Skip to main content
SpringerLink
Log in

Low Mach Number Modeling of Stratified Flows

  • Conference paper
  • First Online:
Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 77))

Abstract

Low Mach number equation sets approximate the equations of motion of a compressible fluid by filtering out the sound waves, which allows the system to evolve on the advective rather than the acoustic time scale. Depending on the degree of approximation, low Mach number models retain some subset of possible compressible effects. In this paper we give an overview of low Mach number methods for modeling stratified flows arising in astrophysics and atmospheric science as well as low Mach number reacting flows. We discuss how elements from the different fields are combined to form MAESTRO, a code for modeling low Mach number stratified flows with general equations of state, reactions and time-varying stratification.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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. Almgren, A.S., Bell, J.B., Nonaka, A., Zingale, M.: Low mach number modeling of type ia supernovae. iii. reactions. Astrophys. J. 684, 449–470 (2008). doi:10.1086/590321

    Article  Google Scholar 

  2. Almgren, A.S., Bell, J.B., Rendleman, C.A., Zingale, M.: Low mach number modeling of type ia supernovae. i. hydrodynamics. Astrophys. J. 637, 922–936 (2006)

    Article  Google Scholar 

  3. Almgren, A.S., Bell, J.B., Rendleman, C.A., Zingale, M.: Low mach number modeling of type ia supernovae. ii. energy evolution. Astrophys. J. 649, 927–938 (2006)

    Article  Google Scholar 

  4. Bannon, P.: Nonlinear hydrostatic adjustment. J. Atmos. Sci. 53(23), 3606–3617 (1996)

    Article  MathSciNet  Google Scholar 

  5. Batchelor, G.K.: The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere. Quart. J. R. Meteor. Soc. 79, 224–235 (1953)

    Article  Google Scholar 

  6. Bell, J.B., Day, M.S., Rendleman, C.A., Woosley, S.E., Zingale, M.A.: Adaptive low mach number simulations of nuclear flame microphysics. J. Comp. Phys. 195(2), 677–694 (2004)

    Article  MATH  Google Scholar 

  7. Botta, N., Klein, R., Almgren, A.: Asymptotic analysis of a dry atmosphere. In: Neittaanmäki et al. (eds.) ENUMATH 99, Numerical Mathematics and Advanced Applications, p. 262. World Scientific, Singapore (1999)

    Google Scholar 

  8. Boussinesq, J.: Theorie Analytique de la Chaleur, vol. 2. Gauthier-Villars, Paris (1903)

    Google Scholar 

  9. Brown, B.J., Vasil, G.M., Zweibel, E.G.: Energy conservation and gravity waves in sound-proof treatments of stellar interiors: part i. anelastic approximations. Astrophys. J. 756(109), 1–20 (2012)

    Google Scholar 

  10. Day, M.S., Bell, J.B.: Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Model. 4(4), 535–556 (2000)

    Article  MATH  Google Scholar 

  11. Duarte, M., Almgren, A.S., Balakrishnan, K., Bell, J.B.: A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations. Submitted for publication, arXiv:1311.4265 (2014)

    Google Scholar 

  12. Durran, D.R.: Improving the anelastic approximation. J. Atmos. Sci. 46(11), 1453–1461 (1989)

    Article  Google Scholar 

  13. Durran, D.R.: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Atmos. Sci. 601, 365–379 (2008)

    MATH  MathSciNet  Google Scholar 

  14. Dutton, J.A., Fichtl, G.H.: Approximate equations of motion for gases and liquids. J. Atmos. Sci. 26, 241–254 (1969)

    Article  Google Scholar 

  15. García-Senz, D., Bravo, E.: Type ia supernova models arising from different distributions of igniting points. Astron. Astrophys. 430, 585–602 (2005). doi:10.1051/0004-6361:20041628

    Article  Google Scholar 

  16. Gilet, C., Almgren, A.S., Bell, J.B., Nonaka, A., Woosley, S., Zingale, M.: Low mach number modeling of core convection in massive stars. APJ 773, 137 (2013)

    Article  Google Scholar 

  17. Gilet, C.E.: Low Mach Number simulation of core convection in massive stars. Ph.D. thesis, University of California, Berkeley (2012)

    Google Scholar 

  18. Gilman, P.A., Glatzmaier, G.A.: Compressible convection in a rotating spherical shell. i. anelastic equations. Astrophys. J. Supp. 45, 335–349 (1981)

    Article  MathSciNet  Google Scholar 

  19. Glatzmaier, G.A.: Numerical simulation of stellar convective dynamos i. The model and method. J. Comp. Phys. 55, 461–484 (1984)

    Google Scholar 

  20. Gough, D.O.: The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456 (1969)

    Article  Google Scholar 

  21. Klein, R., Pauluis, O.: Thermodynamic consistency of a pseudoincompressible approximation for general equations of state. J. Atmos. Sci. 69:961–968 (2012)

    Google Scholar 

  22. Klemp, J.B., Wilhelmson, R.B.: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci. 35, 1070–1096 (1978)

    Article  Google Scholar 

  23. Krueger, B.K., Jackson, A.P., Calder, A.C., Townsley, D.M., Brown, E.F., Timmes, F.X.: Evaluating systematic dependencies of type ia supernovae: the influence of central density. Astrophys. J. 757, 175 (2012). doi:10.1088/0004-637X/757/2/175

    Article  Google Scholar 

  24. Kuhlen, M., Woosley, S.E., Glatzmaier, G.A.: Carbon ignition in type ia supernovae. ii. a three-dimensional numerical model. Astrophys. J. 640, 407–416 (2006). doi:10.1086/500105

    Article  Google Scholar 

  25. Kurowski, M., Grabowski, W., Smolarkiewicz, P.: Towards multiscale simulation of moist flows with soundproof equations. J. Atmos. Sci. 70, 3995–4011 (2013)

    Article  Google Scholar 

  26. Latour, J., Spiegel, E.A., Toomre, J., Zahn, J.P.: Stellar convection theory. i. the anelastic modal equations. Astrophys. J. 207, 233–243 (1976)

    Article  MathSciNet  Google Scholar 

  27. Lipps, F.: On the anelastic approximation for deep convection. J. Atmos. Sci. 47, 1794–1798 (1990)

    Article  Google Scholar 

  28. Lipps, F., Hemler, R.: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 39, 2192–2210 (1982)

    Article  Google Scholar 

  29. Lipps, F., Hemler, R.: Another look at the scale analysis for deep moist convection. J. Atmos. Sci. 42, 1960–1964 (1985)

    Article  Google Scholar 

  30. Livne, E., Asida, S.M., Höflich, P.: On the sensitivity of deflagrations in a chandrasekhar mass white dwarf to initial conditions. Astrophys. J. 632, 443–449 (2005). doi:10.1086/432975

    Article  Google Scholar 

  31. Majda, A., Sethian, J.A.: Derivation and numerical solution of the equations of low mach number combustion. Comb. Sci. Tech. 42, 185–205 (1985)

    Article  Google Scholar 

  32. Malone, C., Nonaka, A., Almgren, A., Bell, J., Zingale, M.: Multidimensional modeling of type i x-ray bursts. i. two-dimensional convection prior to the outburst of a pure he accretor. APJ 728, 118 (2011)

    Article  Google Scholar 

  33. Malone, C., Nonaka, A., Woosley, S., Almgren, A.S., Bell, J.B., Dong, S., Zingale, M.: The deflagration stage of chandrasekhar mass models for type ia supernovae: i. early evolution. APJ 782(1), 11 (2014)

    Article  Google Scholar 

  34. Niemeyer, J.C., Hillebrandt, W., Woosley, S.E.: Off-center deflagrations in chandrasekhar mass type IA supernova models. Astrophys. J. 471, 903\(-\) \(+\) (1996). doi: 10.1086/178017

    Google Scholar 

  35. Nonaka, A., Almgren, A.S., Bell, J.B., Lijewski, M.J., Malone, C.M., Zingale, M.: Maestro:an adaptive low mach number hydrodynamics algorithm for stellar flows. Astrophys. J. Supp. 188, 358–383 (2010)

    Article  Google Scholar 

  36. Nonaka, A., Aspden, A.J., Zingale, M., Almgren, A.S., Bell, J.B., Woosley, S.E.: High-resolution simulations of convection preceding ignition in type ia supernovae using adaptive mesh refinement. Astrophys. J. 745, 73 (2012). doi:10.1088/0004-637X/745/1/73

    Article  Google Scholar 

  37. Ogura, Y., Phillips, N.A.: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179 (1962)

    Article  Google Scholar 

  38. O’Neill, W., Klein, R.: A moist pseudo-incompressible model. Atmos. Res. (2013)

    Google Scholar 

  39. Plewa, T., Calder, A.C., Lamb, D.Q.: Type ia supernova explosion: gravitationally confined detonation. Astrophys. J. 612, L37–L40 (2004)

    Article  Google Scholar 

  40. Rehm, R.G., Baum, H.R.: The equations of motion for thermally driven buoyant flows. J. Res. Natl. Bur. Stan. 83, 297–308 (1978)

    Article  MATH  Google Scholar 

  41. Tapp, M., White, P.: A non-hydrostatic mesoscale model. Q. J. Roy. Meteor. Soc. 102(432), 277–296 (1976)

    Article  Google Scholar 

  42. Timmes, F.X., Brown, E.F., Truran, J.W.: On variations in the peak luminosity of type Ia supernovae. Astrophys. J. 590, L83–L86 (2003). doi:10.1086/376721

    Article  Google Scholar 

  43. Vasil, G.M., Lecoanet, D., Brown, B.P., Wood, T.S., Zweibel, E.G.: Energy conservation and gravity waves in sound-proof treatments of stellar interiors. ii. lagrangian constrained analysis. Astrophys. J. 773, 169 (2013)

    Google Scholar 

  44. Wilhelmson, R., Ogura, Y.: The pressure perturbation and the numerical modeling of a cloud. J. Atmos. Sci. 29, 1295–1307 (1972)

    Article  Google Scholar 

  45. Woosley, S.E., Wunsch, S., Kuhlen, M.: Carbon ignition in type ia supernovae: an analytic model. Astrophys. J. 607, 921–930 (2004)

    Article  Google Scholar 

  46. Zingale, M., Almgren, A.S., Bell, J.B., Nonaka, A., Woosley, S.E.: Low mach number modeling of type ia supernovae. IV. white dwarf convection. Astrophys. J. 704, 196–210 (2009)

    Article  Google Scholar 

  47. Zingale, M., Nonaka, A., Almgren, A.S., Bell, J.B., Malone, C.M., Orvedahl, R.J.: Low mach number modeling of convection in helium shells on sub-chandrasekhar white dwarfs. I. Methodology. Astrophys. J. 764, 97 (2013). doi:10.1088/0004-637X/764/1/97

    Article  Google Scholar 

  48. Zingale, M., Nonaka, A., Almgren, A.S., Bell, J.B., Malone, C.M., Woosley, S.E.: The convective phase preceding type ia supernovae. Astrophys. J. 740, 8 (2011)

    Article  Google Scholar 

Download references

Acknowledgments

The work at LBNL was supported by the Applied Mathematics Program of the DOE Office of Advance Scientific Computing Research under U.S. Department of Energy under contract No. DE-AC02-05CH11231. The work at Stony Brook was supported by a DOE/Office of Nuclear Physics grant Nos. DE-FG02-06ER41448 and DE-FG02-87ER40317 to Stony Brook. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725. The MAESTRO code is freely available from http://bender.astro.sunysb.edu/Maestro/.

Author information

Authors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Ann Almgren, John Bell & Andrew Nonaka

  2. Stony Brook University, Stony Brook, NY, USA

    Michael Zingale

Authors
  1. Ann Almgren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Bell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew Nonaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michael Zingale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ann Almgren .

Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Almgren, A., Bell, J., Nonaka, A., Zingale, M. (2014). Low Mach Number Modeling of Stratified Flows. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation