Application of the finite analytic numerical method to a flowdependent variational data assimilation

An anisotropic diffusion filter can be used to model a flow-dependent background error covariance matrix,which can be achieved by solving the advection-diffusion equation.Because of the directionality of the advection term,the discrete method needs to be chosen very carefully.The finite analytic method is an alternative scheme to solve the advection-diffusion equation.As a combination of analytical and numerical methods,it not only has high calculation accuracy but also holds the characteristic of the auto upwind.To demonstrate its ability,the one-dimensional steady and unsteady advection-diffusion equation numerical examples are respectively solved by the finite analytic method.The more widely used upwind difference method is used as a control approach.The result indicates that the finite analytic method has higher accuracy than the upwind difference method.For the two-dimensional case,the finite analytic method still has a better performance.In the three-dimensional variational assimilation experiment,the finite analytic method can effectively improve analysis field accuracy,and its effect is significantly better than the upwind difference and the central difference method.Moreover,it is still a more effective solution method in the strong flow region where the advective-diffusion filter performs most prominently.

THE ANALYTICAL SOLUTIONS BASED ON THE CONCEPT OF FINITE ELEMENT METHODS

On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.

Similarity solutions of vertical plane wall plume based on finite analytic method

The turbulent flow of vertical plane wall plume with concentration variation was studied with the finite analytical method. The k-epsilon model with the effect of buoyancy on turbulent kinetic energy and its dissipation rate was adopted. There were similarity solutions in the uniform environment for the system of equations including the equation of continuity, the equation of momentum along the flow direction and concentration, and equations of k, epsilon. The finite analytic method was applied to obtain the similarity solution. The calculated data of velocity, relative density difference, the kinetic energy of turbulence and its dissipation rate distribution for vertical plane plumes are in good agreement with the experimental data at the turbulent Schmidt number equal to 1.0. The variations of their maximum value along the direction of main flow were also given. It shows that the present model is good, i.e., the effect of buoyancy on turbulent kinetic energy and its dissipation rate should be taken into account, and the finite analytic method is effective.

THE STABILITY AND CONVERGENCE OF THE FINITE ANALYTIC METHOD FOR THE NUMERICAL SOLUTION OF CONVECTIVE DIFFUSION EQUATION

In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.

HYBRID FINITE ANALYTIC SOLUTIONS OF SHALLOW WATER CIRCULATION

The hybrid finite analytic(HFA) method is a kind of numerical scheme in rectangular element. In order to simulate the shallow circulation in irregular bathymetry by HFA scheme, the model in sigma coordinate system was obtained. The model has been tested against three cases: 1) Wind induced circulation; 2) Density driven circulation and 3) Seiche oscillation. The results obtained in the present study compare well with those obtained from the corresponding analytical solutions under idealized for the above three cases. The hybrid finite analytic method and the circulation model in sigma coordinate system can be used calculate the flow and water quality in estuaries and coastal waters.

Numerical Simulation of Residual Circulation due to Bottom Roughness Variability Under Tidal Flows in A Semi-Enclosed Bay

Nowadays there are some chronic serious environmental problems, such as eutrophication, blue tide and so on, in a complicated coastal zone or a semi-enclosed bay, because the water exchanges between an inner bay and an outer sea is weak compared with the supply of contaminant. Under this situation, a method to improve the water quality by 3-dimensional small unsymmetrical structures has been proposed by Komatsu et al. In this paper, several numerical simulations of the tidal current and concentration for various arrangements of bottom roughness in a semi-enclosed model bay are carfled out with a depth-averaged 2-D numerical model. The model is solved by the hybrid finite analytic method with nonstaggered grid. And the SIMPLES algorithm with Rhie and Chow＇ s momentum interpolation technique is used for the simulation. The effect of Komatsu＇ s method for water purification is examined by numerical simulation. The result of numerical experiment indicates that it is possible to generate a new tidal residual current and to activate a tidal exchange by bottom roughness arrangement only.

Behavior of near-field dilution of thermal buoyant jet discharged horizontally in compound open-channel

The RNG κ-ε model considering the buoyancy effect, which is solved by the hybrid finite analytic method, is used to simulate the mixture of the horizontal round thermal buoyant jet in compound open channel flow. The mixing features near the spout and flowing characteristic of the secondary currents are studied by numerical simulation. Meanwhile, （1） the distribution of the measured isovels for stream-wise velocity, （2） secondary currents, （3） the distribution of the measured isovels for temperature of typical cross-section near the spout, were obtained by the three-dimensional Micro ADV and the temperature measuring device. Compared with experimental data, the RNG κ-ε model based on buoyancy effect can preferably simulate the jet which performs the bifurcation phenomenon, jet reattachment （Conada effect） and beach secondary currents phenomenon with the effect of ambient flow, buoyancy, and secondary currents of compound section and so on.

NUMERICAL SIMULATIONS OF THE VORTEX-SHEDDING IN FLOWS AROUND RECTANGULAR CYLINDERS

Flows around rectangular cylinders with a series of width-to-height ratios are calculated by means of the Improved Finite Analytic Method (IFAM) and the formation, development and shedding of vortices from the cylinders are simulated successfully. According to these results of time-dependent processes the physical phenomena in the flows are investigated in detail, and the discontinuity of Strouhal number is explained in the case of the width-to-height ratio equal to 3.0. The numer- ical solutions here show good agreement with the experimental results. In addition, based on several hundreds of the calculated flow patterns a moving picture is made by the computer image processing technology and recorded on a video tape, and then the vivid pictures of the physical process of vortex-shedding can be replayed later and analysed in detail.

A Layer-IntegratedModel of Solute Transport in HeterogeneousMedia

This study presents a numerical solution to the three-dimensional solute transport in heterogeneous media by using a layer-integrated approach.Omitting vertical spatial variation of soil and hydraulic properties within each layer,a threedimensional solute transport can be simplified as a quasi-three-dimensional solute transport which couples a horizontal two-dimensional simulation and a vertical onedimensional computation.The finite analytic numericalmethod was used to discretize the derived two-dimensional governing equation.A quadratic function was used to approximate the vertical one-dimensional concentration distribution in the layer to ensure the continuity of concentration and flux at the interface between the adjacent layers.By integration over each layer,a set of system of equations can be generated for a single column of vertical cells and solved numerically to give the vertical solute concentration profile.The solute concentration field was then obtained by solving all columns of vertical cells to achieve convergence with the iterative solution procedure.The proposed model was verified through examples from the published literatures including four verifications in terms of analytical and experimental cases.Comparison of simulation results indicates that the proposed model satisfies the solute concentration profiles obtained from experiments in time and space.

BEHAVIOR FOR NON-BUOYANT CIRCULAR SURFACE JETS IN COFLOWING STEAMS BY NUMERICAL SIMULATION

A three-dimensional mathematical model for calculating the steady flow of circular non-buoyant turbulent surface in a coflowing channel is presented in this paper. For closing the control equations, the k turbulence model is used in the mathematical model. The wall-function method is employed to treat the wall boundary condition at the bottom plane. The closed control equations are discretized by using Hybrid Finite Analytic Method (HFAM) on a non-uniform staggered grid system. The numerical results are obtained for different ratio of the jet velocity to the coflowing velocity. The comparison with experimental data shows good agreement.