Content area
Abstract
The problem of the oblique injection of a secondary stream into a free stream of different total pressure (non-isoenergetic) has practical application in many physical situations such as in the film cooling of gas turbine blades. This thesis describes a new method for predicting inviscid non-isoenergetic flow from an arbitrarily inclined slot into a uniform free stream. In the absence of flow separation the solution depends on three parameters: (i) the slot angle $\beta,$ (ii) a parameter, $C\sb{p\sb{t}}$ describing the difference in total pressure between the free stream and slot fluids, and (iii) the ratio of densities, D, between the two fluids. For given values $C\sb{p\sb{t}}$ and $\beta,$ there exists a unique value of $M\sp2/D,$ where M is the ratio of mass flow of the injectant to the main stream.
To guide in the development of the present theory, an isoenergetic solution $(C\sb{p\sb{t}}=0,\ D=1)$ was found using classical theory. Specifically, $$M = \left\lbrack{\lambda\over1 - \lambda}\right\rbrack\sp\lambda\qquad {\rm where,}\qquad \lambda = {\beta\over\pi},\qquad 0 < \lambda < 1$$High curvature of the streamlines near the downstream slot lip produces an unrealistic suction for large slot angles. The addition of source/sink singularities to the model to simulate flow separation improves the results, but adds empiricism to the solution.
The present non-isoenergetic procedure uses the dividing streamline to separate the flowfield into two zones; an internal flow region containing the slot fluid and an external flow region containing the free stream fluid. The technique involves finding, by approximate means, the shape of the dividing streamline that satisfies continuity of static pressure between the regions. The boundary conditions at the upstream slot lip were pivotal in finding a unique solution. The results presented here extend previous work to include arbitrary values of $\beta,\ C\sb{p\sb{t}}$ and D.
For a vertical slot, mass flow rates and discharge coefficients obtained from experiments and viscous numerical predictions are compared to present inviscid calculations. The results show good agreement at reasonable Reynolds numbers provided that the separation region is not large. The effects of separation can be included in the present theory by incorporating the approximate shape of the separated flow region, and the results show a significant improvement, especially at higher mass flow ratios and slot angles. Calculated discharge coefficients for lower slot angles do not compare as well with measured values, but this may be due to the differences in geometries used for the comparison.
The solution algorithm used in the present problem is quite general, and may be applied to other geometries and flow conditions.
