**4. Direct numerical simulation for reacting flows**

Along with the development of computational technology, direct numerical simulation becomes more and more popular in combustion simulation studies. It is a powerful tool for fundamental study and test sources for RANS and LES models.

In this chapter, a DNS of turbulent reacting channel flows with the consideration of the interaction between the velocity and scalars by buoyancy effect is performed using a spectral method. The instantaneous reaction rate is in Arrhenius form. The computational domain and coordinate system are shown in figure 20. x, y, z are the flow direction, normal direction, and span-wise direction separately. The height of the channel is 2H, the length in the stream-wise is 12.6H and the width in the span-wise direction is 6.28H. The flow is fully developed and the reactants mixed sufficiently.

Fig. 20. The computation domain for DNS.

The instantaneous continuity, momentum, species concentration and energy equations of incompressible turbulent reacting flows, with consideration of the buoyancy effect using Boussinesq approximation and taking the Arrhenius expression of one-step kinetics.

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = 0 \tag{12}$$

Turbulent Combustion Simulation

0.0

0.0

averaged reaction factor K profiles is shown in figure 25.

Fig. 22. The averaged value of the correlation of concentration and v velocity

The instantaneous contour of temperature and species are shown in figure 23 and 24. The plenty turbulent structures and near-wall stripe structures can be seen. The statistical

0.2

0.4

0.6

0.8

1.0

+

Fig. 21. The RMS value of velocities.


1.0

1.5

2.0

2.5

RMS

3.0

very close to each other.

by Large Eddy Simulation and Direct Numerical Simulation 221

In the SM DNS simulation results, U=0.063845247, Re=191.54. The two DNS results are

0 50 100 150 200

y +

0 40 80 120 160 200

y +

 FD DNS SM DNS

FD DNS u'<sup>+</sup>

FD DNS v'<sup>+</sup>

FD DNS w'<sup>+</sup>

SM DNS u'<sup>+</sup>

SM DNS v'<sup>+</sup>

SM DNS w'<sup>+</sup>

rms

rms

rms

rms

rms

rms

$$\frac{\partial u\_i}{\partial t} + u\_j \frac{\partial u\_i}{\partial \mathbf{x}\_j} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} [\nu (\frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i})] - F\_i \tag{13}$$

$$\frac{\partial}{\partial t}(Y\_1) + \frac{\partial}{\partial \mathbf{x}\_j}(\boldsymbol{\mu}\_j Y\_1) = \frac{\partial}{\partial \mathbf{x}\_j}(\mathbf{D} \frac{\partial Y\_1}{\partial \mathbf{x}\_j}) - w\_1 / \rho \tag{14}$$

$$\frac{\partial}{\partial t}(Y\_2) + \frac{\partial}{\partial \mathbf{x}\_j}(\mathbf{u}\_j Y\_2) = \frac{\partial}{\partial \mathbf{x}\_j}(\mathbf{D} \frac{\partial Y\_2}{\partial \mathbf{x}\_j}) - w\_2 \ / \ \rho \tag{15}$$

$$\frac{\partial}{\partial t}(T) + \frac{\partial}{\partial \mathbf{x}\_j}(\boldsymbol{\mu}\_j T) = \frac{\partial}{\partial \mathbf{x}\_j}(\frac{\lambda}{c\_p \rho} \frac{\partial T}{\partial \mathbf{x}\_j}) + w\_1 Q\_1 \;/\ \ (\mathbf{c}\_p \rho) \tag{16}$$

The Fi term in momentum equation stands for the effect from buoyancy force which caused by the difference of the temperature, also it induces the scalar and velocity interaction. The Reynolds number Rem defined by the channel half width H and average velocity Um is 3000. The parameters in DNS cases are given in Table 1.

For all cases, the mass fraction of species 1 (fuel) is given as 1.0 at the top wall and 0.0 at the bottom wall, whereas the mass fraction of species 2 (oxidizer) is given as 0.0 at the top wall and 1.0 at the bottom wall. The wall temperature is given as 900K. Periodic boundary conditions are used in the longitudinal and spanwise directions and solid-wall boundary conditions are used on the top and bottom boundaries.


Table 1. Parameters for DNS Cases.

For numerical simulations, the Galerkin-Tau spectral expansion method is adopted. The Fourier transform is used in x and z directions and the Chebyshev transform is used in the y direction. Uniform grid distribution is in x and z directions and the Gauss-Lobatto nonuniform grid distribution is used in the y direction. The number of grid nodes in x and z directions is 128, and in the y direction is 129. This results a total of 2.11 million nodes. The time step used is 0.01H/Um. A third-order scheme is used for time marching.

The statistical RMS value of velocities and a correlation were compared with the literature data (Kawamura 2000), which are shown in figure 21 and 22. The predictions from this chapter are labelled by SM DNS, and the predictions from literature are labelled by FD DNS.

*j i j ij ji*

11 1 *<sup>j</sup>* / *j jj <sup>Y</sup> (Y )+ (u Y ) = (D ) w <sup>ρ</sup> t x xx* 

22 2 *<sup>j</sup>* / *j jj <sup>Y</sup> (Y )+ (u Y ) = (D ) w <sup>ρ</sup> t x xx* 

*j jp j <sup>λ</sup> <sup>T</sup> (T)+ (u T) = ( )+ w Q (c <sup>ρ</sup>) t x xc <sup>ρ</sup> <sup>x</sup>*

The Fi term in momentum equation stands for the effect from buoyancy force which caused by the difference of the temperature, also it induces the scalar and velocity interaction. The Reynolds number Rem defined by the channel half width H and average velocity Um is 3000.

For all cases, the mass fraction of species 1 (fuel) is given as 1.0 at the top wall and 0.0 at the bottom wall, whereas the mass fraction of species 2 (oxidizer) is given as 0.0 at the top wall and 1.0 at the bottom wall. The wall temperature is given as 900K. Periodic boundary conditions are used in the longitudinal and spanwise directions and solid-wall boundary

> Case B E/R (K) Q (kJ/kg) *Fy* (m/s2) 1 0 0 0 0 2 0.1 0 0 0 3 1.0 0 0 0 4 108 15000 100 0 5 108 20000 100 0 6 1010 20000 100 0 7 108 20000 100 -0.5 8 108 15000 100 *Fy* 9 108 20000 100 *Fy*

For numerical simulations, the Galerkin-Tau spectral expansion method is adopted. The Fourier transform is used in x and z directions and the Chebyshev transform is used in the y direction. Uniform grid distribution is in x and z directions and the Gauss-Lobatto nonuniform grid distribution is used in the y direction. The number of grid nodes in x and z directions is 128, and in the y direction is 129. This results a total of 2.11 million nodes. The

The statistical RMS value of velocities and a correlation were compared with the literature data (Kawamura 2000), which are shown in figure 21 and 22. The predictions from this chapter are labelled by SM DNS, and the predictions from literature are labelled by FD DNS.

time step used is 0.01H/Um. A third-order scheme is used for time marching.

 

The parameters in DNS cases are given in Table 1.

conditions are used on the top and bottom boundaries.

Table 1. Parameters for DNS Cases.

*<sup>j</sup> 1 p* <sup>1</sup> /

1

2

(13)

(14)

(15)

(16)

*j i i i*

*uu u p u +u = + [ν( + )] F t x xx x x* 

In the SM DNS simulation results, U=0.063845247, Re=191.54. The two DNS results are very close to each other.

Fig. 21. The RMS value of velocities.

Fig. 22. The averaged value of the correlation of concentration and v velocity

The instantaneous contour of temperature and species are shown in figure 23 and 24. The plenty turbulent structures and near-wall stripe structures can be seen. The statistical averaged reaction factor K profiles is shown in figure 25.

Turbulent Combustion Simulation

Fig. 25. The averaged reaction factor K profiles.

given in the turbulent combustion model.

=

*P λ ρc*

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

, and put 1 1 2 1 2 exp *2 2*

*<sup>E</sup> w =w =B<sup>ρ</sup> YY ( )= <sup>ρ</sup> YYK*

1 1

*x x xx c*

*j j jj P*

2D

*1*

*' '*

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

*2*

2λ

K

by Large Eddy Simulation and Direct Numerical Simulation 223

 Case 4 Case 5 Case 6

reaction term into the equations,

*1*

(17)

(18)


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

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*

*RT*

1 11 1

*j jj*

*' '*

*1 P jj P*

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

*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*

1 1 2

*Y KT EρQ Y KY Y K ρK'Y Y K + ρcTx x Rc T*

*'' '*

*K'Y Y K' <sup>λ</sup> EY K T D( ) x x xx ρcR x x T*

*K'Y K'Y Y K u K'Y + u = u K' u Y + t x x xx*

*j j jj P jj*

1

*j j jj*

1

*' ' ' ' ' '' j 1 1 1*

1 1

*j j jj*

2 2

<sup>112</sup> <sup>2</sup> 2D

1

2

*1 1*

*1 '*

y

Fig. 23. The instantaneous contour of temperature fluctuations in case 5 (a-y+=8.2, b-y+=24.9)

Fig. 24. The instantaneous contour of concentration fluctuation (case 8, y+=8.2)

(a)

Fig. 23. The instantaneous contour of temperature fluctuations in case 5

Fig. 24. The instantaneous contour of concentration fluctuation (case 8, y+=8.2)

(a-y+=8.2, b-y+=24.9)

(b)

Fig. 25. The averaged reaction factor K profiles.
