Abstract
We review our recent studies on the hierarchy of coherent vortices in high-Reynolds-number turbulence of an incompressible neutral fluid, which were conducted through analyses of data obtained by direct numerical simulations of the Navier–Stokes equation. We show results on turbulence under four different boundary conditions: namely, turbulence in a periodic cube, turbulent wake behind a circular cylinder, turbulence between a pair of parallel planes (i.e., turbulent plane Poiseuille flow), and a zero-pressure gradient turbulent boundary layer. By decomposing each of these turbulent fields into different length scales, we show that turbulence is composed of the hierarchy of coherent vortices with different sizes. More concretely, in a region apart from solid walls, each level of the hierarchy consists of tubular vortices and they tend to form counter-rotating pairs. It is a strain-rate field around them that stretches and amplifies smaller vortices. In other words, the energy cascade in turbulence away from walls is not caused by breakups of larger eddies, but vortex stretching of smaller eddies in larger-scale strain-rate fields. In near-wall regions, the sustaining mechanism of vortices depends on their scale, which we need to consider depending on the distance from a wall. Large vortices (i.e., wall-attached eddies), whose diameter is as large as the distance from a wall, are sustained by the mean-flow stretching, whereas smaller vortices (i.e., wall-detached eddies), whose diameter is smaller than the distance, are created by being stretched by larger vortices. The latter mechanism corresponds to the energy cascade similarly observed in wall-free turbulence. Scale decomposition can also reveal the largest vortices in each turbulence, which depends on the boundary condition. It is particularly important that the largest wall-attached eddies in the turbulent boundary layer are hairpin vortices even in downstream regions.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
Notes
We use the definition of \(E_{\parallel }\) as
$$\begin{aligned} \frac{1}{2} u'^2 = \int _0^\infty E_{\parallel }(k) dk \,, \quad \quad \quad \quad \quad \quad (29) ,\end{aligned}$$which is different from \(\phi _1\) adopted by Sreenivasan (1995) with a factor 2; \(\phi _1=2E_{\parallel }\).
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Acknowledgements
We thank Dr Lozano-Durán and Prof Jiménez for providing us with their turbulent channel data. The DNS were conducted under the supports of the NIFS Collaboration Research Programs (08KTBL006, 11KNSS023, 13KNSS043, 15KNSS066, 17KNSS101, 18KNSS108, 20KNSS145, 22KISS010) and the HPCI System Research projects (hp210207, hp210075, hp220232). Our studies were partly supported by JSPS Grants-in-Aids for Scientific Research (16H04268, 20H02068, 20H01819, 20J10399, 21K20403, 23K13253).
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Goto, S., Motoori, Y. Hierarchy of coherent vortices in developed turbulence. Rev. Mod. Plasma Phys. 8, 23 (2024). https://doi.org/10.1007/s41614-024-00161-8
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DOI: https://doi.org/10.1007/s41614-024-00161-8