Non-universal scaling transition of momentum cascade in wall turbulence

As a counterpart of energy cascade, turbulent momentum cascade (TMC) in the wall-normal direction is important for understanding wall turbulence. Here, we report an analytic prediction of non-universal Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}$ ) scaling transition of the maximum TMC located at $y_{p}$ . We show that in viscous units, $y_{p}^{+}$ (and $1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$ ) displays a scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-6/7}$ ) to $Re_{\unicode[STIX]{x1D70F}}^{3/5}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-3/5}$ ) in turbulent boundary layer, in sharp contrast to that from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-2/3}$ ) to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-1/2}$ ) in a channel/pipe, countering the prevailing view of a single universal near-wall scaling. This scaling transition reflects different near-wall motions in the buffer layer for small $Re_{\unicode[STIX]{x1D70F}}$ and log layer for large $Re_{\unicode[STIX]{x1D70F}}$ , with the non-universality being ascribed to the presence/absence of mean wall-normal velocity $V$ . Our predictions are validated by a large set of data, and a probable flow state with a full coupling between momentum and energy cascades beyond a critical $Re_{\unicode[STIX]{x1D70F}}$ is envisaged.