The physics of rotating homogeneous turbulence is studied at various fields. They contain the study on the large-scale structure of rotating homogeneous turbulence to investigate the decay rate of final period of decay of rotating homogeneous turbulence, the velocity-derivative skewness to examine the reduction of spectral transfer by rotation and the turbulence model that can simulates proper decay rate of rotating homogeneous turbulence at high Reynolds number.
In the study on large-scale structure of rotating homogeneous turbulence, the large-scale structure of rotating homogeneous turbulence is investigated with an asymptotic expansion method, and the corresponding final period of decay rate is studied. Extending the hypothesis of Batchelor and Proudman (1956) to rotating homogeneous turbulence, it is possible to determine the non-rotational part and the rotational one of the large-scale structure separately. For the non-rotational part, it is determined by the turbulence generation hypothesis only, and a further insight about spectral structures of the 2$^{nd}$ and the 3$^{rd}$ order velocity spectral tensors is obtained. For the rotational part, it is considered for two cases. The one, a general rotation rate, shows that combined with the results of non-rotational part, the asymptotic orders of velocity and vorticity spectral tensor are preserved as $O(\kappa^2)$ and $O(\kappa^4)$ with respect to non-rotating turbulence but their orders in physical space varies as $O(r^{-5})$ and $O(r^{-7})$, respectively. The other, a vanishing rotation rate, gives an exact representation of the large-scale structure of rotating homogeneous turbulence and verifies the results of general rotation rate. It is also found that the velocity and the voricity spectral tensors are even functions in rotation rate because of the invariant property of Navier-Stokes equation about inversion coordinate transform, and due to the homogeneity of turbulence, the helicity of large-scale structure is zero even if rotation presents. For a general rotation, the energy spectrum function $E(\vec{\kappa}, \vec{\Omega})$ is determined as $E(\vec{\kappa},{\Omega})$=C(\vec {\Omega})\kappa^4+ο(\kappa^5)$ where $C(\vec {\Omega})$ is an even function of rotation rate, and it becomes $(C(t)+ \Omega_x\Omega_y C_{xy}^\Omega(t))\kappa^4+ο(\kappa^5)$ too for a vanishing rotation case. The consequence of rotation about energy spectrum function is disclosed through enstropy. Finally, using the results of large-scale structure, it is found that the decay rate of rotating homogeneous turbulence is $t^{-5/2}$ in final period of decay just as that of non-rotating homogeneous turbulence because the rotation does not change the asymptotic order of velocity spectral tensor with respect to non-rotating turbulence, and an integral length scale of vorticity may increase in final period of decay.
For the velocity-derivative skewness of rotating homogeneous turbulence, it is analytically derived from the turbulent kinetic energy dissipation rate equation by assuming a simple energy spectrum function. It predicts the cut-off of spectral energy transfer at large rotation rate, and the predicted reduction of skewness due to rotation agrees well with the results of direct numerical simulation and eddy-damped quasi normal Markovian closure.
Finally, in order to analysis the decay rate of rotating homogeneous turbulence, the model coefficient $C_{\epsilon2}$ in the dissipation rate equation is theoretically formulated in terms of the macro-scale Rossby number $Ro_l$ and the power exponent m in the energy spectrum function $E(\kappa)\sim \kappa^m$ of small wavenumber range. The proposed model satisfies the asymptotic decay rate of turbulent kinetic energy, which depends on the small wavenumber range energy spectrum, for both extremes of rotation rate. A number of test computations prove that the present model faithfully reproduces the available experimental as well as direct numerical simulation data of rotating homogeneous turbulence.
