Derivation Of Shear Turbulent Models Using Renormalization Group Method

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.