A coupled variational principle for 2D interactions between water waves and a rigid body containing fluid

2017 ◽  
Vol 827 ◽  
Author(s):  
Hamid Alemi Ardakani
Keyword(s):  
Variational Principle ◽  
Potential Flow ◽  
Water Waves ◽  
Variational Principles ◽  
Equations Of Motion ◽  
The Body ◽  
Two Dimensional ◽  
Hydrodynamic Equations ◽  
Gravity Driven ◽  
Complete Set

New variational principles are given for the two-dimensional interactions between gravity-driven water waves and a rotating and translating rectangular vessel dynamically coupled to its interior potential flow with uniform vorticity. The complete set of equations of motion for the exterior water waves, the exact nonlinear hydrodynamic equations of motion for the vessel in the roll/pitch, sway/surge and heave directions, and also the full set of equations of motion for the interior fluid of the vessel, relative to the body coordinate system attached to the rotating–translating vessel, are derived from two Lagrangian functionals.

A variational principle for three-dimensional interactions between water waves and a floating rigid body with interior fluid motion

2019 ◽  
Vol 866 ◽  
pp. 630-659 ◽  
Author(s):  
Hamid Alemi Ardakani
Keyword(s):  
Variational Principle ◽  
Rigid Body ◽  
Water Waves ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Three Dimensions ◽  
Fluid Sloshing ◽  
Gravity Driven

A variational principle is given for the motion of a rigid body dynamically coupled to its interior fluid sloshing in three-dimensional rotating and translating coordinates. The fluid is assumed to be inviscid and incompressible. The Euler–Poincaré reduction framework of rigid body dynamics is adapted to derive the coupled partial differential equations for the angular momentum and linear momentum of the rigid body and for the motion of the interior fluid relative to the body coordinate system attached to the moving rigid body. The variational principle is extended to the problem of interactions between gravity-driven potential flow water waves and a freely floating rigid body dynamically coupled to its interior fluid motion in three dimensions.

Variational principles of guiding centre motion

1983 ◽  
Vol 29 (1) ◽  
pp. 111-125 ◽  
Author(s):  
Robert G. Littlejohn
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Equations Of Motion ◽  
Phase Volume ◽  
Rigorous Derivation ◽  
Volume Energy ◽  
Symmetric Systems

An elementary but rigorous derivation is given for a variational principle for guiding centre motion. The equations of motion resulting from the variational principle (the drift equations) possess exact conservation laws for phase volume, energy (for time-independent systems), and angular momentum (for azimuthally symmetric systems). The results of carrying the variational principle to higher order in the adiabatic parameter are displayed. The behaviour of guiding centre motion in azimuthally symmetric fields is discussed, and the role of angular momentum is clarified. The application of variational principles in the derivation and solution of gyrokinetic equations is discussed.

scholarly journals On the Benjamin–Lighthill conjecture for water waves with vorticity

2017 ◽  
Vol 825 ◽  
pp. 961-1001 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Water Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Constant Depth ◽  
Lipschitz Continuous ◽  
Vorticity Distribution ◽  
Inviscid Incompressible Fluid ◽  
Two Dimensional Flow ◽  
Wave Heights ◽  
Gravity Driven

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

Gravity-driven flow in a horizontal annulus

2017 ◽  
Vol 830 ◽  
pp. 479-493 ◽  
Author(s):  
Marcus C. Horsley ◽  
Andrew W. Woods
Keyword(s):  
Horizontal Well ◽  
Outer Boundary ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Two Dimensional ◽  
Narrow Gap ◽  
Dense Fluid ◽  
Annular Geometry ◽  
Gravity Driven ◽  
Gravity Driven Flow

A theory for the low-Reynolds-number gravity-driven flow of two Newtonian fluids separated by a density interface in a two-dimensional annular geometry is developed. Solutions for the governing time-dependent equations of motion, in the limit that the radius of the inner and outer boundaries are similar, and in the case that the interface is initially inclined to the horizontal, are analysed numerically. We focus on the case in which the fluid is arranged symmetrically about a vertical line through the centre of the annulus. These solutions are successfully compared with asymptotic solutions in the limits that (i) a thin film of dense fluid drains down the outer boundary of the annulus, and (ii) a thin layer of less dense fluid is squeezed out of the narrow gap between the base of the inner annulus and dense fluid. Application of the results to the problem of mud displacement by cement in a horizontal well is briefly discussed.

Quasi-three-dimensional simulation of crescent-shaped waves

2020 ◽  
Author(s):  
Alexander Dosaev ◽  
Yuliya Troitskaya
Keyword(s):  
Water Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Boundary Integral ◽  
Main Direction ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Wave Interactions

<p>Many features of nonlinear water wave dynamics can be explained within the assumption that the motion of fluid is strictly potential. At the same time, numerically solving exact equations of motion for a three-dimensional potential flow with a free surface (by means of, for example, boundary integral method) is still often considered too computationally expensive, and further simplifications are made, usually implying limitations on wave steepness. A quasi-three-dimensional model, put forward by V. P. Ruban [1], represents another approach at reducing computational cost. It is, in its essence, a two-dimensional model, formulated using conformal mapping of the flow domain, augmented by three-dimensional corrections. The model assumes narrow directional distribution of the wave field and is exact for two-dimensional waves. It was successfully applied by its author to study a nonlinear stage of of Benjamin-Feir instability and rogue waves formation.</p><p>The main aim of the present work is to explore the behaviour of the quasi-three-dimensional model outside the formal limits of its applicability. From the practical point of view, it is important that the model operates robustly even in the presence of waves propagating at large angles to the main direction (although we do not attempt to accurately describe their dynamics). We investigate linear stability of Stokes waves to three-dimensional perturbations and suggest a modification to the original model to eliminate a spurious zone of instability in the vicinity of the perpendicular direction on the perturbation wavenumber plane. We show that the quasi-three-dimensional model yields a qualitatively correct description of the instability zone generated by resonant 5-wave interactions. The values of the increment are reasonably close to those obtained from the exact equations of motion [2], despite the fact that the corresponding modes of instability consist of harmonics that are relatively far from the main direction. Resonant 5-wave interactions are known to manifest themselves in the formation of the so-called “horse-shoe” or “crescent-shaped” wave patterns, and the quasi-three-dimensional model exhibits a plausible dynamics leading to formation of crescent-shaped waves.</p><p>This research was supported by RFBR (grant No. 20-05-00322).</p><p>[1] Ruban, V. P. (2010). Conformal variables in the numerical simulations of long-crested rogue waves. <em>The European Physical Journal Special Topics</em>, <em>185</em>(1), 17-33.</p><p>[2] McLean, J. W. (1982). Instabilities of finite-amplitude water waves. <em>Journal of Fluid Mechanics</em>, <em>114</em>, 315-330.</p>

scholarly journals Experiments on the motion of solid bodies in rotating fluids

1923 ◽  
Vol 104 (725) ◽  
pp. 213-218 ◽  
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Steady Motion ◽  
The Body ◽  
Two Dimensional ◽  
Experimental Conditions ◽  
Axis Of Rotation ◽  
Rotating Liquid ◽  
Steady Motions ◽  
Solid Bodies

Some years ago it was pointed out by Prof. Proudman that all slow steady motions of a rotating liquid must be two-dimensional. If the motion is produced by moving a cylindrical object slowly through the liquid in such a way that its axis remains parallel to the axis of rotation, or if a two-dimensional motion is conceived as already existing, it seems clear that it will remain two-dimensional. If a slow three-dimensional motion is produced, then it cannot be a steady one. On the other hand, if an attempt is made to produce a slow steady motion by moving a three-dimensional body with a small uniform velocity (relative to axes which rotate with the fluid) three possibilities present themselves:— ( a ) The motion in the liquid may never become steady, however long the body goes on moving. ( b ) The motion may be steady but it may not be small in the neighbourhood of the body. ( c ) The motion may be steady and two-dimensional. In considering these three possibilities it seems very unlikely that ( a ) will be the true one. In an infinite rotating fluid the disturbance produced by starting the motion of the body might go on spreading out for ever and steady motion might never be attained, but if the body were moved steadily in a direction at right angles to the axis of rotation, and if the fluid were contained between parallel planes also perpendicular to the axis of rotation, it seems very improbable that no steady motion satisfying the equations of motion could be attained. There is more chance that ( b ) may be true. A class of mathematical expressions representing the steady motion of a sphere along the axis of a rotating liquid has been obtained. This solution of the problem breaks down when the velocity of the sphere becomes indefinitely small, in the sense that it represents a motion which does not decrease as the velocity of the sphere decreases. It seems unlikely that such a motion would be produced under experimental conditions.

scholarly journals Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170425 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Constant Curvature ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
The Body ◽  
Two Dimensional ◽  
Many Body ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Hamiltonian Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Symmetry And Conservation Laws ◽  
The Body ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Symmetry Properties

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

Variational principles in continuum mechanics

1968 ◽  
Vol 305 (1480) ◽  
pp. 1-25 ◽  
Keyword(s):  
Variational Principle ◽  
Continuum Mechanics ◽  
General Problem ◽  
Water Waves ◽  
Variational Principles ◽  
Differential Forms ◽  
Lagrangian Description ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Vector Potentials

Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇ X + λ∇ μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.

scholarly journals Spatially Quasi-Periodic Water Waves of Infinite Depth

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Jon Wilkening ◽  
Xinyu Zhao
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Small Scale ◽  
Normal Velocity ◽  
Natural Setting ◽  
Two Dimensional ◽  
Periodic Functions ◽  
Initial Value

AbstractWe formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

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