A reduced dynamical model of convective flows in tall laterally heated cavities

Abstract

Proper orthogonal decomposition (the Karhunen-Loève expansion) is applied to convective flows in a tall differentially heated cavity. Empirical spatial eigenfunctions are computed from a multicellular solution at supercritical conditions beyond the first Hopf bifurcation. No assumption of periodicity is made, and the computed velocity and temperature eigenfunctions are found to be centro-symmetric. A lowdimensional model for the dynamical behaviour is then constructed using Galerkin projection. The reduced model successfully predicts the first Hopf bifurcation of the multicellular flow. Results determined from the low-order model are found to be in qualitative agreement with known properties of the full system even at conditions far from criticality.

Analysis of Chaotic Natural Convection in a Tall Rectangular Cavity With Non-Isothermalwalls

“…These modes indicate that most of the variation in the system occurs near the center of the cavity. The structure of the modes is quantitatively similar to eigenmodes represented in prior work Liakopoulos et al (1997). For Ra = 2.5 × 10 7 the first five modes are used to calculate the temperature and streamfunctions of the system to visualize the error in using a reduced set of modes.…”

“…For example, if the Ra = 2.5 × 10 7 results had been used as the base for a Galerkin projection it seems clear from examination of the modes that the Ra = 18×10 7 would be poorly captured by the projection. In this case, future work based on the Galerkin projection outlined by Liakopoulos et al (1997) will be modified for this problem. A projection will be derived for multiple Rayleigh numbers, representing the two cell, chaotic, and three cell regions of the flow.…”

“…Using the derivation of Liakopoulos et al (1997), the stationary empirical eigenfunctions for the temperature (φ k ) and flow field (ϕ k ) are determined for a specific Gr and P r, (equivalent to a specific Ra) using POD. The input temperature (Θ) and flow field (Û ) are separated into time varying (Θ ,Û ) part and time averaged components prior to performing the POD, and can then be expanded in terms of the calculated…”

Particle Based Simulation of Fluid Flow in Periodically Grooved Channels

“…They presented low-order models successfully which describe the dynamical characteristics of the flow for Re close to the design conditions. They found that far from the design conditions, the reduced models predict quasi-periodic or period-doubling routes to chaos as Re is increased [13]. Greiner et al [14] investigated the three-dimensional flow in a channel with symmetric, transverse grooves on two opposite walls by using the spectral element technique.…”

A Spectral Vanishing Viscosity Method for Stabilizing Low-Dimensional Galerkin Systems

“…POD has been successfully implemented in conjunction with experimental (e.g., [3-6,? ]) as well as with numerical studies (e.g., [7,2,[8][9][10][11][12]) in thermal convection, shear layers, cavity flows and external flows, to mention just a few.…”