Multimodal analysis of weakly nonlinear sloshing in a spherical tank

2013 ◽  
Vol 719 ◽  
pp. 129-164 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Technical Note ◽  
Modal System ◽  
Weakly Nonlinear ◽  
Multimodal Analysis ◽  
Spherical Tank ◽  
Sloshing Frequency ◽  
The Stability ◽  
Asymptotic Time ◽  
Time Periodic ◽  
Good Agreement

AbstractSloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963,Tech. Rep.TN D-1991, NASA Technical Note) and Sumner (1966,Tech. Rep.TN D-3210, NASA). A good agreement is established for$0. 2\leq h\lesssim 1$. Discrepancy for higher liquid depths$1\lesssim h\lt 2$are explained by secondary resonance.

scholarly journals Generalizing the Multimodal Method for the Levitating Drop Dynamics

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
M. O. Chernova ◽  
I. A. Lukovsky ◽  
A. N. Timokha
Keyword(s):  
Free Surface ◽  
Almost Periodic ◽  
Modal System ◽  
Periodic Oscillations ◽  
Weakly Nonlinear ◽  
Modal Theory ◽  
Drop Dynamics ◽  
Finite Dimensional ◽  
Multimodal Method ◽  
Good Agreement

The present paper extends the multimodal method, which is well known for liquid sloshing problems, to the free-surface problem modeling the levitating drop dynamics. The generalized Lukovsky-Miles modal equations are derived. Based on these equations an approximate modal theory is constructed to describe weakly-nonlinear axisymmetric drop motions. Whereas the drop performs almost-periodic oscillations with the frequency close to the lowest natural frequency, the theory takes a finite-dimensional form. Periodic solutions of the corresponding finite-dimensional modal system are compared with experimental and numerical results obtained by other authors. A good agreement is shown.

On the stability of Kelvin waves

1989 ◽  
Vol 206 ◽  
pp. 1-23 ◽  
Author(s):  
W. K. Melville ◽  
G. G. Tomasson ◽  
D. P. Renouard
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Resonance ◽  
Approximate Solutions ◽  
Kelvin Waves ◽  
Kelvin Wave ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Good Agreement

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincaré modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincaré waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d'Hières & Zhang (1987).

scholarly journals Theoretical Analysis and Numerical Simulation of Resonances and Stability of a Piecewise Linear-Nonlinear Vibration Isolation System

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
X. Gao ◽  
Q. Chen
Keyword(s):  
Vibration Isolation ◽  
Theoretical Study ◽  
Piecewise Linear ◽  
Isolation System ◽  
Single Degree Of Freedom ◽  
Vibration Isolation System ◽  
Weakly Nonlinear ◽  
The Stability ◽  
Subharmonic Resonances ◽  
Good Agreement

A methodology is presented to study the resonance and stability for a single-degree-of-freedom (SDOF) system with a piecewise linear-nonlinear stiffness term (i.e., one piece is linear and the other is weakly nonlinear). Firstly, the exact response of the linear governing equation is obtained, and a modified perturbation method is applied to finding the approximate solution of the weakly nonlinear equation. Then, the primary and 1/2 subharmonic resonances are obtained by imposing continuity conditions and periodicity conditions. Furthermore, Jacobian matrix is derived to investigate the stability of resonance responses. Finally, the results of theoretical study are compared with numerical results, and a good agreement is observed.

Nonlinear Sloshing in a Spherical Tank

2013 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Steady State ◽  
Natural Frequency ◽  
Secondary Resonance ◽  
Weakly Nonlinear ◽  
Modal Theory ◽  
Spherical Tank ◽  
Theoretical Results ◽  
Multimodal Method ◽  
Comparison With Experiments ◽  
Good Agreement

Steady-state resonant sloshing in a spherical rigid tank due to horizontal harmonic excitations at the lowest natural frequency is classified by combining the nonlinear multimodal method with the Moiseev-Narimanov asymptotics. The theoretical results are validated by comparison with experiments of Sumner & Stofan (1963) and other already published model tests. A good agreement is found for the depth-to-tank radius ratios 0.2 ≤ h ≲ 1 but, when 1 ≲ h ≲ 2, secondary resonance and splashing limits the applicability of the constructed weakly-nonlinear modal theory.

scholarly journals Technical Note: Quantification of interferences of wet chemical HONO LOPAP measurements under simulated polar conditions

2008 ◽  
Vol 8 (22) ◽  
pp. 6813-6822 ◽  
Author(s):  
J. Kleffmann ◽  
P. Wiesen
Keyword(s):  
Technical Note ◽  
Swiss Alps ◽  
Pollution Level ◽  
Research Station ◽  
Polar Regions ◽  
Alpine Research ◽  
Absorption Photometer ◽  
Present Pilot Study ◽  
Good Agreement

Abstract. In the present pilot study, an optimized LOPAP instrument (LOng Path Absorption Photometer) for the detection of nitrous acid (HONO) in the atmosphere (DL 0.2 pptV) was tested at the high alpine research station Jungfraujoch at 3580 m altitude in the Swiss Alps under conditions comparable to polar regions. HONO concentrations in the range <0.5–50 pptV with an average of 7.5 pptV were observed at the Jungfraujoch. The diurnal profiles obtained exhibited clear maxima at noon and minima with very low concentration during the night supporting the proposed photochemical production of HONO. In good agreement with recent measurements at the South Pole, it was demonstrated, that interferences of chemical HONO instruments can significantly influence the measurements and lead to considerable overestimations, especially for low pollution level. Accordingly, the active correction of interferences is of paramount importance for the determination of reliable HONO data.

OPTIMAL VELOCITY MODEL WITH RELATIVE VELOCITY

2006 ◽  
Vol 17 (01) ◽  
pp. 65-73 ◽  
Author(s):  
SHIRO SAWADA
Keyword(s):  
Nonlinear Analysis ◽  
Relative Velocity ◽  
Velocity Model ◽  
Fundamental Diagram ◽  
Optimal Velocity ◽  
Traffic Jam ◽  
Optimal Velocity Model ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Good Agreement

The optimal velocity model which depends not only on the headway but also on the relative velocity is analyzed in detail. We investigate the effect of considering the relative velocity based on the linear and nonlinear analysis of the model. The linear stability analysis shows that the improvement in the stability of the traffic flow is obtained by taking into account the relative velocity. From the nonlinear analysis, the relative velocity dependence of the propagating kink solution for traffic jam is obtained. The relation between the headway and the velocity and the fundamental diagram are examined by numerical simulation. We find that the results by the linear and nonlinear analysis of the model are in good agreement with the numerical results.

Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth

2002 ◽  
Vol 470 ◽  
pp. 319-357 ◽  
Author(s):  
ODD M. FALTINSEN ◽  
ALEXANDER N. TIMOKHA
Keyword(s):  
Finite Depth ◽  
Modal System ◽  
Full Account ◽  
Inviscid Flows ◽  
State Response ◽  
Linear Damping ◽  
Convergence Problems ◽  
Run Up ◽  
Fourth Order Polynomial ◽  
Good Agreement

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h/l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h/l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tank's surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h/l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Global Stability Analysis of the Wake behind a Circular Cylinder Based on the POD-Chiba Method

2011 ◽  
Vol 137 ◽  
pp. 72-76
Author(s):  
Wei Zhang ◽  
Xian Wen ◽  
Yan Qun Jiang
Keyword(s):  
Stability Analysis ◽  
Circular Cylinder ◽  
Global Stability ◽  
Dimensional Model ◽  
Global Stability Analysis ◽  
The Stability ◽  
Low Dimensional ◽  
Proper Orthogonal ◽  
Calculated Amount ◽  
Good Agreement

A proper orthogonal decomposition (POD) method is applied to study the global stability analysis for flow past a stationary circular cylinder. The flow database at Re=100 is obtained by CFD software, i.e. FLUENT, with which POD bases are constructed by a snapshot method. Based on the POD bases, a low-dimensional model is established for solving the two-dimensional incompressible NS equations. The stability of the flow solution is evaluated by a POD-Chiba method in the way of the eigensystem analysis for the velocity disturbance. The linear stability analysis shows that the first Hopf bifurcation takes place at Re=46.9, which is in good agreement with available results by other high-order accurate stability analysis methods. However, the calculated amount of POD is little, which shows the availability and advantage of the POD method.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

Stability and collapse of holes in liquid layers

2018 ◽  
Vol 855 ◽  
pp. 1130-1155 ◽  
Author(s):  
Cunjing Lv ◽  
Michael Eigenbrod ◽  
Steffen Hardt
Keyword(s):  
Liquid Film ◽  
Power Laws ◽  
Theoretical Models ◽  
Liquid Volume ◽  
Capillary Length ◽  
Minimum Thickness ◽  
The Stability ◽  
Film Profile ◽  
Hole Closure ◽  
Good Agreement

We investigate experimentally and theoretically the stability and collapse of holes in liquid layers on bounded substrates with various wettabilities. It is shown that for a liquid layer with a thickness of the order of the capillary length, a stable hole exists when the hole diameter is bigger than a critical value $d_{c}$. Consequently, a further increase of the liquid volume causes the hole to collapse. It is found that$d_{c}$increases with the size of the container, but its dependence on the contact angle is very weak. The experimental results are compared with theory, and good agreement is obtained. Moreover, we present investigations of the dynamics of the hole and the evolution of the liquid film profile after the collapse. The diameter of the hole during collapse and the minimum thickness of the liquid film shortly after the collapse obey different power laws with time. Simple theoretical models are developed which indicate that the collapse of the hole is triggered by surface tension and the subsequent closure process results from inertia, whereas the growth of the liquid column after hole closure results from the balance between the capillary force and inertia. Corresponding scaling coefficients are determined.

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