Utilization of Whale-Inspired Tubercles as a Control Technique to Improve Airfoil Performance

Abstract

This research exploits the Whale-inspired tubercles as a control technique to improve the performance of airfoils. The flow field of NACA0012 airfoil with spherical leading-edge tubercles was computationally simulated. This airfoil section resembles the flipper of the Humpback whale and is used in many engineering applications. Tubercles, with a diameter of 10% of the airfoil chord (C), are arranged such that the span-wise distance between the centerlines of two adjacent tubercles is 20% C. k- turbulence model was used for a wide range of angle of attack (α = 0o - 25o) and Reynolds number (Re = 65,000 - 1,000,000). Results demonstrated that the presence of tubercles improves the airfoil performance by delaying or even preventing stall in the investigated range of operating conditions (α and Re). Simple active control scheme is proposed to obtain optimum performance (i.e., optimum values of lift and drag coefficients).

Full text

TRANSACTION ON CONTROL AND MECHANICAL SYSTEMS, VOL. 2, NO. 5, PP. 212-218, MAY, 2013.
RECEIVED: 7, MAR., 2013; REVISED: 15, APR., 2013; ACCEPTED: 21, APR., 2013; PUBLISHED: 25, APR., 2013. ISSN: 2345-234X
Abstract: This research exploits the Whale-inspired
tubercles as a control technique to improve the performance of
airfoils. The flow field of NACA0012 airfoil with spherical
leading-edge tubercles was computationally simulated. This
airfoil section resembles the flipper of the Humpback whale
and is used in many engineering applications. Tubercles, with a
diameter of 10% of the airfoil chord (C), are arranged such
that the span-wise distance between the centerlines of two
adjacent tubercles is 20% C. k-

turbulence model was used for
a wide range of angle of attack (α = 0o - 25o) and Reynolds
number (Re = 65,000 - 1,000,000). Results demonstrated that
the presence of tubercles improves the airfoil performance by
delaying or even preventing stall in the investigated range of
operating conditions (α and Re). Simple active control scheme
is proposed to obtain optimum performance (i.e., optimum
values of lift and drag coefficients).
Keywords: Bio-Inspired Engineering, Tubercles, Computational,
Airfoil Performance, Stall, Control Technique.
1
1. INTRODUCTION
A. Background
Bio-inspired engineering is an emerging discipline that
continues to prove competence in many applications. One
of these applications concerns the utilization of
leading-edge tubercles that are found in the flippers of the
Humpback whales, Fig. 1[1]. These naval large mammals
have an unusual maneuvering ability to undertake sharp
movements to catch prey. The tubercles on the leading-edge
act as passive controls that improve flipper performance.
Previous research investigations confirmed that tubercles
cause a delay in the stall angle of attack. Thus, the lift of the
airfoil is kept beyond the regular stall angle of attack, which
represents a big practical achievement. This situation may
be attributed to the vortex generation that energizes the
boundary layer for greater attachment. This type of passive
control is simple, cheap and needs no maintenance. The idea
of using the tubercles on the leading-edge of airfoils/blades
can be utilized in many engineering fields, e.g.
turbomachinery applications.
B. Previous work
There are many experimental and numerical studies that
considered the flippers of the humpback whale themselves.
The studies include those of Miklosovic et al.[2], Fish et
al.[3] and Brown[4] whose experimental tests clarified that
Ahmed F. Abdel Gawad, PhD. Umm Al-Qura University, Saudi Arabia
(afaroukg@yahoo.com)
Please refer to the Parameter Index Table at the end of the paper.
the humpback whale-inspired commercial fans move more
air and use less power than conventional fans.
Fig. 1. Tubercles on the leading-edge of the flippers of the Humpback
whale [1].
Other investigations concerned the whale-inspired
airfoils/wings. Pedro and Kobayashi[5] simulated
numerically two different wings at Re = 5 × 105. One of the
wings displays a scalloped leading-edge and the other one
has a smooth leading edge. Their results revealed a
significant increase in the aerodynamic performance for the
scalloped flipper close to separation. Hansen et al.[6]
investigated experimentally the three-dimensional effects on
various tubercle configurations for a NACA0021 airfoil at
Re = 120,000.
They stated that the effectiveness of tubercles as
a passive flow control mechanism is more dependent on the
Reynolds number than on three-dimensional effects. Kouh
et al.[7] employed CFD to investigate the effects of
leading-edge protuberances (tubercles) of varying amplitude
and wavelength on the performance of NACA0012 airfoil.
Their results showed that, for an infinite wing,
maximum lift coefficient increases with an increase in
protuberance wavelength, whereas, for a finite wing with
fixed aspect ratio, both stall angle of attack and stall lift
coefficient increase with the wavelength. Based on his
research work, Lane[8] emphasized that the technology of
inserting tubercles on airfoils/blades technology can be used
in a huge range of machines (turbines, compressors, pumps,
and fans that use blades or rotors) and in any lifting surface
(airplane wings, windmill blades or sailboat masts). Some
studies considered the blades of wind turbines. As a
practical application, Wind Energy Institute of Canada
(WEICan) carried out a test to determine the performance of
the Canadian WhalePower Corporation’s turbine blades that
had been fitted to a Wenvor 25kW-turbine[9].
Utilization of Whale-Inspired Tubercles as a Control
Technique to Improve Airfoil Performance
Ahmed Farouk Abdel Gawad

AHMED FAROUK ABDEL GAWAD: UTILIZATION OF WHALE-INSPIRED TUBERCLES AS A CONTROL TECHNIQUE TO IMPROVE AIRFOIL PERFORMANCE.
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The blades contain tubercles along most of the leading
edge of the blade. Their site measurements covered power
curves and annual energy production for different wind
speeds and operating conditions. Dewar et al.[10] presented
an invention to enhance the effective capture of force from
wind and other moving fluids to generate electrical power.
Their invention relates to employing a tubercles
leading-edge rotor to enhance lift and reduce drag. Krause
and Robinson[11] focused on designing, simulating, and
analyzing a horizontal-axis wind turbine (HAWT) with
whale-inspired blades. They showed that the blades are
characterized by a superior lift/drag ratio due to greater
boundary layer attachment from vortices energizing the
boundary layer.
Also, hydrofoils and marine applications were
considered in many research works. Custodio[12] examined
the effects of leading-edge sinusoidal tubercles on the lift,
drag, and pitching moments of two-dimensional hydrofoils
in a water tunnel with comparison to the baseline
NACA634-021 hydrofoil. As a ratio of the mean chord
length, the amplitude of the protuberances (tubercles)
ranged from 2.5% to 12%, while the span-wise wavelength
was 25% and 50%. He stated that the amplitude of the
protuberances has a large effect on the performance of the
hydrofoils, whereas, the wavelength has little. Stanway[13]
investigated the hydrodynamic effects of leading-edge
tubercles on the performance of three-dimensional
hydrofoils. He considered both the static foils, such as
rudders or dive planes of marine vehicles, and the dynamic
foils, with application to flapping foil propulsion. His
measurements indicated that stall was delayed on the foil
with tubercles; maximum lift was reduced in almost all
cases. However, he suggested that the vortical structures
generated by the tubercles interfere with the thrust wake of
the wing, and thus performance deteriorates. Gruber et al.[14]
investigated experimentally the impact of the leading-edge
protuberances (tubercles) on the marine tidal turbine blades,
especially at lower tidal flow speeds. They compared three
different blade designs (baseline and two
tubercle-modified). Their results of power coefficients were
presented for a range of tip speed ratios. They illustrated
that, for all test criteria, the tubercle-modified blades
outperformed the smooth leading-edge baseline-design
blades at the lower test velocities, and did not show
degraded performance at the higher velocities.
2. SCOPE OF PRESENT WORK
Based on the above illustration, we can say with
confidence that developing new versions of the
airfoils/blades with leading-edge tubercles is a hot issue.
Yet, there is still a big need for more investigations to
clarify the flow characteristics of this type of airfoils/blades
and demonstrate the possibilities of better control
techniques.
The present work demonstrates the development of a
new airfoil with spherical leading-edge tubercles. The study
depends on the numerical simulation of the flow field
around the airfoil in presence of tubercles. The commercial
software "Fluent 12.0"[15] was used to carry out the
simulation. The NACA0012 profile, Fig. 2, was utilized in
this work. The NACA0012 profile was reported by other
researchers[7,11] as the closest profile to represent the
humpback flipper. Moreover, this profile is commonly used
in many aerodynamic (e.g. wind turbines) and
hydrodynamic applications. A wide range of angle of attack
was tested; from 0o to 25o. The values of Reynolds number
ranged between 65,000 and 1,000,000. The standard k-

model was used as the turbulence modeling technique. The
obtained results with discussions as well as conclusions are
reported in the following sections.
Fig. 2. General view of the present NACA0012 airfoil with spherical
tubercles.
3. GOVERNING EQUATIONS AND K-

MODEL
The equations that govern the flow around the airfoil
model are time-averaged continuity and momentum
equations, which, for the steady flow of a constant-property
fluid, are given, respectively, by
0
i
i
U
x


(1)
1
( - ) -
ii
j i j
j j j i
UU P
U u u
x x x x




   
(2)
In the above, i, j = 1, 2, 3, Ui is the mean-velocity vector
in three directions x, y and z, P is the static pressure,

is the
fluid density and

is its kinematic viscosity. Repeated
indices imply summation. In the k-

turbulence closure, the
unknown Reynolds stresses ( 𝑢𝑖 𝑢𝑗
̅
̅
̅
̅
̅
̅
) in Eq. 2 are assumed to
vary linearly with the local rate of strain, thus
2
( ) -
3
j
i
i j t ij
ji
U
U
u u k
xx



  

(3)
and

t, the eddy viscosity, is evaluated from:
2
tk
C



(4)
The turbulence kinetic energy (k) and its dissipation
rate (

) are obtained from the solution of the transport
equations
( ) -
t
jk
j j k j
kk
UP
x x x


  

  
(5)

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2
12
( ) -
t
jk
j j j
U C P C
x x x k k



   

  

  
(6)
Where, Pk is the rate of production of k.
- ( )
j
i
k i j ji
U
U
P u u xx




(7)
The complete model involves a number of coefficients,
which are here assigned their standard values as C

= 0.09,
C

1 = 1.45, C

2 = 1.9,

k = 1.0,

= 1.3.
4. PROPOSED AIRFOIL MODEL WITH
TUBERCLES
The proposed model is formed by adding spherical
tubercles at the leading-edge of the standard NACA0012
airfoil, Fig. 2. This is an exceptional shape of tubercles.
Although the spherical shape is much similar to the real
ones of the humpback Whale's flipper, it was not
implemented before for research or commercial
applications. Usually, tubercles are employed by forming a
wavy shape (sinusoidal) at the leading-edge of the
airfoil/blade, Fig. 3. The wavy shape is controlled by the
amplitude (A) and wave length (λ).
(a) NACA0012 airfoil[12]
(b) NACA0021airfoil[8]
Fig. 3. Airfoils with tubercles showing amplitude and wavelength
parameters.
The proposed shape of the spherical tubercles is superior
on the wavy shape in the following aspects:
(1) The spherical tubercles are easier to manufacture. The
tubercles are to be manufactured separately and fixed
into pre-prepared places at the leading-edge of the
airfoil.
(2) The spherical tubercles may be fabricated from lighter
material than that of the airfoil. Thus, the overall weight
of the airfoil with tubercles is reduced or at least not
affected.
(3) The spherical tubercles may be fabricated from an
inflatable material. Thus, the amplitude of the tubercle
(A) can be easily controlled. Moreover, the wavelength
(λ) can be controlled also by choosing the sequence of
active tubercles. If a next tubercle is not filled, then, the
wavelength (λ) is doubled and so on, Fig. 4. The original
profile of the airfoil may be completely restored with
proper filling of the tubercles.
(4) For the inflatable tubercles, an integrated system of a
control unit and a suitable air-compressor can be used to
control the amplitude and wavelength of the tubercles.
This control system is expected to be simple and
inexpensive. The proposed control system is to be
descried in a following section.
In the present study, the diameter of the spherical
tubercle equals 10% of the airfoil chord, i.e. dt = 0.1 C. The
amplitude of the tubercle equals the radius of the sphere, i.e.
A = 0.05 C. The wave length is fixed at 20% of the chord,
i.e. λ = 0.2 C as shown in Fig. 4. As this paper concerns
only one value for both λ and A, the present work may be
extended to find the optimum values of λ and A for best
operating conditions.
Side view
Plan (top) view
Fig. 4. Computational domain and boundary conditions, Not to scale.
5. COMPUTATIONAL ASPECT AND
BOUNDARY CONDITIONS
The computational domain and boundary conditions are
shown in Fig. 4. The computational unstructured mesh is
based on tetrahedral-shaped elements, Fig. 5. The mesh is
very fine next to the solid boundary (airfoil). The size of the
elements increases towards the far field away from the
airfoil. SIMPLE algorithm (semi-implicit-method for
pressure-linked equations) of Patanker and Spalding[16] was
used to solve the velocity and pressure fields. For the cells
next to the airfoil surface, the standard wall-function was
prescribed. The applied boundary conditions, Fig. 4, can be
listed as: (i) velocity at upstream boundary is uniform, so
u=Uf, v=w=0. (ii) symmetry boundary condition is applied

AHMED FAROUK ABDEL GAWAD: UTILIZATION OF WHALE-INSPIRED TUBERCLES AS A CONTROL TECHNIQUE TO IMPROVE AIRFOIL PERFORMANCE.
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at far boundaries, so u=Uf, v=w=0. (iii) no-slip condition is
used on the airfoil surfaces, so u=v=w=0. (iv) zero-gradient
condition is assumed for all variables (

) at downstream
boundary, so

/

x = 0.
The value of y+ (=
w
y


) from the wall for the first
node was in the range from 3 to 5. Careful consideration
was paid to approximately eliminate the dependence of
solution on the mesh size by improving the clustering of
cells near solid walls until results are almost constant. The
investigation was carried out using different numbers of
cells, namely: 5105, 7.5105, 10105 and 15105. It was
found that there is almost no change of the results for the
two mesh sizes 10105 and 15105. So, there was no need
to increase the number of cells above 10105.
The computations were carried out for different values
of Reynolds number (Re) to cover a wide range of operating
conditions. The values of Re ranged between 65,000 and
1,000,000. The values of angle of attack (

) ranged between
0o and 25o.
6. NUMERICAL SCHEME VALIDATION
To make sure that the results of the present numerical
scheme are correct and reliable, validation has to be carried
out. The present numerical scheme was used to predict the
lift coefficient (CL) of the NACA0012 airfoil without
tubercles at different values of angle of attack.
In the present work, the airfoil without tubercles is
named "Regular". The present results are compared to the
results of Sheldahl and Klimas[17], Fig. 6, at two values of
Reynolds number. As it is seen in Fig. 6, the present results
compare very well to those of [17]. So, the present scheme
can be used with confidence to predict the flow field around
airfoils with tubercles.
7. RESULTS AND DISCUSSIONS
In this section, results and discussions of the
distributions of pressure coefficient (Cp), streamline
patterns, and lift and drag coefficients are presented for
different operating conditions.
Fig. 5. Computational mesh.
C. Pressure coefficient Distributions and Streamline
Patterns
Figure 7 shows comparisons of the distributions of the
pressure coefficient (Cp) and streamline patterns for the
NACA0012 airfoil with and without tubercles at
Re = 1,000,000. For the airfoil with tubercles, the results are
shown for two sections. The first section (i) passes through
the centerline of the spherical tubercle. The second section
(ii) passes through the midline between two adjacent
tubercles, Fig. 4b. It is noticed in Fig. 7 that the presence of
the tubercles modifies the distributions of Cp especially
around the tubercle itself. For α = 5o, the changes of Cp
distribution is generally small. Starting from α = 10o, there
is a noticeable change of Cp distributions for the
tubercle-airfoil.
Fig. 6. Validation of the present numerical scheme.
Changes include the minimum negative values of Cp and
the zone extension of each value of Cp. These changes
explain the changes of values of lift and drag coefficients as
will be demonstrated in the following section. Streamline
patterns show the appearance of one circulation zone for α =
20o and 25o. For α = 25o, the circulation zone becomes large
and covers most of the upper surface of the airfoil. Slight
differences of this circulation zone are noticed for both the
tubercle-airfoil and regular-airfoil.
D. Lift and Drag Coefficients
The presence of tubercles affects greatly the values of
lift coefficient (CL) at different angles of attack. Tubercles
may delay or even remove stall within the actual operating
range of angle of attack. Fig. 8 shows predictions of CL for
different values of Reynolds number. The present results of
the tubercle-airfoil are compared to the results of [17] for the
regular-airfoil. As can be seen in Fig. 8, the values of CL for
tubercle-airfoil and regular-airfoil are the same till certain
value of α, where, CL of the tubercle-airfoil becomes almost
constant without appearance of stall. These values of α are
7o, 8o and 8o for Re = 65,000, 650,000 and 1,000,000,
respectively. On the other hand, the stall of the
regular-airfoil occurs at α = 8o, 11o and 12o for the
corresponding values of Reynolds number. For both the
tubercle- and regular-airfoil, maximum value of CL
increases with Re. At Re = 1,000,000, there is a noticeable
small increase of CL between α = 15o and 20o for the
tubercle-airfoil. The maximum value of CL decreases by
X
Y
-0.5 0 0.5 1
-0.5
0
0.5
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Lift Coefficient, Cl
Angle of attack, 
Regular - Re=650,000 (Present)
Regular - Re=700,000 [17]
Regular - Re=1,000,000 (Present)
Regular - Re=1,000,000 [17]

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216
about 10% when using the tubercle-airfoil in comparison to
the regular-airfoil. However, this reduction in the maximum
value of CL is compromised by the disappearance of stall till
α = 25o (Maximum investigated angle of attack).
(a)

= 5o
(b)

= 10o
(c)

= 15o
(d)

= 20o
(e)

= 25o
(I) Tubercle-airfoil
(II) Regular-airfoil
Fig. 7. Comparisons of the pressure coefficients (Cp) distributions and
streamline patterns for the NACA0012 airfoil with and without tubercles,
Re = 1,000,000.
Fig. 9 shows a compression between the present
spherical tubercles and the wavy tubercles of [5] for
NACA0012 at Re = 123,000. The results of [5] cover a wide
range of the amplitude (A) and wave length (λ) of the wavy
shape, Fig. 3a. For comparison with the present work, the
value of λ = 0.2 C of [5] is chosen and the values of CL are
redrawn in Fig. 9. For example, A05W20 means that A =
0.05 C and λ = 0.2 C and so on. A00W00 refers to the
regular NACA0012 Airfoil. It is seen in Fig. 9 that the
values of CL of the present tubercle-airfoil fit exactly among
the values of CL of [5]. This gives confidence in the present
results and supports the utilization of the spherical tubercles
with the advantages that were explained before.
Fig. 10 shows the predictions of drag coefficient (CD) in
comparison to the NACA0012 regular-airfoil. Generally, the
values of CD for tubercle-airfoil are greater than those of
regular-airfoil. It seems this is a penalty that may have to be
paid when using tubercles as also reported by others [8,12].
However, in many aerodynamic and hydrodynamic
applications, it is much more important to improve lift
characteristics rather drag. There is almost no change of
values of CD between α = 0o and 5o as well as above 20o.
Fig. 8. Predictions of lift coefficient (CL) for different values of Reynolds
number (Re).
Fig. 9. Comparison of the lift coefficient (CL) between the present spherical
tubercles and the wavy tubercles of [5], Re = 123,000.
Fig. 10. Predictions of drag coefficient (CD) in comparison to the
NACA0012 regular-airfoil.
8. PROPOSED CONTROL SCHEME
Although tubercles are considered a passive
flow-control technique, an active control may be added for
optimum performance. Figure 11 shows the proposed
control scheme for obtaining the optimum performance.
This means the optimum values of CL and/or CD according
Cp: -1 -0.8-0 .6-0 .4 -0.2 0 0.2 0.4 0.6 0.8 1
Cp: -0.8 -0.7 -0.6-0.4 -0.3-0.1 0 0.1 0.3 0.4 0.6 0.7 0.8 0.9 1
Cp: -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0 .2 0 0.2 0.4 0.6 0.8 1
X
0 0.2 0.4 0.6 0.8 1
Cp: -1.8 -1 .6 -1.4 -1.2 -1 -0.8 -0.6 - 0.4 -0.2 0 0.2 0.4 0 .6 0 .8 1
Cp: -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0. 6 0. 8 1
Cp: -2.8 -2. 6 -2.4 -2.2 -2 -1 .8 -1.6 -1.2 -1 -0 .8 -0.4 -0.2 0 0.2 0.6 0.8 1
Cp: -1.8 -1. 6 -1.4 -1.2 -1 -0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Cp: -2.8 -2 .6 -2 -1.8 -1.6 -1.4 -1. 2 -1 -0.8 -0.6 -0.2 0 0 .2 0 .6 0 .8
Cp: -1.2 -1 -0.8 -0.6 -0.4 -0 .2 0 0.2 0.4 0.6 0.8 1
Cp: -2.4 -2.2 -2 - 1.8 -1.6 -1.4 -1.2 -1 -0.8 -0. 6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Lift Coefficient, Cl
Angle of Attack, 
Tubercles - Re=65,000 (Present)
Regular - Re=160,000 [17]
Tubercles - Re=650,000 (Present)
Regular - Re=700,000 [17]
Tubercles - Re=1,000,000 (Present)
Regular - Re=1,000,000 [17]
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25
Lift Coefficient, Cl
Angle of Attack, 
Tubercles (Present)
A00W00
A05W20
A10W20
A15W20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 5 10 15 20 25 30
Drag Coefficient, Cd
Angle of Attack, 
Tubercles - Re=160,000 (Present)
Regular - Re=160,000 [17]
Tubercles - Re=1,000,000 (Present)
Regular - Re=1,000,000 [17]

AHMED FAROUK ABDEL GAWAD: UTILIZATION OF WHALE-INSPIRED TUBERCLES AS A CONTROL TECHNIQUE TO IMPROVE AIRFOIL PERFORMANCE.
TRANSACTION SERIES ON ENGINEERING SCIENCES AND TECHNOLOGIES (TSEST) ©
217
to the application. To apply this scheme a series of
investigations are to be carried out to determine the
characteristics of CL and CD at different operating
conditions (i.e. Uf, α, A, λ).
Fig. 11. Proposed control scheme for the airfoil with spherical tubercles.
9. CONCLUSIONS
Based on the illustrated results and discussions, the
following points can be concluded:
1-Based on validation, the present numerical scheme
compares very well to the experimental/computational
results of others.
2-The present spherical tubercles are superior in comparison
to the wavy tubercles in many aspects of being easier to
manufacture, lighter in weight, and ability to control.
3-The presence of tubercles delays or even prevents stall in
the corresponding range of operating conditions (α and
Re).
4-There is a penalty that may have to be paid when using
tubercles that appears in increase of values of CD in
comparison to regular-airfoils.
5-A series of investigations should be carried to find the
optimum values of λ and A to obtain best operating
conditions of the spherical tubercles.
6-Simple active control scheme of λ and A may be utilized
to obtain optimum performance (i.e. optimum values of CL
and CD).
REFERENCES
[1] http://www.oceanlight.com/html/humpback_whale.html:
March 2013
[2] D.S. Miklosovic, M.M. Murray, L.E. Howle, and F.E. Fish,
"Leading-edge Tubercles Delay Stall on Humpback Whale
(Megaptera novaeangliae) Flippers," Physics of Fluids, vol.
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TRANSACTION ON CONTROL AND MECHANICAL SYSTEMS, VOL. 2, NO. 5, PP. 212-218, MAY, 2013.
TRANSACTION SERIES ON ENGINEERING SCIENCES AND TECHNOLOGIES (TSEST) ©
218
PARAMETER INDEX TABLE
1
A
Tubercle amplitude
2
Apf
Planform area of airfoil
3
C
Airfoil chord
4
CD
Drag coefficient; = drag force/ (0.5

Uf2 Apf)
5
CL
Lift coefficient; = lift force/ (0.5

Uf2 Apf)
6
Cp
Pressure coefficient; = P/(0.5

Uf2)
7
C

,C

1,C

2
Constants of k-

model
8
dt
Tubercle diameter
9
k
Turbulence kinetic energy
10
P
Static pressure
11
Pk
Rate of production of k
12
Re
Reynolds number
13
Uf
Free-stream velocity
14
Ui
Mean-velocity vector in three directions
15
u, v, w
Velocity components in x-, y-, z-direction
16
( )
ij
uu
Reynolds stresses
17
xi
Coordinate vector in x-, y-, z-direction
18
y
Distance normal to wall
19
y+
Dimensionless distance normal to wall
GREEK
20

Angle of attack
21

ij
Kronecker delta
22

Turbulence dissipation rate
23

Any variable (e.g., u, v, w, P,… etc.)
24

Tubercle wavelength
25

Fluid kinematic viscosity
26

t
Eddy viscosity
27

Fluid density
28

k,

Constants of k-

model
29

w
Wall shear stress
ABBREVIATIONS
30
CFD
Computational Fluid Dynamics
31
HAWT
Horizontal-Axis Wind Turbine
32
NACA
National Advisory Committee for Aeronautics
33
PLC
Programmable Logic Controller
34
Re
Reynolds number; = Uf C/

35
SIMPLE
Semi-Implicit-Method for Pressure-Linked
Equations
36
WEICan
Wind Energy Institute of Canada
Ahmed Farouk Abdel Gawad
Professor of Computational Fluid
Mechanics, Mech. Eng. Dept., Umm
Al-Qura Univ. (UQU), Saudi Arabia;
Assoc. Fellow AIAA; Member
ASME, ACS, SIAM, AAAS;
Marquis Who is Who, IBC, ABI
Biographee; ICFDP8, ICFDP9,
ICFD10-Conference General
Coordinator; UQU ASME Student
Section
Advisor; www.drahmedfarouk.net;
abdel_gawada2@asme.org.
Available online at:
http://tsest.org/index.php/TCMS/article/view/158
Download full text article at:
http://tsest.org/index.php/TCMS/article/download/158/91
Cite this work as:
Ahmed Farouk Abdel Gawad, "Utilization of
Whale-Inspired Tubercles as a Control Technique to
Improve Airfoil Performance," TSEST Transaction on
Control and Mechanical Systems, Vol. 2, No. 5, Pp.
212-218, May, 2013.