2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference [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:
\({f}_{\mathit{ij}}=\frac{1}{2\pi }\sqrt{\pi g\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}}\mathrm{tanh}\left[\pi H\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}}\right]}\)
where \(i\) and \(j\) 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:
\({p}_{\mathit{ij}}(x,y,z)=\mathrm{cos}\left(i\pi \frac{2x+l}{2l}\right)\mathrm{cos}\left(j\pi \frac{2y+L}{2L}\right)\left(\mathrm{cosh}\left({\lambda }_{\mathit{ij}}(H-z)\right)+{A}_{\mathit{ij}}\mathrm{sinh}\left({\lambda }_{\mathit{ij}}(H-z)\right)\right)\)
where \({\lambda }_{\mathit{ij}}\) designates the verifying vertical wave number: \({\lambda }_{\mathit{ij}}H\mathrm{tanh}\left({\lambda }_{\mathit{ij}}H\right)=4{\pi }^{2}{f}_{\mathit{ij}}H/g\) and with a verifying coefficient: \({A}_{\mathit{ij}}=-\mathrm{tanh}\left({\lambda }_{\mathit{ij}}H\right)\). The fluctuating sloshing pressures are maximum on the free surface. The height of the free surface level is expressed as: \({z}_{l}(x,y)={p}_{\mathit{ij}}(x,y\mathrm{,0})/(\rho \mathrm{.}g)\).
In the particular case where \(\frac{L}{l}\) is large, the formula is simplified for the longitudinal modes [bib1], [bib2].
The reference [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:
\({f}_{\mathit{ijk}}=\frac{c}{2}\sqrt{\frac{{i}^{2}}{{L}^{2}}+\frac{{j}^{2}}{{l}^{2}}+\frac{{(2k+1)}^{2}}{4{h}^{2}}}\)
where \(i\), \(j\), and \(k\) are the integer orders of the longitudinal, transverse, and vertical modes (number of nodal lines in each horizontal direction and the vertical direction).
2.2. Benchmark results#
For \(L=0.8m\), \(l=0.1m\) and \(h=0.3\phantom{\rule{2em}{0ex}}m\) the first slop modes have the following frequencies:
Mode \(i\), \(j\) |
Frequency (\(\mathit{Hz}\)) |
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 \(L=0.8m\), \(l=0.1m\) and \(h=0.3\phantom{\rule{2em}{0ex}}m\) the first acoustic modes have the following frequencies:
Mode \(i\), \(j\), \(k\) |
Frequency (\(\mathit{Hz}\)) |
0, 0, 0 (1st mode according to \(z\)) |
1166,67 |
1, 0, 0 (1st mode according to \(x\)) |
1458,33 |
2, 0, 0 |
2103,24 |
3, 0, 0 |
2872,58 |
0, 0, 1 |
3500,00 |
2.3. Bibliographical references#
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.
MUTO, KASA, NAKAHARA, ISHIDA « Experimental tests on sloshing response of a water pool with submerged blocks » - ASME, vol. PVP 98, (1985).
BLEVINS R.D. Formulas for natural frequency and mode shape. Ed Krieger