Modal Modeling of Nonlinear Fluid Sloshing in Tanks with Non-Vertical Walls. Non-Conformal Mapping Technique

Abstract

Nonlinear sloshing of an incompressible fluid with irrotational flow is analyzed. The fluid partly occupies a smooth tank with walls having non-cylindrical shape. No overturning, breaking and shallow water waves are assumed. Nonconformal mapping technique by Lukovsky (1975) is developed further. It assumes that tank’s cavity can be transformed into an artificial cylindrical domain, where equation of free surface allows both normal form and modal representation of instantaneous surface shape. Admissible tensor transformations have due singularities in mapping the lower (upper) corners of the tank into artificial bottom (roof). It leads to degenerating spectral boundary problems on natural modes. The paper delivers the mathematical background for these spectral problems and establishes the spectral and variational theorems. Natural modes in circular conical cavity are calculated by variational algorithm based on these theorems. It is shown that the algorithm is robust and numerically efficient for calculating both lower and higher natural modes. Finally, the paper shows that the well-known infinite-dimensional modal systems by Lukovsky (derived for sloshing in cylindrical tank) keep invariant structure with respect to admissible tensor transformations (for translatory motions of the vehicle). This makes it possible to offer the simple derivation algorithm of nonlinear modal systems for the studied case. When using anzatz by Lukovsky we derive the five-dimensional modal system for nonlinear sloshing in circular conic tanks.