scholarly journals Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 4. Oblique forcing and linear viscous damping

2017 ◽  
Vol 822 ◽  
pp. 139-169 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Steady State ◽  
Three Dimensional ◽  
Finite Depth ◽  
Width Ratio ◽  
Viscous Damping ◽  
Response Curves ◽  
Viscous Boundary Layer ◽  
Wave Regime ◽  
Modal Theory ◽  
Nonlinear Sloshing

Faltinsen et al. (J. Fluid Mech., vol. 487, 2003, pp. 1–42) (henceforth, Part 1) examined an undamped nonlinear resonant steady-state sloshing in a square-base tank by developing an approximate (asymptotic) Narimanov–Moiseev-type multimodal theory. The focus was on longitudinal and diagonal harmonic tank excitations. Neglecting the linear viscous boundary-layer damping was justified for model tanks with breadths of the order of metres. However, nonlinear sloshing in clean tanks of smaller size (count in centimetres) may be affected by damping in finite depth conditions. Qualitative and quantitative properties of the damped resonant steady-state sloshing in a square-base tank are now studied by using the modal theory from Part 1 equipped with the linear damping terms. The tank harmonically oscillates along an arbitrary horizontal (oblique) direction. An analytical asymptotic steady-state undamped solution is derived and the corresponding response curves are analysed versus the forcing direction. When the tank width $=$ breadth $=$ $L\sim 10$  cm, the surface tension effect on the free-surface dynamics can be neglected but the linear viscous damping should be included into the Narimanov–Moiseev nonlinear asymptotic modal theory. We analytically show that the steady-state damped sloshing possesses a series of distinguishing features so that, e.g. the square-like standing wave regime fully disappears and becomes replaced by swirling. Typical response curves of the damped steady-state resonant sloshing are studied for the liquid depth-to-width ratio exceeding 0.5. The computational results of the steady-state resonant response amplitudes are in a satisfactory agreement with observations and measurements by Ikeda et al. (J. Fluid Mech., vol. 700, 2012, pp. 304–328), which were conducted with a relatively small laboratory container.

Steady-State Dynamic Response of a Limited Slip System

1968 ◽  
Vol 35 (2) ◽  
pp. 322-326 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Slip System ◽  
External Load ◽  
Initial Conditions ◽  
Viscous Damping ◽  
Steady State Response ◽  
Response Curves ◽  
State Response ◽  
System Nonlinearity

The steady-state response of a system constrained by a limited slip joint and excited by a trigonometrically varying external load is discussed. It is shown that the system may possess such features as disconnected response curves and jumps in response depending on the strength of the system nonlinearity, the level of excitation, the amount of viscous damping, and the initial conditions of the system.

Resonant sloshing in an upright annular tank

2016 ◽  
Vol 804 ◽  
pp. 608-645 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Ivan A. Lukovsky ◽  
Alexander N. Timokha
Keyword(s):  
Steady State ◽  
Harmonic Excitation ◽  
Irregular Waves ◽  
Elliptic Type ◽  
Response Curves ◽  
Modal Theory ◽  
Resonant Forcing ◽  
Sloshing Frequency ◽  
Amplitude Component ◽  
Liquid Depth

Resonant sloshing in an upright annular tank is studied by using a new nonlinear modal theory, which is complete within the framework of the Narimanov–Moiseev asymptotics. The applicability is justified for a fairly deep liquid (the liquid-depth-to-outer-tank-radius ratio $1.5\lesssim h=\bar{h}/\bar{r}_{2}$) and away from the non-dimensional inner radii $r_{1}=\bar{r}_{1}/\bar{r}_{2}=0.08546$, 0.17618, 0.27826, 0.31323, 0.31855, 0.43444, 0.46015, 0.48434, 0.68655, 0.70118. The theory is used to describe steady-state (stable and unstable) resonant waves due to a harmonic excitation with the forcing frequency close to the lowest natural sloshing frequency. We show that the surge-sway-pitch-roll excitation is always of either longitudinal or elliptic type. Existing experimental results on the horizontally excited steady-state wave regimes in an upright circular tank ($r_{1}=0$) are utilised for validation. Inserting an inner pole with the radii $r_{1}\approx 0.25$ and 0.35 ($1.5\lesssim h$) causes that no stable swirling and/or irregular waves exist. The response curves for an elliptic-type excitation are examined versus the minor-axis forcing-amplitude component. Stable swirling is then expected being co- and counter-directed to the angular forcing direction. Passage to the rotary (circular) excitation keeps the co-directed swirling stable for all resonant forcing frequencies but the stable counter-directed swirling disappears.

scholarly journals Resonant Steady-State Sloshing in Upright Tanks: Effect of Three-Dimensional Excitations and Viscosity

2018 ◽  
Author(s):  
Alexander N. Timokha ◽  
Ihor A. Raynovskyy
Keyword(s):  
Steady State ◽  
Order Approximation ◽  
Three Dimensional ◽  
Primary Resonance ◽  
Resonance Phenomenon ◽  
Phase Lag ◽  
Response Curves ◽  
Sloshing Frequency ◽  
Theoretical Results ◽  
Good Agreement

Bearing in mind recent experimental and theoretical results showing that viscous damping can qualitatively affect resonant sloshing in clean tanks, the Narimanov-Moiseev multimodal sloshing theory for an upright circular container is revised to analytically analyze steady-state surface waves when the container performs a small-amplitude sway/roll/pitch/surge prescribed periodic motion with the forcing frequency close to the lowest natural sloshing frequency. The revised theory is applicable for the radius-scaled mean liquid depths h > 1 providing the secondary resonance phenomenon does not occur at the primary resonance zone. A focus is on how the damping influences the phase lag as well as on the amplitude response curves versus the forcing type, which can in the lowest-order approximation be treated as if the container translatory moves along an elliptic orbit in the horizontal plane. The analytical results are compared with existing experiments for longitudinal and circular orbital tank excitations. Whereas a good agreement is found for longitudinal excitations, a discrepancy is detected for the circular orbital forcing. The discrepancy may, most probably, be explained by the wave breaking and mean angular mass-transport (Ludwig Prandtl, 1949) phenomena. Occurrence of the Prandtl phenomenon makes inapplicable the existing analytical inviscid sloshing theories, even if they are modified to account for damping.

scholarly journals The first observation of 4D tomography measurement of plasma structures and fluctuations

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Chanho Moon ◽  
Kotaro Yamasaki ◽  
Yoshihiko Nagashima ◽  
Shigeru Inagaki ◽  
Takeshi Ido ◽  
...  
Keyword(s):  
Steady State ◽  
Three Dimensional ◽  
Axial Direction ◽  
Axial Position ◽  
Entire Cross Section ◽  
Helicon Plasma ◽  
Emission Profile ◽  
Dimensional Measurement ◽  
Tomography System ◽  
Three Dimensional Measurement

AbstractA tomography system is installed as one of the diagnostics of new age to examine the three-dimensional characteristics of structure and dynamics including fluctuations of a linear magnetized helicon plasma. The system is composed of three sets of tomography components located at different axial positions. Each tomography component can measure the two-dimensional emission profile over the entire cross-section of plasma at different axial positions in a sufficient temporal scale to detect the fluctuations. The four-dimensional measurement including time and space successfully obtains the following three results that have never been found without three-dimensional measurement: (1) in the production phase, the plasma front propagates from the antenna toward the end plate with an ion acoustic velocity. (2) In the steady state, the plasma emission profile is inhomogeneous, and decreases along the axial direction in the presence of the azimuthal asymmetry. Furthermore, (3) in the steady state, the fluctuations should originate from a particular axial position located downward from the helicon antenna.

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