effect of air wat interface on vortex shedding frm vertical circular cylindet

8/6/2019 Effect of Air Wat Interface on Vortex Shedding Frm Vertical Circular Cylindet

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The effect of airwater interface on the vortex sheddingfrom a vertical circular cylinder

Jungsoo Suh 1, Jianming Yang, Frederick Stern

IIHR Hydroscience and Engineering, University of Iowa, Iowa City, IA 52242-1585, USA

a r t i c l e i n f o

Article history:

Received 29 December 2009

Accepted 26 August 2010

Keywords:

Airwater interface

Vortex shedding

Turbulence

Large-eddy simulation

Level-set method

Cylinder

a b s t r a c t

The flow past an interface piercing circular cylinder at the Reynolds number

Re=2.7 104 and the Froude numbers Fr=0.2 and 0.8 is investigated using large-eddy

simulation. A Lagrangian dynamic subgrid-scale model and a level set based sharp

interface method are used for the spatially filtered turbulence closure and the airwater

interface treatment, respectively. The mean interface elevation and the rms of interface

fluctuations from the simulation are in excellent agreement with the available

experimental data. The organized periodic vortex shedding observed in the deep flow

is attenuated and replaced by small-scale vortices at the interface. The streamwise

vorticity and the outward transverse velocity generated near the edge of the separated

region, which enforces the separated shear layers to deviate from each other and

restrains their interaction, are primarily responsible for the devitalization of the

periodic vortex shedding at the interface. The lateral gradient of the difference between

the vertical and transverse Reynolds normal stresses, increasing with the Froude

number, is the main source of the streamwise vorticity and the outward transverse

velocity at the interface.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The flow past a circular cylinder has been an important topic in fluid dynamics for a long time because of its wide

applications in engineering and its abundant flow physics including separation, reattachment, vortex shedding, etc. One

prominent feature of this flow is that it presents very different vortical structures in different ranges of Reynolds numbers

(Re, definition deferred to Section 3.2). For extensive discussions on the single-phase flow past a circular cylinder, see

Williamson (1996) and Zdravkovich (2003a, 2003b) among others.

Unlike its single-phase counterpart, the flow past an interface piercing cylinder has received much less attention,in spite of its importance in various applications including offshore structures and surface vessels. Although a few

experimental and numerical studies on the flow around a surface-piercing cylinder are available in the literature, the

detailed hydrodynamics is not well understood yet. In general, the free surface adds great complexities to the flow due to

the generation of waves in various forms and their interaction with the body and vortices, the airwater interfacial effects

like bubble entrainment and surface tension, and the three-dimensional flow separation.

There were several experimental studies on the flow past an interface piercing cylinder with different configurations

and flow conditions. The literature survey of these studies will follow the order of increasing Re, as the Reynolds number is

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jfs

Journal of Fluids and Structures

0889-9746/$- see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jfluidstructs.2010.09.001

Corresponding author. Tel.: +1 319335 5215; fax: +1 319335 5238.

E-mail address: [email protected] (F. Stern).1 Current affiliation: Nuclear Engineering & Technology Institute, Korea Hydro & Nuclear Power Co., Daejeon, Republic of Korea.

Journal of Fluids and Structures 27 (2011) 122
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where ui (i=1, 2, 3) is the velocity in the orthogonal coordinate xi direction and

ri 1

J

@

@xi

J

hi

, 2

following Pope (1978). The Jacobian of the coordinate transformation is defined as J=hihjhk, and hi @xi=@xi with xi a

Cartesian coordinate.

The momentum equation is written as follows:

@ui@t

rjuiuj1

rrjtij

1

r

@p

@xigi Hji ujuj

tijr

Hij uiuj

tijr

, 3

where r is the density, p the pressure, t the time, and gi the gravity vector in the xi direction. In addition,

Hij 1

hihj

@hi@xj

4

and @xi hi@xi following Pope (1978). The viscous stress tensor tij is defined as follows:

tij m@ui@xj

@uj@xj

uiHijujHji 2ulHildij

, 5

where m is the dynamic viscosity and dij is the Kronecker delta function.

The airwater interface has to be solved as a part of the solution. In this study, the interface is tracked as the zero levelset of a signed distance function, f, or the level-set function, by solving the following advection equation:

@f

@t ui

@f

@xi 0: 6

In addition, the reinitialization of the level-set function is required to keep f as a signed distance function in the course of

interface evolution (Sussman et al., 1994).

The density and viscosity are defined according to the level-set function and sharp jumps of the fluid properties occur at

the interface. In this study, the density maintains its jump whereas viscosity is smoothed over a transition band across the

interface (Yang and Stern, 2009).

To handle the fully inhomogeneous turbulence in this study, the Lagrangian dynamic subgrid-scale (SGS) model based

on Sarghini et al. (1999) is adopted as it averages the model coefficient along the flow pathline (Meneveau et al., 1996).

In the LES, the small dissipative eddies are modeled by the SGS model whereas the large, energy carrying, eddies are

resolved by the spatially filtered NavierStokes equations. Hence, Eq. (3) can be rewritten as in the following form:

@u i@t

rju iuj1

rrjtijrj ~t ij

1

r

@p

@xigi Hji ujuj

tijr

~tij

Hij uiuj

tijr

~t ij

7

with t ij mSij and ~tij ntSij with nt the turbulent eddy viscosity, respectively. Note an effective total viscosity cannot bedefined using mrnt because r is discontinuous across the interface in the present study (Yang and Stern, 2009). Hereafterthe filtering sign for LES will be dropped for simplicity.

2.2. Numerical method

A finite-difference method is used to discretize the governing equations on a non-uniform staggered orthogonal grid, in

which the contravariant velocity components ui, uj, uk are defined at centers of cell faces in the xi, xj, xk directions,respectively. All other variables are defined at cell centers. Time advancement of the present study is based on the semi-

implicit four-step fractional step method by Choi and Moin (1994). The diagonal diffusion terms are advanced with the

second-order CrankNicholson method and the other terms by the second-order explicit AdamsBashforth method. The

pressure Poisson equation is solved to enforce the continuity equation.

The convective terms are discretized using the fifth-order HamiltonJacobi Weighted-ENO (HJ-WENO) scheme (Jiang

and Shu, 1996). The other terms are discretized by the second-order central difference scheme. The pressure Poisson

equation is solved using a semi-coarsening multigrid solver from the HYPRE library (Falgout et al., 2006). In general, the

Poisson solver is the most expensive part of the whole algorithm and uses about 8085% of the total computation time.

Both the level-set and reinitialization equations are solved using the third-order TVD RungeKutta scheme (Shu and

Osher, 1988) for time advancement and the fifth-order HJ-WENO scheme (Jiang and Shu, 1996) for spatial discretization.

These equations are solved for the cells in a narrow band about a few grid-cell width to reduce computational overhead

(Peng et al., 1999). An additional damping function is added to the right-hand side of the level-set advection equation to

drive the level-set function toward the desired smooth solution and minimize reflection at the outer boundary ( Vogt andLarsson, 1999).

J. Suh et al. / Journal of Fluids and Structures 27 (2011) 122 3


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3. Verification and validation

3.1. Grid and domain

The computational domain is shown in Fig. 1. The set-up followed the previous numerical studies (Kawamura et al.,

2002; Yu et al., 2008). The total height is 6D with D the cylinder diameter; and the portions of air and water are 2 D and 4D,

respectively. The distance from the center of the cylinder to the outer boundary is 20 D including a buffer zone for damping

wave reflections.Three body-fitted cylindrical grids, coarse (C), medium (M), and fine (F), were used to investigate the influence of grid

resolution. The grid parameters are listed in Table 1. The grid points were clustered near the cylinder surface to capture the

boundary layer and flow separation. The grid close to the interface was also refined to resolve the interface deformations.

In order to investigate the effect of computational domain size, an additional complementary simulation with doubled

height (12D with 4D and 8D for air and water portions, respectively) using grid C1Nr Ny Nz 128 128 256 was

performed.

For simplicity, a Cartesian coordinate system is used for the discussion of results. The origin is the center of the cylinder

at the still free surface level, and the x, y, and z axes are defined in the streamwise, transverse, and vertical direction,

respectively. The instantaneous Cartesian velocity components ~u, ~v, and ~w are defined correspondingly. U, V, and Wand u,

v and w indicate the mean velocity and velocity fluctuation of each Cartesian component, respectively.

3.2. Simulation conditions

In the present study, all variables are nondimensionalized with respect to D and the uniform inflow velocity U1. Hence,

two dimensionless parameters, the Reynolds number and the Froude number, can be defined as

Re U1D

n, 8

with n the kinematic viscosity of the liquid phase, and

Fr U1ffiffiffiffiffiffi

gDp : 9

In this study, the flow at Re=2.7 104 and Fr=0.8 is investigated. The Froude number is related to the deformation of the

interface and the present value 0.8 corresponds to a moderate interface deformation. In addition, a complementary case of

Fr=0.2 is performed using the medium grid, to study the effect of the Froude number on the flow field near the interface

and only partial results, as stated in the text and figures, from the Fr=0.2 case are discussed herein.On the cylinder wall, the no-slip condition was imposed while the slip condition (i.e., zero velocity in the normal

direction to the face and the Neumann condition for the other directions) was used at the bottom (water side) and the top

Fig. 1. Schematic diagram of the computational domain and coordinate system.

Table 1

Grid parameters and hydrodynamic forces on an interface piercing cylinder.

Grid Nr Ny Nz CD CrmsL y

+

C 128 128 128 0.959 0.195 1.927

M 256 128 128 0.983 0.216 0.966

F 512 128 128 0.984 0.220 0.485

Kawamura et al. (2002) 99 161 33 0.97 0.24

J. Suh et al. / Journal of Fluids and Structures 27 (2011) 1224


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(air side) of the computational domain. The radial outer boundary was divided into in- and outflow at y 903 and 2701, in

which y is the tangential angle starting from the downstream direction. A Dirichlet boundary condition was used for the

inflow boundary with uniform streamwise velocity U1 1 and a convective boundary condition (@ui=@tU1@ui=@n 0)

was used for the outflow boundary (Breuer, 1998).

3.3. Verification and validation

Fig. 2 shows the time histories of the drag and lift coefficients from the medium grid, along with the running mean ofthe drag coefficient. The drag and lift coefficients are defined as

CD Drag

12rLU

21DH

, CL Lift

12rLU

21DH

: 10; 11

Note these coefficients are based on the still water depth H=4D and the water density rL in order to be consistent with theprevious studies (Kawamura et al., 2002; Yu et al., 2008).

The convergence of the running mean from the time history of CD was used to define the statistically stationary state.

The criterion for this convergence is that the fluctuations of the running mean are less than 1% of the mean drag coefficient

CD, which was ensured for all cases run in this study. For instance, the simulation on the medium grid reached statistically

steady state around t=80. After the flow was converged, flow field data covering 16 vortex shedding cycles were collected

for statistics. Here the cycle is defined using the Strouhal number St fD=U1 with f the frequency and a typical Strouhal

number in this flow is St=0.2. Fig. 3 shows the FFT of the CD and CL on the medium grid. For the FFT ofCL, the highest peak is

around 0.2, corresponding to the Karman vortex shedding. The FFT of CD has a broad range of frequency content.

Fig. 2. Time histories of the drag and lift coefficients. Dotted line is the running mean of the drag coefficient.

St

Magnitude

0 1 2 310

-4

10-3

10-2

10-1

10

0

101

CDCL

Fig. 3. FFT of the drag and lift coefficients from the medium grid.



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CD and CLrms from three cases are listed in Table 1 with the fine grid results from Kawamura et al. (2002). Both CD and

CLrms show monotonic convergence using refinement ratio rG=2. Table 2 tabulates results from grid studies for CD and CL

rms

following the procedure given by Xing and Stern (2010). For CD, the order of accuracy is pG=2.39 with grid uncertainty

UG=0.09%. For CLrms, the order of accuracy is pG=4.19 and the uncertainities are much lower compared to the mean drag

coefficient with UG=0.01%.

The mean streamwise velocities at x =4.5 and y=0 from the simulation are compared with the available experimental

data (Inoue et al., 1993) and the fine grid solution by Kawamura et al. (2002) in Fig. 4. The present results are in good

agreement with the measurement by a hot-film anemometer (Inoue et al., 1993) and the computation by Kawamura et al.(2002). The decrease of streamwise velocity near the interface is clearly observed. The effect of grid resolution on the mean

velocity is also shown in this figure. The differences between the results from the three grids are not evident. The effect of

computational domain size is not significant either. Since the simulation results between different grids are very similar

and the amount of data from the fine grid is very large for post-processing, numerical results from the medium grid will be

presented in this study unless otherwise mentioned.

Profiles of the mean interface elevation and the rms of the interface fluctuations at two transverse planes are shown in

Fig. 5. They are in very good agreement with the experimental data (Inoue et al., 1993). However, the present simulation

under-predicted the depression of the interface in the near-wake region behind the cylinder. It also confirms the difference

between computational results from different grids or domain size is not significant.

Contours of the mean interface elevation are shown in Fig. 6, side by side with the previous measurement (Inoue et al.,

1993). The generation of bow wave in front of the cylinder, a large surface cavity behind the cylinder, and the diverging

Kelvin wake are evident. The simulation result is in good agreement with the experiment on the overall magnitude and

location of interface waves and the expansion angle of Kelvin wave, although the size of large depressed region behind thecylinder was slightly under-predicted. The feature that the slope from the bow wave crest to the depressed region is almost

constant is preserved (Kawamura et al., 2002).

Fig. 6 also shows the rms of the interface fluctuations, along with that from the previous experiment. A good agreement

between the simulation and the experiment is observed on the overall distribution of the fluctuation intensity and the

location of its peak value. Most of the interface fluctuations are observed in the immediately downstream of the constant

slope region between the crest and the cavity. The peak fluctuation is observed near the edge of the flat bottom of the

surface cavity. In addition to this region, the simulation result gives some slight fluctuations in front of the cylinder,

probably due to the presence of the necklace vortices. Nevertheless, the bow wave in front of the cylinder and the constant

slope region between the crest and the cavity retain a fixed wave shape with negligible fluctuations. But remarkable

interface fluctuations are generated in the near-wake surface cavity region.

Table 2

Grid studies for CD and CLrms.

Parameter e21 e32 RG PG UG (%)

CD 0.004 0.021 0.190 2.39 0.09

CLrms 0.0013 0.0238 0.055 4.19 0.01

U

z

-0.5 0 0.5 1 1.5

-3

-2

-1

0

1Grid C

1

Grid C

Grid M

Grid F

Experiment

Kawamura

Fig. 4. Vertical profiles of the mean streamwise velocity at x=4.5 and y= 0.



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4. Results and discussion

4.1. Instantaneous flow

4.1.1. Airwater interfaceFig. 7 shows the instantaneous airwater interfaces from the simulations. For the Fr=0.8 case, the bow waves in front of

the cylinder and the Kelvin waves displaced away from the cylinder by the separation region are clearly observed. The

wavelength of the Kelvin wave system approximately corresponds to the theoretical value, i.e. 2pFr2 % 4. The roughsurface, immediately up- and downstream of the cylinder, indicates the existence of vortical structures below the interface

(Sarpkaya, 1996). On the other hand, the upstream bow waves, the cavity region in the downstream of the cylinder and the

Kelvin waves are almost unnoticeable in the Fr=0.2 case.

In addition, the mean interface elevation and the rms of the interface fluctuations from the Fr=0.2 case are shown in

Fig. 8. The height of the wave crest in front of the cylinder and the depth of the cavity region behind the cylinder are much

reduced and a nearly flat interface is observed. The location of the region with the most intensive interface fluctuations is

quite similar to the Fr=0.8 case, but the rms value of the interface fluctuations is greatly reduced.

4.1.2. Vorticity field

In order to assess the effect of the interface on the cylinder wake, contours of the instantaneous vertical vorticity atdifferent horizontal planes and the interface are presented in Fig. 9. Organized vortex shedding, which is very similar to

y

0 1 2 3 4 5 6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid C1Grid CGrid MGrid FExperiment

hmean

hrms

y

1 2 3 4 5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid C1Grid CGrid MGrid FExperiment

hmean

hrms

Fig. 5. Profiles of the mean interface elevation and the rms of the interface fluctuations: (a) x=0.9 and (b) x=2.0.

Experiment

0.03

0.15

-0.30

-0.15

0.03

0.15

-0.15

LES x =0 .9 x =2 .0

Experiment

0.01

0.05

0.010.05

LES x =0 .9 x =2 .0

Fig. 6. Comparison of the mean interface elevation (left) and the rms of the interface fluctuations (right) around the cylinder at Fr=0.8.



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4.1.3. Vortical structures

To further identify large-scale flow structures, isosurfaces of the second invariant of the velocity gradient tensor rv(Hunt et al., 1988) from both cases are shown in Fig. 11. At Fr=0.8, the dominant coherent structures are the elongated

quasi-vertical vortices away from the interface, which are originated from the shear-layer instability. As the interface isapproached, the structures are inclined to the outward transverse direction. On the other hand, the prevalent vortical

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

Fig. 9. Instantaneous vertical vorticity at the interface and horizontal planes for Fr=0.8: (a) on the interface; (b) z= 0.5; (c) z= 1; and (d) z= 3.5.

Contour interval is 1.2.

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

Fig. 10. Instantaneous vertical vorticity at the interface and horizontal planes for Fr=0.2: (a) on the interface; (b) z= 0.5; (c) z= 1; and (d) z= 3.5.

Contour interval is 1.2.



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structures inside the separated region are small and parallel to the interface. Most of these structures are located slightly

below the interface, but some structures exist at the interface, which indicates the rotation and deformation of the

interface. This observation is consistent with previous work (Yu et al., 2008). However, the figure for the Fr=0.2 case shows

that the vortical structures parallel to the interface diminish due to the reduced interface deformation and vertical vortices

can maintain their direction even near the interface at a smaller Froude number.

4.2. Hydrodynamic forces

Fig. 12 shows the distribution of sectional CD and CrmsL along the cylinder. The magnitude of CD increases to its local

maximum value at the interface due to the enhanced momentum mixing and turbulence generation, which is consistentwith (Yu et al., 2008). The sectional CrmsL clearly shows the attenuation of vortex shedding at the interface. At Fr=0.2, vortex

shedding is stronger, especially, near the interface; and the enhanced mixing near the interface and the attenuation of

vortex shedding appear in a narrower region than the larger Froude number case.

4.3. Mean flow

4.3.1. Phase-averaged vorticity

In order to investigate the usefulness of mean flow analysis for the present time-dependent flow, vertical vorticities

from two different depths were phase averaged with the deep-level vortex shedding frequency. Fig. 13 shows the phase-

averaged vertical vortices at two planes, z= 0.5 and 3.5, and at two phase angles, jA and jA T=2, where Tis the vortexshedding frequency at the deep region. The phase difference between (a), (c) and (b), (d) is T/2. While the shear layer

pattern in the deep region is in opposite direction between (c) and (d), the change of the vorticity distribution close to theinterface is not significant, especially for the separated shear layer immediately downstream of the cylinder. Hence the

Fig. 11. Instantaneous vortical structures identified by the second invariant of the velocity gradient tensor Q=2.0: (a) Fr=0.8 and (b) Fr=0.2.



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effect of vortex shedding frequency on the flow structures near the interface is convinced to be small, and mean flow

analysis might be helpful for the analysis of flow field near the interface.

4.3.2. Boundary layer

Fig. 14 shows the friction and pressure coefficients

Cf tw

12rLU

21

12

and

Cp pp112rLU

21

13

cD,cLrms

z

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-3

-2

-1

0

1

Grid C

Grid M

Grid FFr=0.2

cLrms c

D

Fig. 12. Sectional drag and lift coefficients.

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

x

y

-2 0 2 4 6-2

-1

0

1

2

Fig. 13. Phase-averaged vertical vorticity at horizontal planes. Contour interval is 1.2: (a) at z= 0.5 and phase jA; (b) at z= 0.5 and phase jA T=2;

(c) at z= 3.5 and phase jA; and (d) at z= 3.5 and phase jA T=2.



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on the cylinder wall at three depths. The boundary layer separation, indicated by a change of Cf from positive to negative, is

moved from f 0:462p in the deep flow to f 0:514p near the interface. The wall pressure distributions in the deep floware in good agreement with the single phase experimental data ( Norberg, 1992) of Re= 2.0 104 up to the separation point,

but the present simulation yields higher pressure values than the single phase experiment at the rear part of the cylinder.

While the pressure in the separated region of the single phase flow shows a slightly reduced value than that at the

separation point, continuously increasing values are observed in the deep flow in the present simulation. In addition, the

pressure distribution near the interface is quite different. The pressure gradient along the cylinder is significantly

decreased due to the negative interface elevation slope near the cylinder. The adverse pressure gradient downstream is

also lessened by the reduced upstream pressure gradient and responsible for the displacement of the separation point

further downstream near the interface. Fig. 15 is the close-up view of the interface elevation near the cylinder boundary.

Along the cylinder circumference, a steeper slope of the interface elevation is observed inside the boundary layer than thatof the outside. Immediately after the position of an angle p=2 from the front stagnation point, the interface elevation startsto increase inside the boundary layer while it is still decreasing outside. The location where the wave elevation starts to

increase is consistent with the boundary layer separation point. This rising interface elevation along the cylinder

downstream made the pressure in this region keep increasing, which is shown in Fig. 14.

4.3.3. Separation pattern

Fig. 16 shows the separation pattern of the mean flow. The separated shear layer was visualized approximately using

the isosurfaces of the stagnation Cp near the interface immediately before the separation, assuming that the flow outside

the separation region is inviscid (Kandasamy et al., 2009). Inside the separated region, the mean vortex core lines are extracted

using the vortex core identification technique (Sujudi and Haimes, 1995). The mean vertical vortices (V1) are due to the vortex

shedding in the deep flow. As the interface is approached, these vortices are inclined and attached on the cylinder wall. In

addition, two kinds of vortex core lines are observed inside the separated region near the interface: (a) the mean streamwisevortices located near the edge of the separation region immediately under the interface (V2); (b) a V-shaped mean vortex

Angle from stagnation point (divided by )

Cf

0 0.2 0.4 0.6 0.8 1-0.01

0

0.01

0.02

0.03

z=-0.5z=-2z=-3.5

Angle from stagnation point (divided by )

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

Exp.(Re=2.0104)

z=-0.5z=-2z=-3.5

Fig. 14. Force coefficients on the cylinder surface: (a) friction coefficient and (b) pressure coefficient.

0.09

-0

.09

-0.1

8

x

y

-0.3 -0.2 -0.1 0 0.1 0.2 0.30.3

0.4

0.5

0.6

Fig. 15. Close-up view of the mean interface elevation around the cylinder boundary.



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observed inside the separated region (V3). V3 is inclined and approaches to V2 as it goes downstream. V3 is located near the

high gradient region of the mean streamwise velocity where a significant amount of turbulent kinetic energy is produced.

4.3.4. Mean velocity

Fig. 17 shows the mean streamwise velocity U from the Fr=0.8 case on the centerline for several different vertical

locations, along with the single-phase measurements at Re= 3900 and 1.4 105, respectively (Cantwell and Coles, 1983;

Ong and Wallace, 1996; Lourenco and Shih, 1993). To the best knowledge of the authors, there is no velocity measurementavailable in the literature on the single phase flow past a cylinder at a similar Reynolds number as the present case. Hence

we will compare the present simulation results with the available experimental data in the subcritical Reynolds number

regime. The far-wake centerline streamwise velocity for single-phase flow shows independence on Re because the

measurement profiles (x44) from two different Reynolds numbers approach almost same value (U=U1 % 0:8). On the

other hand, the near-wake profile varies significantly. The recirculation zone immediately after the cylinder for the high

Reynolds number flow is significantly smaller than in the low Reynolds number case. The near-wake profile of the mean

centerline streamwise velocity away from the interface (z= 3.5) from the present simulation falls between the

experiments of lower and higher Reynolds numbers, whereas the far-wake profile approaches the same value as all

the experiments. In the present simulation, the recirculation region increases substantially as the profile approaches the

interface. The streamwise length of the recirculation region at the interface (x=0.53.56) is more than triple of the deep

flow recirculation zone (x=0.51.33). Its far-wake velocity magnitude is also remarkably smaller than the deep flow region.

In addition, the small positive streamwise centerline velocity zone is observed to exist immediately downstream of the

cylinder (see the top-left close-up view of the interface centerline velocity), which means a complicated interface topologywith the presence of nodal and saddle points (Kandasamy et al., 2009).

Fig. 16. Mean separation pattern with the vortex core lines.

x

U

0 2 4 6 8 10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Interface

z=-1.5

z=-2.5

z=-3.5

Exp. 1 (Re=3900)

Exp. 2 (Re=3900)

Exp. (Re=1.4105)

Fig. 17. Mean streamwise velocity on the centerline in the wake.



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showing the separation at the interface is an open type. Immediately outside the front separated region, the mean

streamwise velocity changes significantly in the transverse direction. In addition, the outward transverse velocities noted

in Fig. 18 are observed in the same region. Note the mean vertical velocity is negligible inside the separated region

although there are significant interface fluctuations in this region. Contours of the mean transverse velocity at the interface

from the Fr=0.2 case are also shown in Fig. 19. As expected, the outward mean transverse velocity is reduced and only

appears near the cylinder and the width of the wake is also reduced at a smaller Froude number.

Fig. 20 shows contours of the mean streamwise velocity at two cross-stream (yz) planes in the near wake. The interface

position and the location of the cylinder are marked in the figure, and only half of the domain is shown. At the x=1 plane,

the width of the wake increases significantly near the interface, whereas the width is almost constant in the deep flow.

Since the x=1 plane is in the recirculation region, negative mean streamwise velocities are observed in the deep flow and

near the interface. However, there is no negative streamwise velocity at the narrow wake region around z= 1. Hence, thewidth of the recirculating zone increases as the interface is approached. At the plane x=2.5, a negative streamwise velocity

is still observed near the interface, whereas the velocities in the deep flow increase to positive values, which confirms the

slower velocity recovery in the wake at the interface. At this cross-stream plane, the width of the wake is substantially

increased near the interface and in the deep flow. The narrowest wake region locates at a slightly lower position (z= 1.1)

in this plane. This observation is consistent with Kawamura et al. (2002).

4.3.5. Mean vorticity

The mean streamwise vorticity is responsible for the attenuation of the vortex shedding and the significantly increased

wake width near the interface. In addition, the large outward mean transverse velocity near the interface shown in Fig. 18

is induced by the streamwise vorticity near the interface. The mean streamwise vorticity is shown in Fig. 20(b). The

locations of the vortices and the shapes of wakes correlate well, indicating the effect of secondary swirl of the vortical

structures on the wake structures (Kawamura et al., 2002). At the x=1 plane, three vortices with alternative rotationdirections are observed behind the cylinder, which are responsible for the complicated serpentine wake pattern. In

addition, a pair of strong counter-rotating vortices is observed across the interface around y = 1. This vortex pair is

responsible for the increased wake width and the large outward transverse velocity near the interface. Weaker vortices are

also observed in the wake of both the air and water regions. On the other hand, at the x =2.5 plane, vortices are only seen

near the interface and in the air region.

Note that the mean transverse vorticity component is significant near the interface only. Since the magnitude of the

mean transverse vorticity is larger than that of the mean streamwise vorticity at the interface between the two surface-

parallel vorticity components, the mean transverse vorticity is more responsible for the interface fluctuations. Contours of

the mean transverse vorticity at the interface are shown in Fig. 21. High vorticity regions near the cylinder and the Kelvin

waves are due to the change of interface elevation. In addition, significant mean transverse vorticity is observed inside the

separated region. The location of this region matches well with the region of high interface fluctuations, which indicates

that the mean transverse vorticity is responsible for the interface fluctuations.

The vertical distributions of the mean vertical vorticity component in the deep flow and the air region are due to theKarman vortex shedding and the shear-layer instability. However, its distribution near the interface is inclined outward in

x

y

-2 0 2 40

1

2

x

y

-2 0 2 40

1

2

x

y

-2 0 2 40

1

2

x

y

-2 0 2 40

1

2

Fig. 19. Contours of the mean velocity at the interface: (a) streamwise velocity for Fr=0.8; (b) transverse velocity for Fr=0.8; (c) vertical velocity forFr=0.8; and (d) transverse velocity for Fr=0.2.



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the transverse direction, due to the outward mean transverse velocity generated near the interface. The location of the high

magnitude vorticity matches with the high gradient region of the mean streamwise velocity.

4.4. Reynolds stresses

Fig. 22 shows contours of the Reynolds stresses at two cross-stream planes in the near wake. For the streamwise

Reynolds normal stress uuuu , at the x=1 plane, peak values are produced near the interface and along the separation region

in the deep flow where the mean streamwise velocity is recirculated. Unlike the mean streamwise velocity, uuuu does not

show a two-dimensional distribution in the deep flow. Note that large uuuu

is observed where the mean streamwise velocitygradients are very high (see Fig. 20), which indicates a region of high turbulent kinetic energy production (uu

luu

kUl,k).

x

y

-2 0 2 40

1

2

Fig. 21. Contours of the mean transverse vorticity at the interface. Contour interval is 1.0.

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

Fig. 20. Contours of the mean flow at two cross-stream planes: (a) streamwise velocity at x=1.0; (b) streamwise vorticity at x=1.0; (c) transverse vorticity

at x=1.0; (d) vertical vorticity at x=1.0; (e) streamwise velocity at x=2.5; (f) streamwise vorticity at x=2.5; (g) transverse vorticity at x=2.5; and

(h) vertical vorticity at x =2.5. Contour intervals are 1.0 except for streamwise vorticity, a contour interval 0.5 is used.



17/22


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vorticity and presumably the mean outward transverse velocity at the interface in the present study. As a consequence, in a

case with a significant Froude number, the formation of the separated fluctuating region behaves as an obstacle, which

accelerates the flow around it (Vlachos and Tellionis, 2008).

Contours ofwuwu at the interface for the Fr= 0.2 case are also shown in Fig. 23. The distribution area and magnitude are

much reduced comparing to the Fr=0.8 case. The lateral gradients of wuwu are small for most of the region except near the

cylinder.

4.5. Vorticity transport

Longo et al. (1998) explained the physical mechanism for the mean streamwise vortices generated in solid-interface

juncture flow using the vorticity transport equation, which can be derived by taking the curl of the time-averaged Navier

Stokes equation. For a steady flow of constant density, the equation can be written as

U@Ox@x

V@Ox@y

W@Ox@z

A

Ox@U

@xOy

@U

@yOz

@U

@z

B

n@2Ox@x2

@2Ox@y2

@2Ox@z2

C

@@x

@uuvu@z

@uuwu@y

D

@2

@y@zvuvuwuwu E

@2

@z2

@2

@y2

vuwu, F 14

where Ox, Oy, and Oz are the streamwise, transverse, and vertical components of the mean vorticity, respectively. Term (A)

in Eq. (14) represents the material derivative of the mean streamwise vorticity. The first term of term (B) is the vorticity

amplification by the streamwise stretching, while the other terms provide vortex-line bending effects. Term (C) suggests

the vorticity damping by the viscous diffusion, and the other terms (D), (E), and (F) are the vorticity production by

inhomogeneities in the Reynolds stress field (Launder and Rodi, 1983). In addition, there are other terms from the density

discontinuity in the present study, but it is expected negligible except close to the interface.

The dominant terms in the mean streamwise vorticity transport equation at the cross-stream plane x=1 are the y and zcomponents of terms (B) and (E) shown in Fig. 24. Two terms by the vortex bending are of a similar magnitude, but of

0.01

0.03

0.08

x

y

-2 0 2 40

1

2

0.010.03

x

y

-2 0 2 40

1

2

Fig. 23. Distribution of the Reynolds normal stress wuwu at the interface: (a) Fr=0.8 and (b) Fr=0.2.



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opposite signs, so the total effect of them is canceled out. Therefore, the remaining (E) term @2=@y@zvuvuwuwu is the main

production mechanism of the mean streamwise vorticity. The vertical and transverse gradients of the difference between

vuvu and wuwu are responsible for the generation of the streamwise vorticity near the interface and presumably cause the

outward transverse velocity at the interface.

The dominant terms for the mean transverse vorticity at the cross-stream plane x=1 are the zcomponent of terms (B),(E), and (F). The zcomponent of term B, Oz@V=@z, is from the bending of vertical vorticity, which is dominant in the deep

y

-2 -1 0

-3

-2

-1

0

1

302418

126

0-6

-12

-18-24-30

y

z

-2 -1 0

-3

-2

-1

0

1

y

-2 -1 0

-3

-2

-1

0

1

75.6

4.22.8

1.40

-1.4

-2.8-4.2-5.6

-7

y

-2 -1 0

-3

-2

-1

0

1

543

21

0

-1-2

-3-4-5

y

-2 -1 0

-3

-2

-1

0

1

y

-2 -1 0

-3

-2

-1

0

1

y

-2 -1 0

-3

-2

-1

0

1

y

-2 -1 0

-3

-2

-1

0

1

z z

z z z

z z

y

z

-2 -1 0

-3

-2

-1

0

1

2

1.6

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

-1.6

-2

30

2418

1260

-6-12-18

-24-30

543

21

0

-1-2

-3-4-5

5

432

10

-1-2-3

-4-5

543

21

0-1-2

-3-4-5

543

210

-1-2

-3-4-5

Fig. 24. Dominant source terms for the mean vorticity at the cross-stream plane x=1.0 for Fr=0.8: (a) y component of term (B) for streamwise vorticity;

(b) zcomponent of term (B) for streamwise vorticity; (c) term (E) for streamwise vorticity; (d) zcomponent 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; and

(i) term (E) for vertical vorticity.



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flow due to the vortex shedding and shear-layer instability, by the outward mean transverse velocity generation near the

interface. In other words, the swirling motion of the vortex shedding and shear-layer instability in the deep flow are

changed to interface fluctuations near the interface via vortex bending. The other sources for the mean transverse vorticity

are terms (E) and (F), @2=@x@zwuwuuuuu and @2=@x2@2=@z2uuwu, respectively.

The dominant terms for the vertical vorticity component are also shown in Fig. 24. In the deep flow, term (F),

@2=@y2@2=@x2uuvu is the dominant source for the mean vertical vorticity. Hence, the present analysis using the vorticity

transport equation confirms that the shear stress is primarily responsible for the Karman vortex shedding and shear-layer

instability shown in the deep flow. The dominant terms for the mean vertical vorticity near the interface are the zcomponent of terms (B) and (E). The z component of term (B), Oz@W=@z, is the vortex stretching of the vertical vorticity.

Term (E), @2=@x@yuuuuvuvu, is from the anisotropy between the streamwise and transverse Reynolds normal stresses.

For the Fr=0.2 case, the streamwise vorticity and the dominant sources in Eq. (14) at x=1.0 are shown in Fig. 25.

Compared with the Fr=0.8 case, the vorticity strength is much reduced, supporting the fact that the outward transverse

velocity is driven by the streamwise vorticity. The strength of the dominant sources for the streamwise vorticity is also

greatly reduced. Same as the Fr=0.8 case, the y and z components of terms (B) and (E) are dominant sources for the

streamwise vorticity. Similarly, two terms from vortex bending are canceled. Hence, the term (E) is the main mechanism

for the streamwise vorticity, but its strength is greatly reduced.

Interestingly, for a flat interface with negligible fluctuations, which corresponds to a zero or very low Froude number

case, Walker (1997) suggested that the outward mean transverse velocity at the interface, or the origin of the surface

current, is owing to the transverse gradient of the Reynolds-stress anisotropy, @=@ywuwuvuvu, near the interface. Then in

Hong and Walker (2000), a set of Reynolds-averaged NavierStokes equations for free-surface flows were derived to

extend the order-of-magnitude analysis in Walker (1997) to high Froude numbers. And their results of two cases for aturbulent jet issuing from a circular nozzle of diameter d in parallel beneath a free surface at a uniform velocity Ue, one at

Fr=1.0 and the other at Fr=8.0 (Fr Ue=ffiffiffiffiffiffigd

p), showed the Reynolds stress anisotropy is smaller in the higher Froude

number case than that in the lower Froude number case, but the much larger free-surface fluctuations in the former case

compensate its weaker Reynolds-stress anisotropy and result in even stronger surface current.

In the present study, instead of the Reynolds-averaged NavierStokes equations, we adopt the Reynolds-averaged

vorticity transport equations, following the lines of our previous study (Longo et al., 1998). But the vorticity generation

terms due to the presence of airwater interface are ignored in the analysis for simplicity. It should be noted that, although

the attenuation of the vortex shedding at the interface can be attributed to the outward mean transverse velocity, as shown

in the previous section, it can also be connected to the mean streamwise vorticity as well and the mechanism behind this

phenomenon can be explained using the mean streamwise vorticity via the mean vorticity transport equation. For the

small Froude number case, we obtain results consistent with Walkers order-of-magnitude analysis of the momentum

equations, as both analyses give the identical term as the main mechanism for the outward mean transverse velocity and

the interface fluctuations are small. However, for the larger Froude number case, our results show a higher Reynolds stressanisotropy as well as a stronger surface current; on the other hand, it is reasonable to expect a higher contribution to the

vorticity generation from the interface fluctuations in the larger Froude number case. Therefore, although the order-of-

magnitude analysis method using the Reynolds-averaged NavierStokes equations in Hong and Walker (2000) is very

general and useful, their conclusions on the origin of surface current in submerged horizontal turbulent jet spreading could

be questionable when extended to other flows, such as the turbulent wakes from a surface-piercing cylinder as studied

y

z

-2 -1 0

-3

-2

-1

0

1

y

z

-2 -1 0

-3

-2

-1

0

1

15

12

9

6

3

0

-3

-6

-9-12

-15

y

z

-2 -1 0

-3

-2

-1

0

1

15

12

9

6

3

0

-3

-6

-9-12

-15

y

z

-2 -1 0

-3

-2

-1

0

1

1.5

1.2

0.9

0.6

0.3

0

-0.3

-0.6

-0.9-1.2

-1.5

Fig. 25. Streamwise vorticity and dominant source terms for Fr=0.2 at x =1.0: (a) streamwise vorticity at x=1.0; (b) y component of term (B) for

streamwise vorticity; (c) zcomponent of term (B) for streamwise vorticity; and (d) term (E) for streamwise vorticity. Contour level of streamwise vorticityis 0.5.



21/22

herein. It has to be pointed out that the two Froude numbers considered in the present study are still relatively low,

although both have non-negligible interface fluctuations. Cases with much higher Froude numbers and terms due to

interface fluctuations in the Reynolds-averaged equations have to be considered for a more complete spectrum of sources

of surface current, which are beyond the scope of the present study.

5. Conclusions

Large-eddy simulation of the flow past an interface piercing circular cylinder at Re= 27 000 and Fr= 0.2 and 0.8 has been

performed using a level-set/ghost-fluid method for the sharp interface treatment of airwater interface and a Lagrangian

dynamic SGS model for the dynamic modeling of the eddy viscosity in inhomogeneous complex flows. At this subcritical

Reynolds number, the latter has some particular advantages over the usual Smagorinsky SGS model for the realistic

prediction of the laminar separation from the cylinder and the transition to turbulence in the wake. The excellent

agreement between the numerical results and the available experimental data demonstrates the accuracy of the

simulation in this work.

The present study shows vortical structures are significantly changed at the interface. In the deep flow, organized

periodic vortex shedding is observed. Near the interface, the organized vortex shedding disappears and small structures

inclined along the interface are observed. The two shear layers deviate from the symmetric vertical plane such that there

are no more direct interactions between them. The flow field is also remarkably changed near the interface. Separation is

enhanced due to the reduced adverse pressure gradient by the negative interface elevation slope along the cylinder. The

distribution of the mean velocity near the interface is also significantly altered from that in the deep flow. In addition, the

size of the separated region is substantially increased in the streamwise and transverse directions. The streamwise

vorticity and outward transverse velocity generated at the edge of the separated region are responsible for the increased

width of the separated region and the attenuation of vortex shedding near the interface. The lateral gradient of the

difference between the vertical and transverse Reynolds normal stresses, increasing with the Froude number, is

responsible for the streamwise vorticity and outward transverse velocity generated at the interface. Turbulence kinetic

energy is also significantly generated at the edge of separated flow region.

The present work provides discussions on the interaction between the vortical structures and the interface, using one

case with experimental data available in the literature. As a first step toward the deep understanding of the detailed

hydrodynamic mechanism, this study is limited with regard to the range of Reynolds and Froude numbers considered and

the neglect of terms related to the fluctuations of the interface position in the Reynolds-averaged vorticity transport

equations. The effect of the Reynolds and Froude number variation and the inclusion of additional terms due to the

interface fluctuations in the Reynolds-averaged equations warrants further study for a more complete understanding of

vortex interface interactions.

Acknowledgments

This work was supported by research Grants N00014-01-1-0073 and N00014-06-1-0420 from the Office of Naval

Research (ONR), with Dr Patrick Purtell as the program manager. The simulations presented in this paper were performed

at the Department of Defense (DoD) Supercomputing Resource Centers (DSRCs) through the High Performance Computing

Modernization Program (HPCMP). We are grateful to Professor Nobuhiro Baba at Osaka Prefecture University, Japan for

providing us with the experimental data figures.

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effect of air wat interface on vortex shedding frm vertical circular cylindet

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