【作者】 刘正锋；

【导师】 王晓宏；

【作者基本信息】 中国科学技术大学 ， 工程热物理， 2008， 博士

【摘要】 基于湍流模式的数值模拟在解决工程实际中湍流问题时发挥了重要的作用。然而这些常用的湍流模型主要是通过量纲分析和数学物理性质的合理性来实现的,模型中的湍流常数利用典型的湍流实验数据来标定。随着湍流模型在工程中的应用日益广泛,湍流模型的合理性、准确性面临着考验。因此从理论上来建立湍流模型,对人们研究和认识湍流有着重要的意义。基于湍流与临界现象的相似性,湍流研究工作者将用于分析相变临界现象的重正化群(Renormalization group)方法应用到湍流研究中来。Forster,Nelson和Stephen(FNS)最早将重正化群方法用来分析Navier-Stokes方程。而后,Yakhot和Orszag(YO)较系统地利用重正化群方法分析了各向同性湍流,并取得了一系列成果。Rubinstein和Barton等开创性地利用Yakhot和Orszag发展的湍流重正化群理论对Reynolds应力剪切模型进行了推导,然而他们的工作中存着一些不合理的地方。本文将利用湍流重正化群理论对一些常用的剪切湍流模型进行分析,主要包括Reynolds应力非线性涡粘性模型和二阶矩封闭模型的分析,并从理论上求得包含在这些模型中的湍流常数。在Rubinstein和Barton等工作的基础上,本文首先直接利用重正化群方法对Reynolds应力的涡粘性模型进行了分析,从理论上得到了一个二阶非线性涡粘性模型;计算上,本文通过积分波数的变换,采用单一的值y=3,即ε=4,回避了Rubinstein和Barton以及Yakhot等工作中的同一参数采用不同取值的不自洽问题。理论分析所得的非线性涡粘性模型中不仅含有平均应变的平方项(▽U)~2,还含有平均速度的二阶导数项U▽~2U,这一项可以看成Reynolds应力“松弛”项,能更好的反映湍流运动的非线性本质,但是,该模型非线性项中少了平均应变率张量与平均涡量张量的相互作用项,这是该理论模型的一个缺陷。本文还对Reynolds应力微分方程进行理论分析,对其中的惯性项、速度与压力梯度关联项以及耗散项予以模拟,并从理论上计算了各湍流常数,得到了完整的二阶矩封闭模型。结果表明重正化群二阶矩模型和著名的LRR模型很接近。需要指出的是,本文通过对Reynolds应力微分方程中耗散项的非线性部分模拟,得到二阶矩模型中的各向同性回归项。工程实际中Reynolds数为有限值的剪切湍流,其大涡运动在空间和时间上存在结构,不再满足自相似性。因此,直接将Yakhot—Orszag湍流重正化群理论推广到剪切湍流场分析时,其理论基础不完善。剪切湍流场中,湍流平均场是缓慢变化的,脉动场是快速变化的,它们具有不同的时间和空间尺度。因此,本文引入Yoshizawa采用的双重尺度展开,区分脉动场(快变量)和平均场(缓变量),并利用尺度比例常数对脉动场围绕其各向同性部分进行展开,得到脉动中的各向异性部分和各向同性的迭代关系,同时将剪切应变率引入脉动场各向异性分量中。在此基础上,利用Yakhot-Orszag湍流重正化群理论中的模式耦合原理,消除脉动场中的各向同性部分高波数分量,对Reynolds应力非线性涡粘性模型和二阶矩模型进行分析。引入双重尺度展开后,新的非线性模型比直接利用重正化群方法分析所得非线性模型有所改进,模型中含有平均应变率张量与平均涡量张量的相互作用项。新模型满足Speziale建议的合理性条件,不含有平均涡量的平方项。通过与其他模型的系数比较,本文所得理论模型与Speziale建议的非线性模型很接近;对于Reynolds应力二阶矩模型,主要通过对Reynolds应力输运方程中的三元速度关联项、脉动与压力梯度关联项以及耗散项的模拟对其进行封闭。本文同样是从耗散项的高阶项(各向异性项)中导出二阶矩模型中的Reynolds应力各向同性回归项。相比于直接利用湍流重正化群理论分析的剪切模型,引入双重尺度展开后,分析过程大大简化。两个典型湍流算例的数值结果表明,基于双重尺度展开所得的剪切模型比标准K-ε模型均有一定程度的改进,其中新的重正化群二阶矩模型具有相当于Gibson-Launder模型的精度。更多还原

【Abstract】 Numerical simulation based upon turbulence models plays an important part in solving the turbulence problems in the engineering project. However, these turbulence models often used are constructed by the dimensional analysis and the invariance principles for the intrinsic of the Navier-Stokes equation; the turbulent constants in these models are determined from the typical experimental data. Along with the turbulence models are widely used in engineering project, their rationality and accuracy face the challenges. So, to develop turbulence models theoretically has its important meanings, for people to study and understand turbulence.According to the analogies between turbulence and critical phenomenon, the renormalization group (RNG) method, which was successfully used in the critical phenomenon study, was applied into the turbulence study by researchers. Forster, Nelson and Stephen (FNS) firstly used the renormalization group method to analyze the Navier-Stokes equation. Later, Yakhot and Orszag (YO) used the renormalization group method to analyze the homogeneous isotropic turbulence systematically, and they obtained fruitful results. Rubinstein and Barton pioneerly used the renormalization group theory of turbulence, which was derived by Yakhot and Orszag, to formulate shear turbulent models, but there were some inconsistency problems in their work. In this paper, the renormalization group theory of turbulence is used to formulate shear turbulent models, including the nonlinear eddy viscosity model and the second-moment closure model of Reynolds stress tensor, and the turbulence model constants are evaluated analytically.Based upon the previous work of Rubinstein and Barton, we directly use the renormalization group method, to construct the Reynolds stress nonlinear viscosity model. During the model elimination procedure, the shifts for wave number are used instead of the shifts for integral area, to avoid the inconsistency problem of the parameter values in their basic works. In this work, all the turbulence constants are calculated just with single parameter value: y = 3 andε= 4. The theoretical nonlinear model consist not only the quadratic terms of the mean strain rate ((?)U)~2, but also a term of the second-order derivates of the mean velocity (U(?)~2U), which can be treated as the relax-effect of Reynolds stress tensor. This term reflect the nonlinear characteristics of turbulence flow better. But in the nonlinear model, there is no such a term as the interaction of mean strain and mean vortex terms, which is the shortcoming of the theoretical model. Besides, a second-moment closure model is obtained through the theoretical analysis of Reynolds stress differential equation. The convection-diffusion term, the velocity-pressure-gradient term and the dissipation term in the differential equation are analyzed, and the proposed turbulence model constants are also evaluated. It is shown that the RNG second-moment closure model is very close to the well-known LRR model. It should also be pointed out that the Reynolds stress isotropic-return term is obtained through the higher order analysis of the dissipation term modeling.In the shear turbulence of finite Reynolds number in the engineering project, the big vortices have structures in space and time, and do not satisfy self-similar property any more. So, the Yakhot and Orszag theory can not be directly used to analyze the shear turbulence because of its shortcomings. In the shear turbulence field, the mean field varies slowly, while the fluctuation field varies fast, and they have different time and space scales. So, in our work, the two-scale expansion proposed by Yoshizawa is adopted to distinguish the slow variable and the fast variable, and the fluctuation parts are expanded around their isotropic parts using the scale ratio parameter, and the iterative equations between the isotropic and anisotropic parts are obtained. At the same time, the mean strain can also be introduced into the anisotropic part. Then using the mode elimination procedure in the renormalization group theory of turbulence derived by Yakhot and Orszag, we eliminate the high wave numbers, and set up the nonlinear eddy viscosity model and the second-moment closure model.Comparing with the former nonlinear model derived using the renormalization group method directly, the new model has some improvements after adopting the two-scale expansion technique, and it contains the interaction term of the mean strain and the mean vortex term. The new model constants also satisfy the consistent condition suggested by Speziale, with no quadratic terms of the mean vortex term. Through the comparison with other theoretical models, we find the new model is very close to the model proposed by Speziale. Besides, with the analysis of the complex terms in the Reynolds stress transport equation, such as the triple-velocity correlation term, the velocity-pressure gradient correlation term and the dissipation term, a new second-moment closure model is obtained. In this work, the Reynolds stress isotropic-return term is also obtained by the higher order analysis of the dissipation term modeling. With the two-scale expansion technique, the mode elimination procedure is much simpler than before. The numerical simulations of two typical turbulent flows show that the new models based on the two-scale expansion technique are better than the stand K-ε|- model, and the second-moment closure model has approximate accuracy of Gibson-Landau model.更多还原