Numerical simulation of the internal wave propagation in continuously densitystratified ocean

A numerical model is proposed to simulate the internal wave propagation in a continuously density-stratified ocean, and in the model, the momentum equations are derived from the Euler equations on the basis of the Boussinesq approximation. The governing equations, including the continuity equation and the momentum equations, are discretized with the finite volume method. The advection terms are treated with the total variation diminishing （TVD） scheme, and the SIMPLE algorithm is employed to solve the discretized governing equations. After the modeling test, the suitable TVD scheme is selected. The SIMPLE algorithm is modified to simplify the calculation process, and it is easily made to adapt to the TVD scheme. The Sommerfeld＇s radiation condition combined with a sponge layer is adopted at the outflow boundary. In the water flume with a constant water depth, the numerical results are compared to the analytical solutions with a good agreement. The numerical simulations are carried out for a wave flume with a submerged dike, and the model results are analyzed in detail. The results show that the present numerical model can effectively simulate the propagation of the internal wave.     

HIGH ACCURATE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOW WITH PRIMATIVE VARIABLE

This paper presents a higher order difference scheme for the computationof the incompressible viscous flows.The discretization of the two-dimensional incompress-ible viscous Navier-Stokes equations,in generalized curvilinear coordinates and tensor for-mulation,is based on a non-ataggered grid.The momentum equations are integrated intime using the four-stage explicit Runge-Kutta algorithm [1]and discretized in space us-ing the fourth-order accurate compact scheme[2]The pressure-Poisson equation is dis-cretized using the nine-point compact scheme.In order to satisfy the continuity constraintand ensure the smoothness of pressure field,an optimum procedure to derive a discretepressure equation is proposed [9][3]The method is applied to calculate the driven cavityflow on a stretched grid with the Reynolds numbers from 100 to 10000.The numerical re-sults are in very good agreement with the results obtained by Ghia et al [7]and includethe periodic solutions for high Reynolds numbers.     

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.     

BASIC THEORY AND MATHEMATICAL MODELING OF URBAN RAINSTORM WATER LOGGING

In this paper, a mathematical model for the urban rainstorm water logging was established on the basis of one- and two-dimensional unsteady flow theory and the technique of non-structural irregular grid division. The continuity equation was discretized with the finite volume method. And the momentum equations were differently simplified and dis-cretized for different cases. A method of “special passage” was proposed to deal with small-scale rivers and open channels. The urban drainage system was simplified and simulated in the model. The method of “open slot” was applied to coordinate the alternate calculation of open channel flow and pressure flow in drainage pipes. The model has been applied in Tianjin City and the verification is quile satisfactory.     

A hybrid DEM/CFD approach for solid-liquid flows简

A hybrid scheme coupling the discrete element method(DEM)with the computational fluid dynamics(CFD)is developed to model solid-liquid flows.Instead of solving the pressure Poisson equation,we use the compressible volume-averaged continuity and momentum equations with an isothermal stiff equation of state for the liquid phase in our CFD scheme.The motion of the solid phase is obtained by using the DEM,in which the particle-particle and particle-wall interactions are modelled by using the theoretical contact mechanics.The two phases are coupled through the Newton’s third law of motion.To verify the proposed method,the sedimentation of a single spherical particle is simulated in water,and the results are compared with experimental results reported in the literature.In addition,the drafting,kissing,and tumbling(DKT)phenomenon between two particles in a liquid is modelled and reasonable results are obtained.Finally,the numerical simulation of the density-driven segregation of a binary particulate suspension involving 10 000 particles in a closed container is conducted to show that the presented method is potentially powerful to simulate real particulate flows with large number of moving particles.     

ANALYSIS OF WATER QUALITY IN SHALLOW LAKES WITH A TWO-DIMENSIONAL FLOW-SEDIMENT MODEL

The governing equation for sediment pollutions was derived based on the turbulent diffusion of pollutants in shallow lakes. Coupled with shallow water equations, a depth-averaged 2-D flow and water quality model was developed. By means of the conservation law, a proposed differential equation for the change of sediment pollutants was linked to the 2-D equations. Under the framework of the finite volume method, the Osher approximate Riemann solver was employed to solve the equations. An analytical resolution was used to examine the model capabilities. Simulated results matched the exact solutions especially well. As an example, the simulation of CODMn in the Wuli Lake, a part of the Taihu lake, was conducted, which led to reasonable results. This study provides a new approach and a practical tool for the simulation of flow and water quality in shallow lakes.     

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.     

THREE-DIMENSIONAL SIMULATION OF MEANDERING RIVER BASED ON 3-D RNG κ-ε TURBULENCE MODEL

A 3-D numerical model for calculating flow in non-curvilinear coordinates was established in this article. The flow was simulated by solving the full Reynolds-averaged Navier-Stokes equations with the RNG κ-ε turbulence model. In the horizontal x-y-plane, a boundary-fitted curvilinear co-ordinate system was adopted, while in the vertical direction, a σ co-ordinate transformation was used to represent the free surface and bed topography. The water level was determined by solving the 2-D Poisson equation derived from 2-D depth averaged momentum equations. The finite-volume method was used to discretize the equations and the SIMPLEC algorithm was applied to acquire the coupling of velocity and pressure. This model was applied to simulate the meandering channels and natural rivers, and the water levels and the velocities for all sections were given. By contrasting and analyzing, the agreement with measurements is generally good. The feasibility studies of simulating flow of the natural fiver have been conducted to demonstrate its applicability to hydraulic engineering research.     

NUMERICAL STUDY OF SQUARE-STERN SHIP WAVE IN SHALLOW WATER

This paper employed shallow water equations with moving pressure to calculate water waves generated by a square-stern ship in shallow water. The moving ship is considered as moving pressure on free surface. The finite element method with moving grids is used to solve the shallow water equations based on wave equation model [3]. A non-reflection boundary condition [5]is imposed on open boundaries surrounding the ship. 3-D surface elevations, depth-averaged horizontal velocities are presented. The numerical solutions are physically reasonable. It is found that wave resistance coefficients, draftchange and pitch angle vary rapidly in neighborhood of critical flow (Fh=u/ gh= 0. 9 -1. 1). The numerical results also indicate that the wave resistance coefficients, draft change and pitch angle of square-stern ship are larger than those of sharp-stern ship with the same hull structure at the same speed.     

Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method

A computationally efficient hybrid finite-volume/finite-difference method is proposed for the numerical solution of SaintVenant equations in one-dimensional open channel flows. The method adopts a mass-conservative finite volume discretization for the continuity equation and a semi-implicit finite difference discretization for the dynamic-wave momentum equation. The spatial discretization of the convective flux term in the momentum equation employs an upwind scheme and the water-surface gradient term is discretized using three different schemes. The performance of the numerical method is investigated in terms of efficiency and accuracy using various examples, including steady flow over a bump, dam-break flow over wet and dry downstream channels, wetting and drying in a parabolic bowl, and dam-break floods in laboratory physical models. Numerical solutions from the hybrid method are compared with solutions from a finite volume method along with analytic solutions or experimental measurements. Comparisons demonstrates that the hybrid method is efficient, accurate, and robust in modeling various flow scenarios, including subcritical, supercritical, and transcritical flows. In this method, the QUICK scheme for the surface slope discretization is more accurate and less diffusive than the center difference and the weighted average schemes.     

CHEBYSHEV FINITE SPECTRAL METHOD FOR 2-D EXTENDED BOUSSINESQ EQUATIONS

In this article,an accurate Chebyshev finite spectral method for the 2-D extended Boussinesq equations is proposed.The method combines the advantages of both the finite difference and spectral methods.The Adams-Bashforth predictor and the fourth-order Adams-Moulton corrector are adopted for the numerical solution of the governing differential equations.An efficient wave absorption strategy is also proposed to effectively absorb waves at outgoing wave boundaries and reflected waves from the interior of the computational domain due to barriers and bottom slopes at the incident wave boundary to avoid contamination of the specified incident wave conditions.The proposed method is verified by a case where experimental data are available for comparison for both regular and irregular waves.The case is wave diffraction over a shoal reported by Vincent and Briggs.Numerical results agree very well with the corresponding experimental data.     

NUMERICAL SIMULATION OF HYDRODYNAMIC CHARACTERISTICS ON AN ARC CROWN WALL USING VOLUME OF FLUID METHOD BASED ON BFC

In the present study, a new algorithm based on the Volume Of Fluid (VOF) method is developed to simulate the hydrodynamic characteristics on an arc crown wall. Structured grids are generated by the coordinate transform method in an arbitrary complex region. The Navier-Stokes equations for two-dimensional incompressible viscous flows are discretized in the Body Fitted Coordinate (BFC) system. The transformed SIMPLE algorithm is proposed to modify the pressure-velocity field and a transformed VOF method is used to trace the free surface. Hydrodynamic characteristics on an arc crown wall are obtained by the improved numerical model based on the BFC system (BFC model). The velocity field, the pressure field and the time profiles of the water surface near the arc crown wall obtained by using the BFC model and the Cartesian model are compared. The BFC model is verified by experimental results.     

A FINITE ELEMENT SIMULATION OF NEW YORK BIGHT TIDES FROM A WAVE EQUATION

A finite element model is used to simulate the tidal circulations in the New YorkBight.In this simulation a generalized wave continuity equation coupled with the primitive mo-mentum equations is used to produce a stable and accurate algorithm.The simulation is carriedout for 30 days to allow for a direct comparison with field measurement.The computed resultsagree well with the observed data.     

A TVD SCHEME FOR INCOMPRESSIBLE FLOW COUPLED WITH DIFFERENT TURBULENCE MODLES ON A GROUND-MOUNTED SQUARE-RIB FLOW

A finite-difference Total Variation Diminishing (TVD) numerical simulation model for coupling the Reynolds Averaged Navier-Stokes (RANS) equations, pressure-relative continuity equation and various k-εturbulence models was developed to solve the incompressible flow based on the pseudo-compressibility method. The hyperbolicity of all these equations was studied and the discretization of the fully coupling equations with all the primal variables and source terms were made in this article. Numerical simulation for modeling the flow around a ground-mounted square rib was implemented and validated by comparing with the published wind tunnel experimental data. It is shown that such a numerical simulation method with a proper turbulence model has a very good accuracy to simulate the flow around a surface-mounted rib. It is concluded that the Renormalization Group (RNG) and Chen-Kim k-εturbulence models have much better ability to predict the characteristics of the vortex structure and flow separation than the standard k-εmodel.     

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations

In this paper, the multi-scale Runge-Kutta_Galerkin method is developed for solving the evolution equations, with the spatial variables of the equations being discretized by the multi-scale Galerkin method based on the multi-scale orthogonal bases in 0(,)mH a b and then the classical fourth order explicit Runge-Kutta method being applied to solve the resulting initial problem of the ordinary differential equations for the coefficients of the approximate solution. The proposed numerical scheme is validated by applications to the Burgers equation(nonlinear convection-diffusion problem), the Kd V equation(single solitary and 2-solitary wave problems) and the Kd V-Burgers equation, where analytical solutions are available for estimating the errors. Numerical results show that using the algorithm we can solve these equations stably without the need for extra stabilization processes and obtain accurate solutions that agree very well with the corresponding exact solutions in all cases.     

A NEW NUMERICAL MODEL FOR THREE DIMENSIONAL SURFACE WATER FLOWS

A three dimensional numerical model based on the Reynolds equations is presented that can be used to predictthe surface water flow in open channels.The model uses a computational mesh that conforms to the free water sur-face and the bottom of the channel so that the accuracy of boundary condition application,code complexity,and e-conomy could be enhanced.The k-ε turbulence model is used to estimate the eddy viscosity coefficient.Instead ofusing the“rigid-lid”approximation a 2-D equation derived from integrating the continuity equation over the totaldepth is adopted to determine the elevation of the free water surface.A new algorithm is presented based on theconventional SIMPLE procedure.The block correction technique is employed to enhance rate of convergence.The model presented is applied to a bottom discharge into a rectangular straight channel for three dimensionalphenomena to obtain the free water surface configuration,velocities and pressure.The computed results are ingood agreement with the previous experimental values.     

Lattice Boltzmann model for shallow water in curvilinear coordinate grid

In this study, a multi-relaxation time lattice Boltzmann model for shallow water in a curvilinear coordinate grid is developed using the generalized form of the interpolation supplemented lattice Boltzmann method. The Taylor-Colette flow tests show that the proposed model enjoys a second order accuracy in space. The proposed model is applied to three types of meandering channels with 180°, 90° and 60° consecutive bends. The numerical results demonstrate that the simulated results agree well with previous computational and experimental data. In addition, the model can achieve the acceptable accuracy in terms of the water depth and the depth-averaged velocities for shallow water flows in curved and meandering channels over a wide range of bend angles.     

MODELING DAM-BREAK FLOOD OVER NATURAL RIVERS USING DISCONTINUOUS GALERKIN METHOD

A well-balanced numerical model is presented for two-dimensional, depth-averaged, shallow water flows based on the Discontinuous Galerkin (DG) method. The model is applied to simulate dam-break flood in natural rivers with wet/dry bed and complex topography. To eliminate numerical imbalance, the pressure force and bed slope terms are combined in the shallow water flow equations. For partially wet/dry elements, a treatment of the source term that preserves the well-balanced property is presented. A treatment for modeling flow over initially dry bed is presented. Numerical results show that the time step used is related to the dry bed criterion. The intercell numerical flux in the DG method is computed by the Harten-Lax-van Contact (HLLC) approximate Riemann solver. A two-dimensional slope limiting procedure is employed to prevent spurious oscillation. The robustness and accuracy of the model are demonstrated through several test cases, including dam-break flow in a channel with three bumps, laboratory dam-break tests over a triangular bump and an L-shape bend, dam-break flood in the Paute River, and the Malpasset dam-break case. Numerical results show that the model is robust and accurate to simulate dam-break flood over natural rivers with complex geometry and wet/dry beds.     

A TVD Scheme for Incompressible Flow Coupled With Different Turbulence Modles on A Ground-Mounted Square-RIB Flow

A finite-difference Total Variation Diminishing (TVD) numerical simulation model for coupling the Reynolds Averaged Navier-Stokes (RANS) equations, pressure-relative continuity equation and various k-ε turbulence models was developed to solve the incompressible flow based on the pseudo-compressibility method. The hyperbolicity of all these equations was studied and the discretization of the fully coupling equations with all the primal variables and source terms were made in this article. Numerical simulation for modeling the flow around a ground-mounted square rib was implemented and validated by comparing with the published wind tunnel experimental data. It is shown that such a numerical simulation method with a proper turbulence model has a very good accuracy to simulate the flow around a surface-mounted rib. It is concluded that the Renormalization Group (RNG) and Chen-Kim k-ε turbulence models have much better ability to predict the characteristics of the vortex structure and flow separation than the standard k-ε model.     

Numerical study of the flow in the Yellow River with non-monotonous banks

A 2-D depth averaged RNG k- ε model is developed to simulate the flow in a typical reach of the Upper Yellow River with non-monotonic banks. In order to take account of the effect of the secondary flow in a bend, the momentum equations are modified by adding an additional source term. A comparison between the numerical simulation and the field measurements indicates that the improved 2-D depth averaged RNG k- ε model can improve the accuracy of the numerical simulation. An arc spline interpolation method is developed to interpolate the non-monotonic river banks. The method can also be reasonably applied for the 2-D interpolation of the river bed level. Through a comparison of the water surface gradients simulated in the seven bends of the studied reach, some analytical formulae are improved to reasonably calculate the longitudinal and transverse gradients in meandering river reaches. Furthermore, the positions of the maximum water depth and the maximum velocity in a typical bend are discussed.