Reynolds and froude number effect on the flow past an interfacepiercing circular cylinder
 Author: Koo Bonguk, Yang Jianming, Yeon Seong Mo, Stern Frederick
 Organization: Koo Bonguk; Yang Jianming; Yeon Seong Mo; Stern Frederick
 Publish: International Journal of Naval Architecture and Ocean Engineering Volume 6, Issue3, p529~561, 30 Sep 2014

ABSTRACT
The twophase turbulent flow past an interfacepiercing circular cylinder is studied using a highfidelity orthogonal curvilinear grid solver with a Lagrangian dynamic subgridscale model for largeeddy simulation and a coupled level set and volume of fluid method for airwater interface tracking. The simulations cover the subcritical and critical and post critical regimes of the Reynolds and sub and supercritical Froude numbers in order to investigate the effect of both dimensionless parameters on the flow. Significant changes in flow features near the airwater interface were observed as the Reynolds number was increased from the subcritical to the critical regime. The interface makes the separation point near the interface much delayed for all Reynolds numbers. The separation region at intermediate depths is remarkably reduced for the critical Reynolds number regime. The deep flow resembles the singlephase turbulent flow past a circular cylinder, but includes the effect of the freesurface and the limited span length for subcritical Reynolds numbers. At different Froude numbers, the airwater interface exhibits significantly changed structures, including breaking bow waves with splashes and bubbles at high Froude numbers. Instantaneous and mean flow features such as interface structures, vortex shedding, Reynolds stresses, and vorticity transport are also analyzed. The results are compared with reference experimental data available in the literature. The deep flow is also compared with the singlephase turbulent flow past a circular cylinder in the similar ranges of Reynolds numbers. Discussion is provided concerning the limitations of the current simulations and available experimental data along with future research .

KEYWORD
Largeeddy simulation , Interfacepiercing circular cylinder , Runup , Wave breaking , Vortical structures

INTRODUCTION
The turbulent flow past a circular cylinder has been investigated extensively for a long time due to its importance in many engineering applications. It shows different features at different Reynolds numbers (
Re ), based on the freestream velocity, cylinder diameter, and kinematic viscosity. There are three states of the flow with differentRe : subcritical (Re =1000 ~ 2×10^{5} ), critical (Re = 2×10^{5} ~ 3.5×10^{6} ), and supercritical (Re > 3.5×10^{6} ) flows. In the subcritical range the boundary layer along the cylinder is laminar and the transition to turbulence happens in the free shear layer downstream of the cylinder. With an increase ofRe , the location of the transition moves upstream (Wissink and Rodi, 2008). In the critical range the base suction and drag decrease dramatically and this is associated with a revitalized boundary layer characterized by events in the order of a laminar separation, a transition to turbulence, a reattachment, and a turbulent separation further downstream with much reduced downstream wake width. In the supercritical regime, the boundary layer along the cylinder becomes turbulent before the separation. The base suction and drag of this regime is low due to the later separation of the turbulent boundary layer (Kravchenko and Moin, 2000).The twophase flow past an interfacepiercing circular cylinder is much more complicated than the singlephase flow due to the interaction between the viscous effect near the body and the airwater interface phenomena, including wave runup, breaking waves, thin liquid sheet formation, and air entrainment. It also exists in many engineering applications such as ship, ocean, coastal, and hydraulic engineering. The effects of the interface on the force distribution on the body, vortex generation and turbulence structures, and airwater interface structures, especially, their changes with the Reynolds and Froude (
Fr ) numbers are not well understood. A better understanding of these effects is also important for the cases when vortex and/or waveinduced vibrations are to be considered.There are several experimental and numerical studies on the flow past a surfacepiercing circular cylinder in mainly subcritical Re regime. Inoue et al. (1993) conducted a towing tank experiment to investigate the characteristics of free surface turbulence for
Re =27,000 andFr =.8 andRe =29,000 andFr = 1. They found that the periodic vortex shedding occurs in the deep flow, while this periodic vortex shedding is attenuated and higher frequency fluctuations are more prominent near the interface. Kawamura et al. (2002) investigated the wavewake interaction about a surfacepiercing circular cylinder using LargeEddy Simulation (LES) based on a Smagorinsky SubGrid Scale (SGS) model atRe =27,000 with three differentFr =0.2, 0.5 and 0.8. At a lowFr , surface deformations were small and the influence on the wake was negligible. On the contrary, the generated surface wave was very steep and strongly unsteady at a highFr . They also predicted significant surface fluctuations inside the recirculation zone immediately after the surface wave crest. In addition, they were able to visualize the attenuation of vortex shedding near the interface. Flows past an interfacepiercing cylinder at variousRe =1×10^{5} and up toFr = 3 simulated using LES based on a Smagorinsky SGS model and a Volume of Fluid (VOF) method by Yu et al. (2008). They also showed that the freesurface attenuates the organized vortex shedding at the interface. At a higher Re, the free surface effect was reduced whereas this effect was enhanced at a higherFr .As Re increases, the mean drag coefficient increases; however, it decreases along withFr . They also showed that the dominant Strouhal number of the lift force decreased along withRe . Recently, Suh et al. (2011) studied the effect of airwater interface on the vortex shedding from a vertical circular cylinder forRe =27,000 andFr =0.2, 0.8 using a highfidelity orthogonal curvilinear grid solver. The organized periodic vortex shedding was observed in the deep flow whereas it was attenuated and replaced by smallscale vortices near the interface. The attenuation of the organized vortex shedding at the interface is caused by the streamwise vorticity and the outward transverse velocity generated near the edge of the separated region. The anisotropy between the vertical and transverse Reynolds normal stresses is the primary source of the streamwise vorticity and the outward transverse velocity near the interface.The abovementioned studies were focused on relatively low
Re andFr in the subcriticalRe /Fr regimes (Fr <1, subcritical flow;Fr =1, critical flow; andFr >1, supercritical flow). Chaplin and Teigen (2003) carried out an experimental study at higher Reynolds numbers up to 4.7×10^{5} and larger Froude numbers up to 1.67 with a constantRe /Fr ratio of 2.79×10^{5} . They found that the total resistance coefficient reached a maximum atFr ~1. They also measured the runup on the front of the cylinder and found that the runup at a givenFr has a strong dependence onRe . However, their study was limited to resistance, runup, and stagnation and base pressure measurements and analyses, many other flow features, such as the overall free surface and wave patterns, breaking waves near the cylinder, flow separation, vortex shedding, and other turbulent quantities, were not addressed. To the authors’ knowledge, this is the only recent study in the literature done on the turbulent flow past a vertical surfacepiercing circular cylinder in the critical and supercriticalRe /Fr regimes. As discussed above, the turbulent flow past a circular cylinder in the critical Re regime has very different flow physics than those in the subcriticalRe regime. Also the bow wave breaks in front of the cylinder for large Fr cases. Near the interface, the interactions of turbulence and interface are more complicated than the subcriticalRe /Fr cases studied in Suh et al. (2011) and the other work. Further investigations are required for better understanding of the effect of Reynolds and Froude numbers on the flow, especially, near the airwater interface.To this end, LES of the twophase turbulent flow past an interfacepiercing circular cylinder at different
Re /Fr with a constant ratio, same as Chaplin and Teigen (2003), are performed in the present study. This work can also be regarded as an extension of the precursory work for subcriticalRe andFr (Suh et al., 2011). The same twophase NavierStokes solver and computational domain, and similar grid sizes reported in Suh et al. (2011) are utilized here. In Suh et al. (2011), the Froude numbers were not high and there were no breaking waves. The level set method could effectively and accurately describe the evaluation of the airwater interface. However, for the supercriticalFr cases in this study, the level set method turns out to be too dissipative and incapable to capture the details of breaking waves. Therefore, a coupled level set and volumeoffluid method (Wang et al., 2009; Wang et al., 2012) is employed to obtain a much improved interface capturing capability. In this study the effect ofRe /Fr on the flow and the nature of unsteady freesurface are presented and discussed in detail. In addition, verification and validation of simulations are discussed using drag/friction/pressure coefficients, wave elevations, runup height on the front face of the cylinder, and depression depth behind cylinder. Mean separation pattern, vorticity, Reynolds stress, and vorticity transport are studied forRe effects and airwater interface structures, vortical structures, and wave breaking phenolmena are investigated forFr effects. Discussion is provided concerning the limitations of the current simulations and available experimental data along with future research.COMPUTATIONAL METHODS
CFDShipIowa version 6.2, a sharp interface orthogonal curvilinear grid solver for twophase incompressible flows recently developed at IIHR (Yang and Stern, 2009; Suh et al., 2011) is used for all computations. The interface is tracked by a Coupled Level Set and VolumeofFluid (CLSVOF) method (Wang et al., 2009; Wang et al., 2012) instead of the level set method used in Suh et al. (2011) to capture detailed breaking waves.
> Mathematical model
An orthogonal curvilinear coordinate system is adopted for the governing equations. The NavierStokes equations for twophase, immiscible, incompressible flows have been written in the orthogonal curvilinear coordinates. The continuity equation is given as follows:
where
u _{i} is the velocity in the orthogonal coordinateξ direction and .J is the Jacobian of the coordinate transformation, defined asJ =h_{i}h_{j}h_{k} , and withx_{i} a Cartesian coordinate.The momentum equations are given as follows:
where
ρ is the density,p is the pressure, t is the time, andg_{i} the gravity vector in theξ direction. Also, and∂ξ (i ) =h_{i} ∂ξ_{i} . The viscous stress tensorτ _{ij} is defined as follows:where
μ is the dynamic viscosity,ν_{t} is the turbulent eddy viscosity, andS _{ij} is the strain rate as defined as below:In the LES approach adopted in this paper, the large, energy carrying eddies are resolved by spatial filtering of the governing equations whereas the smallscale, dissipative eddies are modeled by a SGS model. The Lagrangian dynamic SGS model based on Sarghini et al. (1999) is used in present LES. Eq. (2) is obtained as follows
with and , respectively. For simplicity, the filtering signs are dropped hereafter.
The interface is represented by the LevelSet (LS) function which is corrected using the VOF function to enforce mass conservation. The LS function
φ is defined as a distance function which is negative in the air, positive in the liquid, and zero at the interface. The VOF function F is defined as the liquid volume fraction in a grid cell that gives zero in the air, one in the liquid, and a value between zero and one in an interfacial cell, respectively.The LS function and the VOF function are advanced using following equations respectively.
Each phase of constant density and viscosity can be defined using the LS function in the computational domain and sharp jumps of the fluid properties occur at the phase interface. In this study, the density keeps its sharp jump and the viscosity is smoothed over a transition band across the interface (Yang and Stern, 2009).
> Numerical method
The finitedifference method is used to discretize the NavierStokes equations on a nonuniform orthogonal curvilinear grid. A staggered variable arrangement is adopted, i.e., the contravariant velocity components
u_{i} ,u_{j} ,u_{k} are defined at cell faces in theξ_{i} ,ξ_{j} ,ξ_{k} directions, respectively, and all other variables are defined at cell centers. A semiimplicit time advancement scheme is used to integrate the momentum equations with the secondorder CrankNicolson scheme for the diagonal viscous terms and the secondorder AdamsBashforth scheme for other terms. A fractionalstep method is employed for velocitypressure coupling, in which a pressure Poisson equation is solved to enforce the continuity equation.The convective terms are discretized using the nonconservative fifthorder HamiltonJacobi WeightedENO (HJWENO) (Jiang and Shu, 1996) scheme and all other terms are approximated using the secondorder central difference scheme. A semicoarsening multigrid Poisson solver from the HYPRE library (Falgout et al., 2006) is used. Usually, the Poisson solver takes most of the computation time in a single time step.
The LS advection equation is solved using the thirdorder TVD RungeKutta scheme (Shu and Osher, 1988) for time advancement and the fifthorder HJWENO scheme for spatial discretization. To keep the LS function as a signed distance function, it has to be reinitialized after a certain time of evolution. The CLSVOF method (Wang et al., 2009; Wang et al., 2012) is used to redistance the LS function and improve mass conservation properties of the LS method. In the CLSVOF method, the interface is reconstructed based on the VOF function with the interface normal computed from the LS function. The level set field is then redistanced to reflect the position of the reconstructed interface, which satisfies the volume conservation constraint.
SIMULATION CONDITIONS, VERIFICATION AND VALIDATION
> Grid, computational domain, and geometry
Bodyfitted cylindrical grids were used for all cases and Table 1 shows the number of grid points and distance of the first grid point from the cylinder wall (
y ^{+} ) for each case. Note that three different grids (coarse, medium, and fine) forRe = 458,000 andFr =1.64 were used for a grid convergence study. The grid points are clustered near the cylinder wall to resolve the boundary layer and flow separation. Near the interface the grid was also refined to capture the interface deformation. Same as Suh et al. (2011), the computational domain was set up such that the portions of cylinder in the water and air are of length 4D and 2D with D the cylinder diameter, respectively. Note that this is the same aspect ratio used by Inoue et al. (1993), Kawamura et al. (2002) and Yu et al. (2008). The distance from the center of the cylinder to the outer boundary is 20D including a buffer zone to damp wave reflections as shown in Fig. 1.> Flow conditions, boundary conditions, and initial conditions
The diameter of cylinder,
D and the freestream velocity,U _{∞} , were used for the normalization of all variables and the two nondimensional parameters, Froude number and Reynolds number, are defined as follows:As shown in Fig. 1(a), the outer boundary was divided into inflow and outflow boundaries at
θ = 90° andθ = 270° , in whichθ is the tangential angle starting from the downstream direction a nd a uniform inflow and a convective outflow boundary conditions were used. Noslip boundary conditions were applied on the cylinder wall, while the slip boundary condition was adopted at the bottom and the top of the computational domain as shown in Fig. 1(b). A uniform velocity field same as the upstream velocity is prescribed to the entire computational domain as the initial condition. In the present simulation, a constant CourantFriedrichsLewy (CFL) number of 0.3 was used where the magnitude of the time step varied from 1×10^{−2} to 1×10^{−4}D /U _{∞} depending on the flow conditions.The investigations cover 10 simulation cases, as also shown in Table 1. Cases 13 are for subcritical and cases 410 are for post critical
Re . Cases 1, 2, 4, 5 and 8 are for comparison with Chaplin and Teigen (2003). Cases 68 are for verification study at the highest Re and Fr condition. Cases 3, 4 and 9 are forRe = 27,000, 234,000 and 458,000 study at sameFr = 0.84 and Cases 7, 9, and 10 are forFr = 0.84, 1.24 and 1.64 study at sameRe = 458,000. Cases 1 and 3 are similar to Suh et al. (2011), but at a somewhat larger subcritical Re and Fr, respectively.> Verification and validation of integral variables
The time histories of the drag coefficient
C_{D} and lift coefficientC_{L} with the running mean ofC_{D} andC_{L} are shown in Fig. 2. The drag and lift coefficients are defined asIt should be noted that the still water depth, i.e.,
H = 4D here, and the water density are used for these coefficients (Kawamura et al., 2002; Yu et al., 2008; Suh et al., 2011). The statistically stationary state is defined using the convergence of the running mean from the time history ofC_{D} . When the fluctuations of the running mean are smaller than 1% of the mean value, the flow is considered statistically stationary. After the flow reached steady state, 16 vortex shedding cycles, 80 nondimensional time units, were used for statistics as shown Fig. 2.The grid verification study is performed following the methodology and procedures proposed by Stern et al. (2006) and Xing and Stern (2010) with three different grids (coarse, medium, and fine) for
Re = 458,000 andFr = 1.64 using four parameters (total drag coefficientC_{D} , friction drag coefficientC_{Df} , pressure drag coefficientC_{Dp} , and rms of lift coefficientC_{L}^{rms} ). Verification is a process for assessing the simulation numerical uncertaintyU_{SN} , defined as followed,U_{SN}^{2} =U_{I}^{2} +U_{T}^{2} +U_{G}^{2} , whereU_{I} is the iterative uncertainty,U_{T} is the timestep uncertainty, andU_{G} is the grid uncertainty. It should be noted thatU_{T} was not considered in the present study due to the use of a constant CFL number, i.e. varied time steps.Iterative convergence is assessed by examining the iterative history of drag forces and the iterative uncertainty
U_{I} is estimated as half the range of the maximum and minimum values as shown below;where
S_{U } is the maximum solution andS_{L} is the minimum solution. In Table 2,U_{I} /ε _{12} for three different grids are shown and its magnitude is one order smaller than that of the grid error.For all three grids, the iteration uncertainties are considered to be negligible in comparison to the grid uncertainties. Table 3 shows the iterative uncertainties for all the cases and it is less than 1% of the mean value of
S , which also implies that the simulations are not contaminated by the iterative errors. so that simulation numerical uncertainty .In Table 2, all the parameters show monotonic convergence using the grid refinement ratio
r_{G} = 1.41 and the order of accuracyp_{G} forC_{D} ,C_{Df} ,C_{Dp} , andC_{L}^{rms} is 1.13, 1.71, 1.14, and 0.67 respectively and the grid uncertaintyU_{G} forC_{D} ,C_{Df} ,C_{Dp} , andC_{L}^{rms} is 1.1%S _{1} , 0.13%S _{1} , 1.1%S _{1} , and 0.79%S _{1}, respectively.Validation is a process for assessing the simulation modeling uncertainty by comparing with experimental data
D . The comparison errorE is defined as the difference betweenD and the simulationS . If the absolute value ofE is less than the validation uncertaintyU_{V} , given byU_{V}^{2} =U_{D}^{2} +U_{SN}^{2} , whereU_{D} is the uncertainty in the data, the combination of the all the errors inD andS is smaller thanU_{V} and validation is achieved at theU_{V} level. As presented in Table 4,U_{V} of all the variables is greater thanE althoughU_{D} is not available, which shows the present simulation has achieved theU_{V} level of validation.Fig. 3(a) compares present simulations with available experimental data for single and twophase flow. All data is graphed for the correct Re scale; however, the singlephase data has no
Fr correspondence and Hay (1947) twophase data has a somewhat differentRe / Fr = 2.97×10^{5} , whereas the present simulations and Chaplin and Teigen (2003) results are forRe / Fr = 2.97×10^{5} , as per bothRe andFr scales. Chaplin and Teigen (2003) experiments are for with and without splitter plate to suppress free surface effects, with end plate, for variable aspect ratio from 47, and subject free surface effects forFr >.5 and blockage for allFr . Hay (1947) experiments are for no end plate and aspect ratio 2 and also subject free surface effects. Note that the present simulations are subject free surface effects and for aspect ratio 4. As will be discussed later, for subcriticalRe aspect ratio needs to be larger than all current values to achieve singlephase benchmark conditions for the deep condition. The Chaplin and Teigen (2003) data with splitter plate is substantially smaller/larger than singlephase benchmark data for sub and postcriticalRe , but with similar criticalRe values. Chaplin and Teigen (2003) data without splitter plate and Hay have similarRe / Fr but different aspect ratio and experimental setup nonetheless somewhat similar trends: both showC_{D} max at criticalRe andFr near one (Fr = .75 vs.Fr = .96, respectively) and Hay (1947) has overall smaller values. Present simulations agree Chaplin and Teigen (2003) for sub and post critical Re but do not show increased values for criticalRe and nearFr = 1. Fig. 3(b) compares present simulations with Chaplin and Teigen (2003) data without splitter plate forC_{D} vs. depth. Experiments show bulge inC_{D} below free surface for all Fr with maximum bulge for Fr=0.96. Simulations forFr =0.84 show similar trends experiments forFr = 0.84, but under predictC_{D} values.Fig. 4 shows the Fast Fourier Transform (FFT) of the
C_{D} andC_{L} forRe = 458,000 andFr = 1.64. The dominant Strouhal number (St ) forC_{L} is obvious while the FFT ofC_{L} shows a large range of frequencies. Fig. 5(a) presentsSt with differentRe andFr , including singlephase reference data. The results have good agreement with singlephase data for lowerRe and the dominantSt of theC_{L} increases withRe (andFr ) increase, but not as much as one of the reference data. The dominantSt from three different grids forFr =1.64 are also shown in Fig. 5 and the fine grid simulation predicts closest with one of the experiments. Fig 5(b) shows CL RMS vs.Re , including singlephase reference data. The simulation trends forFr = .84 are similar with the singlephase data in showing decreasing CL RMS with increasingRe . The simulations forFr =.2 and subcritcalRe show larger values than those forFr =.84 and subcriticalRe . The simulations for post criticalRe andFr = 1.64 show larger values than singlephase data for sameRe . Note that the presentC_{D} ,St andC_{L} RMS results forFr = .84 andRe = 234,000 have close values to Suh et al. (2011) results forFr = .8 andRe = 27,000.Fig. 6 shows the runup height and depression depth with different
Re andFr numbers. The runup height is the maximum bow wave height in front of the cylinder and the depression depth is the largest cavity depth behind the cylinder. Both runup height and depression depth increase asFr increases. The runup heights of the present simulations have good agreement with the Bernoulli’s equation (Fr ^{2} / 2 ) and the results from the previous study by Chaplin and Teigen (2003). Fig. 6 also shows the runup height from three grids forRe = 458,000 andFr = 1.64 and the fine grid has the closest to that from the Bernoulli’s equation and previous experiments (Chaplin and Teigen, 2003). The present results for depression depth agree with Hay (1947) for subcriticalFr , but have larger values for supercriticalFr . A ratio between runup height and depression depth forFr = 0.84, 1.24, and 1.64 is 1.375, 0.97, and 1.23, respectively.Flow field validation for Re = 234,000/ Fr = 0.84
In this section, the flow near the interface is compared with previous studies (Suh et al., 2011; Kawamura et al., 2002; Yu et al., 2008; Inoue et al., 1993). Note that the present simulation and the previous work have similar
Fr , but different mostly subcriticalRe .Contours of the mean interface elevation and the rms of the interface fluctuations are shown in Fig. 7 and compared with measurements by Inoue et al. (1993) using
Re = 27,000 andFr = 0.80. Overall, the present results of the mean interface elevation are in good agreement with the previous study (Inoue et al., 1993) including the magnitude, location of interface waves, and the expansion angle of Kelvin wave, although lower interface elevations are observed near the rear parts of the cylinder. The rms of the interface fluctuations from the current simulation also shows a good agreement with the experimental work (Inoue et al., 1993) including the location of the peak fluctuations.Fig. 8 presents the mean interface elevation and the rms of the interface fluctuations at two transverse planes for
Fr = 0.84, comparing with previous studies (Kawamura et al., 2002; Suh et al., 2011) withRe = 27,000 andFr = 0.8. The results of the current simulation with largerRe (= 234,000) are in good agreement with previous studies except for the underprediction of the depression, which is also reported in Suh et al. (2011). In addition, this indicates thatRe has little effect to the mean interface elevation behind the cylinder for subcriticalRe .[Fig. 8] Profiles of mean interface elevation at x = 0.9 and x = 2.0 for Re = 234,000 and Fr = 0.84.
Fig. 9 shows the vertical profiles of the mean streamwise velocity at
x = 4.5 andy = 0 forFr = 0.84. The previous studies from Inoue et al. (1993), Kawamura et al. (2002), and Suh et al. (2011) ofRe = 2.7×10^{4} andFr = 0.8 were compared. Note that the fine grid solutions were compared for Suh et al. (2011). The present results forFr = 0.84 agree with ones from Inoue et al. (1993), Kawamura et al. (2002), and Suh et al. (2011) near the airwater interface and it is clearly shown that the mean streamwise velocity decreases as the interface approaches. However, larger streamwise velocity profile is shown than previous studies in the deep flow.In this section, the flow in the deep water was compared with previous experiments on the singlephase flow past a cylinder at a similar Reynolds numbers as the present
Re (=234,000) which is in the lower criticalRe regime (Achenbach, 1968). Fig. 10(a) shows the mean streamwise velocity on the centerline aty = 0.0 along with the singlephase experimental measurements (Cantwell and Coles, 1983). Overall, the profile of the present simulation has a good agreement with the experimental data. The recirculation region is shown immediately after the cylinder (x = 0.5 tox = 1.4) in the present simulation.Fig. 10(b) and (c) shows the mean streamwise and transverse velocity in the wake at
x = 1. For comparison the data of the experiments atRe = 140,000 (Cantwell and Coles, 1983) are also used. They agree fairly well with experimental data; however, larger streamwise velocity is observed and smaller transverse velocity is shown aty = 0.5 andy = 0.5.Fig. 11 presents the pressure and friction coefficients, defined as follows:
where
τ_{w} is the wall shear stress. Some experimental measurements of the surface pressure and friction coefficients at closeRe in single phase flows are also plotted for comparison and reference. For subcriticalRe , both the pressure distributions and skin frictions in the deep flow have a good agreement with the experiments. However, for postcriticalRe , the predicted surface pressure still exhibits a subcritical behavior, e.g., atRe = 458,000. Issues that may explain this behavior will be discussed later. Note that Suh et al. (2011) showed similar comparisons as Figs. 10 and 11 (for subcritical Re ) forFr = .8 andRe = 27,000 with very similar values.Fig 12(a), (b) and (c) compares present simulations with Chaplin and Teigen (2003) experiments without splitter plate for stagnation pressure, base pressure coefficient vs. depth and in deep water vs.
Re , respectively. The latter case includes benchmark singlephase data. Stagnation pressure increases with Fr similarly as runup height with close agreement simulations and experiments. The experimental base pressure vs. depth shows similar trend asC_{D} vs. depth. The simulation forFr =.84 is similar in shape to the experiments but under predicts the pressure magnitude. The experimental deepwater base pressure vs. depth shows large differences benchmark singlephase data. The present simulations for subcriticalFr show smaller values than the experiments, whereas the values for postcriticalRe agree with the experiments. Another issue with the experiments of Chaplin and Teigen (2003) is that their pressure measurements showed no clear evidence of vortex shedding for all cases.Current results for subcritical
Re with fixed subcriticalFr show similar trends with Suh et al. (2011) on runup height, depression depth, mean surface elevation, and mean streamwise velocity. The drag coefficient in deep water is different with that of Suh et al. (2011) due to the difference inRe . Overall conclusion is that variation in subcriticalRe has small influence for fixed subcriticalFr .REYNOLDS NUMBER EFFECT WITH FIXED FROUDE NUMBER = 0.84
Three different Reynolds numbers from subcritical to postcritical (27,000, 234,000, and 458,000) with
Fr = 0.84 are used to investigate Reynolds number effect on flow structures near the interface.> Mean separation pattern
Fig. 13 shows the separation pattern of the mean flow with vortex core lines, which is obtained using the approach discussed in Kandasamy et al. (2009) and Sujudi and Haimes (1995). The separated shear layer was visualized approximately using the isosurfaces of the stagnation
C_{p} near the interface immediately before separation, assuming that the flow outside the separation is inviscid. Inside the separated region, the mean vortex core lines are extracted using the vortex core identifycation technique. The separated region at the interface decreases asRe increases from subcritical to critical since the separation point is delayed, which is also shown in singlephase flow past a circular cylinder (Achenbach, 1968). Immediately below the interface, the separation region is reduced for allRe and “neck” shape is observed. The neck becomes narrower asRe increases. Three different types of vortices (V1, V2, and V3) were defined in Suh et al. (2011) as the mean vertical vortices, the mean streamwise vortices, and the Vshaped mean vortices and these vortices are observed in allRe . The vertical vortices (V1) are generated from the vortex shedding behind the cylinder and these vortices are inclined and connected to the cylinder wall as the interface is approached. The other two vortices (V2, V3) are observed inside the separation region near the interface as discussed in Suh et al. (2011). It is interesting to note the close resemblance between the mean separation pattern and the profile of the sectional drag coeffcient as a function of depth in Fig. 3(b); however, in present case for fixedFr . Note that the present results forRe =27,00 are very similar Suh et al. (2011) for Fr = .8.Vorticity
For a singlephase flow past a circular cylinder, the regular Karman vortex street and organized wake patterns are prominent for low
Re whereas the separated shear layer becomes unstable and smaller vortices are generated at higherRe . Fig. 14 shows the instantaneous vertical vorticity contours with streamlines at various depths and differentRe . Near the interface atz =0.5 the attenuated vortex shedding. i.e., no organized periodic vortex is observed for allRe due to the streamwise vorticity generated near the separated region. However, the separation point is much delayed with an increase inRe . Suh et al. (2011) also showed that the delayed separation occurs near the interface. Atz = 1 where the “neck” shape is formed, the narrowing of the wake is observed for the critical and postcriticalRe (234,000, 458,000). In addition, the separation point moves downstream asRe increases. The narrow wake region becomes wider fromz = 1 toz = 2 which is correlated to the recovery of the separation region as shown in the mean separation patterns.Fig. 15 shows the mean streamwise vorticity contours for two
Re at three different depths. Near the interface small vortices are dominant instead of the organized vortices and more small vortical structures are observed asRe increases. In addition, the width of wake is significantly large for allRe . The maximum streamwise vortices are observed in the edge of the separation region. As discussed in Suh et al. (2011), three pairs of counterrotating vortices are observed atz =1 forRe =27,000 while only two pairs of counterrotating vortices are shown with much reduced wake region for critical Re due to the delayed separation point and smaller recirculation region. The increased wake region and strong transverse velocity near the interface are generated from these vorticity pairs (Suh et al., 2011). Relatively smaller vortices are shown atz =2 for allRe .The contours of the mean transverse vorticity are presented in Fig. 16. The mean transverse vorticity is significantly large as the interface approaches. High vorticity magnitudes are observed in the place where the separated region is located and it is related to high interface fluctuations inside separation region. At
z = 1 two positive vortices are detached from the cylinder wall and negative vortices are observed near the cylinder forRe = 27,000 while a pair of the symmetric positive vortices forms from the cylinder surrounding the negative vortices for criticalRe , which is the recirculation region. Since the transverse vorticity is responsible for the interface fluctuations, much smaller values are observed atz = 2.Fig. 17 shows the mean vertical vorticity contours. The vertical vorticity is inclined outward in transverse direction from the interface to
z = 1. Similar patterns are observed for criticalRe with steeper inclined angle. It is also shown that in deep flow the vertical vorticity is created by Karman vortex shedding and the unstable shearlayers behind the cylinder.Reynolds stresses
The streamwise Reynolds stress () is shown in Fig. 18. Subcritical Re (27,000) shows much higher magnitude of than critical and postcritical
Re (234,000, 458,000) at all three depths. It is also shown in Singh and Mittal (2005) that subcriticalRe has larger than critical and supercriticalRe . The peak values are concentrated on the separated region where very high mean velocity gradients are observed. They are inclined into transverse direction from the interface to aboutz =1 as shown Fig. 22 for subcriticalRe and criticalRe . The size of wake decreases withRe increase atz = 1 due to the delayed separation point and it gets recovery atz = 2, which is also shown in the mean separation pattern. Fig. 19 presents the mean transverse Reynolds stress (). Near the interface high values of are observed at near separation point and its magnitude shows stronger asRe increases. It is shown that fromz = 1 to deep flow achieves a peak along the center axis while the peaks in other Reynolds stresses are located off the flow axis. Atz = 1 the peak moves toward into the cylinder asRe increases. However, the peak moves downstream with increase inRe atz = 2. Increased magnitude is observed close to the cylinder especially in the region of separated shear layer asRe increases, indicating an increased unsteady activity in that region of the flow, particularly for criticalRe .The vertical Reynolds stress ( ) is shown in Fig. 20. Three peaks in are observed from the interface to the deep flow as shown in Fig. 22. The maximum magnitudes are observed near the interface for all
Re due to the fluctuations of the interface which matches with the first peak from subcriticalRe and criticalRe (Fig. 22). The second one is at aroundz = 1 and this is also seen in Fig. 20 with smaller wake region asRe increases into the critical regime. The features of near the interface are similar with that of .The mean Reynolds shear stress ( ) is shown in Fig. 21. It is shown that the interface reduces the shear stress remarkably in the wake region (Yu et al., 2008; Suh et al., 2011). For critical
Re , much smaller wake region is observed atz = 1 and it recovers atz = 2.Vorticity transport
Suh et al. (2011) studied the mechanism of the vorticity generation at subcritical
Re (27,000) using the Reynoldsaveraged vorticity transport equations which can be obtained by taking curl of the timeaveraged NavierStokes equation:where Ω
_{x} , Ω_{y} , and Ω_{z} are the streamwise, transverse, and vertical components of the mean vorticity, respectively. Term (A) represents the material derivative of the mean streamwise vorticity. The first term of term (B) is the vorticity amplification by the streamwise stretching and the rest provide vortexline bending effects. Term (C) represents the vorticity damping by the viscous diffusion, and (D), (E), and (F) are the vorticity production by inhomogeneity in the Reynolds stresses. It is noted that the effect of the airwater interface including the density discontinuity across the interface was neglected.Dominant source terms for the mean vorticity for
Re = 27,000 andRe = 458,000 are shown in Figs. 23 and 24 respectively. In the streamwise vorticity equation, the dominant terms are y and z components of term B and term E. They andz components of term B create the vortex bending whose have a similar magnitude but with opposite signs so they have little effect. The remaining E term, anisotropy term between transverse and vertical Reynolds stress, is mainly responsible for the generation of the mean streamwise vorticity near the interface. This behavior is observed in allRe from subcritical to criticalRe regime.In the transverse vorticity equation, the dominant terms are
z component of term B, E term, and F term. Due to the vortex shedding and shear layer instability, the bending of the vertical vorticity is generated which is thez component of term B. In deep water, the term F is a dominant for the vertical vorticity equations. Thez component of the term B is the vortex stretching of the vertical vorticity. The anisotropy term between the streamwise and transverse Reynolds stresses is responsible for the term E.Overall, for higher
Re (234,000, 458,000) the separation point is much delayed compared with that of the subcriticalRe (27,000). Due to the delayed separation, much smaller wake region and smaller separation region are shown for post criticalRe .> Overall Froude number effect with different
Re There is little influence of Reynolds number at the airwater interface based on variables related to the airwater interface such as mean interface elevation, runup height, and Kelvin wave patterns, etc. The instantaneous airwater interface elevations for all different Fr with different Re are shown in Fig. 25. For the lowest
Fr (= 0.2), it is hard to see the bow wave in front of the cylinder and small interface deformation is observed in the wake region. For subcriticalFr (= 0.44, 0.84), relatively smaller bow waves are observed in front of the cylinder with Kelvin waves behind the cylinder. There are some freesurface roughness and turbulence in the wake region behind the cylinder. ForFr = 1.24, much increased bow wave is observed and it breaks and wraps around the bow. A lot of splashes and bubbles are intense immediately behind the cylinder. The Kelvin waves are less visible just behind the cylinder due to stronger freesurface oscillations and turbulence. ForFr = 1.64, the bow wave increases remarkably with the largest wake region among fourFr and it also breaks with similar patterns of plunging breakers. Much smaller airwater interface structures including splashes and bubbles are observed behind the cylinder. It is also hard to distinguish the Kelvin waves behind the cylinder due to much larger freesurface oscillations and turbulence.The mean airwater interface elevations (top) are presented in Fig. 26. The timeaveraged free surface shows the bow wave in front of the cylinder, depression region or cavity region behind the cylinder, and the diverging Kevin wave. It is also confirmed that the height of the bow wave and the depression depth increases as
Fr increases. Except forFr = 0.2, the diverging Kelvin wave is evident but it is not shown for highFr (1.24, 1.64) since it is in the far downstream. More details of the diverging Kelvin wave will be discussed later. Fig. 26 also shows the rms of the interface fluctuations (bottom) for differentFr andRe . For lowFr (02.0.84) as shown in Figs. 26(a), (b) and (c) the fluctuation of the interface starts near the end of the slope from the crest to the surface cavity due to the relatively smaller runup height and there is almost no fluctuations in front of the cylinder. However, the higher runup height in front of the cylinder for highFr cases (Fig. 26(d), (e)) causes the fluctuation from the cylinder front to the depression region. For lowFr (0.20.84), the peak value is observed near the edge of the bottom of the depression region, whereas it is shown in further downstream and detached from the cylinder wall for highFr (1.24, 1.64). In addition, small structures of the interface fluctuation are observed for highFr due to highRe .FROUDE NUMBER EFFECT ON FLOW STRUCTURES WITH RE = 458,000
Three different Froude numbers (0.84, 1.24, and 1.64) with
Re = 458,000 are investigated to have better understanding of the effect of the airwater interface on the flow structures in the near wake at different Froude numbers. Different Reynolds numbers with aboveFr are also compared to examine the Reynolds number dependency. The fine grid solutions will be used forFr = 1.64 withRe = 458,000 in this study unless otherwise mentioned.> Airwater interface structures at different
Fr The mean airwater interface elevation contours for all three
Fr in Fig. 27 show diverging and transverse wave patterns. The diverging waves are more dominant than the transverse waves with wave lengths that roughly correspond to the theoretical transverse wave length in a Kelvin wave pattern, i.e.,λ_{t} =2πFr ^{2} . The diverging wave lengths for all threeFr match well with theoretical values as shown in Fig. 28(a). In addition, different Reynolds numbers other thanRe = 458,000 at Fr = 0.84 andFr = 1.24 are compared and they show almost similar wave length with ones ofRe = 458,000.The diverging angles for all three
Fr are also shown in Fig. 28 and they are all larger than that of the Kelvin wave value (19°) which is typically observed in ship flows. Bhushan et al. (2010) studied the flow around the AirCushion Vehicle (ACV) numerically with different Froude numbers and they showed similar Froude number effects that the diverging wave angle of the ACV decreases almost linearly up toFr = 1.0 and is almost constant thereafter which is within 10% of the Kelvin waves. The flow patterns at highFr show similar flow features with a slender body whereas the body acts like a blunt object at lowFr . The diverging angles decreases linearly withFr as shown in Fig. 28(b), which implies that the body looks slenderer asFr increases due to a cavity region immediately behind the cylinder. In Fig. 28(b), different Reynolds numbers thanRe = 458,000 are also added to investigateRe dependency on the freesurface flow and theRe effect is negligible on the diverging wave angle.Fig. 29 presents the closeup view of the mean airwater interface elevation contours with isosurface near the cylinder. Contour levels for high Fr (1.64) is 2 and about 5 times greater than those for medium (1.24) and low
Fr (0.84), respectively. ForFr = 0.84, the Kelvin waves propagate with a wide angle and relatively small depression depth behind the cylinder. ForFr = 1.24, the depression region immediately behind the cylinder is narrow and deep, which makes the cylinder act like a slender body and this causes smaller diverging angle than that of lowFr . ForFr = 1.64, much larger depression depth is observed and it is even narrower and longer than that of both low and mediumFr , implying that the flow past the cylinder atFr =1.64 shows features with a slenderer body.Fig. 30 shows the mean wave profiles at different streamwise locations for three
Fr . There is a small and wide depression region behind the cylinder and its recovery is fast forFr = 0.84. In contrast, it is clearly seen that there is a deep and narrow cavity behind the cylinder forFr = 1.24 andFr = 1.64 which is also shown in Fig. 30(b) and (c). The mean wave profile forFr = 1.24 recovers faster than that forFr = 1.64, as shown in Fig. 30(d), (e), and (f). A small bump is shown on the centerliney = 0.0 atx = 1.0 for bothFr = 1.24 andFr = 1.64 and it grows in the streamwise direction up tox = 3.0 and then it starts spreading out to the transverse direction with reduced wave elevation.> Vorticity and velocity at the interface
Threedimensional instantaneous coherent vortical structures identified by the second invariant of the velocity gradient tensor (Hunt et al., 1998) for the three Fr are shown in Fig. 31. At
Fr = 0.84, the organized shed vortex tubes are present in the deep flow and they attenuate as the airwater interface approaches. This is consistent with previous work (Suh et al., 2011; Yu et al., 2008). Due to the interface deformations, vortical structures of smaller scales are observed at the airwater interface for all threeFr . At medium and highFr (1.24, 1.64) small vortical structures near the airwater interface disappear due to the cavity region behind the cylinder and there are only large structures in the deep flow.Fig. 32 presents contours of the mean vertical vorticity magnitude at different depths for all three
Fr . At the airwater interface, the shear layers stretch out with a smaller angle and smaller vertical vortices are observed behind the cylinder as Fr increases. Atz = 0.5, the shear layers separate from the cylinder wall and diverge immediately in the transverse direction forFr = 0.84; whereas for medium and highFr (1.24, 1.64), the shear layers extend toward the downstream in parallel first and then diverge in the transverse direction. It is clearly shown that a cavity (empty space due to depression depth) behind the cylinder for highFr makes the cylinder together with its shear layers act like a slender body such that the “separation” of the shear layer occurs away from the real cylinder wall. There is no significant difference in the shear layer pattern and regular vortex streets are observed in the deep flow (z = 3.5) for all the Fr cases.Contours of the mean streamwise velocity with the streamlines are shown in Fig. 33. No remarkable difference in the streamline pattern is observed in the deep flow for all the
Fr cases. For lowFr , the recirculation length at the airwater interface is approximately two times of the cylinder diameter while it has much reduced length for highFr and it has similar size of the cavity behind the cylinder. For both medium and highFr , counterrotating vortices are shown immediately behind the cylinder at the interface which is also observed in the deep flow. The streamline patterns for high Fr also confirm that the depression region behind the cylinder acts as a cavity resulting in flow around a slender body.> Wave breaking at different
Fr Fig. 34 shows the isosurface of the airwater interface with slices of the wave profile orthogonal to the cylinder wall for the three
Fr . ForFr = 0.84, small disturbances are shown on the interface; however, no breaking waves are observed. For medium and highFr (1.24, 1.64), the flow becomes unsteady such that small interface structures including splashes and air bubbles are present especially forFr =1.64 which is also shown in Chaplin and Teigen (2003). The waves arise and break down to the cylinder shoulder as shown Fig. 34(b) and (c), which corresponds to a spilling breaker and a plunging breaker, respectively according to the analysis of Galvin (1968). Similar wave breaking patterns are observed in experimental study of the divergent ship bow waves using eight different Froude numbers by Shakeri et al. (2009) and they showed that forFr < 1.0 nonbreaking waves were generated; spilling breaking waves occurred at betweenFr = 1.0 andFr = 1.3; and plunging breakers were generated at 1.4 < Fr < 1.8.The slices of the wave profile cut for
Fr = 1.24 at different circumferential planes, which are presented by the tangential angle θ starting from the downstream direction, are given in Fig. 35 and it demonstrates the spilling wave breaking. In this figure, the flow direction is into the paper which is perpendicular to the cylinder. A small jet grows and starts overturning as shown in Fig. 35(a). It falls into the undisturbed water surface and then a small air entrainment is shown in Fig. 35(b) and (c), respectively. The repeated spilling breaking events including overturning and entrapping air are presented in Figs. 35(d)(f).The plunging wave breaking is described in Fig. 36 using the slices of the wave profile cut for
Fr =1.64. The wave propagates from the center of cylinder to the cylinder shoulder as shown in Fig. 34(c). When the wave crest grows and starts overturning (Fig. 36(a) and (b)), a jet is stretched and breaks up into droplets before it plunges on the undisturbed water surface (Fig. 36(c) and (d)). After the jet touches the water surface, an oblique splash is created (Fig. 36(e)). These plunging wave breaking events are very similar to those observed in the previous plunging wave breaking literatures (Bonmarin, 1989; Peregrine, 1983; Tallent et al., 1990). However, the direction of the plunging wave breaking is different from the mean flow, which is similar to the numerical simulation of the plunging wave breaking over a submerged bump (Koo et al., 2012) and the numerical study of wave breaking around a wedgeshaped bow (Wang et al., 2010). It should be mentioned that the wave plunges forward in the same direction of the mean flow for most previous studies on plunging wave breaking. The shape of the plunging jet is an ellipse with aspect ratio 1.8 which is close to the findings from most previous studies ( ). The thickness of the jet is approximately 0.012 and the similar jet thickness is reported in the experimental studies (Bonmarin, 1989; Grue and Jensen, 2006; Kimmoun and Branger, 2007). The jet angle relative to the undisturbed water surface is about 67° which is larger than that from previous experimental and computational studies on the plunging wave breaking.The velocity vectors of the plunging jet are given in Fig. 37 with the colored vectors by the vertical velocity component. Strong air flows are induced by the overturning motion of the jet as shown in Fig. 37(a) and (b), resulting in a pair of vortices immediately before the overturning jet. After the jet touches the interface, a strong vortex is created as a result of the entrained air and small vortices are also produced due to splashes as shown in Fig. 37(c).
In summary, for the
Fr effect in fixed postcriticalRe = 458,000, it is shown that the length of Kelvin wave increases withFr ; much less vertical structures are obtained near the airwater interface for postcriticalFr ; much larger cavity is formed behind the cylinder forFr =1.24 and 1.64 and the flow behind cylinder becomes very different from that ofFr = 0.84; and wave breaking is observed for higherFr (1.24 and 1.64) while there is no wave breaking forFr = 0.84.CONCLUSIONS AND FUTURE WORK
The twophase turbulent flow past an interfacepiercing circular cylinder at different Reynolds and Froude numbers is studied numerically using largeeddy simulation with a Lagrangian dynamic subgridscale model. Verification and validation studies are performed to show the accuracy of the simulations with comparisons against available experimental data.
It is found that the variation in subcritical
Re has small influence for fixed subcriticalFr . Current results for subcritical Re with fixed subcriticalFr show similar trends with Suh et al. (2011) on runup height, depression depth, mean surface elevation, and mean streamwise velocity.The flow features near the airwater interface show significant changes with different Reynolds numbers from subcritical to post critical regime at
Fr = 0.84. The present simulations show that the interface makes the separation point more delayed for all regimes ofRe . Remarkably reduced separated region below the interface atz = −1 was observed for the criticalRe regime and it is responsible for much reduced wake and recirculation region behind the cylinder and it recovers in the deep flow.The present study shows that the airwater interface structures are remarkably changed with different Froude numbers. For subcritical Fr, relatively smaller bow waves are observed in front of the cylinder with Kelvin waves behind the cylinder and small amount of freesurface roughness and turbulence are also seen in the wake region. For
Fr =1.24 , much increased bow wave is observed and it breaks and wraps around the bow. A lot of splashes and bubbles appear immediately behind the cylinder. Stronger freesurface oscillations and turbulence makes Kelvin waves less visible. ForFr =1.64 , the bow wave increases remarkably with the largest wake region and deepest depression and it also breaks with similar features of plunging breakers which are also shown in previous studies of plunging wave breaking. Much smaller airwater interface structures including splashes and bubbles were observed behind the cylinder. It is also hard to distinguish the Kelvin waves behind the cylinder due to much larger freesurface oscillations and turbulence. AsFr increases, the Kelvin wave angle decreases and deeper and narrower depression region behind the cylinder were observed. This cavity region behind the cylinder makes the cylinder resemble a slender body so that the flow features are significantly different that around the cylinder.Recently, the 27th International Towing Tank Conference (ITTC) Ocean Engineering Committee (2013) conducted a workshop on wave runup and vortex shedding on cylinders. Participants submitted results for specified conditions without access to the experimental validation data prior to the workshop. Subsequently the participants were provided the experimental data and will submit final results. The authors submitted results for both test cases (Yoon et al., 2013; Yeon et al., 2013). Relevant to the present research are the results submitted for the vortexshedding test case, which covered sub, critical and post critical
Re for singlephase conditions using the same computational method as used herein, but without airwater interface and investigations of effects of aspect ratio/span length, conservative vs. nonconservative convection schemes, and grid resolution. Highest accuracy predictions for sub, critical and post criticalRe required aspect ratio of 8 for sub critical vs. 2 for critical and post criticalRe ; conservative convection schemes for critical and post criticalRe ; and much finer grids with 67 million grid points. The present simulations for the deep flow have limitations in this regard along with free surface effects for larger Fr. The available experimental data for critical and post criticalRe andFr is also limited, as previously discussed. Thus, additional twophase surfacepiercing cylinder experimental and numerical studies for critical and post criticalRe andFr are required to confirm the effects of bothRe andFr . However, unless very large experimental facilities are used it will be difficult to achieve sufficiently large aspect that the deep flow is unaffected by the free surface and has sufficient additional span length for canonical singlephase flow without blockage effects, especially for sub and critical Re.

[]

[]

[]

[]

[]

[]

[]

[Table 1] Simulation conditions.

[Fig. 1] Computational domain with grid and boundary conditions: (a) plan view; (b) side view.

[]

[]

[Fig. 2] Time histories and running means for the drag and lift coefficients for Re = 458,000 and Fr = 1.64 with the fine grid.

[]

[]

[]

[Table 2] Verification study with case Re = 458,000 / Fr = 1.64

[Table 3] Iterative uncertainty using drag coefficient (CD ).

[Table 4] Validation study with case Re = 458,000 / Fr = 1.64.

[Fig. 3] Drag coefficients vs. Reynolds and froude numbers (a) and sectional drag coefficients vs. elevation (b).

[Fig. 4] FFT of the drag and lift coefficients for Re = 458,000 and Fr = 1.64 with the fine grid.

[Fig. 5] Strouhal number vs. Reynolds number (a) and CL RMS vs. Reynolds number (b).

[Fig. 6] Runup height and depression depth with various Re and Fr : (a) Fr = 0.84; (b) Fr = 1.24; (c) Fr = 1.64; (d) runup height; and (e) depression depth.

[Fig. 7] Mean interface elevation (a) and the rms of the interface fluctuations (b) for Re = 234,000 and Fr = 0.84. Bottom: measurement by Inoue et al. (1993), Re = 27,000, Fr = 0.8.

[Fig. 8] Profiles of mean interface elevation at x = 0.9 and x = 2.0 for Re = 234,000 and Fr = 0.84.

[Fig. 9] The vertical profiles of the mean streamwise velocity at x = 4.5, y = 0.0 for Re = 234,000 and Fr = 0.84.

[Fig. 10] Mean stream wise velocity on the centerline y =0.0 (deep flow) (a) and streamwise (b) and transverse (c) velocity at x =1 (deep flow).

[Fig. 11] Force coefficients on the cylinder surface (deep flow) (Norberg, 1992) : (a) pressure coefficient; (b) friction coefficient.

[]

[]

[Fig. 12] (a) Stagnation pressure vs. depth, (b) base pressure vs. depth and base pressure deep water vs. Re .

[Fig. 13] Mean separation pattern with vortex core lines: (a) Re = 27,000; (b) Re = 234,000; (c) Re = 458,000.

[Fig. 14] Instantaneous vertical vorticity contours with streamlines.

[Fig. 15] Mean streamwise vorticity contours. contour levels: 5 to 5 with interval 0.25.

[Fig. 16] Mean transverse vorticity contours. contour levels: 5 to 5 with interval 0.25.

[Fig. 17] Mean vertical vorticity contours. contour levels: 5 to 5 with interval 0.25.

[Fig. 18] Contours of streamwise Reynolds stress ().

[Fig. 19] Contours of transverse Reynolds stress ().

[Fig. 22] Reynolds stress at x = 1.0: (a) ; (b) ; (c) ; (d) for Re = 27,000; (e) ; (f) ; (g) ; (h) for Re = 458,000.

[Fig. 20] Contours of vertical Reynolds stress ().

[Fig. 21] Contours of Reynolds shear stress ().

[]

[]

[]

[]

[]

[]

[Fig. 23] Dominant source terms for the mean vorticity at the crossstream plane for Re = 27,000: (a) y component of term (B) for streamwise vorticity; (b) z component of term (B) for streamwise vorticity; (c) term (E) for streamwise vorticity; (d) z component of term (B) for transverse vorticity; (e) term (E) for transverse vorticity; (f) term (F) for transverse vorticity; (g) term (F) for vertical vorticity; (h) z component of term (B) for vertical vorticity; (i) term (E) for vertical vorticity.

[Fig. 24] Dominant source terms for the mean vorticity at the crossstream plane for Re = 458,000: (a) y component of term (B) for streamwise vorticity; (b) z component of term (B) for streamwise vorticity; (c) term (E) for streamwise vorticity; (d) z component of term (B) for transverse vorticity; (e) term (E) for transverse vorticity; (f) term (F) for transverse vorticity; (g) term (F) for vertical vorticity; (h) z component of term (B) for vertical vorticity; (i) term (E) for vertical vorticity.

[Fig. 25] Wave field: 1st column, front view; 2nd column, side view; 3rd column, wake region: (a) Fr =0.2; (b) Fr =0.44; (c) Fr =0.84; (d) Fr =1.24; (e) Fr =1.64.

[Fig. 26] Mean interface elevation (top) and the rms of the interface fluctuations (bottom) around the cylinder: (a) Fr =0.2; (b) Fr =0.44; (c) Fr =0.84; (d) Fr =1.24; (e) Fr =1.64.

[Fig. 27] Mean airwater interface elevation contours: (a) Fr =0.84; (b) Fr =1.24; (c) Fr =1.64.

[Fig. 28] (a) Wave lengths normalized with that of Kelvin waves and (b) diverging wave angle.

[Fig. 29] Closeup of mean airwater interface elevation contours: (a) Fr =0.84; (b) Fr =1.24; (c) Fr =1.64.

[Fig. 30] Wave profiles at different streamwise locations for three Fr : (a) x =1.0; (b) x =2.0; (c) x =3.0; (d) x =4.0; (e) x =5.0; (f) x =6.0.

[Fig. 31] Instantaneous vortical structures identified by the second invariant of the velocity gradient tensor Q= 1.0 for three Fr : (a) Fr =0.84; (b) Fr =1.24; (c) Fr =1.64.

[Fig. 32] Mean vertical vorticity contours at different depths. contour interval is 1.0.

[Fig. 33] Mean streamwise velocity contours with streamlines at different depths.

[Fig. 34] Isosurface of airwater interface with the slices of the wave profile cut: (a) Fr = 0.84; (b) Fr = 1.24; (c) Fr = 1.64.

[Fig. 35] Slices of the wave profile cut for Fr = 1.24: (a) θ =198°; (b) θ =202°; (c) θ =207°; (d) θ =210°; (e) θ =215°; (f) θ =222°.

[Fig. 36] Slices of the wave profile cut for Fr = 1.64: (a) θ =212°; (b) θ =267°; (c) θ =280°; (d) θ =288°; (e) θ =300°

[Fig. 37] Velocity vector fields of plunging jet for Fr = 1.64: (a) θ =267°; (b) θ =280°; (c) θ =300°.