**2. Turbulent combustion models in LES**

The Sub-Grid Scale (SGS) turbulent combustion model is a key point in LES study. There are two methods in modelling: one is to build the turbulent combustion model for all turbulence scale, the other is to build the models for big scale and small scale separately. Generally, there are probability density function models, laminar flame-let models, eddy break up model, and ASSCM sub-grid scale model, etc.

As for probability density function models, there are filtered density function method (FDF), probability density function method (PDF), and filtered mass density function method (FMDF), which are similar methods and all rooted in the probability function. They solve the PDF transport equation by the Lagrangian Monte Carlo scheme without using the assumed PDF functions. Also the detail chemical reaction kinetics can be applied directly without models, while the mixing term and the convection term need to be closed. The value of the filtered scaler, such as the averaged temperature, can be calculated by the integration over the composition space. When the FMDF or FDF method extended into turbulent combustion SGS model, the joint probability density function of the sub-grid-scale (SGS) scalar quantities are obtained by solution of its modelled transport equation. In the work of Colucci et al 1998 and James et al 2000, the FMDF combined with detail reaction gave good prediction in temperature

Turbulent Combustion Simulation

expression is rewritten as:

by Large Eddy Simulation and Direct Numerical Simulation 211

The exact expression of the SGS reaction rate is very complex. Anyway it presents the influence from the turbulent fluctuation under the grid scale. Conceptually, comparison with the Reynolds decomposition and the correlation of the turbulent fluctuation, the

 is the integral value in the control volume. The separate subtract terms in the SGS reaction rate denote the contribution from the 'small' scale turbulence fluctuation, and they can be

The ASSCM sub-grid scale turbulent combustion model is applied to a partly diffusion jet flame and a premixed flame after a bluff body. The SGS turbulent combustion model is verified by the experimental data and the LES database is used to verified the RANS turbulent combustion model. The instantaneous turbulent and flame structures are studied too. The implicit box filter function and the Smagorinsky-Lilly's eddy viscosity turbulent model are used. The Smagorinsky constant is 0.1 in this chapter. The SGS mass flux and heat flux are in gradient models. Second order upwind scheme in space, second order central difference scheme for momentum equation, second order in temporal, and PISO algorithm are used. A random velocity component, satisfying Gaussian distribution is superposed at the inlet boundary. The grid size near inlet is 0.5mm around, and under 2mm in the simulation domain. The time step is 0.1ms. Each time step, it iterated 25 steps. As a

The methane-air jet flame, the 'Flame D', is measured by the Sandia Laboratory. The sketch map of this flame is shown in the figure 1. The central flow consists of 25% methane and 75% dry air, the inlet velocity is 29.7m/s, and the inlet temperature is 294K. The annular flow velocity is 6.8m/s and its temperature is 1880K. The surrounding flow velocity is 0.9m/s, and its temperature is 291K. The exit is located at a distance of 1m from the jet exit.

The specific heat for all the species are using subsection polynomial expression, such as the

*C TT T T* 403.58 9.0575 – 0.014425 1.5805 10 6.3431 10

Figure 2 is the averaged temperature comparison between the RANS unified second order moment (USM) transportation turbulent combustion model prediction results and the LES ASSCM turbulent combustion model prediction results with the experimental results. In

specific heat value of methane when the temperature is between 300K to 1000K:

11 1.3 0.2 <sup>8</sup> *w = fu* 2.119 10 exp 2.027 10 / *YY ( ox fu RT)* (7)

(8)

2 5 3 9 4

The *CK,Yox* is the constant, between 0.05 to 0.005 and *Ls* is the SGS mixing length.

**3. Jet flame simulation and premixed flame simulation using LES** 

closed by the products of the gradient of the filtered values:

comparison, the RANS model is applied for the jet flame too.

The methane air reaction kinetics is taken from Westbrook as:

*p*

(5)

(6)

and species profiles. The SGS PDF method still use Monte Carlo method to solve the transport equation of the SGS PDF function ( Gao et al 1993). Renfo et al 2004 used this method and Smagorinsky model to predict the [OH] time series in H2/N2 jet turbulent flame. The predicted power spectral densities and the [OH] time scale are close to the measurements.

The SGS laminar flame-let model is a quick reaction model, that means it assumes that the chemical time scales are shorter than the turbulent time scales. It uses a conservation scalar, such as the mixture fraction, to define the flow variables, such as the species concentration. Assumed PDF functions, as the double-delta function, the clipped Gaussian function and the Beta-pdf function are commonly used. The filtered mixture fraction and its variance can be calculated by transport equations. The relational expression also can be defined by experimental data. DesJardin et al 1999, using the experiential expressions, simulated the ethyne-air jet flame, the temperature prediction is higher than the experimental data in some regions. Jones 2002 applied this method in jet flame and combustion chamber in gas turbine engine, with Smagorinsky turbulent model. The averaged velocity and mixture fraction are close to the experimental data. In the simulation of the gas turbine chamber, the temperature predictions are close to the experimental data in the down stream regions. The instantaneous vortex structures showed detail flow information in the combustion chamber. The Eddy Break Up (EBU), Eddy Dissipation Concept (EDC) and other similar concept SGS model are rooted from the same type RANS model. The turbulent time scale is rescaled by the Kolmogorov turbulent time scale, and the concentrations were calculated by the filtered values. Yaga 2002 applied this model in a swirl combustion chamber with Smagorinsky turbulent model and three steps reaction kinetics. The prediction results of the temperature and the methane concentration are in good agreement with the experimental data while the CO concentration is not very well.

An ASSCM sub-grid scale model is proposed by author and colleagues. In the filtered species equation, there are two reaction terms:

$$\begin{aligned} \frac{\partial \rho \tilde{Y}\_s}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\rho \tilde{u}\_j \tilde{Y}\_s) &= \frac{\partial}{\partial \mathbf{x}\_j} (\frac{\mu}{\mathbf{S} \mathbf{c}\_s} \frac{\partial \tilde{Y}\_s}{\partial \mathbf{x}\_j}) \\ -\tilde{w}\_{L,s} - w\_{S,s} - \frac{\partial \mathbf{g}\_{Sj,s}}{\partial \mathbf{x}\_j} \end{aligned} \tag{1}$$

The instantaneous reaction expression for s species is in the Arrhenius form:

$$
\sigma w\_s = B \rho^2 Y\_{fu} Y\_{\alpha x} \exp\left(-E \;/\, RT\right) = \rho^2 K Y\_{fu} Y\_{\alpha x} \tag{2}
$$

B is the pre-exponential factor, E is the activation energy and R is the universal gas constant. If the fluctuation of density can be neglected, in order to shorten the expression, the exponential term and the pre-exponential factor can be merged into one parameter K. The filtered 'big' scale reaction rate is defined as:

$$
\tilde{w}\_{L,s} = \rho^2 \tilde{K} \tilde{Y}\_{fu} \tilde{Y}\_{\alpha x} \tag{3}
$$

The filtered SGS reaction rate is:

$$\sim\_{\text{avs}=\rho^2(\text{KY}\_{fu}Y\_{\text{ox}}-\tilde{\text{K}}\tilde{Y}\_{fu}\tilde{Y}\_{\text{ox}})} \tag{4}$$

and species profiles. The SGS PDF method still use Monte Carlo method to solve the transport equation of the SGS PDF function ( Gao et al 1993). Renfo et al 2004 used this method and Smagorinsky model to predict the [OH] time series in H2/N2 jet turbulent flame. The predicted

The SGS laminar flame-let model is a quick reaction model, that means it assumes that the chemical time scales are shorter than the turbulent time scales. It uses a conservation scalar, such as the mixture fraction, to define the flow variables, such as the species concentration. Assumed PDF functions, as the double-delta function, the clipped Gaussian function and the Beta-pdf function are commonly used. The filtered mixture fraction and its variance can be calculated by transport equations. The relational expression also can be defined by experimental data. DesJardin et al 1999, using the experiential expressions, simulated the ethyne-air jet flame, the temperature prediction is higher than the experimental data in some regions. Jones 2002 applied this method in jet flame and combustion chamber in gas turbine engine, with Smagorinsky turbulent model. The averaged velocity and mixture fraction are close to the experimental data. In the simulation of the gas turbine chamber, the temperature predictions are close to the experimental data in the down stream regions. The instantaneous vortex structures showed detail flow information in the combustion chamber. The Eddy Break Up (EBU), Eddy Dissipation Concept (EDC) and other similar concept SGS model are rooted from the same type RANS model. The turbulent time scale is rescaled by the Kolmogorov turbulent time scale, and the concentrations were calculated by the filtered values. Yaga 2002 applied this model in a swirl combustion chamber with Smagorinsky turbulent model and three steps reaction kinetics. The prediction results of the temperature and the methane concentration are in good agreement with the experimental data while the

An ASSCM sub-grid scale model is proposed by author and colleagues. In the filtered

*<sup>s</sup> <sup>s</sup> j s j j s j*

*<sup>ρ</sup><sup>Y</sup> <sup>μ</sup> <sup>Y</sup> + (ρuY )= ( ) t x x Sc x*

B is the pre-exponential factor, E is the activation energy and R is the universal gas constant. If the fluctuation of density can be neglected, in order to shorten the expression, the exponential term and the pre-exponential factor can be merged into one parameter K. The

( ) <sup>~</sup> *S s fu fu ox ox w KY Y KY Y*

 

*L,s S,s*

*w w*

The instantaneous reaction expression for s species is in the Arrhenius form:

, <sup>2</sup>

*Sj,s*

(1)

*g*

*j*

exp / *2 2 w =B <sup>s</sup> fu ox fu ox ρ Y Y E RT = ρ KY Y* (2)

*<sup>2</sup> w = L,s fu ox <sup>ρ</sup> KY Y* (3)

(4)

*x*

power spectral densities and the [OH] time scale are close to the measurements.

CO concentration is not very well.

species equation, there are two reaction terms:

filtered 'big' scale reaction rate is defined as:

The filtered SGS reaction rate is:

The exact expression of the SGS reaction rate is very complex. Anyway it presents the influence from the turbulent fluctuation under the grid scale. Conceptually, comparison with the Reynolds decomposition and the correlation of the turbulent fluctuation, the expression is rewritten as:

$$\mathfrak{uv}\_{S,s} = \rho^2 \left[ \widetilde{\boldsymbol{K}} (\boldsymbol{Y}\_{\boldsymbol{\hat{n}}} \boldsymbol{Y}\_{\boldsymbol{\alpha}} - \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\hat{n}}} \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\alpha}}) + \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\alpha}} (\boldsymbol{K} \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\hat{n}}} - \widetilde{\boldsymbol{K}} \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\hat{n}}}) + \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\hat{n}}} (\boldsymbol{K} \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\alpha}} - \widetilde{\boldsymbol{K}} \widetilde{\boldsymbol{Y}}\_{\boldsymbol{\alpha}}) \right] \tag{5}$$

 is the integral value in the control volume. The separate subtract terms in the SGS reaction rate denote the contribution from the 'small' scale turbulence fluctuation, and they can be closed by the products of the gradient of the filtered values:

$$(\widetilde{\boldsymbol{K}\mathcal{Y}\_{\partial\boldsymbol{X}}} - \widetilde{\boldsymbol{K}}\widetilde{\boldsymbol{Y}}\_{\partial\boldsymbol{X}}) = \boldsymbol{C}\_{\boldsymbol{K},\boldsymbol{Y}\_{\partial\boldsymbol{X}}}\boldsymbol{L}\_{\mathrm{S}}^{2}\frac{\partial\widetilde{\boldsymbol{K}}}{\partial\mathbf{x}\_{j}}\frac{\partial\widetilde{\boldsymbol{Y}}\_{\partial\boldsymbol{x}}}{\partial\mathbf{x}\_{j}}\tag{6}$$

The *CK,Yox* is the constant, between 0.05 to 0.005 and *Ls* is the SGS mixing length.
