Content uploaded by L. Y. Jiang
Author content
All content in this area was uploaded by L. Y. Jiang on Nov 14, 2018
Content may be subject to copyright.
This article was published in an Elsevier journal. The attached copy
is furnished to the author for non-commercial research and
education use, including for instruction at the author’s institution,
sharing with colleagues and providing to institution administration.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Reynolds analogy in combustor modeling
Lei-Yong Jiang
*
, Ian Campbell
Gas Turbine Laboratory, Institute for Aerospace Research, National Research Council Canada, 1200 Montreal Road,
M-10, Ottawa, Ontario, Canada K1A 0R6
Received 9 April 2007; received in revised form 31 October 2007
Available online 5 February 2008
Abstract
The Reynolds analogy concept has been used in almost all turbulent reacting flow RANS (Reynolds-averaged Navier–Stokes) sim-
ulations, where the turbulence scalar transfers in flow fields are calculated based on the modeled turbulence momentum transfer. This
concept, applied to a diffusion flame model combustor, is assessed in this paper. Some of the numerical results, obtained from a flamelet
combustion model with the turbulent Prandtl/Schmidt number varying from 0.25 to 0.85, are presented and compared with a benchmark
experimental database. It is found that the turbulent Prandtl/Schmidt number has significant effects on the predicted temperature and
species fields in the combustor. This is also true for the temperature profile along the combustor wall. In contrast, its effect on the velocity
field is insignificant in the range considered. With an optimized turbulent Prandtl/Schmidt number, both velocity and scalar fields can be
reasonably and quantitatively predicted. For the present configuration and operating conditions, the optimal Prandtl/Schmidt number is
0.5, lower than the traditionally used value of 0.85. This study suggests that for accurate prediction of turbulence scalar transfers in
practical reacting flows, the Reynolds analogy concept should be improved and new approaches should be developed.
Crown Copyright Ó2007 Published by Elsevier Ltd. All rights reserved.
Keywords: Reynolds analogy; Turbulence scalar transfer; Combustor modeling; Schmidt number; Prandtl number
1. Introduction
Accurate prediction of temperature distribution is criti-
cal in the development of advanced combustion systems.
For example, poor temperature profiles at the liner and exit
of a gas turbine combustor can significantly reduce lifetime
of the combustor and turbine vanes and blades behind. In
extreme cases, devastating structural damage to engine
components can occur.
In almost all turbulent reacting flow RANS simulations,
turbulence scalar transfers (mixture fraction, species, and
energy or temperature) are calculated based on the Rey-
nolds analogy concept. In this approach, the turbulent Pra-
ndtl (Pr
t
) and Schmidt (Sc
t
) numbers are used to link the
turbulence scalar transfers in flow fields to the momentum
transfer that is determined by a selected turbulence model.
An existence of an analogy between the wall shear and
heat flux in boundary layers was first postulated by Rey-
nolds over a century ago [1]. This original hypothesis has
been considerably amended and applied to general turbu-
lent heat and species transfers [2,3]. Recently, its applica-
tions to high-Mach-number boundary layers [4], turbine
flows [5] and film cooling [6] have been studied. The Rey-
nolds analogy factors for flow parameters related to hyper-
sonic propulsion and turbines have been determined [4,5].
The suitability of Reynolds analogy to disturbed turbu-
lent thermal boundary flows has been reported by a num-
ber of authors. Choi and Orchard [7] investigated the
heat transfer characteristics over a triangular-profiled riblet
surface, while de Souza et al. [8] studied the large-scale
organization of a boundary layer disturbed by a cylinder
wake flow. They all pointed out that this concept did not
hold in these disturbed boundary flows. Vogel and Eaton
[9] carried out heat transfer and fluid dynamic measure-
ments downstream of a backward-facing step. It was found
that Reynolds analogy failed in the recovering boundary
0017-9310/$ - see front matter Crown Copyright Ó2007 Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2007.12.006
*
Corresponding author. Tel.: +613 993 9235; fax: +613 952 7677.
E-mail address: leiyong.jiang@nrc-cnrc.gc.ca (L.-Y. Jiang).
www.elsevier.com/locate/ijhmt
Available online at www.sciencedirect.com
International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
layer, and it was only valid far downstream of the reattach-
ment point. Time-resolved gas temperature in the oscillat-
ing turbulent flow of a pulse combustor tail pipe was
studied by John and Keller [10]. The results indicated that
the analogy between momentum and thermal transport at
the tail pipe wall was no longer valid.
Since the 1970s, the Reynolds analogy concept has been
further extended into computational simulations of general
turbulent reacting or mixing flows. The main advantage of
this approach is that the turbulence scalar transfers can be
effectively computed from the modeled momentum transfer
without solving a full second moment closure for both
momentum and scalar transportations. Consequently, the
computing time to reach a converged solution is much
reduced.
Numerous experimental studies on Pr
t
and Sc
t
were car-
ried out in the last century, particularly in the period of
1930s–1970s [2,3]. Hinze [2] reviewed a large number of
experimental measurements in pipe and 2D channel flows,
and pointed out that the overall Pr
t
or Sc
t
varied from 0.6
to 0.8. Recently, based on their velocity and concentration
half-width measurements in axisymmetric jets of air and
helium, Panchapakesan and Lumley [12] obtained an aver-
age value of 0.7 for Sc
t
.
In most turbulent reacting or mixing flow simulations, it
has become a common practice to set Le Sc
t
/Pr
t
=1or
Pr
t
=Sc
t
[11]. Traditionally a constant value of Pr
t
=Sc
t
0.85 has been used in jet flows [13,14] and gas turbine com-
bustor modeling [15,16]. However, low Pr
t
and Sc
t
num-
bers from 0.20 to 0.5 have been used by a number of
authors for simulating kerosene-fired gas turbine combus-
tors. Crocker et al. [17] successfully modeled an entire com-
bustor from the compressor diffuser exit to the turbine
inlet, including air split and liner wall temperature predic-
tion. A low value of 0.25 was used for Pr
t
and Sc
t
since
it consistently demonstrated better agreement with the
combustor fuel/air mixing results. Kaaling et al. [18] sys-
tematically studied five RQL (rich burn, quick quench, lean
burn) low-emission combustor designs. The CFD calcula-
tions were calibrated against CARS (coherent anti-Stokes
Raman spectroscopy) temperature measurements, and
good agreement was found by using Pr
t
=Sc
t
= 0.2. Large
eddy simulations (LES) of a Rolls–Royce production gas
turbine combustor were performed by Cannon et al. [19],
where Pr
t
=Sc
t
= 0.5. Moreover, the effect of Schmidt
number on turbulence scalar mixing of a gaseous jet issued
into a cross airflow was investigated by He et al. [20].By
comparison with the available experimental data, Pr
t
=
Sc
t
= 0.2 was recommended.
To provide a benchmark database for the evaluation
and development of various physical models, a series of
experiments were performed on a diffusion flame model
combustor at the National Research Council of Canada
(NRCC). The comprehensive results include mean and
fluctuating velocity components, mean temperature, wall
temperature, radiation heat flux, as well as species concen-
trations [21].
The objectives of the present work are to find out if such
a low value of Pr
t
and Sc
t
is a real physical fact in combus-
tor modeling, and to assess the Reynolds analogy concept
Nomenclature
C
p
specific heat at constant pressure
Dmolecular diffusivity
fmixture fraction
f00 fluctuating component of mixture fraction
Htotal enthalpy
hheat transfer coefficient
kturbulence kinetic energy
Le lewis number
Mmolecular weight
pprobability density function
Pr
l
laminar Prandtl number
Pr
t
turbulent Prandtl number
rradial coordinate
Senergy source term
Sc
t
turbulent Schmidt number
St Stanton number
Ttemperature
Umean axial velocity
U
1
free stream velocity
uaxial velocity component
Vvelocity vector
v00 fluctuating velocity vector
vradial velocity component
u00v00 turbulence shear stress, qu00 v00=
q
xcoordinate along the combustor axis of symme-
try
Yspecies mass fraction
y
+
non-dimensional parameter, ffiffiffiffiffiffiffiffiffiffi
sw=q
py=t
ydistance to the wall boundary
Zmass fraction of element
Greek symbols
C
t
turbulent Prandtl or Schmidt number
eturbulence dissipation rate
llaminar viscosity
l
t
turbulent viscosity
qdensity
s
w
wall shear stress
tkinematic viscosity
/species mass fraction, mixture fraction, or total
enthalpy
/00 scalar fluctuation component
uspecies mass fraction, density or temperature
xspecies source term
1252 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
currently used in turbulent reacting flow RANS simula-
tions. Since the model combustor geometry is much simpler
than practical combustors, its boundary conditions are well
defined and a comprehensive experimental database is
available, the assessment of the above issues is relevant.
2. Experimental measurements
A schematic diagram of the model combustor is shown
on the left side of Fig. 1, including the fuel and air inlets,
combustion chamber and contracted exhaust section (all
dimensions are in mm). Air entered the combustion cham-
ber around a disc flame-holder, while fuel was fed through
the center of the bluff body. The test rig was mounted on a
three-axis traversing unit with an accuracy of ±100 lm.
Fuel used in the experiments was commercial grade pro-
pane, and dry air was delivered from a shop air supply.
Both air and fuel flows were controlled by Sierra side-trak
mass-flow controllers with 2% accuracy of full scale (fuel
100 l/min and air 2550 l/min).
To reduce the heat losses through walls, a 25.4 mm thick
fibre blanket of Al
2
O
3
was wrapped around the combus-
tion chamber. Four narrow slots were cut into the blanket
to allow appropriate physical and optical access to the
chamber interior. Interchangeable sets of stainless steel
and fused silica windows were used, the former for physical
probing with gas sampling probes, radiometers and ther-
mocouples, and the latter for optical probing with a laser
Doppler anemometer (LDA). The viewing area of the win-
dows measured 17 mm in width and 344 mm in length.
Measurements of velocity were made using both a two-
and three-component LDA system operating in a back
scattering mode. In the lower section of the combustion
chamber, limited optical access forced the use of a single
fibre optic head to measure axial and tangential velocities.
In the upper section of the chamber, a complete three-com-
ponent LDA system was used. Gas temperatures were
acquired using an uncoated 250 lm diameter, type ‘‘S”
thermocouple supported by a twin-bore ceramic tube.
Thermocouples embedded in and flush with the combustor
wall were used to measure the wall temperature. Gas
species measurements were made with a sampling probe
connected to a Varian Model 3400 Gas Chromatograph.
The major species measured were CO, CO
2
,H
2
Oand
C
3
H
8
.
3. Numerical simulations
Axisymmetric, steady, turbulent, reacting flows were
considered in the present study, and a commercial software
package, fluent, was used as the platform for all numerical
simulations. The computational domain, governing equa-
tions, selected physical models, boundary conditions and
solution methods are described in the following sub-
sections.
3.1. Computational domain
The computational domain covered the entire combus-
tor flow field from the fuel and air inlets to the exhaust exit,
as shown on the right side of Fig. 1. The internal and exter-
nal conjugate heat transfers from the combustion mixture
to the flame-holder body and insulation walls were also
modeled.
Fig. 1. The model combustor and computational domain.
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1253
Author's personal copy
Since the flow field was axisymmetric, 2D quadrilateral
meshes were used. Fine grids were laid behind the flame-
holder in the combustion chamber in order to resolve the
recirculation region. Fine grids were also generated in the
shear layers between the recirculation region and fuel and
air jets, as well as in the gap between the flame-holder edge
and air inlet chamber wall. Coarse grids were used in the
stainless steel walls and ceramic blanket. A total of
74,100 elements was used for most of the simulations.
The skewness was less than 0.2 in the flow field domain
and the aspect ratio was less than 12 for 99.5% elements.
Efforts were made to keep the wall parameter, y
+
ðffiffiffiffiffiffiffiffiffiffi
sw=q
py=tÞ, in the desired range (30–60). A number of
meshes were tested to ensure mesh independence of the
numerical results.
3.2. Governing equations
The first-moment Favre-averaged conservation equa-
tions for mass, momentum, species, mixture fraction and
total enthalpy, may be expressed in a coordinate-free form
as [22,23]
r
qe
V
¼0ð1Þ
r
qe
Ve
V
¼r
pþrTr qv00v00
ð2Þ
r
qe
Ve
Yi
¼r qDire
Yi
r qv00Y00
i
þxið3Þ
r
qe
V~
f
¼r qDr~
f
r qv00f00
ð4Þ
r
qe
Ve
H
¼r l
Prl
re
H
r qv00H00
þSHð5Þ
In the above equations,
qrepresents mean density, e
Vis the
mean velocity vector, v00 stands for fluctuation velocity vec-
tor, the viscous stress tensor T¼l½r e
Vþðre
VÞT2
3lr
e
VI with Ia unit tensor, qv00v00 denotes Reynolds stresses,
Y
i
is the mass fraction of the ith species, fstands for mix-
ture fraction, Hdenotes total enthalpy, and Dand Pr
l
represent molecular diffusivity and Prandtl number,
respectively.
For closure of the above equations, the species source
term, x
i
in Eq. (3) is obtained from a selected combustion
model. The energy source term, S
H
in Eq. (5) includes vis-
cous heating and radiation heat transfer that is obtained
from a radiation model. As mentioned earlier, Reynolds
stresses, qv00v00 or turbulence momentum transfer in Eq.
(2) are modeled by a selected turbulence model, while
qv00Y00 ;qv00f00;qv00 H00 or turbulence scalar transfers in Eqs.
(3)–(5) are computed based on Reynolds analogy.
3.3. Turbulence modeling and Reynolds analogy
In a previous benchmarking on turbulence modeling
[24], the Reynolds stress model (RSM) produced better
results than three popular two-equation eddy-viscosity
models. Therefore, it was chosen to model turbulence
momentum transfer in the present flow. Since detailed
description of the RSM takes a lot of space and is beyond
the scope of the present paper, it is not given here. It can be
readily found in Refs. [23,25].
For axisymmetric flows, as in the present case, only four
Reynolds stress components need to be considered. The
four transportation equations of these components along
with the turbulence dissipation rate equation are solved
in the combustor flow field. From the solutions of Rey-
nolds stresses and dissipation rate, the turbulent viscosity
(momentum transfer coefficient) is then computed:
lt¼
qClk2=eð6Þ
where C
l
= 0.09, and kand eare turbulent kinetic energy
and dissipation rate, respectively.
Following the Reynolds analogy concept, the turbulence
scalar transfers are modeled as
r qv00/00 ffir lt
Ct
r~
/
ð7Þ
where /stands for species mass fraction, mixture fraction
or total enthalpy, and C
t
represents Pr
t
or Sc
t
. Note that
in Eq. (7), the turbulence scalar transfer coefficients,
l
t
/C
t
, are simply the products of the turbulence momentum
transfer coefficient (l
t
) and 1/C
t
. The isotropic turbulence
transfer (coefficient) assumption is abandoned in momen-
tum transfer; however, it is still used for turbulence scalar
transfers.
The rationale and limitation of Reynolds analogy could
be revealed by reducing the conservation Eqs. (2)–(5) to
axisymmetric steady boundary flows and neglecting the
streamwise pressure gradient, molecular viscous terms,
and source terms. Then the following equations are
obtained:
q~
uo~
u
oxþ
q~
vo~
u
or¼1
r
o
orrlt
o~
u
or
ð8Þ
q~
uo~
/
oxþ
q~
vo~
/
or¼1
r
o
orrlt
Ct
o~
/
or
! ð9Þ
where the turbulent viscosity concept is applied to both
streamwise momentum and scalar transfers. With C
t
=1,
the two-equations become identical. That is, under
appropriate boundary conditions, the solution of all
these flow parameters can be obtained from a single partial
differential equation [11]. For wall boundary flows with
relatively large radius with respect to boundary thick-
ness, the original form of Reynolds analogy can be deduced
[5]:
2St
cf
¼ðh=qCpU1Þ
ðsw=qU2
1Þ1ð10Þ
where St =h/(qC
p
U
1
) is the Stanton number and
cf¼sw=ð0:5qU2
1Þis the wall friction coefficient. From this
equation, the turbulent heat transfer coefficient can be esti-
mated from the measurement of pressure loss due to fric-
tion in the flow.
1254 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
3.4. Combustion modeling
Some of the numerical solutions obtained from a lami-
nar flamelet combustion model are presented and discussed
in this paper. The laminar flamelet model views the turbu-
lent flame as an ensemble of laminar flamelets [26]. For dif-
fusion flames, it approximates the unsteady distorted
turbulent diffusion flame using a steady laminar counter-
flow diffusion flame.
The governing equations for laminar counter-flow diffu-
sion flames can be transformed from physical coordinates
to a mixture fraction coordinate by neglecting some insig-
nificant terms. In the adiabatic case, the species fractions,
density and temperature along the flamelet axis (normal
to the flamelet) can be uniquely determined by two param-
eters, the mixture fraction, f, and the scalar dissipation
rate, v
st
, at the position where fis stoichiometric.
The mixture fraction can be written as
f¼ZiZi;ox
Zi;fuel Zi;ox
ð11Þ
where Z
i
stands for the mass fraction of element, ‘‘i”,Z
i, ox
and Z
i, fuel
denote the elemental mass fraction of ‘‘i”at the
oxidizer and fuel inlets, respectively. The scalar dissipation
rate in turbulent flows is modeled by the following
expression:
vst ¼Cvef002=kð12Þ
where C
v
= 2.0. The coupling between non-equilibrium
chemistry and turbulence is accounted for by the b-func-
tion probability density function, and the time-averaged
flow parameters in the non-adiabatic condition are com-
puted from the following equation:
ui¼Z1
0
p~
f;f002
uiðf;vst;HÞdfð13Þ
where urepresents species mass fraction, density or tem-
perature, and His the total enthalpy. The flamelet library
can be pre-processed and tabulated, which offers tremen-
dous computational saving.
It should be mentioned that fluctuation of the scalar dis-
sipation rate in Eq. (13) is ignored, which has become a
common practice [27]. A number of approaches have been
proposed to account for fluctuation of the scalar dissipa-
tion rate [26]; however, there is no conclusive solution.
Moreover, it may be argued that the turbulence effect on
the scalar dissipation rate could have been, to some extent,
modeled by Eq. (12) and the probability density function of
the mixture fraction in Eq. (13). In short, this is still an
open area of research.
A major advantage of the flamelet model over other
combustion models, such as eddy-dissipation and probabil-
ity density function models, is that detailed and more real-
istic chemical kinetics can be incorporated into turbulent
reacting flows. In the present study, the chemical reaction
mechanism from Stahl and Warnatz [28] for propane–air
flames was employed. It consists of 228 chemical reactions
and 31 species, including O, O
2
, OH, H, H
2
,H
2
O, H
2
O
2
,
HO
2
,N
2
, CO, CO
2
, CH, CH
2
,CH
3
,CH
4
,CHO,CH
2
O,
CH
2
CO, CH
3
CO, CH
3
CHO, C
2
H, C
2
H
2
,C
2
H
3
,C
2
H
4
,
C
2
H
5
,C
2
H
6
,C
3
H
6
,C
3
H
8
,N
*C
3
H
7
,I
*C
3
H
7
and C
2
HO.
3.5. Other physical models
To account for the radiation heat transfer between the
gas mixture and the combustion chamber walls, a discrete
ordinates radiation model [29] was employed. The absorp-
tion coefficient of the gaseous mixture was determined
from local species mass fractions in the flow. An absorp-
tion coefficient of 0.5 was used for the stainless steel wall
and the disc flame-holder. For internal wall boundaries,
as mentioned earlier, effort was made to keep the y
+
value
in the desired range of 30–60. However, there were local
regions where y
+
value was outside this range. To minimise
this effect, an enhanced wall boundary treatment was
applied at all internal walls. In this approach, a two-layer
model is combined with wall functions, and the viscous
and fully turbulent regions are smoothly blended.
Polynomials determined from the JANAF tables [30]
were used to calculate the specific heat of each species as
a function of temperature. For other thermal properties
of the mixture such as molecular viscosity and thermal con-
ductivity, the values of air at 900 K were used. The thermal
conductivity was 25 W/m K for the stainless steel, and
0.1 W/m K for the ceramic insulation.
3.6. Boundary conditions
The fuel mass-flow rate was 16.2 g/s and the airflow rate
was 550 g/s, and the corresponding overall equivalence
ratio was 0.46. For both flows, the inlet temperature was
293 K. The Reynolds number based on the air entry veloc-
ity and flame-holder diameter was 1.9 10
5
. An estimated
turbulence intensity of 10% and hydraulic diameters were
used to estimate Reynolds stress components and turbu-
lence dissipation rate at the fuel and air inlets. A sensitivity
study was performed with the inlet turbulence intensity of
5% and 2%, respectively. The difference in computed turbu-
lence kinetic energy along the combustor centerline is
observable only in the fuel inlet passage and a small region
near x= 80 mm, and the maximum deviation between case
2% and case 10% is only 2.3%. The difference is even smal-
ler for the mean axial velocity and temperature along the
combustor centerline.
The external wall temperatures were defined based on
the experimental measurements. A room temperature of
293 K was assigned to the walls of the inlet section, and
the upstream edges of the combustion chamber and cera-
mic insulation walls. A linear temperature profile from
294 to 405 K was specified along the outer boundary of
the ceramic insulation wall. The temperature of the outer
boundary of the exit section was set to 960 K. The same
temperature was assigned to the downstream edge of the
combustion chamber wall because its heat resistance was
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1255
Author's personal copy
much smaller than the ceramic insulation. A linear
temperature profile from 960 to 405 K was assigned to
the downstream edge of the insulation wall. Finally, the
pressure at the combustor exit was set to the atmospheric
value.
3.7. Solution methods
A segregated solver with a second-order accurate
scheme was used to resolve the flow field. A node-based
method for derivatives was chosen in order to maintain
numerical stability as C
t
approached 0.25. At convergence,
the normalized residuals of flow variables were less or
equal to 10
5
in all test cases. The monitored axial veloci-
ties at two points in the shear layer downstream of the
flame-holder remained unchanged at least for the first four
digits. A 4-node LINUX cluster, 64-bit, 2.6 GHz, dual
CPU and 8 GB RAM for each node, was used to perform
all simulations.
4. Results and discussion
Numerical simulations were performed with C
t
varying
from 0.25 to 0.85, and a large amount of data were pro-
cessed and analyzed. As stated earlier, only some of the
results are presented here. In the following sub-sections,
the velocity field results are first presented, then the temper-
ature and species results are discussed, and finally the Rey-
nolds analog concept is assessed.
4.1. Velocity distributions
The upper half of Fig. 2 shows the axial velocity con-
tours and flow path-lines for C
t
= 0.85. The lower half of
the figure presents the experimental data with the zero axial
velocity lines specified. The flow patterns in the combustion
chamber are excellently predicted. Two recirculation zones
are formed behind the flame-holder although, in the exper-
imental case, the central recirculation zone is not com-
pletely resolved and no flow path-lines are drawn due to
the limited data points. It is significant that both reattach-
ment points or lengths of the two recirculation zones are
well predicted. The central recirculation zone created by
the fuel jet is completely confined within the annular recir-
culation zone generated by the annular air jet. This implies
that the transportation of fuel into the flow field is realized
by laminar and turbulent diffusion only through the annu-
lar recirculation zone.
Although turbulence scalar transfers are calculated
based on the modeled turbulence momentum transfer, the
former also affects the latter since they are coupled. The
effect of C
t
on velocity field is illustrated in Figs. 3 and 4
for C
t
= 0.50 and 0.25, respectively. The differences in
the flow fields between C
t
= 0.85 and 0.50 are minor. For
C
t
= 0.25, the length and volume of the annular recircula-
tion zone are slightly reduced in comparison with those
in Figs. 2 and 3. The numerical results indicate that in
the range of C
t
studied, the effect of C
t
on velocity field is
limited, particularly for C
t
> 0.35.
Fig. 5 gives axial velocity profiles along the combustor
centerline, and compares the results with the experimental
measurements. Superimposed in the figure, in red, are the
estimated error bars of 2%. The numerical results show
good agreement with the experimental data, except that
the peak value of negative velocity is under-predicted.
The effect of C
t
variation from 0.85 to 0.25 on the centerline
velocity profile is small.
The predicted axial velocity profiles at five cross-sec-
tions, two inside the recirculation region, one close to the
stagnation point, and the last two located downstream of
the recirculation region, are presented in Fig. 6, and quan-
titatively compared with the experimental results. In gen-
eral, the profiles are reasonably predicted except in the
local regions at three middle sections, where the discrepan-
cies increase as C
t
decreases. At these three sections, the
flow field is complex, which represents a difficult task for
numerical simulations.
Fig. 7 shows quantitative comparisons of turbulence
kinetic energy between the numerical and experimental
results at four cross-sections from x= 60 to 240 mm. In
the figure, error bars show the measurement accuracy of
±8%. The trends and magnitudes are reasonably well pre-
dicted, except for the magnitudes at three downstream sec-
tions with C
t
= 0.25.
Fig. 2. Axial velocity contours and flow path-lines, C
t
= 0.85.
1256 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
Fig. 8 compares the numerical results of turbulence
shear stress, qu00v00 =
qwith the measured experimental data
at four cross-sections. Here the estimated measurement
accuracy is about 12%. The shear stress is of primary inter-
est for momentum transport in turbulent flows, particu-
larly in shear flows. Again, the trends and magnitudes are
reasonably well predicted at all sections. The effect of C
t
variation is insignificant, particularly in comparison with
the measurement error.
The above numerical results indicate that the effect of C
t
variation on the predicted velocity field is limited, and the
velocity fields predicted by the RSM turbulence model
agree reasonably well with the experimental data for
C
t
> 0.4. Proper prediction of velocity fields (or momentum
transfer) is a prerequisite for adequate evaluation of Rey-
nolds analogy or C
t
effect on the temperature field of turbu-
lent reacting flows. This is because the turbulence scalar
transfers (and then the temperature field) in the flow are
obtained from the modeled momentum transfer using Rey-
nolds analogy. Based on the acceptable velocity fields, the
Reynolds analogy concept is assessed.
4.2. Temperature distributions
The temperature contours for C
t
= 0.85, 0.5 and 0.25 are
presented and compared with the experimental database in
Figs. 9–11, respectively. As expected, the temperature in
the recirculation region is relatively uniform due to strong
turbulent mixing. Intense chemical reaction takes place
around the envelope of the annular recirculation zone.
In comparison with the experimental data, the high
temperature region is shifted downstream for C
t
= 0.85,
Fig. 3. Axial velocity contours and flow path-lines, C
t
= 0.50.
Fig. 4. Axial velocity contours and flow path-lines, C
t
= 0.25.
Fig. 5. Axial velocity profiles along the combustor centerline.
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1257
Author's personal copy
significantly reduced and shifted upstream for C
t
= 0.25,
and best predicted with C
t
= 0.50. In short, the high tem-
perature region moves upstream and becomes smaller with
decreasing C
t
. This is because the turbulence transfer of
fuel into the airflow and then the chemical reaction are
accelerated as C
t
decreases.
In Figs. 9–11, it is found that the predicted temperature
in the high temperature region is higher than the measured
values. The maximum difference is about 170 K. The main
reason may be that the temperature was measured by a
0.25 mm diameter thermocouple, as mentioned earlier.
Owing to the radiation and conduction losses from the
thermocouple, the measurement error could exceed 100 K
over regions where the gas temperature was high and the
flow velocity was low [31].
The effect of C
t
on the predicted flame length is illus-
trated in Fig. 12. The flame region is represented by the
stoichiometric line of the mean mixture fraction
ð~
f¼0:0603Þin the upper of the figure, and by the OH
mole-fraction contour lines in the lower half. The effect
of C
t
on the flame length and region is obvious. As C
t
decreases from 0.85 to 0.25, both the flame length and
region are significantly reduced, and the flame length
decreases about three times from 342 to 110 mm.
The predicted temperature profiles along the combustor
centerline are compared with the experimental data in
Fig. 13, where the measurement error is about 5%. In the
upstream region (x<80 mm), the effect of C
t
is limited;
the numerical results for C
t
= 0.50 and 0.85 agree well with
the experimental data. In contrast, the effect of C
t
is appar-
ent in the downstream region. It is because in the upstream
region the fuel distribution or chemical reaction is mainly
determined by the location and size of the annular recircu-
lation zone, which is formed by complex flow interactions
among the central fuel jet, annular airflow and two recircu-
lation zones. That is, the flow is convection-dominated.
However, in the downstream region, the turbulent diffusion
or transfer plays a dominant role in the fuel spreading
away from the axis of symmetry, where the flow path-lines
are almost parallel to each other as shown in Figs. 2–4.In
Fig. 7. Turbulence kinetic energy profiles at sections, x= 60–240 mm.
Fig. 8. Turbulence shear stress profiles at sections x= 60–240 mm.
Fig. 6. Axial velocity profiles at sections, x= 10–240 mm.
1258 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
Fig. 9. Temperature contours, C
t
= 0.85.
Fig. 10. Temperature contours, C
t
= 0.50.
Fig. 11. Temperature contours, C
t
= 0.25.
Fig. 12. Variation of predicted flame length with C
t
.
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1259
Author's personal copy
Fig. 13,C
t
= 0.50 gives the best results although the pre-
dicted profile shows a peak in the middle portion, while
the measurements tend to be flat. The predicted maximum
temperature reaches 2120 K, while the measured value is
1950 K.
Fig. 14 presents the temperature profiles for C
t
= 0.85,
0.50 and 0.25 at five cross-sections, x= 52–233 mm. As
expected, at all sections, with C
t
decreasing the temperature
profiles become flatter, i.e., the fuel spreading becomes fas-
ter. The numerical results from C
t
= 0.50 agree reasonably
well with the experimental results, except for the regions
near the combustor wall. In these near-wall regions, the
temperature is under-predicted, which may suggest that
the fuel spreading is under-predicted in these local regions.
Poor performance is observed for C
t
= 0.25 at x=52and
233 mm, and C
t
= 0.85 at x= 52 mm. A strong effect of
C
t
is observed at all sections.
Variation of combustor wall temperature with C
t
is
shown in Fig. 15. The numerical results are compared with
the experimental data that have a measurement error of
2.5%. Unsurprisingly, the predicted wall temperature
increases as C
t
decreases. The results of C
t
= 0.40 show best
agreement with the measurements, although the wall tem-
perature is slightly over-predicted in the upstream region
and under-predicted in the downstream region. As noticed,
this C
t
value does not agree with the preferred value of 0.50
for the temperature prediction inside the combustor. This
may suggest that varying C
t
, instead of a constant value,
should be used in turbulent reacting flow simulations.
4.3. Species distributions
Water is one of the major products in propane–air com-
bustion. Fig. 16 presents the H
2
O mole-fraction profiles at
six sections and the results are compared with the experi-
mental data (which have a 5% measurement error). To
illustrate the main features of chemical reactions, the loca-
tions of the six sections are selected as, one across the cen-
tral recirculation zone, another passing through both
recirculation zones, the third cutting through the annular
recirculation zone, the fourth almost passing through the
stagnation point, and then followed by two downstream
sections (see Figs. 2–4).
The H
2
O profile becomes flatter as C
t
decreases, partic-
ularly for downstream sections, which is consistent with the
trends observed for the temperature profiles in Fig. 14.
Stronger effects of C
t
are found at downstream sections,
x> 112 mm, as well as in the central region at the upstream
section, x= 52 mm. The results for C
t
= 0.50 and 0.85 are
close to each other, and C
t
= 0.50 gives the best agreement
with the experimental data at most sections. The numerical
results with C
t
= 0.25 show large deviations from the
Fig. 13. Temperature profiles along the combustor centerline.
Fig. 14. Temperature profiles at sections x= 52–233 mm.
Fig. 15. Temperature profiles along the combustor wall.
1260 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
measurements at two downstream sections (x= 172 and
233 mm) and one upstream section (x= 52 mm).
Carbon dioxide is another major product in propane–air
combustion, while carbon monoxide is a major intermediate
species. The predicted results are presented and compared
with the measurements in Figs. 17 and 18, respectively.
The experimental error bars of 5% are also included in these
figures. As for the H
2
O profiles in Fig. 16, the predicted spe-
cies profiles show less variations along the radial direction
as C
t
decreases, particularly in downstream sections. The
effect of C
t
is significant at downstream sections and in some
local regions at upstream sections. The predicted results
with C
t
= 0.50 show close agreement with the experimental
database, except for section x= 112 mm.
At section x= 112 mm for C
t
= 0.50, although the con-
centration of CO is over-predicted and that of CO
2
is
under-predicted, the sum of the predicted CO and CO
2
agrees well with the sum of the measurements. This indi-
cates a delay in CO oxidization prediction at this section.
The results from C
t
= 0.85 and 0.25 show large discrepan-
cies from the measurements at downstream sections as well
as in some local regions at upstream sections, particularly
for the CO profiles.
Finally, it should be mentioned that in order to thor-
oughly assess the Reynolds analogy issue, numerical simu-
lations were also carried out with the eddy-dissipation
(EDS) and probability density function (PDF) combustion
models. A large amount of numerical results and figures
were generated. The trends and magnitudes of velocity,
temperature and species distributions are similar to those
obtained from the flamelet combustion model [32].
Although the results are not presented in this paper, the
optimized C
t
numbers are given in Table 1 for the purpose
of comparison. During optimization, C
t
number gradually
decreased from 0.85 to 0.25 with an interval of 0.1 in gen-
eral. Near the optimal value, an interval of 0.05 or even
0.01 was used. For temperature and species prediction
inside the combustor, the numerical results were compared
with and judged by the same experimental datasets, includ-
ing velocity, temperature and species distributions. For
wall temperature prediction, the optimal C
t
number was
judged mainly by the experimental temperature profile
along the combustor wall.
As shown in Table 1, the optimal C
t
for the temperature
and species prediction inside the combustion chamber (line
1) is the same for all three combustion models; however, it
(line 2) is different for the wall temperature prediction. This
indicates that the combustion model has some effect on the
near-wall temperature distribution.
4.4. Discussion
As shown in the above results, the optimal C
t
number
for the temperature and species prediction inside the
Fig. 16. H
2
O profiles at cross-sections, x= 22–233 mm. Fig. 17. CO
2
profiles at cross-sections, x= 22–233 mm.
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1261
Author's personal copy
combustor is 0.50 for all three combustion models. This
number is different from 0.25 in [17], 0.20 in [18] for kero-
sene-fired gas turbine combustors, and 0.20 for a cross-jet
flow simulation [20]. However, it is the same as in [19]
for a Rolls–Royce gas turbine combustor LES study.
All these examples reveal two important facts. First, the
optimized value of C
t
is lower than the traditionally used
value of 0.85. Second, the optimal value of C
t
is most
likely dependent on the combustor configuration and pos-
sibly the operating conditions. That is, a priori optimiza-
tion of C
t
is required in order to reasonably predict
temperature and species distributions inside combustors.
Obviously, this type of prior optimization is not practical
in the real world.
The reasons for the above observations may be three-
fold. First, theoretically, Eqs. (8)–(10) are only valid for
boundary layer flows, where the streamwise pressure gradi-
ent and source terms can be neglected. Certainly, its appli-
cation to complex turbulent reacting flows is questionable.
As stated earlier, a number of authors [7–10] have experi-
mentally found that this analogy cannot apply to disturbed
turbulent thermal boundary flows.
Second, the experimental values of C
t
(0.7) are
obtained from fully developed boundary or pipe flows
[2,3,12], and they may not be suitable for practical turbu-
lent reacting flows. Therefore, in the sense of the average
relative strength between the turbulence momentum and
scalar transfers, a low value of C
t
is a true fact and it
may vary with flow configurations.
Third, Eqs. (8)–(10) are based on the gradient-type dif-
fusion assumption which has been questioned by a number
of researchers, particularly for turbulent energy and heat
transfer [2]. Hinze [2] points out that both the gradient-type
diffusion caused by small-scale turbulence and the convec-
tive action of large-scale turbulent motion should be con-
sidered in turbulent scalar transportations. It may be
expected that the gradient-type diffusion approach is suit-
able for turbulent boundary flows; however, it is not suit-
able for complex flow fields inside combustion systems.
In summary, although the Reynolds analogy concept
has been extended to flow field simulations since the
1970s, for accurate prediction of scalar transfers in turbu-
lent reacting flows without prior optimization, this concept
should be improved and new approaches should be devel-
oped. It is authors’ wish that the outcome of this work
could stimulate R&D activities on turbulence scalar trans-
fers in numerical communities, and eventually solve indus-
trial problems, such as temperature pattern factors at the
exit of gas turbine combustors.
5. Conclusions
The effect of turbulent Prandtl/Schmidt number on the
flow field of a propane diffusion flame combustor with
the interior and exterior conjugate heat transfers has been
numerically studied. Presented and discussed in this paper
are some of the results obtained from a laminar flamelet
combustion model with turbulent Prandtl/Schmidt number
varying from 0.85 to 0.25. For completeness, the optimized
C
t
numbers for other two popular combustion models
(EDS and PDF) are also provided.
In comparison with the comprehensive experimental
database, it is found that the C
t
number has limited effect
on the velocity field. In contrast, it shows a strong effect
on the temperature and species fields, particularly down-
stream in the combustor. This is also true for the tempera-
ture profile along the combustor wall.
For the present combustor configuration and operating
conditions, the optimal C
t
for temperature and species pre-
diction inside the combustor is 0.5 for all three combustion
models, and it varies from 0.40 to 0.55 for the combustor
wall temperature prediction. With C
t
= 0.50, the velocity,
temperature and major species fields are reasonably well
predicted, except in some local regions.
Finally, the rationale and limitation of the Reynolds
analogy are discussed. For reliable temperature and species
prediction in turbulent reacting flows without tuning
Fig. 18. CO profiles at cross-sections, x= 22–233 mm.
Table 1
Optimal Prandtl and Schmidt number
Flamelet PDF EDS
T-chamber 0.50 0.50 0.50
T-wall 0.40 0.55 0.5–0.55
1262 L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263
Author's personal copy
turbulent Pr
t
and Sc
t
numbers, the Reynolds analogy con-
cept should be improved and new approaches should be
studied.
Acknowledgements
The authors are grateful to Dr. Bill Wallace for his valu-
able comments and suggestions during the preparation of
this paper. Also the authors want to express many thanks
to Miss Abdelmesih Gewana (co-op student) for her assis-
tance in the post-processing of numerical and experimental
results.
References
[1] O. Reynolds, On the extent and action of the heating surface for
steam boilers, Manchester Lit. Phil. 14 (1874) 7–12.
[2] J.O. Hinze, Turbulence, McGraw-Hill Book Company Inc., NY,
1987, pp.372–753.
[3] F.M. White, Heat and Mass Transfer, Addison-Wesley Publishing
Company, NY, 1988, pp. 320–641.
[4] M.V. Suraweera, D.J. Mee, R.J. Stalker, Reynolds analogy in high-
enthalpy and high-Mach-number turbulent flows, AIAA J. 44 (4)
(2006) 917–919.
[5] J. Bons, A critical assessment of Reynolds analogy for turbine flows,
Trans. ASME, J. Heat Transfer 127 (2005) 472–485.
[6] T.V. Jones, T.W. Walton, Reynolds analogy in film cooling, in: Tenth
International Symposium on Air Breathing Engines, Nottingham,
England, 1991, pp. 1263–1268.
[7] K.S. Choi, D.M. Orchard, Turbulence management using riblets for
heat and momentum transfer, Exp. Therm. Fluid Sci. 15 (1997) 109–
124.
[8] F. de Souza, J. Delville, J. Lewalle, J.P. Bonnet, Larger-scale coherent
structures in a turbulent boundary layer interacting with a cylinder
wake, Exp. Therm. Fluid Sci. 19 (1999) 204–213.
[9] J.C. Vogel, L.K. Eaton, Combined heat transfer and fluid dynamic
measurements downstream of a backward-facing step, J. Heat
Transfer 107 (1985) 922–929.
[10] E.D. John, J.O. Keller, Time-resolved gas temperature in the
oscillating turbulent flow of a pulse combustor pipe, Combust. Flame
80 (1990) 358–370.
[11] P.A. Libby, F.A. Williams, Fundamental aspects and a review, in:
P.A. Libby, F.A. Williams (Eds.), Turbulent Reacting Flows,
Academic Press, NY, 1994, pp. 1–61.
[12] N.R. Panchapakesan, J.L. Lumley, Turbulence measurements in
axisymmetric jets of air and helium, part 2, helium jet, J. Fluid Mech.
246 (1993) 225–247.
[13] C.L. Lubbers, G. Brethouwer, B.J. Boersma, Simulation of the
mixing of a passive scalar in a round turbulent jet, Fluid Dyn. Res. 28
(2001) 189–208.
[14] D.B. Spalding, Concentration fluctuations in a round turbulent free
jet, Chem. Eng. Sci. 26 (1971) 95–107.
[15] X.S. Bai, L. Fuchs, Sensitivity study of turbulent reacting flow
modeling in gas turbine combustors, AIAA J. 33 (10) (1995) 1857–
1864.
[16] M.K. Lai, CFD analysis of liquid spray combustion in a gas turbine
combustor, The IGTI Expo, Orlando, FL, 1997, 1997-GT-309.
[17] D.S. Crocker, D. Nickolaus, C.E. Smith, CFD modeling of a gas
turbine combustor from compressor exit to turbine inlet, The IGTI
Expo 1998, Stockholm, Sweden, 1998, 1998-GT-184.
[18] H. Kaaling, R. Ryden, Y. Bouchie, D. Ansart, P. Magre, C. Guin,
RQL combustor development including design, CFD calculations,
CARS measurements and combustion tests, in: Thirteenth Interna-
tional Symposium Air Breathing Engines, Chattanooga, TN, USA,
1997, ISABE 97-70697.
[19] S.M. Cannon, C.E. Smith, M.S. Anand, LES predictions of
combustor emissions in an aero gas turbine engine, in: Thirty-ninth
AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,
Huntsville, Alabama, USA, 2003, AIAA-2003-4521.
[20] G. He, Y. Guo, A.T. Hsu, The effect of Schmidt number on turbulent
scalar mixing in a jet-in-crossing flow, Int. J. Heat Mass Transfer 42
(1999) 3727–3738.
[21] I. Campbell, D.L. Logan, An experimental study of a combusting
flow past a confined bluff body, Combustion Institute – Canadian
Section, Halifax, Canada, 1997.
[22] J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics,
Springer-Verlag, NY, 2002, pp. 1–10.
[23] Fluent Inc., Fluent 6.2 Documentation, 10 Cavendish Court, Leba-
non, NH 03766, USA, 2006.
[24] L.Y. Jiang, I. Campbell, Turbulence modeling in a model combustor,
Can. Aeronaut. Space. J. 53 (2) (2007) 45–57.
[25] B.E. Launder, G.J. Reece, W. Rodi, Progress in the development of a
Reynolds-stress turbulence closure, J. Fluid Mech. 68 (1975) 537–566.
[26] N. Peters, Laminar diffusion flamelet models in non-premixed
turbulent combustion, Prog. Energy Combust. Sci. 10 (1984) 319–339.
[27] K.N.C. Bray, N. Peters, Laminar flamelets in turbulent flame, in:
P.A. Libby, F.A. Williams (Eds.), Turbulent Reacting Flows,
Academic Press, NY, 1994, pp. 609–638.
[28] G. Stahl, J. Warnatz, Numerical investigation of time-dependent
properties and extinction of strained methane- and propane–air
flamelets, Combust. Flame 85 (1991) 285–299.
[29] G.D. Raithby, E.H. Chui, A finite-volume method for predicting a
radiant heat transfer in enclosures with participating media, J. Heat
Transfer 112 (1990) 415–423.
[30] NIST, NIST-JANAF Thermochemical Tables, fourth ed., Washing-
ton DC, 1998.
[31] J.P. Sislian, L.Y. Jiang, R.A. Cusworth, Laser Doppler velocimetry
investigation of the turbulence structure of axisymmetric diffusion
flames, Prog. Energy Combust. Sci. 14 (2) (1988) 99–146.
[32] L.Y. Jiang, I. Campbell, Prandtl/Schmidt number effect on temper-
ature distribution in a generic combustor, in: The XVIII International
Symposium Air Breathing Engine Conference, ISABE-2007-0178,
September 2–7, 2007, Beijing, China.
L.-Y. Jiang, I. Campbell / International Journal of Heat and Mass Transfer 51 (2008) 1251–1263 1263
