Skip to main content
SpringerLink
Log in

Alternatives to the Beta Distribution in Assumed PDF Methods for Turbulent Reactive Flow

  • Published:
Flow, Turbulence and Combustion Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

This work addresses a shortcoming of the beta distribution when applied to assumed probability density function (PDF) methods for reactive flow, such as the flamelet-progress variable method. Regardless of parameter values, the beta distribution contains the entire sample space of a conserved scalar, which can be shown to violate the strong maximum principle of partial differential equations. To remedy this, a new class of probability distributions is introduced and tested on several conserved scalar mixing problems. The new distributions do not violate the strong maximum principle, and are in better quantitative agreement with transported PDF and DNS solutions than the beta distribution. The effects of filter width and residence time are examined, and extensions to multiple mixture fractions are outlined.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • ANSYS FLUENT 12.0 theory guide (2009).www.afs.enea.it/project/neptunius/docs/fluent/html/th/main_pre.htm

  • FlameletFOAM-2.3.x. (2014) https://github.com/flameletFoam/flameletFoam-2.3.x

  • PSD data repository: raw data for tpdf simulations and source code for the generation of psd and time-augmented PSD distributions (2020) https://github.com/popovSDSU/PSD_distributions

  • Bilger, R.: Reaction rates in diffusion flames. Combustion and Flame pp 277–284 (1977)

  • Bray, K., Libby, P.: Interaction effects in turbulent premixed flames. Phy. Fluids 19, 1687–1701 (1976)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Cha, C.M., de Bruyn Kops, S.M., Mortensen, M.: Direct numerical simulations of the double scalar mixing layer. part i: Passive scalar mixing and dissipation. Physics of Fluids 18 (2006)

  • Chen, H., Chen, S., Kraichan, R.H.: Probability distribution of a stochastically advected scalar field. Physical Review Letters 63 (1989)

  • Coclite, A., Cutrone, L., Pascazio, G., De Palma, P.: Numerical investigation of high-pressure combustion in rocket engines using flamelet/progress-variable models. In: 53rd AIAA Aerospace Sciences Meeting, p 1109 (2015)

  • Doran, E.: A multi-dimensional flamelet model for ignition in multi-feed combustion systems. PhD thesis, Stanford University (2012)

  • Doran, E., Pitsch, H., Cook, D.: A priori testing of a two-dimensional unsteady flamelet model for three-feed combustion systems. Proceed. Combust. Inst. 34, 1317–1324 (2013)

    Article  Google Scholar 

  • Evans, L.: A strong maximum principle for reaction-diffusion systems and a weak convergence scheme for reflected stochastic differential equations. PhD thesis, Massachusetts Institute of Technology (2010)

  • Fiorina, B., Vicquelin, R., Auzillon, P., Darabiha, N., Gicquel, O., Veynante, D.: A filtered tabulated chemistry model for les of premixed combustion. Combust Flame 157(3), 465–475 (2010)

    Article  Google Scholar 

  • Floyd, J., Kempf, A.M., Kronenburg, A., Ram, R.: A simple model for the filtered density function for passive scalar combustion les. Combust. Theory Model. 13, 559–588 (2009)

    Article  Google Scholar 

  • Gao, F.: An analytical solution for the scalar probability density function in homogeneous turbulence. Phy. Fluids A Fluid Dyn. 3, 511–513 (1991)

    Article  Google Scholar 

  • 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 

  • Girimaji, S.: Assumed \(\beta\)-pdf model for turbulent mixing: validation and extension to multiple scalar mixing. Combust. Sci. Technol. 78, 177–196 (1991)

    Article  Google Scholar 

  • Hasse, C., Peters, N.: A two mixture fraction flamelet model applied to split injections in a DI diesel engine. Proceed. Combust. Inst. 30, 2755–2762 (2005)

    Article  Google Scholar 

  • Haworth, D.C.: Progress in probability density function methods for turbulent reactive flows. Prog. Energy Combust. Sci. 36, 168–259 (2010)

    Article  Google Scholar 

  • Ihme, M., See, Y.: Prediction of autoignition in a lifted methane/air flame using an unsteady flamelet/progress variable model. Combust. Flame 157, 1850–1862 (2010)

    Article  Google Scholar 

  • Ihme, M., See, Y.: LES flamelet modeling of a three-stream MILD combustor: Analysis of flame sensitivity to scalar inflow conditions. Proceed. Combust. Inst. 33, 1309–1317 (2011)

    Article  Google Scholar 

  • Ihme, M., Cha, C., Pitsch, H.: Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach. Proceed. Combust. Inst. 30, 793–800 (2005)

    Article  Google Scholar 

  • Ihme, M., Zhang, J., He, G., Dally, B.: Large-eddy simulation of a jet-in-hot-coflow burner operating in the oxygen-diluted combustion regime. Flow, Turbulence Combust. 89, 449–464 (2012)

    Article  Google Scholar 

  • Klimenko, A.Y., Pope, S.B.: The modeling of turbulent reactive flows based on multiple mapping conditioning. Physics of Fluids 15 (2003)

  • Nguyen, T.M., Popov, P.P., Sirignano, W.A.: Longitudinal combustion instability in a rocket engine with a single coaxial injector. J Propul. Power 34, 354–373 (2018)

    Article  Google Scholar 

  • Perry, B., Mueller, M.: Joint probability density function models for multiscalar turbulent mixing. Combust. Flame 193, 344–362 (2018)

    Article  Google Scholar 

  • Peters, N., Williams, F.A.: Liftoff characteristics of turbulent jet diffusion flames. AIAA J 21(3), 423–429 (1983)

    Article  Google Scholar 

  • Pierce, C., Moin, P.: Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. J Fluid Mech. 504, 73–97 (2004)

    Article  MathSciNet  Google Scholar 

  • Pitsch, H., Peters, N.: A consistent flamelet formulation of non-premixed combustion considering differential diffusion effects. Combust. Flame 114, 26–40 (1998)

    Article  Google Scholar 

  • Pitsch, H., Steiner, H.: Large-eddy simulation of a turbulent piloted methane/air diffusion flame (sandia flame D). Phy Fluids 12, 2541–2554 (2000)

    Article  Google Scholar 

  • Pope, S.: Probability Distributions of Scalars in Turbulent Shear Flows, Springer-Verlag, pp 7–16. Turbulent Shear Flows 2 (1980)

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

    Article  Google Scholar 

  • Pope, S.B.: Turbulent flows (2001)

  • Pope, S.B.: A model for turbulent mixing based on shadow-position conditioning. Physics of Fluids 25(11), (2013)

  • Popov, P.P., Pope, S.B.: Large eddy simulation/probability density function simulations of bluff- body stabilized flames. Combust. Flame 161, 3100–3133 (2014)

    Article  Google Scholar 

  • Popov, P.P., Sirignano, W.A.: Transverse combustion instability in a rectangular rocket motor. J Propul Power 32, 620–627 (2016)

    Article  Google Scholar 

  • Réveillon, J., Vervisch, L.: Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model. Combust. Flame 121, 75–90 (2000)

    Article  Google Scholar 

  • Sabel’nikov, V., Soulard, O.: Rapidly decorrelating velocity-field model as a tool for solving one-point fokker-planck equations for probability density functions of turbulent reactive scalars. Physical Review E 72 (2005)

  • Subramaniam, S., Pope, S.: A mixing model for turbulent reactive flows based on euclidean minimum spanning trees. Combust. Flame 115, 487–514 (1998)

    Article  Google Scholar 

  • Vegendla, S., Hasse, C.: Steady flamelet progress-variable (fpv) modeling and simulation of a high-pressure gasifier. Energy fuels 27(12), 7772–7777 (2013)

    Article  Google Scholar 

  • Zhang, Y.: Reacting flow modeling and applications in STAR-CCM+ (2015)

Download references

Acknowledgements

The author would like to thank Dr. B.A. Perry and Prof. M.E. Mueller for sharing their DNS data, which was invaluable for the development and testing of the \(PSD_F\) distributions.

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, San Diego State University, San Diego, CA, 92115, USA

    Pavel P. Popov

Authors
  1. Pavel P. Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel P. Popov.

Ethics declarations

Conflict of Interest

The author declares that he has no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Popov, P.P. Alternatives to the Beta Distribution in Assumed PDF Methods for Turbulent Reactive Flow. Flow Turbulence Combust 108, 433–459 (2022). https://doi.org/10.1007/s10494-021-00275-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10494-021-00275-w

Keywords

Search

Navigation