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1、湍流模型什么是湍流流动？n湍流流动有一系列的特征: 1. 不规则的, 随意的, 混乱的; 包括一系列尺寸不同的漩涡. 2. 扩散能力增加. 3. 大Reynolds数. 4. 三维. 5. 能量耗散; 小漩涡中的运动能转换成内能. 6. 连续介质. 小湍流的尺度也比分子大得多，所以可以认为流动是连续介质. Kolmogorov理论n较大尺寸的漩涡从主流吸取运动能量;n大尺度的漩涡运动能传递到较小尺度的漩涡中;n大尺度漩涡通过瀑布过程将运动能传递到小尺度漩涡中;n对于尺寸非常小的漩涡, 摩擦力 (粘性应力) 变得非常大, 运动能转化 (耗散) 成为内能. nKolmogorov 简历 (网上摘录
2、)根据 B. V. Gnedenko 在 Kolmogorov 70 寿辰时的讲演，Kolmogorov于1903年诞生于俄罗斯的村镇（现在为市）Tambov。父亲是农学家，母亲在生下Kolmogorov后不久便离开人世，他是被叔母等抚养长大的。1920年（17岁）进入莫斯科大学之前，他当过列车上的乘务员，业余时间写了关于牛顿力学定律的论文，论文的原稿未能保存下来，但我们可以想象他是多么早熟的天才。那时，俄国革命（1917）已经爆发，我很想知道他当时所处的环境，很遗憾没有有关的资料。 1920年进入莫斯科大学，最初对俄国的历史感兴趣，还调查了1516世纪的诺布哥罗德的财产登记。以后参加了 V.
3、V.Stepanov的傅里叶级数（三角级数）讨论班，并于1922年（19岁）写出了关于傅里叶级数，解析集合的著名论文，震动了学术界。其后犹如天马行空，连续发表了许多重要的研究成果。1925年莫斯科大学毕业，1931年当大学教授，1933年任大学数学研究所所长，1937年成为苏联科学院院士。至1987年逝世止，对数学的研究教育作出了很多重大的贡献 Kolmogorov 理论湍流尺度 大尺度漩涡是由流动几何形状来确定的,L. 通过量纲分析确定Kolmogorov 尺度 =(n3/e)1/4 ; t =(n/e)1/2; u =(ne)1/4, kolmogorov 长度尺度n, 运动粘性 m2/s
4、e, 能量耗散率, m2/s3 (单位时间单位质量的能量) L/ = Re3/4湍流尺度n大尺度漩涡的能量损失正比例于 l/u, 定义式为: e =O(u2/(l/u) =O(u3/l)湍流能量分布 uiui = 0E(k)dk分解瞬时变量 nU = U + u”nP = P + p”原因:o我们关心平均值，而不是瞬态值.o非实际的精细网格求解所有尺度的湍流及高分辨率平均化过程 f(xi,t) = + f(xi,t) 瞬时值 平均值 变动值时间平均过程W 是权重函数Reynolds平均和平均操作符规则n W=1; f = f + f” with f” =0 质量权重 或 Favre平均 n对于
5、变量 f, 只能通过相应的外延性质得到守恒方程, rf.n对于变密度来说, 这是非线性组合式和 Reynolds平均得到一个新的未知的修正系数, r”f” .n为了避免这个, Favre平均定义为 W = r / r Favre平均 f = rf/ r + f = f + f with f = rf/ r =0湍流封闭问题mb=2/3m, dij Kronecker函数 delta (dij =1 for i=j and dij = 0 for i=j)时间平均时间平均N-S 方程方程时间N-S方程 对于稳态、2维边界层流动:nVU ; d/dx d/dy 湍流模型 平均流动方程, 是瞬时 N
6、-S 方程的时间平均结果, 引入Reynolds 应力的6个应力分量, , , , , , . 怎样把这些未知量和平均流动量联系起来 ?nEddy Viscosity models(涡粘性模型)nReynolds Stress models(Reynolds应力模型)nLarge Eddy Simulation models(大涡模型模型) 涡粘性模型meff = m + mt mt = rCm vt ltBoussinesq假设假设标准k-e模型n k = (uiui) ; n mt = r Cm k2/e = r Cm1/4k1/2 ln l = Cm3/4 (k3/2/e) (m2/s3
7、)标准k-e模型k = m + mt/k; e = m + mt/e(不可压)标准k-e模型n湍流模型常量k-e e 变异变异n线性模型 o k-e 模型o 低Reynolds 数模型: Ce1f1; Ce2f2; mt=Cmfmrk2/e f1,f2, fm f(Re), 对于高 Reynolds数模型,方程 f1,f2, fm 是一样的. o Renormalisation group (RNG) k-e 模型.o Chens k-e模型 (生成时间尺度, k/P 和 耗散时间尺度, k/e)o V2F (V2, 垂直壁面湍流强度)o 两层模型 (k-e + 壁面附近低Reynolds 数
8、)n非线性模型o 二次和三次 k-e 模型o 低 Reynolds数 二次和三次模型o Sugas 二次和三次模型 k-e e 变异模型变异模型nLinear models o k-e modelo low Reynolds number model: Ce1f1; Ce2f2; mt=Cmfmrk2/e f1,f2, fm f(Re), for high-turbulence Reynolds number, the functions , f1,f2, fm tend to unity. o Renormalisation group (RNG) k-e model.o Chens k-e
9、 model (the production time scale, k/P and dissipation time scale, k/e)o V2F (V2, wall-normal turbulence intensity)o Two-layer models (k-e + low Reynolds variant near wall)nNon-linear modelso Quadratic and cubic k-e modelso Low Reynolds number quadratic and cubic modelso Sugas model in both quadrati
10、c and cubic forms RNG k-e e model (a) 模型常量从k-e e 模型中得到不同值; 并且(b) the dissipation-rate transport equation includes an additional source term per unit volume: Se = - raerae2/k where a a = Cm 3 (1- / 0)/(1+b b 3 ), 0 = 4.38, and b b = 0.012. = S*k/e where S2 = 2 SijSij and Sij = 0.5 * (dUi/dXj + dUj/dX
11、i) nThe additional source term becomes significant for flows with large strain rates, i.e. when 0. The parameter is a measure of the ratio of the turbulent to mean time scale. nIn the limit of weak strain where S and tends to zero, the additional source term tends to zero and the original form of th
12、e k-e model is recovered. Other eddy viscosity modelsnK-w w model (w = e/k)nK-l model nSpalart-Allmaras model (one equation model, solving the transport equation for the turbulent viscosity, nt)Wall boundary conditionsnTwo-layer model and Low Reynolds number modelnThe Log-Law wall functionTwo-layer
13、modelnNWL (Near Wall Layer) requires 15 grids which should be specifiednY+ 3nThe switching criterion depends on the low-Reynolds number modelWall functionnMain assumptionsoVariations in velocity etc. are predominantly normal to the wall, leading to one-dimensional behaviour.oEffects of pressure grad
14、ients and body forces are negligibly small, leading to uniform shear stress in the layer.oShear stress and velocity vectors are aligned and unidirectional throughout the layer.oA balance exists between turbulence energy production and dissipation.oThere is a linear variation of turbulence length sca
15、les.The three near-wall regionsVelocity boundary conditionsThe boundary conditions for turbulence quantities k+ = Cm -1/2 (equilibrium) e+ = Cm3/4/k where k+ = kr/tw e+ = ey/k3/2The first grid near the wall Where the Eddy viscosity models failednEVM does not predict all Reynolds stresses (shear stre
16、sses and normal stresses), but predicts only shear stresses, it works when gradient production dominates, so this will rule out many flows involving strong buoyancy, also flows involving separation, recirculations where the local turbulence is convected in from somewhere else.Reynolds stress modelTh
17、e Boussinesq assumption is not used, but an equation for the stress tensor is derived from the Navier-Strokes equation.Reynolds stress modelsnPij is the production of uiuj;nij is the pressure-strain term, which promotes isotropy of the turbulence;neij is the dissipation of uiuj;nCij and Dij are the
18、convection and diffusion, respectively, of uiuj We need models for the unknowns, ij , Dij and eijReynolds stress models vs. Eddy viscosity modelsAdvantages with EVM:1) simple due to the use of an isotropic eddy viscosity;2) work reasonably well for a large number of engineering flows;3) less computi
19、ng cost and easy to converge.Disadvantages with EVM:1) isotropic and thus not good in predicting normal stresses(u2,v2,w2);2) as a consequence of 1), it is unable to account for curvature effects;3) as a consequence of 1), it is unable to account for irrotational strain. Reynolds stress models vs. E
20、ddy viscosity modelsAdvantages with RSM1) the production terms need not be modelled;2) It can selectively augument or damp the stresses due to curvature effects, acceleration/retardation, swirling flow, buoyancy etc.Disadvantages with RSM1) complex and difficult to implement;2) numerically unstable;
21、3) much more computing cost.The LES modelnTo directly resolve the large eddiesnTo model the subgrid small eddiesTo separate the flow fieldnSelect a filter function G (spatial averaging);nDefine the resolved-scale (large-eddy):nFind the unresolved-scale (SGS)Examples of filter functionGaussianExample
22、: a 1-D flow fieldLarge eddiesApply filterReynolds averaging (RAN)Mean value (non-turbulent)Apply ensemble averagingnSGS stresses go to zero if the mesh size goes to zero. LES equationsApply filter GSGS stressesSmagorinsky modelntij = - nt (dui/dxj + duj/dxi) +2/3kdijnnt = Cs 2(sijsij)1/2nk = Ck 2(s
23、ijsij)ne = Ce k3/2/nCk = 0.202nCs = 0.02 varies with applicationnCe = 2Ck-3/2Csn = min(ky,V1/3) (V-mesh volume)LES near wall treatmentn1) Log-law wall function;n2) Van Driest-style damping function mt/m = k yw+ (1-exp(-yw+/A)2 Thank you for your attention and any questions此此课课件下件下载载可自行可自行编辑编辑修改，修改，仅仅供参考！供参考！感感谢谢您的支持，我您的支持，我们们努力做得更好！努力做得更好！谢谢谢谢
