Multi-Group Discontinuous Asymptotic P_1 Approximation in Radiative Marshak Waves Experiments
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b MULTI-GROUP DISCONTINUOUS ASYMPTOTIC P APPROXIMATION IN RADIATIVE MARSHAK WAVESEXPERIMENTS
A.P. Cohen, S.I. Heizler
Department of Physics, Nuclear Research Center-Negev, P.O. Box 9001, Beer Sheva 8419001, ISRAEL
Abstract
We study the propagation of radiative heat (Marshak) waves, using modified P -approximation equations.In relatively optically-thin media the heat propagation is supersonic, i.e. hydrodynamic motion is negligible,and thus can be described by the radiative transfer Boltzmann equation, coupled with the material energyequation. However, the exact thermal radiative transfer problem is still difficult to solve and requires massivesimulation capabilities. Hence, there still exists a need for adequate approximations that are comparativelyeasy to carry out. Classic approximations, such as the classic diffusion and classic P , fail to describe thecorrect heat wave velocity, when the optical depth is not sufficiently high. Therefore, we use the recentlydeveloped discontinuous asymptotic P approximation, which is a time-dependent analogy for the adjustmentof the discontinuous asymptotic diffusion for two different zones. This approximation was tested via severalbenchmarks, showing better results than other common approximations, and has also demonstrated a goodagreement with a main Marshak wave experiment and its Monte-Carlo gray simulation. Here we derive energyexpansion of the discontinuous asymptotic P approximation in slab geometry, and test it with numerousexperimental results for propagating Marshak waves inside low density foams. The new approximationdescribes the heat wave propagation with good agreement. Furthermore, a comparison of the simulationsto exact implicit Monte-Carlo slab-geometry multi-group simulations, in this wide range of experimentalconditions, demonstrates the superiority of this approximation to others. Keywords:
Radiative transfer, Marshak waves, P approximation
1. INTRODUCTION
The problem posed by the transfer of supersonic radiation (Marshak) waves in matter is of considerableimportance in high energy density physics [1, 2]. Its solution greatly influences the ability to achieve inertialconfinement fusion (ICF), and to describe radiation procedures in stars. This problem is modeled via theBoltzmann (transport) equation for photons, coupled to the matter via the energy balance equation [1, 3].In the recent decades, several experiments involving the propagation of supersonic Marshak waves through
Email address: [email protected] (A.P. Cohen)
Preprint submitted to Journal of L A TEX Templates February 11, 2021 ow-density foams have been carried out and reported [4, 5, 6, 7, 8, 9] (for a short review, see [10]). Typically,in these experiments high energy laser beams are shot into hohlraums, which radiate an X-rays into a dilutefoam attached to the hohlraum. These experiments help validate radiative heat theoretical models. Asnoted, the exact solution of the Boltzmann equation is the principal component of these models, besidemicroscopic opacity data.The Boltzmann equation can be solved numerically by Implicit Monte-Carlo (IMC) simulations, whichare exact when the number of histories goes to infinity [11]. Alternatively, a discrete ordinates ( S N ) methodcan be used, which is exact when the number of ordinates goes to infinity, or the spherical harmonics( P N ) approximation, which is exact when the number of moments goes to infinity [3]. However, thesecalculations are still hard to carry out, especially in more than one dimension, and when scanning manyphysical parameters. Hence, there still exists a need for good adequate approximations that are comparativelyeasy to carry out [12].When the matter is optically thick the Boltzmann equation can be approximated by the diffusion equa-tion [3]. However, when the number of mean free paths in the matter is close to one, the diffusion equation isno longer valid. The full P equations, that give rise to the Telegrapher’s equation, has a hyperbolic form, butwith an incorrect finite velocity, c/ √ P equations), yield a gradient-dependent nonlinear diffusion coefficients (or a gradient-dependentEddington factor) [12, 15], and are still not exact enough to be trusted to make a full reconstruction ofexperimental results [16].The discontinuous asymptotic P approximation is being derived to approximate the Boltzmann solutionin heterogeneous media. It rests on two foundations: (1) the asymptotic P approximation [13, 17, 18, 19];(2) the asymptotic jump conditions in the boundary between two different media, assuming flux continuity(and thus energy conservation), and a discontinuity in the energy density [20]. This approximation is basedon an asymptotic thick limit approach , however, due to the jump conditions, converging to the exactsolution much faster than classic diffusion. These modifications allow the use of a P form that is accurateeven in highly-anisotropic scenarios. This approximation was tested via well-known benchmarks, showingbetter results than other common approximations [21]. In addition, it was also compared to the results in oneof the main Marshak wave experiments (using SiO foam [6]) and IMC gray (mono-energetic) simulations ofthe same problem, showing good agreements [16].In the Marshak wave experiments, the radiation flux that breaks out from the end of the sample ismeasured in a specific energy channel. Hence, the gray solution is not sufficient for analyzing the experimentalresults, and we have to solve the complete multi-energy problem. In the present study we developed a non-gray modification of the discontinuous asymptotic P approximation , using multi-group notations. Thisapproximation is validated by comparing it with numerous experiments, in different physical regimes thathave recently been published. 2 . THE MULTI-GROUP DISCONTINUOUS ASYMPTOTIC P EQUATIONS
The radiative-transfer Boltzmann equation in the frequency-dependent case is:1 c ∂I ( ˆΩ , ~r, t, ν ) ∂t + ˆΩ · ~ ∇ I ( ˆΩ , ~r, t, ν ) + ( σ ′ a ( ν, T m ( ~r, t )) + σ s ( ν, T m ( ~r, t ))) I ( ˆΩ , ~r, t, ν ) = σ ′ a ( ν, T m ( ~r, t )) B ( ν, T m ( ~r, t )) + σ s ( ν, T m ( ~r, t ))4 π Z π I ( ˆΩ , ~r, t, ν ) d ˆΩ + S ( ˆΩ , ~r, t, ν ) (1)where I ( ˆΩ , ~r, t, ν ) is the specific intensity of radiation at position ~r propagating in the ˆΩ direction at time t and frequency ν . B ( ν, T m ) is the black-body radiation term, where T m ( ~r, t ) is the material temperature, c isthe speed of light and S ( ˆΩ , ~r, t ) is an external radiation source. σ ′ a and σ s are the absorption (opacity) andscattering cross-sections, respectively. Eq. 1 assumes an elastic isotropic scattering (i.e. Thomson scattering).Along with the equation for the radiation energy, the complementary equation for the material is: C v ( T m ( ~r, t )) c ∂T m ( ~r, t ) ∂t = σ ′ a ( ν, T m ( ~r, t )) (cid:18) c Z π I ( ˆΩ , ~r, t, ν ) d ˆΩ − aT m ( ~r, t ) (cid:19) (2)where C v ( T m ( ~r, t )) is the heat capacity of the material and a is the radiation constant ( aT m = R ∞ dνB ( ν, T m )).Usually, the energy (frequency) dependency is modeled via the multi-group approximation [3]. Inthis approximation the energy space is divided into G discrete groups, defining a group specific intensity I g ( ˆΩ , ~r, t ) = R ν g ν g − dνI ( ˆΩ , ~r, t, ν ). We also defined b g coefficient as b g = R ν g ν g − dνB ( ν, T m ) . aT m ) , and thegroup Rosseland mean opacity:1 σ ′ ag = Z ν g ν g − dν σ ′ a ( ν, T m ) ∂B ( ν, T m ) ∂T m , Z ν g ν g − dν ∂B ( ν, T m ) ∂T m (3)The energy-dependent Boltzmann equation (Eq. 1) is now replaced with coupled G mono-energetic equations:1 c ∂I g ( ˆΩ , ~r, t ) ∂t + ˆΩ · ~ ∇ I g ( ˆΩ , ~r, t ) = σ ′ ag ( T m ( ~r, t )) h b g ac π T m ( ~r, t ) − I g ( ˆΩ , ~r, t ) i + S g ( ˆΩ , ~r, t ) (4)In Eq. 4 we ignore the scattering term since in the 100-300eV range, scattering is negligible. In the presentstudy, we have used 100 groups opacity cross-sections tables using the CRSTA approximation [22]. The foamheat capacities C v were taken from QEOS tables [23].The first two angular moments of the group-dependent specific intensity I g ( ˆΩ , ~r, t ) can be expressed as: E g ( ~r, t ) = 1 c Z π I g ( ˆΩ , ~r, t, ) d ˆΩ (5) ~F g ( ~r, t ) = Z π I g ( ˆΩ , ~r, t ) ˆΩ d ˆΩ (6)where E g ( ~r, t ) is the group energy density, and ~F g ( ~r, t ) is the group radiation flux. Integrating Eq. 4 overall solid angle R d ˆΩ yields the exact conservation law :1 c ∂E g ( ~r, t ) ∂t + 1 c ∇ · ~F g ( ~r, t ) = σ ′ ag ( T m ( ~r, t )) (cid:0) b g acT m ( ~r, t ) − E g ( ~r, t ) (cid:1) + S g ( ~r, t ) c (7)3ntegrating Eq. 4 with R ˆΩ d ˆΩ, assuming the classic P closure (when the specific intensity is decomposed ofits two first moments) yields: A g ( ~r, t ) c ∂ ~F g ( ~r, t ) ∂t + c~ ∇ E g ( ~r, t ) + B g ( ~r, t ) σ ′ ag ( T m ( ~r, t )) ~F g ( ~r, t ) = 0 (8)when A g = B g = 3. In the asymptotic P approximation instead, the coefficients A g ( ~r, t ) and B g ( ~r, t ) aremedia-dependent, which have closed sets of functions (not free parameters), of ω eff ,g ( ~r, t ), the group-dependent mean number of particles that are emitted per collision or source terms and (for the non-scattering case) isdefined by [3, 21]: ω eff ,g ( ~r, t ) = σ ′ ag b g acT m ( ~r, t ) + S g ( ~r, t ) /cσ ′ ag E g ( ~r, t ) . (9)Notice that in the case of LTE, ω eff ,g ≡
1, the asymptotic distribution tends to the diffusion limit. Thecoefficients are derived from an asymptotic derivation, both in space and time, as detailed in [13, 17]. Forthe exact functional dependence of A g ( ω eff ,g ) and B g ( ω eff ,g ), please see [13, 21]. We note that setting A g = 1and B g = 3 reproduces the P / approximation, which yields the correct particle velocity [12].The asymptotic P approximation supplies a good description in isotropic medium, especially in latetimes. It yields the exact steady-state asymptotic solution, similar to the the asymptotic diffusion approxi-mation [24] and the (almost) correct particle velocity. However, in heterogeneous media, for example, in asystem with sharp boundaries between media, this approximation is not good enough.A series of studies yield the exact solution of two adjacent semi-infinite half-spaces problem (the two-region Milne problem) yielding the exact boundary conditions (when both the asymptotic radiation flux andthe asymptotic energy density are discontinuous) [25, 26, 27, 28]. Zimmerman has offered a discontinuousasymptotic diffusion version, based on Marshak-like approximation for the asymptotic jump conditions [20].This approximation assumes continuous radiation flux ~F A = ~F B (between semi-infinite medium A and semi-infinite medium B ) and discontinues energy density, µ A E A = µ B E B , when µ ( ω eff ,g ) is a medium dependentfunction, and derived by applying the Marshak boundary condition on the asymptotic distribution near aboundary between two different media [3]. We [21] expanded these boundary jump conditions to the wholemedium for the time-dependent asymptotic P equation (Eq. 8): µ g ( ~r, t ) A g ( ~r, t ) c ∂ ~F g ( ~r, t ) ∂t + c~ ∇ ( µ g ( ~r, t ) E g ( ~r, t )) + µ g ( ~r, t ) B g ( ~r, t ) σ ′ ag ( T m ( ~r, t )) ~F g ( ~r, t ) = 0 (10)when we have added the index g for the multi-group case.The approximation has been tested versus the known gray Su-Olson Benchmark [29] and the nonlinearOlson benchmark [12], showing better results than other common approximations [21]. Although thesebenchmarks are characterized in highly-anisotropic flux profile, these modifications that are based on theexact asymptotic solutions, yield good results using P -form equations. In addition, a gray comparison in oneMarshak wave experiment (using SiO foam) showed a good agreement between the discontinuous asymptotic P on the one hand, and the IMC and the experimental results on the other [16]. The experimental resultis of course non-gray and thus different, due to the specific measure in a specific energy band. In this workwe expand the testing in fully multi-group treatment, in various of experimental setups.4 . THE EXPERIMENTS The different experiments that are examined in this study all possess a common procedure, which ispresented schematically in Fig. 1. The blue rays represent high energy (1kJ-10kJ) laser beams that are beenshot into a small ( ∼ Figure 1: A schematic diagram of typical Marshak wave experiments. Laser beam (bluelines) heats the hohlraum, which re-emits soft X-ray radiation (red arrows) into a dilutefoam (gray), then, a Marshak wave propagates in the foam. The figure is taken from [30].
When the heat wave front reaches the edge of the foam, the radiation flux is measured as a functionof time by an X-ray steak camera or an X-ray diode (XRD) [5, 6, 7, 9]. The radiation temperature in thehohlraum is measured as a function of time, usually through the laser entrance hall (LEH in Fig. 1). We usethis temperature to estimate the heat wave drive temperature profile T D ( t ), assuming a black body radiationsource (in frequency and direction) with a given drive temperature (for a wide discussion, see [10, 16]). Themain physical phenomena is the approximately one-dimensional (1D) slab-geometry heat wave propagationin the center of the cylinder. This phenomena is examined in the numerical simulations presented in the nextsection. Other effects, such as the energy leakage to the gold walls, the walls ablation and two-dimensionalgeometry effects, are of second importance, than the basic 1D phenomena [10, 16, 31, 32]The properties of the different experiments that are examined in this study are summarized in Table 1.We specify the range of the optical depths (in mfp) in each experiment, which are reported in the eachpaper. In general, the range varies with the foam’s length, i.e. shorter length means smaller optical depth.An exception is the Moore et al. experiment which has a given constant length, when the range of opticaldepth describes the thermodynamic track during the heat propagation through the foam [9]. This parameterallow us to classify the limits of validity of the different approximations, comparing to the IMC solution andexperimental results. 5 he experiment Foam Type density Max. Temp. Optical[mg/cc] [eV] depth Back et al. PRL SiO
10 85 1-2.5Back et al. POP SiO
50 190 0.75-1.75Back et al. POP Ta O
40 190 1.2-5Xu et al. C H , 50 160 0.65-0.85Moore et al. C H Cl 100 310 2.5-7
Table 1: The different experiments studied in this paper. For each experiment, we specifythe material of the foam, its density, and the maximal drive temperature and itsapproximated optical depth.4. BACK Ta O & SiO HIGH- ENERGY EXPERIMENTS
The first experiments we examine are the Back et al. Ta O and SiO high-energy experiments, whichwere carried out in the OMEGA-60 facility [6]. The foam, either SiO at 50mg/cc or 40mg/cc Ta O , wascoated with a gold cylinder. Three different cylinder foam lengths were used for the SiO sample, and fourdifferent lengths in the Ta O case. The maximal drive energy temperature measured in these experimentswas ≈ eV . The emitted radiation flux was measured by an X-ray streak camera, measuring photons inspectral band in about 550eV. The experimental results of the flux are shown in [6] in arbitrary units.Fig. 2 shows a comparison between the experimental results (orange), the 1D IMC solution (green) anddifferent 1D approximations. The calculated flux in the simulation is in energy band of 535-580eV. The IMCnumerical simulations units were scaled such that the IMC results will best fit to the experimental resultsin the 0.25mm long cylinder for the Ta O (Fig. 2(a)) and 1mm long in SiO (Fig. 2(b)). All the othernumerical simulations where scaled in the same factor. First, we note that in long foam lengths (1mm inTa O and 1.25mm in SiO ), all simulations yield faster flux rise than do the experiments. This is due to2D effects, that are broadly explained in [10, 16].It can be seen that the discontinuous asymptotic P approximation (blue curves) fits the IMC solutionnicely, especially in the Ta O experiment. In the SiO experiment, it yields very good agreement inthe breakout times, but lower maximal results, due to the inherent discontinuity [21]). Both the classicdiffusion (black) and classic P approximation (not shown) yields faster results, and the Larsen flux-limiteddiffusion [12] is late, especially in the SiO experiment. In general, the differences between the differentapproximations in the Ta O experiment are small, due to the relative high-opacity of this experiment (seeTable. 1). 6a) (b) Figure 2: (a) The radiation flux that was emitted from the edge of the foam as function oftime in Back et al. high energy Ta O5 experiment [6]. Four different experiments arerepresented at different foam lengths: 0.25, 0.5, 0.75, 1mm. (b) The SiO experiment [6] ispresented using three different foam lengths: 0.5, 1. 1.25 mm. The experimental results arerepresented by the orange curves. The green curves represent the 1D IMC simulations, theclassic diffusion approximation in shown in black curves, the Larsen flux-limited diffusion inred and the discontinuous asymptotic P approximation in blue curves.5. BACK SiO LOW-ENERGY EXPERIMENT
Another important experiment, also carried out in the OMEGA-60 facility, is the low-energy SiO ex-periment (the maximal drive energy temperature was ≈ eV ). Here, SiO foam at 10mg/cc density wascoated with a gold cylinder. Three different experiments were carried out with three different foam lengths:0.5, 1, 1.5mm. The emitted radiation flux was measured as a function of time with a spectral band in about hν =250eV energy. Figure 3: The radiation flux that was emitted out of the edge of the foam as function of timein the
SiO low-energy experiment [5]. Three different foam lengths: 0.5, 1, 1.5 mm arepresented. P approximation yields the closest agreement withthe IMC simulations, by all means. This is due to the fact that this experiment is relatively optically thick,specifically in the larger lengths. In this experiment we have explicitly checked also the P / approximation,when the results were very close to the the classic diffusion approximation, as well classic P approximation.This is not surprising since this problem is similar to the nonlinear Olson benchmark [12].
6. Xu EXPERIMENT
The next experiment examined in this study was carried out in the SG-II facility [7, 8]. Cylinder foammade of 50mg/cc C H foam were used in two different lengths: 300 and 400 µ m. The maximal drivetemperature inside the hohlraum was ≈ eV . The CH foam is optically thin due to its low Z, in respect tothe experiments been discussed earlier, with optical depth that is less than 1 (Table 1). Hence, the radiationflux starts to leak from the edge of the foam, before the material has been significantly heated (as can beseen later in Fig. 6(a)). In the multi group simulations that also cause the heat wave, spatial profile has along ”tail”, and not a sharp front as in the gray simulations.(a) (b) Figure 4: A comparison between the experimental results (orange), and 1D simulations forthe C H experiment [7], in two different energy bands with ( eV - (a) and eV - (b)), in300 µm foam length. Two energy channels were used in the experiment to measure the flux coming out from the edge of thefoam, in the 300 µm length foam. The first, around 210eV where the opacity is small, and around 420eV wherethe opacity is higher. The simulations presented in that paper [7], showed a significant difference betweenthe two channels (though the experimental data is less decisive). This also demonstrates the importance of8sing multi-group simulations (gray simulations do not represent the real experimental picture). The earlierbreak out in the 210eV channel, is also shown in our IMC simulation shown in Fig. 4(a) (green line), whenthe calculated flux is in energy band 169-223eV and in Fig. 4(b) is in 365-468eV energies.A comparison of the radiated flux in the different foam lengths between the experimental results (orange),the 1D IMC solution (green) and the 1D approximations is shown in Fig. 5. Here, as shown in both Fig. 4and Fig. 5, the classic diffusion (black) yields the closest agreement with the IMC simulations and theexperimental results. The Larsen flux-limited diffusion (red) is significantly slower than the IMC simulations.Our discontinuous asymptotic P approximation (blue) yields the correct breakout time (when the flux rises),but does not fit well at the maximum flux area. However, the relative gap between the two lengths is similarin all approximations and in the experiments. The low opacity of this experiment (less than 1 mfp) is thereason that the discontinuous asymptotic P is less successive in this experiment, since it is an asymptoticthick limit approach approximation. Figure 5: A comparison between the experimental results (orange), and 1D simulations forthe C H experiment [7], in two different foam length ( µm and µm ). To demonstrate the low opacity of this experiment we show in Fig. 6(a) that the heat wave front (in T m , the solid line) does not reach to the edge of the foam, at time=0.9nsec while the effective radiationtemperature, T r = ( E/a ) / has reached the edge at that time. In this optically thin medium, there are“holes” of very optically thin energy bands. That mean that the flux that is measured in the 210eV range,and arises at 0.5-0.9nsec, does not represent the main heat wave progress. Note that the average radiationfront profile in the discontinuous asymptotic P is very similar to the IMC one (which explains the goodagreement between IMC and the discontinuous asymptotic P in the breakout times ).This difference, between T m and T r is because the heat wave transfers in an optically thin material. Forcomparison, we compare the heat wave spatial profile close to the break out time in the optically thick Ta O experiment (1.9nsec in 1mm foam length) in the different approximations (Fig. 6(b)). It can be seen that inthis case, the effective radiation temperature, is very close to the material temperature, T m , which means thesystem is close to local thermodynamic equilibrium. Again, the discontinuous asymptotic P approximation9blue curves) heat wave front progresses at almost the same speed as in the IMC instance. .(a) (b) Figure 6: (a) The C H experiment [7] spatial temperature profiles are presented attime=0.9nsec. The green curves represent the 1D IMC simulations, the classic diffusionapproximation in shown in black curves, the Larsen flux-limited diffusion in red and thediscontinuous asymptotic P approximation in blue curves. The material temperature, T m inthe solid curves, and the effective radiation temperature, T r in the dashed curves. (b) Theheat wave spatial temperature profiles, at time=1.9nsec in Back et al. high energy Ta O5 experiment [6].7. MOORE C H Cl EXPERIMENT
The last experiment examined in this study is the most advanced and detailed set of experiments reportedup to date, and are known as the Pleiades experiments [9]. These experiments were conducted in the high-power NIF facility and the drive temperature had reached ≈ H Cl foam (The Cl plays a major role in determining the foam opacity) indifferent densities (of about 100mg / cc).In this study we examine one representative example, a 100mm / cm C H Cl foam. The shot’s energywas 367.3kJ, and the radiation incident flux to the foam was calculated numerically based on the measuredflux radiated from the back of the hohlraum [9]. The 2.8mm long and 2mm diameter cylindrical foam wascoated with Au tube. The 1D simulations cannot describe the experimental results because this experimenthas significant 2D effects [10]. However the 1D simulations still important for estimating the sensitivity tolocal variables, such as the exact opacity of the foam. Thus, we compare the different approximations to a1D IMC simulations.A comparison between the 1D IMC solution (green) and the 1D approximations is shown in Fig. 7, wherethe calculated flux in the simulation is in energy band of 80-969eV (similar to the energy band that wasin use in the experiment). We can see that the classic diffusion (black) yields results that are faster andhigher than the IMC results. The Larsen flux-limited diffusion (red) is significantly slower and lower than the10 igure 7: A comparison between the 1D simulations for the C H Cl experiment, in twodifferent opacity tables (The numerical CRSTA table, and the same opacity multiplied by 1.1(dash curve). IMC simulations. The discontinuous asymptotic P Approximation (blue) yields the closest agreement withthe IMC simulations. Again, this is because this experiment was conducted in a relatively optically-thickmedium (see Table 1).The nominal calculations were carried out with the nominal CRSTA table [22] for C H Cl. It is quitereasonable to assume an error bar of 10%. Hence, we set the same calculations, only with a multiplied 1.1CRSTA opacity factor (dash curves). The result points out that the discontinuous asymptotic P Approxi-mation, also estimates this effect correctly.
8. QUANTITATIVE ANALYSIS
To quantify the level of agreement of the different approximations, we summarize different parametersof the measured fluxes signals: The breakout (burnthrough) times (Table 2), and the maximal value of thesignal (Table 3), comparing to 1D IMC results. The errors of the different approximations are written inparentheses, while the approximation with the smallest error is marked in red, in each experiment. The timeat which the flux starts to rise is termed the breakout time, which we define as the time the flux reaches25% of its maximal value. The maximal IMC flux values were calculated after smoothing (moving averagewith span 5-9, in different experiments).First, in Table 2 the discontinuous asymptotic P approximation yields better results than the otherapproximations in all experiments - except the Xu experiment, which is extremely optically thin (Table 1),as explained (see Sec. 6). Finally, checking the maximal flux values in Table 3, the discontinuous asymptotic P yields the best approximation in the opaque experiments: the Moore experiment, the Ta O experimentand the low-energy SiO Back experiment. In the optically thin experiments, i.e. the Xu and the high-energySiO experiments, it yields maximal flux results that are too low.11xperiment Name Foam’s length Monte-Carlo Diffusion Flux-limited DiscontinuousFoam [mm] diffusion asymptotic P Back PoP 0.5 0.903 0.866 (-4.0%) 0.974 (7.9%) 0.903 (0.02%)SiO O H H Cl 2.8 + 2.303 2.203 (-4.3%) 2.412 (4.8%) 2.233 (-3.0%)10% Opacity
Table 2: Breakout times [nsec] of the different experiments studied in this paper. For eachexperiment, we compare the breakout times (flux reaches of its maximum value) in theIMC simulation with the different approximations. The smallest difference is marked in red. P Back PoP 0.5 0.143 0.150 (5.4%) 0.135 (-5.4%) 0.127 (-10.8%)SiO O H H Cl 2.8 + 0.185 0.232 (25.0%) 0.193 (4.3%) 0.188 (1.2%)10% Opacity
Table 3: Maximal flux values (a.u.) of the different experiments studied in this paper. Foreach experiment, we compare the maximal flux values in the IMC simulation with thedifferent approximations. The smallest difference is marked in red. . CONCLUDING REMARKS Radiative heat (Marshak) waves experiments have been studied for the last three decades, and werecarried out in the world’s leading high energy laser facilities, such as OMEGA-60, SG-II and the NIFfacilities. The basic physical process that is tested is the radiative transfer of thermal X-ray photons insidemedia of several Rosseland mean free paths, generating Marshak waves, as well as the microscopic opacities(that determines the mfp).The main equation that governs this physical process is the transport (Boltzmann) equation for photons.An exact solution for this equation is a heavy computation task and hard to obtain, thus good approximationsmay be very useful. Unfortunately, the classic well-known LTE diffusion approximation yields insufficientresults, due to the relative low-number of mfp that characterizes these experiments (see Table 1). Recently,we have derived a new modified P approximation, called the discontinuous asymptotic P approximation [21,16], which rests on some basic foundations of transport theory, such as the Case et al. asymptotic solutionfor infinite homogeneous transport equation [24], and the jump conditions of the asymptotic solution ofthe two-region Milne problem [25, 26, 27], with no free parameters. This approximation, which was testedin simple gray benchmarks, also yields many of the transport properties found in the exact full transportsolution.In this study we have expanded the discontinuous asymptotic P approximation for multi-group condi-tions, which is essential for calculating the Marshak wave experiments, where the heat front is tracked viathe measure of the radiated flux in specific energy-bands. The new approximation was validated by variousexperiments in different physical conditions. The agreement of the new approximation results with the IMCsimulations in most of the experiments is surprisingly high. As this approximation rests on asymptoticbehaviors of the exact transport equation, i.e. uses an asymptotic thick limit approach , it yields betterresults as the optical depth of the experiment is larger. The Xu et al. experiment [8], when the approximatedmfp is less than 1, is an exception, when the new approximation yields results that are less satisfactory thanother approximations. In those experiments not affected by 2D effects (the shorter lengths), the simulationsalso usually yield close agreement with the experimental results.This study integrates the basic theoretical derivation of an approximation for the transport equation usingasymptotic regimes, for modeling complicated Marshak waves experiments. The approximation that is basedon basic milestone works in transport, and was first tested in analytic benchmarks, ultimately managed todescribe full experimental results. In future studies we plan to examine whether the computational benefitsto be derived from using simple modified Pi equations, as compared to full transport approaches, are moresignificant in multi-dimensions and in complicated radiation energy-flow scenarios. ACKNOWLEDGEMENTS
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