Extension of the Wall-Driven Enclosure Flow Problem to Toroidally Shaped Geometries of Square Cross-Section

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The two-dimensional wall-driven flow in an enclosure has been a numerical paradigm of long-standing interest and value to the fluid mechanics community. In this paradigm the enclosure is infinitely long in the x-coordinate direction and of square cross-section (d × d) in the y-z plane. Fluid motion is induced in all y-z planes by a wall (here the top wall) sliding normal to the x-coordinate direction. This classical numerical paradigm can be extended by taking a length L of the geometry in the x-coordinate direction and joining the resulting end faces at x = 0 and x = L to form a toroid of square cross-section (d × d) and radius of curvature Rc. In the curved geometry, axisymmetric fluid motion (now in the r-z planes) is induced by sliding the top flat wall of the toroid with an imposed radial velocity, ulid, generally directed from the convex wall towards the concave wall of the toroid. Numerical calculations of this flow configuration are performed for values of the Reynolds number (Re = ulidd/ν) equal to 2400, 3200, and 4000 and for values of the curvature ratio (δ = d/Rc) ranging from 5.0 · 10−6 to 1.0. For δ ≤ 0.05 the steady two-dimensional flow pattern typical of the classical (straight) enclosure is faithfully reproduced. This consists of a large primary vortex occupying most of the enclosure and three much smaller secondary eddies located in the two lower corners and the upper upstream (convex wall) corner of the enclosure. As δ increases for a fixed value of Re, a critical value, δcr, is found above which the primary center vortex spontaneously migrates to and concentrates in the upper downstream (concave wall) corner. While the sense of rotation originally present in this vortex is preserved, that of the slower moving fluid below it and now occupying the bulk of the enclosure cross-section is reversed. The relation marking the transition between these two stable steady flow patterns is predicted to be δcr1/4 = 3.58 Re-1/5 (δ ± 0.005).