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?Electronic Journal «Technical Acoustics» http: // www .ejta.org

2009 8 Panagiota Marazioti

Dept. of Energy Technology, Technological Education Institute (TEI), Athens, Greece;

Dept. of Mechanical and Aeronautics Engineering, University of Patras, Greece 26 500, e-mail: Ця електронна адреса захищена від спам-ботів. Вам потрібно увімкнути JavaScript, щоб побачити її.

An aerothermoacoustic model for computation of the combustion noise (roar) radiated by lifted turbulent jet diffusion flames

Received 10.07.2009, published 09.09.2009

A 2D time-dependent phase-averaged Navier-Stokes flow simulation that encompasses aspects from both the large-eddy simulation (LES) formalism and the conventional k-s approaches was employed to calculate the reacting jet flows. A reactedness-mixture fraction two-scalar exponential probability density function (PDF) model based on non-premixed flame arguments was combined with a local Damkohler number extinction criterion to separate between reacting and nonreacting regions. Although the inclusion of the effects of premixed flame propagation could help improve the model, initial comparisons with experimental results suggest adequate qualitative agreement between computations and reported data. The reasonable agreement obtained for the aero-thermodynamic flame characteristics permitted the meaningful computation of the combustion noise (roar) characteristics of the lifted flame in an effort to address the coupled effects of heat release by the flame and turbulent interactions on the autonomous flame noise generation.

Key words: combustion roar, lifted flame, sound spectrum, turbulent combustion modeling.

1. INTRODUCTION

Renewed interest has emerged in recent years in the study of the interaction between the combustion process and the acoustics of the reacting environment [1, 2]. Combustion roar is related to the noise generated directly by the flame due to turbulent fluctuations independently or in combination with acoustic resonance with the reacting environment and usually involves a broadly distributed spectrum [3]. The topic is of interest to practical combustor designs since it is interrelated with combustor acoustic / pressure oscillations or resonance, acoustic pollution of the environment diagnosis of operational variations and faults and ultimately active or hybrid combustor operation control techniques [4, 5]. Practical examples which are interesting from the point of view of research into combustion roar is the combustion chambers of aircraft jet engines, flares and hot-air balloons where minimization of flame noise is desirable.

In the present work the modeling of the autonomous noise generation by the turbulence / chemistry fluctuations in the flame front vicinity in jet diffusion flames is

investigated. To describe the acoustic performance of the jet flame as autonomous source of sound an integral expression is employed that provides the (acoustically one-dimensional) noise spectrum from the flame in terms of an assumed shape turbulence spectrum at the flame front closely following the formulation put forward and derived in closed form by Klein [5].

As a first step towards understanding the phenomenon turbulent non-premixed jet flames lifted from the burner rim are studied as model problems. A diffusion flame attached to the burner nozzle lifts above the jet exit rim or blows out abruptly when the fuel jet velocity steadily increases and exceeds a critical value [6]. Apart from their relevance in the design and safe operation of industrial systems such jet flame stability phenomena provide a useful research tool for experimental and numerical studies of turbulent reacting flow characteristics and related phenomena such as those presently addressed. Non-intrusive measurements by Schneider, et al. [7] suggest that individual theories such as turbulent premixed flame propagation, laminar flameless quenching, large or small scale mixing are all plausible theoretical viewpoints to describe the coupled aerothermochemical phenomena of lift-off and blow-out.

In the described work a 2D time-dependent phase-averaged Navier-Stokes flow simulation method [8] capable of calculating the mean and turbulent properties of the momentum and thermo-chemical fields is employed to study the behavior of axisymmetric co-flowing methane- air jet diffusion flame configurations. Both laminar and turbulent (lifted-off) operational conditions have been investigated to test and develop the model over a range of conditions with increasing complexity. A modular post-processor is then employed for the prediction of turbulent combustion noise exploiting the time-mean and fluctuating thermochemical quantities obtained from the basic reacting flow field predictions.

2. NUMERICAL METHOD

2.1. Aerodynamic model

The reacting flows were calculated with the 2D time-dependent N-S equations governing the temporal and spatial variation of the velocities and pressures e.g. u =<u >+ U 'with < u >= U + u "where u and < u > are the instantaneous and phase-averaged (or resolved) velocities, u and u "are the time-mean and large-scale (resolved) fluctuating components and u 'is the stochastic (subgrid) turbulent fluctuation. The model description given below closely follows the formulations adopted in [8]. For the reacting flows density-weighted

values ​​are used i.e. < f >= p< f > / P. The equation set may be written as follows:

dp d<~ > d<~ > ~ d < ~ > 1 dp d d < ~ n.

----+ ------- = 0, ----- + < u.- >-------------= ----------- 1 --- [v----------< utuj >] + (P - pj) gi

dt dx. ct 1 d j pdxt dx, dx i ^ >

with the Reynolds stresses obtained from the standard eddy-viscosity formula:

- < uiuJ >=< vt >

dx

d < ut > dx.

+ < k >

(1)

(2)

<vt >

In the conventional k - s model < ~ > is related to the turbulence energy < k > and its dissipation rate <s> as < ~ >= C ^< k >2 / <s> where < k > and <s> are obtained from the standard transport equations:

- (U] k) = -

dXj 3 dXj

vt dk Ok dxj j

------dU,

- u, u 3--------

1 dx 1

- s

(3)

dx

vt ds

oS dx

jj

- Csi - u, u j s1 k 13 dx

(4)

where C ^ = 0.009, Cs1 = 1.44, Cs2 = 1.92, ok = 1.0, os = 1.3.

The standard k-s model has been formulated and tested within steady-state calculation procedures against a range of plane shear flows with no distinct peaks in their energy spectrum. When a 2D time-dependent calculation is used part of the energy spectrum is directly resolved by this type of calculation. There is clearly ambiguity as to whether the standard model is capable of partitioning (correctly or at al) the total stress into its stochastic and periodic contributions [8].

In the present time-dependent calculations the spatial filtering due to the employed mesh is also accounted for in an effort to distinguish the directly computed (albeit 2D) turbulent motions, which are resolved by the mesh of size A = (AxAy) 1/2 , from the turbulence already modeled by the k - s model. < ~ > is therefore here evaluated by borrowing the Smagorinsky mixing length model from the large-eddy simulation (LES) formalism and the hybrid turbulence model is:

-|IP, a) 2 (:

2 < S'j >< Sij >

< vt >= C ^< k > / < s >,

1/2

if

if

Lt =

< k >

3/2

< s >

> A and

(5)

Lt <A.

Cs is taken as 0.1. The resulting ~ is fed back into the production of < k > in the k - s equations thereby producing a continuous distribution of ~ t. The implicit scheme used here, is therefore well suited for this hybrid formulation. Calculations with the standard k - s model have been proven clearly inferior to the recent hybrid method predictions both for cold and reacting bluff-body flows, as has been demonstrated in Koutmos et al. [9].

2.2. Combustion model

2.2.1. Basic turbulence / chemistry interaction model

A partial equilibrium scheme corresponding to a two-scalar description employing the mixture fraction, f, and the CO2 concentration, YCOi, were used. The reaction

CO + OH ^ CO2 + H was introduced to allow for non-equilibrium effects and CO2 formation from CO is assumed to proceed as:

rco2 = kfYCOYOH

. ks ,

V s J

yco2yh, kf = 6.76 * 1011 exp

T

тисячу сто дві

and ks is taken from the JANAF (Tables).

Additionally when the mixture strength exceeds the rich flammability limit the composition is taken as that of equilibrium at this limit diluted with pure fuel. The final composition is calculated from the NASA equilibrium code for given f and YCO2 values ​​by

defining YCO2 as an 'element'. The passive, f, and the reactive, YCO2, variables, are calculated from the equations

d (< f >) D (p ~ ~) 5

-X ------ L + --- Ip < u 3 >< f >1 = -

dt dxj ^ 3 'dXj

pD +

Mt

SCt

d< f >

d {p

< yco2 >

)

dt

dx

-(p

< u j >< Yco2 > J =

•)

pD +

Mt

Sc

dXj

+ prco2

(6)

(7)

with gradient transport assumptions for the turbulent fluxes ujf 'and u "Y'Co .

An exponential joint PDF is constructed from the normalized mixture fraction and CO2 concentration values, f * and YpO2, which are used to transform the physically allowable space of f and YCO2 into a normalized square area suitable for integration. The relationships established by this transformation are:

YCO2 = YCO2 / (fYCO2, fuel) ,

f = f + YCO2 / YCO2 air, 1CO2- 1 CO; where YcO2, fuel = nMCO2 / MCNHM, YCO2, air = MCO2 / (MO2 + MN2 /0.259) .

The local PDF is of the form:

Pf, YCo2) = exp [a1 + a2f + a3Ypo, + af2 + a5Ypo2 + a ^ _f * Ypo2],

(8)

(9)

where f and Yp * O are appropriately transformed variables and is calculated through the

coefficients (a1_a6) which depend on the local moments f ', YCo2, fY'Co2 which are

obtained assuming equilibrium between the turbulent production and destruction of these moments in their general form transport equation:

d (pX 'Z') d (pHJX 'Z') d

dt _ 1

dx.

dx.

pD + Mm

v Sct j

dx.

+ 2

A

Sct

dX dZ dx. dx.

(10)

-Cp-X 'Z' + X'Sz + Z 'Sx .

The moments are then obtained from the following expression:

X Z '= -

1

2.0p

2Mt dX dZ

sct dc, dcI

+ X sZ + Z sX

(11)

X =< f > or < Yco2 > and Z =< f > or < Yco2 > , C0 = 2.0 .

Mean quantities and correlations are evaluated by using PDF information e.g.

rCO2 = '

p JJ ~ co2 JfiPif * YCo2) df

* dY,

co2 ,

(12)

where J is the Jacobian of the transformation, Tt is evaluated as follows in line with the

hybrid model formulation: if Lt > A

Lt > A then xt = A / V< k > and if Lt < A then rt =< f > / < s > .

2.2.2. Extinction / Reignition model

The modeling of finite-rate chemistry effects such as partial extinctions and reignitions encountered in the lifted flames follows the approach of Koutmos [9]. Local extinction is predicted when the local Damkohler number, Dat, defined as the ratio of the turbulent time

scale = 3.88rk (rk is the Kolmogorov time scale) to the chemical time scale, Tch, is below a local critical 'limit', Dacr being a function of position and local conditions [10]. The criterion for local quenching then reads:

Da [T / Tch] _ < 1

1/41. (13)

A = -

Da "

Jf

'JIa / r

R

The mean gas state subsequent to extinction, Yq is obtained by convoluting with the local exponential 2D, PDF involving the reactedness, B, and the mixture fraction [11]:

Yq = (1 / p

) JJ Yq pP (f, B) dfdB .

B is here calculated from the following model Lagrange-type transport equation:

d (pif), 5 (pf iB):

d t

dX

(14)

(15)

while its fluctuations, Bf2 and fB ', are obtained by invoking a scale-similarity assumption [12]. Further details on the full model may be found in Refs. [8, 9, 12].

Reignition is allowed when a) the time-scale criterion, Eq. (13), is inoperative and b) the cumulative probability of finding a flammable mixture at this location, defined as

pf =

f2

J P (f ') df *, is greater than 0.65. Then the source term in Eq. (15) is set equal to

(B0 - B) / rid where Tid is a mixing-dependent chemical ignition delay time [8, 9, 10, 11, 12].

T

2.3. Turbulent flame noise model

For the evaluation of the autonomous sound radiation due to the interaction of the turbulent fluctuations with the flame front the model formulation of Klein, [5] has been followed closely. Starting from the basic wave equation for low Mach number flows the onedimensional sound generated by the fluctuating heat release from a turbulent non-premixed flame is evaluated [5] by deriving an integral expression assuming that the instantaneous combustion zone is infinitely thin and that combustion is fast and determined by the mixing of fuel and air. The sound spectrum can then be expressed [5] as a function of a one-dimensional turbulence spectrum (of assumed shape) of the mixture fraction at the flame front. The resulting expression for the sound spectrum for a non-premixed turbulent flame is then given in integral form as function of frequency:

PP (f) = 2n

JJ [{pDBA)

X

2 Lcor @ f "2

--------Ejd

2U

2 /

2U

(16)

where C0 is the speed of sound, S is the area of ​​combustor nozzle, D is the laminar

1 + 1 f s diffusion coefficient, B = - x / -DJ Ed (k) dk, x = 2.0-f'2 is the scalar dissipation, EID is the

one-dimensional turbulence spectrum of assumed shape:

f1, k1 < ks

eid =

.- (5/3)

(*, / K 2)

0, k1 > kkol

ks< k1 < kkol

with k1 = 2nf / 2U, ks = n / lt, kkol = (s / v3), 0f = 1 / Jf is the thickness of the mixing

layer between fuel and air, A = [T / lame - T0] / [Tflamefst (1 - fst ^ and f is the frequency The parameter sound pressure level, SPL, expressed in dB can also be evaluated as:

SPL (f) = 20 log

Vpp (f) 20 * 10-6

(17)

The time-averaged information about the turbulence sound spectrum and other parameters required in the above expression is derived from the basic reacting flow calculation and the computation of the sound spectrum is performed in a post-processing step.

2.4. Numerical details

The cofl owing methane-air jet configuration and computational domain are shown in Fig. 1. The convective condition d $ l dt + U0 (d $ l dt) = 0, (C = U0) was used at the outlet. A

mesh of 205x123 (x, y) grid points was used with an axial expansion ratio of 1.1. For inlet conditions fully developed flow was assumed.

The equations were solved with a finite-volume method based on a staggered mesh, a pressure correction method (SIMPLE) and the QUICK differencing scheme [9]. A second-order scheme was used for temporal integration.

-4 -5

Time steps were of the order 10 ... 10 s depending on fuel Reynolds number. After an initial transient of about 30t0 (t0 = D lU0) statistics were computed over approximately 100t0.

Figure 1. Methane-air coflowing jet flame configuration Investigated R = 27r and the length of

computation Domain is 50r

3. RESULTS AND DISCUSSION

Computations were initially performed for a steady-state, laminar CH4-air diffusion flame for which experimental data by Meier, et al. [13] is available. Figure 2 shows sample results for undiluted CH4 jet flow at 5 cmls through a 1.2 cm-diameter tube, with a coflow of air at 10cmls which are compared against the experimental data of reference [13].

Results are compared with data at the final steady-state which was reached at about 25.000 time-steps (At = 0.06 s). Overall the agreement is satisfactory in both the reactive scalar (temperature) and the momentum field (axial velocity) despite some experimental uncertainties concerning the rig exit conditions and these comparisons lend support for an extension of the basic model to the more complex turbulent lifted flames.

a)

b)

Figure 2. Laminar CH4 jet flame predictions for fuel jet velocity a) 20 mls with a lift-off height 0.12 mm and b) 40 mls with a lift-off height 0.24 mm

Figure 3 displays time-averaged temperature contours for two jet velocities of 20 and 40 mls investigated. The stoichiometric contour is also superimposed on the plot.

By comparison to reported non-dimensional lift-off heights the present simulation seems to underestimate the stabilization position by about 7% and 12% for the two jet velocities respectively.

An instantaneous isotherm snapshot of the reacting flow field is illustrated in Fig. 4. A variation of the lifted flame base with an up and down movement of about 12% was observed in the time-dependent simulations.

Due to their two-dimensional nature the resulting spectra could not allow a reliable evaluation of the spectral behavior of this all important lifted flame base region.

a)

b)

y

Figure 3. Time-average temperature contours for fuel jet velocity a) 20 mls, b) 40 mls

a)

b)

Figure 4. Instantaneous temperature isotherms: a) uj = 20 mls, b) uj = 40 mls

y

x

x

a)

b)

c)

d)

e)

Figure 5. a) Instantaneous plot of velocity vectors (uj = 20 mls)

b) Instantaneous plot of fuel fraction contours (uj = 20 mls)

c) instantaneous plot of V contours (uj = 20 mls)

d) Instantaneous plot of velocity vectors (uj = 40 mls)

e) Instantaneous plot of V contours (uj = 40 mls)

An overall qualitative picture of the velocity field is shown in the form of vector plots in Fig. 5. The mean velocity (Favre-averaged) due to expansion slows down on approaching the flame front and accelerates as it convects past it. Any relevant local scaling is therefore expected to include the effects of expansion on local parameters e.g. the flame front propagation speed in relation to the laminar flame speed.

Some preliminary joint statistics between temperature and mixture fraction, two important scaling parameters for the partially premixed regime [14], have been produced from the time-dependent calculation for the higher fuel jet velocity of 20 m / s and are shown in Fig. 6. The collected points lie close to the stoichiometric contour in the vicinity of the movement of the flame base.

Figure 6.

Mixture fraction (f) -Temperature (T) scatter plot taken in the vicinity of the lifted flame base

Methane is well known for its bimodel approach to extinction from a wide range of previously reported works (e.g [12]). The present plot implies a lower level of bimodality with the scatter points located mostly above the mixing asymptote and below the partial equilibrium levels. The variance of mixture fraction spreads the points over an area slightly broader than the lean limit.

Experimental data of this nature would be very helpful in identifying the detailed nature and flow behavior of the near stabilization region. The reduced bimodality with respect to customary diffusion flames [12, 14] may be attributed to loss of resolution in the present simulation, to deficiencies in the reignition model which is formulated for pure diffusion flames, or to a lower variability of the chemical time- scale arising from omission of the effects of partially premixed flame propagation. This aspect which is not treated explicitly in the model other than through the inclusion of the relevant chemical time-scale.

Encouragingly the present modeling formulation recovered two important experimentally observed trends. Firstly, the linear relationship between flame lift-off height and jet exit velocity that has been extensively verified through global and detailed measurements as well as the correct slope of this linear variation. Secondly it reproduced adequately the increase in lift-off height for a given jet inflow velocity as the fuel flow is diluted e.g. with N2, something expected since the residence time is now longer when the fuel stream is diluted.

The reasonable performance of the computational model for the prediction of the experimentally observed aero-thermodynamic parameter variations and trends lends support to the extension of the method to include and apply the flame noise described previously to enable a meaningful evaluation of the combustion noise radiated by the present flames. Flame noise calculations were performed for the lifted-off flame configuration with exit fuel velocity of 20 m / s.

Equation (16) was discretized to be able to calculate it numerically and the integral is computed numerically at every grid cell. The part in square brackets in the expression of Eq. (16) can be considered as the strength of the local noise source and gives an indication of the spatial distribution of the local noise 'intensity' in the reacting flow field for each selected frequency. Integrating numerically Eq. (16) for all cell volumes and over the full frequency range (A / = 1 Hz, Pref = 20 [xPa) we deduce the sound spectrum expressed in dB sound pressure level from Eq. (17). The turbulent aero-thermochemical data produced by the basic time-dependent computation are time-averaged and supply the required information to evaluate the parameters involved in Eq. (16) for every cell volume and frequency. Figure 7 display contours of the local noise intensity (the term between the square brackets in Eq. (16)) for a frequency of 100 Hz.

Level LNSS 8 3.16141E-05

7 2.95724E-05

6 248659E-05

5 2.04854E-05

4 1.6564E-05

3 1.24095E-05

2 6.81035E-06

1 2.41011E-06

Figure 7. The local noise source intensity (LNSS) distribution for fuel jet velocity, uj = 20 m / s along the flame length: x-axis is the flame length and y-axis the flame width

It is seen that noise generation levels are predominant mainly in the vicinity of the mixing interface between fuel and oxidizer. These attain maximum values ​​downstream of the lifted-off flame base for the lifted flame and the location of the maximum noise intensity coincides with the region of the peak temperatures levels. The lifted flame produces significantly increased maximum values ​​of the local noise source strength and this appears a reasonably predicted trend since the lifted flame produces increased turbulence levels and temperature fluctuations downstream of its lifted base.

The calculated sound spectra for the above discussed flame are plotted in Figure 8. The lifted flame gives elevated energy content in the sound spectrum by about 30% particularly near the lower frequency range at about 80 Hz. This is evidently consistent with the previously discussed predictions of the local noise intensity levels. It should also be noted that the qualitative distribution of the spectral density also depends on the assumed turbulence spectrum used in Eq. (16).

140 p 135 130

_ 125

co

"D

~ 120

Q-

"> 115 110 105

100 -

0

An aspect that merits discussion is whether the assumption of fast chemistry can still be applied for the partially-premixed flame configuration with lift-off when using Eq. (15) and to what extend the expected discrepancies affect the validity of the present prediction. Experimental results would be quite helpful to identify the extent of disagreement in the sound pressure level predictions of this flame and this would help improve the present modeling effort.

Note with standing the above arguments, it is believed that the present procedure has captured the basic behaviors and trends in the aerothermochemical and flame noise generation characteristics of the studied flames although further tests and refinements would be required to enlarge the applicability of the method.

4. SUMMARY

A computational procedure of the time-dependent reactive Navier-Stokes equations has been employed to study the stabilization mechanism and the autonomous noise generation by the flame front due to turbulent fluctuations in lifted methane jet flames. Although the method exploited primarily diffusion flame concepts with the effects of partial extinctions / reignitions embodied, it produced reasonable qualitative agreement with reported data. Despite the fact that these exploratory and preliminary computations are based on axisymmetric configurations and hence the 3-D or the non-symmetric turbulent behavior is excluded the present method captured many important trends and behaviors and allows for an evaluation and further development of the overall methodology . Further extensions along the line of addressing in a more clear-cut manner the impact of partial-premixing are also required.

The adequate accord between computations and experimental observation in the turbulent aerothermodynamic flow and flame parameters allowed a first attempt at evaluating the turbulent combustion noise (roar) characteristics of the complex flame configuration investigated here. The developed methodology provides a basis to address the coupled effects of turbulence interactions, heat release and chemistry, and the autonomous turbulent combustion noise generated at the flame front. These preliminary results suggest that the modeling procedure followed such complex behaviors as the variation of the flame lift-off height with fuel jet velocity and the accompanying increase in the radiated flame noise levels. Further detailed assessment and improvements of the described methodology may however be required to validate and demonstrate its wider applicability.

Figure 8.

The calculated sound spectra for lifted flame plotted in SPL dB (Af = 1 Hz)

5. ACKNOWLEDGMENTS

The research was funded by the program "K. Karatheodori ", Epitropi Ereunon, University of Patras.

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Ключові слова: шум горіння /турбулентний горіння /моделювання

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