Improved test time evaluation in an expansion tube

Abstract

Traditionally, expansion tube test times have been experimentally evaluated using test section mounted impact pressure probes. This paper proposes two new methods which can be performed using a high-speed camera and a simple circular cylinder test model. The first is the use of a narrow bandpass optical filter to allow time-resolved radiative emission from an important species to be captured, and the second is using edge detection to track how the model shock standoff changes with time. Experimental results are presented for two test conditions using an air test gas and an optical filter aimed at capturing emission from the 777 nm atomic oxygen triplet. It is found that the oxygen emission is the most reliable experimental method, because it is shown to exhibit significant changes at the end of the test time. It is also proposed that, because the camera footage is spatially resolved, the radiative emission method can be used to examine the ‘effective’ test time in multiple regions of the flow. For one of the test conditions, it is found that the effective test time away from the stagnation region for the cylindrical test model is at most 45% of the total test time. For the other test condition, it is found that the effective test time of a 54$^\circ$ wedge test model is at most a third of the total test time.

A analytical test time evaluation equations

This appendix provides a short summary of the analytical test time evaluation equations from Paull and Stalker (1992). The reader is directed to their paper for the full derivation of the equations.

When the test time is terminated by the downstream edge of the unsteady expansion, the test time, $T_{\text {usx}}$, is the time between the arrival of the test gas/accelerator gas contact surface at the end of the acceleration tube and the arrival of the aforementioned downstream edge. The contact surface has a velocity of $V_{7}$, and the edge has a velocity of $V_{7}$–$a_{7}$, where state 7 is the state of the test gas after it has been processed by the unsteady expansion. This test time is as follows:

$$\begin{aligned} T_{\text {usx}} = \frac{x_{A}a_{7}}{V_{7}(V_{7} - a_{7})}, \end{aligned}$$

(A.1)

where $x_{A}$ is the length of the acceleration tube.

In the situation where the test time is terminated by the arrival of the reflected $u+a$ wave, the test time can be determined analytically as a function of the elapse time, $t_{0}$, between secondary diaphragm rupture and the arrival of the driver/test gas contact surface at the upstream edge of the secondary diaphragm centred unsteady expansion. An analytical equation for $t_{0}$ is given in Paull and Stalker (1992):

$$\begin{aligned} t_{0} = \frac{l}{V_{\text {cs}}-V_{2,0}} \ln \left( \frac{a_{2}+V_{\text {cs}}-V_{2,0}}{a_{2}}\right) , \end{aligned}$$

(A.2)

where l is the non-ideal separation distance between the shock tube shock and the driver/test gas contact surface when the secondary diaphragm ruptures from Mirels (1963, 1964), $V_{\text {cs}}$ is the non-ideal velocity of the driver/test gas contact surface from Mirels (1964), $V_{2,0}$ is the velocity ‘immediately after’ the shock tube shock from Paull and Stalker (1992) and Mirels (1963, 1964), which is just the ideal post-shock velocity in the shock tube ($V_{2}$), and $a_{2}$ is the post-shock sound speed in the shock tube.

Now, l can be found by first evaluating the maximum separation distance between the shock tube shock and the driver/test gas contact surface, $l_{m}$, from Eq. 2 in Mirels (1963):

$$\begin{aligned} l_m = \frac{d^2}{16\beta ^2}\left( \frac{\rho _{2,0}}{\rho _{w,0}}\right) ^2 \frac{V_{2,0}}{V_w - V_{2,0}} \frac{V_{2,0}}{\nu _{w,0}}, \end{aligned}$$

(A.3)

where d is the shock tube diameter, $\rho$ is density, $\beta$ is a constant found from solutions of the boundary layer development (with various different methods of calculating it found in Mirels (1963), with Eq. 17 from that paper used in this work), $\nu$ is kinematic viscosity, subscript w indicates conditions at the wall, and subscript 0 indicates conditions immediately behind the shock. Note that velocities in this equation are in a shock-fixed reference frame, unlike other equations discussed in this section. In this context, $V_w$ is the same as the shock velocity in a lab-fixed reference frame ($V_{s,1}$).

The parameters $V_{2,0}$ and $\rho _{2,0}$ are readily solved using the normal shock relations. The wall properties $\rho _{w,0}$ and $\nu _{w,0}$ can be found by assuming the wall temperature is fixed (e.g., at 300K) and that pressure $p_{w,0}=p_{2,0}$.

Thus, l can then be found by numerically solving the equation below:

$$\begin{aligned} -\frac{1}{2}X = \ln \left( 1 - T^{\frac{1}{2}}\right) + T^{\frac{1}{2}}, \end{aligned}$$

(A.4)

where $X=V_{2,0}t/l_m$ ($t = x_{S}/V_w$, where $x_{S}$ is the length of the shock tube) and $T=l/l_m$. Mirels (1963) stated that Eq. A.4 is accurate except for W very close to 1.

Then, $V_{cs}$ can be found from Eq. 19 in Mirels (1964):

$$\begin{aligned} (l/l_{m})^{1-n} = 1 - (V_{\text {cs}} - V_{2,0}), \end{aligned}$$

(A.5)

where n is 1/2 for laminar boundary layers and 1/5 for turbulent ones.

The reflected $u+a$ wave then travels through the unsteady expansion, with the time elapsed between secondary diaphragm rupture and it emerging from the unsteady expansion, $t_{0}$, being given by another equation in Paull and Stalker (1992):

$$\begin{aligned} t_{1} = t_{0}(a_{7}/a_{2})^{\frac{\gamma + 1}{2(1-\gamma )}}, \end{aligned}$$

(A.6)

where $a_{7}$ is the sound speed of the unsteadily expanded test gas, and $\gamma$ is the specific heat ratio of the test gas. In Paull and Stalker (1992), $\gamma$ was always ideal (1.4), but, in this paper, the equilibrium specific heat ratio of the post-shock tube gas ($\gamma _{2}$) has been used, as a little variation was seen between $\gamma _{2}$ and $\gamma _{7}$ from equilibrium calculations using PITOT (James et al. 2018).

After emerging from the unsteady expansion, the reflected $u+a$ wave then travels through the unsteadily expanded test gas (state 7) at velocity $V_{7}$ + $a_{7}$ until it reaches either the end of the acceleration tube or the test gas/accelerator gas contact. In the latter case, there is no test time; otherwise, the test time, $T_{{\text {reflected}}\ u+a}$, can be found from the equation below:

$$\begin{aligned} T_{{\text {reflected}}\ u+a} = \frac{a_{7}}{V_{7} + a_{7}} \left( 2t_{1} - \frac{x_{A}}{V_{7}} \right) . \end{aligned}$$

(A.7)