Commutation errors in PITM simulation

We examine the effect of variable filter width in the partially integrated transport modeling (PITM) method in the framework of second moment closure (SMC) and the commutation errors arising from the non-commutativity of the filtering process with temporal or spatial derivatives. We propose a method to account for this effect in PITM numerical simulations. In particular, we model the commutation terms in a practical way to get tractable Navier-Stokes and turbulence equations. We apply a discrete approximation to the top hat filter to compute the commutation terms for the velocities and turbulent stresses by means of a superfilter width. A special attention is devoted to the treatment of the turbulent stresses. Moreover, a physical interpretation of the commutation terms is given in the spectral space involving the flux of energy transfer of the spectrum. For illustration purpose, we perform numerical simulations of the fully developed turbulent channel flow on several grids including a sudden grid step increase in the grid-size in the streamwise direction that is akin to grid discontinuities. The commutation errors are then distributed in a small delimited region of the field. As a result, we found that the impact of the commutation terms on the PITM solution remains moderate even if the grid-size variation is increasing up to 400% in the streamwise direction but the accounting for the commutation errors has however a slight beneficial effect on the solution by improving the flow prediction.