supersonic_ramji_amit_10241445

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

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

∂2u∂x2

+∂2u∂y2

+∂2u∂z2

!

"#

$

%&

ρ∂v∂t+u∂v

∂x+ v∂v

∂y+w∂v

∂z!

"#

$

%&= ρgy −

∂p∂y+µ

∂2v∂x2

+∂2v∂y2

+∂2v∂z2

!

"#

$

%&

ρ∂w∂t

+u∂w∂x

+ v∂w∂y

+w∂w∂z

!

"#

$

%&= ρgz −

∂p∂x+µ

∂2w∂x2

+∂2w∂y2

+∂2w∂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) =2αM 2 −1

−−2αM 2 −1

∴CL =4αM 2 −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! − 1α! +

tc

!

Where t = thickness & c = chord legnth The Moment Coefficient, C! is:

C! =−2αM! − 1

=−2α1.8! − 1

= 1.336 α


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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)

𝐂𝐏(𝐔𝐏) &

𝐂𝐏(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! (!"#$%) =12𝐶! !"#$% !"#$%&'( 0.233

𝐶! (!"#$#%&) =4αM! − 1

0.233

α=𝟑∘

Surf Def (Deg) Def (Rad) 𝐂𝐏 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∘


𝐶! (!"#$% !"#$%&'() 0.093 𝐶! (!"#$%) 0.047 𝐶! (!"#$#%&) 0.047

α=−𝟏∘


𝐶! (!"#$% !"#$%&'() -0.093 𝐶! (!"#$%) -0.047 𝐶! (!"#$#%&) -0.047

α=−𝟑∘


𝐶! (!"#$% !"#$%&'() -0.280 𝐶! (!"#$%) -0.140 𝐶! (!"#$#%&) -0.140

α=−𝟓∘

Surf Def (Deg) Def (Rad) 𝐂𝐏 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


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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.


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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∘

1.72

3

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

Flow

Dire

ctio

n


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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


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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


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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).


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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

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.

𝐶!, Surface ‘a’

Inci

denc

e A

ngle

, α

Theo

ry, 𝐶

!=

!" !! !!

Expe

rimen

tal,

𝐶 !=

!!!

!!! ! !−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

supersonic_ramji_amit_10241445

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