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Research Papers

# Theoretical Model of Buoyancy-Induced Flow in Rotating Cavities

[+] Author and Article Information
J. Michael Owen, Hui Tang

Department of Mechanical Engineering,
University of Bath,
Bath BA2 7AY, UK

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received March 3, 2015; final manuscript received July 29, 2015; published online September 16, 2015. Assoc. Editor: Ardeshir (Ardy) Riahi.

J. Turbomach 137(11), 111005 (Sep 16, 2015) (7 pages) Paper No: TURBO-15-1037; doi: 10.1115/1.4031353 History: Received March 03, 2015; Revised July 29, 2015

## Abstract

The Ekman-layer equations, which have previously been solved for isothermal source–sink flow in a rotating cavity, are derived for buoyancy-induced flow. Although the flow in the inviscid core is three-dimensional and unsteady, it is assumed that the flow in the Ekman layers is axisymmetric and steady; and, as for source–sink flow, the average mass flow rate in the Ekman layers is assumed to be invariant with radius. In addition, it is assumed that the flow in the core is adiabatic, and consequently the core temperature increases with radius and with rotational speed. Approximate solutions are obtained for laminar flow, and it is shown that the Nusselt numbers for the rotating disks and the mass flow rate in the Ekman layers are proportional to $Grc1/4$, where $Grc$ is a Grashof number based on the rotational Reynolds number and the temperature difference between the disk and the core. The equation for the Nusselt numbers, which includes two empirical constants, depends strongly on the radial distribution of the temperature of the disks.

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## References

Owen, J. M. , Pincombe, J. R. , and Rogers, R. H. , 1985, “ Source–Sink Flow Inside a Rotating Cylindrical Cavity,” J. Fluid Mech., 155, pp. 233–265.
Owen, J. M. , and Long, C. A. , 2015, “ Review of Buoyancy-Induced Flow in Rotating Cavities,” ASME J. Turbomach., 137(11), p. 111001.
Childs, P. R. N. , 2011, Rotating Flow, Elsevier, Oxford, UK.
Owen, J. M. , and Rogers, R. H. , 1995, “ Flow and Heat Transfer in Rotating Disc Systems,” Rotating Cavities, Vol. 2, Research Studies Press, Taunton, UK/Wiley, New York.
Tritton, D. J. , 1988, Physical Fluid Dynamics, OUP, New York.
Owen, J. M. , and Pincombe, J. R. , 1979, “ Vortex Breakdown in a Rotating Cylindrical Cavity,” J. Fluid Mech., 90(1), pp. 109–127.
Farthing, P. R. , Long, C. A. , Owen, J. M. , and Pincombe, J. R. , 1992, “ Rotating Cavity With Axial Throughflow of Cooling Air: Flow Structure,” ASME J. Turbomach., 114(1), pp. 237–246.
Farthing, P. R. , Long, C. A. , Owen, J. M. , and Pincombe, J. R. , 1992, “ Rotating Cavity With Axial Throughflow of Cooling Air: Heat Transfer,” ASME J. Turbomach., 114(1), pp. 229–236.
Owen, J. M. , and Powell, J. , 2006, “ Buoyancy-Induced Flow in a Heated Rotating Cavity,” ASME J. Eng. Gas Turbines Power, 128(1), pp. 128–134.
Long, C. A. , Miche, N. D. D. , and Childs, P. R. N. , 2007, “ Flow Measurements Inside a Heated Multiple Rotating Cavity With Axial Throughflow,” Int. J. Heat Fluid Flow, 28(6), pp. 1391–1404.
Long, C. A. , and Childs, P. R. N. , 2007, “ Shroud Heat Transfer Measurements Inside a Heated Multiple Rotating Cavity With Axial Throughflow,” Int. J. Heat Fluid Flow, 28(6), pp. 1405–1417.
Tang, H. , Shardlow, T. , and Owen, J. M. , 2015, “ Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks,” ASME J. Turbomach. (in press).
Bohn, D. , Deuker, E. , Emunds, R. , and Gorzelitz, V. , 1995, “ Experimental and Theoretical Investigations of Heat Transfer in Closed Gas Filled Rotating Annuli,” ASME J. Turbomach., 117(1), pp. 175–183.
Owen, J. M. , and Pincombe, J. R. , 1980, “ Velocity Measurements Inside a Rotating Cylindrical Cavity With a Radial Outflow of Fluid,” J. Fluid Mech., 99(1), pp. 111–127.

## Figures

Fig. 1

Schematic of flow structure in heated rotating cavity with axial throughflow of cooling air [7]

Fig. 2

Simplified diagram of axial throughflow in an isothermal rotating cavity

Fig. 3

Assumed flow structure

Fig. 4

Assumed shape of velocity distributions in Ekman layer

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