Temperature Data over a Grid 

Once the absolute fringe temperatures are obtained, this data must be transferred to a two-dimensional uniform grid over the fluid region. This is required to apply tomographic algorithms for the reconstruction of the three-dimensional temperature field. The data transfer is achieved by interpolation as described below. The nine different points where the fringes intersect the column (fig. 4.32) are first mapped to a uniform rectangular grid using a quadratic polynomial as a basis. Temperature at any point (such as fig. 4.32) can be computed by using two-dimensional quadratic interpolation again. Interpolation using a higher order schemes can produce oscillations in the interpolated data. Consequently, the interpolated value in the interior may exceed the values at boundary points and is thus undesirable. Though rare, this may occur when the data spacing is large. In the present study, such overshoot and undershoot have been taken care of by using the idea of universal limiters. The limiter used is one-dimensional  in the sense that it is applied only along the vertical direction. Once the interpolated value at a point on the superimposed grid is obtained, its value is compared with the two nearest vertically separated fringes. If the interpolated temperature is outside the range of the two fringe temperatures, the limiter is switched on to force the interpolated value to be one of the temperatures closest to the interpolated value. Interpolation errors in the present work were found to be negligible

The collection of thinned images from the Rayleigh-Benard experiment at a 90 degrees projection angle is shown in Figure 4.33. Interpolation has been carried over the entire image by superimposing a two-dimensional grid on it. The grid has 120 point along the horizontal and 21 points along the vertical direction. Once the interpolation is complete isotherms have been drawn to represent the fringes in the original image. This is shown in Figure 4.34. It can be seen here that the temperature data on the grid follows closely the pattern of the original thinned image and interpolation errors are negligible. The isotherms based on the interpolated grid data are seen to capture the lost fringes in the interferogram as well. Hence the isotherms in Figure 4.34 show all the fringes with continuity throughout the width of the cavity.

Figure 4.32: Data transfer from an interferogram to a two dimensional grid.