scholarly journals A THEORY FOR WAVES OF FINITE HEIGHT

2011 ◽  
Vol 1 (7) ◽  
pp. 9
Author(s):  
Charles L. Bretschneider
Keyword(s):  
Stream Function ◽  
Particle Velocity ◽  
Velocity Potential ◽  
Wave Theory ◽  
Stokes Wave ◽  
Particle Position ◽  
Exact Theory ◽  
Bernoulli’S Equation ◽  
Finite Height ◽  
Expanded Form

A theory for waves of finite height, presented in this paper, is an exact theory, to any order for which it is extended. The theory is represented by a summation liarmdnic series, each term of which is in an unexpanded form. The terms of the series when expanded result in an approximation of the exact theory, and this approximation is identical to Stokes' wave theory extended to the same order. The theory represents irrotational - divergenceless flow. The procedure is to select the form of equations for the coordinates of the particles in anticipation of later operations to be performed in the evaluation of the coefficients of the series. The horizontal and vertical components of these coordinates are given respectively by the following; (equations given). From the above equations it is possible to deduce the expressions for velocity potential and stream function. The horizontal and vertical components of particle velocity are obtained by differentiating £ and ^with respect to time. Along the free surface z -1?!a 0 and z = Vs and all expressions reduce to simple forms, which in turn saves considerable work in the evaluation of the coefficients. The coefficients are evaluated by use of Bernoulli's equation. The final form of the solution is given by two sets of equations. One set of equations (same as above) is used to compute the particle position and the second set (the first time derivatives of the above) is used to compute the components of particle velocity at the particle position. That is, the particles and velocities are referenced to the lines of the stream function and the velocity potential. Expanding the two sets of equations, by approximation methods, results in one set of equation for computing particle velocity and no equations are required for the particle position.The unexpanded form requiring two sets of equations, being an exact solution, is more accurate theoretically, than the Stokes or the expanded form to the same order. Coefficients have been formulated for all terms of the order one to five for both the unexpanded and the expanded form of the theory, and are presented in tabular form as functions of d/L, as consecutive equations.

scholarly journals HORIZONTAL WATER PARTICLE VELOCITY OF FINITE AMPLITUDE WAVES

1970 ◽  
Vol 1 (12) ◽  
pp. 19 ◽  
Author(s):  
Yuichi Iwagaki ◽  
Tetsuo Sakai
Keyword(s):  
Particle Velocity ◽  
Wave Length ◽  
Wave Theory ◽  
Approximate Expression ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Wave Crest ◽  
Amplitude Wave ◽  
Water Particle ◽  
Hot Film

This paper firstly describes two methods to measure vertical distribution and time variation of horizontal water particle velocity induced "by surface waves in a wave tank These two methods consist of tracing hydrogen bubbles and using hot film anemometers, respectively Secondly, the experimental results by the two methods are presented with the theoretical curves derived from the small amplitude wave theory, Stokes wave theory of 3rd order, and the hyperbolic wave theory as an approximate expression of the cnoidal wave theory Finally, based on the comparison of the experimental data with the theoretical curves, the applicability of the finite amplitude wave theories, which has been studied for the wave profile, wave velocity, wave length and wave crest height, is discussed from view point of the water particle velocity.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

The Consistence Between the Stokes Wave Theory and General Wave Theory

2009 ◽  
Vol 25 (3) ◽  
pp. N17-N20 ◽  
Author(s):  
C.-M. Liu ◽  
H.-H. Hwung ◽  
R.-Y. Yang
Keyword(s):  
Wave Theory ◽  
The Self ◽  
The Other ◽  
Stokes Wave ◽  
First Order ◽  
General Wave ◽  
Wave Trains ◽  
Mathematical Equivalence ◽  
Nonlinear Solutions

AbstractThe consistence between the Stokes wave theory and general wave theory is examined in this study. As well known, the nonlinear terms of Stokes wave are generated by the self-interaction of first-order wave. On the other side, using the general wave theory one can also obtain the nonlinear solutions according to the interaction of n waves with the same amplitude, frequency and phase. It is found that the inconsistence between these two wave trains arises due to the subharmonic effects included in general wave theory but not considered in the Stokes theory. In conclusion, these two theories are substantially different unless the Bernoulli constants are properly chosen for mathematical equivalence.

scholarly journals A Study on the Minimization of Mooring Load in Fish-Cage Mooring Systems with a Damping Buoy

2020 ◽  
Vol 8 (10) ◽  
pp. 814
Author(s):  
Gun-Ho Lee ◽  
Bong-Jin Cha ◽  
Hyun-young Kim
Keyword(s):  
Numerical Simulations ◽  
Wave Theory ◽  
Line Length ◽  
Stokes Wave ◽  
Mooring Line ◽  
Spring Model ◽  
Mass Spring Model ◽  
Wave Conditions ◽  
Mass Spring ◽  
Fifth Order

This study established the conditions in which mooring load is minimized in a fish cage that includes a damping buoy in specific wave conditions. To derive these conditions, numerical simulations of various mooring contexts were conducted on a fish cage (1/15 scale) using a simplified mass-spring model and fifth-order Stokes wave theory. The simulation conditions were as follows: (1) bridle-line length of 0.8–3.2 m; (2) buoyancy of 2.894–20.513 N for the damping buoy; and (3) mooring-rope thickness of 0.002–0.004 m. The wave conditions were 0.333 m in height and 1.291–2.324 s of arrival period. Consequently, the mooring tensions tended to decrease with decreasing mooring line thickness and increasing bridle-line length and buoyancy of the buoy. Accordingly, it was assumed to be advantageous to minimize the mooring tension by designing a thin mooring line and long bridle line and for the buoyancy of the buoy to be as large as possible. This approach shows a valuable technique because it can contribute to the improvement of the mooring stability of the fish cage by establishing a method that can be used to minimize the load on the mooring line.

scholarly journals Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

2021 ◽  
Vol 2070 (1) ◽  
pp. 012006
Author(s):  
Santanu Koley ◽  
Kottala Panduranga
Keyword(s):  
Gravity Waves ◽  
Water Depth ◽  
Water Waves ◽  
Eigenfunction Expansion ◽  
Velocity Potential ◽  
Wave Theory ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Flexural Gravity Waves

Abstract In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

scholarly journals A Viscous Velocity Potential/Stream Function

2022 ◽  
Author(s):  
Taofiq O Amoloye
Keyword(s):  
Stream Function ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Panel Method ◽  
Theoretical Development ◽  
Cylinder Surface ◽  
Study Results ◽  
Classical Potential Theory ◽  
Reynolds Number Dependence ◽  
Mathematics And Physics

Abstract The motion of fluids presents interesting phenomena including flow separation, wakes, turbulence etc. The physics of these are enshrined in the continuity equation and the NSE. Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, numerically, or theoretically. Theoretical studies using classical potential theory (CPT) have some gaps when compared to experiments. The present publication is part of a series introducing refined potential (RPT) that bridges these gaps. It leverages experimental observations, physical deductions and the match between CPT and experimentally observed flows in the theoretical development. It analytically imitates the numerical source/vortex panel method to describe how wall bounded eddies in a three-dimensional cylinder crossflow are linked to the detached wake eddies. Unlike discrete and arbitrary vortices/sources on the cylinder surface whose strengths are numerically determined in the panel method, the vortices/sources/sinks in RPT are mutually concentric and continuously distributed on the cylinder surface. Their strengths are analytically determined from CPT using physical deductions starting from Reynolds number dependence. This study results in the incompressible Kwasu function which is a Eulerian velocity potential/stream function that captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and turbulence. This Eulerian Kwasu function also theorizes streaklines. The Lagrangian form of the function is further exploited to obtain flow pathlines.

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