scholarly journals Weak formulation of free-surface wave equations

2001 ◽  
Vol 5 (2) ◽  
pp. 75-85
Author(s):  
A. D. Sneyd
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Surface Displacement ◽  
Wave Dispersion ◽  
The Body ◽  
Weak Formulation ◽  
Free Surface Displacement ◽  
Free Surface Wave

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, η=∑n=1∞αn(t)En where the αn(t) are time-dependent coefficients. For a tank with vertical sides the En are eigenfunctions of the eigenvalue problem, ∇2+λ2E=0,  ∇E⋅n^=0 on the tank side walls. Evolution equations for the αn(t) can be obtained by taking inner products of the linearised equation of motion, ρ∂v∂t=−1ρ∇P+F with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.

Nonlinear Helmholtz oscillations in harbours and coupled basins

1981 ◽  
Vol 104 ◽  
pp. 407-418 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Free Surface ◽  
Additional Assumption ◽  
Phase Plane ◽  
Surface Displacement ◽  
Elliptic Functions ◽  
Forced Oscillations ◽  
Amplitude Parameter ◽  
Spatial Uniformity ◽  
Open Sea ◽  
Free Surface Displacement

Free and forced oscillations in a basin that is connected through a narrow canal to either the open sea or a second basin are considered on the assumption that the spatial variation of the free-surface displacement is negligible. The free-surface displacement in the canal is allowed to be finite, subject only to the restriction (in addition to that implicit in the approximation of spatial uniformity) that the canal does not run dry. The resulting model yields a Hamiltonian pair of phase-plane equations for the free oscillations, which are integrated in terms of elliptic functions on the additional assumption that the kinetic energy of the motion in the basin(s) is negligible compared with that in the canal or otherwise through an expansion in an amplitude parameter. The corresponding model for forced oscillations that are limited by radiation damping yields a generalization of Duffing's equation for an oscillator with a soft spring, the solution of which is obtained as an expansion in the amplitude of the fundamental term in a Fourier expansion. Equivalent circuits are developed for the various models.

Oscillatory Free Surface Displacement of Finite Amplitude in a Small Orifice

2004 ◽  
Vol 126 (5) ◽  
pp. 818-826
Author(s):  
Brian J. Daniels ◽  
James A. Liburdy
Keyword(s):  
Free Surface ◽  
Potential Flow ◽  
Surface Displacement ◽  
Experimental Results ◽  
Significant Shift ◽  
Curvature Effect ◽  
Modal Frequency ◽  
Large Curvature ◽  
Curvature Effects ◽  
Free Surface Displacement

The oscillatory free-surface displacement in an orifice periodically driven at the inlet is studied. The predictions based on a potential flow analysis are investigated in light of viscous and large curvature effects. Viscous effects near the wall are estimated, as are surface viscous energy loss rates. The curvature effect on the modal frequency is shown to become large at the higher modal surface shapes. Experimental results are obtained using water for two orifice diameters, 794 and 1180 μm. Results of surface shapes and modal frequencies are compared to the predictions. Although modal shapes seem to be well predicted by the theory, the experimental results show a significant shift of the associated modal frequencies. A higher-order approximation of the surface curvature is presented, which shows that the modal frequency should, in fact, be reduced from potential flow predictions as is consistent with the large curvature effect. To account for the effect of finite surface displacements an empirical correlation for the modal frequencies is presented.

Subharmonic resonant interaction of a gravity–capillary progressive axially symmetric wave with a radial cross-wave

2019 ◽  
Vol 869 ◽  
pp. 439-467 ◽  
Author(s):  
Meng Shen ◽  
Yuming Liu
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Threshold Value ◽  
Resonant Interaction ◽  
The Body ◽  
Wave Radiation ◽  
Axially Symmetric ◽  
Damping Term ◽  
The Cross ◽  
Cross Waves

We theoretically investigate the problem of subharmonic resonant interaction of a progressive (axially symmetric) ring wave with a radial cross-wave in the context of the potential-flow formulation for gravity-capillary waves. The objective is to understand the nonlinear mechanism governing energy transfer from a progressive ring wave to its subharmonic cross-waves through triadic resonant interactions. We first show that for an arbitrary three-dimensional body floating in an unbounded free surface, there exists a set of homogeneous solutions at any frequency in the gravity-capillary wave context. The homogeneous solution depends solely on the mean free-surface slope at the waterline of the body and physically represents a progressive radial cross-wave. Unlike standing cross-waves, a progressive cross-wave loses energy during propagation by overcoming the work done by surface tension at the waterline and through wave radiation. We then consider the subharmonic interaction of a progressive ring wave, which is forced by a radial swelling–contraction deformation of a vertical circular cylinder, with subharmonic cross-waves. We derive the nonlinear spatial–temporal evolution equation governing the motion of the cross-wave by use of the average Lagrangian method. In addition to energy-input terms from the interaction with the forced ring wave, the evolution equation contains a damping term associated with energy loss in cross-wave propagation. We show that the presence of the damping term leads to a non-trivial threshold value of the ring wave steepness (or amplitude) beyond which the cross-wave becomes unstable and grows with time by taking energy from the ring wave. Finally, we extend this analysis to the experimental case of Tatsuno et al. (Rep. Res. Inst. Appl. Mech. Kyushu University, vol. 17, 1969, pp. 195–215) in which asymmetric wave patterns are observed during high-frequency vertical oscillations of a surface-piercing sphere. The theoretical prediction of the threshold value of oscillation amplitude and characteristic features of generated radial cross-waves agrees reasonably well with experimental observations.

scholarly journals A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

2016 ◽  
Vol 472 (2191) ◽  
pp. 20150833 ◽  
Author(s):  
Philippe H. Trinh
Keyword(s):  
Free Surface ◽  
Riemann Surface ◽  
Surface Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Free Surface Flow ◽  
Surface Flow ◽  
The Body ◽  
Bluff Bodies ◽  
Free Surface Waves

The standard analytical approach for studying steady gravity free-surface waves generated by a moving body often relies upon a linearization of the physical geometry, where the body is considered asymptotically small in one or several of its dimensions. In this paper, a methodology that avoids any such geometrical simplification is presented for the case of steady-state flows at low speeds. The approach is made possible through a reduction of the water-wave equations to a complex-valued integral equation that can be studied using the method of steepest descents. The main result is a theory that establishes a correspondence between different bluff-bodied free-surface flow configurations, with the topology of the Riemann surface formed by the steepest descent paths. Then, when a geometrical feature of the body is modified, a corresponding change to the Riemann surface is observed, and the resultant effects to the water waves can be derived. This visual procedure is demonstrated for the case of two-dimensional free-surface flow past a surface-piercing ship and over an angled step in a channel.

scholarly journals Three dimensional numerical simulations for non-breaking solitary wave interacting with a group of slender vertical cylinders

2009 ◽  
Vol 1 (1) ◽  
Author(s):  
Weihua Mo ◽  
Philip L.-F. Liu
Keyword(s):  
Free Surface ◽  
Numerical Model ◽  
Dynamic Pressure ◽  
Laboratory Data ◽  
Three Dimensional ◽  
Time History ◽  
Surface Displacement ◽  
Numerical Results ◽  
Finite Volume Scheme ◽  
Free Surface Displacement

AbstractIn this paper we validate a numerical model for-structure interaction by comparing numerical results with laboratory data. The numerical model is based on the Navier-Stokes(N-S) equations for an incompressible fluid. The N-S equations are solved by two-step projection finite volume scheme and the free surface displacements are tracked by the slender vertical piles. Numerical results are compared with the laboratory data and very good agreement is observed for the time history of free surface displacement, fluid particle velocity and force. The agreement for dynamic pressure on the cylinder is less satisfactory, which is primarily caused by instrument errors.

Equilibrium and oscillations of liquid fuel free surface in coaxial-cylindrical vessels under microgravity conditions

2021 ◽  
Author(s):  
Y. Zhaokai ◽  
A.N. Temnov
Keyword(s):  
Contact Angle ◽  
Free Surface ◽  
Liquid Fuel ◽  
Outer Space ◽  
Surface Displacement ◽  
Ideal Liquid ◽  
Intermolecular Forces ◽  
Hydrodynamic Characteristics ◽  
Free Surface Displacement ◽  
Microgravity Conditions

In the absence of significant mass forces, the behavior of liquid fuel under microgravity conditions is determined by surface tension forces, which are intermolecular forces at the interface of two phases. The paper posed and solved the problem of equilibrium and small oscillations of an ideal liquid under microgravity conditions, and also quantified the influence of various parameters: the contact angle α0, the Bond number, the ratio of the radii of the inner and outer walls of the vessel and the depth of the liquid. For the coaxial-cylindrical vessels, there were obtained expressions in the form of a Bessel series for the potential of the fluid velocities and the free surface displacement field. The study relies on the analytical and experimental data available in the literature and proves the reliability of the developed numerical algorithm. Findings of research show that for and r, with the physical state of the wetted surface being unchanged, the shape of the free surface tends to be flat and the contact angle has little effect on the intrinsic vibration frequency of the free surface of the liquid. The results obtained can be used to solve problems of determining the hydrodynamic characteristics of the movement of liquid fuel in outer space.

On free-surface oscillations in a rotating paraboloid

1963 ◽  
Vol 17 (2) ◽  
pp. 257-266 ◽  
Author(s):  
John W. Miles ◽  
F. K. Ball
Keyword(s):  
Free Surface ◽  
Second Harmonic ◽  
Harmonic Component ◽  
Surface Displacement ◽  
Wave Number ◽  
Rigid Displacement ◽  
Axisymmetric Solution ◽  
Axisymmetric Mode ◽  
Azimuthal Wave ◽  
Free Surface Displacement

Lamb's analysis of small-amplitude, shallow-water oscillations in a rotating paraboloid, interpreted by him in the inconsistent context of an approximately plane free surface, is re-interpreted to obtain results that are valid for $0 \le \omega^2l|2g \; \textless \;1$ (ω = rotational speed, l = latus rectum of paraboloid); no equilibrium is possible for ω2l/2g > 1. It is shown that the frequencies of the dominant modes for the azimuthal wave numbers 0 (axisymmetric motion) and 1 are independent of ω for an observer in a non-rotating reference frame and that the frequencies of all other axisymmetric modes are decreased by rotation (Lamb concluded that they would be increased). An axisymmetric mode of zero frequency, which was over-looked by Lamb, also is found.Exact solutions to the non-linear equations of motion, which reduce to the aforementioned dominant modes for small amplitudes, are determined. The axisymmetric solution is inferred from similarity considerations and is found to contain all harmonics of the fundamental frequency. The finite motion of azimuthal wave-number 1 is a quasi-rigid displacement of the liquid and is found to be simple harmonic except for a second-harmonic component of the free-surface displacement (but the horizontal velocity at a given point remains simple harmonic).

On long-wave propagation over a fluid-mud seabed

2007 ◽  
Vol 579 ◽  
pp. 467-480 ◽  
Author(s):  
PHILIP L.-F. LIU ◽  
I-CHI CHAN
Keyword(s):  
Wave Propagation ◽  
Wave Equations ◽  
Surface Displacement ◽  
Boussinesq Approximation ◽  
Fluid Mud ◽  
Viscous Boundary Layer ◽  
Time Integral ◽  
Long Wave ◽  
Free Surface Displacement ◽  
Viscous Boundary

Using the Boussinesq approximation, a set of depth-integrated wave equations for long-wave propagation over a mud bed is derived. The wave motions above the mud bed are assumed to be irrotational and the mud bed is modelled as a highly viscous fluid. The pressure and velocity are required to be continuous across the water–mud interface. The resulting governing equations are differential–integral equations in terms of the depth-integrated horizontal velocity and the free-surface displacement. The effects of the mud bed appear in the continuity equation in the form of a time integral of weighted divergence of the depth-averaged velocity. Damping rates for periodic waves and solitary waves are calculated. For the solitary wave case, the velocity profiles in the water column and the mud bed at different phases are discussed. The effects of the viscous boundary layer above the mud–water interface are also examined.

On an invariant property of water waves

1971 ◽  
Vol 49 (2) ◽  
pp. 385-389 ◽  
Author(s):  
T. Brooke Benjamin ◽  
J. J. Mahony
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Horizontal Plane ◽  
Wave Motion ◽  
Surface Displacement ◽  
Heavy Liquid ◽  
Invariant Property ◽  
Finite Region ◽  
Free Wave ◽  
Free Surface Displacement

The discussion concerns free wave motions generated from rest in a finite region of an ocean of heavy liquid lying on a horizontal plane. It is shown that the horizontal fist moment of the free-surface displacement varies linearly with time. Hence, if the total volume displaced is not zero and therefore the centroid of the displacement is definable, the centroid travels with a constant horizontal velocity as the wave motion evolves. This conclusion holds exactly for waves of any amplitude and even remains applicable subsequent to the breaking of waves.

Sign in / Sign up

Export Citation Format

Share Document