic energy $\varepsilon_{\mathcal{K}}$ are shown. Total entropy is conserved to machine precision, while total kinetic energy is conserved to within a very small error.
• Figure 5: The numerical solution of DW at $t=t_f$ in terms of the primitive variables $\bm{\rho}_h, \bm{v}_h,\bm{p}_h$, alongside the numerical work term \ref{['eq:kinennumwork']} at cell interfaces (bottom-right). Deviations from exact spatially constant solutions can be observed in the numerical solution (top-right and bottom-left), leading to spurious pressure work (bottom-right).

Theorems & Definitions (4)

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