Skip to main content
SpringerLink
Log in

A Hybrid Scheme for Two-Phase Flow

  • Chapter
  • First Online:
Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Part of the book series: Mathematical Engineering ((MATHENGIN))

  • 2378 Accesses

Abstract

In this chapter, we investigate a two-phase flow generalization of the Euler equations. A symmetrized problem formulation that generalizes previous energy estimates in for the Euler equations is used for the stochastic Galerkin system. The solution is expected to develop non-smooth features that are localized in space. Consequently, we adapt the numerical method to the smoothness of the solution. Finite-difference operators in summation-by-parts (SBP) form are used for the high-order spatial discretization, and the HLL-flux and MUSCL reconstruction are employed for shock-capturing in the non-smooth region. The coupling between the different solution regions is performed with a weak imposition of the interface conditions through an interface using a penalty technique.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abbas Q, van der Weide E, Nordström J (2009) Accurate and stable calculations involving shocks using a new hybrid scheme. In: Proceedings of the 19th AIAA CFD conference, no. 2009–3985. Conference Proceeding Series. AIAA, San Antonio, TX.

    Google Scholar 

  2. Carpenter MH, Nordström J, Gottlieb D (1999) A stable and conservative interface treatment of arbitrary spatial accuracy. J Comput Phys 148(2):341–365. doi:http://dx.doi.org/10.1006/jcph.1998.6114

  3. Chen M, Keller AA, Lu Z, Zhang D, Zyvoloski G (2008) Uncertainty quantification for multiphase flow in random porous media using KL-based moment equation approaches. In: AGU spring meeting abstracts, p A2, Fort Lauderdale, FL.

    Google Scholar 

  4. Chen QY, Gottlieb D, Hesthaven JS (2005) Uncertainty analysis for the steady-state flows in a dual throat nozzle. J Comput Phys 204(1):378–398. doi:http://dx.doi.org/10.1016/j.jcp.2004.10.019

  5. Deb MK, Babuška IM, Oden JT (2001) Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput Methods Appl Math 190(48):6359–6372. doi:10.1016/S0045-7825(01)00237-7, http://www.sciencedirect.com/science/article/pii/S0045782501002377

  6. Debusschere BJ, Najm HN, Pébay PP, Knio OM, Ghanem RG, Le Maître OP (2005) Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J Sci Comput 26:698–719. doi:http://dx.doi.org/10.1137/S1064827503427741

  7. Eriksson S, Abbas Q, Nordström J (2011) A stable and conservative method for locally adapting the design order of finite difference schemes. J Comput Phys 230(11):4216–4231. doi:http://dx.doi.org/10.1016/j.jcp.2010.11.020

  8. Gerritsen M, Olsson P (1996) Designing an efficient solution strategy for fluid flows. J Comput Phys 129(2):245–262. doi:10.1006/jcph.1996.0248, http://dx.doi.org/10.1006/jcph.1996.0248

  9. Gottlieb S, Shu CW (1998) Total variation diminishing runge-kutta schemes. Math Comput 67:73–85

    Article  MATH  MathSciNet  Google Scholar 

  10. Haas JF, Sturtevant B (1987) Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J Fluid Mech 181:41–76. doi:10.1017/S0022112087002003

    Article  Google Scholar 

  11. Harten A (1983) On the symmetric form of systems of conservation laws with entropy. J Comput Phys 49(1):151–164. doi:10.1016/0021-9991(83)90118-3, http://www.sciencedirect.com/science/article/pii/0021999183901183

  12. Harten A, Lax PD, van Leer B (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25(1):35–61. http://www.jstor.org/stable/2030019

  13. Kreiss HO, Scherer G (1974) Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical aspects of finite elements in partial differential equations. Academic, New York, pp 179–183

    Google Scholar 

  14. Li H, Zhang D (2009) Efficient and accurate quantification of uncertainty for multiphase flow with the probabilistic collocation method. SPE J 14(4):665–679

    Article  Google Scholar 

  15. Mattsson K, Nordström J (2004) Summation by parts operators for finite difference approximations of second derivatives. J Comput Phys 199(2):503–540. doi:10.1016/j.jcp.2004.03.001, http://dx.doi.org/10.1016/j.jcp.2004.03.001

  16. Pettersson P, Iaccarino G, Nordström J (2009) Numerical analysis of the Burgers’ equation in the presence of uncertainty. J Comput Phys 228:8394–8412. doi:10.1016/j.jcp.2009.08.012, http://dl.acm.org/citation.cfm?id=1621150.1621394

  17. Pettersson P, Iaccarino G, Nordström J (2013) An intrusive hybrid method for discontinuous two-phase flow under uncertainty. Comput Fluids 86(0):228–239. doi:http://dx.doi.org/10.1016/j.compfluid.2013.07.009

  18. Pettit CL, Beran PS (2006) Spectral and multiresolution Wiener expansions of oscillatory stochastic processes. J Sound Vib 294:752–779

    Article  Google Scholar 

  19. Schwab C, Tokareva SA (2011) High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data. Tech Rep 2011–53, ETH, Zürich

    Google Scholar 

  20. So KK, Chantrasmi T, Hu XY, Witteveen JAS, Stemmer C, Iaccarino G, Adams NA (2010) Uncertainty analysis for shock-bubble interaction. In: Proceedings of the 2010 summer program. Center for Turbulence Research, Stanford University

    Google Scholar 

  21. Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys 27(1):1–31. doi:10.1016/0021-9991(78)90023-2, http://www.sciencedirect.com/science/article/pii/0021999178900232

  22. Strand B (1994) Summation by parts for finite difference approximations for d/dx. J Comput Phys 110(1):47–67. doi:http://dx.doi.org/10.1006/jcph.1994.1005

  23. van Leer B (1979) Towards the ultimate conservative difference scheme. V – a second-order sequel to Godunov’s method. J Comput Phys 32:101–136. doi:10.1016/0021-9991(79)90145-1

    Article  Google Scholar 

  24. Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644. doi:http://dx.doi.org/10.1137/S1064827501387826

Download references

Author information

Authors and Affiliations

  1. Uni Research, Bergen, Norway

    Mass Per Pettersson

  2. Department of Mechanical Engineering and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, USA

    Gianluca Iaccarino

  3. Department of Mathematics Computational Mathematics, Linköping University, Linköping, Sweden

    Jan Nordström

Authors
  1. Mass Per Pettersson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluca Iaccarino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Nordström
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Pettersson, M.P., Iaccarino, G., Nordström, J. (2015). A Hybrid Scheme for Two-Phase Flow. In: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-10714-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10714-1_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10713-4

  • Online ISBN: 978-3-319-10714-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation