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.