AIAA-99-0042
Computational Study of Horizontal Axis Wind Turbines
Guanpeng Xu1 and Lakshmi N. Sankar2
School of Aerospace Engineering
Georgia Institute of technology, Atlanta, GA 30332-0150
ABSTRACT
A hybrid Navier-Stokes potential flow
methodology for modeling three-dimensional unsteady
viscous flow over horizontal axis wind turbine
configurations is presented. In this approach, the costly
viscous flow equations are solved only in a small
viscous flow region surrounding the rotor. The rest of
the flow field is modeled using a potential flow
methodology. The tip vortices are modeled using a free
wake approach, which allows the vortices to deform and
interact with each other. Sample results are presented
for two rotor configurations tested by the National
Renewable Energy Laboratory. Comparisons with
experimental data, full Navier-Stokes simulations and
blade element and momentum theory are given to
establish the efficiency and accuracy of the present
scheme.
INTRODUCTION
Wind energy represents one of the cleanest
sources of energy available to mankind. Recent
advances in airfoil and rotor development, materials
technology,
power
generation
systems
and
manufacturing technology have made wind turbine
systems in general, and horizontal axis wind turbine
(HAWT) systems in particular, economically feasible
alternatives to gas, oil, and coal based power generation
systems. Ref. 1-3 and related publications discuss the
technological and economic aspects of wind energy.
Many of the rotors found on current generation
HAWT systems are designed using a combination of 2D airfoil tools (e.g. Ref. 4-7) and three-dimensional
blade element and momentum (BEM) theory (e.g. Ref.
8, 9). A number of comprehensive computer codes
using this methodology are currently available to the
designer. In these methods, unsteady flow effects are
either ignored, or modeled using a synthesis of 2-D data
(e.g. Ref. 10). As a result, these methods are incapable
of accurately modeling three-dimensional dynamic stall
processes, tower shadow effects, tip relief effects, and
sweep effects. These three-dimensional effects can alter
the airloads, affect the fatigue life, and significantly
influence the cost of ownership of HAWT systems.
Although first-principles based modeling of the
aerodynamics of HAWT systems is a viable approach,
the cost of such detailed simulations based on NavierStokes equations limit their use to explorative studies.
The present work was motivated by the need
for a first-principles based analysis that will faithfully
model the complexities of 3-D unsteady viscous flow
over the rotor, but remain economical enough for
routine engineering use. In the present work, the flow
field is viewed as a combination of viscous regions,
inviscid regions and vortices, as shown in the sketch
below. These regions are modeled using different
methodologies, making the present approach a hybrid
method. The viscous flow over the rotor blades is
usually confined to small regions, even when the flow is
massively separated. The present approach models this
region using 3-D Reynolds-Averaged Navier-Stokes
equations. Much of the flow field surrounding the rotor
is inviscid, incompressible, and irrotational. Modeling
the inviscid region using incompressible flow equations
(e.g. the Laplace's equation) would require a costly
iterative method at every time step for the velocity field.
For this reason, the present approach models these
regions using a compressible flow equation for velocity
potential that is elliptic in space, and hyperbolic in time.
At low tip Mach numbers, the compressible and
incompressible analyses will yield identical results, as
dictated by the physics of the problem. Thus, no errors
are caused by assuming the flow to be compressible.
The tip vortices shed from the blade affect the inflow
ingested by the rotor, and the aerodynamic loads and
torque. These vortices are modeled in the present
method using a free wake model.
The present method has been previously
applied to helicopter rotors in hover and in forward
flight (Ref. 11). It has also been applied to unsteady
viscous flow over oscillating wings and airfoils (Ref.
12). While this method shares some conceptual
similarities with the classical viscous inviscid
interaction model, it does not share any of the
limitations of these methods. For example, the present
method can be applied to 3-D unsteady viscous flows,
whereas conventional viscous-inviscid interaction
1 Graduate Research Assistant
2 Regents' professor, Associate Fellow AIAA
Copyright  1999 American Institute of Aeronautics and
Astronautics, Inc. and the American Society of Mechanical
Engineers. All rights reserved.
1
American Institute of Aeronautics and Astronautics
methods are limited, for the most part, to steady flows.
The present method has no singularities at the
separation point or separation line, whereas the classical
boundary-layer based interaction schemes have a strong
mathematical singularity near separation. Finally, the
present method can model 3-D unsteady interference
effects with nearby bodies (e.g. towers) in a
straightforward manner, using overset grid approaches.
N-S zone
Potential Flow
Zone
Tip Vortex
moving control volume V, surrounded by surface S,
these equations may be written as


 

d
qdV

(
E

q
V
)

n
dS

E
S I G
S V  ndS
dt 
V
(1)


Here VG is grid velocity due to blade rotation, n is the
outward facing unit normal vector, and q is the flow
vector:
  
u 

 

q  v 
w 
 

 e 

(2)
The unknowns are the density , pressure p, the
Cartesian velocity components (u, v, w), and the energy


per unit volume, e. The quantities E I and E V are the
inviscid and viscous fluxes, respectively. For example,
the inviscid flux is given as follows:
E Ix
 
u2  p

  uv
 uw

u(e  p)
 

 uv
,
 2

 E Iy  v  p
 vw



v(e  p)


 
,
 uw


 E Iz   vw

w2  p



w(e  p)







(3)
Equation (1) may be represented in semi-discrete form
on a finite volume grid as follows:
This paper is organized as follows. The
mathematical formulation behind the Navier-Stokes and
potential flow methodology is first described.
Procedures for transferring the flow field information
between the viscous and inviscid domains are next
described. Finally, numerical results are presented for
two rotors tested at the National Renewable Energy
Laboratory under the Combined Experiment Rotor
(CER) program. The first rotor is of rectangular
planform, is untwisted, and is referred to as the Phase II
rotor in NREL documentation. The second rotor, called
the Phase III rotor, has a nonlinear twist distribution.
Comparisons with experiments, full Navier-Stokes
simulations, and blade element and momentum theory
based simulations are presented to establish the
reliability, accuracy and efficiency of the present
method.
MATHEMATICAL FORMULATION
Viscous Zone:
The present method solves the ReynoldsAveraged compressible Navier-Stokes equations in
regions close to the rotor blade. On a deforming or




d
qVd     E I  qVG  nS 
dt
faces
 
E
 V nS
faces
(4)
This equation holds over a discrete control volume Vd
surrounded by cell faces of area S.
In the present work the inviscid fluxes were
computed at the control volume surfaces using a version
of the Roe scheme (Ref. 13) that is third order accurate
in space. A fifth order accurate scheme is available as
an option, but has not been used in the calculations
presented here. The viscous fluxes were computed using
a spatially second order accurate scheme. Higher order
methods for discretization of the viscous terms were not
considered, since the truncation errors associated with
the present second order discretization are quite small,
and are of order O(2/Re), where  is a typical grid
spacing, and Re is the Reynolds number based on tip
speed. The resulting ordinary differential equations may
be formally written at a typical control volume
identified by three indices ( i, j, k) as:
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American Institute of Aeronautics and Astronautics
dq
dt
derivatives. It is assumed that the vorticity in the outer
region causes negligibly small losses in total pressure,
and that the flow is isentropic. Thus,
 Ri , j ,k
i , j ,k
(5)
where R contains the right hand of equation (4). This
ODE is highly nonlinear in q. It is therefore linearized
at each time step 'n+1' using the known information at
the time level 'n' as follows (Ref. 14):
 dq 
 
 dt 
n 1
n
 R 
q n 1  q n

 R n    q n 1  q n
t
 q 


(6)
n+1
The resulting form is linear in q . However,
since each control volume in the Navier-Stokes zone
(i,j,k) is coupled to its neighbor cells (i1, j1, k1), a
coupled system of equations results. Solution of the
coupled system was accomplished by approximately
factoring the matrix associated with the above system
for (qn+1-qn) into three smaller invertible matrices
(Ref.15). Steady state is reached, if one exists, when the
residual R goes to zero.
Potential Flow Zone:
The potential flow equation is a highly
simplified form of the Navier-Stokes equations. The
flow field is assumed to be inviscid and irrotational in
this zone. The mass conservation equation may be
written in conversation form as
 t   u  x   v  y   w z  0
(7)
The velocity in the full potential zone is
decomposed into three parts:
 


V  V    V w
u  u   x  u w
v  v   y  v w
w  w   z  ww
(8)
In the present formulation, the unknown is the

velocity potential . The quantity V  represents
oncoming stream (e.g. wind) which may be unsteady,
and spatially non-uniform. The terms uw, vw, ww are the
vortical velocity components induced by the tip vortices
emanating from the turbine blade tips, and may include
free-stream turbulence, assuming that the turbulence
effects may be quantified.
An auxiliary relation is needed to express the
density  in terms of the velocity potential  and its
1
a 2  1

  
  a 2 
(9)
where a is the speed of sound, given by the energy
equation:
a2
V2
a2
u 2  v 2  w2
 t 
   
 1
2
 1 2
(10)
Using equations (9) and (10), equation (7) may
be written as a second order hyperbolic partial
differential equation for :

 2   tt   x xt   y  yt   z  zt 
a 
 
         V  VW




(11)
This equation was solved at all points on the body-fitted
grid where viscous effects are considered negligible,
using an implicit time marching solution procedure
developed by Sankar et al (Ref. 16).
Navier-Stokes/Full Potential Coupling
The boundaries that separate the two zones
must be carefully handled to allow pressure waves and
vorticity to leave the viscous zone and propagate
through the inviscid zone without false reflections at the
zone interfaces. The following coupling procedure has
been developed.
The sketch below shows the computational
domain and the boundaries that separate the two zones.
For grid generation purposes, the flow field surrounding
the rotor blade is divided into two blocks, one above
and the other below. For each block there are three
interfaces that surround the Navier-Stokes zone. The
plane k = kmatch is an interface between the viscous zone
and the inviscid zone. The two planes given by i =
imatch1 and i = imatch2 are the upstream and downstream
interfaces in the chordwise direction, respectively.
These three planes extend all the way in the radial
direction.
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
the position vector rMar ker was found by numerically
integrating the following equation in time:
block 1
Kmatch
Imatch1
Imatch2
3
2
1
N.-S.
block 2
FP
Interface Conditions for the Viscous Zone
The Navier-Stokes equations are elliptic in
space and require prescription of the flow properties
(density, velocity, and pressure) at the all interface
planes. The potential flow zone and the inviscid zones
overlap each other by one or more cells, so that the
potential flow field is known at the interface k=Kmatch,
etc. The velocity components needed by the inner
region solver are obtained by computing the disturbance
velocities x, y and z and adding freestream and wake
induced velocities, according to equation (8).
The energy equation given by equation (10) is
next used to get the speed of sound, and temperature.
The isentropic law given by equation (9) is finally
applied to get  and p.
Interface Boundary Conditions for the Inviscid Zone
The potential flow zone is governed by the
second order partial differential equation (11) that
requires the specification of a Dirichlet or a Neumann
boundary condition on all boundaries. In the present
study, the normal component of the velocity field from
the potential flow and the Navier-Stokes formulation
was forced to be equal, at the interfaces. That is,
vn 

     
 V  n  VW  n  V Navier Stokes  n
n
The above equation gives an explicit relationship for
n in terms of the viscous flow velocities and the
rotational velocity field.
Tip Vortex Modeling:
The tip vortex that leaves the Navier-Stokes
zone and enters the potential flow zone was converted
into 200 to 300 connected line segments or markers,
which together form a helical shape. The spatial
positions of these markers were subsequently tracked in
time in a Lagrangean fashion. Vortices from all the
blades were modeled in this manner. The vortical
strength each of these markers was assumed to be the
maximum bound circulation at the blade, at the instance
in time when the marker is released into the potential
flow zone. During subsequent time levels, these markers
move at local flow velocity. Their positions, given by


drMar ker 
   VW
dt

The induced velocity VW due to these markers
was computed using the Biot-Savart law at all points in
the inviscid region. Trailing edge vortices shed inboard
were captured only in the Navier-Stokes zone, which
covers a region 3 to 6 chords behind the blade trailing
edge. These vortices are very weak compared to the tip
vortex. Once these inboard vortices leave the NavierStokes zones, they were ignored. This approximation is
consistent with lifting line methods, where the near
wake (excluding the tip vortex) is usually tracked only
for three to six chord lengths.
RESULTS AND DISCUSSION
A number of calculations have been carried for
a three-bladed horizontal axis wind turbine system
tested at NREL (Ref. 17). The configuration was made
of untapered blade sections. Twisted rotors (referred to
as Phase III rotors in NREL literature), and untwisted
rotors (Phase II rotors) both were studied.
Figure 1 shows a typical H-O grid. For wind
directed along the axis of the rotor, the flow properties
are periodic from one blade to the next, and viscous
flow and inviscid flow zones over only a single blade
need to be considered. However, the tip vortices from
all the blades must be modeled. The user may specify
the number of cells along the chord, radial and normal
directions. The grid generation process is extremely
fast, and may be completed in a matter of minutes on
workstations.
The user can choose the size of the NavierStokes zone, and the potential flow region at the start of
the flow analysis. The Navier-Stokes zone should be
large enough to enclose the boundary layer over the
blade, and any separated flow regions. In the cases
studied here, the Navier-Stokes zone extended 1/4
chord ahead of blade leading edge, and 1 to 2 chords
behind blade trailing edge, and 2 radii beyond the blade
tip. The viscous zone extended about 1/3 chord length
above the upper surface, and 1/3 chord length below the
lower surface. The outer boundary of a potential flow
zone was a cylinder of radius equal 2 blade radii, and a
height of 2.2 blade radii (1.1 radii above the rotor disk
and below).
Both the hybrid Navier-Stokes/Full Potential
code, and a stand-alone Navier-Stokes solver are
operational. The Navier-Stokes code uses exactly the
same equations and numerical procedures as the viscous
flow method in hybrid flow solver. The only difference
is that the Navier-Stokes equations are solved over the
entire flow field. Calculations were carried out for the
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Phase II rotor at three wind conditions: 10, 15 and 20
meters per second. Figure 2 shows the Navier-Stokes
code and hybrid code predictions, compared against
measured data. In general, very good agreement with
measured data from NREL aerodynamics experiments
at attached flow conditions (for example, at 10 m/s) and
under stalled conditions was observed. In these
calculations, the transition point was assumed to be at
40% chord, based on the 2-D calculations carried out by
Wolfe et al. [Ref. 18, 19]. Downstream of the transition
location, and in the wake, the eddy viscosity was
calculated using the Baldwin-Lomax turbulence model.
Figure 2 also shows predictions from the
lifting line method embedded in YawDyn (Ref. 9). It
appears that the blade element and momentum method
tends to overpredict the power generation at low wind
speeds.
Figure 3 shows results for the Phase III rotor
from the hybrid code. The Navier-Stokes calculations
are costly, and were done only at a single wind
condition for this case. From figure 2, however, it may
be anticipated that the Navier-Stokes solver would have
yielded results that are quantitatively equivalent to the
hybrid code. As in the case of the Phase II rotor, the
BEM theory predicts the trends in the power generation
well, but the magnitudes are overestimated.
An attempt was made to identify the source of
the discrepancy between the present first-principles
based methods, and the BEM theory code. Figure 4
shows the spanwise lift distribution for the Phase II
rotor from the full Navier-Stokes code and the lifting
line code. The agreement is very good at stations
outboard of 60% blade radius. These outboard stations
operate at high dynamic pressures, and are primarily
responsible for the power generation. In the inboard
regions, the low rotational velocity r of the blade
sections yields low dynamic pressures, exaggerating the
differences in the sectional lift coefficients between the
lifting line theory and the Navier-Stokes code.
Although the BEM theory and the firstprinciple codes have comparable blade loads at the
outboard stations, the induced velocity (or the inflow
through the rotor) was not similar for the two codes.
The BEM theory uses an analytical expression for the
inflow velocity based on the combined blade elementmomentum theory. The Navier-Stokes code and the
hybrid code compute the inflow from first principles.
The power generation depends on the induced drag,
which is roughly the product of the sectional lift forces
and the inflow velocity. Thus the two methods (lifting
line theory and present method) predict somewhat
different values for the power generation.
Figure 5 shows the evolution of power
coefficient with iterations for three wind conditions,
using the hybrid code. It is seen that the calculations
rapidly settle down to steady state values in 2000
iterations or less. A typical iteration requires 19 seconds
on a HP Model 750 workstation with 128 MB of
memory for a 110 x 43 x 40 grid. Thus, reliable steady
state results can be obtained in 10 CPU hours or less on
HP workstation class systems. On faster multiprocessor
machines such as the Cray Y/MP, and the SGI Origin
2000 systems, the CPU time may be effectively reduced
to 2 hours or less by a combination of faster clock
speeds and more processors. One of the goals of this
research is to reduce the CPU time to 2 hours or less on
such advanced computer systems. Such a fast turnaround will be needed for industry use of this firstprinciples based methodology in a design environment.
It may be noted that the hybrid solver converges
quickly, while the full Navier-Stokes simulation
requires 10,000 time steps or more. This apparent
discrepancy is explained by the fact that the hybrid
solver starts with an initial guess for several revolutions
of the tip vortex shed into the wake, and its associated
inflow velocity. In the present simulations this starting
wake is assumed to be a helix. Thus, the hybrid code
spends much of the calculations relaxing the tip vortex
shape, and adjusting the tip vortex strength based on the
blade loads. The full Navier-Stokes calculations, on the
other hand, start from a uniform flow. The blade has to
spin several revolutions (for several thousand time
steps) to generate the tip vortex, and establish the
inflow through the rotor.
In addition to yielding engineering quantities
of interest to the designer, the present first principles
based simulations provide useful information on the
flow features. Features such as flow separation, flow
unsteadiness, and the radial migration of the low
momentum fluid due to centrifugal pumping effects may
be studied. The designer may be able to use these
visualizations to design rotors with minimum flow
separation,. Figures 6 and 7 show the particle traces
over the Phase II rotor, computed by releasing particles
into the flow field, and tracking their subsequent motion
in time. A coordinate system attached to the rotor is
used for clarity. At lower wind conditions (e.g. 10 m/s)
the flow is well attached, over the outer 50% of the
rotor, as shown in figure 6. Only the inboard stations
are separated. The radial migration of the fluid particles
along spiral trajectories, due to centrifugal pumping
effects is also seen. At higher wind conditions (e.g. 20
m/s wind) much of the flow over the rotor is extensively
separated, except near the tip where the strong inflow
due to the tip vortex reduces the angle of attack and
keeps the flow attached. As a result, significant amounts
of power are generated in the tip regions at this wind
condition.
CONCLUDING REMARKS
An efficient first-principles based method has
been developed for predicting unsteady viscous flow
over HAWT systems. The calculations done to date
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American Institute of Aeronautics and Astronautics
have been for axial flow conditions, and good
agreement with measured data have been obtained. The
formulation is general enough so that unsteady wind
conditions and free-stream turbulence effects may be
included.
Work is under progress on inclusion of tower
shadow effects using an overset approach. Empirical
transition models and a one equation eddy viscosity
transport model have also been developed and are being
tested. Additional carefully done validation of the flow
solver for steady, and unsteady wind conditions is
needed.
ACKNOWLEDGMENTS
This work was supported by the National
Renewable Energy Laboratory under Contract No.
XCX-7-16466-02. The authors are thankful to Dr. Peter
Tu of NREL and Dr. Walter P. Wolfe of Sandia
National Laboratories for their guidance and
encouragement in the course of this study.
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17.
Simms, D. A., Robinson, M. C., Hand, M. M.,
and Fingersh, L. J., Characterization and Comparison of
Baseline Aerodynamic Performance of Optimally
Twisted versus Non-Twisted HAWT Blades. Wind
Energy Book VIII: Conference Papers - Proceeding from
Energy Week '96, 29 January - 2 February 1996,
Houston, Texas. Vol. I: pp. 143-148, 1996.
18.
Wolfe, W. P. and Ochs, S. S., "CFD
Calculations of S809 Aerodynamic Characteristics,"
AIAA paper 97-0973. January 1997.
19.
Wolfe, W. P. and Ochs, S. S., "Predicting
Aerodynamic Characteristics of Typical Wind Turbine
Airfoils Using CFD," Sandia Report SAND96-2345,
September 1997.
6
American Institute of Aeronautics and Astronautics
Overall Grid
Near Grid
Fig 1 Body fitted grid over Phase II Rotor
20
20
NREL experiment
Generator Power[kw]
Generator Power[kw]
BEM Theory
15
10
NREL experiment
N-S Solver
Hybrid Code
BEM Theory
5
0
0
5
10
15
20
-5
Hybrid code
15
10
5
25
0
Wind Speeds[m/s]
Fig 2 Computed vs. Measured Power
Generation for Phase II rotor
0
5
10
Wind Speed[m/s]
15
Fig 3 Computed vs. Measured Power
Generation for Phase III rotor
7
American Institute of Aeronautics and Astronautics
20
20
20
16
Power(kw)
CL/(r/R)^2
25
Navier-Stokes Solver
15
12
BEM Theory
10
5
10 m/s
8
8 m/s
4
0
0.0
0.2
0.4
0.6
r/R
6 m/s
0
0.8
1.0
Fig 4 Spanwise loading over Phase II
Rotor with comparisons with RDYNE.
1.2
0
1000
2000
3000
Iterations of code
4000
Fig 5 Evolution of Power with
Iterations for Phase III Rotor.
Fig 6 Streamlines over rotor upper surface at 10 m/s
8
American Institute of Aeronautics and Astronautics
Fig 7 Streamlines over rotor upper surface at 20 m/s
5000