Sketch of a square base tank forced in sway/surge and roll/pitch. Rigid body motions are considered in the moving coordinate system framed with the rigid tank.
Top views of the cubic tanks with wave probes for the first (a) and the second (b) model test series.
Local phenomena in the corner during swirling.
Computed steady-state response curves by the adaptive modal technique in the plane (bold solid lines, is an arbitrary point on the curve), where is either wave amplitude or hydrodynamic force amplitude. Accuracy of the adaptive method is and therefore the actual response curves belong to a neighborhood of the computed data visualized as circles or ellipses depending on the ratio between the axes units.
Transition to steady-state swirling, phases III–V. Nondimensional wave amplitudes (at w3 and w5) and hydrodynamic forces (normalized by , where is the fluid mass). Numerical data for , , , and (forcing period is ). The lower dotted line represents the steady-state amplitude, and and are maximum and minimum amplitudes at phase III, whose mean value is marked by the upper dotted horizontal line.
Diagonal resonant excitations with , , . Response curves for stable diagonal waves. Solid lines (curves , , and ) imply theoretical steady-state wave amplitude (dimensionless elevation at w3) and hydrodynamic forces and (normalized by , where is the fluid mass) vs . Measured amplitudes are represented by minimum (▵) and maximum (▿) during the last of the model tests. The figure includes the dotted lines that encompass the domains for the expected measured steady-state amplitudes. Points and their abscissas denote the endpoints of the theoretical response curves, where the diagonal regime changes stability properties. The dashed line shows extra sub-branches, which exist in the asymptotic limit , but disappear in adaptive modal modeling. Analogously, the range implies theoretical estimate of irregular, chaotic motions as .
The same as in Fig. 7 , but for swirling. Solid lines (curves and ) are parts of branches in Fig. 7 . The curve with endpoints and corresponds to stable swirling, so that is its effective frequency domain. Dashed lines present additional frequency domains of stable swirling established by single-dominant theory [Faltinsen et al. (Ref. 3 )]. However, adaptive modal theory expects irregular, chaotic motions in the ranges and .
The last seconds from model test recordings at w3 and w5. Diagonal forcing with , , , and .
Theoretical results and model test data on amplitudes of stable steady-state motions. Dimensionless horizontal force consists of maximum of and . Diagonal forcing with , , . The notation is the same as in Figs. 7 and 8 .
Longitudinal resonant excitation with , , . Response curves and model test data. Solid lines (curves , , and ) imply theoretical steady-state wave amplitude (dimensionless elevation at w3/g10 and w5/g8) and hydrodynamic forces and (normalized by , where is the fluid mass) vs . The frequency domains and correspond to irregular, chaotic wave motions. The mean amplitudes in phase III are computed and drawn as the branch (crest-and-dotted line). Measured amplitudes are represented by minimum (▵) and maximum (▿) during the last of the model tests. Swirling cases are ⑨–12.
The same as in Fig. 11 , but for . The frequency domains and imply theoretical estimates of irregular, chaotic wave motions. The estimate of changes to , if calculations are made with small initial conditions. Swirling cases are 15 and 16.
The same as in Fig. 11 , but only for dimensionless wave elevations at g10 and g8 and for roll excitation corresponding to , .
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