A Second-Order Theory for the Potential Flow About Thin Hydrofoils

1984 ◽  
Vol 28 (01) ◽  
pp. 55-64
Author(s):  
Colen Kennell ◽  
Allen Plotkin
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Potential Flow ◽  
Order Theory ◽  
Wave Drag ◽  
Second Order ◽  
The Body ◽  
Surface Shape ◽  
Surface Boundary ◽  
Free Surface Boundary Condition

This research addresses the potential flow about a thin two-dimensional hydrofoil moving with constant velocity at a fixed depth beneath a free surface. The thickness-to-chord ratio of the hydrofoil and disturbances to the free stream are assumed to be small. These small perturbation assumptions are used to produce first-and second-order subproblems structured to provide consistent approximations to boundary conditions on the body and the free surface. Nonlinear corrections to the free-surface boundary condition are included at second order. Each subproblem is solved by a distribution of sources and vortices on the chord line and doublets on the free surface. After analytic determination of source and doublet strengths, a singular integral equation for the vortex strength is derived. This integral equation is reduced to a Fredholm integral equation which is solved numerically. Lift, wave drag, and free-surface shape are calculated for a flat plate and a Joukowski hydrofoil. The importance of free-surface effects relative to body effects is examined by a parametric variation of Froude number and depth of submergence.

Numerical Study of Forward-Speed Ship Motion and Added Resistance

2017 ◽  
Author(s):  
D. C. Hong ◽  
T. B. Ha ◽  
K. H. Song
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Numerical Study ◽  
Three Dimensional ◽  
The Body ◽  
Experimental Results ◽  
Surface Boundary ◽  
Forward Speed ◽  
Added Resistance ◽  
Following Seas

The added resistance of a ship was calculated using Maruo’s formula [1] involving the three-dimensional Kochin function obtained using the source and normal doublet distribution over the wetted surface of the ship. The density of the doublet distribution was obtained as the solution of the three-dimensional frequency-domain forward-speed Green integral equation containing the exact line integral along the waterline. Numerical results of the Wigley ship models II and III in head seas, obtained by making use of the inner-collocation 9-node second-order boundary element method have been compared with the experimental results reported by Journée [2]. The forward-speed hydrodynamic coefficients of the Wigley models have shown no irregular-frequencylike behavior. The steady disturbance potential due to the constant forward speed of the ship has also been calculated using the Green integral equation associated with the steady forward-speed free-surface Green function since the so-called mj-terms [3] appearing in the body boundary conditions contain the first and second derivatives of the steady potential over the wetted surface of the ship. However, the free-surface boundary condition was kept linear in the present study. The added resistances of the Wigley II and III models in head seas obtained using Maruo’s formula showing acceptable comparison with experimental results, have been presented. The added resistances in following seas obtained using Maruo’s formula have also been presented.

On higher-order wave theory for submerged two-dimensional bodies

1969 ◽  
Vol 38 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Nils Salvesen
Keyword(s):  
Free Surface ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Drag ◽  
Higher Order ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
Free Surface Waves ◽  
Body Shapes

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.

Review of Approximations to Evaluate Second-Order Low-Frequency Load

2012 ◽  
Author(s):  
Guillaume de Hauteclocque ◽  
Flávia Rezende ◽  
Olaf Waals ◽  
Xiao-Bo Chen
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
The Body ◽  
Difference Frequency ◽  
Surface Boundary ◽  
First Order ◽  
Potential Part ◽  
The Difference

The second order low-frequency loads are one of main sources of excitation for moored systems. These loads are usually decomposed into the quadratic part, contributed only by first order quantities and potential part contributed by the second order potentials. In shallow water the second order incoming and diffracted potentials give a significant contribution to the low frequency forces. Therefore, the accuracy on the determination of this parcel of the low-frequency loads is a key issue for the assessment of mooring lines and operability of systems moored in shallow water area, as for example LNG terminals. Due to the complexity in computing the second order diffraction potential, which would involve a non-homogeneous free surface boundary condition, the so-called Pinkster approximation has been proposed. This approximation is based on the assumption that the major contribution to the potential part of low-frequency loads is given by the second order potential of the undisturbed incoming waves. The methods to compute the wave forces related to the second order potentials are based on scaling of the first order wave induced forces. Another approximation recently formulated in Chen and Rezende consists of developing the second-order bi-frequency load into a series of different orders of the difference frequency. The potential contribution to the term proportional to the difference-frequency can be evaluated efficiently by involving an integral over a small zone on the free surface around the body. In the present paper, the existing approximations are revisited and compared to analytical solution of exact second-order load on a vertical cylinder and for the case of floating body (LNG) in shallow water. Some guidelines in the practical use of different approximations will be derived.

The Second-Order Theory of Heaving Cylinders in a Free Surface

1968 ◽  
Vol 12 (04) ◽  
pp. 313-327
Author(s):  
Choung Mook Lee
Keyword(s):  
Free Surface ◽  
Water Surface ◽  
Vertical Axis ◽  
Order Theory ◽  
Second Order ◽  
The Body ◽  
Two Dimensional ◽  
Forced Motion ◽  
Axis Of Symmetry ◽  
The Waves

A second-order potential solution is sought for a two-dimensional symmetric cylinder placed horizontally in a free surface and forced to oscillate vertically. The forced motion is simple harmonic, and the amplitude is small compared to the beam of the cylinder. The resulting potential-theory problem is solved by placing singularities of all orders at the intersection of the water surface at rest with the vertical axis of symmetry, and by determining their strengths from the boundary condition on the body. The pressure distribution on the cylinder, the force acting upon it, and the waves generated by it are derived through the second order. Numerical computations are made for a circular cylinder and for a U-shaped cylinder, and the results are presented in graphs.

Hydrodynamic Forces on a Submerged Sphere Undergoing Large Amplitude Motion

1994 ◽  
Vol 38 (04) ◽  
pp. 272-277
Author(s):  
G. X. Wu
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Large Amplitude ◽  
Multipole Expansion ◽  
The Body ◽  
Surface Boundary ◽  
Free Surface Boundary Condition ◽  
Submerged Sphere ◽  
Instantaneous Position ◽  
Surface Boundary Condition

The hydrodynamic problem of a sphere submerged below a free surface and undergoing large amplitude oscillation is investigated based on the velocity potential theory. The body surface boundary condition is satisfied on its instantaneous position while the free-surface boundary condition is linearized. The solution is obtained by writing the potential in terms of the multipole expansion.

Analyses of Waves and the Wave Resistance due to Transom-Stern Ships

1969 ◽  
Vol 13 (02) ◽  
pp. 155-167
Author(s):  
Bohyun Yim
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Boundary ◽  
Exact Formulation ◽  
Wave Problem ◽  
Ship Wave ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

Waves and the wave resistance due to a ship with a transom stern are analyzed starting with an exact formulation of the general ship wave problem. Green's theorem is utilized together with the well-known Green's function which satisfies the linear free-surface boundary condition on the free surface. Here it is required to take at least the second-order terms in the formal development. It is found that the immersed transom stern acts to cancel stern waves. Using a limited form of Michell's wave-resistance formula, the wave resistance of the transom-stern ship is analyzed

Comments on Second-Order Wave Effects on Large-Diameter Vertical Cylinder

1978 ◽  
Vol 22 (04) ◽  
pp. 266-268
Author(s):  
Subrata K. Chakrabarti
Keyword(s):  
Free Surface ◽  
Large Diameter ◽  
Vertical Cylinder ◽  
Wave Interaction ◽  
Scattered Wave ◽  
Second Order ◽  
Surface Boundary ◽  
Order Solution ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Conditions

In a recent paper, Raman et al (RJV) [1] 2 presented a second- order solution for the wave interaction with a large-diameter, bottom-mounted, surface-piercing, fixed vertical cylinder. Earlier, Chakrabarti [2,3] obtained an approximate solution for the vertical cylinder due to Stokes's fifth-order wave. Later, Yamaguchi and Tsuchiya (YT) [4] proposed a second-order solution in a closed form for the wave interaction problem with the vertical cylinder which was examined by Chakrabarti [5]. All these solutions are different due to the assumptions regarding the frequency of the scattered wave and the kinematic free-surface boundary condition. RJV [1] claim that their solution is superior to the earlier solutions because, according to them, it satisfies the nonlinear free-surface boundary conditions completely.

The effect of non-linearity at the free surface on flow past a submerged cylinder

1965 ◽  
Vol 22 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Second Order ◽  
Order Effects ◽  
Surface Boundary ◽  
First Order ◽  
Flow Past ◽  
Free Surface Boundary Condition ◽  
Non Linear ◽  
Wave Induced

Plane potential flow past a circular cylinder beneath a free surface under gravity is investigated in order to determine the importance or otherwise of non-linear effects from the free-surface boundary condition. It is shown that non-linear second-order corrections to the first-order linearized expressions for the wave-induced forces on the cylinder are considerably larger than second-order effects which are present even with a linear free-surface condition. Further evidence for the importance of non-linearity is presented in the form of streamline plots of the first-order solution showing strange behaviour at wave crests.

Linear and non-linear potential-flow calculations of free-surface waves with lift and induced drag

2000 ◽  
Vol 214 (6) ◽  
pp. 801-812
Author(s):  
C-E Janson
Keyword(s):  
Free Surface ◽  
Potential Flow ◽  
Lift Force ◽  
Radiation Condition ◽  
Linear Potential ◽  
Surface Boundary ◽  
Model Approximation ◽  
Non Linear ◽  
Lifting Surfaces ◽  
The Waves

A potential-flow panel method is used to compute the waves and the lift force from surface-piercing and submerged bodies. In particular the interaction between the waves and the lift produced close to the free surface is studied. Both linear and non-linear free-surface boundary conditions are considered. The potential-flow method is of Rankine-source type using raised source panels on the free surface and a four-point upwind operator to compute the velocity derivatives and to enforce the radiation condition. The lift force is introduced as a dipole distribution on the lifting surfaces and on the trailing wake, together with a flow tangency condition at the trailing edge of the lifting surface. Different approximations for the spanwise circulation distribution at the free surface were tested for a surface-piercing wing and it was concluded that a double-model approximation should be used for low speeds while a single-model, which allows for a vortex at the free surface, was preferred at higher speeds. The lift force and waves from three surface-piercing wings, a hydrofoil and a sailing yacht were computed and compared with measurements and good agreement was obtained.

SPLASH Free-Surface Flow Code Methodology for Hydrodynamic Design and Analysis of IACC Yachts

1993 ◽  
Author(s):  
Bruce S. Rosen ◽  
Joseph P. Laiosa ◽  
Warren H. Davis ◽  
David Stavetski
Keyword(s):  
Free Surface ◽  
High Performance ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Surface Boundary ◽  
Induced Drag ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
America’S Cup ◽  
Areas Of Interest

A unique free-surface flow methodology and its application to design and analysis of IACC yachts are discussed. Numerical aspects of the inviscid panel code and details of the free-surface boundary condition are included, along with enhancements developed specifically for the '92 America's Cup defense. Extensive code validation using wind tunnel and towing tank experimental data address several areas of interest to the yacht designer. Lift and induced drag at zero Froude number are studied via a series of isolated fin/bulb/winglet appendages. An isolated surface piercing foil is used to evaluate simple lift/free­surface interactions. For complete IACC yacht models, upright wave resistance is investigated, as well as lift and induced drag at heel and yaw. The excellent correlation obtained for these cases demonstrates the value of this linear free-surface methodology for use in designing high performance sailing yachts.

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