Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

An Eulerian stochastic fields (ESF) method for large eddy simulation of turbulent combustion in

compressible flow

Cheng Gong

Division of Fluid Mechanics

Lund University, Sweden

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Motivation and introduction

• Eulerian stochastic fields (ESF) is a Monte-Carlo method to solve the

transported pdf equations in Eulerian frame.

– It is easy to coupled with convectional CFD code;

– It has been used in different combustions, e.g., premixed/non-

premixed jet flame, spray combustions.

• ESF method has only been coupled with pressure based solver.

– Pressure-velocity coupled using SIMPLE or PISO type method;

– Energy is solved in stochastic fields with neglecting the exchange with

kinetic energy;

– Usage is Limited in low speed conditions (Ma<0.3)

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Motivation and introduction

Bo Zhou et al., combust.

Flame, 2015

Sun et al. Exp. Therm. Fluid Sci., 2012

High speed jet flame

Supersonic combustion

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Compressible LES solver

0

j

j

u

t x

,( ) ( )sgs

i j ij i j iji

j j

u u pu

t x x

,

( ) sgsj j sgs sgs

j i j j j ij

j j j

Eu puE ku q H

t x x x

1

2

sgs

i iE e u u k

Transported equations

, ,

sgs

ij i j i j

sgs

j j j j j

sgs

ij j i j j i j

u u u u

H Eu Eu pu pu

u u

Prj

j

hq

x

Unclosed terms: Total energy

Molecular heat flux (unity Lewis number)

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Compressible LES solver

3/2( )

Pr

sgssgs sgsj jsgs sgst

ij d

j j t j j

k u uk kc k

t x x x x

Sub-grid scale models

1 22

3 3

sgs sgs

ij t ij kk ij ijS S k

sgs

t c k

Pr

sgssgs sgs t ij ij i

t j j j

uh kH u

x x x

Closure for momentum equation

Closure for energy equation

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Eulerian stochastic fields method

( ) ( )

1( ) 2 ,

2

n nn n

j t

j i i

nn nt

i

sgs i

d u dt dt dtx x x

C dt dWx

Joint pdf equation

ESF method

1 1

1( ;x, t)

NsNn

sgs

n

PN

1

1 Nn

nN

Impossible to solve directly due to the Multi-dimensions (Ns + 5).

Usually, it is solved using Monte-Carlo approaches.

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Numerical implementation

The LES solver is a modified version of rhoCentralFoam in OpenFOAM. The

convective terms of the equations are discreted usiong the Kurganov and Tadmor

scheme.

LES equations

ESF equations Step 1: Advection-diffusion-Wiener

Step 2: Micro-mixing

n(2) n(1) n(2) (1)0.5 ( )

sgs

tC

Step 3: Energy correction

(3) (2) m(2)

1

1 Nn n

m

h h h hN

Step 4: Chemical reaction

2

sgs

sgs

( ) ( ) ( , ) ( )

nn n n n

ad wien mixt

( ) ( )

( ) 2

1( , ) ( )

2

n nn

ad j t

j i i

nnn t i

wien

i

n n

mix

sgs

ux x x

dW

x t

C

n(1) n (1)

exp

( ) ( ) ( ( ))n n

n ad wien n

implicit Euler licit

t t t t

n n(3) n(3)

1

0

( ) , 0 :

t

nt dt t

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Validation case

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Non-reacting flow field

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Reacting flow field

Time average during 1.2 ms, with NF=8, Cphi=2.0

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Flame stabilization

Two pairs of recirculation zones:

1) Induced by the fuel injection;

2) Induced by the strut.

The flame is stabilized by the strut

induced recirculation zones.

Local extinction/re-ignition behavior

around the fuel injection induced

recirculation zones.

Local extinction re-ignition

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Comparison with WSR model

Most previous studies failed to predict the temperature profile close to the

strut (x= 11 mm);

The ESF method provide good agreement at x= 11 mm, which reveals that the

ESF can capture the local extinction /re-ignition behavior, successfully .

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Conclusions

• The Eulerian stochastic fields method was implemented to a

compressible LES solver in OpenFOAM;

• The LES-ESF solver was validated on the strut-stabilized combustion in

a supersonic combustor;

– The compressible LES solver provide good performance on the

supersonic flow field;

– Comparing to the WSR model, the ESF method can provide much

better prediction for the local extinction and re-ignition behavior in

the present condition.

Cheng Gong, Mar. 30, 2016 Division of Fluid Mechanics, Lund University

Thanks for your attention