Numerical investigation of shear-driven flow in a toroid of square cross-section

Abstract

A numerical investigation has been performed for the 3-D flow of an incompressible fluid in a torus shaped enclosure of square cross-section, where the fluid motion is induced by sliding the top wall of the enclosure radially outwards. The flow in this geometry is characterized by two non-dimensional numbers, the curvature ratio (δ=d/Rc) and the Reynolds number (Re=uwalld/v) where Rc is the radius of curvature of the torus at the center of the cavity, d is the side length of the enclosure cross-section and uwall the velocity of the top wall of the enclosure. Calculations were performed for 3-D flow in an almost straight enclosure with δ = 0.005 at Re = 3200 and a strongly curved one with δ = 0.25 at Re = 2400. The 3-D flow was computed by choosing a small sector of the torus and applying periodic boundary conditions along the circumferential boundary. The 3-D flow calculations were started with axi-symmetric flow as initial condition and perturbed by a small random disturbance to seed the centrifugal instability into the flow. Integral quantites defined using different components of the vorticity were monitored at different cross sectional planes to study the development and dynamics of the 3-D flow. A technique of volume visualization was used to visualize r vorticity and θ vorticity contours through out the computational domain to understand the dynamics of the 3-D flow. The 3-D flow calculated for both cases δ = 0.005 and 0.25 shows span-wise vortices also called Taylor-Gortler-Like vortices. These vortices while being convected around by the primary re-circulating flow in the torus cross-section experience span-wise oscillation resulting from a secondary instability accompanied by their growth and collapse in size. The net effect of this dynamics results in the periodic rearrangement of the vortices, when viewed along the circumferential span. Volume visualization of r-vorticity contours show the existence of two pairs of vortices wrapped around each other as they are convected around by the primary re-circulating flow. The dynamics that induce the periodic rearrangement have been explained from volume visualization of the vorticity components. "Vortex tilting" of theta-component of vorticity is identified as a mechanism for explaining the interaction of the primary re-circulating flow in the span-wise vortices present.