Capturing Turbulence with Numerical Dissipation: a Simple Dynamical Model for Unresolved Turbulence in Hydrodynamic Simulations

TL;DR

We present a semi-implicit Large-Eddy Simulation (SLES) framework to model unresolved turbulence in hydrodynamic simulations by explicitly tracking subgrid turbulent energy $e_{ m turb}$ and implicitly sourcing it from local numerical dissipation. The method is calibrated against decaying supersonic turbulence DNS, demonstrating accurate reproduction of the mean small-scale turbulent energy, its scale dependence, and the density–turbulence correlation, and is applied to isolated galaxy disk simulations where it yields locally variable star formation efficiency comparable to explicit LES. SLES combines the practicality of ILES with the explicit tracking of subgrid turbulence, enabling straightforward integration into common astrophysical codes and providing a useful, cost-efficient tool for subgrid turbulence applications in galaxy formation and ISM studies. While successful in the supersonic, star-forming ISM, the approach requires careful handling of shear and subsonic regimes and invites future extensions to magnetohydrodynamics and more sophisticated cascade closures.

Abstract

Modeling unresolved turbulence in astrophysical gasdynamic simulations can improve the modeling of other subgrid processes dependent on the turbulent structure of gas: from flame propagation in the interiors of combusting white dwarfs to star formation and chemical reaction rates in the interstellar medium, and nonthermal pressure support of circum- and intergalactic gas. We present a simple method for modeling unresolved turbulence in hydrodynamic simulations via tracking its sourcing by local numerical dissipation and modeling its decay into heat. This method is physically justified by the generic property of turbulent flows that they dissipate kinetic energy at a rate set by the energy cascade rate from large scales, which is independent of fluid viscosity, regardless of its nature, be it physical or numerical. We calibrate and test our model against decaying supersonic turbulence simulations. Despite its simplicity, the model quantitatively reproduces multiple nontrivial features of the high-resolution turbulence run: the temporal evolution of the average small-scale turbulence, its dependence on spatial scale, and the slope and scatter of the local correlation between subgrid turbulent velocities, gas densities, and local compression rates. As an example of practical applications, we use our model in isolated galactic disk simulations to model locally variable star formation efficiency at the subresolution scale. In the supersonic, star-forming gas, the new model performs comparably to a more sophisticated model where the turbulent cascade is described by explicit subgrid terms. Our new model is straightforward to implement in many hydrodynamic codes used in galaxy simulations, as it utilizes already existing infrastructure to implicitly track the numerical dissipation in such codes.

Figures (12)

• Figure 1: Schematic overview of different types of subgrid turbulence models: implicit (ILES), explicit (LES), and proposed in this work, semi-implicit Large-Eddy Simulations (SLES). The SLES model combines the explicit modeling of the subgrid turbulent energy and its decay, similar to explicit LES, with the implicit source term provided by numerical dissipation, akin to ILES. The labels on the $x$-axis of the power spectrum cartoon represent the driving scale ($L_{\rm drive}$), resolution scale ($\Delta$), and physical dissipation scale ($\eta$). Green colors illustrate the explicit subgrid turbulence energy, $e_{\rm turb}$, while "cascade" and "decay" represent the source and sink terms $\Sigma$ and $\epsilon$ in Equation (\ref{['eq:eturb']}), respectively.
• Figure 2: Gas density (gray) and small-scale turbulent energy slices (orange) in the coarse-grained high-resolution simulation (DNS; the top set of panels) and low-resolution resimulations of the same ICs with the SLES model (the bottom set of panels). The maps are shown at $t = 2/3\;t_{\rm cross,0}$. The leftmost panel in the top row shows the density slice at the native resolution of the DNS, while the other columns show coarse-grained values, with density being volume-weighted and $e_{\rm turb}$ computed from Equation (\ref{['eq:eturb-coarse']}); the corresponding grid size is indicated at the top. The bottom panels show slices at the same time step from low-resolution resimulations using the corresponding grid sizes, where $e_{\rm turb}$ is predicted using the SLES model. Thus, in total, four simulations are shown in the plot as indicated by the green shaded background: one DNS and three SLES resimulations. Low-resolution SLES resimulations produce smoother solutions than the DNS, but they capture the main global features of the small-scale turbulence, in particular, the correlation with gas density.
• Figure 3: Comparison of the global mean small-scale turbulent energy, $E_{\rm turb} \equiv \langle e_{\rm turb} \rangle_{\rm box}$, in coarse-grained DNS (dashed line) and low-resolution SLES resimulations with different values of the dissipation parameter, $C_\epsilon$ (colored lines). For this comparison, we use the grid size of 32$^3$, i.e., the cell size $\Delta \equiv L_{\rm box}/32$. In the DNS, $e_{\rm turb}$ is computed following Equation (\ref{['eq:eturb-coarse']}) while in the low-resolution resimulations, it is tracked by the SLES model. The mean $E_{\rm turb}$ is normalized to the total initial kinetic energy, i.e., $E_{\rm kin,0} \equiv E_{\rm turb}(<L_{\rm box})$ at $t=0$. The time axis is normalized by the initial box-crossing time of turbulence, $t_{\rm cross,0} \equiv 0.5\;L_{\rm box}/\sigma_0$, or $\approx 0.058$ sound-crossing times for the initial Mach number of $\mathcal{M} \approx 8.6$. The value of $C_\epsilon = 1.6$ provides the best match to the global $E_{\rm turb}$ and therefore we will use it as our fiducial choice in the rest of the paper. The agreement with the DNS can be further improved by making $C_\epsilon$ dependent on the local subgrid Mach number, as shown with the thin line and described in the text.
• Figure 4: Comparison of coarse-grained DNS (dashed lines) with the SLES resimulations at corresponding resolutions (colored lines). The top panel shows the evolution of the global Mach number, $\mathcal{M} = \sqrt{ 2 E_{\rm kin} }$ in the adopted units ($\langle \rho \rangle \equiv 1$). The bottom panel shows the mean small-scale turbulent energy, $E_{\rm turb} \equiv \langle e_{\rm turb} \rangle_{\rm box}$, normalized by the initial total kinetic energy in the box, $E_{\rm kin,0}$. The SLES model captures both the evolution of $E_{\rm turb}$ and its scale dependence. As shown in Appendix \ref{['app:dns-convergence']}, $E_{\rm turb}$ is underestimated for 128$^3$ due to a lack of convergence on these small scales; however, we still show the results for this grid to demonstrate that the scaling of $E_{\rm turb}$ with the grid size remains the same in SLES.
• Figure 5: Cell-by-cell correlation between gas density and small-scale velocity dispersion, $\sigma \equiv \sqrt{ 2 e_{\rm turb} / \rho}$ on a 32$^3$ grid. The left column shows the DNS, i.e., each point shows coarse-grained values described in Section \ref{['sec:turb:setup']}; the right column shows a 32$^3$ resimulation with the SLES model, where each point shows cell values at the native resolution, with $e_{\rm turb}$ being predicted by the SLES model. The color shows the divergence of coarse-grain (for DNS) or resolved (for SLES) velocity. The rows correspond to different time steps: from top to bottom, 1, 3, and $5\;t_{\rm cross,0}$. Solid black lines show the medians and 18$^\text{th}$ to 84$^\text{th}$ interpercentile ranges. The red dashed line shows the adiabatic scaling corresponding to $e_{\rm turb} \propto \rho^{5/3}$, with the normalization being the same in all panels. The SLES model captures the relation between $\rho$ and $\sigma$ in terms of normalization, slope, the magnitude of scatter, and its scaling with the resolved velocity divergence.