Lagrangian transport in a class of three-dimensional buoyancy-driven flows

Abstract

The present study concerns the Lagrangian dynamics of three-dimensional (3D) buoyancy-driven cavity flows under steady and laminar conditions due to a global temperature gradient imposed via an opposite hot and cold sidewall. This serves as the archetypal configuration for natural-convection flows in which (contrary to the well-known Rayleigh–Bénard flow) gravity is perpendicular (instead of parallel) to the global temperature gradient. Limited insight into the Lagrangian properties of this class of flows, despite its relevance to observed flow phenomena as well as scalar transport, motivates this study. The 3D Lagrangian dynamics are investigated in terms of the generic structure and associated transport properties of the global streamline pattern (‘Lagrangian flow topology’) by both theoretical and computational analyses. The Grashof number$Gr$is the principal control parameter for the flow topology: limit$Gr=0$yields a trivial state of closed streamlines;$Gr>0$induces symmetry breaking by fluid inertia and buoyancy and thus causes formation of toroidal coherent structures (‘primary tori’) embedded in chaotic streamlines governed by Hamiltonian mechanisms. Fluid inertia prevails for ‘smaller’$Gr$and gives behaviour that is dynamically entirely analogous to 3D lid-driven cavity flows. Buoyancy-induced bifurcation of the flow topology occurs for ‘larger’$Gr$and underlies the emergence of ‘secondary rolls’ observed in the literature and to date unreported secondary tori for ‘larger’ Prandtl numbers$Pr$. Key to these dynamics are stagnation points and corresponding heteroclinic manifold interactions.