turbulence-radiation interactions in flames: a chaotic-map ...acfd/asme-xumcd.pdfscales. these...

Proceedings of IMECE'02 ASME International Mechanical Engineering Congress & Exposition 2002

November 17-22, 2002, New Orleans, Louisiana, November 17-22, 2002

IMECE 2002 33918

TURBULENCE-RADIATION INTERACTIONS IN FLAMES: A CHAOTIC-MAP BASED FORMULATION

Ying Xu 1 Department of Mechanical Engineering

University of Kentucky Lexington, KY 40506-0108

J. M. McDonough 2 Department of Mechanical Engineering


M. Pinar MengSc: 3 Department of Mechanical Engineering


IMECE '02 ASME International Mechanical Engineering Congress & Exposition

November 17�22, 2002, New Orleans, Louisiana

IMECE2002-xxxxx

ABSTRACT In this paper we report initial efforts in developing large-

eddy 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 sev- eral 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.

NOMENCLATURE CB specific heat Di binary diffusion coefficient FB body force Gr Grashof number I~, spectral intensity k thermal conductivity L length scale p pressure

1Ph.D Student 2Professor; [email protected] 3Professor; mengucQengr.uky.edu

1

Pe P~clet number qR thermal radiation R specific gas constant Re Reynolds number ~' radiation propagation direction Sc chemical source T temperature u u velocity component U referenced velocity magnitude v v velocity component

mass fractions of species i a thermal diffusivity fl thermal volumetric expansion coefficient

anisotropy correction dimensionless temperature absorptivity

# dynamic viscosity ~, kinematic viscosity p density r 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. Nu- merical 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

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 prefer- able to "model" the small-scale fluctuations and solve for the large-scale parameters accurately. This strategy (large- eddy 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 a t tempt 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 investi- gated 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 proba- bility density function (PDF) of mixture fraction and total enthalpy. Mazumder and Modest [14] used a velocity- composition PDF method to investigate TRI; the so-called PDF/Monte 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., Mc- Donough 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 inves- tigates 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

2

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 turbulence- radiation 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) D t - - - - V p + # A U + FB, (lb)

D T p C p - ~ - = k A T + V "qR + So, (lc)

D ( p Y i ) D--T- - V . Di~7(pYi ) + &i, i = 1 , . . . , N s . (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


species production terms. FB, Sc and qR are, respectively body force, chemical source and thermal radiation contribu- tions 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 op- erators D~, V and A are the usual substantial derivative, gradient and Laplacian, respectively, in a chosen coordi- nate system. Further details can be found in, e.g., Libby and Williams [26].

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(Re 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) -/~], (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'+

~0 8 s (8', t)Ibx ( T ( < t)) el.' ' (3)

where I0x is radiation intensity incident at the boundary (s'= 0) of the medium. It should also be noted that ~x de- pends on temperature and species concentrations in a non- trivial 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

3

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.

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) as- pects 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 at tempt to invert this mapping that is em- bodied in every RANS approach, and in most (but not all) LES techniques; namely, at tempt to use statistics (velocity correlations, velocity-temperature correlations, etc.) to de- duce 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 (delib- erately) under-resolved large-scale solutions of LES, rather than discarding these results.

Within this context we can still employ the usual LES decomposition:

Q ( x , t ) = q ( x , t ) + q * ( x , t ) , x E F ~ ~, d = 2 , 3 (4)

with q representing large (computed) scales, and q* denot- ing 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.


[32] the solution ansatz

q* = A ( M , (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 stochas- tic 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 (Mc- Donough 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:

4

1 ut + uuz + vu u = ~ e A u , (6a)

1 Gr vt + uv~ + vv u = ~eAV - ~e20, (6b)

Ot + uO~ + vOu = ~--~eAO. (6c)

Here Gr is the Grashof number, Re the Reynolds number and P e the P~clet number:

I~g6TL 3 U L U L Gr _= u~ ~ , Re =- , P e ~ , (7)

V 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 - T o 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

u(x, y, t) = E ak (t)~ok (X, y), (9a) k

v(x , y, t) = E b} (t)~k (X, y), (9b) k

O(x, y, t) = E ck(t)cpk(X, y). (9C) k

Application of the Galerkin procedure consists of substitut- ing 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 - Clkl 2 Re a, (10a)

+ B(1)b2 + B(2)ab _ C[k[2 b _ Gr (10b) Re ~ 7c'

Clkl~ (10c) d 4- C(1)ac 4- C(2)bc - ~e c,

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 (~)) -72a(n)b (n) +aTC(n), ( l lb )

c (n+l) : (1-- ~uTa(n) -- ~vT b(n)) / ( 1 + S T ) +CO- ( l lc )

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 PDE (similar to the well-known Lorenz equations [42]). But in contrast to those, the single mode is arbitrary, with the mode number embed- ded 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 ] ' i = 1 , 2 ,

and since we have based our derivations on a single wavevec- tor k it is natural to set j31 = j32. By comparing Eqs. (10) and (11) one sees that

TCIkl 2 vGr 8 T = Pe ' also, aT 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

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 d i rec- tions. Values of '71, '72, ~uT and ")'vT are related to rescaled Uy, vx, Ty and Tx, respectively• The 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).

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 effec- tively. 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-


ation intensity time series that can be measured at different flame heights. The intensity fluctuations are a direct man- ifestation 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 small- scale 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 inter- polated. 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 t ime 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 =/72) , 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

6

~o

~8

~6

E

N

~o

8

e

4

2

- l e - 12 ~ ~ 0 4 ~ 12

Radius, r [ cm l

1 8

1 6

E 1 2

~1o e

6

4

2

- 1 6 - ~ 2 - a - 4 o 4 e 1 2 l e

Radius, r [ cm ]

Figure 1. RECONSTRUCTED AXIAL AND RADIAL MEAN VELOC- ITY CONTOURS FROM EXPERIMENTAL DATA OF WECKMAN AND STRONG [25]: (a) AXIAL VELOCITY COMPONENT; (b) RADIAL VELOC- ITY COMPONENT

time series for intensity display a periodic behavior through- out 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 re- sulting 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


~o,

m'

m~

_ _ l a '

4~

2oJ

e,

4:

r~

-ul,

- ~ -8 4 0 4 8 12 16 ~ K

Radius, r [can]

- ~ 2 - e -4 o 4 e 1 2 1 ~

I~zdlus, • [cxnt

-12 ~ -,4 0 4 8 12 1£

aa(l~*, r [cm]

Figure 2. TEMPERATURE CONTOURS; (a) RECONSTRUCTED FROM THE 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, AMPLITUDE FACTOR 0.75.

adjusting the amplitude factor A given in Eq. (5). Figure 4 provides a comparison of intensity of fluctu-

ations 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 assess- ment 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 ampli- tudes are significantly higher, and plotted curves overlap excessively if all four are displayed), calculated exactly as

7

=* o)

8 0 Q O

70QO

600O

5000

400O

80OO

7000

6O0O

500O

400O

8O0O

7000

6O00

500O

4 0 0 0

80OO

7O00

600O

5 O O O

400O

h= 3cnn

h=11crn

h=14cm

h=19cxn

(a) | ! !

h = 3 c m

• h=11orn `zA~AAAaA&AA~AAAA|~LkAAAAAAkAAaAAAAkAAAAA~ahaA~AkAAA~AAa"AaAAAAA~AAa~AA~AAAuAAka~AAAAAAAk"

h=14cm |•••|•|•|•A••A••••|A•••••A••••AA•A••A•••AAA•A•••A•••••••A•AA•A•A•••A/•A•|A•A•u|••••••A•A••A••••AAi r f ~ T ~ y ~ | ~ ; ~ T T ~ T ~ | ~ v ~ T ~ T ~ T ~ T ~ T ~ | ~ T f ~ T ~ v ~ T ~ v ~ | ~

h= lgcm _ _ . - - _ . - : -.- = - - - - : . . . . . . . . . - - : - : - - . - - : - - - - . . . : : . : . : . . . . - - - - : - - : : - : - : - : - - : . - - : - . - . : - - : : - . . . . . . . . . . _ . - : :

(b) | ! !

h = 3 c m

h=11cm

~ ~ i ~ n n ~ T ~ ~ 1

h=14crn

, h= l 9c, m

( c )

i ! !

, -" " ' lq"W "'v V " "y v v v ~ ' ' w l T ' T ' "'~ ,iv vvy,'v ,

~v~,~ . . . . . p|- V ' qYN" 'Y ' "~ vv r , , , - r ~/ q , , w - h=14cm

r-, ' Vv ~"~" ! v-y-V,v-Trvvv,, v"yv,'"Q{y,'v,y", . . . . . . . "v , (o)

m • • |

0 . 3 8 0 0 , 3 8 5 0 . 3 9 0 0.395 0 . 4 0 0

,Sca led T i m e ( A r b i t r a r y U n i t s )

Figure 3. T IME 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


8DQO~

gOOD

8000!

6OO@

4OQO

F ,e F

h=14cm

h=lgcm

0 . 3 ~ 0.38,5 0.390 0.395 0.400

Scaled Time (Arbitranj Units)

Figure 4. TIME SERIES FOR RADIATION INTENSITY CORRESPONDING TO TWO DIFFERENT AMPLITUDE FACTORS: (a) 0.36 (SEE FIG. 2B); (b) 0.75 (SEE FIG. 2C).

interactions implied by Eq. (3), and is a significant effect reported earlier in [21] and [22] in the context of a more detailed (and less efficient) modeling approach.

CONCLUSIONS In this paper we have presented a new approach to SGS

modeling of turbulence-radiation interactions in the context of large-eddy simulation. We outlined the derivation of model equations and then conducted what can be viewed as (a portion of) an a priori test of this new discrete dynamical system technique. Computed temperature and radiation intensity fluctuations were provided to characterize behavior of the model as bifurcation parameters of the DDS were varied. These simulations represent the first application of this approach over a spatially-extended region, and furthermore the first direct use of large-scale results (experimental in the present case) to automatically set bifurcation parameters of the DDS.

The results produced by the model are physically rea- sonable and self consistent; however, the nature and quan- tity of extant experimental data, and absence of detailed DNS data for turbulence-radiation interactions, make it dif-

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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. Exten- sion 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 adaptive- control modalities.

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