AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

DIRECT NUMERICAL SIMULATION OF VIBRATIONINDUCED T-S WAVE ON BOUNDARY LAYER WALL

The spatial growth of the disturbance in the boundary layer is directly numerically simulated, and the receptivity of the Blasius basic flow to the local two-dimensional （2-D） sustainable micro-vibration is investigated. Results show that the disturbance velocity presents the sine vibration features with the change of time, and the vibration period is identical to the vibration of the local wall. The disturbance velocity presents the fluctuation feature downstream, and the streamwise wave length approximates to the results from the Orr-Sommerfeld equation （OSE）. The growth rate from direct numerical simulation（DNS） is a little greater than that from OSE, and their trends are almost consistent. Under the condition of Re= 2 800, the disturbance amplitude gradually grows in the given computational region with the period T=30. However, it firstly increases and then decreases with the period T= 20. The disturbance harmonic of the former is obviously larger than that of the latter. The maximum streamwise and vertical disturbance velocities from DNS do not fully coincide with those from OSE at the vicinity of the local vibration wall, but coincide well with the former when they travel downstream. The 2-D disturbance induced by the local micro-vibration represents the form of Tollmien-Schlichting （T-S） wave on the boundary layer.

Proteome analysis of round-headed and normal spermatozoa by 2-D fluorescence difference gel electrophoresis and mass spectrometry

Globozoospermia is a severe form of teratozoospermia characterized by round-headed spermatozoa with an absent acrosome, an aberrant nuclear membrane and midpiece defects. Globozoospermia is diagnosed by the presence of 100% round-headed spermatozoa on semen analysis, and patients with this condition are absolutely infertile. The objective of this study was to investigate the differences in protein expression between human round- headed and normal spermatozoa. Two-dimensional （2-D） fluorescence difference gel electrophoresis （DIGE） coupled with mass spectrometry （MS） was used in this study. Over 61 protein spots were analysed in each paired normal/round-headed comparison, using DIGE technology along with an internal standard. In total, 35 protein spots identified by tandem mass spectrometry （MS/MS） exhibited significant changes （paired t-test, P 〈 0.05） in the expression level between normal and round-headed spermatozoa. A total of nine proteins were found to be upregulated and 26 proteins were found to be downregulated in round-headed spermatozoa compared with normal spermatozoa. The differentially expressed proteins that we identified may have important roles in a variety of cellular processes and structures, including spermatogenesis, cell skeleton, metabolism and spermatozoa motility.

Comparison of Fully 3-D,Simplified 3-D and 2-D Numerical Simulations on a Circulating Fluidized Bed Boiler

Based on the fully three-dimensional(3-D)and two-dimensional(2-D)comprehensive CFD(Computational Fluid Dynamics)combustion models for a circulating fluidized bed boiler,a simplified 3-D computational domain considering the corrections of furnace side wall openings is proposed.It aims to compensate for the deficiencies of the large amount of computation in the fully 3-D model and improve the air and gas flow treatments at the openings in the simplified 2-D model.Three different computational domains,named as the fully 3-D model,simplified 3-D model and 2-D model,were implemented to perform a comparative CFD analysis in an ultra-supercritical circulating fluidized bed boiler including the hydrodynamics,penetration depth of secondary air,temperature and species distribution.The simulation results computed by the simplified3-D model yield better agreement with the fully 3-D simulation results than those of the 2-D model.The simplified 3-D model is recommended as an alternative computational domain for the conventional 2-D model in the numerical simulation of large-scale circulating fluidized bed boiler.

A 2-D Non-local Closure Model for Atmospheric Boundary Layer Simulations

In this paper a new approach for PBL simulation, the non-local closure scheme based on the transient turbulence theory has been used. It was set up as an alternative to local closure schemes which physical concept is reasonable and distinct. A 2-D non-local closure model was developed in order to study the PBL structure and simulatesome interesting atmospheric processes over non-ulliform underlying surface, especially under the convective and unique weather conditions, such as sea-land circulation and the TIBL structure. The modelled results show good agreement with field measurement.

Application of 2-D sediment model to fluctuating backwater area of Yangtze River

Based on the characteristics of backflow, a two-dimensional mathematical model of sediment movement was established. The complexity of the watercourse boundary at the confluence of the main stream and the tributary was dealt with using a boundary-fitting orthogonal coordinate system. The basic equation of the two-dimensional total sediment load model, the numerical calculation format, and key problems associated with using the orthogonal curvilinear coordinate system were discussed. Water and sediment flow in the Chongqing reach of the Yangtze River were simulated. The calculated water level, flow velocity distribution, amount of silting and scouring, and alluvial distribution are found to be in agreement with the measured data, which indicates that the numerical model and calculation method are reasonable. The model can be used for calculation of flow in a relatively complicated river network.

Numerical Simulation of the Effects on Low Temperature Drainage from LNG on the Ocean Water Environment

In order to research the distribution of thermal field of low temperature discharge from LNG in the ocean, further to formulate reasonable wastewater marine disposal scheme and provide scientific support to the protection of ocean environment, a two-dimensional hydrodynamic model was used to simulate the tide current field in the radial sand ridge group of shallow sea area of Jiangsu maritime space. This model was based on the tide current characteristics near the LNG project and the seasonal design of the typical hydrological conditions in different seasons. This model was used as the hydrodynamic condition that simulated the transport and diffusion for the heat quantity in the change of time and spatial distribution. Considering the comprehensive heat radiation coefficient of water surface, the 2-D water temperature model was established to predict the spatial distribution and temporal variation of the temperature field in the receiving waters. Finally, results showed that the low temperature drainage less affected in the ocean water environment.

2-D NUMERICAL SIMULATION OF RADIONUCLIDE TRANSPORT IN THE LOWER YANGTZE RIVER

The assessment of the radiological impact of the liquid discharges from nuclear power plants is a major issue for the envi- ronmental protection. In this study, a numerical model for the radionuclide transport in the aquatic environment is built, based on the hydrodynamic equations, including the complete set of Saint-Venant equations, the sediment transport equations, with consideration of several different particle sizes and the deposition and erosion of the suspended sediments, and the radionuclide transport equations The exchanges of radionuclides between water, suspended matter and bed sediments are described in terms of kinetic transfer coe- fficients. The model is used to simulate the transport of the radionuclides discharged from a planned nuclear power plant project to be sited along the lower Yangtze River. From the model results, one may see the detailed temporal-spatial evolution of the radio- nuclide contamination in the solution, in the suspended matter and in the bed sediments. The model can be used as a basic tool for studying the environmental impacts of the liquid discharges from nuclear facilities on a river system.

2-D NUMERICAL SIMULATION OF CRUSH BEDROCK RIVER

In this paper, the erosion-resisting coefficient was introduced to computebed deformation in a crush bedrock river. In the case of crush bedrock, there has been no propercontrol equation to describe bed stability, which leads to difficulty in calculation of the beddeformation with conventional methods. The data from field survey were used to give thee-rosion-resisting capability with an appropriate coefficient. After the determination oflongitudinal distribution expressed by polynomial regression and transversal distribution expressedby normal distribution function, the plane distribution of erosion-resisting coefficient in a crushbedrock river was obtained. With the computational results from a 2-D horizontal flow mathematicalmodel, the erosion-resisting coefficient and controlling condition of local stability were employedto compute the values of bed deformation when riverbed is stable. The above method was applied in acase study, and the computational results of flow and bed deformations are in good a-greement withphysical model test data.

2-D NUMERICAL SIMULATION OF FLOODING EFFECTS CAUSED BY SOUTH-TO-NORTH WATER TRANSFER PROJECT

Since the General Channel designed for the South-to-North Water Transfer Project in China has to cross many rivers and streams flowing from west to east, there are potentially serious effects additional flooding on the western side of the project alignment. Therefore, a 2-D numerical model for forecasting basin flood disasters was established and verified using historical flood data. The model was applied to researching the interaction between the proposed Project and flooding events for 5 streams in the Anyang River reach as a representative case study. Simulated results indicate that the model could correctly forecast the flood, submerged area and depths, and water surface elevations along the left side of the channel. The discharge capacity and location of hydraulic structures in the transfer canal alignment were analyzed. Then adjustments to the dimensions and positioning of proposed hydraulic structures were recommended at intersections, especially the addition of a channel to transfer flood water from one stream to another, which can effectively limit the sluice and protect the Anyang City from flooding.

NUMERICAL SIMULATION OF TWO-DIMENSIONAL OVERTOPPING AGAINST SEAWALLS ARMORED WITH ARTIFICIAL UNITS IN REGULAR WAVES

A new mathematical model for the overtopping against seawalls armored with artificial units in regular waves was established. The 2-D numerical wave flume, based on the Reynolds Averaged Navier-Stokes （RANS） equations and the standard κ-ε turbulence model, was developed to simulate the turbulent flows with the free surface, in which the Volume Of Fluid （VOF） method was used to handle the large deformation of the free surface and the relaxation approach of combined wave generation and absorbing was implemented. In order to consider the effects of energy dissipation due to the armors on a slope seawall, a porous media model was proposed and implemented in the numerical wave flume. A series of physical model experiments were carried out in the same condition of the numerical simulation to determine the drag coefficient in the porous media model in terms of the overtopping discharge. Compared the computational value of overtopping over the seawall with the experimental data, the values of the effective drag coefficient was calibrated for the layers of blocks at different locations along the seawalls.

全隐式差分法求解河道平面二维恒定水流运动方程

本文针对河道平面形状窄长的特点，采用全隐式差分法求解河道平面二维恒定水流运动方程，网格划分为曲线正交网络，本文同时提出了一种适合于复杂河道边界的曲线正交网格产生办法．实际计算表明，全隐式方法精度较高，且基本无条件稳定，在普通微机上运行。

基于三次插值函数算法的时间积分方案与二阶时空余差数值模式——以原始大气运动方程与理想全球模拟个例为例

从大气运动原始方程和欧拉算符出发,用泰勒级数展开,给出二阶时空微商余项预报方程。进而讨论用三次插值函数——双三次曲面拟合求上游点的准拉格朗日时间积分方案与相应的二阶时空余差数值模式——"双三次模式"。则双三次模式是通过实现各个大气物理量场的二阶可导,从而可对预报方程做空间非线性("三次")时间离散积分,成为"双三次曲面拟合——时间步积分——双三次曲面拟合——……"一种新算法数值模式。讨论双三次数值模式的数学基础:三次插值函数及其数值分析极性定律用于数值模式。指出:双三次模式和谱模式都具有数学"收敛性";而Coons双三次曲面具有对变量场拟合二阶可导"最优性";和Hermite双三次曲面片具有对网格变量场二阶可导运算"等价性"。又指出:有限差分模式的中央差近似斜率和曲率,分别是三次样条斜率和曲率作"三点平滑"。双三次模式适合采用原始大气运动方程,适合采用准拉格朗日时间积分方案,并给出一个理想全球模拟个例。因大气运动本质上是非线性的,理论上可按变量场双三次曲面曲率判断,以采用符合物理诠释的局域或单点平滑,以保持模式时间积分稳定性。且未来容易实现全球多重/时变套网格双三次数值模式。

Ekman层风速随高度分布方程的数值解试验

根据大气水平运动方程推导出Ekman层风速随高度分布方程,该分布可用二阶线性微分方程表示,用有限差分法求解该二阶线性微分方程的边值问题的数值解,并给出计算程序,输出结果。将结果与经典解析解比较,讨论了经典解的正确性和适用范围。

二维离散数据的数值微分问题的积分算子方法

二维离散数据的数值微分问题是典型的不适定问题。基于GROETSCH的思想,提出了利用积分算子方法来构造近似偏导数和方向导数的方法,并将该方法用于二维离散数据的数值微分问题,并给出了误差估计。数值实验表明该方法是有效的。

NUMERICAL SIMULATION AND ANALYSIS ON THE HEMODYNAMICS OF AN ELASTIC ANEURYSM

Intracranial aneurysms are pathological dilatations which endanger people＇s health. Hemodynamics is thought to be an important factor in the pathogenesis and treatment of aneurysms. To date, the bulk of investigations into hemodynamics have been conducted by making use of mathematically idealized models for rigid aneurysms and associated arteries. However the walls of aneurysms and associated arteries are elastic in vivo. This study shows the differences of the simulation between elastic and rigid wall models. The numerical simulation of elastic aneurysm model is made fi＇om a representative Digital Subtraction Angiography （DSA） image and calculated with CFD software to get the wall deformation and the velocity field. Then the results are analyzed. By comparing the simulation results of the two models from their velocity vectors and shear stress distribution, many differences can be noted. The main difference exists in the distribution of velocity magnitude at some sections, with one outlet having obviously off-center distribution for the elastic wall model. The currents of the distribution of wall shear stress along the wall of aneurysm simulated in rigid and elastic wall models were similar. But there were apparent differences between the two models on the values of wall shear stress especially at the neck of aneurysm. The off-center distribution of velocity magnitude affects the distribution of wall shear stress and the exchange of substance through the wall. The analysis demonstrated clearly that the results of 2-D elastic numerical simulation were in good agreement with the clinical and pathological practice. The results of this study play an important role in the formation, growth, rupture and prognosis of an aneurysm on clinic application.

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. The t9 semi-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.

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.

On a novel tracking differentiator design based on iterative learning in a moving window

Differential signals are key in control engineering as they anticipate future behavior of process variables and therefore are critical in formulating control laws such as proportional-integral-derivative(PID).The practical challenge,however,is to extract such signals from noisy measurements and this difficulty is addressed first by J.Han in the form of linear and nonlinear tracking differentiator(TD).While improvements were made,TD did not completely resolve the conflict between the noise sensitivity and the accuracy and timeliness of the differentiation.The two approaches proposed in this paper start with the basic linear TD,but apply iterative learning mechanism to the historical data in a moving window(MW),to form two new iterative learning tracking differentiators(IL-TD):one is a parallel IL-TD using an iterative ladder network structure which is implementable in analog circuits;the other a serial IL-TD which is implementable digitally on any computer platform.Both algorithms are validated in simulations which show that the proposed two IL-TDs have better tracking differentiation and de-noise performance compared to the existing linear TD.