Research Papers

Actuator Disk Theory—Steady and Unsteady Models

[+] Author and Article Information
Jean-Jacques Chattot

Mechanical & Aerospace Engineering,
University of California Davis,
Davis, CA 95616

Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received March 9, 2013; final manuscript received January 20, 2014; published online March 21, 2014. Assoc. Editor: Yves Gagnon.

J. Sol. Energy Eng 136(3), 031012 (Mar 21, 2014) (10 pages) Paper No: SOL-13-1083; doi: 10.1115/1.4026947 History: Received March 09, 2013; Revised January 20, 2014; Accepted January 27, 2014

In this paper, the classical work on steady actuator disk theory is recalled and the interpretation of the flow as the continuous shedding of vortex rings of constant strength that causes the slip stream is shown to be consistent with the classical model, yet offers a different approach that provides more flexibility for the extension to unsteady flow. The ring model is in agreement with the conservation law theorems which indicate that the axial inductance inside the streamtube, far downstream, is twice that at the rotor disk. The actuator disk relationship between the power and the axial inductance at the rotor plane has been used previously to construct the prescribed wake in the 3D vortex line model (VLM) of the author. The local pitch of the vortex sheet helices is based on the axial inductance along the streamtube, therefore, it is important to be able to calculate the axial inductance in all cases. In unsteady flow, the strength of the shed rings varies with power absorbed by the rotor and a new relationship between the two is derived to first-order, using the unsteady Bernoulli equation, which allows to calculate the inductance at the rotor and in the wake with the Biot-Savart law, even in cases where the Betz limit is exceeded. This more rigorous model can replace the semi empirical models such as dynamic stall and dynamic inflow used in the blade element method (BEM). The proposed new model has been incorporated in the VLM and applied to two sets of experiments, the Tjaereborg in-field tests and the NREL wind tunnel campaign, for which test data has been collected and serves for the comparison. The simulations show an overall good agreement with the experimental data. In some tests, the power coefficient exceeds the Betz limit, in which cases there are no solutions to the steady actuator disk theory, but the new model provides a solution for the inductance even in this case.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Rankine, W. J., 1865, Transactions, Institute of Naval Architects, Adelphi Terrace, London, Vol. 6, p. 13.
Froude, R. E., 1889, Transactions, Institute of Naval Architects, Adelphi Terrace, London, Vol. 30, p. 390.
Spalart, P. R., 2003, “On the Simple Actuator Disk,” J. Fluid Mech., 494, pp. 399–405. [CrossRef]
Betz, A., 1926, Wind Energie und Ihre Ausnutzung durch Windmuhlen, Gottingen, Vandenhoeck.
van Kuik, G. A. M., 2007, “The Lanchester-Betz-Joukowsky Limit,” Wind Energy, 10, pp. 289–291. [CrossRef]
Manwell, J. F., McGowan, J. G., and Rogers, A. L., 2010, Wind Energy Explained: Theory, Design and Application, 2nd ed., Wiley, New York.
Burton, T., Jenkins, N., Sharpe, D., and Bossanyi, E., 2011, Wind Energy Handbook, 2nd ed., Wiley, New York.
Hansen, M. O. L., 2008, “Aerodynamics of Wind Turbines, Earthscan London. Available at: http://www.earthscan.co.uk.
Glauert, H., 1935, “Aerodynamic Theory,” The General Momentum Theory, Volume IV division L-III, W. F.Durand, ed., Springer, Berlin, Germany. Reprinted in 1963 as a Dover edition.
Wilson, R., and Lissaman, P. B. S., 1974, Applied Aerodynamics of Wind Power Machines, Oregon State University, Corvallis, OR.
Sorensen, J. N., and van Kuik, G. A. M., 2011, “General Momentum Theory for Wind Turbines at Low Tip Speed Ratios,” Wind Energy, 14, pp. 821–839. [CrossRef]
Sorensen, J. N., and Myken, A., 1992, “Unsteady Actuator Disk Model for Horizontal Axis Wind Turbines,” J. Wind Eng. Indus. Aerodyn., 39, pp. 139–149. [CrossRef]
Chattot, J.-J., and Braaten, M. E., 2012, “Wind Turbine Pitch Change Simulation With Helicoidal Vortex Model,” Proceedings of ASME Turbo Expo, Copenhagen, Denmark, June 11–15, 2012, ASME, Paper No. GT2012-68294. [CrossRef]
Chattot, J.-J., 2002, “Design and Analysis of Wind Turbines Using Helicoidal Vortex Model,” Comput. Fluid Dyn. J., 11(1), pp. 50–54.
Moriarty, P. J., and Hansen, A. C., 2005, “AeroDyn Theory Manual,” NREL/TP-500-36881. Available at: http://www.nrel.gov/docs/fy05osti/36881.pdf.
Snel, H., and Schepers, J. G., eds., 1994, JOULEI: Joint Investigation of Dynamic Inflow Effects and Implementation of an Engineering Method, ECN-C-94-107, ECN, Petten, The Netherlands.
Suzuki, A., 2000, Application of Dynamic Inflow Theory to Wind Turbine Rotors, Salt Lake City, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT.
Hand, M. M., Simms, D. A., Fingersh, L. J., Jager, D. W., Cotrell, J. R., Schreck, S., and Larwood, S. M., 2001, “Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaigns,” NREL, Golden, CO, Technical Report No. NREL/TP-500-29955.
Chattot, J.-J., 2007, “Helicoidal Vortex Model for Wind Turbine Aeroelastic Simulation,” Comput. Struct., 85, pp. 1072–1079. [CrossRef]


Grahic Jump Location
Fig. 1

Control volume/control surface for conservation theorems

Grahic Jump Location
Fig. 2

Thrust and power coefficients vs. axial induced velocity

Grahic Jump Location
Fig. 3

Vortex rings for the modelization of slip stream

Grahic Jump Location
Fig. 4

Flow field at the actuator disk

Grahic Jump Location
Fig. 5

Elementary contribution of a vortex ring to the inductance at point M

Grahic Jump Location
Fig. 6

Flow in Trefftz plane

Grahic Jump Location
Fig. 7

Evolution of ub(t) (left) and space distributions of u(x, t), for a step change ΔuT = −0.1

Grahic Jump Location
Fig. 8

Evolution of ub(t) (left) and space distribution u(x, t) at t = 50 (right), for a step change ΔCP = 0.0505

Grahic Jump Location
Fig. 9

Steady curve and step change trajectory CP(ub)

Grahic Jump Location
Fig. 10

Power comparison for Tj_II.6

Grahic Jump Location
Fig. 11

Distributions of uT and u (left) and wake shape (right) for Tj_II.6

Grahic Jump Location
Fig. 12

Cp(ub) for Tj_II.6

Grahic Jump Location
Fig. 13

Power comparison for Tj_II.4

Grahic Jump Location
Fig. 14

Distributions of uT and u (left) and wake shape (right) for Tj_II.4

Grahic Jump Location
Fig. 15

Cp(ub) for Tj_II.4

Grahic Jump Location
Fig. 16

NREL turbine thrust comparison

Grahic Jump Location
Fig. 17

NREL turbine torque comparison

Grahic Jump Location
Fig. 18

NREL turbine power comparison

Grahic Jump Location
Fig. 19

Distributions of uT and u (left) and wake shape (right) for NREL rotor

Grahic Jump Location
Fig. 20

Cp(ub) for NREL rotor



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In