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.