Welcome to RadMom1D. RadMom1D is a computational framework for solving the radiative transfer equation in one spatial dimension. It includes a package of C++ subroutines for solving radiative transfer problems using numerical methods (i.e., Computational Fluid Dynamics or CFD). In particular, a Godunov-type finite volume method with piecewise linear reconstruction of the solution within the computational domain is used to solve the system of partial differential equations of interest. All required files are in this directory and the corresponding subdirectories. In this framework, systems of PDEs, resulting from the application of the method of moments to the radiative transfer equation, are solved. The current implementation involves three moment closure techniques:

• M1: the first-order maximum entropy moment closure, which involves solving angular moments up to first order. The unclosed second-order moment in the transport equation for the first-order moment is expressed in terms of the lower-order moments by assuming a distribution that maximizes the radiative entropy and reproduces the lower-order moments;

• P1: the first-order spherical harmonic moment closure, which involves solving angular moments up to first order. The unclosed second-order moment in the transport equation for the first-order moment is expressed in terms of the lower-order moments by expanding the distribution as a series expansion in terms of orthogonal legendre polynomials (1D equivalent of 3D spherical harmonics) with the constraint that the lower-order moments are angular moments of the assumed distribution;

• P3: the third-order spherical harmonic moment closure, which involves solving angular moments up to third order. The unclosed fourth-order moment in the transport equation for the third-order moment is expressed in terms of the lower-order moments by expanding the distribution as a series expansion in terms of orthogonal legendre polynomials (1D equivalent of 3D spherical harmonics) with the constraint that the lower-order moments are angular moments of the assumed distribution.

The RadMom1D code is templated such that it can be used to solve the systems of PDEs arising from moment closure techniques of varying order, which involve different number of independent variables. As an example, the M1 and P1 closures involve only two independent variables in 1D, whereas the P3 closure involves 4 independent variables in one spatial dimension. The type of moment closure technique to be used must be specified prior to compiling the RadMom1D code so as to ensure the proper specialization of the templated classes are compiled.

Copyright (C) Joachim Sarr.

RadMom1D is a free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation (version 2 of the License, or any later version). A copy of the licensing agreement is provided in this directory in the file GNU_license.txt. For additional information contact the Free Software Foundation Inc., 65 Mass Ave, Cambridge, MA 02139, USA.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

Contributors to the development of RadMom1D include: Joachim Sarr

For additional information concerning RadMom1D contact the primary author:

Joachim A. R. Sarr University of Toronto Institute for Aerospace Studies 4925 Dufferin Street, Toronto, Ontario M3H 5T6, Canada

[email protected]

RadMom1D_Input_Parameters: This class includes definition and manipulation of 1D RadMom input variables.

RadMom1D_UniformMesh: contains definitions for RadMom1D computational cell (solution and geometry).

RadMom1D_pState: base class for primitive variable solution state class.

RadMom1D_cState: base class for conserved variable solution state class.

RadMom1D_pState_First_Order: contains primitive variable solution state class definition for first-order moment closures (i.e, M1 and P1). Derived class of RadMom1D_pState.

RadMom1D_cState_First_Order: contains conserved variable solution state class definition for first-order moment closures (i.e, M1 and P1). Derived class of RadMom1D_cState

RadMom1D_pState_Third_Order: contains primitive variable solution state class definition for third-order moment closures (i.e, P3). Derived class of RadMom1D_pState.

RadMom1D_cState_Third_Order: contains conserved variable solution state class definition for third-order moment closures (i.e, P3). Derived class of RadMom1D_cState.

Medium1D_State: contains gas state class definition for an absorbing, scattering participating medium. In particular, it stores radiative properties of the participating medium such as the absorption coefficient and the scattering coefficient. For now, the radiative properties implemented are based on the assumption of gray media (not spectrally dependent radiative properties) and of isotropic scattering. More complex/realistic gas radiative property models will be implemented in the future.

Cell1D_Uniform: contains class definitions for the 1D uniform cell making up the computational domain.

In the RadMom1D.cc file, set the variable "Closure_Type_ptr" to the desired moment closure technique to be used for providing approximate solution to the radiative transfer equation. The possible options are:

• Closure_Type_ptr = "M1": for the M1 moment closure
• Closure_Type_ptr = "P1": for the P1 moment closure
• Closure_Type_ptr = "P3": for the P3 moment closure

In the command line, execute the command "make clean" to clean up all the object files generated from previous compilations of the RadMom1D code.

In the command line, execute the command "make radMom1D" to compile the RadMom1D code. This will generate an executable named "radMom1D".

Once the code compiled, it can be used along with a given set of inputs to solve the resulting set of PDEs and write the solution to an output file (in *.dat format with can be visualized using Tecplot). The command used to run a test case with input file name "test.in" is the following:

./radmom1D -f test1D.in

In the "Test" directory, some test samples are provided for use with the radMom1D exectutable. These tests involve solving the radiative transfer equation using both the M1, P1, and P3 moment closures (see subdirectories "M1", "P1", and "P3" in "Test" directory). The input file name associated with each of these test is "test1D.in". The scripts for executing the RadMom1D code can be found in these subdirectories, i.e., M1_Closure_script.sh for M1, P1_Closure_script.sh for P1, and P3_Closure_script.sh for P3 and can be exectuted in the command line using: ./M1_Closure_script.sh, or ./P1_Closure_script.sh, or ./P3_Closure_script.sh Upon completion of the numerical simulations, an output file "test1D.dat" will be generated.

A Tecplot macro is provided in the "Test" subdirectory which extracts solution profiles for the M1, P1, and P3 closures (Comparisons_E.mcr for the radiative energy density or zeroth-order moment, Comparisons_F.mcr for the radiative heat flux or first-order moment, and Comparisons_Sr.mcr for the radiative source term) for the purpose of comparing the predictive capabilities of the different moment closures.