Temperature variance and turbulent kinetic energy as potentials of the inhomogeneous dissipation

The two-point decomposition of the dissipation rates identifies the Laplacians of the temperature variance $\tv$ and of the turbulent kinetic energy $k$ with the inhomogeneous parts of the respective dissipation rates, $\Dx\tv=4\epsthih/\kappa$ and $\Dx k=2\epsih/\nu$. By Green's third identity both fields are therefore represented exactly, in any flow domain, as boundary terms plus Newtonian potentials of the inhomogeneous dissipations. At isothermal no-slip boundaries the boundary terms vanish and the complete $\tv$ and $k$ fields are potentials of the inhomogeneous parts of their own dissipation rates. At boundaries with fluctuating temperature the wall variance enters as boundary value. The representation is kinematic, with no transport equation entering its derivation, and it contains no adjustable constant. The dynamics of the variance transport, including the mechanism sustaining the turbulence, acts on the variance fields only through the inhomogeneous dissipations. The representation is the exact, parameter-free counterpart of the elliptic-relaxation structure of near-wall turbulence closures. Channel-flow direct numerical simulation data at $\Pran=0.71$, $0.025$ and $0.01$ under isothermal, imposed-flux, Robin and conjugate thermal conditions reproduce the representation to 0.1\% on wall-resolved data at isothermal walls, and to a few percent where the wall variance enters as boundary value or the wall is not resolved. ... mehrAn a~priori reconstruction from published variance profiles traces the sweep from shear-driven mixed convection to the Rayleigh--B\'enard limit, in which the bulk is harmonic. The inhomogeneous fractions $f_\theta=\epsthih/\epsth$ and $f_u=\epsih/\eps$ measure the departure of the variance fields from harmonicity, and the representation provides an exact consistency condition for turbulence closures and data-driven models.