Markstein number

In combustion engineering and explosion studies, the Markstein number (named after George H. Markstein who first proposed the notion in 1951 ^[1] ) characterizes the effect of local heat release of a propagating flame on variations in the surface topology along the flame and the associated local flame front curvature. There are two dimensionless Markstein numbers: ^{[2] [3]} one is the curvature Markstein number and the other is the flow-strain Markstein number. They are defined as:

${\displaystyle {\mathcal {M}}_{c}={\frac {{\mathcal {L}}_{c}}{\delta _{L}}},\quad {\mathcal {M}}_{s}={\frac {{\mathcal {L}}_{s}}{\delta _{L}}}}$

where ${\displaystyle {\mathcal {L}}_{c}}$ is the curvature Markstein length, ${\displaystyle {\mathcal {L}}_{s}}$ is the flow-strain Markstein length and ${\displaystyle \delta _{L}}$ is the characteristic laminar flame thickness. The larger the Markstein length, the greater the effect of curvature on localised burning velocity. George H. Markstein (1911—2011) showed that thermal diffusion stabilized the curved flame front and proposed a relation between the critical wavelength for stability of the flame front, called the Markstein length, and the thermal thickness of the flame. ^[4] Phenomenological Markstein numbers with respect to the combustion products are obtained by means of the comparison between the measurements of the flame radii as a function of time and the results of the analytical integration of the linear relation between the flame speed and either flame stretch rate or flame curvature. ^{[5] [6] [7]} The burning velocity is obtained at zero stretch, and the effect of the flame stretch acting upon it is expressed by a Markstein length. Because both flame curvature and aerodynamic strain contribute to the flame stretch rate, there is a Markstein number associated with each of these components. ^[8]

Clavin–Williams formula

The Markstein number with respect to the unburnt gas mixture was derived by Paul Clavin and Forman A. Williams in 1982, using activation energy asymptotics. ^{[9] [10]} The formula was extended to include temperature dependences on the thermal conductivities by Paul Clavin and Pedro Luis Garcia Ybarra in 1983. ^[11] The Clavin–Williams formula is given by ^{[3] [12]}

${\displaystyle {\mathcal {M}}={\frac {r}{r-1}}{\mathcal {J}}+{\frac {\beta (Le_{\mathrm {eff} }-1)}{2(r-1)}}{\mathcal {I}},}$

where

${\displaystyle {\mathcal {J}}=\int _{1}^{r}{\frac {\lambda (\theta )}{\theta }}d\theta ,\quad {\mathcal {I}}=\int _{1}^{r}{\frac {\lambda (\theta )}{\theta }}\ln {\frac {r-1}{\theta -1}}\,d\theta .}$

Here

${\displaystyle r>1}$	is the gas expansion ratio defined with density ratio;
${\displaystyle \beta }$	is the Zel'dovich number;
${\displaystyle Le_{\mathrm {eff} }}$	is the effective Lewis number of the deficient reactant (either fuel or oxidizer or a combination of both);
${\displaystyle \lambda (\theta )=\rho D_{T}/\rho _{u}D_{T,u}}$	is the ratio of density-thermal conductivity product to its value in the unburnt gas;
${\displaystyle \theta =T/T_{u}}$	is the ratio of temperature to its unburnt value, defined such that ${\displaystyle 1\leq \theta \leq r}$.

The function ${\displaystyle \lambda (\theta )}$, in most cases, is simply given by ${\displaystyle \lambda =\theta ^{n}}$, where ${\displaystyle n=0.7}$, in which case, we have

${\displaystyle {\mathcal {J}}={\frac {1}{n}}(r^{n}-1),\quad {\mathcal {I}}={\frac {1}{n}}(r^{n}-1)\ln(r-1)-\int _{1}^{r}\theta ^{n-1}\ln(\theta -1)d\theta .}$

In the constant transport coefficient assumption, ${\displaystyle \lambda =1}$, in which case, we have

${\displaystyle {\mathcal {J}}=\ln r,\quad {\mathcal {I}}=-\mathrm {Li_{2}} [-(r-1)]}$

where ${\displaystyle \mathrm {Li_{2}} }$ is the dilogarithm function.

See also

• G equation
• Matalon–Matkowsky–Clavin–Joulin theory
• Clavin–Garcia equation

References

1. Markstein, G. H. (1988). Experimental and theoretical studies of flame-front stability. In Dynamics of curved fronts (pp. 413-423). Academic Press.
2. Clavin, P., & Graña-Otero, J. C. (2011). Curved and stretched flames: the two Markstein numbers. Journal of fluid mechanics, 686, 187-217.
3. Clavin, Paul, and Geoff Searby. Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars. Cambridge University Press, 2016.
4. Oran E. S. (2015). "A tribute to Dr. George H. Markstein (1911–2011)". Combustion and Flame. 162 (1): 1–2. Bibcode:2015CoFl..162....1O. doi:10.1016/j.combustflame.2014.07.005.
5. Karpov V. P.; Lipanikov A. N.; Wolanski P. (1997). "Finding the markstein number using the measurements of expanding spherical laminar flames". Combustion and Flame. 109 (3): 436. Bibcode:1997CoFl..109..436K. doi:10.1016/S0010-2180(96)00166-6.
6. Chrystie R.S.M.; Burns I.S.; Hult J.; Kaminski C.F. (2008). "On the improvement of two-dimensional curvature computation and its application to turbulent premixed flame correlations". Measurement Science and Technology. 19 (12): 125503. Bibcode:2008MeScT..19l5503C. doi:10.1088/0957-0233/19/12/125503. S2CID   21642877.
7. Chakraborty N, Cant RS (2005). "Influence of Lewis number on curvature effects in turbulent premixed flame propagation in the thin reaction zones regime". Physics of Fluids. 17 (10): 105105–105105–20. Bibcode:2005PhFl...17j5105C. doi:10.1063/1.2084231.
8. Haq MZ, Sheppard CG, Woolley R, Greenhalgh DA, Lockett RD (2002). "Wrinkling and curvature of laminar and turbulent premixed flames". Combustion and Flame. 131 (1–2): 1–15. Bibcode:2002CoFl..131....1H. doi:10.1016/S0010-2180(02)00383-8.
9. Clavin, Paul, and F. A. Williams. "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." Journal of fluid mechanics 116 (1982): 251–282.
10. Clavin, Paul. "Dynamic behavior of premixed flame fronts in laminar and turbulent flows." Progress in Energy and Combustion Science 11.1 (1985): 1–59
11. Clavin, P., & Garcia, P. (1983). The influence of the temperature dependence of diffusivities on the dynamics. Journal de Mécanique Théorique et Appliquée, 2(2), 245-263.
12. Bechtold, J. K., & Matalon, M. (2001). The dependence of the Markstein length on stoichiometry. Combustion and flame, 127(1-2), 1906-1913.