Three-dimensional turbulent free-surface flow past a submerged horizontal cylinder: combined dependence on Froude number and gap ratio

Free-surface response to flow past a fully submerged circular cylinder is known to be governed by the Froude number italic Fr <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Fr</mml:mtext> </mml:mrow> </mml:math> $\textit{Fr}$ and the gap ratio upper G <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:mi>G</mml:mi> </mml:math> $G$ . However, most previous work has primarily examined this combined dependence in two-dimensional configurations at low Reynolds numbers. Here, three-dimensional high-resolution numerical simulations are conducted to investigate turbulent two-phase free-surface flow past a fully submerged circular cylinder at high Froude numbers ( 0.8 less than or slanted equals italic Fr less than or slanted equals 2.0 <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:mn>0.8</mml:mn> <mml:mo>⩽</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Fr</mml:mtext> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mn>2.0</mml:mn> </mml:math> $0.8 \leqslant \textit{Fr} \leqslant 2.0$ ) and small gap ratios ( 0.025 less than or slanted equals upper G less than or slanted equals 0.4 <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:mn>0.025</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>G</mml:mi> <mml:mo>⩽</mml:mo> <mml:mn>0.4</mml:mn> </mml:math> $0.025 \leqslant G \leqslant 0.4$ ), at a Reynolds number italic Re equals 2000 <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>2000</mml:mn> </mml:math> $\textit{Re} = 2000$ and a Weber number italic We equals 1000 <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:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">We</mml:mtext> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1000</mml:mn> </mml:math> $\textit{We} = 1000$ , corresponding to a fully turbulent regime and small surface tension. The investigated parameter space corresponds to a flow regime where wake-induced interface disruption generates bubbles over a broad range of scales. This mechanism is of practical significance for engineering applications involving naval and offshore structures. The upstream and downstream free-surface flow patterns are classified. Free-surface breaking upstream of the cylinder is reported for 1.1 less than or slanted equals italic Fr less than or slanted equals 1.4 <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:mn>1.1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Fr</mml:mtext> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mn>1.4</mml:mn> </mml:math> $1.1 \leqslant \textit{Fr} \leqslant 1.4$ , and its physical mechanism and occurrence condition are discussed. Downstream, as the breaking pattern transitions from spilling to plunging with increasing Froude number, the bubble number density spectrum exhibits distinct scal