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Line-sampling-based Monte Carlo method

Mathieu Galtier, Frédéric André, Stéphane Blanco, Jérémi Dauchet, Mouna

El-Hafi, Vincent Eymet, Richard Fournier

To cite this version:

Mathieu Galtier, Frédéric André, Stéphane Blanco, Jérémi Dauchet, Mouna El-Hafi, et al.. Line-sampling-based Monte Carlo method. Computational Thermal Radiation in Participating Media V, Apr 2015, Albi, France. �hal-02013900�

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Line-sampling-based Monte Carlo method

M Galtiera,b, F Andr´ea, S Blancoc,d, J Dauchete, M El Hafib, V Eymetg and R Fournierc,d.

aUniversit´e de Lyon, CNRS, CETHIL, UMR5008, F-69621 Villeurbanne, France

bUniversit´e de Toulouse, Mines-Albi, UMR-CNRS 5302, Centre RAPSODEE, Campus

Jarlard, F-81013 Albi CT cedex 09, France

cUniversit´e de Toulouse, UPS, LAPLACE (Laboratoire Plasma et Conversion d’Energie), 118

route de Narbonne, F-31062 Toulouse cedex 9, France

dCNRS, LAPLACE, F-31062 Toulouse, France

e Clermont Universit´e, Institut Pascal, BP10448, F-63000 Clermont-Ferrand, France

f CNRS, UMR 6602, IP, F-63171 Aubire, France

g eso-Star SAS, 8 rue des pˆechers F-31410 Longages, France

E-mail: mathieu.galtier@insa-lyon.fr

Null-collision Monte Carlo algorithms [1, 2, 3, 4] consist in adding a virtual null-collision coefficient to the real extinction one. These collisions, corresponding to pure forward scattering events, have no effect on the radiative transfer equation. A direct consequence of their introduc-tion is that the resulting extincintroduc-tion coefficient ˆβ (defined as the sum of absorption, scattering and null-collision coefficients) can be defined arbitrarily. It can then be chosen as simple as desired to guarantee a rigorous free path sampling whatever the heterogeneity of the medium properties (pressure, temperature, mole fractions). The intermediate step of producing meshes, that can be the source of unquantifiable bias, is no longer required.

In more formal terms, the exponential extinction term does not depend on the real extinc-tion coefficient anymore, but only on the arbitrary field ˆβ. Thereby, the absorption coefficient appears only in a linear form in the recursive integral formulation of radiative transfer equation. Then it becomes possible when studying thermal radiation in a gaseous mixture to decompose the absorption coefficient as the sum of contributions of each molecular transition (or line) for each considered species.

This opens the door to the development of reference methods for which the costly step of producing numerous high-resolution absorption spectra vanishes. The absorption coefficient evaluation is no more required, only some molecular transitions are sampled during the Monte Carlo simulation and their exact contribution to absorption coefficient (for a given wavenum-ber and a given location) are computed from parameters gathered in molecular spectroscopic databases [5, 6, 7]. Nevertheless, it implies that probabilities must be associated to each molec-ular species and to each transition. The choice of these probabilities is fully arbitrary and has only an influence on the convergence rate of the algorithm.

Such a probabilistic model is proposed, and the considered Monte Carlo approach is tested and validated against six unidimensional and non-scattering configurations gathered by Andr´e and Vaillon in [8]. The description of these cases, that cover a wide variety of high-temperature applications, is depicted in Fig. 1.

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x 0 x0 T = 1500K χCO2= 0.5 0.1m T = 500K χCO2= 0.05 1m (a) Case 1 x 0 x0 T = 1500K χH2O= 0.5 0.1m T = 500K χH2O= 0.05 1m (b) Case 2 x 0 x0 T = 400 + 2000x/1.5 K χH2O= 0.2 χCO2= 0.1 1.5m T = 800 + 1600(8− x)/6.5 K χH2O= 0.2 χCO2= 0.1 6.5m (c) Case 3 x 0 x0 T = 400 + 2000x/1.5 K χCO2= 0.1 1.5m T = 800 + 1600(8− x)/6.5 K χCO2= 0.1 6.5m (d) Case 4 x 0 x0 T = 400 + 2000x/1.5 K χH2O= 0.2 1.5m T = 800 + 1600(8− x)/6.5 K χH2O= 0.2 6.5m (e) Case 5 x 0 x0 T = 1500K χH2O= 0.15 0.1m T = 1200K χH2O= 0.12 0.1m T = 900K χH2O= 0.09 0.1m T = 500K χH2O= 0.06 0.1m T = 300K χH2O= 0.03 200m (f) Case 6

Figure 1. Description of the six test cases. Dimensions, temperature and mole-fractions fields are given. Fields are piecewise defined and atmospheric pressure is retained.

For each case, the considered quantity is the spectrally integrated intensity at location x0

resulting from the emission/absorption of the gaseous column [0, x0]. The considered range

for spectral integration is [10 − 15000cm−1]. Results obtained with the proposed approach (denoted Imcm) and 106 independent realizations are given in Tab. 1 with their associated

standard deviations. The computation times1 required to reach a 1% standard deviation are

Table 1. Intensities integrated over the 10 to 15000cm−1range for the six considered test-cases. Estimations obtained by the proposed approach Imcm and 106 realizations are given with their

associated standard deviation. Also displayed are the computation times required to reach a 1% uncertainty. Results can be compared with those of a line by line simulation Ilbl.

Case Imcm (W/m2/sr) σ (W/m2/sr) t1% (s) Ilbl (W/m2/sr) 1 3125.61 4.42 0.97 3126.06 2 3315.11 8.15 1.38 3311.88 3 39223.87 51.56 1.75 39202.5 4 12325.99 16.16 1.26 12320.1 5 38240.31 49.58 1.27 38215.0 6 885.93 3.93 9.86 886.55

also displayed. These results can be compared to those obtained with a line-by-line approach and denoted Ilbl2.

1

The given computation times were obtained on a single core of an Intel Core i7 processor - 2.8GHz. They exclude the loading of spectroscopic data and of associated probabilities.

2

Whereas the line-by-line computation requires, for test-cases 3 to 5, a discretization of the properties fields to be performed, the proposed approach does not need such an approximation. Indeed, the latter is an extension of null-collision algorithms that allow to compute at each collision a quantity depending strictly on the local properties (pressure, temperature, mole-fractions), see [3].

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The computation is performed with the CDSD1000 spectroscopic database for CO2 and with

HITEMP for H2O. A line-wing truncation at 25cm−1 is also assumed.

The specific intensities obtained with the proposed Monte Carlo approach and the determin-istic line-by-line one are fully consistent. The computation times required by the Monte Carlo approach to get a 1% confidence interval are comprised between 0.97 and 9.86 seconds according to the case study. Because of the stochastic nature of the proposed method, these computation times depend little on the spectral integration range or on the size of molecular spectroscopic database.

Beyond prospects of simulation, this approach offers an important versatility and possibility of analysis due to the fact that the computation does not rely on ”rigid” precomputed spectra. For instance, it becomes possible to easily study the effect of a given spectroscopic database, parameter or hypothesis on a radiative observable with the very same algorithm, without having to recompute a large set of spectra to cover the heterogeneity of the medium. Two examples applied to test-case 2 are given in Fig. 2 where the effects of the choice of database and line-wing truncation distance are evaluated.

0 0.5 1 1.5 2 2.5 3 3.5 4 1200 1300 1400 1500 1600 1700 1800 1900 M ea n in te ns it y (W / m 2/sr / cm − 1) Wavenumber (cm−1) HR Deterministic - Hitemp 2010 Monte Carlo - Hitemp 2010 Monte Carlo - Hitemp Monte Carlo - Hitran 2008

(a) 0 0.5 1 1.5 2 2.5 3 3.5 4 1200 1300 1400 1500 1600 1700 1800 1900 M ea n in te ns it y (W / m 2/sr / cm − 1) Wavenumber (cm−1) HR Deterministic - Truncature at 25cm−1

Monte Carlo - No truncature Monte Carlo - Truncature at 25cm−1

Monte Carlo - Truncature at 5cm−1

Monte Carlo - Truncature at 0.5cm−1

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Figure 2. Intensities averaged over several narrowbands, computed for test-case 2, with different spectroscopic databases (a) and different line-wing truncation distances (b). Line-by-line computation are depicted in solid line.

[1] Woodcock E et al. 1965 Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry Proceedings of the Conference on Applications of Computing Methods to Reactor Problems

[2] Skullerud H 1968 Journal of Physics D: Applied Physics 1 1567–1568

[3] Galtier M et al. 2013 Journal of Quantitative Spectroscopy and Radiative Transfer 125 57–68 [4] Eymet V et al. 2013 Journal of Quantitative Spectroscopy and Radiative Transfer 129 145–157 [5] Rothman L et al. 2010 Journal of Quantitative Spectroscopy and Radiative Transfer 111 2139–2150

[6] Tashkun S and Perevalov V 2011 Journal of Quantitative Spectroscopy and Radiative Transfer 112 1403–1410 [7] Rothman L et al. 2013 Journal of Quantitative Spectroscopy and Radiative Transfer 130 4–50

[8] Andr´e F and Vaillon R 2010 Journal of Quantitative Spectroscopy and Radiative Transfer 111 1900–1911

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Figure

Table 1. Intensities integrated over the 10 to 15000cm − 1 range for the six considered test-cases.
Figure 2. Intensities averaged over several narrowbands, computed for test-case 2, with different spectroscopic databases (a) and different line-wing truncation distances (b)

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