The Fundamental Equations of Motion in Anisotropic Turbulent Flow

Abstract

It is defined with x_i (i=1, 2, 3) a orthogonal cartesian co-ordinate system. In turbulent flow the velocity component u_i' are compounded of the mean (temporal) values and the fluctuations, and are writed _??_i=ui+ui'. For the pressure p=_??_+p'. Introducing these values into the Navier-Stokes' equation and neglecting the molecular viscosity and the fluctuation of density ρ, the fundamental equations are expressed as follows. where the index κ (and every index that is found twice in a term) shows the summation from 1 to 3 after the Einstein's expression. In anisotropic trubulence we put Ci_k=coriolis-axiator. _??_i_k=-_??_ apparent (eddy) stresses.
The mean values of §_??_are considered as the components of Prandtl's mixing-length. ε_??_ are the additional terms which represent the turbulence arising from agency other than velocity gradient.
Using (2), the expressions for the apparent (eddy) stresses are introduced as follows: where Austaush coefficients or coefficients of eddy viscosity If we transform the equation (3) as follows, we can obtain new following expressions having the more clearer physical meaning for eddy stresses. where each notation is generally known. According to the expression (4), we can find a new physical improvement that the symmetric stress tensor is caused by rotation in addition to deformation.
Introducing the equation (4) into the frictional term (by the eddy viscosity) in the equation of motion, we can obtain new general expressions for the resistance arising from the eddy transport of momentum in three dimentional anisotropic turbulent flow. When these new equations are applied to the horizontal wind motion whose variation in the horizontal components are negligibly small compared to vertical in the narrow region in the immediate neighbourhood of the ground, the new terms of the transverse resistance which are impossible to be found in the old theory are appeared in the following form. the equations (5) coinside with, the Sakakibara-Isimaru's equations. Moreover in this report the rate of dissipation of energy due to eddy viscosity only is computed, and it is found that the Dissipation Function contains the terms of rotation. Here the physical inspection of the Watanabe's Equation of motion in the “microgyrostatic field” is done.