Using helicity to characterize homogeneous and inhomogeneous turbulent dynamics

The ability of the helicity decomposition to describe compactly the dynamics of three-dimensional incompressible fluids is invoked to obtain new descriptions of both homogeneous and inhomogeneous turbulence. We first use this decomposition to derive four coupled nonlinear equations that describe an arbitrary three-dimensional turbulence, whether anisotropic and/or non-mirror-symmetric. We then use the decomposition to treat the inhomogeneous turbulence of a channel flow bounded by two parallel free-slip boundaries with almost the ease with which the homogeneous case has heretofore received treatment. However, this ease arises from the foundation of a random-phase hypothesis, which we introduce and motivate, that supersedes the translational invariance of a turbulence that is hypothesized to be homogeneous. For the description of this channel turbulence, we find that the three-dimensional modes and the two-dimensional modes having wave vectors parallel to the boundaries each couple precisely as in a homogeneous turbulence of the corresponding dimension. The anisotropy and inhomogeneity is in large part a feature incorporated into the solenoidal basis vectors used to describe an arbitrary solenoidal free-slip flow within the channel. We invoke the random-phase hypothesis, a feature of the dynamics, with closures, such as Kraichnan's direct-interaction approximation and his test-held model, in addition to the one most utilized in this manuscript, the eddy-damped quasi-normal Markovian (EDQNM) closure.