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Description
Background of the study
Recent studies [1, 2] indicate that Navier-Stokes Fourier-based CFD struggles to predict high-temperature hypersonic flows above 50 km, necessitating the inclusion of bulk viscosity (BV) and vibrational nonequilibrium effects [3, 4]. While the Stokes hypothesis often sets BV to zero, it is crucial for hypersonics [5, 6, 8], with potential for negative values in excited plasmas [9, 10, 11]. Plasma wind tunnels are used for testing [12], but {Link: this paper https://researchgate.net} describes a method using Plasma Ball Formation (PBF), a technique allowing precise control over plasma conditions [14, 15] and simulation of re-entry flight data [17].
Methodology
Hollow spherical resonators allow for precise measurement of sound velocity and adiabatic factors [19]. PBF serves as a central acoustic generator, eliminating conventional transducer issues [14]. This allows the study of acoustic dispersion and measurement of BV via its effect on acoustic pressure [14, 20]. The formula used for calculating the BV coefficient ($\eta_b$) is:
\begin{equation} \eta_b = \frac{2 \rho c^2}{\omega Q} - \frac{3 \eta}{4} - \frac{\gamma - 1}{\gamma} \frac{\lambda}{C_v} \end{equation}
with $\rho$ the mass density, $\omega$ the resonance angular frequency, $\eta$ the dynamic shear viscosity, $\lambda$ the thermal conductivity and $C_v$ the isochoric heat capacity per unit volume [14].
Results
Using data from {Link: a previous study https://researchgate.net} [14], the PBF experiment utilised a sulfur-argon mix at $1930\text{ }^\circ\text{C}$ ($1.82\text{ mg/cm}^3$ density, $15.6\text{ cm}^3$ volume). Calculations indicate that the BV ($\eta_b$) is up to 500 times greater than the dynamic shear viscosity ($\eta$), with values ranging from 14.0 to 15.2 $\times 10^{-3}\text{ Pa}\cdot\text{s}$, varying with thermodynamic state (Figure 1). This confirms the significance of the effect [14].
Conclusion
The study finds that for partially dissociated diatomic vapour, bulk viscosity significantly exceeds shear viscosity in high-temperature, non-equilibrium conditions, which is crucial for accurate hypersonic flow modelling. The PBF method offers a cost-effective, precise technique for measuring both sound velocity and bulk viscosity, with potential application for various gas mixtures (air, $\text{CO}_2$) 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.
References
[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.