Vortical Structures and Instability Analysis for Athena Wetted Transom Flow with Full-Scale...

Shanti BhushanAssistant Research Professor

Center for Advanced Vehicular Systems,

200 Research Blvd.,

Mississippi State University,

Starkville, MS, 39759

e-mail: [email protected]

Tao XingAssistant Professor

Department of Mechanical Engineering,

Engineering Physics,

Room 324F, PO Box 440902,

University of Idaho,

Moscow, ID, 83844-0902


Frederick SternProfessor and Research Engineer

IIHR-Hydroscience & Engineering,

C. Maxwell Stanley Hydraulics Laboratory,

The University of Iowa,

Iowa City, IA, 52242-1585


Vortical Structuresand Instability Analysisfor Athena Wetted TransomFlow with Full-Scale ValidationVortical structures and associated instabilities of appended Athena wetted transom flowin full-scale conditions are studied using DES to explain the source of dominant transomflow frequency, including verification and validation using full-scale experimental data.The results are also compared with model-scale bare and appended hull predictions andexperiments. The grid used for the validation is sufficiently fine as it resolves 70% and91% of the experimental inertial subrange and turbulent kinetic energy values, respec-tively. The model-scale bare and appended hull resistance predictions compare within2.5%D and 5.4%D of the experimental data D, respectively. The full-scale appended hullresistance predictions compare within 4.2%D of the extrapolated data using the ITTCline. The averaged comparison error of the full-scale transom wave elevation mean, RMSand dominant frequency predictions and the experimental data is 8.1%D, and the predic-tions are validated at an averaged 11.2%D interval. The transom wave elevationunsteadiness is attributed to the Karman-like transom vortex shedding as both show thesame dominant frequency. The Karman-like instability shows St¼ 0.148 for the bare hulland St¼ 0.103 6 4.4% for model- and full-scale appended hull. The appended hull simu-lations also predict: horseshoe vortices at the juncture of rudder-hull withSt¼ 0.146 6 3.9% and strut-hull with St¼ 0.053 6 2%; shear layer instability at thestrut-hull intersection with St¼ 0.0067 6 3%; and unsteady sinkage and trim induced bytransom vortex shedding with St¼ 2.19. The instabilities do not show significant variationon scale, propeller or motions. The bare hull simulation also predicts flapping-like insta-bility in the wake with St¼ 0.144. [DOI: 10.1115/1.4006173]

Keywords: transom flow, full-scale validation, vortex instability

1 Introduction

Many modern ships have transom sterns for which dry or wet-ted transom flows are observed depending upon the Froude num-ber (Fr). Transom flows show complex physics such as, unsteadybreaking waves including rooster tail, high levels of surfaceroughness, bubble entrainment and vortex shedding. Athena, agunboat of PG 84 class which was converted to a Research Vessel(R/V), has been used in several previous studies to understandtransom flows. Athena bare hull transom flows at Fr¼ 0.25 and0.43 were studied in the Office of Naval Research (ONR) ShipWave Breaking Workshop [1] to assess the capability of computa-tional fluid dynamics in predicting ship generated waves andwave breaking. The experimental data included detailed meanwave elevation (f) around the ship. A wetted transom flow wasobserved for Fr¼ 0.25 which showed a Kelvin wave pattern withstrong diverging shoulder wave from the transom edge. A drytransom flow was observed for Fr¼ 0.43 which showed a break-ing plunging bow waves and a transom shoulder wave travelingtowards the center-plane. The shoulder waves interacted with thewave reflected from the center-plane to form a rooster tail. Theworkshop included simulations from five different solvers includ-ing unsteady Reynolds averaged Navier-Stokes (URANS), poten-tial flow and Euler solvers. One of the URANS submissions wasCFDShip-Iowa V.4 (V4) [2] which compared well with theexperiments, and outperformed other solvers. For Fr¼ 0.25, theV4 results showed an unsteady transom transverse wave confinedbetween the center-plane and the diverging wave, which was

attributed to the Karman-like vortex shedding from the transombottom corner. For Fr¼ 0.43, simulation showed intense bowwave breaking with repeated reconnection of the plunging breakerwith the free surface, resulting in multiple free surface scars. Theinitial overturning bow wave was created due to the thickening ofthe bow wave sheet. The reconnection of the first plunger with thefree-surface produced a strong counter rotating vortex pair result-ing in large cross flow gradients and subsequent plungers.

Waytt et al. [3] performed full-scale experiments for appendedAthena wetted transom flow at Fr¼ 0.25 and measured transomwave elevation mean, standard deviation (RMS) and dominantspectral frequency (f). Experimentalists proposed that the transomwave elevation unsteadiness could be due to either the collision ofshoulder waves traveling towards the center-plane or shipmotions. Latter was eliminated as the trim dominant frequencywas an order of magnitude smaller than that of wave elevation.The data were compared with V4 full-scale fixed appended hullpredictions using detached eddy simulation (DES) with wall-functions (WF), and with an Euler code, Numerical Flow Analysis(NFA) model-scale fixed bare hull predictions. V4 predicted wet-ted transom, whereas NFA predicted dry transom flow. Bothcodes predicted the mean wave elevation well with V4 somewhatunder-predicting and NFA over-predicting. V4 performed betterthan NFA for RMS predictions, as the latter showed rapid attenua-tion of the peak away from the transom due to inadequate turbu-lence modeling. V4 and NFA predicted the dominant frequencywithin 6%D and 39%D of the experimental data D, respectively.In V4 predictions, the transom wave elevation unsteadiness wasattributed to the Karman-like shedding from the transom bottomcorner, whereas NFA predictions did not provide an explanation.

Recently, Drazen et al. [4] measured resistance, mean andunsteady transom wave elevation and void fraction for Fr¼ 0.38

Contributed by the Fluids Engineering Division of ASME for publication in theJOURNAL OF FLUIDS ENGINEERING. Manuscript received May 28, 2011; final manuscriptreceived February 17, 2012; published online March 16, 2012. Assoc. Editor: Z. C.Zheng.

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and 0.43 for a transom stern model. The data were compared withNFA and V4 DES predictions. The experimental data showedwetted transom flow for Fr¼ 0.38 with a recirculation region oneither side of the model centerline, wake spanning the entire tran-som width and breaking waves up to the transom. The resistancemeasurements showed a dominant frequency f¼ 2 Hz, which wasexpected to be induced by the flow. The transom wake was foundto be unsteady with dominant f¼ 1.96 Hz. A dry transom wasobserved for Fr¼ 0.43 with a well-defined and narrow wakewhich quickly steepened to a defined peak where it spilled out in arooster tail region. Breaking waves were observed mostly in therooster tail region. The resistance and transom wave elevationsmeasurements did not show a dominant frequency. NFA predictedpartially wetted and dry transom for Fr¼ 0.38 and 0.43, respec-tively, and steady resistance for both the cases. Overall, the wakeand wave breaking pattern were predicted well including the dom-inant frequency for the wetted case. The dominant frequency wasattributed to the transom breaking waves which exhibited shearlayer instability. The results also showed a dominant frequency inthe rooster tail region. The cause of the instability was notexplained and the finding was not validated as measurementswere not available in that region. V4-DES predicted wetted tran-som with unsteady resistance for Fr¼ 0.38 and dry transom withsteady resistance for Fr¼ 0.43. Overall, the resistance, wave ele-vation mean and dominant frequency were predicted within 3%D, 5.5% D and 0.1% D of the experiments, respectively. Theunsteady resistance and transom wave elevation for the wettedcase were attributed to the Karman-like shedding from the tran-som bottom corner. A dominant frequency was predicted in therooster tail region for the dry transom case in agreement withNFA predictions. This unsteadiness was attributed to the breakingshoulder waves traveling towards the center plane.

The objective of this study is to perform validation of appendedAthena wetted transom flow at full-scale conditions using full-scale experiments, and to confirm the dominant transom flow fre-quency as Karman-type shedding. Grid and time step verification,and domain and turbulence resolution sensitivity studies are per-formed for full-scale at fixed condition to quantify numericaluncertainties. Analysis of the large scale vortical structures andscaling of associated instabilities are performed for model- andfull-scale appended hull and model-scale bare hull.

2 Background for Transom Flow and Instability

Studies

Potential flow solvers have limitations in predicting wettedtransom flows, as the transom corner pressure is assumed to beatmospheric which leads to a dry transom irrespective of Fr. Sev-eral models have been developed [5–7] to address the above limi-tation, such as introduction of nonlinear, damping and advectionterms in the free-surface equation. Such models have been appliedwith partial success because of the absence of universal modelconstants. Model- and full-scale backward facing step with free-surface [8–10] studies have shown that the Reynolds number (Re)plays an important role for the transom wave predictions. In thesestudies, URANS solvers predicted wave elevation mean and RMSbetter than Potential flow solvers when compared with theexperiments.

In URANS solvers the transom corner pressure is solved explic-itly; thus a wetted or dry transom is predicted depending on theFr. However, the free-surface modeling using the surface trackingmethod may have a singular solution at the intersection of free-surface with the hull. This may affect the solution in the transitionFr range [11]. On the other hand, level-set methods have the abil-ity to predict smooth transition from wet to dry transom and viceversa. A precursor of this study is Wilson et al. [2] whereinURANS simulations were performed for fixed model-scale barehull Athena for both wetted Fr¼ 0.25 and dry transom Fr¼ 0.43and 0.62. The simulations were performed using V4 which usesthe level-set method. Results included detailed analysis of the

bow and transom wave breaking and unsteadiness, including com-parison with experimental data [12]. The study reported goodagreement for the mean wave elevation and breaking wave predic-tions for both dry and wet transom flows.

V4 simulations have also been performed for bare andappended hull Athena using URANS with WF for full-scale, butfocused mostly on dry transom flows [13,14]. Xing et al. [13] per-formed verification and validation for model-scale predictionsusing experimental data [15,16]. The simulations included barehull resistance and motions at steady state and single run full-Frcurve, and appended hull full-Fr powering curve. Bhushan et al.[14] performed full-scale simulations for bare hull resistance,sinkage and trim including grid verification, and appended hullpowering, seakeeping and maneuvering for smooth and rough-wall. The results were compared with model-scale experimentaldata extrapolated to full-scale using the ITTC line.

The instability analysis is performed based on the canonicaland ship flow studies, which have shown Karman-like, horseshoevortex, shear layer, flapping and helical vortex instabilities. Kar-man instabilities are caused by the interaction of two opposite vor-tices, initiated by shear layer instability, for non-reattachingseparation such as those in circular cylinder. For wall-boundedseparation bubbles such as those in backward-facing steps, thevortices amalgamate with their mirror image to form large-scalevortices that impinge on the wall just after recirculation and areeventually shed. These vortices scale similar to the Karman insta-bility and are referred to as the Karman-like shedding. They areassociated with Strouhal number StH¼ fH/US based on half wakewidth H and shear layer velocity (US). Sigurdson [17] reportedStH¼ 0.07–0.09 independent of Re and geometry for single phasecanonical flows. The wave induced separation for surface piercingNACA 0024 airfoil at Fr¼ 0.37 predicted StH¼ 0.067 6 2.2%and 0.07 in URANS and DES NACA simulations [18,19], respec-tively. The predictions showed that the free-surface reduces boththe strength and frequency of the vortex shedding resulting inlower StH. These instabilities were reported for ship simulations:the transom vortex shedding for Athena wetted transom flows; forDES of tanker hull KVLCC2 at 30 deg drift without free-surfacedue to the interaction of bow vortices StH¼ 0.0735 [20]; and forDES of Wigley hull at 45 and 60 deg drifts at Fr¼ 0.181 due tothe interaction of hull and tip vortices StH¼ 0.08 �0.087 [21].Overall, Karman-like instability show StH¼ 0.078 6 15.4% range.

Junction flow occurs when a boundary layer encounters an ob-stacle. The obstacle causes adverse pressure gradient leading tothree-dimensional boundary layer separations that wrap aroundthe obstacle, which are referred to as horseshoe vortices. Simpson[22] reviewed the laminar and turbulent flat-plate boundary layerjunction flows for obstacles with principal axis normal to the platesurface. They reported that the horseshoe vortex separation beginsaround ReT¼ 1000 and the dominant frequency of the vortexincreases with ReT, where T is the thickness of obstacle. Thehorseshoe vortex shedding consists of two vortex system whosestrength and stretching depends on the bluntness of the obstacle.For example, a circular cylinder obstacle shows stronger vortexsystem than an airfoil obstacle, but the latter exhibits a vortexcore closer to the wall and smaller spanwise spacing between thevortex legs. An airfoil obstacle with large angle of attack (>10deg) has a stronger vortex leg on the suction side than on the pres-sure side, which resembles a single vortex system. As an obstacleleading edge back sweep angle is increased, the vortex coresmove away from the wall, the vortex intensity is reduced andturbulent activity is spread over a greater region. Reverse trendis observed for the forward sweep angle. These instabilitiesshowed StT¼ fT/U0¼ 0.17� 0.28 based on the largest dominantfrequency.

Boundary layer flow separation begins with shear layer instabil-ity, which is initiated at a critical Re that depends on the spatialrestrictions imposed by the mean length scale. The simulations forseparated laminar boundary layer over an airfoil [23] predicted Sth¼ 0.0056 6 2% based on the momentum thickness of the

031201-2 / Vol. 134, MARCH 2012 Transactions of the ASME


boundary layer at separation (h) and US. Such instabilities nearthe free-surface separation and inside the separation bubble forsurface piercing NACA 0024 studies showed Sth¼ 0.00622 6 1.5% and 0.00384 6 0.5% for laminar and turbulentsimulations, respectively. In KVLCC2 simulation, two shear-layerinstabilities were predicted on the leeward side, one from the bowSth¼ 0.00292 and other from the stern Sth¼ 0.00103. For theWigley hull at 60 deg drift simulation, shear layer instability Sth� 3� 10�4 was predicted on the keel. Kandasamy et al. [19] con-cluded that shear-layer instability St varies inversely with theadverse pressure gradient. This explains the lower St in turbulentflows compared to laminar, at the stern compared to that on thebow, and on the leeward side for ships at large drift angles.

Flapping instability occurs when a wall bounded separationbubble forms a recirculation region which exhibits a periodicenlargement and shrinkage. Such instabilities are scaled using thereattachment length XR and free stream velocity U0 which showsStR ¼ fXR/U0¼ 0.12 for flow past cylinders, 0.1 for backward fac-ing steps and 0.073 for a two-dimensional square rib (refer to Ref.[19] for details). A periodically enlarging and shrinking free-surface separation bubble was predicted in surface-piercingNACA 0024 simulations, which were referred to as flapping-likeinstability. URANS and DES predicted StR¼ 0.13 and 0.28,respectively. The larger StR in DES was primarily due almosttwice XR predictions than URANS.

Helical vortices are similar to the tip vortices formed over adelta wing. For such a vortex, the core frequency decrease down-stream such that the St based on the distance from the separationpoint remains constant. Xing et al. [20] predicted fore-body side,stern and aft-body bilge helical vortices on the leeward side ofKVLCC2 at large drift angles and their scaling agreed well withthe delta wing theory. However, their circulation strengths weremuch higher and decreased rapidly downstream compared to thedelta wing tip vortices. These differences were attributed to thedifference in the Re. Tip-vortices were also predicted for Wigleyhull at large drift angles which showed similar qualitativebehavior.

3 Computational and Analysis Methods

The general-purpose solver, V4, solves the URANS/DES equa-tions in the liquid phase of a free-surface flow. The governingequations are solved in either absolute inertial earth-fixed or rela-tive inertial coordinates for an arbitrary moving but non-deforming control volume. A dynamic overset grid approach isused to allow three degrees of freedom (3DOF) ship motions. Asimplified body force model is used for propeller along with a pro-portional, integral and differential revolution per second (RPS)controller. Readers are referred to Ref. [24] for details, and refer-ences of the PETSc and SUGGAR libraries used in V4.

Modeling. The governing equations for the water phase indimensionless form are:

@Ui

@Xi¼ 0 (1)

@Ui

@tþ ðUj � UGjÞ

@Ui

@Xj¼ � @p̂

@Xjþ 1

Re

@2Ui

@Xj@Xj� @

@Xjuiuj (2)

where, Ui¼ (U,V,W) are the URANS velocity components, UGj

are the local grid velocity in either the absolute inertial earth-fixed or relative inertial Cartesian coordinates Xi ¼ (X,Y,Z),p̂ ¼ pabs=qU2

0 þ z=Fr2 þ 2k=3 is the dimensionless piezometricpressure where pabs is the absolute pressure, uiuj are the Reynoldsstresses, Fr ¼ U0=

ffiffiffiffiffiffigLp

, and k is the turbulent kinetic energy. L isthe ship length and Re is based on L.

The Reynolds stresses in URANS or subgrid-stresses in DESare obtained using an isotropic two-equation blended k-x/k-emodel,

� uiuj ¼ �t@Ui

@Xjþ @Uj

@Xi

� �� 2

3dijk (3)

where dij is the Kronecker delta. The unknown turbulent eddy vis-cosity (�t) is evaluated from k and the specific dissipation rate (x)transport equations. In the DES model, the dissipative term of thek-transport equation is modified based on the local turbulentlength scale and grid spacing. The boundary conditions for the tur-bulence models are specified either using a near-wall or smooth-wall WF approach. The WF approach uses a multi-layer modelwhich smoothly blends the sub- and log-layer regions, and isimplemented using a two-point approach.

The location of the free-surface is given by the ‘zero’ value ofthe level-set function, positive in water and negative in air. Sincethe free-surface is a material surface, the level set function followsa simple advection equation. For stability purposes, a small artifi-cial diffusion term is added to the equation. Negligible shear stressin the air phase is assumed which provides the jump condition atthe free-surface. As a good approximation for air-water interfaces,the pressure in the air is assumed equal to the atmospheric pres-sure. The velocity and turbulent quantities k and x are extendedfrom the air-water interface to air by solving an equation similarto the level set function over the whole air domain.

A simplified body force model is used for the propeller whichprescribes axisymmetric body force with axial and tangentialcomponents. The propeller model requires thrust, torque, andadvance coefficients as input, and provides the torque and thrustforces. These forces appear as a body force term in the momentumequation for the fluid inside the propeller disk. The location of thepropeller is defined in the static condition of the ship, and movesaccording to the ship motion. The propeller RPS required toachieve a target ship speed is obtained by performing a self-propelled simulation starting from static ship conditions using acontroller. The controller solves for the propeller RPS to balancethe propeller thrust and the ship resistance.

The forces and moments are computed in the absolute inertialearth-fixed coordinates from the static pressure, hydrostatic pres-sure and frictional forces acting on the ship surfaces, and propellerthrust and torque assuming that the center of rotation is coincidentwith the center of gravity. The forces and moments are then pro-jected to the non-inertial ship-fixed coordinates to obtain the surgeand sinkage velocities, and trim angular velocity assuming rigidbody motion. The velocities are then transformed back to theabsolute inertial earth-fixed coordinates to predict the evolution ofthe ship location and attitude.

Numerical Methods and High Performance Computing. Thegoverning equations are discretized using finite differenceschemes on body-fitted curvilinear grids. The time marching isdone using the second-order backward difference scheme. Theconvection and diffusion terms are discretized using a hybrid sec-ond/fourth order and second order schemes, respectively. Therigid body equations are solved implicitly using a predictor/cor-rector approach. The pressure Poisson equation is solved using aprojection algorithm to satisfy continuity using the PETSc toolkit.The interpolation coefficients required for the dynamic oversetgrids is obtained using SUGGAR software which runs as a sepa-rate process from the flow solver. Message Passing Interfacebased domain decomposition is used for multiple processorsimulation.

Analysis Methods. Verification studies for the integral andlocal variables are performed following the quantitative methodol-ogy and procedures proposed by Refs. [25–27] to estimate numer-ical uncertainties and confidence interval of a solution.Uncertainties due to the numerical iteration (UI) are estimatedfrom the dynamic range of the running mean oscillations. The firststep for estimation of the grid (UG) and time step (UT) uncertain-ties is the convergence study. For this purpose three solutions are

Journal of Fluids Engineering MARCH 2012, Vol. 134 / 031201-3


obtained using systematically refined grid-spacing or time stepswith refinement ratio:

rG=T ¼Dx2

Dx1

¼ Dx3

Dx2

(4)

where the subscripts 3, 2 and 1 represent the coarse, medium andfine grids, respectively. Dx is either grid or time step spacing. Theconvergence of the solution is checked from the solution (S) onthe three grids,

RG=T¼e12

e23

¼S2�S1

S3�S2

¼

0<RG=T <1 : Monotonicconvergence

�1<RG=T <0 : Oscillatoryconvergence

RG=T�1 : Monotonicdivergence

RG=T��1 : Oscillatorydivergence

8>><>>:

(5)

The ratio of numerical and theoretical order of accuracy P is usedto quantify the distance metric from the asymptotic solution:

P ¼�lnðRG=TÞ=lnðrG=TÞ

pth(6)

where, pth is the theoretical order of accuracy of the numericalmethods. The solutions are expected to be in the asymptotic rangewhen P¼ 1. The uncertainties are estimated using the factor ofsafety method discussed in Xing and Stern [27] and reportedbased on %S1. The differences in the half (SHD) and full domain(SFD) simulations are quantified as domain error:

UDO ¼SFD � SHD

SFD� 100 (7)

The iterative, grid, time step and domain uncertainties provide anestimate of total numerical uncertainty USN for a simulation asbelow:

USN ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2

I þ U2G þ U2

T þ U2DO

q(8)

Fig. 1 (a) Domain, grids, boundary conditions for model- and full-scale appended Athena. TheLES regions are shown for (b) G4 and (c) G1.



The validation uncertainty UV of the study accounts for both nu-merical and experimental (UD) uncertainties:

UV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2

D þ U2SN

q(9)

The validation study provides a confidence interval for the numer-ical predictions by comparing the total uncertainties in the studyUV and the comparison error (E). E is defined by the differencebetween data and simulation values,

E ¼ ðD� SÞD

� 100 (10)

The numerical predictions are validated at UV interval when jEj �UV.

The turbulence resolution uncertainties are quantified by com-paring URANS and DES transom flow predictions, resolved tur-bulent kinetic energy (TKE) predictions in DES, and comparisonof the inertial subrange predictions with Kolmogrov’s and experi-mental spectra. Resolved TKE (kR) is obtained by partitioning theresolved velocity into time averaged Ui and fluctuating compo-nents ui as,

Ui ¼ Ui þ ui : i ¼ 1; 2; 3 (11)

kR ¼1

2uiuj (12)

The vortical structures are identified using isosurfaces of the nor-malized helicity Q-criterion [28]. The analysis methods for theinstability study are discussed in Sec. 7.

4 Experimental Data, Domains/Grids, Boundary and

Simulation Conditions

Full-scale measurements of the Athena R/V transom wake wereperformed in the St. Andrews Bay using the quantitative visual-ization (QViz) system and a light detection and ranging (LIDAR)sensor. The measurements were performed for several speeds cov-ering both wet and dry transom flows Fr¼ 0.14� 0.62. Wyattet al. [3] processed the LIDAR data for Re¼ 2.9� 108, Fr¼ 0.25and reported transom wave elevation: (a) mean and RMS con-tours, (b) mean and RMS profiles at various transom-aft locationsX¼ 1.0447� 1.1245, and (c) frequency spectra of the unsteadi-ness at center plane of the above transom-aft locations. The uncer-tainties in the mean wave elevation measurements due to thesinkage and LIDAR were reported to be 60.05 m (7%f peak) and60.02 m (62.9%f peak), respectively, which providesUD¼ 7.63%D. The mean wave elevation were corrected usingmean sea elevation measured at Fr¼ 0.046. Herein, the data are

Fig. 2 Time history and running mean of (a) the transom wave elevation atX 5 1.0447, Y 5 0 for G1 and (b) CT for G1, G2, G4 and G2-URANS full-scale cases

Fig. 3 Propeller RPS obtained from G5-SP is compared withthe extrapolated experimental data [15]



reported in non-dimensional form using L and U0. The numeri-cal mean wave elevation is shifted up by 2.13� 10�3L to matchthe experimental far-field value.

The study uses half domain bare hull grid G0 consisting of5.9M points, which is the same as that used by Wilson et al. [2].The grid consists of overset hull, multiple bow, transom andfree-surface refinement and background grid blocks. The half do-main appended hull grid G5 consisting of the 2.8M point wasgenerated by adding overset grid blocks for the skeg, rudder, sta-bilizer, propeller shaft and struts to the G0 hull grid as shown inFig. 1(a). The hull and the appendage grids are body-fitted “O”type mesh, except for the stabilizer and shaft caps which have anopen topology. Cartesian refinement grid blocks were added nearthe free-surface, in the transom and the appendage regions forbetter resolution of the transom flow. A Cartesian backgroundgrid is used for specifying the boundary conditions away fromthe hull. The grid G5 was systematically refined with rG¼ 21/4 toobtain grids G4 with 5.3M points, G3 with 8.9M points and G2with 15M points. The grid G1 with 43M points was refined fromG2 using rG¼ 21/2. The full-domain grid G3-FD consisting of17.8M points was generated by mirroring G3 grid blocks alongthe Y¼ 0 plane. The averaged normalized near wall grid resolu-tion decrease from yþ ¼ 115� 50 and 4� 1.8 with the gridrefinement for full- and model-scale simulations, respectively.The validation study using grid G1 was performed using 288 pro-cessors for 6L/U0, which required approximately 384 h of wallclock time or 110 K CPU hours. The results for the last 2.05L/U0

were used to obtain the transom flow validation and instabilityanalysis. The averaging period is expected to be sufficient as thewave elevation running mean fluctuations are< 1% of the meanas shown in Fig. 2, which establishes statistically stationaryunsteady solutions. The sampling period corresponds to 15 peri-ods of the transom wave elevation dominant frequency; thus issufficient to capture the instability frequencies.

The fixed and self-propelled simulations are performed in therelative inertial and absolute inertial earth fixed coordinates,respectively. The hull and appendages have no-slip boundary con-ditions for the model-scale simulations, whereas WF boundaryconditions are applied for the full-scale simulations. The half do-main simulations use a symmetry boundary condition at the Y¼ 0plane. The inlet boundary condition is specified at X-Min plane,convective boundary at X-Max and far-field boundary conditionselsewhere. In 3DOF simulations, all the grid blocks surge, butonly the hull, appendages and appendage refinement blocks sinkand trim together. For self-propelled simulations with predictedsurge shown in Fig. 3, controller thrust, torque, and advance coef-ficients are extrapolated from the model-scale experiment [15].All simulations except one are performed using DES. Instantane-ous DES solutions in Figs. 1(b) and 1(c) show that the LES isactive in the flow separation region encompassing the appendagesand in the wake region aft of the transom, whereas URANS isactive in the boundary layer and outer regions.

The model- and full-scale wetted transom simulations are per-formed at Fr¼ 0.25 as summarized in Table 1. The full-scalesimulations at Re¼ 2.9� 108 include verification, domain andturbulence resolution studies for fixed conditions, validation forself-propelled free to sink and trim conditions, and instabilitystudies for both fixed and self-propelled conditions. The fixedconditions are for sinkage/L¼�3.28� 10�4 and trim¼ 0.0414deg based on the model-scale bare hull data [12]. The grid veri-fication is performed using grid triplets G2, G3 and G4 withrG¼ 21/4 and G1, G2 and G4 with 21/2. The time step verifica-tion study is performed for G3 using rT¼ 21/2, Dt¼ 7.07� 10�4,10�3 and 1.41� 10�3. G3-FD is performed for the domainstudy. URANS predictions (G2-URANS) are compared with G2and G1 for the turbulence resolution sensitivity study.

Self-propelled simulation with predicted surge, sinkage andtrim (G5-SP) is performed to obtain propeller RPS at full-scaleconditions. A coarse grid is used for this case as it is approxi-mately eight times more expensive than the fixed simulations.

Tab

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Condit

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(M)

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deg

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10

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G4

Full

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ded

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sinkag

ean

dtr

imR

e¼

2.9�

10

85.3

�3.2

80.0

414

60%

2.1

8�

21.1

12.8

75.6

61.2

5c

G3

8.9

70%

1.9

5�

8.3

32.6

55.5

63.0

9G

215

85%

2.0

6�

13.3

33.1

05.5

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Self-propelled G1-SP simulation with predicted sinkage and trimis performed for the transom flow validation using full-scale data.G1 and G1-SP predictions are used for the instability study. Themodel-scale bare and appended hull simulations are performed atRe¼ 1.15� 107 for fixed conditions. G0-BH-M is performed forthe instability study. G4-M, G3-M and G2-M are performed fortotal resistance (CT) grid verification and validation using model-scale data [12], and the instability study. The instability studyidentifies the prominent vortical structures and the associated St iscomputed. The predictions are compared with the previous canon-ical and ship flow studies, and the effect of appendages, Re, pro-peller and ship motions are discussed.

5 Verification, Domain and Turbulence Resolution

Uncertainties

The grid and time step verification studies are performed for in-tegral and local variables for model- and full-scale appendedAthena. The integral variables include frictional (Cf), pressure(Cp) and CT resistance coefficients. The local variables includetransom wave elevation mean and RMS profiles and center-planedominant frequency at X¼1.047, 1.115 and 1.245.

Full-Scale Appended Athena. The verification studies usingrG¼ 21/4 shows mixed convergence with fairly large uncertaintiesfor the local variables likely due to ill-conditioned rG �1. Thus,only rG¼ 21/2 results are discussed here and summarized in Table2. The grid verification study shows UI � 0.8%S1 and UI/e12

� 0.36. The results show mostly monotonic convergence 0.3 �RG � 0.6 with an averaged P¼ 0.85. The averaged UG¼ 4.8%S1

for both the integral and local variables. The time step verificationstudy shows UI/e12¼ 1.2 for the integral variables, and mixed os-cillatory and monotonic convergence �0.33 � RG � 0.54 with

averaged P¼ 1.4. The averaged UT¼ 1.6%S1 and 6.4%S1 for theintegral and local variables, respectively.

The full domain predictions show< 2%, 4.5% and< 0.1%asymmetry for the transom wave elevation mean, RMS and domi-nant frequency, respectively. Instantaneous wave elevation pro-files show up to 10% asymmetry in the shoulder wave peak, whereasthe experimental data shows 30% asymmetry. As shown in Figs.4(a)–4(c), the transom vortex shedding is relatively symmetricwith< 0.1% difference in the dominant frequency, and is qualita-tively similar to the half domain predictions. This is expected as thetransom vortices are large scale organized structures aligned alongthe Y-axis. The full-domain predicts resistance coefficient, transomvortex shedding frequency and dominant frequency and associatedamplitude within 2% of the half domain. However, former predicts4.5% and 5 – 10% accurate RMS and high-frequency amplitudesthan the latter, respectively. On the other hand, half domain G2 simu-lation shows 5 – 10% accurate mean, RMS and dominant frequencyamplitude predictions than the full-domain even though both havesimilar computational cost. Thus considering the numerical expense,finer half domain grid is used for the validation study. Full domainsimulations with similar resolution are of future interest for improvedRMS and shoulder wave asymmetry predictions.

The experimental transom wave elevation unsteadiness spectrain Fig. 4(a) shows f�2 scaling close to Kolmogrov’s inertial sub-range f �5/3 scaling. The largest energy containing eddy has a timescale s0¼ 0.12 (f¼ 8.3) and length scale l0¼ 0.012, where the lat-ter is estimated from the peak amplitude. The inertial subrangebegins at sin¼ 0.067 (f ¼15) and extends up to s¼ 0.01 (f¼ 100).The smallest time scale captured in the experiment corresponds to

a length scale l0ss0

� �3=2

¼ 5� 10�3L to 2.9 � 10�4L [29]. All the

simulations, representative figures shown in Figs. 4(b)–4(e), predicta wetted transom flow with unsteady vortex shedding. URANS

Table 2 Summary of verification, domain and validation study for full-scale and grid verification for model-scale appended Athenatransom flow

Study, Case C‡f C†

p C‡T f mean f RMS f frequency

ITTC Model-scale 2.92� 10�3 -Full-scale 1.80� 10�3

Experimental data Model-scale (D) [17] - 6.862� 10�3 -Full-scale data (D) extrapolatedfrom [17] using ITTC

5.732� 10�3

Model-scale bare hull, UD%D 1.5Full-scale [3] Data (D) - Fig. 6 Fig. 7 6.97 – 9.57

UD%D 7.63Full-scale grid study with rG¼H2;G1,G2,G4

UI%S 0.45 0.72 0.21 0.7UI/e12 0.17 0.14 0.36RG 0.47 �0.69 0.6 0.5 0.3 0.33P 0.85 - 0.67 1.2 0.7 0.8UG%S 5.22 3.91 4.93 4.8 7.5 1.36

Full-scale time step study withrT¼H2; G3, DT¼ 7.07� 10�4, 10�3, 1.41� 10�3

UI%S 0.5 0.75 0.6 0.8UI/e12 0.81 0.5 1.2RG 0.21 �0.33 �0.1 0.45 �0.04 0.54P 2.25 - - 1.15 - 0.89UT%S 1.41 1.68 1.79 7.68 6.07 4.5

Full-scale domain Study, G3 and G3-FD UDO%S < 0.1 < 0.1 < 0.1 1.8 2.5 < 0.1Model-scale grid study with rG¼ 21/4;G2-M,G3-M,G4-M

UI%S 0.2 0.7 0.52 0.7UI/e12 0.09 0.25 0.12RG �1.17 �0.31 0.56 �0.72 0.40 0.21P - - 1.02 - 1.29 2.22UG%S - 4.55 5.90 14.36 11.5 7.98

Full-scale validation, G1-SP 2.00� 10�3 3.10� 10�3 5.43� 10�3 Fig. 6 Fig. 7 7.81USN%S 5.43 4.32 5.28 9.25 10.0 4.77UV%D - - 5.49 12.0 12.56 9.0E%D(%ITTC) (-13.89) 4.22 9.0 7.5 �5.9



predicts only the large scale organized structure in Fig. 4(f). Theflow separation occurs from the transom bottom corner as shown inFig. 4(h). In DES, more and more small-scale vortical structuresare resolved with grid refinement. The resolved TKE levels in thetransom region increases up to 95% on G1 as shown in Table 1 andFig. 5. G1 transom wave elevation spectra predicts f�1.85 scalingstarting at f¼ 15 and a peak at s¼ 0.126 (f¼ 7.9). The slope of thespectra is in between those of the experiment and Kolmogrov’s.

The smallest resolved scale, which corresponds to the averagedgrid scale, in the transom region is l¼ 5.6� 10�4 L and s¼ 0.0138(f¼ 71). The resolved scales are an order of magnitude smaller thanthe largest energy containing eddy, and lie well within the inertialsubrange. The grid is sufficiently fine as it resolves 70% of the iner-tial subrange and 91% of kr ð¼

Ð 100

15f�1df ) in the experiment. How-

ever, under predicted RMS and dominant frequency amplitude areexpected as the entire inertial subrange is not resolved.

Fig. 4 (a) Experimental and numerical spectra for the transom wave elevation unsteadiness atX 5 1.115 are compared. Isosurfaces of Q 5 300 obtained for (b) G3, (c) G3-FD, (d) G2, (e) G1, (f)G2- URANS and (g) G2-M. (h) The flow separation at Y 5 0.01 plane is shown for G4.



Comparison of G1 and G1-SP shows that the self-propelledconditions improve transom wave elevation mean, RMS and dom-inant frequency amplitude predictions by 6%, 12% and 10% com-pared to fixed conditions, respectively. It does not, however, show

a significant effect on the transom wave elevation dominant fre-quency as shown in Fig. 4(a) or the transom vortex shedding fre-quency. This is expected as the propeller is located in the deepwater, and the ship motions show four times smaller frequencies

Fig. 6 The mean transom wave elevation (a) contours from the experiment [3], (b) contoursfrom G1-SP, and (c) profiles from the experiment and G1-SP are compared. The transom corneris at X 5 1.0. The profile locations A-H are shown in sub-figure (a).

Fig. 5 (a) Modeled, (b) resolved and (c) total TKE distribution at X 5 1.0447 for G1



than the dominant transom flow frequency. Thus the transom flownumerical uncertainties are not expected to be significantlyaffected by the propeller or motions.

The averaged USN estimated in Table 2 for the validation studyis 5.3%S1, 9.3%S1, 10%S1 and 4.8%S1 for the resistance, waveelevation mean, RMS and dominant frequency, respectively. Thecorresponding validation intervals are UV¼ 5.5%D, 12%D,12.6%D and 9%D. The UI and UG levels predicted for theappended hull are similar to that for the URANS model- and full-scale bare hull predictions [13,14], but UI/e12 is large. The gridand time step uncertainties could be contaminated by large itera-tive errors, the differences in the resolved flow physics and cou-pling of modeling and numerical errors. Verification studies usingfull domain grids and larger pressure-velocity iterations to reduceUI are of future interest in reducing numerical uncertainty levelsand validation intervals.

Model-Scale Appended Athena. As summarized in Table 2,the grid verification study using rG¼ 21/4 shows UI � 0.7%S1 andUI/e12 � 0.25 for the resistance coefficients. Mixed oscillatoryand monotonic convergence �0.79 � RG � 0.56 are predictedwith large variations in P¼ 1.09 � 2.15. Averaged UG¼ 5.9%S1

and 12%S1 for the resistance coefficients and the local variables,respectively. The validation interval for CT is estimated to beUV¼ 6.1%D. The flow shows wetted transom with unsteady

Fig. 7 The transom wave elevation RMS (a) contours from the experiment [3], (b) contoursfrom G1-SP, and (c) profile from the experiment and G1-SP. The transom corner is at X 5 1.0.The profile locations are marked in Fig. 6(a).

Fig. 8 The transom wave elevation unsteadiness spectra inthe experiment [3] and G1-SP are compared atX 5 1.0447–1.245, Y 5 0. The FFT locations are marked in Fig.6(a) with black dots.



vortex shedding in Fig. 4(g), similar to the full-scale simulations.As summarized in Table 1, the resolved TKE levels in the transomregion increases with grid refinement from 50% to 80%. Themodel-scale simulations predict up to 10% lower resolved TKElevels than the full-scale, as the former predicts lower velocity inthe transom bottom separated shear layer than the latter.

UI, UI/e12 and UG levels for the model-scale are similar tothose in full-scale. The uncertainties for the local variables

are 2 times larger than that for the integral variables for bothscales. This is due to the differences in resolved transomflow physics on grid refinement. The appended hull casesshow less convergent solutions and larger uncertainty levelsthan the bare hull cases [13,14]. This could be partly becausethe appended hull simulations are for unsteady wetted tran-som flows, whereas the bare hull simulations were for steadydry transom flows.

Fig. 9 Contours of xy and free-surface wave elevation at X 5 1.0667 at DT 5 0.012L/U obtained using G2-URANS(left panel) and G1-SP (right panel) are compared with experimental data [3] (dotted line). G3-FD wave elevationprofiles are shown in black line on the right panel. The vorticty contour levels are from 2100 to 100 at an intervalof 10. The inset figure shows the isosurface of normalized helicity Q 5 300.



6 Validation for Full-Scale Appended Athena

As shown in Table 2, CT is underestimated by up to 4.2%Dcompared with the extrapolated experimental data using ITTCline. Cf predictions are 14% higher than the ITTC line similar tothose reported by Bhushan et al. [14]. The resistance coefficientsshow< 5% variation due to the effect of propeller and motions.The sinkage and trim predictions lie within the bare and appendedhull experimental data range. The RPS¼ 8.66 prediction in Fig. 3is in close agreement with the extrapolated experimental data.

The experimental mean transom wave elevation contour in Fig.6(a) shows a V-shaped wake, which consists of transverse anddiverging waves. The transverse wave peak is observed atX¼ 1.12, Y¼ 0.0233. The diverging wave originates from thestern edge and has a prominent trough at X¼ 1.02, Y¼ 0.06 andan attenuated peak downstream. The transverse and divergingwave pattern are also clearly evident in the wave elevation profilesin Fig. 6(c). The numerical simulation predicts a transverse wavepeak around X¼ 1.115, Y¼ 0.0227 with 10%D lower elevationthan the experiment in Fig. 6(b). The diverging wave has slightlyshorter wavelength and smaller angle compared to the experiment.The trough and peak of the diverging waves are predicted atX¼ 1.0, Y¼ 0.051 and X¼ 1.15, Y¼ 0.079, respectively. Thetrough location is predicted well, but the elevation is over-predicted by 26%D. As shown in Fig. 6(c), the wave elevations arepredicted within 7%D of the experiment for X¼ 1.0447 – 1.089.For X¼ 1.115–1.245, wave elevation predictions attenuate morerapidly than the experiment resulting in 20%D under prediction.The averaged jEj ¼ 9%D<UV, thus mean wave elevation predic-tions are validated at 12%D interval.

The experimental transom wave elevation RMS contoursin Fig. 7(a) shows prominent unsteadiness in the region X¼ 1 – 1.2,Y¼ 0 – 0.02 with two centerline peaks at X¼ 1.065 and 1.12.

A secondary peak is observed at Y¼ 0.05 due to divergingwaves. The RMS profiles in Fig. 7(c) show peaks close to thecenterline and in the diverging wave region consistent with thecontours. The numerical predictions in Fig. 7(b) show high RMSin the region X¼ 1 – 1.17, Y¼ 0 – 0.015 with a single peak atX¼ 1.11. The secondary peak is predicted at X¼ 1.066 – 1.115,Y¼ 0.05. The primary and secondary peak values are predictedwithin 1%D and 4.5%D of the experiment, respectively. TheRMS profiles in Fig. 7(c) are predicted within 4%D of experi-ment for X¼ 1.0447 – 1.1404 except at X¼ 1.089. For X¼ 1.17– 1.245, the peak attenuates faster than the experiment andthe results are 20%D under predicted. The averagedjEj ¼ 7.5%D<UV, thus RMS predictions are validated at the12.6%D interval.

The experimental spectra of the transom wave elevationunsteadiness in Fig. 8 shows a dominant frequency range of 6.97– 9.57 for the different aft locations. The peak amplitude increasesfrom X¼ 1.0447 – 1.115 and decreases thereafter. The numericalsimulation predicts a small variation in the dominant frequencyaround f¼ 7.81 within 6%D of the averaged data. The peak ampli-tudes grow and decay similar to the experiment, but shows up to9.5%D lower values. The averaged jEj ¼ 7.8%D<UV, thus thedominant frequency is validated at the 9%D interval.

The instantaneous experimental wave elevation profiles in Figs.9(a)–9(e) shows up to 30% asymmetry in the peak. Based on theseprofiles, experimentalists explained the transom wave elevationdominant frequency due to the collision of breaking shoulderwaves traveling from either side towards the centerline. Analysisof the numerical volume solution shows that the shoulder wave ismostly steady, whereas the wave elevation between the centerlineand the shoulder increases (decreases) as xy increases (decreases).The increase and decrease in xy is due to the passage of longitudi-nal transom vortices in between the center plane and the rudder.

Fig. 10 Isosurfaces of Q 5 300 for instantaneous solution from G2. Inset figures are obtained using averaged solution. Threedifferent types (A, B and C) of juncture vortices are marked and associated dominant frequency modes are shown. Contoursare of the absolute pressure with levels from 20.5 to 0.1 at an interval of 0.02.



This behavior is more prominent in URANS than in DES, as latterpredicts numerous small scale turbulent structures. This explainsthe prominent correlation between the wave elevation unsteadi-ness and the Karman-like transom vortex shedding frequencies forall the grids and simulation conditions. Thus the primary cause oftransom wave elevation dominant frequency is the Karman-likeshedding. The full domain predictions suggest that the asymmetryobserved in the experimental data is due to the high frequencywave elevation oscillations.

A strong correlation between the unsteady transom wave eleva-tion and the Karman-like vortex shedding from the transom cornerwere also reported in previous studies, e.g., Athena bare hull wet-ted transom flow at Fr¼ 0.25 [2], backward facing step with free-surface [8], and wetted transom model at Fr¼ 0.38 [4]. Theunsteady transom wave elevation due to traveling and collidingshoulder waves have been reported in dry transom flow studiessuch as, for Athena bare hull at Fr¼ 0.43 and 0.62 [2] and tran-som model at Fr¼ 0.43 [4].

7 Instability Study

A representative plot showing the key vortical structures pre-dicted in the appended hull simulations are marked in Fig. 10. Thefigure is shown for G2, as the large scale vortical structures arenot clearly visible for G1 due to the presence of numerous small-scale structures. Table 3 summarizes the length and velocityscales, and dominant frequency evaluation locations and samplingperiod for the bare and appended hull instabilities. The instabilityperiod sP (period�U0/L) or frequencies for the bare hull instabil-ities are obtained from the time history of the pressure resistancecoefficient as shown in Fig. 11(a) and 11(b) similar to the previ-ous canonical and ship studies. However, for the appended hullsimulations, integral variables show several modulated peaks dueto vortex shedding from the appendages as shown in Figs. 11(a)and 11(c), which make identification of the dominant frequencyfor a particular vortex difficult. Thus, the shedding period/fre-quency is obtained from the hull piezometric pressure fluctuationclose to the shedding location. A representative plot showing thetime history and running mean of the transom corner piezometricpressure fluctuation is shown in Fig. 11(d). Overall, the samplingperiods vary from 5 – 20 vortex turnover time and thus are consid-ered adequate for the study. The instabilities and associated St pre-dicted in this study are summarized in Table 4.

Full-Scale Appended Athena. The phases of the transom vor-tex shedding is shown in Fig. 12. It is expected that the flow sepa-ration at the transom corner would begin as a shear-layerinstability. However, such instability is not predicted, similar tothe bare hull study [2]. Sigurdson [17] and Kandasamy et al. [19]found that the frequencies of shear layer instabilities are an orderof magnitude larger than that of the Karman-like shedding forhigh Re. Thus, even smaller time step size and finer grids arerequired for their prediction. The vortex shedding from the tran-som is identified as the Karman-like shedding similar to those in abackward facing step and is scaled using the wetted transomheight (H) and US to obtain StH. As shown in Fig. 13, sP¼ 0.128for the Karman-like shedding which does not show significant de-pendence on the propeller or ship motions. Simulations at full-scale conditions predicts 7% higher wetted transom height andStH than those at fixed conditions. StH ¼0.104 6 3.5% is 14%higher than the upper limit of the St range in the previous studies.

Three different types of horseshoe vortices are identified asshown in Fig. 10. Type A: at the junction of rudder and hull,where an airfoil obstacle with 15 deg leading edge sweepbackintersects normally a flat-plate boundary layer. Type B: at thejunction of struts and hull, where an airfoil obstacle intersects anear flat plate boundary layer with axis inclined at an angle of 55deg. Type C: at the junction of struts with propeller shaft, wherean airfoil obstacle intersects a curved surface almost normally.Type A shows a symmetric horseshoe vortex with bimodal peak

Tab

le3

Evalu

ati

on

of

len

gth

scale

,velo

cit

yscale

,fr

eq

uen

cie

san

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tasso

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Inst

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Len

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Var

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per

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Vort

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man

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oci

ty(U

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ded

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1024

sam

ple

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98

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Cas

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Obst

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11

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pel

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27

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0.0

533

60.0

0025

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ike

Rea

ttac

hm

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length

(XR

)F

ree

stre

amvel

oci

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0.9

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Anal

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Kel

vin

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.St

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Vort

exin

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dm

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Ship

Len

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(L)

Fre

est

ream

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oci

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Sin

kag

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sam

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L



as shown in Fig. 14(a). The higher period amplitude is 50 –60% lower than the lower period amplitude. The period of thevortex shedding varies only by 2% on propeller and shipmotions and StT¼ 0.152 6 1.6%. Type B shows an asymmetrichorseshoe vortex with stronger vortex on the obtuse angle side.This is expected as the flow in the obtuse angle side is similarto the suction side of the airfoil obstacle at large angle. Forthis case, only one dominant frequency is predicted as shownin Fig. 14(b) which gives StT¼ 0.053 6 2%. The volume solu-tion shows almost symmetric horseshoe vortex for Type C

with several low amplitude (almost half of Types A and B)dominant frequencies as shown in the Fig. 10 inset. Thus, adominant frequency associated with the shedding cannot beidentified. Type A horseshoe vortex StT is 17% lower than thelower limit of the StT range reported for a circular cylinder[22]. A lower value is expected as the rudder is more stream-lined than the latter, and has a 15 deg leading edge sweep-back. However, ReT¼ 8.7� 104 in the present case is largerthan those in the circular cylinder study. StH associated withstrut-hull juncture horseshoe vortex is 70% smaller than the

Fig. 11 (a) Time history and running mean of the pressure resistance coefficient for G1 and G0-BH-M. The spectra of the pres-sure resistance coefficients predicted for (b) G0-BH-M and (c) G2. (d) The piezometric pressure fluctuation at transom cornerX 5 0.998, Y 5 0.01 for G1.

Table 4 Vortical structures instability for model- and full-scale bare and appended hull Athena at Fr 5 0.25

Case Mode Period sP Frequency f Length Scale L St

URANS model-scalebare hull [2]

Karman-like 0.0795 12.58 0.007a 0.088

G0-BH-M Karman-like 0.0539 18.55 0.008 0.148Flapping 0.16 6.25 0.023 0.144

G2-M Karman-like 0.13 7.7 0.0125 0.098Horseshoe (rudder-hull) 0.054 18.5 0.0076 0.14Horseshoe (strut-hull) 0.091 11.0 0.0048 0.053Shear-Layer (strut-hull intersection) 0.053 18.86 3.65� 10�4 0.0069

G1 Karman-like 0.126 (0.120)b 7.9 (8.3)b 0.0128 0.100Horseshoe (rudder-hull) 0.051 19.61 0.0076 0.15Horseshoe (strut-hull) 0.092 10.8 0.0046 0.052Shear-Layer (strut-hull intersection) 0.051 19.6 3.45� 10�4 0.0066

G1-SP Karman-like 0.128 (0.120)b 7.81 (8.3)b 0.0137 0.107Horseshoe (rudder-hull) 0.049 20.41 0.0076 0.155Horseshoe (strut-hull) 0.088 11.4 0.0047 0.053Shear-Layer (strut-hull intersection) 0.052 19.23 3.39� 10�4 0.0065Vortex induced motions 0.456 2.19 1 2.19

aRecomputed to be different from Wilson et al. [2].bExperimental data [3].



circular cylinder StT range. This is probably because struts areeven more streamlined (or smaller width/length ratio) than therudder.

The hull boundary layer separates around the half chord lengthtowards the acute angle side of the struts and hull intersectionleading to vortex shedding, as shown in Figs. 15(a)–15(e). Thisinstability is identified as the shear-layer instability. sP¼ 0.053 ofthe shear-layer instability is the same for both the starboard andport-strut junctures and does not show dependence on the propel-ler or ship motions as shown in Fig. 16. h¼ 3.41� 10�4 61.0%and velocity US¼U0. This gives Sth¼ 0.0066, which is about5.3% higher than the averaged Sth reported for the surface pierc-ing NACA 0024 laminar studies.

Unsteady sinkage and trim are predicted as shown in Fig. 17(a).Previous full-scale dry transom simulations for Fr > 0.36 pre-dicted steady motions [14]. Thus, it is expected that the unsteadymotions are due to the Karman-like transom vortex shedding. Thisinstability is referred to as vortex induced motions. The period orfrequency of the ship motions are obtained from the motions timehistory. The dominant frequency is scaled using L and U0 to

obtain StL¼ fL/U0. The amplitude of the trim is 54.67% of themean¼ 0.0311 deg, whereas for the sinkage amplitude is only6.7% of the mean¼�8.9� 10�4L. Both sinkage and trim showthe same dominant low frequency sP¼ 0.456, as shown in Fig.17(b), which gives StL¼ 2.19.

Model-Scale Athena Bare Hull. DES resolves almost 95% ofthe total TKE, and predicts small-scale vortical structures andhigher transom wave slopes than URANS. DES CT predictions are3% better than the URANS, and compare within 2.47%D with theexperimental data [12]. The bare hull transom wave elevation pre-dictions shows a diverging wave pattern similar to appended hull,but does not predict the transverse wave peak. This difference inthe wave pattern is due to the presence of the rudder, whose lengthis 20% of the transverse wave wavelength. The bare hull predic-tions show a larger extent of high RMS region with peak slightlyfurther away from the transom compared to the appended hull.The transverse wave cuts away from the transom do not show sig-nificant effects of the appendages.

The previous bare hull simulation using URANS predicted onlyKarman-like instability, whereas the DES predicts Karman-likeand flapping-like instabilities, as shown in Figs. 11(a) and 11(b).In the URANS, the averaged wetted transom height is 0.007,which was mistakenly reported to be 0.005 [2]. The correctedheight gives Karman-like shedding StH ¼ 0.088. DES predicts47% and 14% higher shedding frequency and averaged wettedtransom height than the URANS, respectively. StH¼ 0.148 inDES, which is 68% higher than URANS.

The DES wake region in Fig. 18 shows a periodic growth anddecay, which is identified as the flapping-like instability. This fea-ture is not predicted in the URANS bare hull or appended hullsimulations. The period of this instability sP¼ 0.16 is obtainedfrom the periodicity of the Kelvin-wave expansion angle. Theaveraged XR¼ 0.023, which is the length from the transom to thepoint where the separation streamline touches the free-surface, asshown in Fig. 12. StR¼ 0.144 for the flapping-like instability ofthe Kelvin wave, which is 9.7% higher than those predicted in thesurface-piercing NACA 0024 URANS.

Model-Scale Appended Athena. The CT predictions showjEj ¼ 5.4%D<UV¼ 6.1%D, and thus is validated at the 6.1%Dinterval. CT is 8% lower than the full-scale simulation. Cf is 12%higher than the ITTC line similar to the full-scale simulations. Cp

shows< 3.5% variation on scale. The CT, Cp and Cf predictionsare 12%, 38% and 12% higher than the bare hull predictions,

Fig. 13 Fourier amplitudes of piezometric pressure fluctua-tions at the transom corner near the symmetry plane atX 5 0.998, Y 5 0.01 for model- and full-scale appended Athenasimulations

Fig. 12 Phases of Karman-like transom vortex shedding is shown for G2 at Y 5 0.01 cross-section. Contours are of the abso-lute pressure with levels from 20.2 to 0.1 at an interval of 0.006.



respectively. The transom wave elevation pattern does not showsignificant dependence on the scale. The model-scale simulationpredicts 5.2% lower transom wave elevation mean and RMS thanthe full-scale, but the dominant frequency compares within 2.3%.The scaling of instabilities follows the same procedure as that forthe full-scale. As shown in Fig. 13 and Table 4, the Karman-likeshedding period, transom wetted height and StH do not show sig-nificant dependence on scale; however, it predicts a 50% largerperiod and 40% higher wetted transom height than the bare hull.StH¼ 0.098 is 10% higher and 47% lower than the bare hullURANS and DES, respectively.

Type A horseshoe vortex shows a bimodal peak with nearlyequal amplitude, as shown in Fig. 14(a). This is in contrast to thefull-scale where the amplitude of the higher period is almost halfof the lower period. The period sP¼ 0.54 and StT¼ 0.14 of thevortex shedding is 6.1% higher and 7.1% lower than the full-scale, respectively. Type B horseshoe vortex shedding period andStT¼ 0.053 are within 2% of the full-scale predictions, as shownin Fig. 14(b). However, the model-scale shows 30% higher peakamplitude than full-scale.

The period of the shear-layer instability sP¼ 0.053 shown inFig. 16 is the same for both the starboard and port strut junctures.

Fig. 14 Fourier amplitude of piezometric pressure fluctuations at (a) rudder-hull juncture (X 5 0.964, Y 5 0.037) and (b) star-board strut-hull juncture (X 5 0.92, Y 5 0.034) for model- and full-scale appended Athena simulations

Fig. 15 Phases of hull-strut juncture vortex shedding due to shear-layer instability is shown for G2 at cross-sectionY 5 0.0524. The cross-section is shown in the subfigure (a) inset. Contours are of the streamwise velocity with levels from20.25 to 0.1 at an interval of 0.01.



Momentum thickness h¼ 3.65� 10�4 and Sth¼ 0.0069 are 5.8%higher than the full-scale.

8 Conclusions

Vortical structures and associated instabilities of appendedAthena wetted transom flow in full-scale conditions are studiedusing DES to explain the source of dominant transom flow fre-quency, including verification and validation using full-scale ex-perimental data. Time and grid verification studies and

quantification of domain and turbulence resolution uncertaintiesare performed for the integral and local variables for full-scale atfixed conditions. The results are also compared with model-scalebare and appended hull predictions and experimental data.

The grid verification study shows mostly monotonic conver-gence with averaged UG¼ 5%S for both integral and local varia-bles. The time step study shows mixed convergence withaveraged UT¼ 4%S. The full domain simulation predicts 5%Daccurate RMS predictions than the half domain when comparedwith the experimental data. However, a half domain predicts 5 –10% accurate mean and RMS predictions than the full domainwith similar grid sizes. Thus, considering the computational cost,a half domain grid is used for the validation. The grid used for thevalidation study consisting of 43M points is deemed adequate as itresolves 70% and 91% of the experimental inertial subrange andturbulent kinetic energy values, respectively.

The model-scale bare and appended hull CT predictions com-pare within 2.5%D and 5.4%D of the experimental values, respec-tively. The full-scale predictions compare within 4.2%D theextrapolated experimental data using ITTC. The model- and full-scale appended hull Cf predictions are 11–15% higher than the

Fig. 16 Fourier amplitudes of piezometric pressure fluctua-tions on starboard strut X 5 0.927, Y 5 0.0524 close to the hull-strut juncture for model- and full-scale appended Athenasimulations

Fig. 17 (a) Time history of predicted sinkage and trim and (b)their Fourier amplitudes for G1-SP

Fig. 18 Phases of the flappinglike instability (sP 5 0.16) for G0-BH-M. The vortical structures are shown by the isosurfaces ofQ 5 300 and colored by piezometric pressure with levels from 20.5 to 0.1 at an interval.



ITTC line. The full-scale sinkage and trim predictions lie withinthe model-scale bare and appended hull experimental data range.

The full-scale transom wave elevation predictions show a trans-verse and diverging wave pattern similar to the experiments. Thesimulation predicts the transverse wave pattern well, butshows diffused diverging wave compared to the experiment. Boththe experiment and predictions show large RMS values inthe transverse wave region. The experiment shows averageddominant frequency of 8.3, whereas simulation predicts asmall variation around 7.81. The transom wave elevationpredictions show average jEj ¼ 9%D<UV¼ 12%D for themean, jEj ¼ 7.5%D<UV¼ 12.6%D for the RMS and jEj¼ 7.8%D<UV¼ 9%D for the dominant frequency. Overall, theresults are relatively good as average jEj< 8.1%D and are vali-dated at an acceptable 11.2%D interval. Experimentalistsexplained the origin of the dominant frequency due to the colli-sion of traveling shoulder wave. Herein, it is shown that theKarman-like shedding from the transom bottom corner has thesame dominant frequency as that of the transom wave elevation;thus it is identified to be the primary cause of the unsteadiness.

The model- and full-scale appended hull simulations predict:Karman-like shedding from the wetted transom corner, horseshoevortices at the juncture of rudder-hull and strut-hull, shear layerinstability due to the separation of the hull boundary layer at thestrut-hull intersection, and unsteady ship motions due to the tran-som vortex shedding. The Karman-like shedding showsStH¼ 0.103 6 4.4%, which is 14% higher than the values reportedin previous canonical and ship studies. The rudder-hull horseshoevortex shows bimodal frequency spectra with StT¼ 0.14 and0.152 in model- and full-scale, respectively. StT is 9–17% lowerthan those for a circular cylinder obstacle. This is expected as therudder is a more streamlined obstacle than a cylinder. The strut-hull horseshoe vortex is asymmetric as the strut axis is inclined at55 deg and shows averaged StT¼ 0.053 6 2%. The shear layerinstability show Sth¼ 0.0067 6 3%, which is 8% higher than thosepredicted for the surface piercing NACA 0024 case. Both sinkageand trim show StL¼ 2.19. Overall, the instabilities show less than7% dependence on scale, propeller or motions.

The model-scale bare hull study predicts StH¼ 0.148 for theKarman-like shedding, which is 68% larger than the URANS pre-dictions. DES also predicts flapping-like instability due to thegrowth and decay of the wake region. StR¼ 0.144 for this instabil-ity, which is 9.7% higher than that in the surface piercing NACA0024 URANS.

Full domain simulations are desirable for improved RMS andasymmetric shoulder wave peak predictions. Recently, V4 wasoptimized for high performance computing to enable such largegrid simulations with hundreds of millions of grid points [30].Also of interest is investigation of advanced DES models toaddress the modeled-stress depletion issues for large grid compu-tations [31]. Detailed analysis of the instabilities for the transomstern model and comparisons with the Athena predictions are inprogress.

Acknowledgment

The Office of Naval Research under Grant Nos. N00014-01-1-0073 and N00014-06-1-0420, administered by Dr. Patrick Purtell,sponsored this research. Simulations were performed on NAVYHPCMP machines Babbage IBM P5 and DaVinci IBM P6.

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