Frequency-Dependent Material Motion Benchmarks for Radiative Transfer
AANS M&C 2021 - The International Conference on Mathematics and Computational Methods Appliedto Nuclear Science and Engineering · Raleigh, North Carolina · April 11–15, 2021
Frequency-Dependent Material Motion Benchmarks for Radiative Transfer
Ryan G. McClarren , N.A. Gentile Dept. Aerospace and Mechanical Engineering, University of Notre DameNotre Dame, Indiana, USA L-38 Lawrence Livermore National LaboratoryLivermore, California, [email protected], [email protected]
ABSTRACT
We present a general solution for the radiation intensity in front of a purely absorbingslab moving toward an observer at constant speed and with a constant temperature. Thesolution is obtained by integrating the lab-frame radiation transport equation through theslab to the observer. We present comparisons between our benchmark and results fromthe Kull simulation code for an aluminum slab moving toward the observer at 2% thespeed-of-light. We demonstrate that ignoring certain material motion correction termsin the transport equation can lead to 20-80% errors with the error magnitude growing asthe frequency resolution is improved. Our results also indicate that our benchmark canidentify potential errors in the implementation of material motion corrections.
KEYWORDS: high-energy density physics, radiative transfer, material motion corrections, special relativity,verification, benchmark solutions
1. Radiation Spectrum in Front of a Moving Slab
We are interested in solving the problem of the radiation spectrum in front of a slab that is movingat a fast speed that is, nevertheless, slow relative to the speed of light. In our notation, the subscriptL denotes a quantity in the laboratory reference frame, and the subscript F is for the fluid, orco-moving frame.Considering the observation point z = Z , we compute the solution I L ( Z, µ, ν L , t ) . This problemis depicted in Fig. 1. For a photon to reach position Z at time t Z it will have its position z ( t ) givenby z ( t ) = Z + µc ( t − t Z ) . (1)Given that the back of the slab has position z b = vt , a photon that leaves the back of the slab andreaches Z at time t Z will be emitted at time t b that can be determined from the equation z ( t b ) = vt b as t b = (cid:18) µct Z − Zµc − v (cid:19) + , (2)where ( · ) + gives the positive part of its argument (i.e., ( x ) + = x if x > , and ( x ) + = 0 otherwise).Similarly, a photon reaching Z at time t Z emitted from the front of the slab at time t f : t f = (cid:18) L + µct Z − Zµc − v (cid:19) + . (3) a r X i v : . [ c s . C E ] A ug cClarren and Gentile z = 0 Zs θ slab motion L Distance s ismeasuredalong rayslab vacuumray Figure 1. Schematic of the problem: a slab of length L moves in the z direction toward theobservation point at z = Z . The solution I ( Z, µ, ν L , t ) with µ = cos θ is computed byintegrating along the ray over the length s , given by Eq. (4) . Additionally, the length of the ray through the slab for a photon is then s = c ( t f − t b ) (4)Using these relations the solution I ( Z, µ, ν L , t Z ) is I ( Z, µ, ν L , t Z ) = I Z ( µ L , ν L ) = B ( γD L ν L , T )( γD L ) [1 − exp ( − γD L σ a , slab ( γD L ν L ) s )] , (5)which can be found by using an integrating factor.
2. Analytic Results and Code Benchmarking
We consider a slab of aluminum with density of 0.1 g/cm , a thickness of L = 0 . cm., and atemperature of T = 1 keV. The slab is moving at a speed of 0.5994 cm/ns (about 2% of the speedof light), a speed chosen based on a Mach 45 radiating shock benchmark [3]. The observationpoint is Z = 12 cm and time t Z = 10 ns. We compute the multigroup radiation energy densitybased on three group structures: • A coarse structure with 50 logarithmically spaced groups from 0.001 keV to 30 keV, • A medium structure of 89 total groups based on the coarse set where the extra groups are addedbetween 1 and 10 keV, • A fine structure of 124 groups based on the medium set where the extra groups are added between1 and 2 keV. requency-Dependent Material Motion Benchmarks for Radiative Transfer −3 −2 −1 ν (keV)10 κ g ( c m / g )
50 group89 group124 group (a) κ g ν (keV)10 κ g ( c m / g )
50 group89 group124 group (b) Detail near spectral lines
Figure 2.
Group specific opacities, κ g in units of cm /g for the aluminum slab.These group structures were chosen to successively capture the spectral lines in the 1 and 2 keVrange. The opacities were provided by the TOPS opacity database [4] and are shown in Figure 2.Our benchmark solution was computed by using Mathematica to compute the solutions. We canalso investigate the effects of material motion corrections (MMC) by computing the solution fora stationary slab (i.e., v = 0 ), and when the frequencies in the opacity and blackbody emissionterms are not Doppler shifted by γD L (we call this the no ν Doppler solution). Even though in ourproblem v/c ≈ . , we will be able to show that ignoring MMC leads to errors on the order of80% for this problem.We can also compare the benchmark to solutions computed by the implicit Monte Carlo method[5,6] in Kull [7]. We can verify that the MMC is correctly implemented in Kull by comparing itssolution to the benchmark. We can also compare Kull solutions with different MMC terms turnedoff, for example the Doppler shift of ν and show that the benchmark can elucidate that these termsare missing.Results for the 50 group problem are shown in Figure 3. In Figure 3(a) the Kull solution withfull MMC and × particles per time step and the benchmark are compared, and reasonableagreement outside of the low-energy groups with few computational particles is observed. Theimportance of MMC is quantified in Figure 3(b) where the benchmark is compared with solutionswhere the slab is stationary and without the Doppler shift on ν via the percent absolute error. Herewe can see that errors up to 20% occur in the higher energy groups when the Doppler shift of ν isignored. Finally, Figure 3(c) compares the Kull IMC solution with full MMC and the benchmark,as well as the solution lacking the Doppler correction and a solution where the slab is stationary.Figure 3(c) indicates that above photon energies of 1 keV the Kull solution is, on the whole, closerto the benchmark than to the solutions missing MMC corrections. We can conclude from thisfigure that our benchmark is able to verify if certain MMC terms are correctly implemented in acode. cClarren and Gentile −2 −1 ν (keV)10 −7 −6 −5 −4 −3 E r a d , g ( G J / c m - ke V ) BenchmarkKull IMC (a) Benchmark and KULL Result −3 −2 −1 ν (keV)05101520253035 % E rr o r No ν Dopplerv=0 (b) Absolute % Error in Different Models −1 ν (keV)10 −1 % E rr o r BenchmarkNo ν Dopplerv=0 (c) Absolute % Error between KULL solu-tions with MMC and different analytic so-lutions
Figure 3.
Comparisons for the 50 group aluminum slab problem: (a) the benchmark solution andthe KULL IMC solution with all MMC terms, (b) comparison of the benchmark solution with asolution where the slab is stationary ( v = 0 ) and one where the Doppler correction for thefrequency is ignored, and (c) comparison of the KULL IMC solution with full MMC with thebenchmark solution, a solution with frequency Doppler shifts ignored, and with the slabstationary. ACKNOWLEDGEMENTS
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Liver-more National Laboratory under Contract DE-AC52-07NA27344. Lawrence Livermore NationalSecurity, LLC. LLNL-ABS-813186
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