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Development of a benchmark solution in compressible liquid flows: analytical solution to the water shock tube problem

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Abstract

This paper discusses the development of an analytical solution to a pure liquid shock tube problem with water as the working fluid. The solution procedure associated with classical gas shock tube problem has been extended for the case of a compressible liquid. The analytical model employed accounts for the compressibility effects in liquid water using the high-accuracy Modified NASG equation of state. The solution methodology presented in this work is unique in that it provides comprehensive modeling of the complete physics of the water shock tube problem. The work also demonstrates the applicability of analytical solution over a wide range of pressures and temperatures. The solution profiles of the water shock tube problem and the air shock tube problem for the same thermodynamic conditions and shock tube geometry were compared and salient features were discussed. The water shock tube problem is also solved numerically in this study, with a more elaborate PDE-based mathematical model, using the AUSM\(^+\)-up numerical algorithm. The in-depth comparative analysis shows the close agreement of the numerical results with the analytical solution developed. The analytical solution to the liquid shock tube offers multiple flow features, such as shock wave, expansion fan, and contact discontinuity in the solution. These features make the proposed analytical solution a powerful benchmarking tool that could test the various computational capabilities of codes developed for simulation of compressible liquid flow transients. The solution procedure presented in this work is also flexible enough for application to any liquids provided that the relevant equations of state are available.

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  • 14 February 2022

    The reference 13 which was published incompletely has been corrected.

References

  1. Schwacha BG. Liquid cutting of hard materials, US Patent 2,985,050; 1961.

  2. Chadwick R, Robertson J. Energy absorber for high pressure fluid jets, US Patent, 3,730,040; 1973.

  3. Ramezanzadeh H, Ramiar A, Yousefifard M, Ghasemian M. Numerical analysis of sinusoidal and step pulse velocity effects on an im**ing jet quenching process. J Therm Anal Calorim. 2019;140:331–49.

    Article  Google Scholar 

  4. Le Méhauté B, Wang S. Water waves generated by underwater explosion. Singapore: World Scientific; 1996.

    Book  Google Scholar 

  5. Liu M, Liu G, Lam K, Zong Z. Smoothed particle hydrodynamics for numerical simulation of underwater explosion. Comput Mech. 2003;30(2):106–18.

    Article  CAS  Google Scholar 

  6. Li T, Wang S, Li S, Zhang AM. Numerical investigation of an underwater explosion bubble based on FVM and VOF. Appl Ocean Res. 2018;74:49–58.

    Article  Google Scholar 

  7. Gurovich V, Virozub A, Rososhek A, Bland S, Spielman R, Krasik YE. Quasi-isentropic compression using compressed water flow generated by underwater electrical explosion of a wire array. J Appl Phys. 2018;123(18):185902.

    Article  Google Scholar 

  8. Heiderich J, Todd RE. Along-stream evolution of gulf stream volume transport. J Phys Oceanogr. 2020;50:2251–70.

    Article  Google Scholar 

  9. Jamieson AJ. The five deeps expedition and an update of full ocean depth exploration and explorers. Mar Technol Soc J. 2020;54(1):6–12.

    Article  Google Scholar 

  10. Chen W, Chen G, Jeng D, Xu L. Ocean bottom hydrodynamic pressure due to vertical seismic motion. Int J Geomech. 2020;20(9):06020025.

    Article  Google Scholar 

  11. Houqun C. Seismic safety analysis of tall concrete dams, investigation and insights on critical challenges. Earthq Eng Eng Vib. 2020;19(3):533–9.

    Article  Google Scholar 

  12. Deng X, Inaba S, **e B, Shyue K-M, **ao F. High fidelity discontinuity-resolving reconstruction for compressible multiphase flows with moving interfaces. J Comput Phys. 2018;371:945–66.

    Article  Google Scholar 

  13. Re B, Abgrall R. Non-equilibrium model for weakly compressible multi-component flows: the hyperbolic operator. In: di Mare F, Spinelli A, Pini M (eds) Non-Ideal Compressible Fluid Dynamics for Propulsion and Power. NICFD 2018. Lecture Notes in Mechanical Engineering. Springer, Cham; 2020. https://doi.org/10.1007/978-3-030-49626-5_3.

  14. Merouani S, Hamdaoui O, Kerabchi N. Liquid compressibility effect on the acoustic generation of free radicals. J Appl Water Eng Res. 2020;8:247–61.

    Article  Google Scholar 

  15. Pastina B, Isabey J, Hickel B. The influence of water chemistry on the radiolysis of the primary coolant water in pressurized water reactors. J Nucl Mater. 1999;264(3):309–18.

    Article  CAS  Google Scholar 

  16. Li Z, Mazinani A, Hayat T, Al-Rashed AA, Alsulami H, Goodarzi M, Sarafraz M. Transient pool boiling and particulate deposition of copper oxide nano-suspensions. Int J Heat Mass Transf. 2020;155:119743.

    Article  CAS  Google Scholar 

  17. Li Z, Sarafraz M, Mazinani A, Hayat T, Alsulami H, Goodarzi M. Pool boiling heat transfer to CuO-H\(_{2}\)O nanofluid on finned surfaces. Int J Heat Mass Transf. 2020;156:119780.

    Article  CAS  Google Scholar 

  18. Berg A, Iben U, Meister A, Schmidt J. Modeling and simulation of cavitation in hydraulic pipelines based on the thermodynamic and caloric properties of liquid and steam. Shock Waves. 2005;14(1–2):111–21.

    Article  Google Scholar 

  19. Neuhaus T, Dudlik A. Experiments and comparing calculations on thermohydraulic pressure surges in pipes. Kerntechnik. 2006;71(3):87–94.

    Article  Google Scholar 

  20. Yasutomi M. The physics of liquid water. Bei**g: Jenny Stanford Publishing; 2020.

    Book  Google Scholar 

  21. Campbell I, Pitcher A. Shock waves in a liquid containing gas bubbles. Proc R Soc Lond A. 1958;243(1235):534–45.

    Article  Google Scholar 

  22. Campbell A, Davis W, Travis J. Shock initiation of detonation in liquid explosives. Phys Fluids. 1961;4(4):498–510.

    Article  Google Scholar 

  23. Lee I, Finnegan S. Prediction of shock response of pyrotechnic mixtures from thermal analysis. J Therm Anal Calorim. 1997;50(5–6):707–17.

    Article  CAS  Google Scholar 

  24. Ben-Dor G, Igra O, Elperin T. Handbook of shock waves, three volume set. Amsterdam: Elsevier; 2000.

    Google Scholar 

  25. Biance A-L, Chevy F, Clanet C, Lagubeau G, Quéré D. On the elasticity of an inertial liquid shock. J Fluid Mech. 2006;554:47.

    Article  Google Scholar 

  26. Armstrong M, Bastea S. System and method for rapid, high throughput, high pressure synthesis of materials from a liquid precursor, US Patent App, 16/203,177; 2020.

  27. Song X, Li B, **e L. Experimental investigation on the properties of liquid film breakup induced by shock waves. Chin Phys B. 2020;29:086201.

    Article  CAS  Google Scholar 

  28. Debnath P, Pandey K. Numerical analysis of detonation combustion wave in pulse detonation combustor with modified ejector with gaseous and liquid fuel mixture. J Therm Anal Calorim. 2020;. https://doi.org/10.1007/s10973-020-09842-1.

    Article  Google Scholar 

  29. Kaiser J, Winter J, Adami S, Adams N. Investigation of interface deformation dynamics during high-weber number cylindrical droplet breakup. Int J Multiph Flow. 2020;132:103409.

    Article  CAS  Google Scholar 

  30. Ash C. The shattering effect on rain. Science. 2020;369(6501):265–265.

    Article  Google Scholar 

  31. El-Said EM, Awad MM, Abdulaziz M. A comprehensive review on pressurized thermal shock: predictive, preventive and safety issues. J Therm Anal Calorim. 2020;. https://doi.org/10.1007/s10973-020-10030-4.

    Article  Google Scholar 

  32. Li L, Lu XX, Ren XB, Ren YJ, Zhao ST, Yan XF. The mechanism of liquid dispersing from a cylinder driven by central dynamic shock loading. Def Technol. 2020;. https://doi.org/10.1016/j.dt.2020.07.001.

    Article  Google Scholar 

  33. Sheikholeslami M, Jafaryar M. Nanoparticles for improving the efficiency of heat recovery unit involving entropy generation analysis. J Taiwan Inst Chem Eng. 2020;115:96–107.

    Article  CAS  Google Scholar 

  34. Sheikholeslami M, Farshad SA, Shafee A, Babazadeh H. Performance of solar collector with turbulator involving nanomaterial turbulent regime. Renew Energy. 2021;163:1222–37.

    Article  Google Scholar 

  35. Sod GA. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys. 1978;27(1):1–31.

    Article  Google Scholar 

  36. Keith TG, John JE. Gas dynamics. London: Pearson Education Inc; 2006.

    Google Scholar 

  37. Chandran J, Salih A. A modified equation of state for water for a wide range of pressure and the concept of water shock tube. Fluid Phase Equilib. 2019;483:182–8.

    Article  Google Scholar 

  38. Liou MS. A sequel to AUSM, part II: AUSM+-up for all speeds. J Comput Phys. 2006;214(1):137–70.

    Article  Google Scholar 

  39. Tait P. Physics and chemistry of the voyage of HMS challenger, vol II, part IV, SP LXI.

  40. Dymond J, Malhotra R. The tait equation: 100 years on. Int J Thermophys. 1988;9(6):941–51.

    Article  Google Scholar 

  41. Saurel R, Cocchi JP, Butler PB. Numerical study of cavitation in the wake of a hypervelocity underwater projectile. J Propul Power. 1999;15(4):513–22.

    Article  Google Scholar 

  42. Le Métayer O, Saurel R. The noble-abel stiffened-gas equation of state. Phys Fluids. 2016;28(4):046102.

    Article  Google Scholar 

  43. Beda L. Chemical thermodynamics. J Therm Anal. 1995;44(2):513–6.

    Article  CAS  Google Scholar 

  44. Danaila I, Joly P, Kaber SM, Postel M, editors. Gas dynamics: the Riemann problem and discontinuous solutions: application to the shock tube problem. New York: Springer; 2007. p. 213–33.

    Google Scholar 

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  1. Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram, 695547, India

    R. Jishnu Chandran & A. Salih

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  1. R. Jishnu Chandran
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  2. A. Salih
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Jishnu Chandran, R., Salih, A. Development of a benchmark solution in compressible liquid flows: analytical solution to the water shock tube problem. J Therm Anal Calorim 147, 5279–5292 (2022). https://doi.org/10.1007/s10973-021-10871-7

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