Spanwise length scales of turbulent motions essential for sustaining near-wall turbulence

To investigate the spanwise length scales of turbulent motions essential for sustaining near-wall turbulence, we perform numerical experiments in direct numerical simulations of full-scale turbulent channel flows by filtering out motions wider than a prescribed cutoff wavelength lamda Subscript z f <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\lambda _{zf}$ . As lamda Subscript z f <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\lambda _{zf}$ decreases, near-wall short vortical structures are suppressed, while long streaks are amplified. When lamda Subscript z f <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\lambda _{zf}$ is reduced to approximately 200 wall units, the flow relaminarises, indicating that at least turbulent motions twice those of near-wall coherent structures are necessary. To elucidate the underlying mechanism, we derive transport equations for Reynolds shear stresses associated separately with long streaks and short vortical structures, and decompose nonlinear transport terms into contributions from spanwise-wide motions, long streaks, short vortical structures and their interactions. Spanwise-wide motions enhance the energy production of short vortical structures by supplying energy, via nonlinear interactions, to vertical velocity fluctuations; thus, their removal suppresses these structures. Conversely, nonlinear interactions involving spanwise-wide motions act as a major energy sink for long streamwise fluctuations, so their absence amplifies streaks. However, long streaks ultimately depend on short vortical structures, as their energy is extracted from the mean shear by vertical velocity fluctuations, which draw energy from spanwise velocity fluctuations sustained by nonlinear interactions involving both short structures and spanwise-wide motions. Therefore, when short structures are strongly suppressed at sufficiently small lamda Subscript z f <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\lambda _{zf}$ , the flow relaminarises. These findings highlight the central role of short vortical structures and their dependence on ‘inactive’ spanwise-wide turbulent motions, and reveal multiple interconnected energy cycles governing near-wall coherent structures.