Instability of natural convection in a laterally heated cube with perfectly conducting horizontal boundaries
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O R I G I NA L A RT I C L E
Alexander Yu. Gelfgat
Instability of natural convection in a laterally heated cube with perfectly conducting horizontal boundaries
Received: 24 August 2019 / Accepted: 23 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Oscillatory instability of buoyancy convection in a laterally heated cube with perfectly thermally conducting horizontal boundaries is studied. The effect of the spanwise boundaries on the oscillatory instability onset is examined. The problem is treated by Krylov-subspace-iteration-based Newton and Arnoldi methods. The Krylov basis vectors are calculated by a novel approach that involves the SIMPLE iteration and a projection onto a space of functions satisfying all linearized and homogeneous boundary conditions. The finite volume grid is gradually refined from 1003 to 2563 finite volumes. A self-sustaining oscillatory process responsible for the instability onset is revealed, visualized and explained. Keywords Natural convection · Instability · Krylov methods · SIMPLE iteration 1 Introduction Buoyancy convection of air in a laterally heated square cavity is a widely recognized benchmark problem used for validation of numerical methods. It was proposed in [1] and since then is being mainly used for comparison of calculated steady flows. Later, it was extended to compare calculated critical parameters of the primary steady–oscillatory transition. The reader is referred to book [2] for the details and the references. With the growth of computer power, the two-dimensional formulation was replaced with the three-dimensional one, and the benchmark quality results for 3D steady flows are known and cross-verified [3–8]. Study of the primary instability of steady flows requires computation of the critical parameters via the comprehensive linear stability analysis, which involves direct computation of steady flows and eigenvalues of the linearized problem. This task has become affordable for 2D base flows (see [8–12] and references therein), but remains a challenge for fully three-dimensional problems. This study presents the first attempt to study stability of buoyancy convection in a 3D laterally heated cube by direct Krylov-subspace-based Newton and Arnoldi solvers with the goal to obtain grid convergent stability results. We begin with a cubical box with perfectly thermally conducting horizontal boundaries, for which the critical Grashof number is relatively small [13–16]. The spanwise vertical boundaries are considered to be either perfectly thermally conducting or perfectly insulated, so that there are two similar, but different problems to study. A more computationally demanding case with the thermally insulated (adiabatic) horizontal boundaries is left for a separate study and is not reviewed here. Communicated by Vassilios Theofilis. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00162-020-00541-z) contains supplementary material, which is available to authorized users. A. Y. Gelfgat (B) Sc
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