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Flow transitions of a low-Prandtl-number fluid in an inclined 3D cavity. (English) Zbl 1034.76019
This paper presents a numerical solution of natural convection of low-Prandtl-number fluid (Pr \( = 0.025\)) in two-dimensional (\(H \times L = 1 \times 4\)) and three-dimensional (\(1 \times 6 \times 4\)) side-heated cavities. Both the cavities are inclined at \(80^\circ\) with respect to the vertical direction and are heated along the side of length \(L = 4H\). This set of parameters is chosen to promote the interaction between longitudinal and transversal multicellular modes. Also, this inclination angle is chosen from a previous linear stability analysis of the basic (plane-parallel) flow that predicts the same critical Rayleigh number Ra for longitudinal oscillatory and stationary transversal modes. The unsteady governing continuity, Navier-Stokes and energy equations in non-dimensional form is solved numerically using a Chebyshev-collocation pseudospectral method in vorticity-stream function variables. In all the cases studied a grid of \(31 \times 81\) collocation points is found to be enough to obtain accuracies of better than about 2%. It is found that in both two-dimensional and three-dimensional enclosures the first transition gradually leads to a transversal stationary centered shear roll. In the two-dimensional geometry the flow becomes time-dependent and multicellular (3 rolls) at the onset of Hopf bifurcation, followed by subsequent period-doubling. On the other hand, in the case of three-dimensional enclosure, the onset of oscillation is due to a fully three-dimensional standing wave composed of three counter-rotating longitudinal rolls. Further, the authors show that the inclination configuration enables a new dimension in the parameter space, and therefore makes more feasible the study of several types of instabilities and their corresponding interactions for suitable choices of operating parameters.

76E06 Convection in hydrodynamic stability
76R10 Free convection
76M22 Spectral methods applied to problems in fluid mechanics
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