Reconstruction of a Numerically Generated Thermal Field

The second application taken up for analysis comprises of a numerically generated convective thermal field in a horizontal differentially heated fluid layer. For definiteness, the wall temperatures employed are 15⁰C and 30⁰C respectively. The three-dimensional temperature field  has been determined as follows. The stream function, vorticity, and energy equations are solved in two dimensions with symmetry conditions applied on the side walls, by a finite difference method [101]. The solution thus obtained corresponds  to a  system of longitudinal rolls spread over an infinite fluid layer. Such geometries show a polygonal plan form corresponding to a fully three-dimensional temperature field [102]. The three-dimensionality has been simulated in the present work by superimposing a sine variation in the thermal field parallel to the axis of the roll. A surface plot of the resulting temperature field revealed the flow to be organized in the form of cubic cells in the fluid layer.

Figure 4.66: Temperature surface of the midplane of the layer, in the form of cubic cells

The advantages of selecting the field to be reconstructed in the particular manner outlined above are : (1) The field  is continuous and hence reconstruction errors can be expected to be small, as compared to the application with holes.(2). Error with perfect data being small, one can systematically study errors induced by the initial guess, and noise in the projection data. (3)The thermal field begin analyzed is physically realizable.

For reconstruction, the fluid layer has been discretized into 11 planes and each plane into cells. The relaxation factor for the proposed algorithms has been set to unity. Since the algorithms are being tested under conditions of limited data, only two and four projections have been considered. A convergence criterion of has been uniformly employed in the computation. Results obtained using the proposed MART algorithms alone have been reported.