![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-9-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-10-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-11-320.jpg)
![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 [L2
T-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:](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-35-320.jpg)
![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.
Uc
Re (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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-45-320.jpg)
![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.”](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-60-320.jpg)
![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).](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-66-320.jpg)
![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
TotalCL
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
CLflap
Simulation Time [s]
model0
model1.4
model1.8
model2.4
model2.8](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-72-320.jpg)
![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
CLmain
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
TotalCLrms
TotalCL
Average CL Max CL
Min CL CL rms](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-73-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-78-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-79-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-80-320.jpg)
![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('HomeDirectoryDataMain1.txt',' ');
flap1 = importdata('HomeDirectoryDataFlap.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];](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-95-320.jpg)
![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 HomeDirectoryDataMain1.txt -ascii main1
save HomeDirectoryDataMain2.txt -ascii main2
save HomeDirectoryDataFlap1.txt -ascii flap1
save HomeDirectoryDataFlap2.txt -ascii flap2
save HomeDirectoryDataWLE.txt -ascii wle
save HomeDirectoryDataFlapTE1.txt -ascii flapTE1
save HomeDirectoryDataFlapTE2.txt -ascii flapTE2
save HomeDirectoryDataFlapTE3.txt -ascii flapTE3
save HomeDirectoryDataFlapTE4.txt -ascii flapTE4
save HomeDirectoryDataMainTE1.txt -ascii mainTE1
save HomeDirectoryDataMainTE2.txt -ascii mainTE2
save HomeDirectoryDataMainTE3.txt -ascii mainTE3
save HomeDirectoryDataMainTE4.txt -ascii mainTE4
dlmwrite('HomeDirectoryDataData.txt',data,...
'delimiter',' ','precision', 8)](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-96-320.jpg)
![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('HomeDirectoryDataResults.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.](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-97-320.jpg)
![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](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-98-320.jpg)
![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:ProgramDataSolidWorksSolidWorks
2011templatesPieza.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 "HomeDirectoryMain1.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 "HomeDirectoryMain2.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 "HomeDirectoryMainTE4.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)](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-99-320.jpg)
![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");
}](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-102-320.jpg)
![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() {}](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-103-320.jpg)
![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(("HomeDirectoryDataData.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("HomeDirectoryDataResults.txt",true);
BufferedWriter fbw = new BufferedWriter(fstream);](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-105-320.jpg)
![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");
}
}](https://image.slidesharecdn.com/66b58bfa-93a1-4172-a09a-e6c1d3484118-160222101836/85/Flaps-with-Wavy-Leading-Edges-for-Robust-Performance-agains-Upstream-Trailing-Vortices-106-320.jpg)
Flaps with Wavy Leading Edges for Robust Performance agains Upstream Trailing Vortices
- 1. 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
- 2. i
- 3. 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.
- 4. iii
- 5. iv 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.
- 6. v
- 7. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 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
- 12. xi
- 13. xii 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
- 14. 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
- 15. 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
- 16. 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
- 17. 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.
- 18. 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/.
- 19. 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.
- 20. 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.
- 21. 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.
- 22. 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 authors15 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.
- 23. 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 Sousa 19 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).
- 24. 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.
- 25. 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
- 26. Chapter 2: Theoretical Background 10 Or Rate of Increase of Mass – Net Mass Flow out of the Volume The mathematical expression for this is: SV dSnudV 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: fu t f Dt Df (2.3) The equation (2.3) can be rewritten as follows: u Dt 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:
- 27. Chapter 2: Theoretical Background 11 V S S iii 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: iiuu t 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 i ij x u x u S 2 1 (2.11) And then: i j j i ijij x u x u S 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 i ij ij jii xx u xx u x p x u x u p xx uu t u 0 22 (2.13)
- 28. 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 u u t u x u t u x u u t u x uu t u i j i j i Continuityby j j i j i j i j jii 0 (2.15) Finally the Momentum Equation can be written as: TermViscous jj i Termessure i TermLinearNon j i j ii xx u x p x u u 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/.
- 29. 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 p p l U tt l x x U u u i i i i (2.18) The NSE can also be expressed as: jj i ij i j i xx u x p x u u 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.).
- 30. 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.
- 31. 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 i T i dttu T u 0 )( 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.
- 32. Chapter 2: Theoretical Background 16 N n i N i txu N txu 1 ),( 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 i j i xx u x p x uu x u u t u 2 1 (2.26) Or rearranging ji j i jij i j i uu x u xx p x u u 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.
- 33. 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.
- 34. 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 i Tijji S x u x u kuu 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.
- 35. 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 [L2 T-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:
- 36. Chapter 2: Theoretical Background 20 0)( 4 1 UlUmvnP mfpth (2.37) If we then approximate the velocity gradient by: ... 0)( ... )0( TOH l UlU TOH 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 dU lvPP 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.
- 37. 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 section̅̅̅̅ . 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. MaKn~S l 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).
- 38. 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 t b wwtb j j xx c xxd f c fcSfc x u t ~~~ ~1~ ~~)1( ~~ 2 2 22 1 121 (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.
- 39. Chapter 2: Theoretical Background 23 Where 222 3 1 3 3 1 ~~ ~ v v v f d S c f (2.45) And √ is the vorticity magnitude, d is the distance to the nearest wall and i j j i ijttt w w w w v v x u x u Wccf dS rrrcrg cg c gf f f 2 1 exp 10,~ ~ min 1 1 1 2 432 22 6 2 6 1 3 3 6 3 3 1 2 (2.46) And the model constants are: 2 2 1 14 3132 21 1 5.0 2.11.723.0 41.0622.0 3 2 1335.0 bb wt tvww bb cc cc 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.
- 40. 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 x kP Dt kD 1 * (2.48) j t jt xx P Dt D 1 2 1 1 (2.49) Where the production term P and the stress tensor τij are: j i ij x u P (2.50) ijij k k ijtij k x u S 3 2 3 2 2 (2.51) And then the transformed k-ε is: j tk j x k x kP Dt kD 2 * (2.52) jjj t jt xx k xx P Dt D 1 2 22 2 2 2 (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 x kP Dt kD * (2.54) jjj t jt xx k F xx P Dt D 1 12 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)
- 41. Chapter 2: Theoretical Background 25 Where 4 11 argtanhF (2.57) 2 2 21 4 ; 500 ; 09.0 maxminarg 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 ka t (2.59) Where Ω is the absolute value of the vorticity and F2 is given by: 2 22 argtanhF (2.60) 22 500 ; 09.0 2maxarg 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)(SST1Set a (2.62) *2 2 * 22 * 22k2 2 41.009.0 0828.0856.00.1 )-k(Standart2Set (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
- 42. 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).
- 43. 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. S SdFd t 0 (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 SdFd t SdFd t SdFd t SdFd t 321 321 (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 SF t (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.
- 44. Chapter 2: Theoretical Background 28
- 45. 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. Uc Re (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
- 46. 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
- 47. 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.
- 48. 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
- 49. 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
- 50. 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:
- 51. Chapter 3: Problem Description 35 5 x t UC (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.
- 52. Chapter 3: Problem Description 36
- 53. 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.).
- 54. 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
- 55. 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
- 56. 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.
- 57. 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
- 58. 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
- 59. 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 w y 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.
- 60. 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.”
- 61. 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
- 62. 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
- 63. 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
- 64. 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: 2 21 U pp Cp (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
- 65. 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.
- 66. 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).
- 67. 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
- 68. 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
- 69. 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.
- 70. 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 from 42 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.
- 71. Chapter 4: Results 55 sm kk myx L r kkr yyxx rLxxkUvtyxv rLyykUutyxu 6U m36L 35.0,400, 25.0,2, 2 1 exp r with )()0,,( )(0,, 21 00 2 2 2 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
- 72. 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 TotalCL 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 CLflap Simulation Time [s] model0 model1.4 model1.8 model2.4 model2.8
- 73. 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 CLmain 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 TotalCLrms TotalCL Average CL Max CL Min CL CL rms
- 74. 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 CLflaprms CLflap Average CL Max CL Min 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 CLmainrms CLmain Average CL Max CL Min CL CL rms
- 75. 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
- 76. 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
- 77. 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.
- 78. 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
- 79. 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
- 80. 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
- 81. 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
- 82. 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
- 83. 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.
- 84. Chapter 5: Final Remarks 68
- 85. 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
- 86. 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.
- 87. 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
- 88. Appendix B: Domain Size Dependency Study 72
- 89. 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
- 90. 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
- 91. 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.
- 92. 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
- 93. 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 1 SQ (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.
- 94. Appendix C: Validation against S-A Turbulence Model 78
- 95. 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('HomeDirectoryDataMain1.txt',' '); flap1 = importdata('HomeDirectoryDataFlap.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];
- 96. 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 HomeDirectoryDataMain1.txt -ascii main1 save HomeDirectoryDataMain2.txt -ascii main2 save HomeDirectoryDataFlap1.txt -ascii flap1 save HomeDirectoryDataFlap2.txt -ascii flap2 save HomeDirectoryDataWLE.txt -ascii wle save HomeDirectoryDataFlapTE1.txt -ascii flapTE1 save HomeDirectoryDataFlapTE2.txt -ascii flapTE2 save HomeDirectoryDataFlapTE3.txt -ascii flapTE3 save HomeDirectoryDataFlapTE4.txt -ascii flapTE4 save HomeDirectoryDataMainTE1.txt -ascii mainTE1 save HomeDirectoryDataMainTE2.txt -ascii mainTE2 save HomeDirectoryDataMainTE3.txt -ascii mainTE3 save HomeDirectoryDataMainTE4.txt -ascii mainTE4 dlmwrite('HomeDirectoryDataData.txt',data,... 'delimiter',' ','precision', 8)
- 97. 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('HomeDirectoryDataResults.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.
- 98. 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
- 99. 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:ProgramDataSolidWorksSolidWorks 2011templatesPieza.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 "HomeDirectoryMain1.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 "HomeDirectoryMain2.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 "HomeDirectoryMainTE4.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)
- 100. 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)
- 101. 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("HomeDirectorymodel.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.*;
- 102. 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"); }
- 103. 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() {}
- 104. 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(); }
- 105. 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(("HomeDirectoryDataData.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("HomeDirectoryDataResults.txt",true); BufferedWriter fbw = new BufferedWriter(fstream);
- 106. 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"); } }
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