**5. Turbulent combustion model study by DNS**

The DNS database is used for the scalar fluctuation correlations transportation equations' budget. The exact values are compared with the model values, and then the improvement is given in the turbulent combustion model.

Firstly, a RANS transport equation combustion model is studied. In turbulent combustion model, the correlations are important terms. As for incompressible flow, *Pc* is constant, *D*

$$\eta = \frac{\lambda}{\rho c\_p}, \text{ and } \text{ put } \ w\_1 = w\_2 = B\rho^2 Y\_1 Y\_2 \exp(-\frac{E}{RT}) = \rho^2 Y\_1 Y\_2 K \quad \text{reaction term into the equations,} \quad \rho^2 = \frac{\lambda}{\rho c\_p} \rho^2 = \frac{\lambda}{\rho c\_p}$$

then the exact transportation equation for *T'Y '1* , *K'Y '1* and *Y 'Y '* <sup>1</sup> *<sup>2</sup>* are:

1 11 1 1 1 1 <sup>112</sup> <sup>2</sup> 2D *' ' ' ' <sup>j</sup> ' '' j jj j j jj ' ' 1 ' 1 j j jj P Y T' Y T' Y T u Y T' + u = T'u Y u + t x x xx Y T' T' Y <sup>ρ</sup><sup>Q</sup> D( ) + YYYK <sup>ρ</sup>T'Y Y K x x xx c* (17) 1 1 1 1 2 1 1 2 2 2 2D 2λ *' ' ' ' ' '' j 1 1 1 j jj j j jj '' ' 1 j j jj P jj ' ' 1 1 1 P jj P K'Y K'Y Y K u K'Y + u = u K' u Y + t x x xx K'Y Y K' <sup>λ</sup> EY K T D( ) x x xx ρcR x x T Y KT EρQ Y KY Y K ρK'Y Y K + ρcTx x Rc T* (18)

Turbulent Combustion Simulation

Fig. 28. The profiles of the *K'Y '1* .

Fig. 29. The profiles of the *'*

Fig. 30. The budget of the 1 2

by Large Eddy Simulation and Direct Numerical Simulation 225




y

y

 P12 D <sup>12</sup> 12 S12

 Case 4 Case 5 Case 6

y

 Case 4 Case 5 Case 6


Y1 'v' -0.0030 -0.0025 -0.0020 -0.0015 -0.0010 -0.0005 0.0000

*<sup>1</sup> v'Y* .


Y'1Y '2 Case6

*' ' Y Y* equation.

0.0

0.5

1.0

1.5

2.0

Y1 'K'

$$\begin{split} \frac{\partial \overline{\chi\_{1}^{'} \dot{Y}\_{2}^{'}}}{\partial t} + \overline{u\_{j}} \frac{\partial \overline{Y\_{1}^{'} Y\_{2}^{'}}}{\partial x\_{j}} &= -\overline{u\_{j}^{'} Y\_{2}^{'}} \frac{\partial \overline{Y\_{1}}}{\partial x\_{j}} - \overline{u\_{j}^{'} Y\_{1}^{'}} \frac{\partial \overline{Y\_{2}}}{\partial x\_{j}} - \frac{\partial \overline{u\_{j}^{'} Y\_{1}^{'} Y\_{2}^{'}}}{\partial x\_{j}} + \\ D \frac{\partial}{\partial x\_{j}} (\overline{\partial \overline{Y\_{1}^{'} Y\_{2}^{'}}}) - 2D \overline{\frac{\partial Y\_{1}^{'}}{\partial x\_{j}} \frac{\partial Y\_{2}^{'}}{\partial x\_{j}}} - \rho \overline{Y\_{1} Y\_{1} Y\_{2} K} - \rho \overline{Y\_{2} Y\_{1} Y\_{2} K} \end{split} \tag{19}$$

The statistical correlation values are shown in figure 26 to 29. The temperature fluctuation is very important in reaction correlations. Then the budgets are studied and shown in figure 30 to 32. In these results, *K'Y '1* value changes greatly and *T'Y '1* are more important than 1 2 *' ' Y Y* according to the average value.

Fig. 26. The profiles of the 1 2 *' ' Y Y* .

Fig. 27. The profiles of the *T'Y '1* .

Fig. 28. The profiles of the *K'Y '1* .

1 2 12 12 1 2 2 1

*' '' '' '' <sup>j</sup> '' ''*

*YY YY Y Y uYY +u = uY uY + t x x xx*

*YY Y Y D( ) <sup>ρ</sup>YYYK <sup>ρ</sup>YYYK*

The statistical correlation values are shown in figure 26 to 29. The temperature fluctuation is very important in reaction correlations. Then the budgets are studied and shown in figure 30 to 32. In these results, *K'Y '1* value changes greatly and *T'Y '1* are more important than

112 212 2D



y

 Case 4 Case 5 Case 6

y

 Case 4 Case 5 Case 6

*' '*

(19)

*j j jj*

12 1 2

*j j jj*

*x x xx*

1 2

*' ' Y Y* according to the average value.

Fig. 26. The profiles of the 1 2

Fig. 27. The profiles of the *T'Y '1* .

Y1 'Y2 '0.03

> -0.03 -0.02 -0.01 0.00 0.01 0.02

> > *' ' Y Y* .

Y1 'T'

*'' ' '*

*j jj*

Fig. 29. The profiles of the *' <sup>1</sup> v'Y* .

Fig. 30. The budget of the 1 2 *' ' Y Y* equation.

Turbulent Combustion Simulation

by Large Eddy Simulation and Direct Numerical Simulation 227

12 12 1 1 2 1 2

*YY YY Y Y C μ μ Y Y +u = + () t x ρσ x x <sup>σ</sup> x x*

 

*' ' ' '*

The comparison of the DNS exact values and the RANS model values are shown in figure 33 and 34. Generally, the model values of the production term (a), diffusion term (b) and the total dissipation term (c) are close to the exact values from DNS database. The difference between the exact values and the model values is rooted from the model assumption.

*j YY j j j j*



(c)

K'Y1 '

y

Case 5

 DNS RANS

*'' '' ' ' t e <sup>j</sup>*

2 12 3 1 2 1 2 exp

*<sup>ε</sup> <sup>E</sup> C Y Y C [Bρ(Y + <sup>β</sup>Y ) ( )]Y Y <sup>k</sup> RT*

RANS

In which, 1 *C =* 11.5 , KY 0.7 *YY σ = σ =* , 2 *C =* 0.005 , 3 *C =* 0.012 , *β =* 1.0 .


<sup>y</sup> (a)


Fig. 33. The DNS statistical value and RANS model value of terms in 1


0.0000

0.0001

0.0002

0.0003

PK Y1 DNS

KY1 +SKY1 K'Y1'

Case 5


equation in case 5.

YY

(21)

 DNS RANS

Case 5


(b)

y K'Y1 '

*' ' K Y* correlation

Fig. 31. The budget of the *T'Y '1* equation.

Fig. 32. The budget of the *K'Y '1* equation.

From the magnitude and distribution, the chemistry affects a lot in correlations and their budget. In RANS model, the correlation transportation equations need to be closed with the production term, turbulent diffusion term and the dissipation term. As for the production term, the isotropic turbulent model is applied with *Yσ* = 0.7. It is assumed that the dissipation of the correlation is direct proportion to itself. And gradient model is used in the diffusion term.

$$\begin{split} \frac{\partial \overline{K'Y\_1'}}{\partial t} + \overline{u\_j} \frac{\partial \overline{K'Y\_1'}}{\partial x\_j} &= \frac{\mathbb{C}\_1 \mu\_t}{\rho \sigma\_{KY}} \frac{\partial \overline{Y\_1}}{\partial x\_j} \frac{\partial \overline{K}}{\partial x\_j} + \frac{\mu\_e}{\sigma\_{KY}} \frac{\partial}{\partial x\_j} (\overline{\frac{\partial K'Y\_1'}{\partial x\_j}}) \\ - \mathbb{C}\_2 \frac{\varepsilon}{k} \overline{K'Y\_1'} - \mathbb{C}\_3 [B\rho(\overline{Y\_1} + \beta \overline{Y\_2}) \exp(-\frac{E}{RT})] \overline{K'Y\_1'} \end{split} \tag{20}$$



From the magnitude and distribution, the chemistry affects a lot in correlations and their budget. In RANS model, the correlation transportation equations need to be closed with the production term, turbulent diffusion term and the dissipation term. As for the production term, the isotropic turbulent model is applied with *Yσ* = 0.7. It is assumed that the dissipation of the correlation is direct proportion to itself. And gradient model is used in the

1 1

*<sup>ε</sup> <sup>E</sup> C K'Y C [Bρ(Y + <sup>β</sup>Y ) ( )]K'Y <sup>k</sup> RT*

 

*K'Y K'Y Y K C μ μ K'Y +u = + () t x ρσ x x <sup>σ</sup> x x*

*' ' ' 1 1 t e 1*

> *' ' 1 1*

*j KY j j j j*

2 3 12 exp

*j*


0.00

0.05

0.10 PY1

Y'1 K' Case4

KY

y

K DY1 K Y1 K S1,Y1 T S2,Y1 T


(b)

y

(20)


Fig. 31. The budget of the *T'Y '1* equation.

K DY1 K Y1 K S1,Y1 T S2,Y1 T

0.0003 PY1

Y'1 K' Case5


(a)

Fig. 32. The budget of the *K'Y '1* equation.

y


diffusion term.

Y'1

T' Case4

 PY1 T DY1 T Y1 T SY1 T

$$\begin{split} \frac{\partial \overline{Y\_1' Y\_2'}}{\partial t} + \overline{u\_j} \frac{\partial \overline{Y\_1' Y\_2'}}{\partial \mathbf{x}\_j} &= \frac{\mathbb{C}\_1 \mu\_t}{\rho \sigma\_{\mathbf{YY}}} \frac{\partial \overline{Y\_1}}{\partial \mathbf{x}\_j} \frac{\partial \overline{Y\_2}}{\partial \mathbf{x}\_j} + \frac{\mu\_e}{\sigma\_{\mathbf{YY}}} \frac{\partial}{\partial \mathbf{x}\_j} (\frac{\overline{\partial Y\_1' Y\_2'}}{\partial \mathbf{x}\_j}) \\ - \mathbb{C}\_2 \frac{\varepsilon}{k} \overline{Y\_1' Y\_2'} - \mathbb{C}\_3 [\mathbb{B}\rho(\overline{Y\_1} + \rho \overline{Y\_2}) \text{exp}(-\frac{E}{RT})] \overline{Y\_1' Y\_2'} \end{split} \tag{21}$$

In which, 1 *C =* 11.5 , KY 0.7 *YY σ = σ =* , 2 *C =* 0.005 , 3 *C =* 0.012 , *β =* 1.0 .

The comparison of the DNS exact values and the RANS model values are shown in figure 33 and 34. Generally, the model values of the production term (a), diffusion term (b) and the total dissipation term (c) are close to the exact values from DNS database. The difference between the exact values and the model values is rooted from the model assumption.

Fig. 33. The DNS statistical value and RANS model value of terms in 1 *' ' K Y* correlation equation in case 5.

Turbulent Combustion Simulation

reasonableness of ASOM model.

are


0.000

0.005

0.010

Y'1 K'

by Large Eddy Simulation and Direct Numerical Simulation 229

The comparison between DNS exact values and the algebraic model values are shown in figure 35 and 36. The model values are in same trend with the exact values while in some regions, the model values are rather of distortion. The overestimated value in the near wall regions and the underestimations in the main flow regions resulted from the gradient simplification. The model constant in ASOM model is 0.005. These results show the


*YYK YYK* 12 12 *= K(Y Y Y Y )+ Y (KY KY )+ Y (KY KY ) OX Fu OX Fu OX Fu Fu Fu OX OX* (24)

The three quasi-correlation parts in the right hand side, shortened as *Y Y*1 2 *R* , *Y K*<sup>1</sup> *R* and *Y T*<sup>1</sup> *R* ,

<sup>2</sup> ΦΨ *<sup>Φ</sup>,<sup>Ψ</sup> <sup>S</sup>*

Initially, the turbulent model is Smagorinsky-Lilly SGS model with *L =C S SΔ* . While this model get bigger dissipation in flow regions especially in the near wall region, then a

In the spectral method, each variable in flow can be expended as Fourier function, in this chapter, the DNS wave number is 128, then assumed that the cut-off wave number in LES is 64, thus the DNS results can be divided into two parts: the low-pass value represents the resolvable value in LES and the rest high order value represents the sub-grid scale value.

*Φ Ψ <sup>=</sup> ΦΨ =C L*

Fig. 36. The DNS statistical value and RANS model value of correlation *K'Y '1* .

As for the ASSCM SGS model in LES, the small scale reaction is calculated by:

*R*ΦΨ

Using the filtering function, the statistic value of SGS correlation

trend, and they have similar distribution in all domain.

database and SGS model value are shown in figure 37 to 42. 1*<sup>Y</sup> <sup>R</sup> <sup>T</sup>* and

damping correction is used [zhang 2005]:

y

*j j*

(25)

<sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>K</sup>* and

<sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>T</sup>* from DNS

<sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>K</sup>* have same

*x x*

1 exp /*+ + L =C S SΔ[ ( y A )]* , *CS* =0.2, *<sup>+</sup> A* =26 (26)

 DNS ASOM

Case5

Fig. 34. The DNS statistical value and RANS model value of terms in 1 2 *' ' Y Y* correlation equation in case 6.

The algebraic second order moment (ASOM) RANS turbulent combustion model is quite simple but have been applied in jet flame successfully. The model expression is:

$$
\overline{Y\_1'Y\_2'} = C\_{YY} \frac{k^3}{\varepsilon^2} \frac{\overline{\partial Y\_1}}{\overline{\partial x\_j}} \frac{\partial \overline{Y\_2}}{\partial x\_j} \tag{22}
$$

$$\overline{K'Y\_1'} = C\_{KY\_1} \frac{k^3}{\varepsilon^2} \frac{\partial \overline{K}}{\partial \mathbf{x}\_j} \frac{\partial \overline{Y\_1}}{\partial \mathbf{x}\_j} \tag{23}$$

Fig. 35. The DNS statistical value and RANS model value of correlation 1 2 *' ' Y Y*

D12



(c)

The algebraic second order moment (ASOM) RANS turbulent combustion model is quite

*k YY Y 'Y '= C*

3

3

2

*1 KY1 j j k KY K'Y = C*

y

1 2

*j j*

 K=0.0 DNS K=0.1 DNS K=1.0 DNS K=0.0 ASOM K=0.1 ASOM K=1.0 ASOM y

1

*ε x x* 

*ε x x* 


Y1 'Y2 '  DNS RANS

Case 6


(b)

y

*' ' Y Y* correlation

(22)

(23)

*' ' Y Y*

Y1 'Y2 ' Case 6

 DNS RANS


12+S12

(a)

y


Fig. 34. The DNS statistical value and RANS model value of terms in 1 2

simple but have been applied in jet flame successfully. The model expression is:

*'*


Fig. 35. The DNS statistical value and RANS model value of correlation 1 2

Y'1 Y'2 1 *2 YY* 2

Y1 'Y2 ' Case 6

 DNS RANS


equation in case 6.


0.0000

0.0004

P12

The comparison between DNS exact values and the algebraic model values are shown in figure 35 and 36. The model values are in same trend with the exact values while in some regions, the model values are rather of distortion. The overestimated value in the near wall regions and the underestimations in the main flow regions resulted from the gradient simplification. The model constant in ASOM model is 0.005. These results show the reasonableness of ASOM model.

Fig. 36. The DNS statistical value and RANS model value of correlation *K'Y '1* .

As for the ASSCM SGS model in LES, the small scale reaction is calculated by:

$$\widetilde{\mathbf{Y}\_{1}Y\_{2}K} - \widetilde{\mathbf{Y}\_{1}}\widetilde{\mathbf{Y}\_{2}K} = \widetilde{\mathbf{K}(\mathbf{Y}\_{OX}Y\_{Fu} - \widetilde{\mathbf{Y}\_{OX}}\widetilde{\mathbf{Y}\_{Fu}}) + \widetilde{\mathbf{Y}\_{OX}(\mathbf{K}\mathbf{Y}\_{Fu} - \widetilde{\mathbf{K}}\widetilde{\mathbf{Y}\_{Fu}}) + \widetilde{\mathbf{Y}\_{Fu}(\mathbf{K}\mathbf{Y}\_{OX} - \widetilde{\mathbf{K}}\widetilde{\mathbf{Y}\_{OX}})}} \tag{24}$$

The three quasi-correlation parts in the right hand side, shortened as *Y Y*1 2 *R* , *Y K*<sup>1</sup> *R* and *Y T*<sup>1</sup> *R* , are

$$R\_{\Phi\Psi\Psi} = \stackrel{\mathcal{V}}{\Phi\Psi} - \tilde{\Phi}\tilde{\Psi} = \mathcal{C}\_{\Phi,\Psi} L\_S^2 \frac{\partial \tilde{\Phi}}{\partial \mathbf{x}\_j} \frac{\partial \tilde{\Psi}}{\partial \mathbf{x}\_j} \tag{25}$$

Initially, the turbulent model is Smagorinsky-Lilly SGS model with *L =C S SΔ* . While this model get bigger dissipation in flow regions especially in the near wall region, then a damping correction is used [zhang 2005]:

$$L\_S = C\_S \Delta \left[ 1 - \exp(-y^+ \text{ / } A^+ \text{)} \right], \text{ \textdegree C}\_S = 0.2, \text{ \textdegree A}^+ = 26 \tag{26}$$

In the spectral method, each variable in flow can be expended as Fourier function, in this chapter, the DNS wave number is 128, then assumed that the cut-off wave number in LES is 64, thus the DNS results can be divided into two parts: the low-pass value represents the resolvable value in LES and the rest high order value represents the sub-grid scale value. Using the filtering function, the statistic value of SGS correlation <sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>K</sup>* and <sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>T</sup>* from DNS database and SGS model value are shown in figure 37 to 42. 1*<sup>Y</sup> <sup>R</sup> <sup>T</sup>* and <sup>1</sup>*<sup>Y</sup> <sup>R</sup> <sup>K</sup>* have same trend, and they have similar distribution in all domain.

Turbulent Combustion Simulation

RY1Y2


Fig. 40. The DNS value of SGS *Y Y*1 2 *R* with wall correction.


Fig. 41. The DNS value of SGS *Y K*<sup>1</sup> *R* with wall correction.


Fig. 42. The DNS value of SGS *Y Y*1 2 *R* with wall correction

0.00000

0.00004

0.00008

0.00012

0.00016

RY1 Y2 0.000

0.001

0.002

0.003

0.004

0.005

RY1 K

by Large Eddy Simulation and Direct Numerical Simulation 231




y

 K=0.0 DNS K=0.0 Model K=0.1 DNS K=0.1 Model K=1.0 DNS K=1.0 Model

Model-ZN

y

Model-ZN Case 4 Case 5 Case 6  Case 4 Case 5 Case 6

y

Model-ZN

Fig. 37. The DNS value of SGS *Y T*<sup>1</sup> *R* .

Fig. 38. The DNS value of SGS *Y K*<sup>1</sup> *R* .

Fig. 39. The DNS value of SGS *Y Y*1 2 *R* .

 Case 4 Y1 T

 Case 4 Y2 T

 Case 5 Y1 T

 Case 5 Y2 T

 Case 6 Y1 T

 Case 6 Y2 T

> Case 4 Y1 K

> Case 4 Y2 K

> Case 5 Y1 K

> Case 5 Y2 K

> Case 6 Y1 K

> Case 6 Y2 K

> > Case 4 Case 5 Case 6



y


y

y




0.000

0.002

0.004

0.006

DNS

0.008

0.00

0.01

0.02

0.03

0.04 DNS

RY1 T

Fig. 37. The DNS value of SGS *Y T*<sup>1</sup> *R* .

RY1 K

Fig. 38. The DNS value of SGS *Y K*<sup>1</sup> *R* .

DNS RY1Y2

Fig. 39. The DNS value of SGS *Y Y*1 2 *R* .

Fig. 40. The DNS value of SGS *Y Y*1 2 *R* with wall correction.

Fig. 41. The DNS value of SGS *Y K*<sup>1</sup> *R* with wall correction.

Fig. 42. The DNS value of SGS *Y Y*1 2 *R* with wall correction

Turbulent Combustion Simulation

it is closer to the exact model factor value.



Fig. 45. The SGS flux model tested with damping model.

values are all close to the experimental data in the most regions.

**6. Conclusion** 

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.0000

0.0002

0.0004

0.0006

gUY1 K=0.0 DNS

by Large Eddy Simulation and Direct Numerical Simulation 233

moment turbulent model can give reasonable predictions in most regions. The damping modification model can give better prediction results than the constant value model because

> K=0.0 Model K=0.1 DNS K=0.1 Model K=1.0 DNS K=1.0 Model



(b)

The LES and DNS methods are more and more important recently. There are many SGS turbulent combustion models for LES. The algebraic sub-grid scale turbulent combustion is quite simple in expression, while it is successfully applied in partly diffusion jet flame and premixed after bluff body flame. The predicted temperature, species, velocity and RMS

y

(a)

gVY1 K=0.0 DNS

y

 K=0.0 Model K=0.1 DNS K=0.1 Model K=1.0 DNS K=1.0 Model

In ASSCM turbulent SGS combustion model, the model factor is constant, while the exact value for model factor is shown in figure 43. It can be treat as a constant in most regions, but changes a lot in the near wall regions and varies according to different cases.

Fig. 43. The SGS model factor from DNS statistic.

While the model factor value with the near wall damping modification is shown in figure 44.

Fig. 44. The modified SGS model factor (XModel-ZN) divided by constant model value (XModel)

According to the figures above, it is clearly that the modified model (XModel-ZN) is better than the constant value model (XModel) in the near wall regions because its value is closer to the exact DNS statistic value, so the damping model can give more reasonable prediction results.

The SGS scalar flux model is using gradient model with damping modification:

$$\mathcal{S}\_{u\_{\dot{j}}Y\_{1}} = \mathcal{C}\_{u\_{\dot{j}}Y\_{1}} \frac{\mathcal{Y}\_{T}}{\sigma} \frac{\partial \bar{Y}\_{1}}{\partial \mathbf{x}\_{j}} \tag{27}$$

Sum up the main points in this part, briefly, there are typical strip structures in the velocity and scalar field. The chemical reaction enhanced the turbulence especially in high shear regions. In the transportation equations of correlations, the production term and the dissipation term are more important. Comparison with the exact value from DNS, the RANS second order moment turbulent combustion model and the RANS algebraic second order

In ASSCM turbulent SGS combustion model, the model factor is constant, while the exact value for model factor is shown in figure 43. It can be treat as a constant in most regions, but

While the model factor value with the near wall damping modification is shown in figure 44.

Fig. 44. The modified SGS model factor (XModel-ZN) divided by constant model value (XModel) According to the figures above, it is clearly that the modified model (XModel-ZN) is better than the constant value model (XModel) in the near wall regions because its value is closer to the exact DNS statistic value, so the damping model can give more reasonable prediction

The SGS scalar flux model is using gradient model with damping modification:

1



1

*T*

 

*u u j j1 <sup>j</sup> <sup>γ</sup> <sup>Y</sup> g =C Y Y <sup>σ</sup> <sup>x</sup>*

Sum up the main points in this part, briefly, there are typical strip structures in the velocity and scalar field. The chemical reaction enhanced the turbulence especially in high shear regions. In the transportation equations of correlations, the production term and the dissipation term are more important. Comparison with the exact value from DNS, the RANS second order moment turbulent combustion model and the RANS algebraic second order

y

K


(27)

(b)

y

0.0003 Case 4

Case 6 XY1

Case 5

changes a lot in the near wall regions and varies according to different cases.


(a)

Fig. 43. The SGS model factor from DNS statistic.

XModel-ZN/XModel

 Case 4 Case 5 Case 6 y



XY1 Y2

results.

moment turbulent model can give reasonable predictions in most regions. The damping modification model can give better prediction results than the constant value model because it is closer to the exact model factor value.

Fig. 45. The SGS flux model tested with damping model.
