Dynamical stability and flow regimes in a stably stratified, valley-shaped cavity heated from below

We investigate the three-dimensional stability of a stably stratified fluid in a valley-shaped cavity heated along its sloping walls using linear stability analysis and direct numerical simulations. First, we describe the quiescent pure-conduction base state, and derive a necessary condition for instability, valid for any slope angle alpha α $\alpha$ . Next, we examine the sequence of flow regimes for alpha equals 30 Superscript ring α = 30 ∘ $\alpha = 30^{\circ }$ and Prandtl number italic probability equals 7 Pr = 7 ${\textit{Pr}} = 7$ , including two-dimensional steady states, the onset of a Hopf bifurcation, and the development of both steady and oscillatory three-dimensional structures preceding the transition to fully unsteady chaotic flow. Owing to the specific boundary conditions on valley walls and top, flow dynamics across the regimes examined collapse onto a single dimensionless parameter – the composite stratification parameter upper Pi Subscript c Π c $\varPi _c$ . Asymmetric circulation remains the dominant flow pattern; even in the fully unsteady chaotic regime, it persists in the time-averaged field. Finally, heat transfer, as quantified by the Nusselt number italic Nu Nu ${\textit{Nu}}$ , scales as italic Nu tilde upper Pi Subscript c Superscript 0.43 Nu ∼ Π c 0.43 ${\textit{Nu}} \sim \varPi _{c}^{0.43}$ , or equivalently italic Nu tilde italic Ra Superscript 0.275 Nu ∼