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


Calculation method used for the reference solution
--------------------------------------------------------

The reference [:ref:`bib3 <bib3>`] provides the general formula for the sloshing modes of a compressible fluid domain comprising a free surface, contained in a parallelepipedic reservoir with rigid walls:

:math:`{f}_{\mathit{ij}}=\frac{1}{2\pi }\sqrt{\pi g\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}}\mathrm{tanh}\left[\pi H\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}}\right]}`

where :math:`i` and :math:`j` are the integer orders of the longitudinal and transverse modes (number of nodal lines in each horizontal direction). The fluctuating pressure modes have an expression of the form:

:math:`{p}_{\mathit{ij}}(x,y,z)=\mathrm{cos}\left(i\pi \frac{2x+l}{2l}\right)\mathrm{cos}\left(j\pi \frac{2y+L}{2L}\right)\left(\mathrm{cosh}\left({\lambda }_{\mathit{ij}}(H-z)\right)+{A}_{\mathit{ij}}\mathrm{sinh}\left({\lambda }_{\mathit{ij}}(H-z)\right)\right)`

where :math:`{\lambda }_{\mathit{ij}}` designates the verifying vertical wave number: :math:`{\lambda }_{\mathit{ij}}H\mathrm{tanh}\left({\lambda }_{\mathit{ij}}H\right)=4{\pi }^{2}{f}_{\mathit{ij}}H/g` and with a verifying coefficient: :math:`{A}_{\mathit{ij}}=-\mathrm{tanh}\left({\lambda }_{\mathit{ij}}H\right)`. The fluctuating sloshing pressures are maximum on the free surface. The height of the free surface level is expressed as: :math:`{z}_{l}(x,y)={p}_{\mathit{ij}}(x,y\mathrm{,0})/(\rho \mathrm{.}g)`.

In the particular case where :math:`\frac{L}{l}` is large, the formula is simplified for the longitudinal modes [:ref:`bib1 <bib1>`], [:ref:`bib2 <bib2>`].

The reference [:ref:`bib3 <bib3>`] also provides the general formula for the acoustic modes of a compressible fluid domain comprising a free surface contained in a parallelepipedic reservoir with rigid walls:

:math:`{f}_{\mathit{ijk}}=\frac{c}{2}\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}+\frac{{(2k+1)}^{2}}{4{h}^{2}}}`

where :math:`i`, :math:`j`, and :math:`k` are the integer orders of the longitudinal, transverse, and vertical modes (number of nodal lines in each horizontal direction and the vertical direction).


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

For :math:`L=0.8m`, :math:`l=0.1m` and :math:`h=0.3\phantom{\rule{2em}{0ex}}m` the first slop modes have the following frequencies:


.. csv-table::

    "Mode :math:`i`, :math:`j` ", "Frequency (:math:`\mathit{Hz}`)"
    "1, 0", "0.898252"
    "2, 0", "1.384516"
    "3, 0", "1.709525"
    "4, 0", "1.975511"
    "5, 0", "2.208850"
    "6, 0", "2.419691"
    "7, 0", "2.613567"
    "8, 0", "2.794020"
    "0, 1", "2.794020"
    "1, 1", "2.804871"


For :math:`L=0.8m`, :math:`l=0.1m` and :math:`h=0.3\phantom{\rule{2em}{0ex}}m` the first acoustic modes have the following frequencies:


.. csv-table::

    "Mode :math:`i`, :math:`j`, :math:`k` ", "Frequency (:math:`\mathit{Hz}`)"
    "0, 0, 0 (1st mode according to :math:`z`)", "1166,67"
    "1, 0, 0 (1st mode according to :math:`x`)", "1458,33"
    "2, 0, 0", "2103,24"
    "3, 0, 0", "2872,58"
    "0, 0, 1", "3500,00"



Bibliographical references
---------------------------


1. WAECKEL F., LEPOUTERE C. Internal note EDF/DER "Effect of gravity on the free surface of a fluid coupled to a structure", HP-61/93/139.


2. MUTO, KASA, NAKAHARA, ISHIDA "Experimental tests on sloshing response of a water pool with submerged blocks" - ASME, vol. PVP 98, (1985).


3. BLEVINS R.D. Formulas for natural frequency and mode shape. Ed Krieger