A theory of two-dimensional airfoils with strong inlet flow on the upper surface

Abstract

The two-dimensional theory of airfoils with arbitrarily strong inlet flow into the upper surface was examined with the aim of developing a thin-airfoil theory which is valid for this condition. Such a theory has, in fact, been developed and reduces uniformly to the conventional thin-wing theory when the inlet flow vanishes. The integrals associated with the arbitrary shape, corresponding to the familiar Munk integrals, are somewhat more complex but not so as to make calculations difficult. To examine the limit for very high ratios of inlet to free-stream velocity, the theory of the Joukowski airfoil was extended to incorporate an arbitrary inlet on the upper surface. Because this calculation is exact, phenomena observed in the limit cannot be attributed to the linearized calculation. These results showed that airfoil theory, in the conventional sense, breaks down at very large ratios of inlet to free-stream velocity. This occurs where the strong induced field of the inlet dominates the free-stream flow so overwhelmingly that the flow no longer leaves the trailing edge but flows toward it. Then the trailing edge becomes, in fact a leading edge and the Kutta condition is physically inapplicable. For the example in this work, this breakdown occurred at a ratio of inlet to free-stream velocity of about 10. This phenomena suggests that for ratios in excess of the critical value, the flow separates from the trailing edge and the circulation is dominated by conditions at the edges of the inlet.

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