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.

## References

1.
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
.
2.
Owen
,
J. M.
, and
Long
,
C. A.
,
2015
, “
Review of Buoyancy-Induced Flow in Rotating Cavities
,”
ASME J. Turbomach.
,
137
(
11
), p.
111001
.
3.
Childs
,
P. R. N.
,
2011
,
Rotating Flow
,
Elsevier
,
Oxford, UK
.
4.
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
.
5.
Tritton
,
D. J.
,
1988
,
Physical Fluid Dynamics
,
OUP
,
New York
.
6.
Owen
,
J. M.
, and
Pincombe
,
J. R.
,
1979
, “
Vortex Breakdown in a Rotating Cylindrical Cavity
,”
J. Fluid Mech.
,
90
(
1
), pp.
109
127
.
7.
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
.
8.
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
.
9.
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
.
10.
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
.
11.
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
.
12.
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).
13.
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
.
14.
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
.