Investigation of pressure and the Lewis number effects in the context of algebraic flame surface density closure for LES of premixed turbulent combustion

Abstract

Large scale industrial combustion devices, for example, internal combustion engines, gas turbine combustors, etc., operate under high-pressure conditions and utilize a variety of fuels. Unfortunately, the majority of the current numerical combustion modelling approaches are not fully validated for high-pressure and the non-unity Lewis number ($\hbox {Le} =$ thermal diffusivity/mass diffusivity) effects in premixed turbulent combustion. In any case, a numerical model needs to be checked for the effects of these parameters to guarantee generality of the model. In the present study, these two critical features of the models are numerically explored utilizing fundamental elements of several algebraic flame surface density reaction rate closure models accessible in the open literature. The Lewis number impact is likewise examined utilizing LES of recently published subgrid scale fractal flame surface density model, which indicated acceptable results for high and low-pressure methane fuelled applications. The computed numerical results are compared with an extensive experimental dataset for lean methane and propane fuels featuring various flow and turbulence conditions at operating pressures in the range of 1–10 bar. The quantitative results from most of the selected models do not show the experimentally observed trends at high-pressures and for non-unity Le number fuels. Modifications to the models are incorporated to reflect effects of these two important parameters utilizing a broad parametric investigation resulting in a satisfactory agreement with the experimental data.

Appendix 1

This section shows the calculation of pressure exponent n of the explicit pressure correction term for adapted Model-A and Model-Z.

Model-A

For $\varXi \gg 1$, Eq. (13) for Model-A without an explicit pressure correction term can be written as:

$$\begin{aligned} \varXi = \left[ C_\mathrm{A} \varGamma \frac{u_{{\varDelta }}^{\mathrm {'}}}{s_\mathrm{L}^{0}} \right] =\frac{s_\mathrm{T}}{s_\mathrm{L}^{0}}, \end{aligned}$$

(A1)

$$\begin{aligned} \varXi *s_\mathrm{L}^{0} =\left[ C_\mathrm{A} \varGamma \frac{u_{{\varDelta }}^{\mathrm {'}}}{s_\mathrm{L}^{0}} \right] s_\mathrm{L}^{0} = s_\mathrm{T}. \end{aligned}$$

(A2)

From Eq. (A2), the pressure dependence of the predicted turbulent flame speed $s_\mathrm{T}$ from the Model-A is calculated and compared with the experimentally observed pressure dependence of turbulent flame speed, which is $s_\mathrm{T}\propto p^{0.07}$ for methane [29]. For methane using $s_\mathrm{L}^{0}\propto p^{-0.5}$ and $l_{F}\propto p^{-0.5}$, the pressure dependence of the efficiency function from Eq. (10) is found to be $\varGamma \propto p^{1/3}$. Putting the pressure dependence values of $s_\mathrm{L}^{0}$ and $\varGamma$ in Eq. (A2) and we get $\varXi *s_\mathrm{L}^{0}\propto p^{1/3}/p^{-0.5}{*p}^{-0.5}=p^{1/3}$. The pressure exponent of 0.33 is larger than the experimental one 0.07. Now an explicit pressure correction term $\left( p/p_{0} \right) ^{n}$ is introduced in Eq. (A1) with a value of n giving the correct pressure dependence of $s_\mathrm{T}$:

$$\begin{aligned} \left( \frac{p}{p_{0}} \right) ^{n}*\varXi *s_\mathrm{L}^{0}\mathrm {=}\left( \frac{p}{p_{0}} \right) ^{n}\left[ C_\mathrm{A} \varGamma \mathrm {}\frac{u_{{\varDelta }}^{\mathrm {'}}}{s_\mathrm{L}} \right] s_\mathrm{L}^{0}. \end{aligned}$$

(A3)

For Model-A, n is found to be -0.25 which is giving the experimental dependence of $s_\mathrm{T}$close to zero for methane i.e. $\left( p/p_{0} \right) ^{n}*\varXi *s_\mathrm{L}^{0}\propto p^{-0.25}*p^{1/3}/p^{-0.5}{*p}^{-0.5}=p^{0.08}$. It is noted that in above calculations $u_{{\varDelta }}^{\mathrm {'}}$ is taken to be independent of pressure. At high Reynolds number and constant inlet velocity, $u^{'}$ will be approximately constant. However, with increasing pressure and thus increasing Reynolds number the Kolmogorov length decreases. At constant filter width the total subgrid turbulent energy and therefore $u_{{\varDelta }}^{'}$ may increase slightly. However, as discussed in Sect. 6.3, LES simulation using the adapted Model-A with $n=-0.25$ gives very good agreement of the predicted turbulent flame speed with the experimental data, justifying the assumption of a small effect of a pressure variation of $u_{{\varDelta }}^{\mathrm {'}}$ on the turbulent flame speed.

Model-Z

A similar analysis is performed for the Model-Z with Eq. (16) and n is found to be 0.15.