A CIP-Based Numerical Simulation of Free Surface Flow Related to Freak Waves

An enhanced numerical model for simulating two-dimensional incompressible viscous flow with distorted free surface is reported. The numerical simulation is carried out through the CIP （Constrained Interpolation Profile）-based method, which is described in the paper. A more accurate interface capturing scheme, the VOF/WLIC scheme （VOF：Volume-of-Fluid;WLIC：weighed line interface calculation）, is adopted as the interface capturing method. To assess the developed algorithm and its versatility, a selection of test problems are examined, i.e. the square wave propagation, the Zalesak’s rigid body rotation, dam breaking problem with and without obstacles, wave sloshing in an excited wave tank and interaction between extreme waves and a floating body. Excellent agreements are obtained when numerical results are compared with available analytical, experimental, and other numerical results. These examples demonstrate that the use of the VOF/WLIC scheme in the free surface capturing makes better results and also the proposed CIP-based model is capable of predicting the freak wave-related phenomena.

Gravity-capillary waves modulated by linear shear flow in arbitrary water depth

Considering that the fluid is inviscid and incompressible and the flow is irrotational in a fixed frame of reference and using the multiple scale analysis method, we derive a nonlinear Schrodinger equation(NLSE) describing the evolution dynamics of gravity-capillary wavetrains in arbitrary constant depth. The gravity-capillary waves(GCWs) are influenced by a linear shear flow(LSF) which consists of a uniform flow and a shear flow with constant vorticity. The modulational instability(MI) of GCWs with the LSF is analyzed using the NLSE. The MI is effectively modified by the LSF. In infinite depth, there are four asymptotes which are the boundaries between MI and modulational stability(MS) in the instability diagram. In addition, the dimensionless free surface elevation as a function of time for different dimensionless water depth,surface tension, uniform flow and vorticity is exhibited. It is found that the decay of free surface elevation and the steepness of free surface amplitude change over time, which are greatly affected by the water depth, surface tension, uniform flow and vorticity.

Effect of Viscosity on Free-Surface Waves in Oseen Flows

Based on the complex dispersion relation for the two-dimensional free-surface waves generated by a moving body in the steady Oseen flows, the effect of viscosity on wavelength and wave amplitude was investigated by means of an asymptotic method and a numerical analysis. A comparison between the asymptotic and numerical analysis for the viscous decay factor demonstrates the validity of the perturbation expansions for the wave profile. The numerical result shows that the wavelength of viscous wave is slightly elongated in comparison with that of inviscid wave.

Numerical and Experimental Investigation of Interactions Between Free-Surface Waves and A Floating Breakwater with Cylindrical-Dual/Rectangular-Single Pontoon

This paper investigates the hydrodynamic performance of a cylindrical-dual or rectangular-single pontoon floating breakwater using the numerical method and experimental study. The numerical simulation work is based on the multi-physics computational fluid dynamics(CFD) code and an innovative full-structured dynamic grid method applied to update the three-degree-of-freedom(3-DOF) rigid structure motions. As a time-marching scheme, the trapezoid analogue integral method is used to update the time integration combined with remeshing at each time step.The application of full-structured mesh elements can prevent grids distortion or deformation caused by large-scale movement and improve the stability of calculation. In movable regions, each moving zone is specified with particular motion modes(sway, heave and roll). A series of experimental studies are carried out to validate the performance of the floating body and verify the accuracy of the proposed numerical model. The results are systematically assessed in terms of wave coefficients, mooring line forces, velocity streamlines and the 3-DOF motions of the floating breakwater. When compared with the wave coefficient solutions, excellent agreements are achieved between the computed and experimental data, except in the vicinity of resonant frequency. The velocity streamlines and wave profile movement in the fluid field can also be reproduced using this numerical model.

Nonlinear Model of Mean Free Surface of Waves

A nonlinear model of mean free surface of waves or wave set-up is presented. The model is based on that of Roelvink (1993), but the numerical techniques used in the solution are based on the Weighted-Average Flux (WAF) method (Watson el al., 1992), with Time-Operator-Splitting (TOS) used for the treatment of the source terms. This method allows a small number of computational points to be used, and is particularly efficient in modeling wave set-up. The short wave (or primary wave) energy equation is solved by use of a more traditional Lax-Wendroff technique. A nonlinear wave theory (James, 1974) is introduced. The model described in this paper is found to be satisfactory in most respects when compared with the measurements conducted by Stive (1983) except in modeling the mean free surface very close to the mean shoreline.

ON THE RESONANT GENERATION OF WEAKLY NONLINEAR STOKES WAVES IN REGIONS WITH FAST VARYING TOPOGRAPHY AND FREE SURFACE CURRENT

The effect of nonlinearity on the free surface wave resonated by an incident flow over rippled beds, which consist of fast varying topography superimposed on an otherwise slowly varying mean depth, is studied using a WKBJ-type perturbation approach. Synchronous, superharmonic and in particular subharmonic resonance were selectively excited over the fast varying topography with corresponding wavelengths. For a steady current the dynamical system is autonomous and the possible nonlinear steady states and their stability were investigated. When the current has a small oscillatory component the dynamical system becomes non-autonomous, chaos is now possible.

B-Spline Approximation of Ship Waves on the Free Surface

We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.

Oblique Wave-free Potentials for Water Waves in Constant Finite Depth

在这份报纸，与一致有限深度为水在水波浪的 linearised 理论构造倾斜的没有波浪的潜力的一个方法以一种系统的方式被介绍。水把免费表面或一个冰封面作为一个薄有弹性的盘子建模。为免费表面的盒子，表面紧张的效果可以被忽视或考虑。这里，没有波浪的潜力是修改 Helmholtz 方程的单个答案，有在在液体区域的一个点的奇特，他们很快在上面的表面和水区域和腐烂的底部满足条件离开奇特的点。这些在获得解决方案到与象长水平的柱体那样的圆形的剖面图包含身体的倾斜的水波浪问题是有用的沉没或在有作为一块漂浮的橡皮建模的免费表面或一个冰封面的一致有限深度的水里沉浸一半板。最后，与这里构造的没有波浪的潜力有关的上面的表面的形式在很多个数字图形地被描绘设想波浪运动。非倾斜的没有波浪的潜力和上面的表面的结果没有波浪的潜力被获得。这里构造的没有波浪的潜力将在水里在包含无穷地长水平的柱体的水波浪问题的数学学习是有用的、也沉浸一半或完全沉浸。

INTERNAL RESONANT INTERACTIONS OF THREE FREE SURFACE-WAVES IN A CIRCULAR CYLINDRICAL BASIN

The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second_order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface_waves were discussed based on the above. The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non_resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second_order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.

Modeling of the whole process of shock wave overpressure of freefield air explosion

The waveform of the explosion shock wave under free-field air explosion is an extremely complex problem.It is generally considered that the waveform consists of overpressure peak,positive pressure zone and negative pressure zone.Most of current practice usually considers only the positive pressure.Many empirical relations are available to predict overpressure peak,the positive pressure action time and pressure decay law.However,there are few models that can predict the whole waveform.The whole process of explosion shock wave overpressure,which was expressed as the product of the three factor functions of peak,attenuation and oscillation,was proposed in the present work.According to the principle of explosion similarity,the scaled parameters were introduced and the empirical formula was absorbed to form a mathematical model of shock wave overpressure.Parametric numerical simulations of free-field air explosions were conducted.By experimental verification of the AUTODYN numerical method and comparing the analytical and simulated curves,the model is proved to be accurate to calculate the shock wave overpressure under free-field air explosion.In addition,through the model the shock wave overpressure at different time and distance can be displayed in three dimensions.The model makes the time needed for theoretical calculation much less than that for numerical simulation.

Nonlinear Dynamic Behaviors of A Floating Structure in Focused Waves

Floating structures are commonly seen in coastal and offshore engineering. They are often subjected to extreme waves and, therefore, their nonlinear dynamic behaviors are of great concern. In this paper, an in-house CFD code is developed to investigate the accurate prediction of nonlinear dynamic behaviors of a two-dimensional（2-D） box-shaped floating structure in focused waves. Computations are performed by an enhanced Constrained Interpolation Profile（CIP）-based Cartesian grid model, in which a more accurate VOF（Volume of Fluid） method, the THINC/SW scheme（THINC： tangent of hyperbola for interface capturing; SW： Slope Weighting）, is used for interface capturing. A focusing wave theory is used for the focused wave generation. The wave component of constant steepness is chosen. Comparisons between predictions and physical measurements show good agreement including body motions and free surface profiles. Although the overall agreement is good, some discrepancies are observed for impact pressure on the superstructure due to water on deck. The effect of grid resolution on the results is checked. With a fine grid, no obvious improvement is seen in the global body motions and impact pressures due to water on deck. It is concluded that highly nonlinear phenomena, such as distorted free surface, large-amplitude body motions, and violent impact flow, have been predicted successfully.

GLOBAL STABILITY OF WAVE PATTERNS FOR COMPRESSIBLE NAVIER-STOKES SYSTEM WITH FREE BOUNDARY

In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any γ 〉 1 with V being the adiabatic exponent provided that the wave strength is suitably small.

Second-order random interfacial wave solutions for two-layer fluid with a free surface

A previous study （Song. 2004. Geophys Res Lett, 31 （15）：L15302） of the second-order solutions for random interracial waves is extended in a constant depth, two-layer fluid system with a rigid lid is extended into a more general case of two-layer fluid with a top free surface. The rigid boundary condition on the upper surface is replaced by the kinematical and dynamical boundary conditions of a free surface, and the equations describing the random displacements of free surface, density-interface and the associated velocity potentials in the two-layer fluid are solved to the second order using the same expansion technology as that of Song （2004. Geophys Res Lett, 31 （15）：L15302）. The results show that the interface and the surface will oscillate synchronously, and the wave fields to the first-order both at the free surface and at the density-interface are made up of a linear superposition of many waves with different amplitudes, wave numbers and frequencies. The second-order solutions describe the second-order wave-wave interactions of the surface wave components, the interface wave components and among the surface and the interface wave components. The extended solutions also include special cases obtained by Thorpe for progressive interracial waves （Thorpe. 1968a.Trans R Soc London, 263A：563～614） and standing interracial waves （Thorpe. 1968b. J Fluid Mech, 32：489-528） for the two-layer fluid with a top free surface. Moreover, the solutions reduce to those derived for random surface waves by Sharma and Dean （1979.Ocean Engineering Rep 20） if the density of the upper layer is much smaller than that of the lower layer.

Generation of linear and nonlinear waves in numerical wave tank using clustering technique-volume of fluid method

A two-dimensional （2D） numerical model is developed for the wave sim- ulation and propagation in a wave flume. The fluid flow is assumed to be viscous and incompressible, and the Navier-Stokes and continuity equations are used as the governing equations. The standard k-e model is used to model the turbulent flow. The Navier- Stokes equations are discretized using the staggered grid finite difference method and solved by the simplified marker and cell （SMAC） method. Waves are generated and propagated using a piston type wave maker. An open boundary condition is used at the end of the numerical flume. Some standard tests, such as the lid-driven cavity, the constant unidirectional velocity field, the shearing flow, and the dam-break on the dry bed, are performed to valid the model. To demonstrate the capability and accuracy of the present method, the results of generated waves are compared with available wave theories. Finally, the clustering technique （CT） is used for the mesh generation, and the best condition is suggested.

MODELING THE INTERACTION OF SOLITARY WAVES AND SEMI-CIRCULAR BREAKWATERS BY USING UNSTEADY REYNOLDS EQUATIONS

A vertical 2-D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid(VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.

Doubly periodic patterns of modulated hydrodynamic waves:exact solutions of the Davey-Stewartson system

Exact doubly periodic standing wave patterns of the Davey-Stewartson （DS） equations are derived in terms of rational expressions of elliptic functions.In fluid mechanics,DS equations govern the evolution of weakly nonlinear,free surface wave packets when long wavelength modulations in two mutually perpendicular,horizontal directions are incorporated.Elliptic functions with two different moduli （periods） are necessary in the two directions.The relation between the moduli and the wave numbers constitutes the dispersion relation of such waves.In the long wave limit,localized pulses are recovered.

A Higher Order BEM for Wave-Current Action on Structures—Direct Computation of Free-Term Coefficient and CPV Integrals

A higher order boundary dement method （HOBEM） is implemented for wave-current action on structures. The freeterm coefficient and Cauchy principal value （ GPV） integrals are computed by direct methods. Numerical experiments are carried out to validate the computation of free-term coefficient and GPV integrals. The results show that the computation precision of free-term coefficient is very high for various bodies, even with edges and corners, and the convergence speed is fast for CPV integrals for different meshes. The comparison of the second order mean drift force due to wave-current action on a uniform cylinder is made with an analytic solution. It is found that good agreement exists between the present calculation and the analytic solutions. Finally, the numerical code is applied for computing wave-current action on Snorrc TLP.

Higher Harmonics Induced by Waves Propagating over A Submerged Obstacle in the Presence of Uniform Current

To investigate higher harmonics induced by a submerged obstacle in the presence of uniform current, a 2D fully nonlinear numerical wave flume（NWF） is developed by use of a time-domain higher-order boundary element method（HOBEM） based on potential flow theory. A four-point method is developed to decompose higher bound and free harmonic waves propagating upstream and downstream around the obstacle. The model predictions are in good agreement with the experimental data for free harmonics induced by a submerged horizontal cylinder in the absence of currents. This serves as a benchmark to reveal the current effects on higher harmonic waves. The peak value of non-dimensional second free harmonic amplitude is shifted upstream for the opposing current relative to that for zero current with the variation of current-free incident wave amplitude, and it is vice versa for the following current. The second-order analysis shows a resonant behavior which is related to the ratio of the cylinder diameter to the second bound mode wavelength over the cylinder. The second-order resonant position slightly downshifted for the opposing current and upshifted for the following current.

Improvements on Mean Free Wave Surface Modeling

Some new results of the modeling of mean free surface of waves or wave set-up are presented. The stream function wave theory is applied to incident short waves. The limiting wave steepness is adopted as the wave breaker index in the calculation of wave breaking dissipation. The model is based on Roelvink (1993), but the numerical techniques used in the solution are based on the Weighted-Average Flux (WAF) method (Watson et al., 1992), with Time-Operator-Splitting (TOS) used for the treatment of the source terms. This method allows a small number of computational points to be used, and is particularly efficient in modeling wave set-up. The short wave (or incident primary wave) energy equation is solved by use of a traditional Lax-Wendroff technique. The present model is found to be satisfactory compared with the measurements conducted by Stive (1983).

An Application of the Real Free Surface Boundary Condition in Calculation of Ship wave Problem

A real free surface boundary condition,taking the viscous effects and surface tension into account,is applied to the nonlinear calculation of wave making resistance.It may provide more information about the character of the nonlinear ship wave and be helpful to improving the stability,convergence and local wave profile in potential calculation of the nonlinear ship wave.The wave making calculations for Series 60 are presented.