# Hydrodynamic Forces Exerting on an Oscillating Cylinder under Translational Motion in Water Covered by Compressed Ice

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Boundary Conditions

## 3. Solutions of the Stationary Problem

## 4. Solutions of the Radiation Problem

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\rho $ water density |

${\rho}_{1}$ ice density |

${h}_{1}$ thickness of the ice plate |

g acceleration due to gravity |

E Young’s modulus of elastic plate |

$\nu $ Poisson ratio |

$D=E{h}_{1}^{3}/\left[12(1-{\nu}^{2})\right]$ ice rigidity |

Q longitudinal ice stress |

${Q}_{0}$ first critical value of the stress parameter when the anomalous dispersion occurs |

${Q}_{*}=2\sqrt{\rho gD}$ second critical value of the stress parameter when the ice buckling occurs |

$M={\rho}_{1}{h}_{1}$ parameter describing the inertial property of ice |

${c}_{p}$ phase velocity of FGWs |

${c}_{g}$ group velocity of FGWs |

a cylinder radius |

$\mathsf{\Omega}$ frequency of cylinder oscillations |

$Fr\equiv U/\sqrt{ga}$ Froude number in terms of the cylinder radius a |

${\overline{F}}_{s1}\equiv {F}_{s1}/\left(\right)open="("\; close=")">\pi \rho \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}{a}^{2}$ normalised wave resistance in the stationary flow |

${\overline{F}}_{s2}\equiv {F}_{s2}/\left(\right)open="("\; close=")">\pi \rho \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}{a}^{2}$ normalised lift force in the stationary flow |

${\tau}_{jk}$ a complex matrix of radiation forces |

${\mu}_{jk}$ a real matrix of added masses |

${\lambda}_{jk}$ a real matrix of damping coefficients |

$\overline{\mu}={\mu}_{jj}/\left(\pi \rho \phantom{\rule{0.166667em}{0ex}}{a}^{2}\right)$ normalised added mass coefficient |

$\overline{\lambda}={\lambda}_{jj}/\left(\pi \rho \phantom{\rule{0.166667em}{0ex}}{a}^{2}\mathsf{\Omega}\right)$ -normalised damping coefficient |

## Abbreviations

FGW | flexural-gravity wave |

## References

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**Figure 1.**Sketch of the flow around a rigid cylinder in a deep water covered by a flexible ice cover.

**Figure 2.**(Color online.) The wave resistance ${\overline{F}}_{s1}$ and lift force ${\overline{F}}_{s1}$ as functions of the dimensionless velocity ${\mathrm{Fr}}_{h}\equiv U/\sqrt{gh}$. Solid black lines (1) represent the results shown in Figure 6 of [14] for the infinitely deep fluid; red dots are the results of this work for $M=0$; blue lines (2) are the results of this work for $M=922.5$ kg/m${}^{2}$.

**Figure 3.**(Color online.) The wave resistance ${\overline{F}}_{s1}$ and lift force ${\overline{F}}_{s2}$ as functions of the Froude number $\mathrm{Fr}=U/\sqrt{ga}$ for different values of the compression parameter $\tilde{Q}=0,1.2,1.8,1.95.$

**Figure 4.**Configuration of the domains ${G}_{j}\phantom{\rule{0.277778em}{0ex}}(j=1,...,6)$ for different values of the compression parameter $\tilde{Q}:0\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{a}\right);1.2\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{b}\right);1.8\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{c}\right);\mathrm{and}1.95\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{d}\right)$. The thickness of the ice plate was ${h}_{1}=0.5$ m.

**Figure 5.**(Color online.) The dependence of the added mass coefficients: ${\overline{\mu}}_{ij}\phantom{\rule{0.277778em}{0ex}}(i,j)=1,2$ on the dimensionless frequency of cylinder oscillation for $\tilde{Q}=0,1.2,1.8,1.95$ and fixed Froude number $\mathrm{Fr}=0.5$. Crosses show the values of ${\overline{\mu}}_{11}$ and ${\overline{\mu}}_{22}$ for the fluid covered by a rigid lid.

**Figure 6.**(Color online.) The dependence of the added damping coefficients: ${\overline{\lambda}}_{ij}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(i,j)=1,2$ on the dimensionless frequency of cylinder oscillation for $\tilde{Q}=0,1.2,1.8,1.95$ and fixed Froude number $\mathrm{Fr}=0.5$.

**Figure 7.**(Color online.) The dependence similar to those shown in Figure 6 but for $\mathrm{Fr}=1$.

**Figure 8.**(Color online.) The dependence similar to those shown in Figure 6 but for $\mathrm{Fr}=1$.

**Figure 9.**(Color online.) The dependences of the diagonal damping coefficients ${\overline{\lambda}}_{11}$ (

**a**) and ${\overline{\lambda}}_{22}$ (

**b**) on the dimensionless frequency $\tilde{Q}=0,1.2,1.8,1.95$ and Froude number $\mathrm{Fr}=7$.

**Figure 10.**(Color online.) The frequency dependences of the added mass $\overline{\mu}$ (

**a**) and damping coefficient $\overline{\lambda}$ (

**b**) for a few values of the parameter $\tilde{Q}=1.2,\phantom{\rule{0.166667em}{0ex}}1.4,\phantom{\rule{0.166667em}{0ex}}1.8$.

**Table 1.**Dependences of the parameters ${\mathrm{Fr}}_{p}$, ${\mathrm{Fr}}_{g}$ and ${\sigma}^{*}$ on the normalised compression parameter $\tilde{Q}$.

$\tilde{\mathit{Q}}$ | ${\mathbf{Fr}}_{\mathit{p}}$ | ${\mathbf{Fr}}_{\mathit{g}}$ | ${\mathit{\sigma}}^{*}$ |
---|---|---|---|

0 | 1.7124 | 1.1225 | 0.2226 |

1.2 | 1.1231 | 0.3311 | 0.3691 |

1.8 | 0.5715 | −0.6325 | 0.6285 |

1.95 | 0.2870 | −1.2972 | 0.8650 |

$\phantom{.}{\mathit{G}}_{\mathit{j}}\phantom{.}$ | ${\mathit{k}}_{2}^{\left(1\right)}$ | ${\mathit{k}}_{2}^{\left(2\right)}$ | ${\mathit{k}}_{3}^{\left(1\right)}$ | ${\mathit{k}}_{3}^{\left(2\right)}$ | ${\mathit{k}}_{3}^{\left(3\right)}$ | ${\mathit{k}}_{4}^{\left(1\right)}$ | ${\mathit{k}}_{4}^{\left(2\right)}$ | ${\mathit{k}}_{4}^{\left(3\right)}$ |
---|---|---|---|---|---|---|---|---|

${G}_{1}$ | $\phantom{.}x<0\phantom{.}$ | $\phantom{.}x>0\phantom{.}$ | $x>0$ | - | - | $x<0$ | - | - |

${G}_{2}$ | $x<0$ | $x>0$ | $x>0$ | $x<0$ | $x>0$ | $x<0$ | - | - |

${G}_{3}$ | - | - | $x>0$ | $x<0$ | $x>0$ | $x<0$ | - | - |

${G}_{4}$ | - | - | $x>0$ | - | - | $x<0$ | - | - |

${G}_{5}$ | - | - | $x>0$ | - | - | $x<0$ | $x>0$ | $x<0$ |

${G}_{6}$ | - | - | $x>0$ | $x<0$ | $x>0$ | $x<0$ | $x>0$ | $x<0$ |

## Short Biography of Authors

Yury Stepanyants graduated in 1973 with the HD of MSc Diploma from the Gorky State University (Russia) and started to work as the Engineer with the Research Radiophysical Institute in Gorky. Then, he proceeded his career with the Institute of Applied Physics of the Russian Academy of Sciences (Nizhny Novgorod) taking successively the positions of Junior, Senior, and Leading Research Scientist from 1977 to 1997. In 1983 Yury obtained a PhD in Physical Oceanography, and in 1992 he obtained a degree of Doctor of Sciences in Geophysics. After immigration in Australia in 1998, Yury worked for 12 years as the Senior Research Scientist with the Australian Nuclear Science and Technology Organisation in Sydney. Since July 2009 he holds a position of Full Professor at the University of Southern Queensland in Toowoomba, Australia. Yury has published more than 100 journal papers, 3 books, several review papers and has obtained 3 patents. His major scientific interests are in: Hydrodynamics and geophysical fluid mechanics; Theory of nonlinear oscillations and waves; Exact and asymptotic solutions of nonlinear equations; Mathematical modelling and computational physics. | |

Izolda Sturova graduated from Novosibirsk State University in 1964 and began working at the Institute of Hydrodynamics where she works until now. She took the positions of: the Intern-Researcher, Junior Scientist (since 1967), Senior Scientist (since 1976), Leading Research Scientist (since 1988), Head of a Sector (since 1991), and Chief Scientist (from 1998 to present time). In 1972 she obtained a PhD degree in Fluid Mechanics, and in 1995 the degree of Doctor of Sciences in Fluid Mechanics. Izolda is the author of over 100 scientific publications, including one monograph. Her research interests are: surface, internal and flexural-gravity waves, hydroelastic properties of ice and floating platforms. |

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**MDPI and ACS Style**

Stepanyants, Y.; Sturova, I.
Hydrodynamic Forces Exerting on an Oscillating Cylinder under Translational Motion in Water Covered by Compressed Ice. *Water* **2021**, *13*, 822.
https://doi.org/10.3390/w13060822

**AMA Style**

Stepanyants Y, Sturova I.
Hydrodynamic Forces Exerting on an Oscillating Cylinder under Translational Motion in Water Covered by Compressed Ice. *Water*. 2021; 13(6):822.
https://doi.org/10.3390/w13060822

**Chicago/Turabian Style**

Stepanyants, Yury, and Izolda Sturova.
2021. "Hydrodynamic Forces Exerting on an Oscillating Cylinder under Translational Motion in Water Covered by Compressed Ice" *Water* 13, no. 6: 822.
https://doi.org/10.3390/w13060822