(Large Eddy Simulation)/assumed sub-grid PDF - Springer Link

SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

October 2011 Vol.54 No.10: 2694–2707 doi: 10.1007/s11431-011-4518-6

A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF (Probability Density Function) model for supersonic turbulent combustion WANG HongBo1, 2, QIN Ning2, SUN MingBo1, WU HaiYan1 & WANG ZhenGuo1* 1

College of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, China; 2 Department of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD, UK Received April 11, 2011; accepted May 23, 2011; published online August 7, 2011

A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF (Probability Density Function) closure model has been developed for supersonic turbulent combustion. Scalar transport equations for all species in a given chemical kinetic mechanism were solved, which are necessary in the supersonic combustion where the non-equilibrium chemistry is essentially involved. The clipped Gaussian PDF of temperature and multivariate  PDF of composition were used to close the sub-grid chemical sources that appear in the conservation equations. The sub-grid variances of temperature and composition were constructed based on scale similarity approach. A semi-implicit approach based on the PDF model was proposed to tackle the resulting numerical stiffness associated with finite rate chemistry. The model was applied to simulate a supersonic, coaxial H2-air burner, where both the mean and rms (root mean square) results were compared with the experimental data. In general, good agreements were achieved, which indicated that the present sub-grid PDF method could work well in simulating supersonic turbulent combustion. Moreover, the calculation showed that the sub-grid fluctuations of temperature and major species in the combustion region were of the order of 10%–20% of their rms, while the sub-grid fluctuation of hydroxyl might be as high as 40%–50% of its rms. LES (Large Eddy Simulation), assumed PDF (Probability Density Function), supersonic, turbulent combustion Citation:

1

Wang H B, Qin N, Sun M B, et al. A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF (Probability Density Function) model for supersonic turbulent combustion. Sci China Tech Sci, 2011, 54: 26942707, doi: 10.1007/s11431-011-4518-6

Introduction

In the high-Reynolds-number flow inside the scramjet combustor, turbulence and combustion interact intensively with each other. Turbulence influences combustion in two aspects: 1) influencing the mixing process between the fuel and the oxidizer by turbulent transport; 2) influencing the chemical process by inducing fluctuations of temperature and species. Contrariwise, combustion influences turbulence mainly by changing the local Reynolds number. On *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011

one hand, the heat release may increase the molecular viscosity by increasing the temperature, which decreases the local Reynolds number and laminarizes the turbulence. On the other hand, combustion may dilate and accelerate the fluid, which increases the local Reynolds number and enhances the turbulence. Therefore, it is necessary to take the interaction between turbulence and combustion into account so that the combustion flow inside the scramjet combustor can be simulated with good accuracy. In general, the best method is to use DNS (Direct Numerical Simulation) to resolve the turbulence and combustion of all scales, so that the errors resulting from models can be avoided. Due to limited computational resources, tech.scichina.com

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however, it is impossible to carry out DNS in the simulation of a scramjet engine in the foreseeable future. Fortunately, simulations can be carried out by introducing some models to compensate for the unresolved part of the turbulence. There are two kinds of approaches, that is, RANS (Reynolds-Averaged Navier-Stokes) and LES (Large Eddy Simulation). Both RANS and LES introduce some kind of averaging or filtering in the conservation equations. Because of the nonlinearity of the equations, some unclosed terms appear after the averaging or filtering and the modeling of these terms is the main subject in the simulation of turbulent flows. Two types of these terms can be identified in the moment-equation approach [1]. The first type is the turbulent convection term, which is a correlation with the velocity fluctuation and is invariably closed by gradient diffusion assumptions. The second type is the scalar moments, and in particular, the mean or filtered chemical reaction rate, which results from the interaction of turbulence and reaction and needs to be closed by the turbulent combustion model. An attractive option for the turbulence-chemistry closure is to utilize PDF (Probability Density Function) models. The PDF of a random variable (or a set of random variables) contains all of the single point statistical information about the variable(s). Thus, if the PDF of temperature and composition is known, then the unclosed chemical source terms can be directly evaluated [2]. There are two classes of PDF approaches: the transported PDF and the assumed PDF. The first one solves an evolution equation for the PDF [3–6]. Although this approach is believed to be more accurate from a physical point of view, the computational cost is so expensive that it is not a viable engineering tool in study of the scramjet engine at present. The second approach, which involves an assumed form for the PDF using information from a finite number of moments, is computationally less expensive and may be more suited for supersonic combustion or combustion processes that require a large number of different species [7]. The assumed PDF method was first used in the framework of RANS [7–10]. Frankel et al. [8] developed a hybrid RANS/assumed PDF approach and applied it to the study of turbulent combustion in a supersonic mixing layer. The results show modest improvement from previous “laminar-like” calculations. In addition, the results appear to be somewhat independent of the form of the PDF. Girimaji [9] proposed a multivariate  PDF model for the scalar joint-PDF, based on which the unclosed chemical reaction terms can be expressed as simple functions of scalar-means and turbulent scalar energy. Főrster et al. [11] validated an assumed PDF model with a multivariate  PDF for species mass fractions and a Gaussian distribution for temperature. The analysis and calculation showed that the assumed PDF model is well suited for supersonic combustion and can yield excellent results if the variances of both PDFs can be evaluated exactly. Baurle et al. [2, 10, 12–14] applied the

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assumed PDF method together with RANS to simulate supersonic coaxial H2-air jets. In general, good agreements with the experiments were obtained. The assumed PDF method can also be integrated in LES, where the PDF actually becomes a sub-grid (filtered, large-eddy) PDF that describes the stochastic behavior of the scalar inside the grid cell. Because the combustion process occurs on the small scales of the turbulence, there is essentially no resolved part of the combustion process. Although the combustion process has to be modeled entirely, LES has been shown to provide much more accurate solutions for turbulent combustion problems than RANS modeling approaches [15]. The reason is that the small-scale turbulent motions and the small-scale molecular mixing process are very much governed by the large scales of the turbulence, which are typically captured with good accuracy using LES. Pierce et al.[16] also argue that it is because of the inaccurate modeling of the large scales, in particular, large-scale mixing, RANS approaches sometimes fail to predict turbulent reacting flows accurately, so that even with a fairly simple model for the chemistry, LES may be able to outperform Reynolds-averaged computations that employ more sophisticated chemistry models. Another advantage is the modeling of counter-gradient transport phenomena. Since the unresolved scalar flux increases almost linearly with the filter size, its type (gradient or countergradient) does not change. Therefore a portion of the counter-gradient transport phenomena is expected to be directly described through resolved motions since all characteristic length scales are involved in the counter-gradient transport. Accordingly, the counter-gradient transport can be taken into account in LES even when a sub-grid scale gradient-type closure is used. Cook et al. [17] proposed a  PDF of mixture fraction to model the chemical source in the context of LES. Data from both the experiment and the DNS simulation show that the predictions of the model are accurate, given the exact values for the filtered scalar and its variance. Meanwhile, the authors suggested a model for the variance of the scalar based on scale similarity. It was also demonstrated that the assumed  PDF yields reasonably accurate results. Forkel et al. [18] used LES together with  PDF of mixture fraction to simulate a turbulent hydrogen diffusion flame, showing very good satisfactory agreement between the numerical results and the experimental data. Pierce et al. [16] developed an approach for chemistry modeling for LES. Instead of solving transport equations for all species in a typical chemical mechanism and modeling the unclosed chemical source terms, this study adopted an indirect mapping approach, whereby all of the detailed chemical processes were mapped to a reduced system of tracking scalars. Only two such scalars were considered: a mixture fraction variable, which tracks the mixing of fuel and oxidizer, and a progress variable, which tracks the global extent of reaction of the local mixture. However, this approach was incorporated in

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the form of a steady-state one-dimensional flamelet model, which may be only well suited for fast reaction. Although the assumed PDF formulations based on mixture fraction are popular and successful in low-speed applications, they have seldom been used in high-speed flows [19]. This is because these approaches tend to either treat the reacting system as a mixed-is-burned flame sheet or assume that the mixture is in chemical equilibrium. Actually, neither of the infinitely fast chemistry assumptions is appropriate in supersonic flows due to limited flow residence times. Hence, a feasible approach, for incorporating chemistry into LES in supersonic turbulent combustion, would be to solve scalar transport equations for all species in a suitable chemical kinetic mechanism and model the filtered source term in each equation. The objective of the present work is to develop a hybrid LES/assumed sub-grid PDF model for supersonic turbulent combustion. First, the governing equations and involved models are presented. Then the approach is applied to simulate a supersonic, coaxial H2-air jet, where the results are compared with the experiment.

2

Governing equations and numerical method

The Favre filtered compressible Navier-Stokes equations together with the species transport equations can be written as follows:  (  ui )   0, t xi

(1)

(  ui ) (  ui u j ) p ( tij   ij )    , t x j xi x j

(2)

 )  (  E )  (  hu  j   (ui ( tij   ij )  Q j ) , t x j x j

(3)

(  Y ) (  Y u j )    t x j x j

 Y  M j    ,  D    x j   x j 

 ui u j 2 uk     ij  ,  x   j xi 3 xk 

   T   t Q j     (T )   c p t  Prt  x j Sct 

(4)

M j   

 t Y Sct x j

.

(5)

N

 h

s

s

Ys , x j

 

 . 

(8)

In the present study, the fifth order WENO scheme is used for inviscid fluxes and viscous fluxes are discretized by means of a second order accurate-centered scheme. A time-splitting method is used to uncouple the flow and reaction as below.  t   t  Q n 1  Lc   L f (t ) Lc   Q n , 2    2 

(9)

where Lf and Lc denote the solver for flow and reaction, respectively. The temporal discretization is a second order, implicit dual time-step approach, in which the inner pseudo-time iteration is achieved by an LU-SGS method. The reaction equations are solved by a semi-implicit approach based on the PDF model, as shown later.

3 Sub-grid scale turbulence model In the present study, the one-equation Yoshizawa sub-grid scale model [20] is adopted for the closure of the turbulent convection terms.  D k  Pk  Dt x j

 k    (   k t )   Dk , x j  

(10)

k3/ 2 , k   1/ Prt . Pk and Dk are the production and dissipation of the sub-grid turbulent kinetic energy, respectively. Here, the values of C and Cd need to be determined. At equilibrium, the effective Smagorinsky constant implied by the Yoshizawa SGS model [20] is given by

where  t  C k 1/ 2 , Pk  2  t ij ij , Dk  Cd 

 Cs 

2

1

 2C  C / Cd  2 .

(11)

The value of Cs is 0.126 when the default values are used

where

 ij   t 

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(6)

(7)

Here, ‘’ denotes the spatial filtering and ‘~’ denotes the Favre filtering, defined by

for C and Cd, that is, C=0.05 and Cd=1.0. Although there is no universal value for the Smagorinsky constant, the values on the order of 0.2 for homogeneous turbulence and 0.065 for shear flow are commended by the LES community [21]. As a compromise, C=0.02075 and Cd=1.0 are used in the present work, which were tested by Baurle et al. [22]. The test showed that these values have been proved to be too dissipative with the second-order upwind scheme. However, in the present approach, the fifth-order WENO scheme produces much less numerical dissipation and the choice of the same constants may be balanced by the high order discretization of the convective fluxes.

4

Sub-grid combustion model

The law of mass action gives the chemical sources in the

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species transport equations as follows (including the thirdbody effect): 

n

  M   (  j   j )   k fj 

mj

   sj N 1   Y  n kbj  j   s   ,  s 1  M s   j 1

 sj

 Ys   s 1  s 

N 1

 M

where N 1

N 1

s 1

s 1

where Kcj is the equilibrium constant and the terms kfj , k bj , Ifj and Ibj can be closed using the assumed PDF.

4.1 (12)

m j   sj , n j   sj

(13)

N t YN 1 sj  Ys , M N 1 s 1 M s

(14)

Closure for kfj and k bj using Gaussian PDF of

temperature Since kfj and kbj are functions of temperature, we can obtain kfj and kbj using the PDF of temperature B  E / RT kfj   Aj T j e aj PT (T )dT , B j  Eaj / RT k / K cj )PT (T )dT , bj   ( A j T e

and

PT (T ) 

there are three-body reactions, and zero otherwise. If the one-point-one-time joint PDF of temperature and composition, P, is known, the filtered production rate can be determined from

The approaches of Girimaji [9], Baurle et al. [2, 12, 13] and Gerlinger et al. [23] are followed here, assuming a Gaussian distribution for temperature and a multivariate  PDF model for species concentrations. The assumption of the following approach is of statistical independence of temperature, composition, and density. Thus the probability density function can be written as the product P (T , Y1 , , YN )  PT (T ) PY (Y1 , , YN ) (    ).

(16)

The filtered production rate is now given by 

N



s 1

   M   (  j   j )  kfj    M s  j 1

N  sj nj   k bj  Ms  s 1

 sj

  I fj 

  I bj ), 

 k bj  k fj / K cj ,

 N 1, j  N N t   sj  sj , I bj   Ys   Ys  s 1  s 1 M s 

  . 

(20)

PT (T ) 

 (T  T ) 2  1 exp    2  2T  2 2T  ( H (T  Tmin )  H (T  Tmax ))  A1 (T  Tmin )  A2 (T  Tmax ),

(21)

where A1 and A2 are areas of the clipped parts. Here, 2 can be T =100 K and T =3500 K. The variance T min

max

modeled by using the scale similarity approach as below. Since there is no analytical solution for this integral equation, numerical calculation has to be used. To avoid solving this integral at every time step, a look-up table for the filtered forward and backward reaction rates are constructed before the simulation. The reaction rates kfj and k bj may then be interpolated from the tabulated values de-

(17)

pending on the filtered temperature T and the sub-grid 2 / T . The table temperature fluctuation intensity I  T T

consisted of 201 points in T direction (100 K  T  3500 K) and 101 points in IT direction ( 0  IT  0.5 ).

where   N 1, j N N t   sj   B j  Eaj / RT sj  , I fj   Ys   , k fj  Aj T e Ys  s 1  s 1 M s 

 (T  T ) 2 1 exp   2 2T  2 2T

Following Gerlinger et al. [23], a lower and upper temperature integration limit is introduced that still covers the important part of the temperature range. That is, the clipped Gaussian PDF is used. In order not to violate the normalization property of the PDF, the Dirac delta functions are added at the clipped ends.

     (T , Y1 , , YN )P(T , Y1 , , YN )dTdY1  dYN . (15) 

mj

(19)

where

which represents the third-body species with tsj being the corresponding third-body coefficients. The stoichiometric coefficient of reactant N+1,  N 1, j or  N 1, j is one if

n

2697

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(18)

4.2 Closure for Ifj and Ibj using multivariate  PDF of composition Ifj and Ibj are functions of species mass fractions. The multivariate  PDF of Girimaji [24] is used as the PDF of com-

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composition, that is PY (Y1 , , YN ) 

4.3

 ( 1    N )  ( 1 )    (  N ) (22)

The parameters of the model s are functions of the filtered mass fractions Ys and the sub-grid scalar energy  Y 

 Y , that is 2

s 1

s

1 S

 s  Ys 

 Y

N

2 and  Y =  Y 2 Scale similarity model for T s s 1

N  N       Ys s 1   1   Ys  .  s 1   s 1 

N

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  1 , 

(23)

In the hybrid RANS/PDF simulations of turbulent combus2 and the turbulent scalar energy  tion, the variance T

Y

are usually evaluated by solving additional transport equations, which is necessary because the second order information cannot be constructed from the RANS solution. In the present hybrid LES/PDF simulations, however, the variances can be directly reconstructed from the LES-resolved results by using the scale similarity model based on the fractal nature of turbulence, that is, the small- scale statistics can be inferred from larger-scale structures.  2  f 2  c( f 2  f ).

N

2 where S   Ys . s 1

This PDF is only fulfilled if  s  0 , so a constraint should be set as

Y  1 S .

(24)

The investigation of Gerlinger et al. [23] has shown that it is only necessary if the species mass fraction of any species is close to 1. In this case, the calculation of s, and therefore the calculation of the total turbulent production term are believed to be erroneous. Fortunately, in this case the chemical production rates approach zero and the described errors become meaningless. Accordingly,  N 1, j

 N    N tsj    I fj     Ys sj   Ys   s 1  s 1 M s      PY (Y1 , , YN )dY1  dYN  I fj1 I fj 2 ,

(25)

Both terms on the right-hand side are computable by applying a filter of width ˆ   to f . There is no universal criterion for choosing ˆ and c . An alternative is to use dynamic procedure for determining c [24]. Here the method of Forkel et al. [18] is used which does not require manually adjusting the model parameter. If ˆ  2 , the filtered variable in this larger cell can be calculated as below: 1  f ijk  (6 fi , j , k  fi 1, j , k  fi 1, j , k  fi , j 1, k 12  fi , j 1, k  fi , j , k 1  fi , j , k 1 ).

N

 sj

1    f 2 ijk  (6( fi , j , k  f ijk ) 2  ( fi 1, j , k  f ijk ) 2 12    ( f  f ) 2  ( f  f ) 2

I fj1   (  s   sj  r ) s 1 r 1

 ( fi , j 1, k

mj

 (B  m p 1

j

 p) ,

(26)

 N 1, j

 N tsj  I fj 2    (  s   sj )  M  s 1 s 

,

N

where B    s and Ifj2 represents the third-body effect.

(30)

With the same weighting the desired approximation for the sub-grid variance can be calculated by

i 1, j , k

and

(29)

ijk

i , j 1, k

ijk

   f ijk ) 2  ( fi , j , k 1  f ijk ) 2

  ( fi , j , k 1  f ijk ) 2 ).

(31)

The scale similarity model has been successfully used for reaction rate [25] and the variance of mixture fraction [16–18, 26]. Here this model is used for the variances of T and Ys.

s 1

Similarly

5 A semi-implicit approach for the PDF model I bj  I bj1 I bj 2

(27) Here, a semi-implicit approach is proposed based on the present PDF model to tackle the resulting numerical stiffness associated with finite rate chemistry. If   j  0 , one can obtain

and N

 

I bj1   (  s   sj  r ) s 1 r 1

nj

 (B  n p 1

 N 1, j

I bj 2

 N tsj  (  s   sj )     s 1 M s 

.

j

 p) ,

(28)

N

 sj

I fj1   (  s   sj  r ) s 1 r 1

mj

 (B  m p 1

j

 p)

Wang H B, et al.



 sj

N

  (

s 1, s 

r 1

  sj  r ) (     j  r )

 (B  m

   C   D , 

r 1

j

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it can be written in the following form:

  j

s

mj

p 1

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(40)

where

 p)

n N   sj  n j 1  C  M     j  k  bj   M s  I bj j 1   s 1  N     m 1    j  kfj  j   M s sj  I fj  ,  s 1   n N  m     D  M     j  kfj  j   M s sj  I fj j 1   s 1 

  j 1  N  sj     (  s   sj  r )  (     j  r ) r 1  s 1, s  r 1 mj  ( B  m j  p )    p 1   sj   j 1 N      (  s   sj  r )  (     j  r ) r 1  s 1, s  r 1 mj   1 S    1 Y ( B  m j  p)    p 1     Y   I fj1 Y .

N nj   sj   j  k bj  Ms  s 1

(32)

  j  0,

(33)

else,

   I bj  .  

If    

Hence  I fj Y , I fj  I fj1 I fj 2    I fj ,

(41)

n 1  n t

,

(42)

using semi-implicit discretization, one can obtain

n 1   n 1  C t / 2   D t  / 1  C t / 2  .

(43)

where I fj  I fj1 I fj 2

(34)

and   j 1  N  sj I fj1     (  s   sj  r )  (     j  r ) r 1  s 1, s  r 1 mj  1 S  ( B  m j  p)    1 .  p 1    Y 

The 9-species 19-step chemistry mechanism for H2-air combustion is adopted (see Tables 1 and 2). This model is

(35)

Similarly  I bj Y , I bj    I bj ,

  j  0,

(36)

else,

where I bj  I bj 1 I bj 2

(37)

and   j 1  N  sj I bj 1     (  s   sj  r )  (     j  r ) r 1  s 1, s  r 1 nj   1 S  ( B  n j  p)    1 .  p 1    Y 

(38)

Since n

   M   (  j   j )  j 1

N    m    kfj  j   M s sj  s 1 

N   sj  nj   I fj  kbj    M s   s 1

6 Chemical model

   I bj  (39)  

Table 1

Parameters of the solar PTCS Bj

Eaj/R

1

H 2 +O 2  HO 2 +H

1.0×10

8

0.0

28197.38

2

H+O 2  OH+O

2.6×108

0.0

8459.21

3

H 2 +O  OH+H

1.8×10

4

1.0

4481.37

4

H 2 +OH  H 2 O+H

2.2×10

7

0.0

2593.15

5

OH+OH  H 2O+O

6.3×106

0.0

548.84

10

No.

Reaction

Aj

6

H+OH+M  H 2 O+M

2.2×10

2.0

0.0

7

H+H+M  H 2 +M

6.4×105

1.0

0.0

8

H+O+M  OH+M

6.0×10

4

0.6

0.0

9

H+O 2 +M  HO 2 +M

2.1×103

0.0

503.52

10

O+O+M  O 2 +M

6.0×101

0.0

906.34

8

11

HO 2 +H  OH+OH

1.4×10

0.0

543.81

12

HO 2 +H  H 2 O+O

1.0×107

0.0

543.81

13

HO 2 +O  O 2 +OH

1.5×10

7

0.0

478.35

14

HO 2 +OH  H 2 O+O 2

8.0×106

0.0

0.0

15

HO 2 +HO 2  H 2 O 2 +O 2

2.0×10

6

0.0

0.0

16

H+H 2 O 2  HO 2 +H 2

1.4×106

0.0

1812.69

17

O+H 2 O 2  HO 2 +OH

1.4×107

0.0

3222.56

18

OH+H 2 O 2  HO 2 +H 2 O

6.1×106

0.0

720.04

19

H 2 O 2 +M  OH+OH+M

11

0.0

22910.37

1.2×10

The units are moles, seconds, cubic meters and Kelvins.

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Third-body efficiencies relative to N2=1.0 No

H2O

H2

6

6.0

1.0

7

6.0

2.0

8

5.0

1.0

9

16.0

2.0

19

15.0

1.0

taken from Jachimowski’s [27] detailed chemistry mechanism neglecting the oxidation process of nitrogen, and has been successfully used in a number of supersonic reacting flow studies [27–30]. The backward reaction rate constant kbj is evaluated from the forward reaction constant kfj and the equilibrium constant Kcj. kbj  k fj / K cj .

Figure 1

Table 3

The equilibrium constant kcj is calculated from the Gibbs free-energy minimum condition. N

0 s

The entropy S and the enthalpy Hs can be obtained from the specific heat data at the standard state. S s0  

C ps T

dT ,

(46)

H s   C ps dT  H . 0 s

7

Results and discussions

The test case is a supersonic, coaxial burner that was experimentally studied by Cheng et al. [31]. This case is a suitable one for validation of turbulent combustion models because both the mean and rms of temperature and concentrations can be obtained. Unfortunately, there is no information available about the boundary layer thickness and turbulent intensity of the jet inflows, which may influence the results of the simulation more or less. In the present study, the inflow boundary layer thicknesses are set to zero. Moreover, the mean profiles of all variables are fixed at inflow and white noises with relative amplitude of 20% are superimposed onto the mean inflow profiles of velocities and temperature. The schematic diagram of the burner together with the computational dimensions is shown in Figure 1. The inner jet and outer jet mean inflow conditions were specified based on the measured values (see Table 3). The burner lip surfaces and the upstream wall are assumed to be adiabatic and non-catalytic. The upstream grid and main grid are

Coaxial H2-air burner exit conditions

Parameter

(44)

 ( sj  sj )  N  S0 H   1atm  s1 exp   ( sj  sj )  s  s   . (45) K cj       RT   R RT    s 1

Schematic diagram of computational domain and mesh.

Inner jet

Outer jet

Ambient air

Mach number

1.0

2.0

0.0

Temperature, K

545.0

1250.0

300.0

Pressure, kPa Y

112.0

107.0

101.0

H2

1.0

0.0

0.0

Y O2

0.0

0.245

0.233

Y N2

0.0

0.580

0.757

Y H 2O

0.0

0.175

0.01

divided into 4 and 13 blocks of structured mesh, respectively. Each block of the upstream grid consists of 31×31×101 grid nodes and each of the main grid consists of 31×31×161 grid nodes (see Figure 1). 7.1

Instantaneous flowfields

Instantaneous results in three planes are shown, that is, x/Di=0.0, z/Di=10.8 and z/Di=40.0. Figure 2 shows the instantaneous density contour. Due to the velocity differences, two shear layers are formed. One lies between the outer jet and the ambient air and the other between the inner fuel jet and the outer jet. The latter one is more important because the combustion occurs in or around it. For clarity, the former is named the shear layer and the latter one the mixing layer from now on. The Kelvin-Helmholtz instability is developed in the near-field and then large-scale structures are generated. Although there is little reaction in the near-field (see Figures 3 and 4), turbulent mixing in this stage is very important because it has a significant effect on the combustion process occurring downstream, as shown in Figure 5. The large-scale turbulence structures enhance the near-field mixing and part of O2 is entrained in the fuel jet. Because of the low temperature in the fuel jet, however, the entrained O2 is not consumed quickly but continues to mix with the local fuel for quite a long distance, which may result in a rich and partially premixed flame in the downstream region. Figures 3 and 4 show that the mixing layer is ignited somewhere in the upstream region of position z/Di=10.8 while the partially premixed flame in the fuel jet basically occurs between z/Di=10.8 and z/Di=40.0. Moreover, the

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partially premixed flame and the diffusion flame in the mixing layer may combine into a single one further downstream. Figures 6 and 7 show the sub-grid fluctuation intensity of

Figure 2 Density contour.

Figure 3 Temperature contour.

Figure 4 Hydroxyl mass fraction.

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temperature and composition given by the present scale similarity method, respectively. They demonstrate that large sub-grid fluctuations exist in the shear layer, the mixing layer and the flame. In the combustion regions, the sub-grid

Figure 5 Oxygen mass fraction contour.

Figure 6

2 contour. T

Sub-grid temperature intensity

Figure 7 Sub-grid composition intensity

N

2 contour.  Y s s 1

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temperature intensity may be as large as 200 K in some areas and the sub-grid composition intensity of the order of 0.05. The fluctuations existing in the shear layer and the mixing layer are generated by the shear strain and the fluctuations existing in the flame, however, they may be generated by the combustion, which can change the local velocity and temperature as well as the Reynolds number significantly. 7.2

Time mean results

The time mean results of the present simulation together with the experimental data are shown in Figures 8–11. Basically, good agreements with the experimental data are achieved. It should be noted that because of a slight device misalignment, the experimental data are not completely symmetric, which may introduce some additional discrepancies between the simulation and the experiment. At z/Di=10.8, the mixing layer is well developed and ignited. The experiment shows that the ignition just starts at this position and the hydroxyl mole fraction in the mixing layer is of the order of 0.002. Moreover, no apparent increase in the water mole fraction and temperature is found around the mixing layer. However, the present simulation gives higher peaks in hydroxyl mole fraction, water mole

October (2011) Vol.54 No.10

fraction and temperature. One possible reason is that the ignition delay is under-predicted by the present chemical model used here. Another reason may be the too large near-field sub-grid temperature intensity given by the simulation, as can be seen in Figure 6, which may result from the neglect of the inflow boundary thickness. The over-predicted sub-grid temperature fluctuation usually reduces the ignition delay as analyzed by Baurle et al. [2]. Notably, the reaction layer appears to be very thin at this station, and the experimental measurement volume may be insufficient to truly capture such a thin layer. If this is occurring, then this would lead to smaller values being measured. At z/Di=21.5, all the calculated variables agree well with the experimental data. Although the reaction is mainly located in the mixing layer, there may be intermittent partially premixed combustion occurring in the central area of the fuel jet as mentioned above, which can be seen from the distribution of the hydroxyl and temperature. At z/Di=32.3 and z/Di=43.1, the calculated mole fractions of hydroxyl and water are smaller than the experimental data around the central line of the burner. Also, the calculated temperature profiles are a little lower and narrower than the experiment profiles. This means the global extent of reaction at these two stations is under-predicted by the present calculation. It is believed that the neglect of the

Figure 8 Comparisons of mean hydrogen mole fraction.

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October (2011) Vol.54 No.10

Figure 9 Comparisons of mean water mole fraction.

Figure 10 Comparisons of mean hydroxyl mole fraction.

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Figure 11 Comparisons of mean temperature.

inflow boundary thicknesses is, to some extent, responsible for this deviation. Because a low speed region coming from the upstream boundary layers in the mixing layer is missed by neglecting the inflow boundary thicknesses. Consequently, the mean residence time for the flow in the mixing layer is reduced and the Damköhler number is reduced as well. It suggests again that the inflow boundary information is very important to the calculation. As mentioned before, the lack of inflow turbulent boundary layer data makes it difficult to include this effect appropriately in the present numerical study. 7.3

The rms results

The accurate prediction of the scalar fluctuations is very important to the modeling of the chemistry-turbulence interaction. Therefore it is useful to validate the rms results given by the present hybrid LES/assumed sub-grid PDF method. The rms results of the present simulation together with the experimental data are shown in Figures 12–15, where the ‘modeled’ denotes that given by the present scale similarity approach. All the calculated rms results agree well with the experimental data especially in the flame region, where the result concerns the validity of the combustion model. In the outer shear layer region, however, it is difficult to resolve all the energy-carrying structures since so many grid points would be needed to do so in the 3D calculation. As pointed out by Cheng et al. [31] ambient air can be entrained into

the flame and affect the flame properties since this is an open flame. This effect becomes stronger and stronger as the downstream distance increases. Consequently, the scalar fluctuations around the outer shear layer are not well predicted in the downstream region due to the inability to capture the contamination of the ambient air. We just pay attention to the inner flame region which is more important to validate the proposed combustion model. The detailed effect of the ambient air is left for future exploration. The good quantitative agreement with the experiment in the flame region suggests that both the LES algorithm and the sub-grid PDF model adopted here are reasonable for the simulation of supersonic turbulent combustion. In particular, it suggests that the scale similarity method used to construct the variances of the scalars is effective, which is crucial to the PDF combustion model. In the turbulent flow without reaction, it is expected that the LES can resolve most of the turbulence, that is, the sub-grid part is very small. In the reactive flow, however, that is not always the case. The transport of some scalars related to the reaction may not be well resolved by the LES since the reaction layer is too thin. In the present supersonic combustion case, both the experiment and calculation show that the species and temperature fluctuations can be as high as 40% and 20%, respectively, which means the chemistry-turbulence interaction is significant. As can be seen from the present calculation, the sub-grid fluctuations of hydrogen mole fraction, water mole fraction and temperature are of the order of 10%–20% of their rms. Remarkably,

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October (2011) Vol.54 No.10

Figure 12 Comparisons of rms hydrogen mole fraction.

Figure 13 Comparisons of rms water mole fraction.

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Figure 14 Comparisons of rms hydroxyl mole fraction.

Figure 15 Comparisons of rms temperature.

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the sub-grid fluctuation of hydroxyl mole fraction may be as high as 40%–50% of its rms. Hence the sub-grid model of the combustion is so important that the neglect of the sub-grid fluctuation effects on the filtered chemical reaction rates may lead to an erroneous flow description.

8

Conclusion

LES is believed to outperform RANS in turbulent combustion problems because the small scale turbulent motions and the small scale molecular mixing process are very much governed by the large scales of the turbulence, which are better captured using LES. In order to incorporate chemistry into LES in supersonic turbulent combustion, a hybrid LES/assumed sub-grid PDF closure model has been developed, where scalar transport equations for all species in a given chemical kinetic mechanism were solved, which are necessary in the supersonic combustion where the nonequilibrium chemistry is essentially involved. The clipped Gaussian PDF of temperature and multivariate  PDF of composition were used to close the sub-grid chemical sources that appeared in the conservation equations. The sub-grid variances of temperature and composition were constructed based on scale similarity. A semiimplicit approach based on the PDF model was also proposed to tackle the resulting numerical stiffness associated with finite rate chemistry. The model was applied to simulate a supersonic, coaxial H2-air burner, where good agreements with the experimental data were obtained. The comparison of the rms results indicated the good performance of the proposed method and the importance of the sub-grid closure in the supersonic combustion simulation. Meanwhile, the calculation also suggested the importance of the inflow boundary thickness, which should be carefully addressed in future experiments and simulations. This work was supported by the National Natural Science Foundation of China (Grant Nos. 50906098 and 91016028).

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