University of Southern California Dissertations and Theses

RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.

(USC Thesis Other)

RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content RANS SIMULATIONS FOR FLOW-CONTROL STUDY
SYNTHETIC JET CAVITY ON NACA0012 AND NACA65(1)412 AIRFOILS
by
Jocelyn Sarah´ı Mendoza Martinez
A Thesis Presented to the
FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING
In Partial Fulfillment of the
Requirement for the Degree
MASTER OF SCIENCE
(AEROSPACE ENGINEERING PROGRAM)
May 2024
Copyright [2024] [Jocelyn Sarah´ı Mendoza Martinez]

Acknowledgments
I dedicate this work to my friends Sandra, Kelvin, Arturo, Idan, Morgan, Jessica,
Danny, Jorge, Alan P, Alan V, Ashley, and family. My parents did not go to college and
have never understood why I wanted to spend my summers at school and continue with
graduate school, but have always valued the importance of education and dreamed as big
as I can, as the popular phrase says: ‘The size of your dreams must always exceed your
current capacity to achieve them. But if your dreams do not scare you, then they are not
dreaming big enough.’
In addition, I would like to acknowledge all my professors for all their patience, office
hours, and discussion sessions invested in my education and research guidance. Throughout
my graduate school journey, my understanding of diversity and inclusion has been profoundly
reshaped. After facing the barrier of a different educational background, I am still persevering
to get the opportunity to conduct research and pursue a Ph.D.
i

Table of Contents
Acknowledgements i
List of Figures iv
Nomenclature vi
Abstract vii
1 Background 1
1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Computational Methods 9
2.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Application and Test Configurations 14
3.1 Airfoil profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 NACA0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 NACA65(1)412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Thin airfoil theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 The symmetric airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 The cambered airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Xfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Baseline 2D-RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5.1 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5.2 Grid independence study . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Synthetic jet geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6.1 NACA0012 rectangular cavity . . . . . . . . . . . . . . . . . . . . . . 25
3.6.2 NACA65(1)412 circular cavity . . . . . . . . . . . . . . . . . . . . . . 26
3.7 RANS Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Results 29
4.1 Case 1. NACA0012 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Case 2. NACA0012 Synthetic Jet Cavity . . . . . . . . . . . . . . . . . . . . 32
4.3 Case 3. NACA65(1)412 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Case 4. NACA65(1)412 Rectangular Synthetic Jet Cavity . . . . . . . . . . . 38
4.5 Case 5. NACA65(1)412 Circular Synthetic Jet Cavity . . . . . . . . . . . . . 40
ii

5 Discussion and conclusions 44
Bibliography 48
iii

List of Figures
1.1 Pressure coefficient plot at 2° degrees for [10] experiments. . . . . . . . . . . 5
1.2 Vorticity contour plots for the airfoil, baseline case (a) and with a cavity (b),
at 10° degrees showing the separation bubles and wake of the airfoil. [16] . . 6
3.1 NACA0012 geometry.[5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 NACA65412 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Thin airfoil theory example for symmetric and cambered airfoil. . . . . . . . 18
3.4 NACA0012 experimental lift coefficient data compared with Xfoil and thin
airfoil theory at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 NACA0012 experimental drag coefficient data compared with Xfoil data at
3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 NACA65412 experimental lift coefficient data compared with Xfoil at 3 × 106
Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.7 NACA65412 experimental drag coefficient data compared with Xfoil at 3×106
Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.8 NACA0012 coordinate points, from [20]. . . . . . . . . . . . . . . . . . . . . 23
3.9 NACA65412 coordinate points, from [21]. . . . . . . . . . . . . . . . . . . . 23
3.10 2D planar fluid C-shaped region aroung the airfoil. . . . . . . . . . . . . . . 23
3.11 Different grid divisions for meshing operations. . . . . . . . . . . . . . . . . 25
3.12 NACA0012 meshing model with zoom in at the leading and trailing edge. . . 26
3.13 NACA0012 with a rectangular cavity. . . . . . . . . . . . . . . . . . . . . . . 26
3.14 NACA65412 with a circular cavity . . . . . . . . . . . . . . . . . . . . . . . . 27
iv

4.1 Lift coefficient vs angle of attack comparing Xfoil data, documented experimental data, and RANS simulations at 3 × 106 Re. . . . . . . . . . . . . . . 30
4.2 Drag coefficient vs angle of attack comparing Xfoil data, documented experimental data, and RANS simulations at 3 × 106 Re. . . . . . . . . . . . . . . 31
4.3 Velocity magnitude contour for NACA0012 at 3 × 106 Re. . . . . . . . . . . 32
4.4 Lift coefficient vs angle of attack comparing RANS with and without cavity
at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Velocity magnitude contour for NACA0012 with rectangular cavity at 3 ×106
Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Pressure coefficient distribution for the NACA0012 baseline and with cavity
at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Lift coefficient vs angle of attack by comparing RANS simulations and adding
rectangular and circular cavity at 3 × 106 Re. . . . . . . . . . . . . . . . . . 36
4.8 Velocity magnitude contour for NACA65412 baseline case at 3 × 106 Re. . . 37
4.9 Velocity magnitude contour for NACA65412 with rectangular cavity at 3×106
Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.10 Pressure coefficient distribution for the NACA65412 baseline and with rectangular cavity at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.11 Velocity magnitude contour for NACA65412 with circular cavity at 3 × 106 Re. 41
4.12 Pressure coefficient distribution for the NACA65412 baseline and with circular
cavity at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1 Velocity contour comparison at 12° degrees between baseline (a)and with cavity (b) of NACA0012 at 3 × 106 Re. . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Velocity contour comparison 12° degrees between baseline (a), with rectangular cavity (b) and circular cavity (c) of NACA65412 at 3 × 106 Re. . . . . . . 46
v

Nomenclature
Acronyms
AoA Angle of Attack
CFD Computational Fluid Dynamics
NACA National Advisory Committee for Aeronautics
RANS Reynolds Averaged Navier–Stokes equations
Re Reynolds number
SST Shear Stress Transport
UAVs Unmanned Aerial Vehicles
vi

Abstract
Future industry will continue to demand lightweight aircraft. The wing design has
a significant impact on the runway distance, approach speed, climb rate, and operation
range. Then, the general idea is to design wings that delay flow separation and promote
aerodynamic efficiency. During the last few years, the USC Fabulous Fluid Facility has
been addressing this issue by conducting research on the airfoils for small unmanned aerial
vehicles that are designed for short trips or specific missions. Particularly, the NACA0012
and NACA65(1)412 airfoils are used for these applications.
In the flow control area, researchers have been exploring various methods, such as the
use of active and passive flow control techniques, where literature supports the impact of
synthetic jet cavity design. It can accelerate layers of fluid energy through holes or slots on
the surface of the airfoil. The objective of this project is to develop a flow control study of
a synthetic jet cavity by comparing computational and experimental data from NACA0012
and NACA65(1)412 airfoils, the analysis is focused on the following parameters: size, shape,
and angle of attack.
This work compares Xfoil mathematical predictions, wind tunnel experimental data
and RANS analysis for lift and drag coefficients. The RANS data is obtained by modeling,
meshing, and processing flow separation with Ansys Fluent, using the k-omega-SST turbulence model due to its proven effectiveness in adverse pressure gradients and separating
flows. The flow circulation for the boundary layer is analyzed under air standard conditions,
and an implicit scheme is utilized for time-integration stability. Finally, the solutions for
the lift and drag coefficients are determined based on residual differences and when all
simulations are iterated until full convergence is achieved.
The results show that the angle of attack for stall conditions can be modified, the lift
coefficient was reduced 9% on the NACA0012 and increased by 14% on the NACA65(1)412
with the addition of a rectangular cavity. Besides, by changing the cavity shape from
rectangular to circular, the lift coefficient only increased 5% on the NACA65(1)412 with the
synthetic jet cavity.
vii

Chapter 1
Background
1.1 State of the art
Since the initial 12-second flight by the Wright brothers to the modern aircraft industry,
the relationship between size and weight has been a critical factor for the aviation industry,
evolving along with the diversification of aircraft shapes and configurations. This trend
is expected to continue and keep the focus on aerodynamic efficiency and structural integrity.
Future industry will continue to demand lightweight aircraft, but with sufficiently
thick airfoil sections to provide stiffness. However, according to Olsman[1], increasing the
thickness of conventional airfoils may lead to increased flow separation and a decrease in
aerodynamic performance. The general idea is to design wings that delay flow separation
and promote aerodynamic efficiency while having structural strength.
During the last few years, the USC Fabulous Fluid Facility has been addressing this
issue by conducting research on the airfoils for small unmanned Aerial Vehicles that are
designed for short trips or specific missions. These airfoils operate at 102 − 103 Reynolds
numbers, which leads to reduced efficiency performance compared to higher Reynolds
numbers due to laminar boundary layer separation.
1

According to Tank [2], the discrepancy in the low Reynolds number experimental data
can be mainly attributed to the susceptibility of lift and drag to variations in experimental
conditions, such as free-stream turbulence level, surface finish, model mounting technique,
and acoustic environment. Furthermore, there are challenges related to accurately measuring the small aerodynamic forces at low Re, particularly drag, with acceptable levels of
uncertainty. The increased sensitivity to experimental conditions at low Reynolds numbers
is a result of their greater influence on the boundary layer over the airfoil.
Therefore, examining the flow control at the airfoil surface boundary layer is a crucial
aspect to consider. When exposed to an adverse pressure gradient, there is a potential for
a flow reversal close to the wall, which can lead to the separation of this smooth boundary
layer.[3] Consequently, this may cause an increase in pressure drag and a reduced lift. The
behavior of the laminar boundary layer varies based on factors such as pressure distribution;
it may reattach temporarily, creating a separation bubble, and subsequently transitioning
into a turbulent boundary layer. This temporary transition contributes to enhanced aerodynamic performance with localized changes in lift and drag. [4]. At these 102 − 103 Reynold
numbers, the variations between reattachment and complete separation occur intermittently, leading to data hysteresis, where measurements differ depending on angle of attack.[5]
In contrast, at higher Reynolds numbers, 3 × 106
, the viscous forces become relatively
insignificant compared to the nonlinear inertial forces, which are more likely to lead to a
transition - turbulent boundary layer. In this state, the turbulent boundary layer effectively
mixes momentum from the freestream towards the no-slip wall and exhibits enhanced
resistance to flow separation. [6] As a result of these characteristics, turbulent boundary layers tend to remain attached for longer periods compared to laminar boundary layers.[3]
2

In modern times, airfoils are considered suitable shapes that induce a maximum
lift-to-drag ratio. Therefore, airfoil design optimization based on the Reynolds number
has become an interesting research area. To address these challenges, researchers have
been exploring various methods, such as the use of active and passive flow control techniques.
The flow control mechanisms can be categorized into passive and active flow control
devices. The main distinction between these lies in the fact that active flow control relies
on some form of energy input to manipulate the flow, whereas passive flow control methods
manipulate the flow without the need for any external sources of energy.[7]. For example,
active techniques can involve adjusting the frequency of a speaker or synthetic jets, and
passive techniques can introduce static gurney flaps, and blow holes. Synthetic jets are the
primary focus of this study.
In 1998, Glezer et al. [8] defined synthetic jets as the result of the interaction of
distinct vortical structures generated by periodically ejecting fluid through an orifice
at the flow boundary. Unlike traditional continuous or pulsed jets, synthetic jets are
formed using the working fluid of the flow system where they operate, thereby imparting
linear momentum to the flow system without introducing mass across its boundary. As
a result, when synthetic jets interact with an external flow near the boundary, they
can create closed recirculation regions, leading to apparent alterations in the boundary’s
behavior. In short words, Pitita et al. [9] defined synthetic jets, also known as zero net
mass flux, as an orifice, an oscillating membrane, or a cavity that influences the flow behavior.
In most situations, the primary objective of this control flow technique is to eliminate
or accelerate layers of fluid energy through holes or slots on the surface of the airfoil. In the
latter scenario, the high-energy fluid introduced into the boundary layer pushes the velocity
of the fluid away from the wall, consequently reducing friction drag.[10]
3

Numerous experimental and computational studies have been conducted in the literature to investigate the impact of synthetic jet design. In recent times, jet-based synthetic
control devices have demonstrated promising potential for use in industrial applications and
have proven effective in managing flow separation, as stated by Glezer, Amitay, Rumsey,
Wygnanski, and Findanis.[11]
For example, in the same research line, Sarvankar et al. [12] created a numerical
model for the simulation of flow over a NACA0012 airfoil with the commercial software
Ansys Fluent and affirmed that the k − ω − SST model for turbulence emulates the
experimental values of lift and drag coefficients. Furthermore, by adding static gurney
flaps, optimal results were obtained with 0.25% of the chord of the leading edge and the
maximum value of the lift-to-drag ratio increased 4%. That study pointed to the benefits of incorporating a gurney flap and a suction slot to reduce drag and delay flow separation.
In addition, Vladimir Kornilov [10] used blow and suction orifices on the upper and
lower surface at .61 c - .79c on a NACA0012 at 7 × 105
. He compared modified and
unmodified airfoils for constant inlet velocity and showed that the modified airfoils have
better aerodynamic performance due to the increase in lift coefficient. A plot of pressure
coefficients was used to show an increase in pressure difference before the orifices, as
indicated by Figure 1.1.
Moreover, Yousefi et al. [13] studied a tangential blowing slot at 5x105 Re on the
trailing edge and a perpendicular suction slot at the leading edge. The most effective
location for achieving the desirable effects was 2.5% of the chord length for suction at the
airfoil leading edge. Meanwhile, the jet widths of 3.5% and 4% of the chord length were
the most effective. The results of this study demonstrated that when the blowing jet width
4

Figure 1.1: Pressure coefficient plot at 2° degrees for [10] experiments.
increases, the lift-to-drag ratio improves.
Another study by Yousefi and Saleh [14], they explored the impact of increasing the jet
length. As the jet length increased, the lift coefficient increased while the drag coefficient
decreased, resulting in an enhanced lift-to-drag ratio. This leads to a greater improvement
in the lift-to-drag ratio compared to the tip suction. On the other hand, when the jet was
short, the tip suction yielded a higher lift-to-drag ratio. They concluded that for center
suction jet lengths of 0.25 times the chord length, the efficiency ratio increased by 2%, and
in the case of tip suction, the lift-to-drag ratio increased by 9% for jet lengths of 0.25 times
the chord length, respectively.
Also, in [11] D. You and Moin, large-eddy simulation work confirmed that the position
of the synthetic jet at 12 % of the leading edge chord effectively delays the onset of flow
separation and causes a significant increase in the lift coefficient. The drag coefficient was
found to decrease approximately 15 to 18% with the synthetic jet and the stall angle of
attack for which increases from 12° for an uncontrolled airfoil to approximately 18° for the
controlled case.
Exploring a different shape, Olsman et al.[15] examined a circular cavity with their
5

focus being the dynamic response of the airfoil featuring a cavity, rather than a study on
boundary layer separation control. Their findings demonstrated that the pressure variances
were elevated from -20° to 20° angles of attack, a result of the cavity’s location on a
NACA0018 suction side near the leading edge.
Furthermore, the same author in [16] presented a comparison between DNS and
experimental data for a circular cavity situated at 0.21c on a NACA0018 with a Reynolds
number of 20,000. The overarching effect of the cavity is the creation of vortices that lessen
the flow separation.
The most significant increase in the lift-to-drag ratio were observed for 10° degrees,
pictures of the vorticity contours are shown in Figs. 1.2a and 1.2b. Here, the flow across the
airfoil with a cavity separates considerably ahead of the cavity’s forward edge, and the cavity
is located within the separation bubble. They described that the wake of the airfoil with a
cavity has smaller vortices and is narrower. The vortex dipoles in the wake of the airfoil
are associated with fluctuations in the lift force on the airfoil, may reasonably infer that the
wake would become turbulent and modify the coherence and strength of these structures.[16]
(a) (b)
Figure 1.2: Vorticity contour plots for the airfoil, baseline case (a) and with a cavity (b), at
10° degrees showing the separation bubles and wake of the airfoil. [16]
When the angles of attack are significantly positive, the flow separates ahead of
the cavity, and this separated flow interacts with the cavity, resulting in a wake that is
narrower than in the absence of a cavity [16]. A narrower wake at the top surface of an
6

airfoil indicates that the flow behind the airfoil is more concentrated and streamlined.
This often suggests better aerodynamic performance because a narrower wake means
less energy is lost to turbulence and drag. Essentially, it means the airfoil is causing less
disturbance to the airflow behind it, which can lead to improved efficiency and reduced drag.
In this context, the Table 1.1 summarizes the research done in this area.
Author Airfoil Reynolds
number AoA x ∆cl ∆cd ∆cl/cd
Vladimir
Kornilov
2D
NACA
0012
Experiments
7 × 105
-6° - 12°
0.06c to
0.07c 2% 0.86% 2.3 %
Sarvankar
et al.
2D
NACA
0012
RANS
9 × 106 0° - 16°
Slot
0.07c
and gurney flap
at T.E.
N/A N/A 4.4%
Yousefi
et al.
2D
NACA
0012
RANS
5 × 105 10° - 18°
0.04c
and
0.08c
0.8% 1.5% 4%
Yousefi
et al.
3D
NACA
0012
RANS
5 × 105 10° - 18° 0.03c 5% 2% 2% - 9%
D. You
and P.
Moin
3D
NACA
0015
LES
896,000 12° - 18° .12c N/A N/A 4%
Olsman
and
Colonius
3D
NACA
0018
Exp and
DNS
20,000 0° - 15° .21c N/A N/A N/A
Table 1.1: Synthetic jet cavity summary
Then, there is prior research evidence backing the postponement of stall through the
7

detachment delay from the airfoil surface using synthetic jets. The objective of this thesis
is to develop a flow control study of a synthetic jet cavity by comparing computational and
experimental data from NACA0012 and NACA65412 airfoils. The analysis focuses on the
following parameters: size of the cavity, shape and angle of attack.
8

Chapter 2
Computational Methods
In order to achieve the objective of this thesis, the following research is based on computational fluid dynamics, CFD, a numerical analysis used to obtain aerodynamic coefficient
predictions. According to Anderson [5] it can be explained in three steps:
1. A suitable fine computational grid/mesh is required adjacent to the airfoil wall to
obtain values of the flow velocity at several grid points. Then the velocity gradient at
the surface is obtained by using one-sided differencing. This gradient is essential to
calculate the shear stress and skin friction along the airfoil surface.
2. Boundary layers present on real-world vehicles are turbulent, necessitating that any
CFD, analysis of this airflow takes this into consideration. The majority of CFD simulations dealing with turbulent flows use the Reynolds-averaged Navier-Stokes equations
(RANS) and require a turbulence model. Various turbulence models are accessible, all
of which are based to a certain degree on empirical evidence. These turbulence models
introduce considerable ambiguity into drag computations.
3. Determining the positions on an object where the airflow detaches is also subject to
uncertainty. In CFD, identifying separated airflow locations usually requires solutions
based on Navier-Stokes equations; occasionally, solutions from the Euler equations can
provide some indication of flow detachment in specific cases. The characteristics and
9

positions of airflow detachment vary between laminar and turbulent flows, this could
provide uncertainty. However, CFD can be a reliable source of analysis that does not
require an experimental setup, which leads to reduced costs and time.
2.1 The Governing Equations
In detail, the flow is governed by the Navier-Stokes equations that are solved by the mass
(continuity) and momentum conservation mathematical formulations. The mass equation in
the form of a partial differential equation generally holds for the three-dimensional, unsteady
flow of any type of fluid, inviscid or viscous, compressible or incompressible. It is given by
2.1:
∂ρ
∂t + ∇ · (ρV⃗ ) = 0 (2.1)
In which ρ is the density of the flow and V⃗ the velocity vectors.
Similarly, the momentum equation can be written in vector form as 2.2
∂(ρV⃗ )
∂t + ∇ · (ρV⃗ )V⃗ = −∇p + ∇ · ⃗τ + ρ⃗g (2.2)
Where p is the pressure, τ the stress tensor, and g represents the acceleration due to
gravity.
CFD software models incorporate these governing equations to solve the Reynoldsaveraged Navier-Stokes equations. Instantaneous governing equations can be averaged over
time by employing Reynolds decomposition to segregate the variables in the Navier-Stokes
equations into mean and fluctuating parts and subsequently calculate the average velocity
values.[17] Working with laminar and turbulent flows, the models iterate the mass and
momentum. These solutions vary according to the size of the mesh, the fluid element, the
pressure field, the mean body forces, viscous stresses, and Reynolds stresses. Then, the
10

Navier-Stokes equations are simplified by replacing the unknown Reynolds stresses with a
simple model.
∇ · U = −
∇P
ρ
+ µ∇2U − ρu
′
iu
′
j
(2.3)
Here the term u
′
iu
′
j
is referencing the Boussinesq assumption, which states that the turbulent eddy viscosity is proportional to the mean strain rate, simplifying the Navier-Stokes
equations by replacing the unknown Reynolds stresses with a simple model.
Background research [6], [10], [12], [13], and [14] indicated that the Shear Stress Transport, SST, model is adequate to solve the flow around an airfoil shape. According to Ansys
Fluent, a commercial software, this model can blend the robust and accurate formulation of
the k −ω model in the near-wall region and independently of the k −ε model in the far field.
In other words, the SST k − ω model is related to the standard k − ω model, but includes:
• The standard k − ω model and the transformed k − ε model are added together with a
blending function designed to activate the transformed k − ε model at the wall region.
• The SST model includes a damped cross-diffusion derivative term in the ω equation.[18]
• Turbulent viscosity definition is adjusted to consider the transportation of turbulent
shear stress.[18]
These SST k − ω model characteristics provide a solution for several kinds of flows. In
particular, the transportation equations for this model are 2.4 and 2.5 [18]
∂(ρk)
∂t +
∂(ρuik)
∂xi
=
∂
∂xj

Γk
∂k
∂xj

+ Gk − Yk + Sk (2.4)
∂(ρω)
∂t +
∂(ρuiω)
∂xi
=
∂
∂xj

Γω
∂ω
∂xj

+ Gω − Yω + Dω + Sω (2.5)
In the following equations, Gk denotes the generation of kinetic energy of turbulence
resulting from mean velocity gradients, computed as Equation 2.6, with the Boussinesq
11

hypothesis, Gk = µtS
2 where S is the modulus of the mean rate of strain tensor, defined in
the same way as for the model k- ϵ.
Gk = −ρu
′
iu
′
j
∂uj
∂xi
(2.6)
Meanwhile, Gω denotes the generation of ω, the specific dissipation rate, which measures
the rate at which turbulent kinetic energy dissipates due to viscous effects calculated as
Equation 2.7.
Gω = α
ω
k
Gk (2.7)
Here, k represents the turbulent kinetic energy, which measures the energy associated
with turbulent fluctuations. The coefficient α is given by Equation 2.8.Rω and Re t are
constant depending on the flow model.The parameter α
∗
reduces the turbulent viscosity,
leading to a correction for low Reynolds numbers. For high Reynolds numbers, the k- ω
model α = α∞ = 1.
α =
α∞
α∗

α0 + Ret/Rω
1 + Ret/Rω

(2.8)
Γk and Γω denote the effective diffusivity of k and ω, which are determined as Equations
2.9 and 2.10 where σk and σω represent the turbulent Prandtl numbers for k and ω, correspondingly. The turbulent viscosity, µt
, is determined by combining k and ω as Equation
2.11
Γk = µ +
µt
σk
(2.9)
Γω = µ +
µt
σω
(2.10)
12

µt = α
∗
ρk
ω
(2.11)
Next, Yk and Yω denote the dissipation of k and ω caused by turbulence. Then Dω
denotes the cross-diffusion term, as Equation 2.12. Finally, Sk and Sω are the source terms
defined by the user. [18]
Dω = 2(1 − F1)ρσω,2
1
ω
∂k
∂xj
∂ω
∂xj
(2.12)
Therefore, Ansys fluent is the software that this thesis will use to calculate the RANS
simulations. It is documented that flow separation is dependent on its ability to enhance
the stability of the boundary layer by introducing or extracting vortex patterns. Then,
considering that a synthetic jet cavity will be added and analyzed in this study, the
hypothesis is that it can optimize the airfoil performance, the effectiveness of which depends
on factors such as the size, shape, and angle of attack of the cavity.
13

Chapter 3
Application and Test Configurations
A baseline is established to serve as a point of reference for NACA0012 and NACA65412.
In this case, the thin airfoil theory and the vorticity panel method are used to calculate
the ideal values of the lift and drag coefficients. Second, the documented experimental data
from Abbott and Von Doenhoff [4] are reviewed. Next, modeling, meshing, and processing
with Fluent are performed to obtain computational calculations of the airfoil without any
modifications. Then, a synthetic jet cavity is added with a specific cavity size, position,
Reynolds number, and angle of attack to repeat the computational analysis. Lastly, the
results are analyzed and compared with the experimental wind tunnel data.
3.1 Airfoil profiles
The NACA0012 and NACA65(1)412 are analyzed during this study, to compare symmetric
and cambered airfoil profiles.
3.1.1 NACA0012
This airfoil is symmetrical, with an identical shape above and below the chord line. In this
type of airfoil, the chord line aligns with the camber line, and the maximum thickness of
14

the profile is 12% of the chord. This particular airfoil is featured on helicopter blades or,
particularly, on the Lockheed C-5 Galaxy aircraft.
The profile documentation indicates that the maximum thickness is observed at x/c =
0.3, while the minimum pressure point occurs on the surface at x/c = 0.11. It should be
noted that the location of the minimum pressure and, thus, the maximum velocity does not
align with the position of the maximum thickness of the airfoil. [5]
Figure 3.1: NACA0012 geometry.[5]
3.1.2 NACA65(1)412
NACA65(1)412, which is also called NACA65412 for simplicity, is an airfoil with a camber,
having the highest camber at x/c = 0.5 of the chord length. The 6-series airfoils are
specifically created to enhance the area where the airflow stays laminar and decrease drag,
promoting a higher degree of laminar flow compared to the 4- or 5-series airfoils. The
second digit, 5, is the second position indicates the location of the minimum pressure in
tenths of the chord at x/c = 0.5.
The number in parentheses (1) signifies that minimal drag is sustained at lift coefficients
0.1 higher and lower than the designated lift coefficient, 0.4, indicated by the third digit in
tenths. Lastly, the final two digits indicate a thickness of 12% of the chord at 39.9%. This
can be found in turbine blades and supersonic jets.
15

Figure 3.2: NACA65412 Geometry
3.2 Thin airfoil theory
In aerodynamics, theory there are two well-known ways to work with low-speed airfoils, one
is the thin airfoil theory and the other is using the vortex panel method.
3.2.1 The symmetric airfoil
According to the Kutta-Joukowski theorem, an airfoil can be represented by a vortex sheet
positioned along its camber line.[5] For a symmetric airfoil, the camber is considered negligible, as it aligns with the chord line. The fundamental equations required to determine
the lift coefficient are outlined in this section, starting from the total circulation around the
airfoil, as Equation:
Γ = αcV∞
Z π
0
(1 + cos θ)dθ = παcV∞ (3.1)
Where α is the angle of attack, c the chord, V∞ the freestream velocity, and θ an small
angle. Substituting Equation 3.1 into the Kutta-Joukowski theorem, the lift per unit span
can be found as Equation:
L
′ = ρ∞V∞Γ = ρ∞V∞(παcV∞) (3.2)
To finally lift coefficient is obtained as 3.3:
cl =
L
′
q∞S
(3.3)
16

When discussing 2D airfoils as opposed to 3D airfoils, the surfaces are defined by the
chord length. This can be expressed as S = c(1). Substituting Equation 3.2 into 3.3, it
states that the lift coefficient linearly proportional to the angle of attack:
cl =
παcρ∞V
2
∞
1
2
ρ∞V
2
∞c(1) = 2πα (3.4)
Then, the differential of the lift represents the lift slope as 3.5:
Lift slope =
dcl
dα
= 2π (3.5)
3.2.2 The cambered airfoil
Starting with the total circulation equation for a cambered airfoil:
Γ = cV∞

πA0 +
π
2
A1

(3.6)
With A0 = α −
1
π
R π
0
dz
dx dθ0 and An =
2
π
R π
0
dz
dx cos nθ0dθ0
The lift per unit span becomes as 3.7:
L
′ = ρ∞V∞Γ = ρ∞V
2
∞c

πA0 +
π
2
A1

(3.7)
Equation 3.8 results in the lift coefficient being expressed as:
cl =
L
′
1
2
ρ∞V
2
∞c(1) = π(2A0 + A1) (3.8)
Substituting the values of A0 and A1 into 3.8 leads to Equation 3.9.
cl = 2π

α +
1
π
Z π
0
dz
dx(cos θ0 − 1)dθ
(3.9)
Lift slope = dcl
dα = 2π (3.10)
17

Similar to a symmetric airfoil, the theoretical lift slope for a cambered airfoil is 2π,
indicating that it remains constant regardless of the shape of the airfoil. However, there is
a distinction in the expression for Cl between symmetric and cambered airfoils, primarily
due to the presence of the integral terms A0 and A1. These terms make cl = 0 when
α = 0 for a symmetric airfoil and cl a negative value for a cambered airfoil at the same angle.
As an example, Figure 3.3 shows the relationship of Cl and the angle of attack:
Figure 3.3: Thin airfoil theory example for symmetric and cambered airfoil.
3.3 Xfoil
Xfoil is a computational tool designed for developing airfoils suitable for a range of Reynolds
number conditions. It integrates a potential flow panel technique with an integral boundary
layer model to assess airflow around airfoils. The software process starts with providing
airfoil coordinate points, Reynolds number, and Mach number value, so Xfoil can quickly
forecast airfoil behavior. Convergence of the solution is achieved by iteratively adjusting
18

the flow solutions based on the angle of attack.[19]
Furthermore, Xfoil provides the option for the user to define an Ncrit parameter,
to simulate the level of free stream turbulence, turbulence intensity, in the analysis by
emulating a transition point. A higher value of Ncrit models the effects of reduced levels of
free stream turbulence, whereas lower values of Ncrit indicate a higher level of turbulence
in the flow field. The typical Ncrit values suggested by Xfoil for different scenarios are as
follows: Ncrit = 12 − 14 for sailplanes, Ncrit = 11 − 13 for motorglider, Ncrit = 10 − 12
for clean wind tunnel, and Ncrit = 9 for average standard wind. This last one is used in
this study.[19] [6]
Lift and drag coefficients were calculated using Xfoil for NACA0012 and NACA65412.
The values are displayed in the Figures 3.4, 3.5, 3.6, and 3.7. The predicted values are
analyzed in the following section.
3.4 Experimental data
In this section, experimental data are obtained from [4] for NACA0012 and NACA65410,
which is taken as the closest to NACA65412 at 3 × 106 Reynolds number. It is important
to note that the Xfoil software provided all numbers with five decimal values, whereas the
experimental values were obtained by visual analysis of graphs, so there is human-induced
variability.
In Figure 3.4 the experimental lift coefficient is compared with the Xfoil data and
the thin aifoil theory line. Specifically, it is noted that the lift coefficient predictions
by Xfoil (represented by blue diamonds) align with the thin airfoil line (in black) from
0° up to 13° However, beyond 13° the blue line diverges from the black line. This
divergence can be attributed to the onset of a separated flow between 14° and 16° on
19

the upper surface of the NACA0012 airfoil, leading to stall due to flow detachment.
In the same way, the experimental green curve follows the same path as the blue one,
with a major uncertainty at 9° which has the greatest difference of 8% between all the values.
Similarly, Figure 3.5 illustrates the position of the Xfoil predictions (color green) for the
drag coefficient with experimental results (color blue), where uncertainty is inherent in any
experimental study. The difference remains relatively low, an average percentage difference
of 11% from 0° to 8° but it rises to 16% from 9° to 12° possibly due to the influence of
the configuration of the setup and the flow conditions. Particularly, at 9° degrees there
is an evident difference with the experimental value, this may suggest the presence of a
separation bubble, a region where the boundary layer transitions from laminar to turbulent,
with a temporary recirculation zone.
Figure 3.4: NACA0012 experimental lift coefficient data compared with Xfoil and thin airfoil
theory at 3 × 106 Re.
Furthermore, in Figure 3.6 the experimental lift coefficient of NACA65412 is compared
20

Figure 3.5: NACA0012 experimental drag coefficient data compared with Xfoil data at 3×106
Re.
to the Xfoil data. The range was established from zero to 12 since it reflects the available
range in the experiment [4]. Once more, a slight uncertainty can be observed between the
curves, with the higher difference located at 7° and an average percentage difference of 11%.
Meanwhile, Figure 3.7 shows the prediction of the drag coefficient and the experimental
data for the NACA65412 airfoil. The average difference is 7%, and this may be a consequence
of the setup of the experimental data.
3.5 Baseline 2D-RANS
To validate the ANSYS-FLUENT model, the plain NACA0012 and NACA65412 airfoils are
drawn by points shown in Figures 3.8 and 3.9. Both of them have a chord of 1 meter in
length denoted as x=c.
21

Figure 3.6: NACA65412 experimental lift coefficient data compared with Xfoil at 3 × 106
Re.
Figure 3.7: NACA65412 experimental drag coefficient data compared with Xfoil at 3 × 106
Re.
22

Figure 3.8: NACA0012 coordinate points, from [20].
Figure 3.9: NACA65412 coordinate points, from [21].
3.5.1 Computational domain
In order to analyze the flow, the computational domain, took the form of a 2D planar fluid
C-shaped region. The domain dimensions were set at seven times the chord length (7c) to
simulate free-stream conditions and limited the influence of the boundaries. The domain
was divided into segments representing the inlet (on the left side), the airfoil, and the outlet
(on the right side) in Figure 3.10.
Figure 3.10: 2D planar fluid C-shaped region aroung the airfoil.
23

3.5.2 Grid independence study
To maintain consistency in the CFD simulations regardless of the grid size, a grid independence test was performed. The domain was divided into several smaller sections to
determine the required number of divisions. They are dividing the total length of the
domain (7c) The grid independence study was conducted with 4 different types of divisions,
as shown in Figure 3.11, until it was concluded that after 400 divisions (.009c cell length)
there is no mesh dependency.
This is an exponentially changing mesh size applied along the far field to accurately
capture the boundary layer and keep the y+ value lower than 1. A close-up view of the
mesh is depicted in Figure 3.12, and its features are detailed in Table 3.1.
Sizing Value
Growth Rate 1.2
Number of Divisions 400 = .009c cell length.
Curvature Min Size 13.52 mm = 1.35 × 10−2
c
Minimum Edge Length 46.83 mm = 4.68 × 10−2
c
∆ Leading Edge Min .0064 mm = 6.4 × 10−6
c
∆ Upper Surface Min .0063 mm = 6.3 × 10−6
c
∆ Lower Surface Min .0063 mm = 6.3 × 10−6
c
∆ Trailing Edge Min .0064 mm = 6.4 × 10−6
c
Table 3.1: Mesh sizing values.
3.6 Synthetic jet geometry
The previous grid independence study was repeated for all the airfoil models studied, keeping
the same division and sizing parameters. The dimensions of the cavities were specified as
follows.
24

(a) 100 Divisions = .035c cell length. (b) 200 Divisions = .018c cell length.
(c) 300 Divisions = .012c cell length. (d) 400 Divisions = .009c cell length.
Figure 3.11: Different grid divisions for meshing operations.
3.6.1 NACA0012 rectangular cavity
On the top surface of the NACA0012, the rectangular cavity of 0.02c (depth) × 0.05c (length)
was placed in the normalized position of x= 0.05c. The location was determined according
to the wall shear stress, a tangential force per unit area on the airfoil surface, governed by
the slope of the velocity profile. This is because it can provide important information on
the behavior of the boundary layer. Then a zero value of the wall shear stress was chosen to
determine the location of the cavity; when this value is reached, it can signify a transition
25

Figure 3.12: NACA0012 meshing model with zoom in at the leading and trailing edge.
point between different flow regimes or a point where the boundary layer starts to detachment
from the surface of the airfoil.
Figure 3.13: NACA0012 with a rectangular cavity.
3.6.2 NACA65(1)412 circular cavity
Similarly, Olsman [1] conducted a successful test on a circular cavity, which is detailed
here to assess its effects. The NACA65412 cavity has a diameter of 0.1 c at a normalized
position of 0.1 c (Figure 3.14), where the wall shear stress is zero. In this study, the cavity
configuration was altered to investigate the impact of increasing its dimensions twice and
the effect of sharp edges on a rectangular cavity.
The location of the cavity was compared with the literature; Godarzi and Yousefi [22]
26

Figure 3.14: NACA65412 with a circular cavity
and You and P. Moin [11] agreed that the cavity location should be placed between the
leading edge and 0.1c.
3.7 RANS Setup
This part covers the RANS solution setup procedure. These simulations use a steady
pressure-based solver incorporating the k − ω − SST turbulence model and a second-order
upwind scheme to discretize the convective terms.
First, the inflow boundary experienced the average free-stream velocity. Next, the
specified boundary condition closely resembles the traditional definition of the Dirichlet
boundary condition. Here, the standard atmospheric pressure of 101,325 Pa was assigned as
the reference pressure at the outflow boundary. The rest of the free-stream parameters are
shown in Table 3.2.
Parameter Value
Reynolds number 3 × 106 Re
Temperature 288.16 K
Dynamic viscosity 1.80 × 10−5 kg/(ms)
Velocity 45.6 m/s
Density 1.225 kg/m3
Table 3.2: Free-stream parameters for RANS calculations.
The k − ω − SST turbulence model, mentioned in Chapter 2, is used due to its proven
effectiveness in adverse pressure gradients and flow separation. The fluid being analyzed is air
under standard conditions, and an implicit scheme is utilized for time-integration stability.
In addition, to ensure precise resolution of the flow and turbulent kinetic energy, a second27

order upwind scheme is chosen. Lastly, the convergence of the solution for the lift and drag
coefficients is determined based on residual differences and when all simulations are iterated
until full convergence is achieved.
28

Chapter 4
Results
4.1 Case 1. NACA0012 Baseline
To achieve a solution to the problem, it is essential to test the computational algorithm
with the baseline cases. First, the analysis focuses on the baseline NACA0012; the angle of
attack varies from 0° to 16° to analyze the lift coefficient and from 0° to 13° to study the
drag coefficient, based on accessible experimental data.
Figure 4.1 illustrates the fundamental characteristics of NACA0012, contrasting
thin airfoil theory, Xfoil estimations, and experimental data discussed in the previous
section, now incorporating the RANS results (highlighted by red rectangles) derived from
simulations. It can be seen that the red lift curve gradually increases as it approaches
the stall condition, which is documented for NACA0012 to occur at 16° when the
Reynolds number is 3 × 106
. Furthermore, the RANS values agree well with the Xfoil
data; the difference was calculated for each value as 5% from one curve to another. Meanwhile, the experimental data have an average difference of 6% compared to the RANS values.
Moreover, Figure 4.2 describes the drag coefficient baseline behavior of NACA0012,
comparing RANS with Xfoil approximations and experimental data. The average difference
29

Figure 4.1: Lift coefficient vs angle of attack comparing Xfoil data, documented experimental
data, and RANS simulations at 3 × 106 Re.
with the Xfoil data is 15%. However, when the values are compared with the experimental
data, the average deviation is 14%; both significantly higher than the lift coefficient
variations.
To understand the variation in the graph 4.2, it can be compared with the results
of Lombardi et al. [5], who also experience disparities between the experiment data and
the CFD-calculated values for the friction drag of the skin and the total drag coefficient
of NACA0012 at the same Reynolds number. That work suggests that the pressure drag
resulting from flow separation accounts for 15 percent of the overall drag. On a streamlined
two-dimensional object, the drag is predominantly attributed to skin friction, with pressure
drag being relatively minor in comparison. However, using CFD, where the transition point
is artificially created, the drag value obtained is a consequence of skin friction, but may not
calculate the pressure drag related to flow separation very accurately.
30

Figure 4.2: Drag coefficient vs angle of attack comparing Xfoil data, documented experimental data, and RANS simulations at 3 × 106 Re.
In addition, the flow characteristics of the NACA0012 can be studied in greater detail
in Figure 4.3 with a velocity contour plot.
In detail, Figure 4.3-a shows the velocity magnitude distribution at a 0° where the flow
is smooth and attached. The stagnation point is clear in color blue at the leading edge, with
no significant regions of high velocity near the walls, indicating the absence of large pressure
gradients or separation. In figure 4.3-b the flow starts to accelerate more on the upper
surface of the airfoil, as indicated by the change in color toward the green spectrum, which
represents higher velocities. This is typical of increased lift production due to a positive
angle of attack. Next, Figure 4.3-c the velocity magnitude for an 8° degrees as angle of
attack shows a more pronounced high-velocity region on the upper surface, especially near
the leading edge. Finally, in Figure 4.3-d there is a significant high-speed region on the
upper surface near the leading edge. But in general, for angles lower than 12, there is no
flow separation.
31

Figure 4.3: Velocity magnitude contour for NACA0012 at 3 × 106 Re.
4.2 Case 2. NACA0012 Synthetic Jet Cavity
After defining the CFD model and describing the flow characteristics for the plain
NACA0012 the synthetic jet cavity was added. The initial mesh plus adjustments for the
cavity region and the same setup were used to obtain the RANS data for the lift coefficient
present in purple in Figure 4.4. It shows a decrease in the lift coefficient compared to
the experimental and RANS data without the cavity. Numerically speaking, the average
difference between the baseline and the cavity line is 9% lower and the largest decrease is
located at 13° angle of attack.
To address the cavity addition, Figure 4.5 shows the velocity contours. The first notable
characteristic is that the cavity is a low-velocity zone in all four angles of attack displayed.
In particular, Figure Plot 4.5-a shows the velocity magnitude of the modified NACA0012
32

Figure 4.4: Lift coefficient vs angle of attack comparing RANS with and without cavity at
3 × 106 Re.
airfoil at the angle of attack 0°, where the flow remains largely attached throughout the
surface of the airfoil, and the presence of the cavity does not show a significant disturbance
in the flow pattern at this angle. Next, in Figure 4.5-b at 4° the plot reveals a slight
variation in velocity near the cavity, indicated by a local change in color. This might suggest
that the cavity is beginning to affect the flow characteristics.
Later, in Figure 4.5-c at 8° there is a slightly orange region near the leading edge and
extends to the top of the cavity indicating an increase in the magnitude of the velocity. Also,
a blue low-velocity region near the trailing edge of the airfoil is gaining more presence. This
could be indicative of flow separation. Lastly, in Figure 4.5-d at a 12° angle of attack, the
high-velocity region becomes red and extends further around the cavity area, and the blue
region is now more prominent, indicating a detachment zone. When comparing the velocity
contour with (Fig. 4.5) and without the cavity (Fig. 4.3) the more noticeable difference
is the blue low-speed area near the trailing edge of the suction surface. This observation
33

Figure 4.5: Velocity magnitude contour for NACA0012 with rectangular cavity at 3 × 106
Re.
could indicate a location change for the transition point caused by the interaction between
the cavity and the flow. In addition, considering the effect of the cavity on the airfoil’s
performance, changes in the flow pattern could affect lift, drag, and stall behavior, especially
as the angle of attack increases.
To obtain further details about the airflow surrounding the NACA0012 airfoil, the
distribution of the pressure coefficient, Cp, is illustrated in the graph 4.6a at 6° and 4.6b at
12° degrees. The baseline, Case 1, is indicated with solid lines, and Case 2 is indicated with
dashed lines. In both plots, the Cp of the upper surface is colored red, it starts at a high
negative value at the leading edge, which indicates a very low-pressure region, possibly due
to the stagnation point. This is followed by a rapid increase to a less negative value as we
move towards the trailing edge, which suggests that the pressure increases along the surface.
In the meantime, the lower surface (blue line) shows a start at zero value at the leading
34

edge, followed by an increase towards one and then a gradual decrease in both angles of
attack.
(a) 6° (b) 12°
Figure 4.6: Pressure coefficient distribution for the NACA0012 baseline and with cavity at
3 × 106 Re.
On the other hand, the configuration with a cavity, indicated by dashed lines, in both
cases, shows a different pressure distribution. For the upper surface (red), there is a very
large negative spike at the leading edge, even more so than the baseline. Also, there is a
slight drop and then recovery near 0.05-.07c which is located at the same position as the
cavity with a maximum peak value of -2.1 for 6° and -3.9 for 12° degrees, then the curve
continues to decrease towards the trailing edge. Meanwhile, the lower surface (blue) is more
similar to the baseline case, but with a slightly lower pressure.
As a general observation, the pressure difference is lower with the cavity, which implies
a reduction in the lift coefficient. The Cp values at the leading edge are significantly lower
for both surfaces with the cavity, which might be due to the altered flow dynamics caused
by introducing a little disturbance that could help accelerate the boundary layer. At a
Reynolds number of 3 × 106
, viscous effects remain notable, and the cavity is expected to
induce alterations in the boundary layer characteristics compared to the standard airfoil.
35

Then, the distribution suggests that the cavity has a significant effect on the pressure
distribution.
4.3 Case 3. NACA65(1)412 Baseline
Moving on with a different shape, the NACA65412 is used to compare the effect of the
cavity on a cambered airfoil. In Figure 4.7 there are three curves, following the same color
designation, the red is the baseline case, the purple is the case with a rectangular cavity,
and a new yellow corresponds to a circular cavity. Between the baseline curve and the
rectangular cavity curve, there is an average percentage of 14% and the higher difference is
located at 13° degrees. In contrast, the baseline curve and the circular cavity curve have a
difference of 5% with the higher value also located at 13° angle of attack.
Figure 4.7: Lift coefficient vs angle of attack by comparing RANS simulations and adding
rectangular and circular cavity at 3 × 106 Re.
36

Obtaining information about the NACA65412 baseline case, Figure 4.8 shows the
changes in the flow field as the angle of attack increases at a Reynolds number of 3 million.
Figure 4.8: Velocity magnitude contour for NACA65412 baseline case at 3 × 106 Re.
In particular, in Figure 4.8-a at 0° degrees the flow around the airfoil is even and
attached, indicating laminar flow without any signs of separation. Next, 4.8-b at 4° degrees
shows a slight increase in velocity over the upper surface (in color green) which is expected
as the angle of attack increases, and there is no indication of separation yet. Later, 4.8-c
on the upper surface at 8° the velocity near the leading edge is beginning to show a yellow
region with higher velocity, and near the trailing edge a blue region showing lower velocity.
Moreover, there is a starting stall pattern if the trend continues with a further increase in
the angle of attack. Finally, 4.8-d at 12° degrees, the high-velocity region becomes more
evident, now in color red, near the leading edge, and the low-speed region, blue, at the
trailing edge. This airfoil is likely approaching stall conditions, which according to RANS
data was at 13° degrees.
37

Then, Figure 4.9 shows that when the rectangular cavity is added, the image displays
four contour plots of the velocity magnitude around a modified NACA65412 airfoil at
different angles performed with a Reynolds number of 3 × 106
.
4.4 Case 4. NACA65(1)412 Rectangular Synthetic Jet
Cavity
Figure 4.9: Velocity magnitude contour for NACA65412 with rectangular cavity at 3 × 106
Re.
Each plot illustrates a more pronunce impact of the cavity on the normalized velocity
as the angle of attack increases. First, Figure 4.9-a at 0° degrees, the flow appears smooth
and attached across the airfoil surface. The cavity does not seem to significantly disturb
the airflow, indicating a neutral aerodynamic influence, same as 4.8-a. Second, plot 4.9-b
38

at 4° degrees a minor disturbance in the flow can be observed at the cavity location, where
the velocity magnitude is slightly altered. This could suggest that the cavity is beginning
to have a measurable effect on the flow. Third, 4.9-c at 8° degrees, the impact of the
cavity is more pronounced, with the reduction of the low-velocity region (indicated with
color blue) compared to the baseline case. This indicates that the cavity may be reducing
the flow separation. Lastly, 4.9-d plot at 12° degrees the low-velocity region has reduced
considerably, comparing with 4.8-d indicating less flow separation likely caused by the cavity.
Comparing Figures 4.8 and 4.9 the main differences lie in the flow behavior near
the trailing edge. The presence of the cavity appears to delay the flow separation as observed at 12 °degrees, suggesting that modification could have effects on the transition point.
Again, at this point, it is interesting to see the behavior of the pressure coefficient
with the presence of the cavity, then Figure 4.10 shows the upper and lower curves for the
baseline case (case 3) and with the cavity (case 4).
(a) 6° (b) 12°
Figure 4.10: Pressure coefficient distribution for the NACA65412 baseline and with rectangular cavity at 3 × 106 Re.
39

In Figure 4.10-a and 4.10-b the upper surface for the baseline case (red solid line)
exhibits a very low pressure at the leading edge, which rapidly increases, and becomes less
negative as we move toward the trailing edge. This is typical behavior for the suction side
of an airfoil as the angle of attack increases. However, the lower baseline surface (blue
solid line) near the leading edge shows much higher pressure than the upper surface and
decreases towards the trailing edge, reflecting the behavior of the pressure side of an airfoil.
When the rectangular cavity is added, at 6° and 12° degrees, the top surface (red dashed
line) starts with a significantly lower pressure coefficient at the leading edge, indicating a
stronger suction effect than the baseline case. After this, there is a pronounced increase in
pressure, which outlines the rectangular cavity, with a peak at -2.5 at 6° and approximetely
-3 at 12° degrees, marking a pressure difference with the baseline case, before gradually
falling towards the trailing edge. On the lower side (blue dashed curved) it starts similarly
to the case 1 at the leading edge, but shows a slightly lower pressure coefficient as it moves
towards the trailing edge.
Analyzing the presence of the rectangular cavity in the NACA65412 airfoil has a
noticeable impact on the pressure distribution, as the angle of attack increases, the pressure
difference area also increases. This could suggest a possible flow attachment, which could
affect the lift coefficient.
4.5 Case 5. NACA65(1)412 Circular Synthetic Jet
Cavity
With the positive results of the rectangular cavity, to enhance the impact, another shape is
studied, as illustrated in Figure 4.11. In 4.11-a with a circular cavity at no angle of attack,
40

the flow over the airfoil is smooth and uninterrupted by the cavity. Next, plot 4.11-b with a
circular cavity at 4°, the cavity starts to influence the flow pattern, now the velocity is faster
on the top surface, as it changes to green color.
Figure 4.11: Velocity magnitude contour for NACA65412 with circular cavity at 3 × 106 Re.
Next, 4.11-c with a circular cavity at 8° there is a high-velocity region (red) at the
leading edge and a distinct low-velocity region extending toward the trailing edge (blue),
indicating the beginning of flow separation. Lastly, plot 4.11-d with a circular cavity at 12°
the high-speed region has expanded over the cavity, while the low-speed region expanded,
suggesting flow separation. However, this region is smaller than the baseline case.
To depict the flow around the circular cavity and make a comparison with Case 3
and Case 4, Figure 4.12 illustrates the pressure coefficient at 6 ° and 12° degrees. For
the case 3, baseline, in both angles the upper surface is in a red continuous line that
experiences a significant negative pressure coefficient at the leading edge, indicating a
strong suction peak. This is followed by a rapid recovery as it moves towards the trailing
41

(a) 6° (b) 12°
Figure 4.12: Pressure coefficient distribution for the NACA65412 baseline and with circular
cavity at 3 × 106 Re.
edge. The lower surface (blue solid line) begins with a pressure coefficient of zero, which
then rises to 1 before it starts to decline as it approaches the trailing edge, which is expected.
However, at both angles, the circular cavity on the upper surface is represented with
a red dashed line and it shows an extremely negative pressure coefficient at the leading
edge, even more so than the case 3, no cavity. Following this, there is a fluctuation
characterized by a sudden increase in pressure and then a decrease, forming a distinct
dip, with a peak at -2.3 at 6° and -3.2 at 12° before gradually increasing towards the
trailing edge. Meanwhile, the lower surface (blue dashed line) has a behavior similar to
case 3, the baseline. In general, the pressure difference are is bigger in the circular cavity case.
Now comparing the circular cavity with the previous cases, case 3 baseline and case 4
rectangular cavity. The pressure distribution area increases with the cavity as the angle
increases. It also shows more variation than the pressure gradient seen in the baseline case,
due to the cavity disturbance in the flow, especially noticeable at higher angles of attack,
which helps to energize the boundary layer and prevents separation as possible. In the
same way, the rectangular cavity seemed to have a more pronounced effect on delaying flow
42

separation at higher angles of attack compared to the circular cavity. This could be due to
the shape and edges of the rectangular cavity inducing a different flow pattern. However,
both cavities demonstrate that they can influence the separation point.
This section can be concluded with table 4.1, it shows the maximum ratio of lift coefficient to drag coefficient for all the five cases, all the values were calculated with the RANS
lift and drag values.
Case Description (cl/cd)max
1 NACA0012 Baseline 45
2 NACA0012 Rectangular 44
3 NACA65412 Baseline 66
4 NACA65412 Rectangular 78
5 NACA65412 Circular 71
Table 4.1: (cl/cd)max comparison table.
Here for the NACA0012, case 2 shows lower (cl/cd)max, this suggests that the rectangular modification at that position has a detrimental impact aerodynamic efficiency compared
to the baseline, case 1, for this airfoil. In addition, Case 3, Case 4, and Case 5 involved the
NACA65412 airfoil with different configurations. The NACA65412 baseline has a higher
value than the NACA0012 series at 3 × 106Re. However, the NACA65412 with rectangular
cavity shows the highest (cl/cd)max among all cases, the circular one has a value higher than
the baseline but lower than the rectangular case.
The table showed that modifications to the shape could affect the aerodynamic efficiency measured by the lift-to-drag ratio for a cambered airfoils. The NACA65412 airfoil,
particularly with the rectangular modification, exhibits superior performance compared to
the baseline. This suggests that specific design modifications can enhance its efficiency.
43

Chapter 5
Discussion and conclusions
The background, Chapter 1, shows that the introduction of a cavity for the application
of synthetic jets could be used to affect the flow separation over cambered airfoils at high
angles of attack. In particular, the cavity can be strategically placed along the surface of an
airfoil to be beneficial for the aircraft during the take-off and landing phases.
In chapter 2, the computational methods were presented to understand the application
and analysis of the Reynolds-Averaged Navier-Stokes (RANS) simulations in aerodynamic
performance. Then, in Chapter 3 the thin airfoil theory, Xfoil, and experimental configurations were explained to set cases: the NACA0012 and NACA65412 airfoil, both with
and without synthetic jet cavities. Later, chapter 4 showed a successful numerical model
for flow simulation with the he k − ω − SST equations in Ansys software, and it was validated in agreement with documented data, particularly with the Reynolds number of 3×106
.
For the NACA0012 airfoil, Figures 4.1 and 4.2 highlight how RANS simulations
closely match the Xfoil data and experimental findings, showing an average percentage
difference of 5% and 15% for lift and drag coefficients, respectively, suggesting a robust alignment with established aerodynamic theories and models, particularly in the prediction of lift.
44

In addition, detailed velocity magnitude graphs (Figures 4.3 and 4.5) and pressure
coefficient distributions (Figure 4.6) showed the influence of synthetic jet cavities on flow
separation. It is easier to compare both in Figure 5.1 which shows the role of the cavity in
flow separation.
Figure 5.1: Velocity contour comparison at 12° degrees between baseline (a)and with cavity
(b) of NACA0012 at 3 × 106 Re.
In exploring the effects of synthetic jet cavities on a cambered airfoil profile, the
NACA65412, Figures 4.7 to 4.12 reveal that both rectangular and circular cavities impact
the lift coefficient and flow patterns. Specifically, the presence of cavities alters the distribution of the pressure coefficient and flow separation, as shown in the velocity magnitude
graphs for the NACA65412 with rectangular cavities (Figure 4.9) and circular cavities
(Figure 4.11). This comparison is easier to see in Figure 5.2, where case 3, 4 and 5 are
show at 12° degrees. Here, the baseline has the largest separation region, while this zone
is reduced with the presence of the cavity, introducing a little disturbance that may be
energizing the boundary layer.
Hence, the analysis establishes a link between experimental data and computational
fluid dynamics (CFD) models, offering valuable insights into the aerodynamic characteristics of airfoils equipped with synthetic jet cavities. It is important to keep in mind
45

Figure 5.2: Velocity contour comparison 12° degrees between baseline (a), with rectangular
cavity (b) and circular cavity (c) of NACA65412 at 3 × 106 Re.
the limitations of the Reynolds-averaged Navier-Stokes (RANS) equations, which average
turbulent fluctuations over time to yield a steady-state solution and simplify the complexity
of turbulent flows. These simplifications have their limitations, particularly when dealing
with flows that exhibit inherent unsteadiness and temporal variability. In cases of inherently
unsteady phenomena like flow separation, vortex shedding, or buffeting, RANS models
may not accurately capture the transient dynamics of the flow. To address time-varying
phenomena, there are Unsteady RANS (URANS) simulations, which enables the resolution
of larger-scale unsteady effects by solving the RANS equations in a time-dependent manner.
Nonetheless, it still employs the turbulence averaging concept and may not precisely capture
all turbulence scales, especially the smaller ones. In scenarios where unsteady effects play a
critical role, more sophisticated simulations such as Large Eddy Simulation (LES) or Direct
Numerical Simulation (DNS) may be necessary. While these methods are expensive and
computationally demanding, they directly resolve a broader range of turbulent scales and
are better equipped to capture the time-varying behavior of the flow.
While RANS provides a powerful tool for analyzing and designing aerodynamic surfaces
under steady flow conditions, its capability to accurately predict the behavior of inherently
unsteady and time-varying processes is limited. Recognizing these limitations is important
for the appropriate application of RANS models and for deciding when more advanced
simulation techniques are necessary to capture the dynamics of unsteady flows.
46

Future work aims to investigate the impact of synthetic jet cavities at lower Reynolds
numbers, specifically 103
, relevant for UAV applications. Currently, the focus is on
understanding the relationship between velocity contours and Reynolds numbers, which
represent the balance between inertial and viscosity forces. It is anticipated that at lower
Reynolds numbers, the effect of the cavity on the lift coefficient will diminish. Thus, the
next step in research could involve quantifying or estimating the extent to which this reduction in lift coefficient is influenced by the Reynolds number, particularly on cambered airfoils.
In summary, this research studied the impacts of cavities on a symmetric and cambered airfoil, establishing a basis for future investigations on incorporating synthetic jet
mechanisms. This work highlighted the possibility of synthetic jet cavities to control airflow patterns and create opportunities to control stall conditions, which could be an real
application for flight mechanics during landing and take-off operations for 3D airfoils.
47

Bibliography
[1] Willem Olsman. “Influence of a cavity on the dynamical behavior of an airfoil”. In:
Technische Universiteit Eindhoven Thesis (2010), pp. 1–3.
[2] Joe Tank. “Aerodynamics at Low Re: Separation, Reattachment, and Control”. In:
Dissertation to University of Southern California for the Ph.D. Degree (2018), pp. 1–
20.
[3] Frank White. “Viscous Fluid Flow”. In: McGraw Hill. Second Edition (1991), pp. 60–
100.
[4] Ira Abbott and Albert Von Doenhoff. “Theory of wing sections including a summary
of airfoil Data”. In: Dover Publications (1949), 133, 462, 463, 614, and 615.
[5] John Anderson. “Fundamentals of Aerodynamics”. In: McGraw Hill. Sixth Edition
(2017), 346-364 and 1064–1077.
[6] Denise Ann Choi. “Wind Tunnel Experiments on the Flow over a NACA 65(1)-412
Airfoil at a Reynolds Number of 20,000”. In: A thesis presented to the faculty of San
Diego State University (2020), pp. 1–12.
[7] Hacımurat Demir Mustafa Serdar Genc Kemal Koca and Halil Hakan Acıkel. “Traditional and New Types of Passive Flow Control Techniques to Pave the Way for High
Maneuverability and Low Structural Weight for UAVs and MAVs”. In: Intechopen
(2020).
[8] Barton L Smith and Ari Glezer. “The formation and evolution of synthetic jets”. In:
Physics of Fluids. AIP Publishing 10 (1998).
[9] Pititat Itsariyapinyo and Rajnish Sharma. “Large eddy simulation of NACA0015 circulation control airfoil using synthetic jets”. In: Elsevier. Aerospace science and technology 82-83 (2018), pp. 1–3.
48

[10] Vladimir Kornilov. “Combined Blowing/Suction Flow Control on Low-Speed Airfoils”.
In: Flow, Turbulence and Combustion Springer Nature (2020), pp. 9–23.
[11] D. You and P. Moin. “Active control of flow separation over an airfoil using synthetic
jets”. In: Center for Turbulence Research, Stanford University, Stanford. Journal of
Fluids and Structures 24 (2008).
[12] Abhijeet Konwar Sumit Shankar Sarvankar Anindita Apurbaa Phukan and Paragmoni
Kalita. “Modeling, Simulation and Optimization. Effects of Gurney Flap and Suction
Slots on the Aerodynamics of a NACA0012 Airfoil”. In: Springer Nature (2020), pp. 29–
42.
[13] Reza Saleh Kianosh Yousefi and Peyman Zahedi. “Numerical study of blowing and
suction slot geometry optimization on NACA0012 airfoil”. In: Journal of Mechanical
Science and Technology 28, 4 (2013).
[14] Kianoosh Yousefi and Reza Saleh. “Three-dimensional suction flow control and suction
jet length optimization of NACA 0012 wing”. In: Springer Science+Business Media
Dordrecht (2015).
[15] W. Olsman et Al. “Flow around a NACA0018 airfoil with a cavity and its dynamical
response to acoustic forcing”. In: Exp Fluids 51.1 (2011), pp. 1–12.
[16] Olsman and T. Colonius. “Numerical Simulation of Flow over an Airfoil with a Cavity”.
In: AIAA 49.1 (2011), pp. 1–7.
[17] Rami S. Soro. “Synthetic jet control on compressor blade at higher reynolds numbers”.
In: San Diego State University Master’s Thesis (2020).
[18] Ansys. “Shear-Stress Transport SST Model”. In: https://www.afs.enea.it/project/neptu
nius/docs/fluent/html/th/node67.htm (Consulted in March 2024).
[19] Mark Drela. “XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils,” Low Reynolds Number Aerodynamics”. In: Springer, Cambridge (1989), pp. 1–
12.
[20] AirfoilTools. “NACA 0012 Airfoils (n0012-il)”. In: http://airfoiltools.com/airfoil/details
?airfoil=n0012-il (Consulted in August 2023).
[21] AirfoilTools. “NACA 65(1)-412 (naca651412-il)”. In: http://airfoiltools.com/airfoil/deta
ils?airfoil=naca651412-il (Consulted in December 2023).
49

[22] R. Fereidouni M. Godarzi and M. Rahimi. “Investigation of flow control over a
NACA012 airfoil by suction effect on aerodynamic characteristics”. In: Canadian
Journal on Mechanical Sciences and Engineering,3 (2012), pp. 102–109.
50

Abstract (if available)

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Aerodynamics at low Re: separation, reattachment, and control

PDF

Numerical modeling of separated flows at moderate Reynolds numbers appropriate for turbine blades and unmanned aero vehicles

PDF

Modeling and analysis of parallel and spatially-evolving wall-bounded shear flows

PDF

Large eddy simulations of laminar separation bubble flows

PDF

Biomimetics and bio-inspiration for moderate Reynolds number airfoils and aircraft

PDF

Boundary layer and separation control on wings at low Reynolds numbers

PDF

Numerical study of flow characteristics of controlled vortex induced vibrations in cylinders

Asset Metadata

Creator Mendoza Martinez, Jocelyn Sarahi (author)

Core Title RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.

School Viterbi School of Engineering

Degree Master of Science

Degree Program Aerospace Engineering

Degree Conferral Date 2024-05

Publication Date 06/12/2024

Defense Date 04/29/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Spedding, Geoffrey (committee chair), Byahut, Saakar (committee member), Luhar, Mitul (committee member)

Creator Email jsm93775@usc.edu,mendoza.jocelyn.2f@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC11399617E

Unique identifier UC11399617E

Identifier etd-MendozaMar-13078.pdf (filename)

Legacy Identifier etd-MendozaMar-13078

Document Type Thesis

Format theses (aat)

Rights Mendoza Martinez, Jocelyn Sarahi

Internet Media Type application/pdf

Type texts

Source 20240612-usctheses-batch-1167 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu