From b9d51368852cb7fb5c97e8cf04c276206053951b Mon Sep 17 00:00:00 2001
From: Jordan Musser <jordan.musser@netl.doe.gov>
Date: Mon, 23 Sep 2024 14:02:42 +0000
Subject: [PATCH] Add chem eulerian01 test notes (exa/docs!94)

---
 docs/source_docs/conf.py                      |   3 +
 docs/source_docs/index.rst                    |   1 +
 .../verification/chemistry/eulerian01.rst     |  93 ++++++++++++++++++
 .../eulerian01_ForwardEuler_accuracy.png      | Bin 0 -> 12480 bytes
 .../eulerian01_ForwardEuler_solution.png      | Bin 0 -> 67650 bytes
 .../test_benchmarks/verification/index.rst    |  27 +++++
 6 files changed, 124 insertions(+)
 create mode 100644 docs/source_docs/test_benchmarks/verification/chemistry/eulerian01.rst
 create mode 100644 docs/source_docs/test_benchmarks/verification/chemistry/images/eulerian01_ForwardEuler_accuracy.png
 create mode 100644 docs/source_docs/test_benchmarks/verification/chemistry/images/eulerian01_ForwardEuler_solution.png
 create mode 100644 docs/source_docs/test_benchmarks/verification/index.rst

diff --git a/docs/source_docs/conf.py b/docs/source_docs/conf.py
index 82aa9ba..e4bbc5f 100644
--- a/docs/source_docs/conf.py
+++ b/docs/source_docs/conf.py
@@ -86,6 +86,9 @@ todo_include_todos = False
 numfig = True
 
 math_numfig = True
+numfig_secnum_depth = 1
+
+math_eqref_format = "Eq.{number}"
 
 bibtex_bibfiles = ['refs.bib']
 
diff --git a/docs/source_docs/index.rst b/docs/source_docs/index.rst
index 939c4e9..b5be310 100644
--- a/docs/source_docs/index.rst
+++ b/docs/source_docs/index.rst
@@ -81,6 +81,7 @@ and performance:
    test_benchmarks/CITests
    test_benchmarks/NightlyTests
    test_benchmarks/qualitative_bencharks/index
+   test_benchmarks/verification/index
 
 Notice
 ------
diff --git a/docs/source_docs/test_benchmarks/verification/chemistry/eulerian01.rst b/docs/source_docs/test_benchmarks/verification/chemistry/eulerian01.rst
new file mode 100644
index 0000000..0ceb3a1
--- /dev/null
+++ b/docs/source_docs/test_benchmarks/verification/chemistry/eulerian01.rst
@@ -0,0 +1,93 @@
+.. include:: ../../../definitions.rst
+
+Eulerian01: athermal, constant density
+--------------------------------------
+
+Description
+~~~~~~~~~~~
+
+Consider a system with a fluid comprised of two chemical species, :math:`A` and :math:`B`, with the transfer of species given by the reversible homogeneous reaction,
+
+.. math::
+   :label: chem_eulerian01_eq01
+
+   A(g) \leftrightarrow B(g)
+
+If both species have the same molecular weight, specific heat, and enthalpy of formation, then the mixture molecular weight, system temperature, and density are constant.
+The evolution of species mass in the absence of convection and diffusion is given by
+
+.. math::
+   :label: chem_eulerian01_eq02
+
+   \rho\frac{\partial X_{A}}{\partial t} = \dot{\omega}_{A}
+
+where :math:`\rho` is density;
+:math:`X_A` is the mass fraction of species :math:`A`; and
+:math:`\dot{\omega}_{A}` is the net rate of production (consumption if negative) of species :math:`A`.
+
+A manufactured solution for species :math:`A` is
+
+.. math::
+   :label: chem_eulerian01_eq03
+
+   X_A(t) = X_A^0 - \sin(6\pi t)/4
+
+where :math:`X_A^0 = X_A(t=0)` is the initial mass fraction of species :math:`A`. Taking the derivative with respect to time, and substituting the result into :eq:`chem_eulerian01_eq02` gives
+
+.. math::
+   :label: chem_eulerian01_eq04
+
+   \dot{\omega}_A = -\rho 3\pi \cos(6\pi t)/2
+
+which has units of kg\ |cdot|\ m\ :math:`^{-3}`\ |cdot| sec\ :math:`^{-1}`. Two modifications are required to implement this Eulerian reaction rate into MFIX-Exa:
+
+- Eq. :eq:`chem_eulerian01_eq04` needs to be divided by the molecular weight of species A because Eulerian reaction rates in MFIX-Exa are provided in units of mol\ |cdot|\ m\ :math:`^{-3}`\ |cdot| sec\ :math:`^{-1}`.
+- The negative sign is omitted because species :math:`A` is a reactant. Specifically, by inspection of chemical equation :eq:`chem_eulerian01_eq01`, species :math:`A` is a reactant when the rate is positive (forward reaction), and a product when the rate is negative (backward reaction).
+
+The resulting Eulerian reaction rate is
+
+.. math::
+   :label: chem_eulerian01_eq54
+
+   R_A = \frac{\rho 3\pi \cos(6\pi t)}{2 M_A}
+
+Setup
+~~~~~
+
+The domain is a single cuboid cell with 0.0025 m edge length and convection and diffusion terms are set to zero for all governing equations. The fluid has a constant temperature of 300 K at 60,795 Pa, and the initial composition is :math:`X_A=X_B=0.5`. The density is chosen to be 2 kg\ |cdot|\ m\ :math:`^{-3}`; therefore, the component and mixture molecular weights are computed from the ideal gas law :math:`M=\rho RT / p_{therm} \approx 0.0821` kg\ |cdot|\ mol\ :math:`^{-1}`.
+
+
+Results
+~~~~~~~
+
+The test is run at five fixed time-step sizes with a refinement factor of 2: 0.2, 0.1, 0.05, 0.025, 0.0125 milliseconds. The :math:`L_1`\ -norm is computed by dividing the sum of the absolute difference between the analytical and numerical solution over all time steps by the total number of time steps.
+
+.. math::
+
+   L_1 = \frac{\sum_n \| X_A(t_n) - X_A^{\mathrm{\texttt{MFIX-Exa}}}\|}{n}
+
+
+Forward Euler
+'''''''''''''
+
+The analytical solution and absolute error with dt = 0.2 msec are shown in :numref:`fig_chem_eulerian01_ForwardEuler_sol`. The :math:`L_1` norm is used to compute the observed order of accuracy shown in :numref:`fig_chem_eulerian01_ForwardEuler_ooa`.
+
+
+.. _fig_chem_eulerian01_ForwardEuler_sol:
+
+.. figure:: ./images/eulerian01_ForwardEuler_solution.png
+   :width: 85%
+   :align: center
+   :alt: Eulerian01 test numerical solution and absolute error
+
+   Numerical solution for species mass fraction and absolute error.
+
+
+.. _fig_chem_eulerian01_ForwardEuler_ooa:
+
+.. figure:: ./images/eulerian01_ForwardEuler_accuracy.png
+   :width: 45%
+   :align: center
+   :alt: Eulerian01 test observed order
+
+   Observed order of species mass fraction.
diff --git a/docs/source_docs/test_benchmarks/verification/chemistry/images/eulerian01_ForwardEuler_accuracy.png b/docs/source_docs/test_benchmarks/verification/chemistry/images/eulerian01_ForwardEuler_accuracy.png
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diff --git a/docs/source_docs/test_benchmarks/verification/index.rst b/docs/source_docs/test_benchmarks/verification/index.rst
new file mode 100644
index 0000000..6d96dde
--- /dev/null
+++ b/docs/source_docs/test_benchmarks/verification/index.rst
@@ -0,0 +1,27 @@
+====================
+Chemistry code tests
+====================
+
+The test cases presented in this chapter for the MFIX fluid solver are summarized below.
+
+.. "Eulerian01", "C", "A", " ", " ", ":math:`{\checkmark}`"
+
+
+.. |check| replace:: :math:`{\checkmark}`
+
++-------------------+---------+---------+--------+-----------+---------+---------+---------+
+| Test              | Freq\ † | Data\ ‡ | Phases |  Momentum | Density | Species | Energy  |
++===================+=========+=========+========+===========+=========+=========+=========+
+| Eulerian01        | C       | A       | Fluid  |           | |check| |         |         |
++-------------------+---------+---------+--------+-----------+---------+---------+---------+
+
+† C-Incorporated into the continuous integration server; M-Monthly; Q-Quarterly; X-Manual; D-Disabled
+
+‡ A-Analytical solution; P-Published benchmark data; R-Regression data from previous code versions
+
+.. toctree::
+    :numbered:
+    :caption: Validation and Verification
+    :maxdepth: 1
+
+    chemistry/eulerian01
-- 
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