We consider the numerical simulation of Rayleigh–Bénard convection in a 2D square cell filled with water ($\mathit{Pr}=4.3$) at a turbulent Rayleigh number of $\mathit{Ra}=5\times 10^{7}$. We focus on the structures and dynamics of the large-scale intermittent flow. Two quasi-stable flow patterns are identified: one consists of a main diagonal roll with two corner rolls; and the other of two horizontally stacked rolls. These stable flow structures are associated with two types of events, which involve corner flow growth and pattern rotation: reversals, when the main roll rapidly switches signs; and cessations, when it disappears for longer periods. Proper orthogonal decomposition (POD) is applied independently to the velocity field and to the temperature field. In both cases, three principal modes were identified: a single-roll, large-scale circulation; a quadrupolar flow; and a double-roll, symmetry-breaking mode. The large-scale circulation is the kinetic mode with the highest energy. The most energetic temperature mode is associated with the mean temperature and corresponds to a velocity field of quadrupolar nature. The vertical heat flux is concentrated in these two modes. The reversal process is characterized by sharp fluctuations in the amplitudes of all modes. Analysis of the interaction coefficients between the spatial modes leads us to propose a three-dimensional model, based on the interaction of the large-scale circulation, the quadrupolar flow and horizontal rolls. The main dynamics and time scales of reversals and cessations are reproduced by the model in the presence of noise.

We investigate large-scale circulation reversals in a two-dimensional Rayleigh-Bénard cell using a proper orthogonal decomposition (POD)-based, five-mode model. The Rayleigh number considered is Ra=5×10^{7} and the Prandtl number is Pr=4.3. A precursor event, corresponding to the action of a mode L_{*} which disconnects the core region from the boundary layers before the onset of the reversal, is identified in the simulation. The five-mode model predicts correctly the behavior of the POD modes observed in the simulation, and in particular that of mode L_{*}. The presence of mode L_{*}, which was not included in an earlier, lower-dimensional version of the model [Podvin and Sergent, J. Fluid Mech. 766, 172 (2015)JFLSA70022-112010.1017/jfm.2015.15], is found to be instrumental for the reversal dynamics of the model, which suggests that it may also be important for those of the simulation. Reversals can therefore be characterized by three time scales: the transition duration, the interreversal time, and the precursor duration, which separates the precursor event from the onset of the reversal. The distribution of the time scales is found to agree well with the simulation when small-scale intermittency is taken into account through the introduction of noise in the model coefficients.

In this paper, we provide a numerical validation of the ten-dimensional Proper Orthogonal Decomposition-based model constructed by Aubry et al. [J. Fluid Mech. 192, 115 (1988)] for the wall region of the turbulent boundary layer. Under certain conditions, this model was shown to display intermittent features highly reminiscent of the experimental observations of the bursting process in the wall layer, which makes it a potential key player in understanding and possibly controlling the dynamics of wall-bounded flows. In the same spirit as in our previous study [Podvin and Lumley, J. Fluid Mech. 362, 121 (1998)], we carried out a numerical simulation of a channel flow with relatively small horizontal dimensions which matched those in the 10-D model. The closure hypotheses used to build up the model were confronted with numerical results. Time histories of the modes in the model were compared to those of the simulation. Emphasis was put on identifying long-term characteristics such as a “mean” intermittency period. Our model, quite similar to Aubry’s, was found to display the same heteroclinic cycles under conditions consistent with the numerical experiment. The intermittency period in the model was found to agree well with that found in the simulation. However, the well-ordered character of 10-D bursts is significantly different from the simulation. To try and understand this discrepancy, we simulated a model with streamwise modes (32-D) and found evidence of increasing complexity in the bursts displayed.

The proper orthogonal decomposition (P.O.D.) is applied to the flow in a differentially heated cavity. The fluid considered is air, and the aspect ratio of the cavity is 4. At a fixed Rayleigh number, P.O.D. empirical functions are extracted, and low-dimensional models are built and compared to the numerical simulation. Generally speaking, low-D models provide a coarse picture of the flow, which is also quick, cheap, and easy to understand. They can help pinpoint leading instability mechanisms. They are potentially key players in a number of applications such as optimization and control. Our goal in this study is to determine how well the flow can be represented by very low-dimensional models. Two moderately complex situations are examined. In the first case, at some distance from the bifurcation point, the dynamics can still be reduced down to two modes, although it is necessary to account for the effect of higher-order modes in the model. In the second case, farther away from the bifurcation, the flow is chaotic. A ten-dimensional model successfully captures the essential dynamics of the flow. The procedure was seen to be robust. It clearly illustrates the power of the P.O.D. as a reduction tool.

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The proper orthogonal decomposition (POD) is applied to the minimal flow unit (MFU) of a turbulent channel flow. Our purpose is to establish a numerical validation of low-dimensional models based on the POD. The simplest (two-mode) model possible is built for the simplified flow in the minimal unit. The dynamical behaviour predicted by the model is compared with that actually occurring in the direct numerical simulation of the flow. The various modelling assumptions which underlie the construction of low-dimensional models are examined and confronted with numerical evidence. The relationship between intermittency in the MFU and intermittent low-dimensional parameters is investigated closely. The agreement observed is quite satisfactory, especially given the crudeness of the truncation considered. To further demonstrate the adequacy of the model, we develop a dynamical filtering procedure to recover information from realistic (partial) measurements. The success obtained illustrates the versatility of the low-dimensional paradigm.

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.

In this paper we propose a method to reconstruct the flow at a given time over a region of space using partial instantaneous measurements and full-space proper orthogonal decomposition (POD) statistical information. The procedure is tested for the flow past an open cavity. 3D and 2D POD analysis are used to characterize the physics of the flow. We show that the full 3D flow can be estimated from a 2D section at an instant in time provided that some 3D statistical information—i.e., the largest POD modes of the flow— is made available.