Unsteady Two-Dimensional Flows in Complex Geometries: Comparative Bifurcation Studies with Global Eigenfunction Expansions

We present a bifurcation study of the incompressible Navier--Stokes equations in a model complex geometry: a spatially periodic array of cylinders in a channel. The dynamics of the flow include a Hopf bifurcation from steady to oscillatory flow at an approximate Reynolds number R of 350 and the appearance of a second frequency at approximately $R\simeq 890$. The multiple frequency dynamics include a substantial increase in spatial and temporal scales with Reynolds number as compared with the simple limit cycle oscillation present close to R = 350. Numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions. The first basis set is derived through principal factor analysis (Karhunen--Loève expansion) of snapshots from accurate direct spectral element numerical solutions of the Navier--Stokes equations. The second set is obtained from the eigenfunctions of the Stokes operator for this geometry. Finally eigenfunctions are derived from a singular Stokes operator, i.e., the Stokes operator modified to include a variable coefficient which vanishes at the domain boundaries. Truncated systems of ($\sim 100$) ODEs are obtained through projection of the Navier--Stokes equations onto the basis sets, and a comparative study of the resulting dynamical models is performed.