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Description
Introduction
The vast distances between Earth and other celestial bodies within our solar system requires space vehicles to transit between them with very large velocities. This ensures transit time is not prohibitively long, however results in vehicles entering the destination atmosphere with an extremely high velocity. This in turn requires significant deceleration of the vehicle upon arrival, commonly vehicle drag provides the deceleration necessary to slow the vehicle to near subsonic speeds.
The strong bow shock processes the free-stream atmosphere to extremely high pressures and temperatures, resulting in thermochemical non-equilibrium as the gas dissociates and ionises.
These phenomena affect the heat flux (with both convective and radiative contributions), drag, and communication processes of hypersonic vehicles [1].
Hypersonic ground testing facilities enable such entry flow conditions to be interrogated prior to vehicle commissioning. In particular, shock tubes offer the ability to analyse the post-shock environment, with spectroscopic measurements commonly used to investigate the relevant thermochemical processes. However, these facilities are not exactly analogous to a normal shock problem, with viscous effects affecting boundary layer growth and shock structure, and non-ideal behaviour unique to individual experiments and experimental facilities resulting in shock speed variation [2-7].
As a primary use of shock tube datasets is to validate, improve or quantify the uncertainty of thermochemical models, numerical results generated using such thermochemistry models for comparison to experimental datasets must also account for these shock-tube flow behaviours [8]. This requires an a posteriori approach, using experimental measurements to reconstruct the experimental flow behaviour with improved fidelity [9,10]. In particular, recent advances in a posteriori modelling of hypersonic shock tubes enables two-dimensional reconstruction of experimental conditions, accounting for both shock trajectory and resolving the boundary layer growth and properties [11].
A recent work postulated that observed infrared emission lines from atomic species may be more susceptible to two-dimensional effects, with the line-of-sight measurement observing any shock curvature and radial variation in gas properties in the boundary layer [10]. Additionally, it has been identified that these lines are sensitive to variations in shock speed [4]. Therefore this work will analyse the influence of two-dimensional and shock trajectory effects on atomic emission lines in the infrared region.
Methodology
Flow Solvers
Two numerical solvers will be utilised in this analysis, NESS2D and NESS2D-t [11, 12]. NESS2D is a steady, viscous, two-dimensional a posteriori method, with a two-temperature reacting gas thermochemistry model. This method uses an outflow pressure variable to hold the shock at a desired position in the domain, resulting in a two-dimensional flow solution for a constant shock-speed experiment [11].
NESS2D-t is an extension of the NESS2D method, and evolves the test slug down the tube using the Newton-Raphson method to solve both the backward Euler problem and the outflow pressure required to pin the shock at the desired location within the flow domain. NESS2D-t therefore inherently captures both the influence of the transient boundary layer growth on the core flow, the boundary layer properties themselves, and the influence of the variation in shock speed [12].
Radiance Model
NASA's NEQAIR v15.2 [13, 14] is a line by line code developed to estimate radiance emissions, given a specified profile of temperatures and number densities.
This allows an estimate of predicted radiance to be made by coupling a shock tube solver's output of spatially resolved temperature and number density profiles with the NEQAIR program.
To ensure two-dimensional effects are captured, each radial slice at each axial station is solved independently using the non-local, non-Boltzmann line-of-sight (\texttt{N F N}) radiance solution [15]. However, this is a simplified approach as it removes dependence on non-local effects in the axial direction and any optical effects which require a ray tracing algorithm to be captured.
For the comparisons using the centreline result, it is assumed that the centreline solution is homogenous across the diameter of the tube, with the boundary at the last line-of-sight point modelled as a greybody with 0 emissivity and a transmissivity of 1.
Additionally, the vibrational weighting of non-equilibrium excitation was computed using the lowest vibrational level. This is achieved by setting the $\mathrm{NEQ}_{\mathrm{QSS},\mathrm{EXC},V}$ flag as 1, which is the new default behaviour in NEQAIR 15.3.
To enable comparison to experimental results, where possible the experimentally determined spatial resolution functions (SRF) and instrument line shapes (ILS) were used to convolve the NEQAIR simulations.
Experimental and Numerical Setup
The test series chosen for analysis was series 50 from the NASA-EAST test facility, as it was previously identified as being affected by shock speed variation effects [4].
This series ran tests at a variety of pressures through 79% $N_2$ and 21% $O_2$ gas, focussing on lunar return speeds between 8-11 km/s through a 10.16 cm diameter tube.
This work enables the influence of the shock trajectory to be isolated from the two dimensional effects, by comparing the synthetic integrated line-of-sight of the full NESS2D-t result, the NESS2D-t centreline and the steady NESS2D result using the shock speed observed at the viewing window.
For brevity, results will focus on test ES50-40, nominally an 8.56 km/s shock through 26.6 Pa synthetic air. However, there is significant deceleration of the shock wave, shown in Figure 1, where the shock velocity decreased by 2 km/s over approximately 4 m to be 8.56 km/s at the viewing window of the optical emission spectroscopy (OES).
Figure 1. Shock speed profile for ES50-40, a 8.56 km/s shock through 79% $N_2$, 21% $O_2$ gas by volume at 26.6 Pa.
The mesh converged domain used a clustered, rectangular structured mesh [11], with 400 and 150 axial and radial points respectively. The Cruden 2017 thermochemistry model was used as the reaction scheme [16], the collision integrals recommended by Wright [17] were used by the in-house OCEAN library to evaluate the multicomponent transport properties.
Overview of Results
The centreline comparison of temperatures and number density profiles (Figures 2 to 5) immediately demonstrates the importance of including the effect of shock trajectory, with the difference between the steady centreline temperature and the transient formulation totalling 2000 K by 100 mm post shock. Visible departure begins from 30 mm post shock, therefore it is expected that this will be reflected in the synthetic spectra produced using the respective methods.
Figure 2. Temperature profiles for ES50-40, a 8.56 km/s shock through 79% $N_2$, 21% $O_2$ gas by volume at 26.6 Pa.
Similarly, the influence of shock trajectory has a significant influence on the electron number density, growing to be nine times larger compared to the steady case.
Figure 3. Electron number density profiles for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
This increased temperature profile at the rear of the test slug of the NESS2D-t solution is a direct consequence of that gas being processed by a significantly stronger shock.
This also increases the ionisation the transient NESS2D-t result, while the atomic number density of oxygen drops by 30\%, and the number density of atomic nitrogen decreases by 24\% compared to the steady NESS2D result.
Figure 4. Atomic oxygen number density profiles for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
Figure 5. Atomic nitrogen number density profiles for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
As an interesting outcome, the centreline mass loss due to the boundary layer growth shown in Figure 6 demonstrates quite good agreement between the NESS2D steady result, and the transient model of NESS2D-t. However, there is significant departure from the Mirels boundary layer [2] profile, due to the dual effects of shock curvature and the simplifying assumptions made by Mirels regarding the local resizing of the boundary layer [2].
Figure 6. Mirels number profiles for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
The wavelength region selected for analysis is between 700-900 nm, where strong atomic lines are present. Particularly, the radiance centred around 745, 820 and 868 nm corresponds to atomic nitrogen transitions, while the lines centred at 777 and 844 nm correspond to atomic oxygen transitions. These atomic lines are particularly strong in this case, due to the significant dissociation and ionisation present, thus allow the influence of flow two-dimensionality to be investigated at multiple transitions.
We begin with the two atomic oxygen lines, the 777 nm triplet and the 844 nm triplet. Figure 7 depicts the emitted radiance associated with the 777 nm triplet (integrated radiance between 770-785 nm), immediately the issues with using this line for inferring information about the core flow become apparent. The three approaches diverge from approximately 7 mm post shock onward, corresponding to which is the location where the boundary layer becomes sufficiently large to begin absorbing the emission from the core flow. This results in the predicted radiance being approximately 15 times smaller when the boundary layer effects are included, with the experimental value existing between the radially homogenous solution and that which includes boundary layer effects. The thermochemically non-equilibrium flow environment within the boundary layer results will have high uncertainty due to the effects of surface catalycity [10], as well as the optical effects neglected in this analysis. These difficulties limit the utility of the 777 nm line for use in thermochemical model evaluation, and care should be taken if it is to be used for spectral fitting.
Figure 7. Atomic O radiance between 770-785 nm for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
In comparison, the atomic oxygen triplet at 844 nm displays some boundary layer dependence, however the NESS2D-t centreline solution is consistently within 7\% of the full solution. Additionally, the importance of modelling the variable shock speed is clear, with the steady NESS2D simulation completely unable to capture the behaviour of the integrated radiance. This analysis indicates that the 844 nm transition is the more reliable atomic oxygen line to model, despite it being a weaker transition compared to the 777 nm line.
Figure 8. Atomic O radiance between 838-849 nm for ES50-40, a 8.56 km/s shock through 79% $N_2$, 21% $O_2$ gas by volume at 26.6 Pa.
Finally, the three atomic nitrogen lines, the 745 nm, 820 nm and the 868 nm are depicted in Figures 9-11. Agreement between the steady and transient solutions is poor after 20 mm, with the transient modelling significantly improving agreement with the experimental results.
Figure 9. Atomic N radiance between 738-752 nm for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
Furthermore, the absorption of the emitted radiance the boundary layer improves agreement with experiment in each of the Figures, with the effect of the boundary layer becoming more pronounced from 50 mm post-shock.
Figure 10. Atomic N radiance between 810-830 nm for ES50-40, a 8.56 km/s shock through 79% $N_2$, 21% $O_2$ gas by volume at 26.6 Pa.
This culminates in a discrepancy between centreline NESS2D-t and the full NESS2D-t result of 8\% by 100 mm post shock, consistent for all of the atomic nitrogen lines in the infrared region.
Figure 11. Atomic N radiance between 854-876 nm for ES50-40, a 8.56 km/s shock through 79\% $N_2$, 21\% $O_2$ gas by volume at 26.6 Pa.
Conclusion
This work offers novel insights into the influence of two-dimensional effects and shock speed variation on atomic line features in the infrared region. In particular, this work highlights issues with the oxygen 777 nm line being sensitive to the properties of the boundary layer, however the 844 nm oxygen triplet remains relatively unaffected by the boundary layer properties. The remaining atomic features in the infrared region had agreement within 8\% between the full NESS2D-t model and the centreline NESS2D-t result, with the visible divergence increasing steadily from 50 mm post-shock. This work informs both uncertainty quantification of thermochemistry models, and the importance of using wavelength regions which behave quasi-one-dimensionally if using a quasi-one-dimensional approach to spectral fitting or rate optimisation.
Further work to examine the ultra-violet region, and the broader envelope of test-conditions, will further improve understanding of regions and conditions where two-dimensional effects are prevalent.
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Summary
The influence of shock speed variation and non-uniform boundary layer properties on infrared atomic emission lines are assessed for Earth return shock tube experiments.