A homogeneous thermal-to-mechanical time-scale ratio and its near-wall behavior

The dissipation rate $\epsth$ of the temperature variance, and the thermal-to-mechanical time-scale ratio formed from it, vary sharply across the near-wall layer in most flows. Using the two-point correlation method, $\epsth$ is split into a homogeneous part $\epsthh$ and an inhomogeneous part that is absorbed exactly by the molecular diffusion in the transport equation of $\tv$. The variance transport equation then retains $\epsthh$ as its dissipation and defines a homogeneous time-scale ratio $\Rh=\tv\,\epsh/(2k\,\epsthh)$, where $\epsh$ is the homogeneous dissipation rate of the turbulent kinetic energy $k$. Expressed through the mechanical and thermal Taylor microscales, $\lambda$ and $\lambda_\theta$, the ratio becomes $\Rh=(5/C_\theta)\,\Pran\,(\lambda_\theta/\lambda)^2$. At an isothermal wall both microscales are linear in the wall distance, so $\Rh$ is smooth and approaches the molecular limit $\Rh|_w=\Pran$ monotonically; at Prandtl numbers of order one its near-wall variation is reduced relative to the classical time-scale ratio $R$, which departs from and returns to the same wall value. The two ratios are related by the exact factorization $R=\Rh(1-f_\theta)/(1-f_u)$, in which the inhomogeneous fractions of the two dissipations, $f_\theta=\epsthih/\epsth$ and $f_u=\epsih/\eps$, carry the entire difference between them and equal one half at the wall. ... mehrThe factorization locates the crossing of $R$ through its wall value just outside the sign change of the mechanical fraction, at $\yp\simeq6$--$8$ over the Prandtl-number range studied. At walls with fluctuating temperature, constant-flux, Robin or conjugate, both ratios are singular, diverging as $\tv_w\,(\yp)^{-2}$ with an amplitude fixed by wall values. The analysis is assessed against channel-flow direct numerical simulation databases from three independent codes, at $\Pran=0.71$, $0.025$ and $0.01$ and under isothermal, constant-flux, Robin and conjugate thermal conditions. The data confirm the isothermal wall limit, the monotone approach, the location of the crossing, and the singular asymptote with its analytical amplitude. In a wall-anchored a~priori reconstruction of $\epsth$ with no adjustable constant, the constant-$\Rh$ closure is exact through the conductive sublayer and its error is one-signed, while the error of the classical constant-$R$ closure changes sign at the crossing. Being exact, the analysis provides a physical basis for the constant time-scale ratio of turbulent heat-flux models and a constraint for data-driven closures.