Numerical simulation of three-dimensional flow around two circular cylinders in tandem arrangement

附录Numerical Filling Simulation of Injection MoldingUsing Three—Dimensional ModelAbstract：Most injection molded parts are three-dimensional, with complex geometrical configurations and thick／thin wall sections．A 3D simulation model will predict more accurately the filling process than a 2.5D mode1.This paper gives a mathematical model and numeric method based on 3D model，in which an equal-order velocity-pressure interpolation method is employed successfully．The relation between velocity and pressure is obtained from the discretized momentum equations in order to derive the pressure equation．A 3D control volume scheme is employed to track the flow front．The validity of the model has been tested through the an analysis of the flow in cavity．Key words：three dimension；equal-order interpolation；simulation；injection molding1 IntroductionDuring injection molding，the theological response of polymer melts is generally non-Newtonian and no isothermal with the position of the moving flow front．Because of these inherent factors，it is difficult to analyze the filling process．Therefore，simplifications usually are used．For example，in middle-plane technique and dual domain technique[1], because the most injection molded parts have the characteristic of being thin but generally of complex shape，the Hele-Shaw approximation [2] is used while an analyzing the flow, i.e.．The variations of velocity and pressure in the gapwise (thickness) dimension are neglected．So these two techniques are both 2.5D mold filling models，in which the filling of a mold cavity becomes a 2D problem in flow direction and a 1D problem in thickness direction．However, because of the us e of the Hele-Shaw approximation，the information that 2.5D models can generate is limited and incomplete．The variation in the gapwise (thickness) dimension of the physical quantities with the exception of the temperature，which is solved by finite difference method，is neglected．With the development of molding techniques，molded华东交通大学理工学院毕业设计（论文）parts will have more and more complex geometry and the difference in the thickness will be more and more notable，so the change in the gapwise (thickness) dimension of the physical quantities can not be neglected．In addition，the flow simulated looks unrealistic in as much as the melt polymer flows only on surfaces of cavity, which appears more obvious when the flow simulation is displayed in a mould cavity．3D simulation model has been a research direction and hot spot in the scope of simulation for plastic injection molding．In 3D simulation model，velocity in the gapwise (thickness) dimension is not neglected and the pressure varies in the direction of part thickness，and 3 D finite elements are used to discretize the part geometry．After calculating，complete data are obtained(not only surface data but also internal data are obtained)．Therefore, a 3D simulation model should be able to generate complementary and more detailed information related to the flow characteristics and stress distributions in thin molded parts than the one obtained when using a 2.5D model(based on the Hele-Shaw approximation)．On the other hand，a 3D model will predict more accurately the characteristics of molded parts having thick walled sections such as encountered in gas assisted injection molding．Several flow behaviors at the flow front．such as “fountain flow”．which 2.5D model cannot predict, can be predicted by 3D mode1. Meanwhile, the flow simulation looks more realistic inasmuch as the overall an analysis result is directly displayed in 3D part geometry or transparent mould cavity．This Paper presents a 3 D finite element model to deal with the three—dimensional flow, which employs an equa1-order velocity-pressure formulation method [3,4]．The relation between velocity and pressure is obtained from the discretized momentum equations, then substituted into the continuity equation to derive pressure equation．A 3D control volume scheme is employed to track the flow front．The validity of the model has been tested through the analysis of the flow in cavity．2 Governing EquationsThe pressure of melt is not very big during filling the cavity, in addition，reasonable mold structure can avoid over big pressure，so the melt is considered incompressible．Inertia and gravitation are neglected as compared to the viscous force．With the above approximation，the governing equations，expressed in cartesian coordinates，are as following：Momentum equationsContinuity equationEnergy equationwhere, x，y，z are three dimensional coordinates and u, v，w are the velocity component in the x, y, z directions．P，T，ρandη denote pressure，temperature, density and viscosty respectively．Cross viscosity model has been used for the simulations：where，n,γ，r are non-Newtonian exponent，shear rate and material constant respectively．Because there is no notable change in the scope of temperature of the melt polymer during filling，Anhenius model[5] for η0 is employed as following：where B，Tb，β are material constants.3 Numerical Simulation Method3.1 Velocity-Pressure RelationIn a 3D model，since the change of the physical quantities are not neglected in the gapwise (thickness) dimension，the momentum equations are much more complex than those in a 2.5D mode1．It is impossible to obtain the velocity—pressure relation by integrating the momentum华东交通大学理工学院毕业设计（论文）equations in the gapwise dimension，which is done in a 2.5D model. The momentum equations must be first discretized，and then the relation between velocity and pressure is derived from it. In this paper, the momentum equations are discretized using Galerkin’s method with bilinear velocity-pressure formulation．The element equations are assembled in the conventional manner to form the discretized global momentum equations and the velocity may be expressed as followingwherethe nodal pressure coefficients are defined aswhere represent global velocity coefficient matrices in the direction of x, y, z coordinate respectively. denote the nodal pressure coefficients thedirection of x, y, z coordinate respectively. The nodal values for are obtained byassembling the element-by-element contributions in the conventional manner. N，is element interpolation and i means global node number and j , is for a node, the amount of the nodes around it.3.2 Pressure EquationSubstitution of the velocity expressions (2) into discretized continuity equation, which is discretized using Galerkin method，yields element equation for pressure:The element pressure equations are assembled the conventional manner to form the global pressure equations．3.3 Boundary ConditionsIn cavity wall, the no- slip boundary conditions are employed, e.g.On an inlet boundary,3.4 Velocity UpdateAfter the pressure field has been obtained，the velocity values are updated using new pressure field because the velocity field obtained by solving momentum equations does not satisfy continuity equation．The velocities are updated using the following relationsThe overall procedure for fluid flow calculations is relaxation iterative，as shown in Fig.l and the calculation is stable without pressure oscillation．3.5 The Tracing of the Flow FrontsThe flow of fluid in the cavity is unsteady and the position of the flow fronts values with time．Like in 2.5D model, in this paper, the control volume method is employed to trace the position of the flow fronts after the FAN(Flow Analysis Network)[6]. But 3D control volume is a special volume and more complex than the 2D control volume．It is required that 3D control volumes of all nodes fill the part cavity without gap and hollow space. Two 3D control volumes are shown in Fig．2．华东交通大学理工学院毕业设计（论文）4 Results and DiscussionThe test cavity and dimensions are shown in Fig.3(a)．The selected material is ABS780 from Kumbo. The pa rametric constants corresponding to then, γ,B, Tb and β of the five-constant Cross-type Viscosity model are 0．2638, 4．514 ×le4 Pa, 1.3198043×le-7 Pa *S, 1．12236 ×1e4K，0．000 Pa-1．Injection temperature is 45℃，mould temperature is 250℃, injection flow rate is 44．82 cu. cm／s. The meshed 3D model of cavity is shown in Fig. 3(b).“Fountain flow” is a typical flow phenomenon during filling．When the fluid is injected into a relatively colder mould，solid layer is formed in the cavity walls because of the diffusion cooling，so the shear near the walls takes place and is zero in the middle of cavity, and the fluid near the walls deflects to move toward the walls．The fluid near the center moves faster than the average across the thickness an d catches up with the front so the shape of the flow front is round like the fountain．The round shape of the flow front of the example in several filling times predicted by present 3D model and shown in Fig．4(a)，conforms to the theory and experiments．Contrarily, the shape of the flow front predicted by 2.5D model and shown in Fig．4(b) do not reveal the“Fountain flow”．The flow front comparison at the filling stage is illustrated in Fig．5．It shows that the predicted results based on present 3D model agree well with that based on Moldflow 3D mode1．The gate pressure is illustrated in Fig．6，compared with the prediction of Moldflow 3D model．It shows that the predicted gate pressure of present 3D model is mainly in agreement with that based on Moldflow 3D mode1．The major reason for this deviation is difference in dealing with the model an d material parameters．华东交通大学理工学院毕业设计（论文）5 ConclusionsA theoretical model and numerical scheme to simulate the filling stage based on a 3D finite element model are presented．A cavity has been employed as example to test the validity. 3D numeral simulation of the filling stage in injection moulding is a development direction in the scope of simulation for plastic injection molding in the future．The long time cost is at present a problem for 3D filling simulation，but with the development of computer hardware and improvement in simulation technique，the 3D technique will be applied widely．华东交通大学理工学院毕业设计（论文）三维注射成型流动模拟的研究摘要:大多数注射成型制品都是具有复杂的几何轮廓和厚壁或薄壁的制品。

基于多尺度方法的1∶3共振双Hopf分岔分析王万永;陈丽娟;郭静【摘要】利用改进的多尺度方法对一个电路振子模型1∶3共振附近的动力学行为进行了研究。

应用该方法得到了系统的复振幅方程，进而得到一个振幅与相位解耦的三维实振幅系统，通过分析实振幅方程的平衡点个数及其稳定性，将系统共振点附近的动力学行为进行分类，发现了双稳态等动力学现象，数值模拟验证了理论结果的正确性。

%The dynamical behavior near a 1∶3 resonance of an electric oscillator was investigated. By using the method of multiple scale, the complex amplitude equations of the system were obtained. Then a three dimension real amplitude system in which the amplitudes decouple from the phases was given. Ana-lyzing the number of equilibrium and its stability of the real amplitude equation, the dynamical behavior around the resonant point was classified. Some interesting dynamical phenomenon were found, for exam-ple,the bistability. Numerical simulations for justifying the theoretical analysis were also provided.【期刊名称】《郑州大学学报（理学版）》【年(卷),期】2016(048)003【总页数】5页(P23-27)【关键词】电路振子;1∶3共振;多尺度方法;分岔【作者】王万永;陈丽娟;郭静【作者单位】河南工程学院理学院河南郑州451191;河南工程学院理学院河南郑州451191;郑州铁路职业技术学院公共教学部河南郑州450052【正文语种】中文【中图分类】O175.1在非线性动力学的研究中，内共振由于能够反应系统线性模态之间的相互作用，有着非常重要的研究价值.文献[1]通过研究一个两端固支屈曲梁模型的内共振，构建了该模型在1∶1和1∶3内共振情形下的非线性模态.文献[2]研究了一个悬索模型的1∶2内共振，并讨论了三次非线性和高阶修正项对系统解的影响.文献[3]研究了一个极限环振子系统发生的1∶3共振双Hopf分岔，并研究了非线性对共振附近动力学行为的影响.文献[4]通过利用3∶1内共振的性质设计了一个非线性振动吸振器.文献[5]研究了内共振条件下风力发电机风轮叶片的空气动力学行为.在内共振和双Hopf分岔的研究中，常用的方法有中心流形和规范型方法、多尺度方法、摄动增量法、Liapunov-Schmidt约化和奇异摄动法.这些方法都存在一些问题，例如中心流形方法计算过程复杂，奇异性理论更加数学化，晦涩难懂，而多尺度方法得到的强共振的实振幅方程中，平衡点是非孤立的平衡点[6]，因而使稳定性分析和分岔分析无法进行.在本文的研究中，将应用一种改进的多尺度方法，把1∶3共振的规范型化为一个三维的实振幅系统，进而可以研究系统在共振点附近的动力学行为.本文以一个电路振子模型为例，利用改进的多尺度方法研究其1∶3共振点附近的动力学行为.其电路示意图如图1所示[7].其数学模型为[7]：其中：x1=v1，x2=i1，x3=v2，x4=i2是状态变量；η1=1/C1，η2=R，η3=1/L1，ρ1=1/C2，ρ2=1/L2是参数；α1、α2、α3是辅助参数.非线性电路模型的动力学行为是非线性动力学研究的重要内容之一.目前已有不少的文献从实验和理论方面对其进行了研究[8-12]，并发现了次谐波振荡、周期解、概周期解、分岔以及混沌等大量的非线性现象[11].本文将应用改进的多尺度方法对该电路系统的1∶3共振进行研究，计算其振幅方程并分析共振点附近的动力学行为.系统(1)在其唯一平衡点(0，0，0，0)处的线性化系统为,其特征方程为λ4+(-α1η1+η2ρ2)λ3+(η1η3+η1ρ2-α1η1η2ρ2+ρ1ρ2)λ3+(η1η2η3ρ2-α1η1ρ1ρ2)λ+η1η3ρ1ρ2=0.为了研究该系统1∶3共振点附近的动力学行为，设其特征方程有两对纯虚根λ1,3=±iω1和λ2,4=±iω2，其中ω1∶ω2=1∶3.可以求得当,时，特征方程(2)有两对纯虚根和.为了得到1∶3共振的规范型方程,将应用改进的多尺度方法对系统(1)进行分析.首先按照如下形式摄动参数设，则系统(1)可写为其多尺度形式的解具有如下形式将式(3)、(5)带入式(4),并对式(4)的右端进行Taylor展开,令两端ε的各次幂的系数相等,可得方程(6)的解具有如下形式其中：Aj(j=1,2)是复振幅，为时间尺度T2的函数；p1和p2是相应于特征值iω1和iω2的右特征向量；c.c. 表示前面各项的复共轭.将式(9)代入式(7),可求得式(7)的解为其中zij是复系数.将式(9)、(10)代入式(8),令长期项的系数为零,可得到A1和A2关于时间尺度T2导数的两个方程.应用左特征向量消去D2A1和D2A2的系数并吸收参数ε[13],可得Cijk和Ciμ με是复系数.在式(11)中，A1和A2为复振幅，为了将式(11)转化为实数振幅方程，通常将A1和A2设为极坐标形式.但是，在强共振条件下，如果将A1和A2设为极坐标形式，将会得到一个实振幅与相位变量耦合的三维系统，其平衡点将是非孤立的平衡点，平衡点的稳定性将无法研究.为了避免这种情况，将复振幅A1和A2设为一种混合形式(极坐标-笛卡尔形式)[13]，将式(12)代入式(11),分离其实部和虚部,可得到一个振幅与相位解耦的三维实振幅方程，如下:0.210 018uv2-0.532 248v3+0.080 357 1uη1ε-0.139 382vη1ε-0.21967uη2ε+0.168 86vη2ε+ 0.258 519 u η3ε+1.345 23vη3ε,0.210 018u2v+0.532 248uv2-0.210 018v3+0.139 382uη1ε+0.080 3571vη1ε-0.168 86uη2ε-0.219 67vη2ε-1.345 23uη3ε+0.258 519vη3ε.若设,则相应于原系统的状态变量x的Hopf分岔是振幅变量a1、a2的静态分岔. 由前面的分析可知1∶3共振的振幅方程是由3个变量组成的三维系统，并且含有3个分岔参数.为了分析共振点(η1c，η2c，η3c)附近的动力学行为，可以固定其中一个分岔参数，分析系统在二维参数平面上共振点附近的动力学行为.为此，固定参数η3，在η1-η2平面内对系统的动力学行为进行分类.根据实振幅方程的平衡点个数及每个平衡点稳定性的不同, 将平面η1-η2分为6个不同的区域，如图2所示.在Ⅰ区中，其平凡平衡点E0(0,0)是稳定的平衡点，对应于原系统的原点.当参数进入Ⅱ区，一个稳定的单模态平衡点E1(a10,0)出现，而平凡平衡点E0(0,0)变为不稳定的平衡点.当参数进入Ⅲ区，一个不稳定的平衡点E2(0,a20)出现，而平衡点E1(a10,0)保持其稳定性，平衡点E0(0,0)仍然是不稳定的.在Ⅳ区，一个新的不稳定的双模态平衡点E3(a12,a22)产生，而平衡点E1(a10,0)和E2(0,a20)是稳定的平衡点.在Ⅴ区，双模态平衡点E3(a12,a22)消失，平衡点E1(a10,0)失稳，平衡点E2(0,a20)仍然是稳定的.在Ⅵ区，平衡点E2(0,a20)保持稳定性，平衡点E1(a10,0)消失.其中单模态平衡点E1(a10,0)和E2(0,a20)分别相应于原系统频率为ω1和ω2的周期解，双模态平衡点E3(a12,a22)则相应于原系统的一个概周期解.为了验证理论分析的正确性，对原系统进行数值模拟，模拟的结果如图3～图8所示.可以发现，当参数在共振点附近变化时，系统出现两个不同频率的周期解，其频率比值接近1∶3.同时在分类图的Ⅳ区，两个不同频率的周期解同时出现，系统出现双稳态现象.本文研究了一个电路振子模型中发生的1∶3共振双Hopf分岔，通过应用改进的多尺度方法得到了该1∶3共振的规范型方程，进而分析其共振点附近的动力学行为，发现了周期解、双稳态等动力学现象，并通过数值模拟验证了结果的正确性.本文在揭示电路振子系统动力学现象的同时，应用了一种研究1∶3共振的新方法，该方法通过应用多尺度方法的过程，并将1∶3共振的复振幅设为一种混合形式，可以得到1∶3共振实振幅系统，从而能够研究共振点附近的动力学行为.【相关文献】[1] LACARBONARA W，REGA G，NAYFEH A H．Resonant non-linear normal modes．Part I：analytical treatment for structural one-dimensional systems [J]．Int JNon-linear Mech,2003，38(6)：851-872．[2] LEE C L， PERKINS N C．Nonlinear oscillations of suspended cables containing atwo-to-one internal resonance [J]．Nonlinear Dyn，1992，3(6)：465-490.[3] 王万永，陈丽娟．非线性时滞反馈对共振附近动力学行为的影响 [J]．信阳师范学院学报(自然科学版)，2014，27(1)：15-18．[4] JI J C， ZHANG N．Design of a nonlinear vibration absorber using three-to-one internal resonances [J]．Mech Syst Signal Processing，2014，42(1/2)： 236-246．[5] LI L,LI Y H,LIU Q K,et al. 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第32卷第3期 岩 土 力 学 V ol.32 No.3 2011年3月 Rock and Soil Mechanics Mar. 2011收稿日期：2009-12-27基金项目：国家自然科学基金项目(No. 50839004，No.51079110 )；教育部新世纪优秀人才支持计划项目（No. NCET -07-0632）。

第一作者简介：姜清辉，男，1972年生，博士，教授，主要从事岩土力学数值计算方法与岩土工程稳定分析方面的教学与研究工作。

E-mail ：jqh1972@文章编号：1000－7598 (2011) 03－879－06有自由面渗流分析的三维数值流形方法姜清辉1, 2，邓书申2，周创兵2（1. 武汉大学 土木建筑工程学院武汉 430072；2. 武汉大学 水资源与水电工程科学国家重点实验室，武汉430072）摘 要：提出了求解有自由面渗流问题的三维数值流形方法，通过构造任意形状流形单元的水头函数，推导了流形单元的渗透矩阵和无压渗流分析的总体控制方程，并给出了自由面的迭代求解策略和渗透体积力的计算方法。

典型算例的数值分析表明，该方法采用数学网格覆盖整个材料区域，在自由面的迭代求解过程中数学网格保持不变，只考虑自由面以下渗流区的介质，只对自由面以下的流形单元形成总体渗透矩阵，具有精度高、收敛速度快、编程简单等优点，而且能够通过单纯形积分精确计算被自由面穿越单元的渗透作用力，因此，特别适用于有自由面渗流问题的模拟。

关 键 词：三维渗流；数值流形方法；自由面；数学网格；流形单元 中图分类号：O342 文献标识码：AThree-dimensional numerical manifold method for seepageproblems with free surfacesJIANG Qing-hui 1,2 , DENG Shu-shen 2 , ZHOU Chuang-bing 2(1.School of Civil and Architectural Engineering, Wuhan University, Wuhan, 430072, China;2.State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China)Abstract: A three-dimensional numerical manifold method for seepage problems with free surfaces is proposed. The hydraulic potential functions for arbitrarily shaped manifold element are constructed and the element conductivity matrix is derived in detail. The global governing equations for unconfined seepage analysis are established by minimizing the flow dissipation energy. The proposed method employs the tetrahedral mathematical meshes to cover the whole material volume. In the process of iterative solving for locating the free surface, the numerical manifold method can strictly realize the seepage analysis of the saturated domain on the condition of mathematical meshes keeping unchanged. Furthermore, the seepage force acting on the transitional elements cut by the free surface can be accurately calculated by the manifold method. Therefore, the proposed method is featured in high accuracy, fast convergence rate and simple programming, especially applicable to simulate the unconfined seepage problem with free surface. Key words: three-dimensional seepage; numerical manifold method; free surface; mathematical mesh; manifold element1 引 言在采用传统的有限元方法求解无压渗流问题时，主要存在两类解法：调整网格法和固定网格法。

微孔端面机械密封间液膜的CFD数值模拟丁雪兴;王燕;佘志刚;毛亚军【摘要】Three-dimensional spatioal model was founded for the liquid membrane among micro-pore end surfaces by using Pro/E software. The model was latticed by using the Gambit software, a numerical simulation was made with Fluent software for the three-dimension flow field in the internal micro-scale gaps under a special condition of the micro-pores etched on one face of the seal. The distributions of pressure and velocity within the flow field and the liquid leakage were obtained. These three performances were resimulated in the case of different depth to diameter ratios ε. The influence of the depth to diameter ratio on the sealing performance of the end mechanical seals was analyzed in the condition of identical film thick ness. The result showed that when ε = 0. 1, the leakage would be minimal and an optimal sealing effectcould be obtained.%应用Pro/E软件建立微孔端面间液膜的三维立体模型,Gambit软件对模型进行划分网格,F1uent软件对微孔端面间特定工况下的内部微间隙三维流场进行数值模拟,得到流场的压力分布、速度分布以及泄漏量.改变微孔深径比再次模拟,得到不同参数下流场所对应的压力分布、速度分布及泄漏量,分析在同一液膜厚度情况下,微孔深径比对端面机械密封性能的影响.结果表明:当深径比ε=0.1时泄漏量最小,可获得最佳密封效果.【期刊名称】《兰州理工大学学报》【年(卷),期】2011(037)002【总页数】5页(P39-43)【关键词】机械密封;端面微孔;泄漏量;CFD【作者】丁雪兴;王燕;佘志刚;毛亚军【作者单位】兰州理工大学,石油化工学院,甘肃,兰州,730050;兰州理工大学,石油化工学院,甘肃,兰州,730050;兰州理工大学,石油化工学院,甘肃,兰州,730050;陕西延长石油油气勘探公司,陕西,延安,716000【正文语种】中文【中图分类】TB421994年，以色列教授 Etsion［1－2］提出微孔端面机械密封的概念.激光加工多孔端面机械密封是一种动压型机械密封，在密封环端面上加工有规则分布的球形微孔.每一个微孔像一个微动力轴承，当两密封面相对运动时，在孔的上方及其周围区域产生明显的动压效应，可大大降低密封端面间的摩擦扭矩［3］.目前，国内外学者大多利用有限差分法［3－5］，通过公式计算求解雷诺方程的方法研究微孔端面密封动静态特性，采用CFD软件方法进行数值模拟研究的较少.利用CFD软件进行建模，可以考虑任意孔型的结构型式［5］，并且可以更全面、准确、直观地反映微孔端面密封流体动特性［6－7］.随着计算流体力学（CFD）和计算机技术的发展，各种计算流体力学软件日趋成熟，使得对密封环端面微凹腔内流场直接进行数值模拟变成可能［8］.本文利用ANSYS-FLUENT软件对微孔端面间特定工况下的内部微间隙三维流场进行数值模拟，得到流场的压力分布、速度分布以及泄漏量.改变操作参数再次模拟，得到不同参数下流场的压力分布及泄漏量，并分析操作参数对微孔端面机械密封的影响.1 Fluent计算模型建立图1是具有微孔的密封副结构示意图，静环表面采用激光加工出球冠形微孔，微孔沿径向呈放射状分布，沿周向呈等间距分布，密封面不直接接触，密封面间形成一定密度的液膜，假定液膜压力沿液膜厚度方向不变化，密封流体黏度保持不变，并忽略密封曲率影响［4］.图1 具有微孔的密封副结构示意图Fig.1 Micro-pores seal pair structure diagram1.1 几何模型的建立1.1.1 建立几何模型计算模型的三维几何建模采用Pro／E软件，计算区域选定为密封转轴与其配合壁面间隙内的全三维空间，如图2b的一列径向孔栏.利用Pro／E建模后的液膜模型如图3所示，选取的是同一径向相邻的四元体模型.图2 具有半球形微孔的机械密封Fig.2 Mechanical seal with hemispherical micro-pores1.1.2 网格化分本文网格划分采用TGrid单元方案，并采用Gambit非结构化网格划分方法，对同一径向相邻四元体模型直接进行网格划分，如图4所示为计算区域网格.图3 利用Pro／E建模后的液膜模型.3 Liquid membrane model set up by using Pro／E modeling图4 计算区域网格Fig.4 Mesh in computational zone1.2 密封环工作的基本假设机械密封的流体润滑理论的主要内容是流体膜润滑，即由流体膜承载保持密封和润滑的成膜理论，其主要控制方程为雷诺（Reynolds）方程［9－11］.忽略一些对问题研究重点和预期的结果没有影响或影响很小的因素［8］，作者作出以下基本假设：1）忽略体积力的作用，如重力或磁力；2）与黏性力比较，可以忽略惯性力的影响，包括流体加速度的惯性力和流体膜的弯曲的离心力；3）在沿流体膜厚度方向上，不计压力的变化，因为膜厚一般为微米数量级，在膜厚范围内，事实上压力不可能发生明显的变化；4）密封流体介质为牛顿（Newton）流体，即剪切力正比于剪应变率；5）流体在摩擦界面上无滑动，即附着于界面上的流体质点的速度与界面上该点的速度相同；6）流体在密封面间的流动为层流，流体膜中不存在涡流和湍流；7）整个机械密封润滑系统的温度处处相等，因此不考虑润滑剂的黏度和密度随温度的变化；8）两密封面不接触，其间存在液膜，且液膜厚度在密封表面各处相等；9）流体为不可压缩流体，密度不随压力变化.10）不考虑流体的表面张力效应. 1.3 边界条件凹腔开在下面的静环上，故液膜下部有突起.内侧压力为大气压，外侧压力为密封介质压力，流体因为压力差从外径向内径流动，在孔栏的径向边界上，对应半径处的压力分别相等，且液膜压力沿周向的变化率在对应半径处相等［4］，除进、出口以外两侧为周期性边界条件，即本模型进出口分别采用压力入口pressure inlet 及压力出口pressure outlet边界条件；上、下表面采用壁面wall边界条件，上表面为旋转壁面，下表面为静止壁面，除进、出口以外两侧为周期性边界条件.压力的数值大小、壁面的运动形式以及速度值将在FLUENT中具体设定.密封环表面为标准壁面条件，采用速度无滑移条件.采用旋转参考坐标系来模拟动静环之间的相互运动.2 模型计算结果2.1 求解方法求解器选择分离的隐式求解器，压力差值格式为标准差值，压力速度耦合采用SIMPLEC算法.扩散项的离散格式采用中心差分格式，对流项的离散格式采用二阶迎风格式.本模型为含有旋转的流动，压力差值方式选择Presto.本模型采用的收敛准则为默认准则，即小于10－3.2.2 算例给定的密封环端面结构参数和工况参数为：内径ri＝10.8 mm，外径ro＝13.5 mm，环境压力（内压）pi＝pa＝0.101 3 MPa，介质压力（外压）po＝0.60795 MPa，流体黏度μ＝0.015 Pa·s，转速n＝3 000 r／min，微孔密度sp＝0.5，微孔半径rp＝50μm，液膜厚度h0＝3μm，微孔深径比分别选为ε＝0.08，0.10，0.20，0.30，微孔深度分别为hp1＝8μm，hp2＝10μm，hp3＝20μm，hp4＝30μm.分别对4组数据进行建模，网格划分，导入FLUENT进行计算，得到深径比不同时的压力分布和速度分布，如图5和图6所示.2.3 模拟结果分析2.3.1 不同深径比的压力沿径向的分布图7表示微孔深径比对无量纲平均动压力的影响.由图可知，微孔深径比对动压效应有较大的影响，深径比在0.1左右时，平均动压力最大，说明微孔深径比存在最佳值，据Etsion试验研究结果［12］表明，在深径比为0.07时，密封面的承载能力最大，两者比较可知，模拟结果与试验结果接近.2.3.2 不同深径比的速度沿径向的分布由图8可知，当ε＝0.1时，产生的流速最低.这与图7产生的压力结果吻合，说明在ε＝0.1时，存在最佳深径比.2.3.3 不同深径比的泄漏量曲线关系由FLUENT软件直接读出4组不同深径比的泄漏量，由Origin软件绘制曲线图，可得微孔深度对泄漏量的影响规律（见图9）.由图9可知，当液膜厚度h0＝3μm，密封压力和密封环转速不变的情况下，随着微孔深度的增加，泄漏量的变化经历先快速下降再慢速上升的过程.在深径比ε＝0.1，hp＝10 μm时，可得泄漏量值最小.2.4 模拟结果与文献结果对比图10为微孔深径比对无量纲平均动压力的影响.由图可知，微孔深径比对动压效应有较大的影响，深径比ε＝0.1左右时，无量纲平均动压力最大，说明微孔深径比存在最佳值.这与文献［13］的数值模拟结果吻合，因此，当深径比ε＝0.1时泄漏量最小，可获得最佳密封效果.图5 同一径向不同深径比四元体压力分布图（kPa）Fig.5 Diagram of pressure distribution in four-tropic body with different depth／diameter ratio in identical radial direction（kPa）图6 同一径向不同深径比四元体速度分布图（m／s）Fig.6 Diagram of velocity distribution in four-tropic body with different depth／diameter ratio on identical radius（m／s）图7 不同深径比的压力沿径向的分布Fig.7 Radial distribution of pressure for different depth／diameter ratio图8 不同深径比的速度沿径向的分布Fig.8 Radial distribution of velocity for different depth／diameter ratio图9 不同深径比与泄漏量的关系Fig.9 Leakage for different depth／diameter ratio图10 微孔深径比对无量纲平均动压力影响的结果对比Fig.10 Comparison of influence of micro-pores depth／diameter ration on mean dimensionlessdynamic pressure3 结论1）通过对4种不同深径比微孔端面密封所产生的压力、速度以及它们的泄漏量比较可知，当液膜厚度h0＝3μm时，通过FLUENT软件模拟的结果得到：当深径比ε＝0.1，微孔深度hp＝10μm时，泄漏量最小.2）本文只针对同一膜厚情况下，深径比与泄漏量之间的关系，此研究为今后对不同膜厚以及其他几何参数改变的研究提供了依据.致谢：本文得到兰州理工大学博士基金项目（BS05200901）资助，在此表示感谢. 参考文献：［1］ ETSION I，BURETEIN L.A model for mechanical seals with regular microsurface structure ［J］.Tribology Transactions，1996，39（3）：667-683.［2］ ETSION I，MICHEAL O.Enhancing sealing and dynamic performance with partially porous mechanical face seals［J］.Tribology Transactions，1994，37（4）：701-710.［3］于新奇，蔡仁良.激光加工的多孔端面机械密封的性能数值分析［J］.现代制造工程，2004（7）：66-68.［4］李国栋.激光加工多孔端面机械密封性能研究及结构优化［D］.兰州：兰州理工大学，2009.［5］丁雪兴，程香平，杜鹃.机械密封混合摩擦微极流体弹性润滑的数值模拟［J］.兰州理工大学学报，2008，34（4）：70-73.［6］侯煜.CFD环形间隙泄漏量及摩擦力的仿真计算［D］.太原：太原理工大学，2007.［7］叶建槐，刘占生.高低齿迷宫密封流场和泄露特性CFD研究［J］.汽轮机技术，2008（4）：81-84.［8］陈汇龙，翟晓.基于多重网格法和CFD的多孔端面机械密封数值分析比较［J］.润滑与密封，2009（10）：36-40.［9］王福军.计算流体动力学分析［M］.北京：清华大学出版社，2004. ［10］温诗铸，杨沛然.弹性流体动力润滑［M］.北京：清华大学出版社，1992. ［11］杨沛然.流体润滑数值分析［M］.北京：国防工业出版社，1998. ［12］ ETSION I，BURETEIN L.Proceeding of 15th International Conference on Fluid Seal［C］.London：Professional Engineering Publishing Limited，1997：3-10.［13］于新奇，蔡仁良.激光加工多孔端面机械密封的动压分析［J］.华东理工大学学报，2004，30（8）：481-484.。

三维解析仿真的英语作文Three-Dimensional Computational Modeling.Three-dimensional (3D) computational modeling is the process of creating a mathematical representation of a three-dimensional object. This representation can be used to simulate the behavior of the object under different conditions. 3D computational modeling is used in a wide variety of fields, including engineering, medicine, and manufacturing.In engineering, 3D computational modeling is used to simulate the behavior of structures and machines. This information can be used to design structures that are safe and efficient. In medicine, 3D computational modeling is used to simulate the behavior of organs and tissues. This information can be used to diagnose diseases and develop new treatments. In manufacturing, 3D computational modeling is used to simulate the behavior of products during the manufacturing process. This information can be used tooptimize the manufacturing process and reduce product defects.There are many different types of 3D computational modeling software available. The type of software used will depend on the specific application. Some of the most popular 3D computational modeling software programs include ANSYS, COMSOL, and Siemens NX.3D computational modeling is a powerful tool that can be used to simulate the behavior of objects in a variety of different fields. This information can be used to design safer and more efficient structures, diagnose and treat diseases, and optimize the manufacturing process.Benefits of 3D Computational Modeling.There are many benefits to using 3D computational modeling. Some of the most notable benefits include:Increased accuracy: 3D computational models are more accurate than traditional 2D models. This is because 3Dmodels can take into account the effects of all three dimensions of space.Reduced time and cost: 3D computational modeling can save time and cost by reducing the need for physical testing. Physical testing can be expensive and time-consuming, and it is not always possible to test all possible scenarios.Improved communication: 3D computational models can be used to communicate complex designs and concepts more easily. This can help to reduce errors and improve collaboration between different teams.Applications of 3D Computational Modeling.3D computational modeling is used in a wide variety of applications, including:Engineering: 3D computational modeling is used to simulate the behavior of structures and machines. This information can be used to design structures that are safeand efficient.Medicine: 3D computational modeling is used to simulate the behavior of organs and tissues. This information can be used to diagnose diseases and develop new treatments.Manufacturing: 3D computational modeling is used to simulate the behavior of products during the manufacturing process. This information can be used to optimize the manufacturing process and reduce product defects.Future of 3D Computational Modeling.The future of 3D computational modeling is bright. As computer hardware and software continue to improve, 3D computational models will become even more accurate and sophisticated. This will open up new possibilities for using 3D computational modeling in a wide variety of applications.One of the most exciting developments in 3Dcomputational modeling is the use of artificialintelligence (AI). AI can be used to automate the process of creating and running 3D computational models. This will make it easier for engineers, scientists, and other professionals to use 3D computational modeling in their work.Another exciting development in 3D computational modeling is the use of virtual reality (VR). VR can be used to create immersive 3D environments that allow users to interact with 3D computational models. This can make it easier to understand complex designs and concepts.3D computational modeling is a powerful tool that is transforming the way we design, build, and heal. As computer hardware and software continue to improve, 3D computational modeling will become even more powerful and versatile. This will open up new possibilities for using 3D computational modeling in a wide variety of applications.。

中文2900字附录Numerical Filling Simulation of Injection MoldingUsing Three—Dimensional ModelAbstract：Most injection molded parts are three-dimensional, with complex geometrical configurations and thick／thin wall sections．A 3D simulation model will predict more accurately the filling process than a 2.5D mode1.This paper gives a mathematical model and numeric method based on 3D model，in which an equal-order velocity-pressure interpolation method is employed successfully．The relation between velocity and pressure is obtained from the discretized momentum equations in order to derive the pressure equation．A 3D control volume scheme is employed to track the flow front．The validity of the model has been tested through the an analysis of the flow in cavity．Key words：three dimension；equal-order interpolation；simulation；injection molding1 IntroductionDuring injection molding，the theological response of polymer melts is generally non-Newtonian and no isothermal with the position of the moving flow front．Because of these inherent factors，it is difficult to analyze the filling process．Therefore，simplifications usually are used．For example，in middle-plane technique and dual domain technique[1], because the most injection molded parts have the characteristic of being thin but generally of complex shape，the Hele-Shaw approximation [2] is used while an analyzing the flow, i.e.．The variations of velocity and pressure in the gapwise (thickness) dimension are neglected．So these two techniques are both 2.5D mold filling models，in which the filling of a mold cavity becomes a 2D problem in flow direction and a 1D problem in thickness direction．However, because of the us e of the Hele-Shaw approximation，the information that 2.5D models can generate is limited and incomplete．The variation in the gapwise (thickness) dimension of the physical quantities with the exception of the temperature，which is solved byfinite difference method，is neglected．With the development of molding techniques，molded parts will have more and more complex geometry and the difference in the thickness will be more and more notable，so the change in the gapwise (thickness) dimension of the physical quantities can not be neglected．In addition，the flow simulated looks unrealistic in as much as the melt polymer flows only on surfaces of cavity, which appears more obvious when the flow simulation is displayed in a mould cavity．3D simulation model has been a research direction and hot spot in the scope of simulation for plastic injection molding．In 3D simulation model，velocity in the gapwise (thickness) dimension is not neglected and the pressure varies in the direction of part thickness，and 3 D finite elements are used to discretize the part geometry．After calculating，complete data are obtained(not only surface data but also internal data are obtained)．Therefore, a 3D simulation model should be able to generate complementary and more detailed information related to the flow characteristics and stress distributions in thin molded parts than the one obtained when using a 2.5D model(based on the Hele-Shaw approximation)．On the other hand，a 3D model will predict more accurately the characteristics of molded parts having thick walled sections such as encountered in gas assisted injection molding．Several flow behaviors at the flow front．such as “fountain flow”．which 2.5D model cannot predict, can be predicted by 3D mode1. Meanwhile, the flow simulation looks more realistic inasmuch as the overall an analysis result is directly displayed in 3D part geometry or transparent mould cavity．This Paper presents a 3 D finite element model to deal with the three—dimensional flow, which employs an equa1-order velocity-pressure formulation method [3,4]．The relation between velocity and pressure is obtained from the discretized momentum equations, then substituted into the continuity equation to derive pressure equation．A 3D control volume scheme is employed to track the flow front．The validity of the model has been tested through the analysis of the flow in cavity．2 Governing EquationsThe pressure of melt is not very big during filling the cavity, in addition，reasonable mold structure can avoid over big pressure，so the melt is considered incompressible．Inertia and gravitation are neglected as compared to the viscous force．With the above approximation，the governing equations，expressed in cartesian coordinates，are as following：Momentum equationsContinuity equationEnergy equationwhere, x，y，z are three dimensional coordinates and u, v，w are the velocity component in the x, y, z directions．P，T，ρandη denote pressure，temperature, density and viscosty respectively．Cross viscosity model has been used for the simulations：where，n,γ，r are non-Newtonian exponent，shear rate and material constant respectively．Because there is no notable change in the scope of temperature of the melt polymer during filling，Anhenius model[5] for η0 is employed as following：where B，Tb，β are material constants.3 Numerical Simulation Method3.1 Velocity-Pressure RelationIn a 3D model，since the change of the physical quantities are not neglected in the gapwise (thickness) dimension，the momentum equations are much more complex than those in a 2.5Dmode1．It is impossible to obtain the velocity—pressure relation by integrating the momentum equations in the gapwise dimension，which is done in a 2.5D model. The momentum equations must be first discretized，and then the relation between velocity and pressure is derived from it. In this paper, the momentum equations are discr etized using Galerkin’s method with bilinear velocity-pressure formulation．The element equations are assembled in the conventional manner to form the discretized global momentum equations and the velocity may be expressed as followingwherethe nodal pressure coefficients are defined aswhere represent global velocity coefficient matrices in the direction of x, y, z coordinate respectively. denote the nodal pressure coefficients thedirection of x, y, z coordinate respectively. The nodal values for are obtained byassembling the element-by-element contributions in the conventional manner. N，is element interpolation and i means global node number and j , is for a node, the amount of the nodes around it.3.2 Pressure EquationSubstitution of the velocity expressions (2) into discretized continuity equation, which is discretized using Galerkin method，yields element equation for pressure:The element pressure equations are assembled the conventional manner to form the global pressure equations．3.3 Boundary ConditionsIn cavity wall, the no- slip boundary conditions are employed, e.g.On an inlet boundary,3.4 Velocity UpdateAfter the pressure field has been obtained，the velocity values are updated using new pressure field because the velocity field obtained by solving momentum equations does not satisfy continuity equation．The velocities are updated using the following relationsThe overall procedure for fluid flow calculations is relaxation iterative，as shown in Fig.l and the calculation is stable without pressure oscillation．3.5 The Tracing of the Flow FrontsThe flow of fluid in the cavity is unsteady and the position of the flow fronts values with time．Like in 2.5D model, in this paper, the control volume method is employed to trace the position of the flow fronts after the FAN(Flow Analysis Network)[6]. But 3D control volume is a special volume and more complex than the 2D control volume．It is required that 3D control volumes of all nodes fill the part cavity without gap and hollow space. Two 3D control volumes are shown in Fig．2．4 Results and DiscussionThe test cavity and dimensions are shown in Fig.3(a)．The selected material is ABS780 from Kumbo. The pa rametric constants corresponding to then, γ,B, Tb and β of the five-constant Cross-type Viscosity model are 0．2638, 4．514 ×le4 Pa, 1.3198043×le-7 Pa *S, 1．12236 ×1e4K，0．000 Pa-1．Injection temperature is 45℃，mould temperature is 250℃, injection flow rate is 44．82 cu. cm／s. The meshed 3D model of cavity is shown in Fig. 3(b).“Fountain flow” is a typical flow phenomenon during filling．When the fluid is injected into a relatively colder mould，solid layer is formed in the cavity walls because of the diffusion cooling，so the shear near the walls takes place and is zero in the middle of cavity, and the fluid near the walls deflects to move toward the walls．The fluid near the center moves faster than the average across the thickness an d catches up with the front so the shape of the flow front is round like the fountain．The round shape of the flow front of the example in several filling times predicted by present 3D model and shown in Fig．4(a)，conforms to the theory and experiments．Contrarily, the shape of the flow front predicted by 2.5D model and shown in Fig．4(b) do not reveal the“Fountain flow”．The flow front comparison at the filling stage is illustrated in Fig．5．It shows that the predicted results based on present 3D model agree well with that based on Moldflow 3D mode1．The gate pressure is illustrated in Fig．6，compared with the prediction of Moldflow 3D model．It shows that the predicted gate pressure of present 3D model is mainly in agreement with that based on Moldflow 3D mode1．The major reason for this deviation is difference in dealing with the model an d material parameters．5 ConclusionsA theoretical model and numerical scheme to simulate the filling stage based on a 3D finite element model are presented．A cavity has been employed as example to test the validity. 3D numeral simulation of the filling stage in injection moulding is a development direction in the scope of simulation for plastic injection molding in the future．The long time cost is at present a problem for 3D filling simulation，but with the development of computer hardware andimprovement in simulation technique，the 3D technique will be applied widely．三维注射成型流动模拟的研究摘要:大多数注射成型制品都是具有复杂的几何轮廓和厚壁或薄壁的制品。

Three-dimension numerical simulation of discharge flow in a scroll air compressorSchool of energy and power engineeringAbstractScroll compressor is being recognized by industry as being high competitive with conventional compressors. Plenty of publications on this subject prove an interest of the researchers as well. Further increases in efficiency may be realized if the flow losses, particularly in the final compression and discharge region are reduced. Detailed understandings of the flow processes occurring in the discharge region are necessary to analysis and reduce the discharge flow losses, which become more serious with operation at large discharge. Due to the complexity of the processes, the only one way to get the results is solving the equations of continuity and momentum using the numerical method. During the past decade, a number of investigations have been conducted on the performance of the scroll compressor. However, relatively little information are available on the details of the fluid flow characteristics within the scroll compressor chamber. In the paper, in the light of the characteristics of a discharge process, reasonable simplification of actual physical model is made and the three-dimension quasi-steady turbulent flow numerical simulation is carried out to study the flow field in the discharge region in a scroll air compressor. Three dimensionaldistributions of velocity and pressure and typical flow patterns that exits in the discharge region are presented, which gives good understanding about the physical processes in the scroll air compressor.1.INTRODUCTIONScroll compressors are applied widely in the refrigeration, air conditioning and power field as being competitive advantages in terms of high efficiency, reduced part requirement, lower noise, and reduced vibration levels. Three exist various losses when a scroll compressor is running, such as moving resistance losses of the orbiting scroll and Oldham, friction losses and flow losses. The discharge flow is the main part of these losses (approximately 3 percent of the input power is consumed due to the flow losses), especially at large discharge . Understanding of the flow processes occurring in the discharge and the final compression region is necessary to reduce these flow losses, which become more pronounced with operation at increasing speed and large discharge. Therefore, three dimension numerical simulation of discharge flow in scroll air compressor with modified top profile is carried out. The important flow patterns that exist in the discharge and final compression region are presented. The analysis results supply the theory basis for finding the sources caused discharge losses and designing the discharge port of scroll compressor, particularly at large discharge.A scroll air compressor of discharge is studied in this paper. The topprofile is modified with symmetrical arcs and the discharge port is kidney-shape port. The basic parameters and modified parameters of scroll tips are shown in the table 1.Figure 1 shows the schematic region of the scroll compressor.Table 1: The basic parameters and modified parameters of scroll tipsFigure 1: Schematic of scroll compressor discharge region.2.PHYSICAL MODELAND AND NUMERICAL METHOD2.1 Physical modelThe gas is driven and compressed by “squish motion” of the orbiting scroll wrap, and this results that an unsteady compressible viscous flow occurs within the scroll compressor working chamber. Due to high rotating speed and steep velocity gradient near the wrap wall, the turbulence characteristics have to be considered. But the orbiting wall speed is small compared to the gas flow velocities, for example, the wall speed is approximately 5 percent of the average velocity of the discharge flow in a scroll air compressor at discharge studied in this paper, so it appears justified that the quasi-steady approach is made to treat the flow field with stationary wall. That is , to ignore the moving of orbiting scroll wall is justified. Therefore, three dimension steady-state turbulence calculations are performed to predict the flow field in the final compressor and discharge region. Air is injected from two sides of thecentral chamber with the instantaneous flow rate at various crank angles. The volume flow rate at various crank angles during the discharge process is shown in figure 2.The figure 3 shows a computational model at a certain crank angle after onset of discharge.Figure 2: V olume flow rate with orbiting discharge crank angleFigure 3: Three dimensional computation model2.2 Numerical methodTurbulent flow exists in the scroll configurations considered and was treated using a normal k- turbulence model. The governing equations were discretized using finite volume method. The SIMPLE algorithm was employed in order to correct the pressure filed. Near the wall, the improved wall function method was employed. The discretization scheme of convection item and diffusion item are respectively the second-order upwind scheme and the central difference scheme. Tao (2001) shows the details of discretization method. In the light of the geometrical characteristics of computational domain, the geometry scale of different parts of the whole domain differs greatly; the block structure gird method was employed to generate grids of the whole domain being separated into several parts, in which grids were generated by the body-fitted coordinate grid system. Grids are so fine that the numerical results are grid-independent. The computational domains at different crank angles are different, so the grids were generated separately. The boundaryconditions are as the followings:(a) InletThe mass flow rate on each of two inlets is the same and equal to the instantaneous volume change rate multiplied by the density.(b)The outlet is set on the location far away as 5 times of height of discharge port in order to guarantee the constant pressure. The discharge pressure is provided on outlet.(c)Non-slip boundary condition for velocity is provided on walls. Advanced wall function method is employed to tread the near wall domain.3. NUMERRICAL RESULTSIn this paper, 0 is defined as the orbiting discharge crank angle. At the discharge moment, that is the crank angle y=45(x is the discharge crank angle), yis taken as zero. Then, y is changing from 0 to 360 degree during the whole process of discharge. According to this definition, for example, at crank angle of 45 degree after the onset of discharge is described as y=45.In this paper, for the convenience of description, location of the z coordinate equal to zero is define as the inlet of discharge port and is named as the surface of the fixed scroll. Location of the z coordinateequal to h (the height of scroll wrap) is named as the top surface of the fixed scroll.The flow field and its discharge from the central chamber region at several crank angles that correspond to 45, 90,180 degree after the onset of discharge is studied. Flow velocity vectors in different axial sections and three dimensional velocity vectors are detailedly analyzed.3.1 y=45The calculated velocity fields in different axial sections are shown in figure 4 (a)-(c) at orbiting discharge crank angle of 45 degree. The flow velocity in fig.4 indicate that flow being injected in the rear of each half central chamber, being turned as it impinges on the opposing wall of orbiting scroll of fixed scroll and proceeding towards the central region of the central chamber. In the central region, flow enters from both half central chamber, passing through throat region formed by the orbiting and fixed scroll tips and proceeds driven by the inertia. Two large scale vortexes develops in the central region near the scroll tips and some small scale vortexes develops in the rear of central chamber near the outer surface of scroll tips. Compared fig 4 (a), (b) with (c), it is shown that vortex flow develops in all different axial sections and number and scale and location of vortex are different in different axial section. That is to say this basic vortex flow pattern persists in this region throughout the entire axial extent of central chamber. Three dimensional velocity vectorsshown in fig 5 indicate clearly the distribution of axial velocity component. The three dimensional flow tends to move vertically downwards as it approaches the central region of the central chamber which is directly upon the discharge port. The axial velocity component is very large at a small axial distance of 0-10mm from the discharge port (when the height of the profile is 52mm). The axial velocity by the order of magnitude is greater than the radial velocity. In contrast, within the rear region of the central chamber, the flow is essentially two dimensional.From the fig 4 and fig 5, it can be seen that the velocity vectors in the mid axial section characterize the general nature of the flow within the entire central chamber. The flow vectors indicate both the two dimensional and the three dimensional nature of the flow depending upon the location. So, only the velocity vectors in the mid plane are analyzed below.Figure 4 : Velocity fields in different axial sectionsFigure 5: Three dimensional velocity vectors3.2 y=90Similar type of flow calculations have been performed at an intermediate crank angle (2=90). It is shown in fig 6. As the discharge process continues in an actual scroll compressor, the orbiting scroll continues to move away from the fixed scroll. This action is associatedwith a progressively increased opening of the central region to the discharge port. This implies a less occluded opening of the discharge port compared with the throat region at y=45.The velocity vectors in axial sections of mid plane and discharge are different from those at y=45 and the magnitude of velocity reduced. A double vortex was predicted to form at the mid axial section and the scale of the vortexes increased to trend to become a large vortex.Figure 6 : Velocity fields in different axial sections3.3 y=180Figure 7 (a)-(b) shows the velocity vectors in the axial sections at orbiting discharge crank angel of 180 degree. From the figure, it is shown that the velocity vectors field is obviously different from those shown in fig .4 and fig.6. A large scale vortex was developed in the discharge region as the discharge port is opened fully. In addition, a less constrictive flow passage exits on the region of the scroll tips, the velocity magnitude reduces further. The velocity vector field shows the occurrence of some small scale vortexes at the rear region of the central chamber.Figure 7: Velocity field in different axial sections4. NONDIMENSIONAL PRESSURE LOSSES COEFFICIENTTo obtain quantitative data characterizing the pressure losses of the final compression and discharge region, nondimensional pressure losses coefficient was define as below:P is the average pressure in the central chamber, pa; pd is the designed discharge pressure, pa.The variation of pressure losses coefficient * with orbiting discharge crank angel for the whole discharge process under different operation conditions is shown in fig 8. the pressure losses coefficient is very large at orbiting discharge crank angel of 0-60 degree. For example, the losses coefficient at 45 degree of orbiting discharge crank angle is approximately ten times larger than the loss coefficient at 180 degree, indicating that the flow losses are largest at the onset of discharge. This result is not surprising, since this is also the point of maximum constriction of the flow area. Furthermore, the high rotating speed and discharge pressure corroborate that significant flow losses would exist. With increasing the opening discharge port, losses coefficient is decreasing rapidly. The results indicate that discharge flow losses concentrate at the onset of discharge and reduce quickly with increasing opening discharge port.In addition, these results imply that the open-close characteristic of discharge port should be stressed to consider when designing a discharge port, particularly for compressor at large discharge. The easier to open, the better the characteristic of discharge port is. Maximum area of discharge port is possibly not the best.Figure 8: Nondimensional pressure losses coefficient with orbitingdischarge crank angle5: CONCLUSIONSInternational Compressor Engineering Conference at Purdue, July 12-15, 2004Three dimension numerical simulation of the discharge flow in a scroll air compressor was conducted to provide the characteristic of flow field in the final compression and discharge region. Detailed analysis is made of the flow velocity vectors in different axial sections. The numerical results show that complex vortex flow patterns exist in the discharge region, not only in axial sections. On the basis of numerical results, the dismensionless pressure losses coefficient is defined and the pressure losses at various crank angles after onset of discharge is analyzed. It is shown that the discharge flow losses greatly large shortly after the onset of discharge. The results shows that, the easier to open, the better the characteristic of discharge port is. Maximum area of discharge port is possibly not the best.REFFRENCES1.Hirano. T., et al., 1989, Development of High Efficiency ScrollCompressor for Heat Pump Air Conditioners, Mitsubishi Heavy Industries, Ltd., Tech. Rev. V ol. 26, No. 3, p: 512-519.2.Patankar S V, Spalding D B., 1972, A calculation procedure forheat、mass and momentum transfer in three-dimensional parabolicflows, Int. J. Heat Transfer, V ol. 15, No.11, p:1787-1806.3.Wang Yunliang, X u zhong, Miao Yongmiao, 1993, Influence ofDifferent Wall Function Methods on turbulent flow fields, Fluid Engineering, V ol. 21, No. 12, : 26-29.4.Tao Wenquan, 2001, Numrical Heat Transfer (second version),Xi’an Jiaotong University Press, Xi’an, 152p.5.Tao Wenquan, 2001, Advanced Numerical Heat Transfer, SciencePublishing Company, Beijing, 41p.6.Thompson J.F., Warsi Z.U.A., Mastin C.W., 1985, Numerical GridGeneration, Foundation and Application, North-Holland New York.。

沈青等，《可压缩湍流数值计算研究进展概述》131文章编号: (2009)-13可压缩湍流数值计算研究进展概述沈清1张涵信2庄逢甘31中国航天空气动力技术研究院，北京，1000722中国空气动力研究与发展中心，四川绵阳 6210003中国航天科技集团公司，北京，100037摘要本文简单介绍了可压缩湍流的三种主要数值计算方法DNS、LES、RANS。

然后，介绍了我国近年来可压缩湍流数值计算研究的最新进展。

采用DNS方法，分别计算了Mc=0.5、0.8和1.5的超声速混合层失稳过程。

对于Mc=0.5的超声速混合层，获得了二次失稳的演化过程。

对于Mc=0.8的超声速混合层，模拟了涡结构与小激波的相互作用与三维结构的演化过程。

对于Mc=1.5的超声速混合层，模拟了不同频率下出现的三种声辐射涡模态。

对于圆锥小攻角高超声速边界层的稳定性问题，采用DNS方法模拟了边界层失稳的攻角效应，发现了多波干涉失稳现象。

采用LES方法，模拟了超燃冲压发动机进气道内的激波-边界层干扰流动。

采用新型转捩/湍流一体模型，模拟了圆锥高超声速边界层转捩和完全发展湍流流动。

针对高超声速湍流中假设湍流Pr数等于0.9的疑问，探索了一种修正思路，取得了对B-L湍流模式在气动热计算精度上提高的结果。

关键词：可压缩湍流；数值计算；边界层；混合层0 引言可压缩湍流问题来源于航天飞行器的高超声速边界层和超声速混合层流动，具有多尺度、高频脉动、强非线性等特点，实验和计算都十分困难。

面对不同时期航天飞行器发展的需要，自上世纪六十年代起，国外先后在常规风洞、静风洞中开展了圆锥、球钝锥和平板高超声速边界层稳定性和转捩研究，在双喷管试验设备上开展了超声速混合层稳定性的研究。

通过大量的研究，人们对这些问题取得了丰富的认识，但是并没有完全解决这些可压缩湍流问题。

随着计算理论、计算方法和计算机技术的发展，数值计算已成为解决湍流问题的重要研究手段。

直接数值模拟（DNS）、大涡模拟（LES）和基于雷诺平均NS方程的湍流模式计算（RANS）是三种主要的湍流流动计算方法。

河海大学博士学位论文三维自由面湍流场数值模拟及其在水利工程中的应用姓名:王志东申请学位级别:博士专业:水力学及河流动力学指导教师:汪德爟20040322摘要自然界中的流体流动问题一般都具有三维性和湍动性,对于最常见的流体一水则同时还拥有自由表面,水利工程中绕闸墩的泄洪水流、船舶工程中的船体绕流等均具有上述流动的三种特性,这三种特性是计算流体力学(CFD领域非常重要的研究内容和方向。

.本文针对粘流场数值模拟技术的研究现状和发展动态作了比较全面的回顾与展望,重点研究了网格生成技术、数值求解技术、湍流模型技术和动边界模拟技术,在此基础上建立了模拟自由面湍流场的数学模型,并成功地应用N-维溢流坝和带闸墩的三维溢流坝过坝水流的数值计算。

本文完成的具体工作和主要结论如下:1以代数网格生成方法为基础提出了一种简单的、可独立于网格生成方法之外的边界正交化技术;针对分区结构网格系统建立了分区交界面处的数据结构与计算模型: 2利用有限体积方法在非正交同位网格系统中建立了SIMPLE求解算法,对非正交网格系统中的边界条件、延迟修正技术及计算节点的梯度计算等专题进行了深入讨论。

通过对倾斜方腔驱动流、后台阶绕流、圆柱绕流等典型流场的数值模拟,表明本文建立的非正交同位网格系统中的SIMPLE算法是合理的;3研究比较了各种湍流模型的特点及应用范围,以水翼绕流场为例完成了RNGk 一￡模型与标准k—s模型的计算比较,结果表明RNGk—s模型能够更好地模拟流场的湍流特性:4研究了运动边界模拟技术中的VOF方法,详细建立了一种精度更高的自由面重构模型,通过对典型运动界面的数值模拟表明,该方法能够更加准确地确定运动界面的形状和位置:5利用本文建立的非正交网格系统中自由面湍流场数学模型,对二维溢流坝过坝水流进行了数值计算,分析了不同坝顶水头下的水面线位置、坝面压力及流量、流速等水流特性,通过与典型实验资料的对比,水面线高度及坝面负压极值均与实验值具有非常好的一致性。