Bounds on the bulk rate of energy dissipation in body-force-driven steady-state
turbulence are derived directly from the incompressible Navier–Stokes equations. We
consider flows in three spatial dimensions in the absence of boundaries and derive
rigorous a priori estimates for the time-averaged energy dissipation rate per unit mass,
ε, without making any further assumptions on the flows or turbulent fluctuations. We
prove
ε [les ] c1v U2/l2
+ c2 U3/l,
where v is the kinematic viscosity, U is the root-mean-square (space and time averaged)
velocity, and l is the longest length scale in the applied forcing function. The prefactors
c1 and c2 depend only on the functional shape of the body force and
not on its magnitude or any other length scales in the force, the domain or the flow. We also
derive a new lower bound on ε in terms of the magnitude of the driving force F. For
large Grashof number Gr = Fl3/v2, we find
c3 vFl/λ2 [les ] ε
where λ = √vU2/ε is the Taylor microscale in the flow and the
coefficient c3 depends only on the shape of the body force. This estimate is seen to be
sharp for particular forcing functions producing steady flows with λ/l ∼ O(1)
as Gr → 1. We interpret both the upper and lower bounds on ε in terms of the conventional
scaling theory of turbulence – where they are seen to be saturated – and discuss them in the
context of experiments and direct numerical simulations.