Matalon–Matkowsky–Clavin–Joulin theory

The Matalon–Matkowsky–Clavin–Joulin theory refers to a theoretical hydrodynamic model of a premixed flame with a large-amplitude flame wrinkling, developed independently by Moshe Matalon & Bernard J. Matkowsky and Paul Clavin & Guy Joulin,{{cite journal |last1=Matalon |first1=M. |last2=Matkowsky |first2=B. J. |title=Flames as gasdynamic discontinuities |journal=Journal of Fluid Mechanics |volume=124 |issue=–1 |date=1982 |issn=0022-1120 |doi=10.1017/S0022112082002481 |page=239|doi-broken-date=29 November 2024 }}{{cite journal |last1=Clavin |first1=P. |last2=Joulin |first2=G. |title=Premixed flames in large scale and high intensity turbulent flow |journal=Journal de Physique Lettres |volume=44 |issue=1 |date=1983 |issn=0302-072X |doi=10.1051/jphyslet:019830044010100 |pages=1–12|url=https://hal.archives-ouvertes.fr/jpa-00232137/file/ajp-jphyslet_1983_44_1_1_0.pdf }} following the pioneering study by Paul Clavin and Forman A. WilliamsClavin, P., & Williams, F. A. (1982). 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, 251-282. and by Pierre Pelcé and Paul Clavin.Pelce, P., & Clavin, P. (1988). Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. In Dynamics of curved fronts (pp. 425-443). Academic Press. The theory, for the first time, calculated the burning rate of the curved flame that differs from the burning rate of the planar flame due to flame stretch, associated with the flame curvature and the strain imposed on the flame by the flow field.{{cite journal |last=Clavin |first=Paul |title=Dynamic behavior of premixed flame fronts in laminar and turbulent flows |journal=Progress in Energy and Combustion Science |volume=11 |issue=1 |date=1985 |doi=10.1016/0360-1285(85)90012-7 |pages=1–59}}

Burning rate formula

According to Matalon–Matkowsky–Clavin–Joulin theory, if $S_L$ and $\delta_L$ are the laminar burning speed and thickness of a planar flame (and $\tau_L=D_{T,u}/S_L^2$ be the corresponding flame residence time with $D_{T,u}$ being the thermal diffusivity in the unburnt gas), then the burning speed $S_T$ for the curved flame with respect to the unburnt gas is given by{{cite book |last1=Clavin |first1=Paul |last2=Searby |first2=Geoff |title=Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars |publisher=Cambridge University Press |date=2016-07-28 |isbn=978-1-107-49163-2 |doi=10.1017/cbo9781316162453 |page=}}{{pn|date=September 2024}}

: $\frac{S_T}{S_L} = 1 + \mathcal{M}_c \delta_L \nabla \cdot \mathbf{n} + \mathcal{M}_s \tau_L \mathbf{n}\mathbf n: \nabla\mathbf v$

where $\mathbf{n}$ is the unit normal to the flame surface (pointing towards the burnt gas side), $\mathbf{v}$ is the flow velocity field evalauted at the flame surface and $\mathcal{M}_c$ and $\mathcal{M}_s$ are the two Markstein numbers, associated with the curvature term $\nabla \cdot \mathbf{n}$ and the term $\mathbf{n}\mathbf n:\nabla\mathbf v$ corresponding to flow strain imposed on the flame.{{cite journal |last1=Clavin |first1=Paul |last2=Graña-Otero |first2=José C. |title=Curved and stretched flames: the two Markstein numbers |journal=Journal of Fluid Mechanics |volume=686 |date=2011-11-10 |issn=0022-1120 |doi=10.1017/jfm.2011.318 |pages=187–217}}

References

Category:Fluid dynamics

Category:Combustion

Matalon–Matkowsky–Clavin–Joulin theory

Burning rate formula

See also

References