Rigorous bound on the plane-shear-flow dissipation rate

Abstract

The Doering-Constantin approach [Phys. Rev. Lett. 69, 1648 (1992)] to derive an upper bound on the dissipation rate in terms of a properly chosen stationary flow profile can be (slightly) improved by casting it in the form of a variational principle for the profile. Only Schwarz's inequality is needed to prove this. Its solution can be evaluated analytically. The mean dissipation bound is \ensuremath{\varepsilon}\ensuremath{\le}${\mathit{U}}^{3}$${\mathit{L}}^{\mathrm{\ensuremath{-}}1}$/9 \ensuremath{\surd}2 provided the Reynolds number Re=UL/\ensuremath{\nu} exceeds 12 \ensuremath{\surd}2 , and the mean transverse momentum flux is bounded by ${\mathit{J}}_{31}$\ensuremath{\le}(\ensuremath{\nu}${\mathit{L}}^{\mathrm{\ensuremath{-}}1}$${)}^{2}$${\mathrm{Re}}^{2}$/9 \ensuremath{\surd}2 .