The 1877 Boussinesq Conjecture: Turbulent Fluctuations Are Dissipative on the Mean Flow

The 1877 assumption/conjecture of Boussinesq is that turbulent fluctuations have a dissipative effect on the mean flow. The eddy viscosity hypothesis is that this dissipative effect can be modelled by a turbulent viscosity term. Many computational and experimental data sets have shown that the eddy viscosity hypothesis is not generally correct. The inverse cascade of energy to larger scales in 2d flows is thought to imply that both are false for 2d flows. This article gives a proof that for strong solutions in 2d and 3d the Boussinesq assumption does holds in a time averaged sense. Since this proof contradicts commonly held interpretations of the Batchelor-Leith-Kraichnen theory of 2d turbulence, this (extended) report includes a brief discussion of BLK theory in the light of the theorem.