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Three-dimensional potential seakeeping code in frequency domain for advancing ships
Published in C. Guedes Soares, T.A. Santos, Trends in Maritime Technology and Engineering Volume 1, 2022
A. Abbasnia, S. Sutulo, C. Guedes Soares
The solution of Laplace’s equation yields the components of the total velocity potential function and the boundary conditions need to be specified on the computational boundary to obtain the unique solutions of velocity potential functions. The boundary integral of the primary Fredholm equation is calculated for solving the Laplace equation, which is given as 2πσ(P→)+∫Γσ(q→)∂G(P→,q→)∂nqdS=ψ(P→)
Fracture Mechanics and Dynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Sneddon (1951) also demonstrated that dual integral equations can be used to solve fluid mechanics problems. That is not surprising as we have already seen that the dual integral equations are for harmonic functions that satisfy the Laplace equation. Naturally, ideal incompressible flow of fluids (or so-called potential flow) is governed by the Laplace equation through the use of velocity potential. Section 31.3 of Sneddon (1951) discussed the potential flow through a circular aperture in a plane rigid screen using dual integral formulation. The recent book by Duffy (2008) provided a list of mixed boundary value problems that can be solved using dual integral formulation. Sneddon (1951) also solved the indentation problems (both plane and 3-D punches) using dual integral formulation. More discussion of dual integral equations can be found in Keer (1968).
Harmonic Functions, Conformal Mapping, and Some Applications
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
The fluid cannot penetrate the walls of the channel. Hence, the velocity vector V can have no component along the normal to the walls at any point. The velocity vector is given by V = grad ϕ, where ϕ is the velocity potential for the fluid flow. Thus, the gradient of ϕ will have no component normal to the walls of the closed channel. This is ensured if the walls of the channel coincide with streamlines because the velocity vector is tangent to the streamlines. This is part of our boundary conditions, and because it involves the normal component of V = grad ϕ, it is a Neumann problem. The remainder consists of our finding a velocity potential ϕ(x, y) that will satisfy the prescribed condition on the velocity at the center of the bottom of the channel.
A partitioned solution approach for the simulation of the dynamic behaviour of flexible marine propellers
Published in Ship Technology Research, 2020
L. Radtke, T. Lampe, M. Abdel-Maksoud, A. Düster
The method utilised for the simulation of the hydrodynamic effects is the potential theory based approach panMARE (see e.g. Berger et al. 2014; Netzband et al. 2017). Accordingly, the fluid is assumed as incompressible, irrotational and non-viscous. The potential flow can, therefore, be computed by solving the Laplace equationfor the total velocity potential Φ. The unsteady Bernoulli equation is used to obtain the total pressureTherein φ and denote the disturbed and the free stream potential, respectively, such that . The constants and refer to the (atmospheric) reference pressure and the reference height (where prevails, here it denotes the position of the free water surface), respectively. The fluid density is denoted by ρ, g is the gravitational constant.
Weakly nonlinear ship motion calculation and parametric rolling simulation based on the 3DTGF-HOBEM method
Published in Ships and Offshore Structures, 2021
Wen-jun Zhou, Ren-chuan Zhu, Xi Chen, Liang Hong
Assuming the fluid is inviscid and impressible and the flow is irrotational, the velocity of fluid particle is expressed by the gradient of velocity potential. Supposing both the steepness of incident waves and the motions of the ship are small enough, the total velocity potential can be linearised aswhere the first terms correspond to the steady-state potential due to the ship speed and is the incident potential, while and represent the diffraction potential scattered by the ship hull and the radiation potential resulting from the motion in the j-th mode, respectively.
Simulation of the fluid-structure interaction of a floating wind turbine
Published in Ships and Offshore Structures, 2019
Bjarne Wiegard, Lars Radtke, Marcel König, Moustafa Abdel-Maksoud, Alexander Düster
The solution process is based on potential flow theory. In short, the Laplace equation for the velocity potential is solved approximately using the BEM. The surface of the immersed body is represented by sources and doublets. In addition, the free form surfaces of the wake panels are simulated only by a doublet distribution. The pressure field is related to the velocity potential through the unsteady Bernoulli equation. Based on the panel elements, the resulting equation is solved approximately by collocation as usual in BEMs. The collocation points are the element centres.