Multiscale investigation of Kolmogorov flow: From microscopic molecular motions to macroscopic coherent structures

It is extremely expensive to study turbulence using conventional molecular simulation methods such as direct simulation Monte Carlo and molecular dynamics methods, as the molecular scales and the turbulent characteristic scales are significantly separated. To bridge this gap, we employ a particle Fokker-Planck method, namely, the Langevin dynamics simulation method, to study two-dimensional Kolmogorov flow, which is induced by a spatially periodic external force in an unbounded domain. Our simulation results predict that when the Reynolds number (Re) exceeds the critical value, a sequence of bifurcations takes place in the flow as the Reynolds number increases, forming a variety of flow patterns. Correspondingly, the effective diffusion coefficient is enhanced due to convection. Two main regimes of the flow have been observed: the small-scale cellular structure regime (Re_c < Re < 8Re_c), and the large-scale coherent structure regime (Re > 8Re_c). We demonstrate that Langevin dynamics can capture the double kinetic-energy cascade when the large-scale structure is formed in two-dimensional turbulence: the inverse energy cascade has a scaling law of k^-4 due to energy condensation in the large-scale structures, while the direct energy cascade has an exponential decay corresponding to the dissipation mechanism. This work provides strong evidence that Langevin dynamics is a promising multiscale tool to study turbulence from molecular motions to large-scale coherent structures.