Self-similar Solution of the Subsonic Radiative Heat Equations Using a Binary Equation of State

Heizler, Shay I.; Shussman, Tomer; Malka, Elad

doi:10.1080/23324309.2016.1157493

Cited by 15 publications

(19 citation statements)

References 13 publications

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“…In this section we introduce a simple model that is derived from the analytical solutions of one-dimensional radiative heat waves, both supersonic and subsonic [20][21][22][23][24][25][26][27]. First, we introduce a one-dimensional (1D) model, which is based on the analytic solution for the supersonic Marshak waves of Hammer and Rosen (HR) [25], pointing out the importance of the different radiation temperatures that are involved on the problem [5,28,29].…”

Section: Simple Analytic Model For the Calculating The Heat Frontmentioning

confidence: 99%

“…Then, we expand the model to include the two-dimensional (2D) effect of the energy loss to the gold walls. This extension is based on the self-similar one-dimensional subsonic Marshak waves solutions for gold [26,27].…”

Section: Simple Analytic Model For the Calculating The Heat Frontmentioning

confidence: 99%

“…There is a vast literature covering self-similar solutions of both supersonic and subsonic radiative (Marshak) heat-waves, assuming LTE conditions, i.e. E( r, t) ≈ aT 4 m ( r, t) [20][21][22][23][24][25][26][27]. In these solutions, one assumes that the Rossland mean opacity κ (which is connected to the absorption cross-section σ a (T m ( r, t)) = κρ, when ρ is the material's density) and the internal energy e(T, ρ) can be approximated in a power-law form (using [25] notations):…”

Section: A the Heat Wave Profilementioning

confidence: 99%

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Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Cohen¹,

Heizler²

2018

Journal of Computational and Theoretical Transport

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We study the problem of radiative heat (Marshak) waves using advanced approximate approaches. Supersonic radiative Marshak waves that are propagating into a material are radiation dominated (i.e. hydrodynamic motion is negligible), and can be described by the Boltzmann equation. However, the exact thermal radiative transfer problem is a nontrivial one, and there still exists a need for approximations that are simple to solve. The discontinuous asymptotic P1 approximation, which is a combination of the asymptotic P1 and the discontinuous asymptotic diffusion approximations, was tested in previous work via theoretical benchmarks. Here we analyze a fundamental and typical experiment of a supersonic Marshak wave propagation in a low-density SiO2 foam cylinder, embedded in gold walls. First, we offer a simple analytic model, that grasps the main effects dominating the physical system. We find the physics governing the system to be dominated by a simple, onedimensional effect, based on the careful observation of the different radiation temperatures that are involved in the problem. The model is completed with the main two-dimensional effect which is caused by the loss of energy to the gold walls. Second, we examine the validity of the discontinuous asymptotic P1 approximation, comparing to exact simulations with good accuracy. Specifically, the heat front position as a function of the time is reproduced perfectly in compare to exact Boltzmann

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Section: Simple Analytic Model For the Calculating The Heat Frontmentioning

confidence: 99%

Section: Simple Analytic Model For the Calculating The Heat Frontmentioning

confidence: 99%

Section: A the Heat Wave Profilementioning

confidence: 99%

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Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Cohen¹,

Heizler²

2018

Journal of Computational and Theoretical Transport

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“…The classic Marshak wave is a self-similar solution in a single similarity variable, and producing those solutions has been the study of several previous studies [11–13]. There have been extensions to the original problem to include two-dimensional effects [14], non-constant drives [15], non-stationary material [16], binary equations of state [17] and magnetic effects [18]. To date there have been no self-similar solutions presented for a multi-temperature model.…”

Section: Introductionmentioning

confidence: 99%

Self-similar solutions for high-energy density radiative transfer with separate ion and electron temperatures

Bennett

McClarren

2021

Proc. R. Soc. A.

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We consider a non-equilibrium plasma radiatively heated by an external source in the high-energy density regime. It is shown that self-similar solutions resembling the classic Marshak wave exist for the ion and electron temperatures, if the electron–ion coupling coefficient scales inversely proportional to either the time from when the heating begins or to the square of the distance from the surface of the plasma. We discuss how such a coupling coefficient may arise, and demonstrate that these solutions are also useful for verifying simulation codes that treat radiation–electron–ion coupling in high-energy density plasmas.

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“…This problem has long been a subject of theoretical astrophysics research [5,6], and of experimental studies testing radiative-hydrodynamics macroscopic modeling [7,8]. Specifically, the propagating radiative Marshak waves in optically thick media are well described by a simple local thermodynamic equilibrium (LTE) diffusion model, yielding self-similar solutions of both supersonic and subsonic regimes [9][10][11][12]. However, in optically thin media, the diffusion limit fails to describe the exact physical behavior of the problem.…”

Section: Introductionmentioning

confidence: 99%

The Discontinuous Asymptotic Telegrapher’s Equation (P₁) Approximation

Cohen¹,

Perry

Heizler³

2018

Nuclear Science and Engineering

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Modeling the propagation of radiative heat-waves in optically thick material using a diffusive approximation is a well-known problem. In optically thin material, classic methods, such as classic diffusion or classic P1, yield the wrong heat wave propagation behavior, and higher order approximation might be required, making the solution harder to obtain. The asymptotic P1 approximation [Heizler, NSE 166, 17 (2010)] yields the correct particle velocity but fails to model the correct behavior in highly anisotropic media, such as problems that involve sharp boundary between media or strong sources. However, the solution for the two-region Milne problem of two adjacent halfspaces divided by a sharp boundary, yields a discontinuity in the asymptotic solutions, that makes it possible to solve steady-state problems, especially in neutronics. In this work we expand the timedependent asymptotic P1 approximation to a highly anisotropic media, using the discontinuity jump conditions of the energy density, yielding a modified discontinuous P1 equations in general geometry. We introduce numerical solutions for two fundamental benchmarks in plane symmetry. The results thus obtained are more accurate than those attained by other methods, such as Flux-Limiters or Variable Eddington Factor. * avnerco@gmail.com † highzlers@walla.co.il deterministic methods, and they are both exact when N → ∞ [13]. Alternatively, a statistical implicit Monte Carlo (IMC) approach can also be used [14], which is exact when the number of particles (histories) goes to infinity. Although these three methods approach the exact solution, their application requires extensive numerical calculations that might be difficult to carry out, especially in multi-dimensions. Hence, there is an extensive body of literature dealing with the search for approximate models which will be relatively easy to simulate, and yet produce solutions that are close to the exact problem (for example, see [15,16]).The classical (Eddington) diffusion theory, as a specific case of the P 1 is relatively easy to solve and is commonly used [1,4,13]. The diffusion equation is parabolic, and thus yields infinite particle velocities. The full P 1 equations, that give rise to the Telegrapher's equation, has a hyperbolic form, but with an incorrect finite velocity, c/ √ 3 [17]. Possible solutions, such as flux-limiters (FL) solution (in the form of a non-linear diffusion notation), or Variable Eddington Factor (VEF) approximations (in the form of full P 1 equations), yielding a gradient-dependent nonlinear diffusion coefficients (or a gradient-dependent Eddington factor), are harder to solve, especially in multi-dimensions [15,16,[18][19][20][21][22][23][24].In previous work, Heizler [17] offered a modified P 1 approximation, based on the asymptotic derivation (both in space and time), the asymptotic P 1 approximation (or the asymptotic Telegrapher's equation approximation) [17]. In steady state, it tends to the well-known asymptotic diffusion approximation [13,25,26]. This approximation shar...

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Self-similar Solution of the Subsonic Radiative Heat Equations Using a Binary Equation of State

Cited by 15 publications

References 13 publications

Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Self-similar solutions for high-energy density radiative transfer with separate ion and electron temperatures

The Discontinuous Asymptotic Telegrapher’s Equation (P₁) Approximation

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Self-similar Solution of the Subsonic Radiative Heat Equations Using a Binary Equation of State

Cited by 15 publications

References 13 publications

Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Modeling of Supersonic Radiative Marshak Waves Using Simple Models and Advanced Simulations

Self-similar solutions for high-energy density radiative transfer with separate ion and electron temperatures

The Discontinuous Asymptotic Telegrapher’s Equation (P1) Approximation

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Product

Resources

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The Discontinuous Asymptotic Telegrapher’s Equation (P₁) Approximation