Title:

Algebraic Models of Plane Couette Equilibria

Turbulence is one of the most profound unsolved problems in classical mechanics, with applications ranging from aviation to arterial flow. Plane Couette flow, the shearing of a viscous fluid between parallel plates, serves as a fundamental model for studying the transition to turbulence. Invariant solutions to the Navier-Stokes equations are theorized to organize turbulent dynamics, but their computation has historically been cost-prohibitive. In this work, we introduce a novel divergence-free basis designed to accelerate the identification and exploration of these solutions. The basis is designed to obey the symmetries and boundary conditions of the problem, improving performance and reducing the necessary modes compared to traditional spectral methods. We detail the Galerkin projection framework and discuss the completeness of the basis set for representing incompressible fluid flows. By leveraging this reduced-order model to generate high-quality initial conditions, we identified dozens of invariant solutions, several of which are previously undocumented in the literature. Finally, we explore the influence of domain aspect ratios and time-evolution dynamics, demonstrating that this methodology provides a robust path for turbulence research.

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