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