A CONSERVATIVE UNSTRUCTURED STAGGERED GRID SCHEME FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A finite-volume formulation is proposed to solve the three-dimensional, non-hydrostatic Navier-Stokes equations on an unstructured, staggered, z-lever grid, with the goal of simulating non-hydrostatic processes in the free-surface flows. The advection and diffusion terms in the momentum equation are discretized explicitly with the Eulerian scheme, which has the attractive property of being conservative. An integral method of the top- layer pressure is applied to account for the full effects of non-hydrostatic pressure at the free-surface layer. It is shown that the results obtained with a small number of vertical layers (e.g., 2-3 layers) are in good agreements with experimental data or analytical solutions, demonstrating the efficiency and accuracy of the model in simulating a range of free-surface flow problems including wave motion and tide-induced motion.     

A SEMI-IMPLICIT 3-D NUMERICAL MODEL USING SIGAM-COORDINATE FOR NON-HYDROSTATIC PRESSURE FREE-SURFACE FLOWS

A 3-D numerical formulation is proposed on the horizontal Cartesian, vertical sigma-coordinate grid for modeling non-hydrostatic pressure free-surface flows. The pressure decomposition technique and θ semi-implicit method are used, with the solution procedure being split into two steps. First, with the implicit parts of non-hydrostatic pressures excluded, the provisional velocity field and free surface are obtained by solving a 2-D Poisson equation. Second, the theory of the differential operator is employed to derive the 3-D Poisson equation for non-hydrostatic pressures, which is solved to obtain the non-hydrostatic pressures and to update the provisional velocity field. When the non-orthogonal sigma-coordinate transformation is introduced, additional terms come into being, resulting in a 15-diagonal, diagonally dominant but unsymmetric linear system in the 3-D Poisson equation for non-hydrostatic pressures. The Biconjugate Gradient Stabilized (BiCGstab) method is used to solve the resulting 3-D unsymmetric linear system instead of the conjugate gradient method, which can only be used for symmetric, positive-definite linear systems. Three test cases are used for validations. The successful simulations of the small-amplitude wave, a supercritical flow over a ramp and a turbulent flow in the open channel indicate that the new model can simulate well non-hydrostatic flows, supercritical flows and turbulent flows.     

SIMULATION OF SEDIMENT-LADEN FLOW BY DEPTH-AVERAGED MODEL BASED ON UNSTRUCTURED COLLOCATED GRID

The 2-D depth-averaged mathematical model for sediment-laden flows has been widely used in river control and other related engineering problems, and now it is usually solved on structured grids. Since the natural river is usually very complicated in plane boundary, and unstructured grids are more attractive in solving the problems with complicated domains, the following questions about solving 2-D depth-averaged model were discussed in this article: (1) a modified Bowyer algorithm was suggested to generate unstructured grids for natural rivers, (2) the Finite Volume Method (FVM) is employed to discretize the governing equations of the 2-D depth-averaged model and an implicit scheme was suggested with unstructured collocated grids, (3) the observed hydrological data of the Chenglingji Reach in the Yangtze River are used for verification of the presented method. It seems that the suggested numerical scheme works very well, and the simulation results of both hydraulic characteristics and river bed deformation are in good agreement with the observed ones.     

THE INSTABILITY OF NON-HYDROSTATIC STRATIFIED FLOWS

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A RAISED PANEL METHOD BASED ON NURBS FOR FREE-SURFACE POTENTIAL FLOWS WITH FORWARD SPEED

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THE FINITE VOLUME PROJECTION METHOD WITH HYBRID UNSTRUCTURED TRIANGULAR COLLOCATED GRIDS FOR INCOMPRESSIBLE FLOWS

In this article a finite volume method is proposed to solve viscous incompressible Navier-Stokes equations in two-dimensional regions with corners and curved boundaries. A hybrid collocated-grid variable arrangement is adopted, in which the velocity and pressure are stored at the centroid and the circumcenters of the triangular control cell, respectively. The cell flux is defined at the mid-point of the cell face. Second-order implicit time integration schemes are used for convection and diffusion terms. The second-order upwind scheme is used for convection fluxes. The present method is validated by results of several viscous flows.     

A HIGHER-ORDER NON-HYDROSTATIC MODEL FOR SIMULATING WAVE PROPAGATION OVER IRREGULAR BOTTOMS

A higher-order non-hydrostatic model is developed to simulate the wave propagation over irregular bottoms based on a vertical boundary-fitted coordinate system.In the model,an explicit projection method is adopted to solve the unsteady Euler equations.Advection terms are integrated explicitly with the MacCormack's scheme,with a second-order accuracy in both space and time.Two classical examples of surface wave propagation are used to demonstrate the capability of the model.It is found that the model with on...     

基于非结构化网格上的三维微幅自由表面流动非静压数值模型

建立了基于非结构化网格求解三维微幅自由表面流动非静压数值模型。一种有限差分法和有限体积法相结合的方法被用来在非结构化网格上离散控制方程。模型中采用分步法,以使压力项分解为静水压力项和动水压力项来单独处理。在每一时间步,得到关于水位和由压力泊松方程得到的动水压力的稀疏的线性方程组。模型是质量守恒的和对于重力波速、风应力、底摩阻、垂向黏性无条件稳定的。通过三个非恒定微幅自由表面流动算例的数值解和解析解的比较,验证了本文所建立的模型能够准确有效的预测三维微幅自由表面流动。     

C-D无结构网格上的三维自由水面非静水模型:Ⅰ-算法

为能准确地模拟垂向运动尺度较大的水流运动，本文基于压力分裂模式和半隐方法提出了C-D无结构、z坐标网格上的非静水压力模型的构造方法，并详细论述了模型的求解过程。模型先在忽略动水压强项的隐式部分的条件下求解控制方程，获得比较接近的初始解；然后，再考虑动水压强项的隐式部分并求解一个关于它的三维Poisson方程，因此计算容易收敛。本文模型与传统的C网格上的非静水模型相比，增加了对水平切向动量方程的求解，消除了因C网格的简化处理所带来的精度损失。控制方程用有限体积—有限差分法离散，自由水面用水位函数法处理。对水位η和动水压强q分别采用不同的隐式因子以兼顾各自稳定和精度的要求。压力分裂模式、θ半隐方法和欧拉—拉格朗日方法（ELM）等的联合使用使得本文模型具有简单、稳定、高效的特点。     

NUMERICAL MODELLING OF FREE-SURFACE FLOWS WITH BOTTOM AND SURFACE-LAYER PRESSURE TREATMENT

A new non-hydrostatic numerical model with the three-dimensional Navier-Stokes equations on structured grids was constructed and discussed. The algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. The eddy viscosity coefficient was calculated by the efficient k-ε turbulence model. A new surface-layer non-hydrostatic treatment and a local cell bottom treatment were introduced so that the three-dimensional model is fully non-hydrostatic and is free of any hydrostat...     

APPLICATION OF A MODIFIED QUICK SCHEME TO DEPTHAVERAGED k-( TURBULENCE MODEL BASED ON UNSTRUCTURED GRIDS

The modified QUICK scheme on unstructured grid was used to improve the advection flux approximation, and the depth-averaged turbulence model with the scheme based on FVM by SIMPLE series algorithm was established and applied to spur-dike flow computation. In this model, the over-relaxed approach was adopted to estimate the diffusion flux in view of its advantages in reducing errors and sustaining numerical stability usually encountered in non-orthogonal meshes. Two spur-dike cases with different defection angles (90oand 135o) were analyzed to validate the model. Computed results show that the predicted velocities and recirculation lengths are in good agreement with the observed data. Moreover, the computations on structured and unstructured grids were compared in terms of the approximately equivalent grid numbers. It can be concluded that the precision with unstructured grids is higher than that with structured grids in spite that the CPU time required is slightly more with unstructured grids. Thus, it is significant to apply the method to numerical simulation of practical hydraulic engineering.     

NUMERICAL SIMULATION OF SLOSHING IN RECTANGULAR TANK WITH VOF BASED ON UNSTRUCTURED GRIDS

A new method for sloshing simulation in a sway tank is present, in which the two phase interface is treated as a physical discontinuity, which can be captured by a well-designed high order scheme. Based on Normalized Variable Diagram (NVD), a high order discretization scheme with unstructured grids is realized, together with a numerical method for free surface flow with a fixed grid. This method is implemented in an in-house code General Transport Equation Analyzer (GTEA) which is an unstructured grids finite volume solver. The present method is first validated by available analytical solutions. A simulation for a 2-D rectangular tank at different excitation frequencies of the sway is carried out. A comparison with experimental data in literature and results obtained by commercial software CFX shows that the sloshing load on the monitor points agrees well with the experimental data, with the same grids, and the present method gives better results on the secondary peak. It is shown that the present method can simulate the free surface overturning and breakup phenomena.     

A 2-D HYDRODYNAMIC MODEL FOR THE RIVER, LAKE AND NETWORK SYSTEM IN THE JINGJIANG REACH ON THE UNSTRUCTURED QUADRANGLES

The main river,the Dongting Lake and river networks in the Jingjiang reach of the Yangtze River constitute a complex water system,for which a full 2-D hydrodynamic model is established instead of the traditional 1-D or compound models for simulation of such complex systems,based on the latest developments of computer technologies and numerical methods.To better handle irregular boundaries and keep the computation cost well in a reasonable limit,unstructured grids of moderate scale are used. In addition,a dynamic boundary tracking method is proposed to simulate variable flow domains at different floods,especially,when the moderate scale gird can not describe flows in narrow river-network channels at low water levels.Theθsemi-implicit method and the Eulerian-Lagrangian Method(ELM)are adopted,which make the model unconditionally stable with respect to the gravity wave speed and Courant number restrictions.Properties and efficiency of the model are discussed,and it is concluded that the new model is robust and efficient enough for the simulation of a big,complex water system.Validation tests show that the simulation results agree well with field data.It takes about 0.96 h to complete the computation of a 76 d flood,which indicates that the model is efficient enough for engineering applications.     

THREE-DIMENSIONAL NUMERICAL MODEL FOR WINDING TIDAL RIVER WITH BRANCHES

Natural rivers are usually winding with branches and shoals,which are difficult to be simulated with rectangular grids. A 3-D current numerical model was established based on the orthogonal curvilinear coordinate system and vertical σ coordinate system. The equations were discretisized using a semi-implicit scheme. The “predictor” and “corrector” steps were applied for the horizontal momentum equations to meet the basic requirement that the depth-integrated currents obtained from the equations for 2-D and 3-D modes have identical values. And a modification of traditional method of dry/wet discriminance was proposed to determine accurately the boundary and ensure the continuity of variable boundary in the simulation. This model was verified with the data measured in a winding tidal river with branches in April,2004. The simulated data of water levels and velocities agree well with the measured ones,and the computed results reveal well the practical flow characteristics,including the vertical secondary flow in a winding reach.     

APPLICATION OF A MODIFIED QUICK SCHEME TO DEPTH- AVERAGED κ-ε TURBULENCE MODEL BASED ON UNSTRUCTURED GRIDS

The modified QUICK scheme on unstructured grid was used to improve the advection flux approximation, and the depth-averaged κ-ε turbulence model with the scheme based on FVM by SIMPLE series algorithm was established and applied to spur-dike flow computation. In this model, the over-relaxed approach was adopted to estimate the diffusion flux in view of its advantages in reducing errors and sustaining numerical stability usually encountered in non-orthogonal meshes. Two spur-dike cases with different defection angles （90°and 135°） were analyzed to validate the model. Computed results show that the predicted velocities and recirculation lengths are in good agreement with the observed data. Moreover, the computations on structured and unstructured grids were compared in terms of the approximately equivalent grid numbers. It can be concluded that the precision with unstructured grids is higher than that with structured grids in spite that the CPU time required is slightly more with unstructured grids Thus, it is significant to apply the method to numerical simulation of practical hydraulic engineering.     

MATHEMATICAL MODELS AND NUMERICAL SIMULATION FOR DENSE PARTICULATE FLOWS

Sedimentation of particles in inclined and vertical vessels is numerically simulated by the Eulerian two-fluid model. The numerical results show an interesting phenomenon with two circulation vortexes in a vertical vessel but one in the inclined vessel. Sensitivity tests indicate that the boundary layer effect is the key to induce this phenomenon. A numerical method based on 2D unstructured meshes is presented to solve the hard-sphere discrete particle model. Several applications show the numerical method has a good performance to simulate dense particulate flows in irregular domains without regard to element types of the mesh.     

NUMERICAL INVESTIGATION ON NON-NEWTONIAN FLOWS THROUGH DOUBLE CONSTRICTIONS BY AN UNSTRUCTURED FINITE VOLUME METHOD

This article presents a numerical investigation on a steady non-Newtonian flow through a two-dimensional channel with double constrictions. The power-law mode is employed in describing the non-Newtonian behavior of the flow. An unstructured finite volume method combined with a fractional-step projection method is developed for the discretization of incompressible equations governing the non-Newtonian flows. The important flow dynamics related with the arterial diseases, such as the wall shear stress and vortex generation, are also numerically studied in detail. Numerical results reveal that there are marked differences between Newtonian and non-Newtonian models.     

THREE-DIMENSIONAL NUMERICAL SIMULATION OF INTAKE MODEL WITH CROSS FLOW

The hydrodynamics of a pump sump consisting of a main channel,pump sump,and intake pipe is examined using Truchas,a three-dimensional Navier-Stokes solver,with a Large Eddy Simulation(LES) turbulence model.The numerical results of streamwise velocity profiles and flow patterns are discussed and compared with experimental data of Ansar and Nakato.Fairly good agreement is obtained.Furthermore,unlike Ansar et al.'s inviscid solution,the proposed numerical model includes the effect of fluid viscosity and consid...     

C-D无结构网格上的三维自由水面非静水压力流动模型Ⅰ：算法

为准确模拟垂向运动尺度较大的水流运动,本文基于压力分裂模式和半隐方法提出了C-D无结构、z坐标网格上的三维非静水压力流动模型的构造方法,并详细论述了模型的求解过程。模型先在忽略动水压强项的隐式部分的条件下求解控制方程,获得比较接近的初始解;再考虑动水压强项的隐式部分并求解一个关于它的三维Poisson方程,因此计算容易收敛。本文模型与传统的C网格上的非静水模型相比,增加了对水平切向动量方程的求解,消除了因C网格的简化处理所带来的精度损失。控制方程用有限体积-有限差分法离散,自由水面用水位函数法处理。对水位η和动水压强q分别采用不同的隐式因子以兼顾各自稳定和精度的要求。压力分裂模式、θ半隐方法和欧拉-拉格朗日方法(ELM)等的联合使用使得本模型具有简单、稳定、高效的特点。