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1 Proceedngs of the of ASME rd Jont US-European Fluds Engneerng Summer Meetng and 8th Internatonal Conference on Nanochannels, Mcrochannels, and Mnchannels FEDSM2010-ICNMM2010 FEDSM-ICNMM2010 August 1-5, 2-4, 2010, Montreal, Canada FEDSM-ICNMM FEDSM-ICNMM SIMULATION OF TWO-PHASE FLOW PAST A VERTICAL SURFACE-PIERCING CIRCULAR CYLINDER Bonguk Koo, Jungsoo Suh 1, Janmng Yang, and Frederck Stern IIHR-Hydroscence & Engneerng, Unversty of Iowa, Iowa Cty, IA, 52242, USA ABSTRACT Large-eddy smulaton of the flow past a surface-percng crcular cylnder s performed to nvestgate the effects of Reynolds and Froude numbers usng a hgh fdelty orthogonal curvlnear grd solver. The present study extends and supports the conclusons of the precursory work for medum Reynolds and Froude numbers. Organzed perodc vortex sheddng s observed n deep flow. At the nterface, the organzed perodc vortex sheddng s attenuated and replaced by small-scale vortces. The streamwse vortcty and outward transverse velocty generated at the edge of the separated regon cause the weakened vortex sheddng at the nterface. The man source of the streamwse vortcty and the outward transverse velocty at the nterface s the lateral gradent of the dfference between the vertcal and transverse Reynolds normal stresses. INTRODUCTION Two-phase flow past a vertcal surface-percng crcular cylnder s relevant for many engneerng applcatons. The arwater nterface adds sgnfcant complcatons. The effects of the nterface on the force dstrbutons on the cylnder, vortex generaton and turbulent structures, and vortex sheddng, especally, ther changes wth the Reynolds (Re) and Froude (Fr) numbers are not well understood. A better understandng of these effects s also mportant for the cases when vortexand wave-nduced vbratons of the cylnder are to be consdered. Recently, the authors studed the effects of ar-water nterface on the vortex sheddng from a vertcal crcular cylnder for medum Re and Fr cases [1] usng a hgh-fdelty orthogonal curvlnear grd solver [2]. The present study extends and supports the conclusons of the precursory work for medum Re and Fr numbers. Smulatons are performed for two-phase turbulent flow past a crcular cylnder n a free stream wth condtons based on the experments of Chapln and Tegen [3]. The forces on the cylnder, ar-water nterface topology ncludng run-up on the front face of the cylnder and the vortex sheddng pattern behnd the cylnder are compared wth the data. COMPUTATIONAL METHODS The mathematcal model and numercal method n the orthogonal curvlnear coordnates used n ths study are the extenson of CFDShp-Iowa verson 6, a sharp nterface Cartesan grd solver for two-phase ncompressble flows recently developed at IIHR by Yang and Stern [4]. In ths solver, the nterface s tracked by a coupled level set and volume-of-flud (CLSVOF) method [5]. A ghost flud methodology s adopted to handle the jump condtons across the nterface, where the densty and surface tenson effect are treated n a sharp way whle the vscosty s smeared by a smoothed Heavsde functon. Mathematcal Model The governng equatons are the Naver-Stokes equatons for two-phase, mmscble, ncompressble flows n the orthogonal curvlnear coordnate system. The contnuty equaton s gven as follows: where ( )[ u ] 0 (1) u s the velocty n the orthogonal coordnate drecton and ( )[ ] 1 J J h followng Pope [6]. The Jacoban of the coordnate transformaton s defned x as J h hjh k, and h wth x a Cartesan coordnate. The momentum equatons are wrtten as follows: 1 Current afflaton: Nuclear Engneerng & Technology Insttute, South 1 Korea Copyrght 2010 by ASME 1 Copyrght 2010 by ASME

2 u t 1 1 p ( j)[ uu j ] ( j)[ j ] g () j j H ( ) u u H ( ) u u (2) j j j j j Each phase of constant densty and vscosty can be defned usng the LS functon n the computatonal doman and sharp jumps of the flud propertes occur at the phase nterface. In ths study, the densty keeps ts sharp jump and the vscosty s smoothed over a transton band across the nterface, where ρ s the densty, p s the pressure, t s the tme, and g the gravty vector n the drecton. In addton, H ( j) 1 hh h j j defned as follows: and () h as n [6]. j s u u j ( j) ( j) u H ( j) u H ( ) 2 u H ( l) j j j l j where µ s the dynamc vscosty and δ j s the Kronecker dalta functon. In large eddy smulaton (LES), the small dsspatve eddes are modeled by the SGS model whereas the large, energy carryng eddes are resolved by the spatally fltered Naver-Stokes equatons. The Lagrangan dynamc Subgrdscale (SGS) model based on Sarghn et al. [7] s adopted n present LES. Eq. (2) s rewrtten as the followng form: u t 1 1 p ( j)[ uu j ] ( j)[ j ] g () (3) j j H ( ) u u H ( ) u u (4) j j j j j j j wth j S j and j t S j, respectvely. Hereafter the flterng sgn for LES wll be dropped for smplcty. Interface Representatons and Flud Propertes The nterface s represented by the level-set (LS) functon whch s corrected usng the volume of flud (VOF) functon to enforce mass conservaton. The LS functon,, s defned as a dstance functon whch s negatve n the ar, postve n the lqud, and zero at the nterface. The VOF functon, F, s defned as the lqud volume fracton n a grd cell that gves zero n the ar, one n the lqud, and a value between zero and one n an nterfacal cell, respectvely. The LS functon and the VOF functon are advanced usng respectvely. D Dt DF Dt t F t (u ) 0 (5) ( u ) F 0 (6) G L G G L G where the subscrpts G and L represent gas and lqud phase, respectvely, the stepwse Heavsde functon s and the smoothed Heavsde functon s H H (7) 1 f 0 H (8) 0 f 0 1 f 1 1 H 1 sn f (9) 2 0 f Numercal Method The fnte-dfference method s used to dscretze the governng equatons on a general orthogonal curvlnear grd. A staggered varable arrangement s adopted,.e., the contravarant velocty components u, u j, uk are defned at cell faces n the, j, k drectons, respectvely, and all other varables are defned at cell centers. A sem-mplct tme advancement scheme s used to ntegrate the momentum equatons wth the second-order Crank-Ncolson scheme for the dagonal vscous terms and the second-order Adams- Bashforth scheme for other terms. A four-step fractonal-step method s employed for velocty-pressure couplng, n whch a pressure Posson equaton s solved to enforce the contnuty equaton. The convectve terms are dscretzed usng the ffth-order Hamlton-Jacob Weghted-ENO (HJ-WENO) scheme and other terms are approxmated usng the second-order central dfference scheme. A sem-coarsenng multgrd solver from the HYPER lbrary [8] s used for the pressure Posson equaton. The LS advecton equaton s solved usng the thrd-order TVD Runge-Kutta scheme [9] for tme advancement and the ffth-order HJ-WENO scheme [10] for spatal dscretzaton. To keep the LS functon as a sgned dstance functon, t has to be rentalzed after a certan tme of evoluton. The CLSVOF method [11] s used to re-dstance the LS functon and mprove mass conservaton propertes of the LS method. In the CLSVOF method, the nterface s reconstructed based on the 2 Copyrght 2010 by ASME

3 VOF functon wth the nterface normal computed from the LS functon. The level set feld s then re-dstanced to reflect the poston of the reconstructed nterface, whch satsfes the volume conservaton constrant. In the present study, the pecewse lnear nterface constructon scheme for the VOF method presented by Gueyffer et al. [12] s used. Computatonal Setup Body-ftted cylndrcal grds of (radal, armuthal, vertcal drectons, respectvely) were used for all cases, as shown n Table 1. The grd ponts were clustered near the surface of the cylnder to resolve the boundary layer and flow separaton. Near the nterface the grd was also refned to capture the nterface deformaton. The computatonal doman was set up such that the portons of cylnder n the water and ar are of length 4D and 2D wth D the cylnder dameter, respectvely. For Fr = 0 case, the ar part s removed and the free surface s treated as a rgd ld. The dstance from the center of the cylnder to the outer boundary s 20D ncludng a buffer zone as shown n Fg. 1. All varables were non-dmensonalzed wth the dameter of cylnder, D and the freestream velocty, U and the two nondmensonal parameters, Froude number and Reynolds number, are defned as followng: Fr U, Re gd U D (10) As shown n Table 1, dfferent Re and Fr cases were nvestgated and a constant rato of Re and Fr of was used, followng Chapln et al. [3]. No-slp boundary condtons were appled on the cylnder wall, whle the slp boundary condton was adopted at the bottom and the top of the computatonal doman. The radal outer boundary was dvded nto nflow and outflow boundares at θ = 90 and θ = 270, n whch θ s the tangental angle startng from the downstream drecton. As shown n Fg 1, a Drchlet boundary condton and a convectve boundary condton [13] were used for the nflow and the outflow, respectvely. In the present smulaton, a constant CFL number of 0.3 was used where the magntude of the tme step vared from to D/U dependng on the flow condtons. Verfcaton and Valdaton The tme hstores of the drag coeffcent (C D ) and lft coeffcent (C L ) wth the runnng mean of C D are shown n Fg. 2. The drag and lft coeffcents are defned the same as Suh et al. [1]. The statstcally statonary state s defned usng the convergence of the runnng mean from the tme hstory of C D. 16 vortex sheddng cycles were used for statstcs. Fg. 3 presents the FFT of the drag and lft coeffcents for Fr = The domnant Strouhal number for C L wth Fr = 0, 0.20, 0.44, 0.84 s 0.192, 0.190, 0.200, 0.210, respectvely and ths s correspondng to the Karman vortex sheddng, however; a wde range of frequences are shown for C D. The mean drag coeffcents are gven n Table 1 and they agree very well wth expermental data from Chapln et al. [3] and Zdravkovch [14]. Note that there s no avalable data for Fr = 0 and Fr = Fg. 4 shows the vertcal profles of the mean streamwse velocty at x = 4.5 and y = 0 for Fr = 0 to Fr =0.84. Note there s no avalable reference data for present Reynolds numbers so the data from Inoue et al. [15] and Kawamura et al. [16] of Re = were compared. The present results wth Fr = 0.84 agree wth ones from Inoue et al. [15] and Kawamura et al. [16] and t s clearly shown that the mean streamwse velocty decreases near the nterface. However, other Fr cases show no sgnfcant decrease of the mean streamwse velocty near the nterface and smaller mean streamwse velocty s observed n deep flow. Fg. 5 presents the mean nterface elevaton at two transverse planes for both Fr = 0.44 and Fr = 0.84 cases. The case wth Fr = 0.44 has good agreement wth the computaton of Fr = 0.50 by Kawamura et al. [16] at both transverse planes. The results of the Fr = 0.84 case are also n good agreement wth prevous studes except for the under-predcton of the depresson, whch s also reported n Suh et al. [1]. The run-up heght are shown n Table 1 and have good agreement wth the results from the Bernoull s equaton (Fr 2 /2). The mean nterface elevaton for Fr = 0.84 s shown n Fg. 6 and compared wth measurement of Fr = 0.80 by Inoue et al. [15]. Overall, the present results are n good agreement wth prevous study even though the current Re s hgher. OVERVIEW OF MEDIUM RE/FR SIMULATION Suh et al. [1] nvestgated the flow past a surface percng crcular cylnder at Re = and Fr = 0.8 usng large-eddy smulaton wth a level-set/ghost-flud method for the sharp nterface treatment of the ar-water nterface and a Lagrangan dynamc SGS model for the dynamc modelng of the eddy vscosty n nhomogeneous complex flows. In deep flow, organzed vortex sheddng can be observed, whle the organzed large-scale vortex sheddng dsappears near the nterface and only small-scale vortces appear mostly at the edge of the separaton regon. The shear layers from the two sdes of the cylnder dgress from each other and no longer nteract n deep flow. Separaton s delayed due to the reduced adverse pressure gradent by the negatve nterface elevaton slope along the cylnder. The dstrbuton of the mean velocty near the nterface s sgnfcantly changed from the deep flow. The centerlne mean streamwse velocty at far wake s ndependent wth Re, whereas t vares sgnfcantly n near wake area. The recrculaton zone ncreases substantally as the profle approaches to the nterfaces. The streamwse length of the recrculaton regon at the nterface s more than three tmes of the deep flow recrculaton area and the magntude of far-wake velocty at the nterface s also remarkably smaller than the deep flow regon. 3 Copyrght 2010 by ASME

4 The pattern of the mean streamwse and transverse velocty profles at the nterface shows to be qute dfferent from that n deep flow. The mean streamwse velocty profles for the horzontal zones away from the nterface s between low Re and hgh Re experment studes, whereas the mean streamwse velocty at the nterface shows sgnfcantly dfferent profle wth ncreased wake depth. As the nterface approaches, the magntude of the mean transverse velocty ncreases. The ncreased wdth of the separated regon and the attenuaton of vortex sheddng near the nterface s generated by the streamwse vortcty and outward transverse velocty at the edge of the separated regon The lateral gradent of the dfference between the vertcal and transverse Reynolds normal stresses s responsble for the streamwse vortcty and outward transverse velocty generated at the nterface. The 2 term ( vv ww) from the mean streamwse vortcty yz transport equaton s the man producton mechansm for the mean streamwse vortcty. The vertcal and transverse gradents of the dfference between vv and ww are responsble for the generaton of the streamwse vortcty near the nterface and presumably cause the outward transverse velocty near the nterface. RESULTS AND DISCUSSIONS Four dfferent Re and Fr cases are smulated; however, the case wth Re = and Fr = 0.84 s manly presented and compared wth prevous computatonal studes by Suh et al. [1]. Note that present studes used Re n the subcrtcal regme (1,000 to ) so that smlar flow patterns were observed. Hgher Re/Fr ( /1.24 and /1.64) cases wll be performed for further nvestgaton. Instantaneous Flow Instantaneous vertcal vortcty at dfferent depth ncludng the nterface are shown n Fg. 7. In deep flow, organzed vortex sheddng s evdent as shown n Fg.7 (c) and (d). However, small scale vortces appear at the nterface nstead of organzed large vortex sheddng. In addton, necklace vortces are observed n front of the cylnder for Fr = 0.20, 0.44 (not shown) and Fr = 0.84, whch was also reported n Suh et al. [1]. The nstantaneous vortcal structures dentfed by the second nvarant velocty gradent tensor was used to obtan further detals on large-scale flow structures, as shown n Fg. 8. The vortcal structures are nclned as the nterface s approached for Fr = 0.84, whle they are almost parallel to the cylnder wall for Fr = 0, 0.20, and Ths pattern s also shown n the vortex core lnes and t wll be dscussed n later secton. Mean Flow The frcton and pressure coeffcents at dfferent depth for Fr = 0.84 are shown n Fg. 9. The boundary layer separaton pont for deep flow s 0.462π whch s exactly same as that from Suh et al. [1]. Near the nterface, the separaton occurs at 0.487π, whereas the separaton takes place at 0.514π n the case of Suh et al. [1]. The pressure coeffcent has good agreement wth expermental data from Norberg [17] wth Re = up to 0.4π. Near the nterface the pressure coeffcent s very dfferent from the deep flow. The delayed separaton pont near the nterface s due to the decreased adverse pressure gradent whch comes from the negatve nterface elevaton slope near the cylnder. And ths was also observed n Suh et al. [1]. The sectonal mean drag coeffcent and rms of the lft coeffcent for Fr = 0.84 are gven n Fg. 10. The drag coeffcent s n good agreement up to the nterface; however there are dfferences under the nterface. The maxmum drag coeffcent appears at the nterface whch was also reported by Suh et al. [1], whereas Chapln et al. [3] showed the drag coeffcent reaches maxmum approxmately z = The rms of the lft coeffcent approaches almost zero at the nterface and ths ndcates the attenuaton of vortex sheddng at the nterface [1]. The mean separaton pattern wth the vortex core lnes, obtaned from the approach dscussed n Kandasamy et al. [18] and Sujud and Hames [19], s gven n Fg. 11 for Fr = The three dfferent types of vortces defned n Suh et al. [1] are observed: mean vertcal vortces (V1), mean streamwse vortces (V2), and V-shaped mean vortex nsde the separaton regon (V3). In contrast, V2 and V3 do not appear n the Fr = 0, 0.20, and 0.44 and V1 s not attached to the cylnder wall and t s almost parallel to the cylnder wall as the nterface s approached. As the Fr gets smaller, the separaton regon gets flatter. Detals of these vortces are dscussed n Suh et al. [1]. Fg. 12 presents the mean streamwse velocty on the centerlne n the wake of the cylnder. All the mean velocty profles except for at the nterface agree wth the measurements by Lourenco and Shh [20]. As approachng the nterface, the recrculaton regon s dramatcally ncreased and the mean velocty at far wake s much smaller, whch was also observed by Suh et al. [1]. Fg. 13 shows the mean streamwse and transverse veloctes at dfferent depths n the wake of the cylnder for Fr = The pattern of the mean streamwse and transverse velocty profles at the nterface s qute dfferent from that n deep flow. In deep flow, the mean streamwse velocty profles at both locatons (x = 1.06 and x = 2.02) agree wth the measurements by Lourenco and Shh [20], whereas at the nterface t has substantally ncreased wake wdth. The maxmum outward mean transverse velocty appears at the nterface and a very small outward mean transverse velocty appears below the nterface. Karman vortex sheddng at the nterface s attenuated due to ths outward mean transverse velocty at the edge of the separated regon, whch was also mentoned by Suh et al. [1]. The mean velocty contours wth path-lnes at the nterface are shown n Fg. 14. Substantal changes of the mean 4 Copyrght 2010 by ASME

5 streamwse velocty for Fr =0.84 occur near the separaton regon, as shown n Fg. 14 (j). The outward mean transverse velocty for Fr = 0, 0.20 and 0.44 s smaller than that for Fr = 0.84, as shown n Fg. 14 (k). Ths was also reported by Suh et al. [1]. The mean vertcal velocty near the separaton regon becomes negatve, whch ndcates the decreased wave elevaton n that regon. The mean streamwse velocty contours at two crossstream planes n the near wake are shown n Fg. 15 (a) and (e). Only a half doman s shown and the locaton of the nterface (dotted lne) and the cylnder wall (sold and dot lne) are shown. Negatve mean streamwse velocty occurs near the nterface and n deep water at x = 1 plane (recrculaton regon). However, at x = 2.5 plane, negatve velocty presents only near the nterface. The wdth of the wake s constant n deep flow at plane x = 1, whereas the wake wdth near the nterface ncreases substantally. The mean vortcty contours at two cross-stream planes at the near wake area are gven n Fg. 15 (b) - (h). The streamwse vortcty s responsble for the large outward mean transverse velocty near the nterface [1]. In addton, the vortex sheddng near the nterface s attenuated and the wake wdth ncreases remarkably near the nterface due to the mean streamwse vortcty. The mean transverse vortcty occurs only near the nterface at x = 1 and x = 2.5. Ths ndcates that the transverse vortcty nduces the nterface fluctuatons. The mean vertcal vortcty at x = 1 has hgh magntude near the nterface and ths corresponds to the hgh mean streamwse velocty gradents near the nterface. Reynolds Stress Fg. 16 shows the Reynolds stresses at two cross-planes. At x = 1 plane, the streamwse Reynolds stress has hgh magntude at the regon below the nterface where the hgh mean streamwse velocty gradents present and t s the ndcaton of hgh turbulent energy producton. At x = 2.5, smlar pattern of magntude appears near the nterface; however; the magntude decreases n deep flow. In the separaton regon, the transverse Reynolds stress reduces as approachng the free surface. The vertcal Reynolds stress shows hgh magntude at the nterface and ths plays an mportant role for the large changes of the streamwse vortcty and the outward transverse velocty at the nterface. The Reynolds shear stress has a small magntude at the free surface. Fg. 17 shows the contours of the vertcal Reynolds stress at the nterface. Hgh gradent of the vertcal Reynolds stress occurs n the regon wth a hgh outward transverse velocty. It ndcates that the lateral gradents of the dfference between the transverse and vertcal Reynolds stress nduce the streamwse vortcty and the outward transverse velocty, as proposed by Suh et al. [1]. CONCLUSIONS The flow past a surface-percng crcular cylnder has been studed for Re and Fr effects usng large-eddy smulaton. The present study s an extenson and supports the conclusons of a precursory study for medum Re and Fr smulatons, whch nvestgated the effects of ar-water nterface on the vortex sheddng from a vertcal crcular cylnder [1]. Smlar flow features have been obtaned snce the Reynolds numbers for current studes are n the subcrtcal regme (Re = 1,000 to ). Quas-vertcal vortces from the organzed perodc vortex sheddng present n deep flow. Organzed large-scale vortex sheddng s attenuated and only small-scale vortces appear at the nterface. The vortex sheddng attenuaton at the nterface s nduced by the streamwse vortcty and outward transverse velocty generated at the edge of separated regon. The lateral gradent of the dfference between the vertcal and transverse Reynolds stresses s responsble for the streamwse vortcty and outward velocty generated at the nterface. To systematcally study the orgn of the surface current,.e., the outward transverse velocty at the nterface, a seres of cases at dfferent supercrtcal Re/Fr ( /1.24 and /1.64) reported n [3], wll be smulated. The effects of Re/Fr on the flow, ncludng forces/pressure/shear-stress dstrbutons, turbulent structures, and vortex sheddng, wll be nvestgated n detal. Verfcaton and valdaton studes wll be done wth systematc grd refnement. The effect of the nterface densty jump on vortcty transport wll also be nvestgated. Further nvestgatons wll be performed to study the orgn of the mean streamwse vortcty and the outward transverse velocty near the nterface usng the mean streamwse vortcty transport equatons. Lastly, to obtan better understandng of vortex- and wave- nduced vbratons, forced pure sway moton smulatons wll be conducted. ACKNOWLEDGEMENTS Ths work was supported by research grants N and N from the Offce of Naval Research (ONR), wth Dr. Patrck Purtell as the program manager. The smulatons presented n ths paper were performed at the Department of Defense (DoD) Hgh Performance Computng Modernzaton Program (HPCMP) NAVO Computng Center. REFERENCES [1] Suh, J., Yang, J., and Stern, F., 2009, The Effect of Ar- Water Interface on the Vortex Sheddng from a Vertcal Crcular Cylnder, submtted Journal of Fluds and Structure. [2] Yang, J., Bhushan. S., Suh, J., Wang, Z., Koo, B., Sakamoto, N., Xng, T., and Stern, F., 2008, Large-Eddy Smulaton of Shp Flows wth Wall-Layer Models on Cartesan Grds, 27th Symposum on Naval Hydrodynamcs, October 2008, Seoul, Korea. 5 Copyrght 2010 by ASME

6 [3] Chapln, J. R. and Tegen, P., 2003, Steady Flow past a Vertcal Surface-Percng Crcular Cylnder, Journal of Fluds and Structure, 18, pp [4] Yang, J. and Stern, F., 2009, Sharp Interface Immersed- Boundary/Level-Set Method for Wave-Body Interactons, J. Comput. Phys., 228(17), pp [5] Wang, Z., Yang, J., Koo, B. G., and Stern, F., 2009, A Coupled Level Set and Volume-of-Flud Method for Sharp Interface Smulaton of Plungng Breakng Waves, Int. J. Mult. Flow, 35(3), pp [6] Pope, S. B., 1978, The Calculaton of Turbulent Recrculatng Flows n General Orthogonal Coordnates, J. Comp. Phy., 26, pp [7] Sarghn, F., Pomell, U., Balaras, E., 1999, Scale-smlar Models for Large-Eddy Smulatons, Phy. Fluds, 11(6), pp [8] Falgout, R. D., Jones, J. E., Yang, U. M., 2006, Numercal Soluton of Partal Dfferental Equatons on Parallel Computers. Vol. 51. Sprnger-Verlag, Ch. The desgn and mplementaton of HYPRE, a lbrary of parallel hgh performance precondtons, pp [9] Shu, C.W. and Osher, S., 1988, Effcent Implementaton of Essentally non Oscllatory Shock-Capturng Schemes, J. Comput. Phys., 77(2), pp [10] Jang, G. and Peng, D., 1999, Weghted ENO schemes for Hamlton-Jacob equatons, SIAM Journal. on Scentfc Computng, 21(6), pp [11] Sussman, M., Puckett, E. G., 2000, A Coupled Level Set and Volume-of Flud Method for Computng 3D Axsymmetrc Incompressble Two-Phase Flows, J. Comput. Phys., 162, pp [12] Guyeffer, D., L, J., Nadm, A., Scardovell, S., Zalesk, S., 1999, Volume of Flud Interface Trackng wth Smoothed Surface Stress Methods for Three-Dmensonal Flows, J. Comput. Phys., 152, pp [13] Breuer, M., 1998, Large Eddy Smulaton of the Subcrtcal Flow past a Crcular Cylnder: Numercal and Modelng aspect, Int. J. Numer. Meth. Fluds, 28, pp [14] Zdravkovch, M. M., 1997, Flow Around Crcular Cylnder, Vol. 1: Fundamentals, Oxford Unversty Press, New York. [15] Inoue, M., Bara, N., Hmeno, Y., 1993, Expermental and Numercal study of Vscous Flow Feld Around an Advancng Vertcal Crcular Cylnder Percng a Free- Surface, J. Kansa Soc. Nav. Arch., 220, pp [16] Kawamura, T., Mayer, S., Garapon, A., Sørensen, L., 2002, Large Eddy Smulaton of a Flow Past Free Surface Percng Crcular Cylnder, J. Fluds Engng., 124, pp [17] Norberg, C., 1992, Pressure Forces on a Crcular Cylnder n Cross Flow, In: IUTAM Symposum Bluff-Body Wakes, Dynamcs and Instabltes. Göttngen, Germany, pp [18] Kandasamy, M., Xng, T., Stern, F., 2009, Unsteady Free Surface Wave-Induced Separaton: Vortcal Structures and Instabltes, J. Fluds and Struc., 25, pp [19] Sujud, D., Hames, R., 1995, Identfcaton of Swrlng Flow n 3-D Vector Felds, AIAA paper, [20] Lourenco, L. M., Shh, C., 1993, Characterstcs of the Plane Turbulent Near Wake of a Crcular Cylnder. a partcle mage velocmetry study. Data taken from A.G. Kravchenko and P. Mon, 2000, Numercal Studes of Flow over a Crcular Cylnder at Re D =3900, Phy. Fluds, 12(2), pp Fgure 1 Computatonal doman wth grd and boundary condtons Fgure 2 Tme hstory and runnng mean for drag coeffcent and tme hstory for lft coeffcent for Fr = Copyrght 2010 by ASME

7 10 0 C D C L 10-1 Magntude St Fgure 3 FFT of the drag and lft coeffcent for Fr = Fgure 6 Mean nterface elevaton; Top: computaton, Fr = 0.84; bottom: measurement by Inoue et al. (1993), Fr = 0.8 Fgure 4 Vertcal profles of the mean streamwse velocty at x = 4.5, y = 0.0 for Fr = 0, Fr = 0.20, Fr = 0.44, and Fr = (a) (b) (a) (c) (b) Fgure 5 Profles of mean nterface elevaton at x = 0.9 and x = 2.0. (a) Fr = 0.44; (b) Fr = (d) Fgure 7 Instantaneous vertcal vortcty at the nterface and horzontal planes for Fr = (a) On the nterface; (b) z = -0.5; (c) z = -1; (d) z = Contour nterval s Copyrght 2010 by ASME

8 (a) (b) (c) (d) Fgure 8 Instantaneous vertcal structures dentfed by the second nvarant of the velocty gradent tensor Q = 0.5 for (a) Fr = 0; (b) Fr = 0.20; (c) Fr = 0.44; (d) Fr = Fgure 9 Pressure coeffcent for Fr = 0.84 Fgure 10 Sectonal drag and lft coeffcents for Fr = Copyrght 2010 by ASME

9 (a) (b) (c) (d) Fgure 11 Mean separaton pattern wth vortex core lnes for (a) Fr = 0; (b) Fr = 0.20; (c) Fr = 0.44; (d) Fr = Fgure 12 Mean streamwse velocty on the centerlne n the wake of a crcular cylnder for Fr = (a) (b) Fgure 13 Mean velocty at two locatons n the wake of a crcular cylnder at Fr = (a) Streamwse; (b) transverse. 9 Copyrght 2010 by ASME

10 (a) (b) (c) (d) (e) (f) (g) (h) () (j) (k) (l) Fgure 14 Mean velocty contour at the nterface. (a), (d), (g), (j) Streamwse velocty; (b), (e), (h), (k) transverse velocty; (c), (f), (), (l) vertcal velocty for Fr = 0, 0.20, 0.44, and 0.84, respectvely. (a) (b) (c) (d) (e) (f) (g) (h) Fgure 15 Contours of the mean flow at two cross-stream planes for Fr = (a) Streamwse velocty at x = 1.0; (b) streamwse vortcty at x = 1.0; (c) transverse vortcty at x = 1.0; (d) vertcal vortcty at x = 1.0; (e) Streamwse velocty at x = 2.5; (f) streamwse vortcty at x = 2.5; (g) transverse vortcty at x = 2.5; (h) vertcal vortcty at x = Copyrght 2010 by ASME

11 (a) (b) (c) (d) (e) (f) (g) (h) Fgure 16 Dstrbuton of the Reynolds stresses at two cross-stream planes for Fr = (a) uu at x = 1.0; (b) vv at x = 1.0; (c) ww at x = 1.0; (d) uv at x = 1.0; (e) uu at x = 2.5; (f) vv at x = 2.5; (g) ww at x = 2.5; (h) uv at x = 2.5. Fgure 17 Dstrbuton of the vertcal Reynolds stress ww at the nterface for Fr = 0.84 Table 1 Smulaton condtons and hydrodynamc forces on the cylnder and run-up Grd and Re D Fr D doman sze D 4D D 6D D 6D D 6D C D (EFD) C D (CFD) E (%) Max y + Run-up (Bernoull) Run-up (CFD) N/A N/A [3] 1.15 [14] 0.79 [3] 0.89 [14] E (%) Copyrght 2010 by ASME