Evidence on the incompatibility of modern smoothed particle hydrodynamics and eddy viscosity models for large eddy simulations

TL;DR

This study addresses the incompatibility between modern SPH-LES and classical eddy-viscosity models for turbulent flows, arguing that SPH's non-local, overlapping kernel discretization induces significant implicit subfilter stresses ($\tau_{SFS}$) that clash with explicit models. Using a coarse-graining framework and a MLS-SPH-ALE MFM implementation in GIZMO, the authors test Taylor-Green flow at $Re=10^4$, comparing Nicoud's $\sigma$-model SFS with an explicit SFS against the implicit SFS arising from discretization, evaluated via the spectral energy density $E(k)$ and the R-index $R_i$. Their results show that explicit eddy-viscosity SFS degrades turbulence predictions and cannot suppress the implicit SFS, especially when the filter width equals the kernel width $D_K$, while reducing the filter width to $\Delta \le D_K/2$ mitigates some issues but at substantial cost. The findings advocate SPH-tailored subfilter stress models or reliance on implicit SFS, with implications for accurate turbulence modeling in multiphase SPH simulations and potential alternatives such as Vortex Particle Methods or ML-guided closures.

Abstract

In this work, we will present evidence for the incompatibility of modern Smoothed Particle Hydrodynamics (SPH) methods and eddy viscosity models. Taking a coarse-graining perspective, we physically argue that modern SPH methods operate intrinsically as Lagrangian Large Eddy Simulations (LES) for turbulent flows with strongly overlapping discretization elements. However, these overlapping elements in combination with numerical errors cause a significant amount of implicit subfilter stresses (SFS). Considering a Taylor-Green flow at $Re=10^4$, the SFS will be shown to be relevant where turbulent fluctuations are created, explaining why turbulent flows are challenging even for modern SPH methods. Although one might hope to mitigate the implicit SFS using eddy viscosity models, we show a degradation of the turbulent transition process, which is rooted in the non-locality of these methods.

Figures (8)

• Figure 1: Typical distribution of spectral energy density obtained with modern SPH methods for incompressible turbulence. The properly resolved range with large eddies passes into an energy deficit range which is non-locally caused and followed by a Lagrangian noise range. From an optimal SFS model we would expect a reduction of the Lagrangian noise in favor of the deficit range. However, with incompatible classical SFS models the noise is barely reduced and the deficit range exacerbated due to non-locality.
• Figure 2: Illustration of spatial coarse-graining emerging from the generalization of Hardy's theory Hardy_1982Okraschevski_2021_2. Adapted from Okraschevski_2022.
• Figure 3: Visualization of the velocity decomposition in Equation (\ref{['eq:PeculiarVelocity']}). Adapted from Okraschevski_2024.
• Figure 4: Qualitative verification of the implementation of the $\sigma$-model Nicoud_2011 for $N=512^3$. (a,b) Flow structures before and after the dissipation peak for the case without explicit SFS model (WCMFM) and (c,d) for the case with explicit SFS model (WCMFM + SIGMA). In (e,f) the scaled eddy viscosity field is displayed.
• Figure 5: Quantitative effect of the $\sigma$-model Nicoud_2011 in physical and spectral space for different resolutions. (a) Averaged kinetic energy, (b) Averaged dissipation rate, (c) Scaled spectral energy density at $t=14\,\text{s}$ for DNS and WCMFM run ($N=512^3$) without explicit SFS model. (d) Scaled spectral energy density at $t=14\,\text{s}$. For orientation the kernel scale for $N=512^3$ is included.