Numerical study of swirling flows in a cylindrical container with co-/counter-rotating end disks under stable temperature difference

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 ⩽ Re ⩽ 2000; the Richardson number, Ri, 0 ⩽ Ri ⩽ 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 ⩽ s ⩽ 1.0. The cylinder aspect ratio: h = 2.0. For the case of negligibly small temperature difference (Ri ∼ 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 · Ri ∼ 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.

[1]  Juan Lopez,et al.  A Continuation and Bifurcation Technique for Navier—Stokes Flows , 2002 .

[2]  C. C. Jahnke,et al.  Flows induced in a cylinder with both end walls rotating , 1994 .

[3]  J. Hyun,et al.  Convective heat transfer in a cylinder with a rotating lid under stable stratification , 1997 .

[4]  Recirculation Zones in a Cylindrical Container , 1998 .

[5]  P. Zandbergen,et al.  Von Karman Swirling Flows , 1987 .

[6]  R. Iwatsu.  Numerical Study of Flows in a Cylindrical Container with Rotating Bottom and Top Flat Free Surface , 2005 .

[7]  J. Sørensen,et al.  Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface , 2001, Journal of Fluid Mechanics.

[8]  Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom , 1996 .

[9]  A. Delgado.  On the rotationally symmetric laminar flow of Newtonian fluids induced by rotating disks , 2000 .

[10]  G. Batchelor.  NOTE ON A CLASS OF SOLUTIONS OF THE NAVIER-STOKES EQUATIONS REPRESENTING STEADY ROTATIONALLY-SYMMETRIC FLOW , 1951 .

[11]  J. Hyun,et al.  Flow of a stratified fluid in a cylinder with a rotating lid , 1999 .

[12]  J. P. Hartnett,et al.  Advances in Heat Transfer , 2003 .

[13]  R. Iwatsu.  Flow pattern and heat transfer of swirling flows in cylindrical container with rotating top and stable temperature gradient , 2004 .

[14]  J. Pedlosky,et al.  On the steady motions produced by a stable stratification in a rapidly rotating fluid , 1967, Journal of Fluid Mechanics.

[15]  J. Pedlosky,et al.  Linear theory of rotating stratified fluid motions , 1967, Journal of Fluid Mechanics.

[16]  D. Dijkstra,et al.  The flow between two finite rotating disks enclosed by a cylinder , 1983, Journal of Fluid Mechanics.

[17]  Y. Jaluria.  Fluid Flow Phenomena in Materials Processing—The 2000 Freeman Scholar Lecture , 2001 .

[18]  J. Lopez.  Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall , 1998, Journal of Fluid Mechanics.

[19]  K. Stewartson.  On the flow between two rotating coaxial disks , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  P. Linden.  The flow of a stratified fluid in a rotating annulus , 1977, Journal of Fluid Mechanics.

[21]  Vishwanath Prasad,et al.  Transport Phenomena in Czochralski Crystal Growth Processes , 1997 .

[22]  T. Kármán.  Über laminare und turbulente Reibung , 1921 .