Analysis of Some Recently Proposed Modifications to the Eddy Dissipation Concept (EDC)

ABSTRACT

The widespread use of the Eddy Dissipation Concept (EDC) of Magnussen has led to a number of suggestions to modify the model. These modifications are to a varying extent in agreement with the underlying ideas of EDC. This article analyzes 20 such attempts. The original formulations of EDC are reviewed and explained. Often, users tend to neglect the expressions for the fraction of reacting fine structures. This part of the model includes some Reynolds number dependency, which is acting where such effects have been requested. The possibly unintended, but widely used, EDC modification in the Ansys Fluent implementation is discussed and analyzed. It is shown that some of the claimed defects of EDC are caused by this implementation. A total of 12 articles proposing changes of the EDC constants are reviewed and the suggestions analyzed and discussed with respect to the reaction rate, fine-structure model and viscous effects. Models combining Arrhenius and EDC at low turbulence Reynolds numbers and a model based on fractal theory are commented.

KEYWORDS:

• EDC
• turbulent combustion
• turbulence-chemistry interaction
• model constants
• low Reynolds number

Nomenclature

$C_{\mathrm{D}1}$,$C_{\mathrm{D}2}$	=	constants of EDC (-)
$C_{\mathrm{R}}$, $C_{\gamma }$, $C_{\tau }$	=	secondary constants of EDC (-)
$C_{\epsilon 1}$, $C_{\epsilon 2}$, $C_{\mu }$	=	constants of $k$-$\epsilon$ model (-)
$c$	=	extent of reaction (-)
$D_c$	=	fractal dimension (-), Equation 53(53) ${\tau }^{\ast }=0.5\frac{k}{\epsilon }({\gamma }^{\ast }{)}^{\frac{1+\frac{2}{3}-\frac{D_c}{3}}{3-Dc}},$(53)
$g_{\mathrm{E}\mathrm{D}\mathrm{C}}$	=	EDC-factor (-), Equation 29(29) ${\bar{R}}_k=g_{\mathrm{E}\mathrm{D}\mathrm{C}}\frac{\bar{\rho }}{{\tau }^{\ast }}(Y_k^{\ast }-{\tilde{Y}}_k).$(29)
$F$	=	ratio of mean reaction rates (-), Equation 42(42) $F=\frac{{\bar{R}}_{k,\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}{{\bar{R}}_{k,\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{r}.\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}=\frac{(Y_k^{\ast }-{\tilde{Y}}_k)}{(Y_k^{\ast }-Y_k^o)}=1-{\gamma }^{\ast }\chi ,$(42)
$f$	=	mixture fraction (-)
$k$	=	turbulence energy (m${ }^2$/s${ }^2$)
$L$	=	length scale (m)
${\dot{m}}^{\ast }$	=	mass inflow to EDC reactor per reactor mass (-)
$n$	=	turbulence energy decay exponent (-)
$R$	=	time scale ratio (-), Equation 50(50) $R=\frac{{\tau }^{\ast }}{{\tau }_{\mathrm{m}\mathrm{i}\mathrm{x}}}=\frac{{\gamma }_{\lambda }^2}{(1-{\gamma }_{\lambda }^3)},$(50)
$R_k$	=	volumetric reaction rate of species $k$ (kg/(s$\cdot$m${ }^3$))
$Re$	=	Reynolds number (-)
$R{e}_{\mathrm{T}}$	=	turbulence Reynolds number, $k^2/(\nu \epsilon )$ (-)
$r$	=	stoichiometric oxidizer requirement of the fuel, mass based (kg/kg)
$t$	=	time (s)
$T$	=	temperature (K)
$u$	=	velocity scale (m/s)
$u^{\prime }$	=	turbulence velocity scale (m/s)
$v$	=	Kolmogorov velocity scale (m/s)
$w$	=	weighting parameter, Equation 57(57) $\frac{L^{\ast }}{\eta }=\frac{w}{w+R{e}_{\mathrm{T}}}1.75+\frac{R{e}_{\mathrm{T}}}{w+R{e}_{\mathrm{T}}}3.15,$(57) (-)
$Y_k$	=	mass fraction of species $k$ (-)
${\alpha }_H$, ${\alpha }_L$	=	constants in the model of Equation 35(35) $\epsilon ={\alpha }_H\lambda k^{3/2}+{\alpha }_L\nu {\lambda }^{2}k.$(35) (-)
${\beta }^{\ast }$	=	constant in Saffman–Wilcox $k$-$\omega$ turbulence model (-)
${\gamma }^{\ast }$	=	mass of fine structures divided by total mass (-)
${\gamma }_{\lambda }$	=	mass of fine-structure regions divided by total mass (-)
$\epsilon$	=	turbulence energy dissipation rate (m${ }^2$/s${ }^3$)
$\eta$	=	Kolmogorov length scale (m)
$\theta$	=	time scale, $k/\epsilon$ (s)
$\lambda$	=	variable of turbulence model, Equation 35(35) $\epsilon ={\alpha }_H\lambda k^{3/2}+{\alpha }_L\nu {\lambda }^{2}k.$(35) (m${ }^{-1}$)
$\lambda$	=	excess air ratio, reciprocal of equivalence ratio (-)
$\nu$	=	kinematic viscosity (m${ }^2$/s)
$\rho$	=	mass density (kg/m${ }^3$)
$\tau$	=	time scale (s)
$\chi$	=	reacting fraction of fine structures (-)
$\omega$	=	turbulence strain rate or frequency ($s^{-1}$)

Superscripts

—	=	average
$\tilde{}$	=	mass-weighted (Favre) average
${ }^{\ast }$	=	fine-structure (reactor) quantity of EDC
${ }^o$	=	surroundings of EDC fine-structure reactor

Subscripts

BaR	=	batch reactor
fu	=	fuel
ox	=	oxidizer
pr	=	product
st	=	stoichiometric

Acknowledgments

During this study and in previous years, I have had useful discussions with, among others, Michał T. Lewandowski (Polish Academy of Sciences, Gdańsk, Poland), Inge R. Gran (office mate at NTNU in the 1990s, now at Sintef Energy, Trondheim), Bjørn F. Magnussen (previously NTNU, later own company, Trondheim). Supplementary information from Ashok De (IIT Kanpur, India) and Madjid Birouk (University of Manitoba, Canada) is appreciated.

Disclosure statement

No potential conflict of interest was reported by the author.