On the primary instability in the wake of polygonal cylinders

A numerical study was undertaken to investigate the primary instability of the flow past polygonal cylinders, with the side number N∈Z[5,16], and at non-dimensional incidence angles (α*) from corner to face orientation. The primary instability is invariably the result of a Hopf bifurcation, with the bifurcation point determined using the linearized Landau equation. The critical Reynolds number, Recr, shows a nonlinear dependence on N and α*. Analysis of sensitivity to base flow modifications with respect to the circular cylinder revealed that the variation in Recr is related to the direction of backflow modifications, from which a negative correlation is established between Recr and the variation of eigenvalues calculated based on the direct and the adjoint eigenmodes of the perturbed fields. Endogeneity analysis was applied to identify the contribution of the local mechanism to the global growth of the leading mode. The contribution from the production mechanism is shown to be insensitive to N and α*, whereas that from the convection mechanism shows a negative relation to Recr.