Simulation of natural convection in the cooling passage of a low heat rejection engine

In this study the geometry of the LHRE's cooling passage was simplified and the symmetrical flow was assumed. A 24x16x20 grid (Fig. 1) was generated in quarter of the flow domain by using a grid generation method.

      The governing equation were non-dimensionalized by using the following definitions of non-dimensional variables:


The boundary conditions was given as follows. On the walls, all components of the velocity was zaro and the temperature was given by the measurements. On the symmetry plans,


According to the known value of the relative parameters, the Rayleigh number was estimated with the value of about 10⁷. In the calculation, four cases with the Rayleigh number of 10⁶, 10⁷, 2x10⁷ and 3x10⁷ was selected. The Prandtl number was set to a constant value of 0.71.

Fig.2 shows the velocity vectors in two curved surfaces and a section plane at Ra=10⁷. The strong three dimensional flow can be seen from the figure. In the region near the cylinder wall (J=1), the fluid moves upward and in the region near the other  wall (J=16), the fluid moves downwards. So the larger scale circulation was formed.  Four circulations was also visible in the flow domain. Two of them was in the upper region and near the symmetry planes and the fluid in the region close to the sections I=12 and I=13 moved toward the cylinder wall. At the region near the top wall, the fluid near I=12 and I=13 sections moved toward the corner of the cylinder.

Similar flow pattern was gotten at the selected Rayleigh number, but the magnitude of the velocity shown in the figure with a vector increased with the Rayleigh number.

Fig. 3 shows the isotherms in the flatten section of J=9, at different Rayleigh number. The distinct difference of the temperature distribution among the them can be seen from the figure. It is obvious that the flow pattern in the selection in the selected section affected the temperature distribution in same section. Large velocity corresponded to the higher value of temperature at the same position in the cross section plane.

Fig. 4 presents the contours of the local Nusselt number on the cylinder wall at different Rayleigh number. The local Nusselt number is defined as

The locally large heat transfer rate was in the bottom region and the minimum was in the top region. In the region near the symmetry, the heat transfer rate takes about average value.
Overall heat transfer results are presented in terms of average Nusselt number, which is defined as:

Where A is the area of the quarter of the cylinder line. The average Nusselt numbers at different Rayleigh numbers are presented in table 1. The heat transfer rate increased with the Rayleigh. Which was the direst result of the larger circulation at the higher Raylergh number.

Ra	106	107	2x107	3x107
	11.01	16.92	22.27	26.63