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 Re-tau approximate to 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. (c) 2005 Elsevier Inc. All rights reserved.