Consistency requirement of data-driven subgrid-scale modeling in large-eddy simulation

TL;DR

This work investigates why data-driven subgrid-scale closures in large-eddy simulation often perform well in training (a priori) but poorly in actual LES runs (a posteriori). Using a DNS-aided LES framework, the authors quantify the numerical deviation $\delta_i$ arising from discretization and commutation and study two SGS closures: an eddy-viscosity and a complex nonlinear form. Training with $\delta_i$ generally reduces a priori accuracy but improves a posteriori energy spectra fidelity and stability, with the nonlinear closure benefiting most when $\delta_i$ is included; excluding the deviation can lead to severe a posteriori instability. The results emphasize that a physically grounded closure form, appropriate filter width, and explicit accounting for numerical deviation are essential for reliable, data-driven SGS modeling in LES, especially for anisotropic and non-equilibrium flows.

Abstract

Data-driven subgrid-scale (SGS) modeling in the large-eddy simulations (LES) suffers from the inconsistency between the \textit{a priori} tests and the a posteriori tests, which make training accurate SGS models a difficult task. We study the difference in filtered high-fidelity data and LES to identify the numerical deviation between the two cases, which is a combined impact of commutation error, numerical errors, and error coupling. The impact of the numerical deviation is examined through two SGS model formulations: the eddy-viscosity and the complex nonlinear models. By incorporating numerical deviations into model training, we enhance consistency, stabilize simulations, and improve predictions of energy spectra in a posteriori tests. Our findings highlight that data-driven methods introduce significant nonlinearity and equation coupling, exacerbating inconsistencies compared to non-data-driven approaches. Finally, while the impact of the numerical deviation can be generalized, achieving accurate model predictions necessitates a physically grounded model form and an optimal filter width.

Figures (10)

• Figure 1: A sketch of the DNS-aided LES evolving from time step $t_n$ to time step $t_{n+1}$.
• Figure 2: (a) Spectrum of the numerical deviation $u_i\delta_i$ (solid line). Dashed line indicates the -5/3 law. (b) The p.d.f. of the energy transfer for the numerical deviation $u_i\delta_i$ for filter width $\sigma/\Delta_{\textrm{LES}}=2.0$.
• Figure 3: Instantaneous (a,d) target dissipation rate $\varepsilon/\langle \varepsilon_{\textrm{fDNS}}\rangle$ and (b,e) modeled dissipation rate $\varepsilon_{\textrm{ml}}/\langle \varepsilon_{\textrm{fDNS}}\rangle$ on the same $x-y$ plane and (c,f) scatter plot showing the correlation between $\varepsilon_{\textrm{ml}}$ and $\varepsilon_{\textrm{fDNS}}$ for eddy-viscosity neural-network SGS models (a,b,c) excluding and (d,e,f) including the numerical deviation for baseline filter width $\sigma/\Delta_{\textrm{LES}}=2.0$. The dashed line in (c,f) indicates identity.
• Figure 4: Instantaneous (a,d) target dissipation rate $\varepsilon/\langle \varepsilon_{\textrm{fDNS}}\rangle$ and (b,e) modeled dissipation rate $\varepsilon_{\textrm{ml}}/\langle \varepsilon_{\textrm{fDNS}}\rangle$ on the same $x-y$ plane and (c,f) scatter plot showing the correlation between $\varepsilon_{\textrm{ml}}$ and $\varepsilon_{\textrm{fDNS}}$ for complex nonlinear neural-network SGS models (a,b,c) excluding and (d,e,f) including the numerical deviation for baseline filter width $\sigma/\Delta_{\textrm{LES}}=2.0$. The dashed line in (c,f) indicates identity.
• Figure 5: (a) Time evolution of the instantaneous (solid) and average (dashed) kinetic energy for eddy-viscosity SGS model excluding (blue) and including (red) the numerical deviation. (b) Energy spectra of the a posteriori for eddy-viscosity SGS model excluding (blue) and including (red) the numerical deviation using time step of $\Delta t_{\textrm{DNS}}$ (solid), $2\Delta t_{\textrm{DNS}}$ (dashed), and $5\Delta t_{\textrm{DNS}}$ (dotted). Energy spectrum of the filtered DNS is given by the black dot-dashed line.