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Matter and Radiation at Extremes, Volume. 9, Issue 6, 067802(2024)

Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions

Cong-Zhang Gao1, Ying Cai1, Cheng-Wu Huang2, Yang Zhao2, Jian-Wei Yin1, Zheng-Feng Fan1, Jia-Min Yang2, Pei Wang1, and Shao-Ping Zhu1
Author Affiliations
  • 1Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
  • 2Laser Fusion Research Center, China Academy of Engineering Physics, Mianyang, Sichuan 621900, People’s Republic of China
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    Abstract
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    Figures & Tables(14)
    Schematic of radiative transfer in a binary stochastic mixture, including absorption, scattering, emission, and transmission of radiation. A fraction of spherical particles (material 1) are randomly mixed into the background medium (material 2). An isotropic radiation source is implemented near the left surface. Purple wavy lines represent the rays of radiation. Gold and red particles correspond to high and low temperatures, respectively.

    Fig. 1. Schematic of radiative transfer in a binary stochastic mixture, including absorption, scattering, emission, and transmission of radiation. A fraction of spherical particles (material 1) are randomly mixed into the background medium (material 2). An isotropic radiation source is implemented near the left surface. Purple wavy lines represent the rays of radiation. Gold and red particles correspond to high and low temperatures, respectively.

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    Ensemble-averaged radiation transmission flux at the outgoing surface (L = 1 cm) as a function of time for the set 1 parameters20 in Table I. Comparisons are made by varying (a) average particle size, (b) particle size distribution, and (c) mixing probability. Ensemble-averaged physical quantities with uncertainties at the steady-state time (i.e., 5000 ps) are tabulated in Table II.

    Fig. 2. Ensemble-averaged radiation transmission flux at the outgoing surface (L = 1 cm) as a function of time for the set 1 parameters20 in Table I. Comparisons are made by varying (a) average particle size, (b) particle size distribution, and (c) mixing probability. Ensemble-averaged physical quantities with uncertainties at the steady-state time (i.e., 5000 ps) are tabulated in Table II.

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    Same as Fig. 2, but calculated with the set 2 parameters25 in Table I. Ensemble-averaged physical quantities with uncertainties at the steady-state time (i.e., 20 000 ps) are tabulated in Table III. For simplicity, a constant size distribution is used in (a) and (c).

    Fig. 3. Same as Fig. 2, but calculated with the set 2 parameters25 in Table I. Ensemble-averaged physical quantities with uncertainties at the steady-state time (i.e., 20 000 ps) are tabulated in Table III. For simplicity, a constant size distribution is used in (a) and (c).

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    Variation of the enhancement factor (as defined in the text) with system size L for parameters identical to those in (a) Olson20 and (b) Brantley and Martos.25 The gray dashed lines delineate where there is no influence of the stochastic mixture on the radiation fluxes. The solid vertical lines indicate the system size used in Figs. 2 and 3, respectively. Symbols connected by solid lines are to guide the eye.

    Fig. 4. Variation of the enhancement factor (as defined in the text) with system size L for parameters identical to those in (a) Olson20 and (b) Brantley and Martos.25 The gray dashed lines delineate where there is no influence of the stochastic mixture on the radiation fluxes. The solid vertical lines indicate the system size used in Figs. 2 and 3, respectively. Symbols connected by solid lines are to guide the eye.

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    Temperature equilibration between radiation and material for case F in Table II. Radiation and material temperatures are denoted by Tr (dashed curves) and Tm (solid curves), respectively.

    Fig. 5. Temperature equilibration between radiation and material for case F in Table II. Radiation and material temperatures are denoted by Tr (dashed curves) and Tm (solid curves), respectively.

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    For case Ft1 in Table IV, Transmitted radiation flux (left vertical axis) and material temperature (right vertical axis) at the outgoing surface (L = 1 cm) as functions of time. Solid and dashed curves denote results obtained for constant and temperature-dependent opacity, respectively. Note that the ranges of the left and right vertical axes are different.

    Fig. 6. For case Ft1 in Table IV, Transmitted radiation flux (left vertical axis) and material temperature (right vertical axis) at the outgoing surface (L = 1 cm) as functions of time. Solid and dashed curves denote results obtained for constant and temperature-dependent opacity, respectively. Note that the ranges of the left and right vertical axes are different.

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    Snapshots of material temperature distributions for case Ft1 in Table IV. (a)–(d) Are obtained for constant opacity and (e)–(h) for temperature-dependent opacity. The first to last columns represent a sequence of times corresponding to ct = 1.0, 5.0, 12.5, and 40.0 cm. The color bar on the right side ranges from 0 to 300 eV.

    Fig. 7. Snapshots of material temperature distributions for case Ft1 in Table IV. (a)–(d) Are obtained for constant opacity and (e)–(h) for temperature-dependent opacity. The first to last columns represent a sequence of times corresponding to ct = 1.0, 5.0, 12.5, and 40.0 cm. The color bar on the right side ranges from 0 to 300 eV.

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    Normalized transmitted radiation flux (with the values scaled by those of the atomic mix) as a function of β. For clarity, logarithmic scales are used for both the horizontal and vertical axes. Data points are fitted to linear functions in different regimes. The crossing of the two fitted lines can be taken roughly as βc. The horizontal dashed lines label the positions of the atomic and chunk mixes.

    Fig. 8. Normalized transmitted radiation flux (with the values scaled by those of the atomic mix) as a function of β. For clarity, logarithmic scales are used for both the horizontal and vertical axes. Data points are fitted to linear functions in different regimes. The crossing of the two fitted lines can be taken roughly as βc. The horizontal dashed lines label the positions of the atomic and chunk mixes.

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    Material temperature distribution at ct = 0.15 cm for (a) atomic mix, (b) R = 0.0005 cm, (c) R = 0.001 cm, (d) R = 0.0025 cm, (e) R = 0.005 cm, (f) R = 0.0075 cm, (g) R = 0.01 cm, and (h) chunk mix. Temperatures are in units of eV. Parameters in the simulations are the same as those in Fig. 8.

    Fig. 9. Material temperature distribution at ct = 0.15 cm for (a) atomic mix, (b) R = 0.0005 cm, (c) R = 0.001 cm, (d) R = 0.0025 cm, (e) R = 0.005 cm, (f) R = 0.0075 cm, (g) R = 0.01 cm, and (h) chunk mix. Temperatures are in units of eV. Parameters in the simulations are the same as those in Fig. 8.

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    Basic structure of RAREBIT2D code.

    Fig. 10. Basic structure of RAREBIT2D code.

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    • Table 1. Sets of physical and numerical parameters used in the simulations.

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      Table 1. Sets of physical and numerical parameters used in the simulations.

      ParameterSet 1 (Olson20)Set 2 (Brantley and Martos25)
      Domain size L × L (cm2)1.0 × 1.010.0 × 10.0
      Grid representation nx × ny2000 × 20002000 × 2000
      Material 1 density ρ1 (g/cm3)2.72.7
      Material 2 density ρ2 (g/cm3)0.10.1
      Mixing probability p10.02–0.20.05–0.3
      Average particle size ⟨r⟩ (cm)0.001 91–0.019 400.087 54–0.700 28
      Material 1 absorption opacity σa,1 (cm−1)4.6–495.19.1
      Material 2 absorption opacity σa,2 (cm−1)0.10.1
      Material scattering opacity σs00
      Initial material temperature T0m (keV)0.0030.003
      Radiation source temperature T0r (keV)0.30.3
      Level of quadrature set s1616
      Total simulation time τ (ps)500020 000
      Time step dτ (ps)0.10.1
      Convergence value of source iteration ϵ10−610−6
      Number of physical configurations N1010

    • Table 2. Ensemble-averaged physical quantities at the outgoing surface at 5000 ps for the set 1 parameters20 in Table I. In each case, the standard deviations [see Eq. (6)] are evaluated by randomly simulating ten physical configurations. ⟨Erad⟩, ⟨Tmat⟩, ⟨Jreflec⟩, and ⟨Jtrans⟩ are in units of J/cm3, eV, J/(cm2 ps) and J/(cm2 ps), respectively. p(r) denotes the particle size distribution; see the Appendix for more details. p1 is the mixing probability. ⟨r⟩ is the mean particle size. σa,i is the absorption opacity for material i (i = 1 and 2).

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      Table 2. Ensemble-averaged physical quantities at the outgoing surface at 5000 ps for the set 1 parameters20 in Table I. In each case, the standard deviations [see Eq. (6)] are evaluated by randomly simulating ten physical configurations. ⟨Erad⟩, ⟨Tmat⟩, ⟨Jreflec⟩, and ⟨Jtrans⟩ are in units of J/cm3, eV, J/(cm2 ps) and J/(cm2 ps), respectively. p(r) denotes the particle size distribution; see the Appendix for more details. p1 is the mixing probability. ⟨r⟩ is the mean particle size. σa,i is the absorption opacity for material i (i = 1 and 2).

      CaseInitial parametersPhysical observable
      p(r)p1⟨r⟩σa,1σa,2⟨Erad⟩⟨Tmat⟩⟨Jreflec⟩⟨Jtrans⟩
      AConstant0.020.019 445.10.11015.140 ± 3.691222.823 ± 0.212272.870 ± 1.631565.981 ± 1.614
      BConstant0.020.006 3745.10.1904.258 ± 1.143216.504 ± 0.069327.930 ± 0.619510.769 ± 0.612
      CConstant0.020.001 9145.10.1843.889 ± 0.632212.828 ± 0.039357.668 ± 0.319480.793 ± 0.328
      DExponential0.020.009 5545.10.11036.180 ± 26.314223.921 ± 1.416262.292 ± 13.400576.655 ± 13.331
      EConstant0.10.019 19.10.1858.673 ± 2.219213.734 ± 0.137350.191 ± 1.212488.218 ± 1.207
      FConstant0.20.019 14.60.1832.901 ± 0.572212.125 ± 0.036363.239 ± 0.354475.148 ± 0.354
      GConstant0.020.019 4495.10.1827.804 ± 6.922211.559 ± 0.447380.880 ± 4.758461.463 ± 4.645
      HUniform0.020.014 3245.10.11025.900 ± 7.055223.401 ± 0.389268.570 ± 3.474570.269 ± 3.442

    • Table 3. Same as Table II, but for the set 2 parameters25 in Table I. The second column λ1 represents the particles’ mean chord length,16 in units of cm. It is related to the average particle size by36λ1=12π⟨r⟩ (constant size distribution), λ1=23π⟨r⟩ (uniform size distribution), and λ1 = π⟨r⟩ (exponential size distribution).

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      Table 3. Same as Table II, but for the set 2 parameters25 in Table I. The second column λ1 represents the particles’ mean chord length,16 in units of cm. It is related to the average particle size by36λ1=12π⟨r⟩ (constant size distribution), λ1=23π⟨r⟩ (uniform size distribution), and λ1 = π⟨r⟩ (exponential size distribution).

      CaseInitial parametersPhysical observables
      λ1p(r)p1⟨Erad⟩⟨Tmat⟩⟨Jreflec⟩⟨Jtrans⟩
      111/40Constant0.05453.574 ± 3.960182.006 ± 0.342578.877 ± 2.522259.524 ± 2.526
      0.1299.913 ± 5.753163.991 ± 0.813666.830 ± 3.308171.591 ± 3.302
      0.2164.770 ± 2.077140.970 ± 0.511743.600 ± 1.05294.578 ± 1.045
      0.3104.204 ± 1.033125.695 ± 0.326776.814 ± 0.47359.944 ± 0.523
      Uniform0.05458.661 ± 4.334182.586 ± 0.464575.613 ± 2.145262.786 ± 2.144
      0.1304.609 ± 3.927164.602 ± 0.596663.679 ± 2.414174.743 ± 2.404
      0.2168.719 ± 1.661141.799 ± 0.253741.321 ± 1.08996.881 ± 1.090
      0.3106.830 ± 1.176126.373 ± 0.332775.320 ± 0.72961.615 ± 0.706
      Exponential0.05471.568 ± 19.529183.733 ± 1.861568.240 ± 11.555270.171 ± 11.549
      0.1313.602 ± 6.419165.814 ± 0.912658.929 ± 3.461179.508 ± 3.454
      0.2178.623 ± 5.412143.798 ± 1.121736.034 ± 2.994102.228 ± 3.049
      0.3111.129 ± 2.010127.654 ± 0.652773.270 ± 1.05063.778 ± 1.119
      211/20Constant0.05530.439 ± 5.623189.245 ± 0.433533.650 ± 3.745304.756 ± 3.745
      0.1376.795 ± 6.307173.387 ± 0.701622.802 ± 3.394215.631 ± 3.387
      0.2214.617 ± 3.981150.123 ± 0.751714.551 ± 2.641123.877 ± 2.638
      0.3134.409 ± 2.378133.688 ± 0.727759.979 ± 1.71877.956 ± 1.705
      Uniform0.05530.602 ± 10.185189.295 ± 0.833533.792 ± 7.065304.622 ± 7.074
      0.1379.685 ± 9.916173.778 ± 1.167620.332 ± 5.683218.099 ± 5.688
      0.2217.255 ± 6.179151.032 ± 1.107713.212 ± 3.913125.212 ± 3.903
      0.3140.670 ± 4.312135.080 ± 0.821756.687 ± 2.67881.283 ± 2.736
      Exponential0.05540.519 ± 34.791190.042 ± 3.018528.130 ± 20.201310.286 ± 20.189
      0.1392.527 ± 21.586175.243 ± 2.259612.691 ± 12.952225.748 ± 12.959
      0.2221.973 ± 16.554151.600 ± 2.797710.529 ± 9.843127.919 ± 9.848
      0.3138.885 ± 5.286134.581 ± 1.232757.801 ± 3.17380.176 ± 3.212
      311/10Constant0.05599.836 ± 4.694195.180 ± 0.395494.452 ± 2.598343.981 ± 2.595
      0.1469.100 ± 15.941183.202 ± 1.379567.531 ± 9.457270.928 ± 9.459
      0.2282.697 ± 11.985161.234 ± 1.839673.403 ± 7.241165.085 ± 7.232
      0.3179.271 ± 12.713143.288 ± 2.592733.515 ± 7.387104.909 ± 7.377
      Uniform0.05597.151 ± 15.691194.972 ± 1.356495.789 ± 8.860342.646 ± 8.834
      0.1454.195 ± 22.950181.205 ± 2.307575.522 ± 14.680262.993 ± 14.656
      0.2272.724 ± 10.595159.639 ± 1.469679.549 ± 6.300158.972 ± 6.288
      0.3186.063 ± 11.810143.749 ± 1.919730.187 ± 7.376108.222 ± 7.382
      Exponential0.05586.790 ± 26.908194.034 ± 2.375501.621 ± 15.651336.821 ± 15.615
      0.1455.625 ± 36.722181.899 ± 3.627576.228 ± 21.528262.220 ± 21.515
      0.2288.627 ± 48.243161.504 ± 6.316671.754 ± 28.302166.746 ± 28.270
      0.3187.779 ± 24.845144.600 ± 4.476729.932 ± 15.563108.485 ± 15.562

    • Table 4. Physical observables at the outgoing surface (L = 1 cm) at 5000 ps for a single physical configuration (N = 1). Units are the same as in Tables II and III. Parameters are taken from Table 4 in Ref. 20. In the fifth to eighth columns, the numbers inside and outside the parentheses represent data obtained with constant and temperature-dependent opacity, respectively. In line with the previous study,20Cv = 1 is used.

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      Table 4. Physical observables at the outgoing surface (L = 1 cm) at 5000 ps for a single physical configuration (N = 1). Units are the same as in Tables II and III. Parameters are taken from Table 4 in Ref. 20. In the fifth to eighth columns, the numbers inside and outside the parentheses represent data obtained with constant and temperature-dependent opacity, respectively. In line with the previous study,20Cv = 1 is used.

      CaseInitial parametersPhysical observables
      p1σa,1σa,2⟨Erad⟩⟨Tmat⟩⟨Jreflec⟩⟨Jtrans⟩
      Ht10.0510.00.03497.414(1111.490)186.186(227.171)556.593(219.947)283.097(619.133)
      Ht20.0510.00.1374.133(1060.040)173.432(225.305)624.807(244.641)213.642(593.969)
      Ht30.0530.00.3211.257(747.426)150.330(206.401)720.364(413.203)118.020(425.436)
      Ft10.210.00.03156.516(639.327)138.932(197.807)749.141(471.260)90.700(367.226)
      Ft20.210.00.1138.620(623.326)134.867(197.279)758.932(480.560)79.623(358.163)
      Ft30.230.00.387.199(338.691)120.012(169.261)790.228(643.454)48.163(195.373)
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    Cong-Zhang Gao, Ying Cai, Cheng-Wu Huang, Yang Zhao, Jian-Wei Yin, Zheng-Feng Fan, Jia-Min Yang, Pei Wang, Shao-Ping Zhu. Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions[J]. Matter and Radiation at Extremes, 2024, 9(6): 067802

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    Paper Information

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    Received: Mar. 14, 2024

    Accepted: Aug. 3, 2024

    Published Online: Jan. 8, 2025

    The Author Email:

    DOI:10.1063/5.0208236

    Topics

    laser devices and laser physics
    Lasers and Laser Optics
    Laser physics
    laser manufacturing
    Instrumentation, Measurement and Metrology