Dynamic Free Surface Boundary Conditions Research Articles

A Numerical Wave Tank For The Generation And ProgagationOf Bichromatic, Nonlinear, Long-crested Surface Waves

We are studying numerically the problem of generation and propagation of bichromatic gravity long-crested waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite and finite depths. A non-othogonal curvilinear coordinate system, which follows the free surface is constructed and the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm.

Limiting criteria of permanent progressive waves

Three parameters to represent wave nonlinearity are derived from the dynamic free surface boundary condition using two different methods. Relationships among the parameters are given for general coordinates including fixed and moving coordinates and for any definition of wave celerity. It is proved that all the proposed parameters become equal to unity at the limiting wave condition. Two relationships among wave variables at the crest of breaking waves are derived. These relationships are exact from a solitary wave to deep water waves. The nonlinearity parameters and limiting wave relationships are examined for special cases, resulting in other famous nonlinearity parameters and breaking conditions. The limiting wave relationships are used to examine the accuracy of breaking waves indices obtained by Yamada and Shiotani [Bull. Disaster Prev. Res. Inst., Kyoto Univ. 18 (1968) 1], Cokelet [Philos. Trans. R. Soc. London, Ser. A 286 (1977) 183] and Dean [Handbook of Coastal and Ocean Engineering, Vol. 1, Gulf Publishing Company, Houston, TX, Chap. 2, pp. 63–94, 1991]. A method for calculating the wave celerity for any fully nonlinear model as a function of water particle velocity and surface elevation is also given.

Wave Influences on Viscous Flow around a Ship in Steady Yaw Motion

This paper concerns an investigation of wave influences on viscous flow around a ship in steady yaw motion, which is based on detailed analysis of numerical solutions of Reynolds-averaged Navier-Stokes (RaNS) equations. The present numerical method solves the unsteady RaNS and continuity equations for mean-velocity and pressure and wave fields with exact nonlinear kinematic and approximate dynamic free-surface boundary conditions. In the following, an overview is given of the present numerical approach, and results are presented for the Series 60 CB=0.6 at yaw angle 10°for with/without-wave cases, including wherever possible comparison with experimental data. Then discussions are focused on identification of salient feature of wave-included effects on the mean-velocity and pressure fields, body surface pressure and frictional streamlines, and hydrodynamic forces acting on the hull. Lastly, some specific aspects of flow for wave breaking and wave-induced vortices are described.

Viscous free surface effect on coastal embankment hydrodynamics

A finite-differnece method was used to calculate the nonlinear hydrodynamic pressures acting on the coastal embankment faces by seismic-wave actions. The nonlinearity of free surface flow, convective acceleration, viscosity and surface tension of fluid are included in the analysis. The kinematic and dynamic free surface boundary conditions are employed for calculating the horizontal fluid velocity, pressure at the free surface and the surface profile of the fluid. The time-dependent water surface is transformed to the horizontal plane, and the flow field is mapped onto a rectangular, making it convenient to model the complex sea bottom geometry and the wavy water surface by the finite-difference method. Fully nonlinear and weakly nonlinear dynamic free surface conditions are used and compared. The effects of surface tension of fluid are also discussed. The nonslip boundary condition is applied on the most part of the interface between fluid and solid face, except the region near the intersection between free surface and wall face. The numerical results are presented for various water depths and ground motion intensities, and their associate viscous effects on coastal embankment hydrodynamics are discussed.

An Improvement to Stokes Nonlinear Theory for Steady Waves

An improvement to Stokes nonlinear theory for steady water waves is presented. The method is based on the solution of the kinematic and the dynamic free-surface boundary conditions. The expression of the stream function is taken from the Stokes V theory. The numerical method is a very simple iterative procedure. The method is valid at any depth greater than U=HL2/d3= 40, where U is the Ursell number, L is the wavelength, d is the mean depth, and H is the wave height. The error in the free-surface boundary conditions is relatively small. The proposed solution is validated against Stokes V theory, a stream function theory, and experimental data as far as the surface elevation and horizontal velocity distribution are concerned. For high waves, near breaking limit, the method gives better results in comparison with Stokes V theory.

A Large-Domain Approach for Calculating Ship Boundary Layers and Wakes and Wave Fields for Nonzero Froude Number

A large-domain approach is developed for calculating ship boundary layers and wakes and wave fields for nonzero Froude number. The Reynolds-averaged Navier–Stokes and continuity equations are solved with the Baldwin–Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The results are validated through comparisons with data for the Series 60 CB= 0.6 ship model at low and high Froude numbers and results of a precursory interactive approach. Both approaches yield satisfactory results; however, the large-domain results indicate improved resolution of the flow close to the hull and wake centerplane and of the Froude number differences due to near-wall turbulence modeling and nonlinear free-surface boundary conditions. Additional evaluation is provided through discussion of the recent CFD Workshop Tokyo 1994, where both methods were among the best. Last, some concluding remarks are made.

Free-Surface Wave-Induced Separation

Free-surface wave-induced separation is studied for a surface-piercing NACA 0024 foil over a range of Froude numbers (0, .2, .37, .55) through computational fluid dynamics of the unsteady Reynolds-averaged Navier-Stokes and the continuity equations with the Baldwin-Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The flow conditions and uncertainty analysis are discussed. A topological rule for a surface-piercing body is derived and verified. Steady-flow results are presented and analyzed with regard to the wave and viscous flow and the nature of the separation.

Nonlinear response of membranes to ocean waves using boundary and finite elements

The behavior of a highly deformable membrane to ocean waves was studied by coupling a nonlinear boundary element model of the fluid domain to a nonlinear finite element model of the membrane. The hydrodynamic loadings induced by water waves are computed assuming large body hydrodynamics and ideal fluid flow and then solving the transient diffraction/radiation problem. Either linear waves or finite amplitude waves can be assumed in the model and thus the nonlinear kinematic and dynamic free surface boundary conditions are solved iteratively. The nonlinear nature of the boundary condition requires a time domain solution. To implicitly include time in the governing field equation, Volterra's method was used. The approach is the same as the typical boundary element method for a fluid domain where the governing field equation is the starting point. The difference is that in Volterra's method the time derivative of the governing field equation becomes the starting point. The boundary element model was then coupled through an iterative process to a finite element model of membrane structures. The coupled model predicts the nonlinear interaction of nonlinear water waves with highly deformable bodies. To verify the coupled model a large scale test was conducted in the OH Hinsdale wave Research Laboratory at Oregon State University on a 3-ft-diameter fabric cylinder submerged in the wave tank. The model data verified the numerical prediction of the structure displacements and of the changes in the wave field. The boundary element model is an ideal modeling technique for modeling the fluid domain when the governing field equations is the Laplace equation. In this case the nonlinear boundary element model was coupled with a finite element model of membrane structures, but the model could have been coupled with other finite element models of more rigid structures, such as a pontoon floating breakwater.

A Fourier approximation for finite amplitude short‐crested waves

Abstract A numerical scheme using Fourier series approximation is presented to calculate short‐crested waves of finite amplitude in water of arbitrary uniform depth. The numerical model preserves the water surface elevation in an implicit form which retains the nonlinear nature of the dynamic and kinematic free surface boundary conditions. Accurate solutions can be derived for the variations in frequency, wave profiles and properties of pressures. The present model is directly reducible to the two‐dimensional limiting cases of progressive and standing waves.

Nonlinear water-wave structure interaction

A numerical model of the interaction of highly deformable fluid-filled membranes and ocean waves has been developed and verified. The fluid domain is modeled using a boundary element model and the membrane is modeled using a finite element model. The finite element model was developed previously, and required little modification. However, the hydrodynamics in the finite element model was simplistic and the boundary element model of the fluid domain was developed to provide a better representation of the potentially nonconservative hydrodynamic loadings on the membrane. The finite element model predicts the nonlinear dynamic behavior of cable and membrane structures where the nonlinearities arise from large displacements, surface rotations, nonlinear stress-strain relationships and nonconservative loadings. The total Langrangian description and the principle of virtual work are used to formulate the equilibrium equations. Explicit stress-strain relationships are neither assumed nor required. The hydrodynamic loadings induced by water waves on structures are computed using a boundary element model. Large body hydrodynamics and ideal fluid flow are assumed and the diffraction/radiation problem solved. Either linear waves or finite amplitude waves are assumed in the model; thus the nonlinear kinematic and dynamic free surface boundary conditions are treated iteratively. To implicitly include time in the governing field equations Volterra's method was used. The models were coupled in an iterative procedure to model the wave structure interaction. Numerical results were compared to results of large scale test conducted on a 3-ft diameter membrane cylinder.

Nonlinear Local Analysis of Steady Flow About a Ship

A nonlinear local analysis of the steady potential flow at a ship bow and stern, and more generally at any point along a ship waterline, is presented. The hull boundary condition and the nonlinear kinematic and dynamic free-surface boundary conditions are satisfied exactly, at the actual position of the free surface, in this analysis. The bow-flow analysis shows that the free surface at a ship bow is tangent to the stem. This theoretical result appears to agree with existing experimental measurements of steady bow waves of the Wigley hull. Simple analytical expressions defining the fluid velocity at the bow and the stern, and more generally at any point along the wave profile, in terms of the elevation of the free surface at the corresponding point are also given. These analytical expressions and the available experimental measurements of wave profiles along the Wigley hull show that the velocity of the flow disturbance due to this hull is fairly small compared to the hull speed everywhere along the wave profile except in very small regions around the bow and the stern, where the total fluid velocity is nearly equal to the hull speed in magnitude but directed vertically. Nonlinearities therefore appear to be quite important, although only in very small regions surrounding a ship bow and stern. A genuine nonlinear method of calculation must then be able to represent the very rapid variation in the direction of the fluid velocity occurring within small regions around a ship bow and stern. In particular, a sufficiently fine discretization is required in these regions.

GENERALIZED WAVE THEORY FOR A SLOPING BED

This paper discusses the development from first principles of a first-order solution for non-breaking waves on a gently sloping bottom. The theory is derived in a similar fashion as was done by Swart and Loubser (1978) for vocoidal waves on a horizontal bottom. The resulting covocoidal theory was compared to an extensive data set for waves over a sloping bottom (Nelson, 1981) and is tested for analytical validity. It adheres exactly to continuity and the kinematic free surface boundary condition, and shows comparable errors in the dynamic free surface boundary condition to that found for the better, general horizontal bed wave theories.

Dynamic Effects of Liquids on Spin Stabilized Spacecraft

This paper presents two modeling methods for the sloshing dynamics of liquids in partially filled tanks in a spinning spacecraft: the homogeneous vortex flow approximation and the boundary layer model. In the homogeneous vortex approach, the liquid velocity is represented by a rigid rotation part and a correction part for the motion at the free surface. This approach has been improved upon to remove fictitious instabilities which were caused by inconsistent assumptions in the derivation of the dynamic free surface boundary condition. In the boundary layer model three boundary value problems must be solved: the inviscid problem, the boundary layer equations and the equations for the viscous correction. The boundary layer equations are solved analytically. The other boundary value problems are solved by using finite element methods. The computer programs developed for these two models are applied to the INTELSAT VI propulsion system. Natural frequencies of the liquid and of the combined liquid spacecraft system are determined. The viscous corrections to the natural frequencies are analysed.

Surface Particle Orbits in Uniform Waves

A numerical method is developed for calculating free-surface particle velocities in steady-state gravity waves of known profile. The method generates a unique solution for the orbit of a particle in the free surface, such that its motion is compatible with the observed wave profile, wavelength and surface mass transport velocity, and satisfies the dynamic free-surface boundary condition. Velocities of all free-surface particles may be derived from the geometry of the orbit. Velocities computed from the profile of a wave generated in the laboratory are shown to be compatible with simultaneous subsurface velocity measurements by the Laser Doppler Anemometer. The method has proved valuable in calculating impact velocities in wave slamming experiments. In view of the scatter apparent in almost all experimental evaluations of wave force coefficients, it is suggested that the method be applied to check the validity, at the free surface, of wave theories used in deriving coefficients from measured forces.

Stream function representation of nonlinear ocean waves

An analytical stream function expression representing a nonlinear gravity water wave is applied both to the representation of measured wave forms and also to nonlinear theoretical waves. The stream function form is chosen so that it is a solution to the Laplace equation and the bottom boundary condition; the parameters in the stream function expression are chosen by a numerical perturbation procedure that provides a best fit to the kinematic and dynamic free surface boundary conditions. The boundary condition errors associated with the nonlinear stream function representation of four measured wave profiles are compared with estimates of the corresponding errors associated with a linear representation. The stream function method is judged more accurate than a linear method if the wave height is greater than 50% of the breaking height. The stream function method is also applied to represent theoretical waves for which only the wave height and period are available to characterize the wave profile. It is demonstrated that the method represents an improvement over other available nonlinear procedures. The method can be employed to represent wave conditions which include a prescribed uniform steady current and a specified pressure distribution on the free surface.