Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Catamaran cross section and probes for wave elevation. The circular cross section center of each hull is placed at a distance b from the catamaran symmetry plane and is characterized by a radius a corresponding to the depth of the catamaran at rest. Dimensions indicated are expressed in meters.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Grid sensitivity study: coarsest-medium-finest meshes designed to be simulated at 6 rad/s. Three different mesh regions: structured around the free surface, body-fitted structured around the body and unstructured in the far field. Mesh size at cylinder wall d1=3e−4 m for M1 (coarse), d2=1e−4 m for M2 (medium) and d3=2e−5 m for M3 (fine).

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Validation study: CFD results are presented through solid and dashed line for ξ=0.635 cm and ξ=1.27 cm, respectively. Dashed-dot line represents potential flow theory. Experiments have been represented through filled dots when stiffeners have been used during the experiments. (a) Nondimensional added mass coefficient: b=1.5a, (b) nondimensional damping coefficient: b=1.5a, (c) nondimensional added mass coefficient: b=2.0a, (d) nondimensional damping coefficient: b=2.0a, (e) nondimensional added mass coefficient: b=4.0a, and (f) nondimensional damping coefficient: b=4.0a.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional vorticity contour for hull configuration b=1.5a at the piston mode frequency: (a) ξ=0.635 cm, (b) ξ=1.27 cm, (c) ξ=2.54 cm, and (d) ξ=3.81 cm

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional wave elevation at probes located at centerline y = 0 (A), y=b+2a (B), y=b+4a (C), and y=b+8a (D). Hull distance b=1.5a, heave amplitude ξ=0.635 cm, oscillation frequency ω=6.50 rad/s.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional added mass, damping coefficients and inner wave amplitude (probeA). From left to right b=1.5a, b=2a and b=4a. From top to bottom ξ=0.635 cm, ξ=1.270 cm, ξ=2.540 cm and ξ=3.81 cm. Vertical lines correspond to undamped piston mode frequency resulting from Eq. (8) with λ = 1. Nondimensional added mass values are indicated on the left axis while nondimensional damping and inner wave amplitude values are indicated on the right axis.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Hydrodynamic coefficients resulting from a forced heave motion of amplitude ξ=1.27 cm. Symbols on the curves represent computed points: (a) nondimensional added mass coefficient and (b) nondimensional damping coefficient.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional added mass (left axis), damping, inner (probe A) and outer wave amplitude (probe B, C, D) (right axis) for hull distance b=1.5a and motion amplitude ξ=1.27 cm. Damping experiences a relative minimum at nondimensional frequency of 0.903 corresponding to the trapping mode frequency presented in Fig. 9. (a) ω=6.25 rad/s : piston mode frequency, (b) ω=7.25 rad/s : zero damping frequency, and (c) wave elevation plot comparison (circles: piston mode frequency, crosses: zero damping frequency).

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Volume of fluid contour and wave elevation plots for ω=6.25 rad/s and ω=7.25 rad/s. Hull distance b=1.5a and oscillation amplitude ξ=1.27 cm.

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Free decay test of a column of water located at z = a (deck height) for an hull configuration b=1.5a: (a) inner wave elevation and (b) fast Fourier transform (FFT) of inner wave elevation

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional piston-mode frequency trend with respect to heave amplitude for three different gap widths

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional inner wave amplitude trend with respect to heave amplitude for three gap widths

Date of download: 11/5/2017 Copyright © ASME. All rights reserved. From: Amplitude Induced Nonlinearity in Piston Mode Resonant Flow: A Fully Viscous Numerical Analysis J. Offshore Mech. Arct. Eng. 2017;140(1):011101-011101-11. doi:10.1115/1.4037487 Figure Legend: Nondimensional inner wave amplitude trend with respect to gap width for four heave amplitudes