Modeling low-frequency dynamics in turbulent flows using data

In turbulent flows, events correlated over a long time period are observed occasionally. Some of those events take place on a regular basis, but they tend to occur without apparent periodicity or even intermittently. They are manifested usually by a low-frequency oscillation of characteristic variables. Such dynamics is often associated with an abrupt increase (or decrease) of friction drag, mixing, heat transfer, and aerodynamic noise, thus affecting the stability and robustness of flow systems. Describing and predicting such events are not straightforward and dependent on heuristic and statistical approaches, lacking direct connection with the first principle from which they originate. This study combines a data-driven algorithm and a short-time averaging to systematically educe a partial differential equation (PDE) model optimally describing the low-frequency dynamics. The proposed formulation is tested for linear advection--diffusion equation, complex Ginzburg--Landau equation, and Navier--Stokes equations. Upon choosing an appropriate averaging time scale, the effects of fast dynamics are suppressed by the short-time averaging, making a PDE model for slow dynamics more amenable to be discovered at a significantly lower computational cost.