Using helicity to characterize homogeneous and inhomogeneous turbulent dynamics

The ability of the helicity decomposition to describe compactly the dynamic s of three-dimensional incompressible fluids is invoked to obtain new descr iptions 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-mirr or-symmetric. We then use the decomposition to treat the inhomogeneous turb ulence of a channel flow bounded by two parallel free-slip boundaries with almost the ease with which the homogeneous case has heretofore received tre atment. However, this ease arises from the foundation of a random-phase hyp othesis, 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 bo undaries each couple precisely as in a homogeneous turbulence of the corres ponding dimension. The anisotropy and inhomogeneity is in large part a feat ure incorporated into the solenoidal basis vectors used to describe an arbi trary solenoidal free-slip flow within the channel. We invoke the random-ph ase 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 Mar kovian (EDQNM) closure.