  Exact budget equations are derived for the coherent and stochastic
contributions to the second-order structure function tensor. They extend the
anisotropic generalised Kolmogorov equations (AGKE) by considering the coherent
and stochastic parts of the Reynolds stress tensor, and are useful for the
statistical description of turbulent flows with periodic or quasi-periodic
features, like e.g. the alternate shedding after a bluff body. While the
original AGKE describe production, transport, inter-component redistribution
and dissipation of the Reynolds stresses in the combined space of scales and
positions, the new equations, called $\varphi$AGKE, contain the phase $\varphi$
as an additional independent variable, and describe the interplay among the
mean, coherent and stochastic fields at the various phases. The newly derived
$\varphi$AGKE are then applied to a case where an exactly periodic external
forcing drives the flow: a turbulent plane channel flow modified by harmonic
spanwise oscillations of the wall to reduce drag. The phase-by-phase action of
the oscillating transversal Stokes layer generated by the forcing on the
near-wall turbulent structures is observed, and a detailed description of the
scale-space interaction among mean, coherent and stochastic fields is provided
thanks to the $\varphi$AGKE.
