airfoil lift 1
DESCRIPTION
lift of an airfoilTRANSCRIPT
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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