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Hybrid Methods in Engineering

ISSN Print: 1099-2391

Hybrid Methods in Engineering

DOI: 10.1615/HybMethEng.v3.i4.30
56 pages


I. P. Gavrilyuk
Berufakademie Thuringen-Staatliche Studienakademie, 99817, Eisenach, Am Wartenberg 2, Germany
I. A. Lukovsky
National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine
A. N. Timokha
National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine


We develop a nonconformal transformation technique and combine it with the variational modal approach for modeling the nonlinear sloshing of an incompressible fluid with irrotational flow. No overturning, breaking, or shallow fluid waves are assumed; the fluid partly occupies an arbitrary smooth tank with rigid walls having a noncylindrical shape. In its theoretical part, the article assumes that the tank's cavity can be smoothly transformed into an artificial cylindrical domain, where the equation of free surface allows for both normal form and modal (Fourier) decomposition of the instantaneous surface shape. This transformation has singularities in the lower (upper) vertexes of the tank. It leads to degenerating boundary problems, but spectral, variational theorems and modal systems save invariant formulations. The main body of the article derives modal theory for sloshing in a circular conical tank. Linearized and nonlinear problems are examined in curvilinear coordinates. The spectral problem on natural modes is solved by the variational method. Solutions are expanded in a series by solid spheric harmonics satisfying the zero-Neumann condition everywhere on the walls. The algorithm is robust and numerically efficient for calculating both the lower and higher natural modes. Calculations are validated by experimental data by Bauer [4] and Mikishev & Dorozhkin [29]. Derivation of the approximate weakly nonlinear modal theory is based on the variational approach by Lukovsky [24] within modal functions ordered in accordance with the Moiseyev detuning. The theory is valid when the semi-apex angle α of the conical tank is between 25° and 55°. This is due to internal resonance at 6° and 12°, and because surface waves become shallow for α > 60°. The theory distinguishes planar and rotating resonant steady-state waves forced by sway. Hard-soft spring for amplitude response of the rotating wave is observed when α is close to 41°. Three-dimensional animation of wave motions accounts for second-order flows and facilitates the physical treatment.