Problems

10.10.1

Using the result of the actuator disk theorems that the average induced velocity at the Trefftz plane is twice that at the rotor, u2 = 2u, show that the application of Bernoulli’s equation upstream and downstream of the disk provides the thrust force on the rotor as T = — (p+ – p-) A where p+ – p – =< p > represents the pressure jump across the actuator disk. Conclude.

10.10.2

As shown in Fig. 10.6, make sketches of the 1-D stream tubes and sketch the blade element operating conditions in the rotating frame as in Fig. 10.11, in the following cases (assume symmetric airfoil profile for simplicity):

(i) u = 0

(ii) u = — 1

(iii) u > 0

and indicate for each case if the flow situation corresponds to propeller, freewheeling or turbine.

10.10.3

By application of the Biot-Savart formula, show that the axial velocity along the x-axis, induced by the vortex sheets in the Trefftz plane, is twice that induced in the plane of the rotor. Hint: consider a single vortex filament n = const. and assume a perfect helix with constant pitch such as given by

У – = П cos(ak – + ч) = п sin( Шї – + ч)

10.10.4

Derive the remainders Saj, k and Scj, k given in Sect. 10.4.2.

10.10.5

Consider the scheme for the convection equation:

дГ дГ

+ С1 + 2u) — 0

д t дх

given in paragraph 6.1, with в — 2.

(i) Show that the scheme is unconditionally stable

(ii) Show that, on a uniform mesh Ax — (1 + 2u)At the scheme has the “perfect shift” property.

Acknowledgments One of the authors (JJC), acknowledges that part of the material in this chapter was originally published in the International Journal of Aerodynamics, Ref. [21].

References

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2. Rankine, W. J.: Trans. Inst. Nav. Archit. 6, 13 (1865)

3. Froude, R. E.: Trans. Inst. Nav. Archit. 30, 390 (1889)

4. Betz, A.: Wind Energie und Ihre Ausnutzung durch Windmuhlen. Gottingen, Vandenhoeck (1926)

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12. Chattot, J.-J.: Analysis and design of wings and wing/winglet combinations at low speeds. Comput. Fluid Dyn. J., Spec. Issue, 13(3) (2004)

13. Drela, M.: XFOIL: an analysis and design system for low reynolds number airfoils. In: Mueller, TJ. (ed.) Low Reynolds Number Aerodynamics. Lecture Notes in Engineering, vol. 54, pp. 1-12. Springer, Berlin (1989)

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21. Chattot, J.-J.: Wind turbine aerodynamics: analysis and design. Int. J. Aerodyn. 1(3/4), 404-444 (2011). http://www. inderscience. com/jhome. php? jcode=ijad

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