A general methodology for investigating flow instabilities in complex geometries: application to natural convection in enclosures

Abstract

SUMMARYThis paper presents a general methodology for studying instabilities of natural convection flows enclosed in cavities of complex geometry. Different tools have been developed, consisting of time integration of the unsteady equations, steady state solving, and computation of the most unstable eigenmodes of the Jacobian and its adjoint. The methodology is validated in the classical differentially heated cavity, where the steady solution branch is followed for vary large values of the Rayleigh number and most unstable eigenmodes are computed at selected Rayleigh values. Its effectiveness for complex geometries is illustrated on a configuration consisting of a cavity with internal heated partitions. We finally propose to reduce the Navier-Stokes equations to a differential system by expanding the unsteady solution as the sum of the steady state solution and of a linear combination of the leading eigenmodes. The principle of the method is exposed and preliminary results are presented.

Numerical study of three-dimensional instabilities in a hydrodynamic model of Czochralski growth

“…Such a solver, developed for the finite volume discretization scheme, is described here. Similar stability solvers were described in several publications published during the last decade [5,[8][9][10], but none of these solvers seem to be capable of handling the complicated crystal growth-related problems involving complicated geometries and nonlinear boundary conditions. The main blocks of the solver described below can be easily implemented for other discretization schemes.…”

Preliminary experiments on the control of natural convection in differentially-heated cavities

“…1. It is placed horizontally, in the y direction, at the bottom of the hot boundary layer, more precisely at an optimum location (Z ¼ 0.1, X ¼ 0.015) reported by Gadoin et al [24]. Resulting from a stability analysis, this position corresponds to the highest level of amplification of a perturbation.…”

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

“…The first one is connected with the calculation of steady-state flows, whose stability is to be studied. The Jacobian-free [16][17][18][19] or other inexact [20][21][22] and exact [23][24][25][26] Newton methods combined with a Krylov-subspace-based iterative linear solver [27] usually are applied for this purpose. These solvers are very effective when relatively simple benchmark problems are considered, however fail to converge in more complicated cases.…”