Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow

Abstract

Energy transfers between modes obtained from the proper orthogonal decomposition (POD) of a turbulent flow past a backward-facing step are analysed with the aim of providing guidelines for modelling unresolved modes in truncated POD–Galerkin models. It is observed that energy transfers are local in the POD basis, and that the Fourier-decomposition-based concepts of forward and backward energy cascades are also valid in the POD basis, the net effect being a forward energy cascade. General features of the eddy-viscosity representation of kinetic energy transfers are investigated through a priori tests. It is observed that the ideal eddy-viscosity model should exhibit a cusp behaviour near the cutoff mode.

Interscale energy transfer in the merger of wakes of a multiscale array of rectangular cylinders

“…Since we consider a number of simultaneous coherent fluctuations, a new, suitable form of these equations is derived and further applied to the analysis of the gathered data. This approach is similar to an intermodal energy flow study presented by Couplet et al [14] (an energy exchange between a number of POD modes is considered therein) or, more recently, by Hosseini et al [15] (an energy exchange between a slowly-varying base flow; the first and the second harmonic of a wall-mounted pyramid's shedding are studied).…”

Approximate deconvolution reduced order modeling

“…In the ROM literature, a common explanation for this failure of standard ROMs is the violation of the concept of energy cascade [3,8,9,14,18,19]. Indeed, in [20] it was shown that the concept of energy cascade is also valid in a POD setting: The main role of the neglected POD modes is to drain energy from the ROM. Thus, since standard ROMs fail to account for this transfer of energy from resolved to unresolved POD modes, they generally yield inaccurate results.…”

“…Many of these models are similar to the large eddy simulation (LES) closure models. Indeed, to model the effect of the neglected POD modes, some authors start with the filtered equations and propose closure models for ROMs, just as in LES (see, e.g., [3,8,20]). Other authors, however, do not employ filtering and the LES analogy to develop models for the neglected POD modes (see, e.g., [9,[17][18][19]).…”

“…Other authors, however, do not employ filtering and the LES analogy to develop models for the neglected POD modes (see, e.g., [9,[17][18][19]). For the sake of brevity, in what follows we will use ROM closure model as a generic term for models of neglected POD modes [3,9,[17][18][19][20]23,24].…”

“…This model was followed by more accurate ROM closure models, such as the Smagorinsky model [33][34][35], the variational multiscale model [14,22] and the dynamic subgrid-scale model [14]. As already mentioned, a physical motivation for the approach used to develop these functional ROM closure models is the concept of energy cascade [20]. We emphasize that, just as in the LES case, these functional ROM closure models do not employ an explicit spatial filter; instead, they use phenomenological arguments to model the effect of the discarded POD modes.…”