Oblique Shock Visualization and Analysis using a Supersonic Wind Tunnel

Transcription

1 Oblique Shock Visualization and Analysis using a Supersonic Wind Tunnel Benjamin M. Sandoval 1 Arizona State University - Ira A. Fulton School of Engineering, Tempe, AZ, I. Abstract In this experiment, oblique shock waves were generated by subjecting various geometries to an incoming air at a Mach number of 2.4. This was accomplished using the supersonic wind tunnel located at Arizona State University in Tempe. During the supersonic exposures, which generally lasted less than one minute, Schlieren imaging was used to capture and visualize the induced shock and Mach waves. Through analysis of the images, the Mach wave angles (µ) and the experimental shock wave angles (β e ) were obtained. Using the relationship M 1 =1/sin (µ) the Mach wave angles were used to calculate the true upstream Mach number (M 1 ). This was then used along with the known cone and wedge angles (Θ) to calculate the corresponding theoretical shock wave angle (β t ), and the corresponding Mach number after the shock wave (M 2 ). The stagnation pressure before the shock (P 01 ) was also experimentally measured and used in conjunction with M 1 and β in order to obtain a theoretical value for the static pressure after the shock wave (P 2t ). In an attempt to find theoretical inconsistencies, the theoretical values of β and P 2 were compared with those measured experimentally, and discrepancies were documented as experimental VS theoretical error. Overall, there the oblique shock angles for the 10 degree half angles were theoretically modeled relatively accurately (within 10%), but this was not the case for the larger geometries. P 2 was modeled theoretically within 14% of experimental values for all geometries except for the large cone, which was expected due to the inability of a two dimensional equation to model a three dimensional object. Nomenclature Θ = half angle of the cone or wedge β e = experimental shock wave angle β t = theoretical shock wave angle µ = Mach wave angle P 01 = stagnation pressure before the shock P 2e = experimental static pressure after the shock P 2t = theoretical static pressure after the shock M 1 = incoming Mach number M 2 = Mach number after the shock wave M n2 = Mach number normal to the shock wave, after the shock γ = specific heat, constant (1.4 for air) 1 Mechanical Engineering Student 1

2 II. Introduction In order to analyze shock waves (both normal and oblique), it is first necessary to understand the reason for the phenomena. Shock waves are caused when an object that is moving through a fluid is moving faster than said fluid has time to "react" and move out of the way. In the case of air, this speed is known as the speed of sound and is roughly 340m/s. In our case air is being forced past a wedge at over the speed of sound. Because the air molecules are unable to propagate fast enough in response to the body, streams of air (roughly.2 micrometers thick) with extraordinarily high pressure, density, temperature and velocity are formed, creating an avenue for the body to pass. These streams are known as the shock waves, and in the case of a wedge or cone they occur at an oblique angle. 1 The properties of these shock waves have been described theoretically to a high degree of accuracy, but because of the overall randomness of air (caused by its numerous, measurable properties), theory can only be considered a model or prediction, always varying slightly from what is witnessed in any real world or experimental situations. The first objective of this experiment is to apply the two dimensional model of oblique shock waves to a wedge, and from known conditions predict it's theoretical shock angle (β t ) and the pressure after the shock (P 2t ). These will be compared with the measured oblique shock angles (β e ) and post-shock pressure (P 2e ) in order to gauge the inaccuracy of the theoretical model. The second objective is to apply the two-dimensional theoretical oblique shock model to a cone (the threedimensional wedge equivalent) and discrepancies are expected. In this case, the magnitude of the discrepancies are the object of interest, seeking to gauge if (or to what degree of accuracy) the two-dimensional model can be applied to a three dimensional object. III. Setup and Procedure Below is a block diagram schematic of the high speed wind tunnel located in the Engineering Center at ASU. This wind tunnel is capable of generating Mach numbers of 2.4, 3.0, and 3.5, depending on which nozzle blocks are installed. Currently the Mach 2.4 nozzle blocks are installed. Also in the diagram is the schematic of the Schlieren imaging used to capture and visualize the density changes in the air. This is achieved using a series of flat and parabolic mirrors that pass light through the air in the test section, and capture what light remains on the other side. This technique is utilized because as the light passes through air at a higher density, much is blocked or absorbed, creating dark lines in the images. These dark lines represent the high density shock regions almost perfectly. In this case, the test section is roughly.18m wide and each image is approximately 680 pixels wide. Our image analysis software (Gimp) is accurate up to 1pixel, giving us an accuracy (in meters) of E-4m. Because of this, all measurements will be rounded to no more than four decimal places of accuracy. 2

3 Figure 1. Schematic of the Arizona State University supersonic wind tunnel and the Schlieren Imaging setup used to visualize the air densities in this experiment. Descriptions of the main components are given in the schematic. The angles in the Schlieren images were measured using the open source photo software, Gimp. Using the angle tool, the object geometries, the shock angles, and the Mach angles were measured to the accuracy of one pixel. The images and their corresponding angle lines and measurements are in the results section, labeled Fig. 3 - Fig. 10. Various equations (listed below) were used in the calculations of both experimental and theoretical values. Equation 1 (Eq. 1) is used to calculate the actual incoming Mach number based off of the angle of the Mach waves generated by thin pieces of tape that "tripped" the flow. The Mach number calculated is considered accurate and is used as a known variable in subsequent analysis. Equation 2, known as the Θ-β- M relation, and Equation 3 were coded into a program used to calculate the theoretical shock angle (β t ) and the experimental post-shock Mach number (M 2 ). Equation 4 is used to find the number normal to the shock wave located after the shock (M n2 ) which was then used in Equation 5 with the average stagnation pressure (method for obtaining the average found in Fig.2 below) to find a theoretical value of the pre-shock static pressure (P 1 ). Finally, P 1 is used in Equation 6 to obtain a theoretical value of the post-shock pressure (P 2 ). 3

4 Equations: M!! = M! sin β! Eq. 1 Eq. 2 Eq. 3 M!! = M! sin β! Eq. 4 P! =!!"!!!!!!!!!!!!!! Eq. 5 P! = P! 1 +!!!!! M!! 1 Eq. 6 Fig 2. Example Graph of Static and Stagnation Pressure VS Time for 10-Degree Cone. Averages were taken in the region indicated (in each case this was the stable and linear region) to obtain values of the stagnation pressure before the shock (P 01 ) and the experimental static pressure after the shock (P 2e ) 4

5 IV. Results Fig Wedge Full Angle Measurement Fig Wedge Shock Angle Measurements Fig Cone Full Angle Measurement Fig Cone Shock Angle Measurements Fig Wedge Full Angle Measurement Fig Wedge Shock Angle Measurements 5

6 Fig Cone Full Angle Measurement Fig Cone Shock Angle Measurements 10 Wedge 10 Cone 20 Wedge 20 Cone Measured Full Angle Half Angle (Θ) Measured Shock Angle (β e ) Measured Mach Angle (µ) Incoming Mach Number (M 1 ) Figure 9. Summary of All Angle Experimental Measurements and Calculated Half Angle M 1. M 1 was calculated using the equation: M 1 = 1/sin(µ); The angle measurements were taken using the open source, analytical photo software, "Gimp." 6

7 Fig 10. Program output for various inputs of M 1 and Θ Geometry, Θ µ M 1 β t M 2 β e β % error Wedge, % Cone, % Wedge, % Cone, % Figure 11. Summary of the theoretical calculations of β, M 1,M 2 10 Wedge 10 Cone 20 Wedge 20 Cone Experimental P 01 (PSIG) Experimental P 2 (PSIA) Theoretical P 2 (PSIA) P 2 % Error 13.7% 2.2% 13.2% 42.3% Figure 12. Summary of theoretical and experimental pressures, along with the error of P 2. 7

8 V. Conclusion When the half angle of the object was kept small, the theoretical VS experimental error also remained small. The 10 wedge had some error, but was within a reasonable value of the calculated shock angle, showing that the theoretical model is within 10% in this case. The cone and wedge with the larger half angles, however, strayed very far from theory. This was expected for both cones, but not for the wedge. In the case of the 20 wedge, the geometry abruptly changes from angled, to horizontal with the x axis. This causes noticeable expansion waves and is believed to be the cause of the error. In the case of the 10 cone, the error was actually much less than expected. It is widely accepted that the two dimensional flow model does not accurately depict a three dimensional object because of the flow that occurs in the z direction (out of the picture) that circulates around the object. The source of this accuracy is contributed to the small angle and lack of flow in the z direction, thus allowing it to be modeled in two dimensions. For both the pressure and the shock angle, the calculation of M 1 likely contributed to the error. The pressures just after the shock (P 2 ) were all within a reasonable range of the theoretical value, except for the 20 cone, which was expected due to the two dimensional modeling limitation. The two wedges were both within 14% of the theoretical model, showing some discrepancies and indicating some experimental or measurement errors. The 10 cone was the most accurately modeled, which again, was unexpected due to it's three dimensional geometry, however the same assumption made for the shock wave angle is used to explain this unexpected result: The small angle created very little flow in the z direction, allowing the 10 cone to be modeled as a wedge. If I was going to improve this experiment, I would install a pitot tube in the test chamber along with the other pressure sensors in order to obtain a true value for M 1. Hopefully this would increase the accuracy of the numbers entered in the known theoretical equations. The analysis of this experiment (mainly that done on the Shlieren images) allowed for a great deal of insight on how and why an oblique shock occurs, and why it acts as it does (it's strength and it's angles.) Also, the theory was followed to the best of my abilities, however there was still much error. I know now to trust theoretical equations as guidelines, but not as accepted truth. They are just predictions based off of compiled observations. VI. References 1 Fox, Robert W., and Alan T. McDonald. Introduction to Fluid Mechanics. New York: J. Wiley, Print. 2 Anderson, John D., Fundamentals of Aerodynamics, Normal Shock Waves and Related Topics, 5 th ed., New York, NY,