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Contribution of large-scale motions to the skin friction in a moderate adverse pressure gradient turbulent boundary layer
Published online by Cambridge University Press: 01 June 2018
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.
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