Two-dimensional turbulent thermal classical far wake

Abstract

The two-dimensional turbulent thermal classical far wake is investigated. The turbulence is described by the Boussinesq hypothesis for the Reynolds stresses with Prandtl’s mixing length model for the eddy viscosity and eddy thermal conductivity. Two conservation laws are derived for the thermal boundary layer equations for the far wake using the multiplier method and two conserved quantities are obtained from the conserved vectors and boundary conditions. The Lie point symmetry associated with the momentum and thermal conserved vectors is derived. It is found that the momentum and thermal mixing lengths are proportional. In obtaining the invariant solution the cases ν≠0, κ≠0 and ν=0, κ=0 needed to be treated separately, where ν is the kinematic viscosity and κ is the thermal conductivity of the fluid. When ν≠0, κ≠0 the associated Lie point symmetry is determined in full but an analytical solution of the reduced ordinary differential equations could not be derived. The invariant solution is obtained numerically using a shooting method with the two conserved quantities as targets. The width of the wake is infinite. When ν=0, κ=0 the width of the wake is finite. In order to obtain the associated Lie point symmetry in full, Prandtl’s hypothesis that the mixing length is proportional to the width of the wake was imposed. The invariant solution is obtained analytically. The lines of constant mean temperature difference and the streamlines for the mean velocity deficit are plotted. The numerical and analytical solutions agree in the limit ν→0, κ→0.