Low Dimensional Models of Turbulent Plane Couette Flow Using the Proper Orthogonal Decomposition

We derive empirical basis functions from direct numerical simulations (DNS) of turbulent plane Couette flow at the same (low) Reynolds number for two different rectangular domains, one corresponding to the Minimal Flow Unit (MFU), the smallest domain which allows sustained turbulence, the other being significantly larger. The Navier-Stokes equations are projected onto subspaces spanned by the dominant energy-bearing empirical modes, and models for energy transport to neglected modes may be included. In truncations containing only streamwise-invariant modes for the larger domain, we find behaviors including nontrivial equilibria, quasiperiodicity with three distinct frequencies, chaos, intermittency, and attracting heteroclinic cycles. These suggest interesting characterizations of the turbulent state, but their limitations have led us to consider models for the more constrained (and simpler) turbulence in the MFU. In particular, a six-mode model, including three streamwise modes, possesses a limit cycle which agrees qualitatively with the DNS data.