DATA-DRIVEN SPARSE IDENTIFICATION OF GOVERNING EQUATIONS FOR FLUID DYNAMICS

Abstract

It is a challenging and important issue to establish a nonlinear dynamic model of system by use of limited data. The data-driven sparse identification method is an effective method developed recently to identify the governing equations of the dynamic system from data developed in recent years. In this paper, governing equations for different flows are identified by data-driven sparse identification methods. Partial differential equation functional identification of nonlinear dynamics (PDE-FIND) scheme and least absolute shrinkage and selection operator (LASSO) scheme are used to identify the governing equations of two-dimensional flow past a circular cylinder, liddriven cavity flow, Rayleigh-Benard convection and three-dimensional turbulent channel flow. An over-complete candidate library is constructed by direct numerical simulation flow field data in the process of identification. Variables in the library are retained up to second order, variable derivatives are retained up to second order, and nonlinear terms are retained up to fourth order. By comparing the results from the two methods, we find both methods show good performance in identifying governing equation with no nonlinear terms, i.e., vorticity transport equation, heat transport equation and continuity equation. PDE-FIND scheme correctly identified the governing equations and Rayleigh number and Reynolds number for the flow field. But LASSO scheme failed to identify the governing equations which contain strong nonlinear terms, i.e., Navier-Stokes equations. This is because grouping effect may occur among the items in the candidate library and only one item in the group is chosen in such case in LASSO scheme. So PDE-FIND scheme is more effective than LASSO scheme in sparse identification of strongly nonlinear partial differential equation. It is also found that selecting data from regions with abundant flow structures can improve the accuracy of data-driven sparse identification results.