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Numerical study of swirling flows in a cylindrical container with co-/counter-rotating end disks under stable temperature difference. (English) Zbl 1189.76719
Summary: A numerical study was conducted to investigate swirling flows of a Boussinesq fluid confined in a cylindrical container with co-/counter-rotating end disks. A vertically stable temperature gradient is imposed, with the stationary sidewall assumed as adiabatic. Flows are studied for a range of parameters: the Reynolds number, \(Re, 100 \leqslant Re \leqslant 2000\); the Richardson number, \(Ri, 0 \leqslant Ri \leqslant 1.0\); and the Prandtl number, \(Pr, Pr = 1.0\). The ratio of the angular velocity of the top disk to the bottom disk, \(s, - 1.0 \leqslant s \leqslant 1.0\). The cylinder aspect ratio: \(h = 2.0\). For the case of negligibly small temperature difference (\(Ri \sim 0\)) and high \(Re\), interior fluid rotates with an intermediate angular velocity of both end disks when they are co-rotating \((s > 0)\). When end disks are counter-rotating \((s < 0)\), shearing flow with meridional recirculation is created. For the case of large temperature difference \((Pr \cdot Ri \sim O(1))\), the Ekman suction is suppressed and the sidewall boundary layer disappears at mid-height of the cylinder. For all the values of \(s\) considered in the present study, the bulk of the fluid is brought close to rest with the fluid in the vicinity of both end disks rotating in each direction. The secondary flow in the meridional section of the cylindrical container exhibits various types of vortices as the governing parameters are varied. These flow patterns are presented in the form of diagrams on the \((s, Re)\) plane and \((s, Ri)\) plane. The average Nusselt number is computed and presented as functions of \(Ri, Re\) and \(s\).

MSC:
76U05 General theory of rotating fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76M20 Finite difference methods applied to problems in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
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