A Method of Computing Nonlinear Wave Resistance of Thin Ships by Coordinate Straining

1975 ◽  
Vol 19 (03) ◽  
pp. 149-154
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Wave Resistance ◽  
Second Order ◽  
Forward Displacement ◽  
Nonlinear Solution ◽  
First Order ◽  
Body Boundary ◽  
Flow Domain

A systematical procedure is derived for determining the distribution of sources along the centerplane of a thin ship which generate a flow (and the associated wave resistance) satisfying the free-surface and body boundary conditions at second order. The method relies on a mapping of the flow domain onto a new domain by slightly straining the coordinates. The horizontal straining is selected such that the nonlinear solution should be easily derived from the first-order (Michell) solution in the strained domain. The results are shown to be similar to those obtained by Guilloton's method, but new features, such as the forward displacement of the bow singularities, are present.

The Consistent Second-Order Wave Theory and Its Application to a Submerged Spheroid

1970 ◽  
Vol 14 (01) ◽  
pp. 23-50
Author(s):  
Young H. Chey
Keyword(s):  
Free Surface ◽  
Solid Wall ◽  
Three Dimensional ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Phenomena ◽  
First Order ◽  
Theoretical Results

Because of the recognized inadequacy of first-order linearized surface-wave theory, the author has developed, for a three-dimensional body, a new second-order theory which provides a better description of free-surface phenomena. The new theory more accurately satisfies the kinematic boundary condition on the solid wall, and takes into account the nonlinearity of the condition at the free surface. The author applies the new theory to a submerged spheroid, to calculate wave resistance. Experiments were conducted to verify the theory, and their results are compared with the theoretical results. The comparison indicates that the use of the new theory leads to more accurate prediction of wave resistance.

scholarly journals Wave-induced response of a floating two-dimensional body with a moonpool

2015 ◽  
Vol 373 (2033) ◽  
pp. 20140109 ◽  
Author(s):  
Arnt G. Fredriksen ◽  
Trygve Kristiansen ◽  
Odd M. Faltinsen
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Resonance Frequency ◽  
Hybrid Method ◽  
Two Dimensional ◽  
Body Boundary ◽  
Flow Domain ◽  
Wave Induced ◽  
Piston Mode ◽  
Nonlinear Hybrid

Regular wave-induced behaviour of a floating stationary two-dimensional body with a moonpool is studied. The focus is on resonant piston-mode motion in the moonpool and rigid-body motions. Dedicated two-dimensional experiments have been performed. Two numerical hybrid methods, which have previously been applied to related problems, are further developed. Both numerical methods couple potential and viscous flow. The semi-nonlinear hybrid method uses linear free-surface and body-boundary conditions. The other one uses fully nonlinear free-surface and body-boundary conditions. The harmonic polynomial cell method solves the Laplace equation in the potential flow domain, while the finite volume method solves the Navier–Stokes equations in the viscous flow domain near the body. Results from the two codes are compared with the experimental data. The nonlinear hybrid method compares well with the data, while certain discrepancies are observed for the semi-nonlinear method. In particular, the roll motion is over-predicted by the semi-nonlinear hybrid method. Error sources in the semi-nonlinear hybrid method are discussed. The moonpool strongly affects heave motions in a frequency range around the piston-mode resonance frequency of the moonpool. No resonant water motions occur in the moonpool at the piston-mode resonance frequency. Instead large moonpool motions occur at a heave natural frequency associated with small damping near the piston-mode resonance frequency.

ANOMALY FREE SUPERGRAVITY IN D=10: I) THE BIANCHI IDENTITIES AND THE BOSONIC LAGRANGIAN

1988 ◽  
Vol 03 (04) ◽  
pp. 953-1021 ◽  
Author(s):  
RICCARDO D’AURIA ◽  
PIETRO FRÉ ◽  
MARIO RACITI ◽  
FRANCO RIVA
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Variational Equation ◽  
Second Order ◽  
Effective Theory ◽  
Spin Connection ◽  
Bianchi Identities ◽  
First Order ◽  
Interaction Terms ◽  
Starting Point

Using a theorem by Bonora-Pasti and Tonin on the existence of a solution for D=10N=1 Bianchi identities in the presence of a Lorentz Chern Simons term, we find an explicit parametrization of the superspace curvatures. Our solution depends only on one free parameter which can be reabsorbed in a field redefinition of the dilaton and of the gravitello. We emphasize that the essential point which enables us to obtain a closed form for the curvature parametrizations and hence for the supersymmetry transformation rules is the use of first order formalism. The spin connection is known once the torsion is known. This latter, rather than being identified with Hµνρ as it is usually done in the literature, is related to it by a differential equation which reduces to the algebraic relation Hµνρ = - 3Tµνρ e4/3σ only at γ1=0 (γ1 being proportional to κ/g2). The solution of the Bianchi identities exhibited in this paper corresponds to a D=10 anomaly free supergravity (AFS). This theory is unique in first order formalism but corresponds to various theories in second order formalism. Indeed the torsion equation is a differential equation which, in order to be solved must be supplemented with boundary conditions. One wonders whether supplemented with a judicious choice of boundary conditions for the torsion equation, AFS yields all the interaction terms found in the effective theory of the heterotic string (ETHS). In this respect two remarks are in order. Firstly it appears that solving the torsion equation iteratively with Tµνρ = -1/3Hµνρ e-4/3σ as starting point all the terms of ETHS except those with a ζ(3) coefficient show up. (Whether the coefficient agree is still to be checked.) Secondly, as shown in this paper the rheonomic solution of the super Poincaré Bianchi identities is unique. Hence additional interaction terms can be added to the Lagrangian only by modifying the rheonomic parametrization of the [Formula: see text]-curvature. The only assumption made in our paper is that [Formula: see text] has at most ψ∧ψ∧V components (sector (1,2)). Correspondingly the only room left for a modification of the present theory is the addition of a (0, 3) part in the rheonomic parametrization of [Formula: see text]. When this work was already finished a conjecture was published by Lechner Pasti and Tonin that such a generalization of AFS might exist and be responsible for the ζ(3) missing term. Indeed if we were able to solve the [Formula: see text]-Bianchi with this new (0, 3)-part then the torsion equation would be modified via new terms which, in second order formalism, lead to additional gravitational interactions. The equation of motion of Anomaly Free Supergravity can be worked out from the Bianchi identities: we indicate through which steps. The corresponding Lagrangian could be constructed with the standard procedures of the rheonomy approach. In this paper we limit ourselves to the bosonic sector of such a Lagrangian and we show that it can indeed be constructed in such a way as to produce the relation between Hµνρ and Tµνρ as a variational equation.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

1988 ◽  
Vol 32 (02) ◽  
pp. 83-91
Author(s):  
X. M. Wang ◽  
M. L. Spaulding
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Model ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Potential Flow Model ◽  
Good Agreement

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.

A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton’s Method

1975 ◽  
Vol 19 (03) ◽  
pp. 140-148
Author(s):  
F. Noblesse
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Perturbation Analysis ◽  
Order Approximation ◽  
Second Order ◽  
Regular Perturbation ◽  
First Order ◽  
Perturbation Problem ◽  
Sinkage And Trim ◽  
Basic Ideas

A thin-ship perturbation analysis, suggested by Guilloton's basic ideas, is presented. The analysis may be regarded as an application of Lighthill's method of strained coordinates to a regular perturbation problem. An inconsistent second-order approximation in which the Laplace equation is satisfied to first order, and the boundary conditions both at the free surface and on the ship hull are satisfied to second order, is derived. When sinkage and trim, incorporated into the present analysis, are ignored, this approximate solution is shown to be essentially equivalent to the method of Guilloton.

Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect

1975 ◽  
Vol 69 (2) ◽  
pp. 405-416 ◽  
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Perturbation Expansion ◽  
Wave Resistance ◽  
The Body ◽  
Thin Body ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F.Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.

scholarly journals Two-Dimensional Supercavitating Plate Oscillating Under a Free Surface

1965 ◽  
Vol 9 (02) ◽  
pp. 40-55
Author(s):  
C. S. Song
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Region ◽  
Integral Form ◽  
Cavitation Number ◽  
Two Dimensional ◽  
First Order ◽  
Upper Half Plane ◽  
Order Quantities

The problem of a supercavitating flat plate at zero and nonzero cavitation numberoscillating under a free surface is analyzed by a linearized method using the accelerationpotential. The analysis is based on the concept of small velocity perturbations where in all second-order quantities are neglected. The flow is assumed two-dimensional, irrotational, incompressible, and gravitation-free. The potential-flow region is mapped on to an upper half-plane and the solution is expressed in an integral form using Cheng andRott's method. Special attention is given to the effect of approximate wake boundary conditions on the computed force and moment. It was estimated that the effect is of secondorder when the cavitation number is a first-order small quantity.

Nonlinear Wave Forces Acting on Submerged Horizontal Cylinders

1988 ◽  
Vol 110 (1) ◽  
pp. 62-70 ◽  
Author(s):  
R. Inoue ◽  
Y. Kyozuka
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Nonlinear Wave ◽  
Wave Breaking ◽  
Nonlinear Effects ◽  
Second Order ◽  
Experimental Results ◽  
Wave Forces ◽  
Regular Perturbation ◽  
Horizontal Cylinders

This paper is to present experimental results of the first and second-order wave forces acting on three kinds of horizontally submerged cylinders. Wave height, wave frequency and the models’ submergence were varied in the experiments. These results are compared with the numerical calculations based on the regular perturbation theory. Through this study, it was found that the calculations of both the first and second-order wave forces coincide with the experiments when the cylinders are submerged at a sufficient depth. However, in the case that the cylinders are close to the free surface and/or wave amplitudes are relatively large, the experimental results become small compared with the calculations because of nonlinear effects, such as wave breaking observed in the experiments.

Numerical Calculation of Second-Order Wave Resistance

1977 ◽  
Vol 21 (02) ◽  
pp. 94-106
Author(s):  
Young S. Hong
Keyword(s):  
Numerical Calculation ◽  
Steady Motion ◽  
Order Theory ◽  
Iteration Scheme ◽  
Wave Resistance ◽  
Second Order ◽  
Lagrangian Coordinates ◽  
Restricted Range ◽  
First Order ◽  
First Order Theory

The wave resistance due to the steady motion of a ship was formulated in Lagrangian coordinates by Wehausen [1].2 By introduction of an iteration scheme the solutions for the first order and second order3 were obtained. The draft/length ratio was assumed small in order to simplify numerical computation. In this work Wehausen's formulas are used to compute the resistance numerically. A few models are selected and the wave resistance is calculated. These results are compared with other methods and experiments. Generally speaking, the second-order resistance shows better agreement with experiment than first-order theory in only a restricted range of Froude number, say 0.25 to 0.35, and even here not uniformly. For larger Froude numbers it underestimates seriously.

Exact Equations for the Large Inextensional Motion of Elastic Plates

1981 ◽  
Vol 48 (1) ◽  
pp. 109-112 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Space Variable ◽  
Second Order ◽  
Straight Edge ◽  
Elastic Plates ◽  
Natural Coordinates ◽  
Governing Equations ◽  
First Order ◽  
Second Order In Time

The governing equations for plates that twist as they deform are reduced to 14 differential equations, first-order in a single space variable and second-order in time. Many of the equations are the same as for statics. Nevertheless, the extension to dynamics is nontrivial because the natural coordinates to use to describe the deformed, developable midsurface are not Lagrangian. The plate is assumed to have two curved, stress-free edges, one built-in straight edge, and one free straight edge acted upon by a force and a couple. There are 7 boundary conditions at the built-in end and 7 at the free end.

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