Data-driven modeling of multiscale phenomena with applications to fluid turbulence

Abstract

This letter introduces a novel data driven framework for constructing accurate and general equivariant models of multiscale phenomena which does not rely on specific assumptions about the underlying physics. This framework is illustrated using incompressible fluid turbulence as an example that is representative, practically important, reasonably simple, and exceedingly well studied. We use direct numerical simulations of freely decaying turbulence in two spatial dimensions to infer an effective field theory comprising explicit, interpretable evolution equations for both the large (resolved) and small (modeled) scales. The resulting closed system of equations is capable of accurately describing the effect of small scales, including backscatter -- the flow of energy from small to large scales, which is particularly pronounced in two dimensions -- which is an outstanding challenge that, to our knowledge, no existing alternative successfully tackles.

Figures (3)

• Figure 1: Representative flow fields used to generate the test data. Shown is the initial vorticity field $\omega$ .
• Figure 2: The energy flux $\Pi$ describing the flow F2 shown in Figure \ref{['fig:initial_conditions']}(b) at $\nu =10^{-6}$ and cutoff scale $\Delta=\ell/64$ for DNS (a), the NGMR model (b), DS model (c), and DM model (d). All fields were calculated from DNS data.
• Figure 3: The accuracy of the evolution equation \ref{['eq:R']}, quantified by the correlation $C_R$, as a function of $\Delta/\ell$ for F1 (blue), F2 (orange) and F3 (yellow) at $\nu = 10^{-6}$.