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New Godunov-Type Method for Simulation of Weakly Compressible Pipeline Flows with Allowance for Elastic Deformation of the Walls

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Abstract

A new Godunov-type scheme for a complete nonlinear system of equations is developed as applied to the simulation of flows in an elastic deformable pipe. The proposed finite-difference method makes use of a one-dimensional grid, but the grid cells are characterized by two spatial sizes. The multidimensional effects and, primarily, pipe deformations are taken into account in the course of grid cell reconstruction. Difference relations are written for such a reconstructed grid. Due to the grid reconstruction procedure, the elastic deformation of the pipe can be taken into account directly without introducing the velocity of propagation of perturbations in the system “weakly compressible fluid–cylindrical shell.” The algorithm was shown to be highly accurate as applied to the water hammer problem.

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REFERENCES

  1. N. E. Zhukovskii, Water Hammer in Pipelines (Gostekhteorizdat, Moscow, 1949) [in Russian].

    Google Scholar 

  2. L. Allievi, Theoria generale del modo perturbato dell’ acdua nei tubi in pressione (Milan, 1903).

  3. O. Schnyder, “Druckstosse in Pumpensteigletungen,” Schweiz Bauztg. 94, 22–23 (1929).

    Google Scholar 

  4. L. Bergeron, “Etudes des variations de regime dans les conduits d’eau,” Rev. Gen. Hydraulique, Nos. 1–2 (1935).

  5. C. A. M. Gray, “The analysis of the dissipation of energy in water hammer,” Proc. ASCE 119 (274), 1176–1194 (1953).

  6. C. A. M. Gray, “Analysis of water hammer by characteristics,” Trans. Am. Soc. Civil Eng. 119, 1176–1189 (1954).

    Google Scholar 

  7. V. L. Streeter and C. Lai, “Water hammer analysis including fluid friction,” J. Hyd. Div. ASCE 88 (3), 79–112 (1962).

    Google Scholar 

  8. E. B. Wylie and V. L. Streeter, Fluid Transients (McGraw-Hill, New York, 1977).

    Google Scholar 

  9. M. V. Lur’e, Mathematical Simulation of Pipeline Transport of Oil, Oil Products, and Gas (Ross. Gos. Univ. Nefti i Gaza im. I.M. Gubkina, Moscow, 2012) [in Russian].

  10. D. A. Foks, Hydraulic Analysis of Transient Flows in Gas Pipelines (Energoizdat, Moscow, 1981) [in Russian].

    Google Scholar 

  11. V. L. Streeter, “Unsteady flow calculation by a numerical method,” J. Basic Eng. 94, 457–466 (1972).

    Article  Google Scholar 

  12. M. H. Chaudhry and M. Y. Hussaini, “Second-order accurate explicit finite difference schemes for water hammer analysis,” J. Fluid. Eng. 107, 523–529 (1985).

    Article  Google Scholar 

  13. G. K. Nathan, J. K. Tan, and K. C. Ng, “Two-dimensional analysis of pressure transients in pipelines,” Int. J. Numer. Methods Fluids 8 (3), 339–349 (1988).

    Article  MATH  Google Scholar 

  14. J. L. Bribiesca, “A finite-difference method to evaluate water hammer phenomenon,” J. Hydrol 51, 305–311 (1981).

    Article  Google Scholar 

  15. J. K. Tan, K. C. Ng, and G. K. Nathan, “Application of the center implicit method for investigation of pressure transients in pipelines,” Int. J. Numer. Methods Fluids 7, 395–406 (1987).

    Article  MATH  Google Scholar 

  16. M. Arfaie and A. anderson, “Implicit finite-differences for unsteady pipe flow,” Math. Eng. Ind. 3, 133–151 (1991).

    Google Scholar 

  17. K. C. Ng and C. Yap, “An investigation of pressure transients in pipelines with two-phase bubbly flow,” Int. J. Numer. Methods Fluids 9, 1207–1219 (1989).

    Article  MATH  Google Scholar 

  18. V. E. Seleznev, V. V. Aleshin, and G. S. Klishin, Methods and Technologies for Modeling Gas Pipeline Systems (Editorial URSS, Moscow, 2002) [in Russian].

    Google Scholar 

  19. V. Guinot, “Boundary condition treatment in 2 × 2 systems of propagation equations,” Int. J. Numer. Meth. Eng. 42, 647–666 (1998).

    Article  MATH  Google Scholar 

  20. E. B. Wylie and V. L. Streeter, “Network system transient calculations by implicit method,” The 45th Annual Meeting of the Society of Petroleum Engineers of AIME, Paper no. 2963 (1970).

  21. A. Verwey and J. H. Yu, “A space-compact high-order implicit scheme for water hammer simulations,” Proceedings of 25th IAHR Congress (Tokyo, Japan, 1993), Vol. 5, pp. 363–370.

  22. A. Verwey and S. Illic, “A space-compact high-order implicit scheme for 1-d advection simulations,” Proceedings of 25th IAHR Congress (Japan, Tokyo, 1993), Vol. 5.

  23. C. Hirsch, Numerical Computation of Internal and External Flows (Wiley, New York, 1990).

    MATH  Google Scholar 

  24. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer-Verlag, Berlin, 1997).

    Book  MATH  Google Scholar 

  25. V. Joviґc, “Finite elements and the method of characteristics applied to water hammer modeling,” Eng. Model. 8, 51–58 (1995).

    Google Scholar 

  26. J. J. Shu, “A finite element model and electronic analogue of pipeline pressure transients with frequency-dependent friction,” J. Fluid. Eng. 125, 194–199 (2003).

    Article  Google Scholar 

  27. C. Bisgarrd, H. H. Sorensen, and S. Spangenberg, “A finite element method for transient compressible flow in pipelines,” Int. J. Numer. Methods Fluids 7, 291–303 (1987).

    Article  Google Scholar 

  28. Y. G. Cheng, S. H. Zhang, and J. Z. Chen, “Water hammer simulation by the lattice Boltzmann method,” Trans. Chinese Hydraulic Eng. Society, J. Hydraulic Eng. 6, 25–31 (1998).

    Google Scholar 

  29. Y. G. Cheng and S. H. Zhang, “Numerical simulation of 2-D hydraulic transients using lattice Boltzmann method,” Transactions of the Chinese Hydraulic Eng. Society, J. Hydraulic Eng. 10, 32–37 (2001).

    Google Scholar 

  30. Q. Hou, A. C. H. Kruisbrink, A. S. Tijsseling, and A. Keramat, “Simulating water hammer with corrective smoothed particle method,” CASA-Report, Vol. 1214 (Eindhoven University of Technology, Department of Math. and Comput. Sci., Eindhoven, 2012).

  31. Q. Hou, A. C. H. Kruisbrink, A. S. Tijsseling, and A. Keramat, “Simulating transient pipe flow with corrective smoothed particle method,” The 11th International Conference on Pressure Surges (BHR Group, Lisbon, 2012), pp. 171–188.

  32. S. A. Gubin, A. V. Evlampiev, and S. I. Sumskoi, “Computation of transient processes in oil pipelines in standard operational conditions and emergency breakage,” Collected Papers of MIFI-99 Scientific Session, Vol. 1: Ecology and Rational Nature Management, Biophysics, Medicine Physics and Technology, Mathematical Methods in Scientific Research, and Theoretical Problems in Physics (Mosk. Inzh.-Fiz. Inst., Moscow, 1999), pp. 33–35.

  33. V. Guinot, “Riemann solvers for water hammer simulations by Godunov method,” Int. J. Numer. Meth. Eng. 49, 851–870 (2000).

    Article  MATH  Google Scholar 

  34. V. Guinot, “Numerical simulation of two-phase flow in pipes using Godunov method,” Int. J. Numer. Meth. Eng. 50, 1169–1189 (2001).

    Article  MATH  Google Scholar 

  35. Y. Hwang and N. Chung, “A fast Godunov method for the water-hammer problem,” Int. J. Numer. Methods Fluids 40 (6), 799–819 (2002).

    Article  MATH  Google Scholar 

  36. M. Zhao and M. S. Ghidaoui, “Godunov-type solution for water hammer flows,” J. Hydraulic Eng. 130 (4), 341–348 (2004).

    Article  Google Scholar 

  37. S. R. Sabbagh-Yazdi, N. E. Mastorakis, and A. Abbasi, “Water hammer modeling by Godunov type finite volume method,” Int. J. Math. Comput. Simul. 1 (4), 350–355 (2007).

    Google Scholar 

  38. S. Bousso and M. Fuamba, “Numerical simulation of unsteady friction in transient two-phase flow with Godunov method,” J. Water Resource Protect. 5, 1048–1058 (2013).

    Article  Google Scholar 

  39. S.-R. Sabbagh-Yazdi, A. Abbasi, and N. E. Mastorakis, “Water hammer modeling using 2nd order Godunov finite volume method,” Proc. Eur. Comput. Conf., Lect. Notes Electrical Eng. 28, 215–223 (2009).

  40. F. Kerger, P. Archambeau, S. Erpicum, B. J. Dewals, and M. Pirotton, “An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows,” J. Comput. Appl. Math. 235, 2030–2040 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  41. S. I. Sumskoi, A. M. Sverchkov, M. V. Lisanov, and A. F. Egorov, “Simulation of systems for shock wave/compression waves damping in technological plants,” J. Phys. Conf. Ser. 751 (1), 012023 (2016).

    Article  Google Scholar 

  42. S. I. Sumskoi and A. M. Sverchkov, “Modeling of nonequilibrium processes in oil trunk pipeline using Godunov type method,” Phys. Proc. 72, 347–350 (2015).

    Article  Google Scholar 

  43. S. I. Sumskoi, A. M. Sverchkov, M. V. Lisanov, and A. F. Egorov, “Simulation of compression waves/shock waves propagation in the branched pipeline systems with multi-valve operations,” J. Phys. Conf. Ser. 751 (1), 012024 (2016).

    Article  Google Scholar 

  44. S. I. Sumskoi, A. M. Sverchkov, M. V. Lisanov, and A. F. Egorov, “Modeling of nonequilibrium flow in the branched pipeline systems,” J. Phys. Conf. Ser. 751 (1), 012022 (2016).

    Article  Google Scholar 

  45. S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Mat. Sb. 47, 271–306 (1959).

    MathSciNet  MATH  Google Scholar 

  46. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).

  47. C. Bourdarias and S. Gerbi, “A conservative model for unsteady flows in deformable closed pipes and its implicit second-order finite volume discretization,” Comput. Fluids 37 (10), 1225–1237 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  48. A. T. Il’ichev and V. A. Shargatov, “Modeling unsteady flows in elastic pipes: Energy conservation law,” in Problems of Mathematical Physics and Mathematical Modeling: Book of Abstracts of the 6th International Conference (National Research Nuclear University MEPhI, Moscow, 2017), pp. 46–47.

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ACKNOWLEDGMENTS

This work was supported by the Russian Science Foundation, grant no. 16-19-00188.

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  1. National Research Nuclear University “MEPhI”, 115409, Moscow, Russia

    S. I. Sumskoi

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Translated by I. Ruzanova

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Sumskoi, S.I. New Godunov-Type Method for Simulation of Weakly Compressible Pipeline Flows with Allowance for Elastic Deformation of the Walls. Comput. Math. and Math. Phys. 58, 1647–1659 (2018). https://doi.org/10.1134/S096554251810010X

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