by Wey H. Leong, K.G. Terry Hollands and Alfred P. Brunger
CHT'01: ADVANCES IN COMPUTATIONAL HEAT TRANSFER
The 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. Leong et al. (1998) defined three benchmark problems:
The experimental Nusselt number results for the cold face are listed in Table 1 at each combination of Ra and j, along with their corresponding 95% confidence limits of uncertainty.
Table 1: Measured Nusselt
number results for the cold face
at the three angular settings with their corresponding 95%
confidence limits of uncertainty
| Nu | |||
| Ra | j = 0° | j = 45° | j = 90° |
| 104 | 1.246 ± 0.013 | 1.614 ± 0.015 | 1.520 ± 0.015 |
| 4×104 | 2.018 ± 0.017 | 2.560 ± 0.027 | 2.337 ± 0.020 |
| 105 | 3.509 ± 0.035 | 3.492 ± 0.034 | 3.097 ± 0.028 |
| 105 | 3.916 ± 0.042 | - | - |
| 106 | 7.883 ± 0.091 | 8.837 ± 0.101 | 6.383 ± 0.070 |
| 107 | 15.38 ± 0.19 | 17.50 ± 0.21 | 12.98 ± 0.16 |
| 108 | 31.22 ± 0.43 | 34.52 ± 0.42 | 26.79 ± 0.34 |
| 3×108 | - | - | 38.12 ± 0.45 |
Important note:
Erratum: There is a typo
error in Table 1 in Leong et al. (1999) for Nu at j = 45° and Ra = 4×104.
The correct value is given above.
Note: The Nu for Ra = 3×108
and j = 90° is obtained from Mamun et al. (2003).
The results at Ra = 105 and j = 0° are found to fall into two sets, one with Nu approximately equal to 3.5 and one with Nu equal to 3.9. Leong et al. (1999) provide a detailed explanation about this phenomenon with the aids of CFD simulations. Fig. 2 shows two different flow patterns at the central plane (z = L/2) obtained from the CFD simulations of a half cavity with the central plane as a symmetry plane. At the 15×15×7 grid size, the flow pattern in Fig. 2a was obtained; at both the 30×30×14 and 60×60×28 grid sizes, the flow pattern given in Fig. 2b was obtained.
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). 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.
Leong, W.H., K.G.T. Hollands, and A.P. Brunger, 1999. Experimental Nusselt numbers for a cubical-cavity benchmark problem in natural convection. Int. J. Heat Mass Transfer, Vol. 42, pp. 1979-1989.
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 September 1, 2003)