计算流体力学.ppt
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1、计 算 流 体 力 学,Computational fluid dynamics,课时: 40小时 40 hours 教材:王新月, 杨青真 .计算流体力学基础, 西北工业大学讲义,西北工业大学出版社 Textbook: Wang.X.Y, Yang.Q.Z “Foundation of Computational fluid dynamics”,Lecture of NPU. 课程性质: 专业课 Specialty course 适用对象:硕士研究生 for Master Degree,基础要求: Requirements : 学过流体力学、粘性流体力学等专业课基础 Fluid Dynami
2、cs, Foundation Dynamics of Viscous Flow have been studied 学过数值分析、计算方法等数学基础课 Learn Numerical Analysis, Computational Method,主要内容: 1.计算流体力学的基础知识,差分形式逼近流体力学基本方程,包括差分逼近基础,流体力学基本方程的解,差分格式的构造。Includes foundation knowledge of Computational fluid dynamics, FD approach to FD basic Eq, solution of the FD Eqs,
3、 constitution of FD. 2.定常不可压势流的数值解法,包括不可压势流基本方程,源汇流动,旋成体绕流,及椭圆型微分方程数值解。 Numerical solution of steady incompressable potential flow, includes the basic Eqs of steady incompressable potential flow, source and sink flow, flow arround a rotational body.,3.特征线方法的概念和应用 Concept and application of character
4、istic line method 4.跨音速定常小扰动势流混合差分法及隐式近似因式分解Small perturbation method for steady transonic flow and Approximate Factorization(AF) 5.时间推进法:包括守恒的非定常欧拉方程组等 Time march methods, includes conservational unsteady Euler Eqs.,6.Navier Stokes 方程的数值解法,包括湍流模型理论,N-S方程的有限体积法,涡流函数解法。Numerical methods for Navier-St
5、okes Eqs, include turbulence models, finite volume method for N-S Eqs. 7.网格设计:包括集合生成方法,保角变换法,微分方程法,混合方法,动网格设计Mesh design includes geometric meshing method, angle conservation method, TTM method, Vortex streamline method,moving grids 8.流场计算中的新方法,包括TVD方法,ENO方法,NND格式谱方法,自适应网格,并行计算与向量计算,非机构网格及其应用Some ne
6、w methods computing the flow fields ,self adapt grids, parallel methods and vector computing, unstructured grid and its applications.,主要参考资料 References,1.书中各章所列 The references of every chapter. 2.张涵信 沈孟育计算流体力学:差分方法的原理和应用 国防工业出版社,2003年1月 Zhang han kin etc Computational Fluid Dynamics Fundamentals and
7、 Applications of Finite Difference Minitry Industry Press. 2003 BeiJing 4.John D. Anderson, JR. Computational fluid dynamics,the basics with application. MCGraw-Hill Apr.2002 计算流体力学入门,清华大学出版社,2003年4月,第一章 差分逼近基础及流体力学 基本方程的解,Chapter 1, Finite Differential Approach and the solution of the Basic Equatio
8、n of Fluid Dynamics 1-1 差分逼近基础 Element of Finite Differential Approach 一.流体力学问题的解( The solution of Fluid Dynamics question ),泛定方程 描述流动现象的一组封闭方程 The closed equations to describe fluid phenomenon, 描述运动的一般规律,不能确定物体形状和边界条件(初始条件)To describe the normal regulation, not define to a certain geometry and BC /
9、 IC,定解条件: 初始条件 过多则出现无解(不存在)confirm condition : initial condition : too more, BC / IC takes no solution 边界条件过少则出现很多解,即不唯一Boundary condition too less BC / IC load unique solution 解的连续性问题,定解条件的微小变化引起域内解的微小变化Continuity of solution , a little change of BC may lead to little change of solution,差分方程:微分方程的近
10、似逼近、近似FDE approach the PDE 数值解必须条件Condition needed for Numerical solution 适定性问题(有解) Confirmed solution (问题)解的性质 the feature of solution 实用的近似方案 be of a practical approach solution,4.近似方程适定,变量数目与方程数目相同Number of equations equal to number of variables 5.可行的求解代数方程组的方法:迭代方法(iterative method),直接求解( direct
11、ive solving method)Possible / valid method to solve linear equations. 6.具备计算条件(内存,速度等) computation facility 7.稳定性,收敛性和精度(h0,得到精确解)Stability, convergence, accuracy,二.微分方程解的存在性和唯一性 Existence and uniqueness of the PDE solution 1.物理过程:适定性Physic phenomena;fixed 2.数学方程:可解不适定Math equation ; possibility not
12、 fixed 3.原因:近似的数学方程忽略了一些次要影响因素 Reason; approximate math equation usually neglect some unimportant influence,4.数学上适定性问题:只能近似的反应物理现象 A fixed question in math ;can approximately reflect the physic phenomena 5.偏微分方程解的唯一性:数理方程重有详尽叙述 Uniqueness of a PDE, has been descript detailedly in Math 6.适定问题+定解条件 差分
13、方程数值唯一性 Fixed question +confirmed BC/IC the uniqueness of the related FDE 7.若微分方程的精确解是唯一的 稳定收敛解也是唯一的 If the solution of PDE is unique then the solution of FDE is unique,三.差分方程数值解收敛性 相容性和稳定性 Convergence consistency and stability 1.收敛性(convergence) 当时间步长和空间步长()趋于零,若差分方程的问题趋于偏微分方程(相同的适定条件,定解条件)When tim
14、e step and space step tend to zero ,the solution of FDE tend to the solution of PDE Lax等价定理 Lax equipollence theorem,2.相容性(consistency) 差分方程对微分方程的近似程序How approximate is the FDE to PDE 3.稳定性(stability) 描述差分解在计算过程中的发展To indicate the development of the error of FDE 误差对后续计算的形象问题(影响小时或者有界) It reflects th
15、e influence of error of the following computation,稳定:计算过程重误差逐渐消失或者有界 Stable ,the error disappear graduately or keep limited 稳定性分析方法 Methods for analysing stability 直观法(或称离散摄动法):观察计算引入的误差的发展过程In discrete perturbation (direct) method, to investigation the development procedure of computational error,矩
16、阵法(Matrix method) 较严格的方法,考虑了边界条件的影响 Strict method, the BC influence is considered 解得到最完整的稳定性估计 Can gain the most integrating (completed) estimation of stabling 用很多矩阵代数知识,使用困难 Refer to a lot knowledge about maxtix analysis,Von Nenmann方法 Von Nenmann method(Fourrie series) 优点:最常用,方便,可靠Advantage: most c
17、ommon,convenience,reliable 缺点:只能用在常数系数的线性初值问题Disadvatage:Only can be used for linear initial value equation with constant coefficient 变系数非线性及各种不同边界条件问题中的应用受限Limited usage for non-linear BC problem with different coefficient,线性化:局部线性化方程后可以使用 Linearized: usable for linearized equation 在网格点式边界点上可以用它得到有
18、用信息 To get useful message at grids and BC 它不仅提供误差影响的发展信息,而且还展现差分格式对解相位变化的作用 It provides the message development of the error Von Neumann方法揭示了误差发展的内部机理 Von Neumann method discovered the mechanism of the numerical error development,d.Hirt 方法(1968) Hirt method (1968) 改型:将差分方程各项用Taylor级数展开 Reformed type
19、 Eq: Reform the FDE using Taylor series expansion 分析改型方程的稳定性 To analyse the stability of the reformed Eq 优点:简单,对简单问题可以得到与其他方法相同结果 Advantage: simple, can gain the same results as other methods for a simple eq,缺点: 不如矩阵方法和傅里叶方法严谨和完整 Disadvantage: not so strict as Matix method and Fourrie series 方法的某些假定
20、的定义不清楚 Meaning of some assumer is not clear 对复杂问题的实用性尚待研究 The applicability for complex question is still to be investigated. 四 Lax定理 Lax Therem,1-2 流体力学基本方程的解 The solution of the Basic Equation of Fluid Dynamics,Euler 方程组的解 Euler Eqs solution 1.定常不可压流Euler方程 Euler Eqs solution for steady incompress
21、ible flow 无粘、定常 Inviscous, steady flow,2维Euler方程 2D Euler Eqs 拟线性方程组 Quasilinear 特征根 Character root 既不是双曲型,也不是椭圆型 Neither hyperbolic, nor elliptic,类型不确定 Type of the equation is uncertain 不能按某确定的方法给出适定性条件 Could not determine the fit condition using specified method 在无旋流中,可以引入势函数 In irrotational flow,
22、 the potential function can be introduced Where the denotes total pressure,这时的方程为Laplace方程,为椭圆型 The equation becomes a Laplace Eqs, it is elliptic 给定边界条件即可求出, 微分后可得到速度分量 The solution can be gained when the BC is specified, and the components of the velocity can be calculated. 定常不可压Euler方程只有在无旋条件下才有解
23、 Therefore,the solution of Euler Eqs exist only in irrotional flow.,由连续方程和无旋条件 Here with the continuity equation and irrotional flow,the equation for:,用流函数表示有旋流动方程 The equation for rotational flow using stream function 双曲型方程Cauchy边界问题有解 Hyperbolic Eqs with the Cauchy boundary value problem is solvab
24、le. 双曲型方程Dirichlet边界问题无解 Hyperbolic Eqs with the Dirichlet boundary value problem is unsolvable.,2.非定常不可压Euler方程 Euler equation for unsteady incompressible flow 三个自变量 t、x、y Three variables are t,x,y,特征方程 Eigenvalue 速度矢量 特征值矢量 Velocityvector Eigenvalue 类型不确定(不可压非定常流Euler方程),但下列情况有解: The type of the e
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