Interaction of a turbulent boundary layer with a free surface

We investigate the interaction between an initially wall-bounded turbulent boundary layer (TBL) and a free surface in a recirculating water tunnel, where air is injected through a backward-facing step to generate well-controlled flow separation and to form a continuous air layer beneath a flat plate. Simultaneous time-resolved particle image velocimetry and free-surface tracking are employed at three streamwise stations to examine the evolution of TBL, free surface and their linkage across friction Reynolds numbers italic Re Subscript tau Baseline equals 800 minus 2200 <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>800</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> <mml:mn>2200</mml:mn> </mml:math> $\textit{Re}_\tau = 800 {-}2200$ and nominal air-layer thicknesses t Subscript italic AL Baseline equals 2 minus 11 <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>t</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">AL</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> <mml:mn>11</mml:mn> </mml:math> $t_{\textit{AL}} = 2{-}11$ mm. The transition to a free-surface boundary condition leads to rapid TBL relaxation and significant turbulence decay. Consequently, the outer layer develops a velocity gain near the free surface and, interestingly, a deficit farther away. The downward shift of Reynolds stress peaks and increasing subsurface two-point correlation lengths suggest enhanced vertical mixing as the mechanism driving this deficit. For the chosen cases where the air layer remains continuous, free-surface deformations are found to be non-dispersive imprints of the underlying TBL. Conditional averaging links strong positive and negative deformations to spanwise vortical structures of opposite rotation. The expanding size of coherent structures also promotes larger free-surface wavelengths downstream, which asymptotically approach a limiting value as italic Re Subscript tau <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mi>τ</mml:mi> </mml:msub> </mml:math> $\textit{Re}_\tau$ increases towards the air-layer breakup regime. Despite overall turbulence decay, free-surface deformation amplitudes continue to grow at higher- italic Re Subscript tau <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mi>τ</mml:mi> </mml:msub> </mml:math> $\textit{Re}_\tau$ cases due to persistent vertical motion. This subsurface motion is progressively suppressed downstream by the free-surface boundary condition, strengthening correlations between deformations and streamwise rather than vertical velocity fluctuations.