Analysis of coherence in turbulent stratified wakes using spectral proper orthogonal decomposition

We use spectral proper orthogonal decomposition (SPOD) to extract and analyze coherent structures in the turbulent wake of a disk at Reynolds number $Re = 5 \times 10^{4}$ and Froude numbers $Fr$ = $2, 10$. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behavior and the relative contribution of low-rank modes to total fluctuation energy increases with $x/D$. The vortex shedding (VS) mechanism, which corresponds to $St \approx 0.11-0.13$ in both wakes, is active and dominant throughout the domain in both wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS mode, which is a prominent feature of the unstratified wake, is inhibited by buoyancy, particularly for $Fr = 2$. The energy at and near the VS frequency is found to appear in the outer region of the wake when the downstream distance exceeds $Nt = Nx/U = 6 - 8$. Visualizations show that unsteady internal gravity waves (IGWs) emerge at the same $Nt = 6 - 8$. A causal link between the VS mechanism and the unsteady IGW generation is also established using the SPOD-based reconstruction and analysis of the pressure-transport term. These IGWs are also picked up in SPOD analysis as a structural change in the shape of the leading SPOD eigenmode. The $Fr = 2$ wake shows layering in the wake core at {$Nt > 15$} which is captured by the leading SPOD eigenmodes of the VS frequency at downstream locations $x/D > 30$. The VS mode of the $Fr = 2$ wake is streamwise-coherent, consisting of V-shaped structures at $x/D \gtrsim 30$. Overall, we find that the coherence of wakes, initiated by the VS mode at the body, is prolonged by buoyancy to far downstream. Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes are shown to efficiently reconstruct important second-order statistics.