Contribution of large-scale motions to the skin friction in a moderate adverse pressure gradient turbulent boundary layer

Abstract

Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG,$\unicode[STIX]{x1D6FD}=1.45$) is performed to explore the contribution of large scales to the skin friction, where$\unicode[STIX]{x1D6FD}$is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from$Re_{\unicode[STIX]{x1D703}}\approx 110$to 6000 with an equilibrium region in$Re_{\unicode[STIX]{x1D703}}=4000$–5500. The spanwise wavelength ($\unicode[STIX]{x1D706}_{z}$) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ($C_{f}$) by employing two approaches: (i) spanwise co-spectra of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$; (ii) conditionally averaged$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$. The velocity–vorticity correlations$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$are related to the advective transport and the vortex stretching, respectively. Although$\langle v\unicode[STIX]{x1D714}_{z}\rangle$negatively contributes to$C_{f}$, the positive contribution of the large scales ($\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$) is observed in the co-spectra of weighted$\langle v\unicode[STIX]{x1D714}_{z}\rangle$. For the co-spectra of weighted$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, we observe an outer peak at$\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$and the superposition of the large scales in the buffer region, leading to the enhancement of$C_{f}$. The magnitude of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$depends on the large-scale streamwise velocity fluctuations ($u_{L}$). In particular, the negative-$u_{L}$events amplify$\langle v\unicode[STIX]{x1D714}_{z}\rangle$in the outer region, and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$is enhanced by the positive-$u_{L}$events. As a result, the skin friction induced by$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$increases in the present APG TBL.