A Localized Finite-Element Method for Two-Dimensional Steady Potential Flows with a Free Surface

1978 ◽  
Vol 22 (04) ◽  
pp. 216-230
Author(s):  
Kwang June Bai
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Boundary Conditions ◽  
Complex Geometry ◽  
The Body ◽  
Finite Domain ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Test Functions ◽  
Infinite Domain

A numerical method is presented for solving two-dimensional uniform flow problems with a linearized free-surface boundary condition. The boundary-value problem governed by Laplace's equation is replaced by a weak formulation (also known as Galerkin's method) with certain essential boundary conditions. The infinite domain of the fluid is reduced to a finite domain by utilizing known solution spaces in certain subdomains. The bases for the trial and test functions are chosen from the same subspace of the polynomial function space in the reduced subdomain. The essential boundary conditions are properly taken into account by an unconventional choice of the basis for the trial functions, which is different from that for the test functions in other subdomains. This method is applied to two-dimensional steady flow past a submerged elliptic section, a hydrofoil at an arbitrary angle of attack, and a bump on the bottom. In each example the body boundary condition is satisfied exactly. Both subcritical and supercritical flows are treated. We present the numerical results of wave resistance, lift force, moment, circulation strength, and flow blockage parameter. The computed pressure distributions on the hydrofoil and wave profiles are shown. The test results obtained by the present method agree very well with existing results. The main advantage of this method is that any complex geometry of the boundary can be easily accommodated.

Time-Domain Simulations of Radiation and Diffraction Forces

2010 ◽  
Vol 54 (02) ◽  
pp. 79-94 ◽  
Author(s):  
Xinshu Zhang ◽  
Piotr Bandyk ◽  
Robert F. Beck
Keyword(s):  
Free Surface ◽  
Time Domain ◽  
Three Dimensional ◽  
The Body ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Forward Speed ◽  
Time Step ◽  
Surface Boundary Conditions ◽  
Strip Theory

Large-amplitude, time-domain, wave-body interactions are studied in this paper for problems with forward speed. Both two-dimensional strip theory and three-dimensional computation methods are shown and compared by a number of numerical simulations. In the present approach, an exact body boundary condition and linearized free surface boundary conditions are used. By distributing desingularized sources above the calm water surface and using constant-strength flat panels on the exact body surface, the boundary integral equations are solved numerically at each time step. The strip theory method implements Radial Basis Functions to approximate the longitudinal derivatives of the velocity potential on the body. Once the fluid velocities on the free surface are computed, the free surface elevation and potential are updated by integrating the free surface boundary conditions. After each time step, the body surface and free surface are regrided due to the instantaneous changing wetted body geometry. Extensive results are presented to validate the efficiency of the present methods. These results include the added mass and damping computations for a Wigley III hull and an S-175 hull with forward speed using both two-dimensional and three-dimensional approaches. Exciting forces acting on a Wigley III hull due to regular head seas are obtained and compared using both the fully three-dimensional method and the two-dimensional strip theory. All the computational results are compared with experiments or other numerical solutions.

On the Experimental Determination of the Resistance Components of a Submerged Spheroid

1973 ◽  
Vol 17 (02) ◽  
pp. 72-79
Author(s):  
César Farell ◽  
Oktay Güven
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Surface Wind ◽  
Rigid Wall ◽  
The Body ◽  
Viscous Drag ◽  
Viscous Resistance ◽  
Surface Boundary ◽  
Total Resistance ◽  
Viscous Effects

Towing-tank measurements of the viscous resistance of a spheroid model by means of wake surveys together with total resistance measurements show that the proximity of the free surface greatly influences the viscous resistance, which becomes much larger than the deep-submergence resistance as the spheroid approaches the free surface. Wind tunnel measurements reveal a similar effect of a rigid wall on the viscous drag of a body. The values of the wave resistance obtained as the difference between the measured values of total resistance and viscous resistance are found to be in agreement, for the range of Froude numbers investigated, with the analytical results obtained neglecting viscous effects and linearizing the free-surface boundary condition, but satisfying exactly the boundary condition on the surface of the body.

A numerical nonlinear analysis of two-dimensional ventilating entry of surface-piercing hydrofoils with effects of gravity

2010 ◽  
Vol 658 ◽  
pp. 383-408 ◽  
Author(s):  
VIMAL VINAYAN ◽  
SPYROS A. KINNAS
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Numerical Scheme ◽  
Suction Side ◽  
Nonlinear Boundary Conditions ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Surface Boundary Conditions ◽  
Dimensional Boundary ◽  
Nonlinear Free Surface

The presence of the free surface adds an element of difficulty to the development of numerical and theoretical methods for the performance prediction of surface-piercing hydrofoils. Existing methods of analysis for two-dimensional surface-piercing hydrofoils or blade sections of a surface-piercing propeller solve either a linear problem, assuming a thin section and ventilated surface along with linear free-surface boundary conditions, or a nonlinear problem in a self-similar setting. Both these approaches cannot be used when the effects of gravity are important, which is the case when a craft is operating at low speeds. A two-dimensional boundary-element-method-based numerical scheme is presented here that overcomes these drawbacks by solving the fully ventilated flow past a surface-piercing hydrofoil of finite dimensions and includes the whole gamut of nonlinear free-surface interactions. The unique aspect of the numerical scheme is that fully nonlinear boundary conditions are applied on the free surface which allows for the accurate modelling of the jet generated on the wetted boundary and the ventilated surface formed on the suction side as a result of the passage of the hydrofoil through the free surface. Moreover, the effects of gravity can be considered to take into account the influence of the Froude number. Ventilated-surface shapes predicted by the present scheme are compared with existing experimental results and are shown to be in good agreement.

Effects of Waves on the Boundary Layer of a Surface-Piercing Body

1986 ◽  
Vol 30 (04) ◽  
pp. 256-274
Author(s):  
Frederick Stern
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Boundary Conditions ◽  
Turbulent Flow ◽  
Significant Influence ◽  
Small Amplitude ◽  
The Body ◽  
Stokes Wave ◽  
Surface Boundary ◽  
Surface Boundary Conditions

The boundary-value problem for the boundary layer of a surface-piercing body is formulated in a rigorous manner in which proper consideration is given to the viscous-fluid free-surface boundary conditions. Simplifications that are appropriate for small-amplitude waves are investigated. To this end, order-of-magnitude estimates are derived for the flow field in the neighborhood of the body-boundary-layer/free-surface juncture. It is shown that, for laminar flow, the parameter Ak/ϵ, where Ak is the wave steepness and ϵ is the nondimensional boundary-layer thickness, is important for characterizing the flow. In particular, for Ak/ϵ sufficiently large such that the free-surface boundary conditions have a significant influence a consistent formulation requires the solution of higher-order viscous-flow equations. For turbulent flow, these conclusions cannot be reached with the same degree of certainty. Numerical results are presented for the model problem of a combination Stokes-wave/flat plate. For this initial investigation, the usual thin-boundary-layer equations were solved using a three-dimensional implicit finite-difference method. The calculations are for laminar and turbulent flow and both demonstrate and quantify the influence of waves on boundary-layer development. Calculations were made using both the small-amplitude-wave and more approximate free-surface boundary conditions. Both the external-flow pressure gradients and the free-surface boundary conditions are shown to have a significant influence. The former influence penetrates to a depth of about half the wavelength and the latter is confined to a region very close to the free surface.

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.

The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions

2001 ◽  
Vol 449 ◽  
pp. 255-290 ◽  
Author(s):  
M. BROCCHINI ◽  
D. H. PEREGRINE
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Phase Region ◽  
The Body ◽  
Breaking Wave ◽  
Surface Boundary ◽  
Two Phase ◽  
Strong Turbulence ◽  
The Mean

Strong turbulence at a water–air free surface can lead to splashing and a disconnected surface as in a breaking wave. Averaging to obtain boundary conditions for such flows first requires equations of motion for the two-phase region. These are derived using an integral method, then averaged conservation equations for mass and momentum are obtained along with an equation for the turbulent kinetic energy in which extra work terms appear. These extra terms include both the mean pressure and the mean rate of strain and have similarities to those for a compressible fluid. Boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer.A number of ‘new’ terms arise for which closure expressions must be found for practical use. Our knowledge of the properties of strong turbulence at a free surface is insufficient to make such closures. However, preliminary discussions are given for two simplified cases in order to stimulate further experimental and theoretical studies.Much of the turbulence in a spilling breaker originates from its foot where turbulent water meets undisturbed water. A discussion of averaging at the foot of a breaker gives parameters that may serve to measure the ‘strength’ of a breaker.

Nonlinear Ship Motions in the Time-Domain Using a Body-Exact Strip Theory

2008 ◽  
Author(s):  
Piotr J. Bandyk ◽  
Robert F. Beck
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Time Domain ◽  
The Body ◽  
Ship Motions ◽  
Surface Boundary ◽  
Theory Approach ◽  
Offshore Structure ◽  
Time Step ◽  
Strip Theory

Modern offshore structure and ship design requires an understanding of responses in large seas. A nonlinear time-domain method may be used to perform computational analyses of these events. To be useful in preliminary design, the method must be computationally efficient and accurate. This paper presents a body-exact strip theory approach to compute wave-body interactions for large amplitude ship motions. The exact body boundary conditions and linearized free surface boundary conditions are used. At each time step, the body surface and free surface are regrided due to the changing wetted body geometry. Numerical and real hull forms are used in the computations. Validation and comparisons of hydrodynamic forces are presented. Selected results are shown illustrating the robustness and capabilities of the body-exact strip theory. Finally, an equation of motion solver is implemented to predict the motions of the vessel in a seaway.

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