Implementation of a state-to-state analytical framework for the calculation of expansion tube flow properties

Abstract

Expansion tubes are an important type of test facility for the study of planetary entry flow-fields, being the only type of impulse facility capable of simulating the aerothermodynamics of superorbital planetary entry conditions from 10 to 20 km/s. However, the complex flow processes involved in expansion tube operation make it difficult to fully characterise flow conditions, with two-dimensional full facility computational fluid dynamics simulations often requiring tens or hundreds of thousands of computational hours to complete. In an attempt to simplify this problem and provide a rapid flow condition prediction tool, this paper presents a validated and comprehensive analytical framework for the simulation of an expansion tube facility. It identifies central flow processes and models them from state to state through the facility using established compressible and isentropic flow relations, and equilibrium and frozen chemistry. How the model simulates each section of an expansion tube is discussed, as well as how the model can be used to simulate situations where flow conditions diverge from ideal theory. The model is then validated against experimental data from the X2 expansion tube at the University of Queensland.

Appendices

Appendix 1: Setting up and running PITOT

PITOT forms part of the CFCFD code collection at UQ’s Centre for Hypersonics [43], and as such, PITOT relies on installation of the accompanying Compressible Flow Python Library (cfpylib) to run. The latest version of the CFCFD code collection, instructions on how to obtain it, and the dependencies required to use it, can be found at the website found in the accompanying CFCFD reference [43] principally on the page titled “Getting the codes and preparing to run them”. A page with separate specific instructions for PITOT also exists [80]. The authors have written an accompanying Makefile which can be used to install PITOT on a compatible Linux system with the correct dependencies installed. The authors use Ubuntu and are aware that PITOT has been used on other Linux distributions as well. It is surely possible to manually install PITOT on a Macintosh or Windows system, but the authors cannot confirm this. The main obstacle would be getting PITOT to find and use CEA [41, 42] on the other operating systems.

PITOT is written in the Python programming language, and after it is installed, the most common way to run the program is to write a configuration file which conforms to Python syntax and then parse it to the overarching program by entering the line below into the terminal:

$$\begin{aligned} \$ \,\,\texttt {pitot.py} --\texttt {config}\_\texttt {file=filename.cfg} \end{aligned}$$

Example annotated configuration files for various scenarios can be found in the examples folder of the CFCFD code collection covering both simple simulations and more complex ones using custom facilities, custom test gases, and other “advanced” features. PITOT has been built to be modular and easy to script, and the configuration info can also be parsed to the program inside a Python script using a Python dictionary. Several different tools which make use of this have been created by the authors to perform tasks such as analysing air contamination or performing parametric studies of different fill pressures in the facility. These tools are included with the basic PITOT installation. The overall PITOT source code is open source and users can browse the code and make changes themselves if it is required.

Appendix 2: Calculating experimental shock speed uncertainty

This appendix details the current procedure used for the calculation of expansion tube shock speed uncertainty in the Centre for Hypersonics at UQ. It is the procedure used by the shot analysis codes in the laboratory for the calculation of shock speed uncertainty, and it includes uncertainty and error from three different sources:

• 1. Distance uncertainty from the measured sensor locations and the physical size of each sensor.

• 2. Time uncertainty in ascertaining shock arrival on the sensors.

• 3. Sampling rate error from the clocking speed of the data acquisition system.

In a shock tube or expansion tube, a shock travels through the tube at shock speed ${V}_\mathrm{s}$. Over the full length of the tube, depending on the strength of the facility driver and the severity of non-ideal effects such as low-density shock tube (or Mirels) effects [36,37,38], wall friction, or heat losses, there may be attenuation of the shock and it will slow down as a function of distance (${V}_\mathrm{s} (x)$), but as this analysis is interested in the local shock speed between two wall pressure sensors, this will not be considered here, and it will be assumed that ${V}_\mathrm{s}$ remains constant between the two sensors.

Consider a shock moving through the acceleration tube of an expansion tube at shock speed ${V}_\mathrm{s}$. Just in front of the shock are two wall pressure sensors, at1 and at2, mounted at locations ${x}_{1}$ and ${x}_{2}$, measured from a single datum point. This is shown in Fig. 23 and constitutes time ${t}_{0}$.

At a certain time after ${t}_{0}$ called ${t}_{1}$, the shock will pass pressure sensor at1. When this occurs, there will be a step increase in pressure at location ${x}_{1}$, which will be recorded by the sensor and later used to ascertain ${t}_{1}$. Similarly, at a certain time after ${t}_{1}$ called ${t}_{2}$, the shock will pass pressure sensor at2, and the step increase in pressure seen at the location ${x}_{2}$ will be recorded by sensor at2 and later used to ascertain ${t}_{2}$.

Fig. 23

Representation of a moving shock wave about to pass wall pressure sensors at1 and at2 in the X2 expansion tube (not to scale)

Knowing the distance between the two sensors (${x}_{2} - {x}_{1}$), and the time at which the shock passes both locations (${t}_{1}$ and ${t}_{2}$), the nominal shock speed can be found as simply distance ($\Delta x$) divided by time ($\Delta t$):

$$\begin{aligned} V_\mathrm{s} = \frac{x_{2} - x_{1}}{t_{2} - t_{1}} = \frac{\Delta x}{\Delta t}. \end{aligned}$$

(3)

It can be seen that (3) is a function of two distances (${x}_{1}$ and ${x}_{2}$), and two times (${t}_{1}$ and ${t}_{2}$). Therefore, to quantify the uncertainty, the uncertainties on both the distance and the time must be considered.

Three different types of distance uncertainty are considered:

• 1. Uncertainty in the measurement of the sensor locations (${x}_{1}$ and ${x}_{2}$).

• 2. Uncertainty in the response of the pressure sensor due to the physical size of the sensor. (The pressure sensors used on X2 in the acceleration tube are 112A22 50 PSI pressure transducers from PCB Piezotronics with a sensor diameter of 5.54 mm [81].)

• 3. Uncertainty due to the shape of the shock not being planar like it is assumed.

These three uncertainties are encapsulated by a single distance uncertainty ($\delta {x}_{i}$) of $\pm 2.0\times 10^{-3}$ m (2 mm) for each sensor location.

Therefore, because the distance uncertainties are independent measurements, the total distance uncertainty ($\delta \Delta x$) for the shock speed calculation is:

$$\begin{aligned} \delta \Delta x = \sqrt{\delta x_{1}^{2} + \delta x_{2}^{2}}. \end{aligned}$$

(4)

One source of time uncertainty and one source of error are considered:

• 1. Time uncertainty in ascertaining shock arrival on the sensors.

• 2. Sampling rate error from the clocking speed of the data acquisition system.

Pressure transducers have a finite rise time to full signal (${\le }{2.0}\,\upmu \hbox {s}$ for the 112A22 pressure transducers used in X2’s acceleration tube [81]), and the facility’s data acquisition system is recording at a set clock speed (2.5 MHz for all sensors on X2, with most acceleration tube sensors also teed off into a 60 MHz card to reduce sampling rate error), meaning it can be difficult to ascertain exactly when the shock has passed each location. Two separate uncertainties are used to quantify this.

Firstly to remove any large uncertainties created by an automated process on what can sometimes be a relatively noisy signal, shock arrival times are found manually by a graphical interface which experimenters use to select shock arrival times for each signal. Instead of selecting a single time for shock arrival, experimenters are instructed to select the data point just before and just after when they believe the shock has arrived, giving a time range for shock arrival. The analysis code then finds both of the data points, calculates the midpoint, and adds a shock arrival uncertainty ($\delta {t}_{i}$) to the data which is half of the distance between the original two points.

Secondly, to take into account the sampling rate error, an extra time uncertainty is added based on the size of a single sample ($\delta {t}_\mathrm{sr}$) to take into account the fact that the shock could arrive at any point in the sample. The size of a full sample instead of only half of a sample has been used as a conservative measure to take into account the fact that multiple samples are actually involved in the calculation process. Recently, acceleration tube pressure data (where shock speeds are often of the order of 10 km/s) have also been recorded on a high-speed National Instruments PXI-5105 card clocking at 60 MHz to reduce the sampling rate error on the acceleration tube shock speeds after it was found that the largest source of experimental uncertainty on these measurements was caused by the normal data acquisition system clocking at 2.5 MHz.

Therefore, because the time uncertainties are independent measurements, the total time uncertainty ($\delta \Delta t$) for the shock speed calculation is:

$$\begin{aligned} \delta \Delta t = \sqrt{\delta t_{1}^{2} + \delta t_{2}^{2} + \delta t_\mathrm{sr}^{2}}. \end{aligned}$$

(5)

Now that total distance and time uncertainties ($\delta \Delta x$ and $\delta \Delta t$) are known, the total shock speed uncertainty ($\delta {V}_\mathrm{s}$) can be found using the uncertainty formula for the division of independent variables, which is shown below in the form appropriate for calculating the shock speed uncertainty:

$$\begin{aligned} \delta V_\mathrm{s} = V_\mathrm{s} \cdot \sqrt{ \left( \frac{\delta \Delta x}{\Delta x} \right) ^{2} + \left( \frac{\delta \Delta t}{\Delta t} \right) ^{2}}. \end{aligned}$$

(6)