Scale organisation in energy- and dissipation-based modal decomposition of turbulent flows

This thesis concerns various aspects of the proper orthogonal decomposition (POD) applied to turbulent flows and of the length scales characterising flow structures educed using the method. The empirical basis produced when applying POD to a flow data set can be used as a tool for analysing flow structures and for constructing reduced-order models (ROMs) via Galerkin projection. Expanding the flow in a truncated POD basis minimises the mean error as measured by the underlying norm. This optimality does not guarantee that a ROM constructed using such a basis is able to accurately capture the dynamics of the flow, however. One reason for this is that energy optimality emphasises large-scale structures in the flow, at the expense of dissipative small-scale structures, leading to under-modelling of viscous dissipation. The dissipation rate-optimised POD (d-POD) is proposed as an approach to address this problem. The formalism leads to velocity modes optimised with respect to their mean contribution to the rate of viscous dissipation.

The approach is applied to a data set representing a cross-section of a direct numerical simulation (DNS) channel flow, and results are compared to those obtained using energy-optimised POD (e-POD). The lowest several d-POD modes exhibit richer structure in the near-wall region, where both turbulence kinetic energy (TKE) and the viscous dissipation rate are reconstructed more efficiently than using the e-POD. While global TKE converges significantly faster when reconstructed using the e-POD compared to the d-POD, the d-POD shows only a modest lead over e-POD in reconstructing global dissipation. Reconstructions of TKE and dissipation using the d-POD show very similar convergence rates, suggesting that this decomposition captures structures across a range of scales efficiently. This is substantiated by the d-POD per-mode contribution to TKE reconstruction normalised by the modal amplitude which is largely independent of mode number. In contrast, the correspondingly normalised contribution to reconstructed viscous dissipation using e-POD modes tends to increase with mode number, reflecting the tendency of small-scale structures to be predominantly captured by higher e-POD modes.

The d-POD is extended to include thermal dissipation, and is applied to a DNS cubic Rayleigh–Bénard convection flow data set. This data set is enriched using the symmetries of the flow, and via commutation relations between the POD and symmetry operators this is formally linked to POD degeneracies resolving different flow orientations. The results of the d-POD are compared to those obtained using the corresponding e-POD extended to include thermal energy. Each of the lowest several d-POD modes has a structural analogue among the corresponding set of e-POD modes, although the ordering of modes differs somewhat between the decompositions. In particular, a corner mode associated with flow reorientations is promoted to a lower index in the d-POD. The d-POD modes show enhanced contributions to quantities reconstructed in the thermal boundary layers, as well as enhanced asymptotic reconstruction convergence rates.
Reconstruction of kinetic and thermal energy, viscous and thermal dissipation, and convective heat flux using d-POD show consistent asymptotic convergence rates across the reconstructed quantities, further indicating the ability of d-POD to capture a broad range of scales efficiently.

The relationship between POD and Fourier modes is also investigated in the thesis. POD modes reduce to Fourier modes along homogeneous and periodic directions, although the requirement of periodicity is sometimes overlooked in literature. The formal consequences of considering Fourier modes as POD modes for flows represented by locally homogeneous analytic kernels on aperiodic domains are investigated. POD and Fourier spectra are shown to differ, especially at higher indices and for smaller macro/micro scale ratios (MMSRs). However, POD spectra are found to match the asymptotic behaviour of analytic Fourier spectra. POD eigenvalues are analysed in terms of the number of Fourier modes required to reconstruct each eigenvalue. Rather than consistently converging to a single Fourier mode with increasing domain size, the reconstructions investigated are found in some instances to approach e.g. triplets of Fourier modes. Finally, the assumption of periodicity implicit in the use of Fourier modes consistently leads to underestimation of the Taylor microscale, due to deformation of the correlation function. Again this effect is found to be most substantial for small MMSRs. In contrast, Taylor microscales are accurately reconstructed using only a modest number of POD modes. These results indicate that spectral POD (SPOD)-based models of homogeneous flows on aperiodic domains may suffer from inaccuracies, especially at low Reynolds numbers, which could be resolved using POD modes instead.