Jurcisinová E, Jurcisin M, Remecký R
Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Oct;80(4 Pt 2):046302. doi: 10.1103/PhysRevE.80.046302. Epub 2009 Oct 2.
The influence of weak uniaxial small-scale anisotropy on the stability of the scaling regime and on the anomalous scaling of the single-time structure functions of a passive scalar advected by the velocity field governed by the stochastic Navier-Stokes equation is investigated by the field theoretic renormalization group and operator-product expansion within one-loop approximation of a perturbation theory. The explicit analytical expressions for coordinates of the corresponding fixed point of the renormalization-group equations as functions of anisotropy parameters are found, the stability of the three-dimensional Kolmogorov-like scaling regime is demonstrated, and the dependence of the borderline dimension d(c) is an element of (2,3] between stable and unstable scaling regimes is found as a function of the anisotropy parameters. The dependence of the turbulent Prandtl number on the anisotropy parameters is also briefly discussed. The influence of weak small-scale anisotropy on the anomalous scaling of the structure functions of a passive scalar field is studied by the operator-product expansion and their explicit dependence on the anisotropy parameters is present. It is shown that the anomalous dimensions of the structure functions, which are the same (universal) for the Kraichnan model, for the model with finite time correlations of the velocity field, and for the model with the advection by the velocity field driven by the stochastic Navier-Stokes equation in the isotropic case, can be distinguished by the assumption of the presence of the small-scale anisotropy in the systems even within one-loop approximation. The corresponding comparison of the anisotropic anomalous dimensions for the present model with that obtained within the Kraichnan rapid-change model is done.
通过场论重整化群和微扰理论单圈近似下的算符乘积展开，研究了弱单轴小尺度各向异性对由随机纳维-斯托克斯方程支配的速度场平流的被动标量的标度律稳定性和单时结构函数反常标度的影响。找到了重整化群方程相应不动点坐标作为各向异性参数函数的显式解析表达式，证明了三维类柯尔莫哥洛夫标度律的稳定性，并找到了稳定和不稳定标度律之间边界维数(d(c)\in(2,3])作为各向异性参数函数的依赖关系。还简要讨论了湍流普朗特数对各向异性参数的依赖性。通过算符乘积展开研究了弱小尺度各向异性对被动标量场结构函数反常标度的影响，并给出了它们对各向异性参数的显式依赖关系。结果表明，在各向同性情况下，对于克莱奇南模型、具有速度场有限时间关联的模型以及由随机纳维-斯托克斯方程驱动的速度场平流的模型，结构函数的反常维数是相同的（普适的），但即使在单圈近似下，通过假设系统中存在小尺度各向异性也可以区分它们。对本模型的各向异性反常维数与在克莱奇南快速变化模型中得到的结果进行了相应比较。
