DNS of passive scalars in turbulent pipe flow

We study the statistics of passive scalars at $Pr=1$ , for turbulent flow within a smooth straight pipe circular cross section up to $Re_{\tau } \approx 6000$ using direct numerical simulation (DNS) Navier–Stokes equations. While featuring general organisation similar axial velocity field, scalar fields show additional energy small wavenumbers, resulting in higher degree mixing and $k^{-4/3}$ spectral inertial range. The DNS results highlight logarithmic growth inner-scaled bulk mean centreline values with friction Reynolds number, implying an estimated von Kármán constant $k_{\theta 0.459$ which also nicely fits profiles. data are used synthesise modified form classical predictive formula Kader & Yaglom ( Intl J. Heat Mass Transfer vol. 15 (12), 1972, pp. 2329–2351), points some shortcomings original formulation. Universality core profile defect is recovered, very nearly parabolic shape. Logarithmic buffer-layer peak variance found number range under scrutiny, well conforms Townsend's attached-eddy hypothesis, whose validity supported by maps. behaviour Prandtl shows good universality outer wall layer, $Pr_t 0.84$ as previous studies, but closer unity near wall, where existing correlations do not reproduce observed trends.