Markstein number

In combustion engineering and explosion studies, the Markstein number 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. The dimensionless Markstein number is defined as:

${\displaystyle {\mathcal {M}}={\frac {\mathcal {L}}{\delta _{L}}}}$

where ${\displaystyle {\mathcal {L}}}$ is the 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. It is named after George H. Markstein (1911—2011), who 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.[1] 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.[2][3][4] 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.[5]

Clavin–Williams equation

The Markstein number with respect to the unburnt gas mixture for a one step reaction in the limit of large activation energy asymptotics was derived by Paul Clavin and Forman A. Williams in 1982.[6] The Markstein number then is

${\displaystyle {\mathcal {M}}_{u}={\frac {1}{\alpha }}\ln {\frac {1}{1-\alpha }}+{\frac {\beta (\mathrm {Le} -1)}{2}}{\frac {1-\alpha }{\alpha }}\int _{0}^{\alpha /(1-\alpha )}{\frac {\ln(1+x)}{x}}\,dx}$

where

• ${\displaystyle \alpha }$ is the heat release parameter defined with density ratio,
• ${\displaystyle \beta }$ is the Zel'dovich number,
• ${\displaystyle {\rm {Le}}}$ is the Lewis number of the deficient reactant (either fuel or oxidizer)

and the Markstein number with respect to the burnt gas mixture is derived by Clavin (1985)[7]

${\displaystyle {\mathcal {M}}_{b}={\frac {1}{\alpha }}\ln {\frac {1}{1-\alpha }}+{\frac {\beta (\mathrm {Le} -1)}{2\alpha }}\int _{0}^{\alpha /(1-\alpha )}{\frac {\ln(1+x)}{x}}\,dx}$

Second Markstein number

In general, Markstein number for the curvature effects ${\displaystyle {\mathcal {M}}_{c}}$ and strain effects ${\displaystyle {\mathcal {M}}_{s}}$ are not same in real flames[8]. In that case, one defines a second Markstein number as

${\displaystyle {\mathcal {M}}_{2}={\mathcal {M}}_{c}-{\mathcal {M}}_{s}.}$

See also

• G equation
• Prandtl number
• Schmidt number

References

1. Oran E. S. (2015). "A tribute to Dr. George H. Markstein (1911–2011)". Combustion and Flame. 162 (1): 1–2. doi:10.1016/j.combustflame.2014.07.005.
2. 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. doi:10.1016/S0010-2180(96)00166-6.
3. 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.
4. 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.
5. 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. doi:10.1016/S0010-2180(02)00383-8.
6. 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.
7. Clavin, Paul. "Dynamic behavior of premixed flame fronts in laminar and turbulent flows." Progress in Energy and Combustion Science 11.1 (1985): 1–59.
8. Clavin, Paul, and Geoff Searby. Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars. Cambridge University Press, 2016.