1. Amit Ramji – A4 – University of Hertfordshire
1
Experimental, Numerical and Theoretical Analysis of Supersonic Flow
Over A Solid Diamond Wedge.
Amit Ramji
10241445 - University of Hertfordshire - Aerospace Engineering - Year 4 Aerodynamics – 6ACM0011 - 23rd March 2014.
Lab Undertaken: 21st Feb 2014 12:20PM
Abstract
In this paper, the study of boundary layer shockwave effects and pressure coefficients 𝐶! are
studied for supersonic flow past a diamond shaped solid wedge. Experimental steady supersonic Euler
flow is studied moving onto Ackeret’s linear theory, Navier stokes and Numerical methods (CFD-
Computational Fluid Dynamics). It is shown in this text that the shock exists in forward region of the
wedge for a constant shape (semi-wedge angle, 𝜀!
. Measured in radians) with variation of incidence
angle (angle of attack, 𝛼!
. Measured in Radians). This study focuses on 2-Dimensional flows where
one is primarily concerned with flows across the blockage in the free steam direction, thereby
ignoring effects in the span wise direction and any combination directions forming cones.
Introduction
In this paper, the study of boundary layer
shockwave effects and pressure coefficients
𝐶! are studied for supersonic flow past a
diamond shaped solid wedge. Experimental
supersonic investigations capture the pressure
head at face ‘a’ (see Figure 3 through Figure 7),
which in turn is used to calculate the pressure
coefficient. The Pressure coefficient varies with
changes in angle of incidence in a steady flow
snapshot case. This is also compared to
analytical methods to compare the pressure
coefficients and lift coefficients using inclined
plate lift theory. Furthermore, CFD analysis is
also employed to calculate the same pressure
coefficient and replicate the observed
experimental shockwaves.
The limitations of each method and any
discrepancies are discussed in this text thereby
analysing the procedures and accuracy of the
methods.
Preliminary
and
Main
Theorem
The study of supersonic aerodynamics has been
of great importance in the history or aircraft and
spacecraft; one can appreciate the complexity of
analysing such vehicles by historical
achievements. A Simplified study of basic
shapes must be carried out for analytical
purposes of which the physics can be
demonstrated. Later these empirical and
theoretical models can be used in more complex
geometry alongside 3-Dimensional flows using
numerical methods (CFD) by element-wise
discretization representing the laws of
compressible fluid flow.
In short, the compressibility effect of high-
speed flow has been disregarded in sub-sonic
flight speeds, which means variance of density
is negligible and ignored. However when
considering super-sonic flows and hypersonic
flows, this compressibility effect cannot be
ignored as steady incompressible flow theory is
no longer completely valid.
For a researcher, a preparatory point for
analysis is the employment of conservation
laws from fundamental physics. Conservation
of mass, momentum (Euler) and energy must be
preserved hence the use of Continuity,
Newton’s 2nd Law and Navier Stokes equations
of motion.

2. Amit Ramji – A4 – University of Hertfordshire
2
Figure 1 – Continuity of a Finite Volume
In order to analyse the flow characteristics and
conventions, Figure 1 shows how the problem is
discretized into element-wise Cartesian
coordinate form for further analysis (Polar
forms also exist for rotating flows). The control
volume is of dimensions δx, δy, and δz, where
initial velocity in the x direction is given by
u m/s , v m/s in the y direction and
w m/s in the z direction. The density is
denoted by ρ kg/m!
in this literature.
For 3-Dimensional flow with compressibility
effects, the following mass continuity equation
is used to describe the fluid flow, this can also
be written in polar coordinates as described in
section 2.4.3 of [1] by Houghton et al.
∂ρ
∂t
+
∂
∂x
ρu( )+
∂
∂y
ρv( )+
∂
∂z
ρw( )+ ρ
∂u
∂x
+
∂v
∂y
+
∂w
∂z
!
"
#
$
%
& = 0
∂u
∂x
+
∂v
∂y
+
∂w
∂z
!
"
#
$
%
& = 0
For 3-Dimensions, conservation of momentum
and Navier Stokes equations are used to
describe the fluid flow through a control
volume as shown in Figure 1.
∂u
∂x
+
∂v
∂y
+
∂w
∂z
!
"
#
$
%
& = 0
ρ
∂u
∂t
+u
∂u
∂x
+ v
∂u
∂y
+ w
∂u
∂z
!
"
#
$
%
& = ρgx −
∂p
∂x
+µ
∂2
u
∂x2
+
∂2
u
∂y2
+
∂2
u
∂z2
!
"
#
$
%
&
ρ
∂v
∂t
+u
∂v
∂x
+ v
∂v
∂y
+ w
∂v
∂z
!
"
#
$
%
& = ρgy −
∂p
∂y
+µ
∂2
v
∂x2
+
∂2
v
∂y2
+
∂2
v
∂z2
!
"
#
$
%
&
ρ
∂w
∂t
+u
∂w
∂x
+ v
∂w
∂y
+ w
∂w
∂z
!
"
#
$
%
& = ρgz −
∂p
∂x
+µ
∂2
w
∂x2
+
∂2
w
∂y2
+
∂2
w
∂z2
!
"
#
$
%
&
Derivation of Ackeret’s theory originates from
fundamental Prandtl Meyer equilibrium laws
where plate forces above and below are
assumed to be in equilibrium in steady steam
flow. By equating the forces on the upper and
lower surface and assuming the incidence
angles are relatively small, one can ascertain the
following derivation. The linearized theory
provides a simpler method of estimating lift and
drag for supersonic aerofoils. It is found that
due to the first order linear function, the lift
coefficient is more accurate than the pressure
parameters due to the cancellation of higher
order terms. This can be further read upon in
section 6.8.3 of literature [1] by Houghton et al.
Figure 2 - Plate theory pressure derivation
CL = CP(Up) +CP(Down) =
2α
M2
−1
−
−2α
M2
−1
∴CL =
4α
M2
−1
Assuming symmetrical aerofoils, the above
calculation for Coefficient of Lift, C! is valid as
the magnitudes of pressure on each surface is
the same. M! = M = Free Stream Mach No.
The Wave Drag Coefficient, C!" is therefore:
C!" =
4
M! − 1
α!
+
t
c
!
Where t = thickness & c = chord legnth
The Moment Coefficient, C! is:
C! =
−2α
M! − 1
=
−2α
1.8! − 1
= 1.336 α

3. Amit Ramji – A4 – University of Hertfordshire
3
The following parameters are used when
calculating the C! by theoretical means.
Mach No 1.8
Internal Total Angle [2ε] (Deg) 10
Chord (mm) 25.4
Semi-Wedge Angle, ε (Deg) 5
Table 1 - Problem Parameters
Limitations
&
Assumptions
for
Ackerets
Theory
Ackeret’s theory assumes linear flows, where
parameters are disregarded by the order of
magnitude i.e. Small terms are neglected
therefore supersonic linearization.
2-dimensional flows, Steady flow, Invisid flow,
alongside having symmetrical sections are
further limitations to Ackerets Theory.
Linearized theory gives a simple way to
estimate the lift and drag for supersonic
aerofoils, when symmetrical sections are
considered. The lift coefficient is more accurate
than the pressures due to cancellation of second
order terms during linearization as highlighted
previously.
The following table calculates the coefficients
of pressure for each surface and coefficient for
lift for the entire diamond wedge shape.
α=𝟓∘
Surface
Stream
Deflection
(Deg)
Stream
Deflection
(Rad)
𝐂 𝐏(𝐔𝐏)
&
𝐂 𝐏(𝐃𝐨𝐰𝐧)
a 0 0.000 0.000
b -10 -0.175 -0.233
c 10 0.175 0.233
d 0 0.000 0.000
𝐶! !"#$% !"#$%&'(
= 𝐶!" + 𝐶!"
− 𝐶!" − 𝐶!"
0.466
𝐶! (!"#$%) =
1
2
𝐶! !"#$% !"#$%&'( 0.233
𝐶! (!"#$#%&) =
4α
M! − 1
0.233
α=𝟑∘
Surf Def (Deg) Def (Rad) 𝐂 𝐏
a 2 0.035 0.047
b -8 -0.140 -0.187
c 8 0.140 0.187
d -2 -0.035 -0.047
𝐶! (!"#$% !"#$%&'() 0.280
𝐶! (!"#$%) 0.140
𝐶! (!"#$#%&) 0.140
α=𝟏∘
Surf Def (Deg) Def (Rad) 𝐂 𝐏
a 4 0.070 0.093
b -6 -0.105 -0.140
c 6 0.105 0.140
d -4 -0.070 -0.093
𝐶! (!"#$% !"#$%&'() 0.093
𝐶! (!"#$%) 0.047
𝐶! (!"#$#%&) 0.047
α=−𝟏∘
Surf Def (Deg) Def (Rad) 𝐂 𝐏
a 6 0.105 0.140
b -4 -0.070 -0.093
c 4 0.070 0.093
d -6 -0.105 -0.140
𝐶! (!"#$% !"#$%&'() -0.093
𝐶! (!"#$%) -0.047
𝐶! (!"#$#%&) -0.047
α=−𝟑∘
Surf Def (Deg) Def (Rad) 𝐂 𝐏
a 8 0.140 0.187
b -2 -0.035 -0.047
c 2 0.035 0.047
d -8 -0.140 -0.187
𝐶! (!"#$% !"#$%&'() -0.280
𝐶! (!"#$%) -0.140
𝐶! (!"#$#%&) -0.140
α=−𝟓∘
Surf Def (Deg) Def (Rad) 𝐂 𝐏
a 10 0.175 0.233
b 0 0.000 0.000
c 0 0.000 0.000
d -10 -0.175 -0.233
𝐶! (!"#$% !"#$%&'() -0.466
𝐶! (!"#$%) -0.233
𝐶! (!"#$#%&) -0.233
Table 2 - Theoretical Calculations

4. Amit Ramji – A4 – University of Hertfordshire
4
Experimental
Set-­‐up
Procedure
1. A 1 inch chord diamond wedge model with
semi-wedge angle, 𝜀 = 5∘
→ (2𝜀 = 10∘
) to
an incidence angle of 𝛼 = 5∘
was set up
inside the supersonic wind tunnel. This
inclination provided the front surface (a) to
be parallel to the free stream flow (𝑀!).
Figure 3 - Incidence Set-up
Figure 4 - Diamond Wedge Inside Supersonic Wind Tunnel
2. The clamp lever was operated which
controlled the pressure inside the mercury
filled manometer tubes. While ensuring the
valve is completely open (closed position
shown in Figure 5).
Figure 5 - Mercury filled U-Tube Manometer's
3. The supersonic wind tunnel was switched
on to allow the pressure readings to stabilise
after a Mach Number of 1.8 had been
achieved. The valve clamp was later locked
to allow the pressure readings to be
recorded. The tunnel was then switched off
to allow the recording of Static, Aerofoil
and Total pressures 𝑃!, 𝑃 & 𝑃!
respectively.
Ensuring the readings (Inches of mercury)
are obtained from the bottom of the
meniscus in all cases, this is not clearly
visible in most cases hence the use of
measurements at the tube walls in all cases
was carried out for experimental accuracy.
(Note: Aerofoil Pressure tapping is only on
surface (a) as shown in Figure 3)
Figure 6 - Mercury filled U-Tube Manometer
4. The procedures 1 to 3 were repeated and
pressure head readings recorded for
𝛼 = 3∘
𝑡𝑜 𝛼 = −5∘
in increments of 2∘
.
5. The atmospheric pressure head of mercury
(inches), for later analyses was recorded.
6. Observations and sketches of shock wave
formation and expansion using a light
refraction set-up were later carried out. Dark
bands indicated the presence of shock
waves whereas light bands shown as
expansion waves are demonstrated in Figure
8.

5. Amit Ramji – A4 – University of Hertfordshire
5
Figure 7 - Wave Observation Mirror Set-Up
Experimental
Results
As shown in procedures 1 through 5 in the
Experimental Set-up section above, the following
data is the measured output from the supersonic
wind tunnel testing.
Face
(a)
PHead
[P1]
(Inch)
Atm
PHead
[P2]
(Inch)
PAbs=[PHg –
[P1-P2]]
(PHg=29.39
inch.Hg)
α=5∘
𝑃! (Static) 18 18.2 29.59
𝑃 (Aerofoil) 28.8 6.45 7.04
𝑃! (Total) 29.3 5.7 5.79
α=3∘
𝑃! 18 18.2 29.59
𝑃 29.2 6.1 6.29
𝑃! 29.4 5.8 5.79
α=1∘
𝑃! 18 18.2 29.59
𝑃 29.5 5.6 5.49
𝑃! 29.4 5.7 5.69
α=−1∘
𝑃! 18 18.2 29.59
𝑃 29.8 5.3 4.89
𝑃! 29.3 5.7 5.79
α=−3∘
𝑃! 18.1 18.2 29.49
𝑃 30.3 4.7 3.79
𝑃! 29.4 5.7 5.69
α=−5∘
𝑃! 18 18.2 29.59
𝑃 30.3 4.7 3.79
𝑃! 29.3 5.7 5.79
Table 3 - Experimental Data
Mach No from;
!!
!!
= 1 + 0.2𝑀! !.!
Pressure Coefficient;
𝐶! =
!
!!!
!
𝑃
𝑃∞
− 1
α=5∘
1.723
0.1039
α=3∘
0.0416
α=1∘
-0.0169
α=−1∘
-0.0748
α=−3∘
-0.1607
α=−5∘
-0.1662
Table 4 - Calculated Mach No and Cp from Experimental
Data
Flow
Visualization
Figure 8 - Experimental Shockwave formation at α=-3°.
Shock Waves
Expansion Fan
FlowDirection

6. Amit Ramji – A4 – University of Hertfordshire
6
CFD
Set-­‐up
In order to carry out supersonic flow analysis
over the diamond wedge, various models have
been created in CATIA V5 and translated into
STEP (.stp) files to be read in STAR-CCM+®.
This process makes it simple to create geometry
of various shapes and orientations. With the
help of the Java script recording one can create
CFD models in STAR-CCM+® in quick
succession while ensuring all parameters other
than the variable (Incidence angle, α) is
changed at any given time ensuring
consistency. The recorded script also dissects
the 3D model and creates a 2D mesh across the
flow direction. One simply needs to label the
Inlet, Outlet and wall faces for the script to
recognise the geometry, create the consistent
mesh and input the flow parameters, if all the
models need to be updated with new parameters
such as a finer mesh, velocity change or groups
of parameters, the Java script can be modified
using text based formatting and run through
STAR-CCM+® per individual model case.
Geometry
&
Orientation
Geometry has been modelled in CATIA V5,
translated into a STEP file (.stp) that is later
imported into STAR-CCM+®. To ensure the
correct translation and orientation into STAR-
CCM+®, measurements were taken within
STAR-CCM to confirm the geometry and
orientations to that tested in the supersonic
wind tunnel tests.
Mesh
The following mesh variables have been used to
model the diamond wedge and its domain. As
can be seen in Figure 9 the mesh density is
sufficient enough to capture the output
requirements and coarse enough to allow
conververgence of the weighted average
iterations.
Domain Size (Flow
Direction)
9xChord
(Position=2xChord pre-
wedge & 6xChord post-
wedge)
Domain Size (Normal to
Flow Direction)
80xThickness
(Position=Centred)
Base Size 2mm
Number of Prism Layers 10
Prism Layer Stretching 1.5
Prism Layer Thickness 33.3 (absolute=0.66mm)
Surface Growth Rate 1.3
Surface Proximity Gap 2.0
Size Type Relative to Base
Size Method Min & Target
Relative Minimum Size 25% of Base
(absolute=0.5mm)
Relative Target Size 100% of Base
(absolute=2mm)
Wrapper Feature Angle 30
Wrapper Scale Factor 45
Table 5 - Mesh Parameters
Figure 9 - Mesh Visualisation (α=5° Shown)
Enabled
Analysis
Models
Below is a list of selected modelling solvers,
which have been used in all cases:
Segregated Fluid Temperature
Segregated Flow
Ideal Gas
Two-Layer All y+wall Treatment
Realizable K-Epsilon Two-Layer
K-Epsilon Turbulence
Reynolds-Averaged Navier Stokes
Turbulent
Gas
Steady Flow
Gradients
Two-Dimensional
Table 6 - Enabled Simulation Parameters

7. Amit Ramji – A4 – University of Hertfordshire
7
Initial
Conditions
The following parameters have been applied to
all cases of CFD modelling. Including initial
conditions for the domain material in this case
air at room temperature and standard
atmospheric pressure.
Material Air (Values from STAR-
CCM+® 8.04.007)
Pressure (Constant) 99525 Pa (29.39 inch.Hg)
Static Temperature
(Constant)
300 K
Turbulence Specification Intensity & Viscosity Ratio
Turbulent Velocity Scale
(Constant)
1 m/s
Velocity (Constant) [X=585, Y=0, Z=0] m/s
Stop criteria 250 iterations
Table 7 - Initial Condition Parameters
In the later section of this literature; a lower &
higher Mach number than 1.8 is chosen to
investigate the flow visualisation and possible
shapes. Additionally the geometry has been
modified to include a larger wedge angle for the
purpose of interest and further analysis.
CFD
Results
Flow
Visualisations
Below are converged (weighted average) flow
visualizations of the CFD models for α=5° to
α=-5°.
Figure 10 - Flow Visualization for α=5° (Left), α=-5° (Right)
Figure 11 - Flow Visualisation for α=3° (Left), α=-3° (Right)
Figure 12 - Flow Visualisation for α=1° (Left), α=-1° (Right)
The following table is compiled with pressure
coefficient probes from the CFD simulation at
surface ‘a’:
Pressure Coefficient
at surface ‘a’, 𝐶!
α=5∘
0.0000
α=3∘
-0.0387
α=1∘
-0.0812
α=−1∘
-0.1217
α=−3∘
-0.1623
α=−5∘
-0.2029
Table 8 - CFD Output for Cp at Surface 'a'
Considering the pressure distribution of the
CFD models in Figure 10 to Figure 12, it can be
observed that the pressure is higher than the
surrounding atmosphere at the 2 forward-facing
surfaces (‘a’ & ‘c’) as shown in yellow and
amber. Additionally the AFT-facing surfaces
(‘b’ & ‘d’) show a pressure that is less than the
surrounding atmosphere. For a symmetrical
section at α=0° these pressure distributions
should be equal in magnitude as described in
the supersonic aerofoils section of Barnard and
Philpott [2]. Experimental measurements shall
never agree to this completely however will be
in the same order or magnitude and has been
modelled and demonstrated in Figure 14 & Figure
15. Where two wedge shapes have been tested
under 3 velocities for a total of 6 test cases at
α=0°, all showing equal pressure distributions.
Pressure distributions of a double wedge
aerofoil in a supersonic free-stream is simple to
comprehend as a result of symmetry and low
incidence angles. Surfaces ‘a’ through ‘d’ of the
diamond cross-section experiences virtually
constant pressure. Following from the detail
that flow over surfaces ‘a’ & ‘c’ is uniform as
the bow shock waves simply deflect the entire
flow until it becomes parallel with the surface
direction of which further reading can be found
in Chapter 5 of Barnard and Philpott [2].
On the opposing end of the flow field the
expansion fans developed from the zeniths on
the upper and lower surfaces turn the flow so
that it is parallel to the AFT-facing surfaces. As
a result all the surfaces should have a uniform
pressure.
Expansion Fan
Shock Waves

8. Amit Ramji – A4 – University of Hertfordshire
8
Convergence
In order to simulate a real life observation, 3
major steps (comprising of 11 minor stages)
must be undertaken. Firstly pre-processor where
the model, conditions and mesh must be
modelled as described in the CFD Set-up section
above. Secondly the solver where STAR-
CCM+® performs calculations of the model
parameters. The computational time required to
complete this stage is dependant on the number
of pre-set iterations or if the finite volume
calculations converge as introduced in Figure 1
where each element requires 4 equations to be
solved. In this analysis the CFD performed an
iterative approach where convergent graphs
indicate the weighted average differences
between element iterations (see Figure 13).
To obtain CFD convergence over all elements
to a very accurate value is meaningless and
would take a very long time even when using
parallel processing. Convergence to a precise
level allows the researcher to establish
confidence that the error is minimal rather than
if the model did not converge. Overall 200
iterations was plentiful as initial conditions for
velocity had been applied, thereby eliminating
the flow propagation effects from the inlet and
modelling steady flow conditions by
eliminating the acceleration terms in the
conservation of momentum equations.
Figure 13 – Sample convergence graph for α=5° at 585m/s in
200 Iterations.
Further
Parameter
Modelling
Mentioned previously in the Initial Conditions
section of this text, a further analysis of velocity
effects and wedge shape effect has been
completed. Where the steady velocity of 350,
585 and 750 m/s has been modelled for α=0° in
addition to a larger semi-wedge angle of ε=20°
being modelled under the same conditions at
α=0°.
Figure 14 - Flow Visualisation for ε=5° & α=0° at 350m/s
(Top-Left), 585m/s (Top-Right) & 750m/s (Bottom).
Figure 15 - Flow Visualisation for ε=20° & α=0° at 350m/s
(Top-Left), 585m/s (Top-Right) & 750m/s (Bottom).

9. Amit Ramji – A4 – University of Hertfordshire
9
Method
Comparison
–
Results
Experimental
Uniformity of pressure at low incidence angles
has been demonstrated however increased
incidences results in a dissimilar observation. It
is generally accepted for low velocities that,
thin aerofoils and aerofoils with sharp leading
edges will stall at low incidences due to the
stagnation at the leading edge and flow around
the leading contour (see section 6.7.3 of
Houghton et al [1] & Figure 1.13 through 1.15
of Barnard and Philpott [2]). The abrupt change
in surface vectors at the peaks between the front
and rear surface would therefore lead to flow
separation and ultimately increased drag and
stall as shown in Figure 8.6 of literature [2].
The separation gap over the peaks seen in Figure
10 to Figure 12 & Figure 14 to Figure 15 is known as
the expansion fan. Prandtl-Meyer Expansion
waves or fan occurs when supersonic flow is
turned into another direction as described in
detail by Houghton et al in section 6.5 of [1].
Theoretical
Calculated theoretical methods above in Table 2
shows that Ackeret's and CFD methods agree.
Results from Ackerets theory and CFD show
the lines of best fit were very close to each
other. This provides evidence that the CFD
modelling and simulation was close enough to
accurate. Comparing Ackerets theory and CFD
to the wind tunnel results they follow the same
trend. This common trend indicates that when
the incidence angle was increased so did C!.
However the wind tunnel results were not close
to the analytical or numerical methods. Upon
analysis of the results for all three methods
there was an equal anomaly in the difference.
The suspected reason why the experimental
results were not the same to the CFD and
Ackerets is due to the difference in free stream
Mach No. If the Mach number were greater
using Ackeret’s equation then the C! would
decrease. Therefore the airflow through the
wind tunnel was not at Mach 1.8 as used in the
calculated methods in Table 2.
Numerical
(CFD)
A numerical solution such as STAR-CCM+®
allows the researcher to perform thousands of
calculations per minute on a single model or an
assembly to represent fluid flow over and
around a blockage.
Java scripts were utilised in order to carry out
the 7 orientations (including α=0°) and velocity
cases (for a total of 12 Cases). The Java
platform allows the user to carry out pre-
processing and input variables into the model
and repeat this process for a wide variety cases.
Java command based refinement of mesh sizes
and scaling factors among many other
parameter changes (see Table 5 through Table 7)
are simplified using the script approach and
saves pre-processor time and errors when
solving a large problem.
Method
Overview
Three methods for calculating the Coefficient of
Pressure (C!) were used as shown in the above
text, one of which from experimental
Supersonic Wind Tunnel testing, second
method utilising Ackeret’s linear theory and
thirdly a numerical solution from CFD
software, STAR-CCM+® by CD-Adapco.
STAR-CCM+® is comprehensive software
package which can perform fluid mechanic’s
calculations for pressure, speed and temperature
around a blockage. It can also be utilised for
duct and pipe flow using a selection of
modelling techniques, boundary conditions and
solvers.
As can be seen in Figures 8.5 & 8.6 of Barnard
and Philpott [2] the above CFD observation
along with theoretical calculations using plate
theory as per Table 2. An elementary
understanding of aerodynamics allows one to
conclude that the above methods are validated
and agrees with literature [1-5] and
observations for supersonic aircraft aerofoils.
Further high speed compressible flow literature
is cited in reference [6] by Kostoff et al. In
addition Figures 4, 5a & 6 by van Oudheusden
et al [7] and Figure 7 through 14 by Khalid et al
[8] agree with the observations seen in CFD
modelling in this text.

10. Amit Ramji – A4 – University of Hertfordshire
10
Overall the results from the CFD were close to
those found in theoretical and experimental
methods. This allows one to reinstate
confidence and reliance on the CFD modelling
technique for larger and complex problems by
understanding the limitations. The results from
the 3 methods are summarised herewith:
Table 9 – Method Comparison Data
Figure 16 - Method Comparison, α against Cp
Further
Parameter
Discussion
Observation of the flow parameters set to create
the flow field simulations show above in Figure
14 & Figure 15 indicate that the theoretical
outcomes have been demonstrated with the use
of numerical methods.
Figure 14 shows the original tested wedge in
the supersonic wind tunnel at α=0° at 350m/s,
585m/s & 750m/s. The supersonic shockwave
resulted by progressively sharpening towards a
parallel direction to the flow field. The 350m/s
simulation shows a bell shape shockwave at the
flow front propagating from the stagnation
point at the leading edge. On the contrary, the
750m/s simulation shows the shockwave to be
more pronounced which will completely be
hugging the surface when reaching close to
hypersonic speeds.
The simulation in Figure 15 shows the same flow
parameters of α=0° at 350m/s, 585m/s &
750m/s however with a much larger wedge
shape with a semi wedge angle, ε=20°. Initial
analysis of this sort provides evidence that as
the wedge angles are increased, the drag
increases are significantly larger. For this
reason supersonic and hypersonic aircraft
contain a sharp point at their nose.
References
1. Houghton, E.L. and P.W. Carpenter,
Aerodynamics for Engineering Students. 2003:
Elsevier Science.
2. Barnard, R.H. and D.R. Philpott, Aircraft Flight:
A Description of the Physical Principles of
Aircraft Flight. 2010: Pearson Education,
Limited.
3. Elling, V. and T.-P. Liu, Supersonic flow onto a
solid wedge. Communications on Pure and
Applied Mathematics, 2008. 61(10): p. 1347-
1448.
4. Kopriva, D.A., Spectral solution of inviscid
supersonic flows over wedges and axisymmetric
cones. Computers & Fluids, 1992. 21(2): p. 247-
266.
5. Pekurovskii, L.E., Supersonic flow over a thin
wedge intersected by the front of an external
plane compression shock. Journal of Applied
Mathematics and Mechanics, 1982. 46(5): p.
621-625.
6. Kostoff, R.N. and R.M. Cummings, Highly cited
literature of high-speed compressible flow
research. Aerospace Science and Technology,
2013. 26(1): p. 216-234.
7. van Oudheusden, B.W., et al., Evaluation of
integral forces and pressure fields from planar
velocimetry data for incompressible and
compressible flows. Experiments in Fluids,
2007. 43(2-3): p. 153-162.
8. Khalid, M.S.U. and A.M. Malik. Modeling &
Simulation of Supersonic Flow Using
McCormack’s Technique. in Proceedings of the
World Congress on Engineering. 2009.
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IncidenceAngle,α
Theory,𝐶!=
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Experimental,
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!!
−1
CFD
α=−5∘
-0.2332 -0.1662 -0.2029
α=−3∘
-0.1866 -0.1607 -0.1623
α=−1∘
-0.1399 -0.0748 -0.1217
α=1∘
-0.0933 -0.0169 -0.0812
α=3∘
-0.0466 0.0416 -0.0387
α=5∘
0.0000 0.1039 0.0000