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.

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.

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.

Simulation of Free Surface Flows Based on MapReduce in Cloud Computing

Simulating turbulent liquids with breaking waves and splashes is among the most desired features in fluid animation. Lagrangian methods such as Smoothed Particle Hydrodynamics method (SPH) are a promising way to capture such properties. However, the Particle-based liquid surface simulation has not been applied very well since its consumption is way too large. This paper derives the governing equations in SPH approaches and parallelizes the dynamics-based surface simulation with the MapReduce program models which apply the SPH approach in Cloud Computing. Compared to the serial methods, this approach obtained a 3.11 times speedup on the experimental platform.

A New Efficient Finite Volume Modeling of Small Amplitude Free Surface Flows with Unstructured Grid

A staggered finite-volume technique for non-hydrostatic, small amplitude free surface flow governed by the incompressible Navier-Stokes equations is presented there is a proper balance between accuracy and computing time. The advection and horizontal diffusion terms in the momentum equation are discretized by an integral interpolation method on the orthogonal unstructured staggered mesh and, while it has the attractive property of being conservative. The pressure-correction algorithm is employed for the non-hydrostatic pressure in order to achieve second-order temporal accuracy. A conservative scalar transport algorithm is also applied to discretize k - c equations in this model. The eddy viscosity is calculated from the k-c turbulent model. The resulting model is mass and momentum conservative. The model is verified by two examples to simulate unsteady small amplitude free surface flows where non-hydrostatic pressures have a considerable effect on the velocity field, and then applied to simulate the tidal flow in the Bohai Sea.

A Higher-Efficient Three-Dimensional Numerical Model for Small Amplitude Free Surface Flows

A higher-efficient three-dimensional non-hydrostatic model is developed to simulate small amplitude free surface flows based on a staggered unstructured grid. In this model, a fractional step algorithm is adopted to solve the Navier-Stokes equations in two major steps. A top-layer pressure method is proposed to minimize the number of vertical layers and subsequently the computational cost. Three classical examples of small amplitude free surface flows are used to demonstrate the capability and efficiency of the model. The satisfactory results demonstrated the capability and efficiency of modelling a range of small amplitude free surface flows with only a small number of vertical layers.

Free Surface Flow Generated by Submerged Twin-cylinders in Forward Motion Using a Fully Nonlinear Method

The free surface flow generated by twin-cylinders in forced motion submerged beneath the free surface is studied based on the boundary element method. Two relative locations, namely, horizontal and vertical, are examined for the twin cylinders. In both cases, the twin cylinders are starting from rest and ultimately moving with the same constant speed through an accelerating process. Assuming that the fluid is inviscid and incompressible and the flow to be irrotational, the integral Laplace equation can be discretized based on the boundary element method. Fully-nonlinear boundary conditions are satisfied on the unknown free surface and the moving body surface. The free surface is traced by a Lagrangian technique. Regriding and remeshing are applied, which is crucial to quality of the numerical results. Single circular cylinder and elliptical cylinder are calculated by linear method and fully nonlinear method for accuracy checking and then fully nonlinear method is conducted on the twin cylinder cases, respectively. The generated wave elevation and the resultant force are analysed to discuss the influence of the gap between the two cylinders as well as the water depth. It is found that no matter the kind of distribution, when the moving cylinders are close to each other, they suffer hydrodynamic force with large absolute value in the direction of motion. The trend of force varying with the increase of gap can be clearly seen from numerical analysis. The vertically distributed twin cylinders seem to attract with each other while the horizontally distributed twin cylinders are opposite when they are close to each other.

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.

NUMERICAL SIMULATION OF TURBULENT FREE SURFACE FLOW OVER OBSTRUCTION

A two-dimensional hybrid numerical model, FEM-LES-VOF, for free surface flows is proposed in this study, which is a combination of three-step Taylor-Galerkin finite element method, large eddy simulation with the Smagorinsky sub-grid model and Computational Lagrangian-Eulerian Advection Remap Volume of Fluid （CLEAR-VOF） method. The present FEM-LES-VOF model allows the fluid flows involving violent free surface and turbulence subject to complex boundary configuration to be simulated in a straightforward manner with unstructured grids in the context of finite element method. Numerical simulation of a benchmark problem of dam breaking is conducted to verify the present model. Comparisons with experimental data show that the proposed model works well and is capable of producing reliable predictions for free surface flows. Using the FEM-LES-VOF model, the free surface flow over a semi-circular obstruction is investigated. The simulation results are compared with available experimental and numerical results. Good performance of the FEM-LES-VOF model is demonstrated again. Moreover, the numerical studies show that the turbulence plays an important role in the evolution of free surface when the reflected wave propagates upstream during the fluid flow passing the submerged obstacle.

Numerical simulation of 3-D free surface flows by overlapping MPS

An overlapping moving particle semi-implicit （MPS） method is applied for 3-D free surface flows based on our in-house particle solver MLParticle-SJTU. In this method, the coarse particles are distributed in the whole domain and the fine particles are distributed in the local region of interest at the same time. With the fine particles being generated and removed dynamically, an algorithm of generating particles based on the 3-D overlapping volume is developed. Then, a 3-D dam break flow with an obstacle is simulated to validate the overlapping MPS. The qualitative comparison among experimental data and the results obtained by the VOF and the MPS shows that the shape of the free surface obtained by the overlapping MPS is more accurate than that obtained by the UNI-coarse and close to that obtained by the UNI-fine in the overlapping domain. In addition, the water height and the impact pressure at Pi are also in an overall agreement with experimental data. Finally, the CPU time required by the overlapping MPS is about half of that required by the UNl-fine.

NUMERICAL SIMULATIONS OF HIGHLY NONLINEAR STEADY AND UNSTEADY FREE SURFACE FLOWS

A numerical simulation model based on an open source Computational Fluid Dynamics （CFD） package-Open Field Operation and Manipulation （OpenFOAM） has been developed to study highly nonlinear steady and unsteady free surface flows. A two-fluid formulation is used in this model and the free surface is captured using the classical Volume Of Fluid （VOF） method. The incompressible Euler/Navier-Stokes equations are solved using a finite volume method on unstructured polyhedral cells. Both steady and unsteady free surface flows are simulated, which include： （1） a submerged NACA0012 2-D hydrofoil moving at a constant speed, （2） the Wigley hull moving at a constant speed, （3） numerical wave tank, （4） green water overtopping a fixed 2-D deck, （5） green water impact on a fixed 3-D body without or with a vertical wall on the deck. The numerical results obtained have been compared with the experimental measurements and other CFD results, and the agreements are satisfactory. The present numerical model can thus be used to simulate highly nonlinear steady and unsteady free surface flows.

COMPUTATION OF FREE SURFACE FLOWS WITH HIGH RESOLUTION FINITE ELEMENT SCHEME

A high-resolution finite element scheme is developed for the computation offree surface problems. The present higher-order accuracy scheme is obtained through the introductionof the difference of almost equal amount of diffusion and antidiffusion, where the antidiffusion isformulated as approximation to diffusion making use of consecutive gradient. The non-oscillatoryproperties are secured by limiters to preserve positivity constraints. The implement of the schemeis based on semi-discrete form and can be considered as reconstruction of the coefficient matrix ofalgebraic system, so, it can he conveniently used by either structured or unstructured meshes. Thepower and flexibility of the present high resolution VOF-FEM approach is validated by examplesincluding convective transport, dam breaking flow, and stilling basin flow in hydraulic engineering.

THE MACCORMACK SCHEME AND THE BOUNDARY-FITTED COORDINATE SYSTEM FOR NUMERICAL SIMULATION OF SUPER CRITICAL FREE SURFACE FLOWS

This paper is concerned with the mathematical model for the numerical simulation of supercritical surface flows. A boundary-fitted coordinate system was used to overcome the difficulties and inaccuracy associated with the determination of flow characteristics near the fl ow boundaries. The MacCormack scheme was applied for the solution of the transfo rmed system of equations. Comparisions between computed results and experimental data show a satisfactory agreement.

Simulation of violent free surface flow by AMR method

A novel CFD approach based on adaptive mesh refinement（AMR） technique is being developed for numerical simulation of violent free surface flows. CIP method is applied to the flow solver and tangent of hyperbola for interface capturing with slope weighting（THINC/SW） scheme is implemented as the free surface capturing scheme. The PETSc library is adopted to solve the linear system. The linear solver is redesigned and modified to satisfy the requirement of the AMR mesh topology. In this paper, our CFD method is outlined and newly obtained results on numerical simulation of violent free surface flows are presented.

Analysis on MHD Stability of Free Surface Jet flow in a Gradient Magnetic Fields

The simplified modeling for analysis on MHD stability of free surface jet flow in a gradient magnetic fields is based on the theoretical and experimental results on channel liquid metal MHD flow, especially, the results of MHD flow velocity distribution in cross-section of channels (rectangular duct and circular pipe), and the expected results from the modeling are well agreed with the recent experimental data obtained. It is the first modeling which can efficiently explain the experimental results of liquid-metal free surface jet flow.

MHD Effect of Liquid Metal Film Flows as Plasma-Facing Components

Stability of liquid metal film flow under gradient magnetic field is investigated. Three dimensional numerical simulations on magnetohydrodynamics （MHD） effect of free surface film flow were carried out, with emphasis on the film thickness variation and its surface stability. Three different MHD phenomena of film flow were observed in the experiment, namely, retardant, rivulet and flat film flow. From our experiment and numerical simulation it can be concluded that flat film flow is a good choice for plasma-facing components （PFCs）

THREE-DIMENSIONAL NON-HYDROSTATIC MODEL FOR FREE-SURFACE FLOWS WITH UNSTRUCTURED GRID

The three-dimensional Navier-Stokes equations were solved with the fractional step method where the hydrostatic pressure component was determined first, while the non-hydrostatic component of the pressure was computed from the pressure Poisson equation in which the coefficient matrix is positive definite and symmetric. The eddy viscosity was calculated from the efficient k-ε turbulence model. The resulting model is computationally efficient and unrestricted to the CFL condition. Computations with and without hydrostatic approximation were compared for the same cases to test the validity of the conventional hydrostatic pressure assumption. The model was verified against analytical solutions and experimental data, with excellent agreement.

A VERTICAL 2D MATHEMATICAL MODEL FOR HYDRODYNAMIC FLOWS WITH FREE SURFACE IN σ COORDINATE

Numerical models with hydrostatic pressure have been widely utilized in studying flows in rivers, estuaries and coastal areas. The hydrostatic assumption is valid for the large-scale surface flows where the vertical acceleration can be ignored, but for some particular cases the hydrodynamic pressure is important. In this paper, a vertical 2t） mathematical model with non-hydrostatic pressure was implemented in the σ coordinate. A fractional step method was used to enable the pressure to be decomposed into hydrostatic and hydrodynamic components and the predictor-corrector approach was applied to integration in time domain. Finally, several computational cases were studied to validate the importance of contributions of the hydrodynamic pressure.

CALCULATION OF NONLINEAR FREE－SURFACE FLOWS RESULTING FROM LARGE￣AMPLITUDE FORCED OSCILLATION OF A TWO－DIMENSIONAL BODY

The BEM combined with the time- st6pping scheme has been applied to the numerical calculation of fully nonlinear free surface flows generated by large amplitude forced transverse oscillation of two-dimensional body. Particular attention is paid on the compatibility of free surface and body surface conditions at the intersection point, and moving radiation boundary is adopted. A new calculating formula of the exact force on the body is also presented.The results demonstrate some nonlinear phenomena and indicate the stability and correctness of the numerical simulation.

Dispersion in Shallow Moment Equations

Shallow moment models are extensions of the hyperbolic shallow water equations.They admit variations in the vertical profile of the horizontal velocity.This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables.The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction.Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface.The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given.A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.