AbstractNatural convection in an inclined cubical cavity heated from two opposite walls maintained at different temperatures and with adiabatic sidewalls is investigated numerically. The cavity is inclined by an angle $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $ around a lower horizontal edge and the isothermal wall set at the higher temperature is the lower wall in the horizontal situation ($\theta = 0^\circ $). A continuation method developed from a three-dimensional spectral finite-element code is applied to determine the bifurcation diagrams for steady flow solutions. The numerical technique is used to study the influence of ${\theta }$ on the stability of the flow for moderate Rayleigh numbers in the range $\mathit{Ra} \leq 150\, 000$, focusing on the Prandtl number $\mathit{Pr} = 5.9$. The results show that the inclination breaks the degeneracy of the stable solutions obtained at the first bifurcation point in the horizontal cubic cavity: (i) the transverse stable rolls, whose rotation vector is in the same direction as the inclination vector ${\boldsymbol{\Theta}}$, start from $\mathit{Ra} \to 0$ forming a leading branch and becoming more predominant with increasing $\theta $; (ii) a disconnected branch consisting of transverse rolls, whose rotation vector is opposite to ${\boldsymbol{\Theta}}$, develops from a saddle-node bifurcation, is stabilized at a pitchfork bifurcation, but globally disappears at a critical inclination angle; (iii) the semi-transverse stable rolls, whose rotation axis is perpendicular to ${\boldsymbol{\Theta}}$ at $\theta \to 0^\circ $, develop from another saddle-node bifurcation, but the branch also disappears at a critical angle. We also found the stability thresholds for the stable diagonal-roll and four-roll solutions, which increase with $\theta $ until they disappear at other critical angles. Finally, the families of stable solutions are presented in the $\mathit{Ra}-\theta $ parameter space by determining the locus of the primary, secondary, saddle-node, and Hopf bifurcation points as a function of $\mathit{Ra}$ and $\theta $.
Voltage collapse in power systems: Dynamical studies from a static formulation
