柱坐标系和球坐标系下NS方程的直接推导

Derivation of 3D Euler Equation in Cylindrical coordinates Derivation of 3D Euler and Navier-Stokes Equations in Cylindrical Coordinates Dingxi Wang School of Engineering, Durham University Contents 1. Derivation of 3D Euler Equation in Cylindrical coordinates 2. Derivation of Euler Equation in Cylindrical coordinates moving at in tangential direction 3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates 1. Derivation of 3D Euler Equation in Cylindrical coordinates Euler Equation in Cartesian coordinates (1.1) Where Conservative flow variables Inviscid/convective flux in x direction Inviscid/convective flux in y direction inviscid/convective flux in z direction And their specific definitions are as follows , , , Total enthalpy Some relationship We want to perform the following coordinates transformation Because According to Cramer’s ruler, we have (1.2.1) (1.2.2) Where Similar to the above (1.2.3) (1.2.4) In addition, the following relations hold between cylindrical coordinate and Cartesian coordinate , , , , , (1.3) (1.4.1) (1.4.2) Derivation Multiplying the both side of equation (1.1) by and applying equalities (1.4.1) and (1.4.2) gives, (1.5) Differentiating the following w.r.t. time gives , , (1.6.1) (1.6.2) Expanding the term and applying the relationships (1.6) yields, (1.7.1) Expanding the term and applying the relationships (1.6) yields, (1.7.2) Substituting relationships (1.7) into equation (1.5) and rearranging gives, (1.8) As we can see from expressions (1.7), the momentum equations in radial and tangential directions contain velocities in Cartesian coordinate; we need to replace them with corresponding variables in cylindrical coordinate. Writing down the momentum equations in radial and tangential directions as follows, (1.9.1) (1.9.2) Multiplying (1.9.1) by and (1.9.2) by , then summing up and applying expressions (1.6) and rearranging yields (1.10.1) Multiplying (a) by and (b) by , then summing up and applying expressions (1.6) yields, (1.10.2) Replacing (1.10) with (1.9) and rearranging equation (1.8) gives (1.11) Where , ,, , Note: different from Euler equation in Cartesian coordinates, the Euler equation in cylindrical coordinates contains source terms from momentum equations in radial and tangential equations. 2. Derivation of Euler Equation in Cylindrical coordinates moving at in tangential direction Where , , , , , , , , , , , Then equation (1.11) can be written as follows (2.1) Where Equation (2.1) adopts rotating coordinates but the variables are measured in absolute cylindrical coordinates. 3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates 3D Navier-Stokes Equations in Cartesian coordinates (3.1) Where , , , , , , , , , , , , In the following derivation, only viscous terms will be derived from Cartesian coordinates to cylindrical coordinates, those inviscid terms having been derived in section 1 will be not repeated. Replacing with gives (3.2.1) Replacing with gives (3.2.2) Multiplying equation (3.1) by , the viscous terms are gives as follows (omitting the negative sign before it from simplicity), (3.3) (3.4.1) , (3.4.2) , (3.4.3) ,(3.4.4) (3.4.5) (3.4.6) Expanding expression (3.3) gives, (3.5) =〉 (3.6.1) =〉 (3.6.2) (3.6.3) (3.7.1) Divergence in Cartesian Coordinates (3.7.2) Divergence in cylindrical coordinates (3.7.3) (3.8.1) (3.8.2) (3.9.1) (3.9.2) As we can see from the above that viscous terms in expression (3.5) for the momentum equation in axial/x direction and energy equation can be expressed in variables in cylindrical coordinates, while the viscous terms in (3.5) for momentum equations in radial and tangential directions still contain variables in Cartesian coordinates. Similar manipulation to (1.10) will be adopted in the following. Writing out the viscous terms for momentum equations in radial and tangential coordinates as follows, (3.10.1) (3.10.2) Multiplying (3.10.1) by and multiplying (3.10.2) by , then summing up and rearranging gives, (3.11.1) Multiplying (3.10.1) by and multiplying (3.10.2) by , then summing up and rearranging gives, (3.11.2) (3.12.1) (3.12.2) (3.12.3) (3.12.4) Substituting (3.6.1), (3.6.2) and (3.12) into expressions (3.11) and rearranging yields, (3.13.1) (3.13.2) Making use of expressions (3.4.1), (3.6.1), (3.6.2), (3.8.1), (3.8.2), (3.9.1), (3.9.2), (3.13.1) and (3.13.2), we can get the final expression of 3D Navier-Stokes Equation in cylindrical coordinates as follows, 3D Navier-Stokes Equation in cylindrical coordinates ， , , , ， , If the moment of momentum equation is adopted to replace the tangential momentum equation, its expression will be simpler. Now for the moment equation, there is no source term. ， , , , ， , For 2D axisymmetric flow field, the tangential momentum equation or moment equation can be omitted as follows, 2D axisymmetric equation in cylindrical coordinate ， ,, , ， [分享]CFD中的湍流模型 流体力学是力学的一个重要分支，它是研究流体（包括液体和气体）这样一个连续介质的宏观运动规律以及它与其他运动形态之间的相互作用的学科，在现代科学工程中具有重要的地位。宏观上讲，黏性流体的流动形态有三种：层流、湍流以及从层流到湍流的转捩。从工程应用的角度看，大多数情况下转捩过程对流体流动的影响不大可以忽略，层流在很少情况下才出现，而在自然界和工程中最普遍存在的是湍流，因此湍流是科学家和工程师研究的重点。湍流理论的研究主要集中在两个方面：一是湍流的触发；二是湍流的描述和湍流问题的求解。 对于土木工程中出现的湍流问题，其求解方法可归纳为四种：理论分析、风洞实验、现场测试和数值模拟。四种方法相互补充，以风洞实验和现场测试为主，理论分析和数值模拟为辅。数值模拟又称数值风洞，它的出现才十几年却取得迅猛发展，是目前数值计算领域的热点之一，它是数值计算方法、计算机软硬件发展的结果。我们知道，描述流体运动（层流）的流体力学基本方程组是封闭的，而描述湍流运动的方程组由于采用了某种平均（时间平均或网格平均等）而不封闭，须对方程组中出现的新未知量采用模型而使其封闭，这就是CFD中的湍流模型。湍流模型的主要作用是将新未知量和平均速度梯度联系起来。目前，工程应用中湍流的数值模拟主要分三大类：直接数值模拟（DNS）；基于雷诺平均N-S方程组（RANS）的模型和大涡模拟（LES）。 DNS是直接数值求解N-S方程组，不需要任何湍流模型，是目前最精确的方法。其优点在于可以得出流场内任何物理量（如速度和压力）的时间和空间演变过程，旋涡的运动学和动力学问题等。由于直接求解N-S方程，其应用也受到诸多方面的限制。第一：计算域形状比较简单，边界条件比较单一；第二：计算量大。影响计算量的因素有三个：网格数量、流场的时间积分长度（与计算时间长度有关）和最小旋涡的时间积分长度（与时间步长有关），其中网格数量是重要因素。为了得到湍流问题足够精确的解，要求能够数值求解所有旋涡的运动，因此要求网格的尺度和最小旋涡的尺度相当，即使采用子域技术，其网格规模也是巨大的。为了求解各个尺度旋涡的运动，要求每个方向上网格节点的数量与Re3/4成比例，考虑一个三维问题，网格节点的数量与Re9/4成比例。目前，DNS能够求解Re(104)的范围。 基于RANS的湍流模型采用雷诺平均的概念，将物理量区分为平均量和脉动量，将脉动量对平均量的影响用模型表示出来。目前，基于RANS方程已经发展了许多模型，几乎能对所有雷诺数范围的工程问题求解，并得出一些有用的结果。其缺点在于：第一：不同的模型解决不同类型的问题，甚至对于同一类型的问题，对应于不同的边界条件需要修改模型的常数；第二：由于不区分旋涡的大小和方向性，对旋涡的运动学和动力学问题考虑不足，不能用来对流体流动的机理进行描述。 LES介于以上两种方法之间，具有两种方法的优点：将旋涡区分为大涡和小涡，对大涡直接求解，而对小涡采用模型。我们知道，大涡在流场中是能量的主要携带者，对流动具有决定性作用，由于受到边界条件的影响，不同的流场类型差异性很大，需要直接求解；小涡对湍流应力的影响很小，由于受到分子之间黏性的影响具有各相同性，适宜于模型化。这样，相比RANS的模型，LES具有通用型。目前能够直接求解范围Re(106)。随着壁面层（wall-layer）模型的发展，可以求解更高雷诺数的问题。 以上是对CFD中的模型的概述，抛砖引玉，欢迎交流。 参考文献（全为综述文献，欢迎索取：mefpz@sina.com） [1] Shuzo Murakami. Current status and future trends in computational wind engineering. J. Wind Eng. Ind. Aerodyn., 1997, 67&68: 3-34. [2] Theodore Stathopoulos. Computational wind engineering: Past achievements and future challenges. J. Wind Eng. Ind. Aerodyn., 1997, 67&68: 509-532. [3] S. Murakami. Overview of turbulence models in CWE-1997. J. Wind Eng. Ind. Aerodyn., 1998, 74-76: 1-24. [4] A. Mochida, Y. Tominaga, S. Murakami et. al.. Comparison of various models and DSM applied to flow around a high-rise building: report on AIJ cooperative project for CFD prediction of wind environment. Wind and Structures, 2002, 5(2-4): 227-244. [5] Marcel Lesieur, Olivier Metais. New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid. Mech., 1996, 28: 45-82. [6] . U. Piomelli. Large-eddy simulation: achievements and challenges. Progress in Aerospace Sciences, 1999, 35: 335-362. [7] Charles Meneveau, Joseph Katz. Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid. Mech., 2000, 32: 1-32. [8] Ugo Piomelli, Elias Balaras. Wall-layer models for large-eddy simulations. Annu. Rev. Fluid. Mech., 2002, 34: 349-374.