Deriving the Asymptotic Telegrapher’s Equation (P1) Approximation for thermal

Radiative transfer Shay I. Heizler

Department of Physics, Bar-Ilan University, Ramat-Gan

Radiative transfer

Table of Contents

• Introduction and motivation

• P1, diffusion, P1/3 and asymptotic P1

• Derivation of asymptotic P1 for radiation

• Results and comparison with exact solutions (Su-Olson benchmark)

Table of Contents

• Introduction and motivation

• P1, diffusion, P1/3 and asymptotic P1

• Derivation of asymptotic P1 for radiation

• Results and comparison with exact solutions (Su-Olson benchmark)

Motivation

Example: the propagation of a super-Sonic Marshak Wave

inside hohlraums.

C.A. Back et. al., PRL 84:274,2000,

Phys. Plas. 7:2126, 2000.

The wave-front should be described accurately in the optically-thin material (in this case, foam).

Diffusion ApproximationThe 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 (P1)(linear, but wrongfinite velocity – )3c

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

MotivationIn 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 ofthe shape of the mesh).

Motivation

What’s new?

Asymptotic P1:

3D radiation-adjustment to the asymptotic P1 approximation where

What’s this work not?

This is nota comparison between different

approximations for the RTE.

:(Stand on the shoulders of giants)

1

diffusion is still in extensive use. Finding the Green function (assuming LTE) and checking the model against other models in the Su-Olson benchmark.

Table of Contents

• Introduction and motivation

• P1, diffusion, P1/3 and asymptotic P1

• Derivation of asymptotic P1 for radiation

• Results and comparison with exact solutions (Su-Olson benchmark)

Introduction

The energy balance equation for the material

- The specific intensityI

The Radiative Transfer Equation (RTE) is:

- Opacityσ

- External SourceS

- Direction of motionΩ - Black body radiation

- Material TemperaturemT- Speed of lightc

Heat capacity -

P1 ApproximationOperating over the RTE:

First moment:

The conservation law (the “first” P1 equation)

Operating over the RTE

(assuming that the specific intensity is taken as a sum of its two first moments):

Energy Density Radiation Flux

Second moment:

The “second” P1 equation

The Diffusion Approximationthe derivative of the radiation flux

with respect to the time is negligible

The “second” P1 equation yields a Fick’s law form, with a diffusion coefficient

The P1 equations yields a parabolic diffusion equation:

“radiation heat capacity” Thermal conductivity

Radiation Temperature:“Tow-temperature” diffusion equation

The LTE Diffusion ApproximationAssuming Local Thermodynamic

Equilibrium (LTE), Material Energy

We yield a single diffusion equation:

“One-temperature” diffusion equation

“Total” heat capacity:

Asymptotic P1 - The Basic Rationale• the P1 equations are an inherently flux-limited, but with the wrong

velocity –

• the P1 equations consists of two equations; an exact equation (theconservation law) and an approximate equation,which contains the termsthat 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 are free to develop a modified ”time-dependent Fick’s law”. This rationale

motivates us to find a modified P1 equations of this form:

Responsible for the steady state

solution

is responsible for the time-

dependent solution (particle-velocity)

The conservation law

Asymptotic P1, P1/3 approximationThe modified P1 equation may lead to a modified Telegrapher’s equation:

The P1/3 approximation sets ad hoc

Asymptotic Diffusion

Classic Diffusion

Yielding the correct particle velocity.

“One third “-P1, “two thirds” diffusion

One of the purposes of this work is to give some physical support to the P1/3

approximation.

Asymptotic P1 - The Basic RationaleApplying the Laplace transform to the time

dependent Fick’s law yields:

with the following diffusion coefficient:

The procedure is now well understood. Following the well-known 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 .

Table of Contents

• Introduction and motivation

• P1, diffusion, P1/3 and asymptotic P1

• Derivation of asymptotic P1 for radiation

• Results and comparison with exact solutions (Su-Olson benchmark)

The Time-Dependent Fick’s LawThe 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:

The Time-Dependent Fick’s LawFrom now on the procedure is identical to the time-independent case !

Getting an s-dependent eigenvalues,With the modified albedo:

Using the Pierl’s integral equation, we assume a general asymptotic solution for the specific energy:

The s-dependent specific intensity is, using the integral transport equation:

and (the i’th component of) the s-dependent radiation

flux is:

The Time-Dependent Fick’s LawThe 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 ( ).

Asymptotic P1Substituting 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 P1 approximation gives some physical base to the P1/3 approximation, exact for

and partial with increases.

The asymptotic P1 approximation gives some physical base to the P1/3 approximation, exact for

and partial with increases.

Asymptotic P1

Private Case: LTE

D0 can be approximated as:

Solving for and yields:

Using the asymptotic P1 equations using these and , is called a semi-LTE treatment.

(Justification of Zimmerman to work with (LTE case).

Assuming LTE, the asymptotic P1 approximation yields a “P 1/5 approximation”.

Morel’s (Larsen’s) asymptotic accuracy test

• The first P1 equation (the conservation law) and the material energy balance equations are identical to P1 approximation , and thus satisfies the asymptotics

• The P1/3 approximation satisfies the asymptotics accuracy test to the , in the diffusion limit 2.

• The P1 approximation satisfies the asymptotics accuracy test to the , in the diffusion limit 1.

What’s about general ?

1

2

equations are identical to P1 approximation , and thus satisfies the asymptotics accuracy test to the .

Assuming and

yields:

Morel’s (Larsen’s) asymptotic accuracy test

The asymptotic P1 approximation satisfies the asymptotics accuracy test to the order in the diffusion limit (setting ) for any general

as expected.

Identical to RTE asymptotics

Table of Contents

• Introduction and motivation

• P1, diffusion, P1/3 and asymptotic P1

• Derivation of asymptotic P1 for radiation

• Results and comparison with exact solutions (Su-Olson benchmark)

The Green FunctionTo find the asymptotic P1 Green function in the one-dimensional slab-geometry

case, we use similar technique that is used in:

s→ 0

Exact(semi-LTE):

A full LTE adjustment:

The LTE Green Function

Almost exact particle velocity !

Results - Su-Olson benchmark

The asymptotic P1 approximation yields the best LTE approximation to the transport solution, even in intermediate times, especially in

the wave-front area.

Results - Su-Olson benchmark

Far enough from the source area, the asymptotic LTE P1 approximation yields a better approximation than the P1 (NLTE) or the P1/3 approximation

(surprise?) for the transport solution.

(Time dependency is not summarized only in the wave

front).