Test Methods and Conditions
3.2 Test Series
3.2.1 Tunnel Operation
“Tailored interface” operation6 of the shock tunnel was proposed by Wittliff et al.
(1959). Tailored operation of a shock tube means selecting operating conditions so that the gas properties on each side of the contact surface, the interface between driven and driver gas that is immediately behind the shock wave, such that the inter- action between the interface and the reflected primary shock wave does not generate a reflected wave. “Undertailoring” refers to conditions where the reflected wave is an expansion, while “overtailoring” refers to conditions where the reflected wave is a shock. Reddy (1971) derives an analytical expression for the tailored (denoted by subscript T) shock Mach number at this condition, which assumes perfect driver gas but real driven gas in the shock tube:
MsT = a4 a1
2
(γ4−1)(1−ε)
β41/2 β41/2+ (α4ε)1/2
(3.30)
In this equation, α ≡ (γ + 1)/(γ −1), β ≡ (γ − 1)/2γ, and ε = ρ1/ρ2, where state 4 is the driver gas, state 1 is the driven gas, and state 2 is the post-shock state. ε may be determined with a real gas solution for a given Ms and P1, such as that provided by the PostShock eq routine from the Cantera (Goodwin, 2003;
Goodwin, 2009) Shock and Detonation Toolbox (Browne et al., 2008). For each shock tube composition and fill pressureP1, Equation (3.30) may then be tabulated to find
6In practice, the conditions selected in the present study are actually slightly undertailored; see Section 2.3.
a4/a1, as presented in Figure 3.11 for the four gas mixtures used in the present study, calculated for P1 = 100 kPa. P4/P1 is measured and T4/T1 is computed based on adiabatic compression of the driver gas. When the driver gas composition is specified7 the sound speed in the driver gas,a4, may also be calculated. The sound speed in the driven gas, a1, is calculated from the known initial fill conditions of the shock tube.
Figure 3.12 presents the pressure ratio P4/P1 necessary to produce a given reser- voir enthalpy for three values of P1 for each of the four shock tube gas mixtures.
Figure 3.12(a) is calculated for a 100% He driver, used in high-enthalpy cases, and Figure 3.12(b) is calculated for an 84% He, 16% Ar (by mass) driver, commonly used for midrange enthalpy cases, including many in the present study. For each P4/P1, there is one value of a4/a1 which results in tailored operation. While it is often in- convenient to adjust the compression ratio to change P4/P1, tailored operation may be achieved over a range of reservoir enthalpies by adjusting the ratio of Ar to He in the driver gas.
0 5 10 15 20
0 5 10 15
a 4/a 1 (Real Driven Gas, Ideal Shock Tube)
MsT (Tailored) Air
CO250%
CO2100%
N2
Figure 3.11: a4/a1 vs. Ms for ideal tailored operation of a reflected shock tunnel
In practice the shock speed is observed to decrease with increasing distance from
7We have thus far assumed onlyγ= 5/3 for the driver, true for both He and Ar, but to calculate the speed of sound the molecular weights are necessary.
0 5 10 15 20 100
101 102 103 104 105
P 4/P 1 (Real Driven Gas, Ideal Shock Tube)
hres [MJ/kg] (Tailored) 1 kPa P1
10 kPa P1
100 kPa P1
Air CO250%
CO2100%
N2
(a) 100% He driver
0 5 10 15 20
100 101 102 103 104 105
P4/P1 (Real Driven Gas, Ideal Shock Tube)
hres [MJ/kg] (Tailored) 1 kPa P1
10 kPa P1
100 kPa P1
Air CO250%
CO2100%
N2
(b) 84% He, 16% Ar driver
Figure 3.12: (a)P4/P1 vs. hresfor ideal tailored operation, 100% He driver; (b) P4/P1 vs. hres for ideal tailored operation, 84% He, 16% Ar driver. All cases are calculated for four driver gas mixtures at shock tube fill pressure valuesP1 of 1, 10, and 100 kPa.
the primary diaphragm due to the interaction between the boundary layer wave and the mean flow behind the shock (Hornung and Belanger, 1990). This is not ac- counted for by the idealized shock tube model used by Reddy (1971) in deriving Equation (3.30). For the T5 shock tube, L/d = 130, which makes these nonideal effects more important in T5 than in shorter reflected shock tunnels and further re- duces the velocity of the incident shock as compared to the ideal value, which results in a reduction of the reservoir enthalpy (Belanger and Hornung, 1994). Figure 3.13 documents the measured shock speed decrement in air for a set of experiments from the present study, all with 84% He, 16% Ar driver and burst pressure P4 ≈100 MPa.
The results are approximate due to the small number of pressure transducers available for determining arrival time and inferring shock speed. The shock speed measured between the last two transducers in the shock tube is about 92.5% the shock speed calculated fromPostShock eqin the Cantera Shock and Detonation Toolbox. Pre- vious results in T5 (Belanger and Hornung, 1994) indicate that the shock wave is continuously decelerating as it travels through the tube. In the present study, this effect is neglected and shock speeds based on shock arrival times at the last two sta-
tions at 4.8 and 2.4 m from the end wall are used to calculate all reflected shock and reservoir conditions.
2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600
500 700 900 1100 1300
ShockSpeedUs[m/s]
P4(84%He,16%Ar)/P1(Air) Experiments Us(calc) 0.925Us(calc)
Figure 3.13: Shock speed decay in air for a set of experiments all with 84% He, 16%
Ar driver and burst pressure P4 ≈ 100 MPa. The shock speed measured between the last two transducers in the shock tube is about 92.5% the shock speed calculated fromPostShock eq in the Cantera Shock and Detonation Toolbox.
Page and Stalker (1983) considered the effects of tunnel geometry and compression ratio (the ratio of initial to final volumes of the driver gas) on thePres/P4 ratio, finding that measured compression ratios for a given P4 were as much as two to three times higher than calculations based upon isentropic processes in the compression tube, and that losses for “plateau pressure” (Pres) correlated with increasing L/d of the shock tube across different facilities8. Page and Stalker (1983) did not consider the effect of tailored tunnel operation.
For the present set of slightly undertailored experiments, a correlation between Pres and P5 is observed, where P5 is calculated from the measured, not theoretical, shock speed Us. As Morgan (2001) observes, “In practice, free-piston shock tunnels have been found to deliver less reflected shock pressure than the theoretical values”.
This is illustrated in Figure 3.14. The average Pres/P5 ratio for the air tests is 0.63,
8Belanger and Hornung (1994) observed a similar effect in T5, withPres/P4averaging about 0.7 for conditions withhres<12.5 MJ/kg, but it is unclear if the empirical correlation provided in Page and Stalker (1983) holds for all cases.
0.62 for N2, 0.41 for 50% CO2, and 0.34 for 100% CO2.
0 50 100 150 200 250 300
10 20 30 40 50 60 70 80
P5 [MPa]
P res [MPa]
100% CO2
50% CO2 Air N2
Figure 3.14: Pres vs. P5 for the present set of transition experiments.
Using the empirical Belanger and Hornung (1994) average value of 0.7 for Pres/P4 for T5 conditions with hres < 12.5 MJ/kg, and assuming a typical T5 driver of 84%
He, 16% Ar, the relationships presented in Figure 3.12(b) for tailored P4/P1 can be used to find the dependence of Pres onhres and fill pressure P1 for a given shock tube gas mixture:
Pres = Pres P4 P1P4
P1 ≈0.7P1P4
P1 (3.31)
While this relationship should correctly portray the general trend of reservoir pressure and enthalpy increasing together under tailored conditions when initial fill pressure and composition are held constant, it is not a reliable quantitative prediction for the relationship, which is highly empirical. In particular, the modeled Pres-hres curves calculated for several values of P1 in Sections 3.2.2–3.2.4 below would in prac- tice require different and facility-dependent values for P1 to produce the equivalent conditions. This is discussed in Hornung and Belanger (1990); Morgan (2001) sum- marizes the situation as follows: “Real gas effects in the test gas do not change the performance insofar as the post-shock pressure is concerned. However, they seriously alter the filling pressure (P1) required to achieve the required flow speed.” The con-
sequences for T5 are shown by comparing the modeled and experimental Pres-hres results in Sections 3.2.2–3.2.4.
3.2.2 Air
Transition onset was observed for a total of 26 experiments in air, which had reser- voir enthalpies between 5.3 MJ/kg and 11.9 MJ/kg and reservoir pressures between 16.5 MPa and 72.0 MPa. Equation (3.31) is used with the air values for P4/P1 from Figure 3.12(b) to plot an empirical relationship forPresvs. hresfor four shock tube fill pressures (25, 50, 100, and 200 kPa). This is presented in Figure 3.15(a), next to the tunnel parameters for each air experiment presented in Figure 3.15(b). These tunnel conditions resulted in boundary layer edge pressures between 9.1 kPa and 59.6 kPa and Dorrance reference temperatures between 1120 K and 2180 K.
2 4 6 8 10 12 14
0 20 40 60 80 100 120
hres (modeled) [MJ/kg]
P res (modeled) [MPa] P1= 25 kPa
P1= 50 kPa P1= 100 kPa P1= 200 kPa
(a) 84% He, 16% Ar driver,P1 air
2 4 6 8 10 12 14
10 20 30 40 50 60 70 80
hres [MJ/kg]
P res [MPa]
Experiments
(b) Experimental air conditions
Figure 3.15: Tunnel parameters for air (a) modeled for several shock tube fill pressures P1, all with a 84% He, 16% Ar driver; and (b) as calculated for the present air experiments.