flaps with wavy leading edges for robust performance agains upstream trailing vortices
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University of Southampton
Faculty of Engineering and the Environment
School of Engineering Sciences
Master of Science Dissertation
Flaps with Wavy Leading Edges
for Robust Performance against
Upstream Trailing Vortices
By
Rafael Pérez Torró
rpt1g12@soton.ac.uk
A dissertation submitted in partial fulfilment of the degree of Master in Science (MSc)
in Race Car Aerodynamics by taught course
First Supervisor: Dr. Jae Wook Kim
Second Supervisor: Prof. Bharathram Ganapathisubramani
September 2013
i
ii
Acknowledgements
Firstly I would like to thank the University of Southampton for providing the
computational resources needed to this project’s fulfilment and especially to my project
supervisor Dr. Kim who first gave me the freedom to complete this project in the way
that I so wished and ultimately helped me out to improve the quality of the written
work.
Secondly I would like to mention my colleagues with whom I had an amazing time here
in Southampton and who were always willing to help me out.
In third place I have a special mention to the Postinger family who made me feel as if I
was part of their family, cheering me up when I was sad and giving me support when I
was alone. A special mention needs to be dedicated to Hannah Postinger too who
always stood by my side and brought a little bit of love into my life. Thank you very
much Hannah.
Per acabar, m’agradaria dedicar també unes paraules en la meua llengua materna a la
meua família. Moltes gràcies per donar-me el vostre suport incondicional. Jo sé que
vosaltres ho donaríeu tot per a què jo poguera tenir el millor futur possible, però
realment a dins de mi jo també sé que si puc ser la meitat de feliç del que vosaltres sou
podeu estar tranquils, heu fet un gran treball. Moltes gràcies de tot cor, us estime molt.
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Abstract Computational Fluid Dynamic calculations have been carried out using URANS
turbulence models in order to test the performance of a two-element airfoil with Wavy
Leading Edge flap shape against upstream trailing vortices. The GA(W) – 1 airfoil
shape was used for the main element and the NACA 634-021 airfoil shape was used for
the flap element. Because of the lack of any previous studies for this configuration, a
parametric study of the elements relative position was first of all carried out. Once the
relative position of the elements was defined the sinusoidal shape was introduced into
the design with different amplitudes and wavelengths and simulations were performed
with usual clean upstream conditions. Results of this experiment showed that the flaps
with undulated leading edges performed worse than the plain leading edge flaps for the
configuration tested here in both terms of mean lift and drag since the relative position
of the elements was just optimised for the plain leading edge case. A second experiment
was carried out in which the effect of an upstream trailing vortex was modelled using a
vortex-train perpendicular to the streamwise direction. Results of this latter experiment
showed that with the conditions tested, the models with Wavy Leading Edges
performed in a more robust way than the straight leading edge flaps.
v
Contents Acknowledgements .......................................................................................................... ii
Abstract ............................................................................................................................ iv
Contents ........................................................................................................................... vi
List of Figures ................................................................................................................ viii
List of Tables .................................................................................................................... x
Nomenclature.................................................................................................................. xii
Chapter 1 : Introduction ........................................................................................... 1
1.1. Dissertation’s Organisation ............................................................................ 1
1.2. Introduction and motivation of the problem .................................................. 1
1.3. Approach to the Problem................................................................................ 2
1.4. Objectives ....................................................................................................... 3
1.5. Literature Review ........................................................................................... 3
Chapter 2 : Theoretical Background ........................................................................ 9
2.1. Basic Notions and Assumptions ..................................................................... 9
2.2. Turbulence .................................................................................................... 12
2.3. Reynolds Averaged Navier-Stokes Equations ............................................. 15
2.4. Eddy Viscosity Approximation .................................................................... 17
2.5. Linear Eddy Viscosity Models ..................................................................... 22
2.6. Unsteady RANS (URANS) .......................................................................... 25
2.7. Discretising the Equations: the Finite Volume Method (FVM) ................... 26
Chapter 3 : Problem Description ........................................................................... 29
3.1. Ambient Conditions ..................................................................................... 29
3.2. Tools Used.................................................................................................... 29
3.3. Model’s Geometry........................................................................................ 29
3.4. Mesh ............................................................................................................. 31
3.5. Domain Size and Boundary Conditions ....................................................... 33
3.6. Time Resolution ........................................................................................... 34
3.7. Turbulence Model ........................................................................................ 35
Chapter 4 : Results ................................................................................................. 37
4.1. Elements’ Relative Position Study ............................................................... 37
4.2. Experiment A: Flaps with Wavy Leading Edges ......................................... 41
vii
4.3. Experiment B: Flaps with Wavy Leading Edges against Upstream Vortex
Condition ................................................................................................................ 54
Chapter 5 : Final Remarks ..................................................................................... 65
Appendix A: Mesh Dependency Study ...................................................................... 69
Appendix B: Domain Size Dependency Study .......................................................... 71
Appendix C: Validation against S-A Turbulence Model ........................................... 73
Appendix D: Matlab, VBA and JAVA Codes ............................................................ 79
References .................................................................................................................. 91
viii
List of Figures Figure 1.1 Humpback Whale’s Flippers detail ................................................................. 2
Figure 2.1 Water Jet Turbulence: Instabilities quickly develop until fully turbulent flow
is achieved. ..................................................................................................................... 13
Figure 2.2 Typical Hot-Wire velocity measurement ...................................................... 15
Figure 2.3 2D Shear Flow Detail .................................................................................... 19
Figure 2.4 URANS Measurement of a Mean Quantity .................................................. 26
Figure 2.5 Volume Ω divided into three subvolumes Ω1, Ω2, and Ω3 ............................ 27
Figure 3.1 Sketch of the Elements’ relative position ..................................................... 30
Figure 3.2 Gap and Overlap Definition .......................................................................... 30
Figure 3.3 WLE Sketch .................................................................................................. 31
Figure 3.4 Volumetric Controls around the model ......................................................... 32
Figure 3.5 Boundary Layer Mesh Detail ........................................................................ 32
Figure 3.6 Wall y+ contour levels over the model .......................................................... 32
Figure 3.7 Wall y+ Distribution ...................................................................................... 33
Figure 3.8 Boundary Conditions and Domain Size Sketch (not to Scale) ..................... 33
Figure 3.9 Courant Number Detail ................................................................................. 35
Figure 4.1 Optimisation Cycle........................................................................................ 38
Figure 4.2 Response Surface with Red Dot indicating the Final Baseline Model ......... 40
Figure 4.3 Baseline geometry with Coordinate System Indications. From Top-Left to
Bottom-Right: Top View, Isometric View, Front View and Rear View. ....................... 40
Figure 4.4 CL and CD Coefficients for all the Models Tested ........................................ 42
Figure 4.5 CLflap Coefficient for all the Models Tested .................................................. 43
Figure 4.6 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i]
Contours on Several Spanwise Planes: a) model0, b) model1.1, c) model2.1 and d)
model3.1 ......................................................................................................................... 44
Figure 4.7 model0 Base Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity .......................................................................................................................... 45
Figure 4.8 model1.1 Base Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 45
Figure 4.9 model1.1 Peak Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 46
Figure 4.10 model1.1 Valley Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 46
Figure 4.11 model2.1 Base Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 47
ix
Figure 4.12 model2.1 Peak Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 47
Figure 4.13 model2.1 Valley Section, Velocity Slice non-dimensionalised by the Free
Stream Velocity .............................................................................................................. 47
Figure 4.14 CP Contours on the models surface: a) model0, b) model1.1, c) model2.1
and d) model3.1 .............................................................................................................. 48
Figure 4.15 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i]
Contours on Several Spanwise Planes: a) model1.4, b) model1.8, c) model2.4, d)
model2.8, e) model3.4 and f) model3.8 .......................................................................... 50
Figure 4.16 Velocity Slice non-dimensionalised by the Free Stream Velocity: a)
model1.4 Peak Section, b) model1.4 Valley Section, c) model2.4 Peak Section, d)
model2.4 Valley Section, e) model3.4 Peak Section and f) model3.4 Valley Section .. 51
Figure 4.17Velocity Slice non-dimensionalised by the Free Stream Velocity: a)
model1.8 Peak Section, b) model1.8 Valley Section, c) model2.8 Peak Section, d)
model2.8 Valley Section, e) model3.8 Peak Section and f) model3.8 Valley Section .. 52
Figure 4.18 Isosurface of Streamwise Vorticity Ω=±100 s-1
: a) model1.4, b) model1.8,
c) model2.4, d) model2.8, e) model3.4 and f) model3.8 ................................................ 53
Figure 4.19 CP Contours on the models surface: a) model1.4, b) model1.8, c) model2.4,
d) model2.8, e) model3.4 and f) model3.8 ..................................................................... 54
Figure 4.20 Detail of Pressure Calculation from Initial Conditions: a) Vorticity contours
at t=0s, b) CP Contours at t=0s, c) Vorticity Contours at t=0.1s, and d) CP Contours at
t=0.1s .............................................................................................................................. 55
Figure 4.21 Total CL History Comparison ..................................................................... 56
Figure 4.22 CLflap History Comparison ........................................................................... 56
Figure 4.23 CLmain History Comparison.......................................................................... 57
Figure 4.24 Total CL Statistic Values Comparison ........................................................ 57
Figure 4.25 CLflap Statistic Values Comparison .............................................................. 58
Figure 4.26 CLmain Statistic Values Comparison............................................................. 58
Figure 4.27 model0 CP rms contours .............................................................................. 59
Figure 4.28 model1.4 CP rms contours ........................................................................... 59
Figure 4.29 model1.8 CP rms contours ........................................................................... 60
Figure 4.30 model2.4 CP rms contours ........................................................................... 60
Figure 4.31 model2.8 CP rms contours ........................................................................... 60
Figure 4.32 model0 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d)
t=0.5s, e) t=0.6s and f) t=0.7s. ........................................................................................ 62
Figure 4.33 model1.4 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d)
t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 62
Figure 4.34 model1.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d)
t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 63
x
Figure 4.35 model2.4 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d)
t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 63
Figure 4.36 model2.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d)
t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 64
Figure A.1 Mesh Dependency Study Convergence ....................................................... 69
Figure B.1 Domain Size Convergence ........................................................................... 71
Figure C.1 Velocity Contours Comparison .................................................................... 73
Figure C.2 Velocity Profiles Comparison Moving from 1 to 4 in the stream direction . 74
Figure C.3 Pressure Coefficient Distribution Comparison ............................................. 74
Figure C.4 Pressure Coefficient Contours Comparison ................................................. 75
Figure C.5 Skin Friction Coefficient Distribution Comparison ..................................... 76
Figure C.6 Wall Shear Stress X-Direction Contours Comparison ................................. 76
Figure C.7 Q-Criterion Iso-Surfaces Q = 100 Coloured by Dimensionless Velocity X-
Direction Comparison .................................................................................................... 77
List of Tables Table 3.1 Ambient Values .............................................................................................. 29
Table 4.1 Final Baseline Parameters .............................................................................. 39
Table 4.2 Models Parameters in terms of cflap ................................................................ 41
Table 4.3 Experiment A Results ..................................................................................... 42
Table A.1 Mesh Dependency Study Results .................................................................. 69
Table B.1 Domain Size Study ........................................................................................ 71
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Nomenclature
Symbols
Re = Reynolds Number
LC = Lift Coefficient
DC = Drag Coefficient
PC = Pressure Coefficient
LmainC = Lift Coefficient Main Element
LflapC = Lift Coefficient Flap Element
c = Chord
b = Elements’ Span
mainc = Main Element’s Chord
flapc = Flap Element’s Chord
t = Time
= Density
V = Volume
S = Surface
u
= Velocity Vector
f = Scalar Function
wandvu, = Velocity Components
zandyx ,, = Cartesian Coordinates
iu = Velocity Components Tensor Notation
ix = Cartesian Coordinates Tensor Notation
i
= Stress Vector
p = Pressure
ij = Kronecker Delta
xiii
ij = Viscous Stress Tensor
ijS = Rate-of-Strain
= Dynamic Molecular Viscosity
= Kinematic Molecular Viscosity
U = Free Stream Velocity
l = Characteristic Length
k = Turbulent Kinetic Energy
= Kolmogorov Length Scale
= Dissipation Rate of Turbulent Kinetic Energy
0l = Large Scales Length Scale
0u = Large Scale Velocity Scale
u = Turbulence Velocity Fluctuation
u = Time Average Velocity
jiuu = Reynolds Stress Tensor
ij = Reynolds Stress Tensor, General Stress Tensor
ijb = Reynolds Stress Isotropic Part
ija = Reynolds Stress Anisotropic Deviatoric Part
u = Molecular Velocity Fluctuations
xyp = Momentum Flux across XY Plane
T = Turbulent Eddy Viscosity
Tl = Turbulent Length Scale
Tu = Turbulent Velocity Scale
Tt = Turbulent Time Scale
m = Molecular Mass
mfpl = Mean Free Path Distance
thv = Thermal Velocity
xiv
n =
Number of Molecules per Unit Volume, Number of Waves along the
Flap’s Span
A = Wavy Leading Edge Amplitude
LEx = Flap’s Leading Edge x-coordinates
LEz = Flap’s Leading Edge z-coordinate
00 yandx = Flap’s Leading Edge Position
= Flap’s Deflection Angle
= Wavy Leading Edge Wavelength
= Specific Dissipation Rate of Turbulent Kinetic Energy
Kn = Knundsen Number
Ma = Mach Number
= Scalar Function
F
= Flux of a Scalar Function
= Rate-of-Rotation, Volume Control
tandn
= Normal and Tangential Unitary Vectors
x = Grid Resolution in the x-direction
t = Time Step
C = Courant Number
rms = Root-Mean-Square
w = Wall Shear Stress
00 yandx = Vortex Core Initial Position
r = Radial Vortex Coordinate
21 kandk = Vortex Equation Constants
L = Domain Length
)(r = Vortex Function
xv
Abbreviations
CFD = Computational Fluid Dynamics
RANS = Reynolds Averaged Navier-Stokes
CAA = Civil Aviation Authority
AoA = Angle of Attack
FAA = Federal Aviation Administration
AIM = Aeronautical Information Manual
WLE = Wavy Leading Edge
LE = Leading Edge Protuberances
TE = Trailing Edge
URANS = Unsteady RANS
S-A = Spalart Allmaras
SST = Shear Stress Transport
APG = Adverse Pressure Gradient
LEP = Leading Edge Protuberances
DES = Detached Eddy Simulation
LES = Large Eddy Simulation
DNS = Direct Numerical Simulation
LDV = Laser Doppler Velocimetry
AR = Aspect Ratio
RHS = Right Hand Side
LHS = Left Hand Side
NSE = Navier-Stokes Equations
HIT = Homogeneous Isotropic Turbulence
TKE = Turbulent Kinetic Energy
HOT = High Order Terms
LEVMs = Linear Eddy Viscosity Models
FVM = Finite Volume Method
FDM = Finite Difference Method
CAD = Computer Aided Design
PDAS = Public Domain Aeronautical Software
WLEF = WLE Flap
VC = Volumetric Control
VBA = Visual Basic for Applications
HA = High Amplitude
MA = Medium Amplitude
LA = Low Amplitude
HW = High Wavelength
MW = Medium Wavelength
LW = Low Wavelength
WSS = Wall Shear Stress
Chapter 1 : Introduction
1.1. Dissertation’s Organisation
Here we are going to introduce the following chapters and subsections in order to give
the reader an overall view of what can be found in this dissertation.
This chapter, the first one, describes the problem the project is focused on and tries to
give a general overview of the previous related work done. In the second chapter the
author reviews the fundamental equations over which all CFD codes are bases and more
specifically the RANS models. In the third chapter a deeper technical description of
how the problem is going to be tackled is given, including the tools and geometry used
as well as general simulation parameters such as mesh size and domain size. In the
fourth chapter the more relevant results obtained in this project are shown and explained
and finally in chapter five a general conclusion and recommendations can be found.
1.2. Introduction and motivation of the problem
According to the Civil Aviation Authority (CAA) there were nearly three million
airplane movements in 2012 in the United Kingdom1. This means nearly 8,000 flights
per day in the UK. Supposing that the same amount of planes that depart from the UK
destined to other countries equals the number of planes arriving UK from other
countries the number of airport operations (take-off and landing) can be roughly
approximated to 16,000 operations per day, only in UK airports.
Every time an airplane either takes off or lands leaves a turbulent wake behind. This
wake is mostly composed by the trailing vortices produced by the airplane wings. This
phenomenon is more pronounced because of the high-lift devices used during take-
off/landing. Rarely the wake produced by an aircraft can produce structural damages on
the aircraft encountering the wake. However it is more likely the vortices induce rolling
moments which can exceed the roll−control authority of the encountering aircraft.
Additionally the induced velocities could increase the effective angle of attack (AoA)
beyond the stall angle and thus causing a failure in the operation.
Due to the high number of take-off/landing operations the air in the runway’s
surroundings can be highly disrupted, and operations might be postponed until ambient
air has settled down. In fact the United States Federal Aviation Administration (FAA)
dedicates in its Aeronautical Information Manual (AIM) a whole chapter to Wake
Turbulence (AIM Chapter 7, section 3)2.
1
CAA, “Table 3.2 Aircraft Movements 2002 - 2012 (in Thousands),” 2012,
http://www.caa.co.uk/docs/80/airport_data/2012Annual/Table_03_2_Aircraft_Movements_2002_2012.pd
f. 2 United States. Federal Aviation Administration., “Aeronautical Information Manual : Official Guide to
Basic Flight Information and ATC Procedures,” Official Guide to Basic Flight Information and ATC
Procedures Aim (2012): v.
Chapter 1: Introduction 2
Since the ideal case would be a continuous landing/taking-off scenario there exist a
need in reducing the sensitivity of the airplane airfoils/wings to these trailing vortices
and hence increase the number of operations under safety conditions.
1.3. Approach to the Problem
Given that one of the biggest issues associated with trailing vortices is the premature
stall they provoke, the solution to the problem must be so that it delays the stall under
such turbulent conditions. It turns out that, in recent years, researchers such as Fish and
Launder3 have found that wavy leading edge (WLE) such as those the humpback
whale’s flipper possesses (see Figure 1.1) are an efficient passive method for delaying
stall.
Figure 1.1 Humpback Whale’s Flippers detail4
The objective of this project is to implement the WLE in a landing/take-off high-lift
configuration. Nonetheless all the studies published regarding WLE are based on single
element configurations, so in order to implement the WLE in a high-lift configuration
two airfoils in tandem will be studied. The rear airfoil of the tandem will be considered
as a flap, and it will be the one with the leading edge (LE) modifications.
All the tests that will be carried out in this project will be computer simulations of the
flow based on the Unsteady Reynolds Averaged Navier-Stokes (URANS) equations
with different turbulent closure models. These simulations will test if the WLE can
improve the performance of high-lift configurations against upstream trailing vortices
and if they can do it in a robust manner.
The process that will be carried out to test the WLE effect will be broken down into two
major experiments. 1) The first part involves the combination of the main element
followed by a second airfoil (flap) with the modified LE. In this part the effect of the
shape of the WLE is tested. 2) The second experiment tests the performance of different
3 F.E. Fish and G.V. Lauder, “Passive and Active Flow Control by Swimming Fishes and Mammals,”
Annual Review of Fluid Mechanics 38 (2006), doi:10.1146/annurev.fluid. 4
“Humpback Whale Speaks, Says ‘Thank You’,” Sea Monster, accessed September 19, 2013,
http://theseamonster.net/2011/07/humpback-whales-speaks-says-thank-you/.
Chapter 1: Introduction 3
WLE shapes against similar turbulent conditions such as those produced by trailing
vortices. The model is hit by vortex convected by the free stream flow, which emulates
the effect of the trailing vortices present during ground operations.
1.4. Objectives
The main objectives to be accomplished in this project are:
First to implement the promising Wavy Leading Edges shapes that have given
good results on airfoils near stall regime situations into a two-element high-lift
configuration and investigate how can they perform with clean upstream
conditions
Second investigate if the Wavy Leading Edges can be beneficial when used in a
two-element high-lift configuration against an upstream trailing vortex
condition.
1.5. Literature Review
In this project major attention is put into the ability of the WLE to delay stall, however
the leading edge modifications are applied to the second element’s leading edge of a
two-element high-lift configuration airfoil. Therefore it is worthwhile to gather some
information about these two major topics, i.e. High-Lift Aerodynamics and Wavy
Leading Edges.
High-Lift Aerodynamics
The paper published by A.M.O. Smith5 in 1975 might be the most referenced paper
regarding high-lift aerodynamics. The author notes that the use of multielement
configurations for high-lift has been present since early XX century. Although it has
been suggested that a single airfoil shape can obtain the same amount of lift as a multi-
element design, Smith states that properly designed multielement are more convenient
and that generally the greater the number of elements the greater the lift. Because of
Bernoulli equation it is well known that that if the surface is to lift the velocity over it
must be higher than that in the pressure surface. But when the flow reaches the trailing
edge the flow is decelerated to velocities even lower than the freestream. If this
deceleration is too stiff it may cause separation because of the high adverse pressure
gradient (APG). Two major aspects then arise when designing high-lift devices: 1) The
analysis of the boundary layer, prediction of separation, and determination of the kinds
of flows that are most favourable with respect to separation; and 2) analysis of the
inviscid flow about a given shape with the purpose of finding shapes that put the least
stress on the boundary layer.
Smith points out that one of the biggest misconceptions in multielement aerodynamics
is that slots supply a blowing type of boundary layer control. Instead the author
5
A. M. O. Smith, “High-Lift Aerodynamics,” Journal of Aircraft 12 (June 1, 1975): 501–530,
doi:10.2514/3.59830.
Chapter 1: Introduction 4
highlights the five major effects present in a common high-lift configuration using
multielement airfoils.
1) Slat effect: The circulation velocities of a former element, for example, a slat, run
counter to the velocities on a downstream element in its LE zone and so reduce pressure
peaks on the downstream element. The effect of this slat effect is to delay the stall
angle.
2) Circulation effect: The upstream element is also beneficiated by the presence of the
downstream element which causes the trailing edge of the adjacent upstream element to
be in a region of high velocity that is inclined to the mean line at the rear of the forward
element. Such flow inclination induces considerably greater circulation on the forward
element.
3) Dumping effect: Because the trailing edge of a forward element is in a region of
velocity appreciably higher than freestream, hence the flow in the boundary layer does
not suffer such a severe deceleration as it would happened in isolation. The higher
discharge velocity relieves the pressure rise impressed on the boundary layer, thus
alleviating separation problems or permitting increased lift.
4) Off-the-surface pressure recovery: The boundary layer from forward elements is
dumped at velocities appreciably higher than freestream. The final deceleration to
freestream velocity is done in an efficient manner. The deceleration of the wake occurs
out of contact with a wall. Such a method is more effective than the best possible
deceleration in contact with a wall.
5) Fresh-boundary-layer effect: Each new element starts out with a fresh boundary layer
at its LE. Breaking up a flow into several short boundary layer runs reduces the risk of
separation because thin boundary layers can withstand stronger adverse gradients than
thick ones and thus lift can be increased.
Rumsey and Ying6 wrote an exceptional review of different numerical investigations
regarding high-lift aerodynamics of multi-element airfoils. The authors point out that
most of the calculations of multi-element CFD calculations have been carried out using
structured-grid Reynolds Averaged Navier-Stokes (RANS) codes. The turbulence
models S-A and Menter’s SST arise as the best options for closing the RANS equations.
In this paper the work of Godin et al.7 is mentioned in which several calculations over
the GA (W)-1 2-element configuration using both S-A and Menter’s models were
conducted. For this case both the S-A and SST models gave very similar results for this
case, with the SST model giving pressures in the separated region on the flap in
somewhat better agreement with experiment. Velocity profiles, including the main
element wake, were predicted similarly by both models in reasonable agreement with
experiment. Turbulent shear stress profiles were also similar between the two models,
but did not agree quite as well with the experiment, particularly in the main element’s
wake region.
Rumsey and Ying also review some investigations done using the NLR-7301 geometry
(typical take-off configuration). Their major findings among others were that in general
6 Christopher L. Rumsey and Susan X. Ying, “Prediction of High Lift: Review of Present CFD
Capability,” Progress in Aerospace Sciences 32 (2002): 145–180. 7 P. Godin, D. W. Zingg, and T. E. Nelson, “High-Lift Aerodynamic Computations with One- and Two-
Equation Turbulence Models,” AIAA Journal 35 (February 1, 1997): 237–243, doi:10.2514/2.113.
Chapter 1: Introduction 5
terms the CL,max tended to be over predicted. In addition the k-ε model was found to
predict larger values of the skin friction and gave worse agreement with experimental
data. Compressible formulation showed better agreement with experiments in the post-
stall regimes. Finally drag was found to be sensitive to the far-field grid and boundary
conditions whereas lift coefficient tend to converge faster as regards mesh resolution
and domain length. A possible cause of this drag misprediction could be the poor
resolution of the grid resolving the wakes of the profiles as suggested by Anderson et
al.8.
Although most of the CFD studies in high-lift devices have been conducted in 2D
Rumsey and Ying state that 3-D RANS computations have recently gain a higher status.
They seem to generally be able to predict many complex multi-element flow fields at
angles of attack below stall. However, their performance near maximum lift conditions
has been less reliable, depending on particular configurations. The authors state that
more grid refinement is necessary in 3-D so results can be truly trusted. Despite this the
use of 3-D CFD calculations is highly encouraged by the authors since flows near
maximum lift possesses dominant three-dimensional effects.
Wavy Leading Edges
During recent years the astonishing agility and manoeuvrability of such a big maritime
mammal as the humpback whale has caught the attention of many researchers. The
humpback whale or “Megaptera novaeangliae” flipper is unique because of the
rounded protuberances located on its LE. This feature present in the whale flippers,
which are a result of the evolutionary Darwinian “natural selection”, act as a passive-
flow control that improves the flipper’s performance. The protuberances delay the stall
angle of the flipper therefore increasing its lift in the post stall regime without adding
additional drag9. Because of this beneficial effect that the LE tubercles have on the
whale’s flipper, it is thought that they might be used in aerodynamic airfoils to increase
its performance too.
The humpback whale’s flippers, which are the longest of any cetacean10
have been
studied both via experimental procedures and numerical computations.
Experimental investigations were conducted by Hansen et al.11
regarding the effect of
different amplitude and wavelengths of LE tubercles on two different airfoils, NACA
0021 and NACA 65-021 at Re = 1.2e5. Result revealed that protuberances were more
effective on the latter airfoil. It was postulated that this was because the position of the
maximum thickness in the NACA 65-021 airfoil was located further downstream which
extends the percentage of the laminar boundary layer. The tubercles, which according to
the authors act like vortex generators, increasing the rate of momentum exchange in the
laminar boundary layer. The authors found that increasing the amplitude leads to a
8 W. Kyle Anderson et al., “Navier-Stokes Computations and ExperimentalComparisons for Multielement
Airfoil Configurations” (presented at the Aerospace Sciences Meeting, Reno, Nevada, 1993). 9 Frank E. Fish et al., “The Tubercles on Humpback Whales’ Flippers: Application of Bio-Inspired
Technolog,” Integrative and Comparative Biology 51 (2011), doi:10.1093/icb/icr016. 10
F. E. Fish and J. M. Battle, “Hydrodynamic Design of the Humpback Whale Flipper.,” J. Morph 225
(1995): 51–60. 11
Kristy L. Hansen, Richard M. Kelso, and Bassam B. Dally, “Performance Variations of Leading-edge
Tubercles for Distinct Airfoil Profiles,” Journal of Aircraft 49 (2011): 185–194, doi:10.2514/1.J050631.
Chapter 1: Introduction 6
smoother stall; however this is accompanied by a smaller value of CL,max and stall
angles. In contrast, airfoils with smaller protuberance’s amplitude performed much
better than the unmodified profile in the post-stall regime without any significant
difference in the drag production. The paper also shows that although reducing the
wavelength proves to be beneficial, for a given value of the tubercles amplitude there is
a limiting wavelength value for which reducing its value does no longer give improved
characteristics. One key finding in this paper is the fact that for similar values of
tubercles amplitude/tubercle’s wavelength ratio the flow over the airfoil behaves
similarly. The authors suggest that tubercles could be beneficial for use on foils
operating near the stall point or in variable flow conditions.
Lohry et al.12
performed numerical computations over a NACA 0020 with WLE using
RANS models, in particular Menter’s SST model. The authors used both centered and
vertex based schemes developed by their own group in Princeton. Calculations were
performed over a range of Re between 62,500 and 500,000. The k-ω SST model proved
to be able to reproduce the experimental measurements and trends with reasonable
accuracy. This paper illustrates how the leading edge protuberances (LEP) act as vortex
generators which tend to reduce the CL,max but to mitigate the stall. Results indicated
that the variation in thickness along the span due to the WLE modification creates a sort
of channels that create spanwise fences that can be used to increase the CL,max if they are
proper optimised. The Reynolds effect study showed that as postulated by Hansen et
al.13
the WLE can be detrimental for very low Re.
Miklosovic et al.14
performed wind tunnel experiments over a wing model that
resembled to the humpback whale flipper. The model cross section was based on the
NACA 0020 airfoil. A comparison was made between a clean LE configuration and a
modified LE configuration of the model. Results showed that the flipper performance
relies in the presence of the LEP which allows the modified LE model to delay stall to
higher AoA values (40% delayed). The drag of the scalloped model was found to be
lesser than the clean one for most AoA tested leading to an expanded operating
envelope combined with drag reduction. Yet the scalloped wing gave lower values of
CL for some AoA in the pre-stall regime. It was also found that the CL was relatively
insensitive to Reynolds effects for Re > 4e5. Further experiments of the authors
15
included infinite span (2D case) wing models in order to quantify the importance of 3D
effects on the flippers. Results showed that the finite (3D case) span scalloped wings
have a largely 3D benefit that is a function of the planform shape and the Reynolds
number. For the 2D case it was found that the WLE are detrimental in the pre-stall
regime producing less lift and more drag. Nonetheless in post-stall regime the WLE
enhance the airfoil performance delaying and smoothing stall. The authors point out that
the WLE have a utility for lifting devices operating past their stall point.
12
Mark W. Lohry, David Clifton, and Luigi Martinelli, “Characterization and Design of Tubercle
Leading-Edge Wings” (presented at the Seventh International Conference on Computational Fluid
Dynamics (ICCFD7), Big Island, Hawaii, 2012). 13
Hansen, Kelso, and Dally, “Performance Variations of Leading-edge Tubercles for Distinct Airfoil
Profiles.” 14
D. S. Miklosovic et al., “Leading-edge Tubercles Delay Stall on Humpback Whale (Megaptera
Novaeangliae) flippers,” PHYSICS OF FLUIDS 16 (2004). 15
David S. Miklosovic, Mark M. Murray, and Laurens E. Howle, “Experimental Evaluation of Sinusoidal
Leading Edges,” Journal of Aircraft 44 (July 1, 2007): 1404–1408, doi:10.2514/1.30303.
Chapter 1: Introduction 7
Pedro and Kobayashi16
emulated the experimental tests using DES computations based
on the S-A turbulence model at Re of 5e5 for wings inspired in the humpback whale
flipper. The model tested had the same wing planform as the whale flipper with a
diminishing chord length from the root to the tip of the wing, sort of the same
configuration used in Miklosovic et al.17
. The major findings were that: 1) The Re
highly influences the type of separation, producing leading edge separation in the
outboard section and trailing edge separation otherwise. 2) The higher aerodynamic
performance for the scalloped flipper is due to the presence of stream-wise vortices
originated by the tubercles. The spanwise vortices bring momentum to the boundary
layer preventing the trailing edge separation and plus confine the leading edge
separation to the wing tip.
An improved performance at the post-stall regime was also found by Johari et al.18
who
conducted wind tunnel experiments comparing a plain NACA 634-021 airfoil with a
range of WLE modifications of it with different wavelengths and amplitude for the
protuberances. The experiments were carried out at Re = 1.83e5 using Laser Doppler
Velocimetry (LDV) and tufts.
Wind-tunnel tests of wings with low aspect ratios (AR) of 1 and 1.5, rectangular
planforms, and four distinct sinusoidal LE and one straight LE were conducted by
Guerreiro and Sousa19
at Reynolds numbers of 70,000 and 140,000. Various
combinations of protuberance’s amplitude and wavelength were analysed and the results
indicated that gains of the order of 45% in maximum lift can be achieved combining
large amplitude and wavelength, which contrasts with Hansen et al.20
findings, past the
stall angle of the airfoil. It is has to be remarked that the airfoil used in these
investigations was the LS(1) – 0417 which has significant differences with other more
common models used in investigations regarding WLE. As in all the papers reviewed
for Re 140,000 the WLE proved to be clearly beneficial for AoA greater than the stall
angle being even more important for the 1.5 AR models. For lower Reynolds numbers
(Re = 70,000) the performance of the WLE models was fairly maintained whereas the
clean LE model performance was clearly diminished. The authors concluded that
sinusoidal WLE are less prone to performance deterioration.
Malipeddi21
conducted numerical 3D calculations using the commercial CFD package
Fluent over a NACA 2412 airfoil. All wings tested had a chord and wing span of 0.1.
The calculations were performed at a Re of 5.7e5
using DES on SA and k-ω models.
Investigations were carried out as regards the influence of the undulations wavelength
and amplitude. The amplitude was found to be very important whereas the wavelength
played a minor role. Results showed that wings with WLE performed better than the
16
Hugo T. C. Pedro and Marcelo H. Kobayashi, “Numerical Study of Stall Delay on Humpback Whale
flippers” (presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2008). 17
Miklosovic et al., “Leading-edge Tubercles Delay Stall on Humpback Whale (Megaptera
Novaeangliae) flippers.” 18
Hamid Johari et al., “Effects of Leading-Edge Protuberances on Airfoil Performance,” AIAA Journal 45
(November 1, 2007): 2634–2642, doi:10.2514/1.28497. 19
J. L. E. Guerreiro and J. M. M. Sousa, “Low-Reynolds-Number Effects in Passive Stall Control Using
Sinusoidal Leading Edges,” AIAA Journal 50 (February 1, 2012): 461–469, doi:10.2514/1.J051235. 20
Hansen, Kelso, and Dally, “Performance Variations of Leading-edge Tubercles for Distinct Airfoil
Profiles.” 21
Anil Kumar Malipeddi, “Numerical Analysis of Effects of Leading-Edge Protuberances on Aircraft
Wing Performance” (Master of Science Thesis, Wichita State University, 2011).
Chapter 1: Introduction 8
clean wing in post-stall regime giving higher values of CL with almost the same value of
CD. However the clean configuration proved to be better in pre-stall regime. It was
found that the protuberances along the LE produced streamwise vortices which carried
high momentum flown in the boundary layer keeping the boundary layer attached. It
was postulated that the streamwise vortices produced by the protuberances are the
reason why the modified airfoil produced 48% higher lift than the clean airfoil in the
post-stall regime.
Chapter 2 : Theoretical Background
In this chapter the student aims to set up the theory on which the entire project relies on.
As this is projects is based on Computational Fluid Mechanics calculations it is worth
reviewing the main theories this two sciences are based on.
2.1. Basic Notions and Assumptions
Whenever we look to any fluid in motion we see that it tends to develop series of very
complicated structures. One might think that these complicated structures must be
governed by an also complicated mathematical expression or a set of very complicated
expressions and in fact this turns out to be true. But if we look at them in a more general
way we only have to recall a more simple formulation that drives us back to the
secondary school level: Newton’s Second Law which states that:
Mass x Acceleration = Force
But in Fluid Mechanics it is rebuilt in the equivalent form of:
Rate of Momentum Change = Force
So now we find ourselves with an expression that relates the Rate of Momentum
Change with the forces present in the fluid motion. These forces are for a wide range of
Aerodynamic applications reduced to two: Pressure Forces and Viscous Forces. Here
we have obviated the Gravity Forces since the fluid we are dealing with is air for which
Gravity Forces are relatively much smaller than the Pressure and Viscous Forces due to
the air density.
In order to determine the Viscous Forces we have to assume that the fluid we are
dealing with is pure homogeneous, this is the case for air. For these pure homogeneous
fluids the constitutive law is given by the Newtonian Fluid Model which states that:
Viscous Stresses Rate-of-Strain
We also suppose that the air is a continuum matter which also fits with our everyday
experience where we think of fluids like air or water in a continuum perspective.
Although we know that matter is formed by smaller structures called molecules, in most
engineering applications where the scales of motion are much larger than the molecules
the continuum assumption seems to be a fair assumption. Plus in these engineering
applications the physical laws of conservation which refer to Mass, Momentum and
Energy conservation are also applied.
Mass Conservation
The Mass Conservation satisfies the well-known statement that matter cannot be created
or destroyed. For a fluid volume V enclosed by a surface S this means that the rate of
increase of mass inside the volume equals to the rate at which mass enters the volume
through the surface minus the rate at which mass leaves the volume through the surface.
Rate of Increase of Mass = Rate of Mass Entering – Rate of Mass Leaving
Chapter 2: Theoretical Background 10
Or
Rate of Increase of Mass – Net Mass Flow out of the Volume
The mathematical expression for this is:
SVdSnudV
t
(2.1)
Applying the Gauss divergence theorem combining and shrinking V to a point gives:
0
u
t
(2.2)
Given that the material derivative of a function f is given by:
fut
f
Dt
Df
(2.3)
The equation (2.3) can be rewritten as follows:
uDt
D
(2.4)
For incompressible flows where the density does not change following a fluid element
the equation (2.4) can be simplified to:
0 u
(2.5)
Or in tensor notation:
0
i
i
x
u
(2.6)
Equations (2.5) and (2.6) represent the Mass Conservation Equation in vector and tensor
notation respectively.
Momentum Conservation
As mentioned before the main law governing the mechanics of the fluid motion is the
Newton’s Second Law which applied to a volume V bounded by a surface S can be
resumed as:
Rate of Increase of Momentum –Net Flux of Momentum out of the Volume = Sum of
Forces Acting on the Surface
The mathematic expression for this can be written as follows:
Chapter 2: Theoretical Background 11
V S Siii dSndSnuudVuu
t
(2.7)
Where is the stress vector acting on the surface S which is our representation of
unresolved molecular motions, i.e. momentum flux density of molecules crossing dS.
Applying the Gauss and shrinking V to the point equation is written as:
iiuut
u
(2.8)
Or in tensor notation:
i
ji
j
jii
xx
uu
t
u
(2.9)
Now the stress can be split into its normal and its tangential parts. The normal part is
due to the pressure forces whereas the tangential part is due to the viscous forces. The
stress tensor hence can be written as:
ijijij p (2.10)
In addition as the flow is assumed to be Newtonian the viscous stresses are only related
to the local strain rate, which for incompressible flows is:
i
j
j
iij
x
u
x
uS
2
1
(2.11)
And then:
i
j
j
iijij
x
u
x
uS 2
(2.12)
Substituting equation (2.12) in (2.11) and then replacing in (2.10) we have:
bleincompresifor
ji
j
jj
i
i
i
j
j
iij
ij
jii
xx
u
xx
u
x
p
x
u
x
up
xx
uu
t
u
0
22
(2.13)
Chapter 2: Theoretical Background 12
But given that the divergence of the velocity is zero for incompressible flow equation
(2.13) reduces to:
jj
i
ij
jii
xx
u
x
p
x
uu
t
u
2
(2.14)
Now the LHS of the equation can also be simplified using the Continuity equation.
Dt
Du
x
uu
t
u
x
u
tu
x
uu
t
u
x
uu
t
u
i
j
ij
i
Continuityby
j
j
i
j
ij
i
j
jii
0
(2.15)
Finally the Momentum Equation can be written as:
TermViscous
jj
i
Termessure
i
TermLinearNon
j
ij
ii
xx
u
x
p
x
uu
t
u
Dt
Du
2
Pr
(2.16)
The Navier-Stokes Equations
Equations (2.6) and (2.16) all together constitute the Navier-Stokes Equations (NSE).
This set of non-linear equations governs all fluid flows. There are four equations, three
momentum equations plus the continuity equation, for four unknowns, three velocity
components plus the pressure. However, despite the system is closed, the non-linearity
present in them because of the LHS second term in equation (2.16) makes them almost
impossible to solve analytically for most industrial flows. Nonetheless, with the
computational power available nowadays, solutions can be found for simple geometries
and low Re numbers using Computational Fluid Dynamics or most commonly known as
CFD.
2.2. Turbulence
The Oxford Dictionaries defines Turbulence as “violent or unsteady movement of air or
water, or of some other fluid”22
. This violent or unsteady movement is due to the non-
linear terms in the NSE equations. We can see turbulence every day in our lives. When
the smoke comes out of a chimney we can clearly see how fluid structures rapidly
destabilise and break down into smaller structures. This behaviour can be seen in Figure
2.1 where the well-defined jet of water quickly breaks down into vortical structures.
This happens when the viscous term is no longer big enough to balance the non-linear
term in equation (2.16) and then instabilities arise. This imbalance happens when the
22
Oxford Dictionaries, “‘Turbulence’. Oxford Dictionaries.,” n.d., definition/english/.
Chapter 2: Theoretical Background 13
viscous forces are much smaller than the inertial forces, or in other words, when the Re
is too high.
lUlU
ForcesViscous
ForcesInertial
Re
(2.17)
Where U∞ is the freestream velocity, l is a characteristic longitude and
is the
kinematic viscosity.
Figure 2.1 Water Jet Turbulence: Instabilities quickly develop until fully turbulent flow is
achieved.23
One can find defining the dimensionless variables:
2~;~;~;~
U
pp
l
Utt
l
xx
U
uu i
ii
i
(2.18)
The NSE can also be expressed as:
jj
i
ij
ij
i
xx
u
x
p
x
uu
t
u~~
~
Re
1~
~
~
~~
~
~ 2
(2.19)
Which shows that as the Re number increases the viscous term becomes less important
and the non-linear inertial term rules the equation.
23
Milton Van Dyke, “An Album of Fluid Motion,(1982),” Parabolic, Palo Alto (Calif.) (n.d.).
Chapter 2: Theoretical Background 14
Basic Features of Turbulent Flows
Turbulent flows are by nature random and chaotic. But this does only mean that
variables are continuously fluctuating, i.e. they do not have a unique value. It is
unpredictable in detail, but with predictable statistical properties. They are also
inherently 3-Dimensional and unsteady, which does not mean that statistical values
cannot be steady though if they are averaged over large enough population sizes.
Turbulent flow contains vorticity, the curl of the velocity. These vortical and unsteady
properties make turbulence to contain a wide range of time, velocity and length scales,
which is commonly known as Energy Cascade. The famous Richardson’s quote
summarises it perfectly:
“Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.”
― Lewis Fry Richardson
What Richardson was trying to say was that turbulence, or energy is produced by large
scales, or low frequency scales, that then break down into smaller scales transferring
energy until it reaches a point where the viscosity forces are of the same order of
magnitude of the inertial forces and turbulence is then dissipated into heat. This scale,
where turbulence dissipates, is known as the Kolmogorov Scale24
in honour to Andréi
Kolmogórov. For Homogeneous Isotropic Turbulence (HIT) where Turbulent Kinetic
Energy Production ( ) equals the Turbulent Kinetic Energy Dissipation ( ) the
smallest scales happen where Re = 1, i.e. where inertial forces equal viscous forces.
Taking this into account and using some dimensional analysis the Kolmogorov Length
Scale η can be found to be:
4/13
(2.20)
And then the ratio between the largest scales l0, at which production of TKE happens,
and the smallest scales η, where energy is finally dissipated into heat, is:
4/3
4/3
00 Re
lulo
(2.21)
This means that the higher the Re number, the smaller the smallest scales are compared
to the largest scales and wider is the scale’s range. So to fully solve the NSE equations
the full range of scales from the biggest to the smallest has to be resolved. This is called
Direct Numerical Simulation (DNS). Another approach is to resolve the biggest scales
and model the effect of the smallest ones. This is called Large Eddy Simulation. And
24
Andrey Nikolaevich Kolmogorov, “The Local Structure of Turbulence in Incompressible Viscous Fluid
for Very Large Reynolds Numbers,” Proceedings of the Royal Society of London. Series A: Mathematical
and Physical Sciences 434, no. 1890 (1991): 9–13.
Chapter 2: Theoretical Background 15
finally, the last option is to completely model the entire range of scales, which is called
Reynolds Averaged Navier-Stokes (RANS). This latter approach is the one chosen for
this project since it is the less computationally expensive. However this benefit of being
less computationally expensive comes with the drawback that it is also less accurate
than LES and DNS, in that order.
2.3. Reynolds Averaged Navier-Stokes Equations
As aforementioned in this project turbulence is going to be tackled using RANS. The
bases on what these equations are built are the NSE and the Reynolds Decomposition.
Reynolds Decomposition and Average
We have already said that turbulence is a random process for which statistical properties
can be extracted. Figure 2.2 shows a typical Hot-Wire velocity measurement. It can be
noted that velocity u(t) varies randomly in time but once the experiment is concluded
we can obtain an average or mean velocity quantity ū and a velocity fluctuation u’(t). Of
course this infinite limit only attempts to say that the variable has to be averaged over
large amounts of time to have a statistically converged value. Additionally we can
express now the velocity measurement as:
tututu (2.22)
Where the mean quantity ū is defined as:
T
iT
i dttuT
u0
)(1
lim
(2.23)
Figure 2.2 Typical Hot-Wire velocity measurement
This process described here is known as Reynolds Decomposition introduced by
Osborne Reynolds in 189525
. A fluctuating variable is decomposed into its mean and
fluctuating parts.
Although above we have presented a time averaged quantity, this is not the only way of
obtaining averaged quantities. Another option could be the ensemble averaging in
which a variable u is measured in N experiments at the same time t and position x. Then
the average quantity is given by:
25
Osborne Reynolds, “On the Dynamical Theory of Incompressible Viscous Fluids and the
Determination of the Criterion” 186, Philosophical Transactions of the Royal Society of London (1895):
123–164.
Chapter 2: Theoretical Background 16
N
n
iN
i txuN
txu1
),(1
lim),( (2.24)
However this Ensemble averaging approach is more complex to be obtained in a
laboratory where measurements on a single point in space during time are more suitable,
i.e. Hot-Wire Anemometry.
From its definition the following rules can be obtained:
: The time average of an averaged time quantity is the average itself.
: The time average of a fluctuating quantity is zero.
(
)
(
)
: The time average of a time and/or space derivative
is the derivative of the time averaged quantity.
Applying the Reynolds Decomposition and Average to the Navier-Stokes
Equations
Now we have the necessary tools to define the Reynolds Averaged Navier-Stokes
Equations (RANS). Using the definition (2.22) in equation (2.6) and (2.16), applying
the averaging rules and averaging over time the equations we have:
0
i
i
x
u
(2.25)
And
jj
i
ij
ji
j
ij
i
xx
u
x
p
x
uu
x
uu
t
u
21
(2.26)
Or rearranging
ji
j
i
jij
ij
i uux
u
xx
p
x
uu
t
u
1 (2.27)
We note here that equations (2.25) and (2.27) are very similar to the unaveraged NSE
(2.6) and (2.16). The difference is that here in RANS equations we are solving for the
mean flow quantities and we have an extra term . The term , which comes
from the non-linear term of the Navier-Stokes equation, is a symmetric tensor, which
dimensionally speaking has units of stress, and is known as the Reynolds Stress Tensor,
ij.
Chapter 2: Theoretical Background 17
2
2
2
wwvwu
wvvvu
wuvuu
ij
(2.28)
Additional stresses due to the fluctuating turbulent quantities are introduced by this
term. This term is the correlation between the velocity components and can be seen as
the average rate of momentum transfer and accounts for the extra dissipation that
characterises turbulent flows. Because it is symmetric, i.e. this term
introduces also 6 new unknowns which leave us now with 10 (6 components of the
Reynolds Stress Tensor plus 3 mean velocity components plus the mean pressure)
unknowns for still 4 equations. This cannot be a surprise since we have only performed
mathematical and algebraic operations over the N-S equations without introducing any
additional physical principles. This is known as the Closure Problem. The role of the
turbulence models is then to give an expression to model this term and then close the
problem.
The Reynolds Stresses diagonal components represent the normal
stresses whereas the off-diagonal components represent the shear stresses. The
Turbulent Kinetic energy (TKE or k) mentioned above is defined to be half the trace of
iiuuk 2
1
(2.29)
In the principal axis of the Reynolds Stress Tensor the shear stresses are zero and the
normal stresses are the eigen-values, which are non-negative. This means that that the
Reynolds Stress Tensor is a symmetric semi-positive defined tensor.
The Reynolds Stress Tensor can also be decomposed into its isotropic part:
ijij kb 3
2
(2.30)
And its anisotropic deviatoric part:
ijijij ka 3
2
(2.31)
2.4. Eddy Viscosity Approximation
The simplest way of modelling the Reynolds Stress Tensor is to prescribe a linear
relationship between the stress tensor and mean Rate-of-Strain as Boussinesq26
did in
1877. Boussinesq, similarly to the Stress-Rate-of-Strain relation for Newtonian Fluids
(2.12) defined a linear relationship between the anisotropic deviatoric part of the
26
Boussinesq, “Theorie de l’Ecoulement Tourbillant” 23, no. Mem. Presentes par Divers Sa- vants Acad.
Sci. Inst. Fr (1877): 46–50.
Chapter 2: Theoretical Background 18
Reynolds Stress and the mean Rate-of-Strain. But in order to understand why he did so,
we might first consider a two-dimensional flow at the molecular level as illustrative
example, with x-velocity component u>0 and y-velocity component v=0.
For this 2D illustrative flow, at the molecular level, velocity can be Reynolds
decomposed into its mean velocity and its molecular random velocity perturbation so the velocity is expressed as . If the instantaneous flux of any fluid
property is calculated across a given horizontal plane we will have that the flux is
proportional to the velocity perpendicular to the plane, i.e. in this case. The x-
momentum flux dpxy across a differential surface area dS then will be:
dSvuudpxy
(2.32)
And then using an ensemble averaging over all molecules we have:
dSvupd xy
(2.33)
Given that the stress tensor acting on the plane we are looking at is σxy=dPxy /dS and
equation (2.10) we have that the viscous stress tensor τxy at this plane is:
vuxy
(2.34)
We can now clearly see how similar this expression is to the Reynolds stresses (2.28).
Here the difference is that the fluctuating velocity is due to the molecular movement and
in the Reynolds stresses is due to turbulence. In laminar flows, the flow is structured in
layers that slide over one another because of the existence of a velocity gradient
between them. The momentum exchange between layers of flow due to the fluctuating
velocities results into friction and ultimately energy dissipation. We have seen that
turbulence flows are much more dissipative than laminar flow and this is because of the
extra momentum exchange due to the turbulent fluctuations . At the molecular level
this is known as the viscous stresses, which are proportional to the velocity gradients, or
the strain rate, as equation (2.12) shows, whereas at the macroscopic level we can think
of this as the Reynolds stresses. Consequently because they both, i.e. the viscous stress
and the Reynolds stress, similarly extract energy from the flow we can relate the
Reynolds stresses with the velocity gradient as follows:
ijT
i
j
j
iTijji
S
x
u
x
ukuu
2
3
2
(2.35)
Where the positive scalar coefficient νT is the Eddy Viscosity, or the Turbulent
Viscosity.
Some clarifications need to be made at this point about the Eddy Viscosity.
The Molecular Viscosity ν is a constant property of the flow whereas the Eddy
Viscosity νT is a scalar function of space and time.
The Eddy Viscosity hypothesis implies that the anisotropy tensor is aligned with
the mean Rate-of-Strain.
Chapter 2: Theoretical Background 19
If νT can somehow be prescript the Closure Problem is solved and the RANS
equations can be solved.
νT has dimensions of [L2T
-1], hence by dimensional analysis we can estimate the
eddy viscosity as or where c is a non-
dimensional constant, lT is a characteristic length scale (typically the size of the
large scale eddies) and uT is a characteristic velocity scale (typically the velocity
magnitude of the large scale eddies)
Velocity and Length Scales
Further investigation of the 2D molecular example we have just seen can give us an
expression of the molecular viscosity that we can then use to similarly define the eddy
viscosity.
So let’s consider again the 2D simple shear flow shown in Figure 2.3 where lmfp is the
mean-free-path, i.e. the average length travelled by a molecule between collisions. Each
particle coming from point P that crosses the x-axis brings a momentum deficit of
m[U(0)-U(-lmfp)] whereas molecules coming from point Q bring a momentum surplus of
m[U(lmfp)-U(0)] where m is the molecular mass.
Figure 2.3 2D Shear Flow Detail
If we suppose that the average vertical molecular speed is ⁄ , because mainly the
same amount of particles are coming from both sides of the y=0 plane, then the average
number of molecules crossing a unitary area in the positive vertical direction is
, where n is the number of molecules per unit volume and vth is the molecules
thermal velocity. Consequently the momentum flux due to molecules crossing y=0 from
below is given by:
mfpth lUUmvnP )0(4
1
(2.36)
And the momentum flux due to molecules crossing form above is:
Chapter 2: Theoretical Background 20
0)(4
1UlUmvnP mfpth
(2.37)
If we then approximate the velocity gradient by:
...
0)(...
)0(TOH
l
UlUTOH
l
lUU
dy
dU
fmp
mfp
fmp
fmp
(2.38)
And we use the fact that ρ=m·n we can express the viscous stress as:
dy
dU
dy
dUlvPP mfpthxy
2
1
(2.39)
Where
is the molecular dynamic viscosity. In dimensional analysis
grounds we can think of vth and lmfp as the velocity and length scales at the molecular
level. By comparison we can also define the eddy viscosity vT as a product of a
turbulent velocity scale uT and a turbulent length scale lT. This turbulent scales have to
be defined using turbulence quantities in the same way the scales of the molecular
viscosity was defined using molecular quantities.
It seems reasonable to use the turbulent perturbation velocity to define the turbulent
velocity scale, and given that the turbulent kinetic energy is ⁄ , it seems
also reasonable to use the turbulent kinetic transport partial differential equation to
prescribe the turbulent velocity scale. In statistic terms k is nothing but a root mean
square, which gives the intrusion level between neighbour flow layers and thus the
interaction level between flow layers.
But the turbulent length scale remains still undefined. One of the main conclusions of
1980/1981 AFOSR-HTTM Stanford Conference on Complex Turbulent Flows was that
the great amount of uncertainty about turbulence models comes from the transport
equation used to model the length scale. Dimensional analysis has been the main tool to
define this length scale. On dimensional grounds, Kolmogorov (1942), related the
turbulence eddy viscosity, the turbulent kinetic energy, and turbulent length scale as
follows:
k ~k ~k ~ 21 TT l (2.40)
Where ε is the dissipation of turbulent kinetic energy and ω is the specific dissipation
rate of turbulent kinetic energy which has dimensions of (time)-1
. The inverse of the
specific dissipation rate ω-1
can be seen as the time scale on which dissipation of
turbulent kinetic energy occurs. Despite dissipation occurs at the smaller scales, the
dissipation rate is the rate at which energy is transferred from the large scales to the
small scales by the Energy Cascade process and therefore it is set by properties of the
large scales such as k and lT.
Chapter 2: Theoretical Background 21
The Intrinsic and Specific assumptions behind Boussinesq Approximation
The Boussinesq approximation can be seen in two parts. In first place, the intrinsic
assumption that the Reynolds Stress anisotropy aij=σij – 2/3k δij is determined by the
local mean Rate-of-Strain , and secondly that the relation between the Reynolds
stresses anisotropy and the mean Rate-of-Strain are linear related by the scalar νT as
follows.
ijTij Sa 2 (2.41)
This two faces of the same problem are well explained in Pope27
. For the intrinsic
assumption, the example of an axisymmetric contraction is studied. At the contraction
zone the axial mean Rate-of-Strain is constant
⁄ and the lateral
compressive strain rates are ⁄ , whereas at the straight section
. Therefore, if the Boussinesq intrinsic assumption stands, there would only be non-
zero Reynolds stresses at the contraction part where the mean Rate-of-Strain is non-zero
too. However experiments show that at the straight section immediately after the
contraction section the Reynolds stresses created at the contraction are still present. In
the simple 2D shear flow example studied in the previous section the relation between
the molecular time scale vth/lmfp and the shear time scale ⁄ is of the
order of the Knundsen number Kn times the Mach number Ma.
Ma Kn~Sl
v
mfp
th
(2.42)
This is typically very small, which means that the molecular motions almost
instantaneously adapt to what is imposed by the strain and can therefore be directly
related to the local mean Rate-of-Strain. In contrast, for turbulent shear flows such as
the axisymmetric contraction, experiments have shown that the ratio between the
turbulence time scale tT=k/ε and the strain time scale S-1
can be of the order of the unity
or even bigger. This means that turbulence does not rapidly adjust to what the mean
strain says and hence cannot be in general directly related to the local strain rate.
A part from this assumption, the specific assumption is made by establishing a linear
relationship between the mean Rate-of-Strain and the Reynolds stress anisotropy tensor.
This assumption implies that both tensors, i.e. and σij, are aligned but also that the
Reynolds stresses are scaled by the same scalar, i.e. νT. In simple Turbulent Shear flows
the normal Rates-of-Strain are all equal to zero and yet measured
normal Reynolds stresses are not zero but also they are different from each other. This
shows that the specific assumption does not stand even for the simplest flows, where the
Reynolds Stress tensor might be misaligned with the mean Rate-of-Strain tensor.
Furthermore it has to be pointed out that this is not only a problem of different eigen
vectors, but also eigen values, because as just seen Reynolds stress components may
sometimes scale differently each other.
27
S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).
Chapter 2: Theoretical Background 22
2.5. Linear Eddy Viscosity Models
The Closure Models based on the Boussinesq approximation are called Linear Eddy
Viscosity Models (LEVMs) because of the linear relationship between the Reynolds
Stress Tensor and the Main-Rate-of-Strain established by the Boussinesq
Approximation. The goal of these LEVMs is to provide additional equations to either
directly prescript the Eddy Viscosity or prescribe a velocity scale uT and length scale lT
or time scale tT from which the Eddy Viscosity can then be obtained. The LEVMs can
be divided in:
Algebraic / Zero equation Models: the Boussinesq Approximation is used, with
where uT and lT are prescribed using algebraic expressions, e.g.
Baldwin-Lomax.
One-equation Models: The Boussinesq Approximation is used introducing an
extra Partial-Differential Transport Equation, e.g. Spalart-Allmaras model. In
Spalart-Allmaras a transport equation for νT itself is used.
Two-equation Models: The Boussinesq Approximation is used, with or
where two extra Partial-Differential Transport Equations are used to
prescribe uT and lT or tT respectively. Typically the TKE (k) transport equation is
used to prescribe the velocity scale, where √ , and models then differ in
which equation is used to prescribe the length scale or time scale. In the k-ε
model, the transport equation for the dissipation rate ε is used for the time
scale
definition whereas in the k-ω model the specific dissipation rate
ω transport equation is used instead.
In this project we are going to focus on the One-equation Spalart-Allmaras and the
Two-equation Menter’s k-ω SST models since these are the ones used in the
computations carried out.
Spalart-Allmaras Model
As aforementioned the Spalart-Allmaras is a One-equation model that instead of
prescribing νT using velocity and a length or time scales it has its own transport equation
to directly solve νT. Spalart and Allmaras28
describe their Standard model as:
ii
b
jj
tb
wwtb
j
jxx
cxxd
fc
fcSfcx
ut
~~~~1~
~~)1(
~~2
2
22
1121
(2.43)
Where the turbulent viscosity is computed from:
1~
vT f (2.44)
28
P. R. Spalart and S. R. Allmaras, “A One-Equation Turbulence Model for Aerodynamic Flows,”
Recherche Aerospatiale no. 1 (1994): 5–21.
Chapter 2: Theoretical Background 23
Where
222
3
1
3
3
1
~~
~
v
v
v
fd
S
cf
(2.45)
And √ is the vorticity magnitude, d is the distance to the nearest wall and
i
j
j
iijttt
w
w
ww
v
v
x
u
x
uWccf
dSrrrcrg
cg
cgf
ff
2
1exp
10,~
~min
1
11
2
432
22
6
2
6
1
3
3
6
3
3
1
2
(2.46)
And the model constants are:
2
2
114
3132
21
15.0
2.11.723.0
41.0622.03
21335.0
bbwt
tvww
bb
cccc
cccc
cc
(2.47)
Menter’s k-ω SST Model
In his paper, Menter29
proposes two new Two-equation turbulence models. Both models
are based on former Two-equation models, i.e. the k-ε model30
and the k-ω model31
.
Menter takes the best of each of the two models and combines them into a new one with
some modifications. The k-ω model is used for the flow near the walls because, unlike
any other two-equation model, the k-ω model does not involve damping functions and
allows zero initial wall conditions to be specified and, because of its simplicity, the k-ω
model is more numerical stable with respect to other models. However, because of the
strong sensitivity of the k-ω model to freestream boundary conditions, the k-ε is
preferred out of the boundary layer regions. This integration of both models is achieved
29
F.R. Menter, “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA
Journal 32, no. 8 (August 1994): 1598–1605. 30
B. E.Launder and B.I. Sharma, Letters in Heat and Mass Transfer, vol. 1 (New York ; Oxford:
Pergamon Press., 1974). 31
David C. Wilcox, “Reassessment of the Scale-Determining Equation for Advanced Turbulence
Models,” AIAA Journal 26, no. 11 (1988): 1299–1310.
Chapter 2: Theoretical Background 24
by means of blending functions that activates the k-ω model near the walls and switches
to the k-ε in the outer region. To achieve this, the k-ε model is transformed into a k-ω
formulation. An additional cross-diffusion term is added to the original k-ω formulation
and the values of the constants are slightly modified.
The original k-ω is:
j
tk
j x
k
xkP
Dt
kD
1
*
(2.48)
j
t
jt xxP
Dt
D
1
2
11
(2.49)
Where the production term P and the stress tensor τij are:
j
iij
x
uP
(2.50)
ijij
k
kijtij k
x
uS
3
2
3
22
(2.51)
And then the transformed k-ε is:
j
tk
j x
k
xkP
Dt
kD
2
*
(2.52)
jjj
t
jt xx
k
xxP
Dt
D
12 22
2
22
(2.53)
Then equations (2.50) and (2.51) are multiplied by a function F1 and equations (2.52)
and (2.53) are multiplied by (1 - F1) and both sets of equations are added together to
give the new model:
j
tk
j x
k
xkP
Dt
kD
*
(2.54)
jjj
t
jt xx
kF
xxP
Dt
D
112 21
2 (2.55)
Where the constants of the new model ϕ are calculated from a combination of 2 sets of
constants ϕ1 (form the original model) and ϕ2 (from the transformed k-ε model).
2111 1 FF (2.56)
Chapter 2: Theoretical Background 25
Where
4
11 argtanhF (2.57)
2
2
21
4;
500;
09.0maxminarg
yCD
k
yy
k
k
(2.58)
Where y is the distance to the next surface and CDkω is the positive portion of the cross-
diffusion term in equation (2.53).
The term arg1 goes to zero as the distance to the wall increases due to it is in inverse
proportion to the wall distance y. As arg1 goes to zero as it approaches the boundary
layer edge, so does F1 and then the k-ε model is recovered.
Finally, the eddy viscosity for the SST model is defined as:
21
1
;max Fa
kat
(2.59)
Where Ω is the absolute value of the vorticity and F2 is given by:
2
22 argtanhF (2.60)
22
500;
09.02maxarg
yy
k
(2.61)
The sets of constants to be used in (2.56) are:
*2
1
*
11
*
111k1
1
41.009.0
31.00750.05.085.0
:inner) (SST 1Set
a
(2.62)
*2
2
*
22
*
22k2
2
41.009.0
0828.0856.00.1
)-k(Standart 2Set
(2.63)
2.6. Unsteady RANS (URANS)
Because of the unsteadiness nature of both the two-element configuration and the Wavy
Leading Edges airfoils, and the fact that in the second part of the investigations the
effect of a vortex convected by the freestream impinging the models will be tested, the
use of Unsteady RANS equations was a must.
URANS differ from Steady RANS in that the transcient (unsteady) term of the NSE is
retained. So URANS still is based on the RANS Equations (2.25) and (2.27) which
Chapter 2: Theoretical Background 26
means that the Reynolds decomposition (2.22) is used as well with a slightly modified
notation.
uUU (2.64)
Where a second prime has been added to the modelled turbulent fluctuation part.
However the dependent variables are no longer just a function of space but also of time.
That means that the velocity quantities, or the pressure quantity must now be defined as
ui=ui (x,y,z,t) or pi=pi(x,y,z,t) respectively.
URANS then take into account time variations of the mean quantities. Nevertheless we
still care about statistical values of such quantities. This can be easily seen in Figure 2.4
where represents the mean quantity ⟨ ⟩ represents its mean value and represents its
fluctuation around ⟨ ⟩ and the modelled turbulent fluctuation cannot be seen in the
figure.
Figure 2.4 URANS Measurement of a Mean Quantity
Here the mean quantity is time dependent and hence an average of it in time can be
extracted ⟨ ⟩. The difference between them is therefore and the real value U can be
expressed as:
uuUuUU
(2.65)
2.7. Discretising the Equations: the Finite Volume Method (FVM)
If Navier-Stokes Equations want to be numerically solved the equations must be
discretised. The most widely used approach, and the one used in this project is the Finite
Volume Method because of its conceptual simplicity an ease of implementation for
arbitrary structured and unstructured grids32
. The FVM is based on averaged volume
values contained in each cell whereas for example the Finite Difference Method (FDM)
is based on local function evaluations at each grid point. Hence in the FVM, integral
conservation laws are applied to each cell volume analogous to the control volume
concept used in fluid mechanics.
32
Charles Hirsch, Numerical Computation of Internal and External Flows: The Fundamentals of
Computational Fluid Dynamics (Butterworth-Heinemann, 2007).
Chapter 2: Theoretical Background 27
For example, for a scalar φ with on a volume Ω discretised into smaller
volume controls/cells Ω1, Ω2, Ω3, as shown in Figure 2.5, where equation (2.66) applies.
SSdFd
t0
(2.66)
Figure 2.5 Volume Ω divided into three subvolumes Ω1, Ω2, and Ω3
We can have that (2.66) applied to the whole volume Ω equals applying it to the three
smaller volumes and adding them together.
AEDABDEBABCA
AEBCA
SdFdt
SdFdt
SdFdt
SdFdt
321321
(2.67)
Where the internal fluxes cancel because the flux in adjacent cells contributes in the
same amount but with opposite signs.
BAAB
dSdS
(2.68)
We can replace equation (2.66) by its discrete form (2.69) where the volume integrals
are substituted by the average volume values at each cell φi and the surface integral is
replaced by a sum over all the bounding faces of the considered cell volume Ωi.
0
Faces
ii SFt
(2.69)
It is worth mentioning that the coordinates of the cell are not present on equation (2.69)
explicitly and hence the value can be seen as the average value of the quantity inside the
cell, and grid coordinates are only needed to compute the cell volume and the bounding
surfaces. The equation below also expresses that the variation of the average quantity φ
over the time interval ∆t equals to the sum of the fluxes exchanged between
neighbouring cells. Finally, the FVM also allows to easily introducing boundary
conditions because it only needs to prescribe the desired flux values on the boundaries.
Chapter 2: Theoretical Background 28
Chapter 3 : Problem Description
3.1. Ambient Conditions
The models will be simulated at Sea Level conditions using the following values.
Table 3.1 Ambient Values
Density ρ [Kg · m-3] 1.225
Pressure P [Pa] 101325
Dynamic Viscosity μ [Pa·s] 1.7885
Free Stream Velocity U∞ [m·s-1] 6.0
Re 493150
The Reynolds number Re is calculated using equation (3.1) with the chord of the model
c = cmain + cflap =1.2 m.
UcRe
(3.1)
3.2. Tools Used
In this project several software and computing languages have been used to carry out to
accomplish the project’s goals. For the Geometry creation the software used was
SOLIDWORKS because it was the CAD software provided by the University of
Southampton. As regards as the CFD computations STAR-CCM+ was the solver
chosen among the variety of tools provided by the University. This was mainly because
the author already had experience with the software, which is always an advantage
because it takes the learning curve out of the contest, but also because it provides with
an efficient meshing and CAD management environment which enables to have all the
CFD process of Pre-Processing, Solving and Post-Processing in a single tool.
Additionally Matlab was used in the Baseline geometry definition (see Chapter 4:
4.1Elements’ Relative Position Study) for implementing the optimisation scheme and
creating basic geometry that was then used in SOLIDWORKS to create the 3D models.
During this optimisation the need of automating the process lead to the use of 2 codes in
VBA and JAVA that took care of automating the 3D geometry creation and the CFD
simulations respectively.
3.3. Model’s Geometry
As aforementioned (see Chapter 1) the geometry will be defined by two tandem airfoils.
The upstream element will be regarded as the main element in a typical two-element
high-lift configuration whereas the second element will be considered as a flap. The
main airfoil used in the studies will be the GA(W) – 1 which is a typical airfoil used for
Chapter 3: Problem Description 30
landing flap configurations33
. The airfoil geometry used for the flap element is the
NACA 634-021 since it resembles the cross section of the humpback whale flipper34
.
The coordinates for the GA(W)-1 were obtained from the UIUC airfoil coordinates
database35
, whereas the coordinates for the NACA 634-021 were obtained from the
Public Domain Aeronautical Software (PDAS) web page36
. The chord for the main
element cmain is 1 m and the chord of the flap cflap is 0.2 m. The origin of coordinates is
located at the leading edge of the main element as shown in Figure 3.1. The AoA of the
main element remains constant at zero degrees for all simulations.
Figure 3.1 Sketch of the Elements’ relative position
The gap is defined as the minimum distance between the lower point of the main
element trailing edge and the flap profile whereas the gap is defined by the horizontal
distance between the closest point of the flap profile to the lowest point of the main
element trailing edge. A sketch of the gap and overlap definition can be seen in Figure
3.2.
Figure 3.2 Gap and Overlap Definition
The geometry of the Wavy Leading Edge Flap (WLEF) will be based on the baseline
model aforementioned. The leading edge of these models will follow (3.2) where is
the x-coordinate of the Flap’s LE, A is the amplitude of the protuberances, n is the
33
Rumsey and Ying, “Prediction of High Lift: Review of Present CFD Capability.” 34
Fish and Battle, “Hydrodynamic Design of the Humpback Whale Flipper.” 35
Department of Aerospace Engineering UIUC Applied Aerodynamics Group, “UIUT Airfoil
Coordinates Database,” 1995, http://www.ae.illinois.edu/m-selig/ads/coord_database.html#G. 36
Ralph Carmichael, “Public Domain Aeronautical Software,” 31, http://www.pdas.com/index.html.
-0.2
-0.1
0
0.1
0.2
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Y
X
GA(W)-1
NACA 63(4)-021
Chapter 3: Problem Description 31
number of waves present along the Flap’s Span, and zLE is the z-coordinate of the Flap’s
LE.
LELE znAx 2sin (3.2)
From (3.2) one can easily see that the WLE wavelength λ is just ⁄ where b is the
model’s span. It has to be pointed out that the use of a sine function allows the
protuberances to grow from the baseline LE, i.e. for the modified LE the average of
(3.2) still coincides with the baseline geometry with no protuberances at all (see Figure
3.3).
Figure 3.3 WLE Sketch
3.4. Mesh
The mesh strategy chosen for this project was an unstructured meshing algorithm based
on polyhedral elements because it is able to adapt to complicated geometries such as the
two-element plus the additional wavy shape on the Flap’s LE that was investigated in
this project. A generic base size was used for the entire domain and additional mesh size
control volumes were used to produce a finer mesh near the model and in the wake zone
in order to well-capture all the flow features. The volumetric controls (VC) can be seen
in Figure 3.4. Each VC has an element size which is a percentage of the general base
size (16%, 12%, 8% and 4% respectively) so every time the base size is reduced or
increased for purposes such as mesh dependency studies (see Appendix A: Mesh
Dependency Study), so do the VC, refining then the areas where the mesh needs more
resolution.
Chapter 3: Problem Description 32
Figure 3.4 Volumetric Controls around the model
In terms of Near Wall Treatment, both turbulence models used, i.e. Menter’s SST and S-
A, need a well-defined mesh near the wall because no wall functions are used. The first
grid point must then lie on the viscous sub-layer, and hence the y+ value should be less
than 5. Figure 3.5 shows a detailed picture of the prism layer mesh in the boundary
layer zone for both elements.
Figure 3.5 Boundary Layer Mesh Detail
Figure 3.6 and Figure 3.7 show that the y+ value was almost always below the value of
5 unless at the stagnation points where the boundary layer is starting to develop and
hence its value increases considerably.
Figure 3.6 Wall y+ contour levels over the model
Chapter 3: Problem Description 33
Figure 3.7 Wall y+ Distribution
The baseline mesh used for this computational study finally had around 5M elements.
The number of elements for the other meshes used for WLE geometries oscillates
among this value, but due to the geometry modifications, mainly due to curvature
refinement, lead to different amount of cells used.
3.5. Domain Size and Boundary Conditions
In any CFD calculation the real domain has to be truncated to a finite volume control
where ultimately the computations are going to be carried over. This is the
computational domain.
The size of the computational domain used in this project is summarized in Figure 3.8.
The dimensions used are a result of a domain size study (see Appendix B: Domain Size
Dependency Study).
Figure 3.8 Boundary Conditions and Domain Size Sketch (not to Scale)
Because of we are solving a set of partial differential equations; it is needed to provide
some information at the initial time and boundaries in order to have a well-posed
problem.
In Figure 3.8 we can also see the boundary conditions applied to the computational
domain. The inlet of the domain was modelled as a Velocity Inlet for which the
Chapter 3: Problem Description 34
freestream velocity was prescribed, i.e. U∞ = 6 m/s, which gives a Reynolds number
based on the chord of the combined elements c = cmain + cflap = 1.2 m of Re = 493,151 ≈
500,000. Because of the extra equations needed to model the turbulence, also extra
boundary conditions have to be prescribed for the turbulence quantities. Because of two
different turbulence models were used, the turbulent viscosity ratio option was chosen.
The turbulence viscosity is a quantity present in both S-A and SST models and hence can
be used to prescribe the turbulent boundary conditions. In order to have a natural
transition on the model’s boundary layer a turbulent viscosity ratio of ⁄ was
used with a turbulence intensity of 0.5%.
The top and the bottom part of the domain were specified as Symmetry walls which is a
common strategy for defining freestream conditions. Here the flux across these two
boundaries is zero, which is equivalent to say that the velocity component on the normal
direction to these boundaries is set to zero. Furthermore, this implies that the following
gradients vanish.
0
0
0
nUtand
tUn
n
(3.3)
Where are the normal and the tangential unitary vectors to the boundary surface
respectively, φ is a scalar quantity and is the velocity vector.
As regards the outlet, a Pressure Outlet condition was used using the same turbulent
parameters as for the inlet because the flow is supposed to have similar conditions far
away from the model in both sides.
For a viscous flow that passes over a solid wall the relative velocity between the flow
and the solid wall is supposed to be zero at the wall surface. This is known as non-slip
boundary condition and is the main source of shear stress.
Finally, both sides of the domain were defined as periodic boundaries with a null
pressure jump. The solution obtained on one lateral side of the domain is mapped into
the opposite side every iteration. Using this configuration the effect of an infinite wing
is achieved but without taking out the cross-flow components as happens when using
the symmetry condition.
3.6. Time Resolution
As previously stated, in this project Unsteady RANS calculations have been used. When
the solution is allowed to vary on time it is necessary to define a time integration
method. In this project an implicit time scheme has been used. In theory, implicit time
schemes are more computationally expensive than explicit schemes because some
matrix inversion is needed during the solving process. However, while explicit schemes
are at the best conditionally stable as long as a sufficient small time step is used,
implicit time schemes are unconditionally stable, which allows for larger time steps
without risking the stability of the solution. Nevertheless it is very convenient to still
use time steps that are representatives of the flow time scale. For example for a flow
with velocity U∞=5 m/s using a grid resolution of ∆x=1 m and a time resolution of ∆t=1
s the convective courant number C would be:
Chapter 3: Problem Description 35
5
x
tUC
(3.4)
This means that the a fluid particle would go through at least 5 cells in each time step,
which is not desirable to happen. So despite using an implicit time scheme, the time step
was set to ∆t=0.0005 s for all the simulations carried out in this project, and as seen in
Figure 3.9 the Courant number was kept very close to the unit almost everywhere.
However, at some regions where velocities were higher than the free stream velocity,
i.e. in the gap zone between the elements, values above 1 were registered.
Figure 3.9 Courant Number Detail
3.7. Turbulence Model
Simulations carried out in this project used the SST turbulence model described in
section 2.5 because it is recommended in many of the papers reviewed (see Chapter 1)
and because it is supposed to better perform in separate flow conditions such as those
we shall expect in this project. However given that no experimental or computational
data was available to validate the results shown here, it was decided to use the S-A
turbulence model to compare the results and see if there were major differences that
could show up modelling errors. The results of this validation study can be found in
Appendix C: Validation against S-A Turbulence Model.
Chapter 3: Problem Description 36
Chapter 4 : Results
In this chapter we are going to investigate all the results extracted from the investigation
of the models in the different scenarios they have been tested. First we are going to
review the element’s relative position investigation in which the an optimisation scheme
was used to obtain a configuration of the two-element airfoil with high lift performance
such as what it is usually used on landing and taking-off configurations. Second, a study
of different WLEs shapes over the baseline’s flap’s LE was carried out. And finally,
third, a selection of modified WLE models were tested against an upstream vortex
condition and compared with the baseline case to test the robust performance of the
WLEs models in comparison with the baseline.
4.1. Elements’ Relative Position Study
The two airfoils used in this study have never been tested in combination in a two-
element configuration as far as the student knowledge. The initial baseline model tested
for this project can be seen in Figure 3.1 and poorly produced less lift than the main
element on its own. Therefore it was necessary to perform a study of the relative
position of both elements in order to achieve a high-lift configuration which is
ultimately the main goal of a two-element airfoil. Investigating the relative position
involves mainly three variables: 1) The flap’s X-Coordinate position xo, 2) The flap’s
Y-Coordinate position yo and 3) The flap’s deflection angle δ.
In order to efficiently quantify the benefits of one particular configuration a surrogate
modelling approach was taken. The surrogate modelling method used was Kriging37
,
which seems to be the most effective and versatile method for use in engineering design
and more particularly for intensive CFD based calculations38
. The benefits of using such
method is that not only provides a response surface which can accurately predict the
effect of the variables tested, but also can predict the expected improvement that can be
obtained by testing new design points.
It has to be mentioned that the simulations performed to obtain the results needed to fill
the response surface were obtained in 2D Steady RANS CFD simulations. The reason
for that is that exploring a 3D variable space can take long and then the computer
efficiency of 2D Steady RANS calculations, despite its inaccuracy where the final
approach taken. It is also worth mentioning that the surrogate models were built just
based on lift results, i.e. CL coefficient.
Initial Sample
In order to investigate the effect of these three variables, the Dr Andreas Sobester’s
Latin Hypercube39
code was used for designing the initial population for 30 models, 10
37
D. G. Krige, “A Statistical Approach to Some Mine Valuation and Allied Problems on the
Witwatersrand” (MSc thesis., University of the Witwatersrand, 1951), (G00907503). 38
Alex Forrester, “SESG6019 Notes: Design Search and Optimization - Case Study 1:Fast Global
Optimization” (2013). 39
Andreas Sobester, Bestlh.m (Southampton: University of Southampton, n.d.).
Chapter 4: Results 38
per each variable. The Latin Hypercube Sampling is a statistical method for generating a
sample of plausible collections of parameter values from a multidimensional
distribution which is often used to construct computer experiments.
In the context of statistical sampling, a square grid containing sample positions is a
Latin square if (and only if) there is only one sample in each row and each column. This
means that using this sampling method variables are sampled only once in each variable
space given then an initial population that can represent the model with accuracy
without repeating any variable value in the experiments. Applied to this project this
means that the effect of a given flap’s deflection angle is tested just one reducing the
general computational costs since the representation of the response surface obtained
has been examined better than it would have been with a simple random sampling plan
for example.
Automation of the process
In order to have a completely automated optimisation process some code has to be
created to: 1) Create the geometric models, 2) Run the models in a CFD environment, 3)
Extract the results that are going to be analysed, and 4) Analyse the Results, i.e. create
the surrogate model, and suggest new design points.
Figure 4.1 Optimisation Cycle
All the process is summarized in Figure 4.1. First, the base geometry, i.e. the wireframe
of the 3D model was created in Matlab. This enables to modify the relative position of
the flap’s by translating and rotating the flap’s points coordinates, additionally it also
allows to modify the shape of the trailing edge by changing the parameters A and n in
equation (3.2). The wireframe geometry then is stored in coordinate files with extension
.txt so we have a txt file per every line present in the model. Second SOLIDWORKS is
started and a VBA macro code is played so it creates the 3D models without any user
Chapter 4: Results 39
intervention and saves them in a format compatible with STAR-CCM+. These two
programs communicate themselves by means of a main txt file called “Data.txt” which
stores the main characteristics of the model, i.e. the model id, the flap’s X and Y
Coordinate position, the flap’s deflection angle and the value of the parameters A and n
(in this case both A and n are equal to zero because we are working with the baseline
model which does not have WLEs). Third, once the 3D model is ready, it is imported
into STAR-CCM+ and it is prepared and simulated again without user intervention
using a JAVA macro code. This macro names the geometry, creates the computational
domain, meshes, converts the mesh into 2D to save computational power, simulates and
finally extracts the CL coefficient to a text file named “Results.txt”. In this file every
time a design is added, a new line with the model characteristics and results is added to
the end of the file. Finally, the results file is read by Matlab again which updates the
surrogate model and based on it suggests a new design point and the cycle starts again.
Final Baseline Model
After 52 calculations (30 initial samples plus 22 surrogate model updates) the best
model was model 35 with a CL of 1.60 which is 220% better than the single element
configuration which had a CL of 0.50. The geometric values for model 35 where: xo =
0.9955 m, yo = -0.02540 m and δ = 33º. Despite model 35 was the best design model
tested it was not finally chosen to be the baseline model because the flap’s LE was too
close to the main element’s TE so the space left for further flap’s LE shape
modifications would not have been enough. However a very similar configuration was
taken for the final baseline model with the following parameters:
Table 4.1 Final Baseline Parameters
xo 0.995 m
yo -0.028 m
δ 33º
Chapter 4: Results 40
Figure 4.2 Response Surface with Red Dot indicating the Final Baseline Model
Figure 4.2 shows how despite the baseline is not in the predicted maximum zone of CL
it is still a good design near the maximum area. In fact the CL for this Baseline is 1.40
which is still 180% better than the single element configuration and much better than
the initial two-element configuration which produced less lift than even the single-
element one.
The final 3D geometry is shown in Figure 4.3. It can be seen in the front view that the
space left for the WLEs shape is still enough.
Figure 4.3 Baseline geometry with Coordinate System Indications. From Top-Left to Bottom-
Right: Top View, Isometric View, Front View and Rear View.
Chapter 4: Results 41
4.2. Experiment A: Flaps with Wavy Leading Edges
Once the baseline geometry was defined it was time to begin the investigation of
different wavy shapes on the flap’s LE. In total 9 extra cases were prepared to be
simulated using Unsteady RANS with the Menter’s SST closure model. Given that in
the papers reviewed concerning the WLE effect reveal the importance of the amplitude
and the wave length of the protuberances on the performance of the modified airfoil
geometry it was decided to try different combinations of both parameters. Three
different amplitudes, i.e. high, medium and low, and three different wavelengths, i.e.
high, medium and low, were combined to produce 9 models. The High Amplitude (HA)
models had A = 3 cm, the medium amplitude (MA) A = 2 cm and the low amplitude
(LA) had A = 1 cm. The High Wavelength (HW) models had only one wave over the
flap’s span n = 1 which means that λ = 50 cm. The Medium Wavelength (MW) models
had four waves n = 4 with λ = 12.5 m, and the Low Wavelength (LW) had eight waves
n = 8 with λ = 6.25 cm. Table 4.1 summarises all the models parameters in terms of the
chord of the flap cflap including the baseline case (model0) as it is usually done.
However from now onwards we will refer to them by their model names. For each
model calculations were done until results were statistically converged and then
statistical values were taken such as for example the mean velocity or the pressure
coefficient variance.
Table 4.2 Models Parameters in terms of cflap
Model
Name
WLE
Amplitude
A
WLE
Wavelength
λ
model0 0 cflap 0 cflap
model1.1 0.05 cflap 2.5 cflap
model1.4 0.05 cflap 0.625 cflap
model1.8 0.05 cflap 0.3125 cflap
model2.1 0.1 cflap 2.5 cflap
model2.4 0.1 cflap 0.625 cflap
model2.8 0.1 cflap 0.3125 cflap
model3.1 0.15 cflap 2.5 cflap
model3.4 0.15 cflap 0.625 cflap
model3.8 0.15 cflap 0.3125 cflap
Results
In Table 4.3 it can be seen how none of the models tested had a better performance than
the baseline case for clean upstream conditions. All the modified models have less lift
and more drag than the baseline case in this particular configuration. The reason for this
happening is that the baseline model was the result of the elements’ relative position
optimisation. The introduction of the WLE to the flap’s LE modifies the basic
parameters of a multielement wing, i.e. the gap and the overlap, consequently some
Chapter 4: Results 42
sections of the flap are outside that optimal configuration and hence do not perform as
good as the baseline case with a straight LE for the specific configuration tested here.
But not only that, the WLE add additional flow features on the flap zone as we will see
in former sections which promotes flow separation or attachment in different sections of
the flap’s span.
Table 4.3 Experiment A Results
Model CL CD CLflap
model0 1.4059 0.0320 0.1178
model1.1 1.1767 0.0410 0.0862
model1.4 1.2166 0.0427 0.0940
model1.8 1.2471 0.0442 0.1004
model2.1 1.1662 0.0442 0.0884
model2.4 1.1803 0.0466 0.0929
model2.8 1.1751 0.0474 0.0938
model3.1 1.1330 0.0462 0.0866
model3.4 1.1463 0.0498 0.0914
model3.8 1.1081 0.0468 0.0851
Figure 4.4 show that model1.8 is the best model with LE modifications in terms of
overall lift at the same time that is not the worse model in terms of drag. Model1.4,
model 2.4 and model2.8 are the best ones in this order. We can clearly see in Figure 4.5
that the models just mentioned are also the ones with the highest flap’s lift coefficient
although still lesser than model0, the baseline.
Figure 4.4 CL and CD Coefficients for all the Models Tested
Chapter 4: Results 43
Figure 4.5 CLflap Coefficient for all the Models Tested
Wall-Shear Stress Streamwise Direction Contours for n=1 and n=0
In order to investigate the effect of the WLE on the flap element the first magnitude
observed is the Wall Shear Stress (WSS) in the streamwise direction. The WSS can be
expressed as:
0
y
wy
u
(4.1)
Figure 4.6 a) shows the WSS contours on the baseline model. In this figure we can
observe that only a small stripe of flow is attached to the flap’s upper surface at the first
¼ of its chord. Further downstream two reattachment zones can be founded at ¼ and ¾
of the flap’s span. If we observe the vorticity slices just after this reattachment zones we
can isolate a structure with a “V” shape composed by two counter-rotating vorticity
contours. We will see in the following paragraphs that this “V” shape appears also on
the WLE models again and it can be associated with zones of attached flow.
Chapter 4: Results 44
Figure 4.6 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i] Contours on
Several Spanwise Planes: a) model0, b) model1.1, c) model2.1 and d) model3.1
Figure 4.6 b) shows the WSS contours for model1.1 and we can clearly see that the
modification of the flap’s LE has completely modified the flow behaviour on the flap
element. The flow is now detached in the entire chord length at the valley zones, i.e.
where the chord length is lesser, and it is completely attached at the peak zones, i.e.
where the chord length is bigger, which agrees with the papers reviewed40, 41
. We can
observe again the “V” shape on the vorticity contours where the flow is attached. The
flow tends to travel from the peak zones with more airfoil thickness to the valley zones.
This movement creates streamwise vortices that enclosures the flow in-between valleys
and separates zones of attached flow from zones of detached flow. Figure 4.6 c) and
Figure 4.6 d) show that the flow behaviour is very similar between models with the
same WLE’s wavelength but also showing that the area of attached flow is increased
with higher amplitudes. However if we recall Table 4.1we can see that the flap’s lift
increases from the model with A=1 cm (model1.1) to the model with A=2 cm
(model2.1) but the model with the higher amplitude (model3.1) with A=3 cm has
however less lift than model2.1 and hence pointing out that the increment in amplitude
is no longer beneficial for this particular wavelength past a certain value of A>2 cm.
Additionally we can see at the upper-right corner of the aforementioned figures that the
WLE modifications on the flap also affects the main element flow near its trailing edge
where a negative zone of the WSS can be founded. This zone becomes bigger with
higher amplitudes which explains why the overall lift coefficient diminishes as the
amplitude increases for n=1 (λ=2.5 cflap). It can be as well observed that due to the
amplitude rise the two counter-rotating vortices that form this “V” tend to separate
leaving then more space in between them were the flow can remain attached and as a
result the area of attached flow becomes wider.
40
Johari et al., “Effects of Leading-Edge Protuberances on Airfoil Performance.” 41
Malipeddi, “Numerical Analysis of Effects of Leading-Edge Protuberances on Aircraft Wing
Performance.”
Chapter 4: Results 45
Velocity Contours at Peak, Valley and Base Sections for n=1 and n=0
We can see in the following figures (Figure 4.7 to Figure 4.13) how for the baseline
model (Figure 4.7) that the gap between both elements forces the flow to accelerate
giving more momentum and making the boundary layer to attach near the flap’s LE,
something that would not occur if the flap element, recall with more than 30º deflection
angle, would have been tested in isolation. We can also observe that the stagnation point
in the flap element creates a zone of low velocity and hence high pressure under the
main element trailing edge which contributes to a higher overall CL.
Figure 4.7 model0 Base Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Figure 4.8 model1.1 Base Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
In contrast, we can see how in Figure 4.8, Figure 4.10, Figure 4.11 and Figure 4.13 for
the WLE models the sections with shorter chord length, i.e. the base and valley sections,
the flow separates from the very beginning whereas in Figure 4.9 and Figure 4.12 the
flow is attached at the peak sections of the flap. Nonetheless, despite the flow is
attached at the peaks in model1.1 and model2.1 (model3.1 is not shown here because of
its similar behaviour with the other cases shown) the velocity of the flow is lower than
Chapter 4: Results 46
that of the baseline model at its attached zone. This explains how, despite having a
wider area of attached flow, the lift produced by the modified flap models is still lower
than the baseline flap. Finally, the pressure increase created by the presence of the flap
element is less significative for the cases with undulated LEs which explains why the
total lift is lesser.
Figure 4.9 model1.1 Peak Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Figure 4.10 model1.1 Valley Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Chapter 4: Results 47
Figure 4.11 model2.1 Base Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Figure 4.12 model2.1 Peak Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Figure 4.13 model2.1 Valley Section, Velocity Slice non-dimensionalised by the Free Stream
Velocity
Chapter 4: Results 48
Pressure Coefficient Contours for n=1 and n=0
Figure 4.14 clearly shows that the WLE were not particularly beneficial for this
configuration. Here we will investigate the pressure coefficient Cp, which can be
expressed as:
221
U
ppC p
(4.2)
The area covered by low pressure is smaller and unstructured for the flap element and
additionally include detrimental effects on the suction surface of the main element near
its trailing edge which finally leads to, as stated before, a lower performance in both lift
and drag.
If we compare the baseline against the modified models we can see that for the baseline
there exists a zone of very low pressure at the entire flap’s LE span whereas for the
other models, not only the pressure is higher but also it is for a smaller area. Near the
valley zones, the main element experiments a more intense negative pressure gradient.
This pressure increase on the suction surface of the main element becomes even more
intense with higher amplitudes as seen in Figure 4.14 c) and Figure 4.14 d). This is due
to the fact that when the valley is deeper, its stagnation point moves further downstream
getting away from the main element and consequently, the increase in pressure does no
longer only affect the main’s element pressure surface but also its the suction surface.
Figure 4.14 CP Contours on the models surface: a) model0, b) model1.1, c) model2.1 and d)
model3.1
Wall Shear Stress Streamwise Direction Contours for n=4 and n=8
Increasing the number of waves n along the flap’s span also increases the amount of
streamwise vortices. The combined effect of increasing n and A results in more
Chapter 4: Results 49
interaction between the vortices created by the protuberances. We can see in Figure 4.15
a) at the middle section of the flap’s span that inside the characteristic “V” pair of
vortices now there is another pair of counter-rotating vortices. We can see in Figure
4.15 c) how the interaction between these four vortices becomes stronger as the
amplitude is increased and so does the strength of the vortices. The effect of what just
mentioned keeps increasing till it reaches a point where they become so important that
vortex fences are created in the streamwise direction completely separating the positive
zones from the negative zones of WSS as shown in Figure 4.15 e). This behaviour leads
to a more patterned WSS contours that tend to group in pairs of two peaks with a valley
separating them.
As the number of waves n increases, the wavelength λ decreases, joining the vortices
together and forcing them to interact in a more intensive way. Particularly, in Figure
4.15 e) and Figure 4.15 f) we can see how the positive zones of WSS tend to gather for
n=8 into groups of three and two. This pattern is repeated along the spanwise direction.
The flow coming from three peaks at the LE joins into a single strip of positive WSS
that finally reaches the TE. The flow that joins two peaks at the LE however can only
reach the TE together for the higher amplitude case. Nevertheless if we compare the lift
coefficients of these models against the baseline case only model1.8 can barely resist
the comparison. It is hence postulated that the WLEs tend to tidy up the flow in the
streamwise direction due to the fences effect created by the streamwise vortices, but at
the same time it debilitates the flow reducing its streamwise velocity.
Chapter 4: Results 50
Figure 4.15 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i] Contours on
Several Spanwise Planes: a) model1.4, b) model1.8, c) model2.4, d) model2.8, e) model3.4 and
f) model3.8
Velocity Contours at Peak and Valley Sections for n=4 and n=8
Following what was previously done with the n=1 models we now explore the velocity
contours at the most important sections. In Figure 4.16 we can observe that for n=4
models, although the flow might be attached in bigger regions for the WLEs models
than for the baseline, as stated before the velocity in these regions is considerably lower.
For example at Figure 4.16 c) and Figure 4.16 e) it can be seen that a large percentage
of the flap’s chord is having attached flow as the WSS positive sign indicates. We can
also confirm that negative areas of WSS can be associated with separated flow regions
as seen in Figure 4.16 f) where the flow is detached along the entire chord length. This,
results in a more intense and unstable wake and ultimately in a higher drag than the
baseline.
At the same time, due to the introduction of the streamwise vortices produced by the LE
protuberances, the flow is highly three-dimensional. The flow tends to move from the
peak zones to the valley zones promoting reattachment in the valley sections as seen in
Figure 4.16 b) and Figure 4.16 d).
Chapter 4: Results 51
A similar behaviour can be observed for the n=8 models shown in Figure 4.17. Again
we see that velocities at the suction surface of the flap are lower compared to the
baseline model where velocities of nearly 1.5U∞ were reached at the very beginning of
the flap. It can be appreciated that light blue colours are predominant in Figure 4.17,
which means that velocities on the suction surface of the flap are even lower than the
freestream velocity. Because of this reduction on the modified models boundary layer’s
edge velocity, the undulated models show a CLflap drop of almost 30% in the worst case
(model3.8). However there are regions of completely attached flow (from the flap’s LE
to the TE) when the protuberances are introduced (recall that the flap’s deflection angle
is 33º) in contrast with the just almost 1/8 attached zone for the baseline case.
Consequently it can be postulated that the WLEs models would be much more resistant
to upstream perturbations because although they have slower boundary layers they have
zones of attached boundary layers along the entire flap chord.
Figure 4.16 Velocity Slice non-dimensionalised by the Free Stream Velocity: a) model1.4 Peak
Section, b) model1.4 Valley Section, c) model2.4 Peak Section, d) model2.4 Valley Section, e)
model3.4 Peak Section and f) model3.4 Valley Section
Chapter 4: Results 52
Figure 4.17Velocity Slice non-dimensionalised by the Free Stream Velocity: a) model1.8 Peak
Section, b) model1.8 Valley Section, c) model2.8 Peak Section, d) model2.8 Valley Section, e)
model3.8 Peak Section and f) model3.8 Valley Section
Isosurfaces of Vorticity ±100 s -1
for n=4 and n=8
In previous sections it has been stated that the WLEs introduce streamwise vortices
increasing the cross-flow velocity components. Previously we saw vorticity contours in
different sections of the flap, here in Figure 4.18 it can be seen iso-surfaces of constant
vorticity in the streamwise direction. It is shown in the figure below that streamwise
vorticity is only created by the LE protuberances. A pair of counter-rotating vortices is
created in each peak because of the flow moving from the peak to its two adjacent
valleys. If we recall Figure 4.16, where we had vorticity contours on several planes
along the flap’s chord, we can now see how these vortices evolve. We can relate zones
of flow attachment to zones where the vortices tend to join and separated zones where
the vortices tend to separate. Vortices are most likely to be attracted by the attachment
zones where the pressure is lower rather than the separated regions from which they
tend to move away. At the same time we see that the WLEs also introduce streamwise
Chapter 4: Results 53
vorticity on the main airfoil near its trailing edge. This fact increases the cross flow
component of it, which might not be the most desirable effect when high lift coefficients
are pursued.
Figure 4.18 Isosurface of Streamwise Vorticity Ω=±100 s-1
: a) model1.4, b) model1.8, c)
model2.4, d) model2.8, e) model3.4 and f) model3.8
Pressure Coefficient Contours for n=4 and n=8
In Figure 4.19 how the lower pressure zones coincide with the zones where vortices
shown in Figure 4.18 joined. The low pressure zones are always located on the zones
where positives values of the WSS appear. However similarly with what happened with
the high wavelength models the pressure coefficient tends to be higher on the suction
surface of the main element than it was on the baseline model, and given that the main
element is the one that produces the most part of the lift the medium and low
wavelength models are also less efficient than the baseline in terms of overall lift.
Chapter 4: Results 54
Figure 4.19 CP Contours on the models surface: a) model1.4, b) model1.8, c) model2.4, d)
model2.8, e) model3.4 and f) model3.8
4.3. Experiment B: Flaps with Wavy Leading Edges against
Upstream Vortex Condition
In this second experiment 4 modified models plus the baseline case were tested against
an upstream vortex condition. The vortex was added to the Experiment A flow
conditions and then the new solution was used as Initial Condition for Experiment B.
The vortex was based on equations (4.3) extracted from42
with some minor
modifications to adapt to the case studied here. The strength of the vortex is controlled
by the constants k1 and k2 and the initial position is controlled by the constants x0 and y0.
Constant L represents the domain length and U∞ as usual represents the freestream
velocity in the x-direction.
The procedure to incorporate the vortex to previous solutions was tedious but simple.
The solutions were extracted from STAR-CCM+ into coma separated values files and
then modified using a MATLAB code with equations (4.3).
42
J.W. Kim, “Quasi-disjoint Pentadiagonal Matrix Systems for the Parallelization of Compact Finite-
difference Schemes and Filters,” Journal of Computational Physics 241 (2013): 168–194.
Chapter 4: Results 55
sm
kk
myx
L
rkkr
yyxx
rLxxkUvtyxv
rLyykUutyxu
6U
m36L
35.0,400,
25.0,2,
2
1exp
r
with)()0,,(
)(0,,
21
00
2
22
12
2
0
2
0
01
01
(4.3)
It is worth saying that (4.3) does not include modifications to the pressure field, the
kinetic energy field or the specific dissipation field, and hence identical values to the
non-vortex solutions were used. As the solver main variable are velocity’s components,
and all the rest of the variables highly depend on them, the solver computed at the first
iteration values for all the rest quantities needed to initialise the flow field based on the
velocity field provided. This can be seen in Figure 4.20 where we can observe that no
pressure due to the vortex appears at the initial time because the pressure field was not
modified in by the vortex equations. However we see how after some iterations the
solver has computed the new pressure field according to the new velocity field in Figure
4.20 d). The author nevertheless acknowledges that proper calculations of these non-
computed variables would have result in a smoother evolution of the solution.
Figure 4.20 Detail of Pressure Calculation from Initial Conditions: a) Vorticity contours at t=0s,
b) CP Contours at t=0s, c) Vorticity Contours at t=0.1s, and d) CP Contours at t=0.1s
Results
In this section the results obtained for the Experiment B are investigated. We focus on
the total lift coefficient of the model, the main element’s lift coefficient and the flap’s
lift coefficient. Simulations time begins when the vortex has been placed in front of the
model and then is convected downstream until it hits the models. In Figure 4.21 we can
observe that t ≈ 0.325s the vortex hits the LE of the main element, however since the
very beginning the inclusion of the vortex has a direct effect on the lift produced by the
models. On one hand, it is noticed that the performance of the baseline model (model0)
is more affected by the presence of the vortex and quickly starts decreasing its value.
Past t=0.4s the baseline model never recovers the initial CL value of 1.4 and ends up
Chapter 4: Results 56
producing even less lift than model1.8. On the other hand, although the models with
WLE are also affected by the introduction of the vortex condition they seem to have a
more continuous trend during all the simulation time. WLEs models end up having very
similar CL coefficients to the ones they started or even with slightly higher values.
Figure 4.21 Total CL History Comparison
Figure 4.22 CLflap History Comparison
The same trend can be observed in Figure 4.22 for the flap’s lift coefficient. However in
this case all the models with modified LE end up producing even more lift at t=0.7s
than at t=0s and what is more, more lift than the baseline flap in contrast with what
happened with clean freestream conditions on Experiment A. An interesting feature to
look at Figure 4.22 is that the models higher number of waves along the span, or lower
wavelengths, tend to have more stable behaviour after t=0.2s, which is even before the
vortex has encountered the main element.
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Tota
l CL
Simulation time [s]
model0
model1.4
model1.8
model2.4
model2.8
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
CLf
lap
Simulation Time [s]
model0model1.4model1.8model2.4model2.8
Chapter 4: Results 57
As regards the main element, the curves showing the CLmain history in Figure 4.23 are
almost identical to the ones showed in Figure 4.21. Again the WLEs models experiment
less variations than the baseline model during the simulated time frame.
Figure 4.23 CLmain History Comparison
At this point the reader might be interested in how big the variations were experimented
by the models. Figure 4.24, Figure 4.25 and Figure 4.26 show that the difference
between the time averaged CL values and both maximum and minimum values reached
during the tested time frame is much bigger for the baseline. The root-mean-square
(rms) gives us an idea of how big where the variations in the lift coefficient suffered by
the models, and we can clearly see that for the baseline model were considerably higher,
being twice as big as the rest of the models if we look at Total CL and CLmain values.
Figure 4.24 Total CL Statistic Values Comparison
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
CLm
ain
Simulation Time [s]
model0
model1.4
model1.8
model2.4
model2.8
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
model0 model1.4 model1.8 model2.4 model2.8
Tota
l CL
rms
Tota
l CL
Average CL Max CLMin CL CL rms
Chapter 4: Results 58
Figure 4.25 CLflap Statistic Values Comparison
Figure 4.26 CLmain Statistic Values Comparison
Cp rms contours
We can confirm all what was afore stated if we investigate how the pressure has varied
on the surface of the models by investigating the rms of the pressure coefficient.
Comparing the following figures (Figure 4.27 to Figure 4.31) we observe that the
highest values of the CP rms are present on the baseline model at the flap’s LE. If we
recall the last section (4.2 Experiment A: Flaps with Wavy Leading Edges) the flow on
the baseline was just attached at this part of the flap, and because of that we may
conclude that flow finally separated because of the presence of the vortex (this
affirmation will be soon confirmed).
There are although some variations as well on the WLEs models as depicted by Figure
4.28 to Figure 4.31. This is however inevitable because the vortex tends to destabilise
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
model0 model1.4 model1.8 model2.4 model2.8
CLf
lap r
ms
CLf
lap
Average CL Max CLMin CL CL rms
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
model0 model1.4 model1.8 model2.4 model2.8
CLm
ain r
ms
CLm
ain
Average CL Max CLMin CL CL rms
Chapter 4: Results 59
the whole flow field, and yet the ability to maintain the flow attached on the modified
models highly depend on the patterned vortical structure created because of the
sinusoidal protuberances. Nonetheless the CP rms contours on the modified models’
main element show that because of the more regular behaviour of the modified flaps the
flow on the main element is much more stable.
Figure 4.27 model0 CP rms contours
Figure 4.28 model1.4 CP rms contours
Chapter 4: Results 60
Figure 4.29 model1.8 CP rms contours
Figure 4.30 model2.4 CP rms contours
Figure 4.31 model2.8 CP rms contours
Chapter 4: Results 61
Wall-Shear Stress Streamwise Direction Contours
In this section we are going to analyse instantaneous values of the WSS in the
streamwise component in order to clarify what happened when the vortex collided with
the models. For the reader’s sake of ease in the following figures, a part form the WSS
contours a detail of the vorticity contours has been added to show the location of the
vortex at each time location.
In Figure 4.32 we can confirm what aforementioned: the small strip of attached flow
present in the simulations with clean upstream conditions is no longer present when the
vortex condition is introduced. The flow structure on the flap rapidly brakes down and
only the two recirculation zones remain, for a short period of time. Ultimately we
observe that at t=0.6 the WSS on the flap’s suction surface is all negative, which is an
indication of separated flow.
More difficult is to comment on Figure 4.33 because the presence of the vortex in the
upstream flow brakes the well-patterned flow we used to have with clean upstream
conditions. The “V” shape pair of counter-rotating vortices is almost unappreciable and
as a result, regions of attached flow are more irregularly distributed. Nevertheless we
can say that this patterned flow organisation is finally recovered at t=0.6s. At t=0.7s the
counter-rotating pair of vortices is again present in the middle section of the flap’s span,
and consequently positive values of the WSS can be seen. We can conclude that this is
the reason why in Figure 4.21 and Figure 4.23 the models with sinusoidal LEs end up
having almost the same lift value that the one they started with.
If the number of waves is increased we see that the flow features depicted by the WSS
contours are much similar earlier on. For example in Figure 4.34 and Figure 4.36 the
WSS distribution on the flap for model1.8 and model2.8 starts to be qualitatively stable
for t=0.5s and t=0.4s respectively whereas for both model1.4 and model2.4 this
happens at t=0.6s.
Finally, if we have a look at the zone where the WLE are strictly located, i.e. form the
LE with +A to –A (being A the amplitude of the sinusoidal wave), we see that the
contours of the WSS are almost constant during the simulated time. But also if we
compare them to former figures in section 4.3 (Figure 4.15) we see that they are very
alike as well. In contrast, if we have a look at the same percentage of the flap’s chord on
the baseline model, we can depict big differences. Given that this part of the flap
element is the one that higher interacts and influences the flow and performance of the
main element, we can conclude that the more stable performance on the modified flaps’
LE region is the responsible for the overall robust performance of the main element in
terms of lift production.
Chapter 4: Results 62
Figure 4.32 model0 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d) t=0.5s, e)
t=0.6s and f) t=0.7s.
Figure 4.33 model1.4 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d) t=0.5s, e)
t=0.6s, f) t=0.7s
Chapter 4: Results 63
Figure 4.34 model1.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d) t=0.5s, e)
t=0.6s, f) t=0.7s
Figure 4.35 model2.4 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d) t=0.5s, e)
t=0.6s, f) t=0.7s
Chapter 4: Results 64
Figure 4.36 model2.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d) t=0.5s, e)
t=0.6s, f) t=0.7s
Chapter 5 : Final Remarks
Unsteady 3D RANS CFD calculations have been performed over a two-element airfoil
configuration with a flap’s sinusoidal leading edge shape. Models with flap’s LE
modifications were tested in two different environments. In the first part of the
experiments, the modifications were introduced to an already optimised two-element
case and were tested against upstream clean conditions and then compared to the
straight LE case. In the second part of the experiments, some modified models and the
baseline model with straight LE were tested against an upstream upcoming vortex in
order to test if the WLE shapes can contribute to have a more robust performance when
using multi-element airfoils in turbulent conditions such as those usually present in the
airports.
For the first experiment (see 4.2 Experiment A: Flaps with Wavy Leading Edges) the
main conclusion that can be obtained is that for clean upstream conditions the WLEs are
not beneficial in terms of both lift and drag. Consequently the aerodynamic efficiency
drops when the sinusoidal LEs are used.
Positive effects that one may obtain with the modified LEs cannot be compared with the
negative effect that the modifications on the flap’s LE has on the overall performance of
the two-element airfoil. Because of the sinusoidal shape of the flap’s LE, two out of
three main parameters that have direct relation with the lift produced by the airfoil are
also modified, i.e. the gap and overlap between the two elements varies along the flap’s
spanwise direction. These two parameters are the main responsible of the second
element beneficial effect present on multi-element configurations and if they are not set
properly the lift produced by the main element drops.
Additionally, if we just look at the lift produced only by the flap element, the WLE
neither have higher lift coefficients than that of the straight model. This is not a surprise
for the author, because although it has been extensively proven that the WLE perform
much better than straight LE, all the experiments to the date have been carried out with
the WLE in isolation. It is also known that the presence of the upstream wake of the
main element completely modifies the flow field that those WLE are facing.
Nevertheless many of the flow characteristics depicted by former studies of the WLEs,
i.e. the spanwise vortex fences separating the attached regions from the detached
regions, are still present.
At the same time, it might be possible that, what it is an optimum flap position for the
straight LE, might not be an optimum position for the sinusoidal edges. This statement
does not mean that there might not be one, however a proper optimisation of the
elements’ relative position taking into account the amplitude and the wavelength of the
protuberances will require an optimisation of five variables: flap’s x-position, flap’s y-
position, flap’s deflection angle, WLE amplitude, and WLE wavelength. This
optimisation is then nothing but at least computationally expensive because of its three-
dimensional and unsteady nature.
Finally it is worth saying that both the flap’s deflection angle and the overall angle of
attack have remained constant during all the calculations. The author acknowledges that
these two parameters can have a noticeable impact on the flow behaviour and hence
they would have been also tested. However, there existed limited time and
computational resources available for this project, and the student consequently decided
Chapter 5: Final Remarks 66
to carry on with the second part of the experiments because it could at least show if the
WLE could be beneficial when an upstream vortex impinges on the airfoils.
For the second experiment (4.3 Experiment B: Flaps with Wavy Leading Edges against
Upstream Vortex Condition) the major finding was that the wavy shapes on the flap’s
LE make the models more resistant to dirty upstream conditions. The variations the lift
coefficient suffers are smaller for the modified models than for the baseline. In some
cases the lift is even higher once the vortex has completely passed over the entire
model. After careful investigation of the results, the student concludes that reducing the
wavelength of the WLE results into more stable behaviour.
The vortex added in front of the models, highly affects the flow field downstream
immediately because of the flow has been treated as incompressible. The baseline
model, which only has a small percentage of the chord with attached flow, is in a very
unstable condition, and the perturbations added upstream of it leads to finally the flow
to separate all over the flap’s upper surface. When the vortex has passed, the baseline
model is not able to recover its initial condition with partially attached flow near its
leading edge. In contrast, although also affected initially by the upstream perturbations,
the WLEs models show much more robust performance during the simulated time.
Their patterned flow field produced by the sinusoidal LE seems to be more stable, and
consequently the models are able to recover the same flow field they used to have
before the introduction of the vortex.
At the same time, the main element, which has not been modified at any point during
the entire project, benefits form of the WLEs robust performance. The baseline flap
completely flow field completely changes when the upstream vortex condition is faced,
and this change is bigger at its LE. The flap’s LE zone is the one that interacts the most
with the main element and consequently big changes in former will led to big changes
in the latter. In contrast, for the modified models, we have seen that the flow remains
almost exactly the same in the first 10% of the flap’s chord. This almost constant
behaviour results in fewer variations on the main element and ultimately an overall
stable performance.
Major Findings and Conclusions
The WLEs did not perform better than the baseline case for clean upstream
conditions for the configuration studied in this project. However a proper
optimised design that includes the WLE parameters in the optimisation may
show different results.
When an upstream vortex impinges the multi-element airfoil, the models that
use WLE have more robust performance than the baseline with straight LE.
The protuberances at the flap’s LE introduce streamwise vortices that separate
the attached regions from the detached ones. The attached zones can be found
where those vortices tend to join, and the separated zones where the vortices
tend to separate.
The amplitude A of the protuberances is related with the intensity of those
streamwise vortices and the wavelength λ is related with the interaction that
those vortices have with themselves.
The behaviour we see when the WLE are used in a single element configuration
with clean upstream conditions, i.e. flow attached at the peak sections and
Chapter 5: Final Remarks 67
detached at valley section, cannot be assured when they are used in a two-
element configuration because of the effect of the upstream element.
The two-main objectives of the project have been fulfilled. The WLE have been
successfully implemented into a two-element configuration and the student has
gained some insight in how they perform in that situation. The performance
against upstream trailing vortices has been tested via Experiment B and results
showed that WLE offer more robust performance than straight LE.
Further Work and Recommendations
Both the angle of attack α and the flap deflection angle δ, have remained
constant during this project. It is recommended that the effect of these two
variables has to be tested to fully understand the flow produced by the WLE
when used in the flap element.
An optimisation of the models is needed to conclude that there is no way that
WLE can offer more lift than straight models. It is recommended that optimal
values of flap’s relative position and WLE shape are founded and then compared
with the straight LE optimum design.
The flow field created by the WLE shapes has multiple features that might be
too complicated to capture by a RANS model. Especially we have seen that
models based on the linear eddy viscosity assumption can be a poor
approximation of the real flow when sudden changes in the Strain rate appear.
This is the case in the flap region where we might expect massive separation. It
seems then more convenient to use more accurate models such as DES or LES
to confirm the findings obtained in this project.
Given that there are no academic papers that investigate the WLE in a two-
element configuration, a wind tunnel experiment will give at least some data to
validate against.
Finally, we have seen in this project, and in the literature review, that the WLE
can offer higher lift coefficients at high angles of attack than straight LE. The
maximum lift coefficient is nevertheless reduced, but there are some
applications where the author believes that having lift at very high AoA can be
very beneficial. For example in wind turbines, where high AoA means high
momentum produced by the turbine blades. It might be that the maximum lift is
lower for these WLE wind turbine blades; however the component of the force
they produce that contributes to make the wind turbine spin is much bigger
because of the very high AoA.
Chapter 5: Final Remarks 68
Appendix A: Mesh Dependency Study 69
Appendix A: Mesh Dependency Study
In order to quantify the amount of error in the simulations coming from numerical
errors/uncertainties, e.g. round-off error, iterative error or discretization error, a mesh
dependency study was carried on the baseline geometry. Modifying the base size of the
mesh the number of elements used for the calculations was changed for 6 cases from the
coarsest mesh with 3 Million elements to the finest mesh with 12.4 Million elements.
Results show that even for the finest mesh tested there still were minor variations on the
aerodynamic values investigated. Given that the computational power available is not
infinite the mesh with 4.7 Million cells was finally chosen because it was the best trade-
off between accuracy and computational efficiency. The number of simulations that
have been carried out in combination with the fact that because of the nature of
unsteady calculations, the Courant number must always be close to the unit, increased
the importance of having a computationally efficient mesh.
Table A.1 Mesh Dependency Study Results
# Elements (Millions)
Cl Error
% Cd
Error %
Cl main
Error% Cl Flap Error%
3.0 1.3969 -0.9% 0.0328 5.8% 1.2800 -0.9% 0.1169 -0.2%
3.6 1.4008 -0.6% 0.0327 5.4% 1.2836 -0.6% 0.1172 0.0%
4.7 1.4037 -0.4% 0.0322 3.8% 1.2861 -0.4% 0.1176 0.3%
5.8 1.4057 -0.2% 0.0316 1.8% 1.2888 -0.2% 0.1169 -0.3%
7.6 1.4061 -0.2% 0.0312 0.7% 1.2893 -0.2% 0.1168 -0.3%
12.4 1.4089 - 0.0310 - 1.2917 - 0.1172 -
Figure A.1 Mesh Dependency Study Convergence
With a 4.7 Million mesh simulations already took around 4 days to converge in the
worst cases. This means that if the author would have used the finest mesh as a base for
further calculations the cost in terms of time would have been unaffordable. It can be
seen in Table A.1 that the mesh with 4.7 Million cells has an error below 0.5% with
respect to the finest mesh in terms of CL coefficient. The author acknowledges that time
and computational power would not have been a request, they always are, and the best
option would have been taken the most accurate grid. However it has also to be pointed
Appendix A: Mesh Dependency Study 70
out that coarser meshes although would have been more computationally efficient, all
engineering problem need at least a minimum of accuracy so the results, could be
trusted. As a consequence the author decided that the grid with 4.7 Million elements had
to be the baseline mesh for further calculations.
Appendix B: Domain Size Dependency Study 71
Appendix B: Domain Size
Dependency Study
To be sure that the size of the domain was not influencing too much the computational
simulations a domain size study was carried out. Table B.1 shows the three different
domain sizes tested in this study. The first one, the smallest one, with 6 m before the
model and 12 m after the model with a height of 12 m as well. It can be seen that results
do not differ that much in any cases showing that the smallest domain was already a
good choice. However, given that it was tested that the domain with 12 m + 24 m x 24
m was computationally affordable and that it can be translated in chord units to 10c+20c
x 20c which is a common strategy in CFD calculations was the final domain size
selected to be the baseline for further calculations. Into account was also taken that a
vortex was planned to be introduced in the domain, which needs some space to be
placed away from the model and from the inlet because of computational stability.
Table B.1 Domain Size Study
Domain Dimensions
# Elements (Millions)
Cl Error % Cd Error % Cl main Error% Cl Flap Error%
6 + 12 x 12 3.1 1.4058 0% 0.0337 3% 1.2874 0.4% 0.1184 0.8%
12 + 24 x24 4.7 1.4037 0% 0.0322 -2% 1.2861 0.3% 0.1176 0.1%
15 +30 x30 8.0 1.3995 - 0.0328 - 1.2821 - 0.1175 -
Figure B.1 Domain Size Convergence
Appendix B: Domain Size Dependency Study 72
Appendix C: Validation against S-A Turbulence Model 73
Appendix C: Validation against S-A
Turbulence Model
In order to validate the results obtained using the SST model the baseline case was
simulated using the S-A model as well and the results were compared. In Figure C.1 we
can see that results obtained are similar between both turbulence models. However the
S-A model a larger zone of attached flow over the flap and a weaker wake combined
with a higher velocity above the main airfoil with lower velocity below it, which leads
to a higher performance in terms of lift coefficient. But if we have a look at Figure C.2
we can see how inside the boundary layer the velocity profiles are quite similar. Even
though at the very beginning, at Profile 1, there seems that S-A, as mentioned before
predicts a higher flow velocity, further downstream at Profiles 2, 3 and 4 the velocity
profiles are almost identical showing that the modelling of the attached boundary layer
is similar for both turbulence models
Figure C.1 Velocity Contours Comparison
Appendix C: Validation against S-A Turbulence Model 74
Figure C.2 Velocity Profiles Comparison Moving from 1 to 4 in the stream direction
If we have a look at the pressure distribution over the model’s surfaces in Figure C.4 we
can confirm that the same trends stand for both models although as stated before the S-A
model predicts slightly bigger differences between the pressure and the suction surfaces
of the model. In both models it can be appreciated how intense is the pressure gradient
on the flap’s suction surface forcing the boundary layer to detach near ¼ of the flap’s
chord. We can further investigate the pressure on the model by looking at Figure C.3
where it is shown that the S-A pressure coefficient difference between pressure and
suction surfaces is slightly bigger on the main element. The pressure distribution on the
flap element also shows what has been just stated. However the differences are only
quantitative different showing in both cases the same pressure distribution shape with
some differences in the flap element where more turbulence modelling is present due to
the wake of the main element.
Figure C.3 Pressure Coefficient Distribution Comparison
Appendix C: Validation against S-A Turbulence Model 75
Figure C.4 Pressure Coefficient Contours Comparison
Figure C.6 shows the Wall Shear Stress in the streamwise direction. We can see that
again the main differences appear in the flap element where the SST model predicts a
smaller strip of positive Wall Shear Stress and hence a smaller part of the flap’s flow is
attached. These differences can be further observed in Figure C.5 where we can see that
the Skin Friction coefficient predicted by the one equation model remains above zero
for more space at the first ¼ of the flap’s chord. Additionally we can see that the
presence of the flap creates a zone at the last ¼ of the main element’s pressure surface
where the pressure created by the stagnation point of the flap induces an adverse
pressure gradient on the main element flow that deteriorates its boundary layer making
the skin friction coefficient to get very close to zero on the pressure side of the main
element.
Appendix C: Validation against S-A Turbulence Model 76
Figure C.5 Skin Friction Coefficient Distribution Comparison
Figure C.6 Wall Shear Stress X-Direction Contours Comparison
Appendix C: Validation against S-A Turbulence Model 77
Figure C.7 Q-Criterion Iso-Surfaces Q = 100 Coloured by Dimensionless Velocity X-Direction
Comparison
Finally in Figure C.7 we can see the iso-surfaces for the second invariant of the velocity
gradient at a value of 100. Positive values of this Q-Criterion represent the parts of the
flow where rotation dominates strain and hence is a very common tool to identify vortex
regions as can be seen in (C.1) where Ω is the Rate-of-Rotation tensor and S is the Rate-
of-Strain tensor.
22
2
1SQ
(C.1)
We can see that the same structures appear in both cases but with a smaller size in the
case of the SST model. We can clearly see the recirculation created by the presence of
the flap on the pressure side of the main element near its TE and how once the flow is
detached in the flap bigger structures start to develop.
All in all, although minor quantitative differences we see that both structures and trends
predicted by both models are qualitatively similar making the point of this validation
study.
Appendix C: Validation against S-A Turbulence Model 78
Appendix D: Matlab, VBA and JAVA Codes 79
Appendix D: Matlab, VBA and JAVA
Codes
All over this project, but mainly during the baseline elements’ relative position
investigation (see section 4.1), some codes have been used. The most important ones
can be founded in this appendix.
Matlab Code for Creating Base Wireframe Geometry
The code here creates the wireframe geometry needed to build the 3D models. Each
curve is saved in a text file in space separated values. It depends on the function
rotateFlap.m which is not included here.
function
createNew(ID,FlapOriginX,FlapOriginY,FlapAoA,Amplitude,wNumber)
% close all;
main1 = importdata('HomeDirectory\Data\Main1.txt',' ');
flap1 = importdata('HomeDirectory\Data\Flap.txt',' ');
flap2 = zeros(length(flap1),3);
main2 = zeros(length(main1),3);
wleZ = 0:0.001:0.5;
wle = zeros (length(wleZ),3);
modelID = ID;
A = Amplitude;
n = wNumber;
alfa = - FlapAoA;
flap1 = rotateFlap(flap1,alfa);
for i=1:length(wleZ)
wle(i,:) = [A*sin(2*pi*wleZ(i)*n/0.5) 0 wleZ(i)];
end
wle = rotateFlap(wle,alfa);
flapOriginX = FlapOriginX;
flapOriginY = FlapOriginY;
flapOrigin = [flapOriginX flapOriginY 0];
data = [modelID flapOriginX flapOriginY -alfa A n];
for i=1:length(flap1)
flap1(i,:) = flap1(i,:) + flapOrigin;
flap2(i,:) = flap1(i,:) + [0 0 0.5];
Appendix D: Matlab, VBA and JAVA Codes 80
end
for i=1:length(main1)
main2(i,:) = main1(i,:) + [0 0 0.5];
end
for i=1:length(wleZ)
wle(i,:) = wle(i,:) + flapOrigin;
end
flapTE1 = [flap1(1,1) flap1(1,2), flap1(1,3);
flap1(1,1) flap1(1,2),0.5];
flapTE2 = [flap1(length(flap1),1) flap1(length(flap1),2),
flap1(length(flap1),3);
flap1(length(flap1),1) flap1(length(flap1),2),0.5];
flapTE3 = [flap1(1,1) flap1(1,2), flap1(1,3);
flap1(length(flap1),1) flap1(length(flap1),2),
flap1(length(flap1),3)];
flapTE4 = [flap1(1,1) flap1(1,2), 0.5;
flap1(length(flap1),1) flap1(length(flap1),2), 0.5];
mainTE1 = [main1(1,1) main1(1,2), main1(1,3);
main1(1,1) main1(1,2),0.5];
mainTE2 = [main1(length(main1),1) main1(length(main1),2),
main1(length(main1),3);
main1(length(main1),1) main1(length(main1),2),0.5];
mainTE3 = [main1(1,1) main1(1,2), main1(1,3);
main1(length(main1),1) main1(length(main1),2),
main1(length(main1),3)];
mainTE4 = [main1(1,1) main1(1,2), 0.5;
main1(length(main1),1) main1(length(main1),2), 0.5];
save HomeDirectory\Data\Main1.txt -ascii main1
save HomeDirectory\Data\Main2.txt -ascii main2
save HomeDirectory\Data\Flap1.txt -ascii flap1
save HomeDirectory\Data\Flap2.txt -ascii flap2
save HomeDirectory\Data\WLE.txt -ascii wle
save HomeDirectory\Data\FlapTE1.txt -ascii flapTE1
save HomeDirectory\Data\FlapTE2.txt -ascii flapTE2
save HomeDirectory\Data\FlapTE3.txt -ascii flapTE3
save HomeDirectory\Data\FlapTE4.txt -ascii flapTE4
save HomeDirectory\Data\MainTE1.txt -ascii mainTE1
save HomeDirectory\Data\MainTE2.txt -ascii mainTE2
save HomeDirectory\Data\MainTE3.txt -ascii mainTE3
save HomeDirectory\Data\MainTE4.txt -ascii mainTE4
dlmwrite('HomeDirectory\Data\Data.txt',data,...
'delimiter',' ','precision', 8)
Appendix D: Matlab, VBA and JAVA Codes 81
disp(horzcat('Model ',num2str(modelID),' Data.txt Saved'))
Matlab Code for the Optimisation Process
This code here uses Kriging43
and other functions developed by Dr Andreas Sobester44
for creating a response surface and suggest new design points. The code here depends
on many matlab functions which were all obtained from SESG6019: Design Search and
Optimisation module at University of Southampton. A text file with the results is used
to input the data needed to perform the optimisation.
addpath('HomeDirectoryOptimisation');
for index=1:1
clear all; close all;
maxval = inf;
acuracy = 51;
best=0;
scrsz = get(0,'ScreenSize');
global ModelInfo
f=zeros(acuracy,acuracy,acuracy);
ei=zeros(acuracy,acuracy,acuracy);
pos = zeros(3,1);
normpos = zeros(3,1);
results = importdata('HomeDirectoryData\Results.txt',' ');
var = results(:,2:4);
[n,k]=size(var);
Xplot = zeros(acuracy,k);
newPoints=zeros(1,k);
minlim = [0.975 -0.05 0];
maxlim = [1.05 -0.025 40];
for i=1:k
ModelInfo.X(:,i)=(var(:,i)-minlim(i))./(maxlim(i)-minlim(i));
end
val = results(:,5);
for l=1:n
label(l) =cellstr((num2str(results(l,1))));
end
ModelInfo.y = -val(:);
varname = 'Cl';
while (maxval==inf)
% Set upper and lower bounds for search of log theta
UpperTheta=ones(1,k).*2;
LowerTheta=ones(1,k).*-3;
% Run GA search of likelihood
[ModelInfo.Theta,MinNegLnLikelihood]=ga_DSO(@likelihood,k,[],[],[],[],
LowerTheta,UpperTheta);
43
Krige, “A Statistical Approach to Some Mine Valuation and Allied Problems on the Witwatersrand.” 44
Sobester.
Appendix D: Matlab, VBA and JAVA Codes 82
% Put Cholesky factorisation of Psi into ModelInfo
[NegLnLike,ModelInfo.Psi,ModelInfo.U]=likelihood(ModelInfo.Theta);
% Plot surrogate models
X=0:1/(acuracy-1):1;
for i=1:acuracy
for j=1:acuracy
for l=1:acuracy
ModelInfo.Option='Pred';
f(i,j,l)=predictor([X(j) X(l) X(i)]);
ModelInfo.Option='NegLogExpImp';
ei(i,j,l)=predictor([X(j) X(l) X(i)]);
end
end
end
%A=10.^-ei;
A=-f;
for i=1:n
if(val(i)>best)
best=val(i);
bestID=results(i,1);
end
end
[maxval maxloc] = max(A(:));
[pos(2) pos(1) pos(3)] = ind2sub(size(A), maxloc);
maxvalCl=-f(pos(2),pos(1),pos(3));
end
for i=1:k
normpos(i) = (pos(i)-1)/(acuracy-1);
newPoints(i) =normpos(i) * ((maxlim(i))-minlim(i))+ minlim(i);
for g=1:acuracy
Xplot(g,i) = X(g) * ((maxlim(i))-minlim(i))+ minlim(i);
end
end
newPoints
modelID = results(end,1)+1;
disp(horzcat('Cl for Model',num2str(modelID),' expected:
',num2str(maxvalCl)))
disp(horzcat('Best model so far is: model',num2str(bestID),' with Cl =
',...
num2str(best)))
end
Appendix D: Matlab, VBA and JAVA Codes 83
VBA Code for Geometry Creation in SOLIDWORKS
With this code, the wireframe geometry created before in Matlab and stored in text files
was imported into SOLIDWORKS and then using geometric operations the final 3D
models were created.
Sub main()
Set swApp = _
Application.SldWorks
Set Part = swApp.NewDocument("C:\ProgramData\SolidWorks\SolidWorks
2011\templates\Pieza.prtdot", 0, 0, 0)
swApp.ActivateDoc2 "Part2", False, longstatus
Set Part = swApp.ActiveDoc
Dim myModelView As Object
Set myModelView = Part.ActiveView
myModelView.FrameState = swWindowState_e.swWindowMaximized
Set Part = swApp.ActiveDoc
Part.InsertCurveFileBegin
Open "HomeDirectory\Main1.txt" For Input As #1
Do While Not EOF(1)
Input #1, X, Y, Z
boolstatus = Part.InsertCurveFilePoint(X, Y, Z)
Loop
Close #1
boolstatus = Part.InsertCurveFileEnd()
Part.InsertCurveFileBegin
Open "HomeDirectory\Main2.txt" For Input As #6
Do While Not EOF(6)
Input #6, X, Y, Z
boolstatus = Part.InsertCurveFilePoint(X, Y, Z)
Loop
Close #6
boolstatus = Part.InsertCurveFileEnd()
[…] ‘some code has been omitted here referring to adding more
wireframe curves
Part.InsertCurveFileBegin
Open "HomeDirectory\MainTE4.txt" For Input As #6
Do While Not EOF(6)
Input #6, X, Y, Z
boolstatus = Part.InsertCurveFilePoint(X, Y, Z)
Loop
Close #6
boolstatus = Part.InsertCurveFileEnd()
Set Part = swApp.ActiveDoc
boolstatus = Part.Extension.SelectByID2("Curve1", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve12", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("Curve1", "REFERENCECURVES",
0, 0, 0, False, 1, Nothing, 0)
Appendix D: Matlab, VBA and JAVA Codes 84
boolstatus = Part.Extension.SelectByID2("Curve12", "REFERENCECURVES",
0, 0, 0, True, 1, Nothing, 0)
Part.InsertCompositeCurve
boolstatus = Part.Extension.SelectByID2("Curve2", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve13", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("Curve2", "REFERENCECURVES",
0, 0, 0, False, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve13", "REFERENCECURVES",
0, 0, 0, True, 1, Nothing, 0)
Part.InsertCompositeCurve
boolstatus = Part.Extension.SelectByID2("Curve3", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve8", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("Curve3", "REFERENCECURVES",
0, 0, 0, False, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve8", "REFERENCECURVES",
0, 0, 0, True, 1, Nothing, 0)
Part.InsertCompositeCurve
boolstatus = Part.Extension.SelectByID2("Curve4", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve9", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("Curve4", "REFERENCECURVES",
0, 0, 0, False, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve9", "REFERENCECURVES",
0, 0, 0, True, 1, Nothing, 0)
Part.InsertCompositeCurve
boolstatus = Part.Extension.SelectByID2("CompCurve3",
"REFERENCECURVES", 1.18843647886268, 6.04510587053255E-02,
0.09993755764566, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve5", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve6", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve7", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("CompCurve4",
"REFERENCECURVES", 1.214, -0.0232152, 0.5, False, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("CompCurve3",
"REFERENCECURVES", 1.214, -0.0232152, 0, True, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve5", "REFERENCECURVES",
0, 0, 0, True, 4098, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve6", "REFERENCECURVES",
0, 0, 0, True, 8194, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve7", "REFERENCECURVES",
0, 0, 0, True, 12290, Nothing, 0)
Part.FeatureManager.InsertProtrusionBlend False, True, False, 1, 0, 0,
1, 1, True, True, False, 0, 0, 0, True, True, True
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("CompCurve1",
"REFERENCECURVES", 0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("CompCurve2",
"REFERENCECURVES", 0, 0, 0, True, 0, Nothing, 0)
Appendix D: Matlab, VBA and JAVA Codes 85
boolstatus = Part.Extension.SelectByID2("Curve10", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve11", "REFERENCECURVES",
0, 0, 0, True, 0, Nothing, 0)
Part.ClearSelection2 True
boolstatus = Part.Extension.SelectByID2("CompCurve1",
"REFERENCECURVES", 1, -0.00074, 0, False, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("CompCurve2",
"REFERENCECURVES", 1, -0.00074, 0.5, True, 1, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve10", "REFERENCECURVES",
0, 0, 0, True, 4098, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Curve11", "REFERENCECURVES",
0, 0, 0, True, 8194, Nothing, 0)
Part.FeatureManager.InsertProtrusionBlend False, True, False, 1, 0, 0,
1, 1, True, True, False, 0, 0, 0, False, True, True
Part.ViewZoomtofit2
Set Part = swApp.ActiveDoc
boolstatus = Part.Extension.SelectByID2("Loft1", "SOLIDBODY", 0, 0, 0,
False, 0, Nothing, 0)
boolstatus = Part.Extension.SelectByID2("Loft2", "SOLIDBODY", 0, 0, 0,
True, 0, Nothing, 0)
longstatus = Part.SaveAs3("HomeDirectory\model.x_t", 0, 0)
swApp.ExitApp
End Sub
JAVA Code for Simulation Automation in STAR-CCM+
This JAVA code imports the geometry, names it, sets up the case with both mesh and
simulation parameters and finally saves results to a text file. However the code is too
long to be included here and just some parts of it are included here. Hence the empty
methods such as for example createBlock() have been taken out of the code because of
printing expenses
// STAR-CCM+ macro: macro.java
package macro;
import java.util.*;
import java.lang.*;
import java.io.*;
import star.common.*;
import star.base.neo.*;
import star.meshing.*;
import star.resurfacer.*;
import star.dualmesher.*;
import star.prismmesher.*;
import star.vis.*;
import star.flow.*;
import star.segregatedflow.*;
import star.metrics.*;
import star.saturb.*;
Appendix D: Matlab, VBA and JAVA Codes 86
import star.turbulence.*;
import star.material.*;
import star.base.report.*;
public class macro extends StarMacro {
public void execute() {
Simulation mySim = getActiveSimulation();
String[] modelData = readData();
String ID = modelData[0];
mySim.println("Model " + ID + " succesfully loaded" );
String simulationName = "model" + ID;
importGeo(simulationName);
prepareGeo();
createBlock();
performSubstract();
createRegion();
createVolCont();
createMeshCont();
applyVolCont();
createPhyContSA();
applyLocalSet();
createDervParts();
createReports();
createMonitors();
createPlots();
editFieldFun();
parallelMeshing(true);
mySim.saveState(resolvePath("HomeDirectory" + simulationName +
".sim"));
meshModel();
convert2D();
runSimulation(0);
saveResults(modelData);
}
private void importGeo(String modelName) {
Simulation simulation_0 =
getActiveSimulation();
PartImportManager partImportManager_0 =
simulation_0.get(PartImportManager.class);
partImportManager_0.importCadPart(resolvePath("HomeDirectoryCAD\\"
+ modelName + ".stp"), "AllEdges", 30.0, 4, true, true);
simulation_0.println("Geometry imported succesfully");
}
Appendix D: Matlab, VBA and JAVA Codes 87
private void prepareGeo() {
Simulation simulation_0 =
getActiveSimulation();
MeshPartFactory meshPartFactory_0 =
simulation_0.get(MeshPartFactory.class);
CompositePart compositePart_0 =
((CompositePart)
simulation_0.get(SimulationPartManager.class).getPart("Part1"));
CadPart cadPart_0 =
((CadPart)
compositePart_0.getChildParts().getPart("PartBody.1"));
CadPart cadPart_1 =
((CadPart)
compositePart_0.getChildParts().getPart("PartBody.2"));
meshPartFactory_0.combineMeshParts(cadPart_0, new
NeoObjectVector(new Object[] {cadPart_1}));
compositePart_0.explode(new NeoObjectVector(new Object[]
{compositePart_0}));
PartSurface partSurface_0 =
cadPart_0.getPartSurfaceManager().getPartSurface("PartBody");
cadPart_0.splitPartSurfaceByPatch(partSurface_0, new IntVector(new
int[] {34}), "flapElement");
cadPart_0.splitPartSurfaceByPatch(partSurface_0, new IntVector(new
int[] {35}), "flapElementTE");
PartSurface partSurface_1 =
cadPart_0.getPartSurfaceManager().getPartSurface("PartBody 2");
cadPart_0.splitPartSurfaceByPatch(partSurface_1, new IntVector(new
int[] {4}), "mainElement");
cadPart_0.splitPartSurfaceByPatch(partSurface_1, new IntVector(new
int[] {5}), "mainElementTE");
simulation_0.println("Geometry prepared succesfully");
}
private void createBlock() {}
private void performSubstract() {}
private void createRegion() {}
private void createVolCont() {}
private void createMeshCont() {}
private void applyVolCont() {}
private void createPhyContSA() {}
Appendix D: Matlab, VBA and JAVA Codes 88
private void applyLocalSet() {}
private void createDervParts() {}
private void createReports() {}
private void createMonitors() {}
private void createPlots() {}
private void editFieldFun() {}
private void meshModel() {
Simulation simulation_0 =
getActiveSimulation();
MeshContinuum meshContinuum_0 =
((MeshContinuum)
simulation_0.getContinuumManager().getContinuum("Mesh 1"));
MeshPipelineController meshPipelineController_0 =
simulation_0.get(MeshPipelineController.class);
meshPipelineController_0.generateVolumeMesh();
String simulationName = simulation_0.getPresentationName();
simulation_0.println("Model meshed succesfully, Saving 3D
mesh...");
simulation_0.saveState(resolvePath("HomeDirectory" +
simulationName + "_Mesh.sim"));
}
private void runSimulation( int numberIterations) {
Simulation simulation_0 =
getActiveSimulation();
StepStoppingCriterion stepStoppingCriterion_0 =
((StepStoppingCriterion)
simulation_0.getSolverStoppingCriterionManager().getSolverStoppingCrit
erion("Maximum Steps"));
int currentIteration =
stepStoppingCriterion_0.getMaximumNumberSteps();
stepStoppingCriterion_0.setMaximumNumberSteps(numberIterations +
currentIteration);
simulation_0.getSimulationIterator().run();
}
Appendix D: Matlab, VBA and JAVA Codes 89
private void convert2D() {}
private void parallelMeshing(boolean state) {}
private String[] readData(){
Simulation sim = getActiveSimulation();
File f = new File(("HomeDirectoryData\\Data.txt"));
Scanner s;
String[] dataArray = new String[6];
try {
s = new Scanner(f);
while (s.hasNextLine()) {
String linea = s.nextLine();
Scanner sl = new Scanner(linea);
sl.useDelimiter("\\s");
dataArray[0] = (sl.next());
dataArray[1] = (sl.next());
dataArray[2] = (sl.next());
dataArray[3] = (sl.next());
dataArray[4] = (sl.next());
dataArray[5] = (sl.next());
}
s.close();
} catch (Exception e) {
sim.println("file not foud");
}
return dataArray;
}
private void saveResults(String[] modelData){
Simulation mySim = getActiveSimulation();
ForceCoefficientReport cdReport =
((ForceCoefficientReport)
mySim.getReportManager().getReport("Cd"));
ForceCoefficientReport clReport =
((ForceCoefficientReport)
mySim.getReportManager().getReport("Cl"));
double cdValue = cdReport.getReportMonitorValue();
double clValue = clReport.getReportMonitorValue();
double efValue = clValue/cdValue;
String cd = String.valueOf(cdValue);
String cl = String.valueOf(clValue);
String ef = String.valueOf(efValue);
try{
FileWriter fstream = new
FileWriter("HomeDirectoryData\\Results.txt",true);
BufferedWriter fbw = new BufferedWriter(fstream);
Appendix D: Matlab, VBA and JAVA Codes 90
fbw.write(modelData[0] + " " + modelData[1] + " " +
modelData[2] + " " + modelData[3] + " " + cl + " " + cd + " " + ef);
fbw.newLine();
fbw.close();
}catch (Exception e){
mySim.println("Error happened during results saving
operation");
}
mySim.println("Model " + modelData[0] + ": results saved");
}
}
References 91
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