Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T08:59:50.662Z Has data issue: false hasContentIssue false
  • English
  • Français

A spectral closure for premixed turbulent combustion in the flamelet regime

Published online by Cambridge University Press:  26 April 2006

N. Peters
N. Peters
Affiliation:
Institut für Technische Mechanik, RWTH Aachen. 5100 Aachen. Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Premixed turbulent combustion in the flamelet regime is analysed on the basis of a field equation. This equation describes the instantaneous flame contour as an isoscalar surface of the scalar field G(x,t). The field equation contains the laminar burning velocity sL as velocity scale and its extension includes the effect of flame stretch involving the Markstein length [Lscr ] as a characteristic lengthscale of the order of the flame thickness. The scalar G(x,t) plays a similar role for premixed flamelet combustion as the mixture fraction Z(x,t) in the theory of non-premixed flamelet combustion.

Equations for the mean $\overline{G}$ and variance $\overline{G^{\prime 2}}$ are derived. Additional closure problems arise for the mean source terms in these equations. In order to understand the nature of these terms an ensemble of premixed flamelets with arbitrary initial conditions in constant-density homogeneous isotropic turbulence is considered. An equation for the two-point correlation $\overline{G^{\prime}({\boldmath x},t)G^{\prime}({\boldmath x}+{\boldmath r},t)}$ is derived. When this equation is transformed into spectral space, closure approximations based on the assumption of locality and on dimensional analysis are introduced. This leads to a linear equation for the scalar spectrum function Γ(k,t), which can be solved analytically. The solution Γ(k,t) is analysed by assuming a small-wavenumber cutoff at k0 = lT−1, where lT is the integral lengthscale of turbulence. There exists a $k^{-\frac{5}{3}}$ spectrum between lT and LG, where LG is the Gibson scale. At this scale turbulent fluctuations of the scalar field G(x,t) are kinematically restored by the smoothing effect of laminar flame propagation. A quantity called kinematic restoration ω is introduced, which plays a role similar to the scalar dissipation χ for diffusive scalars.

By calculating the appropriate moments of Γ(k,t), an algebraic relation between ω, $\omega,\overline{G^{\prime}({\boldmath x},t)^2}$, the integral lengthscale lT and the viscous dissipation ε is derived. Furthermore, the scalar dissipation χ[Lscr ], based on the Markstein diffusivity [Dscr ][Lscr ] = sL [Lscr ], and the scalar-strain co-variance Σ[Lscr ] are related to ω. Dimensional analysis, again, leads to a closure of the main source term in the equation for the mean scalar $\overline{G}$. For the case of plane normal and oblique turbulent flames the turbulent burning velocity sT and the flame shape is calculated. In the absence of flame stretch the linear relation sT ∼ u′ is recovered. The flame brush thickness is of the order of the integral lengthscale. In the case of a V-shaped flame its increase with downstream position is calculated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 611 - 629
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdel-Gayed, R. G., Al-Kishali, K. J. & Bradley, D. 1984 Turbulent burning velocities and flame straining in explosions.. Proc. R. Soc. Lond. A 391, 393–414.Google Scholar
Abdel-Gayed, R. G. & Bradley, D. 1981 A two-eddy theory of premixed turbulent flame propagation.. Phil. Trans. R. Soc. Lond. A 301, 1–25.Google Scholar
Abdel-Gayed, R. G., Bradley, D., Hamid, H. N. & Lawes, M. 1985 Lewis number effects on turbulent burning velocities In 20the Symp. (Intl) on Combustion, pp. 505–512. The Combustion Institute.
Abdel-Gayed, R. G., Bradley, D. & Lawes, M. 1987 Turbulent burning velocities: a general correlation in terms of straining rates.. Proc. R. Soc. Lond. A 414, 389–413.Google Scholar
Ashurst, W. T. 1990 Geometry of premixed flames in three-dimensional turbulence. Center for Turbulence Research, Proc. Summer Program 1990, pp. 1–7.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K., Howells, I. D. & Townsend, A. A. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity. J. Fluid Mech. 5, 134–319.Google Scholar
Borghi, R. 1990 Turbulent premixed Combustion: Further Discussions on the Scales of fluctuations. Combust. Flame 80, 304–312.Google Scholar
Borghi, R. & Dutoya, D. 1979 On the scales of the fluctuations in turbulent combustion In 17the Symp. (Intl) on Combustion, pp. 235–244. The Combustion Institute.
Bray, K. N. C. 1979 The interaction between turbulence and combustion In 17the Symp. (Intl) on Combustion, pp. 223–233. The Combustion Institute.
Bray, K. N. C. 1990 Studies of the turbulent velocity.. Proc. R. Soc. Lond. A 431, 315–335.Google Scholar
Bray, K. N. C., Champion, M. & Libby, P. A. 1988 Mean reaction rates in premixed turbulent flames In 22nd Symp. (Intl) on Combustion, pp. 763–769. The Combustion Institute.
Bray, K. N. C., Libby, P. A. & Moss, J. B. 1984 Flamelet crossing frequencies and mean reaction rates in premixed turbulent combustion. Combust. Sci. Technol. 41, 143–172.Google Scholar
Cant, R. S. & Bray, K. N. C. 1989 A theoretical model of premixed turbulent combustion in closed vessels. Combust. Flame 76, 243–263.Google Scholar
Catlin, C. A. & Lindstedt, R. P. 1991 Premixed turbulent burning velocities derived from mixing controlled reaction models with cold front quenching. Combust. Flame 85, 427–435.Google Scholar
Clavin, P. 1985 Dynamic behaviour of premixed flame fronts in laminar and turbulent flows. Prog. Energy Combust. Sci. 11, 1–59.Google Scholar
Clavin, P. & Joulin, G. 1983 Premixed flames in large scale and high intensity turbulent flow. J. Phys. Lett. Paris 44, L-1-L-12.Google Scholar
Clavin, P. & Williams, F. A. 1982 Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity. J. Fluid Mech. 116, 251–282.Google Scholar
Corrsin, S. 1951a The decay of isotropic temperature fluctuations in an isotropic turbulence. J. Aeronaut. Sci. 18, 417–423.Google Scholar
Corrsin, S. 1951b On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22, 469–473.Google Scholar
Damköhler, G. 1940 Der Einfluß der Turbulenz auf die Flammengeschwindigkeit in Gasgemischen. Z. Elektrochem. 46, 601–652.Google Scholar
Gouldin, F. C., Bray, K. N. C. & Chen, J. Y. 1989 Chemical closure model for fractal flamelets. Combust. Flame 77, 241–259.Google Scholar
Kerstein, A. R., Ashurst, W. T. & Williams, F. A. 1988 Field equation for interface propagation in an unsteady homogeneous flow field.. Phys. Rev. A 37, 2728–2731.Google Scholar
Magnussen, B. F. & Hjertager, B. H. 1977 On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion In 16th Symp. (Intl) on Combustion, pp. 719–729. The Combustion Institute.
Mason, H. B. & Spalding, D. B. 1973 Prediction of reaction rates in turbulent premixed boundary layer flows. In The Combustion Institute European Symp. (ed. F. J. Weinberg), pp. 601–606. Academic.
Matalon, M. & Matkowsky, B. J. 1982 Flames as gasdynamic discontinuities. J. Fluid Mech. 124, 239–259.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT Press.
Pao, Y. H. 1965 Structure of turbulent velocity and scalar fields at large wave numbers. Phys. Fluids 8, 1063–1075.Google Scholar
Pao, Y. H. 1968 Transfer of turbulent energy and scalar quantities at large wavenumbers. Phys. Fluids 11, 1371–1372.Google Scholar
Pelce, P. & Clavin, P. 1982 Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. J. Fluid Mech. 124, 219–237.Google Scholar
Peters, N. 1984 Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10, 319–339.Google Scholar
Peters, N. 1986 Laminar flamelet concepts in turbulent combustion In 21st Symp. on Combustion, pp. 1231–1256.
Peters, N. & Keller, D. 1992 A consistent flamelet formulation for premixed turbulent combustion. Unpublished.
Searby, G. & Quinard, J. 1990 Direct and indirect measurements of Markstein numbers of premixed flames. Combust. Flame 82, 298–311.Google Scholar
Sivashinsky, G. I. Cascade renormalization theory of turbulent flame speed. Combust. Sci. Technol. 62, 77–96.
Vassilicos, C. 1990 Fractal and moving interfaces in turbulent flows. Ph.D. thesis, Faculty of Mathematics, University of Cambridge.
Williams, F. A. 1985 Turbulent combustion. In The Mathematics of Combustion (ed. John Buckmaster), pp. 97–131. SIAM.
Wu, M.-S., Kwon, S., Driscoll, J. F. & Faeth, G. M. 1991 Preferential diffusion effects on the surface structure of turbulent premixed hydrogen/air flames. Combust. Sci. Technol. 78, 69–96.Google Scholar
Yakhot, V. 1988a Propagation velocity of premixed turbulent flames. Combust. Sci. Technol. 60, 191–214.Google Scholar
Yakhot, V. 1988b Scale invariant solutions of the theory of thin turbulent flame propagation. Combust. Sci. Technol. 62, 127–129.Google Scholar