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:

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

where ${\mathcal {L}}$ is the Markstein length, and $\delta _{L}$ 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. 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. 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.

Clavin–Williams formula

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. The Markstein number then is

${\mathcal {M}}={\frac {r}{r-1}}\ln r-{\frac {\beta (Le_{\mathrm {eff} }-1)}{2(r-1)}}\mathrm {Li_{2}} [-(r-1)]$

where

• $r>1$ is the gas expansion ratio defined with density ratio,
• $\beta$ is the Zel'dovich number,
• $Le_{\mathrm {eff} }$ is the effective Lewis number of the deficient reactant (either fuel or oxidizer or a combination of both)
• $\mathrm {Li_{2}}$ is the dilogarithm function.

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

${\mathcal {M}}_{b}={\frac {r}{r-1}}\ln r-{\frac {\beta (Le_{\mathrm {eff} }-1)r}{2(r-1)}}\mathrm {Li_{2}} [-(r-1)].$

When temperature dependence of transport coefficients are taken into account, the Clavin–Williams formula becomes

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

where

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

Here

$\mu (\theta )=\rho D_{T}/\rho _{u}D_{T,u}$	is the ratio of density-thermal conductivity product to its value in the unburnt gas;
$\theta =T/T_{u}$	is the ratio of temperature to its unburnt value, defined such that $1\leq \theta \leq r$.

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

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

where $L$ is some cut-off at the lower limit since the integral is diverges logarathmically when $L\to 1$.

Second Markstein number

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

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

See also

• G equation
• Prandtl number
• Schmidt number