Reduced Data-Driven Turbulence Closure for Capturing Long-Term Statistics

Abstract

We introduce a simple, stochastic, a-posteriori, turbulence closure model based on a reduced subgrid scale term. This subgrid scale term is tailor-made to capture the statistics of a small set of spatially-integrate quantities of interest (QoIs), with only one unresolved scalar time series per QoI. In contrast to other data-driven surrogates the dimension of the "learning problem" is reduced from an evolving field to one scalar time series per QoI. We use an a-posteriori, nudging approach to find the distribution of the scalar series over time. This approach has the advantage of taking the interaction between the solver and the surrogate into account. A stochastic surrogate parametrization is obtained by random sampling from the found distribution for the scalar time series. Compared to an a-priori trained convolutional neural network, evaluating the new method is computationally much cheaper and gives similar long-term statistics.

Figures (16)

• Figure 1: Spectral filters, white = 1, gray = 0.
• Figure 2: The Gaussian filter and various flow snapshots.
• Figure 3: Tracking QoIs with the predictor-corrector approach.
• Figure 4: Empirical cumulative distribution functions and kernel-based probability density functions for the marginal distribution of $E_{[0,15]}$ for the first 10000 days and full 20000 days of the HF simulation.
• Figure 5: Convergence of marginals for $\Delta Q_i$ and HF $Q_i$. The large panels show the KS-distance between the marginal in the first $x$ days and the reference marginal over the full 20000 days. A KS distance of 0.03 corresponds to the largest value from Table \ref{['tab: KS HF converged']}, below which we consider the PDF to be converged. The small panels show some PDFs for points in the convergence graph, with the marginal from the full 20000 days in black.