EFFECTS OF WALL HEAT LOSS ON SWIRL-STABILIZED NON-

PREMIXED FLAMES WITH LOCALIZED EXTINCTION

Huangwei Zhang* and Epaminondas Mastorakos

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK

*hz283@cam.ac.uk

Abstract Large eddy simulation with three dimensional

conditional moment closure model for combustion is

applied to simulate a swirl-stabilized non-premixed

methane flame with localized extinction. The convec-

tive wall heat loss is modeled through introducing a

source term in the conditionally filtered total enthalpy

equation for the CMC cells adjacent to the wall. The

mean heat flux is high on the middle surface of the

bluff body but low near its edges. A comparison be-

tween the results from adiabatic and heat loss wall

boundary conditions is performed. It is shown that the

wall heat loss has a significant influence on the mean

flame structures of the near-wall CMC cells. This can

be directly linked to the increase of the mean condi-

tional scalar dissipation near the wall in the heat loss

case. Furthermore, the degree of local and instantane-

ous extinction near the bluff body measured by condi-

tional reactedness at stoichiometry is intensified due

to the wall heat loss. The flame stabilization and lift-

off are analyzed and the pronounced weakened reac-

tivity caused by the wall heat loss can be observed.

1 Introduction The effects of wall heat loss on flame dynamics

have been widely investigated in fire safety

(Drysdale, 1999), internal combustion and gas tur-

bine engines (Lefebvre, 1999) and micro-scale com-

bustion (Ju and Maruta, 2012).

Modeling the heat transfer between the flame and

chamber wall in the Large Eddy Simulations (LES)

has been studied. For instance, wall functions based

on the temperature logarithmic law were used for

predicting the wall heat fluxes (Grötzbach, 1987;

Schmitt et al., 2007). However, the inclusion of wall

heat loss into the advanced combustion models is not

straightforward due to the different strategies to

solve the energy equations. The enthalpy defect con-

cept was first proposed by Bray and Peters (1994)

and used for defining the enthalpy deviation from the

adiabatic profiles in mixture fraction space, caused

by radiation and/or boundary heat loss. It was intro-

duced into the flamelet model for wall heat loss by

Hergart and Peters (2002) and the similar approach

was adopted by Song and Abraham (2004) to improve

the modelling of diesel combustion. Recently, the ad-

ditional source term representing the wall heat loss

was included into the total enthalpy equation of three-

dimensional Conditional Moment Closure (CMC)

method by De Paola et al. (2008) who simulated com-

bustion in a direct-injection heavy duty diesel engine.

The above were based on RANS and therefore could

not analyze the unsteady heat transfer between the

wall and flame and how the wall heat loss affects the

flame dynamics. For lean blow-off applications, it has

been shown by Kariuki et al. (2012) that flame ele-

ments lie very close to the bluff body in a bluff body

recirculating premixed flame, and similar observa-

tions were made by Cavaliere et al. (2013) for non-

premixed flames close to extinction. The effects of the

wall heat loss on the stabilization of swirl non-pre-

mixed flames need to be examined and to be included

in the simulation method.

The goal of the current study is to apply LES with

a three-dimensional CMC combustion sub-grid model

to a swirl-stabilized non-premixed methane flame.

The wall heat loss is modeled by introducing a source

term in the conditionally filtered total enthalpy equa-

tion. The mathematical formulation and the infor-

mation about the flow investigated are given in Sec-

tion 2 and Section 3 presents the main results and dis-

cussion, followed by the conclusions in Section 4.

2 Mathematical formulation and flow con-

sidered 2.1 LES and CMC modeling

The LES governing equations for mass, momen-

tum and mixture fraction are derived through Favre

filtering their corresponding instantaneous equations.

The constant Smagorinsky model by Fureby (1996) is

used to close the anisotropic part of the sub-grid stress

tensor in the LES equations, i.e. S−2ρkI 3=− 2μ

tDdev⁄ . Here ρ is the filtered density and

k is the sub-grid scale kinetic energy. I is the identity

tensor and Ddev is the deviatoric part of the filtered

strain rate tensor D. The sub-grid scale dynamic vis-

cosity t is calculated as

μt=ck∆ρk

1/2 (1)

where Δ is the filter width taken as the cube root of the

LES cell volume VLES and the constant ck = 0.02.

Based on the local equilibrium assumption, the alge-

braic relation is obtained for k as shown below

S ∶ D+ ρε = 0 (2)

in which the symbol “:” denotes the double inner prod-

uct of two tensors and the dissipation rate ε is modeled

as ε= cεk3/2

∆⁄ with cε=1.048.

The filtered scalar dissipation rate N is calculated

as (Pera et al., 2006)

N=Nres+Nsgs= D∇ξ ⋅∇ξ⏟ resolved

+ cNμtξ

''/2ρΔ2

⏟ sub-grid

(3)

with Nres and Nsgs being resolved and sub-grid scalar

dissipation, respectively. D denotes the mass diffusiv-

ity and is calculated through 𝐷=𝜇/ρSc with Schmidt

number Sc=1. is the molecular dynamic viscosity

calculated through Sutherland’s law. Here the constant

cN =42 is used for the sub-grid model in Eq. (3)

(Garmory and Mastorakos, 2011). ξ and ξ'' are re-

solved mixture fraction and its variance. In the present

investigation, the latter is modeled as (Pierce and

Moin, 1998)

ξ'' = cvΔ

2∇ξ ⋅∇ξ (4)

in which cv = 0.1.

The conservative 3D-CMC equations for non-pre-

mixed combustion are (Garmory and Mastorakos,

2014; Zhang et al., 2014)

∂Qα

∂t⁄⏟ unsteady

+ ∇ ⋅ U|ηQα⏟

convection

= Qα∇∙U|η⏟

dilatation

+ N|η ∂2Q

α∂

2η2⁄⏟

micromixing

+ ωα|η⏟chemistry

(5)

− ∇∙ [ 𝜌|𝜂 P(η)(UYα|η− U|ηQα)] 𝜌|𝜂 P(η)⁄⏟

turbulent diffusion

where Qα= Yα|η is the conditionally filtered mass

fractions of -th species and is the sample space

variable for mixture fraction ξ. U|η, N|η, ωα|η and

𝜌|𝜂 are the conditionally filtered velocity, scalar dis-

sipation rate, reaction rates and density, respectively.

The assumption U|η = �� is adopted here. The Am-

plitude Mapping Closure (AMC) model (Brien and

Jiang, 1991) is used for the conditionally filtered dis-

sipation rate, N|η = N0G(η), where N0 and G() are

N0 = N ∫ P(η)G(η)dη1

0⁄ and G(η) =

exp (−2[erf−1(2η− 1)]

2), respectively. The first or-

der CMC model is applied such that ωα|η =ωα(Q1

,…Qn,Q

T) where n is the number of species

and QT= T|η represents the conditionally filtered

temperature. P(η) is the Filtered probability Density

Function (FDF) assumed to be beta-function shape

and is calculated with ξ and ξ''. The turbulent diffu-

sion term in Eq. (5) is modelled as −∇⋅(Dt∇Qα)

(Triantafyllidis and Mastorakos, 2010) where Dt is

the turbulent diffusivity modelled as Dt=μt/ρSct with

the turbulent Schmidt number Sct=0.7.

The equation for the conditionally filtered total en-

thalpy Qh= h|η has the same form as Eq. (5) exclud-

ing the chemistry term. To include the convective wall

heat loss in the CMC model, the following term is in-

troduced into the RHS of Qh equation for CMC cells

adjacent to a wall (De Paola et al., 2008; Hergart and

Peters, 2002)

qW,Ω|η

=− h(Q

T− TW) (6)

in which qW,Ω|η

is the conditionally filtered volumet-

ric heat loss and TW is the wall temperature. The heat

transfer coefficient h in Eq. (6) is predicted through h

=qW,Ω

ρ|η ∫ (QT-TW)

1

0⁄ P(η)dη. q

W,Ω (in units of W/m3)

is the filtered volumetric heat loss which is calculated

through volume-averaging of the magnitude of the

wall surface heat flux qw,S

(in units of W/m2) as

qW,Ω

= ∫ qw,S

dS∂Ω

VLES⁄ where ∂Ω denotes the faces of

the LES cell LES. Here the surface heat flux magni-

tude qw,S

is estimated from the LES as

qw,S= −q

w,S

l − qw,S

t =λ∇nT+λt∇nT (7)

In Eq. (7), T is the filtered temperature. The gradient

of filtered temperature ∇nT is aligned with the wall

normal direction. qw,S

l and qw,S

t denote the individual

heat fluxes from laminar and turbulent heat transfer.

qw,S

t has been modelled using the classical Reynolds

analogy in Eq. (7). λ = cPμ/Pr and λt = cPμt/Prt are

the laminar and turbulent thermal conductivities, re-

spectively. cP is the specific heat capacity at constant

pressure. The molecular and turbulent Prandtl num-

bers are assumed to be Pr = 1 and Prt = 0.7, respec-

tively, over the entire flow.

2.2 LES/CMC data coupling strategy

Considering the relatively small spatial variations

of conditional reactive scalars Q, the CMC governing

equations, i.e. Eq. (5), are solved on a coarse mesh dif-

ferent from the fine LES one (Navarro-Martinez et al.,

2005; Triantafyllidis and Mastorakos, 2010). In the

present investigation, finite volume discretization is

employed for both LES and CMC equations and the

CMC cells are reconstructed based on the CMC nodes

(red squares in Fig. 1) through selecting the CMC

faces (red lines in Fig. 1) from the LES faces (blue

lines in Fig. 1). This selection is based on the criterion

that the owner and neighbor LES cells of one face

must have different host CMC cells, which are identi-

fied based on the minimal distance between the LES

cell centroids and CMC nodes. The data coupling be-

tween the fine LES and coarse CMC meshes are

shown in Fig. 1. In particular, the conditionally filtered

scalar dissipation rate for Eq. (5) in the CMC resolu-

tion, N|η*, is estimated through FDF-weighted inte-

grating the conditional scalar dissipation, N|η, over

each LES cell constituting the host CMC cell

N|η*=ℒFDF(N|η) (8)

and the filtered volumetric heat loss in the CMC reso-

lution for calculating h in Eq. (6) is obtained through

��W,Ω∗ =ℒ(q

W,Ω) (9)

For the CMC cells, ξ* and ξ

''2*

are necessary to calcu-

late the FDF. They are estimated as

ξ*= ℒ(ξ) (10a)

ξ''2

*

= ℒ (ξ2)+ ℒ (ξ''2) − ℒ2(ξ) (10b)

In Eqs. (8), (9) and (10), ℒ(x) and ℒFDF(𝑥) are

ℒ(𝑥)=∫ ρΩCMC xdη ∫ ρ

ΩCMC dη⁄ (11a)

ℒ𝐹𝐷𝐹(𝑥)=∫ ρΩCMC P(η)xdη ∫ ρ

ΩCMC P(η)dη⁄ (11b)

denoting the Favre averaging and FDF-weighted Fa-

vre averaging operators, respectively. In addition, as

shown in Fig. 1, the interpolated volume fluxes �� ⋅ 𝑑𝐒

(𝐒 is the CMC face normal vectors) and turbulent dif-

fusivity Dt at the CMC faces should be provided from

LES solver to the CMC solver for the flux calculations

of convection and turbulent diffusion terms in Eq. (5).

The unconditionally filtered quantities f (includ-

ing ��, �� and other scalars) at the LES resolution are

obtained through f = ∫ f |η∗P(η)dη

1

0. f |η

∗ is the con-

ditionally filtered scalars (e.g. 𝜌|𝜂 -1, QT and Q) pro-

vided by the CMC solver.

Figure 1: Schematic showing CMC cell recon-

struction and coupling between LES and CMC solv-

ers. The two-dimensional mesh denotes the slice

through a three-dimensional unstructured LES mesh.

Blue lines are LES edges while red lines CMC ones.

Black circles denote centroids of LES cells while red

squares CMC nodes. The cells enclosed by red lines

are the reconstructed CMC cells.

Figure 2: Schematic of the burner.

2.3 Flow considered and numerical imple-

mentations Figure 2 is a schematic of the burner studied ex-

perimentally at the University of Cambridge

(Cavaliere et al., 2013). A bluff body with diameter Db

= 0.025 m is fitted concentrically in a pipe with Dp =

0.037 m. The fuel injector lies at the center of bluff

body top with diameter Df = 0.004 m. The size of the

rectangular chamber is 0.0950.0950.15 m3. The air

bulk velocity Ua,b at the annular pipe outlet is 19.1 m/s

while that of the non-swirling pure methane jet Uf,b is

29.2 m/s. The temperature for both gases is 294 K. The

swirl number SN is 1.23 following Beer and Chigier’s

formula (Beer and Chigier, 1971). The Reynolds num-

ber for air stream based on diameter difference of the

bluff body and pipe, i.e. DpDb, and Ua,b is about

17700 while that for fuel jet based on Df and Uf,b is

about 4500. The stoichiometric mixture fraction is st

= 0.055.

Mixture fraction space is discretized by 51 nodes

clustered around st. In physical space, the Cartesian

coordinate origin lies at the center of circular fuel in-

jector and x is the streamwise direction. y and z are

normal to the side chamber walls. The LES domain

includes the swirler, annulus, chamber and a down-

stream hemisphere far-field (not shown in Fig. 2)

while the CMC domain consists of a partial annulus

(without swirler) starting 0.02 m upstream of bluff

body top, combustor and far-field. Around 10 million

tetrahedral LES cells are used while the number of the

polyhedral CMC cells is about 140,000. Within the

flame region (0<x/Db<2.4), the mean of LES cells for

one CMC cell is about 95. The number of the CMC

faces is about 4,800,000 accounting for 23% of the

LES faces and is roughly 16.5 times the face number

(about 300,000) when these CMC nodes are used as

cell centroids for generating pure tetrahedral cells. The

mean of the CMC face number per CMC cell is 66.

For CMC boundaries in -space, air and fuel with

temperature 294 K are specified at =0 and =1, re-

spectively. All the CMC cells are initialized by the

fully reactive 0D-CMC solutions (here 0D-CMC cal-

culations refer to solving Eq. (5) only with unsteady,

micromixing and chemistry terms) with medium sca-

lar dissipation, i.e. N0=50 1/s. Inert mixing solutions

in -space are injected into fuel and air inlets. For the

wall boundaries in the CMC domain, inert mixing so-

lutions are specified.

For the LES boundary conditions, at all inlets,

zero pressure gradient conditions are applied. For ve-

locity and mixture fraction, Dirichlet conditions are

enforced. Random noise with 5% intensity is imposed

for the velocities at both air and fuel inlets. The total

pressure is fixed to be atmospheric at the far-field out-

let. No slip condition for velocity and zero gradient for

mixture fraction are enforced at the walls. Also, the

walls are assumed to be chemically inert and cold

(TW=298 K). y+<4 for near-wall LES mesh is basically

satisfied to accurately predict the near-wall tempera-

ture gradient. In this paper only the bluff body wall

heat loss effects are investigated, which are more rel-

evant to the flame dynamics like lift-off and local ex-

tinction. The instantaneous flow and mixing fields are

initialized by the results from an LES where the con-

ditional profiles of reactive scalars are from a steady

0D-CMC calculation.

The LES and CMC governing equations are

solved by OpenFOAM and in-house codes, respec-

tively. The time step for both solvers is t = 1.5106

s. Low Ma number assumption is applied in LES. In

the CMC solver, the convection term is discretized

with first-order upwind scheme while the modeled tur-

bulent diffusion term with second-order central differ-

encing. Full operator splitting strategy is employed for

solving Eq. (5) and the conditional chemical source

term is solved with VODPK (Brown and Hindmarsh,

1989). The ARM2 mechanism by Sung et al. (1998) is

used for the methane/air combustion. The simulations

were run on 80 2.53 GHz Xeon CPUs with 3 GB RAM

for each processor and around 24 hours were needed

for 0.001 s of physical time.

3 Results and discussion

3.1 Velocity statistics

The axial and swirl velocity statistics at four

streamwise positions are compared with the corre-

sponding experimental results (Cavaliere, 2013) in

Figs. 3 and 4 and the mean and r.m.s. from the simu-

lations reasonably reproduce the main features of the

flow (e.g. confined Corner and long Inner Recircula-

tion Zones, i.e. CRZ and IRZ, as well as the solid-body

rotating regions within 0.8r/Db0.8 and x/Db<1

shown in Fig. 4). However, the fuel jet penetration is

under-predicted in the LES which can be observed by

the under-prediction of the central mean axial velocity

at x/Db=0.6 and 2.2 in Fig. 3. This may be caused by

the fact that the streamwise adverse pressure gradients

along the centerline are over-estimated, which is re-

lated to the over-prediction of mean swirl velocities at

x/Db=0.4 and 0.6 shown in Fig. 4. In addition, the r.m.s.

of both axial and swirl velocities at x/Db=0.4 and 0.6

from the LES are over-predicted. The higher r.m.s.

quantities in LES may be affected by the over-predic-

tion of the turbulence near the annulus exit. Generally,

the velocity statistics from the simulation show rea-

sonable agreement with the experimental results.

-1.00

0.00

1.00

0.00

0.50

1.00

-1.00

0.00

1.00

Ua/U

a,b

0.00

0.50

1.00

ua

/Ua,b

-1.00

0.00

1.00

0.00

0.25

0.50

-2 -1 0 1 2

-1.00

0.00

1.00 (d) x/Db=4.4

(c) x/Db=2.2

(b) x/Db=0.6

r/Db

(a) x/Db=0.4

-2 -1 0 1 20.00

0.10

0.20

0.30 LES Experiment

r/Db Figure 3: Radial profiles of mean (left) and r.m.s.

(right) axial velocity at x/Db=0.4, 0.6, 2.2 and 4.4.

-1.00

0.00

1.00

0.00

0.30

0.60

0.90

LES Experiment

(b) x/Db=0.6

(a) x/Db=0.4

-1.00

0.00

1.00

Us/U

a,b

0.00

0.30

0.60

0.90

(c) x/Db=2.2

us

/Ua

,b

-1.00

0.00

1.00

0.00

0.25

0.50

-2 -1 0 1 2

-1.00

0.00

1.00

r/Db

-2 -1 0 1 20.00

0.10

0.20

0.30(d) x/Db=4.4

r/Db Figure 4: Radial profiles of mean (left) and r.m.s.

(right) swirl velocity at x/Db=0.4, 0.6, 2.2 and 4.4.

3.2 Wall heat flux and local extinction Contours of instantaneous and mean temperature

overlaid by the stoichiometric mixture fraction and

zero axial velocity iso-lines at the x-y plane are

demonstrated in Fig. 5. The stoichiometric iso-lines

are basically confined within the upstream of IRZ

(x/Db<1). Similar to the experimental observations

(Cavaliere et al., 2013), the flame base is very close to

bluff body, although the lift-off can be observed occa-

sionally near the edge (e.g. the left flame base branch

in Fig. 5a). Figure 6 shows radial profiles of the mean

heat flux qw,S

and the temperature gradient on the bluff

body surface, i.e. 0.08r/Db0.5. Here the averaging

is performed both in time and in the azimuthal direc-

tion. As the radius increases, the mean wall surface

heat flux qw,S and temperature gradient sharply in-

crease at 0r/Db<0.15 and reach their individual plat-

eaus at r/Db=0.15, and eventually decrease at r/Db>0.4.

Clearly, the high values on the middle section of the

surface (0.15<r/Db<0.4) are attributed to the contact

between the cold surface and the flames as well as re-

circulating hot gases in IRZ while the relatively low

heat fluxes at low and large radii result from the cold

streams of air and fuel jets as shown in Fig. 5.

Figure 6 also presents radial distributions of the

mean laminar and turbulent heat fluxes, i.e. qw,Sl and

qw,St . q

w,St is much lower than q

w,Sl on the whole surface

and the fraction of the mean turbulent heat flux,

qw,St q

w,S⁄ , monotonically increases from about 15% at

r/Db=0.08 to about 45% at r/Db=0.5. The laminar heat

flux qw,Sl basically follows the temperature gradient

variations while, interestingly, qw,St increases mono-

tonically with the radius. Explicitly, the turbulent heat

flux is directly linked to the turbulent viscosity 𝜇t

shown in Eq. (7) and therefore to the sub-grid turbu-

lent kinetic energy k shown in Eq. (1). Since the ve-

locity r.m.s. close to bluff body (e.g. x/Db=0.4 in Figs.

3a and 4a) is over-predicted, it should be acknowl-

edged that qw,St close to r/Db=0.5 is accordingly over-

predicted.

Figure 7 demonstrates the Probability Density

Function (PDF) of the instantaneous bluff body sur-

face wall fluxes qw,S

based on the whole surface at all

simulated time instants. The PDF is close to a Gauss-

ian distribution, although clipped at zero, implying

that cold air and fuel streams are occasionally present

at the wall. This feature is different from the results

reported in Wang and Trouvé (2006) with 2D DNS of

ethylene/air non-premixed flames near a wall, where

the PDF of wall heat fluxes is quite narrow and very

few samples can be seen near zero. This discrepancy

can be attributed to the existence of the cold air and

fuel inlets near the bluff body and also the recirculat-

ing fresh gas from downstream. Furthermore, the

mean and peak heat fluxes in Fig. 7 are approximately

1.5105 and 5105 W/m2, respectively. They show

good agreement with the heat flux estimation in

Lataillade et al. (2002) where the order of magnitude

of 5105 W/m2 was obtained from methane/air flames

at 1 bar and 300 K wall temperature.

Figure 8 shows the variations of the condition-

ally filtered volumetric heat loss qW,Ω|η

and total en-

thalpy Qh from one CMC cell near the bluff body.

Each curve is from one instant and in Fig. 8(a) large

fluctuations of qW,Ω|η

can be seen with respect to its

mean profile (denoted as the red line). The introduc-

tion of the volumetric heat loss into the conditionally

filtered total enthalpy equations makes Qh deviate

from the adiabatic profiles, which is a straight line be-

tween =0 and =1. The considerable fluctuations of

Qh occur at about =st resulting from the large fluctu-

ations of qW,Ω|η

there as shown in Fig. 8(a).

The time records of key conditionally filtered

scalars, scalar dissipation rate and volumetric heat loss

at η = ξst are presented in Fig. 9. Between t=0.04 s and

0.04125 s, the flame is burning based on the values of

q|ξst , YOH|ξst

and T|ξst shown in Figs. 9(a)-(c). How-

ever, at around t=0.04125 s, the flame is quenched

(low q|ξst , YOH|ξst

and T|ξst) due to the large scalar

dissipation as shown in Fig. 9(d). The extinction con-

tinues until t=0.043 s when the stoichiometric scalar

dissipation N|ξst becomes very low and the re-ignition

occurs. Based on Fig. 9(e) and similar results from

other near-wall CMC cells, qW,Ω|ξst

basically follows

the evolutions of T|ξst but with some high-frequency

oscillations. It also shows the pronounced decrease/in-

crease during the onsets of extinction/re-ignition. This

can be expected to facilitate the occurrence of these

two critical flame behaviors.

Figure 5: Distributions of (a) instantaneous and (b)

mean temperature. Black iso-lines: instantaneous or

mean stoichiometric mixture fraction. White iso-

lines: zero instantaneous or mean axial velocity.

0.0 0.1 0.2 0.3 0.4 0.50.0

5.0x104

1.0x105

1.5x105

2.0x105

r/Db

qw

,s, W

/m2

qw, s

qlw,s

qtw,s

gradT

1x106

2x106

3x106

4x106

5x106

6x106

7x106

gra

dT

, K/m

Figure 6: Radial profiles of mean heat flux and tem-

perature gradient on the bluff body surface

(0.08r/Db0.5).

Figure 7: Probability density function of the bluff

body surface heat flux. The data are extracted both in

in time and space using the whole bluff body surface.

0.0 0.2 0.4 0.6 0.8 1.0-5x10

6

-4x106

-3x106

-2x106

-1x106

0

0

2x108

4x108

6x108

8x108

(b)

, [-]

h

, J/k

g

(a)

qW

,

, W

/m3

Figure 8: Variations of conditionally filtered (a) vol-

umetric heat loss and (b) total enthalpy. Red line:

mean conditional heat flux. The data are from a

CMC cell immediately adjacent to the bluff body sur-

face (CMC cell centroid coordinate: x/Db=0.012,

y/Db=0.4 and z/Db=0).

0.0400 0.0425 0.04500

3

6

9(d) N|st, 1/s

time, s

0.000

0.001

0.002

0.003

0.004

0.005(b) YOH|st

800

1000

1200

1400

1600

1800

2000(c) T|st, K

0

500

1000

1500

2000(a) q|st, MJ/m

3s

0.0400 0.0425 0.04501

2

3

4

5

6

7(e) qw,|st10

8, W/m

3

time, s

Figure 9: Time records of conditionally filtered (a)

heat release rate, (b) OH mass fraction, (c) tempera-

ture, (d) scalar dissipation and (e) volumetric heat

loss at =st from the same CMC cell as in Fig. 8.

3.3 Near-wall conditional flame structures

To comparatively investigate the wall heat loss

effects on the swirling non-premixed flames, LES/3D-

CMC simulation with adiabatic wall conditions

(termed as adiabatic case hereafter) was also con-

ducted in which all the other numerical implementa-

tions were exactly the same as those of the present case

(termed as heat loss case hereafter). The statistics

about the velocities and local extinction from the adi-

abatic case have been discussed in Zhang et al. (2014).

Figures 11 and 12 show the comparisons of mean

flame structures in mixture fraction space from heat

loss and adiabatic cases. The mean conditional mass

y/Db

x/D

b

(a)

-1.5 -1 -0.5 0 0.5 1 1.50

1

2

3

4

5

6

500 1000 1500 2000

IRZCRZ

y/Db

x/D

b

(b)

-1.5 -1 -0.5 0 0.5 1 1.50

1

2

3

4

5

6

500 1000 1500

0 1x105

2x105

3x105

4x105

5x105

0

1x10-6

2x10-6

3x10-6

4x10-6

5x10-6

6x10-6

PD

F

qW,S, W/m2

Gaussian Fitting

fractions of reactants (i.e. CH4 and O2 in Figs. 12a and

12b)/products (e.g. H2O in Fig. 12c) are higher/lower

in the heat loss case than those in the adiabatic simu-

lation, indicating incomplete reactions when wall heat

loss is included. The conditional mass fraction of

CH2O in Fig. 11(d) is higher in the heat loss case. The

mean profiles of OH mass fraction in -space, heat re-

lease rate and temperature are shown in Figs. 12(a)-(c).

Compared to the adiabatic results, YOH|η and T|η in

the heat loss case are obviously lower and, conversely,

q|η is much higher. The inclusion of wall heat loss

term in Eq. (6) leads to a considerable enthalpy defect

around =st as shown in Fig. 12(d).

The conditional reactedness (Masri et al., 1996;

Zhang et al., 2014), bα|η, is calculated in the CMC cell

investigated in Figs. 11 and 12 through

bα|η= (Yα|η-Yα,m|η) (Yα,b|η-Yα,m|η)⁄ (12)

In Eq. (12), Yα,b|η and Yα,m|η are the fully burning 0D-

CMC solutions with N0=5 1/s and inert mixing ones,

respectively. The extinction solutions are indicated by

bα|η=0 while the fully burning profiles are reached

when bα|η is close to or larger than unity. Figure 13

presents the PDF of reactedness at =st from temper-

ature and mass fractions of OH, CO and NO from the

heat loss and adiabatic cases. Clearly, in both cases,

bOH|ξst and bT|ξst

in Figs. 13(a) and 13(b) show the

peaks when they are close to unity. However, in the

heat loss case, the peaks of bOH|ξst and bT|ξst

are

shifted towards smaller values which statistically

means the weakened reactivity. For bOH|ξst , another

peak appears approaching zero, indicating the instan-

taneous extinction at the present CMC cell when wall

heat loss is considered. However, for bCO|ξst and

bNO|ξst in both cases, the distributions are wide. The

PDF of bCO|ξst is negligibly affected by the heat loss

while that of bNO|ξst move towards smaller values

when wall heat loss is included. The results in Figs.

11-13 only correspond to one selected CMC cell but

similar findings can also be obtained from other near-

wall CMC cells.

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

(a) CH4

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25(b) O

2

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

, [-]

(c) H2O

0.0 0.2 0.4 0.6 0.8 1.00

3

6

9

, [-]

(d) CH2O10-4

Figure 11: Comparisons of the mean conditional

mass fractions of (a) CH4, (b) O2, (c) H2O and (d)

CH2O between adiabatic (red lines) and heat loss

(black lines) cases. The same CMC cell as in Fig. 8.

0.00 0.02 0.04 0.06 0.08 0.100.0

1.5

3.0

4.5

6.0

(a) OH10-3

0.00 0.02 0.04 0.06 0.08 0.100

100

200

300

400

500(b) q,MJ/m

3s

0.0 0.2 0.4 0.6 0.8 1.00

800

1600

2400(c) T,K

0.0 0.2 0.4 0.6 0.8 1.0

-5

-4

-3

-2

-1

0

1(d) h106,J/kg

Figure 12: Comparisons of the mean condi-

tional (a) OH mass fraction, (b) heat release rate, (c)

temperature and (d) total enthalpy between heat loss

(black lines) and adiabatic (red lines) cases. The

same CMC cell as in Fig. 8.

0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

10

12 (a) T

PD

F0.0 0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

10(b) OH

0.0 0.5 1.0 1.5 2.00

1

2

3

4

PD

F

b |st

(c) CO

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

b |st

(d) NO

Figure 13: Probability density function of reacted-

ness at =st from (a) temperature, mass fractions of

(b) OH, (c) CO and (d) NO. Red lines: adiabatic

case, black lines: heat loss case.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

2

<N

>,

1/s

, [-]

1

3

1: y/Db=0.49

2: y/Db=0.4

3: y/Db=0.18

Figure 14: The mean conditional scalar dissipation

rates of three near-wall CMC cells (x/Db=z/Db=0)

from adiabatic (red lines) and heat loss (black lines)

cases.

The mean conditional scalar dissipation rates ⟨N|η⟩ from the adiabatic and heat loss cases are com-

pared in Fig. 14. For all three cells corresponding to

different radial positions near the bluff body, ⟨N|η⟩ in

the heat loss case is always larger than that in adiabatic

one because when the wall heat loss is considered the

mean mixture fraction gradient magnitude near the

wall is increased although the mass diffusivity D

demonstrates slight decrease (results not shown here).

This difference concerning ⟨N|η⟩ explains the com-

parisons of mean flame structures in Figs. 11 and 12.

Also, the difference of ⟨N|η⟩ between two cases in-

creases with the increased y/Db. This explicitly implies

that the wall heat loss would greatly influence scalar

dissipation near the outer part of the bluff body, where

the iso-surfaces of instantaneous ξst and the flame

base are occasionally attached to. However, the peak

values of ⟨N|η⟩ in these three cells are still well below

the critical peak value from 0D-CMC calculations, i.e.

N0170 1/s. This indicates the relatively low scalar

dissipation rate near the bluff body.

0.000

0.001

0.002

0.003

0.004

0.005

(d)(c)

(b)

YO

H|

, [-

]

(a)

0.00 0.05 0.100.000

0.001

0.002

0.003

0.004

YO

H|

, [-

]

, [-]0.00 0.05 0.10

, [-]

Figure 15: Comparisons of mean conditional OH

mass fractions between adiabatic (red lines) and heat

loss (black lines) cases at four streamwise positions

x/Db = (a) 0.03, (b) 0.17, (c) 0.8 and (d) 1.6 with

y/Db=0.4 and z/Db=0.

Figure 16: Iso-surfaces of instantaneous stoichio-

metric mixture fraction colored by conditional (a)

OH mass fraction and (b) temperature at stoichiome-

try.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0

10

20

b|st

0

5

10

adiabatic case

wall heat loss case

PD

F(b

OH

|st) (b)

PD

F(b

T|

st) (a)

Figure 17: Probability density function of reacted-

ness at =st from (a) temperature and (b) OH mass

fraction corresponding to the three dimensional stoi-

chiometric iso-surface within 0x/Db0.8.

3.4 Flame stabilization and lift-off

The streamwise variations of the mean condi-

tional mass fraction of OH, ⟨YOH|η⟩, are shown in Fig.

15 through comparing the heat loss and adiabatic re-

sults. It can be observed that ⟨YOH|η⟩ in the adiabatic

case is higher than in the heat loss case but the differ-

ence becomes negligible at x/Db=1.6. Iso-surfaces of

instantaneous mixture fraction (from the LES resolu-

tion) colored by Y𝑂𝐻|ξst and T|ξst

(from the host CMC

cells of the local LES meshes) in the heat loss case are

plotted in Fig. 16. At the iso-surfaces close to the bluff

body, large flame holes quantified by low Y𝑂𝐻|ξst

(<0.001) and T|ξst (<1200 K) can be observed. This

explicitly manifests the instantaneous extinction at the

flame base regardless of the local mixing state and

therefore the lift-off near the bluff body, consistent

with the experiment (Cavaliere et al., 2013).

The lift-off phenomenon is also seen in the adia-

batic case as discussed in Zhang et al. (2014) where it

is attributed to the strong convection approaching the

swirling air inlet. To appreciate how the wall heat loss

affects the flame stabilization and lift-off, Fig. 17 pre-

sents the PDFs of reactedness at stoichiometry from

temperature and OH mass fraction, i.e. bT|ξst and

bOH|ξst . Here the samples include 50 time instants and

each instant is extracted from about 3,000 CMC cells

enclosing the three dimensional iso-surfaces of the in-

stantaneous stoichiometric mixture fraction near the

bluff body, i.e. 0x/Db0.8 (see Fig. 16a). In the adi-

abatic case (red lines in Fig. 17), bT|ξst has a single

peak which is centered at 0.92 with some negative

skewness, indicating some degree of instantaneous ex-

tinction at the flame base. The extinction is also char-

acterized by the pronounced bimodularity in PDF of

bOH|ξst in Fig. 17(b). In the heat loss case (black lines

in Fig. 17), the negative skewness of the PDF of bT|ξst

is intensified and the PDF of bOH|ξst tends to be single

peak around zero. Both features concerning bT|ξst and

bOH|ξst statistically indicate the weakened reactivity in

the local CMC cells at the flame base when the bluff

body heat loss effects are taken into account. Physi-

cally, the comparisons in Fig. 17 also imply that the

lift-off is expected to be affected by the wall heat loss

modeling and hence the lift-off heights discussed in

Zhang et al. (2014) where the walls are assumed to be

adiabatic would be somewhat under-estimated.

4 Conclusions Large eddy simulation with three dimensional

conditional moment closure combustion model is ap-

plied to a swirl non-premixed methane flame with lo-

cal extinction. The convective wall heat loss is in-

cluded as a source term in the conditionally filtered to-

tal enthalpy equation for the CMC cells adjacent to

walls. The mean heat flux is high on the middle bluff

body surface but low near its edges. The sub-grid heat

flux based on the resolved temperature gradient is rel-

atively low compared to the laminar counterpart but

increases with the turbulent intensity.

(a) (b)

x/Db=0.8

For the CMC cells immediately adjacent to the

bluff body, the heat loss facilitates the occurrences of

extinction and re-ignition. It has a significant influ-

ence on the mean flame structures, which is directly

linked to the changes of the conditional scalar dissipa-

tion near the wall. Furthermore, the degree of local and

instantaneous extinction measured by conditional re-

actedness at stoichiometry is intensified due to the

wall heat loss. The wall heat loss also shows a signifi-

cant impact on the flame stabilization and lift-off near

the bluff body surface.

Acknowledgments This research was carried out under the financial

support from the EPSRC and Rolls Royce Group

through a Dorothy Hodgkin Postgraduate Award.

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