Turbulent energy density and its transport equation in scale space

The energy spectrum contains information not only on the intensity but also on the scale dependence of the turbulent fluctuations; the spectrum is commonly used to describe the dynamics of homogeneous isotropic turbulence. On the other hand, one-point statistical quantities such as the turbulent kinetic energy are mainly treated for inhomogeneous turbulence. Although the energy spectrum must be useful in describing the scale dependence of inhomogeneous turbulence, the Fourier transform cannot be performed in general cases. In this work, instead of the energy spectrum in the wavenumber space, the energy density in the scale space was introduced on the basis of the two-point velocity correlation in the physical space. The transport equation for the energy density was derived for inhomogeneous turbulence. Direct numerical simulation (DNS) data of homogeneous isotropic turbulence were first used to evaluate the energy transfer in the scale space. The energy density equation was compared with the energy spectrum equation to assess the role of the energy density. DNS data of turbulent channel flow were also used to evaluate the energy density equation for inhomogeneous turbulence. The energy transport in the physical and scale spaces was examined in different regions of channel flow. It was shown that the transport equation for the energy density adequately describes the energy transfer in the scale space. The energy flux from the large to the small scales was observed for both turbulent flows in a similar manner to the conventional energy cascade in the wavenumber space.