Practical Snippets - University Projects

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Supersonic Flow Properties

An investigation was conducted using a supersonic blowdown wind tunnel to determine the half-angle of an oblique shock wave for fixed cones with half-angles of 20°. The range of Mach numbers considered was between 2.5 and 3.3. The cone specimen was seen to be pitching in the test section and the upper and lower half-angles had to be considered separately. The obtained experimental data showed that, as Mach number increased from 2.5 to 3.3, the oblique shock wave half-angle decreased from 42.2° to 33.0° for the upper half-angle of 16.8°, while the oblique shock wave half-angle decreased from 45.7° to 33.7° for the cone lower half-angle of 23.2°. This experimental data was compared to the theoretical predictions from the \(\theta\)-\(\beta\)-\(M\) relation, in an attempt to warrant the applicability of the theoretical relationship and accuracy of its approximation. Subsequently, it was seen that almost all the points were satisfactorily in agreement, when considering the uncertainties present in the experiment, except for the measurements at the Mach numbers of 3.0 and 3.3 for the cone lower half-angle, which should be tested again to examine the repeatability in the values. It should be noted that the largest source of uncertainty and error was from the schlieren visualization, which led to the uncertainty of the experimental half-angles being ±1.6°. Overall, the experiment was accepted as successful given its standard.

Background And Research Question

When the relative velocity between a sharp cone and the incident air flow is supersonic, a shock wave will form due to the deflection and compression of the flow. If the shock wave is inclined relative to the upstream flow direction, the shock wave will be oblique. The half-angle of an oblique shock wave is theoretically predicted with the \(\theta\)-\(\beta\)-\(M\) relation, but it can be experimentally obtained in a supersonic wind tunnel for verification. The oblique shock wave for cone specimens of 20° half-angle and Mach numbers between 2.5 and 3.3 was to be investigated. The oblique shock wave angle was to be revealed using schlieren visualization and digitally measured from photographs. This was done to compare the actual oblique shock wave half-angle against the theoretically predicted oblique shock wave half-angle.

The ultimate aim of the investigation was to answer the following research question: For given cone specimens of 20° half-angle with incident air flowing at chosen Mach numbers between 2.5 and 3.3, how does the actual oblique shock wave half-angle correlate with the theoretical oblique shock wave half-angle, as calculated with the \(\theta\)-\(\beta\)-\(M\) relation?

Oblique Shock Waves

As air flows past a body, the flow is disturbed and deflected around the body. As the relative speed of the flow with respect to the body approaches supersonic, compressibility effects begin to occur and the density of the flow begins to vary locally. As the relative speed is further increased, this behaviour leads to a shock wave, which separates two regions of varying flow where the shock transition occurs instantaneously along the shock wave.

An oblique shock wave occurs when a supersonic and compressible flow encounters a sharp corner or cone. This effectively turns the flow on itself rapidly and leads to the compression of the flow. So, the resulting oblique shock wave is inclined relative to the incident upstream flow direction. It should be noted that, if the shock wave is perpendicular to the flow direction, it has become detached from the body and is classified as a normal shock wave.

Diagram of an oblique shock wave forming as a supersonic flow encounters a sharp cone:

The \(\theta\)-\(\beta\)-\(M\) relation defines the relationship between the half-angle of the oblique shock wave, half-angle of the cone, and Mach number of the incident flow. Although, the \(\theta\)-\(\beta\)-\(M\) equation is technically 2-dimensional, it applies equally to bodies of revolution for 2-dimensional arrangements. In most cases, it is reasonable to assume that the heat capacity ratio for air is equal to ratio of the constant pressure and volume specific heat capacities and is 1.4 under standard conditions.

It should be noted that there are 2 values for the oblique shock wave half-angle for each value of the deflection angle. The smaller value is known as the weak shock solution and larger value is known as the strong shock solution. Additionally, the downstream flow is supersonic for the weak shock solution, while the downstream flow is subsonic for the strong shock solution. Typically, the weak shock solution is naturally favoured, but, if the half-angle of the cone is greater than the maximum deflection angle, the shock wave will be curved and detached.

Definition of the \(\theta\)-\(\beta\)-\(M\) relation between the oblique shock wave, cone, and incident flow:

\begin{gather*} \tan (\theta) = 2 \cot (\beta) \left(\frac{M_1^2 \sin^2 (\beta) - 1}{M_1^2 (\gamma + \cos (2\beta)) + 2}\right) \end{gather*}

Supersonic Blowdown Wind Tunnel

Wind tunnels offer an accurate and convenient way of performing aerodynamic measurements using geometrically and dynamically scaled specimen in a test section. A supersonic blowdown wind tunnel uses high-pressure air which is released through the tunnel, so that the flow is conditioned to have supersonic velocities through the test section. The supersonic velocities are obtained by initially accelerating the subsonic flow to sonic velocity through a converging nozzle and then accelerating the flow further to greater supersonic velocities through a diverging nozzle. A wind tunnel will typically use short blows of air to consume less energy and allow for multiple tests.

Simplification of the basic components of a supersonic blowdown wind tunnel:

For supersonic flow, the area ratio of the cross-sectional area of the test section to the cross-sectional area of the throat is needs to be considered to condition the flow. For air, the primary composition components are oxygen and nitrogen, which are both diatomic gases, and, thus, allows the area ratio to be simplified when the heat capacity ratio is assumed to be approximately constant at 1.4. It should be noted that, when a supersonic blowdown wind tunnel is started, a normal shock wave travels down the tunnel to the second throat, and, when the wind tunnel is shut off, a normal shock wave travels from the second throat to the first throat, so the test specimen should always be parallel to the air flow for safety when the wind tunnel is started and shut off (considered with other potential risk involved in the experiment).

Considerations for the area ratio between the test section and throat to condition the flow:

\begin{gather*} A_R = \frac{A_{test}}{A_{throat}} = \left(\frac{M_1^2 + \frac{2}{\gamma - 1}}{1 + \frac{2}{\gamma - 1}}\right) ^ {\frac{\gamma}{\gamma - 1}} \left(\frac{1 + \gamma}{1 + \gamma M^2_1}\right) \\[9px] A_R = \frac{A_{test}}{A_{throat}} = \left(\frac{M_1^2 + 5}{6}\right) ^ {3.5} \left(\frac{2.4}{1 + 1.4 M^2_1}\right) \end{gather*}

Schlieren Visualization

Schlieren visualization is used to optically capture an image of a 2-dimensional air flow around the surface of a body. This is accomplished due to the changes in the properties, specifically the density and refractive indexes, of the air as a result of the flow. Essentially, because light rays are refracted differently when they encounter different refractive indexes, a knife edge is used to block light rays which pass through areas of high air density and heavy refraction (without affecting light rays which pass through areas of normal air density with unchanged refraction). Thus, the schlieren visualization is sensitive in the derivative of density perpendicular to the knife edge, which will make areas of high density appear darkened to the viewing plane (and areas of normal density appear unaffected to the viewing plane).

Basic arrangement of components to create a schlieren visualization for observation:

Moreover, it should be recognized that the change in the density of the air is very small, which necessitates the need for the schlieren visualization. For illustration, the effect on the density, and subsequent change in the density, is given by the Gladstone-Dale relation, which shows that the refractive index of air is linearly proportional to the density of the air. The Gladstone-Dale constant is typically only between 0.1x10^-3m³/kg and 1.5x10^-3m³/kg for most gases.

Definition of the Gladstone-Dale relation between the density and refractive index of air:

\begin{gather*} n = K \rho + 1 \,\rightarrow\, \rho = \frac{n-1}{K} \,\therefore\, \Delta n = K \Delta \rho \,\rightarrow\, \Delta \rho = \frac{\Delta n}{K} \end{gather*}

Investigation Objectives

With the research question and background, the objectives of the investigation were formally defined as follows:

To measure the variation of the actual half-angle of the oblique shock wave resulting from a cone specimen with a half-angle of 20°, when the incident flow, as air under standard conditions, had different Mach numbers between 2.5 and 3.3. As a secondary outcome, this also looked at the use of schlieren visualization to measure oblique shock wave angles.
To calculate and interrogate the predicted half-angle of the oblique shock wave using the \(\theta\)-\(\beta\)-\(M\) relation for the given cone specimen at and between each of the Mach numbers for the chosen range between 2.5 and 3.3.
To compare the experimental data and theoretical model, with regard to the actual and predicted half-angle of the oblique shock wave for the given cone specimens and Mach numbers of the incident flow as air.

Experiment Apparatus

The experiment was performed in an open-circuit supersonic blowdown wind tunnel. Essentially, the wind tunnel functions by releasing compressed air through a configurable throat, such that it is flowing at supersonic velocities at the test section. The minimum testable Mach number is 2.5, while the maximum testable Mach number is 3.3. Initially, the air is compressed to the required pressure in a compressor, dried to remove moisture in a drier, and then fed to a storage tank - when testing, the pressure of the storage tank must always be above 700kPa. A main shut-off valve contains the air within the storage tank until it is released into the tunnel, but it may also be used to shut down the tunnel in an emergency. A pilot valve regulates the pressure of the air in the following settling chamber to ensure a constant static pressure in the test section. The settling chamber minimizes pressure and volume flow fluctuations caused by a decrease in the pressure of the storage tank and any movements of the pilot valve. There are also screens and wire gauzes fitted within the settling chamber to aid in reducing fluctuations in turbulence.

Subsequently, the first throat features a converging-diverging nozzle, which is primarily used to accelerate the flow to supersonic velocities. Additionally, this throat is manipulated by sliding a lower block relative to a fixed upper block, to change the shape and cross-sectional area of the throat. With regard to the dimensions, the entry height is 380mm, the convergence height can be adjusted to a minimum of approximately 6mm for a Mach number of 3.3, and the exit height is 101.6mm. The position of the sliding block is directly controlled by a manual veeder counter, such that the area ratio can be precisely configured for the desired Mach number of the incident flow.

The test section then encompasses the cone specimens and is fitted with clear glass windows to allow for observations and shleiren visualization. Additionally, to ensure a constant air velocity through the test section and across the cone specimens, the height of the test section is constant, but the walls of the test section have a slightly increasing taper to allow for the growth of the boundary layer and minimize interference from viscous effects. For interest, during testing, the temperature of the test section can decrease to roughly 100K due to the flow.

Finally, the second throat decelerates the flow to subsonic velocities. This is possible because the second throat also has an adjustable cross-sectional area and, with a larger area ratio than the first throat, the velocity of the air will be decreased. The diffuser further decreases the air velocity and allows for the discharge of the air into the atmosphere.

Schematic of the components of the open-circuit supersonic blowdown wind tunnel:

Schematic of the first throat of the open-circuit supersonic blowdown wind tunnel:

Calidration relationship between the Mach number and veeder counter value:

\begin{gather*} M = - 6.922 \times 10^{-10} \times V^3 + 2.514 \times 10^{-6} \times V^2 - 0.004 \times V + 4.118 \end{gather*}

The cone specimens used for the experiment include two symmetrical and vertically mounted cylinders with conical-shaped sharp points. The approximate dimensions of the cone specimens include a length of 100mm and diameter of 20mm with a chamfer of 27.5mm at 20° measured from the leading edge. A simple mounting bracket is used to position the cone specimens in the test section (unfortunately, the cone specimens were seen to be pitching slightly and were not completely parallel with the incident flow during the tests).

Schematic of the cone specimens used to create the oblique shock waves:

The setup of the schlieren visualization, which was monochromatic, consisted of a light source (as a candle flame), collection of five flat and concave mirrors, test section, knife edge, imaging lens, and digital camera as the viewing plane. The light rays from the light source were reflected by a concave mirror and then a flat mirror to the test section. In the test section, the light rays are exposed to air at differing densities due to the flow of the air past the cone specimens and resulting oblique shock wave. Thus, the light rays are refracted at different angles. After this section, the light rays are reflected by a second flat mirror to a second concave mirror, so that the light rays reach a final flat mirror which reflects the light rays towards the knife edge. At the knife edge, the light rays which were previously heavily refracted by the air of higher density are blocked. Finally, the remaining light rays enter the imaging lens and Nikon DR5200 digital camera.

Schematic of the setup of the schlieren visualization for monochromatic photography of the test section:

Collection of photographs of the apparatus of the wind tunnel, cone specimens, and schlieren visualization:

Methodology And Procedure

Using a supersonic blowdown wind tunnel, it is possible to create an oblique shock wave on a cone specimen, where the half-angle of the oblique shock wave can be measured using schlieren visualization. Subsequently, using a digital camera to continuously capture photographs of during each test at different Mach numbers, it is possible to precisely measure the half-angle of the oblique shock wave for each chosen Mach numbers. Thus, this will provide experimental data of the variation of the oblique shock wave half-angle for different Mach numbers. It should be recognized that, for clear and accurate results, the schlieren visualization must be correctly set up and aligned with the cone specimens.

Furthermore, the theoretical half-angle of the oblique shock wave can be calculated for each of the applied Mach numbers using the fixed half-angle of the cone specimens and \(\theta\)-\(\beta\)-\(M\) relation. Thus, the experimentally obtained half-angles of the oblique shock wave can be compared with the theoretically calculated half-angles of the oblique shock wave, in an attempt to convincingly justify the accuracy of the experiment and theoretical predictions.

Finally, it should be noted that the Mach number of the flowing air is the independent variable, while the half-angle of the oblique shock wave is the dependent variable. The controlled variables include the half-angle of the cone specimens, stagnation pressure of the air, and temperature of the air, which are related to the speed of sound in the air. The simplified procedure for conducting the experiment was based on the following steps:

Place the cone specimens parallel to the test section of the wind tunnel and aligned with the schlieren visualization.
Begin the compression of the air in the compressor.
Pause until the required pressure value in the storage tank is reached.
Shut off the compressor.
Adjust the first throat with the veeder counter to give the highest desired Mach number.
Record the current image number of the digital camera.
Start the wind tunnel to perform a short blow of supersonic air.
Immediately begin the continuous photographs of the digital camera.
Allow the wind tunnel to settle to a steady state.
Repeat items 3 to 7 by decreasing to the desired Mach numbers incrementally.
Shut off the wind tunnel and all other active components.

For the safety precautions of the experiment, a risk assessment was completed. Notably, this included ensuring that hearing protections was worn by the operators and nearby bystanders while the wind tunnel was operating, due to the loud cacophony which may cause hearing damage. In addition, it is important to ensure there are no loose particles in the passages of the wind tunnel before starting. The operating precautions of the experiment were as follows:

Begin the tests near the maximum testable Mach number of 3.3 and end the tests near the minimum testable Mach number of 2.5. It was extremely critical that the maximum and minimum Mach numbers were not exceeded.
Ensure the mounting bracket of the cone specimens is securely installed with the cone specimens parallel to the flow.
Begin the continuous photographs of the digital camera almost immediately after the wind tunnel is started in order to clearly capture the oblique shock wave. The tests had very short durations of relevance due to the short blows of air.
Confirm that the wind tunnel has settled to a steady state after each test.

Observations And Data Analysis

The wind tunnel was started for short blows of air and photographs of the oblique shock wave could be captured with the digital camera of the schlieren visualization. A total of 4 short blows were performed, where the veeder counter value was increased to decrease the Mach number in steps from 3.3 to 2.5. The captured photographs were then digitally analysed to find the oblique shock wave half-angles for each test. When completing the experiment, there was a constant stagnation pressure of 250kPa. Additionally, the tank pressure was initially increased to 1,400kPa and decreased as the experiment proceeded. When installed, the cone specimens had a slight pitch with an upper half-angle of 16.8° and lower half-angle of 23.2°. These angles were constant for each test, as the position of the cone specimens was not changed.

Collection of photographs of the experiment results for the tested Mach numbers of 3.3, 3.0, 2.7, and 2.5:

Observations of the experimental Mach numbers and half-angles of the oblique shock wave:

Test	Tank Pressure [kPa]	Veeder Counter [-]	Mach Number [-]	Upper Oblique Shock Wave Half-Angle [°]	Lower Oblique Shock Wave Half-Angle [°]
A	1400	239.0	3.3	33.3	33.7
B	1330	347.5	3.0	33.0	36.6
C	1200	478.3	2.7	36.5	43.2
D	1100	584.0	2.5	42.2	45.7

Using the upper half-angle of 16.8° of the cone specimens and lower half-angle of 23.2° of the cone specimens, the theoretical variation of the oblique shock wave half-angle could be determined through numerical and iterative methods using the \(\theta\)-\(\beta\)-\(M\) relation, where the specific heat capacity ratio for air is approximated as 1.4. Thus, it was also possible to find the percentage difference between the experimental and theoretical values of the oblique shock wave half-angle for each Mach number. To measure the half-angles of the oblique shock wave and cone specimens, a digital protractor was used with an uncertainty of ±0.1°. However, based on the clarity of the photographs with resolutions of 6,000px by 4,000px, it was conservatively estimated that the uncertainty in the angle measurements was ±1.6°. The uncertainty in the Mach numbers was found using the calibration of the veeder counter and uncertainty propagation. Finally, the uncertainty in the half-angle of the oblique shock wave was demonstrated with the uncertainty of the angle measurements.

Calculation of the velocity of a flow given the Mach number and speed of sound in air:

\begin{gather*} v = M c \,\text{ with }\, c = 331.3\text{m/s} \times \sqrt{1 + \frac{T}{273.15^\circ\text{C}}} \end{gather*}

Propagation of uncertainty calculated by the root-mean-square method using the lowest resolution of accuracy:

\begin{gather*} y = f(x_1, x_2, x_3) \implies \Delta y = \sqrt{\left(\frac{\delta y}{\delta x_1} \Delta x_1\right)^2 + \left(\frac{\delta y}{\delta x_2} \Delta x_2\right)^2 + \left(\frac{\delta y}{\delta x_3} \Delta x_3\right)^2} \end{gather*}

Results Discussion

A supersonic blowdown wind tunnel experiment was conducted to determine the variation of the half-angle of an oblique shock wave with Mach number for cone specimens with half-angles of 20°. This was achieved by using a schlieren visualization to capture photographs of the oblique shock waves. So, using digital analysis, it was possible to find the half-angle of the oblique shock wave, which could be compared against theoretical results from the \(\theta\)-\(\beta\)-\(M\) relation.

Experimental and theoretical half-angle of the oblique shock wave at the upper and lower half-angles of the cone:

Test	\(M\) [-]	\(\theta_u\) = 16.8° ± 1.6°			\(\theta_u\) = 23.2° ± 1.6°
Test	\(M\) [-]	\(\beta_{exp}\) [°]	\(\beta_{the}\) [°]	Difference [%]	\(\beta_{exp}\) [°]	\(\beta_{the}\) [°]	Difference [%]
D	2.5	42.2 ± 1.6	39.0	8.29	45.7 ± 1.6	47.4	-3.50
C	2.7	36.5 ± 1.6	36.8	-0.67	43.2 ± 1.6	44.7	-3.30
B	3.0	33.0 ± 1.6	34.2	-3.36	36.6 ± 1.6	41.7	-12.8
A	3.3	33.3 ± 1.6	32.2	3.56	33.7 ± 1.6	39.6	-14.8

The obtained experimental and theoretical data has been summarized and graphically represented. This showed that the half-angle of the oblique shock wave progressively decreased as the Mach number increased for the tested range of Mach numbers between 2.5 and 3.3. For the upper half-angle of the cone at 16.8°, the experimental data showed that the half-angle of the oblique shock wave decreased from 42.2° to 33.0°, while the theoretical data showed that the half-angle of the oblique shock wave should decrease from 39.0° to 32.2°. For the lower half-angle of the cone at 23.2°, the experimental data showed that the half-angle of the oblique shock wave decreased from 45.7° to 33.7°, while the theoretical data showed that the half-angle of the oblique shock wave should decrease from 47.4° to 39.6°. Moreover, the rate at which the half-angle of the oblique shock wave decreased was negative and decayed as Mach number increased. This trend was expected as the half-angle of the oblique shock wave must decrease as the Mach number increases, because of the increased pressure and momentum of the flow of the air past the cone specimens.

Variation in the half-angle of the oblique shock wave for the upper half-angle of the cone at 16.8°:

Variation in the half-angle of the oblique shock wave for the lower half-angle of the cone at 23.2°:

Thus, it is evident that the presented trend in the experimental data was mostly in agreement with the theoretical data. For the upper half-angle of the cone at 16.8°, the percentage difference between the experimental and theoretical data varied from -3.36% and 8.29%, while, for the lower half-angle of the cone at 23.2°, the percentage difference between the experimental and theoretical data varied from -3.30% and -14.8%. Additionally, when considering the uncertainties in the values, the experimental values lay within the confidence bounds of the theoretical predictions for the majority of measurements. However, exceptions included measurements at Mach numbers of 3.0 and 3.3 for the lower half-angle of the cone, which were still slightly underpredicted from the lower theoretical confidence bounds by -6.12% and -3.75% considering the upper limits of experimental uncertainty respectively.

A reason for this difference may have been due to the assumption that the point of each cone is perfectly sharp, but this is not true for the experimental tests due to manufacturing tolerances and wear. Another possible reason for this difference is that the \(\theta\)-\(\beta\)-\(M\) relation is accurate for a wedge but only gives an approximation for a conical shape due to the true nature of the streamlines which are actually curved instead of straight - for a more robust relationship, it would be more appropriate to consider the Taylor-Maccoll equation with more comprehensive detail. A final reason may have been due to the large and compounding uncertainties and errors involved in the experiment.

It should be recognized that the largest source of uncertainty and error was from the schlieren visualisation, which regrettably resulted in impaired clarity and affected the measurements of the experimental half-angles up to ±1.6°. This discrepancy may have formed due to the misalignment of the mirrors and elements in the apparatus set up. Although robustly checked and varying in significance, other sources of uncertainty and error may have arisen from the setting of the veeder counter value, surface roughness along the walls of the wind tunnel which results in boundary layer and viscous effects, digital protractor used to measure the experimental angles, slight changes in air properties as the experiment progressed, possible pressure and volume flow fluctuations, and assumption that the air behaved as an ideal diatomic gas. Unfortunately, the repeatability of the experiment was not directly tested.

Summarized Conclusions

An experiment using a supersonic blowdown wind tunnel was conducted to determine the half-angle of an oblique shock wave for fixed cone specimens with half-angles of 20° and a range of Mach numbers between 2.5 and 3.3. Due to the pitching of the cone specimens in the test section, experimental values were obtained for the different upper and lower half-angles of the cone specimens. The obtained experimental data was also compared against the theoretical predictions, in an attempt to justify the theoretical model and accuracy of the experiment.

With the photographs of the oblique shock waves from the schlieren visualization, the half-angle of the oblique shock wave for the given cone specimens were successfully measured. This showed that the experimental half-angles of the oblique shock wave decreased as the Mach number increased. This trend was discovered for the upper and lower half-angles of the cone specimens. Specifically, for the upper half-angle of 16.8°, the experimental half-angle of the oblique shock wave progressively decreased from 42.2° to 33.0° and, for the lower half-angle of 23.2°, the experimental half-angle of the oblique shock wave progressively decreased from 45.7° to 33.7°.
The theoretical oblique shock wave half-angles were calculated using the \(\theta\)-\(\beta\)-\(M\) relation. Similarly to the experimental data, the theoretical half-angle of the oblique shock wave for the cone specimens progressively decreased at a decaying rate as the Mach number steadily increased. This was expected due to the increasing pressure and momentum as the Mach number increased. For the upper half-angle of 16.8°, the theoretical half-angle of the oblique shock wave progressively decreased from 39.0° to 32.2° and, for the lower half-angle of 23.2°, the theoretical half-angle of the oblique shock wave progressively decreased from 47.4° to 39.6°.
The experimental and theoretical results were compared with regard to the half-angle of the oblique shock wave. This was performed using the percentage difference between the results relative to the theoretical results. For the upper half-angle of 16.8°, the percentage difference varied from -3.36% and 8.29% and, for the cone lower half-angle of 23.2°, the percentage difference varied from -3.30% and -14.8%. When considering the uncertainty in the theoretical results, the experimental results typically lay in this range, so the accuracy of the results was acceptable, although the precision could be improved. However, at the Mach numbers of 3.0 and 3.3 for the lower half-angle, the measurements of the half-angle of the oblique shock wave notably varied from the theoretical results, so it was recommended for the measurements to be examined for repeatability in the results with further testing.
The largest source of uncertainty and error was from the schlieren visualization, which led to the uncertainty of the experimental half-angles being conservatively estimated to be ±1.6°. This will need to be addressed if the accuracy of the experiment is to be improved or if more precise results are required for analysis.