Spectral proper orthogonal decomposition analysis of the turbulent wake of a disk at Re=50 000

The coherent structures in the turbulent wake of a disk at a moderately high Reynolds number (Re) of 50000 are examined using spectral proper orthogonal decomposition (SPOD) which considers all three velocity components in a numerical database. The SPOD eigenvalues at a given streamwise (x) location are functions of azimuthal wave number (m), frequency (St), and SPOD index (n). By x/D = 10, two specific modes dominate the fluctuation energy: (i) the vortex shedding (VS) mode with m = 1, St = 0.135, n = 1, and (ii) the double helix (DH) mode with m = 2, St -> 0, n = 1. The VS mode is more energetic than the DH mode in the near wake but, in the far wake, it is the DH mode which is dominant. The DH mode, when scaled with local turbulent velocity and length scales, shows self-similarity in eigenvalues and eigenmodes while the VS mode, which is a global mode, does not exhibit strict self-similarity. Modes m = 0, 3, and 4, although subdominant, also make a significant net contribution to the fluctuation energy, and their eigenspectra are evaluated. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress, < u(x)'u(r)'>, is evaluated by varying (m, St, n) combinations. Higher SPOD modes contribute significantly to the TKE, especially near the centerline. In contrast, reconstruction of < u(x)'u(r)'> requires far fewer modes: vertical bar m vertical bar <= 4, parallel to St parallel to <= 1, and n <= 3. Among azimuthal modes, m = 1 and 2 are the leading contributors to both TKE and < u(x)'u(r)'>. While m = 1 captures the slope of the shear-stress profile near the centerline, m = 2 is important to capture < u(x)'u(r)'> at and near its peak. SPOD is also performed in the vicinity of the disk to describe the modal transition to the principal contributors in the wake. The leading SPOD modes show a high-frequency shear-layer peak close to the disk and the vortex shedding mode commences its initial dominance of the wake at the end of the recirculation region.