Three-dimensional continuation study of convection in a tilted rectangular enclosure.

Abstract : A continuation method developed from a three-dimensional spectral finite element code is used to study natural convection in a tilted rectangular cavity. The cavity has its length equal to two times the side of its square cross section and it contains a fluid with a Prandtl number Pr=1. A detailed bifurcation diagram is first obtained in the case without inclination in order to get the sequence of the different branches of solutions and determine the stable solutions. The focus is then put on the stable solutions in the inclined cavity, when the tilt occurs around its longest axis. The subtle changes induced by the tilt on the convective system are clarified. Three different stable solutions are obtained: the longitudinal roll L− solution (with the same sense of rotation as the inclination angle), which develops smoothly from zero Rayleigh number on the leading branch; the longitudinal roll L+ solution (with a sense of rotation opposite to the inclination angle), which is on a disconnected branch and is stabilized beyond a secondary bifurcation point; the oblique roll O± solutions (corresponding to transverse roll solutions perturbed by the longitudinal flow induced by the tilt), which quickly appear beyond saddle-node points on new disconnected branches. The domain of existence of these stable solutions is eventually obtained and described in the Rayleigh number-inclination parameter space. Finally, the Nusselt number is determined as a function of the inclination at a constant Rayleigh number for the different stable solutions. The Nusselt number is maximum at an inclination of 49.55∘ for the leading longitudinal roll L− solution.
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Juan Felipe Torres, Daniel Henry, Atsuki Komiya, Shigenao Maruyama, Hamda Ben Hadid. Three-dimensional continuation study of convection in a tilted rectangular enclosure.. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2013, 88, pp.043015. ⟨10.1103/PhysRevE.88.043015⟩. ⟨hal-00931500⟩

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