| Summary and Conclusion. The relaxation method described in this chapter is used to evaluate the invisible two-dimensional rotational flow of both an incompressible and a compressible fluid through a channel with a sharp constriction present. In each case, the stream lines, lines of constant velocity and vorticity are indicated. It is to be expected, that an incompressible flow and a low speed compressible one markedly in their general features. This is backed up by the diagrams of the respective streams lines and constant velocity lines. The limits for and free steams velocity values differ, however, and the incompressible values are five times those case (for example, figures (3) and (10)). In the deflections of the central stream lines, there is no difference in the actual values, to the degree of accuracy used in the calculations. Thus, in view of the feet that solutions for the incompressible rotational flow past cylinders of various cross sections are known (chapter II and III), it may be deduced, that in the low speed compressible rotational flow past these cylinders, the general features, both quantitatively and qualitatively, will not differ markedly from those already obtained for the incompressible case. The velocity diagrams, figures (4) and (11), again display the same general features, namely low velocities in the outsides corners of the constriction, and large velocities at the inner corners, in both cases the region of highest velocity being on the upper wall downstream of the constriction, although displayed by both flows, is much more marked in the compressible cases. From figure (12), this region is also seen to be a region of high vorticity, and in practice could develop into a small dead air region bordered by a free stream line. This will tend to 'soften' the sharpness of the upper corner. Thus, in practice the flow in the compressible case may 'soften' the sharp constriction, making the change in cross section much more gradual. Before the corner were excluded, in the preliminary calculation, very high values of the velocity were obtained at the inner corners of the constriction. The supersonic regions which appear there are of small physical importance in the flow of a real fluid and are omitted in the calculations. They can be obtained, but there is some doubt as to the validity of the final result. It should be noted, that lines of constant vorticity are also lines of constant density, and from Figures (8), lines of constant pressure can be obtained. The drop in pressure across the channel, at the stations other than in the stream, can thus be obtained, if desired. One important features arising from the compressible case is the feat that the presence of a constriction tends to decrease the value of the vorticity, changing it in this case from 0.2 to 0.187 in the free stream. Initially the vorticity is constant across the channel, and finally the vorticity is constant, with a value which differs, of course, from the original value. One further result of importance, is the fact that a large vorticity results in back flow, which in practice would possibly result in break away. Thus, there is a possibility that there can be breakaway in a flow, if high vorticity regions are present. The condition on the non-dimensional. Number N, for no obvious basic flow in the channel, is that it must be numerically greater than, in both the compressible and incompressible cases. In conclusion, it must be remembered that relaxation is an approximate process, and no matter how fine a mosh is used, a discontinuity may be missed, until, however, theoretical solutions are obtained, relaxation constitutes a very powerful technique, since particular ports of a field can be examined in great detail, provided the stream function and, if necessary, its derivatives, are known with sufficient accuracy at the boundaries. The ability to treat parts of the field independently, makes it useful where wind tunnel evidence shows certain features of a flow. The more that is known about a particular problem, the more accurate is the final relaxation solutions. One prohibitive features of a relaxation technique is the fact that it involves much laborious calculation. It is neither possible nor profitable to reproduce the calculations in detail. |
