Thrust

LiftNet Aerodynamic Force

Drag

Weight

P1V1Z1

P2V2Z2

P’1V’1Z’1

P’2V’2Z’2

P”1V”1Z”1

P”2V”2Z”2

A

B

Chapter 2

Physics behind horizontal axis and vertical axis turbines

2.1 Lift force

Fig. 2.1.1. Schematic diagram of a fluid flow around an airfoil with forces acting on it

(Lift Force - Wikipedia, the free encyclopedia).

The fluid flowing around an airfoil exerts an aerodynamic force on it. Lift is defined here

as the component of this force in the direction perpendicular to the oncoming flow whereas drag

force is the component along the flow direction as shown in the fig 2.1.1. The Bernoulli’s

equation describes the lift force acting on the airfoil.

At points A and B, above and below the airfoil the Bernoulli’s energy equation is given by

P1

ρ+ 1

2V 1

2+g Z1=P2

ρ+ 1

2V 2

2+g Z2

(2.1.1)

Contour

dz

U∞

P∞

pdx

pdyx

y

where Pρ

,12

V 2,∧gZ are the pressure head, velocity head and potential head of the system.

So at same potential head,

12(V 1

2−V 22)=

P2−P1

ρ

(2.1. 2)

Since V 1>V 2 , P2>P1.

So there is a net force P2–P1 acting at the bottom (B) of the airfoil causing the lift.

2.2 Blasius Theorem

Fig. 2.2.1. Forces acting on an element of a body (Kundu & Cohen, 2010).

In a general cylindrical body of arbitrary cross-section as shown in the fig 2.2.1, drag D

and lift L are the x and y components of the force exerted on it by the surrounding fluid. Only

normal pressures are acting in inviscid flow, the forces on a surface element dz are

dD=−pdy∧dL=pdx

(2.2.1)

Defining it as complex quantity ‘i'

dD−idL=−pdy−ipdx=−ip (dx−idy )=−ip dz¿

(2.2.2)

where conjugate dz¿=dx−idy

(2.2.3)

after integrating equation 2.2.2

∫0

D

dD−i∫0

L

dL=D−iL=−i∮c

❑

p dz¿

(2.2.4)

where c: counter-clockwise

The Bernoulli’s equation without the gravity (zero potential head)

P∞+ 12

ρ U∞2 =p+1

2ρ(u2+v2)

(2.2.5)

where u, v are the components of velocity in x and y direction.

p=P∞+12

ρU ∞2 −1

2ρ(u+ iv)(u−iv)

(2.2.6)

So,

D−iL=−i∮c

❑

[{P∞+ 12

ρ U∞2 }−1

2ρ (u+iv ) (u−iv )]dz¿=1

2ρi∮

c

❑

(u+iv ) (u−iv )¿ dz¿¿

(2.2.7)

since in a closed loop,

∮c

❑

{P∞+12

ρU ∞2 }dz¿=0

(2.2.8)

D−iL=12

ρi∮c

❑

√ (u2+v2) e iθ √(u2+v2 ) e−iθ ¿|dz|e−iθ¿

(2.2.9)

where (u+iv )=√ (u2+v2 ) e iθ ; (u−iv )=√(u2+v2 ) e−iθ ; dz¿=|dz|e−iθ

(2.2.10)

and dz=|dz|e iθ, so after rearranging,

D−iL=12

ρi∮c

❑

(u−iv)2 dz=¿ 12

ρi∮c

❑

( dωdz )

2

dz¿

(2.2.11)

where

dωdz

=u−iv

(2.2.12)

The above equation is the Blasius theorem and applies to every plane steady irrotational flow.

The theory holds true to any contour surrounding the body provided that there are no

singularities between the body and the contour chosen (Kundu & Cohen, 2010).

Contour C

U∞

P∞

x

y

dz dA

r

2.3 Kutta – Zhukhovsky lift theorem

Fig. 2.3.1. Domain of integration for the Kutta-Zhukhovsky theorem

(Kundu & Cohen, 2010).

From a large distance from the body the flow is considered to be laminar as shown in the

fig 2.3.1. So all singularities are located near the body at z = 0 (Kundu & Cohen, 2010).

The complex potential is considered in the form

ω=Uz+ m2 π

lnz+ iГ2 π

lnz+ µz+ ..

(2.3.1)

where

Uz is uniform flow potential and U is the scale variation of velocity in length scale; m

2 πlnz is

source and sink with m defining the intensity of the velocity in the radial direction at source or

sink; iГ2 π

lnz is clockwise vortex with circulation Г and µz

is doublet with µ=mεπ

as ε → 0 and

x = ± ε.

The mass efflux of the sources is absorbed by the sink as the body contour is closed. So m=0.

Hence,

dωdz

=U +0+ iГ2 πz

− µ

z2+. .

(2.3.2)

and

f ( z )=( dωdz )

2

=U 2−( Г2πz )

2

+2iUГ2 πz

+. .

(2.3.3)

To integrate f ( z ) dz around the contour 0 - 2π, the terms except the coefficient of 1/z becomes

zero.

So,

∮c

❑

f (z ) dz= iUГπ

∫0

2 πdzz

= iUГπ

∫0

2π

−idθ=−2UГ

(2.3.4)

since z=re-iθ

Hence,

D−iL=12

ρi∮c

❑

f (z ) dz=−i ρUГ

(2.3.5)

Since the drag force D is zero so the lift force L = ρUГ and independent of the contour or shape

of the body.

Blade flight path

Streamtube

(a) (b)V∞ V1∞ V2∞V3∞

V1w V2wV3wVw

V1aV2a V3a

Va

2.4 Potential flow models

Potential flow models that have been used for decades as the primary design tool for

vertical axis turbines can be categorized as momentum models and vortex models; a detailed

review of these methods is discussed by Paraschivoiu, (2002) and Nabavi, (2007).

2.5 Momentum models

Momentum models based on Glauert’s Actuator Disc Theory and Blade Element Theory

are that the total change in the axial momentum across the actuator disc equals the aerodynamic

forces exerted on the blades in the axial direction and is also equal to the pressure difference

across the disk (Laoulache). Bernoulli’s energy equation is then used in each stream tube to find

a relation between pressure and velocity in the wake. Since the momentum equation becomes

invalid at high tip speed ratios and high rotor solidities so these models are not functional in

these higher ranges (Paraschivoiu, 2002).

The main momentum models developed are the Single Streamtube model, the Multiple

Streamtube model and the Double-Multiple Streamtube model. The first and simplest Single

Streamtube model was first developed by Templin, (1974, June) for determining the performance

of a vertical axis turbine. This model assumes that the rotor is enclosed in a single streamtube

and the flow velocity within the streamtube is assumed to be uniform as shown in fig 2.5.1(a).

Although this model is elegant in its simplicity and but only predicts overall performance for

lightly loaded blades and is incapable of estimating the heavy loads on the blades with high

solidities and blade tip speed as it requires a more precise knowledge of the variations of flow

velocity across the rotor. To account for these large variations, Wilson et al. used a sinusoidal

method to predict the velocity across the width of turbine.

Fig. 2.5.1. (a) Single streamtube model (b) Multiple streamtube model (Alidadi, 2009, June).

V1∞

V2∞

V3∞

V1D

V2D

V3w

V1U

V2U

V3U

Upstream Downstream

Multiple Streamtube Model developed by Strickland, (1975, October) is an advanced model

where the streamtubes are aerodynamically independent as shown in fig 2.5.1(b). The

momentum balance with identical streamtube velocity is determined individually for each

streamtube. Although the model holds better results than the Single Streamtube Model, the

results are only valid for lightly loaded blades.

In further development, Double-Multiple Streamtube model for the vertical axis turbine shows

the differences between the upwind and downwind passes of each blade by dividing the each

multiple streamtube into two parts: upwind and downwind (Paraschivoiu, 1981, February). The

momentum balance is then determined separately for each half of each streamtube as shown in

the figure 2.5.2. Despite the fact, that this model resembles the calculated values with the

experimental results better as compared to the results of Multiple Streamtube model, this model

appears to have convergence problems, especially on the downstream side and at higher tip

speed ratios (Islam, 2008).

Fig. 2.5.2. Schematic of Double Multiple Streamtube Model

(Nabavi, 2007, and Alidadi, 2009, June).

2.5.1 Single Actuator Disc Analysis in Vertical Axis Turbine

In 1-D single actuator disc double streamtube model, there is an elemental resisting

torque dτ due to elemental drag force dD, the inflow factors a1 and a2 at radius r on either side of

the disc and h is the length at radius r as shown in fig 2.5.1.1. (Newman, 1983, December).

V(1-2a1)

dr

h r

drdr

RdD1

dD2

V

V

V(1-a1)

V(1-a2) V(1-2a1)

h0

Fig. 2.5.1.1. Actuator disc for vertical-axis turbine (Newman, 1983, December).

Since the turbine is rotating in a clockwise direction, so dD1 > dD2. The overall drag force is

given by

dD=d D1−d D2=dr h ρ V ( 1−a1) 2 V a1−drh ρ V (1−a2 ) 2V a2

¿2 dr h ρV 2[a1(1 – a1)– a2(1 – a2)]

(2.5.1.1)

Now the torque is given by

dτ=r dD=2 r dr h ρ V 2[a1(1– a1)– a2(1– a2)]

(2.5.1.2)

Assuming zero loss of energy due to blade section drag

Ω dτ=¿ (1– a1 ) Vd D1+(1 – a2 )Vd D2=2 dr h ρV 3[a1 (1 – a1 )2+a2(1 – a2)2]¿

(2.5.1.3)

From equations 2.5.1.2 and 2.5.1.3

x [ a1 (1– a1 ) – a2 ( 1 – a2 ) ]=[a1 (1– a1)2+a2(1 – a2)2]

(2.5.1.4)

Where x= r ΩV

therefore,

C p=∫0

R

Ωdτdr

dr

12

ρV 3∫0

R

2hdr

C p=2∫

0

λ

x [a1 (1 – a1) – a2 (1– a2 ) ]hdx

∫0

λ

hdx

(2.5.1.5)

where λ is tip speed ratio.

For Cp = + ve, a1 > a2 & Cp max. for each r ( or x )

dd a1

[a1 (1 – a1 ) – a2 (1 – a2 ) ]=0

¿ ,d a2

d a1

=(1−2 a1 )(1−2 a2 )

(2.5.1.6)

also from equation 2.5.1.4,

dd a1

[a1 (1 – a1 )2+a2(1 – a2)2]=0

¿ ,d a2

d a1(1−a2 ) (1−3 a2 )+( 1−a1 ) (1−3 a1 )=0

¿equation (2.5 .1 .6 ) ,(1−a1 ) (1−3 a1 )

(1−2 a1 )=n=

−(1−a2 ) (1−3 a2)(1−2 a2 )

(2.5.1.7)

since, a1 > a2 ; 1/3 ≤ a1≤1/2 and a2 ≤ 1/3

a1

a2

a

x

The solution of the quadratic of a2 is given by

a2=n+2

3−[

(n+2 )2

9–

(1+n )3

]1 /2

(2.5.1.8)

The values of n can be determined from the assumed values of a1 in the equation 2.5.1.7, thereby

determining the corresponding values of a2 from the equation 2.5.1.8. The relation of a1 and a2

with respect to x is plotted in the fig 2.5.1.2.

Cp is determined finally by integration of equation 2.5.1.5 numerically using Simpson rule. The

results depend on the shape of the blade outlines. Three different profiles are identified as

mentioned in the table 2.5.1.1.

Table 2.5.1.1. Turbine Silhouettes (Newman, 1983, December).

h ∫0

λ

hdx

Rectangular h0 h0λ

Fig. 2.5.1.2. Axial induction factor a as a function of x (Newman, 1983, December).

A – A1 AA1

V(1 – a1) V(1 – a2) V(1 – f2)p1 p2 p3 p4

A

p∞ p∞

V(1 – f1)

Fig. 2.5.2.1. Schematic diagram of double actuator disc (Newman, 1983, December).

Parabolic h0(1− xλ)1 /2

2/3 h0λ

Triangular h0(1− xλ) 1/2 h0λ

At very large tip speed ratio, the above theoretical curves are limited to 16/27. The vertical axis

turbine tends to this limit with a slower pace than the horizontal axis turbines with significant

less power output for small tip speed ratio.

2.5.2 Double/multiple Actuator Disc Analysis in Vertical Axis Turbine

Cp

λFig. 2.5.1.3: Comparison of power coefficients between experimental and the ideal Betz Limit (Newman, 1983, December).

In a single rotation of the blades in Darrieus turbine, the torque is greatest when the

blades are in upstream and downstream, so it’s quite logical to represent the turbine with a

double actuator disc (Newman, 1983, December). The one-dimensional analysis of a single disc

with maximum Cp = 16/27, is reformulated with two discs.

The area of each disc is considered as A whereas A1 is the area of the upstream disc as shown in

the figure 2.5.2.1. From continuity theorem,

V (1−a1 ) A1=V (1−a2 ) A

A=(1−a1 )(1−a2 )

A1

(2.5.2.1)

The flow through the inner annulus A1 and outer annulus A – A1 of the front disc is given by,

From Bernoulli’s equation,

p1+12

ρ {V (1−a1 )}2=p∞+12

ρV 2

p2+12

ρ {V (1−a1 )}2=p∞+12

ρ{V ( 1−f 1 )}2

So , p1−p2=ρ V 2 f 1(1− f 1

2 )(2.5.2.2)

Linear momentum equations ignoring side pressure is given by

( p1−p2) A−{ρA (V f 1)}V (1−a1 )=0

thereby , ( p1−p2 )=ρV 2 f 1 (1−a1 )

(2.5.2.3)

From equations 2.5.2.2 and 2.5.2.3,

ρ V 2 f 1(1− f 1

2 )=ρ V 2 f 1 (1−a1 )

Therefore, f 1=2 a1

(2.5.2.4)

which is same as single actuator disc theory.

For the inner flow at A1, the Bernoulli’s equation is given by,

p2+12

ρ {V (1−a1 )}2=p∞+12

ρ{V ( 1−f 1 )}2=p3+12

ρ {V (1−a2 ) }2

p∞+ 12

ρ {V (1−f 2 ) }2=p4+12

ρ {V (1−a2 ) }2

So , p3−p4=12

ρV 2( f ¿¿1−f 2)( f 1+f 2−2 )¿

(2.5.2.5)

Linear momentum equation is then given by

( p¿¿1−p2) A1+( p¿¿3−p4) A=ρ A1 V (1−a1 ) V f 2 ¿¿

From equations 2.5.2.1, 2.5.2.3, 2.5.2.4, and 2.5.2.5

ρ V 2 f 1 (1−a1) A1+12

ρ V 2( f ¿¿1−f 2) ( f 1+ f 2−2 ) A=ρ A1V (1−a1 ) V f 2 ¿

Since f 1≠ f 2 , f 1+ f 2=2a2 ,∨f 2=2(a2−a1)

(2.5.2.6)

The coefficient of power is

C p=( p1−p2) AV (1−a1)+(p¿¿3−p4) AV (1−a2 )

12

ρ AV 3¿

From 2.5.2.3, 2.5.2.4, 2.5.2.5, and 2.5.2.6,

14

Cp

=a1 (1−a1 )2+(1−a2 )2 ( a2−2 a1 )

(2.5.2.7)

For maximum Cp the values of a1 and a2 are found from the equations 2.5.2.8a and 2.5.2.8b,

14

∂C p

∂ a1

=(1−a1 ) (1−3a1)−2 (1−a2 )2=0

(2.5.2.8a)

14

∂C p

∂ a2

=(1−a2 )(1+4a1−3a2)=0

(2.5.2.8b)

which are given as, a1=15∧a2=

35

.

After substituting the values a1 and a2 in Cp , it is found Cp = 16/25, a result that is close to single

actuator disc theory exceeding it by 8% (Newman, 1983, December). For an optimum conditions

of a1 and a2 give A1/A = ½ indicating the disc spacing that is comparable to the diameter of each

disc in one-dimensional flow. The analysis with uniform inflow induction factor through double

actuator discs establishes that the maximum power coefficient for a vertical axis turbine is 16/25

instead of the more conventional value of 16/27 for a single actuator disc. Again for a multiple

actuator disc theory (number of actuator discs greater than six) the power coefficient is found to

be 2/3 and the minimum spacing between the disc below which the one-dimensional theory

begins to fail is 0.5 times the diameter of the disc (Newman, 1986, February 24). So a two

actuator disc model for a Darrieus turbine is found to be satisfactory and the optimum inflow

induction factor at each disc can be used to improve the design and structure of the turbine with

cambered or alternatively canted aerofoils.

2.6 Vortex models

The vortex models calculate the velocity field about the vertical axis turbine from the

vorticity effects in the turbine wake. Vortex models use the vorticity transport and Biot-Savart

equations for modeling the shed wake and its influence on the blades. The Kutta – Zhukhovsky

theorem links circulation to lift and conservation of total circulation (Kelvin’s law) and the

strength of the vortex ring can be determined. The computational work is facilitated by modeling

the wake in a series of vortex points in 2D or 3D as a lattice composed of overlapping vortex

rings. The angle of attack is determined from the wake induced inflow and adding kinematic

motion of the blade and the lift and drag is thereby calculated from a lookup table for a given

section and Reynolds number. Just like momentum models there are also different vortex

models. Larson in 1975 analyzed a cyclogiro windmill using this model, a simplified wake with

only two vortices that shed into the wake at each revolution at the points at which the blades

flipped from positive pitch angle to a negative angle, and calculated an average velocity by

which the vortices proceeded downstream. Holme in 1976 and Wilson in 1978 used a 2D vortex

model in vertical axis wind turbine with straight airfoils designed to produce maximum energy

extraction. The power coefficient and force coefficient had the same limits as that of horizontal

axis wind turbines. Wilson and Walker in 1983 proposed Fixed Wake model in which a vortex

sheet wake was used to distinguish the difference between upwind and downwind flows. The

computational cost in both the momentum and fixed wake models were found to be same.

Fanucci and Walters proposed the first Free Wake Model in 1976 for a straight blade, and was

considered the most complex and accurate vortex model for vertical axis turbines. The wake was

modeled by discrete, force-free vortices that were distributed along the blade camber line,

convecting downstream with local flow velocity. Strickland, et al., in 1979 and Li in 2008

predicted the output power from a vertical axis turbine by replacing the blade by a vortex

filament as shown in the figure 2.6.1.

The 2D and 3D vortex model named as VDART2 and VDART3 respectively was proposed by

Strickland. The code was capable of handling dynamic stall and found to be more accurate than

the momentum models and could represent similar wake shapes as observed in experimental

water tank tests but was more expensive in execution. Another similar model VDART-TURBO

was developed with some concession on accuracy in blade forces but gained significant time

savings (Wilson & Walker, 1983, December). Vortex methods could be used for loaded rotors at

large tip speed ratios and also handle perturbations both parallel and perpendicular to streamwise

velocity unlike momentum models. Also a clear picture could be drawn for designing;

Fig. 2.6.1: A blade modeled by a vortex filament (Alidadi, 2009, June).

positioning blades and their diffusers with support structures, on the basis of the shape of the

near wake.

2.7 Panel methods

Panel methods are another development of vortex methods and model the geometry using

the Laplace equation or the Prandtl – Glauert equation for inviscid flows. In VAWT, panel

methods can handle 3D effects automatically and sped up the pace of development in the design

space. Hess and Smith in 1967 proposed the panel methods at The Douglas Aircraft Company

and found useful in geometry and design analysis with 3D flows. In late 1980s panel methods

continued to mature and became more diversified with the coupling of advanced CFD methods.

Formulations vary mostly on the basis of velocity or velocity potential boundary conditions,

singularity distributions over each panel, Kutta condition implementation method at the trailing

edge, order of panel geometry and discretized wake. In addition to these, significant study has to

be carried out on viscosity in wake roll-up and vorticity diffusion and dissipation in the context

of VAWT.

Dixon, et al., in 2008 proposed a 3D, unsteady, multi-body, free-wake panel model for vertical

axis wind turbine of arbitrary configuration. The model was intended realistically to treat blade-

wake interactions, vortex stretching/contraction and viscous diffusion and validated with

experimentation conducted with 3D-stereo Particle Image Velocimetry (PIV) and smoke trail

studies for a straight-bladed VAWT. In final analysis, the tip vortices from a straight bladed

VAWT were found to move inwards due to wake roll-up behavior along with self induction.

Also wake expansion was found to be asymmetric along the flow downstream and the plane

perpendicular to the flow owing wake self-influence and as a result of the cycloidal motion of the

VAWT blades.

2.8 CFD Models

In VAWT modeling, Reynolds Averaged Navier-Stokes (RANS) or other kinds of Navier-Stokes

equations are involved in solving the design and structure. Many high quality commercial

Computational Fluid Dynamics (CFD) packages are available in the market used for coding and

carrying out validation and verification. Turbulence modeling is an important aspect that RANS

solvers use to establish confidence in the results. The results of the 2D VAWT shows the

application of dynamic stall particularly at low tip speed ratios (Ferreira, et al., 2007). In 3D

VAWT, there is a significant challenge due to its unsteady nature that requires a moving mesh

besides high computational cost for its full solution.

RANS simulations have some advantages over the potential flow models in different simplifying

assumptions, providing valuable analysis in the flow field thereby facilitating the optimization

processes and became popular with exponential increase in the computational speed. Since the

RANS 3-D simulations for vertical axis turbines are very expensive and time consuming, little

work has been done on it so far. In 2007, Guerri, et al., and Jiang, et al., separately studied the

flow phenomenon around a vertical axis turbine with RANS equations and Nabavi, (2007) used

FLUENT to solve RANS equations in different operating conditions of vertical axis turbine.

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