2.1. Expansion tube driver requirements

Morgan & Stalker (Reference Morgan and Stalker1991) and Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001a ,Reference Morgan, Ben-Dor, Igra and Elperin b ) have extensively investigated the driver requirements for shock tunnel and expansion tube operation, particularly for expansion tubes operating at superorbital flight conditions. Referring to figure 1, and noting that across the interface the speed of the expanded driver gas, $u_{3}$ , equals the post-shock test gas velocity, $u_{2}$ , Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001b ) normalised this velocity by the unexpanded driver gas sound speed, $a_{4}$ :

$Ma_{3}^{\ast }$ is termed the ‘driver equivalent Mach number’ (Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001b ). By using velocity instead of driven shock Mach number, and by non-dimensionalising $u_{3}$ in terms of $a_{4}$ , the pressure ratio $p_{3}/p_{4}$ becomes independent of the driven gas composition (Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001b ). Using a perfect gas analysis, where ${\it\gamma}$ is the ratio of specific heats, Morgan arrived at the following driver relation:

where:

$F_{1}$ is a parameter which accounts for the geometric transition from the driver to driven tubes. In figure 1, $Ma_{4^{\prime \prime }}=1$ ; the flow is choked since there is a significant area change, and this is also the highest performance arrangement at the tube transition (Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001b ).

Equation (2.2) indicates the extent to which the driver gas must expand, from its initial pressure, $p_{4}$ , to $p_{3}$ (which is equal to the post-shock pressure, i.e. $p_{3}=p_{2}$ ), to achieve a given $Ma_{3}^{\ast }$ , and is plotted in figure 2. These results are for a facility with an area change at the driver, therefore $Ma_{4^{\prime \prime }}=1$ . Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001b ) notes that the maximum possible $Ma_{3}^{\ast }$ is achieved in the limiting case, when the driver gas expands to zero pressure; i.e.  $p_{3}/p_{4}=0$ . Solving for $Ma_{3}^{\ast }$ :

Figure 2. Pressure ratio required to achieve a given driver equivalent Mach number, $Ma_{3}^{\ast }$ , for $Ma_{4^{\prime \prime }}=1$ (calculated from (2.2) and (2.3)).

For ${\it\gamma}_{4}=5/3$ and $Ma_{4^{\prime \prime }}=1$ , equation (2.4) gives $Ma_{3}^{\ast }=3.46$ ; for helium at room temperature (300 K) this corresponds to $u_{3}=3536~\text{m}~\text{s}^{-1}$ ; for helium at 5000 K, which represents a free-piston driver operating at a high compression ratio, $u_{3}=11\,150~\text{m}~\text{s}^{-1}$ . However, observing figure 2, it can be seen that the ratio $p_{3}/p_{4}$ drops rapidly for $Ma_{3}^{\ast }>2$ ; Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001a ) notes that pressure recovery becomes too low beyond this point; the implications are test gas densities which are impractically low, and test flows which become overwhelmed by viscous effects (Mirels Reference Mirels1963a ,Reference Mirels b ).

A practical limit of $Ma_{3}^{\ast }=2$ corresponds to intermediate test gas flow velocities (region 2 in figure 1) of $2038~\text{m}~\text{s}^{-1}$ and $6445~\text{m}~\text{s}^{-1}$ for a helium driver at 300 K and 5000 K, respectively; this, in turn, sets a limit on final test flow velocity as well, since there are limits to the enthalpy and total pressure increases which are possible across the unsteady expansion (Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001a ) quoting Lucasiewicz (Reference Lucasiewicz1973) notes that up to a doubling of flow velocity can typically be achieved across an unsteady expansion). While these calculations are perfect gas estimates, they nevertheless provide an indication of the fundamental limitations of this type of driver arrangement.

2.2. Expansion tube operation with a shock-heated secondary driver

Figure 3 shows an idealised $x{-}t$ diagram of the flow processes which occur for an expansion tube operating with a secondary driver. To enable this mode, a second thin Mylar diaphragm is used to locate a volume of helium gas between the primary diaphragm and the test gas. This is termed the secondary diaphragm; the downstream diaphragm separating the test and accelerator gases becomes the tertiary diaphragm.

Figure 3. Idealised distance–time ( $x{-}t$ ) diagram showing flow processes for an expansion tube with secondary driver and unsteady expansion at secondary diaphragm: s  $=$  shock; rs  $=$  reflected shock; cs  $=$  contact surface; sx  $=$  steady expansion; usx  $=$  unsteady expansion; $u+a=\text{reflected}$ characteristic.

This additional tube section is referred to as a secondary driver because it becomes the driver for the actual test gas (the combination of primary and secondary drivers is then referred to as a ‘compound driver’). If this section of tube has a sufficiently low initial fill pressure, shock processing of its helium can raise its temperature, $T_{sd2}$ , and therefore sound speed, $a_{sd2}$ , far above the equivalent values of the expanded primary driver gas, $T_{sd3}$ and $a_{sd3}$ . When $a_{sd2}>a_{sd3}$ , the secondary driver is said to be ‘over-tailored’. This can result in the secondary driver generating a stronger shock through the test gas than the primary driver would be able to by itself; this performance increasing potential was the motivation for initial work with these devices, and has previously been analysed and confirmed experimentally (Henshall Reference Henshall1956; Stalker & Plumb Reference Stalker and Plumb1968; Morgan & Stalker Reference Morgan and Stalker1991; Neely & Morgan Reference Neely and Morgan1994; Morgan Reference Morgan1997, Reference Morgan, Ben-Dor, Igra and Elperin2001a ).

Figure 4 (from Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001a ) compares the pressure recovery, $p_{3}/p_{4}$ , as a function of $Ma$ , for three different driver options. Taking into account practical limitations of facility operation, Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001a ) determined that a compound driver can extend the maximum practical driver equivalent Mach number from 2 to 4. This therefore opens up the possibilities of simulating flight enthalpies to $20\,000~\text{m}~\text{s}^{-1}$ and beyond – for example, return from missions to the Far Solar System (entry to Earth’s atmosphere at speeds up to $16\,000~\text{m}~\text{s}^{-1}$ ), or entry to the gas giants (atmospheric entry at speeds exceeding $20\,000~\text{m}~\text{s}^{-1}$ ) – and is why this operating regime is referred to as ‘superorbital’ expansion tube operation. This superorbital mode was first demonstrated by Morgan & Stalker (Reference Morgan and Stalker1991), who achieved shock speeds up to $18.7~\text{km}~\text{s}^{-1}$ ; Neely & Morgan (Reference Neely and Morgan1994) further developed the concept, with test flows up to $13~\text{km}~\text{s}^{-1}$ .

Figure 4. Pressure recovery, $p_{3}/p_{4}$ , as a function of driver equivalent Mach number, $Ma_{3}^{\ast }$ , for different driver options (adapted from Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001a ).

2.3. Secondary driver as acoustic buffer

The first experiments with expansion tubes, which took place in the 1960s and 1970s in the United States, had in common the disappointing feature that most test flows had unacceptable levels of noise (Jones Reference Jones1965; Norfleet, Lacey & Whitfield Reference Norfleet, Lacey and Whitfield1965; Spurk Reference Spurk1965; Miller Reference Miller1977; Miller & Jones Reference Miller, Jones, Archer and Milton1983). Miller (Reference Miller1977), who conducted the first extensive expansion tube study with the NASA Langley Expansion Tube/Tunnel, would later note that only a very narrow range of suitably steady test flows could be achieved (Miller & Jones Reference Miller, Jones, Archer and Milton1983).

In the 1980s, The University of Queensland (UQ) was contracted by NASA Langley to investigate expansion tube operation with a free-piston driver (Stalker Reference Stalker and Kim1990). Paull, Stalker & Stringer (Reference Paull, Stalker and Stringer1988) undertook a systematic study with the new TQ facility to identify the reasons for the test flow unsteadiness which had been observed in the earlier expansion tube experiments. Paull & Stalker (Reference Paull and Stalker1992) would eventually categorise test flows as being either high enthalpy or low enthalpy, and determined that test flow noise was associated with the low-enthalpy flows.

Paull & Stalker (Reference Paull and Stalker1992) analysed the effect of the unsteady expansion on noise in the test gas, and found that the sound speed drop through the unsteady expansion had the effect of focusing the frequency components of lateral acoustic waves into a narrow bandwidth of frequencies, which would later appear as strong disturbances in the test flow. While this phenomenon was present for both high- and low-enthalpy conditions, the high-enthalpy test flows were relatively free from disturbances because the test gas had low levels of noise prior to the unsteady expansion. The critical next step was to determine that, for high-enthalpy conditions, the test gas was relatively free of noise because the sound speed of the shock-processed test gas, $a_{2}$ in figure 1, was higher than the sound speed of the expanded driver gas, $a_{3}$ (Paull & Stalker Reference Paull and Stalker1992).

The driver gas expansion through the driver area change and primary diaphragm is a fundamental source of lateral wave disturbances (radial disturbances in an axisymmetric facility) (Jacobs Reference Jacobs1994). Jacobs (Reference Jacobs1994) performed numerical simulations of the NASA Langley facility and showed that these radial disturbances could indeed propagate into the test gas. The principal conclusion from the Paull & Stalker (Reference Paull and Stalker1992) work was that it was necessary to operate an expansion tube in an over-tailored configuration ( $a_{2}/a_{3}>1.25$ was suggested) in order to achieve an ‘acoustic buffer’ against driver gas disturbances.

The acoustic buffer requirement clearly has implications for expansion tube operation at lower enthalpies; Morgan (Reference Morgan, Ben-Dor, Igra and Elperin2001a ) determined that operating a basic expansion tube with $a_{2}/a_{3}>1.25$ would require $Ma_{3}^{\ast }>1.4$ . While it becomes possible to use alternative facilities, such as RSTs, to simulate these low-enthalpy test flows, there are some applications where the total pressure requirement far exceeds what other facilities can deliver. For example, high-Mach-number scramjet test flows ( $3{-}5~\text{km}~\text{s}^{-1}$ ) can have gigapascal total pressure requirements (Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014); however, RSTs are structurally limited to total pressures of between 150 and 300 MPa (Bakos & Erdos Reference Bakos and Erdos1995).

Noting that over-tailored operation is a fundamental aspect of secondary driver operation, Morgan & Stalker (Reference Morgan and Stalker1991) proposed that for low-enthalpy conditions the acoustic buffer function could be met with the secondary driver, thereby, in theory, restoring the useful operational envelope of expansion tubes back to the initial predictions of Trimpi (Reference Trimpi1962). Thus, these initial years of expansion tube research at UQ led to the discovery of the mechanism responsible for the disappointingly noisy test flows in earlier studies (Paull et al. Reference Paull, Stalker and Stringer1988; Paull & Stalker Reference Paull and Stalker1992), furthermore, the shock-heated secondary driver was introduced as a means to boost performance at superorbital conditions, and was simultaneously identified as a solution to the test flow noise problem (Morgan & Stalker Reference Morgan and Stalker1991; Morgan Reference Morgan, Ben-Dor, Igra and Elperin2001a ).

3.1. Reflected shock at the secondary diaphragm

An important feature in figure 6 is that although the secondary driver can be configured for over-tailored operation across a large part of the performance envelope, this performance increase is lost if a reflected shock arises at the secondary diaphragm. While it will be later shown that this becomes a fundamental feature of secondary driver operation at lower enthalpies, where the device is used as an acoustic buffer, in this section the conditions to produce a reflected shock are first examined.

Whether or not a reflected shock arises at the secondary diaphragm depends primarily on the ratio of initial fill pressures in the secondary driver and shock tubes ( $p_{sd1}$ and $p_{1}$ , respectively), and to a lesser extent on the strength of the resulting shock in the shock tube. For $p_{1}\ll p_{sd1}$ , an unsteady expansion will occur; if $p_{1}$ is increased, eventually a point will be reached where the unsteady expansion has zero strength; if $p_{1}$ is further increased, a reflected shock wave will form. The weakest reflected shock will be a Mach wave, and will produce negligible pressure rise, and negligible deceleration, of the region $sd2$ flow; for this case $u_{sd2}=u_{3}$ and $p_{sd2}=p_{3}$ . If ideal gas assumptions are made, the density ratio associated with this condition, ${\it\rho}_{sd1}/{\it\rho}_{1}$ , can be solved as a function of test gas shock strength, expressed in terms of Mach number $Ma_{s,1}$ . Section A.3 details the procedure to solve for the reflected Mach wave case; the results are shown in figure 7.

Figure 7. Effect of secondary driver/test gas fill density ratio, ${\it\rho}_{sd1}/{\it\rho}_{1}$ , on post-diaphragm rupture flow processes. Secondary driver is helium; test gas is air.

Figure 7 shows two solutions for the reflected Mach wave case. The curve denoted ‘Solution 1’ is the only physical solution. It can be seen that, except for weak shocks, the ratio ${\it\rho}_{sd1}/{\it\rho}_{1}\approx 0.9$ indicates the density ratio below which a reflected shock will form in the secondary driver gas. For ${\it\rho}_{sd1}/{\it\rho}_{1}>0.9$ , the secondary driver gas is sufficiently dense that it will undergo an unsteady expansion into the test gas for all test gas shock speeds.

‘Solution 2’ in figure 7 represents the shock Mach number which the secondary driver will drive through the test gas simply by virtue of its high initial fill pressure; i.e. with no initial velocity, and independent of the upstream driver. It is a non-physical solution, since this expansion process must occur through an expansion wave, not a Mach wave. The correct solution is given by (3.2), which is the basic shock tube equation (equation (7.94) from Anderson Reference Anderson1990), and is the dashed curve shown in figure 7. The two solutions are coincidentally equal for $Ma_{s,1}=1$ , for which the driven shock through the test gas is a Mach wave with no associated pressure rise; the two curves diverge for $Ma_{s,1}>1$ , since only the shock tube equation is physically correct.

For an air test gas, the result observed in figure 7 effectively limits the use of a secondary driver as a performance increasing device to superorbital conditions only. This limitation will be less restrictive for lighter test gases.

5.1. High-enthalpy flight

Expansion tubes provide an experimental tool to conduct studies of radiative heating around blunt bodies. Representative Mars and Far Solar System return re-entry trajectories from NASA (2010) are shown in figure 10. These are the hypervelocity, low density, flight conditions through air which a vehicle returning to Earth’s atmosphere would undergo. These currently can only be produced by an expansion tube, and at the upper limits they may depend on the performance increase provided by the secondary driver.

Figure 10. Representative Mars and Far Solar System return re-entry trajectories (NASA 2010). In each plot, thick curves represent Mars return trajectory properties; thin curves represent Far Solar System return: (a) return trajectories; (b) US standard atmosphere; (c) computed Mach number.

For Mars return at $13~\text{km}~\text{s}^{-1}$ , the altitude is 46.1 km, the Mach number is 39.7, with temperature 267 K and static pressure 130 Pa. To achieve this flow condition with an expansion tube, $p_{1}$ must be exceedingly low in order to attain such a high speed and simultaneously very low temperature; it is impractical to operate at such low test gas pressures in expansion tubes, since viscous effects in the low-pressure acceleration tube corrupt the test flow (Mirels Reference Mirels1963a ,Reference Mirels b ).

Ground testing at these conditions takes advantage of the commonly accepted Mach number independence principle. This states that, at high Mach numbers, aerodynamic characteristics such as flow field structure, pressure coefficients, lift, drag, and so forth, become asymptotically independent of Mach number (Kliche, Mundt & Hirschel Reference Kliche, Mundt and Hirschel2011; Anderson Reference Anderson2006). While the validity of this principle has been questioned for high-Mach-number viscous flows (Kliche et al. Reference Kliche, Mundt and Hirschel2011), this commonly accepted principle relaxes the requirement to simulate re-entry flows at true flight Mach numbers. It becomes necessary only to achieve sufficiently high Mach numbers for the independence principle to apply, which for blunt-body flow is met for Mach numbers greater than five (Anderson Reference Anderson2006, figure 4.2).

Considering the $13~\text{km}~\text{s}^{-1}$ Mars return trajectory, and assuming equilibrium gas properties, if the flow is expanded to just Mach 10, its temperature will be significantly higher (Mach 10 at $13~\text{km}~\text{s}^{-1}$ corresponds to an equilibrium free-stream temperature of 4621 K, with specific heat at constant pressure, $c_{p}=1339~\text{J}~\text{kg}^{-1}~\text{K}^{-1}$ ; calculated using CEA). Noting that total temperature and total enthalpy are conserved across the vehicle bow shock, to conserve total temperature it can be necessary for the experiment to account for the increased static enthalpy of the expansion tube free-stream test flow in comparison to true flight. Equating true flight (‘ $\infty$ ’) and expansion tube (‘ $e$ ’) total enthalpies:

Neglecting the static enthalpy of the true flight free-stream flow ( $c_{p}T_{\infty }\approx 0$ ), the target expansion tube test flow velocity, $U_{e}$ , is calculated as follows:

$U_{e}$ is the flight equivalent velocity, and indicates that expansion tube simulation requires a marginally lower velocity than true flight. For $13~\text{km}~\text{s}^{-1}$ flow at $T=4621~\text{K}$ , $U_{e}$ is only 4 % less than $U_{\infty }$ , indicating that most of the enthalpy is associated with the flow kinetic energy:

Neglecting the difference between $U_{e}$ and $U_{\infty }$ for the purposes of this present study, and assuming absolute viscosity, ${\it\mu}$ , is conserved between true flight and experiment, Reynolds number is preserved for a blunt-body vehicle if the product ${\it\rho}L$ is also conserved, where ${\it\rho}$ is the density behind the shock and $L$ is the characteristic length (for a planetary entry vehicle this might be the nose radius).

For the true flight case, the equilibrium properties behind a normal shock are summarised in table 3. It can be seen that expansion tube simulation requires that the density of the flow behind the shock, ${\it\rho}$ , be equal to $0.0258~\text{kg}~\text{m}^{-3}$ . For the expansion tube test flow with temperature of 4621 K and $U_{e}=13\,000~\text{m}~\text{s}^{-1}$ , this corresponds to a free-stream static pressure of $p_{e}=2380~\text{Pa}$ . Whereas the 130 Pa free-stream static pressure at Mach 39.7 is impractical to achieve in an expansion tube, 2380 Pa at Mach 10 becomes more feasible.

The process detailed above was used to calculate Mach 10 equivalent flow conditions for both Mars and Far Solar System return trajectories; results are summarised in tables 4 and 5. These are fairly representative of the type of high-enthalpy flow conditions for which expansion tubes would typically be used.

5.2. Low-enthalpy flight

The scramjet engine is viewed as a potential source of launch vehicle propulsion between Mach 4 and 15 (Curran Reference Curran2001). Most scramjet trajectory studies propose that the launch vehicle should target an approximately constant dynamic pressure trajectory during the scramjet burn cycle (Olds & Budianto Reference Olds and Budianto1998). The magnitude of this dynamic pressure determines the structural and heat loading to the engine, and several authors consider that the representative upper limit for this is around 2000 psf (95.8 kPa) (Billig Reference Billig1993; Hicks Reference Hicks1993; Olds & Budianto Reference Olds and Budianto1998; Hunt & Martin Reference Hunt and Martin2000).

Table 6 shows scramjet access-to-space flight conditions at Mach 10, 12 and 14, for a 95.8 kPa dynamic pressure trajectory. It can be seen that, for Mach 10 flight, the full-scale equilibrium free-stream total pressure is 128 MPa, which is approaching the upper limit of RST facilities (Bakos & Erdos Reference Bakos and Erdos1995). At Mach 14 the total pressure increases by an order of magnitude, to gigapascal levels.

Scramjet ground testing is typically undertaken using sub-scale engine models. Pressure–length scaling ( $p{-}L$ scaling, which is binary scaling at constant temperature) is applied to maintain similarity for Mach number, enthalpy, Reynolds number and binary reactions (Stalker & Pulsonetti Reference Stalker and Pulsonetti2004). $p{-}L$ scaling requires increasing test flow static pressure in inverse proportion to any decrease in model length, so that sub-scale models require even higher static pressures compared to true flight. This means that the actual test flow total pressures required at high Mach numbers may be several times higher than those presented in table 6.

The expansion tube is currently the only ground testing facility able to produce the gigapascal total pressure flows associated with scramjet access to space beyond Mach 10. UQ’s X2 facility has recently been used to generate scramjet test flows in the range Mach 10–14, up to a maximum total pressure of 10.4 GPa (Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014). X2 is a medium-sized free-piston-driven expansion tube; it has a total length of 23 m, and a driven tube bore diameter of 85 mm. Gildfind et al. (Reference Gildfind, Morgan, McGilvray and Jacobs2014) was a proof-of-concept study undertaken to establish the feasibility of producing these flows in an expansion tube. Major outcomes included identification of the need for tuned operation of the free-piston driver (Gildfind et al. Reference Gildfind, Morgan, McGilvray, Jacobs, Stalker and Eichmann2011) and the close coupling of wave processes at low-enthalpy flow conditions (Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014). These considerations were successfully addressed, and subsequent and ongoing work has been to scale these conditions to the larger 65 m X3 facility, which can provide the longer test times and larger test flow core diameters required for practical engine testing (Gildfind et al. Reference Gildfind, Sancho and Morgan2013, Reference Gildfind, Morgan, McGilvray and Jacobs2014).

Although high-Mach-number scramjet flight takes place at relatively low enthalpies ( $4.5{-}12.5~\text{MJ}~\text{kg}^{-1}$ ; $3{-}5~\text{km}~\text{s}^{-1}$ ) compared to other expansion tube applications ( $18{-}200~\text{MJ}~\text{kg}^{-1}$ ; $6{-}20~\text{km}~\text{s}^{-1}$ ), their massive total pressures make expansion tubes an important tool for ground testing of these engines. The flow conditions detailed in table 6 are predominantly representative of the low-enthalpy flow conditions for which expansion tubes will be used in the coming years.

7.1. High-enthalpy examples: $13$ and $11~\text{km}~\text{s}^{-1}$ Mars return

Table 7 shows the calculated facility operating parameters for the $13~\text{km}~\text{s}^{-1}$ Mars return case (Case 1 in figure 11). The test flow static pressure, $p_{7}$ , is observed to vary with $p_{sd1}$ , and achieves a peak of $p_{7}=1.82~\text{kPa}$ for $p_{sd1}=112~\text{kPa}$ . Table 7 also shows $p_{7}$ for operation without a secondary driver; $p_{7}=1.17~\text{kPa}$ for Case 1d. For this flow condition case, the secondary driver can theoretically provide an increase in test flow static pressure of up to 56 %.

For comparison, the Mach 10 true flight equivalent static pressure is 2.38 kPa (from figure 10), and is therefore marginally higher than what X2 can achieve with this specific free-piston driver condition. Nevertheless, these results confirm that the secondary driver can significantly outperform the basic expansion tube.

Figure 12(a) shows experimental confirmation of these results in X2. Measuring the speed of the primary shock as it traverses the length of the facility is an effective means to validate several of the key flow processes occurring during an experiment. It can be seen that there is generally good agreement with computed shock speeds through both the secondary driver and shock tube, although all except Case 1a have the experimental $u_{s,1}$ slightly less than the computed value.

Figure 12. Experimental results for $13~\text{km}~\text{s}^{-1}$ Mars return flight equivalent conditions; (a) shock speeds along the entire driven tube; (b) shock tube static pressures; static pressure traces are time-referenced to Case 1a, with offsets of $50~{\rm\mu}\text{s}$ , $100~{\rm\mu}\text{s}$ and $150~{\rm\mu}\text{s}$ , for Cases 1b, 1c and 1d, respectively.

It is noted that there is some scatter and uncertainty in the acceleration tube shock speeds; this was due to two reasons: firstly, the pressure sensors in this tube were insufficiently sensitive to precisely identify shock arrival at the transducer for these very low pressure conditions; secondly, low fill pressures could not be reliably achieved, or held, at the time of the experiments. Neither of these issues was of concern, since the important measurements were in the secondary driver and shock tube. However, these issues could be readily addressed in any future study requiring use of the final expanded test flow. Uncertainties in the secondary driver and shock tube are much lower, and in most cases the error bars, which are included in the plot, are too small to identify.

Figure 12(b) shows a tube wall static pressure measurement towards the end of the shock tube, $p_{2}$ , for Cases 1a–1d. Comparing these traces to the computed value of $p_{2}$ in table 7, it can be seen that all four experimental traces are 10–20 % lower in magnitude. This discrepancy probably reflects the slightly lower experimental shock speeds which were measured in the shock tube. However, the important observation is that a significantly larger static pressure is observed for Case 1b compared to the other conditions, as predicted by table 7, confirming that the secondary driver can be used as a test gas static pressure control.

The same analysis was repeated for the $11~\text{km}~\text{s}^{-1}$ Mars return condition (Case 4 in figure 11), with results shown in table 8 and figure 13. This example illustrates how the secondary driver can be used to reduce performance. Referring to table 8, Case 1a theoretically has $p_{7}=2.1~\text{kPa}$ , compared to $p_{7}=5{-}6~\text{kPa}$ for Cases 1b–1e. This is reflected by a much lower static pressure in the shock tube (figure 13 b); the relative difference is not as large as predicted by table 8, and is primarily due to the Case 4b experimental shock speeds being higher than the other conditions.

Figure 13. Experimental results for $11~\text{km}~\text{s}^{-1}$ Mars return flight equivalent conditions; (a) shock speeds along the entire driven tube; (b) shock tube static pressures; static pressure traces are time-referenced to Case 4a, with offsets of $50~{\rm\mu}\text{s}$ , $100~{\rm\mu}\text{s}$ , $150~{\rm\mu}\text{s}$ and $200~{\rm\mu}\text{s}$ for Cases 4b, 4c, 4d and 4e, respectively.

In summary, the 11 and $13~\text{km}~\text{s}^{-1}$ Mars return results validate the driver characterisation detailed in Gildfind et al. (Reference Gildfind, James and Morgan2015), and show that the analytical techniques presented in this paper correctly predict the trends for facility performance, both with and without a secondary driver, and show that test gas static pressure can be controlled using the secondary driver.

7.2. Low-enthalpy examples: Mach $10$ and Mach $14$ scramjet flight

Tables 9 and 10 show calculated facility parameters for the Mach 10 (Case 2) and Mach 14 (Case 3) scramjet flow conditions from figure 11. However, recent experience with these conditions (Gildfind Reference Gildfind2012; Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014) has shown that these calculations become invalid for facility configurations when there is a strong reflected shock through the secondary driver gas. Referring to figure 14 (originally from Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014), when the secondary diaphragm ruptures, the shock-processed slug of helium (region $sd2$ ) has a short length but high sound speed, and is rapidly processed by the reflected shock wave (c in the figure). This wave then reflects off the $sd2/sd3$ interface, and returns as a strong compression wave (d and e in figure 14). The time scales for this phenomenon are very short compared to other flow processes; for most practical conditions, the secondary driver would need to be an order of magnitude longer than the shock tube to sufficiently delay arrival of this wave. These wave processes have been described in Gildfind et al. (Reference Gildfind, Morgan, McGilvray and Jacobs2014); however, the broader implications of those results were not examined at the time.

Figure 14. L1d-computed $x{-}t$ diagram for Mach 10 scramjet flow condition in X2. Contours show log of static pressure (in units log (Pa)). (Points on $x{-}t$ diagram: (a) Primary diaphragm is downstream of driver area change; compression wave is transmitted downstream from this area change. (b) Arrival of compression wave at primary shock increases shock speed. (c) Reflected shock arises at secondary diaphragm due to primary shock arrival at dense test gas. (d) Reflected shock hits contact surface and a second compression wave is sent downstream. (e) Second compression wave arrives at primary shock and causes a second primary shock speed increase.) Adapted from Gildfind et al. (Reference Gildfind, Morgan, McGilvray and Jacobs2014, figure 9).

In practice, at low-enthalpy conditions with a dense test gas, the secondary driver behaviour is no longer adequately described by traditional theoretical models. The helium has negligible effect on the magnitude of test flow properties, and acts as a passive slug between the primary driver and test gases. This effect is shown in figure 15(a,b) for the Case 2 scramjet Mach 10 condition. In both plots, the 1-D computational fluid dynamics (CFD) code L1d (Jacobs Reference Jacobs1994) has been used to calculate shock speeds along the length of X2, where $x=0$ corresponds to the primary diaphragm location. These L1d models use a constant volume driver (with initial temperature and pressure as defined in table 1), instead of a free-piston driver, in order to provide a more direct comparison to the analytical calculations summarised in figure 11. The driver length in the L1d model was made sufficiently long to avoid interference of the reflected driver $u+a$ wave during the useful test time.

Figure 15. Shock speeds for X2 Mach 10 scramjet flow conditions detailed in table 9; (a) shows shock speeds for Cases 2a–2f; solid lines are L1d CFD calculations; dashed lines are calculated using the equilibrium analysis in § 6; datapoints are experimental results; (b) shows L1d shock speeds for Case 2f operated with and without a secondary driver, for a wide range of secondary driver fill pressures.

Referring to figure 15(a), the Case 2a–2e curves show shock speeds when secondary drivers of differing fill pressures are introduced between the primary diaphragm ( $x=0$ ) and $x=3.4~\text{m}$ ; the case 2f curve shows the shock speed for a Mach 10 scramjet condition without a secondary driver. Dashed lines show the corresponding ‘target’ shock speeds at each condition; these are the shock speeds predicted by analytical techniques (see table 9), which should theoretically achieve the flow conditions shown in figure 11. It is clear in figure 15(a) that the L1d-computed shock speeds match the target speeds reasonably well through the secondary driver; however, L1d predicts higher shock speeds through the downstream shock and acceleration tubes.

L1d, which has been shown to reliably predict shock speeds in recent scramjet flow condition studies using UQ’s X2 and X3 facilities (Gildfind Reference Gildfind2012; Gildfind et al. Reference Gildfind, Sancho and Morgan2013, Reference Gildfind, Morgan, McGilvray and Jacobs2014), takes into account the full set of longitudinal wave processes. As with the Gildfind et al. (Reference Gildfind, Morgan, McGilvray and Jacobs2014) study, the reinforcing compression wave from the driver, which is captured by L1d, causes an increase in the shock speed above the target level. Referring to the red ‘ $Ma10$ ’ curve in figure 11, it can be seen that as the secondary driver fill pressure is reduced, the secondary driver is predicted by analytical techniques to become increasingly ineffective at driving a shock into the air test gas. This explains why the downstream fill pressures in table 9, $p_{1}$ and $p_{5}$ , become correspondingly lower in order to achieve the same shock speeds. However, since the reinforcing compression wave (d and e in figure 14) essentially restores the full strength of the driver, then higher shock speeds are predicted at these lower fill pressures in figure 15.

Figure 15(b) demonstrates the ineffectiveness of the secondary driver at lower enthalpies from a different perspective. This figure shows L1d-computed shock speeds for flow condition ‘Case 2f’, which is designed for X2 without a secondary driver. The black curve in figure 15(b) is shown to match the ‘target’ shock speeds very closely (note, for this L1d simulation the shock tube runs from $x=0$ to $x=3.4~\text{m}$ ). The other curves show shock speeds for the same condition, except in each case a helium secondary driver is introduced between $x=0$ and $x=3.4~\text{m}$ . Each of these curves represents a different secondary driver fill pressure, across a range spanning 1–1000 kPa. While computed shock speeds evidently vary through the secondary driver, all curves are seen to collapse to the same speeds further downstream in the shock tube. This result shows that the final magnitude of the test flow properties is unaffected by the inclusion of a helium secondary driver once driver secondary wave processes are accounted for. Thus the theoretical predictions of figure 11 are shown to be invalid. At low enthalpies, where a reflected shock processes the secondary driver gas, the theoretical test gas shock speed is not achieved due to the driver wave coupling.

Referring to figure 15(a), it is noted that shock speeds for Cases 2b–2d have been validated by experiment with X2, and good agreement is observed. Error bars are plotted but are not obvious since experimental uncertainties for these shock speeds are low. The remaining shock speeds in figure 15(a,b) were not experimentally validated due to the impractical shock tube fill pressures required for these conditions (100 Pa for Case 2a; 1551 kPa for all of the Case 2f conditions). The 1551 kPa fill pressure exceeds the current manifold and instrumentation pressure rating for X2, and is only required to be this high in the example to match the high performance of this particular free-piston driver condition.

Figure 16. Shock speeds for X2 Mach 14 scramjet flow conditions detailed in table 10; (a) shows shock speeds for Cases 3a–3e; solid lines are L1d CFD calculations; dashed lines are calculated using the equilibrium analysis in § 6; datapoints are experimental results; (b) shows L1d shock speeds for Case 3e operated with and without a secondary driver, for a wide range of secondary driver fill pressures; datapoints are experimental results.

More comprehensive validation was undertaken for the Mach 14 flow condition. Figures 16(a,b) are Mach 14 equivalents of the figure 15(a,b) Mach 10 plots. Full validation at Mach 14 was possible since the maximum shock tube fill pressure was lower (548 kPa). Identical trends are observed in figure 16, thereby supporting earlier conclusions about the secondary driver operation at low-enthalpy conditions. The computed shock speeds match experiment closely through the secondary driver and shock tube. Case 3e (no secondary driver) also indicates good agreement between L1d and experiment; however, all of the secondary driver conditions have slower experimental shock speeds through the acceleration tube than those predicted by L1d. The cause of this discrepancy was not established, since the secondary driver and shock tube measurements were the most relevant results for this secondary driver study, and examination of acceleration tube flow processes would have required computationally intensive axisymmetric CFD of the facility, across several conditions. However, a consistent difference with the secondary driver conditions was the relatively short acceleration tube, arising due to the addition of a third diaphragm. With a shorter acceleration tube the test flow is more influenced by shock establishment and interface mixing, and the test gas is not fully utilised. This issue would need to be re-examined before X2 was used for these specific operating conditions, and normally extension tubes would be added to lengthen the acceleration tube for such experiments.

7.3. Secondary driver as acoustic buffer, and test time implications

As discussed in § 2.3, Morgan & Stalker (Reference Morgan and Stalker1991) proposed the use of a secondary driver to act as an ‘acoustic buffer’ for these types of scramjet conditions, because it had earlier been shown that over-tailored operation could prevent the transmission of driver disturbances from the expanded primary driver gas to the test gas (Paull & Stalker Reference Paull and Stalker1992). Figure 17 demonstrates this ‘acoustic buffer’ using L1d simulations of X2, incorporating full piston dynamics, for the ‘case 2f’ Mach 10 condition in figure 15 and table 9. Figure 17(a) shows sound speed mapped over an $x{-}t$ diagram for the basic expansion tube; figure 17(a) shows the same result with a helium secondary driver initially at 150 kPa. It can be seen that, without a secondary driver, the ratio $a_{2}/a_{3}\ll 1$ , and driver noise would be expected to transmit to the test gas (Paull & Stalker Reference Paull and Stalker1992); figure 17(b), however, shows that the inclusion of a secondary driver strongly over-tailors the interface to the primary driver gas (i.e. $a_{sd2}/a_{sd3}\gg 1$ ), thereby meeting the acoustic buffer requirement.

Figure 17. Computed sound speed for Mach 10 scramjet condition (a) without and (b) with 150 kPa helium secondary driver.

While this arrangement would theoretically shield the downstream gas from primary driver disturbances, the air (region 2) is still under-tailored with respect to region 3; disturbances arising in the secondary driver gas due to Mylar diaphragm rupture, or the flow’s passage down the tube, could theoretically be transmitted to the test gas this way. Since the test gas itself lacks the intrinsic mechanism to prevent noise ingress, it might be expected that these conditions should be somewhat noisier than conditions where the test gas itself provides the acoustic buffer. Furthermore, for scramjet flow conditions, the secondary driver gas will be processed by a reflected shock, and the effect, if any, of this shock on transmission of acoustic disturbances has not been studied in detail.

Recalling figure 16(b), the Mach 14 scramjet flow condition, Case 3e, was experimentally tested with and without a secondary driver, and at a range of different secondary driver fill pressures. For these experiments a flow impact pressure measurement was made at the tube exit using 15° half-angle cone probes. These probes are described in Gildfind et al. (Reference Gildfind, Morgan, McGilvray and Jacobs2014), and were developed for earlier scramjet flow condition testing. The probe has a conical face in order to reduce heating, pressure loading and impact damage to the transducer sensing face, and to provide a generally less severe measurement than a blunt-body Pitot probe. Two probes were placed at radial offsets of $\pm 9~\text{mm}$ around the tube centreline, in accordance with the configuration shown in figure 18(a); cone pressure measurements are shown in figure 18(b–f) for the cases shown in figure 16(b).

Figure 18. Cone pressures for X2 Mach 14 scramjet flow condition, Case 3e from table 10, operated with and without a secondary driver, for a wide range of secondary driver fill pressures: (a) cone probe configuration; (b) no secondary driver (shot x2s2784); (c) 1 kPa secondary driver (shot x2s2787); (d) 10 kPa secondary driver (shot x2s2788); (e) 100 kPa secondary driver (shot x2s2789); (f) 398 kPa secondary driver (shot x2s2757).

Figure 18(b) shows the cone pressure traces for Case 3e without a secondary driver. A $100~{\rm\mu}\text{s}$ period of approximately steady cone pressure ( ${\approx}100~\text{kPa}$ ) is identified on the two traces. Through this region high-frequency noise can be observed, causing a fluctuation of approximately $\pm 10\,\%$ about the mean. After this period of identified test time, the cone impact pressure rises due to the test gas being underexpanded, higher temperature, lower Mach number, and therefore departing too far from the target flow condition.

The same condition is then shown with a range of secondary driver fill pressures in figure 16(c–f). Unsurprisingly, all of these cone pressure traces have a similar magnitude, which is lower than figure 16(b), and which is consistent with the lower experimental shock speeds shown in figure 16(b). Two differences are apparent; firstly, each cone pressure is increasing during the passage of the test gas, which arrives after the accelerator gas (the accelerator gas is the initial region of low impact pressure). For the purposes of comparison, the region of test time is identified as the region where cone pressures are 50–100 kPa; however, a test gas with this pressure gradient would not be useful for practical experiments. The test time according to this definition is approximately half the duration of the case without a secondary driver.

The second observation is that figures 16(c–e) do not exhibit the high-frequency noise of figure 16(b). This supports the argument that the secondary driver has suppressed disturbance transmission to the test gas. Observing 16(f), this is actually Case 3d; observing figure 11, the driver sound speed ratio is matched at this condition ( $a_{sd2}=a_{3}$ ), therefore this condition would be expected to be noisier, and this is observed.

The results from figure 16 suggest that further analysis is required to validate the acoustic buffer function of the secondary driver. While the secondary driver appears to reduce high-frequency noise in the cone pressure traces, it can be seen in figure 16 that useful test time is significantly reduced, and the impact pressure measurements do not exhibit a suitable period of test gas with steady flow properties. Studying and optimising the acoustics and wave processes of this mode of operation is beyond the scope of the current paper, and is part of an ongoing separate investigation.

The short test times of $50{-}100~{\rm\mu}\text{s}$ are not sufficient for scramjet engine tests, which usually require much longer duration test flows than high-enthalpy planetary entry studies. Anderson, Kumar & Erdos (Reference Anderson, Kumar and Erdos1990) suggests minimum test time requirements to achieve flow establishment inside a scramjet engine as follows:

where $t$ is the test time, $u$ is the flow velocity and $L$ is the overall model length. For a reasonably large 1 m long model at Mach 10 ( $3011~\text{km}~\text{s}^{-1}$ from table 6), the test time would therefore need to exceed $3\times 1.0/3011=1~\text{ms}$ , which is at least an order of magnitude higher than the test times identified in figure 18. However, this does not preclude expansion tubes from scramjet testing.

Figure 19. Details of X3 Mach 10 scramjet test flow experiment: (a) X3 expansion tube facility schematic; (b) X3 Pitot rake arrangement; (c) probe detail; (d) Pinckney probe pressure measurement; (e) cone probe pressure measurements.

The high total pressure capability of expansion tubes allows testing of smaller models through binary ( ${\it\rho}{-}L$ ) scaling, and test time can also be increased through the use of a contoured nozzle (Stewart, Jacobs & Morgan Reference Stewart, Jacobs and Morgan2001). McGilvray et al. (Reference McGilvray, Morgan and Jacobs2010) used X2 in the basic expansion tube mode (no secondary driver) with a contoured nozzle, to test a 2∕5 scale, 0.5 m long 2-D scramjet. These results compared well with tests of the full-scale engine in UQ’s T4 RST facility, at equivalent ${\it\rho}{-}L$ product.

For testing of larger engines, the problem of test time is directly addressed by using a large facility. Scramjet engine testing in X2 (McGilvray et al. Reference McGilvray, Morgan and Jacobs2010) and X2 flow condition development (Gildfind et al. Reference Gildfind, Morgan, McGilvray and Jacobs2014) were both conducted primarily as proof-of-concept studies in preparation for scaling up to UQ’s much larger free-piston-driven X3 facility. At 65 m length, with a 183 mm acceleration tube bore, X3 has equivalent total pressure capability to X2, but can test much larger models, and for significantly longer test times.

Figure 19(a) shows a scaled drawing of X3, which has recently been used to generate Mach 10 scramjet test flows in support of scramjet engine testing (Gildfind et al. Reference Gildfind, Sancho and Morgan2013). As part of the flow condition development process, nozzle exit flow measurements were made with the Pitot rake shown in figure 19(b). This rake was instrumented primarily with the cone probes also shown in figure 18, as well as a quasi-static-pressure ‘Pinckney’ probe (Pinckney Reference Pinckney1975). Referring to figure 19(d,e), the useful test time is approximately 1.25 ms, with a core flow diameter of 200 mm (using X3’s Mach 10 440 mm inner exit diameter nozzle). Noting that this condition used a secondary driver, cone impact pressures are much steadier than those shown in figure 18, although there remains a pressure gradient during the test time.

Figure 19 illustrates the potential for large expansion tubes to achieve an equivalent test time and model size capability to that of large RST facilities, but at much higher total pressures.