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.


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.


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)


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


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].


1. Hand, M. M., Simms, D. A., Fingersh, L. J., Jager, D. W., Cotrell, J. R., Schreck, S., Larwood, S. M.: Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaigns, NREL/TP-500-29955 (2001)

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)

5. Joukowski, N. E.: Travaux du Bureau des Calculs et Essais Aeronautiques del’Ecole Superieure Technique de Moscou (1918)

6. Prandtl, L., Betz, A.: Vier Abhandlungen zur Hydro – und Aero-dynamik, Selbstverlag des Kaiser Wilhelminstituts fur Stromungsforshung. Gottingen Nachr. Gottingen, Germany (1927)

7. Goldstein, S.: On the vortex theory of screw propellers. Proc. R. Soc. Lond. Ser. A 123,440-465 (1929)

8. Betz, A.: Schraubenpropeller mit geringstem Energieverlust, Nach der Kgl. Gesellschaft der Wiss. zu Gottingen, Math.-Phys. Klasse, pp. 193-217; reprinted in Vier Abhandlungen zur Hydro – und Aero-dynamik, by L. Prandtl and A. Betz, Gottingen, 1927 (reprint Ann Arbor: Edwards Bros. 1943), pp. 68-92 (1919)

9. Munk, M. M.: The Minimum Induced Drag of Aerofoils, NACA report. 121 (1921)

10. Chattot, J.-J.: Optimization of wind turbines using helicoidal vortex model. J. Sol. Energy Eng. Spec. Issue: Wind Energy 125(4), 418-424 (2003)

11. Chattot, J.-J.: Computational Aerodynamics and Fluid Dynamics: An Introduction. Scientific Computation. Springer, Berlin (2004). ISBN 3-540-43494-1, Second Printing

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)

14. Chattot, J.-J.: Helicoidal vortex model for steady and unsteady flows. Comput. Fluids 35, 733-741 (2006)

15. Coton, F. N., Wang, T., Galbraith, R. A. McD: An examination of key aerodynamics modeling issues raised by the NREL blind comparison. AIAA paper no. 0038 (2002)

16. Hallissy, J. M., Chattot, J.-J.: Validation of a helicoidal vortex model with the NREL unsteady aerodynamic experiment. Comput. Fluid Dyn. J. Spec. Issue 14(3), 236-245 (2005)

17. Schmitz, S., Chattot, J.-J.: Method for aerodynamic analysis of wind turbines at peak power. J. Propuls. Power 23(1), 243-246 (2007)

18. Chattot, J.-J.: Effects of blade tip modifications on wind turbine performance using vortex model. Comput. Fluids 38(7), 1405-1410 (2008)

19. Chattot, J.-J.: Helicoidal vortex model for wind turbine aeroelastic simulation. Comput. Struct. 85, 1072-1079 (2007)

20. Chattot, J.-J.: Optimization of propellers using helicoidal vortex model. Comput. Fluid Dyn. J. 10(4), 429-438 (2002)

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