by Mohammad Arif Hasan Mamun, Wey H. Leong, K.G. Terry Hollands and David A. Johnson
Important note:
Erratum: There is an error in defining the diamond orientation in Mamun et al. (2003). Please click at the following URL to read the details:
http://www.ryerson.ca/~weyleong/benchmark/benchmark2-erratum.pdf
This is an extension to a previous benchmark problem (also called the cubical-cavity problem) in internal natural convection, as the one shown in Fig. 1, is an air-filled cube with two opposing faces isothermal and the remaining four walls (called the sidewalls) having a linear temperature variation from the cold face to the hot face. The new orientation has the cubical cavity standing on one corner with the diagonal between that lowest corner and the opposite corner slanted about 9.74 degrees from vertical: i.e., j = 35.264° and y = 30° in Fig. 1. (Note that in this orientation, the hot face, which constitutes one of the three lower faces of the cavity, is at 45° from the horizontal plane.) This “doubly-inclined” orientation is, for brevity, referred to as the “diamond” orientation.
The experimental Nusselt number results for the cold face are listed in Table 1 at each Ra for the diamond orientation, along with their corresponding 95% confidence limits of uncertainty.
Table 1: Measured Nusselt
number results for the cold face
at the diamond orientation with their corresponding 95%
confidence limits of uncertainty
| Ra | Nu |
| 104 | 1.676 ± 0.026 |
| 4×104 | 2.763 and 2.818 ± 0.033 |
| 105 | 3.856 ± 0.043 |
| 106 | 8.012 ± 0.092 |
| 107 | 15.77 ± 0.19 |
| 108 | 32.84 ± 0.39 |
| 2×108 | 40.46 ± 0.48 |
| 3×108 | 46.69 ± 0.55 |
The results at Ra = 4×104 are found to fall into two sets, one with Nu approximately equal to 2.76 and one with Nu equal to 2.82.
Workers wishing to compare these results to the predictions of their CFD code are advised to follow certain recommendations relating to fluid property values, which are based upon the findings of Leong et al. (1998) and Mamun et al. (2003). They found that treating the fluid properties as variant with temperature gave results much closer to the measured results than the ones with constant fluid properties evaluated at the mean temperature (Tm) of hot and cold faces.
We recommend that CFD simulations be carried out with Tc = 300 K, Th = 307 K and L = 0.1272 m, and the pressure P equal to that which will give the desired Rayleigh number. However, to test their code against the high Ra of greater than 108, it is recommended to use Tc = 298 K and Th = 322 K. The k, m, b and cp in the defining equations for Ra and Nu should be evaluated at Tm and P, and the ideal gas law should be used to evaluate r at the Tm and the P in question. Essentially any of the literature equations that model the air property-variation may be used in the CFD simulations. That is, provided that for the Rayleigh number evaluation, one uses these same literature equations to evaluate the properties at Tm and P, the result should be independent of which equations were actually chosen.
Leong, W.H., K.G.T. Hollands, and A.P. Brunger, 1998. On a physically-realizable benchmark problem in internal natural convection. Int. J. Heat Mass Transfer, Vol. 41, pp. 3817-3828.
Mamun, M.A.H., W.H. Leong, K.G.T. Hollands, and D.A. Johnson, 2003. Cubical-cavity natural-convection benchmark experiments: an extension. Int. J. Heat Mass Transfer, Vol. 46, pp. 3655-3660.
(This web page was last updated on May 25, 2004)