Time scales of the turbulent heat-flux equation and their near-wall behavior

The dissipation rate $\varepsilon_{i\theta}$ of the turbulent heat flux $\overline{u_i\theta}$ is
split by the two-point correlation method, exactly and at any
Prandtl number, into a homogeneous part $\varepsilon_{i\theta,h}$, the curvature of
the velocity--temperature correlation at coincidence, and an
inhomogeneous part $\tfrac14(\nu+\kappa)\Delta_{x}\overline{u_i\theta}$ absorbed by the
molecular diffusion, the $(\nu-\kappa)$ terms being confined to a
cross-diffusion in divergence form. The homogeneous part vanishes
under local isotropy, measures small-scale anisotropy and is not
sign-definite; the heat-flux time scale defined from the flux
equation itself, $\tau_{i\theta,h}=\overline{u_i\theta}/\varepsilon_{i\theta,h}$, is therefore
signed, and its ratios to the mechanical and thermal time scales
have as quotient the thermal-to-mechanical ratio of the
temperature-variance analysis. The near-wall behavior follows
analytically for the four thermal wall conditions. At an isothermal
wall the two parts of the streamwise dissipation are equal, the
wall-normal slopes are in the ratio $1{:}3$, and both components
share the anchors $2\Pr/(1+\Pr)$ and $2/(1+\Pr)$: the
... mehr
heat-flux time scale is the harmonic mean of the thermal and
mechanical time scales, thermally controlled at low Prandtl number.
At walls with fluctuating temperature the wall-normal dissipation
vanishes through the cancellation of finite parts, the wall-normal
ratio attains the universal anchor $-2\Pr/(1+\Pr)$, the
homogeneous dissipation acting as a source of the wall-normal flux
over a layer whose thickness is a wall-parallel microscale of the
wall-temperature field, and the streamwise ratio diverges as
$(y^{+})^{-1}$ with a parameter-free amplitude. Channel- and pipe-flow direct numerical
simulations from four codes at $\Pr=0.71$, $0.025$ and $0.01$
reproduce the splits and anchors to within $0.4$--$7\%$ and locate
the source layer at $2$--$9$ viscous units; at liquid-metal Prandtl
numbers the cross-diffusion reaches $40\%$ of the peak dissipation
and the heat-flux time scale remains of the order of the thermal one
across the channel. The exact equation and the wall identities are
parameter-free constraints for heat-flux closures and data-driven
representations of the scalar flux, and acceptance tests for their
training data.