Clavin–Garcia equation

Clavin–Garcia equation or Clavin–Garcia dispersion relation provides the relation between the growth rate and the wave number of the perturbation superposed on a planar premixed flame, named after Paul Clavin and Pedro Luis Garcia Ybarra, who derived the dispersion relation in 1983.^[1] The dispersion relation accounts for Darrieus–Landau instability, Rayleigh–Taylor instability and diffusive–thermal instability and also accounts for the temperature dependence of transport coefficients.

Dispersion relation

Let ${\displaystyle k}$ and ${\displaystyle \sigma }$ be the wavenumber (measured in units of planar laminar flame thickness ${\displaystyle \delta _{L}}$) and the growth rate (measured in units of the residence time ${\displaystyle \delta _{L}^{2}/D_{T,u}}$ of the planar laminar flame) of the perturbations to the planar premixed flame. Then the Clavin–Garcia dispersion relation is given by^{[2][3][4][5][6]}

${\displaystyle a(k)\sigma ^{2}+b(k)\sigma +c(k)=0}$

where

${\displaystyle {\begin{aligned}a(k)&={\frac {r+1}{r}}+{\frac {r-1}{r}}k\left({\mathcal {M}}-{\frac {r}{r-1}}{\mathcal {J}}\right),\\b(k)&=2k+2rk^{2}({\mathcal {M}}-{\mathcal {J}}),\\c(k)&=-{\frac {r-1}{r}}Ra\,k-(r-1)k^{2}\left[1-{\frac {Ra}{r}}\left({\mathcal {M}}-{\frac {r}{r-1}}{\mathcal {J}}\right)\right]+(r-1)k^{3}\left[L+{\frac {3r-1}{r-1}}{\mathcal {M}}-{\frac {2r}{r-1}}{\mathcal {J}}+(2Pr-1){\mathcal {H}}\right],\end{aligned}}}$

and

${\displaystyle {\mathcal {J}}=\int _{1}^{r}{\frac {\lambda (\theta )}{\theta }}d\theta ,\quad {\mathcal {H}}={\frac {1}{r-1}}\int _{1}^{r}[L-\lambda (\theta )]d\theta .}$

Here

${\displaystyle r=\rho _{u}/\rho _{b}}$	is the gas expansion ratio; ratio of burnt gas to unburnt gas density; typically ${\displaystyle r\in (2,8)}$;
${\displaystyle \lambda (\theta )=\rho D_{T}/\rho _{u}D_{T,u}}$	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 ${\displaystyle 1\leq \theta \leq r}$;
${\displaystyle L=\rho _{b}D_{T,b}/\rho _{u}D_{T,u}}$	is the transport coefficient ratio, i.e., ${\displaystyle L\equiv \lambda (r)}$
${\displaystyle {\mathcal {M}}}$	is the Markstein number;
${\displaystyle Ra=g\delta _{L}^{3}/D_{T,u}^{2}}$	is the Rayleigh number; ${\displaystyle Ra>0}$ (gravity points towards burnt gas) and ${\displaystyle Ra<0}$ (gravity points towards unburnt gas)
${\displaystyle Pr=\nu _{u}/D_{T,u}}$	is the Prandtl number.

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

${\displaystyle {\mathcal {J}}={\frac {1}{m}}(r^{m}-1),\quad {\mathcal {H}}=r^{m}-{\frac {r^{1+m}-1}{(1+m)(r-1)}}.}$

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

${\displaystyle {\mathcal {J}}=\ln r,\quad {\mathcal {H}}=0.}$

See also

• Clavin–Williams formula