FLUID DYNAMIC MODELLING OF WIND TURBINES

sec. D

Vr 0 D Vt

Vz Vr Vt 3

Relatori: Prof.Ing. Lorenzo BATTISTI Prof.Ing. Piero PINAMONTI

Dottorando: Dott.Ing. Luca ZANNE

Udine 21 Maggio 2010

Summary

Introduction PART I : HAWT analysis HAWT Fluid dynamics A turbomachinery approach Inverse design

Summary

PART II : VAWT analysis VAWT fluid dynamics VAWT experimental analysis VAWT free vortex wake Results and conclusions

Introduction

Wind energy market (EWEA) Installed capacity

Offshore WE market (EWEA)

Aim of the thesis & thesis outline

The aim of the thesis is to analyze the fluid dynamic models of wind energy conversion systems, pointing out the limitations of current engineering models and proposing innovative solutions from the design point of view The research activities have been divided in two main parts, following the different rotor – flow interaction characteristics: 1. Horizontal axis wind turbines - HAWT; 2. Crossflow wind turbines, as vertical axis wind turbines - VAWT.

Part I : HAWT analysis HAWT fluid dynamics

HAWT fluid dynamics is mainly based on the actuator disk concept

HAWT fluid dynamics Actuator disk concept

The turbine generates mechanical work from the kinetic energy of the fluid flow The work exchange between the fluid and the shaft is done by is done by the rotor, which can be modelled as an actuator disk The bladed rotor can be represented with equivalent forces distribuited over a permeable, immaterial disk Infinite number of blades Infinite rotational velocity

HAWT fluid dynamics Actuator disk – momentum theory

Froude applied for the first time the actuator disk concept to a rotor in open flow. He applied it with the 1D momentum balance in axial direction Momentum equation T = ∆p ⋅ Am = ρVz ,3 A3 (Vz ,0 − Vz ,3 )

Energy conservation

Weul = ∆p

Mass conservation

Vz ,m Am = Vz ,3 A3 Vz ,1 ≅ Vz ,2 ≅ Vz ,m

ρ

=

Vz2 − Vz2 ,0 ,3 2

Froude result!

Vz ,m = Vz ,0 + Vz ,3 2

Actuator disk Blade element – momentum theory

Drzewiecki first applied Froude result dividing the rotor in different annular streamtubes : Non uniform loading Vz ,m = Vz ,0 + Vz ,3 2

Raero FN Lift φflow

With the blade element airfoil theory rotor performances can be easily calculated The annuli interaction is neglected No swirl flow, (wake expansion?) Ok lightly loaded rotors

Wind.=[ -a·V0; -a’·ωr ]

FT

Drag z

chord line

-ωr θpitch+βtwist φflow αattack V0 Vrel. y

rotor plane

HAWT fluid dynamics General momentum theory

The general momentum theory should overcome the issues of the swirl flow modelling Momentum equation : axial T = ∫ ( p1 − p2 ) dA = ∫ ρVz ,3 (Vz ,0 − Vz ,3 ) + ( p0 − p3 ) dA Am A3

tangential

M = ∫ ρVθ ,3Vz ,3 r3 dA

A3

radial

p3 − p0

ρ

=

p3 ( r3 ) − p3 rtip ,3

(

ρ

)=−

∫r3

2 rtip ,3 Vθ ,3

r3

dr3

1 + 1 2Vθ ,3 Ωr3 1 + 1 2Vθ ,2 Ωr 2 1 ρ ∫A3 (Vz ,0 − Vz ,3 ) dA = ρΩ ∫A3Vz ,3Vθ ,3r3 − dA 2 Vz ,3 Vz ,m

Solutions: • De Vries • Differential

• GM theory is an integral formulation • It needs the wake solution

8 V2 Weul = ⋅ 0 9 2

Actuator disk – momentum theory limitations

Actuator strip Wake states

Conway exact solution

HAWT fluid dynamics Vortex theory

Vortex theory calculates the flow field of the rotor wake by using the fluid dynamic laws of vorticity (BiotSavart law, Kelvin’s theorem, Helmholtz’s laws) Introduced by Joukowski – Betz – Prandtl Most widespread for propeller analysis and design (both for aerodynamic and marine propellers) and for helicopter rotor performance prediction • Prescribed vortex wake • Free vortex wake

Vortex theory Prescribed vortex wake

Vz ,m =

Vz ,0 + Vz ,3 2

Axial velocity

d Γ = 2π ⋅ d ( rVθ ,2 )

gθ ,m = d Γ r Ω + Vθ ,2 2 Vz , m 2π r d Γ r3Ω + Vθ ,3 2π r3 Vz ,3

Radial velocity

Vr ( r ,0 ) = − 1 r ∂Vz r ( r ,0 )...