  Turbulence remains an unsolved multidisciplinary science problem. As one of
the most well-known examples in turbulent flows, knowledge of the logarithmic
mean velocity profile (MVP), so called the log law of the wall, plays an
important role everywhere turbulent flow meets the solid wall, such as fluids
in any kind of channels, skin friction of all types of transportations, the
atmospheric wind on a planetary ground, and the oceanic current on the seabed.
However, the mechanism of how this log-law MVP is formed under the multiscale
nature of turbulent shears remains one of the greatest interests of turbulence
puzzles. To untangle the multiscale coupling of turbulent shear stresses, we
explore for a known fundamental tool in physics. Here we present how to
reproduce the log-law MVP with the even harmonic modes of fixed-end standing
waves. We find that when these harmonic waves of same magnitude are considered
as the multiscale turbulent shear stresses, the wave envelope of their
superposition simulates the mean shear stress profile of the wall-bounded flow.
It implies that the log-law MVP is not expectedly related to the turbulent
scales in the inertial subrange associated with the Kolmogorov energy cascade,
revealing the dissipative nature of all scales involved. The MVP with reduced
harmonic modes also shows promising connection to the understanding of flow
transition to turbulence. The finding here suggests the simple harmonic waves
as good agents to help unravel the complex turbulent dynamics in wall-bounded
flow.
