Supersonic Flow and Shockwaves

AER 304S Aerospace Laboratory II Supersonic Flow and Shockwaves http://sps.aerospace.utoronto.ca/labs/raal Experiment Duration: 150 min Instructor M. R. Emami Aerospace Undergraduate Laboratories University of Toronto Winter 008

1. Purpose Some basic concepts of supersonic flow are demonstrated using a nominal Mach 1.6 wind tunnel. Impact and static pressure probes are employed to monitor velocities in the tunnel at various points along the flow. A Schlieren camera system is used for the examination of shock waves from objects placed in the test section. A computer-aided data acquisition system is used to collect and record pressure data from the impact and static pressure probes.. Apparatus Half section supersonic tunnel (M = 1.6) with fixed static pressure taps. Traversing Pitot impact probe and knife-edge model. Whispair blower, 1. kj/s at 350 rpm (ΔP = 70 kpa). Axial cooling fan for Whispair blower. Intake filter/silencer. Exhaust muffler. Motorized intake flow ball valve. Probe manipulation system for vertical and longitudinal motion. Honeywell absolute pressure transducers model #14PC15A (0-775 mmhg). 4v Solenoid valve manifold (SMC Pneumatics SY100) Wallace & Tiernan dial pressure gages (0-400 and 400-800 mmhg). Schlieren camera system. Test section aperture 5 mm by 30 mm. Data acquisition system (PCIM-DAS160/16 PCI Board). Power supplies and power control unit. Web cams and audio system. 3. Notation and Constants a speed of sound (m/s) A cross-sectional area (m ) A* throat area (m ) γ specific heat ratio (γair = 1.4) M Mach number (V/a) n refractive index p pressures (mmhg) p 1 static pressure (mm Hg) p 0 Pitot tube stagnation pressure (mmhg) ρ density (kg/m 3 ) μ viscosity R ideal gas constant = 87 J/kg K (dry air) T temperature (K) U flow velocity (m/s) V flow velocity (m/s) 1

4. Experiment Setup The major elements which make up the aerospace undergraduate laboratory supersonic facility are indicated in Fig. 1. The tunnel operates as open circuit, meaning air is drawn from the laboratory and exhausted outside the building. 4.1 Wind Tunnel Figure 1: Aerospace Laboratory Supersonic Facility (M = 1.6) The power required to run a wind tunnel scales roughly as the third power of the flow velocity. This factor is reflected in the test section dimensions which for the supersonic facility are small compared with the other laboratory tunnels. However, the 0 HP (15 kw) motor required for starting and running at supersonic speeds is higher powered than either of our other tunnels, although the latter have much larger test sections. The design of a supersonic tunnel can be tricky, mainly due to viscous effects during the starting process. Compression ratios required to start a high Mach number tunnel are usually at least twice the normal shock pressure ratio and even at M = 1.5 approximately 30% extra power is required. The transient starting phenomena are difficult to evaluate theoretically, so a good mix of empiricism, experience, and some luck is needed in the design and manufacturing of a good tunnel. The laboratory tunnel incorporates a supersonic nozzle contour which has been designed using methods of characteristics [1]. The nozzle coordinates that are listed in Table 1

include an empirical correction for boundary layer growth, which is an important factor at high speeds. The contoured tunnel floor is slotted and sealed to permit the insertion of a traversing Pitot tube. The roof of the test section is plane and corresponds to the centerline of a hypothetical symmetrical tunnel of twice the height. Table 1: Static Pressure Tap Locations Tap y h x (mm) A/A * A * M /A Number (mm) (mm) (theory) 0.0 3.45 6.55 1.58 0.63.405 11.6 8.86 1.14 1.6 0.794.545 1 4.7 1.17 17.83 1.06 0.94.755 36.6 13.1 16.79 1.00 1.000 1.00 3 48.3 1.37 17.63 1.05 0.95 1.6 4 61.0 10.69 19.31 1.15 0.869 1.46 5 73.7 9.40 0.60 1.3 0.815 1.57 6 86.4 8.64 1.36 1.7 0.786 1.63 7 99.1 8.33 1.67 1.9 0.775 1.65 8 118.0 8.18 1.8 1.30 0.769 1.66 9 09.5 8.18 1.8 1.30 0.769 1.66 10 300.1 8.18 1.8 1.30 0.769 1.66 11 diffuser - - - - - Figure : Supersonic Nozzle Geometry Air is sucked through the Laval nozzle of the tunnel depicted in Figs. 1 and, but several other methods of flow drive may be used. For example, a closed circuit tunnel reduces the operating power requirement, permitting the pressure of the whole tunnel to be varied if desired; the necessity of continuously drying air is also eliminated for this design. In a blow-down tunnel, power requirements are reduced by charging a compressed or reduced pressure air storage tank which is then used to power the tunnel on an intermittent basis. These and other variants are discussed in [1]. 4. Data Acquisition System The data acquisition system consists of a Pentium IV workstation equipped with a 3

PCIM-DAS160/16 data acquisition (DAQ) board from Measurement Computing. The DAQ board has 3 digital I/O channels, and a 16 channel analog-to-digital (A/D) converter with 16-bit resolution (i.e. 16 discrete voltage values over the measurement range of 0 to 10V), which is equivalent to a voltage resolution of 0.15 mv. The DAQ board allows the computer software to control electromechanical actuators and collect data from a variety of analog and digital sensors. The position of the flow intake valve (i.e. the percentage that the valve is open; 100% corresponding to fully open) is set through a computer-controlled DC gear-head motor. The position of the valve is measured using a rotary potentiometer that converts the valve position to a voltage that can be read by the DAQ board. Therefore, to move the valve to a desired position the software turns on the motor with a direction corresponding to opening or closing the valve, while simultaneously reading the valve potentiometer voltage. Once the voltage corresponding to the desired valve position is reached the software turns off the motor. The pressure from the impact and static pressure probes is measured using pressure transducers that convert pressure values to a voltage that can be read by the DAQ board. Each pressure probe can also be connected to the Wallace & Tiernan dial gages through an array of computer-controlled solenoid valves. 4.3 Probe Manipulator The position of the impact pressure probe (Pitot tube) in the tunnel test section is changed using a computer-controlled manipulation system that has degrees of freedom, namely x (horizontal) and y (vertical) translation. The manipulator consists of linear stepper motors to position the probe with an x-resolution of 0.006096 mm and y-resolution of 0.00075 mm. Limit switches are used for homing the actuators. The computer software can control each component of the experiment individually, such as in the case of moving the probe to a desired position or it can control multiple components simultaneously allowing complex dynamic experiments to be performed. The computer control system enables the experimenter to not only collect data more accurately, but also perform multiple tasks simultaneously. 4

5. Experiment User Interface 1. Start or stop the wind tunnel. There is a 60-second delay between starting and stopping the tunnel.. Exit the experiment. 3. Turn on or off the light for the Schlieren camera system. 4. Save a snapshot image from the Schlieren camera. 5. Start or stop recording a video from the Schlieren camera. The maximum length is 60 seconds. 6. Move the probe in incremental steps. The number box indicates the incremental size of each step in millimetres. Each button will move the probe one incremental step in the indicated direction. 7. To move the probe to a specific coordinate, enter it in the X and Y number boxes and click Move. 8. Return the probe to the Home position. 9. Animation of the current probe position in the tunnel test section. 10. History of previously executed commands. 11. Live pressure data from the pressure transducers. Select pressure channels to display on the pressure graph using the Graph checkboxes (see 18). 1. Start or stop updating the live pressure data. 13. Save the current live pressure data. 14. Select the pressure transducer to connect to the Wallace & Tiernan dial gages (currently disabled). 15. Select the current pressure units; mmhg, psi, or kpa. 16. Open or close the intake valve. The + and buttons open and close the valve in small increments. 17. To move the valve to a specific position enter the value in the number box and click Move. 18. Animation of the current valve position. 19. Select between the Experiment and Pressure Graph control panels. 5

1. Save the static pressure measurement at each tap along the tunnel centreline (roof).. Select the pressure tap (port) location where the vertical impact pressure measurement is taken. 3. Select the number of points to measure in the vertical impact pressure measurement. 4. Record the vertical impact pressure measurement at the selected tap (port) location. 5. Select the height of the impact tube for the horizontal impact pressure measurement. 6. Record the horizontal impact pressure measurement at the selected impact tube height. 7. Select the position of the valve to start the flow variation pressure measurement. 8. Select the position of the valve to stop the flow variation pressure measurement. 9. Record the flow variation pressure measurement while moving the valve from start to stop. 10. Select the static pressure tap (port) for which the MATLAB file. 11. Select the static pressure measurement dataset for the MATLAB file. 1. Select the vertical impact pressure measurement dataset for the MATLAB file. 13. Select the horizontal impact pressure measurement dataset for which the MATLAB file. 14. Display the help dialog for creating the MATLAB file. 15. Create the MATLAB file for the selected tap (port) and datasets. 16. Select the number of samples to take the average over for each pressure measurement. 1. Start or stop the dynamic pressure graph.. Move the pressure scale up or down. 3. Zoom the pressure scale in and out. 4. Dynamic pressure graph display area. Select the channels to display from Pressure Data box. 6

6. Supersonic Flow Theory Supersonic flow theory is extensive, and perhaps the best text for a first appreciation of the topic is [], which provides adequate information for the purpose of this experiment. A more complete and lucid look at the field in somewhat greater depth is given in [3] (an excellent text). It is recommended that the students consult with at least one of these sources for background material. In the nozzle shown in Fig. 3 sonic flow is assumed to exist at the throat where conditions are denoted by the asterisk. Figure 3: Supersonic Wind Tunnel Nozzle The real flow for this case is nearly isentropic because only small quantities of heat are exchanged through the nozzle walls, and the flow is assumed to be frictionless. The following continuity relation, Equation (1), is necessary and leads to several other useful formulas which are listed in Fig. 4. ρ u A = ρ ua (1) 7

Figure 4: Variation of M, P, T, ρ through a Supersonic Nozzle 6.1 Area-Mach Number and Area-Velocity Relations A A = γ + 1 1 1 1 γ γ 1 + M M γ + 1 () Equation () is called the Area-Mach number relation and leads to the remarkable consequences that since M = f(a/a * ) and A/A * 1: a) For subsonic situations: M increases as A/A * decreases (i.e. the nozzle converges). b) For M= 1, A/A * = 1, sonic conditions prevail at the throat. c) For supersonic situations: M increases as A/A * increases, (i.e. the nozzle diverges). The somewhat counter-intuitive conclusion which is enunciated by (c) may be appreciated more readily by the examination of the Area-Velocity relation: 8

da A dv ( M 1) V = (3) In this expression it is evident that increases in velocity follow automatically from an area da enlargement. Also, if A = 0 the nozzle has a minimum area (the throat) and at that location M = 1. 6. Determination of Isentropic Flow Properties in Nozzles 6..1 Use of the Table of Isentropic Flow Properties A MATLAB file can be generated from the experiment user interface which computes the predicted values for M listed in Table 1, using the tabulated A/A* data from the same table. 6.. Tunnel Flow Measurements (Supersonic Regions) In Fig. 5 the Pitot tube measures the stagnation or total pressure behind the shock. In this diagram p 1 corresponds to the static pressure in a plane which is tangent to the shock. Figure 5: Pitot Impact Tube in Supersonic Flow The following expression, known as the Rayleigh Pitot Relation is usually solved recursively to obtain M once p 1 and p 0 are known. p p 0 1 = γ + 1 M γ M γ + 1 γ γ 1 γ 1 γ + 1 1 γ 1 = 167M 5 ( 7M 1) (4) 9

6..3 Tunnel Flow Measurements (Subsonic Regions) Equation (4) cannot be used for the determination of velocities upstream of the throat since a shock wave cannot exist at these locations. The following expression from Fig. 4 should be used for such cases: p p 0 1 = γ γ 1 1 γ M 1+ M = + 1 5 7 (5) 6..4 Optical Methods for Gas Dynamic Analysis In fluids and solids, pressure and density changes propagate at the velocity of sound in the medium. For an ideal gas the sound velocity is: a = γ RT (6) Shock waves formed about a body in a supersonic flow are created because disturbances at the body surface cannot propagate upstream since the maximum propagation speed is limited to the local speed of sound. This principle is illustrated in Fig. 6, where the crosses represent disturbances traveling at a velocity U in each case. Figure 6: Evolution of a Shock Wave Shock waves are thin (about 0.0001 cm) but special diagnostic techniques are available for visualizing the variations in fluid density which accompanies shock formation. The local change of refractive index due to gas compression interferes with the transmission of an illuminating beam, and this provides a visual manifestation of the shock. Interferometers, Schlieren systems, and shadowgraphs are complementary methods to monitor the gas density ρ, the first derivative of gas density, and the second derivative, respectively. A preliminary description of the Schlieren method will be given but a discussion of other methods is beyond the scope of this manual. Their operating principles are interesting and are well covered in [4]. 6..5 The Schlieren Method 10

Whenever there is a change in the local fluid density a concomitant change in the optical refractive index (n) is observed (remember n = 1.0 only for a vacuum). In Fig. 7(a) a uniform fluid density is assumed to be present throughout the disturbance region, and the resulting deflection of a light beam is shown by the solid line OP. Figure 7: Light Refraction at a Disturbance A Schlieren system responds to the first derivative of the density as implied in Fig. 7(b) where the wedge density increases linearly with distance y. The Schlieren apparatus for this experiment, shown schematically in Fig. 8, uses lenses. However, for larger systems with long focal lengths mirrors are customarily employed, since these are less expensive than lenses for comparable size and optical performance. Figure 8: Schlieren Optical System In Fig. 8 a light source S is imaged by the lens L 1 onto aperture A, which serves to define the source and eliminate any spurious light due to reflections from the source envelope. Lens L collimates the light which then passes through the test section and is focused by L 3 in the plane of a knife edge KE. The knife edge is adjusted so that in the absence of any disturbance in the test section it just occludes all the radiation that would normally pass to the viewing screen. A shock wave or similar perturbation of the fluid density in the test section may then cause light to pass around KE as explained in Fig. 7. If the location of the screen is chosen such that it displays a sharp image of the test section via L 3 and the previous conditions have been met, then shock waves are readily observable. 11

An example where the Schlieren technique has been used to examine the flow conditions around a double wedge is shown in Fig. 9. Intuitively one would suspect the shock angle depicted in Fig. 9 to be proportionally related to the flow velocity, and indeed this is the case. The θ, β, M relation, given in the figure, can be used to determine the flow Mach number, where θ, β, M are defined as indicated. Figure 9: Shock Waves with a Wedge 1

7. Experiment Design Some preparation and research will be required to design your experiments prior to actually performing the tests in the wind tunnel. Each experiment should not be viewed as an independent activity. The results of one experiment may prove useful in defining the parameters of another test. A MATLAB file can be generated by the interface to compute both theoretical and experimental Mach numbers, but you need to complete collecting data for all tests before running the file. The tests to complete for creating the MATLAB file are: 1. Static Pressure Measurement: To save the static pressure (p 1 ) along the tunnel centreline (roof) use the Static Pressure Measurement box under the Experiment tab. Enter the number of samples to read from the pressure transducer at each tap in the Samples box.. Horizontal Stagnation Pressure Measurement: To record the stagnation pressure (p 0 ) at each tap location along the tunnel at a fixed height use the Impact Tube Horizontal Pressure Measurement box under the Experiment tab, and enter the Pitot tube height (30 mm corresponds to the tunnel centreline). Enter the number of samples to read from the Pitot tube pressure transducer at each tap location in the Samples box. 3. Vertical Stagnation Pressure Measurement: To record the stagnation pressure (p 0 ) at a tap location along the tunnel at different heights use the Impact Tube Vertical Pressure Measurement box under the Experiment tab, and enter the tap (port) location and the number of transverse points at which the stagnation pressure is measured. Enter the number of samples to read from the impact tube pressure transducer at each vertical point in the Samples box. Flow Variation Pressure Measurement: To record the pressure from the impact and static pressure probes, while opening or closing the flow intake valve, use the Flow Variation Pressure Measurement box under the Experiment tab. Enter the starting position of the valve and the ending position of the valve over which the pressures will be recorded. This recording mode may take up to several minutes to complete if the valve is moved through its full range. Schlieren Camera Measurements: To record images or video from the Schlieren camera use the Schlieren Camera box on the experiment interface. The Image button saves a snapshot from the camera in JPEG format, whereas the Record Video button records a short video from the camera in AVI format. 7.1 Verify the Theoretical Mach Number Formulation Design an experiment to collect static and stagnation pressure data to compute the Mach number. Compare the results with the Area-Mach number relation (Equation ) and address the questions in Section 8.1. Static pressures are measured at taps 1-11 along the tunnel centerline (roof) that are connected to the pressure transducers. Stagnation pressure is measured using the Pitot impact tube that can be positioned at different 13

coordinates in the tunnel and can also traverse the tunnel horizontally at a constant height. The MATLAB file generated by the interface can be used to compute both the theoretical and experimental Mach numbers for this experiment, but you need to complete collecting data for all tests before running the file. 7. Determine the Vertical Mach Number Profile Design an experiment to collect static and stagnation pressure data to compute the Mach number profile across a vertical section of the tunnel and address the questions in Section 8.. Static pressures are measured at taps 1-11 along the tunnel centerline (roof) that are connected to the pressure transducers. Stagnation pressure is measured using the Pitot impact tube that can be positioned at different coordinates in the tunnel and can also traverse the tunnel vertically at a specific tap (port) location. The MATLAB file generated by the interface can be used to compute the experimental Mach numbers for this experiment, but you need to complete collecting data for all tests before running the file. 7.3 Determine the Effects of Flow Restriction Design an experiment to collect static and stagnation pressure data to determine the effects of flow restriction on Mach number, and to address the questions in Section 8.3. Stagnation pressure is measured using the Pitot impact tube that can be positioned at different coordinates in the tunnel. The flow is restricted by closing the flow intake valve (i.e. closing the valve reduces the intake cross-sectional area of the blower). Write a new MATLAB file to compute the Mach numbers for this experiment using the MATLAB file generated by the interface as a template. Note, MATLAB performs calculations using matrix algebra. You can calculate the Mach number for all data points simultaneously by entering the data as a vector and computing the Mach number for the entire vector. 8. Discussion of Results 8.1 Verify the Theoretical Mach Number Formulation 1. How and why do the static and stagnation pressure measurements vary with the height of the Pitot impact tube?. How does the number of samples per measurement affect the overall results? What is a suitable number of samples to have a statistically relevant measurement? 3. Is the Mach number computed from your measurements at sections far from the tunnel throat smaller or greater than the theoretical value? Why? 14

4. At which tap (port) location is the deviation from the theoretical Mach number maximal? Why? 5. Explain various sources that cause deviation from the theoretical value. 8. Determine the Vertical Mach Number Profile 1. How and why does the Mach number vary with the location of the Pitot impact tube along the tunnel (i.e. different tap locations)?. Discuss the variations in the vertical Mach number profile at tap (port) 7. 3. From the vertical Mach number profile characterize the boundary layer along the top and bottom walls of the tunnel. 4. Use dimensional analysis to show the tunnel power requirements scale as V 3. You may assume that the power required is directly proportional to the throat area A*, and power/area is a function of density ρ, viscosity μ, and velocity V. 8.3 Determine the Effects of Flow Restriction 1. Characterize how the Mach number at the throat and two other locations along the tunnel is affected by restricting the intake flow.. At what valve position and pressures does the flow become supersonic? Does this agree with theory? 3. Identify and explain each regime in a plot of Mach number vs. time at tap (port) 7, for opening and closing the valve. Is there a hysteresis? 4. What is the Mach number at the throat when the tunnel goes supersonic? Explain any discrepancies. 9. References [1] A. Pope and K. Goin, High-Speed Wind Tunnel Testing. New York, NY, USA: John Wiley and Sons, 1965. [] J. D. Anderson, Introduction to Flight. New York, NY, USA: McGraw-Hill Book Company, 1978. [3] J. D. Anderson, Fundamentals of Aerodynamics. New York, NY, USA: McGraw-Hill Book Company, 1984. [4] H. W. Liepmann and A. Roshko, Elements of Gas Dynamics. New York, NY, USA: John Wiley and Sons, 1965. 15