Physical significance of artificial numerical noise in direct numerical simulation of turbulence

TL;DR

This work shows that artificial numerical noise in DNS can be interpreted as a physical disturbance equivalent to thermal fluctuations and stochastic environmental noise. By comparing CNS benchmarks, where numerical noise is suppressed, with DNS results for a 2D Kolmogorov flow, the authors demonstrate that including small fluctuations yields statistics that align with DNS on a fine mesh, while DNS without fluctuations diverges due to numerical noise. The findings imply that different numerical setups probe turbulence under different disturbance levels, and that many DNS results can be physically meaningful despite differing statistics. The work highlights the positive role of numerical noise in turbulence studies and suggests broader implications for CFD reproducibility and the use of CNS as a research tool.

Abstract

Using clean numerical simulation (CNS) in which artificial numerical noise is negligible over a finite, sufficiently long interval of time, we provide evidence, for the first time, that artificial numerical noise in direct numerical simulation (DNS) of turbulence is approximately equivalent to thermal fluctuation and/or stochastic environmental noise. This confers physical significance on the artificial numerical noise of DNS of the Navier-Stokes equations. As a result, DNS on a fine mesh should correspond to turbulence under small internal/external physical disturbance, whereas DNS on a sparse mesh corresponds to turbulent flow under large physical disturbance, respectively. The key point is that: all of them have physical meanings and so are correct in terms of their deterministic physics, even if their statistics are quite different. This is illustrated herein. Our paper provides a positive viewpoint regarding the presence of artificial numerical noise in DNS.

Figures (12)

• Figure 1: Time histories of the spatially averaged (a) kinetic energy dissipation rate $\langle D\rangle_A$ and (b) enstrophy dissipation rate $\langle D_{\Omega}\rangle_A$ of the 2D turbulent Kolmogorov flow: CNS$^*$ (red solid line) and DNS (blue dashed line).
• Figure 2: Probability density functions (PDFs) of (a) the kinetic energy dissipation rate $D(x,y,t)$ and (b) the kinetic energy $E(x,y,t)$ of the 2D turbulent Kolmogorov flow, where the integration is taken in $(x,y)\in[0,2\pi)^2$ and $t \in [100, 300]$: CNS$^*$ (red line) and DNS (blue circle).
• Figure 3: Probability density functions (PDFs) of (a) the enstrophy dissipation rate $D_\Omega(x,y,t)$ and (b) the enstrophy $\Omega(x,y,t)$ of the 2D turbulent Kolmogorov flow, where the integration is taken in $(x,y)\in[0,2\pi)^2$ and $t \in [100, 300]$: CNS$^*$ (red line) and DNS (blue circle).
• Figure 4: (a) Time-averaged kinetic energy spectra $\langle E_k \rangle_{t}$ of the 2D turbulent Kolmogorov flow where the black dashed line corresponds to the -5/3 power law and the black dash-dot line denotes the wave number of external force $k=n_K=16$. (b) Spatiotemporal-averaged scale-to-scale energy fluxes $\langle \Pi^{[l]} \rangle$ of the 2D turbulent Kolmogorov flow where the black dashed line denotes $\langle \Pi^{[l]} \rangle=0$ and the black dash-dot line denotes the forcing scale $l=l_f=\pi/n_K=0.196$. Red solid line is the CNS$^*$ result. Blue circles is the DNS result.
• Figure 5: Spatiotemporal-averaged scale-to-scale enstrophy fluxes $\langle \Pi_\Omega^{[l]} \rangle$ of the 2D turbulent Kolmogorov flow where the black dashed line denotes $\langle \Pi_\Omega^{[l]} \rangle=0$ and the black dash-dot line denotes the forcing scale $l=l_f=\pi/n_K=0.196$. Red solid line is the CNS$^*$ result. Blue circles display the DNS result.