  For turbulence, although the two-point velocity difference u(x+r)-u(x) at
each scale r has been studied in detail, the velocity average [u(x+r)+u(x)]/2
has not thus far. Theoretically or experimentally, we find interesting features
of the velocity average. It satisfies an exact scale-by-scale energy budget
equation. The flatness factor varies with the scale r in a universal manner.
These features are not consistent with the existing assumption that the
velocity average is independent of r and represents energy-containing
large-scale motions alone. We accordingly propose that it represents motions
over scales >= r as long as the velocity difference represents motions at the
scale r.
