Deriving the Asymptotic Telegrapher s Equation (P 1 ) Approximation for thermal Radiative transfer

Transcription

1 Deriving the Asymptotic Telegrapher s Equation (P 1 ) Approximation for thermal Radiative transfer Shay I. Heizler Department of Physics, Bar-Ilan University, Ramat-Gan

2 Table of Contents Introduction and motivation P 1, diffusion, P 1/3 and asymptotic P 1 Derivation of asymptotic P 1 for radiation Results and comparison with exact solutions (Su-Olson benchmark)

3 Table of Contents Introduction and motivation P 1, diffusion, P 1/3 and asymptotic P 1 Derivation of asymptotic P 1 for radiation Results and comparison with exact solutions (Su-Olson benchmark)

4 Motivation Example: the propagation of a super-sonic Marshak Wave inside hohlraums. The wave-front should be described accurately in the optically-thin material (in this case, foam). C.A. Back et. al., PRL 84:274,2000, Phys. Plas. 7:2126, 2000.

5 Diffusion Approximation The main advantage: Much simpler (computer memory, scheme) to solve than the transport equation, especially in 2D/3D. Disadvantage: Failure in time-dependent problems describing the streaming particles front emanating from a source - Infinite Particle velocity Possible Solution (1): Telegrapher s Equation (P 1 ) (linear, but wrong finite velocity c 3 ) Possible Solution (2): widely used, Flux-Limiters, Variable Eddington Factors, defining a gradient-dependent diffusion coefficients (or Eddington factors), but complicated non-linear diffusion coefficient

6 Motivation In addition, in multi-dimensional problems, especially in curve-linear geometries in general meshes, In contrast to linear approximations, it causes to a definition of several diffusion coefficients for each cell, may causing distortions in the shape of the radiation fields (usually because of the shape of the mesh).

7 Motivation Asymptotic P 1 : What s new? 3D radiation-adjustment to the asymptotic P 1 approximation where diffusion is still in extensive use. Finding the Green function (assuming LTE) and checking the model against other models in the Su-Olson benchmark. What s this work not? This is not a comparison between different approximations for the RTE. (Stand on the shoulders of giants):

8 Table of Contents Introduction and motivation P 1, diffusion, P 1/3 and asymptotic P 1 Derivation of asymptotic P 1 for radiation Results and comparison with exact solutions (Su-Olson benchmark)

9 Introduction The Radiative Transfer Equation (RTE) is: The energy balance equation for the material I σ S ˆΩ - The specific intensity - Opacity - External Source - Direction of motion Heat capacity - - Black body radiation c - Speed of light T m - Material Temperature

10 P 1 Approximation Operating over the RTE: First moment: The conservation law (the first P 1 equation) Energy Density Radiation Flux Operating over the RTE (assuming that the specific intensity is taken as a sum of its two first moments): Second moment: The second P 1 equation

11 The Diffusion Approximation the derivative of the radiation flux with respect to the time is negligible The second P 1 equation yields a Fick s law form, with a diffusion coefficient The P 1 equations yields a parabolic diffusion equation: Tow-temperature diffusion equation Radiation Temperature: radiation heat capacity Thermal conductivity

12 The LTE Diffusion Approximation Assuming Local Thermodynamic Equilibrium (LTE), Material Energy We yield a single diffusion equation: One-temperature diffusion equation Total heat capacity:

13 Asymptotic P 1 - The Basic Rationale the P 1 equations are an inherently flux-limited, but with the wrong velocity the P 1 equations consists of two equations; an exact equation (the conservation law) and an approximate equation, which contains the terms that include the factor of 3. The rationale says that we must not change the exact equation, while we are free to develop a modified time-dependent Fick s law. This rationale motivates us to find a modified P 1 equations of this form: The conservation law is responsible for the timedependent solution (particle-velocity) Responsible for the steady state solution

14 Asymptotic P 1, P 1/3 approximation The modified P 1 equation may lead to a modified Telegrapher s equation: The P 1/3 approximation sets ad hoc One third - P 1, two thirds diffusion Yielding the correct particle velocity. One of the purposes of this work is to give some physical support to the P 1/3 approximation. Asymptotic Diffusion Classic Diffusion

15 Asymptotic P 1 - The Basic Rationale Applying the Laplace transform to the time dependent Fick s law yields: with the following diffusion coefficient: The procedure is now well understood. Following the wellknown prescription for solving the time-dependent Boltzmann equation using the Laplace-domain on time, and obtaining a modified (albedo and s -dependent) diffusion coefficient solving for and.

16 Table of Contents Introduction and motivation P 1, diffusion, P 1/3 and asymptotic P 1 Derivation of asymptotic P 1 for radiation Results and comparison with exact solutions (Su-Olson benchmark)

17 The Time-Dependent Fick s Law The mono-energetic RTE is in homogenous media: Applying a Laplace transform, using the definition of the effective albedo: Defining modified total-cross-section and albedo: Substituting the modified (s-dependent) coefficients yields a similar in form to the time-independent RTE:

18 The Time-Dependent Fick s Law From now on the procedure is identical to the time-independent case! Using the Pierl s integral equation, we assume a general asymptotic solution for the specific energy: Getting an s-dependent eigenvalues, With the modified albedo: The s-dependent specific intensity is, using the integral transport equation: and (the i th component of) the s- dependent radiation flux is:

19 The Time-Dependent Fick s Law The relation between the energy density and the radiation flux yields a Fick s law relation: with a s-dependent modified diffusion coefficient: Substituting the definitions of the modified total-cross-section and albedo yields: By comparing this modified s-dependent diffusion coefficient to the s- dependent diffusion coefficient in the basic rationale chapter, we solve for and for a general media ( ).

20 Asymptotic P 1 Substituting in the modified diffusion coefficient yields a involved expression, and we cannot solve for and explicitly. Since we look for the asymptotic behavior in time (s 0), i.e. According to the final value theorem, we expand the inverse of the diffusion coefficient in a Taylor series: The asymptotic P 1 approximation gives some physical base to the P 1/3 approximation, exact for and partial with increases.

21 Asymptotic P 1 Private Case: LTE D 0 can be approximated as: Solving for and yields: Assuming LTE, the asymptotic P 1 approximation yields a P 1/5 approximation. Using the asymptotic P 1 equations using these and, is called a semi-lte treatment. (Justification of Zimmerman to work with (LTE case).

22 Morel s (Larsen s) asymptotic accuracy test The P 1 approximation satisfies the asymptotics accuracy test to the, in the diffusion limit 1. The P 1/3 approximation satisfies the asymptotics accuracy test to the, in the diffusion limit 2. What s about general? The first P 1 equation (the conservation law) and the material energy balance equations are identical to P 1 approximation, and thus satisfies the asymptotics accuracy test to the. Assuming and yields: 1 2

23 Morel s (Larsen s) asymptotic accuracy test Identical to RTE asymptotics The asymptotic P 1 approximation satisfies the asymptotics accuracy test to the order in the diffusion limit (setting ) for any general as expected.

24 Table of Contents Introduction and motivation P 1, diffusion, P 1/3 and asymptotic P 1 Derivation of asymptotic P 1 for radiation Results and comparison with exact solutions (Su-Olson benchmark)

25 The Green Function To find the asymptotic P 1 Green function in the one-dimensional slab-geometry case, we use similar technique that is used in: Exact (semi-lte): s 0 A full LTE adjustment:

26 The LTE Green Function Almost exact particle velocity!

27 Results - Su-Olson benchmark The asymptotic P 1 approximation yields the best LTE approximation to the transport solution, even in intermediate times, especially in the wave-front area.

28 Results - Su-Olson benchmark Far enough from the source area, the asymptotic LTE P 1 approximation yields a better approximation than the P 1 (NLTE) or the P 1/3 approximation (surprise?) for the transport solution. (Time dependency is not summarized only in the wave front).