Clean numerical simulation (CNS) of three-dimensional turbulent Kolmogorov flow

TL;DR

This work uses clean numerical simulation (CNS) to solve the fully 3D Navier–Stokes equations for Kolmogorov forcing and compares the results with conventional DNS at $Re=1211.5$ and $n_K=4$. CNS achieves negligible truncation and round-off errors, yielding a long, reliable trajectory that preserves the initial spatial symmetry and reveals an inverse energy cascade at large scales coupled with a direct cascade at small scales, along with accurate statistics. In contrast, DNS rapidly accumulates numerical noise, breaking symmetry and producing significantly different statistics and energy transfer, despite satisfying traditional resolution criteria; this motivates a statistic-stability criterion for DNS validity and introduces the concept of a critical predictable time $T_c$. The findings imply that DNS can be unreliable for long-time turbulence studies and underscore CNS as a powerful theoretical tool, while also suggesting the need for stochastic turbulence models (e.g., LLNS) to account for physical fluctuations in real flows.

Abstract

Turbulence holds immense importance across various scientific and engineering disciplines. The direct numerical simulation (DNS) of turbulence proposed by Orszag in 1970 is a milestone in fluid mechanics, which began an era of numerical experiment for turbulence. Many researchers have reported that turbulence should be chaotic, since spatiotemporal trajectories are very sensitive to small disturbance. Thus, due to the famous butterfly-effect of chaos, unavoidable numerical noises of DNS might have great influence on spatiotemporal trajectories of turbulence. This is indeed true for a two-dimensional (2D) Kolmogorov turbulent flow, as currently revealed by a much more accurate algorithm than DNS, namely the ``clean numerical simulation'' (CNS). Different from DNS, CNS can greatly reduce both of truncation error and round-off error to any required small level so that numerical noise can be rigorously negligible throughout a time interval long enough for calculating statistics. However, In physics, 3D turbulent flow is more important than 2D turbulence. Thus, for the first time, we solve a 3D turbulent Kolmogorov flow by means of CNS in this paper, and compare our CNS result with that given by DNS in details. It is found that the spatial-temporal trajectories of the 3D Kolmogorov turbulent flow given by DNS are indeed badly polluted by numerical noise rather quickly, and besides the DNS result has significant deviations from the CNS benchmark solution not only in the spatial symmetry of flow field and the energy cascade but also even in statistics.

Figures (11)

• Figure 1: Distribution of the vorticity modulus $|\bm{\omega}(\textbf{x},t)|$ of the 3D turbulent Kolmogorov flow governed by (\ref{['NS']}) subject to the periodic boundary condition and the initial condition (\ref{['initial_condition']}) with the spatial symmetry (\ref{['spatial-symmetry']}) in the case of $n_K=4$ and $Re=1211.5$, given respectively by the CNS benchmark solution (left) and the DNS result (right) at some typical times, i.e. (a)-(b) $t=85$, (c)-(d) $t=95$, (e)-(f) $t=100$, and (g)-(h) $t=500$, respectively.
• Figure 2: Distribution of the vorticity modulus at $y=0$, i.e. $|\bm{\omega}(x,0,z,t)|$, of the 3D turbulent Kolmogorov flow governed by Eq. (1) subject to and the initial condition (2) with the spatial symmetry (3) in the case of $n_K=4$ and $Re=1211.5$ given by the CNS benchmark solution (left) and the DNS result (right), respectively, at some typical times, i.e. (a)-(b) $t=85$, (c)-(d) $t=95$, (e)-(f) $t=100$, and (g)-(h) $t=500$.
• Figure 3: Distribution of the vorticity modulus at $x=0$, i.e. $|\bm{\omega}(0,y,z,t)|$, of the 3D turbulent Kolmogorov flow governed by Eq. (1) subject the periodic boundary condition and the initial condition (2) with the spatial symmetry (3) in the case of $n_K=4$ and $Re=1211.5$ given by the CNS benchmark solution (left) and the DNS result (right), respectively, at some typical times, i.e. (a)-(b) $t=85$, (c)-(d) $t=95$, (e)-(f) $t=100$, and (g)-(h) $t=500$.
• Figure 4: Distribution of the vorticity modulus at $z=0$, i.e. $\Large|\bm{\omega}(x,y,0,t)\Large|$, of the 3D turbulent Kolmogorov flow governed by Eq. (1) subject the periodic boundary condition and the initial condition (2) with the spatial symmetry (3) in the case of $n_K=4$ and $Re=1211.5$ given by the CNS benchmark solution (left) and the DNS result (right), respectively, at some typical times, i.e. (a)-(b) $t=85$, (c)-(d) $t=95$, (e)-(f) $t=100$, and (g)-(h) $t=500$.
• Figure 5: Time histories of the mean squared deviations (between the DNS result and the CNS benchmark solution) of $\delta_u$ defined by (\ref{['delta_u']}) (for the velocity) and $\delta_p$ defined by (\ref{['delta_p']}) (for the pressure) in Appendix C of the 3D turbulent Kolmogorov flow governed by Eq. (1) subject the periodic boundary condition and the initial condition (2) with the spatial symmetry (3) in the case of $n_K=4$ and $Re=1211.5$. Red solid line: $\delta_u$; blue dashed line: $\delta_p$.