ISSN 2658–5782

DOI 10.21662

DOI 10.21662

Electronic Scientific Journal

2022. Vol. 17. Issue 1, Pp. 38–50

URL: http://mfs.uimech.org/mfs2022.1.004,en

DOI: 10.21662/mfs2022.1.004

URL: http://mfs.uimech.org/mfs2022.1.004,en

DOI: 10.21662/mfs2022.1.004

Modeling of dynamic phenomena in aqueous foams (review)

Bolotnova R.Kh., Gainullina E.F.

Mavlyutov Institute of Mechanics, Ufa

Analytical review of investigations aimed at studying the aqueous foam behavior under the shock-wave impact is given. The main approaches and theoretical approximations used in the development of aqueous foams models are analyzed. The two-phase gas-drop model of aqueous foam is considered, based on the laws of conservation of mass, momentum, and energy in accordance with one-pressure, two-velocity, two-temperature approximations. The model describes foam dynamics under a high - intensity impact, which destroys the foam structure into microdroplets, and takes into account interfacial heat transfer, interfacial drag and virtual mass forces and the phenomenon of foam syneresis (deposition). The numerical implementation of the model was carried out using new solvers in the OpenFOAM software. The analysis of the numerical solution of the spherical explosion in aqueous foam is given for the conditions of the literature experimental data. The syneresis influence on the intensity and velocity of shock wave propagation is shown. The two-phase model of aqueous foam is considered, which takes into account its elastic- viscous-plastic properties for conditions of weak impact that does not destroy foam structure. The model takes into account the elastic properties of aqueous foam at small deformations and describes it as a non-Newtonian fluid when the foam changes its state from the elastic to viscoplastic. The weak shock wave propagation dynamics in the layer of aqueous foam with the formation of a two-wave structure, consisting of the main wave and the elastic precursor ahead of it, is analyzed. The process of local aqueous foam compaction zone formation, followed by a gaseous region, behind the shock wave front is shown. The reliability of the obtained results of numerical studies is confirmed by their agreement with the literature experimental data

shock wave,

aqueous foam,

OpenFOAM software,

mathematical and numerical modeling

*Purpose.* Review of existing aqueous foam models and investigations analysis of the strong and weak shock waves dynamics in aqueous foam, taking into account the rheological features of its behavior depending on the intensity of the impact.

*Methodology.* In the considered works, the assumption about the foam structure destruction into microdroplets by a strong shock wave was used, which made it possible to apply the gas-droplet model in the single-pressure, two-velocity, two-temperature approximations to describe the dynamics of aqueous foam, taking into account the interfacial interaction forces, heat transfer processes, and the phenomenon of foam syneresis. When studying the interaction of aqueous foams with weak shock waves, a two-phase model was used, taking into account the foam elastic properties by Hooke's law and viscoplastic flow above the von Mises yield critical value - by the Herschel-Bulkley model. To describe the properties of air and water, the equations of state in the form of Peng-Robinson and Mie-Gruneisen are used. The applied models are numerically implemented using the finite volume method by creating and compiling new solvers in the OpenFOAM software.

*Findings.* In the works noted in the review, numerical study of the powerful spherical explosion dynamics in aqueous foam based on the gas-droplet foam model was performed for the experimental data of E. Del Prete et al. It is shown, that the process of foam syneresis leads to the shock wave front velocity increase in the vertical upward direction due to the foam upper layers density reduction during its deposition. The comparative analysis of the calculated and experimental pressure oscillograms in the shock wave at the positions of the sensors showed, that the numerical solutions, taking into account the foam syneresis, have the best agreement with the experimental data of E. Del Prete et al., and the calculations relative error does not exceed 4%.

In the next reviewed paper, the elastic-viscous-plastic model of aqueous foam is considered for the numerical study of the dynamics of weak air shock waves in a layer of aqueous foam under the experimental data conditions of G. Jourdan et al. and M. Monloubou et al. Numerical solutions in the form of pressure profiles demonstrate a two-wave configuration of the shock pulse in the foam: the main compression wave is ahead of the elastic low-amplitude precursor. Comparative analysis of numerical solutions for the presence of a foam layer in the shock tube showed a decrease in the velocity of the shock pulse front by a factor of 2.5 compared to the gas. The impact of a shock wave on the foam layer leads to the formation of a local zone of increased liquid volume fraction, located behind the front of the shock wave.

Agreement between calculations and experimental data G. Jourdan et al. and M. Monloubou et al. confirms the reliability of the obtained numerical solutions.

*Value.* The review provides the analysis of existing aqueous foam models that describe its behavior under shock-wave impact.

The two-phase gas-droplet mixture model is considered for studying the dynamics of strong shock waves in aqueous foam in accordance with one-pressure, two-velocity, two-temperature approximations, taking into account interfacial heat transfer, interfacial drag force, virtual mass force, and the phenomenon of foam syneresis. It is shown that foam syneresis leads to increase in the shock wave propagation in the vertical upward direction due to the density reduction of the upper foam layers.

The aqueous foam dynamics under the influence of weak shock wave, which do not destroy its structure, is analyzed. The elastic-viscous-plastic behavior of aqueous foam is described in terms of a two-phase mixture model. Elastic properties are taken into account in accordance with Hooke's law. The transition to a viscoplastic flow was controlled by the von Mises yield criterion, and above the critical yield value - by the Herschel - Bulkley law for a non-Newtonian fluid.

The causes for the two-wave structure of the shock wave, which consists of the elastic precursor ahead of the main compression wave, are detailed. Features of the increased liquid volume fraction region formation beyond the shock wave front, behind which there is the foam-free zone formed as a result of the foam boundary movement following the shock wave, are revealed.

Numerical implementation of the models was carried out using new solvers of the OpenFOAM software. There is a satisfactory agreement between the calculations and experimental data.

- Rudinger G. Effect of a finite volume occupied by particles on the dynamics of a mixture of gas and particles // Raketn. Tekh. Kosmonavt. 1965. No. 7. Pp. 1217–1222.
- Borisov A.A., Gelfand B.E., Kudinov V.M., Palamarchuk B.I., Stepanov V.V., Timofeev E.I. and Khomik S.V. Shock waves in water foams // Acta Astronautica. 1978. V. 5, No. 11/12. Pp. 1027–1033.

DOI: 10.1016/0094-5765(78)90007-3 - Vakhnenko V.A., Kudinov V.M., Palamarchuk B.I. Damping of strong shocks in relaxing media // Combust. Explos. Shock Waves. 1984. V. 20. Pp. 97–103.

DOI: 10.1007/BF00749928 - Borisov A.A., Gelfand B.E., Timofeev E.I. Shock waves in liquids containing gas bubbles // Int. J. Multiphase Flow. 1983. V. 9, No. 5. Pp. 531–543.

DOI: 10.1016/0301-9322(83)90016-2 - Panczak T.D., Krier H. Shock propagation and blast attenuation through aqueous foams // J. Hazard. Mat. 1987. V. 14. Pp. 321–336.

DOI: 10.1016/0304-3894(87)85004-5 - Vasil’ev E.I., Mitichkin S.Yu., Testov V.G., Haibo H. Numerical simulation and experimental research on the effect of syneresis on the propagation of shock waves in a gas-liquid foam // Technical Physics. 1997. V. 42, No. 11. Pp. 1241–1248.

DOI: 10.1134/1.1258855 - Liverts M., Ram O., Sadot O., Apazidis N., Ben-Dor G. Mitigation of exploding-wire-generated blast-waves by aqueous foam // Phys. Fluids. 2015. V. 27. 076103.

DOI: 10.1063/1.4924600 - Sembian S, Liverts M., Apazidis N. Attenuation of strong external blast by foam barriers // Phys. Fluids. 2016. V. 28. 096105.

DOI: 10.1063/1.4963243 - Kudinov V.M., Palamarchuk B.I., Gel’fand B.E., Gubin S.A. [Shock wave parameters during detonation of explosive charges in a foam]Dokl. Akad. Nauk SSSR [The Proceedings of the USSR Academy of Sciences]. 1976. V. 228, No. 3. Pp. 555–557 (in Russian).

MathNet: dan40111 - Kudinov V.M., Palamarchuk B.I., Gelfand B.E., Gubin S.A. Shock waves in gas-liquid foams // Appl. Mech. 1977. V. 13, No. 3. Pp. 279–283.

DOI: 10.1007/BF00882681 - Britan A., Shapiro H., Liverts M., Ben-Dor G., Chinnayya A., Hadjadj A. Macro-mechanical modelling of blast wave mitigation in foams. Part I: Review of available experiments and models // Shock Waves. 2013. V. 23. Pp. 5–23.

DOI: 10.1007/s00193-012-0417-4 - Zhdan S.A. Numerical modeling of the explosion of a high explosive (HE) charge in foam // Combust., Explos., Shock Waves. 1990. V. 26, No. 2. Pp. 221–227.

DOI: 10.1007/BF00742416 - Britan A.B., Zinovik I.N., Levin V.A. Breaking up foam with shock waves // Combust. Explos. Shock Waves. 1992. V. 28. Pp. 550–557.

DOI: 10.1007/BF00755733 - Britan A.B., Vasil’ev E.I., Kulikovskii V.A. Modeling the process of shock-wave attenuation by a foam screen // Combust. Explos. Shock Waves. 1994. V. 30, No. 3. Pp. 389–396.

DOI: 10.1007/BF00789435 - Surov V.S. Propagation of Waves in Foams // High Temperature. 1996. V. 34, No. 2. Pp. 280–287.

MathNet: tvt2731 - Surov V.S. Interaction of Pressure Waves with an Inhomogeneous Shield of Foam // High Temperature. 1997. V. 35, No. 1. Pp. 21–26.

MathNet: tvt2425 - Rakhmatulin Kh.A. [Gas and wave dynamics] Gazovaya i volnovaya dinamika. M.: Moscow University Publishing House, 1983. P. 200 (In Russian).
- Baer M., Nunziato J. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials // Int. J. of Multiphase Flow. 1986. V. 12, No. 6. Pp. 861–889.

DOI: 10.1016/0301-9322(86)90033-9 - Saurel R., Abgrall R. A multiphase Godunov method for compressible multifluid and multiphase flows // J. Comput. Phys. 1999. V. 150. Pp. 425–467.

DOI: 10.1006/jcph.1999.6187 - Saurel R., Le Metayer O. A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation // J. Fluid. Mech. 2001. V. 431. Pp. 239–271.

DOI: 10.1017/S0022112000003098 - Chinnayya A., Daniel E., Saurel R. Modelling detonation waves in heterogeneous energetic materials // J. Comput. Phys. 2004. V. 96, No. 2. Pp. 490–538.

DOI: 10.1016/j.jcp.2003.11.015 - Abgrall R., Saurel R. Discrete equations for physical and numerical compressible multiphase mixtures // J. Comput. Phys. 2003. V. 186, No. 2. Pp. 361–396.

DOI: 10.1016/S0021-9991(03)00011-1 - Nigmatulin R.I. Dynamics of Multiphase Media. New York: Hemisphere, 1990. P. 532.
- Bolotnova R.Kh., Agisheva U.O. [Features of shock wave propagation in aqueous foams with non-uniform density] Trudy Instituta Mexaniki im. R.R. Mavlyutova Ufimskogo Nauchnogo Centra Rossijskoj Akademii Nauk [Proceedings of the Mavlyutov Institute of Mechanics]. 2012. V. 9, No. 1. Pp. 41–46 (in Russian).

DOI: 10.21662/uim2012.1.007 - Bolotnova R.Kh., Gainullina E.F. [A research of damping properties of aqueous foam under the impact of spherical shock waves] Izvestiya VUZov. Povolzhskij Region. Fiziko-Matematicheskie Nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2017. No. 2. Pp. 108–121 (in Russian).

DOI: 10.21685/2072-3040-2017-2-9 - Bolotnova R.Kh., Gainullina E.F. Influence of Heat-exchange Processes on Decreasing an Intensity of a Spherical Explosion in Aqueous Foam // Fluid Dynamics. 2019. V. 54, No. 7. Pp. 970–977.

DOI: 10.1134/S0015462819070024 - Nigmatulin R.I., Bolotnova R.Kh. Wide–Range Equation of State of Water and Steam: Simplified Form // High Temperature. 2011. V. 49, No. 2. Pp. 303–306.

DOI: 10.1134/S0018151X11020106 - Bolotnova R.Kh., Gainullina E.F., Nurislamova E.A. [Modeling of the spherical explosion attenuation process using aqueous foam] Mnogofaznye sistemy [Multiphase systems]. 2019. V. 14, No. 2. Pp. 108–114 (in Russian).

DOI: 10.21662/mfs2019.2.015 - Bolotnova R.Kh., Gainullina E.F. Modeling the Dynamics of Shock Impact on Aqueous Foams with Account for Viscoelastic Properties and Syneresis Phenomena // Fluid Dynamics. 2020. V. 55, No. 5. Pp. 604–608.

DOI: 10.1134/S001546282005002X - Gainullina E.F. [Influence of aqueous foam syneresis on the shock wave propagation velocity] Mnogofaznye sistemy [Multiphase systems]. 2020. No. 3. Pp. 152–160 (in Russian).

DOI: 10.21662/mfs2020.3.126 - Bolotnova R.Kh., Gainullina E.F. Influence of the dissipative properties of aqueous foam on the dynamics of shock waves // Journal of Applied Mechanics and Technical Physics. 2020. V. 61, No. 4. Pp. 510–516.

DOI: 10.15372/PMTF20200402 - Bolotnova R.Kh., Gainullina E.F. Modeling of weak shock waves propagation in aqueous foam layer // J. Phys.: Conf. Ser. 2021. V. 2103. 012217.

DOI: 10.1088/1742-6596/2103/1/012217 - Gainullina E.F. [Influence of viscoelastic properties of aqueous foam on the dynamics of weak shock waves] Vestnik Bashkirskogo universiteta [Bulletin of Bashkir University]. 2021. V. 26, No. 3. Pp. 548–553 (in Russian).

DOI: 10.33184/bulletin-bsu-2021.3.1 - Landau L.D., Lifshitz E.M. Fluid Mechanics. Oxford: Pergamon Press, 1987. P. 554.
- Schiller L., Naumann Z. A Drag Coefficient Correlation // Z. Ver. Deutsch. Ing. 1935. V. 77. Pp. 40—65.
- Ranz W.E., Marshall W.R. Evaporation from Drops // Chem. Eng. Prog. 1952. V. 48, No. 22. Pp. 141–146.
- Peng D.Y., Robinson D.B. A new two–constant equation of state // Industrial and Engineering Chemistry: Fundamentals. 1976. V. 15. Pp. 59–64.

DOI: 10.1021/i160057a011 - OpenFOAM. The Open Source Computational Fluid Dynamics (CFD) Toolbox.

http://www.openfoam.com <

li>Del Prete E., Chinnayya A., Domergue L., et al. Blast Wave Mitigation by Dry Aqueous Foams // Shock Waves. 2013. V. 23, No. 1. Pp. 39–53.

DOI: 10.1007/s00193-012-0400-0 - Vetoshkin A.G. Physical foundations and technology of foam separation processes. Moscow: Infra-Inzheneriya, 2016. 404 p.
- Dollet B., Raufaste C. Rheology of aqueous foams // Comptes Rendus Physique. 2014. V. 15. Pp. 731–747.

DOI: 10.1016/j.crhy.2014.09.008 - Monloubou M., Le Clanche J., Kerampran S. New experimental and numerical methods to characterise the attenuation of a shock wave by a liquid foam // Actes 24eme Congres Francais de Mecanique. Brest: AFM. 2019. 255125.

https://cfm2019.sciencesconf.org/255125/document - Jourdan G., Marian C., Houas L. et al. Analysis of shock-wave propagation in aqueous foams using shock tube experiments // Phys. Fluids. 2015. V. 27. 056101

DOI: 10.1063/1.4919905