  Energy transfer mechanisms for vortex shedding behind a 2D cylinder at a
Reynolds number of Re=100 are investigated. We first characterize the energy
balances achieved by the true cylinder flow -- both for the flow as a whole and
for each of its most energetic harmonic frequencies. It is found that viscous
dissipation balances production when each is considered over the entire flow
field and therefore that linear mechanisms achieve an energy balance on their
own, thus respecting the Reynolds--Orr equation. Suitable energy conservation
laws reveal that while nonlinear mechanisms neither produce nor consume energy
overall, they nevertheless account for an important transfer of energy to
higher frequencies. The energy balance for DNS is compared to that predicted by
resolvent analysis. Although a suitable energy balance is achieved for each
harmonic, resolvent analysis does not correctly model energy transfer between
temporal frequencies. We further investigate the detailed roles of energy
transfer from the viewpoint of nonlinear triadic interactions by considering a
finite number of harmonic frequencies. It is shown that the nonlinearity plays
a critical role in the redistribution of energy to higher harmonic frequencies.
We observe not only an energy cascade from low frequencies to high frequencies,
but also a considerable inverse cascade from high frequencies to low
frequencies. The true nonlinear energy transfer mechanisms provide insight into
closure models for improved resolvent-based predictions of harmonic modes in
shear flows.
