Instabilities in flow through and around a circular array of cylinders

This paper presents results of two-dimensional direct numerical simulations (DNS) and global linear stability analyses (based on mean flow and base flow) of a viscous incompressible flow past a circular array of cylinders with six-fold rotational symmetry. Six cylinder arrays, with varied patch density $\phi = N_c (d/D)^2$ (with $N_c$ cylinders of diameter $d$ within a patch of diameter $D$) is investigated by adjusting the $N_c$ and its arrangement. The simulations cover a wide parameter space, with $\phi$ ranging from $0.043$ to $0.315$, and the free-stream flow Reynolds numbers ($Re_D = U_{\infty} D / \nu = < 300$ based on $D$ and the uniform incoming velocity $U_{\infty}$ and the kinematic viscosity $\nu$). We focus on the onset of vortex shedding with variant $\phi$, since the onset of global instabilities in such arrays has not been discussed earlier in the literature. For the patch diameters and solid volume fractions considered here,three distinct regimes are identified: (I) a low-$\phi$ regime where cylinders behave nearly independently, the flow is stable, forming steady wake without vortex street; (II) an intermediate-$\phi$ regime where $Re_c$ varies logarithmically with $\phi$, resembling a porous medium; and (III) a high-$\phi$ regime where $Re_c$ approaches that of a solid cylinder ($\phi = 1$).