| Flow past rectangular cylinders represents an idealized flow around bluff bodies.Knowing the hydrodynamic properties of rectangular cylinders in a steady flow is of practical importance for many engineering applications.Although the unbounded flow past rectangular cylinders has been broadly studied,investigations on the flow around rectangular cylinders close to the free surface are limited.Particularly,the relationship between the effects of the depth-to-length ratio(d/l)and the width-to-length ratio(w/l)of rectangular cylinders near a free surface has not been studied;classifications of the flow around rectangular cylinders close to the free surface depending on d/l and w/l have not been tackled;Reynolds-number(Re)effect on flow past a square cylinder in the presence of a free surface has not been well understood.In this study,two-dimensional multiphase simulations of flow past rectangular cylinders close to the free surface were conducted using the Reynolds-averaged Navier-Stokes(RANS)approach.The Mentor shear stress transport(SST)k-?model and the volume of fluid(VOF)model were adopted.The following four topics were investigated:(1)The applicability of the Mentor SST k-?model and the VOF model in the multiphase simulations of flow past rectangular cylinders near the free surface was studied.Three aspects were considered in the numerical validation:1)The velocity and pressure distributions around a square cylinder located at d/l=4.5;2)The force coefficients of rectangular cylinders with various w/l ranging from 0.1 to 5.0 located at d/l=4.5;and 3)The vortex shedding flow around rectangular cylinders with various w/l ranging from 0.1 to 5.0 located at d/l=4.5.The numerical validation test shows that the present multiphase simulations based on the Mentor SST k-?model and the VOF model can provide reasonable predictions of the flows and forces of rectangular cylinders close to the free surface when w/l is in the range of 0.7?w/l?5.0.(2)The effects of d/l and w/l on rectangular cylinders near a free surface were studied.Results show that the effect of d/l on the hydrodynamic properties of rectangular cylinders is affected by w/l and becomes significant when d/l is lower than 1.0.The effect of w/l on the flow past rectangular cylinders near the free surface is different from that in an infinite domain.(3)The flow past rectangular cylinders near a free surface was classified depending on d/l and w/l.In order to identify the vortex shedding patterns,multiphase simulations over a broad range of d/l(0.3-4.5)and w/l(0.7-5.0)were conducted.Flow results show that the anti-symmetric vortex shedding becomes asymmetric and then suppressed as d/l decreases.Three patterns of asymmetric flow depending on w/l were observed:“asymmetrically separated”,“one-sidedly reattached”and“asymmetrically reattached”.The boundaries between different flow patterns were defined based on the velocity and force variations.The critical d/l,below which the vortex shedding is suppressed,falls within the ranges from 0.5 to 0.6 and from 0.8 to0.9 for rectangular cylinders with 0.7?w/l<1.5 and 2.8<w/l?5.0,respectively.The critical d/l for rectangular cylinders with 1.5?w/l<2.8 varies slightly with w/l and is located between0.7 and 1.0.(4)The effect of Re on square cylinders near a free surface was investigated.In this study,multiphase simulations of flow past a square cylinder located at four d/l ratios(0.3,0.7,1.0 and4.5)over the Re range from 2.0×10~4 to 9.9×10~4 were conducted.Strong Reynolds-number effect on the flow was observed at shallow and moderate depths.The surface jet,forming as the result of surface curvature when Re increases,is related to the local Froude number(Fr).The critical Re values for the surface jet formation at the three d/l ratios,i.e.,0.3,0.7 and 1.0,are 3.0×10~4,4.0×10~4 and 4.5×10~4,respectively.The suppression of vortex shedding at shallow depths is explained as the detachment of shear layers together with the modifications of vorticity fluxes.The restoration of vortex shedding when Re is high is due to the surface jet interference which forces the recontact of the shear layers across the sharp corners to couple with each other. |
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