Large-eddy simulation of correlation moments in turbulent combustion

and validation of the RANS-SOM combustion model

F. Wang *, L.X. Zhou, C.X. Xu

Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China

Received 7 June 2005; received in revised form 8 November 2005; accepted 10 November 2005

Available online 15 December 2005

Abstract

The three-dimensional large-eddy simulation (LES) is carried out for a piloted methane–air jet flame (Flame C), measured in Sandia National

Laboratory, and its statistical results are validated by experimental data. The LES statistically-averaged time-averaged temperature, the root mean

square (RMS) value of temperature, and time-averaged methane and oxygen concentration are compared with those obtained using the Reynolds-

averaged Navier–Stokes (RANS) equations with second–order moment (SOM) combustion model. Also, the cross-correlations in the time-

averaged reaction rate expression of SOM model are given by the LES statistics. It is found that there is a similarity between the distribution of the

correlation moments and the distribution of the products of corresponding averaged variable’s gradients, the closure assumptions made in the

algebraic second–order moment (ASOM) combustion model for RANS model is approximately valid.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Turbulent combustion; Large-eddy simulation; Correlation moments

1. Introduction

The turbulent combustion modeling in engineering is a

challenging problem at the present time. It is found that the

eddy-break-up (EBU), EBU-Arrhenius and simplified prob-

ability density function (PDF) models widely adopted in

commercial software frequently lead to remarkable errors in

the predictions. The PDF transport equation model is more

reasonable, but has not yet been widely used, since it needs

large computation time for complex engineering flows. The

laminar flame-let and conditional moment closure models

are encouraging, but are not yet well validated by

experiments for various cases. Recently, a second-order

moment (SOM) combustion model was proposed by the

present authors1.

The unconditional second-order moment turbulence-chem-

istry models, based on the idea of second-order moment

turbulence models, exhibit both the reasonability and the

economy, and hence are considered as perspective models in

engineering application. The early-developed second-order

moment model2 gives the time-averaged reaction rate by

0016-2361/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.fuel.2005.11.006

* Corresponding author. Tel.: C86 1062782231; fax: C86 1062855007.

E-mail address: fang-wang03@mails.tsinghua.edu.cn (F. Wang).

making an approximation of E/RT$T 0/T/1 and omitting the

higher order terms in the series expansion of the non-linear

exponential term. However, in many practical combustion

processes, in particular NO formation, E/RT[1, and T 0/T is

not much smaller than unity, therefore the series expansion

approximation gives serious errors. In recent years, we did

systematic studies on developing second-order moment

turbulence-chemistry models. A version of second-order

moment-PDF (SOM-PDF) model3 and a unified second-order

moment (USM) model4 were proposed. In the SOM-PDF

model, the concentration–concentration correlation is closed

using the second-order moment method, while the tempera-

ture–concentration correlation is closed using the presumed

PDF. In the USM model all of correlation moments, including

the concentration–concentration correlation and the correlation

of the reaction rate coefficient fluctuation with the concen-

tration fluctuation, are closed using the conservation equations

in the same form. The USM model was used to simulate

methane–air jet diffusion combustion and NO formation

(Flame D) measured by the Sandia National Laboratory, and

methane–air swirling combustion and NO formation measured

in our laboratory.1,5 Simulation results and the comparison

with experiments show that the proposed model is more

reasonable than the EBU-Arrhenius (E-A) and simplified PDF

model.

However, the closure assumptions made in the RANS-

SOM model, that is, for the production term in the transport

Fuel 85 (2006) 1242–1247

www.fuelfirst.com

Nomenclature

Cs empirical model constant

B pre-exponential factor

d distance to the closest wall

E activation energy

g SGS mass flux

h enthalpy

k turbulent kinetic energy

K reaction rate coefficient

Ls mixing length for sub-grid scales

P pressure

Pr Prandtl number

q SGS heat flux

R gas-law constant

Sc Schmidt number

S strain-rate tensor

t time

T temperature

u velocity component

V volume of the computational cell

w reaction rate

x space coordinate

Y mass fraction

Greek symbols

3 dissipation rate

k Von Karman constant

m dynamic viscosity

mt sub-grid scale turbulent viscodity

r density

t sub-grid scale value

Subscripts

Fu, fu fuel

i, j, k component

Ox, ox oxygen

s s species

Superscripts

w filtered value

F. Wang et al. / Fuel 85 (2006) 1242–1247 1243

equation model, or simply in the algebraic model, the

correlations of the reaction-rate coefficient fluctuation

with the concentration fluctuation is proportional to the

product of corresponding time-averaged values, need to be

theoretically justified. The large-eddy simulation (LES) of

turbulent combustion attracts more and more attention in

recent years, since it needs less computation requirements

for simulating high Re flows and can give the turbulence

structures in combustion, hence can help us to understand

the mechanism of turbulence-chemistry interaction. How-

ever, up to now it is less used to validate RANS

combustion models.

There are various sub-grid scale (SGS) combustion

models in LES, such as the EBU model, laminar flame-let

model, the G-equation model, linear-eddy model, PDF

equation model and so on. As for turbulent reaction jet

flow simulation, various flame-let models were mostly used.

Pitsch6 applied the Lagrangian flame-let model in a

methane–air jet flame and achieved good predictions.

While, Mattsson et al.7 used the SGS laminar flame-let

model to simulate a propane–air jet flame, but the predicted

statistical results near the inlet region are not in agreement

with the laser holography measurement results. DesJardin

et al.8 did the similar LES and found that the simulation

over-predicts the combustion products. Liu et al.9 used a

dynamic similarity sub-grid-scale SGS combustion model to

simulate a jet diffusion flame; the predicted temperature

distribution is in agreement with that obtained using direct

numerical simulation (DNS), but its comparison with

experimental results is not reported. Meanwhile, a SGS-

SOM combustion model is proposed by Hu et al.10 and is

used to simulate swirling combustion; the statistical results

are in good agreement with experimental results.

In this paper, the SGS-SOM combustion model is

adopted for LES of a piloted jet flame measured in the

Sandia National Laboratory, USA.11 The LES statistical

results will be validated by experiments and the statistical

results of various auto-correlations and cross-correlations,

such as the auto-correlation of temperature fluctuation,

the cross-correlation of concentration fluctuation and the

cross-correlation of the fluctuation of the reaction-rate

coefficient with the concentration fluctuation, will be

compared with those obtained using the RANS model,

including the SOM model, for verifying and improving the

RANS-SOM model.

2. Mathematical model and numerical procedure

The filtered conservation equations for LES are:

Continuity equation

vr

vtC

v

vxi

ðr ~uiÞ Z 0 (1)

Momentum equation

v

vtðr ~uiÞC

v

vxj

ðr ~ui ~ujÞ Zv

vxj

mv ~ui

vxj

� �K

v ~P

vxi

Kvtij

vxj

(2)

where the sub-grid-scale stress tij hrguiujKr ~ui ~uj is closed by

the Smagorinsky–Lilly eddy-viscosity model tij ZK2mt~SijC

1⁄3tkkdij, mt ZrL2Sj~Sj, LSZminðkd;CSV1=3Þ, CsZ0.1.

Fig. 1. The jet combustor.

F. Wang et al. / Fuel 85 (2006) 1242–12471244

Species equation

vr ~Ys

vtC

v

vxj

ðr ~uj~YsÞ Z

v

vxj

m

SCs

v ~Ys

vxj

� �K ~wsKwS;sK

vgSj;s

vxj

(3)

where ~ws and wS,s are the filtered and sub-grid scale reaction

rates, respectively. The sub-grid scale mass flux is

gSj;s Zmt

Sct;s

v ~Ys

vXj

:

Energy equation

vr ~h

vtC

v

vxj

ðr ~uj~hÞ Z

v

vxj

m

Pr

v ~h

vxj

� �K

vqSj

vxj

(4)

where the sub-grid scale heat flux is qSj Zmt

Prt

v ~Tvxj

� �The SGS mass flux and heat flux are also closed by the

Smagorinsky–Lilly model.

The filtered reaction rate ~ws is ~KgYOXfYFu , where

~KZBr2Ð

expðKE=R ~TÞpð ~TÞd ~T .

For the SGS-SOM model, the sub-grid scale reaction rate

wS,s is

wS;Fu Z ~KðgYOXfYFu KgYOX

fYFu ÞC gYOXð gKYFu K gKYFu Þ

C fYFu ð gKYOX K ~KgYOXÞ (5)

Fig. 2. Time-averag

It expresses the effect of small-scale turbulence on the

reaction rate. The correlations are closed using the gradient

modeling. For example, the correlation of the fluctuation of the

reaction-rate coefficient with the concentration fluctuation is

given by

ð gKYOX K ~KgYOX Þ Z CK;YOXL2

S

v ~K

vxj

vgYOX

vxj

(6)

The RANS-SOM model is given in Ref. [1]. The laminar

chemical kinetics of methane–air combustion is given by

Ref. [12]

wfu Z 2:119!1011Y1:3ox Y0:2

fu expðK2:027!108=RTÞ (7)

For the numerical procedure in LES, the grid size is taken as

0.5 mm near the inlet and 2 mm in other regions; the time steps

are 1 ms for non-reacting flows and 0.1 ms for reacting flows.

The second-order difference scheme is adopted, and PISO

algorithm is adopted in numerical solution. A random

fluctuation of Gaussian distribution is superposed to the inlet

velocity. For 2D RANS model the grid sizes are 2–5 mm. For

3D LES running a case on the PC with a 3.0G Intel Xeon CPU

takes about 72 h for non-reacting flows and 480 h for reacting

flows. The 2D RANS-SOM model on the same PC takes about

0.5 h.

3. Results and discussion

Fig. 1 gives the geometrical configuration and sizes of the

piloted methane–air jet flame, measured in Sandia National

Laboratory, USA.11 The central jet consists of 25% methane

and 75% air in volume, and its inlet velocity is 29.7 m/s. The

annular jet is hydrogen–air combustion products with an inlet

temperature of 1880 K and inlet velocity of 6.8 m/s. The co-

flow air velocity is 0.9 m/s.

Fig. 2 shows the comparison of predicted time-averaged

temperature by LES-SOM and RANS-SOM model with the

experimental results. In most regions, the LES-SOM results

ed temperature.

Fig. 5. Time-averaged oxygen concentration.

Fig. 3. RMS values of temperature fluctuation.

Fig. 4. Time-averaged methane concentration.

F. Wang et al. / Fuel 85 (2006) 1242–1247 1245

Fig. 6. Correlation and product of gradients of time-averaged values.

F. Wang et al. / Fuel 85 (2006) 1242–12471246

and RANS-SOM results are in agreement with the

experimental results. Comparatively, the LES-SOM model

is better than the RANS-SOM model predictions, in

particular at the cross-sections of xZ7.5, 15 and 30, d.

The RANS-SOM model over-predicts the temperature at

these positions.

Fig. 3 gives the RMS value of temperature fluctuation. The

LES-SOM model is in good agreement with the experimental

results, and is better than the RANS-SOM. The RANS-SOM

model over-predicts the temperature fluctuation in most cross-

sections. However, in predicting the time-averaged tempera-

ture, which is more interested in engineering application, the

RANS-SOM model is preferred, considering that it can save

much more time than the LES.

Figs. 4 and 5 show the time-averaged methane and oxygen

concentration, respectively. In most regions, both LES-SOM

and RANS-SOM model give good results.

The correlation of reaction-rate coefficient fluctuation with

the methane concentration fluctuation and the product of

Fig. 7. Time-averaged temper

Fig. 8. Instantaneous temper

gradients of corresponding time-averaged values given by

LES, are shown in Fig. 6. It is found that although these

correlation moments are not exactly proportional to the

products of gradients of corresponding averaged variables,

but there is a similarity between the distribution of the

correlation moments and the distribution of the products of

corresponding averaged variable’s gradients, the general trends

of their distributions are similar to each other: their values go

up and down at the same regions and their peak values are

located at the same locations. Thus, the gradient modeling

assumptions made in the algebraic second-order moment

(ASOM) combustion model for RANS model is approximately

valid. Furthermore, the empirical constants in RANS-SOM

model can be determined using the LES results.

Figs. 7 and 8 are time-averaged and instantaneous

temperature maps obtained by RANS-SOM and LES-

SOM, respectively. The physical features of an outer

premixed flame and an inner diffusion flame can be seen

in Fig. 7. The instantaneous temperature map in Fig. 8 gives

the coherent structures of turbulence in the jet flame,

showing the wrinkled flame front.

4. Conclusions

(1) There is a similarity between the distribution of the

correlation moments and the distribution of the products

of corresponding averaged variable’s gradients, the

closure assumptions made in the algebraic second

order moment (ASOM) combustion model is approxi-

mately valid.

(2) For time-averaged temperature and concentration the

LES-SOM results are near to RANS-SOM results.

(3) For RMS values of turbulence properties, LES is

obvious better than RANS modeling.

(4) Considering acceptable accuracy in simulation of time-

averaged variables and much less computation time, the

ature map (RANS-SOM).

ature map (LES-SOM).

F. Wang et al. / Fuel 85 (2006) 1242–1247 1247

RANS-SOM model is preferred to be used in

engineering.

Acknowledgements

This study was sponsored by the 985 Project of Tsinghua

University.

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