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Modelling Sub-Grid Passive Scalar Statistics in Moderately Dense Evaporating Sprays


Spray evaporation in spatially decaying turbulence is simulated using carrier-phase direct numerical simulations (CP-DNS). The CP-DNS cover a much wider parameter range than earlier fully resolved DNS of regular droplet arrays that were used to calibrate scaling laws for sub-grid closures of the distribution of mixture fraction and its conditionally averaged scalar dissipation. The scaling laws include the effects of sub-grid interactions between droplet evaporation and turbulence, and here they are assessed by direct comparison with the statistics from the CP-DNS. Two issues can be observed: Firstly, care must be taken when interpreting the CP-DNS statistics as the lack of resolution of the point particle surface could impact on the values in DNS cells that are located within the (unresolved) quasi-laminar wake. Secondly, the scaling laws present similar agreement with CP-DNS data as had been observed before for the fully resolved DNS. The scaling law for the scalar dissipation approximates the DNS statistics well, independent of the droplet number density, Stokes number and turbulence intensity. The estimated mixture fraction distribution (the PDF) is good for the mixture fraction values larger than a suitable average value but deteriorates for smaller mixture fractions due to inherent model limitations. The data corroborate that the scaling laws for turbulent micro-mixing can potentially serve as sub-grid closures for mixture fraction based combustion models such as flamelet and conditional moment closure approaches in large eddy simulations and may provide better approximations than existing expressions derived from single-phase non-premixed combustion.

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  1. Menon, S., Patel, N.: Subgrid modeling for simulation of spray combustion in large-scale combustors. AIAA J. 44(4), 709–723 (2006)

    Article  Google Scholar 

  2. Lackmann, T., Kerstein, A.R., Oevermann, M.: A representative linear eddy model for simulating spray combustion in engines (RILEM). Combust. Flame 193, 1–15 (2018)

    Article  Google Scholar 

  3. Boileau, M., Pascaud, S., Riber, E., Cuenot, B., Gicquel, L.Y.M., Poinsot, T.J., Cazalens, M.: Investigation of two-fluid methods for large eddy simulation of spray combustion in gas turbines. Flow Turbul. Combust. 80(3), 291–321 (2008)

    Article  MATH  Google Scholar 

  4. Ma, L., Roekaerts, D.: Modeling of spray jet flame under MILD condition with non-adiabatic FGM and a new conditional droplet injection model. Combust. Flame 165, 402–423 (2016)

    Article  Google Scholar 

  5. Bojko, B.T., DesJardin, P.E.: On the development and application of a droplet flamelet-generated manifold for use in two-phase turbulent combustion simulations. Combust. Flame 183, 50–65 (2017)

    Article  Google Scholar 

  6. Ukai, S., Kronenburg, A., Stein, O.T.: Simulation of dilute acetone spray flames with LES-CMC using two conditional moments. Flow Turbul. Combust. 93(3), 405–423 (2014)

    Article  Google Scholar 

  7. Ukai, S., Kronenburg, A., Stein, O.T.: Large eddy simulation of dilute acetone spray flames using CMC coupled with tabulated chemistry. Proc. Combust. Inst. 35 (2), 1667–1674 (2015)

    Article  Google Scholar 

  8. Giusti, A., Mastorakos, E.: Detailed chemistry LES/CMC simulation of a swirling ethanol spray flame approaching blow-off. Proc. Combust. Inst. 36(2), 2625–2632 (2017)

    Article  Google Scholar 

  9. Hasse, C., Peters, N.: A two mixture fraction flamelet model applied to split injections in a DI Diesel engine. Proc. Combust. Inst. 30(2), 2755–2762 (2005)

    Article  Google Scholar 

  10. Navarro-Martinez, S., Kronenburg, A., Di Mare, F.: Conditional moment closure for large eddy simulations. Flow Turbul. Combust. 75(1-4), 245–274 (2005)

    Article  MATH  Google Scholar 

  11. Bilger, R.W.: Turbulent flows with nonpremixed reactants. In: Turbulent reacting flows, pp 65–113. Springer, Berlin (1980)

  12. O’Brien, E.E., Jiang, T.L.: The conditional dissipation rate of an initially binary scalar in homogeneous turbulence. Phys. Fluids A 3(12), 3121–3123 (1991)

    Article  MATH  Google Scholar 

  13. Réveillon, J., Vervisch, L.: Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model. Combustion and flame 121(1-2), 75–90 (2000)

    Article  Google Scholar 

  14. Schroll, P., Wandel, A.P., Cant, R.S., Mastorakos, E.: Direct numerical simulations of autoignition in turbulent two-phase flows. Proc. Combust. Inst. 32(2), 2275–2282 (2009)

    Article  Google Scholar 

  15. Wacks, D.H., Chakraborty, N., Mastorakos, E.: Statistical analysis of turbulent flame-droplet interaction: a direct numerical simulation study. Flow Turbul. Combust. 96(2), 573–607 (2016)

    Article  Google Scholar 

  16. Sreedhara, S., Huh, K.Y.: Conditional statistics of nonreacting and reacting sprays in turbulent flows by direct numerical simulation. Proc. Combust. Inst. 31(2), 2335–2342 (2007)

    Article  Google Scholar 

  17. Wang, H., Luo, K., Fan, J.: Direct numerical simulation and CMC (conditional moment closure) sub-model validation of spray combustion. Energy 46(1), 606–617 (2012)

    Article  Google Scholar 

  18. Seo, J., Huh, K.Y.: Analysis of combustion regimes and conditional statistics of autoigniting turbulent n-heptane sprays. Proc. Combust. Inst. 33(2), 2127–2134 (2011)

    Article  Google Scholar 

  19. Ukai, S., Kronenburg, A., Stein, O.T.: LES-CMC Of a dilute acetone spray flame. Proc. Combust. Inst. 34(1), 1643–1650 (2013)

    Article  Google Scholar 

  20. Bilger, R.W.: A mixture fraction framework for the theory and modeling of droplets and sprays. Combust. Flame 158(2), 191–202 (2011)

    Article  Google Scholar 

  21. Klimenko, A.Y., Bilger, R.W.: Conditional moment closure for turbulent combustion. Prog. Energy Combust. Sci. 25(6), 595–687 (1999)

    Article  Google Scholar 

  22. Zoby, M.R.G., Navarro-Martinez, S., Kronenburg, A., Marquis, A.J.: Turbulent mixing in three-dimensional droplet arrays. Int. J. Heat Fluid Flow 32(3), 499–509 (2011)

    Article  Google Scholar 

  23. Wang, B., Kronenburg, A., Dietzel, D., Stein, O.T.: Assessment of scaling laws for mixing fields in inter-droplet space. Proc. Combust. Inst. 36(2), 2451–2458 (2017)

    Article  Google Scholar 

  24. Wang, B., Kronenburg, A., Tufano, G.L., Stein, O.T.: Fully resolved DNS of droplet array combustion in turbulent convective flows and modelling for mixing fields in inter-droplet space. Combust. Flame 189, 347–366 (2018)

    Article  Google Scholar 

  25. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30(4), 299–303 (1941)

    MathSciNet  Google Scholar 

  26. Salazar, J.P., Collins, L.R.: Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech. 41, 405–432 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tufano, G.L., Stein, O.T., Wang, B., Kronenburg, A., Rieth, M., Kempf, A.M.: Coal particle volatile combustion and flame interaction. Part II: effects of particle Reynolds number and turbulence. Fuel 234, 723–731 (2018)

    Article  Google Scholar 

  28. Kronenburg, A., Bilger, R.W.: Modelling of differential diffusion effects in nonpremixed nonreacting turbulent flow. Phys. Fluids 9.5, 1435–1447 (1997)

    Article  Google Scholar 

  29. Kronenburg, A., Bilger, R.W.: Modelling differential diffusion in nonpremixed reacting turbulent flow: model development. Combust. Sci. Technol. 166(1), 195–227 (2001)

    Article  Google Scholar 

  30. Xu, H., Hunger, F., Vascellari, M., Hasse, C.: A consistent flamelet formulation for a reacting char particle considering curvature effects. Combust. Flame 160(11), 2540–2558 (2013)

    Article  Google Scholar 

  31. Hunger, F., Dietzsch, F., Gauding, M., Hasse, C.: A priori analysis of differential diffusion for model development for scale-resolving simulations. Phys. Rev. Fluids 3(1), 014601 (2018)

    Article  Google Scholar 

  32. Batchelor, G.K.: Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5(1), 113–133 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  33. Elghobashi, S.: On predicting particle-laden turbulent flows. Appl. Sci. Res. 52 (4), 309–329 (1994)

    Article  Google Scholar 

  34. Miller, R.S., Bellan, J.: Direct numerical simulation of a confined three-dimensional gas mixing layer with one evaporating hydrocarbon-droplet-laden stream. J. Fluid Mech. 384, 293–338 (1999)

    Article  MATH  Google Scholar 

  35. White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (1974)

    MATH  Google Scholar 

  36. Turns, S.R.: An introduction to combustion. McGraw-Hill, New York (1996)

    Google Scholar 

  37. Faeth, G.M.: Current status of droplet and liquid combustion. Prog. Energy Combust. Sci. 3(4), 191–224 (1977)

    Article  Google Scholar 

  38. Green, D.W., Perry, R.H.: Perry’s chemical engineers’ handbook. McGraw-Hill, New York (2008)

    Google Scholar 

  39. Billson, M., Eriksson, L.E., Davidson, L.: Jet noise prediction using stochastic turbulence modeling. In: 9th AIAA/CEAS aeroacoustics conference and exhibit, p 3282 (2003)

  40. Sulabh, K.D., Jacob, E.T., Driscoll, J.F.: Unsteady aspects of lean premixed-prevaporized (LPP) gas turbine combustors: flame-flame interactions. AIAA g010-1148 (2010)

  41. Williams, F.A.: Combustion theory, 2nd edn. Perseus Books, Reading, Massachussetts (1985)

  42. Sirignano, W.A.: Fluid dynamics and transport of droplets and sprays. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  43. Vo, S., Kronenburg, A., Stein, O.T., Hawkes, E.R.: Direct numerical simulation of non-premixed syngas combustion using OpenFOAM. In: High Performance Computing in Science and Engineering ’16, pp 245–257. Springer (2017)

  44. Yeung, P.K., Pope, S.B.: Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586 (1989)

    Article  MathSciNet  Google Scholar 

  45. Shekar, S.: Direct numerical simulation of isotropic, decaying turbulence. ITV report, University of Stuttgart (2011)

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We acknowledge the financial support by the Chinese Scholarship Council (NO 201406020093) (B. Wang) and the computational resources by HLRS Stuttgart.

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Appendix: Derivation of the Mixture Fraction PDF in Inter-droplet Space

Appendix: Derivation of the Mixture Fraction PDF in Inter-droplet Space

Here, spherical symmetry and constant thermodynamic properties are assumed for simplification. The continuity and mixture fraction transport equation for the droplet evaporation in the spherical coordinate system can be written as

$$ \frac{1}{r^{2}}\frac{d}{dr}\left( \rho u r^{2} \right) = 0, $$
$$ \frac{1}{r^{2}}\frac{d}{dr}\left( \rho u r^{2} f - \rho r^{2} D \frac{df}{dr}\right) = 0. $$

We introduce the evaporation rate of the droplet, Jm, and the integral of Eq. A.1 is then given by

$$ \rho u r^{2} = \frac{J_{m}}{4\pi}. $$

Equation A.2 is rewritten by substitution of Eq. A.3 as

$$ \frac{df}{dr}=\frac{d}{dr}\left( \frac{4\pi \rho D r^{2}}{J_{m}}\frac{df}{dr}\right). $$

The integral of Eq. A.4 from the droplet surface at rs to a distance r is

$$ f - f_{s} = \frac{4\pi \rho D r^{2}}{J_{m}}\frac{df}{dr} - \frac{4\pi \rho D {r_{s}^{2}}}{J_{m}}{\frac{df}{dr}} |_{r=r_{s}}, $$

where fs is the mixture fraction at the droplet surface. The boundary condition of f at the droplet surface is given by

$$ f_{d}J_{m} = f_{s}J_{m} - 4\pi \rho D {r_{s}^{2}} {\frac{df}{dr}} |_{r=r_{s}}, $$

where fd is the mass fraction of the evaporating fuel in the droplet and it equals 1 for a single component droplet. Substituting Eq. A.6 into Eq. A.5 yields

$$ \frac{df}{dr} = \frac{J_{m}\left( f-f_{d}\right)}{4\pi r^{2} \rho D}. $$

The scalar dissipation Nf can be derived using Eq. A.7 as

$$ N_{f} = D\left( \frac{df}{dr}\right)^{2} = \frac{J_{m}\left( f_{d}-f\right)}{4\pi r^{2} \rho} \left|\frac{df}{dr}\right|. $$

The PDF of mixture fraction is evaluated by an identity formula

$$ P_{f} df \equiv c4\pi r^{2} dr, $$

where c is the droplet number density. Equation A.9 is rewritten with the substitution of Eq. A.8, viz.

$$ P_{f} = \frac{4\pi r^{2} c}{\left|df/dr\right|} = \frac{4\pi r^{2} c}{N_{f} \frac{4\pi r^{2} \rho}{J_{m}\left( f_{d}-f\right)}}=\frac{cJ_{m}\left( f_{d}-f\right)}{\rho N_{f}} \approx \frac{cJ_{m}\left( f_{d}-f_{2}\right)}{\rho N_{f}}. $$

It is important to note that the identity in Eq. A.9 assumes lack of interaction between the mixing fields of the different droplets and accounts for multiple droplets by simple multiplication by c. This will be - as shown in this paper - a sufficiently accurate approximation for dilute sprays but will lead to larger inaccuracies for Pf with f smaller than the average mixture fraction value, \(f < \tilde {f}\), if the droplet loading is high as for these values the mixing fields will interact.

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Wang, B., Kronenburg, A. & Stein, O.T. Modelling Sub-Grid Passive Scalar Statistics in Moderately Dense Evaporating Sprays. Flow Turbulence Combust 103, 519–535 (2019).

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  • DNS
  • Turbulent sprays
  • Mixture fraction
  • Scalar dissipation
  • PDF