Body Surface Boundary Condition Research Articles

Vertical force prediction of planing craft in calm water based on variable section slamming model

This paper proposes a prediction method based on a variable section slamming model for the precise calculation of the vertical force of planing craft. Using the idea of 2D+t theory, the three-dimension planing craft hydrodynamic problem is transformed into a two-dimensional slamming problem. For better hydrodynamic performance in practical engineering, the transverse geometrical section of monohull planing craft is usually designed to be variable in the longitudinal direction. In the 2D+t point of view, this problem is numerically treated as a water impact problem of a section with constantly deforming shape. However, previous conventional slamming models cannot directly take the section variation effect into account. To address this deficiency, this paper develops a new Modified Logvinovich Model (MLM). By revising the body surface boundary condition, the newly proposed model can be applied to calculate the slamming force of two-dimensional variable transverse section. The vertical force distribution and the attitude of planing craft are determined by 2D+t theory and the computational fluid dynamics (CFD) method. The results of 2D+t theory considering the variable transverse section are in good agreement with the CFD results.

A comparative study of several boundary conditions on the body surface for the meshless electromagnetic scattering method

In order to accurately calculate the radar cross sections (RCS) of stealth aircrafts by using the meshless method, several typical body surface boundary conditions from finite-volume time-domain (FVTD) methods, which are suitable for the meshless method, have been compared. The bistatic RCS of a 2-D cylinder irradiated by transverse magnetic wave (TM wave) or transverse electric wave (TE wave) and a 3-D sphere irradiated by different polarization waves are calculated by the meshless method based on different boundary conditions. The numerical results show that the bistatic RCS calculated by the first boundary condition is closest to the series solution, which indicates that the scattering electric field on the surface of perfect conductors is only related to the incident electric field, and the scattering magnetic field is related to the scattering magnetic field near the body surface and the variation of the normal scattering electric field on the body surface. Finally, based on the selected optimal body surface boundary condition, the paper present the electromagnetic scattering field and the bistatic RCS for a 3-D stealth aircraft model, which shows the ability of the meshless method in dealing with 3-D practical problems to a certain extent.

Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack

Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack are considered based on the linearized velocity potential theory together with multipole expansion. The solution starts from the potential due to a single source, or the Green function satisfying both the ice sheet condition and the crack condition, as well as all other conditions apart from that on the body surface. This is obtained in an integral form through Fourier transform, in contrast to what has been obtained previously in which the Green function is in the series form based on the method of matched eigenfunction expansion in each domain on both sides of the crack. The multipole expansion is then constructed through direct differentiation of the Green function with respect to the source position, rather than treating each multipole as a separate problem. The use of the Green function enables the problem of wave diffraction by the crack in the absence of the body to be solved directly. For the circular cylinder, wave radiation and diffraction problems are solved by applying the body surface boundary condition to the multipole expansion, through which the unknown coefficients are obtained. Extensive results are provided for the added mass and damping coefficient as well as the exciting force. When the cylinder is away from the crack, a wide spacing approximation method is used, which is found to provide accurate results apart from when the cylinder is quite close to the crack.

Finite element analysis of second order wave radiation by a group of cylinders in the time domain

A finite element based numerical method is employed to analyze the wave radiation by multiple or a group of cylinders in the time domain. The nonlinear free surface and body surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a Finite Element Method (FEM). The matrix equation of the FEM is solved through iteration and the initial solution is obtained from the result at the previous time step. The three-dimensional (3-D) mesh required is generated based on a two-dimensional (2-D) hybrid mesh on a horizontal plane and its extension in the vertical direction. The hybrid mesh is generated by combining an unstructured grid away from cylinders and two structured grids near the cylinder and the artificial boundary. The fluid velocity on the free surface and the cylinder surface are calculated by using a differential method. Results for various configurations including the cases of two cylinders and four cylinders and a group of eighteen cylinders are obtained to show the joint influences of cylinders on the first- and second-order waves and forces, including the effects of spacing ratios and wave frequency on the second order waves and the mean force, in particular.

Hydrodynamics of a submerged hydrofoil advancing in waves

The hydrodynamic problem of a hydrofoil travelling at constant speed in water waves has been investigated through velocity potential theory. The boundary conditions on the free surface have been linearized, and the effects are accounted for through the Green function. The overall problem is decomposed into the steady forward speed problem and periodic wave radiation and diffraction problems. Each of these problems is solved using the boundary integral equation over the hydrofoil surface together with a vortex sheet behind the trailing edge. The body surface boundary condition is imposed on its mean position. As a result the steady potential will contribute a well-known mj term to the body surface boundary condition on the radiation problem. The numerical difficulty in dealing with this term is effectively resolved through a difference method. The effects of the thickness on the wave radiation and diffraction are investigated. The applicability of various reciprocity relationships in this problem is discussed.

SEMI-ANALYTICAL SOLUTION FOR RADIATION OF A HOLLOW SPHERE WITH AN OPENING HOLE IN FINITE DEPTH

An investigation is conducted on radiation of a submerged hollow sphere with an opening hole in finite water depth in this article. Based on the linear theory, the method of multiple expansions is used to obtain the fluid velocity potential in the form of double series of the associated Legendre functions with the unknown coefficients of an infinite set. In terms of the body surface boundary condition and the matching condition between the inner and outer flows at the hole, the complex matrix equations for the coefficients of the series are established. The infinite sets of matrix equations are solved by truncating the series at a finite number. Subsequently, the added mass and radiation damping, associated with the periodic heave motion of a submerged sphere, are evaluated numerically.

Second-order wave radiation by multiple cylinders in time domain through the finite element method

A time domain finite element based method is employed to analyze wave radiation by multiple cylinders. The nonlinear free surface and body surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a finite element method (FEM). The matrix equation of the FEM is solved through an iteration and the initial solution is obtained from the result at the previous time step. The three-dimensional (3D) mesh required is generated based on a two-dimensional (2D) hybrid mesh on a horizontal plane and its extension in the vertical direction. The hybrid mesh is generated by combining an unstructured grid away from cylinders and two structured grids near the cylinder and the artificial boundary, respectively. The fluid velocity on the free surface and the cylinder surface are calculated by using a differential method. Results for various configurations including two-cylinder and four-cylinder cases are provided to show the mutual influence due to cylinders on the first and second waves and forces.

Analytic solution for wave diffraction of a submerged hollow sphere with an opening hole

An investigation is carried out on the interaction of surface waves with a submerged sphere having an opening hole in finite-depth water in this article. Based on the linear wave theory, the method of multipole expansions is used to obtain the fluid velocity potential in the form of double series of the associated Legendre functions with the unknown coefficients of an infinite set. In terms of the body surface boundary condition and the matching condition between the inner and outer flows at the hole, the complex matrix equations for the coefficients of the series are established. The infinite sets of matrix equations are solved by truncating the series at a finite number. The hydrodynamic pressure on the structure surface and the exciting forces acting on the structure are graphically presented. The dynamic pressure on the wave front surface of the sphere varies slightly with angle of opening hole increasing, while that on the wave back surface does obviously. When the angles of opening hole are increasing, the absolute values of the complex exciting forces tend to fall as a whole.

Wave radiation and diffraction by a submerged sphere in a channel

Wave radiation and diffraction by a submerged sphere is analysed based on the linearized velocity potential theory. The solution is obtained based on the multipole expansion which satisfies the boundary conditions on the free surface, at infinity and on side walls. The coefficients in the expansion are obtained by imposing the body surface boundary condition. Results are obtained for the added mass, the damping coefficient, the exciting force and the drift force. The effect of the trapped mode is also discussed

A Nonlinear Simulation Method of 3-D Body Motions in Waves (1st Report)

A full nonlinear method to simulate three dimensional body motions in waves is presented. This is a time domain method to simulate Euler's equation of ideal fluid motion coupled with the equation of solid body motions.Introducing Prandtl's nonlinear acceleration potential, whose gradient gives acceleration of the fluid, Euler's differential equation of the ideal fluid motion is converted to the integral equation of the acceleration potential. The boundary condition of the acceleration potential on the body surface is systematically derived from the kinematic relation between the acceleration of the solid body and the acceleration of the fluid on the body surface. Since this kinematic boundary condition is a function of the body acceleration, the boundary values on the floating body can not be evaluated explicitly. To overcome this point, the unknown acceleration of the free floating body is eliminated by substituting the equation of body motion into kinematic condition, then implicit body surface boundary condition is derived. This is the kinematic and dynamic condition which guarantees dynamic equilibrium of forces between ideal fluid and the solid body at any instance. With the free-surface boundary condition of the acceleration potential, the formulation of the boundary value problem for the acceleration field is completed.Although this formulation of the acceleration field is mathematically correct, this is not appropriate to numerical computation, because Prandtl's nonlinear acceleration potential does not satisfy Laplace's equation. Therefore, the nonlinear part is shifted from the governing equation to the boundary condition, then the alternative formulation for the numerical computation is derived. The computational flow of the nonlinear simulation method based on this alternative formulation is also given. In order to show the accuracy of this new method, two dimensional numerical results are presented. They show that the conservation of mass, momentum and energy are satisfied excellently.

Hydrodynamic Forces on a Submerged Sphere Undergoing Large Amplitude Motion

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.

Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude motion

The hydrodynamic problem of a circular cylinder 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 condition is linearized. The solution is obtained by writing the potential in terms of the multipole expansion. Various interesting results associated with the circular cylinder are obtained.

Time-Domain Second-Order Wave Radiation in Two Dimensions

This paper presents a time-domain second-order method to study the nonlinear wave radiation problem in two dimensions. A time-stepping scheme is adopted to obtain the resulting flow development which satisfies the nonlinear free-surface boundary conditions and the radiation condition to second order, and the numerical procedure utilizes a boundary integral equation method based on Green's theorem to calculate the field solution at each time step. The body surface boundary condition is expanded about the mean body position to second order by a Taylor series. The method is applied to the cases of a semi-submerged circular cylinder and a rectangular cylinder undergoing sinusoidal sway, heave and roll motions. For the case of the circular cylinder, comparisons of the computed hydrodynamic forces at first and second order are made with previous theoretical and experimental results and a favorable agreement is indicated. The importance of second-order effects in the calculation of the hydrodynamic force is discussed.

Experimental Study of Wave Pressure on VLCC Running in Short Waves

A free-run experiment of a VLCC model ship was carried out in short regular waves. The wave length and the heading angle were varied in range of 0.2≤λ/L≤1.5 and 0°≤χ≤180° respectively. The ship motion, vertical & lateral bending moment at midship, relative water level and wave pressure were measured. Among them, wave pressure on the side wall near the load water line was the particular interest of this experiment.The first harmonics of the measured data were plotted with calculations of the strip method. We tried two strip methods to compare the accuracy. The first one is so called new strip method “NSM” and the second one is improved NSM we have developed. The solution method for the diffraction problem is improved to satisfy the exact body surface boundary condition. The plots of the measured wave pressure with the calculated value show that the accuracy of NSM is poor when λ/L≤0.5. On the other hand, the accuracy of the improved NSM is still good even when λ/L=0.2.Another difficulty for the estimation of wave pressure amplitude near the load water line comes from the exposure of wall in the air. This phenomenon is essentially non-liner. However, we found a very simple mehod to correct the pressure given by NSM. This method is based on the experimental fact that quasistatic pressuure, which is identical to the static pressure measured from water surface, can be substituted for the wave pressure near the load water line.

Multigrid Euler calculations over complete aircraft

A three-dimensional Euler code is applied to a series of configurations of increasing complexity. Comparisons are made to experiments, or to other computations when the former is not available. The method uses the multigrid approach on sets of equally spaced Cartesian grid cells. A unique and robust implementation of the body surface boundary condition on grid cells not aligned with the surface provides accurate results on relatively coarse grids. All computational results are compared to experimental data with good agreement exhibited over a wide range of flight conditions. The solution accuracy is assessed for the Onera M-6 wing, with errors due to grid resolution of less than 1% being achieved for the lift and pitching moment coefficients.

Wave resistance and lift on cylinders by a coupled element technique

This paper presents a new application of one of the most promising methods in ship hydrodynamics to free surface flow problems. It concerns a body having uniform forward speed in an incompressible and inviscid fluid. The method combines localized finite elements in the near field with representation by a boundary integral equation in the far field. The free surface condition is linearized but the body surface boundary condition is satisfied exactly. Calculations for different submerged cylinders are made, as a means of assessing the effectiveness of the new approach. Agreement with existing results is excellent, and it is concluded that the method is suited to further development for the case of an arbitrary body progressing in waves.