Proper orthogonal decomposition analysis and modelling of large-scale flow reorientations in a cubic Rayleigh–Bénard cell

Abstract

This paper investigates the large-scale flow reorientations of Rayleigh–Bénard convection in a cubic cell using proper orthogonal decomposition (POD) analysis and modelling. A direct numerical simulation is performed for air at a Rayleigh number of $10^{7}$ and shows that the flow is characterized by four quasi-stable states, corresponding to a large-scale circulation lying in one of the two diagonal planes of the cube with a clockwise or anticlockwise motion, with occasional brief reorientations. Proper orthogonal decomposition is applied to the joint velocity and temperature fields of an enriched database which captures the statistical symmetries of the flow. We found that each quasi-stable state consists of a superposition of four spatial modes representing three types of structures: (i) a mean-flow mode consisting of two stacked counter-rotating torus-like structures; (ii) two large-scale two-dimensional rolls (pair of degenerated modes) which form large-scale diagonal rolls when combined together; and (iii) an eight-roll mode that transports fluid from one corner to the other and strengthens the circulation along the diagonal. In addition, we identified three other modes that play a role in the reorientation process: two boundary-layer modes (pair of degenerated modes) that connect the core region with the horizontal boundary layers and one mode associated with corner rolls. The symmetries of the different POD modes are discussed, as well as their temporal dynamics. A description of the reorientation process in terms of POD modes is provided and compared with other modal approaches available in the literature. Finally, Galerkin projection is used to derive a POD-based reduced-order model. Unresolved modes are accounted for in the model by an extra dissipation term and the addition of noise. A seven-mode model is able to reproduce the low-frequency dynamics of the large-scale reorientations as well as the high-frequency dynamics associated with the large-scale circulation rotation. Linear stability analysis and sensitivity analysis confirm the role of the boundary-layer modes and the corner-rolls mode in the reorientation process.

Reduced-order modelling of radiative transfer effects on Rayleigh–Bénard convection in a cubic cell

“…Uncoupled results have been integrated over the same time period in the asymptotic regime and have been discussed in a previous study (Soucasse et al 2019).…”

“…Because velocity fluctuations are proportionately larger than temperature fluctuations in coupled calculations, this factor is greater when coupling with radiation: we obtain γ coupled = 1.421 and γ uncoupled = 1.303. In the following, the POD analysis of coupled DNS will be compared to the POD analysis of uncoupled DNS discussed by Soucasse et al (2019). The POD eigenvalue spectrum is shown in figure 5.…”

“…where L A nm is an adjustable parameter, k is the temporal average of the energy of the fluctuating modes in the truncation and (t) is a random noise perturbation of amplitude σ. If the coefficients L A nm are not known before-hand, it is possible to determine them from the observed dynamics, as was done in Soucasse et al (2019). We will get back to this point in section 4.2.1.…”

Reversals of coherent structures in turbulent mixed convection

“…Especially, the occurrence of a secondary roll branch raises further questions with regard to possible links to corner roll-driven reversals in RBC; see Sugiyama et al (2010) and Soucasse et al (2019). For a stable mixed convection case, Kühn et al (2012) also found Y-parallel vortical structures, which occurred in pairs and were otherwise similar to the structure of the secondary branches.…”

“…As for the influence on the flow, the coefficient's changes of sign at t ≈ 15.2 × 10 3 s and t ≈ 17.3 × 10 3 s imply that the main thermally induced structure reverses twice during a reconfiguration event. Interpreting this in the context of super-positioned angular momenta, a reversing second mode reflects a change in the diagonal alignment of the main convection roll by rotation (see Soucasse et al 2019;Vasiliev et al 2019).…”