Flow in a shock tube

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Scot Baker
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1 Flow in a shock tube April 30, 05 Summary In the lab the shock Mach number as well as the Mach number downstream the moving shock are determined for different pressure ratios between the high and low pressure side of the membrane. The speed of the shock is determined by measuring the time needed for the shock to move a certain distance along the tube. The Mach number of the flow behind the moving shock is determined from the inclination angle of the stationary oblique shock caused by a wedge inserted in the test section of the shock tube. Experimental values are compared with theoretical values based on idealized shock tube flow. Experimental rig and test equipment The set-up of the experiment is sketched in figure. The total length of the shock tube is approximately.5 m. The driver section where the pressure is high <0 bar and the driven section where the pressure is low ( mbar) are separated by a circular membrane of aluminum. In the entrance of the driven section, there is a cross of curved knives onto which the membrane is forced as the pressure difference between the sections is increased. At a given pressure difference the membrane bursts and a shock wave is formed which moves into the low-pressure gas. The opening time of the membrane is short, 40 ± 0 µs. The driving pressure is changed by varying the thickness of the membrane (0.3 mm < d < 0.9 mm). Figure shows the relation between membrane thickness and bursting pressure.the variation in pressure at which the membrane bursts is very small for a given membrane thickness, only about %. Downstream the membrane the circular cross-section (diameter 8 cm ) changes its shape to a rectangular cross-section (5 0 cm, height width) whereas the area is constant. The transforming section is 0.3 m long. The gas supplied to the driver section comes from a highpressure gas bottle with a pressure reduction valve. The pressure in the driver section is limited to 3 bar (safety reasons) and in the driven section the pressure should be kept below atmospheric pressure before the shot. The driven section ends in a dump vessel used to damp out shock reflections and limits the final pressure in the the shock tube after a shot. A pressure transducer inside a pressure indicator unit (Druck DPI 50) is connected to the driver section. Through the hold-max function of the unit the pressure value at the time of firing is displayed on the pressure indicator. The gas in the driven section is evacuated by a two-stage vacuum pump and the lowest pressure possible is about 0.0 mbar. The pressure in the driven section is monitored in similar way as the pressure in the driver section. A second pressure indicator unit (Druck DPI 50) continously displays the pressure in the driven section.

2 KTH Mekanik E G L M P B C D O J J A F Q A. High-pressure gas bottle B. Manometer (differential) C. Outlet valve D. Pressure gauge E. Voltmeter F. Needel valve G. Manometer (absolute) H. Oscilloscope I. Time delay unit J. Temperature sensor with preamplifier K. Computer L. Mercury lamp M. Schlieren optics N. CCD camera O. Pirani gauge P. Vacuum instrument Q. Vacuum pump H I N M K Figure : Shock tube and measurement equipment. Opening pressure (atm) mm Membran thickness Figure : Opening pressure as a function of membrane thickness. After evacuating the air in the driven section, ambient air is let in through a small vacuum valve and the pressure is adjusted to the right level. During pumping, the temperature of the gas changes in accordance with figure 3. It is thus necessary to wait about 5 minutes until a temperature balance is established and the speed of sound a in the the gas is known. The driven section of the tube is equipped with a number of shock sensors mounted in the

3 KTH Mekanik 3 Temp. ( C) Time (min) Figure 3: Gas temperature as a function of time during the evacuation phase. side wall. The sensing element is a glass plug with a thin stripe of platinum painted on the end surface of the plug. The surface is mounted flush with the inner side of the tube. When the shock passes over the glass surface the resistance of the platinum film is changed and transformed via an electric circuit to a voltage pulse. Due to the low heat capacity of the film, the sensor is very fast and the response time is less than µs. Two sensors (no and ), 5 cm apart, are connected to a digital storage oscilloscope. The time for the shock to move from sensor to is determined from the oscilloscope traces. The test section is located downstream of the sensors and contains a wedge with an angle of 6. The test section has vertical glass windows to facilitate optical access and the edge of the wedge is aligned horizontal and normal to the glass surfaces. The lower surface of the wedge is parallel with, and cm above the lower wall of the tube. To visualize the oblique shock at the wedge schlieren technique is used. A continuous light source, a LED lamp, is used and the shock system is recorded by a CCD camera. The camera is triggered by sensor but the signal is delayed by a variable time delay unit to be able to take a photo at the appropriate time. Figure 4 shows the electric circuit diagram of the set-up. 3 Theoretical calculations The theoretical calculations are based on idealized shock tube flow. This incorporates the following assumptions:. Ideal gases in driver and driven sections.. Instant opening of the membrane. 3. Flow without friction and heat conduction. An x t diagram of the flow is shown in figure 5. The figure is restricted to times before the waves are influenced by the end walls. The double-line between region and shows the motion of the shock and the dashed line between and 3 is the motion of the contact surface between hot and cold gas determined by equal velocity and equal pressure on both sides. Region 3 contains the isentropic expansion fan. The angle of each characteristic i.e. the propagation velocity matches the local speed of sound minus the gas velocity.

4 KTH Mekanik 4 Temp. sensor 3 Temp. sensor 4 Preamplifier Preamplifier Oscilloscope CCD Camera Time delay unit Figure 4: Schematic of the electric circuit. From boundary conditions stated above a relation between the shock Mach number M s and initial pressure ratio (p 4 /p ) over the membrane could be expressed: p 4 /p = γ M s (γ ) γ + [ γ 4 a γ + a 4 (M s Ms )] γ 4 γ 4. The function is plotted in figure 6 for helium or air as driving gas and air in the low-pressure section. In all cases, the same gas temperature is assumed. The Mach number behind the moving shock, M = u /a, is determined using normal shock relations. The continuity equation gives: ρ c s = ρ (c s u ), where c s is the velocity of the shock, gives the velocity behind the moving shock: ( u = ρ ) c s. ρ The relation between velocity of sound at both sides of the shock is obtained from: T a = a. T Using the temperature and density relations from the table of straight shock relations at a given M s gives the sought M.

5 KTH Mekanik 5 4 Experiments and evaluation of data In each experiment the time between the sensor pulses (the elapse time) as well as the pressure in the driver and driven sections of the tube at the bursting instant are registered. At the lowest pressure ratio, also a photo of the standing oblique shock is taken. The temperature of the different gases is assumed to be equal to room temperature. Shock velocity and pressure ratio are determined from these data. The camera is trigged by the signal from sensor. The signal has to be delayed until the flow over the wedge is stationary. In figure 5 the time between the shock passes the sensor and the wedge is denoted t and the time between the shock passes the sensor and the contact surface reaches the wedge is denoted t. From the figure we hence obtain the following relation: t = l c s, t = l u l l c s In the figure, l = 55 cm and l = 0 cm. The time delay is chosen between t and t. From the pictures taken, the angle of the oblique shock is readily found and hence M from the θ β M diagram. t 3 t 4 t l l X Sensor Figure 5: x-t diagram of the flow.

6 KTH Mekanik 6 5 Discussing the assumptions Ideal gas The assumption of ideal gases is probably relevant at the low Mach numbers (M s < 3) used in the lab exercise, because only minor variations in the heat capacity will occur. Instant membrane opening The time it takes until stationary shock speed is reached depends on the membrane opening. The membrane opening also influences the character of the contact surface. In the lab exercise, the shock wave is fully developed and plane when it passes the sensors and reaches the wedge. But the time delay due to the finite opening time causes the shock curve in the x t diagram to be shifted upwards and some precautions has to be taken in the setting of the spark delay time. We also have a to consider the mixing in the contact zone due to an imperfect membrane opening and thermal gradients which decrease the time of stationary flow around the wedge. Boundary layer formation Behind the shock, the gas temperature and velocity are different compared to the situation along the walls and hence boundary layers start to develop as soon as the membrane bursts. The boundary layer development causes the shock to attenuate, the contact surface to accelerate and the available time for which the flow is constant to diminish. However, these effects are of minor 0 M s a =0.34 (T /T 4 = High pressure: He Low pressure: Air) 4 3 a =0.9 Air a =.0 Air a =. Air Log(p 4 /p ) Figure 6: Shock Mach number M s as a function of the pressure ratio p 4 /p.

7 KTH Mekanik 7 importance as the shock Mach number is low and the tube is short in our case. 6 Presentation of the results The protocol should contain the following: Registered data:. Pressures (p, p 4 ). Temperature of the gases (T, T 4 ) 3. Elapse time between the two sensors ( t) Calculated values:. Delay times (t, t ).. Pressure ratios (p 4 /p ) 3. Shock Mach numbers (M s ) The calculated values should be plotted in the curve M s = f(p 4 /p ) handed out. Along with the pictures taken of the oblique shock, you should give the following data:. Time delay used for the camera.. Measured oblique shock angle (β). 3. Calculated Mach number behind the moving shock (M ).

A & AE 520 Background Information, Adapted from AAE334L, last revised 10-Feb-14 Page 1 1. SUPERSONIC WIND TUNNEL

A & AE 520 Background Information, Adapted from AAE334L, last revised 10-Feb-14 Page 1 1. SUPERSONIC WIND TUNNEL A & AE 50 Background Information, Adapted from AAE334L, last revised 10-Feb-14 Page 1 1.1 BACKGROUND 1. SUPERSONIC WIND TUNNEL 1.1.1 Objectives: This handout is adapted from the one once used in AAE334L