Transitional cylindrical swirling flow in presence of a flat free surface

Abstract

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type-no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.

Unsteady transitional swirling flow in the presence of a moving free surface

“…7, in two orthogonal meridional planes. The contours of these four averaged quantities are found to be extremely close to those obtained in the flat-and-fixed freesurface case reported by Bouffanais and Lo Jacono, 8…”

“…This transition region is in relationship with the interfacial zone ⌼ where the instabilities are concentrated-see Ref. 8: for Re= 900 almost no elevation of the free surface is observed, while at Re= 1500 an additional small crest is observed followed radially by a second small trough before reaching the largest crest ring.…”

“…Conversely, the instability modes for shallow systems-corresponding to ⌳Յ1-are concentrated in the interfacial region denoted by ⌼ in Ref. 8 and where the surface radial jet of angular momentum impinges onto the inner cylindrical core in a state of quasi-solid-body rotation. As it is shown in the sequel, these unstable modes yield free-surface deflection, which becomes nonaxisymmetric at sufficiently large Reynolds number.…”

“…In terms of initial conditions, the steady rotation is started from the steady state ͑statistically steady state͒ reached in the fixed-and-flat freesurface simulations ͑a͒ and ͑b͒ ͓͑c͔͒ from Ref. 8. The steady ͑statistically steady͒ state for cases ͑a͒ and ͑b͒ ͓͑c͔͒ is reached after a transient of very short duration.…”

Two-fluid confined flow in a cylinder driven by a rotating end wall

“…There has been much interest in the flow in a cylinder driven by the rotation of the bottom when the top is a free surface [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Experimentally, the free surface is the interface between a liquid (typically water) and air.…”