Navier-Stokes equations with

application to wind energy problems

Alexander Rauh, Institute of Physics, University of Oldenburg, Germany

Content:

I.) Why are the NSE a strong theory?

II.) Computational fluid dynamics (CFD): Some strategic points

III.) Open problems in the hydrodynamics of wind energy

With thanks for the support of the Oldenburg colleagues

Joachim Peinke

(Hydronamics and ForWind Institute)

Detlev Heinemann

(Energy meteorology)

and

Jürgen Dreher

(Faculty of Physics and Astronomy, Univ. Bochum)

IIa.) Lattice gas automata

Particles move on a lattice, and collide conserving mass and momentum

Advantages: Easy parallelization.

No rounding errors.

Easy tagging of particles.

Disadvantage: Additional storage and CPU to form

fluid parcels.

Sinkink of heavy fluid into a leighter one.

By courtesy of Jürgen Dreher, Ruhr-Universität Bochum

http://www.tp1.ruhr-uni-bochum.de/%7Ejd/racoon/example.html

Display of grid refinement

For parallelization, cell groups are formed

Application to wind energy

An economic incentive

Installed electricity power in Germany 2006

75 GW conventional 20 GW wind

Suppliers

Consumers

EEX Bourse

Estimates of spotmarked for next-day

delivery

Assume an offer of 10h wind power from wind turbines with 5 GW capacity.

This is an offer of 5x107 kWh at the EEX (European Energy Exchange Bourse)

Qualitity assessment by energy meteorology

can make a price difference of the order of

1cent/kWh

or 5x105 Euro per day.

What can Fluid Dynamics contribute to wind power

assessment?

1.) Calculate the efficiency of a single wind turbine (WT) from the Navier-Stokes equation (NSE) for steady winds by CFD.

2.) Elaborate the response of the WT to fluctuating winds by CFD.

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3.) Estimate the effect of shadowing in wind parks by CFD. 4.) Work out a CFD code to consider the orographic influence at a given site.

5.) Improve the present theory to infer wind velocity distribution in a larger area, including height dependence, from measurements at sparsly distributed points.

Is there an optimum efficieny of WD analogous to the Carnot

efficiency?

Definition of efficiency: η = L/PW,

where L is the mechanical power transferred by the wind to the turbine

and PW = ρ/2 AR u3 the available wind power flowing through the rotor area AR; u is velocity component vertical to rotor plane and sufficiently far away from the turbine.

Betz “Carnot” - optimum (1927 ): η = 16/27

Control volume of Betz theory.

There is momentum flow through the mantle surface.

This was neglected: There is no condition on the efficiency.

For a measured efficiency higher than Betz, see

Abe K., Nishida M., Sakurai A, et al.

Experimental and numerical investigation of flow fields behind a small wind turbine with flanged diffuser,

J. of Wind Eng. Ind. Aerodynamics, 93, 951 (2005).

Measured power as function of wind speed over 24h

(Tjareborg 2MW wind turbine)

How can one get a Power Curve from such fluctuating measurements?

How can one estimate the day-ahead power output?

Conclusion

This is to suggest the application of the adaptive mesh refinement method (AMR), including parallelization, to wind energy technology :

- Efficiency of wind turbines (WT) from NSE simulation

- Modelling the (delayed) response of WT to

fluctuating wind.

- …….

.Comment:

The pioneering work on AMR of M.J. Berger and P. Colella, J. Comp. Phys. 82, 64 (1989)

has almost 400 citations until now.

Most in Astrophysics, then comes Plasmaphysics.

No citation in the wind energy problems mentioned.

Closest paper, maybe, is on turbomachinary flow.

An exotic example among the 400 citations:

Simulation of penetration of a sequence of bombs into

granite rock.

Textbooks to Navier-Stokes Equations

L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon Press 1989.

C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford Univ. Press 1997.

U. Frisch, Turbulence, Cambridge Univ. Press 1995.

Ch.R. Doering and J.D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Univ. Press 1995.

O.A. Ladyshenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, N.Y. 1969.

References to Computational Fluid Dynamics

U. Frisch, D. d’Humiere et al., Lattice-Gas Automata for the Navier-Stokes Equation, Complex Systems 1, 75-136 (1987).

C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, Vol.1 and 2, Springer 1997.

M.J. Berger and P. Colella, Local Adaptive Mesh Refinement for Shock Hydrodynamics, J. Comp. Phys. 82, 64 (1989).

J. Dreher and R. Grauer, A Parallel Mesh-Adaptive Framework for Hyperbolic Conservation Laws, Parallel Computing 31, 913 (2005).

References to Hydrodynamics and Wind Energy

K. Abe, M. Nishida, A. Sakurai et al., Experimental und numerical investigations of flow fields behind a small wind turbine with a flanged

diffuser, J. Wind Eng. Ind. Aerodyn. 93, 951 (2005).

A. Rauh and W. Seelert, The Betz Optimum Efficiency for Windmills,

Applied Energy 17, 15 (1984).

A. Rauh and J. Peinke, A phenomenological model for the dynamic response of wind turbines to turbulent wind,. Wind Eng. Ind. Aerodyn. 92, 159 (2004).

A. Rauh, E. Anahua, St. Barth, J. Peinke, Phenomenological Response Theory to Predict Power Output, in: Wind Energy, Proceedings of the Euromech Colloquium, Eds. J. Peinke, St. Barth, P. Schaumann, p.153, Springer 2007.