Abstract
We use spectral proper orthogonal decomposition (SPOD) to extract and analyse coherent structures in the turbulent wake of a disk at Reynolds number $ {\textit {Re}} = 5 \times 10^{4}$ and Froude numbers $ {\textit {Fr}} = 2$ , 10. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behaviour and the relative contribution of low-rank modes to total fluctuation energy increases with $x/D$ . The vortex shedding (VS) mechanism, which corresponds to $ {\textit {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 $ {\textit {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 $ {\textit {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 $ {\textit {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.
Highlights
Turbulent wakes are ubiquitous both in nature and man-made devices
The Kárman vortex street associated with vortex shedding (VS) from the body at a specific frequency is a well-known feature of unstratified bluff body wakes which arises from the global instability of the m = 1 azimuthal mode, as was demonstrated for a sphere by Natarajan & Acrivos (1993) and Tomboulides & Orszag (2000)
Visual inspection shows that the spatial coherence in the wake core, which is a characteristic of dominant spectral proper orthogonal decomposition (SPOD) modes, is lost for the high-n and high-Strouhal number (St) modes similar to the result in the snapshot proper orthogonal decomposition (POD) study of Diamessis et al (2010)
Summary
Turbulent wakes are ubiquitous both in nature and man-made devices. From flow past moving vehicles (Grandemange et al 2015) to flow past topographic features (Puthan, Sarkar & Pawlak 2021) in oceans, they play an important role in transporting momentum and energy across large distances from the wake generator. With the huge amount of numerical and experimental data becoming available, data-driven modal decomposition techniques have seen an unprecedented rise in their use to understand the dynamics and role of coherent structures in turbulent flows These techniques have been used to construct reduced-order models of these flows. In the present work we use spectral proper orthogonal decomposition (SPOD), originally proposed by Lumley (1967, 1970) and recently revisited by Towne, Schmidt & Colonius (2018), to identify and analyse the coherent structures in the turbulent stratified wake of a disk at Re = 5 × 104.
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