DRAFT
IMECE '02
ASME International Mechanical Engineering Congress & Exposition
November 17–22, 2002, New Orleans, Louisiana
Proceedings of IMECE'02
ASME International Mechanical Engineering Congress & Exposition 2002
November 17-22, 2002, New Orleans, Louisiana, November 17-22, 2002
IMECE2002-xxxxx
IMECE 2002
33918
T U R B U L E N C E - R A D I A T I O N INTERACTIONS IN FLAMES: A C H A O T I C - M A P
BASED FORMULATION
Ying Xu 1
J. M. McDonough 2
M. Pinar MengSc: 3
Department of Mechanical Engineering
University of Kentucky
Lexington, KY 40506-0108
Department of Mechanical Engineering
University of Kentucky
Lexington, KY 40506-0108
Department of Mechanical Engineering
University of Kentucky
Lexington, KY 40506-0208
ABSTRACT
Pe
qR
R
Re
~'
Sc
T
u
U
v
In this paper we report initial efforts in developing largeeddy simulation (LES) subgrid-scale (SGS) models capable of
treating turbulence-radiation interactions in sufficient detail to
permit calculation of radiation intensity fluctuations on small
scales. These models are constructed with a fluctuating component consisting of a discrete dynamical system (chaotic map)
and are thus completely deterministic. We present an outline
of the development of this formulation and then employ experimental data to generate large-scale behavior permitting what
might be viewed as part of an a priori test of the SGS model.
We display spatially extensive instantaneous fluctuating temperatures produced by the model as well as time series of fluctuating
intensity calculated from the radiative transfer equation at several heights in a pool fire. We conclude that such results are
physically realistic (and very efficiently computed) and warrant
continued investigations, but we have at this time not yet completely validated the approach due to lack of detailed laboratory
data.
a
fl
#
~,
p
r
NOMENCLATURE
CB
Di
FB
Gr
I~,
k
L
p
specific heat
binary diffusion coefficient
body force
Grashof number
spectral intensity
thermal conductivity
length scale
pressure
P~clet number
thermal radiation
specific gas constant
Reynolds number
radiation propagation direction
chemical source
temperature
u velocity component
referenced velocity magnitude
v velocity component
mass fractions of species i
thermal diffusivity
thermal volumetric expansion coefficient
anisotropy correction
dimensionless temperature
absorptivity
dynamic viscosity
kinematic viscosity
density
time scale
INTRODUCTION
Most industrial scale flames are strongly radiating and
turbulent in nature; they can be viewed as time dependent dynamical systems. Turbulence-radiation interactions
(TRI) need to be accounted for thoroughly in such flames in
order to include the underlying physical mechanisms. Numerical modeling of turbulent diffusion flames requires at
least qualitatively accurate simulation of small-scale fluctuations of velocity, temperature, and concentration fields,
in addition to the corresponding large-scale values. Most
1Ph.D Student
2Professor; jmmcd@uky.edu
3Professor; mengucQengr.uky.edu
1
Copyright (~) 2002 by ASME
of the time-dependent information is lost if flow field simulations are not carried out in detail. Even though some
modern techniques such as the direct numerical simulation
(DNS) are mathematically rigorous and do yield a high level
of accuracy, they are not computationally feasible for application to complex practical systems. Instead, it is preferable to "model" the small-scale fluctuations and solve for
the large-scale parameters accurately. This strategy (largeeddy simulation, LES) allows us to save significant amounts
of CPU time, and brings us to the realm of simulating turbulent flames of practical importance.
The first attempt to include effects of radiation on
flow and temperature fields in a turbulent environment was
made by Townsend [1]. Later studies included those by
Cox [2], Tamanini [3], Kabashnikov and Myasnikova [4],
Grosshandler and Joulain [5], Fischer et al. [6], Fischer
and Grosshandler [7], as well as Faeth and his students (see
Gore et al. [8] [9], and Faeth et al. [10] for an overview).
One of the earliest accounts of TRI in the heat transfer
literature is that by Song and Viskanta [11] who investigated a turbulent premixed flame inside a two-dimensional
furnace. More recently, Gore et al. [12] and Hartick et
al. [13] solved the problem for diffusion flames using a k-a
model. They discussed a closure for the governing equations and correlated gas radiative properties using probability density function (PDF) of mixture fraction and total enthalpy. Mazumder and Modest [14] used a velocitycomposition PDF method to investigate TRI; the so-called
P D F / M o n t e Carlo method they developed was accurate,
yet computationally demanding. Li and Modest [15] used
a similar approach; they assumed the species concentrations and temperature/enthalpy are the random variables
and applied this to a methane-air diffusion flame.
It is now widely believed that turbulence in a flow field
is not random, but deterministically chaotic in light of the
seminal work of Ruelle and Takens [16]. Moreover, it is
possible to model the turbulent fluctuations via numerical
time series calculated from a linear combination of chaotic
maps, with different parameters and weights (see, e.g., McDonough et al. [17]) with the maps derived from the governing equations (see McDonough and Huang [18], and Yang
et al. [19]). Here, we will present an analysis that investigates TRI using a system of chaotic maps. Fluctuations
of absorption coefficient will be calculated as a function of
temperature modeled in terms of these maps, and the emission from flames will be quantified in terms of parameters
of the chaotic maps.
The concept of chaotic-map based TRI has been discussed before by Mengfi~ and McDonough [20], McDonough
et al. [21], Mengfi~ and McDonough [22], and details of the
chaotic map formulation were reported elsewhere (Mukerji
et al. [231; Hylin and McDonough [24]). Our focus in this
paper is on the effect of chaotic map induced fluctuations
on radiation emission at various locations in a flame, which
can be measured experimentally.
We follow this introduction with a fairly detailed section
containing the mathematical formulation of the governing
equations, an overview of typical treatments of turbulenceradiation interactions, and the formalism based on chaotic
maps (discrete dynamical systems, DDSs) that we will employ. We then present and discuss results of applying DDSs
in the context of a pool fire flame studied by Weckman and
Strong [25]. Conclusions drawn from this initial study are
then presented, the main one being that the use of chaotic
maps as subgrid-scale (SGS) models provides a rational
approach to efficiently introducing physically-realistic behavior on subgrid scales, but considerable research is still
needed to develop this into a predictive tool.
FORMULATION AND ANALYSIS
We begin this section with introduction of the system
of governing equations needed for the complete discription
of combustion physics. Then we provide a brief treatment
of past approaches to accounting for turbulence-radiation
interactions, indicating their shortcomings. Following this
we state and justify the assumptions to be employed for the
present study and present the chaotic map formalism.
Governing Equations
The equations governing most combustion processes
consist of the Navier-Stokes (N.-S.) equations in a form appropriate for non-constant density (but low-Mach number)
flows, a thermal energy equation and species concentration
equations, along with an equation of state. These can be
expressed as
p, + v . ( p u ) = 0,
(la)
D(pU)
Dt-- - V p + # A U + FB,
(lb)
DT
p C p - ~ - = k A T + V "qR + So,
(lc)
D(pYi)
D--T- - V . Di~7(pYi) + &i,
i = 1,..., Ns.
(ld)
In these equations U K (u, V) T is the two-dimensional velocity vector; p, p, T are, respectively pressure, density
and temperature related through an equation of state, say
p = p R T , with R being the specific gas constant. Tile Yi
are mass fractions of species i, and cbi are corresponding
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Copyright (~) 2002 by ASME
species production terms. FB, Sc and qR are, respectively
body force, chemical source and thermal radiation contributions to momentum and thermal energy, and #, k, D~ and
Cp are transport and thermodynamic properties: (dynamic)
viscosity, thermal conductivity, binary diffusion coefficient
and specific heat, respectively. Finally, the differential operators D~, V and A are the usual substantial derivative,
gradient and Laplacian, respectively, in a chosen coordinate system. Further details can be found in, e.g., Libby
and Williams [26].
the chemical source term in Eq. (lc), but report that their
approach is not computationally feasible for anything but
the simplest kinetic models. Hence, no details of effects of
species concentration can be deduced, although more global
effects of TRI are produced. It is important to recognize,
however, that there is no true instant-to-instant interaction
between the velocity field and the temperature field, and
thus no detailed turbulence-radiation interaction. Such details cannot be produced when the governing equations are
averaged (RANS) or filtered (usual LES), and then closed
with statistical models representing phenomena on precisely
the scales at which the interactions must occur.
Treatments for TRI
At present the main alternatives available for analysis
of radiating turbulent flows are DNS, RANS and LES, with
the latter two often coupled with PDF methods, specifically
to treat chemistry and/or radiation (see, e.g., [27]). It is
not hard to check that DNS is not possible for simulation of
large-scale practical problems due to the O(Re 3) arithmetic
operation scaling for fluid flow alone. Formally, LES and
RANS are similar, although the details of LES are more
rigorous. Furthermore, the ~ O ( R e 2) arithmetic required
by LES will be feasible on the next generation of parallel
supercomputers, and probably LINUX clusters, up to Re at
least 106 .
The starting point for treatment of radiation is the
radiation transfer equation (RTE) consisting of the radiant energy balance for a non-scattering, absorbing-emitting
gas confined to a solid angle d~t within a spectral interval
[I, 1 + d l ] :
(K" V)Ix = ~X[/bx(T) -/~],
Chaotic Map Subgrid-Scale Models
The principal difficulty in all approaches to turbulence
(except DNS) is dealing with the "closure problem." What
seems not to have been recognized is that the most difficult
(and least sound from a pure mathematical viewpoint) aspects of this problem would never have occurred had we
not averaged (or, in the LES case, filtered) the governing equations. This process results in a need for statistical models that inherently suffer from a basic mathematical inconsistency: the mapping from physics to statistics
is many-to-one, implying that it is noninvertible. Yet, it
is precisely an attempt to invert this mapping that is embodied in every RANS approach, and in most (but not all)
LES techniques; namely, attempt to use statistics (velocity
correlations, velocity-temperature correlations, etc.) to deduce physics (velocity, temperature and composition fields).
Thus, the context into which we will introduce chaotic
map models is a modification of LES based on the following key ideas: i) filter solutions rather than equations; ii)
model physical variables instead of their statistical correlations; iii) directly use model results to enhance the (deliberately) under-resolved large-scale solutions of LES, rather
than discarding these results.
Within this context we can still employ the usual LES
decomposition:
(2)
where ~is the radiation propagation direction; ~x is absorptivity at wavelength I, and Ix is the corresponding spectral
intensity. The formal solution to Eq. (2) is (Viskanta and
Mengfi~ [28]; Modest [29])
Ix(s, t) = Ioe- fo~~(s',t)ds'+
Q(x,t)=q(x,t)+q*(x,t),
~0 8
(8', t)Ibx (T(< t)) el.'s
'
(3)
x E F ~ ~, d = 2 , 3
(4)
with q representing large (computed) scales, and q* denoting the small scales that must be modeled, of any dependent variable vector Q = (Q1,Q2,"" ,QN,) T, where Nv is
the number of variables. In earlier work (e.g., McDonough
and Bywater [30] [31]) components of q* were obtained as
solutions to low-dimensional systems of ordinary differential equations (ODEs) corresponding to high-wavenumber
Fourier modes arising from a Galerkin procedure applied to
the small-scale system of additively split partial differential
equations (PDEs). But beginning with McDonough et al.
where I0x is radiation intensity incident at the boundary
(s'= 0) of the medium. It should also be noted that ~x depends on temperature and species concentrations in a nontrivial way, so Eq. (3) is considerably more complicated than
is explicitly apparent. In turn, this shows that application of
Reynolds averaging to Eq. (2) leaves the entire right-hand
side unclosed. Li and Modest [15] describe the use of a
PDF formalism to treat both the radiation source term and
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Copyright (~) 2002 by ASME
1
[32] the solution ansatz
q* = A ( M ,
ut + uuz + vu u = ~ e A u ,
(6a)
1
Gr
vt + uv~ + vv u = ~eAV - ~e20,
(6b)
Ot + uO~ + vOu = ~--~eAO.
(6c)
(5)
was introduced. Here, A is an amplitude factor computed via Kolmogorov scaling (see, e.g., Frisch [33]; ~ is
an anisotropy correction obtained from scale similarity, as
employed in the dynamic SGS models of Germano et al.
[34]; the factor M is analogous to Kolmogorov's stochastic variable, but in the present context is a deterministic
discrete dynamical system.
These ideas are implicitly contained in [32], and much
of the theory required for modeling subgrid-scale physical
variables is presented in [24]. Furthermore, many of these
ideas were applied specifically to the TRI problem in [22].
In these earlier studies, chaotic maps having no particular
connection to the appropriate physics were employed (to
some extent successfully) as subgrid-scale models. More
recently, McDonough and Huang [18] [35] have devised a
method whereby discrete dynamical systems (i.e., chaotic
maps) can be derived directly from the governing PDEs of
essentially any physical system. To date these have been
studied in a comparison of free and forced convection (McDonough and Joyce [36]) and in association with three different H2-O2 and H2-air kinetic mechanisms (McDonough
and Zhang [37] [38]; Zhang et al., [39]). But in all of these
studies emphasis has been placed on the time series of SGS
behavior at a single spatial location; moreover, the bifurcation parameters needed to evaluate the DDSs were chosen
somewhat arbitrarily.
In the present study we employ experimental data from
a spatially extended set of measurements for a pool fire to
automatically set required bifurcation parameters over the
entire physical domain of the experiments. To keep the
problem at a relatively simple level we will not model Eq.
(ld) for the species mass fractions. Hence, the chemical
source term in Eq. (lc) will be ignored (but see the above
cited references for treatments of this). Furthermore, because the pool fire of the Weckman and Strong [25] data is
burning methanol, and soot formation will be at a relatively
low level, we also ignore the radiation source term in Eq.
(lc) - - despite its general importance. This permits us to
focus on the effects of fluctuating temperature in producing
radiation intensity fluctuations without needing to separate
the feedback from thermal radiation via the radiation source
term in Eq. (lc).
Thus, as in [36] we employ the following system of 2-D
equations, utilizing the Boussinesq approximation to obtain the form of the body force term in Eq. (lb) and Leray
projection [40] to remove the pressure gradient from this
equation. This produces the following system written in
dimensionless form:
Here G r is the Grashof number, R e the Reynolds number
and P e the P~clet number:
G r _=
I~g6TL 3
u~~
,
R e =-
UL
,
Pe ~
V
UL
,
(7)
C~
where a is thermal diffusivity, /3 is thermal volumetric expansion coefficient, and u is kinematic viscosity.
The velocity components (u, v) have been scaled with
a reference velocity magnitude U, and distances are scaled
by a fixed length scale L. The dimensionless temperature
is defined as
T-To
0 =- Ti- ~ 0 '
(8)
where To and T1 are reference temperatures, and ~T = T1 To. The subscripts x, y, t denote partial differentiation
with respect to the spatial coordinates (x,y) and time t
respectively, and A is the Laplacian.
We apply the Galerkin procedure to this system and
represent all dependent variables by Fourier series as
(9a)
u(x, y, t) = E ak (t)~ok (X, y),
k
(9b)
v(x, y, t) = E b} (t)~k (X, y),
k
O(x, y, t) = E
(9C)
ck(t)cpk(X, y).
k
Application of the Galerkin procedure consists of substituting Eqs. (9) into Eqs. (6), and then computing inner products with each of the countably infinite (pks. The result of
this (see [36]) is reduced to
/~+ A(1)a 2 + A ( 2 ) a b -
+ B(1)b2 + B(2)ab _
C[k[2 b _
d 4- C(1)ac 4- C(2)bc 4
Clkl 2
R e a,
Re
Gr
~ 7c'
Clkl~
~e c,
(10a)
(10b)
(10c)
Copyright @ 2002 by ASME
by retaining only a single Fourier mode. Eqs. (10a, b) have
been treated in detail in [18] without the buoyancy term
on the right-hand side of (10b). The process introduced
in [36] consists of a forward Euler numerical integration of
the ODEs (10a,b), with backward Euler applied to (10c),
leading to
a (~+1) = 81a (n) ( 1 - a (n)) -'Tla(n)b (n),
(lla)
b(~+1) : 82b (~) ( 1 - b (~)) - 7 2 a ( n ) b (n) +aTC(n),
(llb)
It should be noted that Eq. (11c) is linear in c, implying that unless a reference level is prescribed, the DDS
will either approach zero or infinity after sufficiently long
time. This is remedied with the term Co, which is set by
the high-pass filtered resolved-scale solution in the context
of a complete LES implementation. In the present study, it
is assigned a fixed value.
Choosing Parameter values
In the preceding subsection, we introduced numerous
bifurcation parameters. Here, we describe the assignment
of their values• Since natural convection is important in
a pool fire, aT is assigned the value 0.06 rather than 0
used in the case of forced convection. In addition, co = 0.1
and ~T : 0.5 are chosen• The relation of 81 and 82
to the Reynolds number gives us the rationale to choose
8 = 81 = 82 such that 8 increases in radial and axial directions. Values of '71, '72, ~uT and ")'vT are related to rescaled
Uy, vx, Ty and Tx, respectively• T h e absorption coefficient
n is determined by using Planck-mean absorption coefficients of CO2 and H20 (see [29])• The ranges of absorption
coefficients for C02 and H20 in the current temperature
field are 0.146--0.324cm -1 and 0.031--0.078cm -1 respectively. Hence, we estimated n = O.lcm -1 as the large-scale
absorption coefficient of mixture. We also assume the absorption coefficient is a function only of temperature, so
the small-scale absorption coefficient can be determined by
fluctuating temperature. Therefore, radiation intensity is
computed by directly integrating Eq. (2).
c (n+l) : (1-- ~uTa(n) -- ~vT b(n)) / ( 1 + S T ) +CO- ( l l c )
We remark that Eqs. (11) can be viewed as the simplest
possible shell model (see Bohr et al. [41]); it consists of a
single Fourier mode for each original P D E (similar to the
well-known Lorenz equations [42]). But in contrast to those,
the single mode is arbitrary, with the mode number embedded in the bifurcation parameters.
One might question the accuracy, even validity, of a
single mode representation, but we emphasize that these
representations are local. Hence, one should expect that
at most a few modes would be needed to capture the local physics. Indirect evidence of this can be seen in [23]
and more recently, and more directly, in Yang et al. [19]
in which Eqs. ( l l a , b) are directly fit to high-pass filtered
(hence, analogous to subgrid scales) experimental data for
two velocity components measured via LDV.
Considering the physical meaning of the various 8s and
7s in Eqs. (11), it is shown in [18] that both ~l and 82 are
related to the Reynolds number via
~=4
1
Re
]'
RESULTS AND DISCUSSION
Our main overall objective in this research program is
to devise a high-fidelity model for the small-scale behavior of radiating turbulent flames. Such a model will allow
us to predict the fundamental physical and chemical phenomena that take place in a flame, and availability of such a
model is likely to allow us to control such flames more effectively. To this end, we propose to represent the small-scale
fluctuations with chaotic maps, as discussed above. Here,
we will present a series of results that model temperature
and radiation intensity fluctuations as observed outside the
flame• We will discuss the affect of chaotic map parameters on these results and comment on the robustness of the
approach.
Below, we will present fluctuating temperature contours
for two different amplitude factors (see Eq. 5). Physically,
these factors are simply the magnitude of the energy contained in the unresolved scales, local in space and time.
Using these factors, we obtain the temperature fluctuations
within the entire domain. After that, we calculate the radi-
i=1,2,
and since we have based our derivations on a single wavevector k it is natural to set j31 = j32. By comparing Eqs. (10)
and (11) one sees that
8T =
TCIkl 2
Pe '
also, aT
vGr
Re 2
In these equations, C is a normalization constant (set to
unity in the present case), and 7 is a time scale. The various
7s all correspond to gradients of the velocity vector and
the temperature• These are obtained from the Galerkin
triple products that produce the coefficients A (i), A (2), etc.,
appearing in Eqs. (10) in combination with numerical time
step parameters to produce 71, 72, 7~T, %T of Eqs. (11).
5
Copyright (~) 2002 by ASME
~o
ation intensity time series that can be measured at different
flame heights. The intensity fluctuations are a direct manifestation of radiation-turbulence interactions.
In a typical turbulent flow field simulation, small-scale
calculations are the most time consuming and usually the
least accurate. If these calculations can be replaced with
reasonably accurate ones that can be obtained faster and
more reliably, more accurate flow field simulations can be
realized. This requirement is particularly important for
chemically reacting flows, where radiation, turbulence and
chemical kinetics interactions need to be accounted for.
In order to perform the chaotic map based calculations
in lieu of more detailed differential equation based smallscale simulations, we need to have spatially extended resolved scale velocity and temperature fields. These can be
obtained from any available computational or experimental
source, e.g., resolved-scale LES results. In this study, we
chose to use data from Ref. [25].
Time-averaged axial and radial velocity data reported
in [25] were read directly from the figures and then interpolated. Readings from the experimental data plots were
obtained on a 17 × 10 grid; then via (linear) interpolation
the grid resolution was increased to 33 × 19. The corresponding velocity profiles are depicted in Fig. 1. The same
approach was carried out for the mean temperature profile,
and plotted in Fig. 2a. By obtaining the entire velocity and
temperature fields, we are able to estimate the velocity and
temperature gradients at every location in the flame. These
gradients are directly used to calculate the bifurcation parameters, ~ , '~2, ~uT, ~vT at every point of the computational grid as indicated in the previous section.
Results for the instantaneous fluctuating temperature
are displayed in Figs. 2b and 2c. This is the first time these
maps (see Eqs. 11) have been used in a spatially extended
simulation. In the past, time series were constructed only at
single points. This provides a proof of concept of the overall
discrete dynamical systems approach to subgrid-scale modeling, and the results presented here can be viewed as a first
step in what is often termed a priori testing of SGS models.
Here, however, we have employed experimental data rather
than DNS results.
Noting that the Reynolds number, defined based on
unit height, is increasing along the flame axis, we choose
a value of/313(=/31= / 7 2 ) , and then (linearly) increase it along
the flame. Figure 3 depicts fluctuating intensity time series
as computed at different heights in the flame. Four different
cases are plotted, corresponding to different ranges of the
bifurcation parameter/3. Fig. 3a shows that for low/3 values, (i.e., low Re) changing from 2.7 at the base to 2.9 at the
flame tip produces no fluctuations in intensity at any flame
height. Fig. 3b has a higher range, which varies from 3.0 at
the flame base to 3.2 at the flame tip. The corresponding
~8
~6
E
N
~o
8
e
4
2
-le
-12
~
~
0
Radius, r
4
~
12
4
e
12
[cml
18
16
E
12
~1o
e
6
4
2
-16
-~2
-a
-4
o
Radius, r
le
[cm]
Figure 1. RECONSTRUCTED AXIAL AND RADIAL MEAN VELOCITY CONTOURS FROM EXPERIMENTAL DATA OF WECKMAN AND
STRONG [25]: (a) AXIALVELOCITYCOMPONENT;(b) RADIALVELOCITY COMPONENT
time series for intensity display a periodic behavior throughout the flame. If the range of/3 is increased further from
3.4 (base) to 3.6 (tip), quasiperiodic intensity fluctuations
are observed, as seen in Fig. 3c. Finally, if the/3 range is
from 3.6 (base) to 3.8 (tip), the behavior is chaotic (see Fig.
3d). This change in qualitative behavior corresponds to the
well-known Ruelle and Takens [16] bifurcation sequence.
In Figs. 3, the level of fluctuations in radiation intensity is higher at the base than at the tip of the flame. The
reason is that the width across which intensity is measured,
is greater at the base. In these figures, we also show the
mean radiation intensity (in red) based on the mean temperature and absorption coefficient values. Note that, if
turbulence-radiation interactions are accounted for, the resulting intensities are larger. This result is expected and
consistent with those published earlier (see e.g., [10], [11],
[15]). It should be understood, however, that the exact
effect of TRI on mean radiation intensity is a function of
flame conditions considered, and may vary from one flame
to another.
Another parameter that affects the extent of TRI is the
magnitude of small-scale fluctuations in the turbulent flow
field, which is a function of the actual flow conditions. Here,
however, we investigate the affect of fluctuations by simply
6
Copyright (~) 2002 by ASME
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Figure 2. TEMPERATURE CONTOURS; (a) RECONSTRUCTED FROM
T H E EXPERIMENTAL DATA OF REF. [25]; (b) FROM CHAOTIC MAPS,
A M P L I T U D E FACTOR 0.36; (c) FROM CHAOTIC MAPS, A M P L I T U D E
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Vv ~"~" ! v-y-V,v-Trvvv,, v"yv,'"Q{y,'v,y", ....... "v ,
(o)
m •
0.380
0,385
•
0.395
0.390
,Scaled Time
adjusting the amplitude factor A given in Eq. (5).
Figure 4 provides a comparison of intensity of fluctuations for two different values of the amplitude factor A
in Eq. (5). We note that in a complete LES this factor
would be set automatically utilizing the approach described
by McDonough [43] in conjunction with the scale similarity hypothesis employed in dynamic SGS models (see [34]).
But here we employ experimental data, so the amplitude
has been set somewhat arbitrarily simply to allow assessment of effects on intensity fluctuations when it is changed.
Part (a) of Fig. 4 displays results at four locations through
the flame with an amplitude of 36% of the mean applied to
temperature fluctuations. This is a repeat of Fig. 3(d). Fig.
4(b) presents results for only two locations (because amplitudes are significantly higher, and plotted curves overlap
excessively if all four are displayed), calculated exactly as
|
(Arbitrary
0.400
Units)
Figure 3. TIME SERIES OF RADIATION INTENSITY ALONG FLAME
AXIS; (a) FOR ,8 RANGE OF 2.7 (AT FLAME BASE) TO 2.9 (AT FLAME
TIP); (b) ~ RANGE 3.0--3.2; (c) ,/3 RANGE 3.4--3.6; (d) /~ RANGE 3.6-3.8.
for part (a) except the amplitude is set at 75%. These time
series are taken from detailed results as shown in Figs. 2(b)
and (c) for a single instant of time.
Also shown in Figs. 4 are the mean intensities (shown
in red) calculated from data corresponding to Fig. 2(a),
i.e., with no turbulent fluctuations. It is clear from the
figures that although the temperature fluctuates about its
mean (not shown), the intensities fluctuate about a different
(higher) mean value. This is a consequence of the nonlinear
7
Copyright (~) 2002 by ASME
F
,e
F
ficult to provide a complete a priori test. Nevertheless, the
results displayed herein show significant promise for this
DDS approach to provide high-fidelity models, not only for
LES (and possibly RANS), but also for use in real-time
control strategies where very high-speed computations are
necessary. We remark that run times for results presented
herein for the complete 33 × 19 grid were a fraction of a
second on a workstation with only a 260 MHz clock.
This approach can be used easily, and in a
computationally-economical way, to determine the effect
of local turbulence-radiation interactions in chemically reacting flames, and will allow more detailed studies of
turbulence-radiation-chemical kinetics interactions. Extension of this methodology will yield prediction of soot formation variations in industrial flames based on the input
flame structure properties. Such a computational tool is
very much required for the development of smart adaptivecontrol modalities.
8DQO~
h=14cm
gOOD
8000!
6OO@
h=lgcm
4OQO
REFERENCES
0.3~
0.38,5
0.390
0.395
0.400
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Scaled Time (Arbitranj Units)
Figure 4. TIME SERIESFOR RADIATION INTENSITY CORRESPONDING
TO TWO DIFFERENT AMPLITUDE FACTORS: (a) 0.36 (SEE FIG. 2B); (b)
0.75 (SEE FIG. 2C).
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CONCLUSIONS
In this paper we have presented a new approach to SGS
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Copyright (~) 2002 by ASME
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