Reduced Order Modeling of Rotationally Driven Flows

“It seems doubtful whether we can expect to understand fully the instability of a fluid flow without obtaining a mathematical representation of the motion of the fluid in some particular case in which instability can actually be observed.” – G. I. Taylor (1923)

Flows of interest in astrophysical and atmospheric applications are often driven by rapid rotation. While their large scale makes direct numerical simulation of such flows impossible, their rotational nature enforces high degrees of symmetry which are attractive for mathematical analysis. Taylor-Couette(TC) flow (the flow between two independently rotating cylinders or spheres) has historically been used as a model problem for studying the fundamental physics and mathematical structure of such flows. TC flow itself displays a rich array of distinct flow regimes, which are both mathematically well-defined and observable in the laboratory. However, despite decades of study the mechanism by which these transitions occur has yet to be fully understood. Our work focuses on deriving predictive models from the governing equations to approximate the relevant physics in TC flow and rotational flows in general. We are particularly interested in self-organization of these flows into distinct coherent structures, which if modeled accurately could serve as an efficient basis for future simulations. To do so, we utilize the Resolvent analysis pioneered by McKeon and Sharma (2010) and used extensively in our group. Resolvent analysis allows us to exploit properties of the linear system dynamics to estimate the relevant coherent structures present in a given flow. In theory the full flow field can then be approximately reconstructed through the nonlinear interaction of a limited subset of Resolvent modes. 

Figure: A representation of Taylor vortex flow using only one singular function of the linear resolvent operator. Note that the distinctive “Taylor roll” vortices, which are known to be the dominant structure in this particular flow are clearly captured by only a single basis function. 

Benedikt Barthel and Beverley McKeon 

Funding 
ONR Grant # N00014-17-1-3022