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<div class="math notranslate nohighlight">
\[\newcommand{\half}{\frac{1}{2}}
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<section id="enforcing-inflow-outflow-solvability">
<h1>Enforcing inflow-outflow solvability<a class="headerlink" href="#enforcing-inflow-outflow-solvability" title="Link to this heading"></a></h1>
<p>This routine enforces solvability for inflow-outflow boundaries,
which have both inflow and outflow cells.
A flux correction factor, <span class="math notranslate nohighlight">\(\alpha_\text{fcf}\)</span> is introduced
which scales the outflow velocities to match the inflow:</p>
<div class="math notranslate nohighlight">
\[\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area} + \alpha_\text{fcf} \sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area} = 0\]</div>
<p>The new flux-conserving velocities to be used for the MAC/nodal projections,
<span class="math notranslate nohighlight">\({\bf u}_\text{fc}\)</span>, are calculated from the correction factor as:</p>
<div class="math notranslate nohighlight">
\[\alpha_\text{fcf} = \frac{-\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area}}{\sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area}},\]</div>
<div class="math notranslate nohighlight">
\[{\bf u}_\text{fc} = \alpha_\text{fcf} \cdot {\bf u}, \ \forall {\bf x} \in \partial\Omega_\text{out}.\]</div>
<p>It must be noted that this routine currently only accounts for boundaries
with the math BC <code class="docutils literal notranslate"><span class="pre">BCType::direction_dependent</span></code>, which is to be used for
an inflow-outflow boundary. It does not compute or correct the boundary velocities
over other math BC types such as those representing pure inflow or pure outflow.</p>
</section>


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55 changes: 55 additions & 0 deletions amrex-hydro/docs_html/Utilities.html
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<li class="toctree-l3"><a class="reference internal" href="#extensive-fluxes">Extensive fluxes</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#enforcing-inflow-outflow-solvability">Enforcing inflow-outflow solvability</a></li>
</ul>
</li>
</ul>
Expand Down Expand Up @@ -396,8 +397,62 @@ <h3>Extensive fluxes<a class="headerlink" href="#extensive-fluxes" title="Link t
s_{i-\frac{1}{2},j,k}^{{n+\frac{1}{2}}} + s_{i+\frac{1}{2},j,k}^{{n+\frac{1}{2}}}
+ s_{i,j-\frac{1}{2},k}^{{n+\frac{1}{2}}} + s_{i,j-\frac{1}{2},k}^{{n+\frac{1}{2}}}
+ s_{i,j,k-\frac{1}{2}}^{{n+\frac{1}{2}}} + s_{i,j,k-\frac{1}{2}}^{{n+\frac{1}{2}}} )\]</div>
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</section>
</section>
<section id="enforcing-inflow-outflow-solvability">
<h2>Enforcing inflow-outflow solvability<a class="headerlink" href="#enforcing-inflow-outflow-solvability" title="Link to this heading"></a></h2>
<p>This routine enforces solvability for inflow-outflow boundaries,
which have both inflow and outflow cells.
A flux correction factor, <span class="math notranslate nohighlight">\(\alpha_\text{fcf}\)</span> is introduced
which scales the outflow velocities to match the inflow:</p>
<div class="math notranslate nohighlight">
\[\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area} + \alpha_\text{fcf} \sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area} = 0\]</div>
<p>The new flux-conserving velocities to be used for the MAC/nodal projections,
<span class="math notranslate nohighlight">\({\bf u}_\text{fc}\)</span>, are calculated from the correction factor as:</p>
<div class="math notranslate nohighlight">
\[\alpha_\text{fcf} = \frac{-\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area}}{\sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area}},\]</div>
<div class="math notranslate nohighlight">
\[{\bf u}_\text{fc} = \alpha_\text{fcf} \cdot {\bf u}, \ \forall {\bf x} \in \partial\Omega_\text{out}.\]</div>
<p>It must be noted that this routine currently only accounts for boundaries
with the math BC <code class="docutils literal notranslate"><span class="pre">BCType::direction_dependent</span></code>, which is to be used for
an inflow-outflow boundary. It does not compute or correct the boundary velocities
over other math BC types such as those representing pure inflow or pure outflow.</p>
</section>
</section>


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28 changes: 28 additions & 0 deletions amrex-hydro/docs_html/_sources/InOutSolvability.rst.txt
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.. include:: CustomCommands.rst

Enforcing inflow-outflow solvability
------------------------------------

This routine enforces solvability for inflow-outflow boundaries,
which have both inflow and outflow cells.
A flux correction factor, :math:`\alpha_\text{fcf}` is introduced
which scales the outflow velocities to match the inflow:

.. math::
\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area} + \alpha_\text{fcf} \sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area} = 0
The new flux-conserving velocities to be used for the MAC/nodal projections,
:math:`{\bf u}_\text{fc}`, are calculated from the correction factor as:

.. math::
\alpha_\text{fcf} = \frac{-\sum_{\partial\Omega_\text{in}} {\bf u} \cdot {\bf \area}}{\sum_{\partial\Omega_\text{out}} {\bf u} \cdot {\bf \area}},
.. math::
{\bf u}_\text{fc} = \alpha_\text{fcf} \cdot {\bf u}, \ \forall {\bf x} \in \partial\Omega_\text{out}.
It must be noted that this routine currently only accounts for boundaries
with the math BC ``BCType::direction_dependent``, which is to be used for
an inflow-outflow boundary. It does not compute or correct the boundary velocities
over other math BC types such as those representing pure inflow or pure outflow.
1 change: 1 addition & 0 deletions amrex-hydro/docs_html/_sources/Utilities.rst.txt
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Expand Up @@ -38,3 +38,4 @@ Note that EB is only an option for Cartesian geometries as of this writing.

.. include:: AdvectiveTerm.rst

.. include:: InOutSolvability.rst
7 changes: 4 additions & 3 deletions amrex-hydro/docs_html/_static/searchtools.js
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Expand Down Expand Up @@ -328,13 +328,14 @@ const Search = {
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1 change: 1 addition & 0 deletions amrex-hydro/docs_html/index.html
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Expand Up @@ -132,6 +132,7 @@
<li class="toctree-l2"><a class="reference internal" href="Utilities.html#eb-slopes">EB Slopes</a></li>
<li class="toctree-l2"><a class="reference internal" href="Utilities.html#computing-fluxes">Computing Fluxes</a></li>
<li class="toctree-l2"><a class="reference internal" href="Utilities.html#constructing-the-advective-term">Constructing the advective term</a></li>
<li class="toctree-l2"><a class="reference internal" href="Utilities.html#enforcing-inflow-outflow-solvability">Enforcing inflow-outflow solvability</a></li>
</ul>
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</ul>
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