A shift transformation for fully conservative methods: turbulence simulation on complex, unstructured grids

Operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries. These “shift” transformations permit conservative, skew-symmetric convective operators and symmetric, positive-definite diffusive operators to be obtained for staggered variables using collocated operators. Shift transformations are not limited to uniform or structured meshes, and this formulation leads to a generalization of the works of Perot (J. Comput. Phys. 159 (2000) 58) and Verstappen and Veldman (J. Comput. Phys. 187 (2003) 343). A set of shift operators have been developed for, and applied to, a time-adaptive Cartesian mesh method with a fractional step algorithm. The resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time. A mass conserving interpolation is used for the variables during mesh adaptation; the interpolation conserves momentum and energy to second order in space. Turbulent channel flow simulations were conducted at Reτ ≈ 180 using direct numerical simulation (DNS). The DNS results from the adaptive method compare favourably with spectral DNS results despite the use of a (formally) second-order accurate scheme.