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Aerodynamics for engineering students - part 5

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cánh có tỉ lệ cao, trực giác sẽ đề nghị rằng dòng chảy trên hầu hết các cánh cư xử như thể nó là hai chiều. Rõ ràng điều này sẽ không được một xấp xỉ tốt ở gần đầu cánh mà sự hình thành của các xoáy dấu dẫn đến dòng chảy cao ba-dunensional. | 230 Aerodynamics for Engineering Students Fig. 5.20 Modelling the displacement effect by a distribution of sources wings having high aspect ratio intuition would suggest that the flow over most of the wing behaves as if it were two-dimensional. Plainly this will not be a good approximation near the wing-tips where the formation of the trailing vortices leads to highly three-dimensional flow. However away from the wing-tip region Eqn 5.23 reduces approximately to Eqn 4.103 and to a good approximation the Cp distributions obtained for symmetrical aerofoils can be used for the wing sections. For completeness this result is demonstrated formally immediately below. However if this is not of interest go directly to the next section. Change the variables in Eqn 5.23 to X x - Xi c Z1 Z jc and z z - Zi c. Now provided that the non-dimensional shape of the wing-section does not change along the span or at any rate only changes very slowly St d yt c dx does not vary with z and the integral I in Eqn 5.23 becomes To evaluate the integral Ỉ2 change variable to X 1 z so that Fig. 5.21 The relation between spanwise load variation and trailing vortex strength Finite wing theory 231 giving 5.24 Z1C s zi For large aspect ratios s c so provided Z1 is not close to 5 i.e. near the wing-tips 2 - . J í Z1C 1 and V Z1 Thus Eqn 5.23 reduces to the two-dimensional result Eqn 4.103 i.e. 2 fc xi dyt 1 Cp djc 7T JXl OXX-Xl Lifting effect To understand the fundamental concepts involved in modelling the lifting effect of a vortex sheet consider first the simple rectangular wing depicted in Fig. 5.21. Here the vortex sheet is constructed from a collection of horseshoe vortices located in the y 0 plane. From Helmholtz s second theorem Section 5.2.1 the strength of the circulation round any section of the vortex sheet or wing is the sum of the strengths of the T-8r Curve defining the spanwise variation in strength of the combined bound vortex filaments - 232 Aerodynamics for Engineering Students .