Kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow

The collocated-mesh scheme is often favored over the staggered-mesh scheme for turbulence simulation in complex geometries due to its simpler form in curvilinear coordinates. The collocated mesh scheme does not conserve kinetic energy however, and few careful checks of the impact of these errors have been made. In this work, analysis is used to identify two sources of kinetic energy conservation error in the collocated-mesh scheme: (1) errors arising from the interpolations used to estimate the velocity on the cell faces, and (2) errors associated with the slightly inconsistent pressure field used to ensure mass conservation for the cell face volume fluxes. It is shown that the interpolation error can be eliminated through the use of first-order accurate centered interpolation operators with mesh-independent weights. The pressure error appears to be intrinsic to the scheme and it is shown to scale as O(Δ t²Δ x²). The effects of the conservation errors are investigated numerically through simulations of inviscid flow over an airfoil and in large eddy simulations of turbulent channel flow. Neither the interpolation error nor the pressure error appear to lead to significant problems in the channel flow simulations where viscous dissipation is present and where the Cartesian mesh is stretched in only one direction. The inviscid airfoil simulations performed on a curvilinear mesh show a much greater sensitivity to the interpolation error. The standard second-order centered interpolation is shown to lead to severe numerical oscillations, while the kinetic energy-conserving first-order centered interpolation produces solutions that are almost as smooth as those obtained with a third-order upwind interpolation. When compared in channel flow simulations, however, the first-order centered interpolation is shown to be far superior to the third-order upwind interpolation, the latter being adversely affected by numerical dissipation. Only slight differences were noted when comparing channel flow simulations using first-order and second-order centered interpolations. These results suggest that numerical oscillations can be controlled in curvilinear coordinates through the use of properly-constructed non-dissipative centered interpolations.