基于DLR型k-ε模型的AMG方法对渐扩管路内紊流数值仿真
|
摘要
对紊流的数值仿真是流动、传热传质中最基本的课题.何永森等研究者通过一系列数值实验发现,用DLR型k—ε紊流模型·BFC法(边界拟合曲线坐标变换法),能够对总扩散角为8~0、扩散度为4的锥形渐扩管路内完全发展的不可压粘性紊流场较精确地数值仿真(又称数值模拟,计算机仿真)。多重网格方法(Multi-Grid Method,简称MGM)也称为多格子方法或多层网格法,是求解偏微分方程(组)大规模离散化方程最有效的方法,它一般可以分为几何多重网格方法(Geometric Multigrid Method,简称CMC)和代数多重网格方法(Algebraic multigrid method,简称AMG)。近年来,由于大型计算机的迅速发展和功能的日趋完善,从而使得多重网格方法做为最优的算法将理论用于现实。由于实际应用问题的错综复杂性,以及数值商业软件对“即插即用”型求解器的要求,使得几何多重网格方法的应用变得越来越困难,而代数多重网格方法的高效性和稳健性(robustness“鲁棒性”)使之成为了当今多重网格方法的研究热点.
本文针对边界拟合曲线坐标贴体网格系统下代数多重网格方法在紊流数值预测中的应用问题,在何永森教授工作的基础上,详细探讨了代数多重网格方法在数值预测紊流中的实施过程,并围绕如何提高锥形渐扩管路内紊流数值仿真效率与精度展开研究,进行了一系列数值实验及与物理实验结果的比较,得到了一些有意义的成果,丰富和充实了代数多重网格方法,拓宽了代数多重网格方法在一些领域中的应用。具有理论和工程应用价值.主要内容和成果包括:
(一)原来的程序软件判断数值收敛与否是根据数值计算结果的情况分次沿时间步推进求解并以人工方式进行.本文对这一情况进行了改进,通过添加了一个判断收敛的子程序,变人工方式为自适应控制收敛方式.数值实验表明,用这种方式控制迭代次数,当迭代结束时,各种流动参数的误差(前后两时间步的差)曲线处于平稳并且很小,计算结果与实验结果较好符合。
(二)将AMG方法引入到紊流数值预测的有限差分计算领域,详细探讨了代数多重网格方法在紊流数值预测中的实施过程.研究了不同有限差分格式下,紊流模型离散得到的大规模代数系统对应系数矩阵的整合及边界条件的嵌入方法,讨论了系数矩阵的“三元组”压缩存储方式,利于节约内存及便于利用AMG方法求解。编制了代数多重网格方法数值求解紊流模型离散得到的大规模代数离散系统的程序接口;将代数多重网格方法与DLR、DHR型k—ε紊流模型.BFC法结合,应用编制的程序DLRAMG和DHRAMG对锥形渐扩管路内紊流进行了数值预测,将数值计算结果与物理实验结果进行了比较,多重网格方法的计算结果与实验结果符合较好。并且与POINT-SOR方法相比,可以节约近三分之一的CPU时间,提高了数值预测效率。
(三)针对原来十三点格式所用的模板节点多,方程组系数矩阵的带宽大,非零元多,数值求解费时问题,设计了一种基于五节点模板的新五点差分格式,并将其与AMG方法结合,进一步提高了紊流数值预测的效率.数值实验结果表明,在AMG方法求解的条件下,新五点差分格式比原十三点差分格式可以节约近三分之一的CPU机时。
(四)本文提出了一种可以同时实现控制网格正交性和任意控制边界网格间距的一种BFC网格生成的新方法。该方法可对生成的网格边界间距大小任意控制,同时生成的BFC网格还具有边界及内部较好的正交性.应用实例的计算结果表明,该方法能够对复杂边界的单连通域或多连通域生成较理想的BFC网格。
(五)另外,研究了k-ε紊流模型对数值预测正弦波壁流动的应用。基于有限体积法结合非正交同位网格系统,压力与速度耦合采用SIMPLE方法,对该种流动进行了数值预测,得到了与实验结果符合较好的计算结果。
Numerical predictions(or numerical simulations) for turbulent flow is the base project in fluid flow and heat transform.He yongsen et al find that fully developed incompressible turbulent flow in a conical diffuser having a total divergence of 8~0 and an area ratio of 4:1 can be simulated by a DLR turbulent model and it's BFC(Boundary-Fitted Coordinates) method,but the predictions for turbulent flow have low efficiency.Algebraic multi-grid methods are by far the most,efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs.Generally speaking,there are two types of multigrid methods:geometric-based approach and algebraic approach.The large computers have developed very fast and the function of computer have been improved in recent years,which make the multigrid methods vary from optimal algorithm theoretically to optimal algorithm in practice.Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares,it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid method(AMG) has become the hotspot due to the high performance and robustness.
In this paper,we make some in-depth studies for applying the AMG algorithms in numerical predictions for turbulent flow.A detailed discussion of implement procession of applying AMG methods to simulate the turbulent flow.Focus on improving the efficiency of simulation,we put forward some feasible methods.A number of numerical experiments have been performed.The computational results are compared with the results of experimental results,and some crucial numerical results are obtained.These researches will make the AMG algorithms richer and apply the AMG methods to more researching fields.The main contents and results are listed as followings:
1.A convergence criterion is posed in numerical computation and thus the steps of time are made by the mode of self-adaptive control.Compared with experimental results,the numerical results can be accepted.In the picture about the curve of convergent history,the errors of time step n+1 and time step n keep little variation.
2.By the precondition of finite difference method,We use the AMG methods in the field of numerical prediction of turbulent flow.A detailed implement process of the AMG method is given in this paper.The process of generating coefficient matrix of different difference schemes is also given in this paper and the method is introduced in detail.The method about how the boundary conditions are embed in the algebraic systems is studied,we develop the joint programs of the AMG method and the program DLRAMG、DHRAMG.Under the same computational condition and control precision,the AMG method is more efficient,than the SOR method which was often used in the past and about one-third of the total CPU time can be saved.
3.Focus on the problem that the number of nodes of 13 points difference scheme is much and the computation cost much time,we design a new 5 points difference scheme.This new 5 points scheme has less non-zero elements than the 13 points scheme in the their systems and then the cost time for computation can be dccrcascd.Numerical experiments show that one third of CPU time can be saved.
4.In this paper,a new method is posed to gcncratc the BFC gird with the adjustable boundary mesh intervals and orthogonality.The real examples show that the grids can be generated by the method for the simple connected and mutli-connected regions with complicated boundary.
5.In addition,in this paper,a body-fitted non-orthogonal collocated grid system is generated. The standardκ-εmodel and the wall function are adopted.Numerical simulation is done for turbulent flow around 2-D single sinusoidal hill and two sinusoidal hills.Non-orthogonal collocated grid-based SIMPLE algorithm is adopted for solving the coupling system of the velocity and pressure equations.Simulation results agree well with the cxperimental data.
本文针对边界拟合曲线坐标贴体网格系统下代数多重网格方法在紊流数值预测中的应用问题,在何永森教授工作的基础上,详细探讨了代数多重网格方法在数值预测紊流中的实施过程,并围绕如何提高锥形渐扩管路内紊流数值仿真效率与精度展开研究,进行了一系列数值实验及与物理实验结果的比较,得到了一些有意义的成果,丰富和充实了代数多重网格方法,拓宽了代数多重网格方法在一些领域中的应用。具有理论和工程应用价值.主要内容和成果包括:
(一)原来的程序软件判断数值收敛与否是根据数值计算结果的情况分次沿时间步推进求解并以人工方式进行.本文对这一情况进行了改进,通过添加了一个判断收敛的子程序,变人工方式为自适应控制收敛方式.数值实验表明,用这种方式控制迭代次数,当迭代结束时,各种流动参数的误差(前后两时间步的差)曲线处于平稳并且很小,计算结果与实验结果较好符合。
(二)将AMG方法引入到紊流数值预测的有限差分计算领域,详细探讨了代数多重网格方法在紊流数值预测中的实施过程.研究了不同有限差分格式下,紊流模型离散得到的大规模代数系统对应系数矩阵的整合及边界条件的嵌入方法,讨论了系数矩阵的“三元组”压缩存储方式,利于节约内存及便于利用AMG方法求解。编制了代数多重网格方法数值求解紊流模型离散得到的大规模代数离散系统的程序接口;将代数多重网格方法与DLR、DHR型k—ε紊流模型.BFC法结合,应用编制的程序DLRAMG和DHRAMG对锥形渐扩管路内紊流进行了数值预测,将数值计算结果与物理实验结果进行了比较,多重网格方法的计算结果与实验结果符合较好。并且与POINT-SOR方法相比,可以节约近三分之一的CPU时间,提高了数值预测效率。
(三)针对原来十三点格式所用的模板节点多,方程组系数矩阵的带宽大,非零元多,数值求解费时问题,设计了一种基于五节点模板的新五点差分格式,并将其与AMG方法结合,进一步提高了紊流数值预测的效率.数值实验结果表明,在AMG方法求解的条件下,新五点差分格式比原十三点差分格式可以节约近三分之一的CPU机时。
(四)本文提出了一种可以同时实现控制网格正交性和任意控制边界网格间距的一种BFC网格生成的新方法。该方法可对生成的网格边界间距大小任意控制,同时生成的BFC网格还具有边界及内部较好的正交性.应用实例的计算结果表明,该方法能够对复杂边界的单连通域或多连通域生成较理想的BFC网格。
(五)另外,研究了k-ε紊流模型对数值预测正弦波壁流动的应用。基于有限体积法结合非正交同位网格系统,压力与速度耦合采用SIMPLE方法,对该种流动进行了数值预测,得到了与实验结果符合较好的计算结果。
Numerical predictions(or numerical simulations) for turbulent flow is the base project in fluid flow and heat transform.He yongsen et al find that fully developed incompressible turbulent flow in a conical diffuser having a total divergence of 8~0 and an area ratio of 4:1 can be simulated by a DLR turbulent model and it's BFC(Boundary-Fitted Coordinates) method,but the predictions for turbulent flow have low efficiency.Algebraic multi-grid methods are by far the most,efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs.Generally speaking,there are two types of multigrid methods:geometric-based approach and algebraic approach.The large computers have developed very fast and the function of computer have been improved in recent years,which make the multigrid methods vary from optimal algorithm theoretically to optimal algorithm in practice.Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares,it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid method(AMG) has become the hotspot due to the high performance and robustness.
In this paper,we make some in-depth studies for applying the AMG algorithms in numerical predictions for turbulent flow.A detailed discussion of implement procession of applying AMG methods to simulate the turbulent flow.Focus on improving the efficiency of simulation,we put forward some feasible methods.A number of numerical experiments have been performed.The computational results are compared with the results of experimental results,and some crucial numerical results are obtained.These researches will make the AMG algorithms richer and apply the AMG methods to more researching fields.The main contents and results are listed as followings:
1.A convergence criterion is posed in numerical computation and thus the steps of time are made by the mode of self-adaptive control.Compared with experimental results,the numerical results can be accepted.In the picture about the curve of convergent history,the errors of time step n+1 and time step n keep little variation.
2.By the precondition of finite difference method,We use the AMG methods in the field of numerical prediction of turbulent flow.A detailed implement process of the AMG method is given in this paper.The process of generating coefficient matrix of different difference schemes is also given in this paper and the method is introduced in detail.The method about how the boundary conditions are embed in the algebraic systems is studied,we develop the joint programs of the AMG method and the program DLRAMG、DHRAMG.Under the same computational condition and control precision,the AMG method is more efficient,than the SOR method which was often used in the past and about one-third of the total CPU time can be saved.
3.Focus on the problem that the number of nodes of 13 points difference scheme is much and the computation cost much time,we design a new 5 points difference scheme.This new 5 points scheme has less non-zero elements than the 13 points scheme in the their systems and then the cost time for computation can be dccrcascd.Numerical experiments show that one third of CPU time can be saved.
4.In this paper,a new method is posed to gcncratc the BFC gird with the adjustable boundary mesh intervals and orthogonality.The real examples show that the grids can be generated by the method for the simple connected and mutli-connected regions with complicated boundary.
5.In addition,in this paper,a body-fitted non-orthogonal collocated grid system is generated. The standardκ-εmodel and the wall function are adopted.Numerical simulation is done for turbulent flow around 2-D single sinusoidal hill and two sinusoidal hills.Non-orthogonal collocated grid-based SIMPLE algorithm is adopted for solving the coupling system of the velocity and pressure equations.Simulation results agree well with the cxperimental data.
引文
[1]Lander B E,Spalding D B.The Numerical Computation of Turbulent Flows.Compu.Methods in Appl.Mech.and Eng.3,New York:Northholl and Publishing Company,(1974),269-280
[2]何永森,刘邵英.机械管内流体数值预测[M].北京:国防工业出版社(第一版),1999:201-223
[3]He Yongsen,Liu Shaoying,Study on Diagnostic System for Numerical Prediction of Turbulent Flow in a Conical Diffuser:Effect of Model Constant and Model Function[C],Proceedings of the Fifth Asian International Conference of Fluid Machinery,Seoul,1997,247-252
[4]He Yongsen,Liu Shaoying,Kobayashi T,Lu Xiaoyong,Luo Jian,Numerical Prediction of Turbulent Flow in a Conical Diffuser:Overview of Each Parameter of the Flow[C],Proceedings of the sixth Asian International Conference of Fluid Machinery,MALAYSIA,2000,198-203
[5]He Yongsen,Kobayashi T,Morinishi Y,Numerical Prediction of Turbulent Flow in a Conical Diffuser Using k-ε Model[J],Acta Meshanica Sinica 8(2):117-126,1992
[6]He Yongsen,Kobayashi T,Morinishi Y,Numerical Prediction of Turbulent Flow in a Conical Diffuser Using Model for Near wall and Low Re Number[J],Suzhou,CMES,1993,593-599
[7]蒋光彪,用DLR型k-ε模型对逆压梯度内部紊流的数值预测:模型函数,网格配置和雷雷诺数的影响,2006.6
[8]孙勇,管路内逆压梯度流场数值仿真差分方法和网格条件影响的研究.湘潭大学硕士学位论文,2005.6
[9]何永森.蒋光彪,锥形渐扩管内紊流数值预测诊断系统的研究--模型函数和网格配置和雷诺数的影响[J].湘潭大学自然科学学报,2006,(2)
[10]A.Brant,S.F.McComick and J.W.Ruge.Algebriac Multigrid Methods(AMG)for sparse matrix equations.In D.J.Evans,editor,Sparsity and Its Applications.Cambridge University Press,Cambridge,1984
[11]M.Brezina,A.J.cleary,R.d.Falgoout,V.E.Henson,J.E.Jones,T.A.Manteuffel,S.F.McCormick and J.W.Ruge.Algebraic multigrid based on element interpolation (AMGe).SIAM J.Sci.Comput.22(5):1570-1592(electronic),2000
[12]舒适.几类基于部分几何和分析信息的代数多重网格方法及其应用(博士论文),2oo4
[13]K,Stuben,Algebraic Multigrid(AMG):An Inroduction with Applications,GMD Report 70,Nov,(1999),also available as appendix in the book"Multigrid" by U.Trottenberg,C.W.Oosterlee,A.Schuller,Academic Press,PP.413-532,2001
[14]肖映雄,代数多重网格算法研究及其在固体力学中的应用(博士论文),湘潭大学土木工程与力学学院,2006
[15]刘超群.多重网格方法及在计算流体力学中的应用[M],清华大学出版社,1995(3)
[16]X.zheng,C.Liao,C.Liu,C.H.Sung,T.T.Huang.Multigrid Computation of Incompressible Flows Using Two-Equation Turbulent Models:Part Ⅰ-Numerical Method,Journal of Fluids Engineering,December,vol.119/893(6),1997
[17]X.zheng,C.Liao,C.Liu,C.H.Sung,T.T.Huang.Multigrid Computation of Incompressible Flows Using Two-Equation Turbulent Models:Part ⅡApplications,Journal of Fluids Engineering,December,vol.119/893,1997
[18]邓启红,汤广发.多重网格方法在通风空调气流数值模拟中的应用,应用基础与工程科学学报,Vol6.NO.1 1998(4)
[19]刘朝霞,常谦顺.由扩散张量导出的各向异性扩散模型的隐式数值模拟,计算物理,22(4):365-368,2005
[20]徐树方,矩阵计算的理论和方法[M],北方大学出版社,1995
[21]任安禄.不可压缩粘性流场计算方法.北京:国防工业出版社(第1版),2003:36-50
[22]章梓雄,董曾南.粘性流体力学[M].北京:清华大学出版社(第一版),13-41,1998
[23]Thompson J F,Thames F C,Martin C W,TOMCAT-A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Any Number of Arbitrary Two-Dimensional Bodies,J.Comp.Phys(24)(1977),274-302.
[24]Thompson J F,Thames F C,Martin C W,Automatic Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Any Number of Arbitrary Two-Dimensional Bodies[J],J.Comp.Phys 24(15),299-310,1974
[25]Steger J L,Sorenson R L,Automatic Mesh-Point Clustering Near a Boundary in Grid Generation with Elliptic partial Differential Equations[J],J.Comp.Phys (33),405-410,1979
[26]Thomas J K,Middlecoeff J F.Direct control of the grid point distribution in meshes generated by elliptic equations AIAA J,1980.18:652-656.
[27]Hilgenstock A.,Elliptic generation of three dimensional grids for internal flow calculations[J].AIAA Journal.1981,10:153-157.
[28]陶文铨.计算传热学的近代进展[M].科学出版社,2001
[29]F.H.Harlow,J.E.Welch,Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.The Physics of Fluids,8(12):2182-2189,1965
[30]Okwuobi P.A.C,Azad R.S,Turbulcnt in a Conical Diffuser with Fully Developed Flow at Entry[J],J.Fluid.Mesh 57(3):603-622,1973
[31]Singh D,Azad R.S,Turbulent Kinetic Energy Balance in a Conical Diffuser[J],Proc.of Turbulent,21-33,1981
[32]Laufer J.The Structure of Turbulent in Fully Developed Pipe Flow[M].NACA Rep.1174 Washington:U.S.Government Printing Office,1955
[33]陈矛章.粘性流体动力学理论及紊流工程计算[M]北京:北京航空学院出版社(第一版),1986:234-264.
[34]张也影.流体力学[M].北京:高等教育出版社(第一版),1986:213-226
[35]吴子牛.计算流体力学基本原理[M].北京:科学出版社(第一版),2001:128-171
[36]陆金浦,关治.偏微分方程数值解法[M].北京:清华大学出版社(第二版),2004:126-141
[37]章本照,印建安,张宏基.流体力学数值方法[M].北京:机械工业出版社(第一版),2003:140-152
[38]刘顺隆,郑群.计算流体力学[M].哈尔滨:哈尔滨工程大学出版社(第一版),1998:190-310
[39]刘儒勋,舒其望.计算流体力学的若干新方法[M].北京:科学出版社(第一版),2003:13-25.
[40]王承尧,王正华,杨晓辉.计算流体力学及其并行算法[M].长沙:国防科技大学出版社,2000:36-82
[41]水鸿寿.一维流体力学差分方法[M].北京:国防工业出版社(第一版),1998:28-42
[42]顾尔柞.流体力学有限差分法基础[M].上海:上海交通大学出版社(第一版),1988:94-105
[43]李万平.计算流体力学[M].武汉:华中科技大学出版社(第一版),2004:161-204
[44]计算流体力学及应用,中国人民解放军总装备军事训练教材编辑委员会[M]北京:国防工业出版社(第一版),1998:121-138。
[45]于明,二维自适应结构网格的变分生成方法,计算物理21(1)(2004),27-34
[46]徐士良.FORTRAN常用算法程序集[M].北京:清华大学出版社(第一版),1992:26-39
[47]何光渝,高水利.Visual Fortran常用数值算法集[M].北京:科学出版社(第一版),2003:62-65
[48]黄卫星,陈文梅.工程流体力学[M].北京:化学工业出版社,2001:163-181
[49]Rousseau A.N.,Albright L.D.,and K.E.Torrance K.E.,A Short Comparison of Damping Functions of Standard Low-Reynolds-Number k-ε Models,Journal of Fluids Engineering[J],460/Vol.119,JUNE 1997
[50]HATTORI H.AND NAGANO Y.,Calculation of Turbulent Flows with Pressure Gradients Using a k-ε Model[J],JSME,SeriesB,Vol.38.No.4,1995
[51]Tomoya Houra,Toshihiro Tsuji,Yasutaka Nagano,Effects of adverse prssure gradient on quasi-coherent structures in turbulent boundary layer[J],Journal of Heat and Fluid Flow,21(2000) 304-311
[52]Yasutaka Nagano,Toshihiro Tsuji,Tomoya Houra,Structure of turbulent boundary layer subjected to adverse pressure gradient[J],Journal of Heat and Fluid Flow,19(1998) 563-572
[53]Laufer J,The Structure of Turbulent in Fully Developed Pipe Flow[J],NACA Rep(1174),Washington:U.S.G.P.O,(1955),6-15
[54]Nagano Y.and Tagawa,M.,An Improved From of the k-ε Model for Bundary Layer Flows[J],Trans.ASME,J.Fluids Eng.1990,112,33
[55]Myong,H.K.and Kasagi,N.,A New Approach to the Improvement of k-εTurbulent Model for wall-Bunded shear Flows[J],ASME,Int.J.SerⅡ,(33)(1),63,1990
[56]Miner,N.N.,Kim,J.,and Moin,P.,Near-wall k-ε Turbulent Moddeling[J],AIAA Journal,27(8).1068,1989
[57]Nagano Y,Hishida M.Improved From of the k-εModel for Wall Turbulent Shear Flows[J].TRrans.ASME,J.Fluids Eng.1987:109-156
[58]何勇,黄社华,张晓元,基于低雷诺数k-ε双层模型正弦波壁流动的数值模拟,武汉大学学报(工学版)Vol 37 No.4,Aug.2004
[59]帅石金,胡欲立,刘永长.改进TM法控制贴体网格节点分布的研究,Vol.25,No.3,Mar,1997.
[60]张涵信,沈孟育.计算流体力学差分方法的原理和应用[M].北京:国防工业出版社(第一版),2003:372-384
[61]傅德薰,马延文.计算流体力学[M].北京:高等教育出版社(第一版),2002:191-199
[62]马铁犹.计算流体力学[M].北京:北京航空学院出版社(第一版),1986:95-103
[63]Temam T.On an approximate solution of the Navier-Stokes equations by the method of fractronal steps[J].Archiv Ration Mech Anal,1969,32:377 385
[64]Fortin M,Peyret R,Temam R J.,Calcul des ecoulements D'un fluide visqueux[J],incomperssibl Mec.1971(8):337 342
[65]陈素琴,黄自萍,沈剑华,顾明.两串列方柱绕流的干扰数值研究.Vol 29 No.3Mar.2001
[66]彭文启,刘培斌,二维及三N-S方程的混合有限分析多重网格方法.[J]华北水利水电学院学报,Vol.19 No.3 Sept.1998(1)
[67]傅凯新,黄云清,舒适.数值计算方法,湖南科学技术出版社,2002
[68]黄云清,舒适,陈艳萍,金继承,文立平.数值计算方法,科学出版社,北京,2009
[69]J.Yan,F.Thiele,L.Xue.A modified full multigrid algorithm for the NavierStokes equations,Computers Fluids 36(2007) 445-454
[70]Feng Liu and XiaoQing Zheng.A Strongly Coupled Time-Marching Method for Solving the Navier-Stokes and k-ω Turbulent Model Equations with Multigrid Journal of computational Physics,128,289-300,Article NO.0211,1996
[71]Siddharth Thakur,*Jeffrey Wright,*Wei Shyy,*Jian Liu,*HongOuyang* and Thi Vu,Prog.Aerospace Sci.Development of Pressure-Based composite Multigrid Methods for Complex Fluid Flows.Vol.32,pp.313-375,1996 Copyright 1996
[72]U.Goldberg,O.Peroomian,S.Shakravarthy.A Wall-Distance-Free k-ε Model With Enhanced Near-Wall Treatment,Journal of Fluids Engineering,September 1998,vol.120:457-462
[73]Carl F.Ollivier-Gooch.Multigrid Acceleration of an upwind Euler Solver on unstructured Meshed,J,AIAA,33-10 P1822-1827,1995
[74]葛永斌,基于时间的驱动方腔的高精度多重网格方法数值模拟,工程热物理学报,2003:24(2):216-219.
[75]Hackbusch W and Trottenberg.Multigrid Methods.Springer-Verlag.Berlin,1982
[76]Brandt A.Multigrid Techniques:1984 Guide with Application to Fluid Dynamics.GMD Studien no.65,GMD,St.Augustin,Germany,1984.(17)Briggs W L.A Multigrid Tutorial.SIAM,Philadelphia,1987
[77]J.W.Ruge and K.Stuben.Algebraic Multigrid in Multigrid Methods.S.Mocormick,ed.k,SIAM,Philadelphia,PA,1987
[78]S.F.Mocormick.Multigrid Methods.SIAM,Philadelphia 1988
[79]J.Xu,Theory of Multilevel Methods.PHD thesis,Cornell University,Ithaca,N.Y.1989
[80]M.Griebel,T.Neunhoeffer and H.REgler.Algebraic Multigrid Methods for the solution of the N-S equations in complicated Domains.Int.J.Numer.Methods for Heat and Fluid,26,1998
[81]V.E.An Algebraic Multigrid Tutorial.LLNL,CASC,Apirl 1999(23)U.Trottenberg,C.W.Oosterlee and A.Schuler.Multigrid.Academic Press,2001
[82]G.Barles P.E.Souganidis,Convergence of Approximation Schemes for Fully Nonlinear Second Order Eqs.,Asymptotic Analysis 4(1991),271-283
[83]Michael G.Crandall,Hitoshi Ishii,and Pierre-Louis Lions,User's Guide to Viscosity Solutions of Second Order Partial differential Equations,Bulletin(New Series)of the American Mathematical Society,Vol.27,No.1,July 1992
[84]Zhou Y L.Applications of discrete functional analysis to the finite difference method.Beijing,International Academy Publishers,1999
[85]刘朝霞,常谦顺,吴声昌.带有时滞与正则项的图像处理模型的近似格式的收敛性,高等学校计算数学学报,2005,27(1):7-16
[86]Hsu C.A curvilinear-coordinate method for momentum,heat and mass transfer of irregular geometry[D].Ph D thesis,University of Minnesota,1981
[87]Prakash C.A finite element method predicting flow through ducts with arbitrary cross sections[D].Ph D thesis,University of Minnesota,1981
[88]Rhie C M.A Numerical Study of the flow past an isolated airfoil with separation,Urbana-Champaign[D],1981
[89]Rhie C M,Chow W L.A Numerical Study of the turbulent flow past an isolated airfoil with trailing edge separation[J].AIAAJ,21:1525-1552,1983
[90]A.D.Ferreira,A.M.G.Lopes,D.X.Viegas,A.C.M.Sousa.Experiment and numerical simulation of flow around two-dimensional hills[J].Journal of Wind Engineering and Industrial Aerodynamics,54/55 173-181,1995
[91]Muzaferija S.Adaptive finite volume method for flow predictions using unstructured meshes and multigrid approach[D].University of London,1994
[92]Ferziger J H,Peric M.Computational Methods for Fluid Dynamics[M].Berlin:Springer-Ver51ag,11983,21:1525-1552,2002
[93]何勇等.基于低雷诺数双层模型正弦波壁紊流的数值模拟[J].武汉大学学报(工学版).2004,37(4):11-15
[2]何永森,刘邵英.机械管内流体数值预测[M].北京:国防工业出版社(第一版),1999:201-223
[3]He Yongsen,Liu Shaoying,Study on Diagnostic System for Numerical Prediction of Turbulent Flow in a Conical Diffuser:Effect of Model Constant and Model Function[C],Proceedings of the Fifth Asian International Conference of Fluid Machinery,Seoul,1997,247-252
[4]He Yongsen,Liu Shaoying,Kobayashi T,Lu Xiaoyong,Luo Jian,Numerical Prediction of Turbulent Flow in a Conical Diffuser:Overview of Each Parameter of the Flow[C],Proceedings of the sixth Asian International Conference of Fluid Machinery,MALAYSIA,2000,198-203
[5]He Yongsen,Kobayashi T,Morinishi Y,Numerical Prediction of Turbulent Flow in a Conical Diffuser Using k-ε Model[J],Acta Meshanica Sinica 8(2):117-126,1992
[6]He Yongsen,Kobayashi T,Morinishi Y,Numerical Prediction of Turbulent Flow in a Conical Diffuser Using Model for Near wall and Low Re Number[J],Suzhou,CMES,1993,593-599
[7]蒋光彪,用DLR型k-ε模型对逆压梯度内部紊流的数值预测:模型函数,网格配置和雷雷诺数的影响,2006.6
[8]孙勇,管路内逆压梯度流场数值仿真差分方法和网格条件影响的研究.湘潭大学硕士学位论文,2005.6
[9]何永森.蒋光彪,锥形渐扩管内紊流数值预测诊断系统的研究--模型函数和网格配置和雷诺数的影响[J].湘潭大学自然科学学报,2006,(2)
[10]A.Brant,S.F.McComick and J.W.Ruge.Algebriac Multigrid Methods(AMG)for sparse matrix equations.In D.J.Evans,editor,Sparsity and Its Applications.Cambridge University Press,Cambridge,1984
[11]M.Brezina,A.J.cleary,R.d.Falgoout,V.E.Henson,J.E.Jones,T.A.Manteuffel,S.F.McCormick and J.W.Ruge.Algebraic multigrid based on element interpolation (AMGe).SIAM J.Sci.Comput.22(5):1570-1592(electronic),2000
[12]舒适.几类基于部分几何和分析信息的代数多重网格方法及其应用(博士论文),2oo4
[13]K,Stuben,Algebraic Multigrid(AMG):An Inroduction with Applications,GMD Report 70,Nov,(1999),also available as appendix in the book"Multigrid" by U.Trottenberg,C.W.Oosterlee,A.Schuller,Academic Press,PP.413-532,2001
[14]肖映雄,代数多重网格算法研究及其在固体力学中的应用(博士论文),湘潭大学土木工程与力学学院,2006
[15]刘超群.多重网格方法及在计算流体力学中的应用[M],清华大学出版社,1995(3)
[16]X.zheng,C.Liao,C.Liu,C.H.Sung,T.T.Huang.Multigrid Computation of Incompressible Flows Using Two-Equation Turbulent Models:Part Ⅰ-Numerical Method,Journal of Fluids Engineering,December,vol.119/893(6),1997
[17]X.zheng,C.Liao,C.Liu,C.H.Sung,T.T.Huang.Multigrid Computation of Incompressible Flows Using Two-Equation Turbulent Models:Part ⅡApplications,Journal of Fluids Engineering,December,vol.119/893,1997
[18]邓启红,汤广发.多重网格方法在通风空调气流数值模拟中的应用,应用基础与工程科学学报,Vol6.NO.1 1998(4)
[19]刘朝霞,常谦顺.由扩散张量导出的各向异性扩散模型的隐式数值模拟,计算物理,22(4):365-368,2005
[20]徐树方,矩阵计算的理论和方法[M],北方大学出版社,1995
[21]任安禄.不可压缩粘性流场计算方法.北京:国防工业出版社(第1版),2003:36-50
[22]章梓雄,董曾南.粘性流体力学[M].北京:清华大学出版社(第一版),13-41,1998
[23]Thompson J F,Thames F C,Martin C W,TOMCAT-A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Any Number of Arbitrary Two-Dimensional Bodies,J.Comp.Phys(24)(1977),274-302.
[24]Thompson J F,Thames F C,Martin C W,Automatic Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Any Number of Arbitrary Two-Dimensional Bodies[J],J.Comp.Phys 24(15),299-310,1974
[25]Steger J L,Sorenson R L,Automatic Mesh-Point Clustering Near a Boundary in Grid Generation with Elliptic partial Differential Equations[J],J.Comp.Phys (33),405-410,1979
[26]Thomas J K,Middlecoeff J F.Direct control of the grid point distribution in meshes generated by elliptic equations AIAA J,1980.18:652-656.
[27]Hilgenstock A.,Elliptic generation of three dimensional grids for internal flow calculations[J].AIAA Journal.1981,10:153-157.
[28]陶文铨.计算传热学的近代进展[M].科学出版社,2001
[29]F.H.Harlow,J.E.Welch,Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.The Physics of Fluids,8(12):2182-2189,1965
[30]Okwuobi P.A.C,Azad R.S,Turbulcnt in a Conical Diffuser with Fully Developed Flow at Entry[J],J.Fluid.Mesh 57(3):603-622,1973
[31]Singh D,Azad R.S,Turbulent Kinetic Energy Balance in a Conical Diffuser[J],Proc.of Turbulent,21-33,1981
[32]Laufer J.The Structure of Turbulent in Fully Developed Pipe Flow[M].NACA Rep.1174 Washington:U.S.Government Printing Office,1955
[33]陈矛章.粘性流体动力学理论及紊流工程计算[M]北京:北京航空学院出版社(第一版),1986:234-264.
[34]张也影.流体力学[M].北京:高等教育出版社(第一版),1986:213-226
[35]吴子牛.计算流体力学基本原理[M].北京:科学出版社(第一版),2001:128-171
[36]陆金浦,关治.偏微分方程数值解法[M].北京:清华大学出版社(第二版),2004:126-141
[37]章本照,印建安,张宏基.流体力学数值方法[M].北京:机械工业出版社(第一版),2003:140-152
[38]刘顺隆,郑群.计算流体力学[M].哈尔滨:哈尔滨工程大学出版社(第一版),1998:190-310
[39]刘儒勋,舒其望.计算流体力学的若干新方法[M].北京:科学出版社(第一版),2003:13-25.
[40]王承尧,王正华,杨晓辉.计算流体力学及其并行算法[M].长沙:国防科技大学出版社,2000:36-82
[41]水鸿寿.一维流体力学差分方法[M].北京:国防工业出版社(第一版),1998:28-42
[42]顾尔柞.流体力学有限差分法基础[M].上海:上海交通大学出版社(第一版),1988:94-105
[43]李万平.计算流体力学[M].武汉:华中科技大学出版社(第一版),2004:161-204
[44]计算流体力学及应用,中国人民解放军总装备军事训练教材编辑委员会[M]北京:国防工业出版社(第一版),1998:121-138。
[45]于明,二维自适应结构网格的变分生成方法,计算物理21(1)(2004),27-34
[46]徐士良.FORTRAN常用算法程序集[M].北京:清华大学出版社(第一版),1992:26-39
[47]何光渝,高水利.Visual Fortran常用数值算法集[M].北京:科学出版社(第一版),2003:62-65
[48]黄卫星,陈文梅.工程流体力学[M].北京:化学工业出版社,2001:163-181
[49]Rousseau A.N.,Albright L.D.,and K.E.Torrance K.E.,A Short Comparison of Damping Functions of Standard Low-Reynolds-Number k-ε Models,Journal of Fluids Engineering[J],460/Vol.119,JUNE 1997
[50]HATTORI H.AND NAGANO Y.,Calculation of Turbulent Flows with Pressure Gradients Using a k-ε Model[J],JSME,SeriesB,Vol.38.No.4,1995
[51]Tomoya Houra,Toshihiro Tsuji,Yasutaka Nagano,Effects of adverse prssure gradient on quasi-coherent structures in turbulent boundary layer[J],Journal of Heat and Fluid Flow,21(2000) 304-311
[52]Yasutaka Nagano,Toshihiro Tsuji,Tomoya Houra,Structure of turbulent boundary layer subjected to adverse pressure gradient[J],Journal of Heat and Fluid Flow,19(1998) 563-572
[53]Laufer J,The Structure of Turbulent in Fully Developed Pipe Flow[J],NACA Rep(1174),Washington:U.S.G.P.O,(1955),6-15
[54]Nagano Y.and Tagawa,M.,An Improved From of the k-ε Model for Bundary Layer Flows[J],Trans.ASME,J.Fluids Eng.1990,112,33
[55]Myong,H.K.and Kasagi,N.,A New Approach to the Improvement of k-εTurbulent Model for wall-Bunded shear Flows[J],ASME,Int.J.SerⅡ,(33)(1),63,1990
[56]Miner,N.N.,Kim,J.,and Moin,P.,Near-wall k-ε Turbulent Moddeling[J],AIAA Journal,27(8).1068,1989
[57]Nagano Y,Hishida M.Improved From of the k-εModel for Wall Turbulent Shear Flows[J].TRrans.ASME,J.Fluids Eng.1987:109-156
[58]何勇,黄社华,张晓元,基于低雷诺数k-ε双层模型正弦波壁流动的数值模拟,武汉大学学报(工学版)Vol 37 No.4,Aug.2004
[59]帅石金,胡欲立,刘永长.改进TM法控制贴体网格节点分布的研究,Vol.25,No.3,Mar,1997.
[60]张涵信,沈孟育.计算流体力学差分方法的原理和应用[M].北京:国防工业出版社(第一版),2003:372-384
[61]傅德薰,马延文.计算流体力学[M].北京:高等教育出版社(第一版),2002:191-199
[62]马铁犹.计算流体力学[M].北京:北京航空学院出版社(第一版),1986:95-103
[63]Temam T.On an approximate solution of the Navier-Stokes equations by the method of fractronal steps[J].Archiv Ration Mech Anal,1969,32:377 385
[64]Fortin M,Peyret R,Temam R J.,Calcul des ecoulements D'un fluide visqueux[J],incomperssibl Mec.1971(8):337 342
[65]陈素琴,黄自萍,沈剑华,顾明.两串列方柱绕流的干扰数值研究.Vol 29 No.3Mar.2001
[66]彭文启,刘培斌,二维及三N-S方程的混合有限分析多重网格方法.[J]华北水利水电学院学报,Vol.19 No.3 Sept.1998(1)
[67]傅凯新,黄云清,舒适.数值计算方法,湖南科学技术出版社,2002
[68]黄云清,舒适,陈艳萍,金继承,文立平.数值计算方法,科学出版社,北京,2009
[69]J.Yan,F.Thiele,L.Xue.A modified full multigrid algorithm for the NavierStokes equations,Computers Fluids 36(2007) 445-454
[70]Feng Liu and XiaoQing Zheng.A Strongly Coupled Time-Marching Method for Solving the Navier-Stokes and k-ω Turbulent Model Equations with Multigrid Journal of computational Physics,128,289-300,Article NO.0211,1996
[71]Siddharth Thakur,*Jeffrey Wright,*Wei Shyy,*Jian Liu,*HongOuyang* and Thi Vu,Prog.Aerospace Sci.Development of Pressure-Based composite Multigrid Methods for Complex Fluid Flows.Vol.32,pp.313-375,1996 Copyright 1996
[72]U.Goldberg,O.Peroomian,S.Shakravarthy.A Wall-Distance-Free k-ε Model With Enhanced Near-Wall Treatment,Journal of Fluids Engineering,September 1998,vol.120:457-462
[73]Carl F.Ollivier-Gooch.Multigrid Acceleration of an upwind Euler Solver on unstructured Meshed,J,AIAA,33-10 P1822-1827,1995
[74]葛永斌,基于时间的驱动方腔的高精度多重网格方法数值模拟,工程热物理学报,2003:24(2):216-219.
[75]Hackbusch W and Trottenberg.Multigrid Methods.Springer-Verlag.Berlin,1982
[76]Brandt A.Multigrid Techniques:1984 Guide with Application to Fluid Dynamics.GMD Studien no.65,GMD,St.Augustin,Germany,1984.(17)Briggs W L.A Multigrid Tutorial.SIAM,Philadelphia,1987
[77]J.W.Ruge and K.Stuben.Algebraic Multigrid in Multigrid Methods.S.Mocormick,ed.k,SIAM,Philadelphia,PA,1987
[78]S.F.Mocormick.Multigrid Methods.SIAM,Philadelphia 1988
[79]J.Xu,Theory of Multilevel Methods.PHD thesis,Cornell University,Ithaca,N.Y.1989
[80]M.Griebel,T.Neunhoeffer and H.REgler.Algebraic Multigrid Methods for the solution of the N-S equations in complicated Domains.Int.J.Numer.Methods for Heat and Fluid,26,1998
[81]V.E.An Algebraic Multigrid Tutorial.LLNL,CASC,Apirl 1999(23)U.Trottenberg,C.W.Oosterlee and A.Schuler.Multigrid.Academic Press,2001
[82]G.Barles P.E.Souganidis,Convergence of Approximation Schemes for Fully Nonlinear Second Order Eqs.,Asymptotic Analysis 4(1991),271-283
[83]Michael G.Crandall,Hitoshi Ishii,and Pierre-Louis Lions,User's Guide to Viscosity Solutions of Second Order Partial differential Equations,Bulletin(New Series)of the American Mathematical Society,Vol.27,No.1,July 1992
[84]Zhou Y L.Applications of discrete functional analysis to the finite difference method.Beijing,International Academy Publishers,1999
[85]刘朝霞,常谦顺,吴声昌.带有时滞与正则项的图像处理模型的近似格式的收敛性,高等学校计算数学学报,2005,27(1):7-16
[86]Hsu C.A curvilinear-coordinate method for momentum,heat and mass transfer of irregular geometry[D].Ph D thesis,University of Minnesota,1981
[87]Prakash C.A finite element method predicting flow through ducts with arbitrary cross sections[D].Ph D thesis,University of Minnesota,1981
[88]Rhie C M.A Numerical Study of the flow past an isolated airfoil with separation,Urbana-Champaign[D],1981
[89]Rhie C M,Chow W L.A Numerical Study of the turbulent flow past an isolated airfoil with trailing edge separation[J].AIAAJ,21:1525-1552,1983
[90]A.D.Ferreira,A.M.G.Lopes,D.X.Viegas,A.C.M.Sousa.Experiment and numerical simulation of flow around two-dimensional hills[J].Journal of Wind Engineering and Industrial Aerodynamics,54/55 173-181,1995
[91]Muzaferija S.Adaptive finite volume method for flow predictions using unstructured meshes and multigrid approach[D].University of London,1994
[92]Ferziger J H,Peric M.Computational Methods for Fluid Dynamics[M].Berlin:Springer-Ver51ag,11983,21:1525-1552,2002
[93]何勇等.基于低雷诺数双层模型正弦波壁紊流的数值模拟[J].武汉大学学报(工学版).2004,37(4):11-15