The evaluation of the long-term behavior of the fluid is well represented by the transient average temperature 0a(x). Thus, by writing the global energy balance inside the tank, it has been shown that the temperature can be characterize in terms of the three control parameters of the flow (Ra, Pr and A) as follow:

2. Numerical results

The approach for verifying the scaling relation eq.4 is firstly done by examining the dependence of these scaling relation on individual control parameters Ra, A, and Pr respectively, which has been achieved by carrying out a series of numerical simulations with several selected values of a specific parameter while keeping the other control parameters unchanged with selected values, and then by examining the combined dependence of the scaling relations on all control parameters, by combining the three sets of individual numerical results. Specifically, numerical simulations with Ra=6.106, 6.107, 6.108, 6.109, and 6.1010 while keeping A=1 and Pr=7 unchanged to show the dependence of the scaling relations on Ra; simulations with A=1/3, 1/2, 1, 2, and 3 while keeping Ra=6.108 and Pr=7 unchanged will be carried out to show the dependence on A; and simulations with Pr=1, 7, 50, 200, and 1000 while keeping Ra= 6.108 and A=1 unchanged to show the dependence of the scaling relations on Pr.

Fig. 2 shows the numerically mean temperature въ (т) obtained for all the numerical cases to prove the dependence of the scaling relation eq.1 on the control parameters Ra, A and Pr =7. The scaling relation eq. 1 shows that the dependence of вa (т) on (ARa) goes like (ARa)-14, and the time series of в3 (т) are well defined with this scaling.

The data set from the several computations carried out has been fitted to the scaling relation eq. 4. In order to find the specific values of the proportionality constant C for each run, the fitting process has been done by mean of the GNU Regression, Econometrics and Time-series Library (gretl) [13]. This tool provides the solution to the non-linear function, minimising the sum of the squares of the deviations.

From the curve-fitting method, the constant C for each run with the minimal standard deviation (denoted as sdi), are listed in Table 1. It is noted that the variation in the C values is of the order of 8%, indicating that a single C value will provide a good representation of the behavior of the flow. This general value of C is found in the same fashion for all 13 sets of data by combining them into a single average set, as C = 1.119.
*

3. Conclusion

The transient process of cooling-down a fluid has been investigated numerically. In order to decribe the long term behaviour of the fluid inside the storage tank, the transient profile has been represented by the mean the average fluid temperature.

This analysis has led to the identification of the significant parameters that define the transient natural convection phenomena inside the storage: the Rayleigh number Ra, the aspect ratio H/D, and the Prandtl number variations Pr.

Variations on the different relevant parameters have been taken into account. The average fluid temperature at each instant of all numerical simulations, have been fitted to a scaling relation proposed in terms of the identified relevant parameters. The non-dimensional temperature (0a) is well represented by the correlation obtained. The similar correlation has been obtained in [14].

References

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[12] L. Davidson, Calculation of the turbulence buoyancy-driven flow in a rectangular cavity using an efficient solver and two different low Reynolds number к— turbulence models. Numer. Heat Transfer Part A, 18 (1990) 129-147.

[13] A. Cottrell, R. Lucchetti, (2005). Gretl, GNU Regression, Econometrics and Time-series Library, version