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Description
Background of the study
According to the recent paper of Asokakumar Sreekala et al. [1], the discrepancy between Navier Stokes Fourier-based Computational Fluid Dynamics and aero thermodynamic flight test data becomes noticeable from altitudes above 50 km. These researchers argue that bulk viscosity (BV) is required to complete the Navier-Fourier model and that the effect of vibrational nonequilibrium must be considered to accurately predict the high-temperature flow field of diatomic and polyatomic gases, in line with the results obtained by Klimov et al. two decades earlier [2]. This can be accounted for by methods such as state-to-state chemical kinetic approaches, which can be applied to arbitrary deviations from local thermodynamic equilibrium, but require much more computation [3, 4]. In the classical form of the Navier-Stokes equations, the Stokes hypothesis is applied, which is equivalent to assuming that the BV coefficient is zero. This is used in cases where compressibility is important [5, 6] because the BV is extremely difficult to measure [7]. However, during hypersonic re-entry it can have an influence on the shock wave structure [8]. According to Molevich et al [9], this parameter can even become negative in non-equilibrium media such as vibrationally excited plasma, leading to acoustic amplification [10, 11].
At ground facilities, very high enthalpy levels are achieved with plasma wind tunnels for extended test times [12]. The non-equilibrium vibrational kinetics in hypersonic flows is similar in many respects to that of electrical discharges in gases, but as mentioned in [13], “the main difference is that in the latter case the vibrational quanta are primarily pumped by electrons while during reentry they are pumped by recombination processes...” This difference may however be suppressed in our setup, thanks to the plasma ball formation (PBF), an acoustic plasma confinement mode we have discovered during the development of a pulsed sulfur plasma lamp [14], also observed by a team at UCLA [15]. Indeed, the molecular dissociation is then governed by the pure vibrational mechanisms [14], i.e. by the collisions between vibrationally excited molecules, rather than by electron impacts [16], p. 228. Another similarity between hypersonic plasmas and the plasma in our device is the radiative energy transfer resulting from the relaxation of the excited electronic states of the molecules, resulting in an optical emission spectrum that deviates significantly from Planck's law. In addition, the wall temperature, heat flow and plasma temperature are close to the STS-2 re-entry flight data at an altitude of about 50 km [17]. In a previous ESA project, the physics were established for the design of an experimental PBF device to provide a technique for testing radiation models in non-equilibrium plasma with control of the plasma translational and vibrational temperatures thanks to the power control parameters [18]. It has been pointed out that the PBF can pave the way for the measurement of the BV in addition to the velocity of sound. In the present study we determine how this coefficient can be calculated from the PBF measurements.
Methodology
Hollow spherical resonators are excellent tools for measuring the sound velocity c in the filling, derived from the resonant frequency, and the thermophysical properties that determine this velocity, such as the adiabatic factor γ [19]. The sphere is of particular interest because it has the highest degree of symmetry and the smallest surface-to-volume ratio. The PBF is also particularly interesting because it acts as a well-centred spherical acoustic generator, eliminating the need for acoustic transducers and thus avoiding the technical difficulties associated with mechanical coupling to the shell. Moreover, the PBF takes place in a spherical mode [14], eliminating acoustic losses due to friction against the wall. In addition, there is significant acoustic dispersion due to vibrational relaxation, allowing the study of this effect [14]. The BV can be investigated through its effect on the acoustic pressure, as the so-called dynamic pressure arises proportional to the particle velocity divergence. The spherical resonator is of particular interest because in spherical modes this parameter peaks at the centre. In the fundamental spherical mode, the acoustic pressure can be measured from the oscillation of the light emission [20]. Furthermore, since the experiments show that the quality factor Q is not significantly affected by the amplitude of the light emission modulation, it is possible to use the physical models developed in photo-acoustics to calculate Q, which in our case was measured by means of long power interruption tests [14]. From the formula given in reference [19], p. 103, using simple algebra, we obtain the following expression for the BV coefficient
η_b=(2 ϱ c^2)/(ω Q)-(3 η)/4-(γ-1)/γ λ/C_v
with ϱ the mass density, ω the resonance angular frequency, η the dynamic shear viscosity, λ is the thermal conductivity and C_v is the isochoric heat capacity per unit volume.
Results
Using the data given in [14], we proceed with the numerical application of this equation. The mass density ϱ is calculated to be 1.82 〖 mg/cm〗^3 from the bulb volume of 15.6〖 cm〗^3, and the filling conditions: 28.1 mg of sulphur, solid at room temperature, plus argon gas at low pressure to ignite the lamp by electrical breakdown of the gas.
The parameters C_v, γ and c are assumed to vary between their equilibrium and frozen values at the average temperature in the bulb, 1930 °C, i.e. 469 – 660 J∙K^(-1)∙〖kg〗^(-3), 1.19 - 1.28, and 586 - 608 m/s respectively. In this approach, the Chapman-Enskog theory was applied for single component gas, diatomic sulphur, to determine η and λ as functions of temperatures. At 1930 °C, the values obtained for η_b in 〖 10〗^(-3) Pa∙s are between 14.0 and 15.2, going from the frozen state to equilibrium (cf. Figure 1), i.e. with a magnitude 500 times greater than η.
Figure 1 Bulk viscosity coefficient η_b of the PBF at 1930°C (mainly diatomic sulphur vapour). From left to right, the thermodynamic state varies from fully equilibrium to fully frozen.
Conclusion
The results show that the bulk viscosity of a partially dissociated diatomic vapour can be 500 times greater than its dynamic shear viscosity at high temperatures and under strong vibrational conditions; neglecting it could therefore lead to serious errors in the calculation of hypersonic flows at boundary layers. The experiment was carried out with a mixture of sulphur vapour and argon; it would be worth trying other diatomic or light polyatomic gases such as air and CO2. The PBF phenomenon opens the way to a new technique for measuring bulk viscosity and sound velocity, providing an opportunity to improve low-cost numerical simulations of hypersonic flows.
[1] Asokakumar Sreekala, V., Chourushi, T., Sengupta, B., & Myong, R. S. (2022). Effects of bulk viscosity, vibrational energy, and rarefaction on flow and vorticity fields around simple bodies at hypersonic speeds. In AIAA SCITECH 2022 Forum (p. 1065).
[2] Klimov, A., Bityurin, V., & Serov, Y. (2001). Non-thermal approach in plasma aerodynamics. In 39th Aerospace Sciences Meeting and Exhibit (p. 348).
[3] Colonna, G., Armenise, I., Bruno, D., & Capitelli, M. (2006). Reduction of state-to-state kinetics to macroscopic models in hypersonic flows. Journal of thermophysics and heat transfer, 20(3), 477-486.
[4] Kustova, E., Nagnibeda, E., Oblapenko, G., Savelev, A., & Sharafutdinov, I. (2016). Advanced models for vibrational–chemical coupling in multi-temperature flows. Chemical Physics, 464, 1-13.
[5] Chikitkin, A. V., Rogov, B. V., Tirsky, G. A., & Utyuzhnikov, S. V. (2015). Effect of bulk viscosity in supersonic flow past spacecraft. Applied Numerical Mathematics, 93, 47-60.
[6] Kosuge, S., & Aoki, K. (2018). Shock-wave structure for a polyatomic gas with large bulk viscosity. Physical Review Fluids, 3(2), 023
[7] Taniguchi, S., Arima, T., Ruggeri, T., & Sugiyama, M. (2018, May). Shock wave structure in rarefied polyatomic gases with large relaxation time for the dynamic pressure. In Journal of Physics: Conference Series (Vol. 1035, No. 1, p. 012009). IOP Publishing.
[8] Galkin, V. S., & Rusakov, S. V. (2005). On the theory of bulk viscosity and relaxation pressure. Journal of applied mathematics and mechanics, 69(6), 943-954.
[9] Molevich, N., Galimov, R., Makaryan, V., Zavershinskii, D., & Zavershinskii, I. (2013, June). General nonlinear acoustical equation of relaxing media and its stationary solutions. In Proceedings of Meetings on Acoustics (Vol. 19, No. 1). AIP Publishing.
[10] Molevich, N. (2004). Acoustical properties of nonequilibrium media. In 42nd AIAA Aerospace Sciences Meeting and Exhibit (p. 1020).
[11] Molevich, N., & Riashchikov, D. (2021). Shock wave structures in an isentropically unstable heat-releasing gas. Physics of Fluids, 33(7).
[12] Hermann, T., Löhle, S., Zander, F., & Fasoulas, S. (2017). Measurement of the aerothermodynamic state in a high enthalpy plasma wind-tunnel flow. Journal of Quantitative Spectroscopy and Radiative Transfer, 201, 216-225.
[13] M. Capitelli, C. M. Ferreira, B. F. Gordiets, A. I. Osipov, "Plasma Kinetics in Atmospheric Gases", Springer-Verlag, 2000, p. 3
[14] G. Courret, P. Nikkola, S. Wasterlain, O. Gudozhnik, M. Girardin, J. Braun, S. Gavin, M. Croci, and P. W. Egolf, "On the plasma confinement by acoustic resonance", The European Physical Journal D, 71(8):1–24, 2017
[15] J. P. Koulakis, S. Pree, A. L.F. Thornton, and S. Putterman, "Trapping of plasma enabled by pycnoclinic acoustic force", Physical Review E, 98(4):043103, 2018
[16] M. Capitelli et al., Fundamental Aspects of Plasma, "Chemical Physics, Kinetics", Springer, 2016.
[17] Shinn, J., Moss, J., & Simmonds, A. (1982, June). Viscous-shock-layer heating analysis for the shuttle windward-symmetry plane with surface finite catalytic recombination rates. In 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference (p. 842).
[18] Courret, G., & Nikkola, P. (2022). Plasma ball formation: an experimental technique to test radiative models in non-equilibrium plasmas. In Proceedings of the 9th International Workshop on Radiation of High Temperature Gases for Space Missions, 12-16 septembre 2022, Santa Maria, Portugal.
[19] Bailey, R. T., Bernegger, S., Bicanic, D., Bijnen, F., Blom, C. W. P. M., Cruickshank, F. R., ... & Zuidberg, B. (2012). Photoacoustic, photothermal and photochemical processes in gases (Vol. 46). Springer Science & Business Media.
[20] Courret, G., Nikkola, P., Croci, M., & Egolf, P. W. (2020). Investigation of a molecular plasma from its acoustic response. IEEE Transactions on Plasma Science, 49(1), 276-284.
Summary
In the classical form of the Navier-Stokes equations, the Stokes hypothesis is applied, which is equivalent to assuming that the bulk viscosity (BV) coefficient is zero. However, this is used in cases where compressibility is important because the BV is extremely difficult to measure. At ground facilities, plasma wind tunnels are used to achieve very high enthalpy levels for extended test times. The non-equilibrium vibrational kinetics in hypersonic flows is similar in many respects to that of electrical discharges in gases, but the main difference is that in the latter case the vibrational quanta are primarily pumped by electrons, whereas during re-entry they are pumped by recombination processes. However, this difference can be suppressed in our setup thanks to plasma ball formation (PBF), an acoustic plasma confinement mode that we discovered during the development of a pulsed sulphur plasma lamp. The molecular dissociation is then governed by pure vibrational mechanisms. Using this phenomenon, we have shown that the bulk viscosity can be 500 times greater than the dynamic shear viscosity at high temperatures and under non-equilibrium conditions; neglecting it can therefore lead to serious errors in hypersonic computational fluid dynamics. The experiment was carried out with a mixture of sulphur and argon; it would be worth trying other diatomic or light polyatomic gases such as air. The PBF phenomenon opens the way for a new technique to measure bulk viscosity in addition to sound velocity under controlled conditions, providing an opportunity to improve low-cost numerical simulations of hypersonic flows.
