Renormalization Group Analysis For Rotating Turbulent Flows

Rotating turbulence is a complex phenomenon, which exists in both nature and engineering project. Rotating effects have a crucial influence on large-scale atmospheric and oceanic flows. Researchers have revealed the fact that rotating effects will change both the average velocity and fluctuating velocity, which leads to the significant effects upon the statistical properties of turbulence. According to the results from the experiments and numerical simulations, turbulence tends to be two-dimensional under high angular velocity. Energy cascade from large to small scales will be suppressed as the result of the nonlinear interaction generated by rotation. Numerical simulation based on turbulence models is widely used to solve the problems in the engineering project. But most of turbulence models are constructed by dimensional analysis.Besides, turbulent constant of the turbulence models are determined by experimental data or experience. As a result of the rationality and accuracy of the turbulent model being challenged, investigating how rotating effects influent turbulence theoretically become important and meaningful.Energy spectrum of turbulence in the inertial range is universal and small-scale vortex is statistical steady and balanced.These features suggest that characteristic scale doesn’t exist in the inertial range.Based on these similarities with critical phenomenon, the renormalization method,which was successfully used to study the critical phenomenon, was applied into the study of turbulence. Forster, Nelson and Stephen (FNS) firstly analyze the Navier-Stokes equation through the renormalization group method. Later, Yakhot and Orszag (Y-O) used the renormalization group method to analyze the homogeneous isotropic turbulence systematically. They obtained the turbulent K-ε model with no adjustable constant. After YO’s work, Rubinstein and Barton generalized the renormalization group method to analyze the weak anisotropic turbulence modified by random force with special correlation function,explaining how anisotropic random force influent the energy spectrum with nonlinear interaction.In this article, the dynamic renormalization group analysis is applied to the investigation for the the rotating turbulence. As the complexity of rotating turbulence,two extreme situations are studied to investigate how rotating effect change the behavior of the turbulent flow:weak rotating flow Ωâ†'0and rapidly rotating flow Ωâ†'∞. Based upon the method which Rubinstein&Barton suggested to study the anisotropical turbulence, weak rotating is studied through a modified N-S equation. A group of anisotropical propagator is set to modify the N-S equation to match the term generated by the Coriolis force during the coarsening process. According to the result of the coarsening procedure for weakly anisotropy, anisotropic part of the random force was considered to be a perturbation of the isotropic part. A group of differential equations with several fix point viscosity are built. Stability of each fix point is investigated through the stability principle of differential equations. Only one fix point is found to be steady. Energy spectrum near the steady fix point is calculated.Several significant features are observed in the rapidly rotating flows, such as the inhibition of energy cascade and two-dimensionalization trend. In this paper, rapidly rotating turbulence is investigated through the renormalization group method. N-S equations are solved to obtain the explicit expression of fluctuate velocity depending on random force, which are applied in the coarsening procedure to avoid the deviation caused by weak anisotropical approximation. The expression of anisotropical renormalization viscosity which consisted of anisotropical integral function is obtained. Through magnitude analysis and integral transformation the expression is solved mathematically.It can be proved that anisotropical renormalization viscosity will tend to be infinitesimal when angular velocity is infinity, which leads to the inhibition of energy transfer between the turbulent motion of different scales. Spherical energy spectrum function and energy density depending on frequency are calculated. The work of this thesis predicts the two-dimensionalization trend and also explains how rotating effect redistribute the energy through nonlinear interaction.