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Soret and Dufour Effects on Unsteady Free Convection Fluid Flow in the Presence of Hall Current and Heat Flux
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作者 Abdul Quader Md. Mahmud Alam 《Journal of Applied Mathematics and Physics》 2021年第7期1611-1638,共28页
Unsteady MHD natural convective heat and mass transfer flow through a semi-infinite vertical porous plate in a rotating system have been investigated with the combined Soret and Dufour effects in the presence of Hall ... Unsteady MHD natural convective heat and mass transfer flow through a semi-infinite vertical porous plate in a rotating system have been investigated with the combined Soret and Dufour effects in the presence of Hall current and constant heat flux. It is considered that the porous plate is subjected to constant heat flux. The obtained non-dimensional, non-similar coupled non-linear and partial differential equations have been solved by explicit finite difference technique. Numerical solutions for velocities, temperature and concentration distributions are obtained for various parameters by the above mentioned technique. The local and average shear stresses, Nusselt number as well as Sherwood number are also investigated. The stability conditions and convergence criteria of the explicit finite difference scheme are established for finding the restriction of the values of various parameters to get more accuracy. The obtained results are illustrated with the help of graphs to observe the effects of various legitimate parameters. 展开更多
关键词 Soret and Dufour Effects on Unsteady free Convection fluid flow in the Presence of Hall Current and Heat Flux
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Evaluation of gas wettability and its effects on fluid distribution and fluid flow in porous media 被引量:11
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作者 Jiang Guancheng Li Yingying Zhang Min 《Petroleum Science》 SCIE CAS CSCD 2013年第4期515-527,共13页
The special gas wettability phenomenon of reservoir rocks has been recognized by more and more researchers.It has a significant effect on efficient development of unconventional reservoirs.First,based on the preferent... The special gas wettability phenomenon of reservoir rocks has been recognized by more and more researchers.It has a significant effect on efficient development of unconventional reservoirs.First,based on the preferentially gas-covered ability and surface free energy changes,definition and evaluation methods have been established.Second,a method for altering rock wettability and its mechanisms have been studied,surface oriented phenomena of functional groups with low surface energy are the fundamental reason for gas wettability alteration of rock.Third,the effect of gas wettability on the surface energy,electrical properties and dilatability are investigated.Last,the effects of gas wettability on capillary pressure,oil/gas/water distribution and flow are investigated with capillary tubes and etchedglass network models.The gas wettability theory of reservoir rocks has been initially established,which provides theoretical support for the efficient production of unconventional reservoirs and has great significance. 展开更多
关键词 Gas-wetting fluorocarbon copolymer contact angle capillary pressure surface free energy surface property fluid flow in porous media
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Lie Group Analysis for the Effects of Variable Fluid Viscosity and Thermal Radiation on Free Convective Heat and Mass Transfer with Variable Stream Condition 被引量:2
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作者 P. Loganathan P. Puvi Arasu 《Engineering(科研)》 2010年第8期625-634,共10页
Natural convective boundary layer flow and heat and mass transfer of a fluid with variable viscosity and thermal radiation over a vertical stretching surface in the presence of suction/injection is investigated by Lie... Natural convective boundary layer flow and heat and mass transfer of a fluid with variable viscosity and thermal radiation over a vertical stretching surface in the presence of suction/injection is investigated by Lie group analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. An exact solution is obtained for translation symmetry and numerical solutions for scaling symmetry. The effects of fluid viscosity and thermal radiation on the dimensionless velocity, temperature and concentration profiles are shown graphically. Comparisons with previously published works are performed and excellent agreement between the results is obtained. The conclusion is drawn that the flow field and temperature profiles are significantly influenced by these parameters. 展开更多
关键词 Scaling Group of Transformations free CONVECTIVE flow TEMPERATURE-DEPENDENT fluid Viscosity Suction/Blowing Thermal Radiation
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GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY 被引量:1
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作者 Nerming MUJAKOVIC Nelida CRNJARIC-ZIC 《Acta Mathematica Scientia》 SCIE CSCD 2016年第6期1541-1576,共36页
In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a s... In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time. 展开更多
关键词 micropolar fluid flow initial-boundary value problem free boundary finitedifference approximations strong aBd weak convergence
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NUMERICAL SIMULATION OF FREE MIXING LAYER WITH CROSS SHEAR
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作者 Lin Jianzhong Wang Canxing Li Feng(Department of Mechanics, State Key Laboratory of Fluid Power Transmissionand Control, Zhejiang University, Hangzhou, 310027, China) 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 1997年第1期14-17,共4页
A spectrum method is used to simulate the time-developing free mixing layerwith cross shear which is introduced in different stages. The results show that the properties of flow are nearly the same for situations whet... A spectrum method is used to simulate the time-developing free mixing layerwith cross shear which is introduced in different stages. The results show that the properties of flow are nearly the same for situations whether the cross shear is introduced in theinitial time or in early stage. If cross shear is introduced in the stage that the roll-up ofmixing layer occurs, the turbulent intensities of now will increase and mixture of now willbe enhanced. 展开更多
关键词 free flow mixing layers (fluids) shear flow cross shear numerical simulation properties of flow
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COMPUTER SIMULATION FOR ELECTRICALLY DRIVEN MOTION OF FLUID
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作者 崔铮 《Journal of Southeast University(English Edition)》 EI CAS 1989年第1期69-76,共8页
In this paper,an equation system of electrohydrodynamics(EHD)based onthe fluid motion equations is discussed.Emphasis is put on the effects of electrical forceand surface tension upon the fluid motion.The Marker and C... In this paper,an equation system of electrohydrodynamics(EHD)based onthe fluid motion equations is discussed.Emphasis is put on the effects of electrical forceand surface tension upon the fluid motion.The Marker and Cell method is used to set up acomputational simulation program of electrically driven motion of fluid.With the help ofthe program,a cylindrical fluid under the influences of electrical field and surface tensionhas been calculated.The result is in good agreement with the experimental observation. 展开更多
关键词 ELECTROHYDRODYNAMICS fluid flow free surface/electrically DRIVEN MOTION MARKER and CELL method
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Numerical Study on Flow Around Modern Ship Hulls with Rudder-Propeller Interactions
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作者 Md.Mashud Karim Nabila Naz 《Journal of Marine Science and Application》 CSCD 2019年第4期400-416,共17页
Reducing the fuel consumption of ships presents both economic and environmental gains. Although in the past decades,extensive studies were carried out on the flow around ship hull, it is still difficult to calculate t... Reducing the fuel consumption of ships presents both economic and environmental gains. Although in the past decades,extensive studies were carried out on the flow around ship hull, it is still difficult to calculate the flow around the hull while considering propeller interaction. In this paper, the viscous flow around modern ship hulls is computed considering propeller action. In this analysis, the numerical investigation of flow around the ship is combined with propeller theory to simulate the hull-propeller interaction. Various longitudinal positions of the rudder are also analyzed to determine the effect of rudder position on propeller efficiency. First, a numerical study was performed around a bare hull using Shipflow computational fluid dynamics(CFD) code to determine free-surface wave elevation and resistance components.A zonal approach was applied to successively incorporate Bpotential flow solver^ in the region outside the boundary layer and wake, Bboundary layer solver^ in the thin boundary layer region near the ship hull, and BNavier-Stokes solver^in the wake region. Propeller open water characteristics were determined using an open-source MATLAB code Open Prop, which is based on the lifting line theory, for the moderately loaded propeller. The obtained open water test results were specified in the flow module of Shipflow for self-propulsion tests. The velocity field behind the ship was recalculated into an effective wake and given to the propeller code that calculates the propeller load. Once the load was known, it was transferred to the Reynolds-averaged Navier-Stokes(RANS) solver to simulate the propeller action. The interaction between the hull and propeller with different rudder positions was then predicted to improve the propulsive efficiency. 展开更多
关键词 free-surface wave WAKE Self-propulsion characteristics Ship flow Computational fluid dynamics Zonal approach RANS Rudder-propeller interaction Ship hulls
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Buoy Relay Method for Instantaneous Fluid Flow with Free Surface
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作者 王士敏 王照林 李俊锋 《Tsinghua Science and Technology》 EI CAS 2000年第1期34-38,共5页
Several methods have been used to approximate free surface boundaries in finite difference numerical simulations. Each of these methods has its advantages and disadvantages. This paper presents a new technique for th... Several methods have been used to approximate free surface boundaries in finite difference numerical simulations. Each of these methods has its advantages and disadvantages. This paper presents a new technique for the numerical solution of transient incompressible free surface fluid flows. This powerful method, which is based on the concepts of “Buoy positioning” and “Buoy relaying\', successfully represents the free surface using a Lagrangian method on a Eulerian grid by directly solving the free surface evolution equation. The Eulerian finite\|difference forms of the full Navier\|Stokes equations are solved by the Successive over Relaxation (SOR) method with a set of buoys to keep track of the free surface. The capabilities of the analysis procedure are demonstrated through viscous free surface fluid flow examples. The method is simpler and more efficient than other methods especially in treating complicated free boundary configurations. 展开更多
关键词 fluid flow with free surfaces computational fluid mechanics buoy relay method
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A NUMERICAL METHOD FOR THE SIMULATION OF FREE SURFACE FLOWS OF VISCOPLASTIC FLUID IN 3D
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作者 Kirill D. Nikitin Maxim A. Olshanskii +1 位作者 Kirill M. Terekhov Yuri V. Vassilevski 《Journal of Computational Mathematics》 SCIE CSCD 2011年第6期605-622,共18页
In this paper we study a numerical method for the simulation of free surface flows of viscoplastic (Herschel-Bulkley) fluids. The approach is based on the level set method for capturing the free surface evolution an... In this paper we study a numerical method for the simulation of free surface flows of viscoplastic (Herschel-Bulkley) fluids. The approach is based on the level set method for capturing the free surface evolution and on locally refined and dynamically adapted octree cartesian staggered grids for the discretization of fluid and level set equations. A regu- larized model is applied to handle the non-differentiability of the constitutive relations. We consider an extension of the stable approximation of the Newtonian flow equations on staggered grid to approximate the viscoplastic model and level-set equations if the free boundary evolves and the mesh is dynamically refined or coarsened. The numerical method is first validated for a Newtonian case. In this case, the convergence of numerical solutions is observed towards experimental data when the mesh is refined. Further we compute several 3D viscoplastic Herschel-Bulkley fluid flows over incline planes for the dam-break problem. The qualitative comparison of numerical solutions is done versus experimental investigations. Another numerical example is given by computing the freely oscillating viscoplastic droplet, where the motion of fluid is driven by the surface tension forces. Altogether the considered techniques and algorithms (the level-set method, compact discretizations on dynamically adapted octree cartesian grids, regularization, and the surface tension forces approximation) result in efficient approach to modeling viscoplastic free-surface flows in possibly complex 3D geometries. 展开更多
关键词 free surface flows Viscoplastic fluid Adaptive mesh refinement Oetree meshes.
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缝洞型油藏的离散缝洞网络流动数学模型 被引量:34
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作者 姚军 黄朝琴 +2 位作者 王子胜 李亚军 王晨晨 《石油学报》 EI CAS CSCD 北大核心 2010年第5期815-819,824,共6页
针对现有油藏渗流理论在描述缝洞型油藏流动机理上存在的困难,基于缝洞型油藏储层的特点,提出了离散缝洞网络流动数学模型。该模型将缝洞型油藏划分为岩块系统、裂缝系统和溶洞系统;其中裂缝和溶洞嵌套于岩块中,并相互连接成网络;岩块... 针对现有油藏渗流理论在描述缝洞型油藏流动机理上存在的困难,基于缝洞型油藏储层的特点,提出了离散缝洞网络流动数学模型。该模型将缝洞型油藏划分为岩块系统、裂缝系统和溶洞系统;其中裂缝和溶洞嵌套于岩块中,并相互连接成网络;岩块和裂缝系统视为渗流区域,溶洞系统视为自由流动区域。阐述了模型的基本原理,推导了相应的有限元数值计算格式,通过算例验证了其正确性。算例分析结果表明,离散缝洞网络流动数学模型能够正确地描述出缝洞型油藏的流动特征。 展开更多
关键词 缝洞型油藏 离散缝洞网络 自由流动 耦合流动 有限元数值计算
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水口侧孔倾角对大方坯结晶器流场和液面波动的影响 被引量:23
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作者 王维维 张家泉 +1 位作者 陈素琼 董进明 《北京科技大学学报》 EI CAS CSCD 北大核心 2007年第8期816-821,844,共7页
基于湍流模型与多相流模型的耦合,实现了对结晶器流动与自由液面波动行为的计算模拟.分析研究了不同侧孔倾角对对称多侧孔水口浇注大方坯结晶器内流场及自由液面波动的影响规律.结果表明:水口侧孔向下角度每增加5°,结晶器宽面和窄... 基于湍流模型与多相流模型的耦合,实现了对结晶器流动与自由液面波动行为的计算模拟.分析研究了不同侧孔倾角对对称多侧孔水口浇注大方坯结晶器内流场及自由液面波动的影响规律.结果表明:水口侧孔向下角度每增加5°,结晶器宽面和窄面冲击点位置分别下降约13mm和10mm;一定浸入深度下,钢水出口向下倾角的大小对大方坯结晶器液面波动影响不大,在3mm以内;结晶器壁面附近的钢液较水口附近钢液活跃.推荐水口侧孔向下倾角为20°. 展开更多
关键词 结晶器 大方坯 流场 液面波动
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动态水力旋流器圆管螺旋流场特性研究 被引量:15
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作者 刘晓敏 蒋明虎 +2 位作者 王尊策 赵立新 李枫 《石油学报》 EI CAS CSCD 北大核心 2003年第2期89-93,共5页
对动态水力旋流器圆管螺旋流场结构及形成原理作了简要描述。在深入研究动态水力旋流器分离机理及分离性能的基础上 ,运用圆管螺旋流动方程式进行了动态水力旋流器内部螺旋流动分布理论研究 ,得到旋流器内部流场按强制涡、自由涡、组合... 对动态水力旋流器圆管螺旋流场结构及形成原理作了简要描述。在深入研究动态水力旋流器分离机理及分离性能的基础上 ,运用圆管螺旋流动方程式进行了动态水力旋流器内部螺旋流动分布理论研究 ,得到旋流器内部流场按强制涡、自由涡、组合涡及螺旋涡的分布特性。同时利用直圆管柱模型 ,模拟计算出了圆管螺旋流的理论分布结果。研究和掌握动态水力旋流器内部流体螺旋流的运动及分布规律 ,为今后深入进行其分离机理、流场测试及分离性能研究打下了一定的基础。 展开更多
关键词 动态水力旋流器 螺旋流 强制涡 自由涡 组合涡 流场分布 非牛顿流体
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不可压流体自由表面流动的SPH数值模拟 被引量:19
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作者 李梅娥 周进雄 《机械工程学报》 EI CAS CSCD 北大核心 2004年第3期5-9,共5页
根据SPH方法的原理提出了一套模拟流体自由表面流动的方法,场变量及其导数通过核函数插值求取,不需进行差分,也不需要网格,时间积分采用分数步长法,避免了不可压条件带来的压力计算不稳定问题。对流项通过粒子的运动求解,完全消除了数... 根据SPH方法的原理提出了一套模拟流体自由表面流动的方法,场变量及其导数通过核函数插值求取,不需进行差分,也不需要网格,时间积分采用分数步长法,避免了不可压条件带来的压力计算不稳定问题。对流项通过粒子的运动求解,完全消除了数值扩散和自由表面模糊问题。可以模拟飞溅、融合等复杂自由表面现象,并对水坝坍塌这一典型自由表面流动问题进行了模拟,模拟结果与试验吻合良好。 展开更多
关键词 不可压流体流动 自由表面 光滑质点流体动力学 分数步长法 数值模拟
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自由液面流体流动问题的数值分析研究 被引量:14
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作者 王建军 李其汉 陆明万 《计算力学学报》 CAS CSCD 北大核心 2003年第1期101-108,共8页
对国内外具有自由液面的流体流动问题研究的数值求解方法进行了较为全面的评述。文中首先讨论了自由液面流体流动问题的数值求解方法,包括差分法、有限元法和边界元法等,然后说明了自由液面的跟踪和网格更新等问题。
关键词 流体流动 数值方法 自由液面
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格子Boltzmann方法模拟双液滴撞击液膜的流动过程 被引量:13
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作者 周轶 郭加宏 陈红勋 《计算物理》 EI CSCD 北大核心 2010年第1期31-37,共7页
采用单相自由面格子Boltzmann方法中流场计算,自由面条件和表面张力处理方法,研究双液滴撞击液膜的流动过程,实现对垂直间隔的两液滴相继撞击液膜流动过程的模拟.数值结果表明,液滴相继冲击液膜时,液滴间上下距离对液膜状态有很大的影响... 采用单相自由面格子Boltzmann方法中流场计算,自由面条件和表面张力处理方法,研究双液滴撞击液膜的流动过程,实现对垂直间隔的两液滴相继撞击液膜流动过程的模拟.数值结果表明,液滴相继冲击液膜时,液滴间上下距离对液膜状态有很大的影响,液滴间距不同,液膜形状会出现很大差异.数值结果与实验结论定性一致. 展开更多
关键词 格子BOLTZMANN方法 自由面 两相流 液滴
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功能表面材料与流体界面相互作用对垂直降液膜流动特性的影响 被引量:10
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作者 马学虎 高大志 +4 位作者 兰忠 张宇 白涛 杨俊玲 陈嘉宾 《高校化学工程学报》 EI CAS CSCD 北大核心 2004年第3期269-274,共6页
通过调节水温度、添加表面活性剂以及铝合金壁面表面改性处理来改变降膜流体与固体表面之间的表面自由能差值,运用JDC-2000型精密测微仪测定垂直降液膜的厚度,研究固体表面和液体间相互作用对流体垂直降膜流动特性的影响;考察了液膜雷... 通过调节水温度、添加表面活性剂以及铝合金壁面表面改性处理来改变降膜流体与固体表面之间的表面自由能差值,运用JDC-2000型精密测微仪测定垂直降液膜的厚度,研究固体表面和液体间相互作用对流体垂直降膜流动特性的影响;考察了液膜雷诺数、流体温度、添加表面活性剂、固体表面材料物理化学性质等因素对垂直壁面降液膜流动特性的影响规律。实验结果表明:改变固体表面与降液膜流体的物理化学特性,即改变固液界面的相互作用能够改变流体的降膜流动特性。降液膜平均厚度随液固表面自由能差的增大而减小。 展开更多
关键词 固液表面自由能差效应 降液膜流动 功能表面 液膜厚度
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水口倾孔倾角对异型坯结晶器流场和液面波动的影响 被引量:2
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作者 张彩军 李增玉 +2 位作者 关开 李建生 朱立光 《过程工程学报》 CAS CSCD 北大核心 2009年第S1期374-378,共5页
基于湍流与多相流模型的耦合,分析了三侧孔水口的侧孔倾角对异型坯结晶器内流场和自由液面波动的影响.结果表明,采用侧开孔型水口时,随着水口侧孔向上倾角的增大,冲击点位置向上移动,结晶器内表面流速和液面波高相应增大;水口倾角由-3&#... 基于湍流与多相流模型的耦合,分析了三侧孔水口的侧孔倾角对异型坯结晶器内流场和自由液面波动的影响.结果表明,采用侧开孔型水口时,随着水口侧孔向上倾角的增大,冲击点位置向上移动,结晶器内表面流速和液面波高相应增大;水口倾角由-3°增加到15°时,结晶器z方向表面流速由0.18m/s增加到0.24m/s,x方向由0.10m/s增加到0.21m/s;水口倾角由-3°增加到15°时,结晶器z方向波高差由3.6mm增加到5.3mm,x方向由3.3mm增加到6.9mm. 展开更多
关键词 结晶器 异型坯 流场 液面波动
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上向流水轮机中自由液面流动的数值模拟 被引量:1
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作者 杨魏 刘竹青 +1 位作者 吴玉林 高德瑜 《排灌机械工程学报》 EI 北大核心 2013年第4期305-308,共4页
为了更加准确地研究上向流水轮机内部流动情况,评价其水力性能,指导水力设计,建立了上向流水轮机(包括蜗壳和转轮)的全流道几何模型.利用非结构化网格划分技术进行流道的网格划分,在转轮叶片、蜗壳导叶等壁面进行网格加密,网格尺度的选... 为了更加准确地研究上向流水轮机内部流动情况,评价其水力性能,指导水力设计,建立了上向流水轮机(包括蜗壳和转轮)的全流道几何模型.利用非结构化网格划分技术进行流道的网格划分,在转轮叶片、蜗壳导叶等壁面进行网格加密,网格尺度的选择保证满足湍流计算对壁面y+的要求.基于RNG k-ε湍流模型进行水轮机内部流场的求解,利用流体体积法进行自由液面捕捉,得到上向流水轮机在考虑自由液面时的内部流动模拟结果.基于数值计算结果进行了水轮机内部流动的损失分析,研究发现:在水轮机固定导叶和转轮进口之间的流道存在较大的损失,占总损失的73%;内部流动细节显示该区域的台阶结构是损失的直接诱因,同时造成了转轮进口流态恶化;可通过改进上向流水轮机过渡段的设计消除台阶,以改善上向流水轮机能量性能. 展开更多
关键词 上向流水轮机 流体体积法 数值模拟 自由液面
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具有弹性支承输流管路的振动分析 被引量:6
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作者 赵千里 孙志礼 柴小冬 《振动.测试与诊断》 EI CSCD 北大核心 2017年第6期1222-1226,共5页
为研究截面内流速不均以及弹性支承对输流管路振动问题的影响,利用伽辽金法对考虑了流动模型修正因子的具有弹性支承的两端固定式输流管路的振动微分方程进行了推导,得到了一般形式下输流管路强迫振动稳态响应的表达式。结果发现:流动... 为研究截面内流速不均以及弹性支承对输流管路振动问题的影响,利用伽辽金法对考虑了流动模型修正因子的具有弹性支承的两端固定式输流管路的振动微分方程进行了推导,得到了一般形式下输流管路强迫振动稳态响应的表达式。结果发现:流动形式为层流时管路的固有频率和临界流速均小于理想流动模型(平推流模型)的值;前者的发散临界流速比后者小约13.4%,与理论结果一致;固有频率随弹簧安置位置的增加而呈现振荡的变化,但始终大于无弹性支承时的值。该方法可作为设计支承形式的基础,可以推广用来研究其他类型管路的振动问题,并为设计人员提供精确的计算结果。 展开更多
关键词 伽辽金法 自由振动 强迫振动 输流管路 流动模型修正因子
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绕椭圆柱管束的流动与传热特性 被引量:4
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作者 梁才航 杨永旺 黄斯珉 《科学技术与工程》 北大核心 2013年第13期3592-3597,共6页
椭圆柱管束换热器在工业领域中得到广泛的应用。采用一种简单合理的自由表面模型深入研究了流体绕流椭圆柱管束的流动与传热问题。通过适体坐标转换方法在非规则流道截面上求解出动量与热量传递控制方程,获得了等壁温条件下的流道内的... 椭圆柱管束换热器在工业领域中得到广泛的应用。采用一种简单合理的自由表面模型深入研究了流体绕流椭圆柱管束的流动与传热问题。通过适体坐标转换方法在非规则流道截面上求解出动量与热量传递控制方程,获得了等壁温条件下的流道内的阻力系数和努塞尔数等准则数,并且和文献值进行了对比。文献值和模拟值之间的最大绝对误差小于3.5%,表明所建立的数学模型是正确的。利用所建立的数学模型分析了流动方向、椭圆柱横截面形状、管束结构参数以及壁面边界条件对绕流椭圆柱管束的流动与传热特性的影响。获得了等壁温和等热流密度条件下的阻力系数和平均努塞尔数。这些准则数为椭圆柱管束的设计和优化提供理论指导。 展开更多
关键词 椭圆柱管束 流体流动 传热 自由表面模型
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