======
Theory
======


Overview
--------

In this section the theoretical model is summarized, a detailed description is presented in the initial FLUNED paper (`De Pietri et al., 2023 <https://doi.org/10.1016/j.cpc.2023.108807>`_).

The concentration of a radioactive species :math:`N_\alpha` is treated as a
passive scalar that does not alter the underlying flow or transport
processes. In addition, only radioisotopes generated by single-step reactions and with constant parent concentration are considered.

With these assumptions, the equations implemented in the FLUNED solver are reported here.

Governing Conservation Equation
-------------------------------

.. math::
   :label: eq1

   \frac{\partial N_\alpha}{\partial t}
   + \nabla \!\cdot\! \bigl(\mathbf{U}\,N_\alpha\bigr)
   \;=\;
   \nabla \!\cdot\!\bigl(D_\alpha\,\nabla N_\alpha\bigr)
   + S_\alpha(N,\phi)

* :math:`\mathbf{U}` - coolant velocity vector  
* :math:`D_\alpha` - molecular diffusivity of species :math:`\alpha`  
* :math:`S_\alpha` - volumetric source term (neutron interactions + decay)

Reynolds Decomposition
----------------------

The Reynolds decomposition for turbulent flows is considered:

.. math::
   :label: eq2

   N_\alpha(\mathbf{x},t) \;=\; \langle N_\alpha \rangle
   \;+\; N_\alpha'

By substituting (2) in (1) gives:

.. math::
   :label: eq3

   \frac{\partial\langle N_\alpha\rangle}{\partial t}
   + \nabla\!\cdot\!\bigl(\langle\mathbf{U}\rangle\,\langle N_\alpha\rangle\bigr)
   + \nabla\!\cdot\!\langle \mathbf{u}' N_\alpha' \rangle
   \;=\;
   \nabla\!\cdot\!\bigl(D_\alpha\,\nabla\langle N_\alpha\rangle\bigr)
   + \langle S_\alpha(N,\phi)\rangle

Turbulent Closure - Gradient Diffusion Hypothesis
-------------------------------------------------

.. math::
   :label: eq4

   \langle \mathbf{u}' N_\alpha' \rangle
   \;=\;
   -\,D_t\,\nabla\langle N_\alpha\rangle
   \;=\;
   -\,\frac{\nu_t}{\mathrm{Sc}_t}\;\nabla\langle N_\alpha\rangle

* :math:`D_t` - turbulent diffusivity  
* :math:`\nu_t` - turbulent viscosity  
* :math:`\mathrm{Sc}_t \approx 0.7` - turbulent Schmidt number  
* Valid only under isotropic-turbulence assumptions.

General Source Term
-------------------

.. math::
   :label: eq5

   \sum_{i\neq\alpha}
        (\langle\phi\sigma\rangle_{i\rightarrow\alpha} + \lambda_{i\rightarrow\alpha})\,N_i(t)
   \;-\;
   \sum_{i\neq\alpha}
        (\langle\phi\sigma\rangle_{\alpha\rightarrow i} + \lambda_{\alpha\rightarrow i})\,N_\alpha(t)

* First (positive) sum - production of :math:`\alpha` from parents :math:`i`.  
* Second (negative) sum - loss of :math:`\alpha` via decay or transmutation.  
* :math:`\lambda_{x\rightarrow y}` - instantaneous decay constant.  
* :math:`\langle\phi\sigma\rangle` - neutron-flux-weighted reaction rate.

Simplified Source Term (Single-Step Transmutation + Decay)
----------------------------------------------------------

Under the study's assumptions (constant parent isotopes, single-step
generation + decay):

.. math::
   :label: eq6

   \langle S_\alpha(N,\phi)\rangle
   \;=\;
   \mathrm{RR}_\alpha(\phi)\;-\;\lambda_\alpha N_\alpha

* :math:`\mathrm{RR}_\alpha` - neutron-induced reaction rate producing
  isotope :math:`\alpha`.  
* :math:`\lambda_\alpha` - decay constant of :math:`\alpha`.

Laminar-Flow Limit
------------------

For laminar regimes, (3) reduces to (1) with  

* :math:`D_t = 0 \quad (\nu_t = 0)`  
* all variables in their instantaneous form (no ensemble averaging).