The double velocity correlation function of homogeneous turbulence with constant mean velocity gradient

Abstract

In this article, as the velocity gradient is taken as a constant value, we obtain the solutions of the equation of fluctuation velocity after Fourier transform. When the mean velocity gradient is small, they represent the picture of eddies, of which the homogeneous turbulence (both isotropic and nonisotropic) of the final period is composed. By using the eddies of these types at different times, we may compose the steady turbulent field with the constant velocity gradient and this field may represent the turbulent field in the central part of the channel flow or pipe flow approximately. Then we may obtain the double velocity correlation function of this turbulent field, which involves both longitudinal correlation coefficient$f\left( {\frac{r}{\lambda }} \right)$ and the transversal correlation coefficient$g\left( {\frac{r}{\lambda }} \right)$. We compare these theoretical coefficients with the experimental data of these coefficients at initial period and final period of isotropic homogeneous turbulence. And then we obtain the relationship between the turbulent double velocity correlation coefficient$f\left( {\frac{r}{\lambda }} \right)$ and the mean velocity gradient. Finally, we get the expressions of the Reynolds stress and the eddy viscosity coefficient.