Benchmark solution
=====================


Calculation method
------------------


Calculation of the conservation of air mass
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The conservation of the mass of gas is written as:


.. image:: images/Object_14.svg
    :width: 185
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.. _RefImage_Object_14.svg:


.. _RefEquation 2.1.1-1:

                     eq 2.1.1-1

We write that the total mass of water and the total mass of air are preserved (because there is no water or gas flow at the edge) and we get:


.. image:: images/Object_15.svg
    :width: 185
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.. _RefImage_Object_15.svg:

therefore


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.. _RefImage_Object_16.svg:


.. _RefEquation 2.1.1-2:

                 Eq 2.1.1-2


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    :height: 42


.. _RefImage_Object_17.svg:

and

.. image:: images/Object_18.svg
    :width: 185
    :height: 42


.. _RefImage_Object_18.svg:


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    :height: 42


.. _RefImage_Object_19.svg:

Speed calculation:


.. image:: images/Object_20.svg
    :width: 185
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.. _RefImage_Object_20.svg:


.. _RefEquation 2.1.1-3:

                                 Eq 2.1.1-3

since

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    :width: 185
    :height: 42


.. _RefImage_Object_21.svg:

and

.. image:: images/Object_22.svg
    :width: 185
    :height: 42


.. _RefImage_Object_22.svg:

and


.. image:: images/Object_23.svg
    :width: 185
    :height: 42


.. _RefImage_Object_23.svg:

with

.. image:: images/Object_24.svg
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    :height: 42


.. _RefImage_Object_24.svg:

Like

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    :height: 42


.. _RefImage_Object_25.svg:


.. image:: images/Object_26.svg
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.. _RefImage_Object_26.svg:

[:ref:`éq 2.1.1-1 <éq 2.1.1-1>`] can then be simplified to the following form:


.. image:: images/Object_27.svg
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    :height: 42


.. _RefImage_Object_27.svg:

with

.. image:: images/Object_28.svg
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    :height: 42


.. _RefImage_Object_28.svg:

Heat equation whose result is known.


Benchmark results
----------------------

With the previous numerical values, we find:


.. image:: images/Object_29.svg
    :width: 185
    :height: 42


.. _RefImage_Object_29.svg:


.. image:: images/Object_30.svg
    :width: 185
    :height: 42


.. _RefImage_Object_30.svg:

and

.. image:: images/Object_31.svg
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    :height: 42


.. _RefImage_Object_31.svg:


.. image:: images/Object_32.svg
    :width: 185
    :height: 42


.. _RefImage_Object_32.svg:

The constants of the heat equation are then:


.. image:: images/Object_33.svg
    :width: 185
    :height: 42


.. _RefImage_Object_33.svg:


Uncertainties
------------

The uncertainties are quite large because the analytical solution is an approximate solution due to the linearization of the equations.