On steady non-breaking downstream waves and the wave resistance – Stokes’ method

2017 ◽  
Vol 815 ◽  
pp. 388-414
Author(s):  
Dmitri V. Maklakov ◽  
Alexander G. Petrov
Keyword(s):  
Integral Equations ◽  
Wave Amplitude ◽  
Wave Resistance ◽  
Quadratic System ◽  
System Of Equations ◽  
Two Dimensional ◽  
Method Of Integral Equations ◽  
Analytical Formulae ◽  
The Mean ◽  
The Waves

In this work, we have obtained explicit analytical formulae expressing the wave resistance of a two-dimensional body in terms of geometric parameters of nonlinear downstream waves. The formulae have been constructed in the form of high-order asymptotic expansions in powers of the wave amplitude with coefficients depending on the mean depth. To obtain these expansions, the second Stokes method has been used. The analysis represents the next step of the research carried out in Maklakov & Petrov (J. Fluid Mech., vol. 776, 2015, pp. 290–315), where the properties of the waves have been computed by a numerical method of integral equations. In the present work, we have derived a quadratic system of equations with respect to the coefficients of the second Stokes method and developed an effective computer algorithm for solving the system. Comparison with previous numerical results obtained by the method of integral equations has been made.

Nonlinear Aspects of Subcritical Shallow-Water Flow Past Two-Dimensional Obstructions

1978 ◽  
Vol 22 (04) ◽  
pp. 203-211
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Flow Problem ◽  
Free Stream ◽  
The Body ◽  
Two Dimensional ◽  
Third Order ◽  
Submerged Body ◽  
The Mean ◽  
The Waves

Some nonlinear aspects of the two-dimensional problem of a submerged body moving with constant speed in otherwise undisturbed water of uniform depth are considered. It is shown that a theory of Benjamin which predicts a uniform rise of the free surface ahead of the body and the lowering of the mean level of the waves behind it agrees well with experimental data. The local steady-flow problem is solved by a numerical method which satisfies the exact free-surface conditions. Third-order perturbation formulas for the downstream free waves are also presented. It is found that in sufficiently shallow water, the wavelength increases with increasing disturbance strength for fixed values of the free-stream-Froude number. This is opposite to the deepwater case where the wavelength decreases with increasing disturbance strength.

On the Heaving Motion of Cylinders of Shallow Draft

1961 ◽  
Vol 5 (04) ◽  
pp. 34-43
Author(s):  
R. C. MacCamy
Keyword(s):  
Integral Equations ◽  
Circular Disk ◽  
Finite Width ◽  
Two Dimensional ◽  
Virtual Mass ◽  
Method Of Integral Equations ◽  
First Approximation ◽  
Strip Theory ◽  
Considerable Simplification ◽  
Special Case

A perturbation procedure is developed for the two-dimensional motion produced by a long ship in heave when the draft is assumed small. The procedure reduces the shallow draft problem to a series of problems for a "raft" of zero draft. A considerable simplification in the method of integral equations is found to occur. For the first approximation, that is, a raft of finite width, the integral equations are solved numerically to determine pressure, virtual mass and damping. The problem of heave of a circular disk of zero draft is treated by the same methods so that an evaluation of strip theory in this special case is possible.

scholarly journals Wave resistance: Some cases of three-dimensional fluid motion

1919 ◽  
Vol 95 (670) ◽  
pp. 354-365 ◽  
Keyword(s):  
Surface Pressure ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Wave Resistance ◽  
The Other ◽  
Two Dimensional ◽  
Pressure System ◽  
Submerged Body ◽  
Submerged Sphere ◽  
The Waves

1. Calculations of wave resistance, corresponding to a pressure system travelling over the surface, have hitherto been limited to two-dimensional fluid motion; in those cases, the distribution of pressure on the surface is one-dimensional, and the regular waves produced have straight, parallel crests. The object of the following paper is to work out some cases when the surface pressure is two-dimensional and the wave pattern is like that produced by a ship. A certain pressure system symmetrical about a point is first examined, and more general distributions are obtained by superposition. By combining two simple systems of equal magnitude, one in rear of the other, we obtain results which show interesting interference effects. In similar calculations with line pressure systems, at certain speeds the waves due to one system cancel out those due to the other, and the wave resistance is zero; the corresponding ideal form of ship has been called a wave-free pontoon. Such cases of perfect interference do not occur in three-dimensional problems; the graph showing the variation of wave resistance with velocity has the humps and hollows which are characteristic of the resistance curves of ship models. Although the main object is to show how to calculate the wave resistance for assigned surface pressures of considerable generality, it is of interest to interpret some of the results in terms of a certain related problem. With certain limitations, the waves produced by a travelling surface pressure are such as would be caused by a submerged body of suitable form. The expression for the wave resistance of a submerged sphere, given in a previous paper, is confirmed by the following analysis. It is also shown how to extend the method to a submerged body whose form is derived from stream lines obtained by combining sources and siuks with a uniform stream; in particular, an expression is given for the wave resistance of a prolate spheroid moving in the direction of its axis.

A Wave Amplitude Transition in a Quasi-Linear Model with Radiative Forcing and Surface Drag

2009 ◽  
Vol 66 (11) ◽  
pp. 3479-3490 ◽  
Author(s):  
Orli Lachmy ◽  
Nili Harnik
Keyword(s):  
Potential Vorticity ◽  
Radiative Forcing ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Vertical Shear ◽  
Mean Flow ◽  
Horizontal Scale ◽  
Zonal Wavenumber ◽  
The Mean ◽  
The Waves

Abstract A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave amplitude bifurcation—in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution can be changed from one type of equilibration to the other.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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