A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper,we present a conservative semi-Lagrangian scheme designed for the numeri-cal solution of 3D hydrostatic free surface flows involving sediment transport on unstruc-tured Voronoi meshes.A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume.This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms.The presented method is fully conservative by construction,since the transported quantity or the vector field is integrated for each cell over the deformed vol-ume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element.The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations.Furthermore,a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance,hence yielding a stability condition which depends only on the explicit discretization of the viscous terms.A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment,which is assumed to be passive.The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solu-tion whenever is known.The new numerical scheme can reach up to fourth order of accu-racy on general orthogonal meshes composed by Voronoi polygons.

SUPPORT SYSTEM FOR FREE SURFACE GRINDING BASED ON PATCH DIVISION

To obtain accurate forms and surfaces in free surface grinding, it is important to provide grinding conditions suitable for a curved surface. A grinding support system for the free surface （GSX-F） is proposed to help the operator grind a free surface with the high accuracy and the high productivity. To succeed in free surface grinding, the property of a ball type wheel must be known. Therefore, a basic study of free surface grinding with a ball type wheel is carried out based on the grinding center （GC）. Some working points for achieving sufficient accuracy in free surface grinding are discussed. GSX-F is constructed using the patch division method and is used to test grinding. Reasonable results are obtained.

Numerical study of three-dimensional free surface dynamics

The dynamic problem of three-dimensional free surface is numerically studied in this paper. The ALE （Arbitrary Lagrange-Euler） kinematic description is introduced into the control equation system. The ALE description method is used to track free surface. Accurate formulations for calculating the normal vector on the free surface are presented. The discrete numerical equations by finite element method are developed by Galerkin weighted residual method. The boundary condition about free-surface tension is represented in the form of weak integration that can be computed by a differential geometry method derived in the present paper. The effect of contact angle is incorporated in the numerical algorithm. Furthermore, the numerical computations are performed and the comparison between computational and analytical results validated the effectiveness of the method. The results of this paper provide a fundamental understandings of the dynamics of liquid free surfaces, in which the surface tension and contact angle boundary conditions are taken into account. Finally, numerical simulation of largescale amplitude sloshing of liquid in a cylindrical container is performed and a numerical analysis of the effect of an annular ring-shaped rigid damping baffle on liquid sloshing oscillations in a cylindrical tank is also carried out.

Numerical and experimental investigation of rock breaking method under free surface by TBM disc cutter

To study the rock breaking method under the free surface induced by disc cutter,the rock breaking simulations were first conducted based on the discrete element method,and the dynamic process of rock breaking under the free surface was studied including stressed zone,crush zone,crack initiation and propagation.Then the crack propagation conditions,specific energy,etc.under different free surface distance(S)were also investigated combined with linear cutting experiments.The results show that the rock breaking process under the free surface induced by disc cutter is dominated by tension failure mode.There exists a critical S to promote crack propagation to free surface effectively.And this rock breaking method can improve the rock breaking force and breaking efficiency significantly when proper.

Prolonged simulation of near-free surface underwater explosion based on Eulerian finite element method

n the area of naval architecture and ocean engineering,the research about the underwater xplosion problem is of great significance.To achieve prolonged simulation of near-free surface underwater explosion,the underwater explosion transient numerical model is established in this paper based on compressible Eulerian finite element method(EFEM).Compared with Geers Hunter formula,EFEM is availably validated by simulating the free-field underwater xplosion case.Then,the bubble pulsation and flow field dynamic characteristics of the cases with different underwater explosive depth are compared in this work.Lastly,the height of the water hump and the pressure of flow flied are analyzed quantitatively through the simulation results.

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.

Numerical Simulation of Sloshing with Large Deforming Free Surface by MPS-LES Method

Moving particle semi-implicit （MPS） method is a fully Lagrangian particle method which can easily solve problems with violent free surface. Although it has demonstrated its advantage in ocean engineering applications, it still has some defects to be improved. In this paper, MPS method is extended to the large eddy simulation （LES） by coupling with a sub-particle-scale （SPS） turbulence model. The SPS turbulence model turns into the Reynolds stress terms in the filtered momentum equation, and the Smagorinsky model is introduced to describe the Reynolds stress terms. Although MPS method has the advantage in the simulation of the free surface flow, a lot of non-free surface particles are treated as free surface particles in the original MPS model. In this paper, we use a new free surface tracing method and the key point is ＂neighbor particle＂. In this new method, the zone around each particle is divided into eight parts, and the particle will be treated as a free surface particle as long as there are no ＂neighbor particles＂ in any two parts of the zone. As the number density parameter judging method has a high efficiency for the free surface particles tracing, we combine it with the neighbor detected method. First, we select out the particles which may be mistreated with high probabilities by using the number density parameter judging method. And then we deal with these particles with the neighbor detected method. By doing this, the new mixed free surface tracing method can reduce the mistreatment problem efficiently. The serious pressure fluctuation is an obvious defect in MPS method, and therefore an area-time average technique is used in this paper to remove the pressure fluctuation with a quite good result. With these improvements, the modified MPS-LES method is applied to simulate liquid sloshing problems with large deforming free surface. Results show that the modified MPS-LES method can simulate the large deforming free surface easily. It can not only capture the large impact pressure accurately on rolling tank wall but also can generate all physical phenomena successfully. The good agreement between numerical and experimental results proves that the modified MPS-LES method is a good CFD methodology in free surface flow simulations.

Three-dimensional numerical simulation of crown spike due to coupling effect between bubbles and free surface

The motion of gas bubbles beneath a free surface will lead to a spike of fluid on the free surface. The distance of the bubbles to the free surface is the key factor to different phenomena. When the inception distance varies in some range, crown phenomenon would happen after the impact of weak buoyancy bubbles, so this kind of spike is defined as crown spike in the present paper. Based on potential flow theory, a three-dimensional numerical model is established to simulate the motion of the free-surface spike generated by one bubble or a horizontal line of two in-phase bubbles. After the downward jet formed near the end of the collapse phase, the simulation of the free surface is performed to study the crown spike without regard to the toroidal bubble＇s effect. Calculations about the interaction between one bubble and free surface agree well with the experimental results conducted with a high-speed camera, and relative error is within 15%. Crown spike in both single- and two-bubble cases are simulated numerically. Different features and laws of the motion of crown spike, depending on the bubble-boundary distances and the inter-bubble distances, have been investigated.

Vertical 2D Modeling of Free Surface Flow with Hydrodynamic Pressure Using SIMPLE Arithmetic in σ Coordinates

A numerical model for shallow water flow has been developed based on the unsteady Reynolds-averaged Navier-Stokes equations with the hydrodynamic pressure instead of hydrostatic pressure assumption. The equations are transformed into the σ-coordinate system and the eddy viscosity is calculated with the standard k-ε turbulence model. The control volume method is used to discrete the equations, and the boundary conditions at the bed for shallow water models only include vertical diffusion terms expressed with wall functions. And the semi-implicit method for pressure linked equation arithmetic is adopted to solve the equations. The model is applied to the 2D vertical plane flow of a current over two steep-sided trenches for which experiment data are available for comparison and good agreement is obtained. And the model is used to predicting the flow in a channel with a steep-sided submerged breakwater at the bottom, and the streamline is drawn.

Experimental Investigation of the Effect of Bow Profiles on Resistance of an Underwater Vehicle in Free Surface Motion

In this paper, towing tank experiments are conducted to study the behavior of flow on a model of the underwater vehicle with various shapes of bows, i.e. tango and standard bows in free surface motion tests. The total resistances for different Froude numbers are considered experimentally. The towing tank is equipped with a trolley that can operate in through 0.05-6 m/s speed with ±0.02 m/s accuracy. Furthermore, the study is done on hydrodynamic coefficients i.e. total, residual and friction resistance coefficients, and the results are compared. Finally, the study on flow of wave fields around bows is done and wave filed around two bows are compared. The Froude number interval is between 0.099 and 0.349. Blockage fraction for the model is fixed to 0.005 3. The results showed that the residual resistance of the standard bow in 0.19 to 0.3 Froude number is more than the tango bow in surface motion which causes more total resistance for the submarine. Finally, details of wave generated by the bow are depicted and the effects of flow pattern on resistance drag are discussed.

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.

Validation of a CIP-based tank for numerical simulation of free surface flows

A constrained interpolation profile CIP-based numerical tank is developed to simulate violent free surface flows.The numerical simulation is performed by the CIP-based Cartesian grid method,which is described in the present paper.The tangent of hyperbola for interface capturing（THINC） scheme is applied for capturing complex free surfaces.The new model is capable of simulating a flow with violently varied free surface.A series of computations are conducted to assess the developed algorithm and its versatility.These tests include the collapse of water column with and without an obstacle,sloshing in a fixed tank,the generation of regular waves in a tank,the generation of extreme waves in a tank.Excellent agreements are obtained when numerical results are compared with available analytical,experimental,and other numerical results.

The interaction between multiple bubbles and the free surface

The flow is assumed to be potential, and a boundary integral method is used to solve the Laplace equation for the velocity potential to investigate the shape and the position of the bubble. A 3D code to study the bubble dynamics is developed, and the calculation results agree well with the experimental data. Numerical analyses are carried out for the interaction between multiple bubbles near the free surface including in-phase and out-of-phase bubbles. The calculation result shows that the bubble period increases with the decrease of the distance between bubble centres because of the depression effect between multiple bubbles. The depression has no relationship with the free surface and it is more apparent for out-of-phase bubbles. There are great differences in dynamic behaviour between the in-phase bubbles and the out-of-phase bubbles due to the depression effect. Furthermore, the interaction among eight bubbles is simulated with a three-dlmensional model, and the evolving process and the relevant physical phenomena are presented. These phenomena can give a reference to the future work on the power of bubbles induced by multiple charges exploding simultaneously or continuously.

Far-Field Noise Induced by Bubble near Free Surface

The motion of a bubble near the free surface is solved by the boundary element method based on the linear wave equation, and the influence of fluid compressibility on bubble dynamics is analyzed. Based on the solution of the bubble motion, the far-field radiation noise induced by the bubble is calculated using Kirchhoff moving boundary integral equation, and the influence of free surface on far-field noise is researched. As the results, the oscillation amplitude of the bubble is weakened in compressible fluid compared with that in incompressible fluid, and the free surface amplifies the effect of fluid compressibility. When the distance between the bubble and an observer is much larger than that between the bubble and free surface, the sharp wave trough of the sound pressure at the observer occurs. With the increment of the distance between the bubble and free surface, the time of the wave trough appearing is delayed and the value of the wave trough increase. When the distance between the observer and the bubble is reduced, the sharp wave trough at the observer disappears.

A Higher-Efficient Non-Hydrostatic Finite Volume Model for Strong Three-Dimensional Free Surface Flows and Sediment Transport

In order to accurately simulate strong three-dimensional （3-D） free surface flows and sediment transport, the fully 3- D non-hydrostatic pressure models are developed based on the incompressible Navier-Stokes equations and convection-diffusion equation of sediment concentration with the mixing triangle and quadrilateral grids. The governing equations are discretized with the unstructured finite volume method in order to provide conservation properties of mass and momentum, and flexibility with practical application. It is shown that it is first-order accurate on nonuniform plane two-dimensional （2-D） grids and second-order accurate on uniform plane grids. A third-order approximation of the vertical velocity at the top-layer is applied. In such a way, free surface zero stress boundary condition is satisfied maturely, and very few vertical layers are needed to give an accurate solution even for complex discontinuous flow and short wave simulation. The model is applied to four examples to simulate strong 3-D free surface flows and sediment transport where non-hydrostatic pressures have a considerable effect on the velocity field. The newly developed model is verified against analytical solutions with an excellent agreement.

High productive technique for polishing free surface with elastic ball type wheel on grinding center

Polishing is an important finishing process in die and mold manufacturing. Hand polishing takes long time and much labor. Efforts are made to automate the polishing process while keeping accuracy. Recently grinding centers have been developed, which are used for free surface polishing in the present work. The new polishing technique applies the same cutting locus as used in the cutting process to remove only cusp height effectively, keeping the form accuracy generated in the cutting process.

Modified surface tension model for free surface flow with single-phase lattice Boltzmann method

A single-phase lattice Boltzmann model with modified surface tension is developed in this paper to solve the problem of high-density-ratio free surface flow.The computational efficiency and accuracy are both enhanced.The restriction to the relaxation factor （which needs to be smaller than 1） is circumvented by the new surface tension algebra,due to its rational physical nature compared with the treatment of Xing,Buther and Yang in their paper （Comp.Mater.Sci.,2007,39（2）：282-290）.The proposed stable surface tension scheme is applied to simulate the free deformation of a square droplet with surface tension effect and the process of a droplet impinging on a liquid film.The numerical solution for free deformation of a droplet agrees well with thermodynamic principles,and also achieves high accuracy in comparison with Xing,et al.＇s model.Three typical impinging modes are successfully obtained with the new scheme,and another particular mode found by Wang and Chen is also successfully simulated.The evolutions of liquid crown agree well with the power law related to time.

THE NONLINEAR INTERACTION OF VORTEX RINGS WITH A FREE SURFACE

Nonlinear interactions of vortex rings with a free surface are considered in an incompressible, ideal fluid using the vortex contour dynamics technique and the boundary integral equation method. The flow is axisymmetric and the vorticity is linearly distributed in the vortex. Effects of the gravity and the surface tension as well as the initial geometric parameter of the vortex on the interaction process are investigated in considerable detail. The interaction process may be divided into three major stages: the vortex free-traveling stage, the collision stage, and the vortex stretching and rebounding stage. Time evolutions of both the vortex and free surface under various conditions are provided and analyzed. Two kinds of waves exist on the free surface during interaction. In a special case where the gravity and surface tension are very weak or the vortex is very strong, an electric-bulb-like 'cavity' is formed an the free surface and the vortex is trapped in the 'cavity' for quite a. long time, resulting in a large amount, of fluid above the mean fluid surface.

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.

Interaction of a submerged horseshoe vortex with a free surface

In this paper a submerged horseshoe vortex under a free surface is discussed and the algebraic expression of the wave elevation is obtained. From this expression, some characteristics of the ship wave are described. There exists a smooth region nearθ=0°, but when the uniform stream passes the other singularities （source, sink, doublet, etc.） there is no smooth region. The mechanism of synthetic aperture radar （SAR） images of the narrow ship wakes is also explained.