A Slender-Ship Theory of Wave Resistance

1983 ◽  
Vol 27 (01) ◽  
pp. 13-33
Author(s):  
Francis Noblesse
Keyword(s):  
Froude Number ◽  
Wave Resistance ◽  
Successive Approximations ◽  
Zeroth Order ◽  
Remarkable Effect ◽  
Ship Wave ◽  
First Order ◽  
Wigley Hull ◽  
Low Froude Number ◽  
The Waves

A new slender-ship theory of wave resistance is presented. Specifically, a sequence of explicit slender-ship wave-resistance approximations is obtained. These approximations are associated with successive approximations in a slender-ship iterative procedure for solving a new (nonlinear integro-differential) equation for the velocity potential of the flow caused by the ship. The zeroth, first, and second-order slender-ship approximations are given explicitly and examined in some detail. The zeroth-order slender-ship wave-resistance approximation, r(0) is obtained by simply taking the (disturbance) potential, ϕ, as the trivial zeroth-order slender-ship approximation ϕ(0) = 0 in the expression for the Kochin free-wave amplitude function; the classical wave-resistance formulas of Michell [1]2 and Hogner [2] correspond to particular cases of this simple approximation. The low-speed wave-resistance formulas proposed by Guevel [3], Baba [4], Maruo [5], and Kayo [6] are essentially equivalent (for most practical purposes) to the first-order slender-ship low-Froude-number approximation, rlF(1), which is a particular case of the first-order slender-ship approximation r(1): specifically, the first-order slender-ship wave-resistance approximation r(1) is obtained by approximating the potential ϕ in the expression for the Kochin function by the first-order slender-ship potential ϕ1 whereas the low-Froude-number approximation rlF(1) is associated with the zero-Froude-number limit ϕ0(1) of the potentialϕ(1). A major difference between the first-order slender-ship potential ϕ(1) and its zero-Froude-number limit ϕ0(1) resides in the waves that are included in the potential ϕ(1) but are ignored in the zero-Froude-number potential ϕ0(1). Results of calculations by C. Y. Chen for the Wigley hull show that the waves in the potential ϕ(1) have a remarkable effect upon the wave resistance, in particular causing a large phase shift of the wave-resistance curve toward higher values of the Froude number. As a result, the first-order slender-ship wave-resistance approximation in significantly better agreement with experimental data than the low-Froude-number approximation rlF(1) and the approximations r(0) and rM.

Preliminary Numerical Study of a New Slender-Ship Theory of Wave Resistance

1983 ◽  
Vol 27 (03) ◽  
pp. 172-186
Author(s):  
C. Y. Chen ◽  
F. Noblesse
Keyword(s):  
Experimental Data ◽  
Froude Number ◽  
Numerical Study ◽  
Wave Resistance ◽  
Remarkable Effect ◽  
First Order ◽  
Wave Potential ◽  
Wigley Hull ◽  
Low Froude Number ◽  
Large Phase

Results of various numerical calculations of wave resistance designed to evaluate the new slender-ship approximations obtained in Noblesse [1]3 are presented. Specifically, three main wave-resistance approximations are evaluated and studied. These are the zeroth-order slender-ship approximation r(0), which is compared with the classical approximations of Hogner and Michell; the first-order slender-ship low Froude-number approximation rIF(1), which is shown to be practically equivalent: to the Guevel-Baba-MaruoKayo low-Froude-number approximation rIF; and the first-order slender-ship approximation r(1), which is evaluated for the Wigley hull and compared with existing experimental data, corrected for effects of sinkage and trim, and with numerical results obtained by using the theory of Guilloton, the low-speed theory, and Dawson's numerical method. The approximations r(1) and rIF(1) are obtained by taking the velocity potential in the Kochin free-wave amplitude function as the first-order slender-ship potential Φ(1) and its zero-Froudenumber limit Φ0(1) respectively. A major difference between the potentials Φ(1) and Φ0(1) resides in the wave potential ΦW(1) that is included in Φ(1), but is ignored in the zero-Froude-number potential Φ0(1). It is shown that the wave potential ΦW(1) may not be neglected in comparison with the potential Φ0(1) and in fact has a remarkable effect upon the wave resistance. In particular, the wave potential ΦW(1) causes a very large phase shift of the wave-resistance curve, which results in greatly improved agreement with experimental data.

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.

Convergence of a Sequence of Slender-Ship Low-Froude-Number Wave-Resistance Approximations

1984 ◽  
Vol 28 (03) ◽  
pp. 155-162
Author(s):  
Francis Noblesse
Keyword(s):  
Froude Number ◽  
Wave Resistance ◽  
Beam Length ◽  
Remarkable Property ◽  
Ship Hulls ◽  
Low Froude Number ◽  
Vertical Cylinders ◽  
Elliptical Cylinders ◽  
Double Hull ◽  
Special Case

Convergence of the sequence of slender-ship low-Froude-number wave-resistance approximations /"/, n > 0, obtained as a particular case of the slender-ship theory of wave resistance recently proposed by the author, is proved for the special case of ship hulls in the form of vertical cylinders with elliptical waterlines. Specifically, it is shown that we have where b is the thickness (beam/length) ratio of the cubical cylinder, Fis the Froude number, and r lf(b,F) is the Guevel-Baba-Maruo-Kayo low-Froude-number wave-resistance approximation associated with the exact zero-Froude-number (double-hull) potential. Vertical elliptical cylinders thus have the remarkable property that the ratio iipj{b,F)/rpp(b,F) is independent of the Froude number, that is, depends only on the thickness ratio of the cylinder.

On the Numerical Radiation Condition in the Steady-State Ship Wave Problem

1987 ◽  
Vol 31 (01) ◽  
pp. 14-22
Author(s):  
Peter Schjeldahl Jensen
Keyword(s):  
Green Function ◽  
Three Dimensional ◽  
Radiation Condition ◽  
Function Technique ◽  
Wave Problem ◽  
Ship Wave ◽  
Rankine Source ◽  
Wigley Hull ◽  
The Green Function ◽  
The Waves

The waves created by a thin ship sailing in calm water are examined. The velocity potential of the ship in the zero Froude number case is known and the additional potential due to the waves is calculated by the Green function technique. The simple Green function corresponding to the Rankine source potential is used here. Two major problems exist with this method. In the Neumann-Poisson boundary-value problem- probably the first iteration toward a full nonlinear solution to the ship wave problem _it is necessary to impose a radiation condition in order to get uniqueness. This problem is related to the second one, which arises due to the existence of eigensolutions. The two-dimensional situation is here analyzed first, thereby easing the three-dimensional analysis. A numerical scheme is constructed and results for the twodimensional waves generated by a submerged vortex and for the three-dimensional waves due to the Wigley hull are presented.

Potential of Michell’s Integral for Ship Wave Resistance

2014 ◽  
Author(s):  
Takashi Tsubogo
Keyword(s):  
Integral Equation ◽  
Froude Number ◽  
Computing Time ◽  
Wave Resistance ◽  
Boundary Integral ◽  
Hull Form ◽  
Number Range ◽  
Gradient Effect ◽  
Depth Direction ◽  
Wigley Hull

The Michell’s integral (Michell 1898) for the wave making resistance of a thin ship has not been used widely in practice, since its accuracy is questioned especially for a Froude number range about 0.2 to 0.35 for conventional ships. We examine calculations by Michell’s integral for some ship forms, e.g. a parabolic strut, Wigley hull and so on. As a result, one reason of the disagreement with experiments is revealed. It must be the gradient of hull form in the depth direction. Then a thin ship theory including the hull gradient effect in the depth direction is presented, which improves slightly in low Froude numbers but needs more computing time than Michell’s integral so as to solve a boundary integral equation.

Simple Calculation of Sheltering Effect on Ship-Wave Resistance and Bulbous Bow Design

1980 ◽  
Vol 24 (04) ◽  
pp. 232-243
Author(s):  
B. Yim
Keyword(s):  
Free Surface ◽  
Simple Calculation ◽  
Wave Resistance ◽  
Beam Length ◽  
Ship Wave ◽  
Surface Pressure Distribution ◽  
Wigley Hull ◽  
Bulbous Bow ◽  
Sheltering Effect ◽  
Optimum Volume

The sheltering effect on the ship wave resistance is treated by the free-surface pressure distribution inside the ship. The theory is developed and the numerical values of wave resistance are obtained for parabolic and sinusoidal hulls. The corrected wave resistance is shown to be reduced more than 20 percent from the Michell's wave resistance even for a Wigley hull with beam-length ratio 0.1. However, the humps and hollows of wave resistance still remain with the magnitude closer to the experimental results. The optimum volume of bulb for sinusoidal ships is also shown to be reduced from 7.6 percent to 18 percent of the values without sheltering effect for the range of Froude numbers from 0.15 to 0.35.

Comparison Between Theoretical Predictions of Wave Resistance and Experimental Data for the Wigley Hull

1983 ◽  
Vol 27 (04) ◽  
pp. 215-226
Author(s):  
C. Y. Chen ◽  
F. Noblesse
Keyword(s):  
Experimental Data ◽  
Froude Number ◽  
Resistance Coefficient ◽  
Wave Resistance ◽  
Number Range ◽  
Theoretical Predictions ◽  
Wigley Hull ◽  
Sinkage And Trim ◽  
Theoretical Results ◽  
Good Agreement

A number of theoretical predictions of the wave-resistance coefficient of the Wigley hull are compared with one another and with available experimental data, to which corrections for sinkage and trim are applied. The averages of eleven sets of experimental data (corrected for sinkage and trim) and of eleven sets of theoretical results for large values of the Froude number, specifically for F 0.266, 0.313, 0.350, 0.402, 0.452, and 0.482, are found to be in fairly good agreement, in spite of considerable scatter in both the experimental data and the numerical results. Furthermore, several sets of theoretical results are fairly close to the average experimental data and the average theoretical predictions for these large values of the Froude number. Discrepancies between theoretical predictions and experimental measurements for small values of the Froude number, specifically for F = 0.18, 0.20, 0.22, 0.24, and 0.266, generally are much larger than for the above-defined high-Froude-number range. However, a notable exception to this general finding is provided by the first-order slender-ship approximation evaluated in Chen and Noblesse [1],3 which is in fairly good agreement with the average experimental data over the entire range of values of Froude number considered in this study.

Numerical Study of a Slender-Ship Theory of Wave Resistance

1985 ◽  
Vol 29 (02) ◽  
pp. 81-93
Author(s):  
Francis Noblesse
Keyword(s):  
Froude Number ◽  
Numerical Study ◽  
Near Field ◽  
Field Potential ◽  
Analytical Approximation ◽  
Wave Resistance ◽  
Theoretical Predictions ◽  
Wigley Hull ◽  
The Green Function ◽  
Hull Forms

This study is a continuation of the previous numerical study by Chen and Noblesse [1]2 of the slender-ship theory of wave resistance presented in Noblesse [2]. Results of systematic calculations of wave resistance are presented for three simple sharp-and round-ended strut-like hull forms having beam/length and draft/length ratios equal to 0.15 and 0.075, respectively. Numerical results are presented for the first order slender-ship approximation and for seven closely related wave-resistance approximations. The nondimensional wave-resistance values associated with these eight approximations are plotted versus the Froude number in the range 1 ≥ F ≥ 0.18. The Kochin wave-energy function corresponding to four approximations is also depicted for three Froude-number values. The wave potential is shown to have more pronounced effects upon the wave resistance, causing large phase shifts in particular, than the nonoscillatory near-field potential. A simple analytical approximation to the near-field term in the Green function is proposed. Finally, theoretical predictions are compared with experimental data for the Sharma strut and the Wigley hull.

scholarly journals New gravity–capillary waves at low speeds. Part 1. Linear geometries

2013 ◽  
Vol 724 ◽  
pp. 367-391 ◽  
Author(s):  
Philippe H. Trinh ◽  
S. Jonathan Chapman
Keyword(s):  
Froude Number ◽  
Fourier Transforms ◽  
Capillary Flow ◽  
Asymptotic Series ◽  
Bond Number ◽  
Capillary Waves ◽  
Low Froude Number ◽  
Linearized Theory ◽  
The Waves ◽  
Exponential Asymptotics

AbstractWhen traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.

scholarly journals Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

2021 ◽  
Author(s):  
Amarjot Singh Bhullar ◽  
Gospel Ezekiel Stewart ◽  
Robert W. Zimmerman
Keyword(s):  
Approximate Solution ◽  
Pore Pressure ◽  
Constant Pressure ◽  
Initial Permeability ◽  
Order Perturbation ◽  
Zeroth Order ◽  
One Dimensional ◽  
First Order ◽  
Outflow Boundary ◽  
Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

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