1 Introduction

Accurate analysis of experimental data from hypersonic impulse facilities is difficult because of the limited amount of free-stream flow property data that can be measured at the nozzle exit. Reflected shock tunnels (RSTs) are particularly difficult because of short test periods and high total pressures. To overcome the lack of knowledge of the gas properties at nozzle exit in RSTs, experimenters typically calculate the quasi-steady gas properties based on assumptions that the test gas is processed, in a cascaded set of idealised processes that are essentially only forward coupled. Two examples of codes that are currently used to calculate free-stream conditions are ESTCj (similar to STN [1]) and NENZF [2] which both account for high-temperature gas effects, using equilibrium and non-equilibrium chemistry effects, respectively.

The processes modelled in this simplified analysis are the initial test gas is shocked, using the experimentally measured shock speed at the end of the shock tube; a reflected shock stagnates the test gas flow; the flow is isentropically relaxed to the experimentally measured mean nozzle supply region pressure; and the test gas is expanded to the experimentally measured mean Pitot pressure during the test time. The complex interactions of the waves, driver mechanics, diaphragm rupture mechanics, high temperature gas effects and boundary layers are not accounted for in this simplified analysis. Also, the flow exiting the nozzle can have both temporal and spatial non-uniformities which may need to be correctly accounted for in detailed analysis of experimental data. Generally, uncertainties quoted for these derived freestream properties only account for the inherent uncertainty from the experimental quantities that are used as inputs and do not indicate the uncertainty in the underlying assumptions. Thus, experimenters cannot make true assessments where freestream conditions are critical.

There have been several different strategies to more completely simulate the physical processes in a reflected shock tunnel. Three of these are (a) modelling from the near-stagnated supply region (using initial properties from the simplified calculation) to the nozzle after the reflected shock, in a steady state axisymmetric calculation; (b) a transient calculation modelling the same region from the incident shock using steady post-shock conditions [3]; and (c) transient modelling of the full facility from nominal fill conditions [4]. By having a calculation which correctly models all the transient and spatially distributed processes, added information such as defined duration of test time, unsteadiness of the test flow, time and level of driver gas contamination becomes available. However, this type of calculation currently comes at a extremely large computational cost and grid independence is difficult to achieve. Also, shot to shot experimental differences in measured properties cannot be accounted for using this technique, although they could be included in an overall uncertainty using a perturbation type sensitivity analysis.

This research investigates if the simplified analysis used by RST experimenters to generate flow properties is adequate in light of the many underlying assumptions. To do this, this paper extends the full-facility simulation method applied by Goozee et al. [4], to numerically simulate the flow in a complete free-piston reflected shock tunnel facility using coupled transient calculations. The present study uses both one- and two-dimensional axisymmetric modelling to simulate the processes from the initial piston motion to the flow into the experimental model in the test section. Not many transient facility simulations have used a hybrid approach, which only models the piston dynamics of the free piston compression process and primary diaphragm processes with L1d. This alters the simulation procedure used by Goozee et al. [4], which did not have a free piston driver, and thus used axisymmetric modelling for the entire simulation. To verify that the flow is being correctly modelled, measured histories of pressures at various key locations throughout the facility are compared to calculated histories. The data produced via this full facility numerical simulation is then investigated and compared with that from the simplified calculations.

2 Experimental facility, instrumentation and condition

Experimental data for the following work has been obtained for a low-enthalpy condition in the \(T^2\) free-piston reflected shock tunnel, located at The University of Queensland. This facility was operated with a conical Mach 8 nozzle, during a boundary layer/shock wave interaction experiment [5]. Time accurate measurements were made of compression tube pressure, stagnation pressure, nozzle exit static and Pitot pressures, as well as incident shock speed during the test campaign. Figure 1 shows a schematic of both the facility and the experimental instrumentation used in this study. The wall static pressure measurement was taken 0.129 m from the entry of the cylindrical duct model (inner diameter of 0.0381 m), positioned on the centreline at nozzle exit. The Pitot pressure probes use an inlet and flow diverter with an inner diameter of 2 mm and a length of 20 mm to direct the flow onto a PCB transducer. This ensures that there is no line of sight for diaphragm fragments to impact the transducers and increases the radial spatial resolution of Pitot pressure measurements. However, this has the undesirable effect of decreasing the experimental response of the Pitot pressure measurement due to viscous effects (as seen in Sect. 4.1).

Fig. 1
figure 1

Schematic of the \(T^2\) reflected shock tunnel facility, test section model geometries and instrumentation. Dimensions are in millimetres

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The experimental condition was tuned to give a nominal test flow Mach number of 8, to test hypersonic boundary/shock wave separation. The initial fill conditions are presented in Table 1. The experimental results have been averaged over nominal runs of the experiment and are presented in Table 2. The experimental test period definition is explained in Sect. 4.

Table 1 Fill conditions and primary diaphragm rupture pressure for Mach 8 condition in the \(T^2\) reflected shock tunnel
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Table 2 Experimental measurements and related uncertainties over the test period for the Mach 8 condition in \(T^2\) reflected shock tunnel
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3 Numerical simulation

Full transient and spatial numerical modelling of a free-piston reflected shock tunnel and test model require transient calculations of flow processes with strong discrete waves, moving boundaries, long tubes and influential boundary layer interactions. To transiently simulate the entire process from piston compression to flow through the test model, the calculation was broken into several smaller calculations, each having an appropriate level of complexity/computational cost needed to resolve the dominating flow features. Figure 2 shows how the calculation of the complete facility and model is separated. The following is a description of each of the calculations and the respective numerical codes (Sects. 3.1–3.4): 1. L1d simulation of the free piston compression and primary diaphragm rupture; 2. mb_cns2 simulation of incident shock in the shock tube to the secondary diaphragm rupture; 3. mb_cns2 simulation of the reflected shock process in the shock tube and expansion through the nozzle; 4. Eilmer3 simulation of initial cylindrical duct of the boundary/shock wave model. The calculations use the geometry and fill conditions described in Section 2, an ideal mixture of perfect gas (reservoir and driver gas) and equilibrium gas (shock tube/nozzle), and a fixed wall temperature of 298 K.

Fig. 2
figure 2

Schematic of simulation procedure and codes used for modelling the transient flow production in \(T^2\) and flow through the boundary/shock wave model. Red arrows show where transient flow properties are recorded and applied as inflow conditions to the next simulation

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3.1 Free piston compression and primary diaphragm rupture

The L1d code [6] was used, which is a quasi one-dimensional Lagrangian formulation of the gas dynamics coupled with empirical relations for viscous effects, area changes and point mass dynamics for piston motion. The simulation of the reservoir (800 gas cells), piston (front seal friction), driver gas (800 cells) and the first 0.511 m of the shock tube (100 cells) were undertaken using viscous effects. Loss regions were applied at both the piston launcher and primary diaphragm, where the rupture pressure was set at 48 MPa. Simulations were run with decreasing cell size until variations in incident shock speed were less than 1 %.

3.2 Incident shock in shock tube to secondary diaphragm rupture

The mb_cns2 code [7] was used (predecessor of the current Eilmer3 code), which is a Navier–Stokes transient compressible flow solver using a cell-centred, finite volume formulation for 2D/axisymmetric calculations. The axisymmetric simulation of the shock tube to the secondary diaphragm (2,325 cells axially, 115 cells radially, wall/throat clustered) started 0.511 m from the primary diaphragm, using transient inflow conditions from the first simulation. The Baldwin–Lomax turbulence model was applied. We have found that it is difficult to achieve grid independence for this type of simulation due to the strong waves, high densities and high Reynolds numbers. The cell-based Reynolds number in the stagnation region (Fig. 4) was significantly greater than the rule of thumb of unity which is used for shock tube simulations [8]. However, the stagnation pressure was seen to remain within the amplitude of the short duration fluctuations (less than 5 \(\mu \)s) when the grid was decreased by factors of 1.5 and 2 (see Fig. 3).

Fig. 3
figure 3

Comparison of critical measurements for increase in grid resolution of 33 \(\%\)

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Fig. 4
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Cell based Reynolds number for stagnation location in shock tube simulation

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3.3 Reflected shock in shock tube and nozzle expansion

To initialise the simulation, the secondary diaphragm wall was removed and the nozzle was included. The nozzle had a mesh resolution of 900 cells axially, 85 cells radially, clustered toward the wall and throat. Again, the Pitot pressure was seen to remain within the amplitude of the short duration fluctuations (less than 5 \(\mu \)s) when the grid was decreased by factors of 1.5 and 2 (Fig. 3).

3.4 Simulation of initial cylindrical duct model at nozzle exit

This was run using the newer Eilmer3 code [9], which is an extended version of the previous mb_cns2 codes to include three dimensional modelling and various useful functions. In particular, the ability to have user-defined boundary conditions was used to include both the spatial (interpolated between 25 \(\mu \)s data and 30 spatial points) and temporal (interpolated between 1 \(\mu \)s data on centreline) variations of inflow into the cylinder. The initial part of the cylinder was modelled up to the first pressure transducer (300 cells axially, 90 cells radially, clustered toward the wall). A laminar boundary layer was assumed, due to the low Reynolds number (on the order \(10^6\) throughout test time) at the first transducer. Simulations with the grid resolution decreased by a factor of 1.5 in both directions showed less than 1 \(\%\) difference in static pressure at the measurement location.

4 Results and discussion

4.1 Full facility simulations

A comparison of the experimental measurement histories and the numerical results at the same spatial locations are presented in Fig. 5. Note that the experimental data was taken from several different experimental runs, all with similar nozzle supply region pressures (within 5 \(\%\)). The overall timing is taken from the numerical simulations and initial shock location (peak pressure for compression tube) is used as a common reference time between experiments and simulations. To determine the test period, we used the criterion that the nozzle supply pressure (offset by a flow transit time) divided by the Pitot pressure has to initially plateau to signify the start and then ends when this varies by more than 10 \(\%\). The transit time was calculated by integrating the inverse of the wave speed (axial velocity plus the speed of sound) for the centreline data from the nozzle supply pressure measurement location to nozzle exit. Using this normalised pressure history, the test period was determined to be between 2,100–2,800 \(\mu \)s. The shock speed, measured between the stagnation sensor and a sensor 110 mm upstream, is \(1{,}081\pm 71\) m/s, which is within 1 \(\%\) of the simulation value of \(1,068\pm 11\) m/s.

Fig. 5
figure 5

Comparison of experimental measurements with numerical simulations at various key locations in the facility

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The comparison shows close agreement between the simulation and experimental histories for both the compression tube pressure and stagnation pressure. This indicates the simulation is adequately modelling the flow processes in both the free piston compression process and the test gas shock reflection processes, however the experimental stagnation pressure shows some large magnitude fluctuations (50 \(\mu \)s period, 15 MPa amplitude) which are not seen in the simulated data.The comparison of Pitot pressure at nozzle exit is close in the period after start up and over the final 1.2 ms shown, however, the experimental history has a sudden drop of 40 \(\%\) at 2.3 ms which is not present in the numerical calculation. The experimental Pitot probe is known to have a long response time (greater than 60 \(\mu \)s) because of viscous effects in the inlet and flow diverter. This explains the initial lag in rise time. Although the stagnation pressure history (both experimental and simulation) does not show a similar drop at 2.06 ms (related by the transit time), 100 \(\mu \)s earlier, a drop of 25 \(\%\) occurs in the wall static pressure in the cylindrical model. This behaviour is not consistent and no plausible explanation can be put forward. Apart from the period around the initial overshoot, the static wall pressure measured in the cylindrical model matches reasonably well with the simulated pressure.

4.2 Flow development

Figure 6 shows the flow development of the interaction of the reflected shock with the test/driver gas contact surface (interface) and expansion through the nozzle over the test period. The first frame (before the test period) shows the bifurcated reflected shock moving back towards the contact surface. Upstream of this, the remaining test gas has moved out towards the wall through the boundary layer. Large vortical structures can already be seen in the supply region test gas. Strong axial rarefaction waves are present in the upstream gas due to non-ideal driver performance. In the second frame, the reflected shock has passed the contact surface, causing driver gas leakage downstream through vortices. This behaviour is similar to that described by Chue and Eitelberg [3] and Goozee et al. [4] in simulations of reflected shock tunnels which had zero to weak fluctuations propagating from the driver. The last frame (after the test period) shows the flow just behind the reflected shock having larger vortical structures, due to a lower level of rarefaction waves. Driver gas arrival does not pre-emptively end the test period for this condition. Between the three frames the nozzle shows start-up with boundary layer development, discrete waves moving down with a fairly uniform flow, and radial non-uniformities coming from the throat propagating downstream.

Fig. 6
figure 6

Flow development of reflected shock and nozzle expansion of simulated \(T^2\) Mach 8 condition at 2, 2.5 and 3 ms (top to bottom). Above the centreline: log root mean square of density gradients; Below the centreline: air to Nitrogen mass fractions. Radial direction scaled by a factor of 3

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Table 3 Computed values of flow properties and variations at nozzle exit across the core flow
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4.3 Spatial and temporal variations in core flow

Table 3 presents the numerically calculated values over the test period, initial 25 \(\mu \)s and final 25 \(\mu \)s. As indicated from the time histories presented in Fig. 5, there are significant changes of flow properties over the test period. This does not necessarily mean that the flow cannot be used to undertake experiments, as the flow may be considered quasi-steady assuming the convective derivatives are greater than the temporal derivatives for a given point location measurement. The calculations presented here can be used to justify this assumption. The standard deviation of properties for the core flow data is presented for space and time (\(\sigma _y\) and \(\sigma _t\), respectively), using the average of the property. This provides a clear definition of the variations over the test period and indicates that the temporal variations dominate all parameters except for radial velocity.

4.4 Simplified calculations

Calculations based on empirical relations to derive flow properties are mainly used by experimenters to determine the flow conditions at nozzle exit. As these calculations use simplifying assumptions and time averaged data from singular spatial locations in the nozzle supply region and nozzle exit, it is not clear how well these reflect the true flow processes to nozzle exit. Therefore, to gauge how well these calculations reflect the true flow, freestream properties are calculated in ESTCj using the estimated values of shock speed, stagnation pressure, Pitot pressure and values at measurement locations averaged over the test period, first 25 \(\mu \)s and last 25 \(\mu \)s. Note that these estimated values come from the simulation data. Using the numerically derived properties at measurement locations allows for a direct comparison of calculation methods and provides the experimenter with an insight into the use the empirical relations. The flow properties at nozzle exit from the ESTCj calculations are presented in Table 4. Compared to the core flow properties from the full numerical calculation, presented as percentage differences to the ESTCj values in the same table, agreement is quite good. The discrepancies between the two sets of data are all less than the uncertainty in ESTCj due to input uncertainties from the experimental measurements as calculated by Dann [5]. For this particular condition and facility, ESTCj can give estimates of average flow properties at nozzle exit that lie within the experimental uncertainties.

Table 4 Flow properties at nozzle exit calculated using ESTCj using simulated values of stagnation pressure, Pitot pressure and shock speed
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5 Conclusion

Simulations were presented of a complete free-piston driven shock tunnel operating a low enthalpy condition. The simulations were validated by detailed comparison with experimental data at various key locations throughout the facility and the test model. Flow development occurs with non-uniform driver conditions in a similar manner to that presented previously in the literature. Spatial and temporal non-uniformities in the core flow at nozzle exit were investigated and the results demonstrate the advantages of performing these computationally costly calculations when experimenters require this information. By comparing flow properties at nozzle exit derived from ESTCj simplified calculation to the spatially averaged data from the full simulation, it was shown that the simple method using simple wave processes could sufficiently model the flow within experimental uncertainties. However, full facility simulations as presented here can allow reflected shock tunnel experimenters to either show that temporal and spatial variations can be ignored or provide accurate information to further analyse their experimental measurements. This type of verification study is required over a range of facilities and conditions to assure the free-stream properties calculated using the accepted simplified calculation can provide suitable levels of accuracy.

Modelling higher enthalpy conditions will require a more involved solution process, as the inclusion of finite-rate chemistry and possibly multi-temperature modelling is required to correctly replicate the nozzle expansion process. Thus, a further calculation can be added by breaking the shock tube calculation, which could still assume equilibrium chemistry, from the nozzle calculation at the throat. Facilities with larger diameters and/or higher total pressure flow conditions will have boundary layers that are relatively thin in comparison to the shock tube diameter. In these situations, the computational power required to adequately resolve the boundary layer will need to be substantially increased in order to predict the effects it has on the test flow at the nozzle exit.