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Unsteady two-dimensional flows in complex geometries: Comparative bifurcation studies with global eigenfunction expansions. (English) Zbl 0873.58058

Summary: 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.

MSC:

35Q30 Navier-Stokes equations
35B32 Bifurcations in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

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