Thru st Li ft Net Aerodynamic Force Dra g Weigh t P1V1Z 1 P2V2Z 2 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
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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
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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