Supersonic Testing

Analyze supersonic flow over rocket geometries and practice formal supersonic tunnel testing procedures.

### Involvement

- Wind Tunnel Testing
- Imagery
- Analysis
- Report

**This lab served as an introduction to running tests and collecting data in a supersonic wind tunnel using Schlieren photography and numerical data analysis. Using a converging-diverging nozzle to accelerate air flow to supersonic velocities, shockwaves were generated using a cone at the end of a rod as the test model in the test chamber of the wind tunnel. The shockwaves were visually inspected using Schlieren imaging, which enables cameras to pick up on density changes in the air due to shockwave interactions. Additionally, using the pitot tubes and pressure gauges in the wind tunnel to gather numerical data, calculations were made to verify the mathematical models presented in class (Equations 1-7). **

**Using only the Schlieren images that were taken of the test section, it is possible to see shockwaves, however they are very faint. In future testing, it would be advantageous to run trials with different camera settings to try to pick up more distinct shockwaves to further the utility of visual analysis. **

**Nomenclature**

A = Area of Nozzle Exit

A* = Area of Nozzle Throat

k = Kinetic Energy

M = Mach Number

P = Pressure

R = Gas Constant

T = Temperature

ß = Shock Angle

c = Speed of Sound

ε = Rate of Dissipation of Kinetic Energy

γ = Heat Capacity Ratio

ρ = Density

**I. Introduction**

This lab was an introduction to Cal Poly’s Supersonic Wind Tunnel and compressible flow. When a fluid is compressible, the fluid density varies with its pressure. Fluids are usually treated as compressible when Mach numbers are greater than 0.3. However, shocks are only possible when the flow is supersonic. In this lab, air speeds hovered close to Mach 3 during tests.

When an aircraft is traveling at a subsonic velocity, the air ahead of the aircraft's nose flows out of the way before the aircraft reaches it. But as the aircraft reaches a sonic, and later a supersonic speed, the air ahead of it receives no "warning" of the aircraft's approach. The aircraft then pushes through the air and creates a shock wave. A shock wave is a type of propagating disturbance that is caused by a body moving faster than the speed of sound in a fluid. As air flows through the shock wave, the pressure, density, and temperature all increase abruptly.

When attempting to visualize the flow of fluids in a wind tunnel test, Schlieren photography is an effective technique. Schlieren photography relies on the fact that light rays are bent whenever they encounter changes in density of a fluid. In Schlieren photography, light rays are created by a light source and are focused by a lens into a beam through a test area. A second lens is set on the opposite side of the test area to keep the light rays focused in a camera. Once the tunnel is running, and the flow is supersonic, the camera takes images of the fluid.

The experiment guided the team to visualize and measure shock angles using Schlieren photography. A cone was mounted as the test model to resemble the front of an aircraft and was exposed to supersonic flow. Shock waves were produced from the test model and flow, and images of the shock were captured. Students predicted the Mach number of the test section from the measured shock angles. The experimental results and visible shock angles were then compared with theoretical, computational results, and with a CFD simulation of the same test model.

### II. Methodology

**A. Supersonic Wind Tunnel Setup**

**B. Schlieren photography Setup **

### C. Test Procedure

**Review all relevant safety information. Ensure all tunnel measurements are recorded.****During the experiment, take note of all possible sources of error. Taking pictures during and after the experiment is advantageous.****Assign roles to all personnel operating the test: manual valve operator, electro-pneumatic operator, valve relay operator, camera operator, data recorder.****Check all components of the setup to ensure working order: schlieren setup, camera and remote, valves, etc.****Take pictures no less than 3 seconds after the run has begun. The camera operator will take photos for about 10 seconds until the test is completed.****Now follow the SSWT Activity Procedure.****Run the test.****After all post run procedures described by the SSWT procedures document have been completed, retrieve the camera and view the photos. If no shock waves are visible, setup the camera/schlieren again, ensuring focus has been setup properly.****Repeat steps 6 through 9 to perform more runs.**

### III. Results

**A. Experimental Results**

* 1. Calculations*

Using the Mach-Area relation equation (Equation 1) in the appendix, a Mach number of 3.17 was calculated. In the experiment guidelines, it was stated that the actual value of the Mach number would be Mach 3. The calculated value was extremely close to the given Mach number, yielding a percent error of 5.67 percent. The calculated Mach number of 3.17 was then compared with the theta-beta-M relation (oblique shock-shock angle-Mach number relation) provided in Cengel’s and Cimbala’s Fluid Mechanics Fundamentals and Applications (Figure 10). This Mach number provided a shock angle of 20 degrees. The expected shock angle was 19 degrees which gave an error of 5.26 percent.

* 2. Pressure Tap*

The experimental data gathered from the observations of the pressure tap in tandem with the pressure data recorded via Lab-View was used in the Isentropic Flow equation (Equation 2). With an overall plenum pressure of 64.569 psig and a static pressure of 2.32 psi, the data produced a Mach number of 3.087. The resulting Mach number was again very close to the expected value of 3.0 and yielded an error of 2.9 percent. According to the theta-beta-M relation (Figure 10), the Mach Number of 3.087 corresponded to an estimated Shock angle (beta) of about 17.8 degrees, differing from the expected 19 degrees. This corresponded to a percent error of 6.23 percent.** **

* 3. Observations*

The pictures obtained by Schlieren Imagery had the greatest error when comparing the observed/calculated Mach number and Shock angle () to the expected values. When observing the Shock angle directly from the Schlieren imagery, a faint outline of a shock can be seen forming on the upper edge of the cone in the left image of Figure 4. This was observed as a value of about 28.96 degrees differing from the expected 19 degrees which corresponded to a very large percent error of about 52.41 percent. Because the test section is not immensely larger when compared to the size of the test article, it is hypothesized that the error is a result of incident shock waves bouncing off the walls of the test section and interacting with the shockwave forming on the test article.

The deflection angle (theta) of the test article as seen in Figure 3 was a value of about 0.8 degrees. When using Equation (5), a Mach number of only 2.066 was calculated which greatly differed from the expected value of 3.17 calculated in Part I. The difference between the two Mach numbers is also relatively large with a percent error of about 53.88 percent. Because the Mach number value is directly related to the shock angle value, it is safe to say that whatever caused the discrepancy between the shock angles also caused the discrepancy between the Mach numbers. In this case, the cause would be the additional shockwaves within the test section, as was hypothesized above.

**A. CFD Results**

* 1. Mesh*

Starting with a base mesh size of 0.15 inches and 20 prism layers, the mesh was refined further behind the cone. Using wake refinement, more cells were created near the shock off of the cone resulting in a total of about 45,000 cells. While the only limiting factor in cell count is compute power, both in cell creation and simulation run time, the cell wake refinement percentage base was lowered from 40.0 in the manual to 20.0. Similarly, the scene base size was lowered to .1 inches resulting in a total of 175,000 cells. This increase in cell count lowered the residuals by an order of magnitude.

* 2. Turbulence Models*

K Epsilon Model:

Solving for both kinetic energy, k, and the rate of dissipation of kinetic energy, this model is often a ‘go-to’ in general cfd applications. Though there are limitations to this model, difficulty solving for epsilon, adverse pressure gradients, jets flows, they are well known. For example, “K” goes to zero at boundaries/walls in reality, though meshes do not resolve this because it takes place so close. This is overcome with a damping addition to the model. Due to k-epsilon models performing superior in free-shear layer flows with small pressure gradients, internal flow low speed flow is modeled well unlike inlets and compressors in which pressure gradients are high. This model proved to converge well with plausible results in this scenario. Furthermore, the K - Epsilon model has proven to perform well in many applications, especially considering relatively lower computational power required. This model also improves the boundary layer within challenging pressure gradients and separation.

Reference: Brown, James L. "Shock wave impingement on boundary layers at hypersonic speeds: computational analysis and uncertainty." AIAA Paper 3143 (2011).

K Omega Model:

The K Omega turbulence model is convenient in that its method for approximating the Navier-Stokes equations uses two more simplified equations ( & ). The two equations used in this method consist of using partial derivatives where similarly to the K Epsilon Model, is the kinetic energy of the turbulence, where is the rate of dissipation (rate at which kinetic energy is transformed to thermal energy). According to NASA’s background and description of this particular model, it is recommended that initial conditions not be used as the method is sensitive to them and may not converge as easily if given certain conditions. Even though the K Omega model is extremely similar to the K Epsilon model, the main advantage of using both separately is that K Omega better models near-wall interaction but adversely, happens to have a more difficult time of converging when compared to K Epsilon.

Reference: Wasserman, S., “Choosing the Right Turbulence Model for Your CFD Simulation > ENGINEERING.com,” Engineering.com Available: https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/13743/Choosing-the-Right-Turbulence-Model-for-Your-CFD-Simulation.aspx.

Reynolds Stress Model:

The Reynolds Stress Model (RSM) is the most complete physical representation of turbulence. However this model is computationally strenuous because solutions to 7 equations are required, as well as high quality modeling and mesh. This model completely avoids the eddy-viscosity hypothesis, which is what K-Epsilon and K-Omega models are based on, and each individual component of the Reynolds stress is directly computed. The model relies on the Reynolds Stress Transport equation, which uses differential transport equations for the Convection, Turbulent Diffusion, Molecular Diffusion, Stress Production, Pressure Strain, Dissipation, and Production by System Rotation terms. Three of these terms -- Turbulent Diffusion, Pressure Strain, and Dissipation -- require modeling for closing the equation, while the rest of the terms do not. While the RSM is much more computationally expensive, this model offers significantly better accuracy than the K-Epsilon and K-Omega models, and requires only initial and boundary conditions to be supplied when computing. The RSM is also the most general of all turbulence models and works well for most flows.

Reference: A. Bosco, B. Reinartz, S. Muller, “Reynolds Stress Model Implementation for Hypersonic Flow Simulations,” RWTH University, 52056 Aachen, Germany. https://www.igpm.rwth-aachen.de/Download/reports/mueller/Bosco_Reinartz_Mueller_08.pdf

Spalart-Allmaras Model:

The Spalart-Allmaras (SA) Model is a simple turbulence model that can be solved relatively quickly. It involves solving only one equation that has been optimised for transonic flow over airfoils. This is a convenient turbulence model to use if only rough estimations required. The simple model works well for low quality meshes; however, it will often under-predict separation and is less accurate than other more complex turbulence models. The SA Model also has trouble computing accurate models for shear flows and decaying turbulence. This is due to lack of accuracy of having a single equation to compute all of the values needed. More equations allow for more degrees of freedom and therefore more accurate results when it comes to more complicated flows. If the test conditions are transonic and only a rough preliminary estimation of the true fluid behavior over a simple geometry is required, the SA turbulence model is a good choice to get a quick solution. Using the SA turbulence model for this lab produced results comparable to those using other CFD turbulence models. For this reason, SA is a good option as it solves quickly and gives results consistent with more complex turbulence models.

Reference: Wasserman, S., “Choosing the Right Turbulence Model for Your CFD Simulation > ENGINEERING.com,” Engineering.com Available: https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/13743/Choosing-the-Right-Turbulence-Model-for-Your-CFD-Simulation.aspx.

Since changes in the modeles are negligible, no single model is best for this specific simulation. While the Reynolds Stress model is the most complete model, it did not perform significantly better than the other models. Similarly, though K-Omega and K-Epsilon are used widely in industry, this simulation worked equally well under all models. The simplicity and geometry of this scene is likely causing the differences in among the models to be very small. More complex geometries and situations would likely benefit from certain models.

* 3. Density Plot*

### IV. Discussion

As seen in the table above, there are several potential sources of error listed that may have contributed to the discrepancy seen between the results that were achieved experimentally and the expected values. This can be seen clearly in the Observations section of the Results that reference the pictures that resulted from the Schlieren Imaging apparatus. The result achieved via Schlieren Imagery produced a Mach number (M = 2.066) and Mach angle of (= 28.96 degrees), as opposed to the expected Mach number (M = 3.0) and expected Mach angle (= 19 degrees). This yielded relatively large percent errors: 34.39 percent error for the Mach angle and 42.81 percent error for the Mach number. Because the majority of the sources of error are estimated to have a rather low effect on the final outcome/results received, it is hypothesized because of the relatively small ratio of the size of the test article to the size of the test section, additional shockwaves/anomalies in the test section may have interacted with the main shock to cause the error. This hypothesis is further affirmed by the fact that the calculations of Mach number and angle yield a very low percent error when compared to the results produced by the Schlieren Imaging apparatus. It is most likely that this hypothesis in conjunction with the low-visibility/ambiguity of shocks on the actual Schlieren image produced such a large error in results. In order to mitigate such an error in the future, it would be recommended to either use a larger ratio of the size of the test section to the size of the test article, or to utilize a method of image capture that produces much sharper and clearer images.

### V. Conclusion

The purpose of this lab was to provide the class with an introductory understanding of Supersonic Wind Tunnels, Schlieren Imaging, and Computational Fluid Dynamics (CFD). It can be concluded that after analyzing the results, the Fluid Dynamics Equations used in conjunction with the four Turbulence Models, proved to be very reliable in approximating the expected results for Mach Number and Mach Angle of the test article (conic section). However, the Schlieren Imaging method of image capture may lead to errors in approximating experimental results received from operating the supersonic wind tunnel. This is displayed in the large percent error seen in the results section. The results achieved via CFD confirm previous expected reliability in approximating fluid flow in various real-world applications. Overall, this lab has been extremely helpful in providing a basic understanding of wind-tunnels used for replicating supersonic fluid flow, as well as possible methods for visualizing supersonic flow.

### VI. Appendix

(1) A/A =(1/M) * [ (2/(+1))*(1+((-1)/2)*M2)](+1)/2*(-1)

(2) P/Pt = (/t) = (T/Tt)/-1 = (1+0.5*(-1)*M2)-/-1

(3) c = RT

(4) P/k = constant

(5) = sin-1(1/M)

(6) (T2/T1) = (P2/P1)*(1/2)

(7) T02/T01 = 1

### VII. Acknowledgements

Dr. Caprico

### VIII. References

“Aerodynamics Test Facilities - Part II,” NPTEL Available: http://nptel.ac.in/courses/101103004/module8/lec2/1.html.

Schmidt, B., “Schlieren Visualization,” Schlieren VisualizationAvailable: http://shepherd.caltech.edu/T5/Ae104/Ae104b_handout2015.pdf.

“Turbulence Modeling Resource,” NASA Langley Research CenterAvailable: https://turbmodels.larc.nasa.gov/wilcox.html.

Brown, James L. "Shock wave impingement on boundary layers at hypersonic speeds: computational analysis and uncertainty." AIAA Paper 3143 (2011).

Wasserman, S., “Choosing the Right Turbulence Model for Your CFD Simulation > ENGINEERING.com,” Engineering.com Available: https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/13743/Choosing-the-Right-Turbulence-Model-for-Your-CFD-Simulation.aspx.

Bosco, B. Reinartz, S. Muller, “Reynolds Stress Model Implementation for Hypersonic Flow Simulations,” RWTH University, 52056 Aachen, Germany. https://www.igpm.rwth-aachen.de/Download/reports/mueller/Bosco_Reinartz_Mueller_08.pdf