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Two-dimensional cross-spectrum of the streamwise velocity in turbulent boundary layers

  • Rahul Deshpande (a1), Dileep Chandran (a1), Jason P. Monty (a1) and Ivan Marusic (a1)


In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity ( $u$ ) component and use it to test the notion of self-similarity in turbulent boundary layers. The primary focus is on the cross-spectrum ( $\unicode[STIX]{x1D6F7}_{cross}^{w}$ ) measured across the logarithmic ( $z_{o}$ ) and near-wall ( $z_{r}$ ) wall-normal locations, providing the energy distribution across the range of streamwise ( $\unicode[STIX]{x1D706}_{x}$ ) and spanwise ( $\unicode[STIX]{x1D706}_{y}$ ) wavelengths (or length scales) that are coherent across the wall-normal distance. $\unicode[STIX]{x1D6F7}_{cross}^{w}$ may thus be interpreted as a wall-filtered subset of the full 2-D $u$ -spectrum ( $\unicode[STIX]{x1D6F7}$ ), the latter providing information on all coexisting eddies at $z_{o}$ . To this end, datasets comprising synchronized two-point $u$ -signals at $z_{o}$ and $z_{r}$ , across the friction Reynolds number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{4})$ , are analysed. The published direct numerical simulation (DNS) dataset of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109) is considered for low- $Re_{\unicode[STIX]{x1D70F}}$ analysis, while the high- $Re_{\unicode[STIX]{x1D70F}}$ dataset is obtained by conducting synchronous multipoint hot-wire measurements. High- $Re_{\unicode[STIX]{x1D70F}}$ cross-spectra reveal that the wall-attached large scales follow a $\unicode[STIX]{x1D706}_{y}/z_{o}\sim \unicode[STIX]{x1D706}_{x}/z_{o}$ relationship more closely than seen for $\unicode[STIX]{x1D6F7}$ , where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.


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Two-dimensional cross-spectrum of the streamwise velocity in turbulent boundary layers

  • Rahul Deshpande (a1), Dileep Chandran (a1), Jason P. Monty (a1) and Ivan Marusic (a1)


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