  In large-eddy simulations (LES) a computational-domain translation velocity
can be used to improve performance by allowing longer time-step intervals. The
continuous equations are Galilean invariant, however, standard
finite-difference-based discretizations are not discretely invariant with the
error being proportional to the product of the local translation velocity and
the truncation error. Even though such numerical errors are expected to be
small, it is shown that in LES of buoyant convection the turbulent large-scale
flow organization can modulate and amplify the error. Galilean invariance of
global flow statistics is observed in well-resolved direct numerical
simulations (DNS). In LES of single-phase convection under an inversion, flow
statistics are nearly Galilean invariant and do not depend on the order of
accuracy of the finite difference approximation. In contrast, in LES of cloudy
convection, flow statistics show strong dependence on the frame of reference
and the order of approximation. The error with respect to the frame of
reference becomes negligible as the order of accuracy is increased from second
to sixth in the present LES. Schemes with low resolving power can produce large
dispersion errors in the surface-fixed frame that can be amplified by
large-scale flow anisotropies, such as strong updrafts rising in a
non-turbulent free troposphere in cumulus-cloud layers. Interestingly, in the
present large-eddy simulations, a second-order discretization in the proper
Galilean frame can yield comparable accuracy as a high-order scheme in the
surface-fixed frame.
