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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.

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