Numerical Simulation of Three dimensional Turbulent Flow through a Francis Turbine Runner

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。

COMPUTATIONAL MECHANICSISCM2007,July 30-August 1, 2007, Beijing, China©2007 Tsinghua University Press & SpringerNu m erical Si m ulation of Three Di m ensional Turbulent Flow in Double-Suction Centrifugal Pu m psMin Yang*, Fujun Wang, Guohui CongDepartment of Fluid machinery, College of Water Conservancy & Civil Engineering, China Agricultural University, Beijing, 100083 ChinaE mail: yangmin0113@AbstractDouble-suction centrifugal pumps are widely used in various fields because the flow rate of those is two times as much as single suction pump with the same diameter impeller and it is theoretically in axial balance. Along with the construction of many large projects, such as Water Transfer Project from South to North in China, the size and capacity of double-suction pump is constantly increasing. The design of those machines is mainly based on experience, and can not fully guarantee the performance of the pump. From the operation of large pump stations, there are general phenomena such as vibration and cavitation. Therefore, it is very imperative to study the internal flow within a double-suction centrifugal pump and provide scientific reference for its structure design and stable operation.This paper describes a numerical study in a double-suction centrifugal pump by using the Reynolds Averaged Navier–Stokes (RANS) approach with the Shear Stress Transport turbulent model (SST) at the different conditions (rated, small and large flow rate). A symmetry condition is employed to provide an easier handling of the geometry. The hydraulic performance is predicted and it is in very good agreement with experimental data. The velocity and pressure distribution in pump are analyzed at the design point and off-design points. Changes of the velocity and the pressure with flow rate are investigated at impeller eye and impeller exit.Figure: A 3D model with double-suction centrifugal pumpREFERENCES1.Wang Hong, Tsukamoto Hiroki. E xperimental and numerical study of unsteady flow in a diffuserpump at off design conditions. J. J ournal of Fluids E ngineering, 2003; 125(6): 767- 778.2.Chen Hongxun. Research on turbulent flow within the vortex pump. J. J ournal of Hydrodyn. Ser. B,2004; 16(6): 701-707.3.Ferreira VG et al. Numerical simulation of turbulent free surface flow with two-equation k-εeddy-viscosity models. J. I nternational J ournal for Numerical Methods in Fluids, 2004; 44(4): 347-375.— 242 —。

MathematicalModeling理论建模及实际应用数学建模（Mathematical Modeling）是一种将实际问题转化为数学问题，并通过数学方法对问题进行分析和解决的方法。

它既是数学的一种应用，也是一种研究问题并解决问题的工具。

数学建模在各个领域都有广泛的应用，如物理学、经济学、生物学、环境科学等等。

本文将从理论建模和实际应用两个方面来介绍数学建模的基本概念、方法以及一些实际应用案例。

在数学建模中，理论建模是首要的一步。

理论建模是指对实际问题进行分析和抽象，从中提取出数学模型的基本要素和关系。

对于一个复杂的实际问题，我们需要通过对问题的认识和理解，找出其中的关键因素和变量，并确定它们之间的数学关系。

这些关系可以是线性的、非线性的、离散的或连续的，可以用代数方程、微分方程、差分方程或概率统计等形式来表示。

理论建模需要深入地了解问题的背景和相关领域的知识，同时还需要灵活运用数学方法和工具来描述问题和解决问题。

数学建模的方法主要包括定性分析、定量分析和验证分析。

定性分析是指通过观察和分析问题的特征和特性，对问题进行描述和理解，找出问题的关键因素和变量，并确定它们之间的关系。

定量分析是指通过运用数学方法和工具，对问题进行计算和求解，得出问题的数值结果和解决方案。

验证分析是指对数学模型的有效性和可靠性进行检验和验证，通过与实际数据进行对比和比较，评估模型的拟合程度和预测能力。

这些方法相互补充和支持，共同构建了一个完整的数学建模流程。

数学建模在实际应用中有着广泛的应用。

以物理学为例，物理学中的很多问题都可以通过数学建模来解决。

比如，天体物理学中的行星运动、星系演化等问题可以通过数学建模来描述行星和星系的位置、速度和质量等参数，进而研究它们的运动规律和相互作用。

在经济学中，数学建模可以用来描述和分析经济系统中的供需关系、利润最大化、成本最小化等问题，从而指导经济政策和决策。

在生物学中，数学建模可以用来描述生物种群的增长、遗传变异、物种竞争等问题，为生态保护和资源管理提供科学依据。

课程名称：工程力学数值模拟Numericalsimulation in engineering mechanics 课程代码：24410044学分：2学时：32 （其中：课堂教学学时：32）先修课程：理论力学、材料力学、弹性力学、计算力学适用专业：工程力学教材：ANSYS有限元分析基础、李汉龙、国防工业出版社、2017年第1版一、课程性质与课程目标（-）课程性质工程力学数值模拟是一门用以培养学生利用商业有限元软件解决工程中的力学问题的专业选修课，本课程主要研究工程中的静动力学问题及相关领域的数值模拟问题。

通过数值模拟技术的学习，能够利用商业有限元软件对工程中的力学问题进行建模，掌握其前后处理和求解方法，使学生具备一定的分析能力和初步的实践能力。

（二）课程目标课程目标1：掌握数值模拟的基本概念及基础理论；课程目标2：将商业有限元软件应用到简单的工程问题分析中；课程目标3：掌握数值模拟技术使用的一般流程，具有进行数值实验研究的初步能力；课程目标4：对计算结果具备一定的分析及判断能力，了解数值模拟技术的新理论和新方法。

（三）课程目标与专业毕业要求指标点的对应关系本课程支撑专业培养计划中毕业要求指标点2、5、6,对应关系如下：1.毕业要求2: 了解力学的发展历史、学科前沿和发展趋势；认识力学在经济社会发展与科学技术创新中的先导性和重要推动作用，初步了解力学交叉研究开拓新的学科生长点的能力；2.毕业要求5：掌握基本的力学实验与数值计算技能。

前者包括理论力学及材料力学等基础力学实验，结构健康监测和微纳米测试等方面的最新实验技术，后者要求具有使用数值模拟软件或编程计算典型工程力学案例的能力；.毕业要求6：初步掌握力学研究的基本方法和手段，具有初步的工程实践能力，初步具备发现、提出、分析和解决与力学有关的工程技术问题的能力。

注：课程目标与毕业要求指标点对接的单元格中可输入“「'，也可标注“H、M、L-o 二、课程内容与教学要求（按章撰写）第一章有限元分析与ANSAS（一）课程内容1.1有限元分析的基本概念1. 2 ANSYS的功能1.3结构分析1.4热分析1.5流体分析（二）教学要求教学内容L1-1.5要求了解。