Skip to main content
SpringerLink
Log in

Probability Density Function Modeling of Turbulent Spray Combustion

  • Conference paper
  • First Online:
Experiments and Numerical Simulations of Turbulent Combustion of Diluted Sprays

Part of the book series: ERCOFTAC Series ((ERCO,volume 19))

Abstract

Spray processes play a crucial role in liquid fueled combustion devices such as Diesel or fueled rocket engines and industrial furnaces. The combustion occurs under turbulent conditions, and a wide dynamic range of length and time scales characterize these processes, where the scales of the flow field and chemical reactions typically differ considerably. Moreover, a strong interdependence of liquid breakup and atomization, turbulent dispersion, droplet evaporation, and fuel-air mixing makes the spray modeling a challenging task. In the present chapter, a one-point one-time Eulerian statistical description of a joint mixture fraction—enthalpy probability density function (pdf) model for the gas phase is derived and modeled. A Lagrangian Monte Carlo method is used to solve the high-dimensional joint pdf transport equation. Two different mixing models, the interaction-by-exchange-with-the-mean and an extended modified Curl model, are employed in order to evaluate molecular mixing in the context of two-phase reacting flows. Moreover, a modified β function for application in turbulent spray flames, which has been proposed in an earlier study of non-reacting spray flows, is discussed in comparison with the standard β function and the transported pdf method. The modified β function is defined through two additional parameters compared to the standard form, and the choice of these parameters is discussed in the present study. A steady, two-dimensional, axisymmetric, turbulent liquid fuel/air spray flame is investigated, where both methanol and ethanol are studied. The numerical results are compared and discussed in context with the experimental data.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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. Abramzon, B., Sirignano, W. A.: Droplet vaporization model for spray combustion calculation. Int. J. Heat Mass Transfer. 32, 1605–1618 (1989).

    Article  Google Scholar 

  2. Anand, G., Jenny, P.: Stochastic modeling of evaporating sprays within a consistent hybrid joint PDF framework. J. Comput. Phys. 228, 2063–2081 (2009).

    Article  MATH  Google Scholar 

  3. Atkins, P., Paula, J. D.: Atkins Physical Chemistry, Oxford Higher Education, 7th Edition, (2001).

    Google Scholar 

  4. Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge University Press, London, (1967).

    MATH  Google Scholar 

  5. Cao, R. R., Wang, H., Pope, S. B.: The effect of mixing models in PDF calculations of piloted jet flames. Proc. Combust. Inst. 31, 1543–1550 (2007).

    Article  Google Scholar 

  6. Crowe, C. T., Sharma, M. P., Stock, D. E.: The particle-source-in cell (PSI-Cell) model for gas-droplet flows. J. Fluids Eng. 99, 325–332 (1977).

    Article  Google Scholar 

  7. Curl, R. L.: Dispersed phase mixing: 1. Theory and effects in simple reactor. AIChE J. 9(2), 175–181 (1963).

    Article  Google Scholar 

  8. De, S., Lakshmisha, K. N., Bilger, R. W.: Modeling of nonreacting and reacting turbulent spray jets using a fully stochastic separated flow approach. Combust. Flame. 158, 1992–2008 (2011).

    Article  Google Scholar 

  9. Demoulin, F. X., Borghi, R.: Assumed PDF modeling of turbulent spray combustion. Combust. Sci. Technol. 158, 249–271 (2000).

    Article  Google Scholar 

  10. Dopazo, O., O’Brien, E. E.: An approach to the auto-ignition of a turbulent mixture. Acta Astronaut. 1, 1239–1266 (1974).

    Article  MATH  Google Scholar 

  11. Durand, P., Gorokhovski, M., Borghi, R.: An application of the probability density function model to diesel engine combustion. Combust. Sci. Technol. 144, 47–78 (1999).

    Article  Google Scholar 

  12. Düwel, I., Ge, H.-W., Kronemayer, H., Dibbel, R., Gutheil, E., Schulz, C.: Experimental and numerical characterization of a turbulent spray flame. Proc. Combust. Inst. 31, 2247–2255 (2007).

    Article  Google Scholar 

  13. Eckstein, J., Chen, J. Y., Chou, C. P., Janicka, J.: Modeling of turbulent mixing in opposed jet configuration: one-dimensional Monte Carlo probability density function simulation. Proc. Combust. Inst. 28, 141–148 (2000).

    Article  Google Scholar 

  14. Garg, R., Narayanan, C., Subramaniam, S.: A numerically convergent Lagrangian-Eulerian simulation method for dispersed two-phase flows. Int. J. Multiphase Flow. 35, 376–388 (2009).

    Article  Google Scholar 

  15. Ge, H.-W., Gutheil, E.: PDF simulation of turbulent spray flows. Atomization Sprays. 16, 531–542 (2006).

    Article  Google Scholar 

  16. Ge, H.-W., Gutheil, E.: Simulation of a turbulent spray flame using coupled PDF gas phase and spray flamelet modeling. Combust. Flame. 153, 173–185 (2008).

    Article  Google Scholar 

  17. Ge, H.-W., Düwel, I., Kronemayer, H., Dibble, R. W., Gutheil, E., Schulz, C., Wolfrum, J.: Laser based experimental and Monte Carlo PDF numerical investigation of an ethanol/air spray flame. Combust. Sci. Technol. 180, 1529–1547 (2008).

    Article  Google Scholar 

  18. Ge, H.-W., Hu, Y., Gutheil, E.: Joint gas-phase velocity-scalar PDF modeling for turbulent evaporating spray flows. Combust. Sci. Technol. 184, 1664–1679 (2012).

    Article  Google Scholar 

  19. Gordon, R. L., Masri, A. R., Pope, S. B., Goldin, G. M.: A numerical study of auto-ignition in turbulent lifted flames issuing into vitiated co-flow. Combust. Theor. Model. 11, 351–376 (2007).

    Article  MATH  Google Scholar 

  20. Gutheil, E.: Structure and extinction of laminar ethanol-air spray flames. Combust. Theor. Model. 5(2), 131–145 (2001).

    Article  MATH  Google Scholar 

  21. Gutheil, E., Sirignano, W. A.: Counterflow spray combustion modeling with detailed transport and detailed chemistry. Combust. Flame. 113, 92–105 (1998).

    Article  Google Scholar 

  22. Heye C. R., Raman, V., Masri, A. R.: LES/probability density function approach for the simulation of an ethanol spray flame. Proc. Combust. Inst. 34, 1633–1641 (2013).

    Article  Google Scholar 

  23. Hollmann, C., Gutheil, E.: Modeling of turbulent spray diffusion flames including detailed chemistry. Proc. Combust. Inst. 26, 1731–1738 (1996).

    Article  Google Scholar 

  24. Hollmann, C., Gutheil, E.: Flamelet-modeling of turbulent spray diffusion flames based on a laminar spray flame library. Combust. Sci. Technol. 135, 175–192 (1998).

    Article  Google Scholar 

  25. Hubbard, G. L., Denny, V. E., Mills, A. F.: Droplet evaporation: effects of transient and variable properties. Int. J. Heat Mass Transfer. 18, 1003–1008 (1975).

    Article  Google Scholar 

  26. Kung, E. H., Haworth, D. C.: Transported probability density function (tPDF) modeling for direct-injection internal combustion engines. SAE Paper 2008-01-0969; SAE Int. J. Engines. 1, 591-606 (2009).

    Google Scholar 

  27. Liu, Z. H., Zheng, C. G., Zhou, L. X.: A joint PDF model for turbulent spray evaporation/combustion. Proc. Combust. Inst. 29, 561–568 (2002).

    Article  Google Scholar 

  28. Lundgren, T. S.: Model equation for non-homogeneous turbulence. Phys. Fluids. 12, 485–497 (1969).

    Article  MATH  Google Scholar 

  29. Luo, K., Pitsch, H., Pai, M. G., Desjardins, O.: Direct numerical simulations and analysis of three dimensional n-heptane spray flames in a model swirl combustor. Proc. Combust. Inst. 33, 2143–2152 (2011).

    Article  Google Scholar 

  30. Masri, A., Gounder, J.: Details and Complexities of Boundary Conditions in Turbulent Piloted Dilute Spray Jets and Flames. In: Bart, Merci, Dirk, Roekaerts and Amsini, Sadiki (Eds.), Experiments and Numerical Simulations of Diluted Spray Turbulent Combustion, 41–68, New York, Springer, (2011).

    Google Scholar 

  31. McDonell, V. G., Samuelsen, G. S.: An experimental data base for computational fluid dynamics of reacting and nonreacting methanol sprays. J. Fluids Eng. 117, 145–153 (1995).

    Article  Google Scholar 

  32. Mehta R. S.: Detailed Modeling of soot formation and turbulence-radiation interactions in turbulent jet flames. Ph.D. thesis, The Pennsylvania State University, (2008).

    Google Scholar 

  33. Menon, S., Fureby, C.: Computational Combustion. In: Encyclopedia of Aerospace Engineering, Wiley, (2010).

    Google Scholar 

  34. Miller, R. S., Bellan, J.: On the validity of the assumed probability density function method for modeling binary mixing/reaction of evaporated vapor in gas-liquid turbulent shear flow. Proc. Combust. Inst. 27, 1065–1072 (1998).

    Article  Google Scholar 

  35. Mortensen M., Bilger R.: Derivation of the conditional moment closure equation for spray combustion. Combust. Flame. 156, 62–72 (2009).

    Article  Google Scholar 

  36. Naud, B.: PDF modeling of turbulent sprays and flames using a particle stochastic approach. Ph. D. Thesis, TU Delft, (2003).

    Google Scholar 

  37. Olguin, H., Gutheil, E.: Influence of evaporation on spray flamelet structures, Combust. Flame (2013), http://dx.doi.org/10.1016/j.combustflame.2013.10.010.

  38. Pai, G. M., Subramaniam, S.: A comprehensive probability density function formalism for multiphase flows. J. Fluid Mech. 628, 181–228 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  39. Peters, N.: Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10, 319–339 (1984).

    Article  Google Scholar 

  40. Pope, S. B.: The relationship between the probability approach and particle models for reaction in homogeneous turbulence. Combust. Flame. 35, 41–45 (1979).

    Article  Google Scholar 

  41. Pope, S. B.: Transport equation for the joint probability density function of velocity and scalars in turbulent flow. Phys. Fluids. 24, 588–596 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  42. Pope, S. B.: PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192 (1985).

    Article  MathSciNet  Google Scholar 

  43. Raju, M. S.: Application of scalar Monte Carlo probability density function method turbulent spray flames. Numer. Heat Transfer A. 30, 753–777 (1996).

    Article  Google Scholar 

  44. Richardson, J. M., Howard, H. C., Smith, R. W.:The relation between sampling-tube measurements and concentration fluctuations in a turbulent gas jet. Proc. Combust. Inst. 4 814–817, (1953).

    Article  Google Scholar 

  45. Rumberg, O., Rogg, B.: Full PDF modeling of reactive sprays via an evaporation-progress variable. Combust. Sci. Technol. 158, 211–247 (2000).

    Article  Google Scholar 

  46. Schiller, L., Neumann, A. Z.: A drag coefficient correlation. VDI Zeitschrift 77, 318–320 (1933).

    Google Scholar 

  47. Wang, H., Pope, S. B.: Lagrangian investigation of local extinction, re-ignition and auto-ignition in turbulent flames. Combust. Theory Modeling. 12, 857–882 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  48. Zhu, M., Bray, K. N. C., Rumberg, O., Rogg, B.: PDF transport equations for two-phase reactive flows and sprays. Combust. Flame. 122, 327–338 (2000).

    Article  Google Scholar 

Download references

Acknowledgement

The authors gratefully acknowledge the financial support of Heidelberg School of Mathematical and Computational Sciences for their financial support. YH acknowledges funding through the China Scholarship Council.

Author information

Authors and Affiliations

  1. Interdisciplinary Center for Scientific Computing, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    Rana M. Humza, Yong Hu & Eva Gutheil

Authors
  1. Rana M. Humza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eva Gutheil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eva Gutheil .

Editor information

Editors and Affiliations

  1. Flow, Heat and Combustion Mechanics, Ghent University, Ghent, Belgium

    Bart Merci

  2. Interdisciplinary Center for Scientific, Heidelberg University, Heidelberg, Germany

    Eva Gutheil

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Humza, R., Hu, Y., Gutheil, E. (2014). Probability Density Function Modeling of Turbulent Spray Combustion. In: Merci, B., Gutheil, E. (eds) Experiments and Numerical Simulations of Turbulent Combustion of Diluted Sprays. ERCOFTAC Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-04678-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-04678-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-04677-8

  • Online ISBN: 978-3-319-04678-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation