The elementary energy transfer between the two-point velocity mean and difference

In this paper, the elementary energy transfer between the two-point average of a velocity field and the two-point difference has been examined. The equations related to the two-point quantities are derived from the Navier-Stokes equations applied to incompressible flows. Some interesting aspects of this simple transfer of energy are discussed both from the point of view of the filtered equations associated with the two-point velocity average and from the point of view of the properties of the structure functions associated with the two-point velocity difference. A new inertial relation valid for homogeneous flows is finally presented.