A Modified Streamtube Model for Vertical Axis Wind Turbines MULTI-SCIENCE PUBLISHING COMPANY

A Modified Streamtube Model for Vertical Axis WindTurbines

by

Moti Keinan

REPRINTED FROM

WIND ENGINEERINGVOLUME 36, NO. 2, 2012

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WIND ENGINEERING VOLUME 36, NO. 2, 2012 PP 145-180 145

A Modified Streamtube Model for Vertical Axis WindTurbines

Moti KeinanHarofe 36, Haifa 34367, [email protected]

ABSTRACTThis work introduces a modified double-multiple streamtube model for Vertical Axis Wind

Turbines (VAWT) that takes into account some parameters that were neglected till now. For

example, the model considers the directions of the flow, rather than assuming that the flow

is parallel to the free-wind. Also a formulation is established to calculate the effect of

inclination of the blades in any direction. The modifications are facilitated by considering the

momentum conservation in a direction perpendicular to the force (which was abandoned in

the traditional models). The model results are compared to other models and to

experimental results with a fair agreement.

ABBREVIATIONSFS Flow Strip

FSF (The system of) Flow Strip Force

HAWT Horizontal Axis Wind Turbine

Re Reynolds number

SNL Sandia National Laboratories

TSR Tip Speed Ratio

VAWT Vertical Axis Wind Turbine

WF Wake Function

Notations

A Area of the cross-section of the FS

AP Area for the computation of the pressure drop

C Dimensionless force, or force coefficient

C Chord length

CD Drag coefficient

CL Lift coefficient

CM Moment coefficient

CN, CC, Czb Projections of the dimensionless blade forces on the axes of the blade system

CNC [CN CC]

CP Power coefficient

cp Specific power coefficient

CR, CT, CZ Projections of the dimensionless blade forces on the axes of the polar system

CRT [CR CT]

CX, CY Projections of the dimensionless blade forces on the axes of the global slice

system

CXX, CYY Projections of the dimensionless blade forces on the axes of the FSF system

D Drag force or the axis in the effective-wind system, in the drag direction

F Force applied on the rotor by the FS at the intersection point

FBXX, FBYY Projections of the blade force on the FSF system

Ff Force figure

f Angle of F relative to axis R

FX, FY Projections of F on the axes of the slice system

FXX, FYY Projections of F on the axes of the FSF system

G Dimensionless (normalized by Ufs) FS speed at intersection point

G0 Dimensionless FS speed at the local upwind the of intersection point

G0XX, G0YY G0 projected on the axes of the FSF system

G0XY [G0XX G0YY]

GEF Dimensionless effective-wind speed in the cross-section plane of the blade

GN, GC Dimensionless GEF projected on the axes of the blade system

GNC [GN GC]

GNI, GCI Dimensionless GEF projected on the axes of the blade inclined system (without

twist)

GR, GT G projected on the axes of the polar system

GRT [GR GT]

GXR, GYR Dimensionless FS speed of local upwind projected on the local slice axes

GXX, GYY G projected on the axes of the FSF system

GXYR [GXR GYR]

GW Dimensionless FS speed at the local wake of intersection point

GWXX, GWYY GW projected on the axes of the FSF system

HX, HY Ratios between the speeds at FSF system and GEF

h Height

h Increment in the vertical axis

INC Matrix of transformation from the polar system to the system of inclined blade

k Reduced frequency

L Lift force or the axis in the effective-wind system, in the lift direction

N Number of blades of the rotor

P Mechanical output power of the rotor

Pat The atmospheric pressure

Pd Pressure at the downwind side of an intersection point

Pdet Energy rate that is detracted from the FS in the interaction with the rotor

Pfs Energy rate of free-wind that sweeps the cross-section of the rotor

PM Total mechanical output power of the rotor

pM Mechanical output power of a segment of the rotor that interacts with a FS

Pu Pressure at the upwind side of an intersection point

P Pressure drop at an intersection point

R Radius of the rotor, (or radial axis of polar system)

S Solidity of the rotor (Nc/2R)

U FS speed at intersection point (induced speed)

U0 Local upwind speed at intersection point

U0XX, U0YY U0 projected on the axes of the force system

UEF Effective-wind speed that is experienced by a blades, in the plane of the cross-

section of the blade. It is also commonly termed as the incident wind speed.

Ufs FS speed at the rotor upwind, at a free-stream flow regime (free-wind speed)

UL, UD UEF projected on the axes of the effective-wind system

UN, UC UEF projected on the axes of the blade system

UNI, UCI UEF projected on the axes of the blade inclined system (without twist)

146 A MODIFIED STREAMTUBE MODEL FOR VERTICAL AXIS WIND TURBINES

UR, UT U projected on the axes of the polar system

UXR, UYR FS local upstream speed projected on the local slice axes

UW Local wake speed at intersection point

UWXX, UWYY UW projected on the axes of the force system

u Angel of U relative to axis R

W Wake function

w Width of cross-section of the FS normal to the direction of the flow

wP Width of cross-section of the FS normal to the direction of the force

Angle of attack

Angle of inclination

Angle of twist (pitch)

Angle of tilt, the angel of inclination plane

Angle of the inclination axis relative to the rotor X axis

Tip speed ratio (TSR), (R/U0)

Angel of the trajectory of the flow inside the rotor relative to axis X

Polar (azimuth) angel in the rotor

P of the previously calculated FS

Increment of the polar (azimuth) angel

P of the previously calculated FS

Angular speed of the rotor

Axes of Systems

I Axis of inclination

L, D Axes of system of the local wind. D is the direction of the free wind, which is the

drag axis, L is the lift axis.

N, C, Zb Axes of system that are attached to the cross-section of the blade. C coincides

with the chord, N perpendicular, Zb span-wise

NI, CI Axes of system of a blade (as above) that undergo inclination, while twist is not

considered yet.

R, T, Z Axes of polar system. R Radial, T Tangential, Z vertical

X, Y Cartesian axes of the global slice system

XR, YR Cartesian axes of the local slice system

XX, YY Axes of FSF the system of the FS forces. Axis XX is the direction of (positive) F

Subscripts

0 Local upwind

1 Entrance point

2 Exit point

B, b Blade

C Chord-wise

EF effective

fs free-stream, or the upwind for the rotor

N Normal

R Radial

T Tangential

W Local wake

Z Vertical

Zb Span-wise


1. INTRODUCTIONA comprehensive survey of the methods in modeling the aerodynamics of Darrieus VAWT is

given in [1]. The most-used one is the double-multiple streamtube model, due to its relative

simplicity and easiness of implementation. The earlier descriptions of the method are

introduced in [8] and [9]. The updated versions are introduced in [2] and [3]. The modified

model proposed in this work contains the following features that were neglected or not

considered in the commonly applied traditional models:

1. While in traditional models the induced speed U in the rotor surface is assumed to be

parallel to the free-wind speed Ufs, in the modified model the real direction of U is

computed (and generally found far from being parallel).

2. In some traditional models (like [9]) the magnitude of U is calculated by solving a one-

dimensional momentum and Bernoulli’s equations. Other works apply an even simpler

procedure by taking the magnitude as equal to the upwind speed. In the frontal surface of

the rotor, for example, U is taken as equal to Ufs. This may express an understanding that

the aerodynamic coefficients of an airfoil are defined with relation to the “speed at

infinity”. But the true speed that should be considered is the induced speed, which is the

average speed that the blade element “sees” at the rotor surface. This speed is equivalent

to the average speed in a wind-tunnel section where the airfoil data are determined.

(Indeed, in the Blade Element Method that is applied for HAWT, the induced speed is

calculated, rather than taking the upwind speed). In the modified model, U is computed for

both magnitude and direction as 2-dimensional case.

3. In commonly traditional models the trajectory of the flow inside the rotor is also assumed

to be a straight line, parallel to the free-wind, as is depicted in Figure 1 [2, 4] for the codes


Downwimd

CARDAA CARDAAV

CARDAAX

Streamtube

Upwind

yy

x

v

v ′

v ″v ′

v ∞

v

v l ″l ′ l

l∞

w ′

w

dl ″dl ′

dl dl∞dl∞

v

v ′

ω

ω r

ωr

α

α′

α

5

2

3

4

1

v

x

wFT

FNX

V∞i

V∞i

V∞i

V ∞l∞

θθθ

θθ

Blade elementflight path

r

r

J + 1J

Figure 1: Layouts of the flow in various Double-Multiple Stream-Tubes models [2, 4].

CARDAA and CARDAAV. In the code CARDAAX, the deflection of the flow due to the

continuity principle is computed. Still, the directions of the speed vectors were assumed to

be parallel to the free-wind, thus not parallel to the flow lines. This incompatibility is

corrected in the modified model.

4. The modified model includes a possible inclination of each blade in an arbitrary direction.

5. The modified model includes a possible pitch angle of the blades.

Regarding points 4 and 5 above: In VAWTs that are in use today, some have zero inclination

and pitch angles, like the H-shaped machines, and some with nonzero angles, as the “eggbeater”

shaped rotor (Figure 2), with inclination in the radial direction. But this is not the only possible

shape of inclination. For example, we find rotors with helical-shaped blades, like the rotor of

‘Turby’ (Figure 3). In this configuration the blades are inclined edgewise in tangential direction. In

some innovative concepts the blades may be inclined in any direction, for example [11]. Also a

pitch angle may be caused due to the aerodynamic torsion moment on flexible blades. The

modified model was established to take into account pitch and inclination angles in any direction.


Figure 2: Darrieus machine of “Eggbeater” structure.

Figure 3: VAWT with helix blades (from “Turby” manual).

2. AXES SYSTEMS, DEFINITIONS AND ASSUMPTIONSThe rotor is divided into horizontal ‘slices’. The flow regime in each slice may differ from the

others due to variations in the free-wind speeds or in geometry. Still, the flow equations for

each slice assume that the flow is horizontal, namely possible vertical flows are ignored. Thus

the model ignores any interaction between the slices, or effects of the upper and bottom

edges of the rotor. Still, vertical forces on the blades are possible, due to inclination. Each slice

is further divided into Flow-Strips (FSs), as are depicted in Figure 4. (The term ‘Flow-Strip’ is

used in this work for the usual term ‘streamtube’, which was derived from models of HAWT).

A global Cartesian system X,Y is attached to each slice, as depicted in Figure 4.

A schematic view of the course of a FS is depicted in Figure 5. The centerline of each FS

intersects the rotor at the entrance point (point 1) and the exit point (point 2). Each

intersection point is characterized by its local parameters: , U0, P0, Pu, Pd, U, UW, PW. The course

of a FS in the vicinity of an intersection point is outlined as an arc of constant radius, and as a

straight line elsewhere. But the computational process does not include a direct computation

of the course. Instead, control-volume equations relate the input and output of each

intersection point, regardless of the course. The speeds as are projected on the local polar

systems (axes R, T, Z) attached to each intersection point are depicted in Figure 6. The

projections of F on the FSF (Flow Strip Force) system are FXX on axis XX and FYY on axis YY,

as are depicted in Figure 7. Axis XX is defined as coincide with F, Thus: FXX F, FYY 0. The

orientation of the FSF system is defined as rotation by an angel f relative to the polar system,

about their common Z axis. The projections of the FS speeds on the FSF system, the rotor

speed R (in reverse direction) and the projections UXR, UYR of the speeds on axes XR,YR of

the local slice system are depicted in Figure 8.

After calculating the entrance point, whose upwind speed is known and 1 is arbitrary, the

next step is to calculate the exit point. This calculation requires the local upwind speeds U0XX2,


Exit pointEdge 2

Ufs Y

Extreme FS

Edge 1

Upwind

Downwind

TheCentral

FS

∆ 2θ

∆ 2pθ

∆ 1θ

1θ

2θ

2pθ

X

Entrance point

+

+

+

+

+

++

++

++++++++

+

Figure 4: The FSs crossing the rotor.

U0YY2 and 2. The magnitude of the speed is assumed to be unchanged in the course inside the

rotor, since no force is applied in the direction of the flow, thus:

U02 UW1

The direction of U02 and 2 are calculated according to the assumed factors that determine

the trajectory of a FS. Two optional assumptions are described herein.


U2

F2

F1

UW 2 , PW 2

U02 , P02

UW1 , PW1

Ufs , Pfs

U01 , P01

P2d

P1d

P1u

P2u

U1

Figure 5: Speeds, pressures and forces along the centerline of a FS.

UT1,T1

UT 2,T2

UR 2,R2

UR1,R1

U1

U2

u1

u2

Z

+ Z

ΩR

ΩR

1θ

2θ

R

Figure 6: The polar system and blade speed.

One assumption is that the side forces that are applied on a FS in the course from the wake

of the entrance point to the upwind of the exit point are negligible, thus the direction of the

speed is also unchanged. This assumption is depicted in Figure 8, as the wake speed of the

entrance point is projected simultaneously on the FSF systems of both intersection points. At

the entrance point it is the local wake speeds UWX1, UWY1 and at the exit point it is the local

upwind speeds U0XX2, U0YY2. The trajectory for this assumption is a straight line. This

assumption seems reasonable for the central FS (1 0), whose two neighboring FS are close

to be symmetrical, thus their resultant force on the central FS is close to zero. The central FS

is the first to be computed, and its course is depicted in Figure 9 with the real and

approximated 2.

The second assumption that is applied for the other FSs is that 2 relative to 1 is

determined by the continuity principle. After the central FS (including its 2 and 2) is

established, the other FSs are calculated sequentially from the center to both edges. The

calculation of each FS is relied on the previously calculated FS, by (see Figure 4):

(2–1)

An iterative process is required here because 2 is needed in advance for calculating 2.

The initial guess is:

Then 2 is calculated by (2-1), the parameters of the exit point are calculated (including

2) and so on. Generally the process converges. Runaway may occur when approaching the

∆ ∆θ θ2 2 P

2 2 2 2 2 P P ( )/


T1

T2

R2

FYY1 = 0

FYY2 = 0

Fxx1

Fxx2

XX1

XX2

YY1

YY2

f1

f2

R1

Figure 7: The FSF axes and forces.

edge, when 2 becomes too large to be physically valid. Eventually sin 2/2 becomes larger

then 1 and 2 becomes complex. In fact when 2 becomes too large, the calculation is

stopped and the edge segment is calculated uniquely.

It is also assumed:

1. The trajectory from the wake of the entrance point to the upwind of the exit point may be

approximated by a straight line.

2. The size of the zones of influence of the intersection points is very small compared to the

size of the rotor, so the trajectory is approximately parallel to the line between the

entrance point and the exit point.

By these assumptions, the angle of the upwind speed of the exit point is approximated as

approx in Figure 10.


Figure 8: The velocities projections on the force axes along the FS.

The blade system (Figure 11) is attached to the cross-section of the blade. The position of

the blade system is defined relatively to the polar system by 3 angles (Figures 12, 13, 14):

1. The tilt angle (rotation about axis Z) that defines the plane of inclination, or the direction

of axis of inclination I.

2. The inclination angle .

3. The pitch angle .

As an intermediate system between the polar system to the blade system we define the

inclination system (axes CI, NI, ZI in Figure 12) that is inclined according to angles and ,

with zero pitch.


Central FS

µ2θ θ2approx

+

Figure 9: The assumed trajectory and approximated 2 for the central FS.

Central FS

2θ

θ− 1

µ

µapprox

+

Figure 10: The assumud trajectory and approximated for the general FS.

Herein the dimensionless parameters are detailed.

The induced speed normalized relative to the free-wind speed:

(2–2)

The TSR is the dimensionless speed of the blades:

(2–3)

The aerodynamic forces of the blade element are introduced in the dimensionless form

(also known as the aerodynamic coefficients), by normalizing relative to the stagnation

pressure and relative to the element area ch/cos:

R

U fs

G U U fs


C1

C2Zb1

Zb2

N1

N2

Figure 11: The blade system.

T

R

NI

CI

I

Polar blade

Inclination plane

Inclined blade

β

φ

Figure 12: The axes of the blade system in a vertical blade and inclined blades.


T1

T2

X

R1

R2

I

2γ

1γ

I

2θ1θ

1φ

2φ

1β

2β

Figure 13: The inclination angles relative to the polar system and relative to the rotor system.

C2

CI1

N2

NI1

NI2

C1

1δ

2δ

N1

CI2

Figure 14: The pitch angles at the intersection points.

(2–4)

where the effective-wind speed UEF is the wind that is experienced by the blade profile and

determines its aerodynamic forces. UEF is composed from the speed of the FS and from the

blade speed R (in reverse direction), projected on the plane of the blade section. For

example the aerodynamic coefficients of the airfoil:

(2–5)

(2–6)

(2–7)

The dimensionless forces in the FSF system, polar system and blade system are:

Where FBXX, FBYY, FBR, FbT, FBZ, FBN, FBC, FBZb are the aerodynamic forces on the segment of

a blade that is confined in a slice, projected on axes XX, YY, R, T, Z, N, C, Zb, of the force system,

polar system and blade system. FBYY 0 by definition. FBZb 0, since the aerodynamic force

in the span-wise direction is neglected. These forces are projected from the lift and drag forces,

as are depicted in Figures 15 and 16.

The orientation of the local axes systems that are attached to the intersection points (for

example, the blade system as is defined in Figure 11) are different for the entrance and exit

points. As a result, all the equations for the exit point are identical to the equations of the

entrance point, with no need to change signs. But eventually, the data are introduced in the

common global X, Y system (or the related polar system), and the sign of some parameters

should be inverted.

(2–8)

CF

U c h G

C

CF

XXBXX

fs EF

YY

RBR

12

2 2

12

1

0

⋅ /cos

UU c h G

CF

U c h G

fs EF

TBT

fs

2 2

12

2

1

1

⋅

⋅

/cos

/cos

EEF

ZBZ

fs EF

NBN

fs

CF

U c h G

CF

U

2

12

2 2

12

1

⋅ /cos

22 2

12

2 2

1

1

⋅

⋅

c h G

CF

U c h G

EF

CBC

fs EF

/cos

/cos

CCZb 0

CM

U c h GM

fs EF

=⋅1

2

2 2 2

1

ρ β∆ /cos

CL

U c h GL

fs EF

12

2 2

1

⋅ /cos

CD

U c h GD

fs FE

12

2 2

1

⋅ /cos

CForce

U c h

Force

U c hEF fs

12

2 12

2 ⋅ ⋅ /cos /coss

12GEF



UC2

UC1

UN1

UN2

N1,CN1

−CC2 N2,CN2

UEF1

UEF2

C2

C1

α1

α2

CD1

CD2

−CC1

CL1

CL2

Figure 15: The speeds and the forces in the blade system.

f1

f2

FBXX1,CXX1

FBXX2,CXX2

−CT1

−CT2

CYY1

CYY2

CR1

CR2

Figure 16: the projections of the aerodynamic forces on the polar system and on the FSF system.

3. THE CROSS-SECTIONS OF FS AT INTERSECTION POINTFigure 17 depicts the FS while crossing the entrance point. (There is no difference at the exit

point.). The width and area of the FS in cross-section normal to U :

(3–1)

The area A is used for the volumetric flow AU. Another area of interest is AP, related to the

pressure drop at the intersection point. The pressure drop in the intersection point (Figure 5):

(3–2)

The force F that the FS exerts on the rotor is attributed to this P (as well as to the

aerodynamic forces on the blades). Thus:

(3–3)

The area AP is normal to axis XX. Its width and area are:

(3–4)

Substituting (3–4) into (3–3), the pressure drop:

(3–5)PF

Au fXX cos( )

w w u f

A A u f

P

P

/cos

/cos

( )( )

PF

A

F

AP

XX

P

P P Pu d

w R u

A R u h

22

22

sin cos

sin cos


wP1

w1

U1

u1

FXX1

f1

R

1θ

θ∆ 1

Figure 17: The cross of a FS through intersection point.

When u approaches ± 90° and f approaches zero, w, A, AP become small and P may

become too large to be physically reasonable. This condition is faced at the edges, when the

flow becomes tangent to the rotor circle, or when the flow speed is very small compared to the

rotor speed. In fact the general model becomes invalid because it assumes that the whole

section of the FS is crossing the rotor. According to (3–4) the width of the FS becomes very

small at the edges. But the actual FS diverges from rotor circle, and in reality is influenced by

the surrounding free-stream. The real flow regime is deferent from the usual FS and should be

treated differently. For example an approximation that may be taken at the edges:

(3–6)

4. RELATIONSHIPS BETWEEN FORCESHerein the aerodynamic forces are transformed from the blade system to the FSF system. The

spanwise aerodynamic force (CZb) is neglected. The forces in the blade system (Figure 15):

(4–1)

The transform from the blade system to the inclined system (Figure 14):

(4–2)

The transform from inclined system to the polar system (Figure 12):

(4–3)

(4–4)

where:

(4–5)

Combining (4–1), (4–2), (4–3), (4–4):

(4–6)

(4–7)

By Figure 16:

(4–8)

(4–9)

where from (2–8):

fC

CT

R

tan 1

C C CXX R T 2 2

CZ

sin sin cos

cos sin

sin

( ) ( )(( ) ( )

cos

C

CL

D

C INCRT

( ) ( )( ) (

cos sin

sin cos

))

C

CL

D

INC I ( )

12

coscos sin cos

sin cos

sin2

C C CZ C I NI sin sin cos ⋅ ⋅( )

CC

CINC

C

CRTR

T

NI

CI

C

C

C

CNI

CI

N

C

cos sin

sin cos

CC

C

C

CNC

N

C

L

D

−

cos sin

sin cos

G 1 1sgn( )


(4–10)

where FBXX is the force on a blade element within a slice, located at the centerline of the FS.

The average number of blades that are within the FS:

where is in radians. The force that the FS exert on the rotor at an intersection point is equal

to the sum of the forces on the blades that are within the FS:

(4–11)

With the common definition for solidity:

(4–12)

(4–13)

The force figure is defined as a dimensionless form of FXX that is normalized by the free-

wind speed:

Substituting (3–1) and (4–13), for infinitesimal (sin/2→/2):

(4–14)

Substituting in (3–5), the pressure drop:

(4–15)

5. RELATIONSHIPS BETWEEN SPEEDSHerein the effective-wind speed is transformed from the polar system to the blade system.

The vertical component of the speed UZ is neglected. Thus the components of the effective-

wind in the polar system are UR and UT R (Figures 6 and 8). The transform of the effective-

wind speed from the polar system to the inclined system (Figure 12):

(5–1)

The transform from the inclined-blade system to the blade system (like (4–2) but inverted):

(5–2)U

U

U

UN

C

NI

CI

cos sin

sin cos

1

U

UINC

U

U RNI

CI

R

T

P U F u ffs f 2 cos( )

FF

AU

SG C

uf

XX

fs

EF XX 2

2

2 cos cos

FF

AUf

XX

fs

2

F SR h U G CXX fs EF XX

22 2 /cos

SNc

R

2

F N F N Uc h

G CXX BXX fs EF XX

2 42 2

cos

N

2

CF

U c h GXX

BXX

fs FE

12

2 2

1

⋅ /cos


The speeds in the directions of lift and drag (like (4–1) but inverse):

(5–3)

Inverting the transform:

(5–4)

By the definition, UEF is collinear with the drag force:

(5–5)

Also:

(5–6)

Substituting (5–5) and (5–6) in (5–4):

(5–7)

Transformation of speeds from the polar system to the FSF system (Figure 7):

(5–8)

Substituting (5–7) in (5–8):

(5–9)

where:

(5–10)

GYY is determined by the local upwind speed. By the conversation of the momentum, the

speed of the FS in YY axis, which is perpendicular to the force, is constant, namely:

(5–11)

Substituting (5–11) in (5–9):

(5–12)GG f

HEF

YY

Y

0 cos

G G GYY YY WYY0

HH

H

f f

f fINCXY

X

Y

cos sin

sin cos

( )( )

1 sin

cos

G

G

H

HG

f

f

XX

YY

X

Y

EF

sin

cos

G

G

f f

f f

G

GXX

YY

R

T

cos sin

sin cos

GG

GINCRT

R

T

( )( )

1 sin

cos

GEF

0

U

U R

G

GUR

T

R

T

f

0

ss

U

U UG UL

D EF

EF fs

0 0

1

U

U RINCR

T

( ) ( )1 cos sin

sin

( ) ( )

cos

U

UL

D

U

U

U

UL

D

N

C

cos sin

sin cos

1

−

cos sin

sin cos

cos sin

sin

1

ccos

1

INCU

U RR

T


(5–13)

Finally, the speeds in the polar system are calculated by the inverse of (5–8), and with

(5–11) and (5–13):

(5–14)

The direction and magnitude of the induced speed (figure 6):

(5–15)

(5–16)

For the computation of an intersection point, the local upwind speeds in the FSF system are

required. These speeds are transformed from the speeds in the slice system, as is depicted in

Figure 8, for both the entrance and the exit points:

Or:

(5–20)

The terms for the upwind speeds, given in the axes of the slice system, are different for the

entrance and exit points, as is depicted in Figure 8. For the entrance point:

(5–21)

In the exit point, GXYR2 depends on the assumption that was used for the trajectory of the FS

between the entrance and exit points. For the central FS the applied assumption is that the

speed is unchanged inside the rotor. According to Figures 8 and 9:

(5–22)

(5–23)

(5–24) 2 2 2 approx

tan 1 2

2

G

GYR

XR

GG

G

f fXYR

XR

YR

22

2

1 1 1 1

( ) ( )cos sin

si

nn cos 1 1 1 1

1

1 f f

G

GWXX

WYY( ) ( )

( ) ( )( ) ( )

cos sin

sin cos

f f

f f

G1 1

1 1

12 ccos u f G

G

XX

YY

1 1 0 1

0 1

( )

GG

GXYR

XR

YR

11

1

1

0

Gf f

f fXY0

cos sin

sin cos

( ) ( )( ) ( )

GXYR

G

G

f f

f

XX

YY

0

0

( ) ( )( )

cos sin

sin cos

f

G

GXR

YR( )

G G G G GR T XX YY 2 2 2 2

uG

GT

R

tan 1

Gf f

f f

G

GRT

XX

YY

cos sin

sin cos

0

G H G fXX X EF sin


where for the central FS: 1 0, GWYY1 is according to (5–11), and:

(5–25)

where equation (5–25) is developed in section 6.

For the general FS, the assumption is according to figure 10:

(5–26)

where 2 is guessed in advance in the iterative process of the computation.

The projections of the upwind speed on the local slice axes:

(5–27)

where G02 is the magnitude of the exit upwind speed. The speed magnitude is assumed to be

unchanged in the internal course, thus with (5–11) and (5–25):

(5–28)

6. THE PROCESS EQUATIONS6.1. The continuity equationThe continuity equation between the entrance point and the exit point:

(6–1)

Substituting (3–1):

(6–2)

6.2. The momentum equationThe momentum equation is introduced for the closed control volume that includes the FS from

upwind to the wake of an intersection point. It is assumed that in the vicinity of the intersection

point, the forces that are applied on the FS by the neighboring FSs are negligible compared to

the forces that are applied by the rotor. According to this assumption, the force that is applied

on the FS is FXX. The momentum equation in the FSF system (with the velocities as depicted

in Figure 8):

(6–3)

(6–4)

( (6–4) is a relationship that was abandoned in the traditional model).

F G G GYY YY YY WYY 0 0→

F A U U UXX XX WXX 0( )

21 1

2

1

2

122

sincos

cossin

G

G

u

u

AG A G1 1 2 2

G G G G

G u f G

W WXX WYY

XX

02 1 12

12

1 1 1 0 122

= ( )( cos ))2

0 12G YY

G GXYR 2 02

cos

sin

approx

tansin sin

cos cos1 2 1

2 1

G G u f GWXX XX1 1 1 1 0 12 cos( )


Combining (6–3) and (4–14), the dimensionless wake speed:

(6–7)

6.3. The energy equationBernoulli’s equation may be derived from the first law of thermodynamic, applied to two

control volumes along the FS, one from upwind to the rotor surface, and the second from the

rotor surface to the wake. Following the trace of the FS (Figure 5):

(6–8)

Combining with (3–2) and assuming that the pressure inside the rotor (far enough from the

rotor circle) is the atmospheric pressure:

(6–9)

(6–8) is reduced to:

(6–10)

Equation (6–9) stems from the assumption that the pressure gradient in direction

perpendicular to the FSs trajectories is small. This is true even at the edges since the angle of

attack there is nearly zero. Thus the atmospheric pressure penetrates through the edges to

the space inside the rotor. Also in most of the rotors shape, the atmospheric pressure probably

penetrates through the upper or bottom sides of the rotor.

Pythagoras theorem:

(6–11)

Substituting (6–11) into (6–10), with (6–4):

(6–12)

Substituting P from (4–15):

(6–13)

6.4. The combined momentum-energy equationEquating Ff from (6–7) and (6–13):

(6–14)G G u f GWXX XX 2 0cos( )

G G F u fWXX XX f2

02 2 cos( )

U UP

XX WXX02 2

2

/

U U U

U U U

XX YY

W WXX WYY

02

02

02

2 2 2

ρ= ∆

U UP

–/ 2W0

2 2

P P PW at0

P U P U

P U P U

u

d W W

012 0

2 12

2

12

2 12

2

G G F GWXX XX f 0


If one approximates that the force is parallel to the speed:

one gets from (6–14):

(6–15)

(6–15) is identical to the momentum-energy (or momentum-Bernoulli) relationship of the disk

theory, which is commonly applied in the analysis of wind turbine.

6.5. The wake equationThe values of GWXX that are resulted from either (6–7), (6–13) or (6–14) should be equal. By

equating GWXX from either two of these equations:

(6–16)

Substituting (4–14) one get the wake function:

(6–17)

W is termed the wake function (WF), and (6–17) as the wake equation. The wake equation

may also be interpreted as momentum equation, as Ff in (6–16) represents the force that is

applied on the FS and the term 2G [G cos(u f)G0XX] represents the change of the momentum.

Indeed, with the traditional assumption that u f 0, and with (6–15), (6–16) becomes

identical to (6–7).

6.6. The power output at an intersection pointFor a point of intersection of the FS with the rotor, multiplying (6–10) by the volumetric flow

AU :

(6–19)

pdet is the rate of energy that is detracted from the FS. The output mechanical power for this

segment of the rotor:

(6–20)

Where FT was defined as the tangential force that the FS exert on the rotor, with the

positive direction counter to the speed of the blades. Substituting FT from (4–16) and R from

(2–3):

(6–21)

We can state that always:

(6–22)

This means that not all the kinetic energy that is detracted from of the FS is converted to a

mechanical work. The process is not adiabatic and some energy is eventually converted to

heat. For example in a standing rotor, Pdet is relatively small, but PM is zero, thus Pdet is entirely

converted to heat.

p pM det

p S R h U G CM fs EF T

0

3 2

360/cos

p F RM T ( )( )

P AU AU U U PW⋅( ) ( )( ) 12 0

2 2 det

W SG C u G G u f GEF XX XX 204 0 cos cos cos( )

F G G u f Gf XX 2 00cos( )

GG GXX WXX

0

2

u f 0


6.7. The efficiency of a sliceThe swept area of a slice is 2Rh. The free-stream power, namely the rate of flow of kinetic

energy that sweeps this area:

(6–23)

Unfortunately, not all this energy can be converted into useful power, namely to

mechanical work. The actual mechanical output power for the slice is:

(6–24)

where CP is the efficiency of the slice of the rotor, or the power coefficient. The practical CP is

even less than the theoretical Betz limit due mainly to the inequality (6–22). CP is calculated by

summation of the mechanical powers of all the intersection points. The specific efficiency is

defined for a single intersection point:

(6–25)

Substituting (6–21) and (6–23) in (6–25):

(6–26)

The total efficiency of the slice:

(6–27)

7. THE COMPUTATION PROCESSThe code SLICE was developed for computing a slice of the rotor according to the modified

model.

On the level of the intersection point, a helpful fact is that once a value of is guessed, all

the other parameters can be computed straightforwardly. Thus values of are scanned in a

region of physically valid solutions and the value that satisfies the wake equation is chosen.

The computation process for a guessed is the following: CL, CD are extracted from the airfoil

data, CR, CT, CZ, CXX, f are calculated according to section 4, GEF, GXX, GRT, G, u are calculated

according to section 5, W is calculated by (6–17) and checked when W() 0.

Firstly the central FS is computed. Then the other FSs are computed sequentially from the

center toward the edges. For each FS the entrance point is computed with upwind according

to (5–21). Then , 2, 2 and the other parameters are calculated in an iterative process.

8. COMPARISON OF THE RESULTS OF THE MODIFIED MODEL TO TRADITIONALMODELSIn this section the results of the SLICE code are compared with traditional models. The

comparison is introduced for a vertical rotor (no inclination), with two sets of solidity and TSR.

The first comparison is for S 0.073 and 4.6, which are identical to the experimental data

introduced in the next section. The second comparison is for S 0.14 and 4.0, values that

are closer to current operational VAWTs.

Cp

PcpP

M

fs

∑ ∑

0

360

cp S G CEF T 2 /cos

cpp

PM

fs

⋅0 360

P p C PM M P fs ∑

P U R hfs fs 12

3 2 ( )


The traditional models are established by inserting the simplifications that are commonly

applied in these models. Two levels of simplifications were described in section 1. In the first,

which is designated herein as ‘APP1’ (approximation No. 1), the direction of the induced speed

and the trajectory between the entrance and exit points are assumed to be parallel to the free-

wind. In the second, which is designated herein as ‘APP2’ (approximation No. 2), the

magnitude of the induced speed is also assumed to be equal to the free-wind. Herein the

formulas of the models are given.

The approximations of APP1:

The trajectory from the entrance point to the exit point is determined by:

(8–1)

The direction of the induced speed is determined by:

u (8–2)

The one-dimensional momentum equation is given by substituting (4–14) and (8–2) in

(6–7), and 0:

(8–3)

Here CX replaces CXX to indicate a force in X direction, the wake speed GWX replaces

GWXX, and the upwind speed G0X replaces G0XX. CX is transformed from the polar system

(Figure 16):

(8–4)

The upwind speed for the entrance point:

(8–5)

The upwind speed for the exit point:

(8–6)

The induced speed is derived from the momentum-Bernoulli relationship, equation (6–15):

(8–7)

In APP1 there is no straightforward procedure to calculate and an iterative process is

required. The possible steps of the iterative process may be as follows. The first step is

calculation of and GEF with the assumptions of APP2 herein, by (8–8) and (8–9). In the next

steps CX is calculated by (8–4), GWX is calculated by (8–3), G is calculated by (8–7), and GEF

are then re-calculated, and so on. In calculating the new and GEF the procedure of (8–8) and

(8–9) of APP2 can be applied after G0X is inverted by G.

GG GX WX

0

2

G GX WX0 2 1

G X0 1 1

C C CX R T cos sin

G GSG C

GWX X

EF X 0

2

2 cos

2 1


The approximations of APP2:

All the above approximations of APP1 are also inserted in APP2, with the additional

relationships:

(8–8)

(8–9)

where uw and Guw are the solutions of a procedure that assumes that U is also equal to the

upwind speed. For the entrance point:

(8–10)

For the exit point:

(8–11)

(At the exit point GWX2 is insignificant.).

APP2 is simpler than APP1 because as and GEF are readily calculated (no iterative process

is required), the forces at the intersection points can be evaluated with no need to apply the

momentum or Bernoulli’s equations. The momentum equation is required only for calculating

the wake speed of the entrance point, as this is also the upwind speed for the exit point.

Results of the modified model and the two unmodified models are depicted in Figures 19–26.

The results versus the azimuth angle are introduced according to the conventions in [6], as are

defined in figure 18, in order to conveniently compare between the models and the

experimental results in the next section. The angle of attack (Figure 19) is introduced as is

defined in figure 15, which is reversal to [5] in the downwind side.

One sees that the differences between APP2 and SLICE are significantly larger than

between APP1 and SLICE, especially when regarding CT (which determines the output

power). The differences between APP1 and SLICE are negligible in the down-wind side but

larger in the up-wind side. Regarding GEF significant differences between the unmodified

G GSG C

X WXEF X

0 2 1

2

12

cos

G

G

X0 1

1

1

1

G GEF EFuw

uw


270°/−90°

90°

180°0°

CRCT

Ufs

θ

Figure 18: The conventions for the azimuth angle and the forces [6].


−100 0 100 200T (deg) azimuth angle

Rear half of the rotor(down-wind side)

Front half of the rotor(down-wind side)

A (

deg)

(an

gle

of a

ttack

)12

4

10

8

6

2

0

SLICEAPP1APP2

Figure 19: A comparison of for the modified and simplified models, S 0.073, 4.6.

Ct (

blad

e ta

ngen

tial f

orce

)

0.25

0.2

0.15

0.1

0.05

0

T (deg) azimuth angel−100 −50 0 50 100 150 200 250

SLICEAPP1APP2

Figure 20: A comparison of CT for the modified and simplified models, S 0.073, 4.6.


T (deg) azimuth angel

−100 −50−1

−0.5

0

0

0.5

1

50 100 150 200 250

SLICEAPP1APP2

Cr

(bla

de n

orm

al c

oeffi

cien

t)

Figure 21: A comparison of CR for the modified and simplified models, S 0.073, 4.6.


−100 −50 0 50 100 150 200 250

Gef

(ef

fect

ive

win

d)

5.5

5

4.5

4

SLICEAPP1APP2

Figure 22: A comparison of GEF for the modified and simplified models, S 0.073, 4.6.


T (deg) azimuth angel−100 0

15

10

5

0100 200 300

A (

deg)

(an

gle

of a

ttack

)SLICEAPP1APP2

Figure 23: A comparison of for the modified and simplified models, S 0.14, 4.0.

Ct (

blad

e ta

ngen

tial f

orce

)

0.25

0.2

0.15

0.1

0.05

0

T (deg) azimuth angel−100 −50 0 50 100 150 200 250

SLICEAPP1APP2

Figure 24: A comparison of CT for the modified and simplified models, S 0.14, 4.0.


SLICEAPP1APP2


Cr

(bla

de n

orm

al c

oeffi

cien

t)

−100−1

−0.5

−50 0

0

0.5

1

50 100 150 200 250

Figure 25: A comparison of CR for the modified and simplified models, S 0.14, 4.0.

SLICEAPP1APP2


Gef

(ef

fect

ive

win

d)

−100 −50 0 50 100 150 200 250

5

4.5

4

3.5

3

Figure 26: A comparison of GEF for the modified and simplified models, S 0.14, 4.0.

models and SLICE are near the edges, at T ≈ ± 90°. The curve of SLICE seems “ugly”

relatively to the unmodified, but is probably more realistic, since in the region of the edge and

ahead the flow speed is dropped and directed outward due to the widening of the FSs.

For S 0.14, 4.0 one can see that the prediction of APP2 is probably unrealistic,

especially in figure 24. The differences between APP1 and SLICE are also not negligible. The

difference between the models is greater in this case than in the previous case because as S is

larger, the changes that are induced on U (relative to the upwind) are greater.

If we assume that the modified code SLICE provides a realistic prediction, we conclude that

the simplified models provide a good prediction as long as S and/or are sufficiently small.

Under conditions of large S and large the deviations of the simplified models become

significant, especially when both the direction and the magnitude of the flow are approximated.

9. COMPARISON TO EXPERIMENT AND OTHER MODELSFor validating a computational model, some researchers compare the computed output

power (like introducing Cp versus TSR) against measured values. But this is a limited

validation since Cp is an integral performance property. A stronger validation is a comparison

of the aerodynamic forces on the blades as function of the azimuth angle. The difficulty to

implement such a comparison is the rarity of available experimental results.

Herein the experimental results from the SNL 17 m machine [5] are introduced against

SLICE code, GARDAV code and the 3-D viscous code [6]. In this experimental test, the

pressures were measured on a blade surface at the equator circle of the “eggbeater” shaped

Darrieus machine. The data were processed to introduce the normal CN and tangential CT

coefficients. Since the measurements were performed on the equator, the run of the SLICE

code was performed for the case of vertical blades, namely 0 and 0. (When computing

a slice that is not at the equator, one should substitute 0° and the local ). For this case of

vertical blades, CR and CT are identical to CN and CC , as are defined in (2-8).

The data of the turbine in the experiment:

Rotor speed: 38.7 rpm

Airfoil: NACA-0015

While comparing it should be remembered that the following error factors may be

important.

Model errors:In a practical run of the model one problem is the inconsistency between the various sources

of airfoil data, and the dependency on variety of conditions, like the surface roughness. In this

comparison, Eppler data [7] were used, which are regarded fair till stall, and problematic at

stall and deep stall.

Another issue is the effect of dynamic stall. The current SLICE code does not include the

dynamic stall effect. The phenomena is related to the rate of change of α, and its strength

depends on the reduced frequency, that is defined as:

R m

c

N

SNc

R

e

8 36

0 612

2

20 073

1 36 6

.

.

.

Re .


(9–1)

By approximating UEF ≈ R:

(9–2)

At high the approximation is fair, and k is nearly uniform. At low , k varies strongly with

the azimuth angle. Still (9–2) is useful to obtain the order of magnitude of k, which is sufficient

for our purpose. From (9–2) and the experiment data we find that:

k 0.037 .

This value is too high to be neglected. (This can be concluded for example from

Theodorsen’s functions [10], and so was treated in [6].) The dynamic pitch may be neglected,

however, when is bellow stall, or slightly exceeding stall. Thus it has been chosen to compare

the experiments with the higher , which were 3.7 and 4.6.

Experimental errors:The main error results from the method of measuring the force of the blade. The force is

derived from pressure transducers at the blade surface, thus the skin friction is ignored.

Ignoring the skin friction causes the derived CT to be larger (toward the positive side) than the

real value, especially at ≈ 90°. Its effect on CN may be negligible.

The comparisons of the models and the experiment results are given in Figures 27–34.

kc

R

S

N

2

kc

U EF

2


14

12

10

8

6

4

2

0−100

A (

deg)

(an

gle

of a

ttack

)

T (deg) azimuth angle−50 0 50 100 150 200 250

Experim

SLICE

Figure 27: A comparison of the angle of attack from SLICE codes to the experimental results [5]

for 3.7.


1

0.5

0

−0.5

−1

−100 −50 0 50T (deg) azimuth angle

Cn

(bla

de n

orm

al c

oeffi

cien

t)

100 150 200 250

ExperimSLICE

Figure 28: A comparison of the normal coefficient from SLICE codes to the experimental results [5]

for 3.7.

0.3

0.25

0.2

0.15

0.1

0.05

0

−100 −50 0 50T (deg) azimuth angle

Ct (

blad

e ta

ngen

tial f

orce

)

100 150 200 250

ExperimSLICE

Figure 29: A comparison of the tangential coefficient from SLICE codes to the experimental results [5]

for 3.7.


0.5

0.4

0.3

0.2

0.1

0

−0.1−90 −60 −30 0 30 60

Downwind zone Upwind zone

90 120 150 180 210 240 270

Experimental data

3-D viscous modelGARDAAV

Sandia − 17 m

Profil NACA 001538.7 rpm

Azimuth angle

Tan

gent

ial f

orce

coe

ffici

ent

Figure 30: A comparison [6] of the tangential coefficient from 3-D viscous and GARDAAV codes to the

experimental results [5] for 3.7.

Experim

−100 −50 0 50 100 150 200 2500

2

4

6

8

10

T (deg) azimuth angle

A (

deg)

(an

gle

of a

ttack

)

SLICE

Figure 31: A comparison of the angle of attack from SLICE codes to the experimental results [5]

for 4.6.

The stall angle of attack according to Eppler: stall 12°. For 3.7, most of the angles are

below this value, except for the region T 165° 220°. For 4.6 there is no stall.

The figures show that and CN as computed by SLICE code are fairly close to the

experimental results. The experimental CT is larger than the computational, as was expected.

This is also true for the CARDAAV code, although with different deviations between the


SLICE

Experim0.25

0.2

0.15

0.1

0.05

0

−100 −50 0 50 100 150 200 250T (deg) azimuth angle

Ct (

blad

e ta

ngen

tial f

orce

)

Figure 32: A comparison of the tangential coefficient from SLICE codes to the experimental results [5]

for 4.6.

1

0.8

0.6

0.4

0.2

0

−0.8−100 −50 0 50 100 150 200 250

−0.6

−0.4

−0.2

Cn

(bla

de n

orm

al c

oeffi

cien

t)


SLICEExperim

Figure 33: A comparison of the normal the coefficient from SLICE codes to the experimental results [5]

for 4.6.

computational and experimental data. By comparing CN (CR) of Figure 34 to Figure 25 one

sees that CARDAAV is closer to APP1 while the 3-D viscous code is closer to SLICE.

It should be noted that such a single comparison may ensure that there is no essential

mistake in a model, but is of no help as a tool for assessing the accuracy of a model, due to the

errors on both the experiment instrumentation and the run of the model, as were detailed at

the beginning of this section.

Also it should be noted that CN and CT do not completely determine the forces on the blade,

which also depend on E 2EF . Thus the comparison that has been done in this section is only

partial. Full comparison of forces may not be performed currently due to lack of data.

10. CONCLUSIONSThe modified model is supposed to provide a more accurate description of the interaction of

the wind with the rotor of VAWT, since it takes into account features that are ignored in the

traditional models. The model simulates the flow across the rotor as a 2-dimensional process,

where the magnitude and direction of the flow are calculated, rather than approximating it to

be parallel to the free-wind. In addition the model accepts any angular orientation of the blades.

Although there is a similarity between the modified model and the traditional models, a

quantitative difference is observed and becomes significant for larger solidity and TSR. The

comparison of the modified model to experimental data and other models validates the

concretness of the modified model, but its accuracy cannot be verified by the current

comparison.


3-D viscous modelGARDAAVExperimental data

Upwind zone

Nor

mal

forc

e co

effic

ient

Downwind zone

Azimuth angel

−90 −60 −30 0 30 60 90 120 150 180 210 240 270

−0.8

1.2

0.8

0.4

0

−0.4

−1.2

Sandia - 17 m

38.7 rpm

Profil NACA 0015

Figure 34: A comparison [6] of the tangential coefficient from 3-D viscous and GARDAAV codes to the

experimental results [5] for 4.6.

For applying the modified model in conditions of large and .

, dynamic stall should be

included in the code.

REFERENCES[1] Islam, M., Ting D.S.K. and Fartaj A., Aerodynamic Models for Darrieus-type Straight-

Bladed Vertical Axis Wind Turbines, Renewable and Sustainable Energy Reviews, 2008,

12, 1087–1109.

[2] Paraschivoiu, I., Double-Multiple Stream-Tube model for studying Vertical Axis Wind

Turbine, Journal of propulsion and power, 1988, 4(4), 370–377.

[3] Paraschivoiu, I., Wind Turbine Design with Emphasis on Darrieus Concept, Montreal

Polytechnic International Press, Montreal, 2002.

[4] Brahimi, M. T., Allet, A. and Paraschivoiu I, Aerodynamic Analysis Models for Vertical-

Axis Wind Turbines, International Journal of Rotating Machinery, 1995, 2(1), 15–21.

[5] Akins, R. E., Measurements of Surface Pressure on an Operating Vertical-Axis Wind

Turbine, Sandia National Laboratories (SNL), 1989, SAND89-7051.

[6] Allet, A. and Paraschivoiu, I., Viscous Flow and Dynamic Stall Effects on Vertical-Axis

Wind Turbine, International Journal of Rotating Machinery, 1995, 2(1), 1–14.

[7] Sheldahl, R. E. and Klimas P. C., Aerodynamic Characteristic of Seven Symmetrical

Airfoils Sections Through 180-degree Angle of Attack for use in Aerodynamic Analysis

of Vertical Axis Wind Turbines, Sandia National Laboratories (SNL), 1981, Report

SAND80-2114.

[8] Templin, R. J., Aerodynamic Performance Theory for the NRC Vertical-Axis Wind

Turbine, National Research Council of Canada, June 1974, Report LTR-LA-160.

[9] Strickland, J. H., The Darrieus Turbine: A Performance Prediction Model Using Multiple

Streamtubes, Sandia National Laboratories (SNL), October 1975, Report SAND80-0431.

[10] Gormont, R. E., A Mathematical Model of Unsteady Aerodynamics and Radial Flow for

Application to Helicopter Rotors, EUSTIS Directorate, U.S. Army Air Mobility Research

and Development Laboratory and Boeing-Vertol Company, May 1973, USAAMRDL

technical report 72–67.

[11] US patent 4,165,468.


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A Modified Streamtube Model for Vertical Axis Wind Turbines MULTI-SCIENCE PUBLISHING COMPANY